The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

The Nucleon-Nucleon Interaction in a
Chiral Effective Field Theory
Dissertation
zur Erlangung des Grades eines
Doktors der Naturwissenschaften in der
Fakultät für Physik und Astronomie der
Ruhr-Universität Bochum
von
Evgeny Epelbaum
aus St. Petersburg
Bochum 2000

Zusammenfassung und Ausblick
In dieser Arbeit haben wir emlge Anwendungen der Methode der effektiven Feldtheorien zur
Beschreibung der Streu- und Bindungszustände zweier Nukleonen betrachtet. Wir möchten jetzt
die erzielten Ergebnisse zusammenfassen:
1. Nach einer Einleitung, die den derzeitigen Stand der Forschung beschreibt und die Fragestellungen
angibt, haben wir als erstes im Kapitel 2 das quantenmechanische Problem
zweier Nukleonen betrachtet und haben gezeigt, wie man eine effektive Niederenergietheorie
basierend auf der Methode unitärer Transformationen konstruieren kann, wenn man von
einem beliebigen realistischen Zweinukleonenpotential im Impulsraum ausgeht. Dies wird
erreicht durch die Entkopplung der Unterräume kleiner und großer Impulskomponenten, die
zusammen den gesamten Impulsraum bilden. Die unitäre Transformation kann durch einen
Operator A parametrisiert werden, siehe GI. (2.65), der der nicht linearen Integralgleichung
(2.69) gehorcht. Diese Gleichung kann numerisch gelöst werden. Die Observablen können
dann allein in dem Unterraum kleiner Impulskomponenten berechnet werden. Abgesehen
davon, daß diese Methode an sich interessant ist, haben wir sie auch im Zusammenhang
mit der chiralen Störungstheorie (CHPT) für das Zweinukleonensystem betrachtet, indem
wir eine Reihe von Fragen untersuchten, die im Rahmen unserer exakten effektiven Theorie
für niedrige Impulse eindeutig gelöst werden können. Natürlich ersetzt dies nicht eine realistische
Rechnung im Rahmen der CHPT, wie sie ja ebenfalls von uns in der vorliegenden
Arbeit durchgeführt wurde. Sie kann aber als Leitfaden benutzt werden. Die Hauptergebnisse
unserer Untersuchung können wie folgt zusammengefaßt werden:
• Wir haben analytisch gezeigt, daß die auf den Niederimpulsraum (mit Impulsen kleiner
als ein gegebener Wert A) projizierte Theorie genau die gleiche S-Matrix liefert wie
die ursprüngliche Theorie, die im gesamten, nicht eingeschränkten Impulsbereich definiert
ist. Dies setzt voraus, daß die Randbedingungen für die Streuzustände geeignet
gewählt sind. Insbesondere sind die hohen Impulskomponenten der transformierten
Streuzustände für Anfangsimpulse, die kleiner als der Abschneideparameter A sind,
exakt Null.
• Ausgehend von einem S-Wellen NN Potential, welches aus einem anziehenden und einem
abstoßenden Teil besteht, entsprechend den Austäuschen leichter (/1L � 300 MeV)
und schwerer (/1H � 600 MeV) Mesonen, siehe GI. (2.113), haben wir die nichtlineare
Gleichung für den Operator A rigoros (in einem numerischen Sinn) gelöst. Dabei haben
wir gezeigt, daß die Observablen, die den gebundenen und Streuzuständen entsprechen,
in der ursprünglichen und effektiven Theorie genau übereinstimmen. Genauer gesagt,
haben wir in beiden Fällen jeweils einen gebundenen Zustand mit der Bindungsenergie
2.23 MeV. Diese Ergebnisse sind unabhängig vom Abschneideparameter, der von
200 Me V bis 5.5 Ge V variiert wurde. Wir haben argumentiert, daß der natürlichste
Wert des Abschneideparameters A ca. 300 MeV ist. Das effektive Potential kann sich

11
vom ursprünglichen wesentlich unterscheiden. Die gilt insbesondere für die Werte von
A im unteren Teil des angegebenen Bereiches.
• Wir haben auch einen alternativen Weg zur Bestimmung des Operators A betrachtet,
der die Kenntniss der "half-off-shell" N N T-Matrix voraussetzt. Diese Methode
führt zu der linearen Gleichung (2.94), die ebenfalls numerisch gelöst wurde.
Für alle gewählten Werte des Abschneideparameters A haben wir eine perfekte
Übereinstimmung zwischen den A's, die durch Lösung der linearen und nichtlinearen
Gleichungen gewonnen wurden, erhalten.
• Wir haben eine Niederimpulsentwicklung des effektiven Potentials untersucht. Dafür
haben wir den Anteil, der dem Austausch des leichten Mesons entspricht, ungeändert
beibehalten und den verbleibenden Teil des effektiven Potentials in eine Reihe von Kontaktwechselwirkungen
wachsender Dimension entwickelt, siehe (2.115), (2.116). Die
entsprechenden lokalen Operatoren sind aus geraden Potenzen der Impulse aufgebaut.
Die dazugehörigen Kopplungskonstanten Ci können durch den Vergleich mit der exakten
Lösung genau bestimmt werden. Für A =300 (400) MeV sind ihre Werte in der
Tabelle 2.1 angegeben. Wir haben gezeigt, daß sie die Eigenschaft der "natürlichen"
Größe haben. Das heißt, sie sind von der Ordnung eins in Bezug auf die Massenskala
Ascale = 600 MeV. Wir haben auch die Beziehung zwischen dieser Skala und der Masse
des schweren ausintegrierten Mesons diskutiert, sowie die Konvergenzeigenschaften
dieser Niederimpulsentwicklung. Dabei stellte sich heraus, daß man Operatoren von
relativ hoher Ordnung explizit mitzunehmen hat, um die Bindungsenergie mit einer
Genauigkeit von einigen Prozent zu bekommen, siehe GI. (2.116). Dies ist wegen der
unnatürlichen Kleinheit dieser Größe im Vergleich zu jeder anderen hadronischen Skala
zu erwarten. Die 3 Sl-Streuphasen können bis zur Laborenergie l1ab
:::: 120 MeV mit
den ersten drei Termen dieser Entwicklung nach Kontaktwechselwirkungen gut reproduziert
werden.
• Basierend auf unserer Niederimpulsentwicklung haben wir die Konstanten Ci auch direkt
durch die Anpassung an die Streuphasen gefunden. Dies ist äquivalent zu der
Vorgehensweise in einer effektiven Feldtheorie. Wir konnten zeigen, daß die Werte dieser
Konstanten nahe den exakten sind, solange man keine Terme sechster Ordnung im
Potential betrachtet. Die Bindungsenergie wird innerhalb 2% reproduziert. Die Mitnahme
von Termen sechster Ordnung führt zu keinen stabilen Ergebnissen. Dies kann
durch die Tatsache erklärt werden, daß der Beitrag solcher Terme zu den entsprechenden
Streuphasen sehr klein ist (bei niedrigen und mittleren Energien) und deswegen
nicht richtig festgelegt werden kann.
• Wir haben auch die Erwartungswerte des entwickelten Potentials in den Bindungsund
Streuzuständen studiert. Für A = 300 MeV ist der Entwicklungsparameter von
der Ordnung 1/2. Dies führt zu einer schnellen Konvergenz für die Erwartungswerte
des entwickelten Potentials in den Bindungs- und niederenergetischen Streuzuständen,
wie aus der Tabelle 2.3 zu entnehmen ist. Wie erwartet, wird die Konvergenz für
Streuzustände bei höheren Energien langsamer.
• Wir haben die Parameter in der effektiven Reichweitenentwicklung ausgehend von dem
entwickelten Potential (2.116) berechnet und die Vorhersagekraft unserer effektiven
Theorie demonstriert. Speziell führt die Bestimmung der freien Parameter im Potential
durch die Forderung, daß die ersten n Koeffizienten in der effektiven Reichweitenentwicklung
korrekt wiedergegeben werden, zu einer Vorhersage für den nächsten Koeffizienten.
Dies ist anders in einer effektiven Theorie ohne Pionen, die im Abschnitt 2.2

etrachtet wurde. In diesem Fall sind solche Vohersagen nicht möglich.
• Im Modellraum niedriger Impulskomponenten kann man auch die Nichtlokalitäten im
Ortsraum studieren. Wir haben gezeigt, daß für typische Werte des Abschneideparameters
das effektive Potential V(x, x') hochgradig nichtlokal ist und daß es vom
ursprünglichen lokalen Potential sehr abweicht. Nur für sehr große Werte des Abschneideparameters
bekommt man das ursprüngliche lokale Potential wieder.
Während diese Doktorarbeit aufgeschrieben wurde, wurde eine ähnliche Arbeit von Bogner
und Mitarbeitern [214] durchgeführt. Dies spricht für eine große Aktivität in diesem Feld.
In dieser Arbeit hat man effektive Potentiale im Niederimpulsbereich unter Anwendung
der Methode gefalteter Diagramme ("folded-diagrams") abgeleitet, die von Kuo, Lee and
Ratcliff [215] entwickelt wurde. Als Ausgangspunkt wurden das Bonn-A und das Paris
Potential gewählt. Diese Methode führt zu nicht hermiteschen effektiven Potentialen und
zur Erhaltung der "half-shell" NN T-Matrix.
2. Zweitens haben wir den realistischen Fall der Kernwechselwirkung betrachtet und eine neue
Methode, nämlich den Projektionsformalismus, zur Herleitung der Kräfte zwischen mehreren
(zwei, drei, ... ) Nukleonen aus effektiven chiralen Lagrangdichten vorgestellt. Dazu
mussten wir zuerst die Regeln für das Abzählen der Impulspotenzen ("power counting") modifizieren,
die ursprünglich von Weinberg vorgeschlagen wurden. Diese Modifizierung war
notwendig, da im Projektionsformalismus der gesamte Fockraum in die Unterräume mit bestimmter
Anzahl von Nukleonen und Pionen aufgespalten wird. Während in zeitgeordneter
Störungstheorie die resultierenden Wellenfunktionen nur bis zu einer bestimmten Ordnung
in der chiralen Entwicklung orthonormal sind, taucht dieses Problem im Projektionsformalismus
gar nicht auf. Ferner hängt das Zweinukleonenpotential in den vorausgehenden
Rechnungen im Rahmen der zeitgeordneten Störungstheorie explizit von der Gesamtenergie
zweier Nukleonen ab. Diese explizite Energieabhängigkeit des Potentials führt zu Schwierigkeiten
bei Anwendungen auf Systeme mit drei und mehr Nukleonen (obwohl sich diese
Energieabhängigkeit mit bestimmten N-Teilchen Wechselwirkungen zur führender Ordnung
wegkürzt). Wir fassen jetzt die Hauptergebnisse des dritten Kapitels zusammen:
• Wir sind von dem allgemeinsten chiral invarianten Hamiltonoperator für nichtrelativistische
Nukleonen und Pionen ausgegangen und zerlegten den vollen Fockraum in zwei
Unterräume r] and A. Der erste Unterraum enthält nur rein nukleonische Zustände,
während im zweiten alle übrigen Zustände enthalten sind. Um einen effektiven Hamiltonoperator
herzuleiten, der nur im r]-Raum wirkt, haben wir eine unitäre Transformation
durchgeführt, die durch einen Operator AAr] parametrisiert ist, siehe (3.190).
Nach einer Projektion A i auf die Zustände mit i Pionen wird aus der Entkopplungsgleichung
(3.192) ein unendliches System (3.205) von gekoppelten Gleichungen. Wir haben
bewiesen, daß dieses Gleichungssystem störungstheoretisch nach Potenzen der Skala
Q kleiner Impulse gelöst werden kann. Dazu haben wir entsprechende Abzählregeln
für Impulspotenzen entwickelt und damit die Ordnungen aller Terme in der Gleichung
(3.192) analysiert. Wir konnten eine rekursive Vorschrift zur Lösung dieses Gleichungssystems
formulieren, so daß die Operatoren A i Ar] für jede endliche Zahl i der Pionen
und zu jeder benötigten Ordnung in der Q-Entwicklung berechnet werden können.
Der effektive, nur im r]-Unterraum wirkende Hamiltonoperator Heff kann dann aus der
Gleichung (3.197) mit Hilfe dieser Regeln (3.223)-(3.229) hergeleitet werden.
• Unter Anwendung des Projektionsformalismus haben wir die expliziten Ausdrücke für
das Zweinukleonenpotential zur nächstführenden Ordnung angegeben (die führende
111

IV
Ordnung des Potentials besteht aus dem Einpionaustausch und zwei Kontaktwechselwirkungen).
Insbesondere haben wir auch die Selbstenergie- und Vertexkorrekturen zu
solchen Kontaktwechselwirkungen berechnet, die früher nicht betrachtet wurden.
• Wir diskutierten ausführlich Ähnlichkeiten und Unterschiede zwischen unserem und
dem aus der zeitgeordneten Störungstheorie hergeleiteten Potential. Speziell stellte
sich heraus, daß in unserem Formalismus der isoskalare spinunabhängige zentrale und
der isovektorielle spinabhängige Anteil, die zu dem Zweipionaustausch gehören, sich
wegheben. Darauf wurde zuerst in der Referenz [108] hingewiesen, wobei in dieser Arbeit
ein anderer Formalismus benutzt wurde. Der Wegfall der oben angegebenen Beiträge
tritt unter Benutzung der zeitgeordneten Störungstheorie nicht auf. Jedoch die
wichtigsten Eigenschaften des Potentials im Projektionsformalismus sind seine Energieunabhängigkeit
und die Orthonormalität der entsprechenden Wellenfunktionen.
• Wir haben die Renormierung des Zweinukleonenpotentials zu der nächstführenden Ordnung
(NLO) durchgeführt. Insbesondere haben wir gezeigt, daß alle auftretenden ultravioleten
Divergenzen durch eine entsprechende Umdefinition der axialen Kopplungskonstanten
gA sowie der verschiedenen Kontaktwechselwirkungen beseitigt werden können.
Die renormierten Ausdrücke für das NLO Potential stimmen mit den von der Münchner
Gruppe überein, wobei die letzteren mit Hilfe von Feynmann Diagrammen abgeleitet
wurden.
• Im Abschnitt 3.8.1 haben wir auch die Struktur des Potentials in der nächst-nächstführenden
Ordnung (NNLO) im Rahmen unseres Formalismus diskutiert. Dabei hat
sich herausgestellt, daß die Ergebnisse in beiden Methoden (im Projektionsformalismus
und in der zeitgeordneten Störungstheorie ) gleich sind. Das NNLO Potential wurde mit
einer anderer Methode von der Münchner Gruppe berechnet (die aber zu den gleichen
Ergebnissen führt) [108].
• Im Abschnitt 3.8.3 haben wir den Projektionsformalismus verallgemeinert, um die Effekte
der virtuellen Anregung der ß-Resonanz im Rahmen der "small scale expansion"
berücksichtigen zu können. Wir diskutierten die führenden Beiträge der intermediären
ß-Anregungen zum Potential. Auch in diesem Fall liefern die beiden Methoden zur
Herleitung des Potentials die gleichen Ergebnisse.
• Desweiteren haben wir die führenden Beiträge zu der Dreinukleonenkraft betrachtet.
Wie in der zeitgeordneten Störungstheorie kürzen sich die verschiedenen Beiträge
führender Ordnung gegenseitig weg, so daß die führende Dreinukleonenkraft verschwindet.
Der Mechanismus dieses sich Weghebens ist aber in unserem Formalismus anders.
Er hängt nicht mit dem Wegheben von Termen zusammen, die durch die Iteration
des energieabhängigen Zweinukleonenpotentials erzeugt werden, wie es in der zeitgeordneten
Störungstheorie der Fall ist. Stattdessen kann dieses Wegheben in unserem
Formalismus auf das Auftreten von den "reduziblen" Diagrammen zurückgeführt werden,
deren gen aue Bedeutung im Abschnitt 3.8.1 erklärt ist. Diese Diagramme dienen
zur Sicherstellung der Orthonormalität der Wellenfunktionen und werden deswegen
manchmal als Wellenfunktionsrenormierungsgraphen bezeichnet.
• Wir haben auch die allgemeinste Lagrangdichte d�/.;S.3) für Kontaktwechselwirkungen
mit vier nukleonischen Beinchen bis zur Ordnung ßi = 3 konstruiert, die reparametrisierungsinvariant
ist (Reparametrisierungsinvarianz ist eine Folge der Lorentzinvarianz
der ursprünglichen Theorie, siehe [182]). Die in den vorangehenden Rechnungen
[74], [76], [78], [127] verwandte Lagrandichte enthält vierzehn freie Parameter

CL.,14
und führt auf Zweiteilchenkräfte, die vom Gesamtimpuls P zweier Nukleonen
abhängen. Während solche P-abhängigen Kräfte die Rechnungen im Zweiteilchensystem
im Schwerpunktsystem nicht beeinflussen, würden sie für Prozesse mit
zusätzlichen Teilchen sowie für drei und mehr Nukleonen sehr wichtig werden. Wir
haben gezeigt, daß die Forderung nach der Reparametrisierungsinvarianz für solche
Kontaktwechselwirkungen zu gewissen Bedingungen an die Parameter Ci führt. So
sind nur 7 dieser 14 Parameter wirklich unabhängig. Als Folge davon hängen die NLO
und NNLO Potentiale nicht von dem Gesamtimpuls ß ab.
3. Drittens haben wir im Kapitel 4 die Kernkräfte und verschiedene Eigenschaften des Zweinukleonensystems
basierend auf der chiralen effektiven Feldtheorie und dem Projektionsformalismus
berechnet. Die Ergebnisse dieser Untersuchung können wie folgt zusammengefaßt
werden:
• Wir haben das Zweinukleonenpotential in NNLO, hergeleitet mit dem Projektionsformalismus,
betrachtet. Dieses besteht aus Ein- und Zweipionenaustäuschen, die auch
die Wechselwirkungen aus dem Hamiltonoperator der Dimension zwei (ßi = 2) enthalten.
Die entsprechenden Niederenergiekonstanten (LEes) wurden aus einer U ntersuchung
der Pion-Nukleon Streuung genommen. Zusätzlich tragen auch zwei und sieben
Kontaktwechselwirkungen jeweils ohne und mit zwei Ableitungen zum Potential bei.
Die entsprechenden Kopplungskonstanten sind durch die Anpassung an die Daten zu
bestimmen.
• Für große Impulse wird das Potential unphysikalisch und muß regularisiert werden.
Wir haben diese Regularisierung auf dem Niveau der Lippmann-Schwinger Gleichung
mit Hilfe einer scharfen oder exponentiellen Abschneidefunktion durchgeführt, siehe
Abschnitt 4.1. In NLO hängt die Physik nicht von der Wahl des Abschneideparameters
im Bereich zwischen 400 und 650 MeV ab. In NNLO wird dieser Bereich größer,
nämlich er liegt dann zwischen 650 und 1000 MeV. Dies kann durch den chiralen Zweipionaustausch
erklärt werden, der in NNLO mr-Korrelationen enthält und eine neue
Massenskala größer als die doppelte Pionenmasse mit sich bringt.
• Wir haben gezeigt, daß die NLO Kopplungskonstanten der Kontaktwechselwirkungen
so kombiniert werden können, daß in jeder Partialwelle jeweils nur eine Kombination
von Parametern auftaucht, siehe (4.16)-(4.23).1 Genauer gesagt können die neun
Kopplungskonstanten mit vier nukleonischen Beinchen durch die Anpassung an die zwei
S-Wellen, vier P-Wellen sowie an den Mischungswinkel EI für Laborenergien kleiner als
100 MeV eindeutig bestimmt werden. Dies vereinfacht die Anpassung der Parameter
beträchtlich im Vergleich zu dem Vorgehen in Referenz [78], in welcher eine globale
Anpassung an alle niedrigen Partialwellen durchgeführt wurde. Wie aus dem "power
counting" zu erwarten ist, verbessern sich die Ergebnisse bei den Übergängen von LO
zu NLO und zu NNLO, siehe Fig. 4.1.
• In NNLO werden die S-Wellen mit einer sehr hohen Genauigkeit (für Laborenergieen

VI
sagen für die Schwerpunktsimpulse 150 MeV und höher sind weitaus besser als die
entsprechenden NLO und NNLO Ergebnisse aus der KSW Methode.
• Alle übrigen Partialwellen sind frei von Parametern. Die D-Wellen, insbesondere die
3 D1 und 3 D3, werden sehr gut beschrieben. Wir haben auch die Abhängigkeit unserer
Ergebnisse in diesen Kanälen von dem Wert des Abschneideparameters diskutiert. Das
Potential in NNLO ist zu stark in den Spin-1 F -Wellen. Für die höheren Partialwellen
haben wir die Ergebnisse der Münchner Gruppe [108] reproduzieren können. Diese
besagen, daß der Einpionaustausch in den meisten Fällen gut funktioniert und der
chirale NNLO Zweipionaustausch für eine deutliche Verbesserung der Ergebnisse in
einigen Partialwellen, wie zum Beispiel in den 3G5-, 3 H5- oder 3 h-Kanälen, sorgt.
• Die Eigenschaften des Deuterons werden in NLO und NNLO meist gut beschrieben,
siehe Tabelle 4.7. In NNLO zeigt die Wellenfunktion des Deuterons eine interessante
Struktur, die durch die Anwesenheit zweier tiefgebundener Zustände in diesem Kanal
zustande kommt. Dies ist eine Folge der NNLO Nährung. Diese unphysikalischen
Zustände haben keine Auswirkung auf niederenergetische Eigenschaften der effektiven
Theorie und können vollständig herausprojiziert werden. Unsere Wellenfunktion kann
zur Untersuchung der Pion-Photoproduktion, der Pion-Deuteron Streuung sowie der
Compton-Streuung am Deuteron benutzt werden (die kombinierte Methode von Weinberg
[114] bleibt jedoch immer noch hilfreich).
• Wir haben in unserer Theorie auch den b..-Freiheitsgrad explizit eingebaut. Mit der
Ausnahme von Kanälen, die auf pionische skalare-isoskalare Korrelationen empfindlich
sind, wie z.B. 3 D3, führt die NNLO-b.. Vorgehensweise zu Ergebnissen, die den NNLO
Ergebnissen ohne explizite b..-Teilchen ähnlich sind. Daraus schließen wir, daß durch die
Berücksichtigung der Effekte der b..-Anregungen mittels Saturierung der LECs in Dimension
zwei sich die wesentlichen mit dieser Resonanz zusammenhängen Phänomene
gut beschreiben lassen. Dennoch ist eine systematischere Studie der Pion-Nukleon
Streuung im Rahmen einer effektiven Feldtheorie mit expliziten b..'s notwendig, um die
quantitativen Effekte weiter spezifizieren zu können.
4. Schließlich haben wir im Kapitel 5 die elektromagnetischen und starken isospinbrechenden
Effekte in der Nukleon-Nukleon Streuung bei niedrigen Energien betrachtet. Dabei haben
wir den KSW Formalismus benutzt, der in Ref. [91] entwickelt wurde. Wir fassen jetzt die
Ergebnisse dieser Untersuchung zusammen.
• Zuerst haben wir die isospinbrechenden Teile der effektiven Lagrangedichte betrachtet.
Die mit dem pionischen und pionisch-nukleonischen System zusammenhängenden
Terme wurden in den Referenzen [199], [222]-[224] diskutiert. Die isospinbrechende
Lagrangedichte im Zweinukleonensektor wurde von van Kolck in Ref. [75] ausgearbeitet.
Wir haben die entsprechenden Terme unter Benutzung eines anderen Formalismus,
nämlich der Methode der äußeren Felder, aufgeschrieben.
• Nach einer kurzen Einführung in den KSW Formalismus haben wir die führenden isospinbrechenden
Effekte in der 1 So Welle diskutiert. Die führende Brechung der Ladungs
invarianz entsteht aus einer Kombination der neutralen und geladenen Pionenmassenunterschiede
im Einpionaustausch und einer elektromagnetischen N N Kontaktwechselwirkung.
Obwohl die zu dieser Kontaktwechselwirkung gehörende Kopplungskonstante
wie Q-2 skaliert, wird sie durch die explizite Anwesenheit der Feinstrukturkonstante
a rv 1/137 numerisch zusätzlich unterdrückt. Wir haben gezeigt, wie man

das KSW Abzählschema ("power counting") in Anwesenheit der isospinbrechenden
Operatoren modifizieren muß .
• Wir haben die 1 So Streuphase für das np, das nn und für das Coulomb-bereinigte pp
System zu nächst führender Ordnung explizit berechnet. Zusätzlich wurde das allgemeine
Klassifizierungsschema verschiedener CIB und CSB Korrekturen angegeben. Dies
erlaubt es, einige phänomenologisch gefundene Ergebnisse zu erklären.
Nach unserem besten Wissen wurde die exakte Projektion eines Nukleon-Nukleon Potentials im
Impulsraum, die im Abschnitt 2.3 ausgearbeitet wurde, vorher noch nie durchgeführt. In dieser
Arbeit haben wir diesen Formalismus zum Studium einiger Probleme angewandt, die im Kontext
einer effektiven Feldtheorie für das NN-System auftauchen. Diese Methode kann jedoch auch
nützliche Einsichten in Bezug auf viele anderen interessanten Fragenstellungen der Kernphysik liefern.
Insbesondere können relativistische Effekte in einer konsistenten Weise studiert werden, da
die Impulskomponenten der Ordnung der Nukleonenmasse und höher ausintegriert sind und deshalb
eine konvergente relativistische Entwicklung zu erwarten ist. Außerdem wäre es interessant,
die Methode auf Drei- und Mehrnukleonensysteme zu verallgemeinern.
Desweiteren läßt sich sagen, daß die Methode einer unitären Transformation (Projektionsformalismus)
noch nie vorher im Kontext der effektiven chiralen Feldtheorie angewandt wurde. Frühere
Berechnungen der Zweinukleonenwechselwirkung [73], [74], [76], [78] aus chiralen effektiven Lagrangdichten
basierten auf der zeit geordneten Störungstheorie und führten zu energieabhängigen
Potentialen. Wie oben schon betont wurde, besteht der wichtigste Vorteil unseres Formalismus gegenüber
der zeitgeordneten Störungstheorie in der Energieunabhängigkeit des Potentials und in der
Orthonormalitätseigenschaft der entsprechenden Wellenfunktionen. Die Abzählungsvorschriften
("power counting rules"), die eine Rechnung in beliebiger Ordnung in der Niederimpulsentwicklung
ermöglichen, wurden im Abschnitt 3.6 und in den Anhängen A, B und C ausgearbeitet.
Unsere Ergebnisse fur die Nukleon-Nukleon Wechselwirkung, abgeleitet aus dem allgemeinsten
chiralinvarianten Hamiltonoperator, zeigen, daß die von Weinberg vorgeschlagene Methode nicht
nur qualitativ sondern sogar viel besser als erwartet, nämlich quantitativ, funktioniert. Sie erweitert
die erfolgreichen Anwendungen der effektiven Feldtheorie (chiraler Störungstheorie ) in
den Pion und Pion-Nukleon Sektoren auf Systeme mit zwei und mehr Nukleonen. Es liegt nun
nahe, auch diejenige Prozesse erneut zu betrachten, die bisher im Rahmen eines kombinierten
Formalismus von Weinberg [114] berechnet wurden (1f - d-Streuung [114], [217], "(d ---+ 1f o d [115],
"(d ---+ "(d [116]). Die entsprechenden Prozesse in Systemen mit mehr als zwei Nukleonen sollten
auch betrachtet werden. Zusätzlich sollten auch ladungs- und isospinbrechende Effekte neu
untersucht werden (in Bezug auf frühere Studien siehe [218], [99]).
Ein sehr wichtiger und aktueller Untersuchungsbereich beinhaltet die Anwendung der chiralen
Störungstheorie auf 3N-Wechselwirkungen. Während die ersten Ergebnisse im Rahmen der
KSW Methode und einer effektiven Theorie ohne Pionen schon publiziert wurden, siehe [119],
[120], [121], [122] , wurde bisher noch keine komplette Berechnung im Potentialformalismus unter
Berücksichtigung der führenden Dreiteilchenkraft im Rahmen der CHPT durchgeführt. Es wäre
sehr interessant zu sehen, ob die Mitnahme von den Dreiteilchenkräften dieses Typs zur Lösung
des Ay-Problems in der elastischen nd-Streuung führen kann. Für eine weitere Diskussion dieses
Problems siehe u.a. [196].
Wir möchten auch darauf hinweisen, daß die Renormiering des effektiven Hamiltonoperators nach
Eliminierung der pionischen Freiheitsgrade sorgfältiger untersucht werden muß, als es im Abschnitt
3.8.2 gemacht wurde. Insbesondere sollte man auch die Null- und Einteilchenoperatoren
Vll

Vlll
betrachten, um eine komplette Renormierung in NLO und NNLO durchführen zu können. Ferner
sollte man das Problem der Renormierung in einer beliebigen Ordnung in der Entwicklung
nach Impulsen untersuchen. Nach unserem besten Wissen wurde dieses Problem im Rahmen des
Hamiltonformalismus noch nicht im Detail ausgearbeitet (auch nicht im Kontext einer effektiven
Feldtheorie) .
Zu guter Letzt haben wir gezeigt, daß die Isospinbrechung in die Methode der effektiven Feldtheorie
für das Zweinukleonensystem in der KSW Formulierung systematisch eingebaut werden
kann. Dafür muß man die allgemeinste isospinbrechende Lagrangedichte konstruieren und das
Abzählschema ("power counting") entsprechend modifizieren. Wir haben gezeigt, daß diese Methode
eine systematische Klassifizierung verschiedener Beiträge zur Ladungsunabhängigkeits- und
Ladungssymmetriebrechung (CIB und CSB) erlaubt. Speziell ermöglicht das Abzählschema zusammen
mit der Dimensionsanalyse das Verständnis, warum die Beiträge einer möglichen Ladungsabhängigkeit
der Pion-Nukleon Kopplungskonstante unterdrückt sind. Es wäre interessant,
diesen Formalismus zu anderen Partialwellen sowie zu höheren Energien zu erweitern, um z.B. Isospinbrechung
in der Pion-Produktion zu untersuchen.

Contents
1 Introduction
2 Low-momentum effective theories for two nucleons 11
2.1 What is effective? . . . . . . . . . . . . 11
2.2 Two nucleons at very low energies . . . . 15
2.3 Going to higher energies: a toy model . . 23
2.3.1 Method of unitary transformation 24
2.3.2 Regularization of the decoupling equation 29
2.3.3 Numerical realization of the projection formalism . 32
2.3.4 Unitary transformation for a potential of the Malfliet-Tjon type 33
2.3.5 Implication for effective theories 37
2.3.6 Coordinate space representation . . . . . . . . . . . 46
3 The derivation of nuclear forces from chiral Lagrangians 48
3.1 Chiral symmetry . . . . . . . . . . . . 48
3.2 Effective Lagrangians . . . . . . . . . . 55
3.3 Chiral perturbation theory with pions 69
3.4 Including nucleons . . . . . . . . . . . 73
3.5 Bloch-Horowitz scheme and the method of unitary transformation 78
3.6 Application to chiral invariant Hamiltonians . . . . . . . . . 81
3.7 Nuclear forces using the method of unitary transformation . 87
3.8 Two-nucleon potential . . . . . . . . . . . . . . . . . . 91
3.8.1 Expressions and discussion . . . . . . . . . . . . . . 91
3.8.2 Renormalization of the N N potential at NLO . . . .
3.8.3 Phenomenological interpretation of some of the 7fN LECs and the role of
100
the ß(1232) . . . 105
3.9 Three-nucleon potential . . . . . . . . . . . .
113
4 The two-nucleon system: numerical results
4.1 Bound and scattering state equations .
4.2 The fits . . . .
4.3 Phase shifts . .
4.3.1 S-waves
4.3.2 P-waves
4.3.3 D- and F-waves
4.3.4 Peripheral waves
4.4 Deuteron properties . .
4.5 Results for the NNLO-ß approach
IX
1
119
119
122
128
128
133
136
140
144
149

x
5 Isospin violation in the two-nucleon system
5.1 Isospin violating effective Lagrangian . . . . . . . . . . . .
5.2 Brief introduction into the KSW approach . . . . . ... .
5.3 The leading CIB and CSB effects in the N N 1 So channel
5.4 Numerical results . .
5.5 Classification scheme
6 Summary and outlook
A Dimensional analysis of the projected decoupling equation (3.206)
B Operators contributing to eq. (3.206) at order r
C Small scale expansion within the projection formalism
D Divergent loop integrals
E Anti-symmetrization of the contact interactions
F The complete set of N N contact interactions with two derivatives
G Partial wave decomposition of the N N potential
H Formulae for the deuteron properties
152
153
156
159
162
163
165
172
175
177
180
181
184
192
194

Chapter 1
Introduction
Nuclear physics as a science has a long history of almost 100 years. In 1911 Rutherford demonstrated
that large-angle alpha-particle scattering can only be explained in terms of a positively
charged nucleus of a radius of about 10-12 cm, which is much smaller than the typical atomic
radii of the size rv 10-8 cm. This important discovery allowed to make a crucial progress in
understanding the atomic structure and stimulated the development of new theoretical concepts
[1], [2]. A few years later Heisenberg [3] and Schrödinger [4] formulated rigorously the principles
of quantum mechanics.
In spite of advances in the knowledge of atomic systems, the concept of the structure of the nucleus
remained unclear until 1932 when a neutral particle, the neutron, was discovered by Chadwick [5].
After that it was proposed that nuclei are composed of neutrons and protons which have nearly
the same mass. Already at that time Heisenberg suggested that the neutron and proton can be
looked upon as corresponding to two states of the same particle [6]. Four years later Cassen and
Condon introduced the concept of isotopic spin to describe these two states [7].
The basic problem in nuclear physics is determining the nature of the interactions between neutrons
and protons. Only if these forces are known we can understand various properties of nucleL
The first and very successful concept of nuclear forces was proposed by Yukawa [8], who assumed
that nucleons interact due to the exchange of massive scalar particles (mesons).
Although the meson-exchange theory of nuclear forces has undergone many developments and
modifications within the last 60 years, Yukawa's fundamental idea about a meson exchange origin
of nuclear forces is still valid today. In order to keep the manuscript consistent, we would like to
briefly summarize basic historical developments of the meson exchange models of nuclear forces.
More detailed historical reviews can be found in [10], [11].
First modifications of Yukawa's theory were due to Proca [12] and Kemmer [13]. They extended
Yukawa's model to pseudoscalar and pseudovector particles. A few years later models including
combinations of pseudoscalar and pseudovector fields were considered by M�ller and Rosenfeld
[14] and by Schwinger [15]. In 1946 Pauli [16] predicted the existence of an isovector pseudoscalar
meson since the exchange of particles with these quantum numbers could correctly explain the
sign of the quadrupole moment um of the deuteron which was measured in 1939 [17] . Few years
later 1l"-mesons were observed experimentally [9] and identified with the particles predicted by
Pauli.
In 1951 Taketani, Nakamura and Sasaki [18] introduced a new concept. The whole range of
nucleon-nucleon interactions is divided into three regions: a long range part (r 2: 2 fm), an
intermediate region (1 fm � r � 2 fm) and a short range or core part with r � 1 fm. Since the
1

2 1. Introduction
nuclear force of Yukawa type due to exchange of particles with the mass m is proportional to
exp (-mr)/(mr), where r denotes the distance between two nucleons, exchanges of the heavier
particles are, in general, of shorter range. While the long range part of the nuclear force is
dominated by one-pion exchange, two-pion exchange as well as exchanges of heavier mesons
become important in the intermediate region. In the core region nucleons are overlapping with
each other. So, the "classical" meson-exchange picture of the nuclear force is not adequate any
more. Taketani and the co-workers proposed a phenomenological treatment of the short-range
nuclear force.
After, in the 1950s, the long-range part of the nuclear interactions due to the one-pion exchange
became well established, more attention has been paid to the two-pion exchange contributions.
Various approaches have been proposed to attack this problem [19], [20]. Soon it became clear
that the two-pion exchange itself can not account for a sufficiently strong spin-orbit force, whose
evidence was transparent from the data.
The situation has changed after the experimental discovery of heavy mesons in 1969s, which has
motivated developments of the one-boson exchange models (OBE) of the nuclear force. One assumed
in such models that the two- and more-pion exchanges can be parametrized in terms of
multi-pion resonances. With only few parameters one could achieve a quite remarkable quantitative
agreement with the existing nucleon-nucleon scattering data [21], [22], [23].
Apart from the one-boson exchange models other concepts like dispersion relations and field theoretical
approaches were suggested to describe the nucleon-nucleon interaction. In particular,
dispersion theory was applied to calculate the two-pion exchange contributions to the N N amplitude
starting from 7r N and 7r7r scattering data. The most detailed work in this direction was
performed by the Stony Brook [24], [25] and the Paris [26], [27] groups. The short range part
of the nuclear force in these models was parametrized by OBE and by some phenomenological
terms. The nucleon-nucleon interaction developed by the Paris group became known as the Paris
potential. The field theoretical approach to the 27r-exchange contributions using the technique of
Feynman diagrams was considered by Lomon and collaborators [28].
One of the most detailed and successful works within the meson-exchange models was performed
by the Bonn group. Starting from the OBE model [23] based on relativistic time-ordered perturbation
theory, Erkelenz, Holinde, Machleidt and Elster calculated two- and some of the threeand
four-pion exchange diagrams and extended the model to take into account effects of virtual
isobar excitations. The so constructed Bonn-potential, which includes apart from these terms
exchanges of heavy mesons, allows for an accurate description of the nucleon-nucleon scattering
data. Recently, few attempts have been undertaken by the J ülich group to incorporate also
correlated meson exchanges (like 7r7r, 7rp) [30], [31], [32].
A series of modern high-quality potentials based on the pure one-boson exchange model was
constructed by the Nijmegen group [33], who also performed a partial-wave analysis (PWA) l of
all pp and np scattering data [36].2 With 15 free parameters they were able to achieve with the
Nijmegen 93 potential a X2 / Ndata of about two. Fitting some of the free parameters in each partial
wave separately in the case of the Nijmegen I,II potentials allows to describe the data perfectly
with X2 / Ndata rv l.
Another quite successful modern high-quality N N force is the so-called CD-Bonn potential,
[36].
1 Another partial-wave analysis has been performed by Arndt and collaborators [34], [35].
2 The Nijmegen database consists of 1787 pp and 2514 np scattering date in the energy range from 0 to 350 MeV

which is also based on the one-boson exchange model [29]. This potential also takes into account
charge dependence of the nuclear force.
Also, a more phenomenological approach of the Argonne group to keep in the potential explicitly
only the one-pion exchange and to represent all remaining contributions in a general operator form
led after fitting of 40 adjustable parameters to a quite accurate description of the two-nucleon
scattering data with an excellent X 2 per datum of 1.09 [37].
All these boson-exchange models appear to be very successful in describing the two-nucleon
scattering data as weIl as the deuteron properties. In spite of this success in the pure twonucleon
sector there exist so me situations in which the OBE models do not allow far satisfactory
explanations and description of data. Since usually the two-nucleon scattering data are used to fix
the free parameters in the potentials, only on-energy-shell physics is reproduced correctly. Offenergy-shell
effects can only be tested in reactions with more than two nucleons and in processes
with external probes.3 Consequently, due to this off-shell ambiguity, the value of, for instance,
the triton binding energy varies remarkably when calculations are performed with the different
two-nucleon forces. None of the existing so-called realistic potentials lead to a correct value of
the triton binding energy. Typically, one observes underbinding of about 5-10%, which can be
explained in terms of a missing three-nucleon force (3NF). Indeed, the inclusion of a three-body
force allows to describe the triton binding energy correctly. One should point out that up to
now much less is known about the nature of the three-body forces compared to the two-body
interactions. This has partly historical reasons: only relatively recently it became possible to
solve the three-body Faddeev equations exactly for any type of realistic two-body force, after all
necessary technical and computational tools were worked out and sufficient computer powers were
available [38], [39], [41]. In addition to these technical and computational difficulties, the effects
of the three-body force in most cases are small and require precise experimental measurements of
the observables.
At presence several models for the three-body force are available. Some of them like the Fujita­
Miyazawa [42] or the Tucson-Melbourne [43] forces are based on the two-pion exchange with one
intermediate ß excitation. Such a two-pion exchange interaction represents the longest range
part of the 3NF. Another model proposed by the Brazil group includes in addition 7f-P and p-p
exchanges [44]. The Urbana-Argonne group has worked out a purely phenomenological 3NF [45].
Finally, the Tucson-Melbourne force has been extended to take into account also the 7f-P and
p-p exchanges (the so-called Tucson-Melbourne model) [46], [47]. For various combinations of
the two- and three-nucleon interactions one can always adjust parameters (typically the values of
the cut-off in the 3NF) to exactly reproduce the triton binding energy [40]. For some observables
the inclusion of the 3NF does, however, not lead to an improved description compared to the
calculations with the purely two-body interactions [41]. The most prominent example of such an
observable is an analyzing power Ay in elastic nd scattering at low energies. The purely twonucleon
calculations yield results which are about 25% off the experimental values. Inclusion of
3 To avoid misunderstanding we note that, in general, only on-energy-shell effects can be observed. One should
understand the "off-energy-shell effects" in this context as follows. Assume, one has some definite two- and morebody
forces. Then, one can always unitarily transform the full Hamiltonian and obtain new two- and more-body
interactions. The old and new two-body potentials are phase-equivalent and give the same two-body S-matrix.
However, to calculate three- and more-body observables one needs the off-shell two-body T-matrix, which is, in
general, modified after performing the unitary transformation. This is precisely what we understand under such
"off-energy-shell effects" . The differences in the off-shell T-matrices are compensated by the modified many-body
interaction and the final result for on-shell quantities is, clearly, the same before and after performing the unitary
transformation.
3

4 1. Introduction
the Thcson-Melbourne 3NF does not lead to a significant improvement. This problem still remains
unsolved [196].
In spite of a very successful description of most of the experimental data the modern realistic
N N potentials are quite phenomenological approaches. For instance, one of the basic ingredients
of these models is a fictitious (J' (550 MeV) boson, which is needed to produce the rather strong
attraction in the central part of the potential. This attraction is evident from the N N scattering
data, see also discussion in ref. [48]. Such a (J'-meson, however, has never been observed experimentally.
For a recent discussion about the experimental evidence of the (J'-meson see ref. [49].
Another common feature of the boson-exchange models is the ad hoc introduction of the strong
form-factors4 at each meson-nucleon vertex in order to correct the large momentum behavior of
the potential and to allow for practical calculations. Such form-factors are usually interpreted
in terms of higher-order (in a coupling constant) meson-nucleon and meson-meson interactions,
dressing each vertex. Because of the non-perturbative nature of such interactions one is not able to
directly calculate these form-factors and approximates them typically by some smooth functions
of momenta. Only in the case of the so-called Ruhr-potential (RuhrPot) [50] the form-factors
are generated dynamically by solving the corresponding integral equations. Another interesting
feature of this potential is its manifest energy independence resulting from the method of unitary
transformation [51], [53] which has been used to define the nuclear force. On the contrary, the
standard approach based on the time-ordered perturbation theory leads necessarily to an effective
Hamiltonian which depends on the initial energy of the nucleons (i.e. to an operator which
depends explicitly on its own eigenvalue). Such an energy dependence of the nuclear forces causes
many complications in practical calculations for more than two nucleons. Another problem of the
various meson-exchange potentials is how to systematicaHy improve these models and whether
the one-boson exchange approximation is well justified. Indeed, if one takes the meson-exchange
picture of the nuclear force seriously, one would expect that two- and more-meson exchanges are
also important (because of the strong meson-nucleon coupling). Usually one argues at this point
that the force based on the two- and more-meson exchanges are, in general, of shorter range than
the one-meson exchange force and thus are less relevant for the energy region of nuclear physics
(with exception of pion-exchange interactions). Although quite plausible, such arguments should
be considered more qualitative than quantitative.
The main conceptual problem of the OBE models, however, can be illustrated in terms of the
following "classical" picture [54]: ass urne that hadrons are hard spheres. The charge radius of the
proton is .J(rI) rv 0.6 fm, while the typical size of light mesons is about 0.5 fm. Then mesons
can not mediate the nuclear force at distances below rv 2 x 0.6 fm + 2 x 0.5 fm = 2.2 fm. Even if
this picture is very much simplified and does not take into account quantum mechanical effects,
it becomes clear that the traditional meson-exchange picture should not be adequate to describe
the nuclear matter phenomena at distances below 2 fm.
One hopes that quantum chromodynamics (QCD), the fundamental theory of the strong interaction,
can help to avoid these problems of the OBE models and provide us with a deeper understanding
of the nature of the nuclear forces. QCD is a SU(3)color gauge theory of the strong
interaction, formulated in terms of quarks and gluons. The structure of nucleons as weH as interactions
between them are, in principle, completely determined by QCD. Direct calculations of the
nuclear force from QCD are not possible up to now. This is because at low energies the interaction
is too strong to apply the usual perturbative methods. However, many attempts were made
to incorporate more information from QCD in models of the N N interactions. In the so-called
4 Such form-factors are anyway not well-defined in quantum field theory.

quark models one typically represents nucleonic degrees of freedom in terms of 3-quark states.
This allows to fit simply the quantum numbers of the nucleons as well as to provide their color
neutrality. The property of confinement which is an important phenomenon of QCD is usually
simulated by some phenomenological confining potentials [55].
Recently a combined model of the nuclear force, the so-called Moscow potential, has been proposed
[56]. This can be considered as furt her extension and improvement of earlier models of the
combined type, which have been developed by Faessler et al. [57], [58]. There the interactions
at large separations are described in terms of meson-exchanges. At short distances one notes
that the six-quark state no longer resembles two distinct nucleons (Le. two three-quark nucleon
clusters), as it happens for large separations. Such considerations lead to a suppression of the local
repulsive co re of the nuclear force, which is an obligatory attribute of boson-exchange models.5
The effects of the repulsive core are simulated by additional deeply-lying bound states, which are
generated by the Moscow-type potentials. As a consequence, additional internal nodes appear
in the deuteron and scattering wave functions. Furthermore, such a suppression of the repulsive
co re leads to a sm aller value for the wN N coupling constant compared to boson-exchange models,
which is consistent with the one predicted from SU(3)-symmetry. The suppression of the repulsive
core in quark model calculations was also pointed out earlier by Harvey [59].
A more systematic attempt to include information from QCD is based on effective field theories
(EFT). Already 20 years aga Weinberg [60] pointed out that requiring the (approximate) SU(2) x
SU(2) chiral symmetry, which is evident from the QCD Lagrangian, leads to a model-independent
and systematic low-energy expansion in momenta for the S-matrix. Chiral symmetry of the QCD
Lagrangian is not a symmetry of the physical vacuum, which is only invariant under a smaller
SU(2) subgroup. Thus, chiral symmetry is spontaneously broken, which can also be verified
from the hadronic spectrum. As a consequence, according to Goldstone's theorem [61], [62], one
observes three light pseudoscalar bosons, which would be massless in the exact chiral limit and can
be identified with pions. Weinberg illustrated his idea with a systematic low-energy expansion
of the amplitude on the example of 11'11' scattering. Such an expansion is possible, since in the
exact chiral limit only derivative interactions between pions are allowed. Thus, for vanishing
momenta pions become free particles. The original idea of Weinberg has been worked out in
detail in calculations by Gasser and Leutwyler for 11'11' scattering and many other processes [63],
[64]. For a review article see ref. [65] . Once the coupling constants of mesonic interactions in
the most general chiral invariant Lagrangian are fixed from some processes, various predictions
can be made for other reactions and observables, allowing for non-trivial tests of the Standard
Model. Even two-loop calculations have been performed recently [66]. Further, external fields can
be incorporated quite naturally and systematically. This opens the possibility to study various
processes including photons.
It is well-known how to couple non-Goldstone degrees of freedom in a chiral invariant manner
[67]. In 1988 the technique of the effective Lagrangian has been applied to calculate various pionnucleon
amplitudes to one-loop [68]. In this approach, the relativistic treatment of nucleon fields
in loops caused an intrinsic problem with chiral power counting. This is because an additional
scale, namely the nucleon mass, which remains finite in the chiral limit is introduced due to
baryon propagators. The problem was successfully solved within the heavy baryon formulation
of chiral perturbation theory (CHPT) [69], [70]. The basic idea of this scheme is to integrate
out the "heavy" component of the baryon field. This can be carried out explicitly in a Lorentz
covariant way in terms of velocity-dependent fields. This allowed numerous applications for the
5The local repulsive core results in the boson-exchange models from exchanges of vector wand p mesons.
5

6 1. Introduction
pion-nucleon system as well as for the processes with electroweak probes [71]. For arecent review
see ref. [72].
Motivated by successful applications of CHPT in the 7f7f and 7f N sectors Weinberg proposed in
1990 to extend the formalism to the N N interaction [73]. The crucial difference is, however,
that the nucleon-nucleon interaction is non-perturbative at low energies: it is strong enough to
bind two nucleons in the deuteron. Thus, direct application of CHPT to the N N amplitude will
necessarily fail. The way out of this problem, proposed by Weinberg, is to apply CHPT not to
the amplitude but to a kernel of the corresponding integral Lippmann-Schwinger (LS) equation.
Such a kernel (or potential) can be defined within time-ordered perturbation theory as a sum over
all irreducible diagrams with two incoming and outgoing nucleon lines. Irreducible means here
that no pure 2N intermediate states are allowed. Reducible diagrams are then generated due to
iterations of the kernel in the LS equation. Weinberg has shown that a systematic power counting
for the potential can be derived in a similar manner as in the case of the 7f7f and 7f N systems.
The corresponding Lagrangian should be extended to allow apart from the 7f7f and 7f N also N N
contact interactions, which are not constrained by chiral but only by isospin symmetry.
These ideas have been extensively studied by the Texas-Seattle group [74], [75], [76], [77], [78].
They have obtained an energy dependent two-nucleon potential at next-to-next-to-leading order.
With altogether 26 free parameters6 they were able to achieve a qualitative agreement with the
N N scattering data and some deuteron properties. Also three-body forces have been considered
by this group. Applying this formalism to many-body problems allows to establish a beautiful
hierarchy of the two- and many-body forces: the three-body forces should be weaker than the
two-body ones, the four-body forces should be weaker than the three-body ones and so on.
Since that time the derivation of the nucleon-nucleon interaction using the technique of effective
theories became a subject of quite remarkable interest and intense discussions. Cohen and collab
orators [79], [80], [81], [83] considered in detail an effective theory with pions integrated out.
In such a case simple analytic calculations for the two-nucleon S-matrix can be performed, since
all interactions between nucleons are of the contact type. They examined different regularization
schemes for divergent integrals in the LS equation like dimensional and cut-off regularizations and
made some interesting observations. First, it turned out that in such a pionless effective theory
the cut-off could be taken to infinity only if the effective range parameter of the interaction is
negative. This conclusion follows, in fact, from a theorem which had been proven by Wigner long
time aga [84] and is based only on such general principles like causality and unitarity. The theorem
says, that if a potential vanishes beyond some range R, then the rate d6(k)/dk at which the phase
shift can change with energy is bounded from below by some function of R, k and 6(k). Secondly,
an effective theory with a finite cut-off was found not to be "systematic" in the sense, that higherorder
terms in the potential do not get systematically smaller as the order is increased. Thus, the
expansion does not seem to converge. We will comment more on that in the chapter 2. Finally,
the use of cut-off schemes and dimensional regularization was shown to lead to different results
for the scattering amplitude. The scattering amplitude calculated with dimensional regularization
only maps to the effective range expansion for on-shell momenta k « 1/ JaTe, where a and Te
are the scattering length and the effective range. For such small momenta both schemes produce
identical results in agreement with the effective range expansion. The experimental values for the
S-wave np scattering lengths and effective ranges are
6 Some of these parameters are redundant.
as = (-23.758 ± 0.010) fm ,
at = (5.424 ± 0.004) fm ,
Ts = (2.75 ± 0.05) fm ,
Tt = (1.759 ± 0.005) fm .
(1.1 )
(1.2)

Thus, the effective theory with dimensional regularization has a quite small range of validity
and provides an extremely poor fit to the data for the ISO partial wave [85]. The problem with
the dimensional regularization is that it does not account for power law divergences of the type
Jooo dnq qm, which turn out to be important for the renormalization of the LS equation. In dimensional
regularization divergent integrals in four dimensions are extended to an arbitrary number
of dimensions D. After the regularization is performed, they are typically expressed in terms of
f-functions which depend on D. Ultraviolet divergences correspond to poles of these functions.
They have to be subtracted in order to provide a finite result. In the minimal subtraction scheme
(MS), which is frequently used in field-theoretical calculations with renormalizable theories, all
1/(D-4) poles are subtracted before taking the limit D --+ 4. This allows to remove all logarithmic
divergences. In the effective theory for N N scattering without pions no such logarithmic divergences
appear and the regularized integrals in four dimensions remain finite and scale independent.
The only divergences that appear in such an effective theory are the power law divergences, which
vanish after performing dimensional regularization. Kaplan, Savage and Wise (KSW) proposed to
subtract also poles in D = 3 dimensions to keep trace of power law divergences [90]. They called
this formalism power divergence subtraction (PDS). Using PDS, Kaplan et al. worked out a new
power counting scheme (KSW approach) for systems with a large scattering length and extended
it to take into account pionic degrees of freedom. Equivalent formalisms with different regularization
schemes are discussed in refs. [86], [87], [88]. The leading and non-perturbative contribution
to the scattering amplitude in these methods is due to N N contact interactions without derivatives.
Pion exchanges start to contribute at subleading order and can be treated perturbatively.
That is why analytic calculations are possible. The corresponding S-matrix is (approximately)
unitary7 and does not depend on the renormalization scale which is introduced due to PDS. The
expansion for the S-matrix is expected to break down at momenta k rv 300 MeV. For a series
of interesting applications of the KSW approach to various processes see references [90]-[99]. In
spite of a successful description of many observables at low energies the formalism fails badly to
describe the shape parameters in the ISO and 3S1 _3 D 1 channels, as has been shown by Cohen and
Hansen [100], [101]. They have pointed out that predictions for these quantities would provide
a sensitive test for the correct inelusion of pionic physics. Note furt her that different potential
models give very elose predictions for the shape parameters. This is, according to ref. [102] , due
to the correct treatment of the one-pion exchange. Thus, it remains unelear, whether such a
perturbative treatment of the pion exchange is justified8 [103].
An interesting work within the power counting scheme proposed by Weinberg has been performed
by Park et al. [104], [105]. They concentrated on np scattering for the ISO partial wave and the
deuteron channel and were able to reproduce the empirical scattering phase shifts up to momenta
k rv 300 Me V. The potential they considered contained apart from the contact N N interaction
without and with two derivatives also the leading OPE term. The cut-off dependence of the
results was investigated. Quite recently they also ineluded the leading two-pion exchange as weH
as contact interactions with four derivatives [106]. For the application of the effective field theory
technique to the solar proton burning process p + p --+ d + e+ + Ve see reference [107].
The Munich group considered recently the peripheral np partial waves within chiral perturbation
theory [108]. No contact terms without and with two derivatives contribute in such partial waves
(starting from the D-waves) because of the large values of angular momenta. The interaction is
7 A different approach has been recently proposed by Lutz [89], which preserves unitarity of the S-matrix exactly
and allows far the description of the N N phase shifts at high er energies.
8 Certainly, for very small energies E « M" � 140 Me V one can even completely integrate out pionic degrees of
freedom.
7

8 1. Introduction
completely dominated by pion exchanges. That is why parameter-free predictions are possible,
since the 7r N low-energy coupling constants are known from pion-nucleon scattering processes.
Furt her , the interaction in these channels is much weaker than in the S- and P-waves. That
opens the possibility to use standard perturbation theory. Kaiser et al. have used the technique
of Feynman diagrams to define the two-nucleon T-matrix. They obtained renormalized expressions
for leading and subleading two-pion exchanges. It was found that the model-independent
27r-exchange corrections are too large for the D- and F-waves. This indicates the increasing
importance of shorter range effects in these channels. For G- and higher waves the 27r-exchange
corrections bring the chiral predictions close to empirical phase shifts. In a following publication
[109J also some two-loop diagrams were considered as weIl as the contributions from virtual
�-excitations and from exchange of vector p and w exchange. These corrections improved the
predictions for the F-waves and for the D-waves below 50-80 MeV laboratory energy. Quite
recently also so me classes of the two-loop diagrams have been ca1culated by Kaiser [110], [111].
Similar investigations for some peripheral partial waves but using a different technique were performed
by the Brazilian group [112]. Recently they also analyzed some three-pion exchange
contributions [113] and found quite surprisingly that they have a long range of about 1.5 fm and
tend to enhance the OPEP.
Instead of trying to describe many-body processes completely in terms of effective field theory
Weinberg proposed in 1992 to apply chiral perturbation theory to generate the irreducible kernel
and combine these with phenomenological external nuclear wave functions [114]. These ideas were
adopted by Beane et al. to ca1culate neutral pion photoproduction on deuterium [115]. The results
agree weIl with experiment and allowed for a first quite successful prediction from CHPT in the
two-nucleon sector . A similar formalism was applied recently to study Compton scattering on the
deuteron [116] and neutral pion electroproduction off deuterium [117].
Also many-body interactions became a subject of intense investigations in the context of effective
theories. While Friar et al. [118] tried to constrain the possible form of the three-nucleon force from
chiral symmetry, a different strategy has been chosen by Bedaque and collaborators. Motivated
by the successful application of the KSW approach to two-nucleon problems, they considered the
three-body system at very low energies with pions integrated out [119], [120], [121]. BasicaIly,
they found a good agreement with experiment in the J = 3/2 channel without having any free
parameter. Here, the three-body force does not play any significant rule since, due to the Pauli
principle, particles can not be very close to each other. The situation is different in the J = 1/2
channel. Here it was shown that a three-body short-range force is required in order to eliminate
the cut-off dependence of the amplitude. Recently the quartet S-wave phase shift in nd scattering
was ca1culated including perturbative pions [122].
The purpose of this thesis is to investigate in detail the two-nucleon system within a cut-off
effective theory. The pioneering ca1culations along this line performed by Ord6iiez et al. [78]
leave much room for improvements. First, as already stressed before, some of the parameters
corresponding to the contact interactions in this work are redundant. As we will show later, one
can eliminate them due to anti-symmetrization of the potential. Secondly, we will use the values of
the 7r N low-energy constants consistent with CHPT ca1culations in the one nucleon sector instead
of determining them from the fitting to the N N data. Thus, the number of free parameters in
calculations to the same order in the low-momentum expansion decreases dramatically from 26
as it was the case in ref. [78] to 9 in the present work. We hope that this will allow to make
clearer statements about the role of chiral symmetry in the N N dynamies. Thirdly, we will use
a different formalism for deriving the NN interaction, which leads to an energy-independent

potential. Furt her , we will apply cut-off functions, which do not introduce any additional angular
dependence. That is why the contact interactions will only contribute to the S- and P-waves and
to the E1 mixing angle. The results for higher partial waves are parameter free predictions. That
establishes a connection with the recent work by the Munich group [108], [109]. Furthermore, we
will demonstrate how to redefine the contact interactions in order to have separate parameters
in each low partial wave. This allows for an enormous simplification of the fitting procedure.
FinaIly, our results for the S-waves and for various deuteron properties will be shown to achieve
the level of precision of modern phenomenological potentials. Thus, we believe, that quantitative
calculations for the N N system using the effective field theory approach are possible.
The manuscript is organized as follows. In chapter 2 we will give an introduction to effective
theories. For a better understanding of the philosophy of effective theories the quantum mechanical
two-body scattering problem will be considered. We will apply the method of unitary
transformation [51], [53] to a model two-nucleon Hamiltonian in order to construct from it an
effective potential, which operates in a subspace of momenta below a given cut-off A [123], [124].
For concrete numerical investigations we will restrict ourselves to a potential of the Malfiiet-Tjon
type [125], [126]. This force consists of two terms, an attractive one due to the exchange of a light
meson and a repulsive term parametrized in terms of a heavy meson exchange. The S-matrices
in the full space and in the subspace of low momenta after performing the unitary transformation
will be shown to be identical. Since we are interested only in the region of low momenta, weIl
below the mass of the heavy meson, we can "integrate out" the heavy meson. More precisely,
we will keep in the potential the light meson exchange unchanged and represent effects of the
heavy meson exchange in terms of the N N contact interactions. This i,s rat her similar to what
is usually done for the real two-nucleon system. The crucial advantage of our model is, however,
that the exact effective Hamiltonian for low momenta is known from the unitary transformation.
This allows us to compare these two effective theories, the one with contact forces and the exact
one, and to study effects of fine tuning which are important for reproducing the large value of the
scattering length. FinaIly, we will address some furt her issues related to the chiral perturbation
theory approach of the two-nucleon system.
We will begin chapter 3 with a brief introduction to chiral symmetry and construct the twoand
three-nucleon potential, based on the most general chiral effective pion-nucleon Lagrangian
using the method of unitary transformations [127]. For that, a power counting scheme consistent
with this projection formalism has been developed. The details are discussed in appendices A, B.
Using this power counting scheme it has been shown how to calculate the effective potential to
an arbitrary order in the small moment um scale Q. In appendix C we have also generalized the
above approach to the case of the so-called "small scale expansion" for the potential, in which
not only the pion mass and external momenta of the nucleons but also the tlN mass splitting
are considered as small quantities and are treated in the same footing. Such an expansion allows
to take into account effects in the potential due to intermediate excitations of the tl-isobar. To
the best of our knowledge, this is the first time that the method of unitary transformation is
systematically applied in the context of chiral effective field theory. We discuss in detail the
similarities and differences to the existing chiral nucleon-nucleon potentials and show that, to
leading order in the power counting, the three-nucleon forces vanish lending credit to the result
obtained by Weinberg using old-fashioned time-ordered perturbation theory. We will also consider
the problem of the renormalization of the potential. For that we will use the express ions for the
divergent integrals presented in appendix D and perform anti-symmetrization of various contact
interactions as explained in appendix E. Finally, we have analyzed the contact terms in the
effective Lagrangian with four nucleon legs and two derivatives and found that seven of fourteen
9

10 1. Introduction
such interactions written down by Ord6iiez et al. [74], [77], [78] are redundant or have coefficients
suppressed by two powers of the inverse nucleon mass, see appendix F. This has no consequences
for the two-nucleon scattering or bound state problem but becomes quite important for systems
with three and more nucleons andjor for two-nucleon problems with external probes.
In chapter 4 we will concentrate on practical applications of the derived potential for studying
bound and scattering states in the two-nucleon system [128]. To that aim we will determine
the ni ne parameters related to the contact interactions by a fit to the np S- and P-waves and
the mixing parameter EI far laboratory energies below 100 MeV. The corresponding partial wave
decomposition of the potential is explained in appendix G. An alternative determination of some
of these constants from the leading effective range parameters is considered as weIl. Other lowenergy
constants are obtained from pion-nucleon scattering. We will discuss numerical results
for phase shifts and show that the predicted partial waves and mixing parameters for higher
energies and higher angular momenta are mostly well described for energies below 300 MeV. We
will also consider various deuteron properties following from our potential. The role of virtual
�-excitations is also discussed.
In the chapter 5 we will consider charge symmetry and charge independence breaking in an effective
field theory approach for the two-nucleon system. We first discuss various terms in the effective
Lagrangian, which lead to isospin violating effects. For that investigation we use the formalism
proposed by Kaplan, Savage and Wise [91] and calculate the nn, np and pp I So phase shifts.
We do not explicitely include virtual photons in this analysis. The charge dependence observed
in the nucleon-nucleon scattering lengths is due to one-pion exchange and one electromagnetic
four-nucleon contact term. This gives a parameter free expression for the charge dependence of
the corresponding effective ranges, which is in agreement with the rat her small and uncertain
empirical determinations. We also compare the low energy phase shifts of the nn and the np
system with data. Finally, we summarize our findings in chapter 6.

Chapter 2
Low-momentum effective theories for
two nucleons
2.1 What is effective?
All known interactions in nature can be classified in terms of four fundamental types: strong,
weak, electromagnetic and gravitational. The first three of them have been unified and build the
basis of the Standard Model. The crucial role in the development of this model has been played
by the requirement of renormalizability. One can easily check the property of renormalizability of
a theory by introducing a parameter ßi characterizing an interaction of type i, which contains fi
(bi) fermionic (bosonic) fields. In four dimensions it is defined by
ß· = 4-d·--f· -b·
2 - 2
3
2 2 2 , (2.1)
where di is the number of derivatives. This quantity simply defines the mass dimension of a
related coupling constant.1 This allows for the following classification of all interactions: those
with ßi 2: 0 are called renormalizable, whereas interactions with ßi < 0 are non-renormalizable
[132]. One defines furthermore a subclass of superrenormalizable interactions with ßi > O. For
instance, the interaction of electrons with photons in quantum electrodynamics (QED), -e1{;j'ljJ,
is renormalizable since according to eq. (2.1) one has ß = 4 - 0 - 3/2 x 2 - 1 = 0, whereas the
electron mass term -m1j;'ljJ is an example of a superrenormalizable interaction. Obviously, only
a finite number of renormalizable theories exists since ßi is bounded from above. It has been
shown that the theory is renormalizable (i. e. that ultraviolet divergences arising in calculations
of observables cancel as so on as all parameters2 of the theory are expressed in terms of physical
or renormalized quantities) if all interactions are renormalizable, see e.g. [133].
For quite a long time it was commonly believed that renormalizability is a fundamental principIe.
Indeed, a non-renormalizable quantum field theory is not well-defined if a finite number of
interactions enters the corresponding Lagrangian. Although one can always perform tree-Ievel
calculations, quantum corrections will produce ultraviolet divergences that cannot be absorbed in
are-definition of the parameters in the Lagrangian. In fact, renormalization can also be carried
out in the case of non-renormalizable theories if one includes in the Lagrangian all interactions
(an infinite number), that are consistent with symmetry principles of the corresponding theory
[133]. Such symmetry principles restrict, of course, the structure of possible interactions, but
1 In this thesis we will always use "natural" units with 1i = c = l.
2There is always a finite number of parameters in such renormalizable theories.
11

12 2. Low-momentum effective theories for two nucleons
these constrain in the same way the ultraviolet divergences. This makes it possible to absorb all
ultraviolet divergences arising from loops in parameters of the Lagrangian and the theory becomes
well-defined.
At first sight, such non-renormalizable field theories seem to be completely useless, because an
infinite number of vertices in the Lagrangian would lead to an infinite set of possible diagrams,
which contribute to a definite process. In such a case, practical calculations would be, of course, not
possible. However, as we will see below, in some cases in the low-energy limit non-renormalizable
interactions become weak and are strongly suppressed compared to renormalizable ones. This is
because coupling constants of non-renormalizable interactions can be expressed like Ci / An, w here
A is so me mass scale, n > 0, and Ci is so me dimensionless number. Thus, if no additional mass
scale appears in the corresponding theory or if all mass sc ales are much smaller than A, one
observes in the low-energy limit E ---+ 0 a strong suppression of non-renormalizable interactions
due to factors E / A. 3 So, in spite of infinitely many parameters entering the Langangian, the
theory possesses a predictive power. Indeed, if calculations are performed to a certain finite level
of accuracy, only a finite number of interactions is relevant. As so on as all relevant couplings are
fixed from so me processes, predictions can be made for other processes and observables. This is
w hat is usually understood under effective field theories (EFTs).
Figure 2.1: The leading l/m! contribution to photon-photon scattering at
second order in the fine structure constant CI' can be represented by a contact
interaction (filled cirele).
There are many physical situations, in which EFTs provide a very useful and sometimes the only
available way of performing practical calculations. If the fundamental theory is known, EFT can
be useful as its low-energy approximation. For example, the leading photon-photon scattering
diagram in QED, shown in fig. 2.1, can be calculated from the QED Lagrangian at one loop order
where 'I/J denotes the electron fields, Fftv == 8ft AV - 8V Aft is the field strength tensor, me and
CI' = e 2 /(47r) rv 1/137 denote the electron mass and electromagnetic fine structure constant,
respectively. In eq. (2.2) we have not shown gauge fixing terms. Already in 1936 Euler, Kockel
3 This statement obviously holds at the tree-level. Quantum corrections include loop integrals, which may
be ultraviolet divergent and should be regularized and renormalized. This requires more careful considerations.
Weinberg has shown that quantum corrections are also suppressed [60).
(2.2)

2.1. What is eHective 13
and Heisenberg proposed to use an effective Lagrangian for calculating photon-photon scattering
at energies much smaller than the electron mass [134], [135], [136]
where F/LV = (-1/2) EJtvpu Fpu is the dual of the field strength tensor FJtv The ellipsis represent
further corrections ofhigher order in 1/me and a. We learn from the effective Lagrangian (2.3) that
photon-photon scattering at second order in a resulting from the process, which includes creation
and destruction of electron-positron pairs, can be represented at very low energies by simple fourphoton
contact interactions without knowing anything about the structure of photon-electron
interaction. This is because one cannot probe details of photon-electron coupling at energies
much smaller than the electron mass. In fact, the effective Lagrangian (2.3) is quite general, since
it includes all possible terms with four field strength tensors and without additional derivatives,
which are allowed by Lorentz and gauge symmetry. All information about the structure of photonelectron
interactions in QED is hidden in the values of the coupling constants of the (F,wFJtV) 2
and (F'WFJtV) 2 contact interactions.4 In QED one can derive these perturbatively to all orders of
a. If the fundamental theory (QED) would be not known, these coefficients could be fixed from
experiment. This simple example illustrates one of the basic principles of the effective field theory
method: the dynamics of a system at low energies is not sensitive to details of the interaction at
high energies.
The method of effective Lagrangians allows to calculate not only the leading low-energy approximation
of the scattering amplitude but also to systematically account for corrections. In the
case of photon-photon interactions this corrections are of two forms. First, even at order a 2
there are additional interactions containing at most four field strength tensors FJtv, FJtv and any
positive number of derivatives,5 like F Jtv (O/m�)FJtv FQßFQß , FJtv (0 2 /m�)FJtv FQßFQß , . . . . Secondly,
there are vertices at order a > 2 with and without derivatives. Some of them have the
same structure as the terms at order a = 2 but also many new interactions like, for instance,
1/m�(F Jtv FJtV)3 with more photons appear. Apart from various tree corrections loop diagrams
will contribute at higher orders. At each order in the low-momentum expansion one can absorb
all ultraviolet divergences arising from loops by corresponding counter terms in the effective Lagrangian.
Thus, calculations with an arbitrary accuracy can be performed within such an effective
theory. For one-loop calculations with the effective Lagrangian of Euler et al., see ref. [138].
In the above example for scattering of light by light the use of effective Lagrangians might be
useful, since it allows to simplify calculations at low-energies. For much more complicated full
QED calculation of the same process see the reference [139]. In some cases calculations within
an underlying fundamental theory are not possible: either the theory is not known or standard
perturbative methods are not available. This happens, in particular, far QCD at low energies. In
such situations effective field theories provide a simple and powerful way to perform a systematic
analysis of low-energy phenomena.
Effective field theory is not only an extremely useful tool for performing practical calculations,
which is alternative to standard methods of renarmalizable field theories. Presumably, it is the
only kind of physical theories that we have at present. Indeed, there is no evidence that the
4 Such a situation, when the strengths of couplings is controlled by some (heavy) resonances, happens quite often
in various effective theories. This is called resonance saturation. For some particular examples see ref. [137)
5 There are no such terms with more than four field strength tensors at order Q2 since in this case one would
need to couple at least six electrons, which requires three powers of Q (or more).

14 2. Low-momentum effective theories for two nucleons
fundamental theories like QED or the Standard Model are really fundamental in the sense that they
remain valid for arbitrary high energies. Rather, these renormalizable theories may be low-energy
approximations of some other and more fundamental underlying theories. As already pointed
out before, non-renormalizable interactions are usually suppressed by inverse powers of energy,
at which new physics appear. For example, the interactions in the effective QED Lagrangian
(2.3) are suppressed by four powers of the inverse electron mass. In principle, there are two
ways of testing new physical effects beyond the Standard Model: either one can try to perform
more precise measurements of observables to test effects of non-renormalizable interactions or one
can increase the energy. At present, no experimental inconsistencies of the Standard Model to
data have been observed. This indicates that the energies at which new physics might become
significant are quite high and, probably, in the TeV region [140].
Up to now we have only considered effective field theories. The discussed ideas find also many
interesting applications in quantum mechanics and in other fields of physics. For example, the
standard multipole expansion of a complicated current source J(r, t) of size d, which generates
radiation with large wavelengths A » d, already uses the basic idea of effective theories: large
distance physics should be not very sensitive to details of the structure at short distances. That
is why one usually observes quick convergence of such multipole expansions.
In the next sections of this chapter we will concentrate, basically, on purely quantum mechanical
problems. All ideas and methods of effective theories, that we have discussed in the context of
relativistic quantum field theories, remain, of course, valid in this case. In particular, as already
stressed above, the low-energy behavior of a theory is not sensitive to details of the short-distance
interaction. 6 Following Lepage's lecture [141], we would like to enumerate here the basic steps of
constructing the effective theory:
• First, one should correctly incorporate the low-energy physics into the effective theory. This
must be known from the underlying theory. Note that if the underlying interaction has a
finite range, one can always restrict an effective theory to very low energies, at which the
details of the interaction are not important.7 In such a case one does not need to know
anything about the underlying theory.
• Secondly, one should introduce an ultraviolet cut-off to exclude high-moment um states. The
cut-off furthermore provides finiteness of various integrals arising by performing calculations.
• Finally, the missed short-range physics is represented in form of local correction terms with
arbitrary coefficients to the effective Hamiltonian. These coefficients have to be fixed from
the data. It is important that one should keep all local terms allowed by the symmetries of the
underlying theory. Putting new correction terms systematically removes the dependence of
the low-energy observables on the cut-off, which is compensated by "running" of parameters
in the effective Hamiltonian. Any given precision can be achieved with a finite number of
correction terms.
60ne needs, in general, higher energies to probe shorter-distance physics. This follows immediately from Heisenberg's
uncertainty relation.
7 A counter example is the Coulomb potential which has an infinite range. In the language of quantum field
theory, the Coulomb force results from the exchange of photons. Since photons are massless, they can not be
completely integrated out even at very low energies. One can, however, separate photons into "soft" and "hard"
on es and keep only "soft" photons explicitly within an effective theory.

2.2. Two nuc1eons at very low energies
2.2 Two nucleons at very low energies
In this section we would like to apply the ideas formulated above to the scattering problem of two
nucleons at very low energies. We will consider proton-neutron scattering and concentrate only
on the strong interaction between these two particles. Possible electromagnetic corrections due
to the magnetic moments of neutron and proton are very small and will be neglected. Here we
will, basically, follow the considerations of refs. [81], [83]. A detailed analysis of the two-nucleon
system within a pionless effective theory at next-to-next-to-Ieading order in the low-momentum
expansion can be found in ref. [98].
The longest range part of the nuclear force is due to the exchange of pions. Since we are interested
in very low energies, even pionic degrees of freedom can be integrated out. Thus, the effective
Hamiltonian entirely consists of contact interactions with four nucleon legs. These terms may
contain any even number of derivatives (terms with an ocid number of derivatives would break
parity invariance) and insertions of spin and isospin matrices. Apart from parity invariance, one
should also provide rotational invariance as weIl as symmetry under time reversal operation. In
many cases it appears to be very useful to require also exact isospin invariance of the effective
Hamiltonian. Because of the low-energy limit, a nonrelativistic treatment of nucleons is weIl
justified. Relativistic corrections may be systematically incorporated in the same way as the
corrections to interaction between nucleons, i.e. by putting local one-nucleon terms with more
than two derivatives in the Hamiltonian.
In what follows, we will only consider S-waves. For nonrelativistic nucleons, one can express the
effective Hamiltonian in the two-nucleon center of mass system in the form
H(p',p) == (S p'lHIS p) = J(p' -p) + V(p',p) ,
m
where m is the nucleon mass. Here, we have cancelled in the first term in eq. (2.4) the factors
p2 resulting from the kinetic energy operator and from the three-dimensional J-function. The
potential V(p',p) is given by
p ,p = 0 + 2 P + p + 4 P + p + 4P P + . .. , (2.5)
V( ' ) C C ( ,2 2 ) C1( ,4 4 ) C 2 ,2 2
where the C's are (unknown) coefficients.8 Note that terms like p'p do not arise in the S-channel
after performing a partial wave decomposition. This can be seen from the formulae of appendix
G. For instance, the term p' . p, which could lead to such a structure, vanishes after projecting
onto the S-state because of its angular dependence. The off-shell g two-body T-matrix can be
found from the Lippmann-Schwinger integral equation
T(p',p;k) = V(p',p) + m (OO q2dqV(p',q) k2 \ .
Jo -
q T(q,p;k) ,
+ ZE
where k is the off-shell momentum and E ---+ 0+. An iterative solution to this equation is represented
diagrammatically in fig. 2.2. Clearly, this equation is not well-defined if one uses the potential
(2.5): each iteration would produce ultraviolet divergences. As already stressed in the last section,
one should cut off the integral entering the LS equation (2.6). Alternatively, one can introduce a
regularized potential, for instance, like
8 This definition of C's differs by the factor 21r2 from the one given in [81], [83).
9 The matrix TUb, eh , q) is called on-shell if Iql I = 1il2 I = q, half-shell if Iq 2 1 = q i=- ql and off-shell otherwise.
15
(2.4)
(2.6)
(2.7)

16 2. Low-momentum effective theories for two nucleons
x
+ +
• • •
Figure 2.2: The diagrammatic representation of the integral equation (2.6).
The shaded blob corresponds to the T-matrix and the filled circles denote the
effective potential.
where the regulator function JA(p 2 ) is 1 for p « A and 0 for p » A.lO We will now solve the
equation (2.6) analytically with the renormalized potential (2.7). We will not restrict ourselves to
a concrete choice of the function JA(p 2 ). The solution of the Lippmann-Schwinger equation can
be expressed as
T(p' ,p; k) �
IA(P"
) (�o P'''Yij(k)P'j) JA(p') . (2.8)
The number n depends on how many terms are kept in the potential (2.5). We will consider only
the first two terms in the expansion (2.5) and hence take n = 1. It is convenient to express the
potential v;.eg (pi, p) in the form
where the matrix )..ij is given by
The LS equation (2.6) can now be expressed in a matrix form like
with
I(k)
r(k) = ).. + )"I(k) r(k) ,
roo q
2 dq q2 fJ. (q2)
JO k
)
2 -q2 +ic
2 d q4 fJ.(q2)
00
Io q q kLq2+ic
(2.9)
(2.10)
(2.11)
(2.12)
lOThe function JA (p2) must decrease fast enough far large p to ensure that all ultraviolet divergences in the LS
equation are removed.

2.2. Two nucleons at very low energies
Here In and I(k) are defined by
Note that In and I(k) depend on A. Solving the linear matrix equation (2.11) one obtains
r (k) = � ( -Co - Ci(h + k 2 h t k 4 h � k 6 I(k)) C2( -1 + C2(h + k 2 h + k4 I(k))) )
w C2( -1 + C2(h + k h + k I(k))) -Ci(h + k 2 I(k))
with the constant w given by
w = - (-1
+ C2h) 2 + (h + k 2 I(k)) (Co + C2(C2h - k 2 (-2 + C2h)))
Using eq. (2.8) one finds for the on-shell T-matrix Ton(k) == T(k, k; k):
Ton (k) = fÄ (k 2 )
Co + C2(C2h + k 2 (2 - C2h))
(-1 + C2h)2 - (h + k2I(k))(Co + C2(C2h + P(2 - C2h)))
17
(2.13)
(2.14)
(2.15)
(2.16)
This is the final result for the on-shell T-matrix at next-to-Ieading order, which follows from our
effective theory. ll
We are now in the position to fix the unknown constants Co and C2 from the data. One way to do
that is to fit the phase shift b corresponding to eq. (2.16) to the experimental one. Alternatively,
one can require that the scattering length a and the effective range r, which are the leading two
coefficients in the effective range expansion for the S-wave phase shift
1 1
k cot(b) = --+ -rk 2 + v2k 4 + v 3 k 6 + v4k8 + O(k10) ,
a 2
(2.17)
are exactly reproduced. In this formula the v's denote the so-called shape parameters. We prefer
here the second way, since it allows for analytic expressions of Co and C2• For the derivation of the
effective range expansion of phase shifts in scattering theory the reader is referred to refs. [142],
[143]. We can easily obtain the effective range expansion for the T-matrix using the definition of
the on-shell S-matrix projected onto the S-wave:
Thus, the phase shift b can be expressed as
Applying the relation
with z = i - 2j(7rkmTon) to eq. (2.19) leads to
so n (k) = exp(2ib(k)) = 1 - i7rkmTon(k) .
b = ; i In (1 - i7rkmTon(k)) .
_ 1 (iZ - 1) In = arccot(z) ,
2i iz + 1
1 7rm
-T (kcotb-ik)
-- -- + - rk + V2k + V3k + V4k + O(k ) - zk
2 a 2
7rm ( 1 1 2 4 6 8 10 · )
11 At leading order one should keep in the potential (2.5) only the first constant term.
(2.18)
(2.19)
(2.20)
(2.21)

18 2. Low-momentum effective theories for two nucleons
Let us now consider the on-shell T-matrix (2.16). The imaginary part of its inverse is given by
Here we have used the equality
k2 _
P i7r
q2 - 2k
i
= -7rkm .
2
8(k - q) ,
(2.22)
(2.23)
where P stays for the principal value integral. Thus, the imaginary parts of the T-matrix in
eqs. (2.21) and (2.16) agree automatically. This has to be expected from the unitarity of the Smatrix.
Before fixing the constants Co and C 2 we have to define the low-momentum expansions
of iX(k2) and of the principal value integral P(I):
1 + hk2 + i 4 k 4 + O(k6) ,
m (to + t 2 k2 + t4 k4 + O(k6))
(2.24)
(2.25)
where fi and ti are some constants. No negative powers of k2 in eq. (2.25) are allowed since
otherwise P (fo oo dq fÄ ( q2) / (k2 - q2)) would not exist for k = O. Expanding the inverse of the
T-matrix (2.16) and comparing the first two terms of this expansion with those in eq. (2.21) we
obtain the following two conditions for Co and C 2 :
7rm
2a
7rmr
4
Solving the second equation leads to two solutions for C 2 :
where
C 2 = � ±
2ah + 7rm
13 hVi5
D = (7r2m2 + 2am7r(2h - hh) + a2( 41l + rm7r h - 4mhto))
For the constant Co one obtains from eq. (2.26)
C _
Is (2ah + m7r)( -2a1j + 2ahIs + Ism7r ± 2Vi5Is)
_ _
0 - 12 3 D12 3
(2.26)
(2.27)
(2.28)
(2.29)
(2.30)
Let us now consider the conditions, under which real solutions for the constants Co and C 2 exist.
For that we should require
D > 0. (2.31)
From the definition (2.13) we see that
(2.32)
where 0: is non-negative dimensionless constant. This simply follows from dimensional reasons.
For the expansion coefficients tn, in in eqs. (2.24), (2.25) we have:
1
in

2.2. Two nucleons at very low energies 19
For large A the term a2rm7fh in the expression (2.29) becomes dominant. Obviously, for positive
effective range r > 0 an upper bound for the values of A exists, above which no real solutions
for Co and C2 are available. This statement is quite general but not new: it can be derived from
Wigner's theorem [84], which is based on such general principles like causality and unitarity. The
theorem says that the rate dJ(k)jdk by which the phase shift can change with energy is bounded
from below by some function of the range R of the potential, the moment um k and the phase
shift J. Thus, since the effective range in the 1 So and 3 SI channels is positive, one cannot take
the cut-off A to infinity. At first sight, this might look like a failure of the cut-off regularization.
Indeed, the regularization scale (cut-off) is usually taken to infinity if the standard renormalization
technique is used for renormalizable field theories. However, in an effective theory the cut-off
becomes physically relevant. Introducing a cut-off in an effective theory provides finiteness of
the amplitude. At the same time, it leads to suppression of all high-moment um virtual states,
for which no information is available. That is why in an effective theory one should take the
cut-off below the scale, at which new physics is expected to appear. In the low-energy nucleonnucleon
effective theory such new physics is associated with pions. We cannot expect that our
derivative expansion of the potential will correctly represent effects of pion exchanges at energies
rv Mn . Thus, if we would take the cut-off A much larger than the pion mass Mn , we would simply
incorporate wrang physics. This explains why the cut-off should remain finite and of the order of
Mn ·
Up to now we have only considered a general scattering problem in the S-channel. We have
neither restricted ourselves to a concrete regularization scheme nor to a particular partial wave.
Let us now consider the 1 So channel and compare different regularization schemes for that case.
We will consider cut-off and dimensional regularizations. Let us start with the cut-off theory and
specify the function JA (p2). For simplicity, we will take
This leads to
(k) = -i7fm m 1 A + k
I
2k + 2k n A - k .
Further, one finds for the particular choice (2.34) of the function JA(p2) that
Ii = 0, ti = (i + l )Ai+l 1
.
For the constants Co and C2 one obtains
with
Co
9D + m(7f - 2aA) (m(97f - 8aA) ± 18VD)
5DAm
- 3 ( m (7f - 2aA))
mA3 1 ± VD '
D = �m2 (37f2 - aA (127f + aA(7frA - 16))) .
For the inverse of the T-matrix we get from eq. (2.16) a quite simple expression:
1 = Am (1 + (7f - 2aA)2 ) _
To n (k) 2aA(7f - 2aA) - a2(4 - 7frA)k2
I(k)k2 .
(2.34)
(2.35)
(2.36)
(2.37)
(2.38)
(2.39)
(2.40)

20 2. Low-momentum effective theories for two nucleons
One can immediately check that the first two terms in the effective range expansion (2.21) are
correctly reproduced. Let us now take a look at the experimental data in the 1 S o channel. All
information we need to this order of ca1culation is contained in the scattering length a and the
effective range r. One could first try to estimate their values from simple dimensional analysis:
since the longest range part of the nuclear force is given by the OPE, it is natural to assume that
the scale entering the values of the coefficients in the effective range expansion is of the order of
the pion mass M'/[ . The experimental values of this quantities are given in eq. (1.1). Our naive
estimation works fairly well for the effective range r:
2
r = (2.75 ± 0.05) fm '" - ,
M'/[
but it fails to describe the scattering length, for which we have
17
a = (-23.758 ± 0.010) fm '" - M'/[ .
(2.41 )
(2.42)
The scattering length takes an unnatural large value in the 1 So channel. As already stressed
before, we have to take the cut-off A of the order of the pion mass. That is why we can make use
of the relation
aA ' " aM'/[ » 1 ,
(2.43)
to simplify the express ions (2.37)-(2.40). From eq. (2.39) we obtain
The coupling constants Co and C2 take the values
Co '"
1 ( 192 - 97frA 36� )
mA 80 - 57frA ± 5V16 - 7frA '
__
3 ( ) 2�
_ 1±
mA3 V16 - 7frA
.
(2.44)
(2.45)
(2.46)
As already discussed above, the inverse of the on-shell T -matrix can be represented at very low
energies in terms of the effective range expansion (2.21). This follows from analytic properties of
the amplitude. From the expression (2.40) we see that the effective range expansion is reproduced
provided that
2A(7f - 2aA)
k2 <
' " 4A2
.
(2.47)
a (4 - 7fr A) 7fr A - 4
Since the cut-off A as well as the inverse of the effective range are of the order of the pion mass,
one obtains from the inequality (2.47)
(2.48)
In these estimations we do not care about the numerical factors like 4/7f. The condition (2.47)
shows that the range of applicability of our effective range theory is in accordance with our
expectation: we reproduce physics correctly at momenta smaller than the cut-off A. This holds
also for a --+ 00.

2.2. Two nuc1eons at very low energies 21
In quantum field theories dimensional regularization (DR) appears to be a simple and powerful
calculational tool, which allows to preserve all known symmetries. We will now try to apply it to
the low-energy 1. effective theory for two nueleons. Since we will not use the cut-off we should put
JA(p2) = Further, all integrals In have to be generalized to an arbitrary number of dimensions
d:
1000 d3q 1000 ddq
I - -2m7f2 -- q n __
-3 ---+ -2m7f2J-L3-d q n-3
n -
(2.49)
0 (27f)3 0 (27f)d '
where f.L is the mass scale introduced due to the regularization. It turns out that all power law
divergences like those in eq. (2.49) vanish after performing the dimensional regularization, see, for
example, [82]. This will be of crucial importance for our furt her considerations. Thus, we obtain
a very simple expression for the inverse
1
of the T-matrix from eq. (2.16):
(2.50)
Again, we can fix the constants Co and C2 requiring that the first two terms in the effective range
expansion (2.21) of the on-shell T-matrix are exact reproduced. This lead to
Co = 2a
7fm
,
(2.51)
The inverse T-matrix (2.50) with the constants Co, C2 given by eq. (2.51) maps to the effective
range expansion only for:
(2.52)
This has to be compared with the corresponding result (2.47) for cut-off regularization. Dimensional
regularization provides a very small domain of validity of the effective theory if the
scattering length is large. For the case a ---+ 00 such a theory even becomes completely useless.
The failure of dimensional regularization can easily be understood if one considers a simple model
[83] with an S-wave separable potential given by
2) -1/2
( 2 12) -1/2 ( V (pI, p) = - : 1 + � 1 + 7f m 2 � 2 ' (2.53)
where 9 is a dimensionless coupling and A is a scale that characterizes the range of the interaction.
The on-shell T-matrix can be obtained from the LS equation (2.6),
Ton(k) = V(k, k) [1 - m (oo q2 dq k2 V(q ; q) . ] -1 = _� [�(1 + A k :) - (A + ik)] -l . (2.54)
io - q + ZE 7fm 9
Comparing this result with the effective range expansion (2.21) allows to read off the scattering
1 length and effective range:
-1 9 2
- Ag ' r = gA . (2.55)
Both quantities are inversely proportional to the range of the potential A, as it follows from
dimensional reasons. Further, the scattering length takes an unnatural large value if the coupling
constant 9 is elose to one. This can be seen as a result of the cancelation between the two
terms in the first bracket in eq. (2.54). The first term results from the tree-contribution to the
amplitude, whereas the second one comes from virtual excitations in the loops, see fig. (2.2). In the
a - - -

22 2. Low-momentum effective theories for two nucleons
effective theory the potential is represented in form of an expansion in powers of momenta. Thus,
loop contributions can be expressed in terms of the power law divergences In and the integral
I, which yields the imaginary part of the S-matrix. For cut-off regularization these power law
divergences In are given by eq. (2.13) and are, in general, proportional to A n . It is crucial, that all
power law divergences vanish if one uses dimensional regularization, see eq. (2.49). That is why
the cancelation between tree- and loop-contributions to the amplitude, which is responsible for
large scattering length, becomes impossible in this case. This explains the failure of dimensional
regularization of the L8 equation for a ---+ 00. 12
Let us now make one more remark concerning the predictive power of our effective theory. The
question is, whether we are doing better than the effective range expansion (2.21). With other
words, once we have fixed free parameters from the first two terms of the effective range expansion,
can we make predictions for higher terms in this expansion? To answer this quest ion we can simply
calculate the next coefficient V2 in eq. (2.21) within the cut-off approach (2.34). We obtain from
eq. (2.40):
(2.56)
In the last equation we made use of the limit of a large scattering length. Thus, the value of
V2 obtained from the effective theory at next-to-leading order depends strongly on the cut-off
A. Clearly, the coefficient V2 cannot be predicted within our effective theory. This is because we
have not incorporated any additional information about the nucleon-nucleon interaction apart
from its parity and rotational invariance as weIl as symmetry under time revers al operation. Our
effective theory should equally weIl describe low-energy properties of any underlying theory, which
satisfy these constraints. For instance, the experimental value of the shape parameter V2 for the
neutron-proton 1 So channel is V2 rv -0.48 fm3. However, if the nucleon-nucleon interaction would
be as simple as the one in eq. (2.53), we would have V2 = O. Various symmetries that we have
incorporated in the effective theory do not allow to extract additional information about higherorder
terms in the effective range expansion from its first two coefficients. In some sense, the
effective theory considered in this section is trivial, since it only reparametrizes the effective range
expansion. Effective theory becomes non-trivial if one can incorporate more information from
the underlying theory. In the next section we will consider such an example. In chapters 3 and
4 we will show how to systematically incorporate the property of chiral symmetry into the N N
effective theory and demonstrate its remarkable predictive power.
Before finis hing this section we would like to comment on the convergence of the effective theory.
To show that the derivative expansion (2.5) of the effective potential converges one should demonstrate,
that the higher terms in this expansion give sm aller corrections to low-energy observables.
The authors of refs. [81], [83] have analyzed the convergence of the expansion (2.5) in the 1 S0 channel for cut-off values satisfying
1 4
- « A « - .
(2.57)
a Kr
12 Kaplan, Savage and Wise [90] pointed out that this is not a problem with dimensional regularization, but rather
with the subtraction scheme. Usually, only such poles of regularized expressions are subtracted, which correspond
to physical number of space-time dimensions d (i.e. d = dphys = 4). This allows to keep track of logarithmic
divergences. In the case of the LS equation with contact interactions the regularized expressions remain finite
since there are no logarithmic divergences at all. In order to account for linear divergences one can subtract poles
corresponding to d = dphys - 1 dimensions. Such power divergence subtraction scheme (PDS) looks, in some sense,
like a compromise between cut-off and dimensional regularizations: for n > 1 one has In --7 ° like in dimensional
regularization with the standard subtraction scheme, whereas h cx A, similar to cut-off approach.

2.3. Going to high er energies: a toy model 23
To estimate the quantum averages of the various terms in the expansion (2.5) they used the leading
order wave function of the low-energy bound state. (In the real world no bound state exists in this
partial wave.) Such a wave function corresponds to the potential V(p' ,p) = Co()(A2_p'2)e(A 2 _p2)
and is proportional to
'IjJ (O)(p) cx ()(A2 -p2)
mB+p2 ,
(2.58)
where B denotes the binding energy. For this case it has been shown that the quantum averages of
all terms in the expansion (2.5) with two and more powers of momenta are of the same size. Here
we refrain from doing similar estimations. We do not want to restrict ourselves to cut-off values
much smaller than 1/r as it is guaranteed by the second equality in (2.57). Requiring only aA » 1
does not lead to simple scaling properties of an coupling constants, which would be present for
rA « 1, as it follows from eqs. (2.45), (2.46). Also the leading order wave function of the bound
state might be a too naive approximation for calculating quantum averages. Rather , one should
take low-energy scattering states. A more careful investigation of convergence of the expansion
(2.5) would necessarily require a more complicated algebra and applying numerical methods. l3 In such a case the outcome might be different. We will co me back to this question in the next
section, where a more interesting example will be considered and the outcome is indeed different.
To summarize, we have illustrated the ideas discussed in sec. 2.1 with the example of two-nucleon
scattering at low energies. The effective potential consists of contact interactions with any number
of derivatives. Such an effective theory works as wen as the effective range expansion for the
inverse of the T -matrix. Cut-off regularization is an appropriate tool to provide finiteness of the
amplitude. One should keep the cut-off finite and of the order of the pion mass. There exists
an upper bound for the cut-off, above which no real solutions for the constants Ci are available.
Dimensional regularization with the standard subtraction scheme fails to provide an adequate
description of the amplitude if the scattering length is very large.
2.3 Going to higher energies: a toy model
In the last section we have considered the effective theory for nucleon-nucleon scattering at very
low energies. This analysis was quite general and applicable, in principle, to any underlying
theory. The only information we used is that the underlying interaction has a finite range of
order rv Such a theory can clearly fit the data for nucleonic three-momenta sm aller
1/M7r•
than M7r• It was shown that the effective theory in this case reproduces the effective range
expansion for the amplitude but cannot go beyond it. We will now consider another example
and go to higher energies. Our starting point is a simple model of the S-wave nucleon-nucleon
interaction, which contains long and short range parts. The parameters of this model are adjusted
to reproduce phase shifts in the 1 So and 3 SI channels. This model is our "fundamental" theory.
We will construct an effective theory for that case and keep explicitly the long range part of
the interaction. This is different to the considerations of the last section. Short range physics
is again represented by contact interactions. To make the scattering amplitude finite we will
introduce a sharp cut-off A in the momentum space, which is chosen between the two scales
corresponding to long and short range parts of the underlying force. Thus, our effective theory
is defined on the subspace of momenta below A. An interesting and new point is that we are
ahle to explicitly integrate out high moment um components from the underlying theory [124].
For that we divide the moment um space into two subspaces, spanning the values from zero to A
1 3 For instance, at higher orders in the derivative expansion one has more complicated non-linear conditions for
fixing the constants Ci .

24 2. Low-momentum effective theories for two nuc1eons
and from A to 00, respectively. Consequently, the two-nucleon Hamiltonian can be regarded as
a two-by-two matrix connecting the two momentum subspaces. By an unitary transformation it
can be block-diagonalized decoupling the two subspaces. In this mann er one can construct an
effective unitarily transformed Hamiltonian14 acting only in the low moment um subspace. The
so constructed Hamiltonian comprises the full physics for low-Iying bound and scattering states
with appropriate boundary conditions. Specifically, for the scattering states the initial momenta
should belong to the low momentum subspace. Nevertheless, and this is an important remark, the
physics of the high momenta is by the very construction included in the effective low moment um
theory. Since the projection formalism in the moment um space has not yet been worked out in
the literature, we will give a rather detailed description of this method in the next sections.
We will furt her try to approximate the unitarily transformed Hamiltonian by the effective one.
This is justified since both are now defined on the same range of momenta. To be more precise,
we will keep the long range part of the interaction unchanged and cast the remaining short range
forces into the form of a string of contact interactions of increasing powers in the momenta. The
low-momentum theory obtained by the exact moment um space projection plays the role of QCD
and the expansion in terms of contact interactions for the heavy meson exchange is our model of
the effective field theory. For a consistent power counting to emerge, the coefficients accompanying
these terms should be of natural size. In other words, there should be a momenturn scale (the
naturalness scale Ascale) such that the properly normalized coefficients are of order one. In fact,
we can precisely calculate these coefficients from fitting the phase shifts or directly from the
unitarily transformed Hamiltonian. This will allow to study scaling properties of these coefficients.
A furt her issue that will be addressed is the investigation of the convergence properties of the
expansion in terms of local operators.
The effective interaction is naturally constructed in moment um space, but one can consider it
in configuration space as weIl. Clearly, one has to expect the effective potential to look totally
different compared to the original potential and as a consequence, the deuteron wave function
will also change. More precisely, the projection of the original potential into the subspace of
small momenta induces non-Iocalities in momentum space which in turn lead to very complicated
co-ordinate space expressions. We will show so me instructive examples of this phenomenon.
2.3.1 Method of unitary transformation
In this section we develop in detail the formalism which allows to study the nucleon-nucleon
interaction in a space of momenta below a chosen cut-off. Besides being interesting in itself, such
a theory in a space of low momenta can also be used to study various aspects of effective field
theory approaches to the nucleon-nucleon interaction. The formalism is quite general and can be
applied to any potential.
To be specific, consider a momenturn space Hamiltonian for the two-nucleon system of the form
H(p,p') = Ho (p)J(p -p') + V(p, p') , (2.59)
where Ho stands for the kinetic energy and V for the spin-independent model force. We introduce
the projection operators
! d3p lp) (pl
! d3p Ip) (pi
Ipl � A,
Ipl > A,
(2.60)
(2.61)
14 We stress that the interaction resulting from the unitary transformation is not an effective interaction considered
in the last sections. This important distinction should be kept in mind.

2.3. Going to higher energies: a toy model 25
where A is a momenturn cut-off which separates the low from the high moment um region. Its
precise value will be given below. Apparently, 7]2 = 7], .>..2 = .>.., 7]'>" = '>"7] = 0 and 7] + .>.. = 1. Using
the 7] and .>.. projectors, the Schrödinger equation takes the form
(2.62)
Obviously, the low and the high momentum components of the state \[f are coupled. Our aim is to
derive a Hamiltonian acting only on low moment um states and which furthermore incorporates aH
the physics related to the possible bound and scattering states with initial asymptotic momenta
from the 7] states. This can be accomplished by an unitary transformation U
H -+ H' = UtHU (2.63)
such that
7]H'.>.. = .>..H'7] = 0 . (2.64)
We choose a parametrization of U given by Okubo15 [51],
where A has to satisfy the condition
-At(1 + AAt)-lj2
'>"(1 + AAt)-lj2
(2.65)
A = '>"A7] .
(2.66)
Different transformations as weH as the relations between the resulting effective Hamiltonians are
discussed in ref. [52]. One can easily check that U given by eq. (2.65) is indeed unitary provided
that the operator A satisfy the condition (2.66):
t _
(
7](I +AtA)-lj2 (1 + AtA)-lj2At ) ( 7](I +AtA)-lj2 -At(I +AAt)-lj2 )
U U - -(1 + AAt)-lj2 A '>"(1 + AAt)-lj2 A(1 + At A)-lj2 '>"(1 + AAt)-lj2
( 7](1 + AtA)-lj2(1 + AtA)(1 + AtA)-lj2 0 )
o '>"(1 + AAt)-lj2(1 + AAt)(1 + AAt)-lj2
(2.67)
It is then straightforward to recast the conditions (2.64) in a different form,
.>.. (H - [A, H] - AHA) 7] = O . (2.68)
This is a nonlinear equation for the operator A, which takes the explicit form
V(p, ij) J d3 q , A( p, � q �')V(�' q ,q �) + / d3 p , V( p,p � �')A(�' p ,q �)
J d3 q , d3 p , A( p, � q �')V(�' q ,p �')A(�' p ,q �)
(Eq - Ep) A(p, ij) (2.69)
15Note that one obviously can perform additional unitary transformations in 1)- and A-subspaces. We do not
consider such transformations here. The condition (2.66) shows that we are looking for those transformations that
map 1)- and A-subspaces into each other. We refrain from a further discussion ofthe generality of the parametrization
(2.65)

26 2. Low-momentum effective theories for two nucleons
Here we denoted the momenta from the TJ- and ),-spaces by if and p, respectively, and Eq, Ep stand
for the corresponding kinetic energies. Once A and thus U have been determined, the effective
Hamiltonian in the TJ-space takes the form
This interaction is by its very construction energy-independent and hermitian [51], [53] .
(2.70)
After this block-diagonalization, the Schrödinger equation separates into two uncoupled equations
in the respective subspaces. According to eq. (2.63) the connection between the eigenstates of H
and H' is
w' = utw ,
(2.71)
so that the transformed problem separates into
TJH'TJW'
),H' ),w'
ETJW' ,
E),w' .
Let us now define the potential after performing the unitary transformation via
v' == H' -Ho .
(2.72)
(2.73)
(2.74)
Note that in this definition we have used the free Hamiltonian Ho, which is not unitarily transformed.16
That is why
v' i= U t VU . (2.75)
In what follows, we will nevertheless refer, for brevity, to V' as to a unitarily transformed potential.
The inequality (2.75) should always be kept in mind.
Let us now consider the transformed scattering states corresponding to an asymptotic momentum
if:
IWq'
� (+) ) = lim iEG'(Eq + iE) lif) ,
(2.76)
f--+O
where Iq) is a momentum eigenstate, Ho Iq) = Eq Iq) and G'(z) is the transformed resolvent
operator
o
(z - ),H' ),)-1 )
. (2.77)
Obviously, z should not be in the spectrum of H. Note that the resolvent operator of the transformed
Hamiltonian H' is block-diagonal. As a consequence, the scattering states IW:Z(+)) lie in
the TJ- (),-) space, if the corresponding asymptotic momentum if belongs to the TJ- (),-) space,
respectively. Immediately, the crucial question arises about the connection between the scattering
states Iw�+)) == limHo iEG(Eq + iE) lif) and Iw:z(+)). It is not obvious that eq. (2.71) holds also
for these scattering states, which are defined through specific boundary conditions, see eq. (2.76).
Note that if the transformed and original scattering states are not connected via an unitary operator,
then the S-matrix would change after performing the unitary transformation. We sketch
now a proof, showing that eq. (2.71) is indeed valid in this case and that the relation
(2.78)
16The unitary operator U does, in general, not commute with the free Hamiltonian Ho . In this sense, equation
(2.74) corresponds to a non-trivial definition of the potential V' . We will comment more on that later on.

2.3. Going to higher energies: a toy model 27
is satisfied. The following steps appear highly plausible but do not replace a mathematically
rigorous proof. Consider the left hand side of this equation:
lim iEG'(Eq + iE) Iv = lim iEUt G(Eq + iE) U Iq')
€-+o €-+o
Ut !� iEG(Eq +iE) ((77 + >'A77) (1 + AtA)- 1/2
+ (>' - 77At>.) (1 + AAt) - 1/2 ) Icf) .
In the last line we used eq. (2.65). Let us define the operators B and C by
Then eq. (2.79) takes the form
with
B
C
( ) - 77B77 = 77 1 + At A 1/2 - 'TJ ,
>'C>' = >. (1 + AAt) - 1/2 _ >. .
(2.79)
(2.80)
(2.81)
(2.82)
(2.83)
Note that the operator A and, as a consequence, B and C depend on the interaction V and are at
least linear in V to lowest order. This is obvious from eq. (2.69). Further it can be excluded that
the matrix element (q'IFIq') is infinitely large. This is because we ass urne that the operator F can
be represented in terms of a (convergent) expansion in powers of the smooth potential V and that
is why its matrix element (plFlcf) does not contain terms like 8(p - cf). Stated differently, it is
not possible that Flq') = alq') + ... , where a is some finite constant and the ellipses correspond to
other states. Therefore, no pole cx 1 j (iE) can be generated in the application of G (Eq + iE). This
can be seen using a perturbative argument and expanding the full resolvent operator as follows:
lim itG(Eq + iE) F Icf) = lim iE (f (Go(Eq + iE)V)i) Go(Eq + iE) F Iq')
(2.84)
€-+o €-+o
i=O
Because of the above mentioned property of F it is clear that (p lFlq') does not contain a term
proportional to 8(p - cf) and therefore one can not generate a pole term 1j(iE) through the
application of Go(Eq + iE) onto FIif). That is why each single term of the series on the right hand
side of this equation equals zero. As a consequence, one obtains
lim iEG(Eq + iE) F Iq') = 0 ,
€-+o
(2.85)
and thus eq. (2.78) is indeed satisfied. We thus have shown that the scattering states \ji�+) initiated
by momenta cf from the low momentum 77-space are connected to the transformed states \ji:r(+)
via the unitary operator U. The corresponding >.-components of \ji:r(+) are strictly zero.
After considering the scattering states, we now turn our attention to the bound states. Again,
the obvious question is: Where do the transformed bound states of H reside? If the cut-off A is
sufficiently large then A goes to zero. This is a simple consequence of the fact that V is assumed
to fall off sufficiently fast for high momenta. Consequently, >.H' >. is approximately equal to >.H >.

28 2. Low-momentum efIective theories for two nucleons
which contains only a small portion of V and, thus, can not support bound states at all. The
transformed bound states have therefore to be solutions of eq. (2.72) and the 'x-components of
the transformed bound states have to be exactly zero. It is not known to us whether this remains
true choosing sm aller and sm aller cut-off values. For the physically reasonable choices used in
next seetions this turns out to be true. Trivially there can not be solutions to the same bound
state energy for both equations (2.72) and (2.73), since this would contradict the non-degeneracy
assumption for the bound states of H.
One can argue just in the same way to see the validity of the equation (2.71) for the states
Iwq(-)) == limHo ifG(E q - if) Iq) and Iwif(-)) == limt----to ifG'(E q - if) Iq). Therefore, the Smatrices
in the original and transformed problem are the same:
(2.86)
As a consequence, the on-shell T-matrix element evaluated by means of the Lippman-Schwinger
(LS) equation
T' = V' + V'GoT' , (2.87)
yields exactly the same matrix elements as gained via the LS equation of the original problem
T = V + VGoT . (2.88)
Note that in eq. (2.88) one integrates over the whole (infinite) moment um range whereas in
eq. (2.87) only momenta up to the cut-off A are involved.
An important observation is that after performing the unitary transformation most local operators
become non-Iocal. For an arbitrary local operator O(ih,ih) = O(pd 8(Pl -P2) one obtains in the
transformed space
(2.89)
which, in general, contains the usual 8-function part but in addition also a strong non-Iocal piece.
These non-Iocalities, which are easy to handle, are nothing but the trace of the high momentum
components from the full space. Note, however, that the momentum operator P of a particle
and, as a consequence, the free particle Hamiltonian Ho are required to be unchanged, in order
to define free asymptotic states. Certainly, one could also unitarily transform the operators Ho
and p. This would not yield any new aspects since the original and the unitarily transformed
problems would be trivially identical, the unitary transformation only leads to change of basis.
We shall not consider this trivial case any furt her. Physical observables such as the deuteron
binding energy and phase shifts are identical in the original and transformed problems as shown
before. We remark that for instance the average momentum in the deuteron will change in the
transformed problem because the moment um operator is not unitarily transformed. This is not
a problem since it is not direct observable. From the other side, all current operators have to be
unitarily transformed and this guarantees the equivalence of all observables.
Let us now discuss an alternative way of determining the operator A, as exhibited in ref. [52J.
That method uses the knowledge of the scattering states to the original Hamiltonian H. Let
us now look in some more detail at this formalism. As it was already pointed out ab ove , the
connection between the scattering states in the original and transformed problem is given by
u Iwif(+))
(2.90)
((1J + ,XA1J) (1 +AtA)- 1 /2 1J+ ('x- 1JAt,X) (1+AAt)-1 /2 ,X) Iwif(+))

2.3. Going to higher energies: a toy model 29
For momenta ij from the 1]-space this equation takes a simpler form
(2.91)
since the resolvent operator of the transformed Hamiltonian H' is block-diagonal as already
pointed out before, cf eq. (2.77). Now we act with the operator (1] + )'A1]) on both sides of
this equation. Because of the trivial relation (1] + )'A1]) (1] + )'A1]) = 1] + )'A1] we conclude from
eq. (2.91) for all momenta ij E 1] that
), Iw�+) ) =
q E 1] " q E Ti
)'A'l1 Iw�+) )
Projecting this equation onto the state (pi, p > A, and making use of the relation
(2.92)
(2.93)
where T(Pl ,P 2 , z) is the ordinary off-shell T-matrix, we end up with the following linear integral
equation for the operator A:
A(� ;1\ _ T(p, ij, Eq) _ ! d3 , A(p, ij') T(ij',ij,Eq)
.
p, q ) - E E q E E
q - p q - q' + Zf
(2.94)
Here the integration over q ' goes from 0 to A. Consequently, the dynamical input in this method
is the T-matrix T(ql,q""2,Eq2 ). Note that this is not a usual Lippmann-Schwinger equation, since
the position of the pole, Eq, in the integration over q' is not fixed but moves with q. It is the
second argument in A which varies, the first one is a parameter for the integral equation.
To end this section we would like to rewrite the decoupling equation (2.69), which is given in the
three-dimensional vector space of momenta, in a more convenient partial wave decomposed form.
For two particles with spin 1/2 one obtains, using the standard Ilsj) representation, the following
equation
Vi:! (p, q)
� 0
InA [00 . . .
'2 ' '2 ' SJ , sJ " sJ ,
q dq P dp All (p, q )Vil' (q ,p )Al'l'(p ,q)
A
11'
(Eq - Ep) A:f, (p, q )
(2.95)
where Vi:!(p,q ) == (lsj,plVll'sj,q) and A:f,(p,q) == (lsj,pIAll'sj,q). In the uncoupled case 1 is
conserved and equals j. In the coupled cases it takes the values 1 = j ± 1. Analogously, we can
partial wave decompose the linear equation (2.94):
sj ( ) A AS2( ') T!j( , E)
ASj ( ) - Tl!' p,q, Eq _ " In '2 d ' 11 p,q 11' q , q, q
q p _
11' p, q -
E - E � q q E - E +'
I
0 q q' Zf
2.3.2 Regularization of the decoupling equation
(2.96)
The factor (Eq - Ep) in eq. (2.69) multiplying A(p, ij) requires some caution when p == IPI and
q == l

30 2. Low-momentum effective theories for two nucleons
powers of the interaction V(p, ij), then one can show that A(p, ij) becomes infinite at each order
in V(p, ij) if both p and q approach A. For instance, the leading approximation for A(p, ij) is
given by
A( � ;;'l V (p, ij)
p,q =
(2.97)
} E q - E p '
from which this singularity can easily be read off. Note that also the linear equation (2.94) is not
well-defined if both p ---7 A, q ---7 A. We proceed by regularizing the original potential V (k', k). We
multiply it with some smooth functions f(k') and f(k) which are zero in a narrow neighborhood
of the points k' = A and k = A and one elsewhere. The precise form of this regularization does,
in fact, not matter [123]. For the actual calculations presented here, we choose
f(k)
f(k)
f(k)
f(k)
1 ,
1 ( Cr(k -A+a)))
"2 1 + cos
1 ( Cr(k - A - a)) )
"2 1 + cos
0,
b
b
'
'
for
for
for
for
kSA-a and k'2A+a,
A-aSkSA-a+b, (2.98)
A+a-bSkSA+a,
A-a+bSkSA+a-b,
with a and b parameters of dimension [energy]. This modification of the potential V in a onedimensional
case is depicted in fig. 2.3. The operator A(p, ij) based on that modified potential is
weIl defined far p, q ---7 A.
Further, we would like to quantify the effect of this regularization. The unregularized and regularized
T-matrices satisfy the following LS equations:
T(ql, q2) V( - � ) + / d3 k V(ql, ql,q2
k)T(k, q2)
E E + .
q2 - k 2E
f(qt)V(ql, q2)f(q2)
+ f(qt) / d3k V(ql,k)f(k)Treg�k, q2) .
Eq2 - Ek + ZE
(2.99)
(2.100)
Let us choose, for simplicity, b = o. Then the function f can be interpreted as the projection
operator in the moment um space
f = 1 -g, (2.101)
where 1 is the unit operator and 9 is the projector onto the moment um states A - a < p < A + a.
The projectors 9 and f satisfy the usual algebra g2 = g, f2 = f, fg = O. Subtracting eq. (2.99)
from eq. (2.100) and projecting this difference on the f-subspace we obtain the following integral
equation for f(tlT)f == f(Treg - T)f
f(tlT)f = -fVg GoTf + fVfGo(tlT)f . (2.102)
To simplify the notation we have used here the operator form. As in the usual LS equation,
f(tlT)f is a half-shell quantity. Note that the first term on the right-hand side of this equation
is of the order 0 ( a) beca use of the insertion of the pro j ector g. In general, each insertion of 9
leads to an additional power of a. The formal solution of this equation can be written as
f(tlT)f = - (1 - fV fGO)-l fV 9 GoTf . (2.103)
Assuming the existence of the inverse operator (1-fV fGO)- l with a finite norm for positive energies
one can estimate the upper bound for the quantity f(tlT)f. The most important observation

2.3. Going to higher energies: a toy model
2
o
-2
-4
-6
-8
-10
o
q
0.6
0.4
q ' [GeV]
Figure 2.3: Modification of the potential due to the regularization.
which follows from eq. (2.103) is that the difference between the regularized and unregularized
T -matrix is linear in a.
This means that for any fixed q < A - a, q > A + a and any finite E > 0 one can choose a such
that ITreg(q, q) - T(q, q)1 < E.
Similarly, the leading correction to the binding energy E of a bound state \)1 due to the regularization
of the potential can be obtained by cakulating the expectation value of (\)1 IV - V reg I \)1):
ll'
31
(2.104)
where we have used the partial wave decomposed form. Again, for any fixed q < A - a, q> A + a
and any finite E > 0 one can choose a such that IEreg - EI < E. Later we will give numerical
examples of the effects caused by that potential modification in some specific cases. We will show
numerically that the effective potential V' (q, q') is affected only within the width a for q, q' ---+ A.
There both V' and V go to zero.

32 2. Low-momentum effective theories for two nucleons
2.3.3 Numerical realization of the projection formalism
Let us now consider in more detail the numerical realization of the projection formalism. To
simplify the notation we will omit all indices l, s and j and consider the uncoupled case. The
generalization to a coupled case is straightforward. The nonlinear equation (2.95) can only be
solved numerically. We do this by iteration starting with the value of A(p, q) given by17
p,q V(p, q)
E - E .
A( ) =
q p
(2.105)
Certainly, the iteration method does not always lead to a convergent solution. We have found
that one can achieve a better convergence if one slightly modifies the usual iteration method:
after each four iterations we perform an averaging over the values of the operator A with suitably
chosen weight factors. These weight factors are found numerically for each particular value of
the cut-off A requiring that the number of iterations, needed to calculate the function A(p, q)
with so me given precision, is minimal. This scheme allows to obtain the operator A(p, q) even in
those cases, when the usual iteration method fails to provide the convergence. Of course, also this
algorithm does not work in all cases.
Alternatively, we have derived the operator A(p, q) from the linear equation (2.96). For that
we have first calculated the usual half-shell T-matrix by solving the corresponding LS-equation
(2.100), which can be rewritten for the S-channel as
(2.106)
where we have omitted the indices l, s and j and have not shown explicitly the regularizing
function f. The term -mJooodkq�V(ql, q2)T(q2,q2)/(q� - k2) which equals zero is added to the
right-hand side of this equation in order to replace the principal value integration by the ordinary
one, which can easily be handled numerically. We have solved this equation using the standard
methods, which are described in ref. [39]. In particular, we discretize it using the ordinary Gauss­
Legendre quadrature rules. Choosing n quadrat ure points {q{} leads to n + 1 coupled equations
for the matrix elements T(qL q2) and T(q2, q2), which can also be expressed in a matrix form. We
have solved this matrix equation by inversion using standard FORTRAN routines. Note that the
on-shell point q2 should, clearly, not belong to the set of quadrat ure points {qD.
Once the half-shell T-matrix is calculated, one can obtain the operator A(p, q) for each fixed
value of p from the linear equation (2.96) . As already pointed out before, this equation has a socalled
moving singularity, which makes it more difficult to handle than the Lippmann-Schwinger
equation. Indeed, one has to discretize both q and q' points in this equation. But this does not
allow to solve this equation as in the last case. The difference Eq - Eql can be exactly zero, since q
and q' belongs now to the same set of quadrat ure points. Thus, one cannot calculate the principal
value integral in the same manner as for the LS equation. To solve this equation we have used
a method proposed by Glöckle et al. [144] . The idea is to expand the function A(p, q) into cubic
splines Si(q)
(2.107)
I 7Rere and in what follows we will always consider the regularized potential vreg(qI, eh) = !(qI) V(qI , q2 ) !(q2)
and suppress the index "reg" for brevity.

2.3. Going to higher energies: a toy model 33
based on a set of suitably chosen grid points {qi} in the interval [0, A]. Inserting this expression
for the unknown function A(p, q) in eq. (2.96) and choosing q = qi one obtains for each i
A( .) - T(p,qi) _" A( .)1000 '2 d ,Sj(q')T(q',qi)
p, qt -m 2 _ 2 � m p, % q q 2 _ + qi P j 0 qi q '2 ZE
' .
(2.108)
The integral can easily be performed, since the q-dependence of A(p, q) is now given by the known
functions Si(q). For each interval qj � q � qj+l these functions are expressed by cubic polynomials
(splines), which satisfy Si(qj) = Oij. Their precise form can be found in the reference [144]. This
procedure leads to a system of linear equations for the expansion coefficients A(p, qi), which can
be solved by the standard methods.
It is important to note that for each p the second derivative of the function A(p, q) has to vanish
at the ends of the interval of interpolation:
d2A(p, q)
d 2 .
(2.109)
I =0
q q=A
These conditions are necessary for the spline method to work. Clearly, only one of these conditions
is satisfied, namely those for q = A. This is guaranteed by the regularization of the potential, the
details of which are described in the last section. Practically, if the function A(p, q) is smooth
enough, one can simply ignore these requirements and obtain a good approximate solution taking
a large number of grid points.
Having calculated the operator A, we are now in the position to consider observables. First, we
need the transformed Hamiltonian H'. The determination of H' according to eq. (2.70) requires
the calculation of (TI + At A) -1/2. This is done in the following manner: as already described in
section 2.3.2, we first define the function B(q, q') as given in eq. (2.80). Putting the projector TI
onto the left-hand side of this equation and taking the square of it we end up with the following
equation:
(2.110)
This can be rewritten (for the S-channel) as the following nonlinear integral equation for the
function B(q, q'):
B(q, q') = -1100 p2 dp A(p, q)A(p, q') -1 fo A
- 100 p2 dp fo A
;p dij B(ij, q)B(ij, q') (2.111)
ij2 dij A(p, q)A(p, ij) (B(ij, q') + fo A
ij, 2 dij' B(ij, ij')B(ij', q'))
Note that q,q' E [O,A]. The function B(q,q') defined in eq. (2.80) is obviously symmetrie in the
arguments q, q': B(q, q') = B(q', q). This is because the function A(p, q) is real, which can be seen
from eq. (2.69). We have solved the equation (2.111) by iteration, using the same Gauss-Legendre
quadrat ure points as discussed before and starting with B(q, q') = -(1/2) If p2 dp A(p, q)A(p, q').
We have found a very fast convergence of the iteration method in this case. The integrations
present in eq. (2.70) to determine H' are performed by standard Gauss-Legendre quadratures
and we end up with an effective potential V' (q', q) defined for q, q' � A.
2.3.4 Unitary transformation for a potential of the Malfliet-Tjon type
We will now present the results obtained using the formalism introduced in the last sections. Our
starting point is a model potential which captures the essential features of the nucleon-nucleon

34
2. Low-momentum effective theories for two nuc1eons
(N N) interaction. We choose a moment um space N N potential with an attractive and a repulsive
part corresponding to the exchange of a light and a heavy scalar meson:
1 ( VH VL
ql, q2 = )
21r2 t + J-Lk - t + J-LI '
� �
V( )
(2.112)
with t = (1]1 - 1]2)2. Here, J-LL and J-LH are the masses of the light and heavy mesons, respectively.
The strengths of the meson exchanges and the masses of the corresponding mesons are choosen
as given in ref. [126]: VH = 7.291, VL = 3.177, J-LH = 613.7 MeV and J-LL = 305.9 MeV.18 That
(Eq - Ep)A(p, q)
[GeV-2]
1
0.8
0.6
0.4
0.2
o
-0.2
-0.4
o
q [GeV]
.5
p [GeV]
Figure 2.4: The function (Eq - Ep)A(p, q) for A = 400 MeV.
potential supports one bound state at E = -2.23 MeV and leads to S-wave phase shifts in the 3 SI
partial wave in fair agreement with results from N N partial wave analysis. Thus, this potential
captures some essential features of the N N interaction. Next, we have to select a value for the
cut-off A. In principle, A could take any value, in particular also above the larger of the two
effective meson masses in the potential. We shall comment on that below. We are, however,
mostly interested in an effective theory with small momenta only and thus will choose the cut-off
between A = 300 and A = 400 MeV. To be more precise, by "small" we mean a scale which is
below the mass of the exchanged heavy particle, so that one can consider the situation with a
propagating light meson and the heavy meson integrated out and substituted by a string of local
contact terms (as will be discussed in more detail below) .
18 For the light meson, we could have chosen the pion mass. However, since nuclear binding is largely due to
correlated two-pion exchange, a somewhat larger value was chosen. All conclusions drawn in what follows are,
however, invariant und er the precise choice of this number.

2.3. Going to higher energies: a toy model 35
In what follows, we will restrict our considerations to N N S-waves. Because of no spin dependence
one can work out the corresponding S-wave potential simply by integrating over angles. This leads
to
(2.113)
with ql,2 = ItZi,21. Correspondingly, we have to set s = [ = [' = j = 0 in the decoupling equation
(2.95).
As already pointed out before, we first need to regularize the equation (2.95) by multiplying
the potential V(ql, q2) by functions f(ql), f(q2)' The presice form of the function f is given in
eq. (2.98). The nonlinear integral equation (2.95) is solved by the iteration method as described
in the last section. This leads to a matrix equation. Typically, we choose a =
20 keV, b = 10 keV
and 100 Gauss-Legendre points to discretize eq. (2.95). The shift in the binding energy due to the
regularization is 0.012 keV, which is a 0.01 permille effect. Thus, the effects of the regularization
are quite small and can be made smaller if so desired. For the value of the cut-off A = 400 MeV
we show the resulting function (Eq - Ep)A(p, q) in fig. 2.4. Here we have multiplied the function
A(p, q) by (Eq - Ep) in order to avoid a peak at p, q ---+ A. A typical total number of iterations is
40-100 to achieve an accuracy of 0.0001 Ge V-3 for the function A(p, q) related to typical values
of A 0f the order 1 GeV -3.
Having calculated the operator A, we are now in the position to consider observables. After the
function B(q, q') is evaluated from equation (2.111), as described in the last section, we perform
the integrations present in eq. (2.70) to determine H' using again Gau�s-Legendre quadratures
and end up with an effective potential V' (q', q) defined for q, q' ::; A. It is displayed in the left
panel of fig. 2.5 in comparison to the original underlying potential for A = 400 MeV. Note that
the very small region of the regularization is again not shown to keep the presentation elearer.
One finds that the effective and the original potentials have a similar shape for momenta below
the cut-off. The main effect of integrating out of high moment um components at the level of the
potential seems to be given in this case just by an overall shift. Indeed, from the right panel of
fig. (2.5) one can see that the variation of the difference between the two potentials is only about
8% for all values of momenta q, q'. The solution of the effective LS equation (2.87) is now very
simple since the integration is confined to q ::; A. The effective bound state wave function obeys
a corresponding homogeneous integral equation
2 A
�'lt(q) + r iP dijV'(q,ij)'lt(ij) = E'lt(q) , (2.114)
mN Ja
with mN = 938.9 MeV the nueleon mass. Using 40 quadrature points the resulting binding energy
agrees within 9 digits with the result gained from the corresponding homogeneous equation driven
by the original potential V and defined in the whole moment um range. Furthermore, the S-wave
phase shifts agree perfectly solving either eq. (2.88) in the full moment um space or eq. (2.87) in the
space of only low momenta. This is shown in fig. 2.6. Note that due to the regularization the phase
shifts go to zero for q ---+ A. Further, the phase shifts are shown as a function of the kinetic energy
in the lab frame and the zero occurs at 7lab = 2A2/mN = 341 (85) MeV for A = 400 (200) MeV.
One can repeat this numerical exercise choosing different cut-off values. Setting for instance A = 2
GeV, the corresponding function (Eq - Ep)A(p, q) is shown in fig. 2.7 and the effective potential
V'(q',q) turnes out to be rather elose to the original one, [V(q',q) - V'(q',q)l/V(O,O) rv 0.02.
Here we have choosen a =200 keV and b =100 keV, which leads to the same shift in the deuteron

36
2
o
-2
-4
-6
-8
-10 �
-12
t-

2.3. Going to high er energies: a toy model
180
160
140
120
bO 100

38
(Eq - Ep )A(p, q)
[GeV-2]
2
1.6
1.2
0.8
0.4
q [GeV]
2. Low-momentum effective theories for two nucleons
Figure 2.7: The function (Eq - Ep)A(p, q) for A =
p [GeV]
2
GeV.
our case given by the exchange of the light meson, and to represent the effects of the heavy
meson exchange by adding a string of local contact interactions to the Hamiltonian. Proceeding
in this way one needs to introduce a cut-off to remove all ultraviolet divergences caused by the
contact interactions. Since a precise choice of the regulating function is of no relevance for the
low-energy observables, we can regard the same cut-off function as in the projecting formalism
discussed above. Speaking more precisely, we take the sharp cut-off. Now both the Hamiltonian
within the effective theory approach and the one derived from the unitary transformation are
defined over the same range of small momenta. That is why we can obtain the values of the
coupling constants accompanying the contact terms from matching the effective Hamiltonian to
the unitarily transformed one. It is commonly believed that the coupling constants, scaled by
some effective mass parameter Ascale, should have the property of "naturalness", which me ans they
should be of the order of one. Only then this expansion makes sense and the power-counting is
self-consistent. The value of the scale Ascale is obviously closely related to the radius of convergence
of this expansion. Let us first consider the simpler case when the pionic degrees of freedom are
integrated out. In that case, the low-energy N N interaction can be described entirely in terms of
contact terms and one expects the scale Ascale to be of the order of the pion mass m-rr . Therefore, all
physical parameters which describe the N N phase shifts up to center of mass momenta of the order
m-rr , such as the scattering length a and the effective range re , are expected to scale like appropriate
inverse powers of m7f' This is, however, not the case in nuclear physics: the NN scattering length
in the lSo-channel takes an unnatural large value, a = (-23.714 ± 0.013) fm » l/m7f. The
physics of this phenomenon is well understood (there is a virtual bound state very near zero

2.3. Going to higher energies: a toy model
(Eq - Ep)A(p, q)
[GeV-2]
3
-1
-3
-5
q [GeV]
p [GeV]
Figure 2.8: The function (Eq - Ep)A(p, q) for A = 200 MeV.
energy) and amounts to a fine tuning between different terms when the corresponding phase shifts
are calculated [83], [90]. To achieve such cancelations in the effective field theory calculations one
has to "fine tune" the parameters. We shall see below how this "fine tuning" works in our model.
The situation is more complicated in the effective theory with pions since different scales appear
explicitly. That is why it is not clear a priori what scale enters the coefficients in the Hamiltonian.
Using a modified dimensional regularization scheme and renormalization group equation
arguments, the authors of [90] have argued that Ascale rv 300
39
MeV. On the other hand, it was
stressed by the Maryland group [83] that no useful and systematic effective field theory (EFT)
exists for two nucleons with a finite cut-off as a regulator. A systematic EFT is to be understood
in the sense that the contributions of the higher-order terms in the effective Lagrangian to the
observables at low momenta (i.e. the quantum averages of such operators) are small and, therefore,
truncation of such operators at some finite order is justified. Within our realistic model, we
can address this question in a quantitative manner as shown below.
Let us now apply the concept of the effective theory to our case. The original potential plays the
role of the underlying theory of N N interactions in analogy to QCD underlying the true EFT
of N N scattering. Integrating out the momenta above some cut-off A < JLH, we arrive via the
unitary transformation at the potential V'(q, q') defined on a subspace of small momenta. Now
we would like to represent this unitarily transformed potential by the one obtained within the
effective theory procedure:
(2.115)

40 2. Low-momentum effective theories for two nuc1eons
V (VI) [GeV-2]
-2
-4
9
-6
-8
7
-10
5
-12
-14
3
[GeV]
q [GeV] q [GeV]
Figure 2.9: Left panel: effective two-nucleon potential VI (solid lines) in comparison with the
original potential V (dashed lines) for momenta less than 200 MeV. Right panel: difference between
the original and effective potential V - VI as a function of momenta q, ql .
Bere, V�ontact is a string of local contact terms of increasing dimension, which is caused by the
heavy mass particle and the high momenta p > A. We define the piece Vifght to be the light
meson exchange contribution (for ql, q < A): Vifght = Viight , w here Viight denotes the second term
in eq. (2.113). Specifically,
with
VI = V(O) + V(2) + V(4) + V(6) +
contact
V(O)
V(2)
V(4)
V(6)
Co ,
C 2 (q12 + q2)
C 4(q12 + q2)2 + C�q12q2 ,
. . . ,
C 6 (q12 + q2)3 + C� (qI2 + q2)q12q2 ,
(2.116)
(2.117)
and the superscript I (2n)' gives the chiral dimension (i.e. the number of derivatives). Thus the first
term on the right hand side of eq. (2.115) is the purely attractive part of the original potential due
to the light meson exchange. In this way, we have an effective theory for NN interactions, in which
the effects of the heavy meson exchange and the high momentum components are approximated
by the series of N N contact interactions and the light mesons are treated explicitly. Note that the
constants Ci , CI in eq. (2.117) correspond to renormalized quantities since the effective potential
VI (ql, q) is regularized with the sharp cut-off A. The first question we address in our model is:
what is the value of the scale Ascale and its relation to the cut-off A? Of course, a priori these two
scales are not related. For the kind of questions we will address in the following, we can, however,
derive some lose relation between these two scales as discussed below. Since we know VI (ql , q)

2.3. Going to higher energies: a toy model 41
numerically, we can determine the constants Ci by fitting eq. (2.116) to V'(q', q) - VIight (q', q). This
is done numerically using the standard FORTRAN subroutines for polynomial fits to functions of
one variable and taking the corresponding polynomials typically of order ten to eleven. Once this is
done for the functions V(q, 0) and V(q, q) all constants Ci, C: in eq. (2.117) can be easily evaluated.
For the potential parameters used so far and the choices of A = 400 MeV and A = 300 Me V, the
resulting constants are given in table 2.1:
Co C' 4
6
1 A = 400 MeV 11 11.23 1 -32.27 1 86.73 1 113.9 -913.6
I A = 300
-265.2
MeV 11 11.15 I -32.93 I 85.76 I 115.6 -232.6
C'
-783.7
Table 2.1: The values of the coupling constants Ci, q in [GeV(-2-i )] for two choices of the cut-off
A.
Naturalness of the coupling constants Ci, q me ans that
(2.118)
C2n
- = anAscale 2 ,
C 2n+2
where the an are numbers of order one. Indeed, as one can read off from table 2.1, such a common
scale exists, namely
Ascale =
600
MeV . (2.119)
This is a reasonable value in the sense that it is very elose to the mass /1H of the heavy mesons,
which is integrated out from the theory. Stated differently, the value for Ascale agrees with naive
expectations. As it turns out, even for cut-off values like A = 300 MeV, there is only a small
difference between the heavy meson mass and the ensuing natural mass scale.
Another observation is that the values of the Ci, CI depend very weakly on the concrete choice
of the cut-off A. Clearly, for such an expansion of the heavy mass exchange in terms of contact
interactions to make sense, A has to be chosen below /1H. Furthermore, since we explicitly keep
the light meson, A should not be smaller than the mass /1L. If one were to select such a value
for the cut-off, one could also consider the possibility of expanding the light meson exchange in a
string of contact terms. We do, however, not pursue this option in here. Therefore, we conelude
that one should set A < Ascale but it is not possible to find a more precise relation. In fig. 2.10
MeV.
we show how weIl the potential V' (q', q) is reproduced by the truncated expansion eq. (2.116) for
A = 300
We have also calculated the two-body binding energy and the phase shifts using the form
eq. (2.116) with the constants given in table 2.1. The corresponding results are shown in fig. 2.11
and in table 2.2. The agreement with the exact values is good. However, one sees that terms
of rat her high order should be kept in the effective potential Vcontact in order to have the binding
energy correct within a few percent. Note, however, that the value of the binding energy is
unnaturally small compared to a typical hadronic scale like the pion mass or the scale of chiral
symmetry breaking. The phase shifts are described fairly weIl for kinetic energies (in the lab) up to
about 100 MeV if one retains the first three terms in the expansion eq. (2.116). For A = 400 MeV,
one can not expect any reasonably fast convergence for the binding energy any more. This can
be traced back to the fact that one is elose to the radius of convergence for momenta elose to
the cut-off. More specificaIly, with q = q' = 400 Me V, the pertinent expansion parameter is
(q,2 + q2)j A;cale ::::' 0.9. The ensuing very slow convergence is exhibited in table 2.2.
Although we have found that Ascale rv 600 MeV and, therefore, the expansion of V' in terms of
contact terms seems to converge for the chosen value of the cut-off A = 300 MeV, the ultimative

42
-3
-5
-7
-9
2. Low-momentum effective theories for two nuc1eons
0.3
0.8
0.6
004
0.2
0
-0.2
-004
q [GeV] 0.3 0
Figure 2.10: Left panel: effective two-nucleon potential V' (solid lines) in comparison with the
truncated expansion (2.116) (dashed lines) for A = 300 MeV. Right panel: difference ßV between
the effective potential V' and the truncated expansion as a function of momenta q, q ' .
I E [MeVJ
I E [MeVJ
V( o ) V(O) + V(2) I V(O) + V(2) + V(4) I V( o ) + V(2) + V(4) + V(6) I V�ontact I
0.46 3.18 1.95 2.29 2.23 I
0.67 7.15 1.82 3.15 2.23 J
Table 2.2: The values of the binding energy calculated with V�ontact' eq. (2.116), for A = 300
(second row) and A =
400
MeV (third row).
0.3
Me V
test of convergence of the expansion (2.116) should be to calculate the quantum averages of
operators V(O), V(2), V(4), V(6) , ... for the low-energy scattering and the bound states, as it
was proposed in [81], [83J. There, the expectation values of various contact interactions in a
pionless theory were estimated for the deuteron using the leading-order approximation (2.58) for
the deuteron wave function. This allows only for a very rough estimate of the size of the operators
in eq. (2.116). Furthermore, the value of the cut-off was chosen much below the scale, at which the
effects of new physics appear. l9 With these assumptions it was found that all contact operators
starting from the terms with two derivatives are of the same order. In our model we can perform
numerically exact calculations of these quantities using not only the bound-state wave function
but also the scattering wave functions without any additional and unnecessary assumptions. Using
the relation eq. (2.93) one obtains for an arbitrary operator 0
(2.120)
19 In the pionless theory considered in [81], [83] this scale is associated with the pion mass m" . In our model this
scale is given by the mass of the heavy meson.

2.3. Going to bigber energies: a toy model 43
150
100
bO 50
Q)
�
�
0
-50
:
'.
��:' ,
, , , , , ,
"
order 0 --- - --order
2 -- - - -- - .
order 4 --- --order
6 _._._.exact
--
- -
_._._.-::... .��
bO
Q)
�
�
150
100
50
0
-50
order 0 - - - ---order
2 ,- ----..
order 4 -----exact
--
-'-_-'- -- -- -1_-L_-' -- -- -'-_-'-_ L---
-L.- --I
-1 00 -1 00 L-o
20 40 60 80 1 00 120 140 160 1 80 200 0 20 40 60 80 1 00 120 140 160 1 80 200
T1ab [MeV] l1ab [MeV]
Figure 2.11: Phase shifts from the effective potential V' (solid line) and the truncated expansion
(2.116) as a function of the kinetic energy in the lab frame for A =300 MeV. Left (right) panel:
the constants Ci, C; are fitted to the effecti ve potential (to the NN phase shift).
with
11 J 12 d 'O( ') TI (q', q,Eq)
q q q, q ,
E E +'
q - q'
ZE
J 1 2 d I 1 2 d I T'* (q�, q,Eq)
O
( I ' ) T'(q�, q,Eq)
h = q1 q1 q2 q2 E E
. q1 ' q2 + '
- ZE E q - E q� ZE
(2.121)
The results for quantum averages of the operators v(2n), (n = 0,1,2,3), are shown in table 2.3.
q - q �
Note that the matrix elements for the bound and scattering states have different units. This
is a consequence of different normalization of those states. One observes a good convergence in
agreement with the naive expectation. Indeed, since the cut-off is chosen to be A =300 MeV and
the scale A s cale ,...,.,600 MeV, one expects the expansion parameter to be of the order AI A s cale � 1/2.
Such a value agrees wen with the one extracted from the results shown in table 2.3. Note furt her
that for higher energies, the convergence is slower, which is also rather natural.
deuteron 25.94 MeV -4.23 MeV 1.07 MeV -0.38 MeV
E1ab = 10 MeV 31.34 GeV-2 -6.19 GeV-� 1.64 GeV-� -0.58 GeV-2
E1ab = 50 MeV 11.54 GeV 2 -3.28 GeV -",;! 1.11 GeV 2 -0.44 GeV 2
E1ab = 100 MeV 7.79 GeV-2 -3.09 GeV-� 1.40 GeV-2 -0.71 GeV-2
Table 2.3: The quantum averages of the operators V(O), V(2), V(4) and V(6) for the bound (second
row) and the scattering states (third to fifth rows) for A = 300 MeV.
Having calculated the quantum averages of the operators V(O), V(2), V(4) and V(6) we can now
better understand the quite slow convergence of the effective theory expansion (2.116) observed

44
2. Low-momentum effective theories for two nucleons
for the two-body binding energy as indicated in table 2.2. Indeed, the binding energy is given by
E(i) = - (w(i) IHo lw(i)) - (w(i) IViigh tlw(i)) (2.122)
- (w(i) IV(O) lw(i)) - (w(i) IV(2)lw(i) ) - ... - (w(i)lV(i)lw(i) ),
where Ho is the free Hamiltonian and the superscript i denotes the order of calculation. Now
putting the correct wave function (qlw) == (qlw(oo)) obtained from the unitarily transformed
potential instead of (qlw(i)) we end up using the values of (WIV(i)lw) from table 2.3 with the
following numerical estimation:
E = -9.3
MeV + 33.9 MeV
-25.9 MeV + 4.2 MeV - 1.1 MeV + 0.4 MeV + ...
2.3 MeV .
(2.123)
From this estimations we see that the rat her small value for the binding energy results from the
cancelation of various terms in the expansion (2.122). This is a similar sort of cancelation to that
observed for the scattering length of the model potential eq. (2.53). In fact, the convergence for
the binding energy is not slow but simply moved to higher order: whereas already the leading
order (LO) term (w(O)IV(O) lw(O)) provides a good approximation (±1O%) for the matrix element
(wIVcontactlw) , only the NNLO result for the binding energy shows a similar quality. Thus, we
expect a fast convergence, like those for the terms (w(i) lV(i) Iw(i)), for the binding energy starting
from NNLO. For instance,
IE - E(6)1
'" 0.21 .
(2.124)
IE -E (4) I
In the real world to which one applies the effective field theory, one does not know the true effective
potential V' and therefore also the constants Ci , CI are unknown. They have to be determined by
fitting such a type of effective potential to the N N data, like the low energy phase shifts andj or the
deuteron binding energy. We can perform this exercise also in our case. Keeping again Viight (q' , q)
as a separate piece we have determined by trial and error the constants Co, C 2 , •
. • by
solving the
inhomogeneous LS equation and truncating the series at various orders. We have performed three
different fits with V(O), V(O) + V(2) and V(O) + V(2) + V(4) representing the LO, NLO and NNLO
results, respectively. To achieve stable values for C i 's we have performed the fits below 10 Me V at
LO and below 25 and 100 MeV at NLO and NNLO.20 The results for the values of the coupling
constants Co, C 2 , C 4 and C� are summarized in table 2.4. We have not performed the sixth
LO 9.49 (9.63) - - -
NLO 10.91 (11.01) -23.09 (-24.80) - -
NNLO 10.82 (10.86) -25.79 (-27.74) 104.1 (121.9) -251.7 (-253.9)
Table 2.4: The values of the constants C i determined from fitting the phase shift (to the effective
range parameters).
order (N3LO) fit since already the fourth (NNLO) order result gives an excellent reproduction
of the phase shifts as shown in the right panel of fig. 2.11. Alternatively, one can determine the
C i 's requiring that the first terms in the effective range expansion (2.17) of the phase shift are
2° This is different to the procedure of the ref. [124], where a larger range of energies was used to fix the coupling
constants. Therefore the values of C;'s given here and in [124] are slightly different.

2.3. Going to higher energies: a toy model 45
exactly reproduced.21 For example, at NLO one can fix the constants Co and C 2 to reproduce
the scattering length and the effective range corresponding to the original potential (2.112). As
shown in table 2.4, both methods give similar results for the coupling constants.
Let us now make two observations concerning the obtained results. First, one observes a much
better convergence for the low-energy phase shift with the couplings from table 2.4 than with the
true values of Ci ' So This has to be expected, since now their values are optimized to obtain a good
description for the phase shifts at low-energies or for the effective range parameters and not to
precisely reproduce the true potential for low momenta, which is, in principle, not an observable.
The same holds also for the two-body binding energy given in table 2.5, which now shows a
much better convergence than with the true values of couplings. This indicates that the "fine
tuning" of the coupling constants may significantly improve the convergence for such "unnatural"
quantities22 like the small binding energy (or, equivalently, large scattering length). Indeed, small
changes in the values of the Ci's lead to large variations of the binding energy E, as follows from
the estimation (2.123), and may allow for E to take its correct value already at 'leading orders.
Secondly, one observes that all Ci'S (with the exception of C�) are within 25% of their exact values.
The deviation of the fitted value for C� from the exact one reminds us of the fact that such fine
tuning can produce sizeable uncertainties in higher orders in such type of cut-off schemes and thus
the interpretation of such values has to be taken with so me caution. This is, in fact, even a more
deep problem since the potential itself is not an observable. Unitary transformations pro duces
phase equivalent potentials without changing the observables. Although we have found only one
solution for the Ci 's shown in table 2.4, fixing the coupling constants from a fit to phase shifts or,
equivalently, to the first terms in the effective range expansion may, in general, produce multiple
solutions. An example of such multiple solutions was given in section 2.2. The precise connection
between the unitarily transformed and effective Hamiltonian is still to be clarified. This might
be quite important for the comparison between various transformed realistic N N interactions
acting only below a certain moment um cut-off and the effective potential derived using chiral
perturbation theory.
One can also perform a similar analysis for the two-nucleon 1 So channel, which is expected to
be even more troublesome than the 3 SI channel for the effective field theory approach since, as
already stated before, the scattering length takes an unnatural large value. We will not consider
here this case but note that all conclusions drawn above hold also in this particular channel. For
more details the interesting reader is referred to reference [124].
To end this section we would like to discuss one furt her issue. In section 2.2 we have considered an
effective theory for two nucleons at very low energies. In such a case even the longest range part
of the nuclear force can be considered as a short range effect. Thus, the effective potential entirely
consists of the contact interactions with increasing number of derivatives. It was demonstrated
that such an effective theory reproduces the effective range expansion but cannot go beyond it.
More precisely, only those first coefficients in the effective range expansion are reproduced which
have been used to fix the free parameters of the potential. No predictions for furt her coefficients
could be made. In our current model we have explicitly incorporated the long range part of the
underlying interaction in the effective Hamiltonian. As a consequence, the predictive power of the
effective theory increases. In table 2.5 we show the values of the binding energy and effective range
21 At NNLO we obtain elose but not exact values of the first four coefficients in the effective range expansion.
22 "Unnatural" means in this context that the corresponding observable takes a very small or very large value as
the result of a cancelation between some large numbers. For instance, the scattering length given in eq. (2.55) is
natural for all values of the coupling constant 9 not very elose to one and takes an unnaturally large value if 9 is
accidentally elose to one.

46 2. Low-momentum effective theories for two nucleons
LO 2.40 5.514 2.084 0.345 0.210 0.063
NLO 2.23 5.514 1.893 0.105 0.005 -0.055
NNLO 2.23 5.514 1.896 0.130 0.059 -0.018
true values 2.23 5.514 1.893 0.130 0.059 -0.026
Table 2.5: The values of the binding energy E and effective range parameters from the unitarily
transformed potential V' and the truncated expansion (2.116) for A =300 MeV. The couplings
are fixed by the requirement that the first terms in the effective range expansion are exactly
reproduced.
parameters calculated with the truncated expansion (2.116) using the values of the couplings given
in table 2.4, which are determined requiring that the first terms in the effective range expansion
are exactly reproduced. For example, at NLO the two constants Co and C2 are fixed requiring
that a and r are exactly reproduced. This allows to make a prediction for V2.
2.3.6 Coordinate space representation
Up to now we have only considered the potentials in momentum space. It is also interesting to
see how the unitarily transformed potential looks like in coordinate space, which usually provides
more physical insights. Denote by Vi(p' ,p) the effective moment um space potential for angular
momentum l. The corresponding r-space expression is obtained from
Vi(x', x) = / p 2 dpp, 2 dp' jl(pX) Vi(P' ,p) jl(P'X') ,
(2.125)
with jl(Y) the conventional l t h spherical Bessel function. The S-wave potential (l = 0) for A =
400 MeV is shown in the left panel of fig. 2.12. It is, of course, symmetrie under the interchange of
x and x', but looks very different from the original local potential. In particular, for A = 400 MeV
the unitarily transformed potential is strongly non-local and purely attractive for short distances.
Of course, if one increases the value of the the cut-off A, a peak along the diagonal x = x'
resembling the delta function should develop and the superposition of the two Yukawa potentials
related to the light and heavy meson exchanges, respectively, should appear along the diagonal.
This is indeed the case as demonstrated in the right panel of fig. 2.11 for A = 5.5 GeV. Note also
that in this case, where U rv 1, the range of the potential is essentially given by the inverse of the
light meson mass. One can also construct the momentum and coordinate representations of the
deuteron wave function. For a momenturn space picture, we refer to ref. [123].
The unitarily transformed potential shows an oscillating behavior in coordinate space. This is
because it is defined only in a certain range of momenta. Due to the cut-off we unavoidably
introduce discontinuities in higher derivatives of the potential over momenta. For instance, the
second derivative of the smooth regulating function f(k) given in eq. (2.98) has discontinuity
points at k = A - a, k
= A - a + b, k = A + a - b and k = A + a. This problem somewhat
rest riets the applicability of the described projection method in moment um space. In particular,
after performing the unitary transformation one could not calculate such quantity for the deuteron
like the quadrupole momentum, which requires the knowledge of second derivatives of the wave
function. However, one can always choose the regulating function f (k) such that the first n
derivatives are continuous.

2.3. Going to high er energies: a toy
V(x, x'); [fm-4] V(x, x') [fm-4]
0
-0.05
- 0 .1
-0.15 -1
-0.2
model 47
X ' [fm] [fm]
Figure 2.12: Coordinate space representation of the effective S-wave potential Vo(x, x') . Left
panel: A = 400 MeV and 0 ::; x,x' ::; 4fm. Right panel: A = 5.5GeV and 1.2 ::; x,x' ::; 2fm.

Chapter 3
The derivation of nuclear forces from
chiral Lagrangians
3.1 Chiral symmetry
Let us start with the QCD Lagrangian for Nf quarks q
LQCD
1
qif/Jq - 4 G�vGQ' JlV - qMq (3.1)
L�CD + L�CD '
where L�CD is the Lagrangian for massless quarks given by the first two terms in the equation
(3.1) and L�CD denotes the quark mass term. To simplify the notation we do not show explicitly
the color indices of quark fields. Note that the Nf x Nf fiavor matrix M can be brought into a
diagonal form by an appropriate unitary transformation on quark fields in the fiavor space. The
covariant derivative of the quark field DJlq is defined via
D Jlq = (OJl + igA� >'
2 Q) q (3.2)
where >' Q are 3 x 3 matrices in the color space. They satisfy the SU(3) Lie algebra
Furt her , 9 is the strong coupling constant and AJl is the gluon field. Whereas quarks belong to
the fundamental representation of the SU(3)col group (color triplet), gluon fields form the adjoint
representation of that gauge group (color octet). The field strength tensor G JlV is defined as
GQ _ !Cl AQ !Cl AQ jQ ß 'YA ß A'Y
JlV - UJl v - Uv Jl - 9 Jl v '
where jQ ß 'Y are the SU(3) structure constants introduced in eq. (3.3).
The QCD Lagrangian for massless quarks L�CD can be rewritten in terms of left- and righthanded
quark fields qL == PLq, qR == PL q, where PR,L are the corresponding projectors
as follows:
48
(3.3)
(3.4)
(3.5)
(3.6)

3.1. Chiral symmetry 49
The left- and right-handed fields are the eigenstates of the chirality operator 1'5 to the eigenvalues
-1 and 1. From eq. (3.6) one immediately realizes that the Lagrangian .L:QCD has the global
symmetry
The axial U(1)A symmetry turns out to be broken at the quantum level due to the Abelian anomaly
[153], i. e. the corresponding Noether current is not conserved but receives a contribution arising
from quantum corrections. Thus, U(1)A is not a symmetry of QCD. The vector U(1)v symmetry
is connected with the baryon number conservation. Its conserved charge is the total number of
quarks minus antiquarks. The symmetry group SU(Nf)L x SU(Nf)R is called the chiral group.
The chiral group transformations of the quark fields can be expressed by
where the Nf x
Nf
(3.7)
(3.8)
matrix generators t i are fundamental representations of SU(Nf) and (Bdi,
(eR)i are the corresponding angles. Here, the index i varies from 1 to NJ -1. Note that one can
skip the indices L and R on the quark fields in eq. (3.8) without changing the result because of
the PR , L in the exponentials. Alternatively, one can parametrize the chiral symmetry group in
terms of SU(Nf)V x SU(Nf)A as follows:
q ---- -+ q' = exp (iOv . t)q, (3.9)
Let us now calculate Noether currents corresponding to an arbitrary symmetry of the Lagrangian
density. For that consider an infinitesimal group transformation
where
(3.10)
(3.11)
Here the constants cabe are the structure constants of the group and ra are the abstract group
generators. 1 Since we consider a global transformation, fa does not depend on space-time coordinates.
The field

50 3. The derivation of nuc1ear forces from chiral Lagrangians
The derivative of the Noether current (3.13) can be expressed as
öJlJ� = -i ((öJl ö(�;(Pi)) fia(cP) + Ö(�;cPi) öJlfi(cP))
-i (��ft(cP) + Ö(�;cPi) öJlfi(cP))
where we have used the Euler-Lagrange equation
(3.15)
(3.16)
Comparing equations (3.14) and (3.15) we conclude that the Noether current is conserved if the
Lagrangian density is invariant under symmetry transformations, since then:
(3.17)
One can now calculate the Noether currents corresponding to the transformations (3.8) and (3.9).
The functions fia( q) are
SU(Nf)v :
SU(Nf)A :
SU(Nf)L:
SU(Nf)R:
fi(q) = tijqj ,
fia(q) = 'Y5tijqj ,
fia(q) = tijPL qj ,
fia(q) = tijPRqj .
(3.18)
(3.19)
In all these cases the quark fields form the basis of the fundamental representations of the corresponding
symmetry groups.2 For the symmetry transformations (3.8) and (3.9) we find, using
eq. (3.13)
and
respectively.
JaJl A - = q'Y5'Y Jlta q ,
The charges associated to the Noether currents can be found via
Note that the charge is a constant of motion
since
dQ
dt =0,
1 d3 X öJl J Jl - 1 d3 X V . J
-I dS ·J
O.
(3.20)
(3.21)
(3.22)
(3.23)
(3.24)
2 A representation of the algebra is simply its linear realization, i. e. the functions li (rp) are linear in the fields
rp.

3.1. Chiral symmetry 51
Here we made use of the fact that the surface term at infinity is negligibly small and that the
current is conserved. Using the definition (3.13) of the Noether current one can express the
conserved charge in the form
(3.25)
where 7ri(X) is the canonical conjugate to cpi (X). Let us now consider a linear realization (representation)
of the Lie algebra (3.11):
where the matrices ta satisfy
= [ta, tb ] iCabctc .
One can use the equal-time commutation relations of CPi (X, t), 7ri (X, t)
[7ri(X, t), CPj(Y, t)] -io(x - y)Oij ,
[cpi(X, t), CPj(Y, t)] 0,
[7ri (X, t), 7rj(Y, t)] 0,
= [Qa, Qb] iCabcQc .
and the Lie algebra (3.27) to obtain the commutation rules for the charges:
(3.26)
(3.27)
(3.28)
(3.29)
(3.30)
Thus, the charges satisfy the same algebra as the corresponding group generators. Another useful
commutation relation is
(3.31)
We will now show that the Noether charges are the generators of an infinitesimal transformation
(3.12) (with fia(cp) given in eq. (3.26)). If one parametrizes an arbitrary group element 9 in the
{

52
and
for the SU(Nf h x SU(Nf)R
[Q�, Qtl
[Q�, Q�l
[Q�, Q�l
for the SU(Nf)V x SU(Nf)A
like the ones in eq. (3.3) for SU(3).
3. The derivation of nuc1ear forces from chiral Lagrangians
iijkQt
iijkQ�
iijkQt ,
(3.36)
symmetry group. The fijk are the structure constants of SU(Nf)
A classieal symmetry can be realized in quantum field theory in two different ways depending on
vacuum transformation properties with respect to the symmetry group under consideration. The
Wigner-Weyl mode is characterized by a symmetrie vacuum:
U-110) =
10) . (3.37)
This condition can be equivalently expressed in terms of conserved charges Qa as
(3.38)
In such a case one should observe in the physieal spectrum degenerate multiplets corresponding
to irreducible representations of the symmetry transformation group. To see this degeneracy one
can consider an arbitrary field operator

3.1. Chira,l symmetry
, ,
:: - =:\
I \
... ... ... �--- ---"', ...
/.. ,;
.. ,
,
Figure 3.1: Spontaneous breaking of the symmetry.
Here, we do not sum over the values of a. Performing a space-time translation and using the fact
that the vacuum state is translational invariant we can rewrite eq. (3.43) as follows:
This matrix element has to be different from zero and hence is infinite.
53
(3.44)
Spontaneous breakdown of a continuous symmetry implies, according to the Goldstone theorem
[61], [62], the existence of massless spinless particles called Goldstone bosons. The number of
Goldstone bosons equals the number of the broken generators. The (abstract) generator ra of
a symmetry group is broken if the corresponding symmetry charge Qa does not annihilate the
vacuum as shown in eq. (3.42). We will now sketch the proof of the Goldstone theorem given by
Guralnik et al. [155] and show the presence of the massless particles in the physical spectrum. A
very important point is that there exists some field operator 'l/Ji such that its vacuum expectation
value does not vanish:
(3.45)
We can illustrate the existence of such an operator with a simple classical example shown in
fig. 3.1. There, the potential V is given as a function offields () and 7r. The potential is rotationally
invariant. The point (() = 0, 7r = 0) does not correspond to a minimum of the potential. In fact,
there is an infinite number of points ( (), 7r) for which the potential takes the minimal value. All
those points underlie the requirement
(3.46)
where A i- 0 is some constant. The physical vacuum in that classical picture corresponds to one

54 3. The derivation of nuclear forces from chiral Lagrangians
of those points, say, to (0" = A, 7r = 0). 5 Thus, the vacuum is characterized by some non-zero
value of the field 0".
Let us now co me back to the Goldstone theorem and consider the vacuum expectation value of the
operator 'l/Ji (x) . U sing the translational properties of field operators and assuming translational
invariance of the vacuum, we obtain
(3.47)
where TJi is some (non-zero) constant. Assuming furt her that -tfjTJj = K,f i- 0 if some of the 7]'S
do not vanish, we obtain using eq. (3.31)
K,f (OI[Qa, 'l/Ji(O)]IO) (3.48)
I d3 x (OI[Jö(x,
t), 'l/Ji(O)]IO)
I: I d3 x {(OIJö(O)e-iP,xln)(nl'l/Ji(O)IO) - (OI'l/Ji(O)ln)(nleiP.X Jö(O)IO)}
n
I:(27r)38(Pn) {(OIJö(O)ln)(nl'l/Ji(O)IO)e-iEnt - (OI'l/Ji(O)ln)(nIJö(O)IO)eiEnt}
n
i- 0,
where we have inserted a complete set of intermediate energy eigenstates Ln In)(nl and used the
translational invariance of the vacuum. Here Pn and En denote the momentum and energy of the
state In). Note that Qa should correspond to a broken generator because otherwise the right-hand
side of eq. (3.48) vanishes. Since K,f does not depend on time, differentiating the equation (3.48)
with respect to time leads to
I:(27r)38(Pn)En {(OIJö(O)ln)(nl'l/Ji(O)IO)e-iEnt + (0 I 'l/Ji (0) In) (nIJö(O)IO)eiEnt} = 0 . (3.49)
n
The positive and negative frequency parts can not mutually cancel. Therefore each of both terms
in the square bracket in eq. (3.49) must vanish for all states In) except those where En = 0
for Pn ---t O. Furthermore, the latter ones must exist since otherwise the equation (3.48) will
not be satisfied. Thus, the physical spectrum should contain massless states In) of spin zero
and the same parity and internal quantum numbers as Jö, whieh are called Goldstone bosons.
For a more rigorous and mathematically correct proof of the Goldstone theorem the reader is
referred to the original publications [61], [62] as well as to the references [155], [156] and [157] .
The plausible physical interpretation of the Goldstone theorem can be obtained from the simple
classieal example considered above and illustrated in fig. 3.1. There, the vacuum state corresponds
to the point (0- = A, 7r = 0), whieh is clearly not invariant under rotations in the 0"-7r surface. The
potential V (0", 7r) takes its minimum value for the points lying on the ring (j2 + 7r2 = A 2• Thus, no
additional energy is needed to move from one such point to another. The modes corresponding
to such transformations can be identified with the Goldstone bosons.6
Let us now co me back to the QCD Lagrangian. To decide whether the physieal vacuum is chiral
symmetrie or not we first note that the charges of SU(Nf)A carry negative parity. This is because
the ')'5 matrix enters the expression (3.21) for the axial-vector current Jc;t. Thus, if the chiral
symmetry would be realized in the Wigner mode, i. e. if the vacuum would be symmetrie, one
5In principle, the vacuum could also be given by a linear combination of those points. This possibility can,
however, be excluded for quantum fields in a space of infinite volume [133].
GIn this particular example such a mode can be defined using the polar angle in the surface CJ-7r.

3.2. Effective Lagrangians 55
would observe parity doublets Ih) and Q'Alh) in the hadron speetrum. No sueh parity doubling
is observed in the real world. Another argument in favor of spontaneously breaking of the chiral
symmetry is the presenee ofvery light pseudoscalar mesons (pions in the case of Nf = 2), which ean
be associated with the Goldstone bosons. In fact, spontaneous breaking of the chiral symmetry is
also evident for other reasons like, for instance, large differenees in vector and axial-vector spectral
functions, results of lattice ealculations and so on [158]. Clearly, only SU(Nf)A is spontaneously
broken.7 Otherwise one would also observe scalar Goldstone bosons. A more evident reason is the
presenee of SU(Nf)V multiplets in the hadron spectrum. For instanee, SU(Nf = 2)v corresponds
to an ordinary isospin transformation group. The high quality of the isospin symmetry of the
strong interaction may be demonstrated by comparing the proton and neutron masses mp = 938.27
MeV and mn = 939.57 MeV, which are indeed very elose to eaeh other.
In faet, the ehiral symmetry is broken not only spontaneously but also explicitly due to the presenee
of the quark mass term .c�CD in the QCD Lagrangian (3.1). Let us now be more eonerete and
regard Nf = 2.
Furthermore, we will assume that the matrix M is already in diagonal form:
M ( � u �d )
�(mu + md) (� �) + �(mu -md) ( � �1 )
1 1
2(mu + md)I + 2(mu -md)T3 , (3.50)
where I is the unit matrix in 2 dimensions and T3 a Pauli matrix. Whereas the term proportional to
the quark mass difference breaks both SU(2)v and SU(2)A symmetries (or, equivalently, SU(2)L
and SU(2)R), the one proportional to mu + md remains SU(2)v invariant. It turns out that the
quark masses mu rv 3 --;-7 MeV and md rv 7 --;- 15 MeV are mueh smaller than the typieal hadronie
mass seale of the order of rv
as a small perturbation.
3.2 Effective Lagrangians
1
Ge V. Thus, the quark mass term .c�CD indeed ean be considered
In the last seetion we have discussed the general symmetry aspeets of QCD. Now we would like to
eoneentrate on the two fiavor case. Let us furt her switeh from quark and gluon to hadron degrees
of freedom. At present, one is not able to direetly determine the strueture and dynamics ofhadrons
from QCD. Therefore, it is useful to apply the powerful method of effective field theory (EFT)
technique to analyze the interactions between the hadrons at low energies. Some examples of
such a teehnique were already given in the previous ehapter. Clearly, one would like to extraet all
possible information from QCD in order to minimize the number of free parameters in the effective
Lagrangian. One expects that the ehiral symmetry of the QCD Lagrangian is very important for
EFT eonsiderations, sinee the lightest hadronic degrees of freedom, the pions, are elosely related
to its spontaneous breaking. We will see below that the spontaneously broken ehiral symmetry
indeed provides strong eonstraints on the strueture of pion-pion and pion-baryon interaetions.
The starting point in the EFT program is the construction of the effective Lagrangian. Apart
from the usual requirements like loeality, invarianee und er Lorentz transformations, parity, timereversal
invarianee and hermeticity the effective Lagrangian should also be chirally symmetrie.
7 In fact, Vafa and Witten have proven that the global SU(Nf)v symmetry of QCD can not be spontaneously
broken [159). This holds more generally for any vector-like (gauge) theory.

56 3. The derivation of nuc1ear forces from chiral Lagrangians
The procedure of implementing the broken symmetry transformation on Goldstone fields was first
discussed by Weinberg [160] in 1968 in the context of the chiral SU(2)v x SU(2)A group broken to
SU(2)v and generalized one year later by Coleman, Callan, Wess und Zumino (CCWZ realization)
[67] to the case of a general compact group G broken to an arbitrary subgroup H. Here we will
basically follow this last work.
An arbitrary element 9 of the group G can be parametrized ass
where {A} and {V} are the sets of generators of the group G satisfying the Lie algebra
[Vi, Vj]
[Aa, Ab]
[Vi, Aa]
CijkVk,
CabcAc + Cabk Vk ,
CiabAb·
(3.51)
(3.52)
Here, the C's are the antisymmetric structure constants of the group G. Clearly, the chiral Lie
algebra (3.36) is a particular case of eqs. (3.52). Note that the generators Vi form a closed Lie
algebra. Therefore, the transformation h = eS 'v corresponds to the subgroup H of G, which is
required to be unbroken. The CCWZ realization is defined by the mappings
where ( and s ' have to satisfy
� � ( = ( (�, go) ,

3.2. Effective Lagrangians 57
The authors of the reference [67] argued that the parametrization (3.51) of an arbitrary group
element 9 E G is unique in some neighborhood of the origin. Consequently, we conclude from
eqs. (3.56), (3.57) and (3.58) that
I '
( = t , s = s ,
(3.59)
and thus he�·A = e {Ah. Equation (3.58) defines a linear transformation of the e's:
(3.60)
where '])( b ) (h) is some representation of h E H. The precise form of this representation depends
on the structure of the group G. For instance, if G is the group SU(N)v x
SU(N)v, i.e.:
with the
SU(N)A
Lie algebra (3.36) and H is the subgroup SU(N)v, then 1)(b)(h) is the adjoint representation of
(3.61)
where the (2N - 1) x (2N - 1) matrix T�b equals the SU(N) structure constant hab' This can
be shown using the chiral Lie algebra, eq. (3.33) and the normalization condition for the SU(N)
generators.10 Note furt her that for the

58 3. The derivation of nuclear forces from chiral Lagrangians
Thus, the most general chiral invariant effective Lagrangian for �, cjJ can be constructed solely
from the cjJ's. This simplifies the procedure enormously. Indeed, according to eq. (3.54), the cjJ's
transform "covariantly" under G: the only difference between the G- and H- transformations is
that in the first case the parameters s are complicated nonlinear functions of the ts and of the
group element. Therefore, any H-scalar built up from the cjJ's is also G-invariant.
Since we would like to describe the interaction between those fields � and cjJ within the EFT approach,
we have also to include derivatives of �, cjJ into the Lagrangian. Although these derivatives
transform linearly under the subgroup H, as it follows from (3.60) and (3.63), it is clear that the
field derivatives, in general, do not belong to the CCWZ realization (3.53)-(3.55). The number of
the ts is equal to the number of the A's and, therefore, the field derivatives cannot belong to the
set of the ts. In addition, they do not belong to the set of the cjJ's. For example, the derivative
0/lcjJ transforms as
(3.66)
The first term in this equation does not vanish in general, since the parameters s ' depend on
�. Therefore, eq. (3.66) violates eq. (3.54). In particular, eq. (3.64) is not valid any more for
derivatives. Although in that case s' = 0,
the derivative of s ' is different from zero. Consequently,
we can not proceed in the above way to construct the effective Lagrangian including the field
derivatives. To generalize the formalism introduced above to include the field derivatives, let us
first consider the following problem. In addition to the set of the ts we introduce fields \)I such
that the group G is realized (nonlinearly) on (�, \)I) in the following manner: the �'s form the
CCWZ realization of G and transform via eq. (3.53) and the \)I's transform linearly under H and
in some arbitrary way under the whole group G:
go (�, \)I) = (t, \)I'), (3.67)
where e are given by eq. (3.55) and \)I' are (nonlinear) functions of go, � and \)I. How to construct
the CCWZ realization for both � and \)I? We first note that for the specific group transformation
go = e-f A eq. (3.67) takes the form
(3.68)
where the �'s are the particular choice of \)I' corresponding to the transformation go = e-�·A .
Following ref. [67], we define a pair (�, �)* by
- _ _ -
(�, \)I) * = (� ,\)I) - e �·A (0, \)I) . (3.69)
We will now show that (�, �)* belongs to the CCWZ realization. For an arbitrary group element
go E G we have:
(3.70)
Now we have to apply the transformation eS'.v on both quantities in the bracket in the last
equality of eq. (3.70). We first note that the point � = 0 is invariant und er H-transformations.
This follows from eq. (3.55) and from the uniqueness of the parametrization (3.51). Further, the
�'s transform linearly under H in the same way as the \)I's, for which a linear transformation
property under H has been required. Thus, we obtain for eq. (3.70):
(3.71)

3.2. Effective Lagrangians 59
where e and s' are given by eq. (3.55). Therefore, the new set (�, �)* indeed defines the CCWZ
realization.
It is now clear how to construct the effective Lagrangian from the es,

60 3. The derivation of nuc1ear forces from chiral Lagrangians
Let us now make a few comments about eq. (3.74). This is, in fact, a crucial result for EFT,
since it says, that the interaction between the Goldstone bosons vanishes at low energies. Indeed,
it is seen in eq. (3.74) that one can always rewrite the Lagrangian in such a form, that only
derivative interactions between the Goldstone bosons are present. We will show below using the
power counting arguments that, as a consequence, the scattering amplitude for any process with
an arbitrary number of Goldstone particles is proportional to some positive power of external
momenta. We point out again that this statement concerning the scattering amplitude is independent
of any particular choice of the realization of the group G. In principle, we could equally
well use an arbitrary nonlinear realization (�, cjJ) to construct the Lagrangian, where cjJ denotes a
set of matter fields, their derivatives as well as derivatives of Goldstone fields. However, insisting
on the chiral invariance of the Lagrangian density would be a non-trivial problem because of the
complicated nonlinear transformation rules of the fields and their derivatives. Using the CCWZ
realization to construct the efIective Lagrangian via eq. (3.74) has the advantage that all building
blocks transform covariantly under G, Le. in the way shown in eq. (3.54), guaranteeing that any
H-scalar is also G-invariant.
To complete the discussion about the construction of a chiral invariant Lagrangian let us apply
now the chiral SU(Nf)v x SU(Nf)A group as a particular example of the general Lie algebra
(3.52). In that case all indices in eq. (3.52) vary from 1 to Ni - 1 and the structure constants G's
are represented in terms of the structure constants f's of the SU(Nf) group as follows:12
Gabe = 0, (3.75)
For example, for the SU(2)v x SU(2)A group we have fijk fijk, where fijk is the ordinary
three-dimensional totally antisymmetric tensor with f123 = 1. Note that one can represent the
chiral SU(Nf)V x SU(Nf)A group by SU(Nf)L x SU(Nf)R defining the corresponding generators
TL and TR as
(3.76)
where Vi and Ai are the generators of the SU(Nf)V and SU(Nf )A, respectively. Then SU(Nf)L
and SU(Nf)R build two independent subgroups of the whole chiral group with the Lie algebra
[(TL ) i , (TL)j]
[(TR) i , (TR)j]
[(TL ) i , (TR)j]
Let us now be more specific and concentrate on the chiral SU(2)v x
the following representation of the generators Vi and Ai fijk (TL h ,
fijk (TRh ,
o.
A - _ _ ilT i
l - 2 '
SU(2)A
(3.77)
group. We choose
where T i is a Pauli matrix and 1 is some flavor scalar matrix quantity with the properties:
12 = 1 , I t = I .
(3.78)
(3.79)
For example, 1 can be equal to the ')'5 matrix. Then, one has the same representation for the
SU(2)A generators as in the case of the QCD Lagrangian (3.1), as can be seen from eq. (3.9).
1 2 Here we use a definition of the chiral Lie algebra, which differs by the factor i from the one given in the last
section , eq. (3.36). This factar can be absorbed by the corresponding redefinition of the generators.

3.2. Effective Lagrangians 61
The representation (3.78) of the 8U(2)v group generators Vi is preferable because we will apply
the formalism to the nucleon fields, which form an isospin doublet, but this is not necessary in
general.13 Note that a particular choice of I in eq. (3.78) is not relevant, since this quantity will
not enter the explicit expressions for the effective Lagrangian, as will be shown below. It is easy
to verify that the Lie algebra (3.52), (3.75) is satisfied with fijk = Eijk if one defines V and A via
eq. (3.78). Using eq. (3.78), one can express the quantity e2�.A in the form:
cos � - ilnm sin � , (3.80)
e2�.A =
where � == lei and ni = �d�. The chiral 8U(2) x 8U(2) group is isomorph to 80(4). It can be
shown, see e.g. ref. [161], that the coordinates ( 0', *) defined via
0'
- == - cos (�),
(3.81)
fn
where fn is some constant, transform as the cartesian components of a 80(4) vector.14 Clearly,
the 80(4) group describes rotations on the four dimensional sphere
(3.82)
Therefore and as it is also obvious from eq. (3.81), the coordinates 0' and * are not independent
from each other. Note furt her that the �i and, therefore, the 7ri are pseudoscalar quantities:
(3.83)
This is because the Noether charges of the 8U(2)A are pseudoscalars, as can be seen from eq. (3.21)
in the case of the QCD Lagrangian (3.1). Furt her , the parameters € in eq. (3.32) should have the
same properties under parity transformation Pas the Noether charges Q, since otherwise the fields
cjJ' in eq. (3.34) would transform under P differently than the cjJ's. Therefore, the transformation
parameters � related to 8U(2)A must be pseudoscalar quantities.
The so-called stereographical coordinates (0', 'Fr ) proposed by Weinberg [160] are defined as
2* (3.84)
Für that particular parametrization one finds the covariant derivative D p, of the pion field (up to
some irrelevant numerical factor) [160], [75], see also [161]:
D _ -
_ � ßp,'Fr P, fn D
'
(3.85)
with
D == 1 + (�)2 2fn .
(3.86)
For the nucleon field \[I, which is an isospin or, equivalently, 8U(2)v doublet, one obtains the
covariant derivative in the form
(3.87)
13 In particular, one has to choose another representation far the V's if one considers matter fields, that form a
different isospin multiplet.
14 Since we consider the group SU(2)v x SU(2)A in some neighborhood of the unity, the mapping � .;==} ir is
unique.

62 3. The derivation of nuc1ear forces from chiral Lagrangians
where the quantity EJ.t is defined as
(3.88)
A more detailed discussion concerning the stereographical parametrization can be found in [161].
The use of the stereographical coordinates requires sometimes rat her tedious calculations. We will
now briefly discuss another more elegant way, which is commonly used in the analysis of processes
with one nucleon. Let us first introduce the standard notation and define the matrix u as
(3.89)
For the generators V and A we will again use the representation (3.78) with l = 1'5 , To establish
the transformation properties of the u we will apply eq. (3.55), which defines how the Cs transform
under go E SU(2)v X SU(2)A. An arbitrary group element go needs not necessarily to be
parametrized in terms of s and � via eq. (3.51). Equally weIl we can write
(3.90)
Here, eV,A are transformation parameters of the SU(2)V,A groups. We can now rewrite eq. (3.55)
using the parametrization (3.90) as
(3.91)
where s' is, as usual, a function of � and eV,A. Using the projectors PR and PL onto the states
with a definite chirality defined in eq. (3.5) we can express (ev . V + eA . A) as
The parameters e R,L are defined by
(3.92)
(3.93)
and VR,L == VPR,L are the generators of SU(2)R,L. Projecting eq. (3.91) onto states with adefinite
chirality we obtain the following equations:
PRi'R" V e�' v
PLih' V e-�' v
Here we have used the properties of the projectors PR,L:
and the following equalities:
P e·v Re e s'·V ,
P -e·v
Le e s'·V .
(3.94)
(3.95)
(3.96)
(3.97)
Since PR,L commute with the V, one can completely drop these projectors in eqs. (3.94), (3.95).
We can now easily read off the transformation properties of the u from eqs. (3.94), (3.95):
90 , h h- 1 h h- 1
U -=---7 U = RU = U L . (3.98)

3.2. Eifective Lagrangians 63
Here hR,L == iJR,L'V denote the SU(2)v-like15 transformations16 given by the matrices e(}R,L'V
and the so-called compensator field h is defined via
(3.99)
Let us repeat once more the logic to make this presentation more dear. An arbitrary group
element go E SU(2)v X SU(2)A can be parametrized via eq. (3.51) in terms of s and � or by a pair
eA, ev as in eq. (3.90) (or, equivalently, by eR and eL given in eq. (3.93)). The quantities s' and,
therefore, also the matrix h can be determined from eq. (3.55). To obtain the transformed fs (or
u' defined in eq. (3.89)), one can again make use of eq. (3.55) or, equivalently, apply eq. (3.98).
The second way is more preferable for our furt her consideration. Note that the matrix h depends,
in general, on space-time due to its explicit dependence on the fs whereas the hR,L in eq. (3.98)
do not.
Let us introduce another useful matrix U as
From eq. (3.98) we see that it transforms as
(3.100)
(3.101)
Consider now the quantity
uJ1, == iut (8J1,U) ut = i (ut 8J1,u - u8J1,ut) = -i ((8J1,ut)u + u8ttut) = -iu (8J1,Ut) u = u1 . (3.102)
Note that because of the definition (3.78) for the SU(2)v generators one has ut = u-1. In the
second and third equalities we have used the relation
(3.103)
One verifies that utt corresponds to the matrix representation of the covariant derivative of �. For
that one recalls that covariant derivatives are characterized by their transformation properties
under the chiral group. Speaking more precisely, the covariant derivative Dtt� should transform in
the same way as � under the subgroup H, which is in our case SU(2)v. The transformation of Dtt�
under the whole group SU(2)v x SU(2)A should have the form of the SU(2) v-transformation.
The only difference is that now the transformation parameters (the s" s entering h) are functions
of the group element and of the fs, given by eq. (3.55). Since we use the matrix form u defined in
eq. (3.89) for the fs, we have to consider the transformation properties of the u and not of the fs
under SU(2)v. As we already know from eq. (3.59), for a SU(2)v-transformation h = e(}v'
v
EH
one has h = h. Furthermore, as follows from eq. (3.93), eR = eL = ev since eA = O. Therefore,
hL = hR = h. Consequently, the matrix u transforms under h E H = SU(2)v as
u ---+ u' = huh-1 • (3.104)
One can easily check using eqs. (3.104) and (3.33) that the �'s are the basis of the adjoint representat
ion of SU(2)v as they should be, see eq. (3.60). For the covariant derivative of � in the matrix
15It should be kept in mind that the (h,L are, in general, not scalars with respect to parity operation as the
transformation parameters of an ordinary SU(2)v rotation.
lßWe refrain here from introducing the commonly used notation, in which the matrices hR,L are denoted by 9R,L.
In our opinion, this leads to confusion, since hR,L do not belong to the corresponding representation of the groups
SU(2)R,L.

64 3. The derivation of nuclear forces from chiral Lagrangians
form we should require the same transformation property (3.104) under go E SU(2)v X SU(2)A'
Using eqs. (3.98), (3.100) and (3.101) one verifies that uJ.L indeed satisfies this condition, i.e. that
and thus corresponds to the matrix representation of the covariant derivative of �.
(3.105)
The pion fields can be defined by a particular parametrization of U. Choosing again the 2 x 2
matrix representation (3.78) of the group generators Ai, Vi we can parametrize U, for example,
as follows:
Z'Fr • T
U=exP ( h ) , (3.106)
where 7ri are pseudoscalar fields (pions) and f1f is some constantP This also uniquely defines the
quantity uJ.L in terms of the pion fields. It is easy to construct chiral invariant terms consisting of
u/s. One simply has to build traces ( ... ) in flavor space of any product of the uJ.L's. For instance,
the term (uJ.Lu J.L ) is clearly chiral invariant:
(3.107)
Let us now introduce the nucleon field \]J. We will require that the nucleon field belongs to the
CCWZ realization and transforms under go E G via eq. (3.54):
(3.108)
Since we have chosen the representation (3.78) of the group generators, h(go, 'Fr) is a 2 x 2 matrix
in the flavor space. Note that h(go, 'Fr) becomes independent on 'Fr for all go E SU(2)v and forms
the fundamental representation of SU(2)v. To define the covariant derivatives of the nucleon field
it is convenient to introduce the so-called connection r J.L:
It follows from eq. (3.98) that r J.L transforms as:
Here we made use of the similar trick as in eq. (3.103):
Therefore, the covariant derivative of the nucleon defined as
transforms in the same way as the field \]J (covariantly):
(3.109)
(3.110)
(3.111)
(3.112)
(3.113)
Thus, any flavor scalar built up from isospinors \]J, D J.L \]J with insertions of uJ.L is chiral invariant.
17 It can be shown that in the chiral limit f" is the pion decay constant.

3.2. Effective Lagrangians 65
As an illustration, let us now construct the leading Lagrangian for pions, i. e. the Lagrangian with
the minimum number of derivatives. We will first work out all chiral symmetrie terms and skip,
for simplicity, the external sources. It is clear from the above discussion that the most general
chiral invariant Lagrangian can be built up from the covariant derivatives of the pion fields. In the
matrix notation used above, such a covariant derivative of the first order is given by eq. (3.102).
One can easily obtain analogous expressions for higher derivatives. For instance, the quantity
8JLuv clearly does not transform in a covariant way as the ufl in eq. (3.105):
(3.114)
To form a covariant object we need again the chiral connection r fl defined in eq. (3.109): (3.115)
Let us now establish the properties of the building blocks (ufl' UJLV, . . . ) und er Lorentz and parity
transformations. To simplify the notation, we will work with the dimensionsless field c/J instead of
7r, which is defined as18
(3.116)
where T is the Pauli matrix. Clearly, observables do not depend on a partieular parametrization
of the quantity u. This follows from Haag's theorem [162], [67]. Using the explicit parametrization
eq. (3.116) of u, we can express u8JLu t and U t 8flu in terms of the pion field c/J:
(3.117)
where

66 3. The derivation o{ nuclear {orces {rom chiral Lagrangians
Therefore,
U ---7
(3.122)
We are now in the position to construct the lowest order Lagrangian for pions. The building blocks
are (uft, UftY, . . . ). Any SU(2)v (isospin) invariant quantity is automatically chirally invariant.
To provide the isospin invariance one has to take traces in flavor space of any product of uft'
UftY, . . .. Clearly, no terms with a single derivative can enter the Lagrangian, which is required to
be a Lorentz scalar. One can write down three SU(2)v and Lorentz scalars with two derivatives:
(3.123)
where g fty is the metric tensor. The last term is not invariant under the parity transformation.
Furthermore, the second and the third terms do vanish anyway, since uft and therefore also ufty
are traceless, as can be read off from eq. (3.118). The invariance of the first term under the charge
conjugation is obvious. Thus, the leading order chiral invariant Lagrangian for pions is given by
a single term
(3.124)
where the coefficient /;/4 is chosen to reproduce the usual free Lagrangian for the pion field 7r.
In the real world, chiral symmetry turns out to be broken not only spontaneously but also explicitly,
due to non-vanishing quark masses. To systematically incorporate the symmetry breaking
terms within the effective field theory one can use the method of external fields [63]. Those fields
do not correspond to dynamical objects but can be used to generate Green functions of currents.
For our purpose of constructing the symmetry breaking terms due to the quark masses we only
need to consider a scalar source 8.19 We can formally require QCD to be a theory of massless
quarks interacting with a scalar field 8 with the corresponding Lagrangian given by
LQCD = L�CD -
ij S q . (3.125)
Clearly, we recover the original QCD Lagrangian (3.1) by freezing the field s and by setting20
s=M. (3.126)
To systematically incorporate the symmetry breaking in the effective Lagrangian caused by nonvanishing
quark masses one has to build up all non-invariant terms having the same transformation
properties with respect to the chiral group as the quark mass term in eq. (3.1). This can be achieved
requiring the Lagrangian (3.125) to be chiral invariant, which leads to definite transformation
properties of s, and taking into account the scalar source s by construction of the most general
chiral invariant effective Lagrangian. At the end, one has to set 8 = M in order to recover the
physically observed situation. The scalar source s in the QCD Lagrangian (3.125) has the following
19 0ne should not eonfuse the matrix field s with the transformation parameters of the group H defined in
eq. (3.51).
20 Note that in general s = M + . .. , where the dots denote other seal ar sources that may enter the QCD Lagrangian.

3.2. Effective Lagrangians 67
properties under the chiral transformation 90 E G, which are required by the chiral invariance of
the Lagrangian eq. (3.125):
90 I h h-1 h h-1 S -"-+ S = L S R = RS L , ( 3.12 7)
Note that the external field S is required to be hermitian, st = s, to provide the hermiticity of
eq. (3.125). The quark masses are treated as a small perturbation. Therefore, the leading symmetry
breaking terms in the effective Lagrangian should contain a minimal number of derivatives and
just one insertion of s, which will be put to M at the end in accord with eq. (3.126). To work out
the symmetry breaking terms, it is convenient to use the quantity U defined in eq. (3.100) and its
derivatives, that transform precisely in the same way as the external field s. Furt her , demanding
charge conjugation invariance for the QCD Lagrangian (3.125) leads to
Since the pions are pseudoscalars, we obtain for the parity transformation:
(3.128)
U ----+ U P = ut . (3.129)
Therefore, a single term with no derivatives and with one insertion of U that satisfies all desired
symmetry properties is
LSB = c(s(U + ut)) , (3.130)
where the real constant cis usually parametrized as c = {;B /2. Note that the term with two traces
like (s)((U + ut)) is, in general, not chiral invariant. However, for the chiral SU(2)v x SU(2)A
group this term is closely related to the one in eq. (3.130). Taking into account that the U's are
SU(2) matrices, we note that they can be parametrized in terms of two complex numbers a and
b, a 2 + b 2 = 1, as
(3.131)
Then, it is easy to verify that U + ut = (U)I, where I is the unity matrix. This enables us to
rewrite the symmetry breaking Lagrangian as
j2B
LSB = _7r_ (s)(U) .
2 s=M
I
(3.132)
In order to obtain a physical interpretation of the symmetry breaking term one can express
eq. (3.130) in terms of pion fields using eq. (3.106). We obtain
(3.133)
The first term is constant and does not contribute to the S-matrix. It is, however, still important
since it parametrizes the non-trivial structure of the vacuum [63], [64]. The second one can be
identified with the pion mass term -(1/2)M;7r2 , where M; = (mu + md)B. Note that to leading
order in mu, md one has equal masses for all pions 7f +, 7f - and 7f0. The SU (2) v symmetry is not
broken.
It can be demonstrated, see for instance refs. [63], [133], that the constant B is related to the
value of the quark condensate as
(OluuIO) = (OlddIO) = -f;B(l + O(M)) . (3.134)

68 3. The derivation of nuc1ear forces from chiral Lagrangians
The corrections O(M) in eq. (3.134) correspond to higher order symmetry breaking terms that
include two and more insert ions of the quark mass matrix M. Eq. (3.134) allows one to express
the pion mass in terms of the quark condensate as follows
This expression is known as the Gell-Mann-Oakes-Renner relation [163].
(3.135)
Let us finally make aremark concerning the constant in: . For that we can calculate the axial-vector
current corresponding to the Lagrangian (3.124), that can be equivalently rewritten as
(3.136)
To calculate the symmetry current we use eq. (3.13). Note that in the case of a SU(2)A transformation
one has (Jv = 0, (JR = (JA and (JL = -(JA, see eqs. (3.92), (3.93). Consequently, an
infinitesimal axial-vector transformation 90 E SU(2)A of the matrices U, ut is given by
The related axial-vector current iS21
U � U' = hRUhR = U - �(J� (TiU + UTi) ,
ut' - h-1Uth-1 R R - ut + �(Ji A (TiUt + UtTi) ,
2
(3.137)
(3.138)
where the ellipses stay for terms with more pion fields. Clearly, the axial current connects the
vacuum state with the one-pion state:22
(3.139)
Here, n hys 23 denotes the experimentally observed value of the pion decay constant, which can be
calculated from the rate for pion decay. The dots correspond to corrections, which are proportional
to powers of p2 / i; . 24 Consequently, these corrections are proportional to the pion mass squared
M; , which is zero in chiral limit. Thus, in the chiral limit in: is not renormalized and in: = n hys.
To perform later the expansion in powers of small momenta Q we need an ordering scheme for
various interactions in the effective Lagrangian, which can be viewed as the derivative expansion
and the expansion in powers of quark masses corresponding to the symmetry breaking terms.
One commonly counts the squared pion mass M; rv (140MeV)2 or, equivalently, the quark mass
insertion as two powers of small external momenta (two derivatives). Therefore, the complete
Lagrangian to leading order is given by
(3.140)
21
Note that BA corresponds to _ca in the notation of eq. (3.12).
22This is a general consequence of Goldstone theorem.
23
0ne commonly uses a different notation, in which OUf Irr (I!:hys) is denoted by F or I (Irr).
24This can be demonstrated from Lorentz invariance and the power counting scheme, that will be introduced in
the following section.

3.3. Chiral perturbation theory with pions 69
where the superscript (2) stands for the number of derivatives and the quark mass matrix insertions
as explained above. Proceeding in the same way as we did for [)2) one can construct the nextto-Ieading
order Lagrangian C(4) , which contains 8 independent terms25 [63], [64]:
c(4) LI (OJ1ut oJ1U)2 + L 2 (0J1UOvut) (oJ1Uo v ut) + L 3(0J1U0J1ut OvUo v ut)
+ L4(0J1U0J1ut) (M(U + ut)) + L5 (OJ1UoJ1ut (MU + UtM)) (3.141)
+ L6(M(U + Ut))2 + L7((Ut - U)M)2 + L8((U M)2 + (ut M)2) ,
where LI,. " ,L8 are the so-called low energy constants (LEC's), that have to be fixed from
experiment.26
3.3 Chiral perturbation theory with pions
In this section we will outline the basic ideas of chiral perturbation theory on an example, which is
the elastic pion-pion scattering. The starting point is the most general Lagrangian constructed as
described in the last section. Apart from the chiral symmetrie part that can be classified in powers
of derivatives acting on the pion fields, various symmetry breaking terms with insertions of the
quark mass matrix and containing any number of field derivatives enter the efIective Lagrangian.
Based on that Lagrangian, one can calculate the tree diagrams for arbitrary processes with any
number of pions. Furthermore, the leading approximation to the scattering amplitude can be
obtained using the interactions from c(2), since insert ions of vertices from higher order Lagrangians
lead to a suppression in powers of momenta or quark masses. Such an expansion clearly makes
sense only if the momenta of the pions are small.
The efIective field theory technique allows to go beyond the tree approximation and to systematically
calculate the quantum corrections represented by loop graphs. All ultraviolet divergences
can be absorbed in a redefinition of the coupling constants accompanying the interactions in the
Lagrangian order by order in powers of low external momenta and quark masses. To see how this
works in practice, we need to establish the power counting rules [60] for an arbitrary scattering
process. The on-shell S-matrix element for a reaction with N external pions can be expressed
as27
(3.142)
where PI , P2 , ... , PN are momenta of external pions and TI is a product of phase space factors corresponding
to the incoming and outgoing particles. The transition amplitude M can be rewritten
in the form
(3.143)
M == M(Q,g,fL) = QV f( Q ,g) .
Here, Q denotes a generic external momentum, g corresponds to a combination of the pertinent
coupling constants and fL is the renormalization scale. To calculate 1/ we have to count all derivatives,
pion propagators, moment um integrations and 6-functions for a specific diagram, since these
are the only dimensionsfull elements apart from the coupling constants. For a process with I inner
pion lines we have:
1/ = L Vi di - 21 + 4L , (3.144)
25This is the most general Lagrangian for SU(3). For the two-fiavor case, not all of these terms are independent
from each other.
26 In order to be able to perform renormalization in the presence of external sources one also needs to include in
eq. (3.141) furt her terms, which do not contain the pion fields. For more details see e.g. refs. [63], [64].
27Here we will only consider connected diagrams.
fL

70 3. The derivation of nuclear forces from chiral Lagrangians
derivatives of
the pion field or pion mass insertions. We count the inner lines twice because the pion propagator
scales as 1/ q2 . The total number of independent integrations (loops) equals the number of inner
lines reduced by the number of delta functions corresponding to each vertex:
where L is the number of loops and Vi the number of vertices of the type i with di
L = 1 - (2: Vi - 1) , (3.145)
where we took into account the overall delta function in eq. (3.142). Therefore we obtain the
following result for the total scaling dimension LI of the transition amplitude M:
(3.146)
Weinberg made in 1979 a crucial observation [60] that this number LI is bounded from below.
Indeed, due to the spontaneously broken chiral symmetry and the Lorentz invariance of the Lagrangian,
no interactions with di � 2 are allowed.28 Furthermore, including loops leads to a
suppression by at least two powers of external momenta against the tree graph with the same
vertices. According to eq. (3.146), the dominant diagrams with LI = 2 are the tree ones with all
vertices from [J2). The first corrections with LI = 4 are given by tree diagrams with one vertex
from .c(4) and l-loop graphs with the leading order vertices.
1['b (Pb) 1['c (Pc)
1['a (Pa)
/
,
1[' d' (Pd)
+
+
,
,
• • •
/
'e / ,
,
•
/
/
/
\ /
\ I
I
, I
\ I
I \
I \
I \
/ \
+ • •
Figure 3.2: Low-energy expansion for pion-pion scattering. The small filled circles
denote the leading order vertex from [P) and the filled square corresponds to vertices
from .c(4).
Let us illustrate the above ideas on an example with pion-pion scattering following the original
work of Weinberg [60]. One first intro duces the Mandelstarn variables s, t and u:
S = (Pa + Pb) 2 = (Pe + Pd) 2 ,
28This argumentation is valid not only in the chiral limit, but also for chiral symmetry breaking terms, since we
count Mrr � Q.
, - -

3.3. Chiral perturbation theory with pions
t
u
(Pa - Pe) 2 = (Pd - Pb) 2 ,
(Pa - Pd) 2 = (Pe - Pb) 2 .
71
(3.147)
For simplicity, we will consider the chiral limit with Mn = O. The Mandelstarn variables then
satisfy 82 + t2 + u2 = O. The S-matrix element can be expressed in the form
The transition amplitude M for the pion-pion scattering can be parametrized as
(3.148)
(3.149)
in terms of a single function A. To leading order (v = 2) this function is completely determined
by the inter action term
_ 1 _(1r . 8 1r)(1r . 8J11r)
21; J1 , (3.150)
corresponding to the leading order Lagrangian (3.136), see eq. (3.106). Note that the chiral
symmetry fixes the coefficient 1/(2{;) for this term. For the function A one obtains
A (2) (8,t,U) _- � J; . (3.151)
As shown in fig. 3.2, at next-to-Ieading order one has to evaluate the one-Ioop diagram with both
vertices (3.150) as weH as the tree graph with the interactions
(3.152)
Here we use the notation of reference [133] . Clearly, the couplings q, c� are some linear combinations
of the LECs defined in eq. (3.141). For the function A one finds at next-to-Ieading
order
where A is the ultraviolet cut-off. Introducing the renormalized couplings
allows to express the amplitude in the form
1 ( 1 2 ( 8 ) 1 2 2 2 ( t )
-- - - 8 In - - -- (U - 8 + 3t ) In -
(21 n)4 27r2 /12 127r2 /12
R R' 1 2 2 2 ( U ) C4 C4 2 2 2 )
- -- (t - 8 + 3u ) In - - -8 - - (t + U ) .
127r2 /12 2 4
(3.153)
(3.154)
(3.155)

72 3. The derivation o{ nuclear {orces {rom chiral Lagrangians
Thus, the function A(4) is suppressed by two powers of external momenta Q as compared to
the leading order result A (2). Apart from a polynomial part with undetermined constants, the
amplitude contains also logarithmic terms, that have the coefficients given by eq. (3.155) as definite
functions of frr. In fact, the structure ofthe non-polynomial part of A(4) can be obtained using the
renormalization group technique even without performing any explicit one-Ioop calculations [60].
Requiring that the amplitude A at each order does not depend on f-l allows to uniquely predict
the non-polynomial terms. To find the numerical coefficients of the logarithms one needs to
perform complete loop calculations.29 Note further, that an additional overall numerical coefficient
'" 1j(47r)2 appears in A(4), that arises from the loop integrals. Therefore, the pertinent expansion
parameter is Qj(47rfn.), that is remarkably smaller then the naive expected one Qjfn. Manohar
and Georgi have shown [166] that the scale which enters the values of the renormalized coupling
constants is Ax ::; 47r f 7r '" 1200 MeV. Their argumentation is based on the observation, that
the change in the renormalization scale of order one would change the effective couplings by
o(ljA�) '" 1j(47rf7r)2. Therefore, if the scale Ax would be very large for one particular value
f-lo of the renormalization scale, 1jA� « 1j(47rf7r)2 , it would be no more the case for another
choice of f-l. A consistent power counting requires the assumption Ax '" 47r f 7r. In that case the
quantum corrections are of the same order of magnitude as the contributions from the renormalized
interaction terms.
One can straightforwardly extend all these results to account for the explicit chiral symmetry
breaking. The amplitude A becomes also a function of the pion mass and does not vanish at
the threshold s = 4M;. Already in 1966 Weinberg calculated the 7r7r scattering lengths from the
leading order amplitude A(2) [167] using the current algebra techniques. Corrections of higher
order in m7rj(47rf7r) seem to improve the agreement with the experimental values [168].
To end this section we would like to make a remark concerning the so-called chiral anomaly. It
turns out that the effective Lagrangian [)2) + .c(4) implies a stronger symmetry than QCD does.
The absence of the terms with an insertion of the total antisymmetric tensor EIl-Vpu together with
the parity conservation requires that all terms in the effective Lagrangian are even in the pion
fields. This rules out such experimentally observed processes like, for instance, 7r0 ----7 rr. This
problem for the SU(3) case has been solved in 1971 by Wess and Zumino [169]. They have found
a term in the action of the effective field theory, the so-called Wess-Zumino-Witten term, that
preserves the chiral symmetry of the action but cannot be expressed as the integral of the chiral
invariant density over spacetime. Its explicit form (in the absence of external sources) is given by
(3.156)
where the integration goes over the five-dimensional sphere whose surface is spacetime, Eijklm
is the totally antisymmetric tensor in 5 dimensions and Ne = 3 is the number of colors. The
geometrical interpretation of this anomalous term was given by Witten [170], who had also shown
that the coefficient of this term is not a freely adjustable parameter. A further discussion of the
chiral anomaly goes beyond the scope of this manuscript. The interested reader is referred to the
original publications [169], [170], [171].
29 These coefficients have been derived in 1972 using unitarity instead of the phenomenological Lagrangian [164],
[165].

3.4. Including nucleons 73
3.4 Including nucleons
In the previous section we have considered chiral perturbation theory for 1r7r scattering. Based on
the most general chiral invariant Lagrangian, one obtains the amplitude in form of an expansion
in powers of small external momenta. The spontaneously broken chiral symmetry guarantees
the weakness of the interactions between the Goldstone modes (pions). This allows to perform
perturbation theory for the S-matrix. At each order in the low momenta one has a finite number
of diagrams to be considered. More complicated graphs with more loops contribute, in general,
to higher orders in this expansion. This scheme can be naturally extended to include processes
with nucleons. The starting point is again the most general chiral invariant effective Lagrangian
for Goldstone bosons interacting with the matter fields, that can be constructed along the lines
introduced in the section 3.2.
The first systematic calculation of pion-nucleon scattering in the framework of chiral perturbation
theory have been performed by Gasser et al. in 1988 [68]. The nucleons have been treated in
this approach fully relativistically. In such a formulation one loses the one-to-one correspondence
between the number of loops and the power of small extern al momenta (pion four-momenta or
mass or nucleon three-momenta).30 The origin of this unwanted feature is that the inclusion of
nucleons unavoidably introduces a new mass scale in the problem, the nucleon mass. This scale is
not a soft one (like the pion mass). It cannot be regarded small and does not vanish in the chiral
limit.
An elegant solution of this problem has been proposed by Jenkins and Manohar in 1991 [69],
see also Bernard et al. in 1992 [70]. The idea is to use a simultaneous expansion in powers of
Q / Ax rv q / ( 41r f 1r) and Q / m, w he re m is the nucleon mass. Let us now briefly explain how such
an expansion may be performed. The four-momentum of the nucleon has the form
(3.157)
where vJl is the four velo city satisfying v 2 = 1. For a heavy nucleon it is convenient to parametrize
PJl as
(3.158)
where lJl is a small residual momentum v·l «m. The advantage of such a parametrization is that
the trivial kinematic dependence of PJl on the large term mVJl is now explicit. In the rest-frame
with vJl = (1,0,0,0) the three-momentum is entirely given by f and PJl = cj m2 + [2, T). Note
that in the limit m ---+ 00 the nucleon four-velocity becomes a conserved quantum number since
its change by a finite amount would lead to an infinite moment um transfer between the incoming
and outgoing nucleons [172]. To get rid of the nucleon mass term in the Lagrangian for the free
Dirac field
(3.159)
one can introduce the eigenstates H and h of the velo city operator p via
H =
p+ eimv.x'l!
v ,
(3.160)
where E� is the projector operator onto the eigenstates of the four-velocity operator v corresponding
to the eigenvalues ±1:
30 This is true for the conventional regularization schemes like, for example, dimensional regularization.
(3.161)

74 3. The derivation o{ nuc1ear {orces {rom chiral Lagrangians
The exponential faetor in (3.160) eliminates the trivial kinematic dependenee of the nucleon field
on the moment um mvw The quantities H and h are usually ealled the large and small eomponents,
respeetively. The free field Lagrangian (3.159) ean be expressed in terms of H, h as
.c = fI(iv . ö)H - h(iv . ö + 2m)h + fIi�h + hi�H
This leads to the following equations of motions for H and h:
(v · ö)H
(v . ö)h
From the second equation one sees that
-Pv+�h ,
2imh + Pv - �H .
Therefore, the large eomponent field H obeys the free equation of motion
(v · ö)H = 0 ,
(3.162)
(3.163)
(3.164)
(3.165)
up to l/m eorreetions. The nucleon mass does not enter this equation of motion to leading order.
Note that in the nucleon rest-frame with vJ.l = (1,0, 0,0), H and h ean be identified with the usual
upper and lower eomponents of a Dirae spinor modulo the faetor eimt. The small eomponent field
h ean now be eompletely eliminated from the Lagrangian (3.162) using the equations of motion
(3.163). A more elegant path integral formulation of this nonrelativistie reduetion ean be found
in the referenee [173].
For the ease of nucleons interaeting with pions one ean proeeed in a similar way as for free
nucleons to eliminate h. The equations of motion (3.163) are then modified by terms including
pion and nucleon fields. Sinee the ehiral symmetry only allows for derivative pion-nucleon and
pion-pion eouplings, these additional terms ean be regarded as small eorrections of order 1/ Ax to
the leading order equations ofmotions (3.163).31 There is one important problem in this approach:
time derivatives of the nucleon fields eontribute large factors of order m. In fact, it ean be proven
[68], [174], [175] that the leading order Lagrangian .cS 1 Jv for nucleons 'l1 interacting with pions 7r,
Le. the Lagrangian of the minimal low-energy dimension, is given by
r(l) _ ,1" ( ' J.l D
9A J.I ) 'T'
'-'7rN - 'i' n J.I - m + TI' I' 5 UJ.l 'i'. (3.166)
Here, the low-energy dimension of an operator has to be understood in the following sense: let Q be
a generic four-momentum of (external) pions or a generie three-momentum of (external) nucleons.
Then, U, 'l1, DJ.I'l1, m count as order one, whereas, for example, uJ.l ' öJ.lU, M7r , (i--y J.l DJ.I - m)'l1 as
order Q. Consequently, one ean apply the equation of motion
(i�- m)'l1 = ... , (3.167)
31 Clearly, one should not understand this statement ab out the smallness of the higher derivative terms too literal:
the operators with more field derivatives are, of course, by no means "smaller" than those with less derivatives.
Rather, one should understand this jargon in the way it was explained in the section 3.3. For a purely pionic
effective theory, the interactions in the Lagrangian with more field derivatives lead to low-energy observables (8matrix
elements) that are suppressed by additional powers of small external momenta. This can be seen from the
power counting (3.146). In this sense, the operators with larger index d - 2 that was defined in eq. (3.146) are of
higher order and called small corrections.

3.4. Inc1uding nuc1eons 75
to eliminate derivatives of the nucleon field in higher order Lagrangians. In eq. (3.167) the dots
correspond to terms of order Q and higher. Using this equation we can rewrite derivatives acting
on the nucleon field in interactions like � ... 82n fJk'J! in terms of other operators of the same
and higher orders. Such terms are anyway present in the effective Lagrangian, which contains all
possible chiral invariant interactions. Thus, eliminating the nucleon time derivatives only modifies
the values of couplings accompanying the interactions in L. With other words, one can simply
drop time derivatives acting on the nucleon field in all interaction terms, leaving a single time
derivative in the free Dirac Lagrangian. Another more elegant way to eliminate the derivatives of
the nucleon field is to use (nonlinear ) field transformations [133]. For a recent discussion on that
approach as well as for concrete examples see the reference [71]. For various problems related to
the inclusion of higher derivative operators in the Lagrangian see refs. [176], [177], [178], [179].
As soon as one has solved the problem with the time derivatives acting on the nucleon fields, the
nonrelativistic reduction can be performed in a similar way as for free nucleons. The only difference
is that the equations of motion for the large and small component fields contain corrections from
higher order terms. Eliminating h from the Lagrangian yields then coefficients of 1/ m corrections.
Equivalently, the effective Lagrangian can from the beginning be derived in terms of nonrelativistic
nucleon fields. In such a case not all of the coupling constants are arbitrary: so me of them have
definite coefficients proportional to inverse powers of the nucleon mass. Those coefficients are
fixed by the Lorentz invariance of the original relativistic Lagrangian. The simplest example of
such terms is the nonrelativistic kinetic energy Nt\!2/ (2m)N. Technically, one can obtain those
fixed coefficients in the easiest manner requiring the so-called reparametrization invariance of the
Lagrangian [182]. Consider the heavy baryon effective field theory discussed at the beginning of
this section. Instead of the usual fields for nucleons it is convenient to introduce the large and
small component fields according to eq. (3.160). If the nucleon mass is very large, the four velo city
becomes a conserved quantum number. As can be seen from eq. (3.160), the fields H and h are
velocity dependent and should be, in principle, denoted by the index v.32 The effective Lagrangian
is then given by
Leff = L Lv (Hv, vf.t, ... ) , (3.168)
v
where dots correspond to other ingredients like, for instance, covariant derivatives of the nucleon
and pion fields. Reparametrization invariance says that we could equally well chose another
parametrization of the four moment um to describe the same physics:
(3.169)
where qf.t has to satisfy (vf.t + qf.t/m)2 = 1. Luke and Manohar [182] proposed to construct the
effective Lagrangian in terms of Hv defined as33
where
- 1 + #
Hv = Hv ,
J2(1 +w·v)
(3.170)
(3.171)
32We did not do that in the above discussion since the velo city is anyway conserved.
33In a more general case when one has to incorporate external fields and/or pions the derivative of the nucleon
field in eqs. (3.170), (3.171) should be replaced by a covariant one.

76 3. The derivation of nuclear forces from chiral Lagrangians
The advantage of this formulation is that the new object Hv transforms in a covariant way under
the reparametrization (3.169):
H- w - eiq'xHv
· (3.172)
The requirements on the l/m-coefficients following from the reparametrization invariance of Leff
can now easily be worked out. A more detailed discussion on the construction of the effective
Lagrangian for pions and nucleons can be found in references [73], [75], [78], [161]. It goes without
saying that as soon as one has Leff, the corresponding Hamiltonian can be obtained applying the
usual rules of the canonical formalism [75].34 We will not discuss the construction of the effective
Lagrangian (Hamiltonian) for pions and nucleons any furt her and simply adopt the corresponding
expressions from refs. [75], [77], [78]. Below, we will show explicitly all terms in the Hamiltonian
we need to calculate the N N potential to a required order and consider some of these structures
in more detail.
Let us now repeat the arguments of Weinberg [73] and derive the power counting rules for a
general process with N nucleons and an arbitrary number of extern al pions. According to the
nonrelativistic treatment and to the fact that the low component nucleon fields are integrated
out, no nucleon-antinucleon pairs can be created or destroyed. In his original work [73], Weinberg
used the covariant perturbation theory (Feynman diagram technique) to estimate the power v
of small external momenta Q for an arbitrary N-nucleon scattering process. We will now use
time-ordered35 perturbation theory to calculate v, since it makes more transparent the problems
associated with the perturbative treatment of the few-nucleon problem. We will proceed similarly
to the case of pion-pion scattering considered in the last section. Instead of covariant propagators
one has in old-fashioned perturbation theory energy denominators and phase-space factors. One
also has to remember that all integrations are now three dimensional and the particles are always
on their mass shell but can be off the energy shell in the intermediate states. The power of
momenta v can be found via
L L ( Pi) Ep
v = 31 - 3 11,. - D + v: d· - - + - . z . z Z
+ 3
(3.173)
Z Z
2 2
where Dis the number of energy denominators (or, equivalently, the number of intermediate states
between time-ordered vertices), 1 is the number of internal lines and Ep denotes the number of
external pion lines. Each of the Vi vertices of type i contains di derivatives or pion mass insertions
and Pi pions. For the moment, we assurne that all energy denominators scale as l/Q. The first
term in eq. (3.173) gives the number of moment um integrations. Not all of these momentum
integrations are independent of each other because of the 6-functions accompanying vertices. We
take into account the reduction of the v due to such 6-functions subtracting the second term in
eq. (3.173). Further , we do not count the phase-space factors of external pions and, therefore,
add Ep/2 to the right-hand side of eq. (3.173). Finally, we also do not count the overall fourdimensional
6-function in the S-matrix elements. However, we should add the factor 3 and not 4
to the right-hand side of eq. (3.173), since this expression for the power v of momenta corresponds
to aT-matrix element, which does not include the overall energy conserving 6-function present in
the S-matrix. The meaning of the last term in eq. (3.173) is now clarified. To make the formula
(3.173) suitable for practical calculations we need few topological identities. First, note that the
number of intermediate states D can be expressed as
D=L Vi-l. (3.174)
34 It was shown by Ostrogradski [180] how to derive a Hamiltonian from a Lagrangian that includes higher
derivatives. See also [178], [181].
35 Sometimes it is also called "old-fashioned" perturbation theory.
'

3.4. Including nucleons 77
Next, we write the number of loops in the form
L=I- :L Vi+C. (3.175)
Here, we modified the earlier expression (3.145) to be valid also for C disconnected diagrams.
Finally, the numbers of internal and external particle lines l and E = En + Ep , where En = 2N,
can be collected via the following identity:
(3.176)
where ni is the number of nucleon fields at the vertex of type i. Using eqs. (3.174)-(3.176) and
eq. (3.173) we end up with the result
where
v = 4 - N + 2(L - C) + :L Vi�i , (3.177)
(3.178)
As already discussed above, the corresponding indices �i are not negative for the purely pionic
Lagrangian.36 Analogously, �i are not negative for pion-nucleon as weIl as for nucleon-nucleon
interactions. In particular, the minimal possible value �i = 0 is achieved for the vertices with two
nucleons and one derivative (and any number of pion fields) or for the four nucleon interactions
without derivatives and pion mass insert ions (and, therefore, also without pion fields). Thus, v
in eq. (3.177) is again bounded from below and, therefore, it seems that the scattering amplitude
für the process N N ----7 N N can be calculated perturbatively in powers of low external threemomenta
of nucleons in a similar way as in the case of pion-pion scattering discussed in the last
section. This already sounds suspicious, since the presence of the low-energy bound state in the
np 3S 1 _3 D 1 channel clearly signals the failure of perturbation theory .
.. J .. .. J 11 J ..
p ! I p' P I l I p'
't
I!
I !
I
.. .. .. .. i ..
-p _p' -p -f -I
-p
a) b)
Figure 3.3: Irreducible and reducible time-ordered diagrams. The vertical dotdashed
lines are the energy cuts. The solid (dashed) lines correspond to nucleons
(pions).
To solve this paradox let us take a closer look at old-fashioned perturbation theory. In the above
derivation of the power counting rule (3.177) we made an assumption that all energy denominators
36 More precisely, it is zero for the Lagrangian for free pions and positive für interactions (in the absence of external
sources).

78 3. The derivation o{ nuclear {orces {rom chiral Lagrangians
scale as l/Q. Let us now consider the two different cases shown in fig. 3.3. In the two-nucleon
center-of-mass system the energy denominator in the graph a) in fig. 3.3 is given by
1 1
Ei - E p2 Im -pl2 Im -vi q2 + M; (3.179)
where Ei is the initial energy of two nucleons and q = p' -p and p = 1P1, p' = Ip'l, q = 1iJ1. This
estimation is valid modulo 11m corrections. Consider now the second diagram. For the energy
denominator corresponding to the energy cut in fig. 3.3 b) we have:
1 1 m m .
Ei - E p2 Im -[2 Im + iE [2 (3.180)
p2 _
+ iE' rv Q2
Thus, the energy denominator in the second case is mlQ times larger than for the first diagram.
In the limit m ---+ (X) it becomes even infinitely large. Therefore, the power counting (3.177) is
not valid any more. To solve the problem with the large energy denominators arising from the
intermediate states with only two nucleons, Weinberg proposed to apply the technique of CHPT
not directly to the scattering amplitude but to the effective potential. The latter is defined as a sum
of all irreducible diagrams, i. e. those diagrams without pure nucleonic intermediate states. The
S-matrix elements are obtained by putting this potential into a Lippmann-Schwinger equation.
In fact, many equivalent schemes for deriving the effective potentials in nuclear and many-body
physics are known like that due to Bloch and Horowitz [183] or the Tainm-Dancoff approximation
[184]. The effective potential, derived using old-fashioned time-ordered perturbation theory,
possesses one unpleasant property: it is, in general, explicitly dependent on the energy of the
incoming nucleons and, as a consequence of this, not hermitian. Furthermore, the nucleonic wave
functions are not orthonormal in this approach [185].
One can avoid these problems by using the method of unitary transformation. It was already
applied successfully in cases where one has an expansion in a coupling constant, such as the
pion-nucleon coupling, see e.g. refs. [186]. In the following sections we will show how to apply
this method to the case of chiral perturbation theory for pions and nucleons, in which the small
momenta of extern al particles play the role of the expansion parameter. In fact, our considerations
are more general since they can be applied to any effective field theory of Goldstone bosons coupled
to some massive matter fields.
3.5 Bloch-Horowitz scheme and the method of unitary transfor­
mation
The method of unitary transformation (projection formalism) was already applied in the chapter 2
to decouple the spaces of small and large momenta in the quantum mechanical two-body system.
Here we would like to repeat the main points, mostly to establish our notation and keep the
section self-contained. We will also compare it with the Bloch-Horowitz scheme of deriving the
effective interactions.
A system of an arbitrary number of interacting pions and nucleons can be completely described
by a Schrödinger equation
(3.181)
with Ho (Hf) denoting the free (interaction) part of the Hamiltonian. In order to solve this
equation for nucleon-nucleon scattering, it is advantageous to project it onto a subspace
{IN), INN), INNN),
I

3.5. Bloch-Horowitz scheme and the method of unitary transformation 79
apply the standard methods of few-body physics to obtain the S-matrix. We will denote the
remaining part of the Fock space by 1'ljJ): I\J!) = 11» + 1'ljJ). Let 'rJ and A be projection operators on
the states 11» and 1'ljJ) which satisfy 'rJ 2 = 'rJ, A 2 = ,\, 'rJA = A'rJ = 0 and A + 'rJ = 1. Eq. (3.181) can
now be written in the form
( 'rJH'rJ 'rJHA ) ( 11» ) = E ( 11» )
AH'rJ AHA 1'ljJ) 1'ljJ)
One can express the state 1'ljJ) from the second line of this matrix equation as
1
1'ljJ) = E _ AHA AH'rJI \J!) .
(3.182)
(3.183)
Putting this into the first line of eq. (3.182) leads immediately to a single equation of Schrödinger
type for the state 11»
with an effective potential Veff(E) given by
(Ho + Veff(E)) 11» = EI1» (3.184)
(3.185)
This decoupling scheme has been first proposed by Bloch and Horowitz [183]. Expanding the
denominator in eq. (3.185) in powers of H[ leads to
(3.186)
In this form it is obviously identical with the result obtained from old-fashioned perturbation
theory [187, 11]. Note that the states 11» are not orthonormal:
(3.187)
Let us now take a look into the method of unitary transformation. We first introduce new states
Ix ) and Icp) , which are related to 11» and 1'ljJ) by a unitary transformation
( ) ( )
Ix )
= ut 11» (3.188)
Icp) 1'ljJ)
Then one can rewrite eq. (3.182) in an equivalent form
ut HU ( Ix ) ) = E ( I x ) )
Icp) Icp)
(3.189)
The two subspaces for I X) and I cp) can be decoupled by choosing U such that the operator ut HU
is diagonal with respect to the two subspaces. We again adopt the ansatz of Okubo [51], as in
chapter 2, in which the unitary operator U is parametrized in terms of a single operator A as
follows
(3.190)

80 3. The derivation of nuclear forces from chiral Lagrangians
The operator A has only mixed non-vanishing matrix elements:
A = >'A1] . (3.191)
The requirement for ut HU no longer to couple the two subspaces leads to the following nonlinear
equation for A:
>. (H - [A, H] - AHA) 1] = 0 , (3.192)
which is sometimes referred to as decoupling equation. Once the two spaces are decoupled one
can choose the Icp) component to be zero and the nucleonic states are orthonormal:
(3.193)
In the last step we have used the fact that Icp) = o.
In the case when the interaction Hamiltonian Hf can be treated as a small perturbation, it is
possible to solve Eq. (3.192) perturbatively to any given order. For instance, for the Hamiltonian
H represented by
00
H=Ho + LHn n==l
(3.194)
with the index n denoting the power of the coupling constant, one assurnes the operator A to be
of the form
00
(3.195)
The solution of eq. (3.192) to order n is then given by
(3.196)
Here, we denote the free-particle energy of the state 11]) by [. One can see from eq. (3.196) that
it is possible to find An for every n recursively, starting from Al.
As soon as the operator A is known, one can obtain the effective Hamiltonian, which operates
solely in the subspace Ix), via
(3.197)
as it follows from eqs. (3.189), (3.190). Expanding (1 + AtA)-1/2 and using eqs. (3.194), (3.195),
(3.196) one can obtain the effective Hamiltonian to any order in the coupling constant.
Several modifications are necessary by applying the formalism described above to effective Lagrangians
(Hamiltonians) and in particular to chiral invariant Lagrangians. First, the expansion
in powers of a coupling constant must be replaced by the expansion in powers of small momenta.
For doing that, power counting rules are necessary. Furthermore, one expects the operator >'A1]
to consist of an infinite number of terms to any order of Q caused by infinite number of vertices
in the Hamiltonian. We now show how these problems can be solved.

3. 6. Application to chiral invariant Hamiltonians
3.6 Application to chiral invariant Hamiltonians
We first want to recall the structure of the most general chiral invariant Hamilton density for
pions and nucleons,
(3.198)
As already noted before, the nucleons are treated nonrelativistically, as it has also been done in
[73], [76]. Consequently, the purely nucleonic part of 1-l0 is nothing but the kinetic energy
V2
1-lNO = -Nt -N
(3.199)
2m
At higher orders in small momenta, which we will not treat here, relativistic corrections to the
kinetic energy (3.199) must be taken into account.
The free Hamilton density for the pion fields (in the interaction picture) is given by
1-l = !ir2 + ! (V1t' )2 + !m2 1t'2
11"0 2 2 2 (3.200)
11"
where the ,., denotes the time derivative and m1l" the pion mass. We split the interaction Hamilton
density into three parts:
(3.201)
The first piece describes self-interactions of pions and contains an even number of derivatives and
pion field operators and any number of M;-factors. The terms with two derivatives (or one M;)
have at least four pion fields. The second piece in eq. (3.201) contains terms with four or more
nucleon fields and any number of derivatives. The terms denoted by 1-l1l"N have any number of
pion fields and at least two nucleon fields and one derivative or factor M1I" ' The structure of the
effective Hamiltonian is now clarified. Later we will explicitly give the leading terms in the effective
Hamiltonian, that we will need to calculate the potential. We now show how to eliminate pions
and to obtain the effective potential for nucleons for the most general chiral invariant Hamiltonian
(3.201) .
Let us first note that because of the baryon number conservation and the absence of anti-nucleons,
the subspaces of the Fock space with different number of nucleons are automatically decoupled.
That is why we define the state I

82 3. The derivation of nuclear forces from chiral Lagrangians
Because of the property eq. (3.191) of the operator A each single equation in eqs. (3.205) can be
expressed as
00 00
). iHrfJ + L ). iHj). j AT] + ). iHo). iAT] - ). iAT]HjT] - ). iAT]HoT] - L ). iAT]Hj). j AT] = 0
j=l j=l
(3.206)
The system of eqs. (3.205) is, clearly, too complicated to be solved exactly. In what follows, we will
apply the usual philosophy of effective theories. We are interested only in low-energy processes.
Therefore, we will expand the matrix elements of an effective potential in powers of the small
moment um scale Q, as it was proposed by Weinberg [73]. To calculate the operator A from the
system of coupled equations (3.205), we will again make use of the expansion in powers of Q. In
particular, the matrix elements of A will be classified by powers of Q by use of simple dimensional
analysis:
(3.207)
Here Q is again the mass scale corresponding to three-momenta of nucleons and four-momenta
of pions. The effective Hamiltonian acting on the purely nucleonic Fock space is defined in
eq. (3.197) and can be found from the original one if the operator A is known. It will be clear
from the power counting arguments that the matrix elements (3.207) with larger 1I A lead to an
effective potential suppressed by additional powers of Q. At first sight, this appears to be a
miracle, since (�I and 1

3.6. Application to chiral invariant Hamiltonians 83
where Pi (ni) is the number ofpion (nucleon) fields at a vertex oftype i. Now we use the topological
identity (3.175) with I = In + Ip to cast eq. (3.208) into the form
vA = 3 -3N - Ep + L l'i /'\,i (3.210)
where
(3.211)
Again we point out that this result takes its form due to the ansatz that the projected operators
)..i A17 consist of an equal number of vertices and energy denominators. Also, it should be mentioned
that we use the number of external pion lines to express the counting index VA instead of the
number of loops and of separately connected pieces as it has been done in [73], [74], [76], [78].
This is more natural in the projection formalism employed here. Let us take a doser look at vertex
dimension /'\,i, which is related to the canonical field dimension. It is weIl known, see e.g. ref. [132],
that all interactions can be classified with respect to /'\,i. Those with /'\,i < 0 are called relevant
(superrenormalizable), with /'\,i = 0 marginal (renormalizable) and with /'\,i > 0 irrelevant (non­
renormalizable). The last ones are "harmless" and can be wen treated within low-energy effective
field theories. In contrast to these, the relevant and marginal interactions lead to complications
with the power counting. This becomes immediately clear if one takes a look at eq. (3.210):
an infinite number of diagrams contributes to a given process at a given order. Furthermore, for
relevant interactions the number VA is even not bounded from below. That is why the perturbative
treatment in powers of the moment um scale Q is not possible in this case. For more details about
the role of such interactions in effective field theories see ref. [192]. Chiral symmetry does not allow
any relevant or marginal interactions in the interaction Hamiltonian Hf. The minimal possible
value of /'\,i for vertices in Hf is one. Such a vertex with /'\,i = 1 contains two nucleons, one pion
and one derivative. In general, for vertices with two nucleons and Pi pions one has /'\,i 2 Pi. The
interactions with pions only have the vertex dimension /'\,i 2 2, where /'\,i = 2 corresponds to the
interaction with four pion fields and two derivatives. The purely pionic interactions have an even
number Pi of pions and /'\,i 2 max(2, -2 + Pi).
Now we would like to determine the minimal value of VA for )..a A17. For that it is convenient to
express vA in terms of L, C and ßi defined in eq. (3.178). Making use of the relations (3.208),
(3.209), (3.175) we obtain
(3.212)
where En is again the number of external nudeon lines. The value of VA is one less than the one
of v for an irreducible time-ordered graph in Weinberg's formula eq. (3.177). This difference is
because the number of energy denominators equals now the number of intermediate states plus
one, as follows from our ansatz about the structure of )..a A17. It is now easy to find the minimal
value of VA. Different to the case of few-nudeon scattering considered in the section 3.4, VA is
not bounded from below for a fixed number En of external nudeon lines. This is because each
additional disconnected purely pionic tree subdiagram with all vertices with two derivatives yields
the factor -2, as can be seen from eq. (3.212). Therefore, the minimal value of VA requires
min(VA) = 3 -3N -2k (3.213)
the maximal possible number of such disconnected pieces. Correspondingly, one finds for matrix
elements of the operator )..4k+i AT] with k a positive integer or zero and i = 0,1,2,3:

84
.... .
- - .... \ .
•
A
.... .
. - - - - .... \
•
A
.. _ - - - - - ,
•
a)
3. The derivation of nuclear forces from chiral Lagrangians
- - - _. - .... .
.... .
.... .
\
. .... \ \
• • •
b)
\
- -- - -, ...
'"'
.... .
- - - - \
•
Figure 3.4: Some leading order contributions to matrix elements of the operators
.x a A'T} with a = 9 in the left pannel and a = 3 in the right pannel. All vertices have
ßi = o.
Here, the parametrization 4k + i of the number of pions in the state .x is convenient, since the pion
selfinteractions with ßi = 0 have at least four pion field operators. For example, one particular
topological structure of the matrix element .x 9 A'T} with the minimal possible value VA = -4 is
shown in fig. 3.4, a). Here, the two disconnected purely pionic subdiagrams lead to the negative
value of VA. Another example corresponding to the leading order matrix element .x3 A'T} is shown
in fig. 3.4, b). Note that no inner pion lines appear in the leading order diagrams apart from those
corresponding to .x 4k +i A'T} with i = 3. In that case one intermediate state with one pion may aiso
occur.
Let us count the power of Q for the operator .x4k +iA'T} starting from its minimal value. For that
we define the order l (l = 0, 1,2,3, ... ) of the operator .xa A'T} to be
(3.214)
and introduce the following notation: .x4k + i A/'T}. We will now find the power v of Q for every term
in eq. (3.206). By HK, we denote a vertex from the interaction Hamiltonian HJ, with the index K,
given by eq. (3.211). Details are relegated to appendix A. It can be seen from eqs. (A.7), (A.10),
(A.12), (A.15) and (A.16) that the minimal possible value of v for the various terms in eq. (3.206)
is given by
min(v) = 4 - 3N - 2k (3.215)
In these cases the number of energy denominators is one less than the number of vertices. This
explains the difference between eqs. (3.213) and (3.215).
We now show how to solve the system of eqs. (3.205) perturbatively. For that, we define the order
of r :::: : 0 of eq. (3.206) via
r = v - min(v) . (3.216)

3.6. Application to chiral invariant Hamiltonians
number
of pions
12 /
,
\
I
,
11 \
\
,
,
\
\
10 /
I
,
9 /
I
,
8 \
\ /
�
7
6 /
,
,
5 /
I
,
4 \
3
2
1
�
0
,
,\
•
•
/
-- - '
,
\
'.
/
I
,
\ .,
\ /
�
, , /
\ e-
, --
/
,
\
'.
/
I
,
\ .,
\ /
�
, , /
\ e-
,
/.
/
,
.... .
,
\
\
• •
• •
• •
I • •
/
I
.,
•
,
e-
-' \ ,
, •
\
,
\
/
--
,
\
'.
/
,
'.
/
I
,
\ .,
\ /
I
,
\ .,
\ /
�
, ,
e-
/
\
�
\ e-
/ . - ,
, , /
, ,
--
/
/ --
,
, \
' '::: .
\ ,
, \
.... �.
2 3
,
\
\
• • •
• • •
• • •
• • •
• • •
• • •
• • •
• • •
.,
/
• •
I
, /
\ e-- ,
,
• •
\
,
/ --
\ ,
, \
'�.
/ --
\ ,
, \
....
�.
•
,
-->-- ..
4 5 6
order r
Figure 3.5: Recursive prescription for calculating the operator ,\4k+i Ar]. The dashed
line shows a sequence of recursive steps. The large circles correspond to states with
4k pions (i = 0).
Thus, we again count the power of Q for the terms entering eq. (3.206) starting from its minimal
value. In appendix B we check what kind of operators ,\4k+i AIr] contribute to eq. (3.206) at each
fixed order r. That equation can be expressed as
85
(3.217)
Here, E('\ 4k+i) denotes the free energy of particles in the state ,\ 4k+i. In the curly brackets we
have written in symbolic form all remaining terms of eq. (3.206). For i = 1,2 they contain only
operators of the type ,\4k +i AIr] with
4k + i < 4k + i for l = r , or l < r (3.218)
As an example, we regard ,\4k+iH,\4 k +iAlr] with i = 1, 2. For 4k + i < 4k
+ i it follows from

86 3. The derivation of nuclear forces from chiral Lagrangians
eqs. (B.2), (B.3) that l ::; r. For 4k + i = 4k + i, eq. (B.4) leads to l ::; r - 2. Finally, for
4k + i > 4k + i eqs. (B.5)-(B.8) require l ::; r - 2.
Eq. (3.218) is also valid for i = 3 apart from the single term _,\4k +3 Hl,\4( k+l) Al=r'l]. This is due
to eq. (B.7). In the ease i = 0, only the operators ,\4 k + i Al'l], restrieted by the eonditions
i = 0, k < k for l = r , or l < r (3.219)
ean enter the right hand side of eq. (3.217). Furthermore, the value of k is bounded from above
in all eases by the inequality
- 1
k ::; k + 6 (r - l + 8) ,
(3.220)
as ean be deferred from the seeond inequality in (B.8) and the equality in (B.16).
Now it is clear how to deal with the system of equations (3.205). The equations have to be solved
order by order, starting from r = O. At eaeh fixed order r one solves the equations with inereasing
number 4k + i starting from 4k + i = 1 to obtain all operators ,\4k + iAl=r'l]. This requires the
knowledge of operators ,\ 4k + i Al'l], 4k + i < 4k + i, of the same order l = r and a finite number
of operators ,\(4k + i ) Al 'I] at lower orders l < r. The only exeeptions of this proeedure are the
equations with arbitrary k's and i = 3, that require the knowledge of the operators ,\ 4( k +l) Al=r'l].
Such equations have, therefore, to be solved after those ones with 4( k + 1) external pions. The
number of equations to be solved at each order can be estimated by use of eq. (3.220) and the
inequalities of appendix B. After solving the required number of equations at order r one ean go
to the next order r + 1. These rules for the reeursive solution of eqs. (3.205) are summarized onee
more graphically in fig. 3.5.
To justify our ansatz about the strueture of the operator A, eonsider the starting equations at
order r = 0, given by
E(,\4),\4Ao'l] = ,\4H 2 '1]
E(,\l),\l Ao'l] = ,\ 1 Hl'l]
(3.221)
(3.222)
From eq. (3.217) one can see that the unknown operator ,\4k + i Al=r'l] has the structure which we
assumed at the beginning of this seetion, if and only if the already known operators '\A'I] entering
the right hand side of this equation have precisely this form. Therefore, to proof our ansatz
recursively for all operators '\A'I] it is sufficient to see that it holds for the eorresponding starting
operators in eqs. (3.221) and (3.222), whieh is obviously the ease.
Having ealculated the operators ,\ 4k + i A'I] it is straight forward to obtain the expansion in powers
of Q for the transformed Hamiltonian eq. (3.197). To do that we have to estimate the order of all
terms on its right hand side. Introducing the operators
and their ehiral power (the power of Q)
with
L = 'I1At \ 4kt '11
t - ., lt /\ +it A l't "
Vt = 3 - 3N + Vt
Vt = 4kt + 2it + lt + l' t
(3.223)
(3.224)
(3.225)

3. 7. Nuc1ear forces using the method of unitary transformation
evaluated via eqs. (3.210) and (A.l) one gets contributions of the following types:
1. [II Lt]
t
v = 3 -3N + LVt
t
2. [II Lt]r,H" 77 [II Ls]
t s
v = 4 -3N + K, +
L Vt + L v s
t s
3. [II Lt] 77AL,\ 4krr.+irr. H,,77 [II Ls]
t s
and [II Lt]r,H",\4krr.+imAlm77[II Ls]
t s
V = 4 - 3N + K, + 2km + im + Zm + L Vt + L Vs (3.228)
t s
4. [II Lt]r,AL,\4km+irr. H",\4kn+in A1n77[II Ls]
87
(3.226)
(3.227)
t
V = 4 -3N + K, + Zm + Zn + 2km + 2kn + im + in + L Vt + L vs · (3.229)
t
Bere, v denotes the corresponding power of Q. In case of the unity operator in eq. (3.197) one
has to drop the v's. The effective potential can be easily read off from the effective Bamiltonian
Via
(3.230)
In the following seetion we will give concrete examples and calculate the leading orders of the two
nuc1eon potential.
3.7 N uclear forces using the method of unitary transformation
We will now apply the formalism described in the last seetion and derive an effective Hamiltonian
acting on the purely nuc1eonic subspace of the full Fock space at leading and next-to-leading
orders. As a starting point we use the effective chiral invariant Hamiltonian for nuc1eons and
pions [127] . It is based on the effective Lagrangian given in ref. [78].38 It reads:
9A t �
(3.231)
-N TCf· ·'\17rN
2j'lr
1
- 2j;
(7r ' 8/17r)(7r ' 8/17r)
+ _1_NtT . (7r x ir)N
4j;
'
+ �CT (NtCfN) . (NtCfN) + �Cs (NtN) (NtN) ,
j \ Nt (2c 1 M;7r2 - C3( 8 /1 7r . 8/17r) + C4 Cijk Eabc aiTa ('\1 j'lrb) ('\1 k'TrC)) N ,
'Ir
D1 (NtN) (NtTCf' :�7rN) + D2 [(NtTCfN) x
4h
8h
2
x (NtTCfN)]
. '�7r
38 For the contact interactions with four nucleon legs we have used the set of terms given in appendix F.
(3.232)
(3.233)
(3.234)
(3.235)
(3.236)

88 3. The derivation of nuclear forces from chiral Lagrangians
+ �(\ [(NtVN)2 + (VNtN)2] + ((\ + 02 )(NtVN) . (VNtN)
+ �02 (NtN) [NtV2N + V2NtN]
+i�63{ [(NtVN) . (VNt x iJN) + (VNtN) . (NtiJ x VN)]
+ (NtN)(VNt . iJ x VN) + (NtiJN) . (VNt x VN)} (3.237)
1 ( 1 - -
+ 2 2 C 4 (6ik6jl + 6il6kj) + C56ij6kl
x [(OiOjNt17kN) + (Nt17kOiOjN)] (Nt171N)
+ (�06 (6ik6jl + 6il6kj) + (05 - 67)6ij6k1) (Nt17kOiN)(ojNt171N)
1 ( 1 - -
-
+ 2 2 (C4 - C6) (6ik6jl + 6i/6kj) + C76ij6kl
x [(OiNt17kOjN) + (ojNt17kOiN)] (Nt171N) ,
1i5 �El (NtN) (NtrN) . (NtrN) + �E 2 (NtN) (NtriJN) . . (NtriJN)
+ �E3 [(NtriJN) x x (NtriJN)] .. (NtriJN) .
)
)
(3.238)
Here we have shown explicitly only the operators 1i". leading to non-vanishing TjH".Tj, ),1 H".),l,
),1 H".Tj, ),2 H".Tj, ),2 H".),l, ),4H".Tj and h. c., which we will need in our furt her calculations. The
corresponding index /'l, is defined in eq. (3.211). The operators with nucleon fields and three or
. . more pion fields are irrelevant. The symbols ' ' ' , x x
' me an that the appropriate products in
co-ordinate and isospin space have to be taken. Note that one has, in principle, four possible
contact terms (two more involving r) in eq. (3.234). However, they can be reduced to two after
performing anti-symmetrization of the potential. This will be explained below. Furt her , the
Hamiltonian used in this paper is always taken in normal ordering. Finally, we have used the
standard (in the one-nucleon sector) notation for the Cl , C3 and C4 terms, which is different from
the corresponding one given in the reference [78] 39 and also our El, 2 ,3 couplings differ from those
defined in [77]. In particular, EI = EU4, E2 = E!J./4 and E3 = Ej/8, where we denote by E* the
corresponding parameters from reference [77].
The Hamiltonian given in [78] contains apart from the terms enumerated above the two additional
interactions
(3.239)
which lead to significant contributions to the two-nucleon potential at next-to-leading order.
However, as argued in [127], no corresponding terms appear in the relativistic Lagrangian after an
appropriate nucleon field redefinition is performed. Therefore, terms of such type in the nonrelativistic
Lagrangian may only represent l/m-corrections, that are irrelevant for our calculations
because of the counting eq. (A.14).
39 In ref. [78] these are the EI-E3 terms.

3.7. Nuc1ear forces using the method of unitary transformation 89
To obtain the effective potential via eq. (3.230), we need to know the operators >..a AZ77 that can
be evaluated along the lines described in the last section. Let us start with >..1 A077. The leading
contributions to eq (3.206) with i = 1 appear at order r = 0, as follows from eq. (3.216). We can
at leading order. Concrete, we obtain for this equation at r = 0:
(3.240)
now make use of formulae of appendix B to determine all terms that contribute to the decoupling
equation (3.206) with i = 1
Bere and in what follows, we denote the contribution from the free Bamiltonian Ho by the pertinent
nucleonic and pionic free energies, E and w, respectively. For >..1 A077 we obtain:
(3.241)
Proceeding analogously and performing the recursion in the direction indicated in fig. 3.5, we find:
>..4 Ao77
>..1 Al77
>.. 2 Al77
>..4Al77
>..1 A277
o ,
0,
---H277
>..2
-
>..2
w1 + w2 w1 + W2
H1>..1 A077 ,
---- >..4 -- -H277 ,
w1 +W2 +W3 +W4 >..2
---H377 ,
w1 +W2
>..1 >..1 >..1
--H1>.. 2 w
A077 - -H2>..1
w
A077 + -A077H1>..1
w
Ao77
+ -A077H277
>..1 >..1 >..1
- 6' w -A077 w + -Ao77E w ,
>..1 >..1 >..1
--H177 - -H1>.. 2 w w
Al77 - -H3>..1
w
Ao77 .
Bere we have used the fact that the following operators vanish:
(3.242)
(3.243)
(3.244)
(3.245)
(3.246)
(3.247)
(3.248)
(3.249)
Note that only those operators >..a AZ77 are shown explicitly in eqs. (3.241)-(3.248) that we will
need in furt her calculations. In particular, we do not show >.. 3 Ao77 and >..2 A277 that do not vanish.
The effective potential defined in eq. (3.230) can be found using eq. (3.197) and the dimensional
analysis rules (3.223)-(3.229). The minimal possible value of 1/ turns out to be 1/ = 6 - 3N.
Correspondingly, the leading order N-body potential can be written in the form:
(6-3N) ( At 1 1A At d A )
Veff = 77 H2 + 0>" H1 + H1>" 0 + 0/\ W 0 77 · (3.250)
Note that in the last term only the pionic free energy w contributes at lowest order. No contribution
to the potential appears at order 7 - 3N because of parity invariance. This is also clear from the

90 3. The derivation of nuc1ear forces from chiral Lagrangians
fact that there is no non-vanishing operator Al H2'f}. At next-to-leading order, 8-3N, one obtains
a more complicated expression for the potential:
Ve�-3N) = 'f}(H4+AbA4H2 + H2A4Ao +A�AlHl +HlAlA2
t l l 1 tl 1 + AOA H2A Ao - "2 AOA AO'f}H2 - "2 H2'f}AoA tl Ao
+ AbA\€Ao - 1AbAl AoE - 1EAbAlAo + A�AlwAo + AbAlwA2
+ AbAl HlA2 Ao + AbA2 HlAl Ao (3.251)
Itl tl Itl 1
1 - "2 AOA AO'f}AoA Hl - "2 AOA AO'f}HlA Ao - "2 AOA t l Hl'f}AoA t l Ao
1 - 1 t l 1 t l t l I
"2 HlA AO'f}AoA Ao - "2 AOA AO'f}AoA wAo - "2 AOA t l wAO'f}AoA tl Ao
+ AbA2 H2 + H2A2 Ao + AbA2(Wl + w2)Ao)'f} .
Here the E's denote the nucleonic free energies related to the accompanying projection operators
(A or 'f}). At next-to-next-to-leading order (NNLO), v = 9 - 3N, one finds:
��-3N) = 'f} ( H5 + AbA2 H3 + H3A2 Ao + At A2 H2 + H2A2 Al
+ A�Al Hl + HlAl A3 + AbAl HlA2 Al + At A2 HlAl Ao
(3.252)
+ AbAlwA3 + A�AlwAo + AbAl H3Al Ao + AbAlH4 + H4Al AO)
Solving eqs. (3.241)-(3.248) recursively one finds the express ions for the operators Aa AI'f} in terms
of the HK,'s. Inserting these into eqs. (3.250), (3.251) and (3.252) and performing straightforward
algebraic manipulations, we obtain the potential as
V(6-3N) eff
V (8-3N)
(3.253)
eff
V (9-3N)
eff
(3.254)
(3.255)

3.8. Two-nucleon potential 91
3.8 Two-nucleon potential
3.8.1 Expressions and discussion
In the last section we have applied the method of unitary transformation to the most general chiral
invariant Hamiltonian for nucleons and pions and obtained the formal expressions (3.253)-(3.255)
for the effective potential. We will now explicitly evaluate all these contributions inserting the
vertices from eqs. (3.231)-(3.238) and reshufRing the operators into normal order. For that, we
will switch to the interaction picture. The free Hamiltonian is chosen according to eqs.
(3.200).
(3.199),
The pion and nucleon field operators in the interaction picture satisfy the free-field
equations of motion:
(0 + m;) 7r 0,
(WO + :�) N o.
Correspondingly, one can decompose the pion field operators 7r via
7r +(t, x)
7r-(t, x)
7r0 (t, x)
! d3k 1 1 [e-ik'Xa (k) + eikoxat (k)]
(27r)3/2 v'2W
! d3k 1 _1_ [e-ikoXa_(k)
(27r )3/2 v'2W
+ +
+ eikoXa�(k)]
! d3k (27 r�3/2 vk- [e-ikoXao(k) + eikoXab(k)] ,
where w = ko =
and neutral pions. The cartesian components of the pion field are given by
(3.256)
(3.257)
(3.258)
Vk2 + m; and ato (a±,o) are the creation (destruction) operators of charged
7r2
= --=
/n-
The equal-time commutation relations of the pion fields and their conjugate require
[a(k), a(k')] = [at(k), at(k')] = 0 ,
v2i
[a(k), at(k')] = o3(k - k') .
For the nucleon field N one has the decomposition:
(3.259)
(3.260)
(3.261)
(3.262)
Here, v is a Pauli spinor, Eis an isospinor and Po = p2j(2m) = E. Further, bt(p, s) (bt(p, s))
is the destruction (creation) operator of a nucleon with the spin and isospin quantum numbers s
and t and moment um p. v and E are normalized via
vt(s)v(s)
Et(t)E(t)
1 ,
1.
(3.263)
The creation and destruction operators b1 (p, s) and bt(p, s) satisfy the anti-commutation relations:
(3.264)

92 3. The derivation of nuc1ear forces from chiral Lagrangians
2
,
, I
I
,
3
,
I
,
',.
I
,
4
I
, , ... , I , ,
I
I , ,
I , ,
I , ,
I I I
, I I
I
, ,
, , I , ,
.... '.' ..
Figure 3.6: Some of the one- and zero-partide processes that will not be considered.
Solid and dashed lines are nudeons and pions, respectively. The heavy dots denote
vertices from eqs. (3.231)-(3.238) with ßi = O.
5
I
(3.265)
Using the decompositions (3.258), (3.262) of the field operators one can express the interaction
Hamiltonian that corresponds to the density (3.231)-(3.238) in terms of creation and destruction
operators. The calculation of the effective potential (3.253)-(3.255) is performed at a fixed time
t = 0, at which the Heisenberg, Dirac (or interaction) and Schrödinger pictures coincide. To
obtain the final result one has to proceed in a way similar to time-ordered perturbation theory,
see, for example, [187]. The only difference is in the energy denominators and in the coefficients of
the operators entering eqs. (3.253)-(3.255). A detailed calculation of the effective potential within
a time-ordered perturbation theory as wen as the relevant matrix elements of the Hamiltonian
(3.231)-(3.238) can be found in the reference [161].
Figure 3.7: Leading order (LO) contributions to the NN potential: one-pion exchange
and contact diagrams. Graphs which result from the interchange of the two
nudeon lines are not shown. For notations see fig. 3.6.
Let us take a dos er look at various terms entering the expressions (3.253)-(3.255), before we
will give the explicit result for the effective potential. For that, we will use the diagrammatic
technique. It should be kept in mi nd that the graphs we will show below only represent the
2

3.8. Two-nucleon potential 93
topological aspects of the processes. Different to time-ordered perturbation theory, one cannot
directly read off the corresponding matrix element from the diagram. To do that, one has to insert
the appropriate energy denominators from eqs. (3.253)-(3.255).
Figure 3.8: First corrections to the NN potential. Contact diagram at next-toleading
order (NLO). The filled diamond denotes the vertices with L::.i = 2 (with two
derivatives). For remaining notations see fig. 3.6.
We first point out that we will not furt her discuss the zero- arid one-particle diagrams, that
describe vacuum fiuctuations and self-energy contributions. Few examples of such diagrams are
shown in fig. 3.6. Consider now the leading order potential, eq. (3.253). It consists of two
terms, the one-pion exchange rv H1 ().. 1 /w)H1 and the contact interactions with four nucleon legs
subsumed in H2 . The corresponding graphs are shown in fig. 3.7. This potential obviously agrees
with the one obtained in time-dependent perturbation theory.40
More interesting is the first correction given in eq. (3.254) representing the next-to-leading order
(NLO) result. The diagrams corresponding to the various terms are shown in figs. 3.8-3.11 and
3.13. The first term refers to the graph of fig. 3.8, the remaining terms in the first line lead to
diagrams 1, 2, 3 in fig. 3.9 and 1, 2 in fig. 3.13 and in the second line to graph 4 of fig. 3.9.
We should mention that all graphs containing vertex corrections with exactly one 7r7r N N -vertex,
which are also contained in the first line, give no contributions, because only odd functions of
the loop moment um enter the corresponding integrals. The first term in the third line subsumes
graphs 5 to 8 offig. 3.9 plus the irreducible self-energy diagrams depicted in fig. 3.10. The next two
terms in the third line refer to graphs 9 and 10 in fig. 3.9 plus the "reducible" self-energy diagrams
offig. 3.11. Such "reducible" diagrams are typical for the method ofunitary transformation and do
not occur in old-fashioned time-ordered perturbation theory. They should not be confused with
truly reducible diagrams, one example being shown in fig. 3.12 (1). In that figure, the horizontal
dot-dashed lines represent the states whose free energy enters the pertinent energy denominators.
In time-ordered perturbation theory, such reducible diagrams are generated by iterations of the
potential in a Lippmann-Schwinger equation, with the potential being defined to consist only of
truly irreducible diagrams. In contrast to the really reducible graphs like the one in fig. 3.12 (1),
the ones resulting by applying the projection formalism do not contain the energy denominators
corresponding to the propagation of nucleons only. In the same notation as used for fig. 3.12
4° Speaking more precisely, only the non-iterated potential is the same in both time-ordered perturbation theory
and projection formalism. This is because the energy denominator entering the corresponding expression for the
potential in old-fashioned perturbation theory depends explicitly on an initial energy of the two nucleons. Such
an energy dependence vanishes if one considers nucleons as infinitely heavy particles (static limit). Then the two
potentials are indeed the same.

94
,
,
,
,
,
,
,
,
,
,
,
,
,
, ,
, ,
,
,
,
,
"
,
,
, - -
" "
- , , ' ,
- , ,
, ,
, ,
,
2
6 7
,
,
3. The derivation of nuclear forces from chiral Lagrangians
\� ,
,
\ ,
,
\ ,
\ ,
3
\
\
,
,
,
,
,
,
,
,
,
,
,
,
,
,
H
,
,
,
,
I , ,
-
4
9
,
,
,
,
,
'<
,
"
,
,
,
Figure 3.9: First corrections to the NN potential in the projection formalism: twopion
exchange diagrams. For notations see fig. 3.6.
(1), one can e.g. express diagram 9 of fig. 3.9 as a sum of two graphs as depicted in fig. 3.12
(2),(3). All diagrams involving contact interactions, shown in fig. 3.13, follow from the fourth line
and, as already noted above, from the second term in the first line of eq. (3.254). We remark
that the three terms in the fifth line add up to zero. We have nevertheless made them explicit
here since in time-ordered perturbation theory, these terms are treated differently and lead to the
recoil correction, i.e. the explicit energy-dependence, of the two-nucleon potential. The last term
in eq. (3.254) corresponds to the vacuum fluctuation diagram shown in fig. 3.6 and will not be
discussed below.
Let us now consider the next-to-next-to-leading order corrections (NNLO) represented by the
equation (3.255). The corresponding diagrams have the same structure as the graphs 1-4 in
fig. (3.9) except that now one 7r7rNN vertex contains two derivatives (or two pion mass insertions),
i. e. one more than the diagrams in fig. (3.9). The two last terms in the first line of eq. (3.255)
are referred to the graphs 4 and 5 in fig. 3.15. The diagrams 1-3 in this figure represent the
terms in the second line of eq. (3.255). The corresponding vertex correction diagrams are shown
in fig. 3.14.
We will now give explicit expressions for the N N potential. As already noted ab ove , its leading
part is given by just one pion exchange with both vertices coming from eq. (3.231) plus contact
interactions corresponding to eq. (3.234). As pointed out in ref. [78] in the context oftime-ordered
perturbation theory, one has to take the static limit for the nucleons in the one-pion exchange
term, since the recoil and relativistic corrections lead to higher order contributions. The same
holds true in the method of unitary transformation, see eq. (3.253). Thus, both methods give
the same result for the leading order potential. In what follows, we will use the slightly different
notation from refs. [78], [127]. In particular, the spin and isospin matrices a and T satisfy the
5
10

3.8. Two-nucleon potential
,, '
\
1
,
, ,
, ,
,
,
,
,
,
,
\
, I
"
,
,
2
5 6
\
1
1 ,
\ 1
1 1
1
\
\
\
1
Figure 3.10: First corrections to the NN potential: irreducible self-energy and vertex
correction graphs. For notations see fig. 3.6.
,
,
,
, 1
I
, \
'I
1
\
I, '
,
,
, , , \
, ,
1
3
7
, ,
, , ,
\ \ , , ,
1 1 , ,
2 3 4
Figure 3.11: First corrections to the NN potential: reducible self-energy graphs. For
notations see fig. 3.6.
following equations:
Oij + iEijkTk
Oij + iEijkO"k
4
\
\
95
(3.266)
(3.267)
The initial (final) moment um of the nucleons in the c. m. system is denoted by p (p'). The
transferred and average momenta are given by
� -I -+
q = p -p, 2
(3.268)
�
p+ p'
k= --

96
_ . . -.:;: " :,,,_ ._ . . _.
3. The derivation of nuc1ear forces from chiral Lagrangians
_. . _ .:: . .,. :",_. _. . _ .
_ . . _._. __ ._ . . _.
_ . . _._."._._ . . _.
2
- . . _._. __ ._ . . -.
_ . . _._.04._._ . . _.
_. __ . :: ,., 1""_ ._ . . _.
Figure 3.12: Reducible graphs. In (1), a truly reducible diagram is shown. In the
projection formalism, one has graphs like (2) and (3). These correspond to diagram
9 in fig. 3.9. The horizontal dot-dashed lines count the free energies of the particles
cut.
respectively. The leading order potential can then be written as:
2 3 4 5 6
Figure 3.13: NLÜ one-loop corrections to the four-nucleon contact interactions in
the projection formalism. For notations see fig. 3.6.
3
(3.269)
The first corrections to this result appear at order two. We now enumerate all such corrections.
Since we want to compare both schemes, we will first discuss the results obtained within oldfashioned
perturbation theory and then present the potential derived using the method of unitary
transformation. We start with the tree-diagram contribution given by the graph of fig. 3.8 with
various contact interactions from eq. (3.237). This leads to the potential
(2) VNN,tree C �2 C k2 (C �2 � �
C k2 )(
1 q + 2 + 3 q + 4 0"1 • 0"2 + Z 5 . x q)
2
+ C6 (if· o\) (if· i'h) + C7 (k . ih)(k . (2 ) , (3.270)
) ' C 0\ + ih (k ;;'I

3.8. Two-nucleon potential
where the Ci 's are given by
Cl ,
-4C 2 ,
- 1 -
-C7 - 2 C5 '
4C7 - 2C5 , (3 .271)
C5 -2C3 ,
C6 -C 4 + C6 ,
C7 -4C6 .
Different to the reference [78], no one-pion exchange diagrams appear at this order, since the
I
\
Figure 3.14: Next-to-next-to-leading order (NNLO) diagrams that contain vertex
corrections. The filled squares denote the 7r7r N N vertex with ßi = 1. One should
sum over all possible time orderings. Graphs resulting from time revers al operation
are not shown. For remaining notations see fig. 3.6.
7r N N vertices with three derivatives can be eliminated from the Hamiltonian or contribute at
higher orders, as pointed out in the last section.
, ,
,
,
,
,
,
,
,
1 ,
1 ,
1 ,
1 '
1 '
'
I,
"
2
"
"
" ,
, ,
, ,
, ,
, '
,
3
2
I
I
I ,
1
"
,
1 1
1
"
4
,
, I
I
I
,
,
\
\
'
"
,
,
\
,
, \
,
5
Figure 3.15: Next-to-next-to-leading order (NNLO) corrections to the NN potential.
For remaining notations see figs. 3.6, 3.14.
97

98 3. The derivation o{ nuc1ear {orces {rom chiral Lagrangians
Further corrections arise from irreducible one-loop diagrams 1-8 in fig. 3.9:
where
vY 211", ) 1-1oop, Irr. .
(3.272)
We have also calculated the contributions from various irreducible one-loop diagrams 1-8 in
fig. 3.10 and 1-2 in fig. 3.13, which involve self-energy insertions and vertex corrections. These
have not been considered in [74], [76]. The corresponding contributions are given by
where
V(2) 111",1-1oop,Irr. .
and by
V(2) .
NN,1-1oop,Irr.
4 3 { � �
g A / dl T1'T2 l .q( ( � l �) ( � �) ( � �) ( � l �))
(2111")4 (27r )3 wlw� � 0'1' 0'2 . q + 0'1' q 0'2 '
(� + �) [2 (0\ . q) (eh , q)} , (3.274)
Wz wq
p,ur.
(3.275)
(3.276)
We remark an unpleasant feature of V1(2)1_100 . . With t = _q2 one can rewrite the term
7r,
proportional to l/w� as (M; -t)-3/2. This function has a cut starting at t = M;. Physically,
this does not make sense since all cuts should be produced by multi-particle intermediate states.
We co me back to this later on.
Finally, the corrections arising by explicitly keeping the nucleon kinetic energy in one-pion exchange
tree diagrams with both vertices from the leading order Lagrangian represents the energydependent
part of the N N potential [76] given by
V(2) _
g
E - (21
2 E _ l (f 2 + l(2)
11" q2 + M;
A ( �
)2 Tl . T2 0'1' q 0'2 ' q
�) ( � �) m 4
( )3/2 '
(3.277)

3.8. Two-nucleon potential 99
where E is an initial energy ofthe two nucleons (full energy). One should stress that on-shell, this
contribution vanishes. This on-shell center-of-mass kinematics is often used as an approximation
for calculations of N N scattering or few-nucleon forces. However, if one iterates this potential in
the Lippmann-Schwinger equation, the energy E can no longer be simply related to the momenta
k and ij. A similar comment applies to using this recoil correction in the calculation of few-nucleon
forces (for N � 3).
Let us now look at the corrections arising in the framework of the projection formalism. The ones
from the tree diagrams with contact interactions with two derivatives do not change and are again
given by eq. (3.270). Apart from the corrections from irreducible one-Ioop graphs 1-8 in fig. 3.9
given by eq. (3.272) one obtains contributions from reducible diagrams 9 and 10, which can be
expressed by
(3.278)
V(2 )
211",1-1oop,red.
Summing up eqs. (3.272) and (3.278) we obtain the two-pion exchange contributions to the potential
within the projection formalism:
(2 )
V211", 1-1oop
(3.279)
As was first noted in ref. [108] using a different formalism, the isoscalar spin independent central
and the isovector spin dependent parts of the two-nucleon potential corresponding to the two-pion
exchange adds up to zero. It is comforting that we find the same result. Note that this is different
from the energy-dependent potential derived within time-ordered perturbation theory.
The contributions from one-Ioop "reducible" diagrams 1-4 in fig. 3.11 and 3-6 in fig. 3.13, which
involve the nucleon self-energy and loop corrections to the four-fermion interactions, are given by
(3.280)
and by
(3.281)
V(2 )
NN, 1-1oop,red.

100 3. The derivation o{ nuc1ear {orces {rom chiral Lagrangians
respectively. Note that Vl�: 1-1oop,red. again contains a term rv l/wg. Combining eqs. (3.274) and
(3.280), this unphysical contribution vanishes. Summing up the corrections eqs. (3.274), (3.276)
corresponding to irreducible graphs, which are again the same as in old-fashioned time-dependent
perturbation theory and those (3.280), (3.281), corresponding to "reducible" diagrams, one gets
the complete result, representing the one-Ioop contributions, which involve self-energy insert ions
and vertex corrections within the projection formalism:
Vb,1-1oop
(2)
VNN,1-1oop
(2)
The complete potential at NLO in the projection formalism is given by the express ions (3.270),
(3.279), (3.282) and (3.283). The formal expressions for the NN potential shown above contain
ultraviolet divergences. In the next section we will show how such ultraviolet divergences can be
removed by redefinition of the coupling constants.
Let us briefly discuss the NNLO potential given in eq. (3.255). The related diagrams are shown
in fig. 3.15. In that case no purely nucleonic intermediate states are possible. Correspondingly,
one obtains the same contributions in time-ordered perturbation theory and in the projection
formalism. This is also evident from the explicit form of the operators in eq. (3.255). The formal
express ions of the NNLO potential were presented in [78]. In the next section we will show the
renormalized contributions given by Kaiser et al. [108].
Finally, let us point out once more that the crucial difference between the two formalisms, timeordered
perturbation theory and method of unitary transformation, results in the treatment of the
energy-dependent term eq. (3.277). As already noted ab ove , it does not appear in the method of
unitary transformation. We note that many of the results derived here have already been found in
[108], where one- and two-pion exchange graphs were calculated by me ans of Feynman diagrams
and using dimensional regularization. The potential was applied to N N scattering in peripheral
partial waves (with orbital angular momentum L 2: 2), where it does not need to be iterated.
Our aim is to apply this potential to all partial waves and to the bound state problem. We will
comment more on that in the next chapter.
3.8.2 Renormalization of the N N potential at NLO
The effective N N potential at NLO derived in the last section is not well-defined, since it contains
ultraviolet divergent integrals. Here, we would like to show how to renormalize this potential.
Since we are dealing with an effective field theory, we can use the standard order-by-order
renormalization machinery for the Lagrangian, which was first developed in the context of chiral
perturbation theory. All these divergent contributions (at NLO) can be renormalized in terms of
three divergent (momentum space) loop functions,41
/00 dl
Jo = (3.284)
J1 T'
41 Clearly, all ultraviolet divergent integrals here and in what follows have to be understood as the limits A -+ 00
of the corresponding integrals over the finite range of momenta 0 < I < A.

3.8. Two-nuc1eon potential 101
Clearly, J2 (J4) is quadratically (quartically) divergent while Jo diverges logarithmically for large
momenta and p, is some quantity with the dimension of mass (renormalization scale). Note that
we could introduce other forms of divergent integrals, which depend on a single dimensionsfull
scale (denoted here by p,). Another choice of defining these divergent loop integrals would give
different finite subtractions but leads to the same non-polynomial terms in the potential. We will
comment more on that later on.
Consider first the one-Ioop corrections to the OPEP, eq. (3.282). The relevant divergent integral
lS /
d3Z ZiZj
(2n)3 w( = a Oij , (3.285)
where i, j = 1,2,3 and a is some (infinite) constant. No other two-dimensional symmetrical (in i
and j) tensors apart from Oij can enter the right-hand side of this equation, since the integrand
depends only on Z and the integration is performed over the whole space. Multiplying both sides
of eq. (3.285) by Oij and performing the summation over i,j we obtain
1/ d3Z Z2
a = - ---
3 (2n)3 w(
. (3.286)
Using eqs. (3.285), (3.286) and (D.1) we can now rewrite the OPEP (3.282) as:
4
. 2
V1�\-lOOP = 48�� 1: (Tl ' T2)(0'1 . if)(0'2 . if) �� {5M; + 3M; In :� + 4h -6M;Jo} , (3.287)
in terms ofthe two divergent loop functions JO,2 defined in eq. (3.284). Therefore, this contribution
has exactly the same form as the OPEP (renormalized OPE),
(3.288)
provided we redefine the coupling constant g� (the superscript "0" denotes the leading term in
the chiral expansion) in the following way:42
(gT)2 = (gO)2 _
{5M2 + 3M 2 In M; + 4J _
(g�)4
A A 12n2 fir 11: 11: 4p,2
2 11:
6M2 J }
0 (3.289)
Clearly, gA and g� differ by terms of second order in the chiral dimension. Consequently, all NLO
one-Ioop corrections to the OPEP can be taken care off by renormalization of g�.
In addition, there are the one-Ioop corrections to the lowest order four-fermion interactions shown
in eq. (3.283). Proceeding analogously to eqs. (3.285), (3.286) and making use of eq. (D.1) one
can express the divergent integral entering eq. (3.283) in terms of the loop functions JO,2 as
(2)
V NN,1-1oop -6::1 ; C� (3 - Tl . T2) (0\ . 0'2) / d3ZZ4 wi3 (3.290)
- g� j2 C� (3 - Tl ' T2) (0'1 . 0'2) {5M; + 3M; In M
4 � + 24n 4h -6M;Jo}
11: p,
or in a more compact notation
(3.291)
42 Note that this expression may change if one performs a complete renormalization (including the wave function
renormalization) .

102 3. The derivation o{ nuclear {orces {rom chiral Lagrangians
with
(3.292)
Performing anti-symmetrization as described in appendix E allows to map the two spin-isospin operators
appearing in eq. (3.291) onto the two non-derivative operators used in v�o�, cf. eq. (3.269).
Therefore, as follows from eq. (E.8), the effect of the one-Ioop corrections to the lowest order contact
interactions with four nucleon legs can be completely absorbed by renormalization of the
constants C� and C� ,
Cs = C� -3S, Cf = C� -3S . (3.293)
We note that furt her renormalization of these couplings is due to the two-pion exchange, as
discussed below.
Consider now the proper two-pion exchange contribution (3.279). The first two integrals can be
immediately expressed in terms of the divergent loop functions JO,2,4 using h , h and h defined
in eqs. (D.2), (D.3) and (D.6). To calculate the last integral we use the following identity:
1 ä 1 (3.294)
Now it is easy to express the integrals entering the last two lines of eq. (3.279) in terms of the
loop functions JO,2,4. Specifically, we have:
(3.295)
Here, we have set q == Iql. Putting pie ces together, the expression for V2�:1-100P takes the form
V2�:1-100P
with
= vJl�p + (Sl + S2 q2) (7"1 . 7"2) + S3 [(0\ . q) (ih . if) - (51 . 52) q2] ,
1 { 2 4 2 M7r 2 4
2f
2
4 3847r 7r -18M7r(5gA -2gA) In - -M7r(61gA - 14gA + 4)
ft
+18M;(5g� -2g�)Jo -3(3g� -2g�)J2 } 1 { 4
,
2 M7r 1 4 2
+(23g� -lOg� - l)Jo },
3g� { M7r 1
647r2 -Jo }
f; ft 3 '
3847r2 f; (-23gA + 10gA + 1) In -;- - 2(13gA + 2gA)
- ln- + -
and the non-polynomial part vJlJP given by
(3.296)
(3.297)
(3.298)
(3.299)

3.8. Two-nuc1eon potential
with
L(q) = � 14M2 + q2 In J4M; + q2 + q .
q V 7r 2M7r
103
(3.301)
The polynomial terms, obviously, renormalize the coupling constants of the dimension zero and
two contact terms with four nucleon legs. Again, we perform anti-symmetrization to map the
terms appearing in eq. (3.296) via eq. (E.13) onto the basis used in the polynomial part of the
V(O ) and in the V�2h,tree' cf. eqs. (3.269), (3.270). Including the contribution from eq. (3.293),
the complete renormalization of the four-nucleon couplings takes the form
Cf
cr
cr
C� - 38 -81 , Cs = C� -38 -281 ,
Cp - 82 , C� = cg - 482 , Cf = cg -83 ,
482 , C� = cg , C6 = cg + 83 , C; = C$ .
c2 -
(3.302)
Note that C5 and C7 do not get renormalized to this order.
Now we would like to comment on the obtained results. The effective N N potential at NLO is
given by the non-polynomial and the polynomial parts shown in eqs. (3.269), (3.300) and (3.270).
The coupling constants Ci 's are renormalized via eq. (3.302). Note that each of these two parts, the
polynomial and the non-polynomial ones, separately does not depend on the renormalization scale
fl,. This fl,-independence is not accidental. Indeed, assume that the potential can be expressed as
(3.303)
where CT and gT correspond to sets of renormalized four-nucleon and remaining couplings, which
in our case do not depend on fl" Q denotes generic nucleon three-momenta and VI and V2 are
non-polynomial and polynomial parts of the potential. For simplicity, we consider here the case
m -+ 00. Since V2 is a polynomial, there exists some non-negative number nmin such that for all
n > nmin: 43
dnV2( Q, M7r, fl" gT, CT)
=
dQn 0 . (3.304)
We now require that the whole effective potential V does not depend on fl,:
dV
dfl, = 0
Therefore, we obtain for all n > nmin:
(3.306)
Here we assume the usual continuity properties of the function V2( Q, M7r, fl" gT, CT). Consequently,
one can express VI as
(3.307)
Further, the function V? (M7r , fl" gT, CT) should vanish since VI does not contain a polynomial part.
That is why both VI and V2 should be separately independent on fl,. In our case it turns out that
the explicit fl,-dependence even vanishes completely if both VI and V2 are expressed in terms of the
renormalized coupling constants. All these couplings should be fixed from fitting to data. This will
.
(3.305)
43We use here a symbolic notation for derivatives. It should be kept in mind that Q represents vectorial quantities.

104 3. The derivation of nuclear forces from chiral Lagrangians
be described in detail below. For gA and i1f we will take the values known from the one-nucleon
sector. Note, however, that we have not performed here a complete renormalization ofthe potential
at NLO, which would also require a calculation of the contributions from disconnected diagrams
(zero- and one-nucleon operators). Therefore, the values of gA and i1f may, strictly speaking,
differ from the corresponding ones obtained from the 1fN scattering by order Q2 corrections and
should, in principle, be extracted from the N N scattering data. It would be interesting to perform
a complete renormalization of the effective potential to obtain more rigorously the values of these
constants. As we have found, the N N scattering data can be reproduced quite accurately using
the values i1f = 93 MeV and gA = 1.26.
Let us now make another comment concerning the choice of the loop functions JO,2,4, eq. (3.284).
Clearly, any other choice of defining this divergent loop integrals would give different finite subtractions
but leads to the same non-polynomial terms (3.300) in the potential. For example, to
express the potential in terms of
Ja = J2/L
T ' - 100 2/L 3
- r�o dl
J4 = 1 dl ,
(3.308)
we have to replace Ja, J2 and J4 by Ja + In 2, J2 + 2f-L2 and J4 + 4f-L4, respectively, in eqs.
(3.292)
(3.289),
and (3.297)-(3.299). This only modifies the definitions of the renormalized couplings
gA and Gi'S but does not change the expressions (3.300) and (3.270) for the effective potential.
Furthermore, to reproduce the results obtained using dimensional regularization we have to skip
the power-Iaw divergences, i.e. to set J2 = 0, J4 = O. Also in that case we get the same expressions
for the effective potential.
The effective potential at NNLO corresponding to eq. (3.255) can be renormalized analogously.
As can be seen from eq. (3.255), the contribution to the effective potential in the projection
formalism is the same as the one obtained with time-ordered perturbation theory (in the static
limit for nucleons). Furthermore, the related graphs shown in fig. 3.15 involve all possible time
orderings. As a consequence, the contribution to the effective potential can be obtained using
covariant perturbation theory and the technique of Feynman diagrams. This has been done by
Kaiser et al. using dimensional regularization [108]. Here, we will not perform explicit calculations
and simply adopt their result. The TPEP at NNLO reads:
where
(3) _
V21f,1-loop - VNNLO + P(k, q) , (3.309)
TPEP
-
VJJtJ -l��;i { -16m(���+ q2) + (2M;(2Cl -C3) -q2 (C3 + ::�)) (2M; + q2)A(q) }
128��ii (Tl ' T2) { -4�1��2 + (4M; + 2q2 -g�(4M; + 3q2))(2M; + q2)A(q)}
� �
+ 51;:�ii ((0\ . ij)(ih . ij) -q2(ih .52)) (2M; + q2)A(q)
3J:ii (Tl ' T2) ((51 ' ij)(52' ij) -l(5l .52))
X {(C4 + _1 4m _)(4M; + q2) - 8m
g� (10M; + 3q2)} A(q)
3g� . ( al � � ) ( � I �) (2M2 2)A()
641fmii z
+ a2 . P x P 1f + q q

3.8. Two-nucleon potential
Here,
g� (1 - g�) ( ) .
641fmf� Tl ' T2 Z (Tl + (T2 . P x P 1f + q q .
( � � ) ( � I �) (4M 2 2)A( )
A(q) =
1 q
2q 2M1f
- arctan -- .
Furt her , P(k, if) in eq. (3.309) is a polynomial in momenta of at most second degree and has the
same structure as the expressions in eq. (3.270).44 Thus, its explicit form is of no relevance here,
since it only contributes to the renormalization of the couplings Cs , CT , Ci . Again, no explicit
dependence on the renormalization scale appears in the expression (3.310) für the non-polynomial
terms. In ref. [108] it is also pointed out that all NNLO one-pion exchange diagrams containing
vertex corrections do not yield any pion-nucleon form factor and only lead to renormalization of
the 1fN coupling gA (i.e. they have the same form as the proper one-pion exchange). Clearly, the
same holds true for all NNLO graphs representing vertex corrections to short range interactions.
For the LECs Cl,3,4 entering eq. (3.310) we should take the values obtained from fitting 1fN phases
in the threshold region, see e.g. ref. [71] or, alternatively, from fitting the invariant amplitudes
inside the Mandelstarn triangle, i.e. in the unphysical region [195]. The so determined parameters
are only slightly different, but these small differences will play a role later on. For example, the
LECs CI,3,4 from fit 1 of ref. [71] are Cl = -1.23 GeV-l, C3 = -5.94 GeV-l and C4 = 3.47 GeV-l.
A recent investigation of the subthreshold amplitudes [195] leads to slightly different values, Cl =
-0.81 GeV-l, C3 = -4.70 GeV-l and C4 = 3.40 GeV-l. It is this latter set we will use in the
following. These values are also consistent with the re cent determination from the proton-proton
interaction based on the chiral two-pion exchange potential [198].
Kaiser et al. also included l/m corrections into the expression (3.310) for the two-pion exchange
potential an NNLO. This is consistent with the power counting scheme in the one-nucleon sector.
In our power counting scheme with Q/m rv Q2 / A�, where Q corresponds to the low moment um
scale, such l/m terms appear first one order higher. Nevertheless, we have decided to keep these
corrections, since otherwise one cannot directly use the values of the LECs Cl,3,4 as determined
from the 1fN sector in the presence of the l/m terms. By doing so we only include some sm aller
terms of higher order in the potential. It can easily be checked, that those terms are indeed small
as compared to the Cl,3,ccontributions. For example, the constant C4 enters eq. (3.310) only in
the combination C4 + 1/ ( 4m). Correspondingly, the pertinent C4 contribution is numerically more
than 10 times larger than the l/m correction.
To end this section we would like to point out once more that eqs. (3.269), (3.300), (3.270) and
(3.310) define the unique expressions for the effective potential up to NNLO, i.e. the results do
not depend on the regularization scheme, as discussed above.
105
(3.310)
(3.311)
3.8.3 Phenomenological interpretation of some of the 1f N LEes and the role
of the ß(1232)
It is well known that the values of some of the LECs can be understood phenomenologically in
terms of the resonance saturation hypotheses. This is discussed in detail in the reference [199]
for the purely meson sector and in ref. [200] for pion-nucleon interactions. The idea of such
a phenomenological interpretation is very simple. Consider the most general chiral invariant
Lagrangian for pions, nucleons and their excitations. The resonances can be integrated out from
44 More precisely, after performing the partial wave decomposition, fCk, if) leads in each partial wave to polynomials
in p and pi of at most second degree.

106 3. The derivation of nuclear forces from chiral Lagrangians
the Lagrangian if one regards their excitation energies to be very large (or, equivalently, if one
considers 7r N scattering at energies much smaller than the excitation energies). This generates a
I
I
+
,
/
,
/
, /
, /
, /
, /
..
Figure 3.16: Ll-resonance saturation of the 7rN LECs C3 and C4. Double solid line
represents Ll-isobar. For remaining notations see figs. 3.6, 3.14.
series of local pion-nucleon operators of increasing dimension with the coupling constants fixed by
details of the pion- (nucleon-) resonance interactions. Certainly, one expects that the resonances
with the lowest excitation energy give the dominant contributions to the values of the LECs. In
particular, the Ll(1232) with the excitation energy of only about two pion masses is expected
to be quite important for the low-energy pion-nucleon dynamics. This expectation is confirmed
by a phenomenological analyses [200]. It turns out that the value of C3 is dominated by the Ll
contributions. The Ll-isobar also leads to a sizable contribution to C4, whereas the value of Cl is
mainly saturated by the scalar-isoscalar meson contributions.45
In many approaches to the N N interactions the effects of the Ll-excitations in the intermediate
states are explicitly included. Because of the relative small value of the LlN-mass splitting (293
MeV) such effects might be quite important for the low-energy N N dynamics. Already in their
pioneering work [78], Ord6iiez et al. incorporated the leading effects due to the intermediate Llexcitations
treating the LlN-mass splitting as a small quantity on the same footing as the pion
mass. Although the LlN-mass splitting does not vanish in the chiral limit, such a phenomenological
extension of chiral perturbation theory is quite useful in many situations and may be
formulated in a systematic fashion (the so-called "small scale expansion") [201]. In this approach
one starts from the most general chiral invariant Lagrangian for relativistic pions, nucleons and
deltas. To describe the spin- and isospin-3/2 field corresponding to the Ll, one typically uses the
Rarita-Schwinger formalism [202], i.e. one introduces a vector-spinor field W jl (x) that satisfies the
equation of motion
(3.312)
with the subsidiary condition
(3.313)
Furthermore, the Rarita-Schwinger spinors for the spin 3/2 field are constructed by coupling spin-
1 vector to spin-1/2 Dirac spinor fields via Clebsch-Gordon coefficients. One then introduces a
complete set of the projection operators (p3/2)jll/ and (PiY\w with i, j = 1,2 onto the spin-3/2
and spin-1/2 components that satisfy the algebra [201]
( P 3/2) jll/ + ( Pn 1/2 ) jll/ + ( P22
1/2 ) jll/ = g jll/ , (3.314)
(3.315)
45 In the one-boson exchange models of the nuclear interaction one typically intro duces the effective (J meson with
a mass of ab out 600 MeV to parametrize the strong pionic correlations observed in this channel.

3.8. Two-nuc1eon potential 107
where i, j, k, 1 = 1 for (p3/2)J-tv, Using these projection operators one can decompose the Rarita­
Schwinger field \[! J-t into one spin 3/2 and two spin 1/2 components. To perform the nonrelativistic
reduction one can proceed in the same way as for the pion-nucleon Lagrangian. One factors out
the exponential factor exp (imv . x) from the nucleon and delta fields, where m is the nucleon mass,
and introduces the the small and large component fields via eq. (3.160). This leads to altogether
six components for the field \[! J.t' Only the large component TJ-t of the spin 3/2 projection of the
Rarita-Schwinger spinor \[! J-t corresponds to the light field with the mass ß = mf::, - m. The
large mass scale m enters the free equations of motion for the remaining five components. Thus,
all these fields can be integrated out, which leads to local l/m corrections to the Lagrangian.
Proceeding in this way one ends up with the free Lagrangian (in the rest frame)
where i, j = 1,2,3
( 'l1 ,, )s:ij TV
Lf::, O - - i Zuo - L.l. u gJ-tv j , (3.316)
r _ TJ-tt
are the isospin indices.46 The projection formalism introduced in sections
3.5, 3.6 can be generalized to incorporate the .6.'s. The subspace Ai of the full Fock space is
now enlarged to include the states with any number of ß's apart from the nucleons and i pions.
Different to the pure pion-nucleon interactions, one also has to include the projection A 0, that
contains any positive number of .6.'s but no pions. This requires an appropriate generalization
of the projection formalism. We first note that the structure of the interactions with the .6. 's in
\
I
\
\
- - -
T
5
\
I
2
\
I
\
I
3 4
\
\ \
- - -
- - -
T T
6 7
Figure 3.17: First corrections to the NN potential with .6.-excitations: vertex corrections
to the one-pion exchange. A class of diagrams is shown for which no purely
nucleonic intermediate states are possible. For notations see figs. 3.6, 3.16.
the effective Lagrangian is the same as for the corresponding nucleonic vertices. The spin-isospin
structure is, of course, different but this does not affect the power counting arguments, for which
only the number of derivatives is of relevance. Therefore, all results and conclusions of appendices
46 The isospin 3/2 field T is described by three isospin doublets T l , T 2 , T 3 that satisfy the condition r i T i (x) = O.

108 3. The derivation of nuclear forces from chiral Lagrangians
A and B are valid also in this case. Furthermore, one has to include states contained in A ° with
.6.'s and without pions. This is discussed in appendix C. Let us now take a closer look at these
modifications.
\ /
\
\
\
,
,
,
\ I I /
, /
\
I
\
\
/
/
/
/
/
/
,
,
\ ,
I
2 3 4
\ \
, " I I
, "
, " "
\ \
I I
5 6 7 !\
Figure 3.18: First corrections to the NN potential with .6.-excitations: irreducible
(1-4) and reducible (5-8) vertex and self-energy corrections to the one-pion exchange.
For notations see figs. 3.6, 3.16.
For the projected operator Aa A1] we make the same ansatz as in the case without .6., namely that
it consists of an equal number of vertices and energy denominators. Since the inclusion of the .6.
does not affect the power counting scheme, equations (3.210) and (3.212) are not modified. Here
and in what follows, we will denote by N the baryon number. Consequently, the minimal value
of VA is again given by eq. (3.213). Furthermore, min(vA) = 2 for the operator AOA1], as follows
from eq. (3.210). Therefore, we will again define the order l of the operator Aa A1] via eq. (3.214).
As discussed in appendix C, the inclusion of the ß's does not change the minimal value of the
chiral power v for the projected decoupling equation (3.206). Therefore, we can apply eqs.
(3.216)
(3.215),
without any changes. Thus, the system of equations (3.205) can be solved perturbatively
precisely in the same way as in the case without ß. The only modification is that at each order
r one has to start by solving the equation (3.217) projected with AO • It is shown in appendix C
that all operators Aa A/1] of the right-hand side of this equation are of higher orders l 2: r + 2.
The resulting operator A ° Ar 1] is then used to solve the remaining equations at order r projected
onto the states with pions.
Since equation (3.217) with 4k + i = 0 vanishes at orders r = 0 and r = 1, the starting equations
are again (3.221), (3.222). Therefore, the ansatz about the structure of the operator A can be
justified in the same way as in sec. 3.6. Finally, all expressions (3.225)-(3.229) can be applied
without any changes to obtain the effective potential.
We are now in the position to calculate the effective potential including .6.-excitations. We will
" "

3.8. Two-nuc1eon potential
2 3 4 5
6 7 8 9 10
Figure 3.19: First corrections to the NN potential with ß-excitations: irreducible
vertex corrections to the four-nucleon contact interactions. For notations see
figs. 3.6, 3.16.
restrict ourselves to the leading and next-to-leading orders. Then we need the following operators:
).1 A01], ).2 A01], ).4A01], ). 0 A21] and ).1 A21]. Note that the operators ). 0 AO,l1] and ).1 A11] vanish, as
discussed above. Obviously, we obtain the same expressions (3.241), (3.242) and (3.243) for the
zeroth order operators, where the energy denominators are modified appropriately:
(3.317)
(3.318)
(3.319)
Here, E is a sum of the pion energies and N ß mass splittings. For instance, E = W1 + W2 + 2ß
for a state with two pions and two deltas. For the remaining operators ). aAlT] at order l = 2 we
have
109
(3.320)
(3.321)
The effective potential at LO and NLO can now be obtained by straightforward calculations. We

110 3. The derivation of nuclear forces from chiral Lagrangians
find the following expressions:
V (6 -3N )
elf
V(8-3N)
elf
(3.323)
We will now discuss these expressions in more detail. Obviously, the leading order contribution
Figure 3.20: First corrections to the NN potential with ß-excitations: reducible
self-energy corrections to the four-nucleon contact interactions. For notations see
figs. 3.6, 3.16.
(3.322) to the effective N N potential is not changed by the explicit inclusion of the ß.47 At
NLO one has many additional contributions to the two-pion exchange diagrams and to the vertex
corrections and self-energy graphs. The first four lines of eq. (3.323) give formally the complete
NLO potential in the case without ß. In addition to the one-pion exchange diagrams containing
self-energy insertions and vertex corrections discussed in sec. 3.8.1 and shown in figs. 3.10, 3.11
one has to take into account the graphs in figs. 3.17, 3.18. In the diagrams shown in fig. 3.17,
which are related to the first term in the second line and to the last term in eq. (3.323), no purely
nucleonic intermediate states appear if one sums up over all possible time orderings. Therefore, the
projection formalism yields the same result as time-ordered perturbation theory, which agrees after
summation over all time orderings with the one obtained using the Feynmann diagram technique.
Again one should keep in mind that this statement holds true because of the static approximation
for the nucleons. Note that we have depicted in fig. 3.17 the corresponding Feynmann diagrams
without showing all time orderings. The one-pion exchange diagrams of fig. 3.18 have a different
topology. The irreducible graphs 1-4 correspond again to the first term in the second line of
47 Note that there are new contributions to the one-nucleon operators due to the diagram with an intermediate
delta-excitation, which will not be considered in what folIows.

3.8. Two�nucleon potential
2 3
Figure 3.21: First corrections to the NN potential with Ll�excitations: One�loop
diagrams without intermediate pions. For notations see figs. 3.6, 3.16.
eq. (3.323) and represent the correction obtained with time�ordered perturbation theory. In
the projection formalism one has to take into account also "reducible" diagrams 5�8 in fig. 3.18
resulting from the last two terms in the second line of eq. (3.323). Note that all diagrams containing
vertex corrections with one 1r1r N Ll vertex give no contributions for the same reason as in the case
without Ll.
There are many new vertex and self�energy corrections with the intermediate Ll 's that contain
contact interactions. In fig. 3.19 we show the dass of irreducible diagrams that correspond to the
second term in the first line and to the second and third terms in the last line of eq. (3.323). This
is the complete contribution in time�ordered perturbation theory. In the projection formalism one
has an additional correction resulting from two "reducible" graphs of fig. 3.20, which are related
to the terms in the third line of eq. (3.323).
, /
, /
,
,
'( '(
/
/
,
/
,
/
/
/
/
,
/
2 3 4 5
Figure 3.22: Leading two�pion exchange contributions with single and double Ll�
excitations to the effective potential. For notations see fig. 3.6, 3.16.
A completely new type of the one�loop diagrams with only contact interactions is related to the
first operator in the last line of eq. (3.323). The three graphs with single and double Ll�excitations
are shown in fig. 3.21. Note that these diagrams yield a purely short�range contribution that
renormalizes the N N contact interactions without derivatives. This is because no momentum
dependence is introduced by the corresponding vertices and energy denominators. The only
,
111

112 3. The derivation of nuclear forces from chiral Lagrangians
moment um dependence is due to the loop integration ensuring that the result is given solely by a
power-Iaw divergent integral. Note that no analogous diagrams with purely nucleonic intermediate
states contribute to the effective potential in our approach.
We now discuss the most interesting type of correction, the two-pion exchange diagrams with
.6.-excitations. Different to the purely nucleonic case, one has no "reducible" graphs and, consequently,
obtains the same results with time-ordered perturbation theory and the projection
formalism. The TPE graphs with the .6.'s at NLO corresponding to eq. (3.323) are shown in
fig. 3.22. The tri angle diagram 1 in this figure is related to the second, third and fourth terms in
the first line of eq. (3.323). The first term in the second line and the last term in eq. (3.323) refer
to graphs 2-5 in fig. 3.22.
We will not calculate the complete NLO potential since the inclusion of the .6. is performed,
basically, to interpret the physics associated with the low-energy pion-nucleon constants. Such a
calculation has been performed by the Munich group [109]. They have found that all one-Ioop selfenergy
and vertex corrections with .6.-isobar excitations lead only to mass and coupling constant
renormalization and have the same structure as the OPE and N N contact interactions. The only
new interaction in the Hamiltonian, that includes the .6.-field and is relevant for calculation of
the NLO potential is
where the 2x4 spin and isospin transition matrices Si and Ti are normalized via
�
(28" - if."kO'k)
3
1 .
-(28·· - Zf."kTk)
3
1) 1) ,
1) 1) •
(3.324)
(3.325)
(3.326)
An explicit form for the matrices S and T can be found, for example, in ref. [189]. In eq. (3.324) we
adopt the same value for the coupling constant corresponding to the 7r N .6. vertex with one derivative
as in [109]. The TPE diagrams with .6.-excitation were also evaluated (but not renormalized)
by Ord6iiez et al. [78] using time-ordered perturbation theory. For completeness, we collect here
the pertinent formulae for the renormalized TPEP with .6.'s worked out by the Munich group
[109]. There are three distinct contributions .
.6.-excitation in the tri angle graphs:
with
s
L(q)
D(q)
.6.
E
V4M; + q2 ,
� In s + q
q 2M1': '
1 100 dJ.l
arctan
mt::. - m = 293 MeV ,
2M; + q2 _ 2.6.2 .
..6. 2M" J.l2 + q2
V J.l2 - 4M2
2.6.
1':
(3.327)
(3.328)
(3.329)
(3.330)
(3.331)
(3.332)

3.9. Three-nucleon potential
Single ß-excitation in the box graphs:
VTPEP A,box-1 =
3g�
(2M2 + q2)2 A(q)
327f f: 7f ß
19::2 f: (Ti · T2) {(12ß2 - 20M; - Ul)L(q) + 6E2 D(q)} (3.333)
12��� f : ((51 . if)(52 . if) - q2(51 . (2)) { -2L(q) + (s2 -4ß2)D(q)}
128��: (Tl · T2) ((51 . if)(52 · if) -q2(51 · 52)) s2 A(q) ,
ß
with A(q) given in eq. (3.311).
Double ß-excitation in the box graphs:
VTPEP A,box-2 = -6 :::f: { -4ß2 L(q) + E[H(q) + (E + 8ß2)D(q)]}
with
38 ::2 f: (Tl · T2) { (12E - s2)L(q) + 3E[H(q) + (8ß2 -E)D(q)]}
51�� f: ((51 . if)(52 . if) - q2(51 . (2)) {6L(q) + (12ß 2 - s2)D(q)} (3.334)
102��2 f: (Tl · T2) ((51 . if)(52 · if) - q2(51 · 52)) {2L(q) + (4ß2 + s2)D(q)}
113
(3.335)
Note that we have not shown the polynomial part of the potential since it has the structure given
by eq. (3.270). Further note that some of the contributions, in particular, those ones corresponding
to the first and the last lines in eq. (3.333) have the same structure as the corresponding NNLO
terms in eq. (3.310) with -C3 = 2C4 = g�/(2ß). This is discussed in detail in ref. [109]. However,
the remaining contributions show a non-trivial dependence of the N N potential on the N ß mass
splitting. In the next chapter we will furt her discuss effects of the explicit inclusion of the ß-isobar
excitations for the low-energy N N observables. It would also be interesting to perform NNLO
analysis of the TPEP with explicit ß's. In that case one should refit the LEC's Cl , C3 and C4 from
pion-nucleon scattering using the small scale expansion. This work is under way [190].
3.9 Three-nucleon potential
Let us consider the three-nucleon force. Obviously, the leading part of the effective potential,
Ve�-3N in eq. (3.253), does not involve three-nucleon operators. The first contributions to the 3N
force start at order 8-3N = -1. Let us now take a closer look at eq. (3.254). The first operator in
this equation corresponds to contact interactions with four nucleon legs and without derivatives.
It does not lead to a three-body force. The next three terms in the first line of eq. (3.254) yield
the two-pion exchange three-nucleon interaction represented by the graph 9 in fig. 3.23. Here, one
has to sum over all possible time orderings. It is interesting that after performing such summation
this three-nucleon force vanishes completely, as has been pointed out by Weinberg [73]. This can
be shown in an elegant way [114] without performing the explicit calculation. Indeed, one obtains

114
/
/
/
/
/
/
/
5
/
/
\
\
\
\
I
I
I
I
3. The derivation of nuclear forces from chiral Lagrangians
2 3
6 7
9 10
I
I
I
I
I
I
/
/
/
/
\
\
\
\
Figure 3.23: Leading contributions to the three-nucleon potential: irreducible twopion
exchange diagrams and irreducible one-pion exchange graph with the NN contact
interaction. Graphs which result from the interchange of the nucleon lines and
from the application of time reversal operation are not shown. In the case of diagram
9, one should sum over all possible time orderings. For remaining notations
see fig. 3.6.
in this case the same result within the projection formalism and time-ordered perturbation theory
and can use the Feynmann diagram technique, since no reducible topologies appear. Further, the
7r7r N N vertex contains a time derivative of the pion field, that has been counted as one power
of small momentum. But since the energy is conserved at each vertex when one calculates a
Feynmann diagram, this time derivative yields a difference of nucleon kinetic energies, which is of
higher order.
The first term in the third line of eq. (3.254) subsurnes the irreducible diagrams 1-8 in fig. 3.23,
representing time-ordered perturbation theory result for the two-pion exchange 3N force. Analogously,
the se co nd term in the first line of eq. (3.254) refers to the irreducible one-pion exchange
3N force involving contact interactions with four nucleon legs and without derivatives. The corresponding
graph is shown in fig. 3.23 (10). The diagrams depicted in fig. 3.23 define the complete
4
/
/
/
/
/
/
/
/

3.9. Three-nucleon potential 115
leading-order 3N force within time-ordered perturbation theory. It is weIl known that the static
2 3 4
5 6
Figure 3.24: Leading contributions to the three-nucleon potential: reducible twopion
exchange diagrams and reducible one-pion exchange graphs with the NN contact
interaction. For notations see figs. 3.6, 3.23.
TPE 3N force cancels against the recoil corrections of the nucleons to the static OPE 2N potential,
when the latter is iterated in the Lippmann-Schwinger equation [191], [46]. The same sort
of cancelation happens for the leading TPE and OPE 3N forces related to diagrams 1-8 and 10
in fig. 3.23, as has been found by van KoIck [77]. Since the energy dependence of the NLO 2N
potential is entirely given by recoil corrections of the nucleons to the static OPE, one can describe
systems of three or more nucleons using the energy-independent part of the 2N potential and
without explicit three- and many-nucleon forces (at next-to-Ieading order in the chiral expansion
for the potential). However, such type of cancelations does not help to remove the problems
in the two-nucleon sector due to the explicit energy-dependence as noted before (i.e. that the
wavefunctions are only orthonormal to the order one is working).
Let us now take a closer look at the 3N force obtained with the projection formalism. As already
stated above, for the TPE graph 9 in fig. 3.23 one obtains the same result using both schemes.
Apart from the irreducible TPE diagrams 1-8 and OPE diagram 10 in fig. 3.23, one also has to
take into account "reducible" graphs shown in fig. 3.24. The diagrams 1-4 in this figure refer to the
last two terms in the third line of eq. (3.254), whereas the graphs 5 and 6 involving the N N contact
interactions arise from the terms in the fourth line of this equation. Obviously, the cancelation of
the leading 3N force with the iterated energy dependent part of the 2N potential, as observed in
the context of old-fashioned perturbation theory, is now not possible because the potential in the
projection formalism is energy independent. Nevertheless, the leading-order 3N force completely
vanishes in the method of unitary transformation, although the mechanism of the cancelation
is different from the above case. In the projection formalism, the 3N force vanishes because of

116 3. The derivation o{ nuc1ear {orces {rom chiral Lagrangians
2 3
Figure 3.25: Three-nucleon force: TPE, OPE and contact interaction. In the case
of diagrams 1 and 2, all possible time orderings should be considered. For notations
see figs. 3.6, 3.23 and fig. 3.14.
an exact cancelation between the contributions from the "reducible" and irreducible OPE and
TPE diagrams. This cancelation was recently pointed out in [185] for the case of an expansion
in the pion-nucleon coupling constant and adopting the static approximation for nucleons. To
understand why the leading 3N force vanishes, let us take a closer look at the terms in the third
line of eq. (3.254). The contributions from the irreducible TPE diagrams 1-8 in fig. 3.23 can be
expressed schematically as:
(3.336)
where we pulled out the common factor M representing the spin, isospin and momentum structure,
which is obviously the same for all graphs 1-8 in fig. 3.23. The contribution from the "reducible"
diagrams 1-4 in fig. 3.24 can be expressed as
[ 2 2 ] Wl + W2
+ -2- w1 w2 WIW2
--2 M=2 w1 22M. w2
(3.337)
The cancelation is now evident. The same sort of cancelation can be observed for the irreducible
diagram 10 in fig. 3.23 and the "reducible" graphs 5 and 6 in fig. 3.24 involving contact interactions.
We conclude that there is no three-nucleon force at the order v = -l.
Let us now consider the 3N force at order v = 0 related to the effective potential (3.255). The
terms in the second line of eq. (3.255) refer to the two-pion exchange diagram 1 in fig. 3.25. The
first term in this equation corresponds to the contact force shown in fig. 3.25 (3). Finally, the
one pion exchange graph 2 involving contact interactions with four nucleon legs arises from the
second and third terms in eq. (3.255). In all cases the method of unitary transformation does not
introduce any new aspects, since it yields the same result as time-ordered perturbation theory.
Within the last approach, this 3N force has been calculated and discussed by van Kolck [77].
In that reference a complete expression for the leading chiral 3N force is given. Unfortunately,
the inclusion of the leading 3N interaction intro duces several new parameters. While the 1f N N
couplings D1 and D2 in eq. (3.236) can, in principle, be determined from the processes like 1fdeuteron
scattering or 1f production and absorption on N N system, the remaining three contact
interactions El,2,3 can only be fixed from data involving systems with more than two nucleons.
At present, this 3N force has not yet been applied in systematic calculations of the three- and

3.9. Three-nuc1eon potential
2 3
Figure 3.26: Leading three-nuc1eon force with b,-isobar excitations. All possible
time orderings should be considered. For notations see figs. 3.6, 3.16 and fig. 3.23.
more-nuc1eon system. First steps in this direction have been done in refs. [118], [196]. In the
first work, Friar et al. consider the constraints on a possible form of the TPE force arising from
requirement of its (approximate) chiral symmetry. They pointed out that any TPE 3N force can
be expressed as
where
117
(3.338)
(3.339)
Here ij and ij' are the moment um transfers between the nuc1eons 1, 3 and 3, 2. Note that these
expressions are valid modulo l/m corrections. The authors of ref. [118] extracted the coefficients
a, b, C and d from various existing 3N forces, which are based on different models, and compared
them with the predictions obtained within the framework of chiral perturbation theory without
explicit b,'s. In the latter case, the leading order values of these coefficients can be calculated
from the 3N force shown in fig. 3.25 (1). The values of the relevant 7r7rNN couplings Cl , C3 and
C4 are known from the pion-nuc1eon sector, as discussed in sec. 3.8.2 and thus the coefficients a,
b, c and d can be predicted without free parameters. It turns out that although the signs of these
coefficients are always the same, their values differ substantially (by factors of two) for different
three-nuc1eon forces. Furthermore, to leading order in low-momentum expansion CHPT leads to
the largest absolute values for the coefficients a, b and d48 and to c = O. The c-term does not
vanish for the Tucson-Melbourne (TM) force [43] and thus violates chiral constraints. Therefore
in ref. [118] it is recommended to drop this c-term in the TM force.
In the second work, Hüber et al. analyse the effects of the 3N force corresponding to the second
graph in fig. (3.25) in elastic neutron-deuteron scattering at low-energies. They found that the
inclusion of such a short-range-long-range force might considerably improve the agreement for
the analyzing power Ay• This study is, however, merely qualitative, since only a part of the
leading-order 3N force resulting from CHPT has been included and no attempt was made to
obtain a reasonable fit to data.
An important observation has been done by van Kolck in ref. [77]. He pointed out that several
vertices entering the expressions for the leading 3N force are saturated by virtual b,-isobar ex-
4 8 Note, however, that the authors of ref. [118) used the values for Cl,3,4 as given in [71) that are somewhat larger
than the ones from the later publication [195], which we adopt in this work.

118 3. The derivation of nuclear forces from chiral Lagrangians
citations. The situation here is similar to the NNLO two-nucleon potential. In that case one
has three independent 1f1fNN vertices, two of which, the C3- and q-couplings, are governed by
intermediate Ll-excitations. An explicit inclusion of the Ll's leads to NLO corrections to the 2N
potential, if the delta-nucleon mass splitting is treated as the small quantity, on the same footing
with M1f' We will now consider the situation with the 3N force if the Ll's are included. Then,
many additional contributions to the leading 3N force at v = -1 arise from eq. (3.323), that
correspond to diagrams with intermediate Ll-excitation. The first term in the second line and
the last term in this equation refer not only to the TPE graphs of fig. 3.23 but also to the TPE
3N force with one intermediate Ll shown in fig. 3.26 (1). The second term in the first li ne of
eq. (3.323) together with the second and third terms in the last line lead to OPE diagram with
virtual Ll-excitation shown in fig. 3.26 (2). Finally, the first term in the last line of this expression
refers to diagram 3 in fig. 3.26. Note that in all cases one gets the same result from the projection
formalism and time-ordered perturbation theory, since reducible topologies are impossible
for diagrams shown in fig. 3.26. Furthermore, one verifies that if the mass of the Ll is regarded
to be large, the graphs of fig. 3.26 go into the diagrams shown in fig. 3.25. The saturation of the
corresponding vertices due to intermediate Ll-excitations is now transparent.
One should stress that not all these 3N forces are new. For example, the TPE 3N interaction
with one intermediate delta is included in many phenomenological models, like for instance, the
Fujita-Miyazawa [42] or the Urbana [197] force. The remaining 3N forces 2 and 3 in fig. 3.26 that
include contact interactions are, certainly, less common, since most of the "realistic" 3N forces
are based on meson-exchange models. However, implicitly, such short range contact interactions
are also present in form of the heavy meson-exchange.
Finally, we would like to discuss another kind of forces, which may aIso appear in the threebody
calculations: two-body interactions, that depend on the total moment um P of a twonucleon
subsystem. Such three-body-like two-nucleon forces do not affect the pure two-nucleon
calculations in the cms, where P = 0, but become important for processes including additional
particles (for example, pion production on the deuteron) and for three and more nucleons, where
the total moment um P of the two-nucleon subsystem is not conserved any more. The Lagrangian
eq. (F.13) given in refs. [77], [78] for the contact interactions with two derivatives will necessarily
lead to the forces of this kind. Moreover, fitting the N N phase shifts will only fix seven parameters
of the fourteen entering eq. (F.13), see e.g. [74], [76], [78], [127]. Thus, the P-dependent forces
appear to have unknown coefficients. Such a situation is, clearly, not satisfactory, since Lorentz
invariance would be violated, see appendix F. In this appendix we have analyzed all possible
contact interactions with four nucleon legs up to order Lli = 3, requiring their reparametrization
invariance. It turns out that only seven independent coupling constants enter the Lagrangian
L�;.,= 2), which all can be fixed from nucleon-nucleon scattering. Furthermore, the N N potential
at NLO does not depend on the total momentum P (as it should because of Galilean invariance).
We have also found that no 1/m-corrections to the contact terms with two derivatives appear in
the effective Lagrangian and that no terms enter L�;.,=3). Consequently, also our NNLO potential
does not depend on P. The P-dependent N N forces may only appear at order N3LO and will
not introduce new unknown coefficients beyond the ones needed in the 2N cms.

Chapter 4
The two-nucleon system: numerical
results
4.1 Bound and scattering state equations
In the last chapter we have given the explicit expressions for the N N force up to NNLO. To
make this presentation more transparent, we will now summarize .our findings and enumerate all
contributions to the effective potential V = VLO + VNLO + VNNLO:
V(O) + VOPEP ,
( 4.1)
CS +CT,
VOPEP
V;TPEP
NLO
( 9A ) 2 o\ ·ifih·if
- 21 7f 'Tl . 'T2 q 2 + M2 7f '
V(O) + V(2) + VOPEP + vJlJP ,
Cl if2 + C2 P + (C3 if2 + C 4 P)(51 · 52) + iC5 � (51 + 52) ' (ifx k)
+ C6 (if· 51)(if· 52) + C7 (k· 5t)(k· 52) ,
- 'Tl . 'T2 { 2 4 2 2 4 2 48g�M; }
3847r2 I ; L(q) 4M7f(59A - 4gA - 1) + q (239A - 10gA - 1) + 4M; + q2
3g� L( ) { � � � � 2 � � }
- 647r2 I ; q a1 ' q a2 . q - q a1 ' a2 ,
VNNLO V(O) + V(2) + VOPEP + vJlJP + vJJf6' , (4.3)
vJJf6'
- 1��;; { - 16m(���+ q2) + (2M;(2C1 - C3) - q2 (C3 + ::�)) (2M; + q2)A(q)}
128�� 1; ('Tl ' 'T2) { - 4�1:� 2 + (4M; + 2q2 - g�(4M; + 3q2))(2M; + q2)A(q)}
+ 51;:� 1; ((51 . if)(52 . if) - q2(51 . 52)) (2M; + q2)A(q)
3;:1 ; ('Tl ' 'T2) ((51 ' if)(52 · if) - q2(51 . 52))
119
( 4.2)

120 4. The two-nuc1eon system: numerical results
where
and q=p' - p.
4 39� r i (0\ + eh) . (p ' x ff) (2M; + q2)A(q)
6 7fm 'Ir
647fm f: Tl' T2 2 0"1 + 0"2 •
x {(c4 + 4 �)(4M; + q2) - :� (10M; + 3q2)} A(q)
g � (1 - g �) ( ) . ( � � ) ( �' ;;'\ (4M2 2)A( )
L(q)
A(q)
q V 'Ir + q n
1 q
P X p) 'Ir + q q ,
� . /4M2 2 I V 4M; + q2 + q
2M'Ir '
-arctan --
2q 2M'Ir '
Now we want to apply this potential to calculate the two-nucleon observables. The scattering
states are described by the Lippmann-Schwinger integral equation. The LS equation (for the
T-matrix) projected onto states with orbital angular momentum I, total spin s and total angular
moment um j is
InOO d " ,,2
sj , sj , P P sj ,,, m sj
Tl'
"
I(P , p) = Vi, I (P , p) + L
(2 )3 Vi, 1" (P , p ) 2 ,,2 . Tl" , ,
I (P , p) ,
I" ° 7f ' P - P + 2'TJ '
with 'TJ --+ 0+. In the uncoupled case, I is conserved. The partial wave projected potential Vi��(p' , p)
can be obtained using the formulae of the appendix G. The relation between the on-shell 8- and
T-matrices is given by
S � ( )_ r
2
T� ( )
1'1 P - Ul'l - 87f2 pm 1'1 P ,
where P denotes the two-nucleon center-of-mass three-momentum. The phase shifts in the uncoupled
cases can be obtained from the S-matrix via
SOj ( 2 ' rOj )
JJ J ' Slj (2·r1j)
.. = exp 2u ·
(4.4)
(4.5)
(4.6)
jj = exp w j , (4.7)
where we have used the notation 8t j . Throughout, we use the so-called Stapp parametrization
[203] of the S-matrix in the coupled channels (j > 0):
S =
)
2 . sm
.
(2 f ) exp ( W' r1j
j_ 1 ;-. 2Uj+1
. r1j
) .
(4.8)
cos (2f) exp (2i8 j� 1)
For the discussion of the effective range expansion for the 3 S I partial wave we will use the different
parametrization of the S-matrix, namely the one due to Blatt and Biedenharn [204]. The
connection between these two sets of parameter is given by the following equations:
8j-1 + 8j+1
sin(8j_ 1 - 8j+1 )
8j-1 + 8j+1 ,
tan(2f)
tan(2E') ,
sin(2f)
sin(2E') ,
(4.9)

4.1. Bound and scattering state equations 121
where J and E denote the quantities in the Blatt-Biedenharn parametrization.
The bound state is obtained from the homogeneous part of eq. (4.5) and obeys
where s = j = 1, [ = [' = 0,2 and Ed denotes the bound-state energy.
( 4.10)
The potential (4.1)-(4.3) is only meaningful for momenta below a certain scale. This is obvious
both from a conceptual and a practical point of view. Indeed, since we are working within
the effective field theory approach, only the low-momentum matrix elements can be derived
systematicaHy. Our perturbative expansion far the potential breaks down for momenta of the
nucleons comparable with the scale Ax. Also from the practical point of view, the expressions
(4.1)-(4.3) are, clearly, not acceptable for large values of the moment um transfer q. In particular,
since the potential grows for large momenta, it leads to ultraviolet divergences in the LS equation
(4.5). As it is appropriate in effective theory, we regularize the potential. This is done in the
following way:
(4.11)
where fR(iJ) is a regulator function chosen in harmony with the underlying symmetries. In what
follows, we work with two different regulator functions,
ftarp(p)
f �xpon (p)
B(A2 _ p2) ,
exp( _p2n / A 2n) ,
( 4.12)
(4.13)
with n = 2,3, ... , p = Ipl and p' = Ip'l. Such a regularization of the effective potential is also
consistent with the discussion in chapter 2. We must introduce a cut-off in order to exclude
the high-momentum intermediate states in the LS equation, that can not be described correctly
within the low-energy effective theory.
In eqs. (4.12), (4.13) we show the two different choices of the regulator function. The sharp cut-off
is most appropriate for comparison with realistic phenomenological potentials. To enable such a
comparison one first needs to integrate out the high-momentum components from the realistic
potentials, since the effective one is defined only for momenta p, p' < A. Eliminating the highmoment
um components can be performed via the unitary transformation, as described in section
2.3. We already performed that step but leave the discussion of the results to a furt her study. The
sharp cut-off regularization scheme is suitable for deriving the phase shifts but leads to troubles
when calculating some deuteron properties. This is because discontinuities in the moment um
space are introduced at p, p' = A. In that case one should use a smooth regularization like the
exponential one given in eq. (4.13). For very large integers n the exponential cut-off approximates
the sharp one. Throughout, we work with n = 2. To the order we are working, the choice n = 1
has to be excluded since the terms of order p2,p /2 would be modified. For n = 2, the error we
make is beyond the accuracy of the order we are calculating. We would like to point out that the
low-energy observables should not be sensitive to the choice of the regulating function fR as weH
as to an exact value of the cut-off. We will also confirm this statement by explicit calculations.
The remaining (weak) dependence of the observables on the choice of fR and on the value of A
can be systematically removed by including the higher order terms [141]. In the following seetions
we will discuss these uncertainties in more details.

122 4. The two-nuc1eon system: numerical results
4.2 The fits
In this section we, basicaIly, concentrate on the determination of the various coupling constants.
For Mn:, in: and gA we use the following values:
Mn: = 138.03 Me V , in: = 93 MeV , gA = 1.26 . ( 4.14)
Furthermore, as already stated above, the values of C 1 ,3,4 are
Cl = -0.81 GeV -1 , C3 = -4.70 GeV-1 , C4 = 3.40 GeV-1 . ( 4.15)
Thus, all parameters entering the non-polynomial part of the potential are fixed.
Consider now the polynomial part vcontact of the potential represented by contact interactions
vcontact = V(O) + V(2). To pin down the nine parameters Cs, CT, Cl , ... ,C7 we do not perform
global fits as done in ref. [78]. Rather we introduce the independent new parameters C2s+1l.,
J
02S+1l. via the following equations:
J
v contact (3 SI )
41f (Cs - 3CT) + 1f (4C1 + C2 - 12C3 - 3C 4 - 4C6 - C7)(p2 + p'2)
- 2 2
C1S0 + C1S0 (p + p' ) , (4.16)
41f (Cs + CT) + i (12C1 + 3C 2 + 12C3 + 3C4 + 4C6 + C7)(p2 + p'2)
- 2 2
C3S1 +C3S 1 (p + p' ), (4.17)
2 ; (-4C1 + C2 + 12C3 - 3C4 + 4C6 - C7) (p p') = Cl PI (pp') , (4.18)
2 ; (-4C1 + C2 - 4C3 + C4 + 2C5 + 4C6 + C7) (p p')
C3P1 (pp') ,
2 ; (-4C1 + C2 - 4C3 + C 4 + 2C5) (pp') = C3P2 (pp') ,
21f
3 (-4C1 -
+ C2 4C3
) ,
+ C4 + 4C5 + 12C6 - 3C7 (pp )
C1 Po (pp') ,
2.j21f ( ,2 ,2
- 3 - 4C6 + C7) p = C3Dl-3S1 P ,
2.j21f 2 2
- 3 - (4C6 + C7)p = C3Dl-3S1 P .
(4.19)
( 4.20)
(4.21)
(4.22)
(4.23)
Here, Ves+1lj) denotes the matrix element (lsjlVllsj) and Ves+1lj _2s+1lj) is (lsjlVll'sj). The
equations (4.16)-(4.23) can be obtained with the help ofthe formulae of appendix G for the partial
wave decomposition. The main advantage of using the new parameters C2s+1[., 02s+1l. instead of
J J
the old ones Cs, CT and Cl , ... , C7 is that now we can fit each partial wave separately. This not
only makes the fitting procedure extremely simple, but also leads to unique results. To leading
order, the two S-waves are depending on one parameter each. At NLO, we have one additional
parameter for 1 So and 3 SI as weIl as one parameter in each of the four P-waves and in EI' At
NNLO (NNLO-Ll), l we have no new parameters, but must refit the various contact interactions
due to the TPEP contribution in all partial waves. We have used two different methods to fix the
1We will denote by NNLO-ß the potential
(4.24)

4.2. The fits
10000
1000
100
10
0.1
0.01
0.001
0.0001
, ,
,
1e-05
0.001
1000
100
10
1
0.1
0.01
0.001
0.0001
1e-05
1e-06
1e-07
0.001
,
, ,
,
,
,
,
,
,
,
,
,
,
,
,
0.01
,
,
,
,
,
,
,
,
0.01
I SO
1Pl
0.1
/
--
0.1
/ -, :
1000 r-���r-��������
100
10
1
0. 1
0.01
0.001
0.0001
1 e-05 ,--� .... .. ..
0.001 0.01 0.1
"
:
,
,",-,-,""-�-,-,-,-,-,-,-

124 4. The two-nuc1eon system: numerical results
LEes of the contact interactions. First, we fit to the phase shifts of the Nijmegen partial wave
analysis [33] for laboratory energies smaller than (50) 100 MeV at (NLO) NNLO. 2 Alternatively,
- -
3
we used the effective range parameters for the phases ISO, SI and EI to fix ClSO ' ClSO ' C3Sp C3Sp
and C3Dl -3Sl ' This is completely analogous to the fitting procedure in the case of the effective
1 5
5
-5
�
-1 5 0.7 0.8 0.9
1 5
-5
r l------
-- -=-::. . -:...-= =-:::- - - ------
--- -- - - ---
-1 5 0.7 0.8 0.9
A [GeV]
Figure 4.2: Running of the four-nucleon coupling constants in the 1 So (upper panel)
and 3 SI _3 Dl (lower panel) partial waves. In the upper panel, the solid (dashed)
line refers to Cl So (C\ s o ) . In the lower panel, the solid (dashed) line refers to C3 SI
( 03 sJ . The dashed-dotted line refers to the constant in the coupled 3 SI - 3 Dl
system. The units are the same as in the table 4.1.
theory considered in section 2.2. In what follows, we will mark the corresponding potentials by a
where v( O ), V(2), VOPEP, vJlJ1P are given in eqs. (4.1), (4.2) and V:LO corresponds to the expressions (3.327)­
(3.334). Although this potential corresponds to the NLO result in the "small scale expansion" , we will consider
it merely in the context of the saturation of the NNLO Cl,3,4 corrections. Therefore, we denote it by NNLO-ß
potential.
2Note that equally weil we could fit directly to the data. However, for easier comparison with other EFT
calculations, we use the Nijmegen PSA to simulate the data.
-.
I
1
I
I

4.2. The fits 125
"*" if the LECs were fixed from the effective range parameters. We have found that the observables
(phase shifts) depend rather weekly on what fitting procedure is used. The values of the LECs do
not change much except for some C's at NLO, where significant variations were observed. Note
furt her that in the S-waves, especially in 1 So, isospin breaking effects like e.g. the charged to
neutral pion mass difference, are known to be important. We do not consider such effects in this
work and thus take an average value for the pion mass. The actual values of the S-wave LECs
see m to be rat her sensitive to the choice of the pion mass. For the phases 1 So, 3 Sl and E1 we use
the errors as given in ref. [33], for all other partial waves we assign an absolute error of 3%. This
number is arbitrary, but taking any other value would not change the fit results, only the total
X2• To perform the fits, we have to specify the value of the cut-off A in the regulator functions
as defined in eqs. (4.12), (4.13). At NLO, for any choice between 380 MeV and 600 MeV, we get
very similar fits (a very shallow X2 distribution in each partial wave). At NNLO, this shallow
distribution turns into a plateau, which shifts to higher values of the cut-off. We can now use
values between 600 MeV and 1 GeV. The results obtained using the sharp and the exponential
cut-off are very similar. For illustration, we consider the sharp cut-off with A = 500 Me V at NLO
and A = 875 MeV at NNLO. We show in fig. 4.1 the absolute quadratic deviations of the fit to
the Nijmegen phase shift analysis (PSA), defined by (8fit -
8PSA)2.
Consider first the S-waves.
Both in 1 So and 3 Sl, we observe a clear improvement when going from LO to NLO to NNLO
(NNLO-Ll). A similar pattern holds for the P-waves and E1 , although the differences between
NLO and NNLO are somewhat less pronounced. Note that the 3 Po wave is very sensitive to the
value of the pion mass, therefore the slightly better NLO fit should not be considered problematic.
The corresponding parameters of the coupling constants are collected in table 4.1.
Cl So
Cl So
C 3S1
C3S1
Cl PI
C3P1
C3Po
C3P2
C3Dl-3S1
NLO 1 NLO* 1 NNLO 1 NNLO* 1 NNLO-Ll 11
-0.1337 -0.09276 -4.249 -4.246 -5.731
1.822 2.125 11.95 11.84 13.82
-0.1304 -0.1022 -6.508 -6.655 -0.9767
-0.3933 0.02432 11.29 11.28 3.446
0.3442 0.3442 -2.045 -2.045 -2.494
-0.3941 -0.3941 -7.061 -7.061 -8.188
1.335 1.335 -2.832 -2.832 -2.993
-0.1907 -0.1907 -8.056 -8.056 -8.431
-0.03170 -0.03568 -3.424 -3.516 -0.5623
Table 4.1: The values of the LECs as determined from the low partial waves. We use the sharp
cut-off with A = 500 MeV and 875 MeV at NLO and NNLO, respectively. The Ci are in 104
GeV-2 while the others are in 104 GeV-4. The parameters of the NLO, NNLO and NNLO­
Ll potentials are obtained from fitting to the Nijmegen PSA. The LECs of the NLO*, NNLO*
potentials in the 1S0 and 3S1 _3 D1 channels are fixed to reproduce exactly the effective range
parameters.
To illustrate the dependence on the cut-off, we show in fig. 4.2 the running of the two (three)
couplings in the 1S0 eSd channels at NNLO (note that the third parameter in 3S1 comes in via
the mixing with the 3 D1 wave). We notice that the variation of these LECs over a wide range of
cut-offs is rat her modest. We also mention that using the 7rN parameters from refs. [200, 205, 71]

126 4. The two-nuc1eon system: numerical results
leads to a considerably worse X 2 in the fits. We take that as an indication that the determination
of the Ci based on the method employed in ref. [195] is more reliable than fitting to 7rN phase
shifts (as long as one works to third order in the chiral expansion). We remark that using the
parameters of ref. [195], the deuteron binding energy Ed comes out as
NLO Ed = -2.175 MeV ,
NNLO Ed = -2.208 MeV , (4.25)
Le. the NNLO result is already within 7.5 permille of the experimental number. Fine tuning
in the parameters in the deuteron channel would allow to get the binding energy at the exact
value of -2.224575(9) MeV without leading to any noticeable change in the phase shifts. We
later consider the deuteron channel separately with an exponential regulator. This will lead to an
improved binding energy but no attempt is made to match the exact value in all digits.
At this point we would like to comment on the increase in the cut-off values when going from
NLO to NNLO (NNLO-b.). Consider first the leading order result. Lepage [141] has pointed
out that the inclusion of the one-pion exchange does not lead to a remarkable increase in the
cut-off values compared to a pionless theory. In order to fit the phase shifts in the S-waves one
should choose the cut-off below 500-600 MeV even if the contact interactions with two derivatives
are taken into account (within our power counting scheme such contact interactions contribute
first at next-to-Ieading order and are of the same size as the lading two-pion exchange terms).
Lepage assumed that such a low value of the cut-off is due to the missed physics associated with
the two-pion exchange. So, naively, one would expect that the inclusion of the leading two-pion
exchange contributions at NLO would allow to take larger cut-off values. However, that does not
happen. This was also pointed out in refs. [105], [106]. According to our analysis, only at NNLO,
after the subleading two-pion exchange contributions are taken into account, one can increase the
cut-off up to 800 to 1000 MeV. The inclusion of the dimension two operators of the pion-nudeon
interaction at NNLO encodes some information about heavy meson exchange as weIl as virtual
isobar excitations, as discussed in detail in ref. [200]. In the present work we were able to separate
the leading effects of the b.-resonance (NNLO-b.). The clear increase in the cut-off values when
going from NLO to NNLO-b. indicates the importance of physics associated with heavier mass
states like e.g. the b.-resonance. Our NNLO (NNLO-b.) TPEP is sensitive to moment um scales
sizably larger than twice the pion mass (as it would be the case for uncorrelated TPE) and deltanucleon
mass splitting. Consequently, the cut-off has to be chosen safely above these scales, say
above 500 MeV (with the sharp regulator). The upper limit of about 1 GeV is related to the
cancelations in the S-waves (fine-tuning), since for too large values of A it is no longer possible
to keep this intricate balance. It is, however, comforting to see that including more physics in the
potential leads indeed to a wider range of applicability of the EFT.
FinaIly, we need to discuss one furt her topic. Performing the fits, we have found two minima in
both the ISO and the 3S1 channel. This is not unexpected. Indeed, the same happens in the case
of the pionless theory considered in the section 2.2. In particular, the requirement of reproducing
exactly the scattering length and the effective range within the effective theory with the sharp
cut-off leads to the equations (2.37), (2.38) for the constants Co and C2. Thus, one observes two
equivalent solutions for Co and C2. Such a situation with two solutions appears also in the NLO
and NNLO theory with pions. At NLO, we find very similar predictions for the phase shifts and
observables as weIl as a very similar quality of the fits in the ISO and 3S 1 _3 D1 channels for
both solutions, see the upper panel in fig. 4.3. So, there is no real criterion to prefer one of these
solutions. However, at NNLO, the behavior of the phase shifts at higher energies differs quite
remarkably as it is illustrated in the lower panel of fig. 4.3. Also, the X 2 for these two solutions

4.2. The nts
60
0; 40
CD
�
0 20
Cf)
�
I
0
0
-l
z
-20
-40
0 0.1
70
0; 50
CD
�
0
Cf) 30
•
0.2 0.3
I - - -
- - - - - - -
0
-l
(\J
z
10
-1 0
0 0.1 0.2 0.3
E1ab [GeV]
Figure 4.3: Phase shifts for the two solutions in the 1 So-wave as dicussed in
the text. At NLO (upper panel), these are indistinguishable. At NNLO (lower
panel), one of the solutions (dashed line) shows an unacceptable behaviour at
higher momenta and is discarded.
127

128 4. The two-nuc1eon system: numerical results
differs typically by factors of 2 ... 10. In what follows, we will only discuss the best solution at
NNLO.
4.3 Phase shifts
4.3.1 S-waves
In fig. 4.4 we show the two S-waves at LO, NLO and NNLO for the cut-off values given above.
Clearly, the lowest order OPEP plus non-derivative contact terms is insufficient to describe the 1 So
phase (as it is weIl-known from effective range theory and previous studies in EFT approaches).
The much more smooth 3 SI phase is already fairly weIl described at leading order. For energies
above 100 MeV, the improvement by going from NLO to NNLO is clearly visible. The corresponding
values of the S-wave phase shifts at certain energies are given in tables 4.2,4.3. For
comparison, we also give the results of the Nijmegen and VPI PSA [206] and of three modern
potentials (Nijmegen 93 [33], Argonne V18 [37] and CD-Bonn [29]). Our NNLO result for 1 So is
visibly better than the one obtained in ref. [207].
It is also of interest to consider the scattering lengths and effective range parameters as it was
done in the cases of the pionless theory and in the model calculations in chapter 2. The effective
range expansion takes the form (written here for a genuine partial wave)
( 4.26)
with p the nucleon cms momentum, a the scattering length, r the effective range and V2, 3,4 the
shape parameters. It has been stressed in ref. [208] that the shape parameters are a good testing
ground far the range of applicability of the underlying EFT since a fit to say the scattering length
and the effective range at NLO leads to predictions for the Vi. In table 4.4, we present our results
for the S-waves in comparison to the ones obtained from the Nijmegen PSA. Note that in the
coupled channel we have used the Blatt and Biedenharn parametrization of the S-matrix in order
to be able to compare our findings with those of ref. [102]. For both S-waves we observe a good
agreement with the data (apart from the last coefficient in the ISO channel). In general, the results
for the 1 So partial wave are not as precise as for the 3 SI chan ne 1. This has to be expected because
of the unnaturally large value of the scattering length in the first case. At this point we would
like to remind the reader of the similar analysis for the effective range coefficients performed for
the pionless theory in sec. 2.2 and for our model in sec. 2.3. In the first case one can not obtain
any predictions for the effective range coefficients, that are not used to fix the LECs. This is
because one describes a purely short-range physics in such a theory. In the second example we
included the known long-range force into the effective Hamiltonian. It is natural to assurne that
the values of the effective range coefficients are governed by the lowest mass scale associated with
the longest range part of the interaction. In the model considered in sec. 2.3 we had, however, no
large separation between the scales related to the long- and short-range parts of the underlying
force. Therefore, only the first effective range (shape) parameter not used in the fit could be
predicted accurately. The fairly precise description of the shape parameters shown in table 4.4
may be considered as a clear indication of large separations between the relevant scales in the case
of the realistic N N interactions. Furthermore, we would like to point out a clear improvement for
all quantities with exception of V2 for the 1 So partial wave when going from NLO to NNLO. Note
that in both cases we had 2 (3) free parameters in the 1 So e Sl-3 Dd channel (and the cut-off).
In our opinion, this is an important argument in favor of a good convergence of our expansion for
the effective potential (which survives the iteration).

4.3. Phase shifts
80
60
40
20
o
-20
-40
o 0.05 0.1
160
140
120
100
80
60 � � - - - _
40
20
o
-20
-40
o 0.05 0.1
ISO [deg]
0.15 0.2 0.25 0.3
3S 1 [deg]
0.15 0.2 0.25 0.3
Figure 4.4: Predictions for the S-waves (in degrees) for nucleon laboratory
energies Elab below 300 MeV (0.3 GeV). The dotted, dashed and solid curves
represent LO, NLO and NNLO results, in order. The squares depict the
Nijmegen PSA results. In the upper and the lower panel, 1 So and 3 SI, respectively,
are shown.
129

130 4. The two-nuc1eon system: numerical results
1 E1ab [MeV] 11 NNLO* NNLO 1 Nijm PSA 1 VPI PSA 1 Nijm93 1 AV18 CD-Bonn
h 147.735 147.727 147.747 147.781 147.768 147.749 147.748
2 136.447 136.450 136.463 136.488 136.495 136.465 136.463
3 128.763 128.781 128.784 128.788 128.826 128.786 128.783
5* 118.150 118.196 118.178 118.129 118.240 118.182 118.175
10* 102.56 102.67 102.61 102.41 102.72 102.62 102.60
20 85.99 86.21 86.12 85.67 86.35 86.16 86.09
30 75.84 76.14 76.06 75.46 76.40 76.12 75.99
50* 62.35 62.79 62.77 62.12 63.36 62.89 62.63
100* 42.33 43.06 43.23 42.98 44.33 43.18 42.93
200 19.54 20.68 21.22 20.88 22.82 21.31 20.88
300 4.15 5.58 6.60 5.08 8.44 7.55 6.70
Table 4.2: 3 Sl np phase shift for the global fit at NNLO (sharp cut-off, A = 875 MeV) compared
to phase shift analyses and modern potentials. The parameters of the NNLO potential are fixed
by fitting the Nijmegen PSA at six energies (E1ab = 1,5, 10,25, 50(, 100) MeV). These energies are
marked by the star. The parameters of the NNLO* potential are chosen to reproduce exactly the
scattering length and the effective range as described in the text.
To show that our results are stable and do not depend on the fitting procedure we present in
table 4.5 the NLO* and NNLO* predictions. Here, the LECs are fixed directly from the first
effective range parameters, as discussed above. Overall, the agreement between the numbers in
tables 4.4 and 4.5 is quite satisfactory. The NLO results turn out to be slightly more sensitive
to the fitting procedure. As is pointed out in ref. [208], one also can perform the moment um
expansion for E1 in a similar way as for the S-waves. The authors of the reference [208] show the
values of the corresponding parameters extracted by themselves from the Nijmegen PSA. They
use, presumably, the different (Stapp) parametrization of the S-matrix in that case. We have not
found any original reference of the Nijmegen group concerning the moment um expansion for E1 .
We thus refrain from discussing this issue any further.3
For completeness, we collect the experimental values for the S-wave scattering lengths and effective
ranges:
as = (-23.758 ± 0.010) fm , Ts = (2.75 ± 0.05) fm ,
at = (5.424 ± 0.004) fm , Tt = (1.759 ± 0.005) fm .
( 4.27)
( 4.28)
Quite recently the results for the NNLO calculation of the phase shifts in the S, P and D channels
within the KSW scheme [91] have been published [211]. Here we would like to compare our
findings with those from ref. [211]. The leading and non-perturbative contribution to the N N Smatrix
in S channels results in the KSW approach from summing up an infinite series of the contact
3Note, however, that in order to fix the LEes in the 3S1 _3 D1 channel we have used the leading coefficient
91 = 1.66 fm3 in the momentum expansion of the EI . This value agrees with the one given in ref. [208]. In principle,
we could take apart from the 3 SI scattering length and effective range the deuteron binding energy as the third
quantity to fix the three free parameters in the potential. However we refrain from doing that because of a strong
correlation between the deuteron binding energy and the 3 SI effective range parameters.

4.3. Phase shifts
1 Elab [MeV] 11 NNLO*
h 62.071
2 64.472
3 64.671
5* 63.659
10* 60.02
20 53.66
30 48.55
50* 40.49
100* 26.30
200 7.63
300 -6.41
NNLO 1 Nijm PSA I VPI PSA 1 Nijm93 1 AV18 CD-Bonn I
62.063 62.069 62.156 62.065 62.015 62.078
64.469 64.573 64.573 64.460 64.388 64.478
64.671 64.762 64.762 64.650 64.560 64.671
63.663 63.708 63.708 63.619 63.503 63.645
60.03 59.96 60.00 59.94 59.78 59.97
53.68 53.57 53.77 53.54 53.31 53.56
48.58 48.49 49.00 48.42 48.16 48.43
40.54 40.54 41.66 40.38 40.09 40.37
26.38 26.78 27.86 26.17 26.02 26.26
7.76 8.94 7.86 7.07 8.00 8.14
-6.24 -4.46 -5.55 -7.18 -4.54 -4.45
Table 4.3: ISO np phase shift for the best fit at NNLO (sharp cut-off, A = 875 MeV) compared
to phase shift analyses and modern potentials. The parameters of the NNLO potential are fixed
by fitting the Nijmegen PSA at six energies (E1ab = 1,5, 10, 25,50, 100 MeV). These energies are
marked by the star. The parameters of the NNLO* potential are chosen to reproduce exactly the
scattering length and the effective range as described in the text.
I SO NLO -23.555 2.64 -0.58 5.4 -31
I SO NNLO -23.722 2.68 -0.61 5.1 -30
ISO NPSA -23.739 2.68 -0.48 4.0 -20
3S 1 NLO 5.434 1.711 0.075 0.77 -4.2
3S 1 NNLO 5.424 1.741 0.046 0.67 -3.9
3S 1 NPSA 5.420 1.753 0.040 0.67 -4.0
Table 4.4: Scattering lengths and range parameters for the S-waves at NLO and NNLO (global
fits) compared to the Nijmegen PSA (NPSA). The values for V2, 3 , 4 in the 1 So channel are based
on the np Nijm II potential and the values of the scattering length and the effective range are
taken from the ref. [209]. The effective range parameters for the 3S 1 _3 D 1 channel are discussed
in [102].
interactions without derivatives. The first corrections are given by dressing the one-pion exchange
and the contact interactions with two derivatives by the leading order amplitude. This is the
crucial difference to our power counting scheme, in which the OPE diagrams are of the same size as
the leading contact interactions and thus should both be treated non-perturbatively. The NNLO
corrections in the KSW scheme are given by various diagrams including contact interactions with 0,
2 and 4 derivatives as weH as pion exchange graphs. For more details see ref. [211]. Because of the
perturbative treatment of the pion exchanges, the authors of ref. [211] could perform an analytic
calculation of the amplitude including its renormalization. The resulting S-matrix satisfies the
perturbative unitarity condition in the sense of the KSW power counting. Furthermore, at each
131

132 4. The two-nuc1eon system: numerical results
I SO NLO* -23.739 2.68 -0.52 5.3 -30
ISO NNLO* -23.739 2.68 -0.61 5.1 -29.7
3S 1 NLO* 5.420 1.753 0.110 0.73 -3.9
3S 1 NNLO* 5.420 1.753 0.057 0.66 -3.8
Table 4.5: Scattering lengths and range parameters for the S-waves at NLO and NNLO. The
corresponding LEes are obtained from the fit to a and r in the S-channel and from the first
effective range parameter in the moment um expansion of E I , as described in text.
order the amplitude is independent on the renormalization scale. Let us now comment on the
results presented in ref. [211]. Imposing the constraints provided by the perturbative solution of
the renormalization group equations (RGE's) and requiring that unphysical poles are removed
from the amplitude at low momenta leave one free parameter at NLO and two parameters at
NNLO for the 1 So and 3 SI channels, that were fixed by a fit to the phase shifts of the Nijmegen
PSA. For the 1 So phase shift one observes a visible improvement when going from LO to NLO and
from NLO to NNLO. At NNLO, the ISO phase shift goes very close to the one from the Nijmegen
PSA up to cms momenta rv 300 MeV (E1ab rv 192 MeV), see fig. 3 in [211]. Such visible agreement
of the phase shift with the data does, however, not yet allow to conclude about the quality of the
results. For example, the authors of ref. [211] point out that the the 1 So phase shift at the cms
momentum p = Mn is offits experimental value4 by 17% at NLO and by less then 1% at NNLO. On
the other side, the effective range expansion (4.26) with the first two coefficients (a and r) yields at
p = Mn the value for the phase shift, which differs from the experimental one by only rv 2%. Thus,
this information is not enough to definitely conclude about improvement of the results obtained
from the effective theory with explicit pions relative to the calculations within the pionless theory.
A better testing ground for that is given by the shape parameters in the effective range expansion
(4.26), which are expected to be sensitive to the pion physics, as discussed above, see also [100],
[101]. At NLO, predictions for V2,3 , 4 totally disagree with the values derived from the Nijmegen
PSA. The NNLO predictions for r and the v's, given in ref. [211] are r = 2.63 fm, V = 2 -1.2
fm3, V = 3 2.9 fm5 and V4 = -0.7 fm7, which still significantly differ from the experimental values
shown in table. 4.4 and are considerably worse than our predictions indicated in the same table.
Such dis agreement with the data is surprising since performing NNLO calculations without pions
and choosing a finite cut-off of the order of Mn , see sec. 2.2, one expects to be able to reproduce
the first four effective range parameters (a, r, V2 and V3 ) exactly. Therefore, no clear improvement
relative to the pionless theory can be observed.
For the 3 SI channel the situation turns out to be much worse than for the 1 So channel for the
KSW scheme. Whereas the phase shift is described quite accurately at NLO, NNLO corrections
are large and destroy the agreement with the data already at the cms momenta of the order of
Mn [211]. This is shown in fig. 4 of that reference. The failure of EFT at NNLO is found to be
due to large contributions resulting from the iteration of the pion exchanges, which is missed in
the KSW approach.
We would also like to comment on the recent work by Hyun et al. [106]. There, the 1 So channel is
considered within an approach similar to ours. In particular, the authors of this reference consider
the potential, which consists of the OPE and leading TPE contributions given in eqs. (4.1), (4.2)
4 As usual, we consider the values of the phase shifts from the Nijmegen PSA as experimental data.

4.3. Phase shifts 133
and a series of contact interactions. The obtained results are similar to ours at NLO. A furt her
important observation made in this work is that the inclusion of the NNLO contact interactions
alone does not allow to improve predictions for the phase shift relative to the NLO calculation.
As is demonstrated in our work, only the inclusion of the complete NNLO corrections (4.3) to the
potential given by the subleading TPE contribution and the contact terms allows to come closer
to the Nijmegen PSA.
Finally, let us briefly comment on the earlier work by Ord6nez et al. [76J. They performed the
LO, NLO and NNLO calculations5 within the potential approach using time-ordered perturbation
theory to obtain the N N force. The Cl, 3 ,4 couplings were not taken from 1f N scattering but instead
considered as free parameters. Furthermore, the anti-symmetrization of the potential was not
performed. As a consequence, many additional redundant parameters corresponding to contact
interactions enter the expressions for the effective potential. A global fit in all lower partial waves
has been performed to fix the 26 parameters at NNLO. This makes it difficult to separate the
effects on the phase shifts of the individual contributions to the potential. The results for the 1 So
and 3 SI phase shifts shown in figs. 6 and 7 in this paper are of the same quality as ours (our 1 So
partial wave looks slightly better at intermediate energies). We can not compare the predictions
for the effective range parameters, since no corresponding analysis has been done in ref. [78J.
4.3.2 P-waves
In fig. 4.6 we show the corresponding partial waves together with the mixing parameter EI for
the best global fit. In some cases, the differences between NLO and NNLO are modest, in 1 PI
and 3 PI NLO is even somewhat bett er. That means that the chiral TPEP is too strong in these
phases. Note also that in the 3 PI phase OPEP is dominant. Thus, the inclusion of the contact
interaction does not lead to a visible change. In 3 P2, NNLO is still too strong but the prediction is
considerably better than the NLO one. The energy dependence of EI is fairly precisely described
at NLO and NNLO. These results are visibly better than the ones obtained in ref. [78J or in
refs. [210J and [211], the latter being a NNLO calculation in the KSW scheme. This is shown
in detail in fig. 4.5, where EI is plotted versus the cms nucleon moment um (p < 350 MeV) in
comparison to the Nimegen PSA and the results from ref. [210J. The most important difference
between our approach and the KSW one is, as already pointed out, that in the last one the pions
are treated perturbatively. Thus, our results might be considered as an indication that in the
two-nucleon system, pions have to be treated non-perturbatively (if one intends to describe data
above p� 150 MeV).6
In the 1 PI and 3 Po channels the phase shift at leading order (iterated OPE without contact
interactions) describes the data only at very low energies. Also, the phase shifts are sensitive to
the choice of the cut-off. This is because no contact interactions appear in the potential at this
order, that could compensate this cut-off dependence. Therefore, the LO approximation in these
channels should not be taken seriously. In fig. 4.7 we show apart from the LO and NLO phase
shifts also the curve, which results if one adds the contact term with two derivatives to the OPE
potential. This is, certainly, not the complete NLO potential, since the leading TPE is missing.
The results for such incomplete and complete NLO potentials are similar and very much improved
relative to the LO calculation.7 Thus, we conclude that this improvement when going from LO
5 They also included the leading effects of intermediate D.-excitations.
6 This is, however, not the only difference between the two schemes. For example, in the KSW formalism the
unitarity of the S-matrix is only given perturbatively.
7 A visible closeness of the phase shifts from the incomplete NLO and from NNLO is accidental.

134
10
8
4
2
o o
• Nijmegen PSA
NNLO
NLO
LO
NNLO (KSW)
NLO (KSW)
0.1
4. The two-nuc1eon system: numerical results
/
/
/
/
"
"
0.2
P [GeV]
/
/
/
/
/
/
"
"
I
I
I
I
I
/
/
/
/
/
/
/
J
//1
Figure 4.5: Predictions for the mixing parameter 1:1 for nucleon cms momenta p
below 350 MeV. The short-dashed, long-dashed and solid curves represent our LO,
NLO and NNLO results, in order. For comparison, the NLO [91J and NNLO [210J
results in the KSW scheme are also shown. The filled squares depict the Nijmegen
PSA results.
to NLO is not due to inclusion of the leading TPE but mainly because the leading short range
contact term with two derivatives is taken into account.
The P-waves have also been recently calculated in the KSW scheme, see fig. 9 of ref. [211J. The
leading non-vanishing contributions to the phase shifts come out in this approach at NL08 from
the (non-iterated) OPE and the first corrections (NNLO) correspond to a single iteration of the
OPE. The contact interactions with two and more derivatives are expected to contribute at higher
orders. No free parameters appear in the calculation of the phase shifts in these channels. The
results shown in ref. [211 J should, in principle, be compared to our LO calculations. The difference
is that we iterate the OPE infinitely many times (and not just one time as in ref. [211 J). Comparing
the NLO and NNLO phase shifts presented in [211 J with each other and with the phase shifts
shown in fig. 4.6 we conclude, that the one-pion exchange becomes non-perturbative in spin triplet
channels at momenta comparable with M7r • As already stressed before, one needs to include the
leading short range effects to obtain a reasonable description of the phases at intermediate energies
in the 1 P1 and 3 Po channels. This observation is also confirmed by the analysis of ref. [211],
8 At leading order these phase shifts are zero.
0.3
I

4.3. Phase shifts
I PI [deg]
0 _---,- -_. .. .- -,_-_ __._-_...
.-_ ____,
-5
-10
-15
-20
-25
-30
-35 '---
o 0.05 0.1 0.15 0.2 0.25 0.3
3 PI [deg]
o �---,- --. .. .- -,- -- --.- --. .. .- -- -- -,
-5
-10
-15
-20
-25
-30
-35
-40 '---
o 0.05 0.1 0.15 0.2 0.25 0.3
--'- -- -- -'- --'- -- --'- -- -- -'- -- -'
--'- -- --'- --'- -- --'- -- --'- -- -- -'
EI [deg]
3 Po [deg]
135
80 r--,-
70
60
50
40
30
20
10
o
-10
-20
14 .--- -,- --,- -,- -�- -�-�
12
10
8
6
4
2
-30 '---
o 0.05 0.1 0.15 0.2 0.25 0.3
3 P 2 [deg]
60 ,--,- --,- -,- -,- --,- -,
50
40
30
20
10
0.05 0.1 0.15 0.2 0.25 0.3
o
//

136 4. The two-nucleon system: numerical results
0
-5
-10
I PI
-1 5 , , , " , , ,
-20
-25
-30
-35
0 0.05 0.1 0.15
[deg]
- - ------- -----
,
,
,
,
,
'0'" ,
,
,
'-

4.3. Phase shifts 137
12
10
1 D2 [deg] 3Dl [deg]
, ,
, ,
�,
, , -10 '
, ,
��':--''':,:::-...
8
, )v
, -15
', ........ , ........ '
/ ,
, , -20
"::�:-. ..........
6
, , ,
, ,
,
, , ,
-25
,
, , , , , , , / , ,
4 , , , , -30
,
, , , , ,
, , ,
,
, ,
-35
2 , / -- -------
, , , ,
- - -- -40
t":'-"'.
0 -45
0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3
3D2 [deg] 3D 3 [deg]
40 6
35 4 0
30
25 //"

138 4. The two-nuc1eon system: numerical results
20
.-.
950 MeV
.... 15
0>
Q)
::2..
10
(\)
0
875 MeV
800 MeV
• Nijm PSA
•
I 5
0
--.J
(\)
z 0
-5
20
0 0.1 0.2 0.3
.-. ....
0>
15
950 MeV
875 MeV
Q)
::2..
10
(\)
800 MeV
• Nijm PSA •
0
�
I 5
0
--.J
C')
z 0
-5
- --- ------------::.= -
0 0.1 0.2 0.3
E1ab [GeV]
Figure 4.9: The 1 D2 partial wave at NNLO for three different values of the
cut-off A (upper panel) and at N3LO (lower panel). The diamonds are the
result of the Nijmegen partial wave analysis.
- -
- -

4.3. Phase shifts 139
-0.5
1 F3 [deg]
3 F2 [deg]
O �--,- --�- --.- -- -.- --�- --. 6 .---�- -- -.- --�- --.- -- -�- --.
-1
-1 .5
-2
-2.5
-3
-3.5
-4
-4.5
o
-1
-2
-3
-4
'\:" '\"-,:,
"�·

140 4. The two-nucleon system: numerical results
More precisely, we have one independent parameter in each D-wave. Thus the cut-off independence
will be restored by the running of the corresponding LECs. To illustrate this, we show
in the lower panel of fig. 4.9 the partial N3LO results for the 1 D2 channel. The corresponding
potential consists of the NNLO terms plus one N3LO contact interaction. As expected, the cut-off
dependence of the phase shift is very much reduced compared to the NNLO result.9 Of course, this
illustrative example can not substitute for a complete N3LO calculation, but one should expect
very similar results. Note that according to our findings, the NNLO potential in all the D-wave
channels is not weak enough to be treated perturbatively, as it has been done in ref. [108]. The
potential has to be iterated to all orders in the LS equation. Only then one obtains a reasonable
description of the phase shifts in these partial waves. Concerning the F-waves, which are shown
in fig. 4.10, 1 F 3 and 1: 3 are well described, whereas the NNLO TPEP is visibly too strong in 3 F2,
3 F 3 , and 3 F4• This can be cured at higher orders by contact interactions. More precisely, a N3LO
calculation should be sufficient. Our phases in the F-waves look very similar to those shown in
ref. [108]. Consequently, a perturbative treatment of the potential in these channels is justified.
The leading non-vanishing contributions to the phases in the D-waves in the KSW sheme result
at NLO from the non-iterated OPE.1° The first corrections come out from a single iteration of
the OPE at NNLO. The results for the 1 D2 and 3 D2 phases shown in fig. 10 of [211] are rather
elose to our LO approximation. In the case of the 1 D2 channel, the OPE term projected onto the
corresponding partial wave is quite weak, so that iterating the potential in the LS equation does
not improve the result obtained with the Born approximation. For the 3 D2 partial wave, iteration
of the OPE is more important. The NNLO result of ref. [211] improves the Born approximation
significantly, see fig. 10 in [211], and is elose to our LO prediction. In the case of the 3 D 3 partial
wave, the authors of [211] observe a good agreement with the data at NNLO, i.e. with the Tmatrix
given by sum of the OPE and its single iteration. According to our analysis, we stress
that this agreement might be fortituous, because the terms resulting from furt her iterating the
OPE are of the same size as those ones ineluded in the NNLO calculation in [211]. In particular,
after summing up an infinite series of the iterated OPE we obtain the phase shift, see LO result
in fig. 4.8, which is totally different to the NNLO calculation of [211]. Furthermore, as already
pointed out above, the dominant effect in this channel is due to the correlated two-pion exchange,
which starts to contribute in the KSW scheme at higher orders and is not incorporated in the
NLO and NNLO calculations in [211].
Comparing our results to the ones obtained by Ord6iiez et al. [78] we note that our predictions for
the 1 D2 and 3 D 1 phase shifts and for the mixing parameter 1:2 are slightly better at intermediate
energies, whereas those one for 3 D2 are a somewhat worse. Unfortunately, the 3 D 3 partial wave
is not shown in ref. [78].
4.3.4 Peripheral waves
In figs. 4.11, 4.12 and 4.13 we show the G-, H- and I-waves together with the mixing parameters
1:4,5,6' These partial waves were first discussed in detail by the Munich group [108]. Their calculation
was perturbative and based on dimensional regularization of the TPE graphs. However,
for these partial waves the iteration becomes unimportant and our findings confirm their results.
The description of IG4, 3G 3 , 3G5, 3 H5, 3 H6, 3 h and 3 h is visibly improved by the NNLO TPEP.
Only in 116 the NLO result is better than the NNLO one. Of course, for the peripheral partial
9The situation here is very much similar to the one with the 1 H and 3 Po phases at leading order considered in
the preceding section.
10
As in the case of the P-waves, these phase shifts are zero at leading order.

4.3. Phase shifts 141
I G4 [deg]
2 °
1.8 -0.5
1.6
1.4 -1 .5
1.2
1 0
0
.-- ;---"
-1
-2
-2.5
3 G3 [deg]
0,8 -3 '�:.' �
- - - -
"
,
"
,
,
,
'�" d>
"�;��" 0
.
0,6
-3,5
0.4
-
.... .... . -:.
-:.",:,-::- '
-4
'���"
"'�§'-
0,2 -4.5
"
�'� "
' , ° -5
:>. ,
° 0.05 0,1 0, 15 0.2 0,25 0.3 ° 0,05 0.1 0,15 0,2 0,25 0,3
3 G4 [deg]
8 0,2
7 .' - °
6
:J>o-;:/
-;>;�
.;.
5 / -0.4 '- , :.�- ,
---
. .;;;:.",
-0.2
4 ,p:'-"'" -0.6
,/
"
3 -0.8
2 -1
1 -1 .2
3 G5 [deg]
, , , ,
0
, , ,
,
,
, , , , ,
,
,
,
,
,
,
,
,
, , ,
, , , ,
° -1.4
0 0.05 0,1 0.15 0.2 0.25 0.3 ° 0,05 0.1 0.15 0.2 0.25 0.3
-0.2
-0.4
-0,6
-0.8
E4 [deg]
O �::;c- -.- -- -.- -- --,- -.- -- -.- -- -,
-1
-1.2
-1 .4
'--_-'--_--'-_---L __ -'--_-'-_-'-'
-1 .6
° 0.05 0.1 0.15 0.2 0.25 0,3
Figure 4.11: Predictions for the G-waves and the mixing parameter E4 (in degrees) for nucleon
laboratory energies E1ab below 300 MeV. For notations, see fig. 4.4.

142 4. The two-nuc1eon system: numerical results
-0.2
IH5 [deg]
0 0.6
-0.4 0.5
3H4 [deg]
-0.6 0.4 , ,f«'
-0.8 , //
0.3 ,4'
-1
,
, ,/'
-1.2 "''''', 0.2 ,�
.
,
"'

4.3. Phase shifts
0.45 .---.,.- -- -,- -- -- -r- -.,- --r- --,
0.4
0.35
0.3
0.25
0.2
0.15
0. 1
0.05
.l. .. .- -- --'- -- -- -'- --'- -- --'- -- -'
O ��...
o 0.05 0.1 0.15 0.2 0.25 0.3
2.5 r--- -,- -_,- -.- -- -,- -- -,- -- -,
2
1.5
0.5
0 �4=::.
o 0.05 0.1 0.15 0.2 0.25 0.3
0
-0. 1
-0.2
-0.3
-0.4
-0.5
-0.6
0 0.05 0.1
.. .L. .- -- -l.- --'- -- --'- -- -'- -- -- -'
E6 [deg]
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
143
o ��=---.-�- -�-�- -,
-1
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
-0.35
o 0.05 0.1 0.15 0.2 0.25 0.3
O ���-_r-_,--.-
-._- -.
-0.4 "--_ _'___-'-_ ___'_ _ __'_ __ L----I
o 0.05 0.1 0.15 0.2 0.25 0.3
0.15 0.2 0.25 0.3
Figure 4.13: Predictions for the I-waves and the mixing parameter E6 (in degrees) for nucleon
laboratory energies E1ab below 300 MeV. For notations, see fig. 4.4

144 4. The two-nuc1eon system: numerical results
waves OPE does already a fairly good job, but the improvement in some of these phases due to
the NNLO TPEP clearly underlines the importance of chiral symmetry in a precise description of
low-energy nuclear physics.
The most interesting situation in the peripheral partial waves appears for the 3 G5 phase,u In
these channel the OPE and the leading TPE are not sufficient to describe the phase shift correctly
at energies higher than 100 MeV and the largest discrepancy of our NLO predictions with the
data is observed. Adding the subleading TPE exchange (NNLO potential) that incorporates the
leading effects of the heavier meson exchanges and intermediate ..6.-excitations, which is hidden in
the values of the couplings Cl,3,4 allows to improve the NLO result considerably and to obtain an
excellent parameter free and cut-off independent description of the phase shift. Performing the
calculations with the NNLO-Ll potential we were even able to separate the leading effects of the
..6.'s. We will comment on that later on. It is comforting to have such a clear indication that the
values for Cl,3,4 fixed from 7f N scattering are consistent with N N calculations.
4.4 Deuteron properties
We now turn to the bound state properties. At NNLO (NLO), we consider the exponential
regulator (4.13) with n = 2 and A = 1.05 (0.60) GeV, which reproduces the deuteron binding
energy within an accuracy of about one third of a permille (2.5 percent).We make no attempt to
reproduce this number with better precision.12 The results for the phase shifts, which correspond
to these values of the exponential regulator, are very similar to those obtained with the sharp
cutoff A = 0.875 (0.50) GeV. For completeness, we list in table 4.6 the values of the coupling
constants in the 3 SI - 3 Dl channel corresponding to the exponential regulator.
NLO -0.0363 0.186 -0.190
NNLO -14.497 15.588 -4.358
NNLO-..6. -8.637 7.264 -0.447
Table 4.6: The values of the LECs as determined from the 3 SI _ 3 Dl channel. We use an
exponential cut-off with A = 0.6 GeV and 1.05 GeV at NLO and NNLO (NNLO-..6.), respectively.
The LEC 0 3 5 1 is in 104 GeV-2 , while the others are in 104 GeV-4 . The parameters of the NLO,
NNLO and NNLO-..6. potentials are obtained from fitting to the Nijmegen PSA.
In table 4.7 we collect the deuteron properties in comparison to the data and two realistic potential
model predictions (the pertinent formulae are given in app. H). We give the results for NLO and
NNLO. We note that the deviation of our prediction for the quadrupole moment compared to
the empirical value is slightly larger than for the realistic potentials. The asymptotic D / S ratio,
called ry, and the strength of the asymptotic wave function, As, are weIl described. The D-state
probability, which is not an observable, is most sensitive to small variations in the cut-off. At
NLO, it is comparable and at NNLO somewhat larger than the one obtained in the CD-Bonn or
ll Note that the 3Zj=I+I-phases show, in general, the largest effects when going from NLO to NNLO and thus are
most sensitive to the short range physics. The three extreme cases are the 1 P 2 -, 3 D 3 - and 3G5-waves. We assurne
some sort of cancelation in the potential in these channels, wh ich leads to a suppression of the OPE and the leading
TPE contributions.
12Note that the deuteron binding energy is not used to fit the free parameters in the potential.

4.4. Deuteron properties 145
the Nijmegen-93 potential. This increased value of PD is related to the strong NNLO TPEP. At
N3LO, we expect this to be compensated by dimension four counterterms. Altogether, we find
a much improved description of the deuteron as compared to ref. [78], where the binding energy,
magnetic moment and quadrupole moment were used in fits. Our results are almost as precise as
the ones obtained in the much more complicated and less systematic meson-exchange models.
11 Ed [MeV]
Qd [fm2]
77
rd [fm]
As [fm-l/2]
PD[%]
NLO 1 NNLO 1 NNLO-Ll 11 Nijm93 1 CD-Bonn 11
-2.1650 -2.2238 -2.1849 -2.224575 -2.224575
0.266 0.262 0.268 0.271 0.270
0.0248 0.0245 0.0247 0.0252 0.0255
1.975 1.967 1.970 1.968 1.966
0.866 0.884 0.873 0.8845 0.8845
3.62 6.11 5.00 5.76 4.83
Exp.
-2.224575(9)
0.2859(3)
0.0256(4)
1.9671(6)
0.8846(16)
-
Table 4.7: Deuteron properties derived from our chiral potential compared to two "realistic"
potentials (Nijmegen-93 and CD-Bonn) and the data. Here, rd is the root-mean-square matter
radius. An exponential regulator with A = 600 MeV and A = 1.05 Ge V at NLO and NNLO
(NNLO-Ll), in order, is used.
It is also interesting to compare our findings with the results reported by Park et al. [105]. There,
np scattering in the 3 SI-3 Dl channel as well as various deuteron properties are investigated. The
potential considered in this work consists of the leading OPE plus contact interactions without
and with two derivatives. To fix three free parameters corresponding to these contact terms the
authors of [105] use the deuteron binding energy and the experimental values for As and 77. For
the quadrupole moment they obtain values from 0.261 to 0.274 fm2 depending on the choice of
the cut-off. The D-state prob ability varies from 3.16% to 5.39%.
Kaplan et al. have recently reported on the NLO calculation of the deuteron charge radius and
the quadrupole moment in the KSW scheme [92]. They found analytic expressions for these
quantities. Having fixed three free parameters from the 3 SI phase shift, deuteron binding energy
and magnetic moment, they made the following predictions for the deuteron charge radius and
the quadrupole moment: reh = 1.89 fm 13 and Qd = 0.40 fm2. The large discrepancy for the
quadrupole moment is because this quantity is zero at LO in this formalism. Thus, the (formally)
NLO correction for Q d yields the first non-vanishing contribution.
The coordinate space S- and D-state wave functions obtained in our approach are shown in
fig. 4.14. At NLO they look qualitatively quite similar to the ones obtained from various potential
models. At NNLO one obtains a lot of structure in the wave functions below 2 fm. This is because
two additional spurious (unphysical) very deeply bound states appear in the 3S1 _3 Dl channel.
The binding energy of these states varies strongly by changing the cut-off. For the exponential
regulator with A = 1.05 GeV we get binding energies of EI = 47.1 GeV and E 2
= 2.5 GeV,
respectively. This values correspond to center-of-mass momenta of about a few Ge V which is
clearly out of the applicability range of the low-momentum effective theory. Furthermore, because
of such huge values of the binding energy these unphysical states obviously do not influence physics
in the energy region below 350 MeV that we are interested in and can, in principle, be integrated
1 3 The experimental value of the charge radius is 2.1303±O.0066 fm [212].

146
---
.... .
�
-
---
.... .
.... .. .. .
:::J
0.5
0
0.7
0.2
-0.3
0
o
/
/'
1
1
/" --
/ " "
---
2 3
2 3
4
4. The two-nuc1eon system: numerical results
- - -
5
S-wave
--- D-wave
- - - - - - --
6 7 8 9
S-wave
--- D-wave
- - - - - - - - - - - --- -
10
4 5 6 7 8 9 10
r [fm]
Figure 4.14: Coordinate space representations of the S- (solid line) and Dwave
(dashed line) deuteron wave functions at NLO (upper panel) and NNLO
(lower panel).

4.4. Deuteron properties
---
.... .
---
3:
�
---
.... .
---
:::J
---
.... .
---
3:
---
.... .
---
:::J
2
1 .5 S-wave
1
0.5
0
-0.5
-1
2
.... .. .
-... .. .
./
-... .. .
./
--- - ---
- - -
- - - D-wave
0 1 2 3
1.5 S-wave
1
0.5
0
-0.5
-1
�
- - - D-wave
0 1 2 3
r [fm]
Figure 4.15: Coordinate space representations of the S� (solid line) and D�
wave (dashed line) for the two unphysical boundstates.
147

148
�
....
---
�
�
....
---
::J
0.7
0.5
0.3
0.1
-0.1
-0.3
0.2
0. 15
0.1
0.05
0
-0.05
0
1 2
r\
I >-�A"
I '\
I / ""
I I '
I I "�
I �
I
I ,�
I I
.... �
f /
.1..- '
3
4 5
o 1 2 3 4 5 6
r [fm]
4. The two-nuc1eon system: numerical results
_____________ 0 CD-Bonn
NNLO
6 7 8 9 10
--- NNLO
---- CD-Bonn
- --
- ---. ._ --
7 8 9 10
Figure 4.16: Coordinate space representations of the S- (upper panel) and
D-wave (lower panel) deuteron wave functions at NNLO compared to the one
from the CD-Bonn potential.

4.5. Results for the NNLO-f:. approach 149
out.14 A very similar situation for the spurious states happens at NNLO also in other S-, P- and
D-waves. Note that these unphysical states are purely short range effects: as one can see from the
fig. 4.15 the wave-functions corresponding to such states become negligable for distances above 2
fm. The corresponding root-mean-square matter radii are (ri)1/2 = 0.27 fm and (r�)1/2 =
0.40 fm.
For separations above 2 fm the NNLO deuteron wave-function is very elose to the one obtained
with the CD-Bonn potential.15 This is shown in fig. 4.16. For a discussion on such deeply bound
states in effective field theories, see ref. [141]. We end this paragraph with the following remark:
According to Levinson's theorem, the difference between the phase shift at the origin and at
infinity is given by mf, with n the number of bound states. Thus, the phase shifts in the S-, Pand
D-waves should become unphysical at large energies. This is, however, of no relevance for
the EFT since we do not attempt to correctly reproduce (or predict) the phase shift behavior for
all energies (from threshold to infinity).
4.5 Results for the NNLO-ß approach
We would like now to discuss the results obtained with the NNLO-f:. potential. While the LECs
C 3 and C4 are dominated by the f:., see ref. [200], most of the correlated two-pion exchange is
parametrized in Cl . Here, we are mostly interested in investigating the role of the f:. in all partial
waves and therefore keep the cut-off A fixed at 875 MeV but refit the LECs C i (see table 4.1 for
their values). More precisely, a best global fit leads to a very similar cut-off value as in NNLO.
Note that, as expected, the precision of the fits is better than NLO in the theory without f:. but
somewhat worse than the corresponding NNLO fits. This is due to the absence of contributions
from higher mass states encoded in the LEes C 1,3 ,4 not present in the NNLO-f:. approach discussed
here. We again point out that it would be interesting to calculate the NNLO corrections with
explicit f:. in the framework of the EFT expansion as detailed in ref. [201]. This would also require
refitting of the LECs C1, 3 ,4' In fact, a study of pion-nueleon scattering in that framework is not
yet available and thus the corresponding LECs are not determined.
Let us now discuss the results of the NNLO-f:. approach. Formally, we follow the Munich
group [109] (for details, see sec. 3.8.3). Again, it is important to stress that we iterate our
potential to all orders. We refrain from showing all partial waves but rat her discuss some particular
examples, collected in fig. 4.17. The two S-waves shown in that figure are not very different
from the NNLO result, although the description of 3 Sl is slightly worse at higher energies. All
P-waves are very similar in NNLO and NNLO-f:., the most visible difference appears in E1, as
can be seen in the figure.
The most dramatic effects appear in the D-waves. This is expected since these are parameter-free
predictions and we had already pointed out the cut-off sensitivity in sec. 4.3.3. Interestingly, the
description of 1 D 2 is almost identical in the two approaches, consequently any important isobar
effect in this partial wave can be weIl represented by contact interactions with their strength given
by the coupling of the f:. to the 7f N system. In 3 D 3 (also shown in fig. 4.17) the absence of the
scalar-isoscalar two-pion correlations is elearly visible. Our result thus confirms a finding made
14 As soon as one is dealing with only the two-nucleon system there is no need to integrate out such unphysical
states, since this does not modify the low-energy observables. We refrain here from the discussion of the complicatians
which may arise in three- and more-body calculations due to such spurious states. Note, however, that
accarding to aur power counting one has an additional three-body force at NNLO which possibly can compensate
the effects of the spurious states.
1 5 We would like to thank Hiroyuki Kamada for supplying us with the deuteron wave-function calculated with
the CD-Bonn potential.

4.5. Results for the NNLO-ß approach 151
already in the Bonn potential, namely that this particular partial wave is essential dominated by
correlated TPE [213]. We also note that the description of 3 D1 and 3 D2 is worse in NNLO-ß than
in NNLO. For the F -waves, the differences between the two approaches are very small, with the
exception of 3 F4, which is improved in the presence of the isobar. The most significant effect in
the higher partial waves shows up in 3G5 as shown in fig. 4.17. Clearly, two-pion correlations not
related to the ß play an important role to bring the prediction for this partial wave in agreement
with the data. The deuteron properties for an exponential regulator with A = 1.05 Ge V are listed
in table 4.7. Most observables come out in between the NLO and NNLO results for the NNLO-ß
approach. The sole exception is the quadrupole moment, which is improved.
An important conclusion following from comparison of the NNLO and NNLO-ß results is that
simply using resonance saturation to encode the physics of the isobar in the dimension two pionnucleon
LECs turns out to be a sufficient procedure in the two-nucleon sector. We also note that
explicit isobar degrees of freedom were considered in ref. [78]. A direct comparison with their
work is, however, not possible since no separate result for fits with and without delta are given.

Chapter 5
Isospin violation in the two-nucleon
system
In this chapter we would like to discuss charge symmetry and charge independence breaking in
an effective field theory approach for few-nucleon systems. In particular, we will concentrate on
the 1 So channel, in which the largest isospin violating effects are observed. Further , we will use
the formalism proposed by Kaplan, Savage and Wise (KSW) [91] throughout this chapter.
It is weIl established that the nucleon-nucleon interactions are charge dependent (for a review, see
e.g. [216] ). For example, in the ISO channel one has for the scattering lengths a and the effective
ranges r (n and p refers to the neutron and the proton, respectively)
.6.rcIB
1
2" (ann + app) - anp = 5.7 ± 0.3 fm ,
1
2" (rnn + rpp) - rnp = 0.05 ± 0.08 fm . (5.1)
These numbers for charge independence breaking (CIB) are based on the Nijmegen potential
and the Coulomb effect for pp scattering is subtracted based on standard methods.1 The
charge independence breaking in the scattering lengths is large, of the order of 25%, since
anp = (-23.714 ± 0.013) fm. In addition, there are charge symmetry breaking (CSB) effects
leading to different values for the pp and nn threshold parameters,
.6.acsB
.6.rcsB
app - ann = 1.5 ± 0.5 fm ,
rpp - rnn = 0.10 ± 0.12 fm . (5.2)
Both the CIB and CSB effects have been studied intensively within potential models of the
nucleon-nucleon (NN) interactions. In such approaches, the dominant CIB comes from the charged
to neutral pion mass difference in the one-pion exchange (OPE), rv .6.agrBE 3.6±0.2 fm. Additional
contributions come from 17f and 27f (TPE) exchanges. Note also that the charge dependence in the
pion-nucleon coupling constants in OPE and TPE almost entirely cancel. In the meson-exchange
picture, CSB originates mostly from p - w mixing, .6.a�s� rv 1.2 ± 0.4 fm. Other contributions due
to 7r - 7], 7f - 7] ' mixing or the proton-neutron mass difference are known to be much smaller.
Within QCD, CSB and CIB are of course due to the different masses and charges of the up and
down quarks. Such isospin violating effects can be systematically analyzed within the framework
of chiral effective field theories. In the two-nucleon sector, a complication arises due to the
1 In particular, the magnetic inter action had also been to be considered.
152

5.1 Isospin violating effective Lagrangian 153
unnaturally large S-wave scattering lengths. This can be dealt with in different manners. One
is the approach proposed by Weinberg, in which chiral perturbation theory is applied to the
kernel of the Lippmann-Schwinger equation (effective potential). This approach is very similar
to the Hamiltonian formalism used in this work and explained in detail in chapter3. This route
was followed by van Kolck et al. , see refs [75], [74], [76], [78]. A different fashion to deal with
low-energy nucleon-nucleon scattering is the approach recently proposed by Kaplan, Savage and
Wise (KSW) [91].2 Essentially, one resums the lowest order local four-nucleon contact terms rv
Co (Nt N)2 (in the S-waves) to generate the large scattering lengths and treats the remaining effects
perturbatively, in particular also pion exchange. This means that most low-energy observables
are dominated by contact interactions. The chiral expansion for NN scattering entails a new
scale ANN of the order of 300 MeV, so that one can systematically treat external momenta up
to the size of the pion mass. There have been suggestions that the radius of convergence can
be somewhat enlarged [88], but in any case ANN is considerably sm aller than the typical scale of
about 1 Ge V appearing in the pion-nucleon sector . For recent calculations of various properties of
the two-nucleon system within this formalism see e.g. refs. [87], [88], [92], [95], [98], [210], [211]. In
this context, it appears to be particularly interesting to study CIB (or in general isospin violation)
which is believed to be dominated by long range pion effects. That is done here.3 First, we write
down the leading strong and electromagnetic four-nucleon contact terms. It is important to note
that in contrast to the pion or pion-nucleon sector, one can not easily lump the expansion in
small momenta and the electromagnetic coupling into one expansion but rat her has to treat them
separately.4 Then we consider in detail CIB. The leading effect starts out at order aQ-2, where Q
is the generic expansion parameter in the KSW approach. It sterns from OPE plus a contact term
of order a with a coefficient aE� l ) of natural size that scales as Q-2. Similarly, the leading CSB
effect results from contact terms with four nucleon legs of order a and order E == mu - md, which
also scale as Q-2. While in the case of E�l) this scaling property is enforced by a cancellation
of a divergence, the situation is a priori different for CSB. However, for a consistent counting of
all isospin breaking effects related to strong or electromagnetic (ern) insertions, one should count
the quark mass difference and virtual photon effects similarly. Note, however, that these CIB and
eSB terms are numerically much smaller than the leading strong contributions which scale as
Q-l because a« 1 and (mu - md)/Ax «1. The corresponding constants, which we call E�1,2),
together with the strong parameters (as given in the work of KSW) can be determined by fitting
the three scattering lengths a pp , ann, an p and the np effective range.5 That allows to predict the
moment um dependence of the np and the nn 1 So phase shifts. Based on these observations, we
can in addition give a general classification for the relevant operators contributing to eIB and
eSB in this scheme. Additional work related to long-range Coulomb photon exchange is necessary
in the proton-proton system. We do not deal with this issue here but refer to recent work using
EFT approaches in refs. [219] , [220], [221].
5.1 Isospin violating effective Lagrangian
Let us first discuss the various parts of the effective Lagrangian underlying the analysis of isospin
violation in the two-nucleon system. To include virtual photons in the pion and the pion-nucleon
2 There exist by now modifications of this approach and it has been argued that it is equivalent to cut-off schemes.
We do not want to enter this discussion here but rather stick to its original version.
3For a first look at these effects in an EFT framework (based on the Weinberg power counting) , see the work of
van Kolck [75]. Electromagnetic corrections to the one-pion exchange potential have been considered in [218].
4 This is because of the different power counting schemes in these cases.
5Whenever we talk of the pp system, we assume that the Coulomb effects have been subtracted.

154 5. Isospin violation in the two-nuc1eon system
system is by now a standard procedure [199], [222]-[227]. The lowest order (dimension two) pion
Lagrangian takes the form
(5.3)
with f7r = 92.4MeV the pion decay constant, V' ft the (pion) covariant derivative containing the
virtual photons, ( ) denotes the trace in flavor space. Further, X contains the light quark mass
matrix M. Speaking more precisely, X can be expressed in the most general case in terms of the
scalar and pseudoscalar external sour ces s and p,6 respectively, as follows:
X(x) = 2B(s(x) + ip(x)), (5.4)
where the constant B is defined in eq. (3.134). Clearly, one has to set p = 0, s = M to end up with
the usual QCD Lagrangian. Further, the last term in eq. (5.3) contains the nucleon charge matrix
Q=ediag(I,0),7 and leads to the charged to neutral pion mass difference, 8M2 = M;± - M;o,
via 8M2 = 87fcxC/ f;, i.e. C = 5.9· 10-5 GeV4. Note that to this order the quark mass difference
mu - md does not appear in the meson Lagrangian (due to G-parity). That is chiefly the reason
why the pion mass difference is almost entirely an electromagnetic (ern) effect. The equivalent
pion-nucleon Lagrangian to second order8 takes the form
Nt (iDo - g; a . ü) N + Nt { �� + Cl (X+) + (C2 - :�) U6 + C3UftUft C str 7rN
+ l (q + 4�) [O"i,O"j]UiUj + C5 (X+ - �(X+)) +., .} N , (5.5)
which is the standard heavy baryon effective Lagrangian in the rest-frame vft = (1,0,0,0).
Further, m is the nucleon mass and Uft the chiral viel-bein introduced in eq. (3.102), uft '"
-ÖftrP/ f7r + ... , The quantities X± are defined via
(5.6)
and transform covariantly under chiral rotation, i.e. via
90 I h h-l X± -"--'- -t X± = X± ,
(5.7)
where the compensator field h is defined in eq. (3.99). The contact interactions to be discussed
below do not modify the form of this Lagrangian (for a general discussion, see e.g. ref. [71]).
Strong isospin breaking is due to the operator '" C5. Electromagnetic terms to second order are
given by [226]:9
C;� = f;Nt {h (Q� - Q�) + hQ+(Q+) + h(Q� + Q�) + f4(Q+)2} N , (5.8)
6The extern al fields s and p are introduced in the QCD Lagrangian via the term -q(s - ip/5)q. The seal ar
sour ce s has already been discussed in sec. 3.2. The transformation properties of the p under charge conjugation,
parity transformation and chiral rotation can be obtained requiring invariance of the above term in the same
way as it has been done for s in sec. 3.2. In particular, one obtains: p ---+ pe = pT, P ---+ pP = -p and
p � pi = hLphj/ = hRph"[,l, where hR,L are the same matrices as in in eq. (3.98). Note further that the field p is
hermitian: pt = p.
70r equivalently, one can use the quark charge matrix e( � + T3 ) /2.
8 Here and in what follows we count the nucleon mass in the same way as the scale Ax ' Note furt her that this
Lagrangian was used in the context of the NNLO two-nucleon potential in sec. 3.8 as weil as in the discussion of
the three-nucleon interactions in sec. 3.9.
gOne of these four terms can be eliminated using the matrix relation given in ref. [227].

5.1 Isospin violating effective Lagrangian 155
with Q± = 1/2( uQut ± ut Qu) and A = A - (A) /2 projects onto the off-diagonal elements of the
operator A. It can be checked [226] that the Q± transform covariantly under chiral SU(2) v x
SU(2)A rotation, Le.:
(5.9)
Further, under parity and charge conjugation transformations one finds:
Q± ---+ Q� =
±Q�
. (5.10)
Evidently, the charge matrices always have to appear quadratic since a virtual photon can never
leave a diagram. The last two terms in eq. (5.8) are not observable since they lead to an equal em
mass shift for the proton and the neutron, whereas the operator rv h to this order gives the em
proton-neutron mass difference. In what follows, we will refrain from writing down such types of
operators which only lead to an overall shift of masses or coupling constants. We note that in the
pion and pion-nucleon sector, one can effectively count the electric charge as a small moment um
or meson mass. This is based on the observation that Mn / Ax rv e/...(47r = fo rv 1/10 since
Ax :::: 41ffn = 1.2 GeV. It is thus possible to turn the dual expansion in small momenta/meson
masses on one side and in the electric coupling e on the other side into an expansion with one
generic small parameter. We also remark that from here on we use the fine structure constant
a = e2 /41f as the em expansion parameter.
We now turn to the two-nucleon sector, i.e. the four-fermion contact interactions without pion
fields. Consider first the strong terms. The effective Lagrangian takes the form
Cf:tN
h(Nt N)2 + l 2 (NtiJN)2 + l 3 (Nt (X+)N)(Nt N) + l4(Nt X+N)(Nt N)
+ l5(Nt iJ(X+)N)(Nt iJN) + l6(Nt iJx+N) (Nt iJN) + . .. , (5.11)
w here the ellipsis denotes terms with two (or more) derivatives acting on the nucleon fields.
Similarly, one can construct the em terms. The ones without derivatives on the nucleon fields
read
CNN
Nt {r1 (Q� - Q�) + r2 Q + (Q +) } N (Nt N)
+ NtiJ {r3(Q� - Q�) + r4Q+(Q+)} N(NtiJN)
+ Nt {r5Q+ + r6(Q+)} N(Nt Q+N) + NtiJ {r7Q+ + r8(Q+)} N(Nt iJQ+N)
+ rg(Nt Q+N)2 + rlO(Nt iJQ+N)2 . (5.12)
There are also various terms resulting from the insertion of the Pauli isospin matrices i in different
Nt N binomials. Some of these can be eliminated by Fierz reordering (or anti-symmetrization, see
appendix E), while the others are of no importance for our considerations. Note that from now on
we consider em effects. The leading CSB rv mu - md has the same structure as the corresponding
em term and thus its contribution can be effectively absorbed in the value of E62) , as defined below.
We re mark since ANN is significantly smaller than Ax' it does not pay to treat the expansion in
the generic KSW moment um Q simultaneously with the one in the fine structure constant (as it is
done e.g. in the pion-nucleon sector ). Instead, one has to assign to each term a double expansion
parameter Qnam, with n and m integers. Lowest order charge independence breaking is due to
a term rv
(Nt
T3 N)2 whereas charge symmetry breaking at that order is given by a structure
rv (NtT3N)(NtN). In the KSW approach, it is customary to project the Lagrangian terms on the
pertinent NN partial waves. Denoting by ß the 1 So partial wave for a given cms energy Ecms, the

156 5. Isospin violation in the two-nuc1eon system
Born amplitudes for the lowest order eIB and eSB operators between the various two-nucleon
states takes the form
_ (�:)
_ (�:)
a (E�l) + E�2))
a (E�l) _
E�2))
(5.13)
where we will determine the coupling constant E�1,2) later on and also derive the scaling properties
of the E�;,2). The terms with the superscript '(1)' refer to eIB whereas the second ones relevant
for eSB are denoted by the superscript '(2)'. Higher order operators are denoted accordingly.
There is, of course, also a eIB contribution to the np matrix element. To be consistent with the
charge symmetrie calculation of ref. [91J, we absorb its effect in the constant D2, i.e. it amounts
to a finite renormalization of D2 and is thus not observable. In eq. (5.13), p = JmEcms is the
nucleon cms moment um.
5.2 Brief introduction into the KSW approach
We will now give a very brief introduction into the KSW formalism and repeat so me basic points of
ref. [91J. It is weIl known that the scattering lenghts a in both np S-channels take unnatural large
values. Let us first consider a pionless theory. In such a case one can calculate the two-nucleon
T-matrix analytically, see sec. 2.2, since all interactions in the effective Lagrangian are of contact
type. As discussed in section 2.2, the range of applicability of such an effective theory is zero
in the case of infinitly large scattering length, if ordinary dimensional regularization is used. In
what follows we will adopt the notation of ref. [91J and work with the amplitude A instead of the
T-matrix defined in eq. (2.6). The connection between the amplitude and the on-shell T-matrix
Ton (p) is gi yen by
= A( ) _ Ton(p) p 27r2 '
The tree level amplitude in the S-channels can be expressed as
Atree =
00
- L C2np2n ,
n=O
(5.14)
(5.15)
where C2n are some constants. Using dimensional regularization with the minimal subtraction
= scheme (MS), which amounts to subtracting poles in the physical dimension (D 4), one finds
the amplitude A in the form:
(5.16)
Here we have used the fact that the power law divergent integrals (like those one in eq. (2.49))
vanish after performing dimensional regularization. If the scattering length would be of a natural
size, i.e. of the order a rv 1/ A, where A is ascale entering the values of the remaining effective range
and shape parameters, which is comparable with the pion mass, one could rewrite the effective
range expansion (2.21) for the inverse T-matrix into the expansion for the amplitude A:
A =
�
An =
00
- ---;
7ra . ar 2 2 P
4 [ ( )
;: 1 -
zap
( 3)]
+ 2 - a p + 0
A 3 . (5.17)

5.2 Brief introduction into the KSW approach
Matching eq. (5.16) to eq. (5.17) yields the following scaling of the C's:
41f 1
C2n rv
mA A2n .
157
(5.18)
Consequently, the effective theory is perturbative: the leading order amplitude Ao = -Co is given
by the tree graph with Co at the vertix; Al = iCgmp/(41f) results from the one-Ioop diagram
with both Co-vertices and so forth.
In the case of a very large scattering length, lai » 1/ A, the expansion (5.17) has a very small
range of validity p < I/lai- Kaplan et al. proposed in that case to expand the amplitude in powers
of p/ A while retaining the factors ap to all orders:
A _ [ 1
_ 41f 1 r/2 2 (r/2)2 4 V 4
- 1+ P + P + 2 P + ... (5.19)
m l/a +ip l/a+ip (l/a +ip)2 l/a +ip
Therefore, for momenta p > I/lai the amplitude can be expressed as
00
A = L An ·
n=-l
(5.20)
As explained in ref. [91], using dimension regularization with the MS scheme does not lead to a
consistent effective theory for p > I/lai- This is also demonstrated in sec. 2.2. To deal with this
problem, the authors of [91] proposed to use dimensional regularization and the so-called power
divergence subtraction (PDS), i.e. to subtract the poles from the dimensionally regulated loop
integrals not only in the physical dimension, but also in the dimension one lower than the physical
one. This allows to take into account linear divergences. In a pionless effective theory one has to
deal with the (ultraviolet divergent) integrals of the form
(5.21)
where q == Igl and E = p2/m is the initial energy of the nucleons. Applying dimension regularization
with PDS leads tolO In --+ -(mEt (:) (ft+ip) . (5.22)
Chosing ft of the order of p, ft rv
couplings C 2n (ft):
p
» I/lai, leads to the following scaling properties of the
(5.23)
Further , each loop contributes a factor of p and the derivatives at vertices scale as p as weIl. With
these rules one can calculate the amplitude A to any order in the expansion (5.20). In particular,
the leading term A-I is given by sum of bubble diagrams with Co vertices; the next-to-Ieading
order term Ao results from dressing the C2-interaction to all orders by Co and so forth. Note
that since one uses a perturbative expansion for the amplitude A, the corresponding S-matrix is
unitary up to the calculated order. To find the phase shift one can use the relation
lOIn MS one would obtain the same result with /-l = O.
(5.24)

158 5. Isospin violation in the two-nucleon system
leading to
.. ..
.. ...
Figure 5.1: Resummation of the lowest order (NtN)2 contact terms ", Co.
1 ( . mp )
6 = 2i In 1 + 2 21f A .
+
• • •
(5.25)
Expanding both sides in the small moment um scale Q '" p (for the amplitude A one uses the
expansion (5.20)) one obtains:
(5.26)
where
1 ( mp )
60 = ---; In 1 + i-A-1 ,
22
21f
6 = 60 + 61 + ... ,
(5.27)
Up to now we have not yet incorporated pions into the effective theory. The tree contribution to
the amplitude due to the one-pion exchange is given by
2 -+
4f2 �2 M2
-+ -+ -+
gA 0"1 . q 0"2 . q
n q + n
Tl . T2 ,
(5.28)
which sc ales as 0(1) and thus occurs at the same order as an insertion of C2p2. Consequently,
pions can be treated perturbatively and the leading contributions to the amplitude due to the OPE
are given by graphs 2, 3 and 4 in fig. 5.2. The last graph in this figure depicts the contribution from
the contact interaction D2M;, which can be, in principle, absorbed by the Co vertex.u To study
at which energies such a perturbative treatment of pions would be no more adequate, the authors
of ref. [91] used the renormalization group technique. We will not repeat their arguments here
and refer the interested reader to the original publication. We only note that the scale entering
the low-momentum expansion of the amplitude turns out to be
161f l;
ANN = - 2 - '" 300 MeV .
gAm
This leads to the expansion parameter Q/ ANN '" Mn/ ANN '" 0.46.
(5.29)
To end this section we would like to write down the expressions for A�I)_A�V) obtained in ref. [91]:
11 However, one needs to keep this term explicitly in order to be able to perform consistent renormalization of the
dressed OPE in the fourth graph in fig. 5.2.

5.3 The leading eIB and eSB effects in the np 1 So channel 159
+ + 2
+ +
Figure 5.2: The subleading amplitude Ao in the isospin symmetrie ease. The shaded ovals are
defined in fig. 5.1. The filled square (diamond) eorresponds to the eontaet vertices with two
derivatives (one insertion of M;).
where the leading term in the expansion of the np I SO amplitude [91] is given by:
5.3 The leading CIB and CSB effects in the N N lS0 channel
(5.30)
(5.31)
Consider now the effeet of the eharged to neutral pion mass differenee 8M2, see the upper left
diagram in fig. 5.3. OPE between two neutrons, neutron and proton or two protons ean obviously
involve eharged and neutral pions. The mass differenee ean be treated in the following way: as
proposed in ref. [224], one ean modify the pion propagator,
(5.32)
with e the pion four-momentum and 'i,j' isospin indices. Sinee we are interested only in the
leading eorreetions rv 8M2 rv ü, it suffiees to work with the expanded form of eq.(5.32),
(5.33)

160 5. Isospin violation in the two-nucleon system
�AII
1,-2
.. I ..
$ + 2
.. ..
+ +
Figure 5.3: Relevant graphs contributing to charge independence breaking at leading order aQ -2.
The open (filled) circle denotes a pion mass insertion rv 8M2 (an insertion of the leading fournucleon
operators rv aEb 1 )). For charge symmetry breaking, the leading contribution is given by
the last diagram with the filled circle denoting an insertionrv aEb2) .
From this we conclude that OPE diagrams with different pion masses have the isospin structure
T(I)i��T(I)j and lead to CIB since
012 =
= (a + b) T(I) . T(2) - bT(31) T(3 2)
(ppIOdpp)cs
(npIOdnp)
(nnI012Inn) = a ,
+ (npI012Ipn) = a + 2b , (5.34)
for the various isospin components of the two-nucleon system and 'Cs' stands for Coulombsubtracted.
Obviously, these effects are of order aQ-2. The np amplitude was already calculated
by KSW. We have worked out the leading corrections �A = Ann - Anp = A�: - Anp due to the
pion mass difference. The pertinent diagrams are shown in fig. 5.3. We follow the notation of
KSW and call these corresponding three amplitudes Ai�_,.IiI,IV where the first (second) subscript
refers to the power in a (Q) and the superscripts to the first three diagrams of the figure. We find
r =
r [
4 � 2 In (1 + �;) - M; : 4p2] ,
(mMnA_l) [ 1 2p i ( 4p2) 1 + � ]
47r pMn Mn 2pMn M; M; + 4p2
r -arctan -+ -- In 1 +- _ 7T
r (mMnA_l) 2 [ i 2p 1 ( M; + 4p2) I 1 1 1 + � ]
47r M2 M 2M2 1/2 M2 2 M2 + 4p2
2
n n n ,.., n n
- arctan - - -- n + - - - ---- -c:------"---;o-
-8M2 �;; , 8M2 = M;± -M;o , M; = M;o + 28M2 , (5.35)
with A-1 == AO,-1 the leading term in the expansion of the np I SO amplitude [91], see eq. (5.31)
Here, J.l is the PDS regularization scale. Note while the diagrams II and III are finite, the corresponding
integral in the �A{v:..2 diverges logarithmically. Therefore, the Lagrangian must contain
,

5.3 The leading CIB and CSB effects in the np 1 So channel 161
a counterterm of the structure E61) (J.l)a(Nt T3 N)(Nt T3 N) (cf. sec. 5.2) since it is needed to make
the amplitude scale-independent. Note that for the graph IV we have used the same subtraction
as performed in [91] . Consequently, for operators of this type with 2n derivatives we can establish
the scaling property E�� rv Q-2+n. This does not contradict the KSW power counting for the
isospin symmetrie theory since a « 1. Stated differently, the leading CIB term of order aQ-2 is
numerically much smaller than the strong leading order contribution rv Q-l. The insertion from
this contact term is shown in the last diagram of fig. 5.3 and leads to an additional contribution to
ßA. In complete analogy, we can treat the leading order CSB effect which is due to an operator
of the form aE6 2) (J.l) (Nt T3N) (Nt N). This term is, however, finite. Putting pieces together, we
get
ßA V 1 = -a
(E(l) + E(2)) [A_l] 2
1,-2,pp 0 0 Co ' ßAV 1 = _ (E(l
_ ) E(2)) [A-l] 2
1,-2,nn a 0 0 Co '
where the coupling constants E61,2) (J.l) obey the renormalization group equations,
(5.36)
(5.37)
Note that from here on we do no longer exhibit the scale dependence of the various couplings
constants E61,2),CO,2,D2. We can now relate the pp and nn scattering lengths to the np one (of
course, in the pp system Coulomb subtraction is assumed),
1
1
For the effective ranges, we have only CIB
(5.38)
(5.39)
Note that this last relation is scale-independent and that it does not contain any unknown parameter.
We re mark that for the CIB scattering lengths difference the pion contribution alone is
not scale-independent and can thus never be uniquely disentangled from the contact term contribut
ion rv While the leading OPE contribution resembles the result obtained in meson
E61).
exchange models, the mandatory appearance of this contact term is a distinctively new feature
of the effective field theory approach. It is easy to classify the leading and next-to-leading em
corrections to these results. At order aQ-l, one has the contribution from two potential pions
with the pion mass difference and also contact interactions with two derivatives. Effects due to the
charge dependence of the pion-nucleon coupling constants, i.e. isospin breaking terms from .c�N,
only start to contribute at order aQo . Such effects are therefore suppressed by two orders of Q
compared to the leading terms. This finding is in agreement with the various numerical analyses

162 5. Isospin violation in the two-nuc1eon system
performed in potential models. We now turn to eSB. Here, to leading order there is simply a
four-nucleon contact term proportional to the constant E�2). Its value can be determined from a
fit to the empirical value given in eq. (5.2). First corrections to the leading order eSB effect are
classified below.
5.4 N umerical results
We now turn to the numerical analysis considering exclusively the 1 So channel. At leading order,
all parameters can be fixed from the pertinent scattering length. Already in ref. [91] it was pointed
out that there are various possibilities of fixing the next-to-leading order constants. One is to stay
at low energies (Le. fitting a and r) or performing an overall fit to the phase shift up to momenta
of about 200 MeV. In this latter case, however, the resulting low energy parameters deviate from
their empirical values and also, at p rv 150 MeV, the correction is as big as the leading term. Since
we are interested in small effects like eIB and eSB, we stick to the former approach and use the
1 So np phase shift around p = 0 to fix the parameters as shown by the dashed curve in fig. 5.4.
More precisely, we fit the parameters CO,2, D2 to the np phase shift under the condition that the
r--"1
0
'---'
60
r--- 0
0 40
Cf)
.-I
'--"
V:)
20
0
0 0
0
oe
o���������������������
o 50 100 150 200
P [Me VJ
Figure 5.4: ISO phase shifts for the np (dashed line) and nn (solid line) systems versus the nucleon
cms moment um. The emipirical values for the np case (open squares) are taken from the Nijmegen
analysis [36]. The open octagons are the nn "data" based on the Argonne V18 potential.

5.5 Classification scheme 163
scattering length and effective range are exactly reproduced. The two new parameters E� I,2) are
determined from the nn and pp scattering lengths. The resulting parameters at f-l = Mn are of
natural size,
-3.46 fm2 , C2 = 2.75 fm4 , D 2 = 0.07 fm4 ,
-6.47 fm2 , E� 2) = 1.10 fm2 . (5.40)
To arrive at the curves shown in fig. 5.4, we have used the physical masses for the proton and the
neutron. We stress again that the effect of the em terms rv E� I, 2) is small because of the explicit
factor of a not shown in eq. (5.40). Having fixed these parameters, we can now predict the nn ISO
phase shift as depicted by the solid line in fig. 5.4. It agrees with nn phase shift extracted from
the Argonne V18 potential (with the scattering length and effective range exactly reproduced) up
to momenta of about 100 MeV. The analogous curve for the pp system (not shown in the figure) is
elose to the solid line since the CSB effects are very small. The leading CIB effect for the effective
range, ßrclB = 1/2(rnn + rpp) - rnp, is given by the last term in eq. (5.39). With the values of
the parameters shown in eq. (5.40) we obtain
ßrClB = 0.01 fm , (5.41)
which agrees with the small value observed experimentally for this quantity, see eq. (5.1).
5.5 Classification scheme
In the framework presented here, it is straight forward to work out the various leading (LO), nextto-leading
(NLO) and next-to-next-to-leading order (NNLO) contributions to CIB and CSB,
with respect to the expansion in Q, to leading order in a and the light quark mass difference. Here,
we will simply enumerate the pertinent contributions which appear at a given order. This also
naturally gives an estimate about their relative numerical importance based on simple dimensional
analysis. Let us first consider CIB. In the elassification scheme given below, TPE/3PE stands for
two / three-pion-exchange.
LO
aQ-2
NLO
a Q-l , c Q-l
NNLO
aQo
Pion mass difference in OPE,
four-nueleon contact interaction with no derivatives.
Pion mass difference in TPE and in the exchange of one radiation pion,
four-nueleon contact inter action with two derivatives,
insertions proportional to the np mass difference,
Pion mass difference in 3PE and TPE,
four-nueleon contact interaction with four derivatives,
CIB in the pion-nueleon coupling constants,
Note that there are no CIB effects due to the light quark mass difference linear in c = mu - md
from OPE and from the four-nucleon contact terms. These type of terms can only appear to
second order in c. Effects due to the charge dependence of the pion-nucleon coupling constants,
i.e. isospin breaking terms from .c��, only start to contribute at order aQo. Such effects are
therefore suppressed by two orders of Q compared to the leading terms, Le. by a numerical factor
of about (1/3)2. This finding is in agreement with the various numerical analyses performed in

164 5. Isospin violation in the two�nuc1eon system
potential models (if one assurnes that there is indeed some charge dependence in the pion�nucleon
coupling constants, see e.g. the discussion in ref. [216].) We remark that the TPE contribution is
nominally suppressed by one order in the small parameter, Le. one would expect its contribution
to be one third of the leading OPE with different pion masses.
It is also instructive to see how the corrections due to the np mass difference come about. For
that, let us write the nucleon mass as m + 6m, where 6m subsurnes the em and strong contribution
to the np mass splitting, Le. these terms are 0(0:') and O( E) . Differentiating the leading isospin�
symmertic amplitude eq. (5.31) with respect to the nucleon mass shift gives
from which it follows that
8A�1 A�1 .
-- = --(/1+Zp) ,
8m 47f
We now turn to CSB. The pattern of the various contributions looks different:
LO
Q�2 0:' ,E Q�2
NLO
Q�1 Q�1 0:' ,E
Electromagnetic four�nucleon contact interactions
with no derivatives.
Ern four-nucleon contact terms with two derivatives,
strong isospin�breaking contact terms with two derivatives,
insert ions proportional to the np mass difference.
(5.42)
(5.43)
We remark that the leading order CSB effects do not modify the effective range but the corresponding
scattering length. Note furt her that we have not considered effects due to virtual
photons (like, for instance, 7f"( exchanges), see ref. [228]. Such diagrams were calculated within
the Weinberg power counting approach in ref. [218]. The ISO np low�energy parameters appear
to be very little affected.

Chapter 6
Summary and outlook
In this thesis we have considered some applications of the effective (field) theory techniques to the
problem of nucleon-nucleon scattering and bound states. Now we would like to summarize the
pertinent results of our investigation:
1. After an introduction, which describes the current status of research and formulates the
problems, we started in chapter 2 to consider the quantum mechanical two-nucleon problem
and have shown how to construct an effective low energy theory based on the method of
unitary transformations based on an arbitrary realistic two-nucleon potential (in moment um
space). This is achieved by a decoupling of the low and high moment um subspaces of the
whole moment um space. The unitary transformation can be parametrized by an operator A,
see eq. (2.65), which obeys a nonlinear integral equation (2.69). This equation can be solved
numerically and any observable can then be calculated in the space of small momenta only.
While the method is interesting per se, we have also made contact to chiral perturbation
theory (CHPT) approaches to the two-nucleon system by studying a series of questions,
which can be addressed unambiguously within the framework of our exact low moment um
theory. Clearly, this should not be considered a substitute for a realistic CHPT calculation,
which we have also performed in this work, but might be used as a guide. The salient results
of this investigation can be summarized as follows:
• We have demonstrated analytically that the theory projected onto the subspace of
momenta below a given moment um space cut-off A leads to exactly the same S-matrix
as the original theory in the full (unrestricted) moment um space provided appropriate
boundary conditions for the scattering states are chosen. In particular, the components
of the transformed scattering states with initial momenta below the cut-off A are strictly
zero in the subspace of momenta above the cut-off A .
• Starting from a S-wave N N potential with an attractive light (/LL � 300 MeV) and repulsive
heavy meson exchange (/LH � 600 MeV) given in eq. (2.113), we have rigorously
solved in numerical sense the nonlinear equation for the operator A and demonstrated
that the observables related to the bound and scattering state spectra agree precisely
for the effective and the full theory up to the cut-off A. In particular, we have just one
bound state with a binding energy of 2.23 MeV. These results are independent of the
value of the cut-off, which was varied from 200 MeV to 5.5 GeV. We have argued that
the most natural choice is A about 300 MeV. The effective potential can differ substantially
from the original one (for values of A on the low side of the range mentioned
before).
165

166 5. Summary and outlook
• We have also considered an alternative way of determining the operator A from the
half-shell two-nucleon T-matrix. This leads to the linear equation (2.94), which has
been solved numericaIly. For all selected values of the cut-off A the solutions of the
linear and nonlinear equations for A agree perfectly with each other.
• We have examined a low-momentum expansion of the potential. For that we have
expanded the heavy meson exchange term in a string of local operators with increasing
dimension but kept the light meson exchange unchanged, see eqs. (2.115), (2.116).
The corresponding coupling constants accompanying these local operators, which are
monomials of even power in the momenta, can be determined precisely from the exact
solution. For A =300 (400) MeV their values are given in table 2.1. We have
shown that they are of "natural" size, i.e. of order one, with respect to the mass scale
Ascale = 600 MeV. We have also discussed the relation of this scale to the mass of the
heavy meson, which is integrated out, and the convergence properties of such type of
expansion. In particular, to recover the binding energy within a few percent, one has
to retain terms of rather high order in this expansion, cf. eq. (2.116). This is to be
expected due to the unnatural smallness of this energy on any hadronic mass scale. The
3 SI scattering phase shift can be weIl reproduced up to kinetic energies llab ':::' 120 Me V
with the first three terms in the contact term expansion.
• Based on the expanded heavy meson exchange term, we have also determined the
constants Ci directly from a fit to the phase shifts. This is equivalent to the procedure
performed in an effective field theory approach. We could show that as long as one does
not include polynomials of order six (or higher), the resulting values of these constants
are close to their exact ones. Furthermore, the binding energy is reproduced within
2%. Including dimension six terms, the fits become unstable. This can be traced back
to the fact that the contribution of such terms to the phase shifts are very small (at
low and moderate energies) and thus can not really be pinned down.
• We have also studied the quantum averages of the expanded potential in the bound and
scattering states. For A = 300 MeV, the expansion parameter is of the order of 1/2 and
we find fast convergence for the bound and the low-Iying scattering states, as shown
in table (2.3). As expected, for scattering states with higher energy, the convergence
becomes slower.
• We have determined the effective range parameters using the expansion (2.116) for
the potential and demonstrated the predictive power of such an effective theory. In
particular, having fixed the free parameters in the potential from the first n terms in
the effective range expansion one obtains a prediction for the next coefficient, which has
not been used in the fit. This is different from the pionless effective theory considered
in the seetion 2.2, where no predictions are possible.
• In the model space of small momenta only, one can also study the non-Iocalities in
the coordinate space representation. We have shown that for typical cut-off values, the
effective potential V(x, x') is highly non-Iocal and looks very different from the original
one. For very large values of the cut-off, one recovers the original local potential.
While this thesis was being written down, a similar work done by Bogner et al. [214] has
been reported, showing a considerable research activity in this field. In this work, effective
low-momentum potentials defined below a given moment um space cut-off A are obtained
from the Bonn-A and Paris potentials using the folded-diagram method of Kuo, Lee and
Ratcliff [215]. The method leads to non-hermitian potentials and preserves the half-shell
NN T-matrix.

2. Secondly, we considered the realistic case of the nuclear interaction and presented a novel
approach, the projection formalism, to the problem of deriving the forces between few (two,
three, ... ) nucleons from effective chiral Lagrangians. For the case at hand, we first had to
modify the power counting rules proposed originally by Weinberg [73], since in the projection
formalism one decomposes the Fock space of nucleons and pions into subspaces with definite
nucleon and pion numbers. While in old-fashioned time-ordered perturbation theory the
resulting wave functions are only orthonormal to a certain order in the chiral expansion,
this problem does not occur in the projection formalism. Furthermore, in the previous
calculations based on time-ordered perturbation theory, the two-nucleon potentials turn
out to be energy-dependent, which is a severe complication for applying these in systems
with three or more nucleons (although to leading order, these recoil corrections are cancelled
by certain N-body interactions). We now summarize the results obtained in chapter 3.
• We start from the most general chiral invariant Hamiltonian for nonrelativistic nucleons
and pions and divide the full Fock space into the two subspaces T} and )... The first one
contains only purely nucleonic states, whereas the second one involves all remaining
states with nucleons and pions. To obtain an effective Hamiltonian acting on the T}space
we perform an appropriate unitary transformation, parametrized in terms of
the operator )"AT} via eq. (3.190). After projecting onto the states )..i with i pions,
the decoupling equation (3.192) turns into an infinite system of coupled equations
(3.205). We have proved that this system of equations can be solved perturbatively in
powers of the small momentum scale Q. For that we have developed an appropriate
power counting rule and analyzed the power of Q for all terms in eq. (3.206). We
have worked out the recursive prescription for solving the system of equations (3.205),
which allows to determine the operators )..i AT} for any finite i to any required order
in the Q-expansion. The effective Hamiltonian Heff acting on the T}-subspace can be
obtained via eq. (3.197) using the power counting rules (3.223)-(3.229).
• We have applied the projection formalism and have given the explicit expressions of
the two-nucleon potential to next-to-leading order (The lowest order potential comprises
the leading one-pion exchange and contact terms). In particular, we have also
calculated self-energy type corrections and one-loop corrections to these four-fermion
operators not considered before.
• We have discussed in detail the similarities and differences of the resulting potential
with those obtained from time-ordered perturbation theory. In particular, it turns
out that in our approach the isoscalar spin independent central and the isovector spin
dependent parts of the two-nucleon potential corresponding to the two-pion exchange
add up to zero. This was first noted in ref. [108] using a different scheme. Note that
in the energy dependent potential of the time-ordered perturbation theory there is no
such cancellation. However, the most salient features of the potential in the projection
formalism is its energy--independence and the orthonormality of the corresponding wave
functions.
• We have performed renormalization of the two-nucleon potential at NLO. In particular,
we have shown that all ultraviolet divergences can be removed by an appropriate
redefinition of the axial coupling constant gA and various couplings of the contact interactions.
The renormalized expressions for the N N potential at NLO agree with the
ones given by the Munich group which were obtained using the Feynmann diagram
technique.
167

168 5. Summary and outlook
• In sec. 3.8.1 we have also discussed the structure of the NNLO potential in the method
of unitary transformation. It turns out that in that case one obtains the same result
in both the projection formalism and time-ordered perturbation theory. The NNLO
potential has been calculated within a different (but leading to the same result) scheme
by the Munich group [108].
• In sec. 3.8.3 we have generalized the projection formalism to include effects of the
virtual Ll-isobar excitation within the "small scale expansion" and discussed the leading
contribution to the N N potential from the intermediate Ll 'so In that case, both the
method of unitary transformation and time-ordered perturbation theory lead to the
same result.
• Furthermore, we have considered the leading contributions to the three-nucleon potential.
As in time-ordered perturbation theory, we find that these sum up to zero.
However, the mechanism for the cancelation between so me of the graphs is distinctively
different in the projection formalism since it is not related to an intricate cancelation
of terms generated by the iteration of the energy-dependent two-nucleon potential in
old-fashioned time-ordered perturbation theory. Rather, in the approach here, these
cancelations can be traced back to the appearance of "reducible" graphs whose precise
meaning is explained in section 3.8.1. These diagrams are in fact responsible for the
orthonormality of the wave functions and are thus sometimes called "wave function
re-orthonormalization" graphs.
• We have also constructed the most general reparametrization invariant effective Lagrangian
for the contact interactions with four nucleon legs up to order Lli = 3, L}J;;5�3)
(reparametrization invariance is a consequence of Lorentz invariance of the underlying
theory, see ref. [182]). The Lagrangian used in previous calculations, see e.g. [74], [76],
[78], [127], contains fourteen free parameters Ci,...,14 and leads to two-nucleon forces,
which depend on the total moment um P of two nucleons. Whereas such P-dependent
forces do not affect calculations of the two-nucleon system in the c.m.s., they would be
very important for processes including other particles and for three and more nucleons.
We have shown that requiring the reparametrization invariance for the contact terms
in the effective Lagrangian yields several constraints on the parameters Ci. Only seven
of these fourteen parameters are really independent. The resulting NLO and NNLO
two-nucleon potentials do not depend on the total moment um P.
3. Thirdly, in chapter 4 we have calculated nuclear forces and properties of the two-nucleon
system based on a chiral effective field theory and the projection formalism. The results of
this investigation can be summarized as follows:
• We considered the two-nucleon potential at NNLO resulting from the projection formalism.
It consists of one- and two-pion exchange diagrams, including dimension two
(Lli = 2) insertions from the pion-nucleon Hamiltonian. The corresponding LEes have
been taken from an investigation of rrN scattering [195]. In addition, there are two and
seven contact interactions without and with two derivatives, respectively. The coupling
constants of these terms must be fixed by a fit to data.
• For large momenta, the potential becomes unphysical and has to be regularized. We
performed this regularization on the level of the Lippmann-Schwinger equation, as
explained in sec. 4.1 using either a sharp or an exponential regulator function. At
NLO, physics does not depend on the cut-off in the range between 400 and 650 Me V. At

NNLO, this range is larger and extends from 650 to 1000 MeV. This can be understood
from the chiral TPEP, which at NNLO includes 1f1f correlations. These introduce a
new mass scale well above twice the pion mass.
• We have shown that the NLO couplings can be combined in such a way that each
combination feeds into one partial wave, see eqs. (4.16)-(4.23).1 More precisely, the
nine couplings with four nucleon legs can be determined uniquely by a fit to the two Swaves,
four P-waves and the mixing parameter E1 for nucleon laboratory energies below
100 MeV. This simplifies the fitting procedure enormously as compared to ref. [78],
where a global fit in all low partial waves has been performed. As expected from the
power counting underlying the EFT, the fits improve when going from LO to NLO to
NNLO, compare fig. 4.l.
• At NNLO, the resulting S-waves are of very high precision (for nucleon laboratory
energies below 300 MeV), see e.g. tables 4.2,4.3 and fig. 4.4. The so-called range
parameters collected in table 4.4 agree with what is found in the phase shift analysis.
The P-waves are mostly well described, in particular the mixing parameter E1 is in
good agreement with the phase shift analysis. We also note that above nucleon cms
momenta of about 150 MeV, our NLO and NNLO results are far better than the ones
obtained in the KSW scheme at NLO and NNLO.
• All other partial waves are free of parameters. The D-waves, in particular 3 D1 and 3 D 3
are very well described. We have also discussed the cut-off sensitivity of these results.
The NNLO TPEP is too strong in the trip let F-waves. For the peripheral waves, we
recover the results of the Munich group [108], namely that in most cases OPE works
well but chiral NNLO TPEP clearly improves the description of some partial waves like
e.g. 3G5, 3 H5 or 3 h.
• The deuteron properties are mostly well described, at NLO and NNLO, compare table
4.7. At NNLO, the deuteron wave functions show some interesting structure due
to the appearance of two very deeply bound states. These are an artifact of the NNLO
approximation. They have no influence on low energy properties and can be projected
out completely from the theory. Our precise deuteron wavefunctions can be used for
pion photoproduction, pion-deuteron scattering or Compton scattering off deuterium
(still, the hybrid approach proposed by Weinberg [114] remains a useful tool).
• We have also considered an approach with explicit D.. degrees of freedom in the TPEP.
This NNLO-D.. approach leads to results very similar to the ones at NNLO in the
theory without isobars, with the exception of the partial waves that are sensitive to
pionic scalar-isoscalar correlations like e.g. 3 D 3 • We conclude that the inclusion of the
D.. via resonance saturation of dimension two 1f N LECs capture the essential physics of
the isobar in the two-nucleon system. We note, however, that a more systematic study
of pion-nucleon scattering in an EFT including the D.. is needed to further quantify
these statements.
4. Finally, we have considered electromagnetic and strong isospin violation in low-energy
nucleon-nucleon scattering in the effective field theory formalism developed in ref. [91].
We now summarize the results of this investigation.
• We first considered isospin violating parts of the effective Lagrangian. The terms
related to the pion and the pion-nucleon system were discussed in [199], [222]-[224].
lThis was also noted by Kaplan et al. [91].
169

170 5. Summary and outlook
The isospin violating Lagrangian in the two-nucleon sector has been worked out by
van Kolck in ref. [75]. We have reconsidered the corresponding terms using a different
approach, namely the method of external fields .
• After a brief introduction into the KSW formalism, we have discussed the leading
isospin violating effects in the 1 So channel. In particular, the leading charge independence
breaking effect is due to a combination of the neutral to charged pion mass
difference in one-pion exchange diagrams together with an electromagnetic N N contact
term. Its corresponding coupling constant scales as Q-2 but is numerically suppressed
by the explicit appearance of the fine structure constant a rv 1/137. We have shown
how the KSW power counting has to be modified in the presence of isospin violating
operators .
• We explicitely evaluated the 1 So phase shifts for the np, nn and Coulomb-subtracted
pp systems at next-to-Ieading order. In addition, we have given a general classification
of the various CIB and CSB corrections. This allows to understand so me phenomenologically
found results.
To the best of our knowledge the exact moment um space projection of the nucleon-nucleon interaction
discussed and performed in sec. 2.3 has never been done before. In this thesis we have
applied this formalism to some problems arising in the context of an effective field theory for the
N N system. The method, however, might also provide new insights into many other interesting
quest ions in nuclear physics. In particular, it opens the possibility of studying relativistic effects
in a consistent and convergent manner, since the moment um components of the order or higher
than the nucleon mass can be integrated out. Furthermore, it would be interesting to generalize
the above formalism to three- and more-nucleon system.
Further, the method of unitary transformation (projection formalism) has never been applied
in the context of the chiral effective field theory. Previous calculations of the 2N interaction
[73], [74], [76], [78] from chiral effective Lagrangians are based on time-ordered perturbation
theory and lead to an energy-dependent potential. As already stressed above, the most important
advantages of our formalism against time-ordered perturbation theory is energy-independence of
the corresponding potential and the orthonormality of the related wave functions. Appropriate
power counting rules, which allow to perform calculations within the projection formalism to any
required order in the low-momentum expansion, have been worked out in sec. 3.6 and appendices
A, B and C.
Our results for nucleon-nucleon interactions derived from the most general chiral invariant Hamiltonian
do not only show that the scheme originally proposed by Weinberg works qualitatively, it
even works much better than it was expected, namely quantitatively. It extends the successful
applications of effective field theory (chiral perturbation theory) in the pion and pion-nucleon
sectors to systems with more than one nucleon. Clearly, one should now reconsider processes,
which have been evaluated using Weinberg's hybrid approach [114] (?f - d scattering [114], [217],
"(d -+ ?f o d [115], "(d -+ "(d [116]) and extend these considerations to systems with more than two
nucleons. In addition, a fresh look at charge symmetry and charge independence breaking in the
Hamiltonian formalism is called for (for earlier studies, see e.g. refs. [218], [99]).
A very important and actual research field comprises the application of CHPT to 3N interactions.
Whereas the first results in the KSW scheme and in the pionless effective theory were already
presented for the three-body system, see e.g. refs. [119], [120], [121], [122], no calculations within
the potential approach including the complete leading 3N force predicted by CHPT have yet been
performed. It would be interesting to see whether the inclusion of the 3N forces of such type

might be responsible for solving the Ay problem in the elastic nd scattering at low energies. For
more discussion on that see ref. [196J.
We also point out that problems related to renormalization of the effective Hamiltonian after
eliminating the pionic degrees of freedom have to be investigated more carefully than it has been
done in sec. 3.8.2. In particular, one should include also the zero- and one-body operators in order
to perform a complete renormalization at NLO and NNLO. Furthermore, it would be interesting
to study the problem of carrying out the renormalization at an arbitrary order in the moment um
expansion. As far as we know, the problem of renormalization within the Hamiltonian approach
has never been worked out in detail (also in the context of an effective field theory).
Last not least we have shown that isospin violation can be systematically included in the effective
field theory approach to the two-nucleon system in the KSW formulation. For that, one has to
construct the most general isospin violating effective Lagrangian and extend the power counting
sehe me accordingly. We have shown that this framework allows one to syste�atically classify
the various contributions to charge indepedence and charge symmetry breaking (CIB and CSB).
In particular, the power counting combined with dimensional analysis allows one to understand
the suppression of contributions from a possible charge-dependence in the pion-nucleon coupling
constants. It would be interesting to extend this formalism to other partial waves and to higher
energies so as to investigate e.g. isospin violation in pion production.
171

Appendix A
Dimensional analysis of the projected
decoupling equation (3.206)
In this appendix, we work out the chiral power of various terms appearing in eq. (3.206). Any
one of these can be characterized by a counting index v, i.e. it is proportional to QV. This index
should not be confused with the one giving the chiral dimension of the matrix-elements (w i IAI

derivative. Further helpful equalities are
1);2:1 ,
1);2:-2+p
173
(A.5)
(A.6)
Whereas (A.5) is generally valid since no renormalizable interactions are allowed by chiral symmetry,
(A.6) follows from simple inspection of eq. (3.211): for n = 0 there are at least two derivatives
and for n 2: 2 it is immediately apparent.
Below we will examine the chiral power v for all terms entering the left-hand side of eq. (3.206).
• )..4k +iHK'fl
v = 4 - 3N - 4k - i + I); • (A.7)
This follows from eq. (3.210) if one notes, that now there is one energy denominator less.
We then obtain from eqs. (A.3), (A.4), (A.6):
v 2: 4 - 3N, when k = 0 ,
v 2: 2 - 3N, when k > 0
• )..4k +i HK)..4Q+ j Az'fl
v = 4 - 3N - 4k - i + 2q + j + I); + l .
The following possibilities are to be considered:
(A.8)
1. q::;k-l
In this case the number of pions at the vertex is restricted by p 2: 4k + i - 4q - j.
Therefore, the inequality (A.6) takes the form
It then follows from eq. (A.8), that
2. q = k
Here we have:
I); 2: -2 + 4k + i - 4q - j
v 2: 4 - 3N - 2k
p 2: Ij - il
(A.9)
(A.I0)
(A.11)
Applying eq. (A.4) and using the (obvious) inequality j - i 2: -Ij - il one can justify
eq. (A.I0) also for that case.
3. q 2: k + 1
One can easily prove eq. (A.I0) far this case using eq. (A.5) and noting, that j -i 2: -3.
• E()..4k +i ) .. 4k+i Az'fl
v 2: 4 - 3N - 2k + l 2: 4 - 3N - 2k . (A.12)
This follows immediately from eq. (3.214). Here, the first inequality becomes the exact
equality when only the pion kinetic energy is taken into account in E()..4k +i ).
• )..4k +iAZ'flHK'fl
v = 4 - 3N - 2k + l + I); • (A.13)
There is no non-vanishing operator 'flHK'fl with I); < O. That is why the inequality (A.I0) is
again valid.

174
• ).. 4k+i AI''lE( Tl)
A. Dimensional analysis of the projected decoupling equation (3.206)
We will count the nucleon mass in the same way as it has been done by Weinberg [73]:
(A.14)
The nucleon kinetic energy must then be counted as Q3 and we obtain the following result:
v ;::: 6 - 3N - 2k + l ;::: 6 - 3N - 2k (A.15)
v = 4 - 3N - 2k + 2q + j + h + l2 + K, ;::: 6 - 3N - 2k (A.16)

Appendix B
Operators contributing to eq. (3.206)
at order r
In this appendix, we work out in detail which operators ).a Ar" actually appear in eq. (3.206) at
order r, which is defined in eq. (3.216). For that we will examine below all terms that enter this
equation. We find
• ).4k+iHK,).4 q + j A/1]
One obtains from (A.8) and (3.216) the following identity:
Let us first consider the case i # 0:
1. q :::; k-1
Using eqs. (A.9), (B.1) one gets
2. q = k
l = r + 2(k - q) + i - j - K,
l :::; r - 2k + 2q + 2 :::; r .
(a) j < i
It follows from eqs. (A.ll), (B.1), that
(b) J = Z
In this case eq. (B.1) takes the form
l :::; r - Ij - i 1 + i - j :::; r .
due to the inequalities (A.3), (A.4) and (A.6).
(c) j 2:: i + 1
The first inequality of eq. (B.3) leads to
3. q = k + 1
l :::; r - 2j + 2i :::; r - 2 .
175
(B.1)
(B.2)
(B.3)
(B.4)
(B.5)

176 B. Operators contributing to eq. (3.206) at order r
(a) i = 1,2
It can be seen from the inequalities (AA), (A.6) that in this case K, � 2. Using the
inequality i - j :S 2 we get from eq. (B.I)
l:Sr-2 .
(b) i = 3
The general inequality (A.5) leads immediately to
4. q � k + 2
1. j=O : l:Sr ,
11. j � 1 : l:S r - 1
(B.6)
(B.7)
Using eq. (A.6) with the number of pions p given by p = 4q + j - 4k - i we obtain from
eq. (B.I)
l :S r - 6(q - k) + 2 + 2i - 2j :S r - 6(q - k) + 8 :S r - 4 . (B.8)
The case i = 0 can be considered analogously:
1. q:Sk-2=?l:Sr-2,
2. q = k - 1 .
(a) j � 1
Putting p = 4 - j in eq. (AA) we see from eq. (B.I), that
l:Sr-2.
(b) j = 0
In this case we have K, � 2, as it follows from eq. (A.6). We obtain
3. q = k
It follows from eqs. (A.3) and (AA), that
l :S r .
l

Appendix C
Small scale expansion within the
projection formalism
In this appendix we would like to generalize the consideration of appendices A and B to include
the (intermediate) Ll-isobar excitations. As discussed in the text, the Ll is treated in a similar
way as the nucleons, i.e. as the light particle (but not massless as N) with the mass Ll = mf:l. - m1
and with both spin and isospin quantum numbers equal 3/2. Now we will generalize the results
obtained in appendices A and B to include also the states related to >.0.
We start with the dimensional analysis of the decoupling equation (3.206) projected by >.0 and
examine the chiral power v for all terms entering the left-hand side of this equation .
• >.0 HK,Tj
v = 4 - 3N + r;, 2: 6 - 3N . (C.1)
Note that here and in what follows, N corresponds to the number of baryons (also Ll's).
This inequality follows because the transition between purely baryonic states requires contact
vertices with four baryon legs or relativistic correction terms. The minimal possible value
of r;, for such interactions is 2.
The following possibilities are to be considered:
v = 4 - 3N + 2q + j + r;, . (C.2)
1. q = 0, j = 0
In that case we have r;, 2: 2, l 2: 2. Consequently, one obtains:
v 2: 8 - 3N . (C.3)
2. 4q + j # 0
In that case 2q + j 2: 1. Prom the inequality (A.5) we obtain
v 2: 6 - 3N . (C.4)
v 2: 6 - 3N . (C.5)
I This is because a trivial momentum dependence of the f:l.-field due to the nucleon mass is factored out, see
eq. (3.160).
177

178 C. Small scale expansion within the projection formalism
• )..0 Al TJHt;, TJ
Equation (A.13) leads to the following inequality:
v 2 8 - 3N , (C.6)
where we have used the facts that min (l) = 2 for )..0 A1TJ and min (I\;) = 2 for TJHt;,TJ.
v 2 8 - 3N . (C.7)
v = 4 - 3N + 2q + j + h + l 2 + I\; 2 8 - 3N . (C.8)
To complete the modifications of the appendix A, we also need to consider ).. O -intermediate states.
Only two terms in eq. (3.206) may have such intermediate states:
• )..4k+iHt;,)"oA1TJ
We consider two different cases:
v = 4 - 3N - 4k - i + I\; + 1 . (C.9)
1. k = 0, i f. 0
Using the inequality (A.4) and the fact that 1 2 2 we obtain
2. k f. 0
In that case eq. (A.6) leads to
v 2 6 - 3N . (C.10)
v 2 4 - 3N . (C.ll)
v = 4 - 3N - 2k + h + l 2 + I\; 2:: 8 - 3N - 2k . (C.12)
We now switch to appendix B and consider which operators )..a A1TJ enter the decoupling equation
(3.206) at order r projected onto the state )..0•
• )..0 Ht;,)..4 q +j A1TJ
Again, we have to consider two cases:
1. q=O
One can use eq. (A.4) if j f. 0 and the fact that I\; 2 2 if j = 0 to obtain
l

179
(C.16)
l�r-2. (C.17)
h + l2 = r - /'i, - 2q - j � r - 2 . (C.18)
In the last inequality we have used the same arguments as for the case ,\0 HK,,\4 q +j Al'r), from
w hich follows that 2q + j + /'i, 2': 2 for all q, j.
We also examine the cases with ,\ o -intermediate cases:
1. k = 0, i # 0
The inequality (A.4) leads to
2. k> 1
From eq. (A.6) we obtain
l = r + 2k + i - /'i, • (C.19)
l � r . (C.20)
l � r. (C.21)
h + l2 = r - /'i, � r - 2 .
(C.22)

Appendix D
Divergent loop integrals
In this appendix we give the divergent loop integrals that enter the expressions for the NLO
potential in terms of divergent integrals Jo , J 2 and J4 defined in eq. (3.284).
h ==
J d3[ [2 1 { 2 M;
2 2 }
(27r)3 wr = 87r2 3Mn: In 4JL2 + 5Mn: + 4J 2 - 6Mn: Jo , (D.l)
(D.2)
(D.3)
(D.4)
(D.5)
with s = J4M; + q2 and w± given in eq. (3.273). All other integrals appearing in the 1�loop
NLO TPEP can be deduced from these expressions by taking proper linear combinations or
differentiation with respect to M;, as explained in the text. Note that both left� and right�hand
sides of these equations do not depend on JL. The explicit logarithmic dependence of the terms on
the right�hand side is compensated by the JL�dependence of Jo .
180

Appendix E
Anti-symmetrization of the contact
interactions
In this appendix we will entirely concentrate on contact interactions and their contributions to
the efIective N N potential. Let us start with the simplest case of the contact interactions without
derivatives. Only two such terms enter the Hamiltonian (3.234). In principle, one can construct
two additional interactions that satisfy all required symmetry principles by insert ions of the isospin
matrices T'S. Weinberg pointed out [73] that only two of these four contact terms are independent.
The remaining ones can be eliminated from the Hamiltonian applying the Fierz transformation,
see, for instance, [132]. Another way to see that is to perform anti-symmetrization ofthe potential.
We will now explain this in detail following the logic of the reference [108].
The contribution of all four contact interactions to the efIective potential is
Vo = Ü1 + Ü2 0\ .
The anti-symmetrized potential v� is given by
where the exchange operator A is defined via
eh + Ü3 Tl ' T2 + Ü4 (0\ . (2) (Tl ' T2) .
Vo - A[vo]
2
and where m i (mD denotes symbolically spin, isospin and momentum quantum numbers of the
nucleon. According to [108], in the two nucleon center-of-mass system the exchange operator A involves
a left-multiplication with the isospin exchange operator (1 + Tl ' T2) /2, a left-multiplication
with the spin exchange operator (1 + 51 . (2)/2 and a substitution p' -+ _p'.1 Using these rules
one obtains the following identities:
A[l] 1 (1 + (51 . 52) + (Tl ' T2) + (51 ' (2)(T1 ' T2)) ,
1 (3 - (51 . 52) + 3(T1 ' T2) - (51 ' (2)(T1 ' T2)) ,
1 (3 + 3(51 . 52) - (Tl ' T2) - (51 ' (2)(T1 ' T2)) ,
1 (9 - 3(51 . 52) - 3(T1 . T2) + (51 ' (2) (T1 . T2)) .
I p and pi are the nucleon initial and final ems momenta, respeetively.
181
(E.1)
(E.2)
(E.3)
(E.4)

182 E. Anti-symmetrization of the contact interactions
Now it is easy to perform anti-symmetrization of eq. (E.1):
where
1
4 (a1 - a2 - a3 - 3(4) ,
1
4( -al - 3a2 + 5a3 + 3(4) . (E.6)
Note that only two independent constants ß1 and ß2 enter the expression (E.5) for the antisymmetrized
contribution. Therefore, we can choose only two operators of the four contact terms
in eq. (E.1) as the basis. For example,
Vo = C s + Cr eh . eh . (E.7)
Requiring that the anti-symmetrized contributions of eqs. (E.1) and (E.7) are the same we obtain
using eqs. (E.5), (E.6)
Cr (E.8)
One can proceed in the same way to consider the contact interactions with two derivatives. Altogether
fourteen such interactions contribute in the two-nucleon cms system (seven terms in
eq. (3.270) plus the the corresponding ones with insert ions of the isospin matrices). For our purpose
it is sufficient to consider a subclass of these interactions, namely those depending on q 2, k2
(since the other contact terms requiring anti-symmetrization enter eq. (3.296)):
V2 =
,1
q 2 + ,2 ih . eh q 2 + ,3 Tl . T2 q 2 + ,4 (0\ . eh) (T 1 . T2) q 2
+ ,1 k 2 + ,2 ih . ih k 2 + '3 Tl ' T2 k 2 + '4 (51 . (2) (Tl ' T2) k 2 .
Note that A[g] = -2k, A[k] = -g/2. Therefore, we obtain using eq. (E.4):
v2 W1 q - 12 W1 + W3 + W4 0"1 ' 0"2 q + W2 Tl . T2 q .
(E.9)
A 2 1 ( 4 3 ) - - 2 2 (E 10)
1
-
12
( 4W2 + W3 - W4) (51 . (2) (Tl ' T2) q2 + W3 k2 -1 ( W3 + 4W1 + 12w2 ) 51 . 52 k2 ( W4 + 4W1 - 4W2 ) (51 . 52 ) (Tl . T2) k2 ,
+ W4 Tl . T2 k2 -1
where W1,2,3,4 are given by
1 1
2'1 - 32 (,5 + 3,6 + 3'7 + 9,8 ) ,
1 1
2'3 - 32 (,5 + 3,6 - ,7 - 3'8 ) ,
1
2(,5 - ,1 - 3'2 - 3'3 - 9'4 ) ,
1
2(,7 - ,1 - 3'2 + '3 + 3'4) .
(E.ll)

Consequently, only four of the eight contact terms in eq. (E.9) are independent. Choosing those
terms to be
(E.12)
and requiring that the anti-symmetrized contributions of eqs. (E.9) and (E.12) are the same, we
end up with the following result using eqs. (E.10), (E.ll):
3"(8
"(7
"(1 - "(3 - 4 - 4 '
-4"(3 - 12"(4 + "(5 - "(7 ,
"(7 "(8
"(2 - "(4 - 4 + 4 '
-4"(3 + 4"(4 + "(6 - "(8 .
183
(E.13)

Appendix F
The complete set of N N contact
interactions with two derivatives
The most general effective Lagrangian for nucleons given in refs. [76], [78] contains 14 contact
interactions with two derivatives, which are multiplied by the coefficients C 1 , ... 14. In this appendix
we would like to show that 7 of these 14 terms are not independent from the others and have fixed
coefficients. Moreover, considering the couplings accompanying these terms as apriori unknown
and non-vanishing quantities would violate general symmetry principles of the theory.1 The
situation he re is similar to the familiar case of the nucleon kinetic energy term Nt\l 2 j(2m)N in
the Lagrangian corresponding to the Hamiltonian (3.199). The Lorentz invariance of the theory
would be broken if one allows an arbitrary coefficient for the term Nt\l 2 N.
As explained in the text, the nucleons in the effective theory can be described by velocity dependent
fields Hv and hv, see eq. (3.160). 2 The total momentum Pp. of the nucleon is then parametrized
by a pair (Vp., Ip.) via eq. (3.158). The small component field hv can be eliminated from the
theory, which leads to 1jm-corrections in the effective Lagrangian. Clearly, such correction terms
have fixed coefficients, defined by the structure of the original relativistic Lagrangian. Explicitly
integrating out the small component field hv has not yet been worked out for nucleon interactions
with four and more legs. Instead of explicitly integrating out hv, it is easier in such a case to write
down all possible terms consisting of Hv , vp. and Sp., which is defined by
and of covariant derivatives of the nucleon and pion fields.3 Note that the spin operator Sp. obeys
the following relations (in four dimensions):
2 3 1
S· v = 0, S = -4' {Sp., Sv} = 2 (vP.vv - 9p.v) , [Sp., Sv] = ifp.vpuV P S u , (F.2)
where we have used the convention f0 1 2 3 = 1. One can require the reparametrization invariance
[182] of the effective Lagrangian to fix the coefficients for the 1jm-corrections and to eliminate
the terms violating Lorentz symmetry. The effective Lagrangian expressed in terms of velo city
dependent fields is reparametrization invariant if it preserves its form under the reparametrization
(F.1)
(Vp., Ip.) +-t (Wp., kp.) = (Vp. + qp. , Ip. - q p.) , (F.3)
m
1 In particular, the Galilean invarianee would be broken.
2 In sec. 3.4 we have suppressed the label v for those fields.
3 AH Dirae bilinears fIv rHv (r = {I, 'Y5 , 'YI" 'Y5'YI" O"l' v }) ean be expressed in terms of (fIvHv), (fIvSI' Hv ), v!'
and of the anti-symmetrie Levi-Civita tensor El'vpu , see e.g. [72].
184

where qJ-! is chosen to satisfy (v + q/m)2 = 1, and, consequently, v . q = O(q2/m). As pointed
out in section 3.4, it is more convenient to work with the fields Hv defined in eq. (3.170) 4 instead
of Hv , since the first ones transform covariantly, via eq. (3.172), und er the reparametrization
transformation (F.3), see ref. [182]:
Hw = eiq.x Hv . (F.4)
At order l/m the two fields Hv and Hv are connected via:
- ( i Hv = 1 +
2m {J ) Hv · (F.5)
In what follows, we will not be interested in the explicit calculation of the l/m-corrections to
the contact interactions. This is because in our power counting scheme one has: Q / m '" Q2 / A�.
Consequently, all l/m-terms are suppressed against the corresponding ones, whose coefficients
scale by inverse powers ofAx' As will be shown below, such l/m-corrections to contact interactions
with four and more nucleon legs contribute at higher order and are irrelevant for our calculations.
Therefore, we can set
Further, we will need various properties for the building blocks of the effective Lagrangian, which
are listed in table F.1.
I Operator 11 H C x --+ x ' = Ax
vp + - AJ-!v V v
Hv Hv + + HvHv
HvSJ-!Hv + + det(A)AJ-!v HvSvHv
Table F.l: Transformation properties for the building blocks of the effective Lagrangian
under hermitian (H) and charge (C) conjugations and homogeneous Lorentz transformation
(x --+ x ' = Ax).
Consider first the contact interactions without derivatives and pion fields. Here and in what
follows, we will not consider the contact interactions with insert ions of the isospin matrices, since
those ones can be eliminated via Fierz reshuffiing or anti-symmetrization of the corresponding
N N potential, as detailed in appendix E. The two possible terms are
where CS,T are some constants. All other terms can be reduced to those two using the first
equality in (F.2) and the fact that v2 = 1. The terms in eq. (F.7) are, clearly, reparametrization
invariant.5
4 In the general case of nucleons interacting with pions and/or external fields one has to replace the derivative
8" in eq. (3.170) by the covariant one. In this appendix we will, however, not consider such a general situation.
5 As is obvious from eq. (F.4), only terms with derivatives of Hv can break the reparametrization invariance if
no l/m corrections are considered.
185
(F.6)
(F.7)

186 F. The complete set of the N N contact interactions with two derivatives
There are also two terms with one derivative:
Note that the terms with just one insertion of the spin-operators S/1 like, for instance,
are not parity and charge conjugation invariant, see table F.l. The same holds true for the term
(F.8)
(F.9)
(F.10)
Both terms in eq. (F.8) are redundant and can be eliminated from the effective Lagrangian applying
the leading order equation of motion (EOM) (3.165) of the field Hv : (v · ö)H = 0. 6 Note, however,
that the terms in (F.8) are reparametrization invariant. This can easily be checked using eqs. (F.4)
and (F.6).
Let us now consider the contact interactions with two derivatives. The following terms appear in
the effective Lagrangian:
.c�� = �Ü1 [ (Hv 7J /1 Hv)(Hv 7J/1 Hv) + (Hv 8 /1Hv)(Hv 8/1 Hv )]
+ ü 2 (Hv 7J /1 Hv )(Hv 8/1 Hv )
+ ü3(HvHv)(Hv( 82 + 7J2 )Hv)
+ ü4(HvHv)(Hv 8 /1 7J/1 Hv)
1 [ - ---itv - ,="p - ,="v - ---itp ]
+ 2i Ü5 E/1VpuV /1 (Hv iJ Hv )(Hv 'ä SU Hv ) - (Hv 'ä Hv)(Hv SU iJ Hv)
+ i Ü6 E/1vpuv /1 (HvHv)(Hv 8v s p7Ju Hv)
- - '="P---itu
+ i Ü7 E/1vpuv/1(HvSV Hv)(Hv 'ä iJ Hv)
+ �i ÜS E/1VpuV /1 [(Hv 7Jv Hv )(HvS p7Ju Hv) - (Hv 8v Hv )(Hv 8u S P Hv)]
1
+ 2 (ügg/1Pgvu + ü10g/1ugvP + üllg/1vgpu)
X [(HvS pa/1 Hv)(Hvs u7Jv Hv) + (Hv 8/1 S P Hv)(Hv 8V sU Hv)]
+(Ü1 2 g/1pgvu + Ü13g/1ugvp + Ü14g/1vgpu ) (HvS p7J/1 Hv )(Hv 8v s u Hv )
+ � (�Ü15 (9/1Pgvu + g/1ugvp) + Ü169/1V9pu)
x [(Hv 8/1
s
p7JV
Hv)(HvSU Hv) + (Hv 8v
sP7J/1 Hv )(HvSU Hv)]
1 1 ( ) - '="/1'="V ---it/1---itv P ---itv -
(F.ll)
+ 2 2Ü17(9/1Pgvu + g/1ugvp) + Ü1Sg/1vgpu (Hv( 'ä 'ä + iJ iJ )S iJ Hv)(HvS Hv)
where Ü1, ... ,lS are constant coefficients. Here we have not shown terms which include the operator
v . Ö, since they can be eliminated applying the equation of motion (3.165) for the field Hv .
6The EOM (3.165) is only valid modulo higher order terms. If no pion fields are considered, such higher order
terms may be divided into two classes: the on es with just one field operator Hv and those ones involving more field
operators (like, for instance, the term (HvHv)Hv). The leading correction of the first kind is given by the term
-iß2/(2m). The higher order corrections to eq. (3.165) of the second kind would lead to contact interactions with six
and more nucleon legs and are irrelevant for our discussion. Thus, eliminating the terms (F.8) using the EOM for Hv
would modify the contact interactions with four nucleon legs at order .6.i = 3 (terms like l/m(Hv 7J. a Hv )(HvHv)).
For further discussion on such redundant terms see e.g. refs. [133], [71].
U

To keep the presentation self-contained, we also show the contact interactions written in ref. [78]:
-�CS(NtN)(NtN) - �CT(NtaN) . (Nt aN) ,
- C� [(Nt V N)2 + (V Nt N)2] - C�(NtV N) . (V Nt N)
- C�(NtN) [NtV2N + V2NtN]
- iC� [(NtVN) ' (VNt x aN) + (VNtN) . (Nta x VN)]
- iC�(NtN)(VNt . a x VN) - iC�(NtaN) . (VNt x VN)
- (C�OikOjl + C�OilOkj + C�OijOkz)
x [(NtO"kOiN)(NtO"IOjN) + (OiNtO"kN)(OjNtO"IN)]
- (C�OOikOjl + C�10ilOkj + C�
2
0ijOkl) (NtO"kOiN)(ojNtO"IN)
( 1 I
- 2"C13 (OikOjl + OilOkj) + C140ijOki
, )
x [(OiNtO"kOjN) + (OjNtO"kOiN)] (NtO"IN) .
187
(F.12)
(F.13)
To be able to compare the Lagrangian (F.11), which is expressed in terms ofthe velo city dependent
field Hv , with eq. (F.13), one has to go in eq. (F.11) into the rest-frame system of the nucleon by
choosing
= (1,0,0,0) . (F.14)
Then, the spin operator takes the form SJ.t = (0, 8)7, where Si is given by8
vJ.t
(F.15)
Note furt her that the four-derivative öJ.t (oJ.t) reads öJ.t = (öD, -V) (ö J.t = (ÖD, V)) . It is easy to
demonstrate that the zeroth component of the derivative operator does not appear in such contact
interactions. For that we express öJ.t using the third equality in eq. (F.2) as:
(F.16)
Now, all terms that involve the operator v . 0 acting onto the nucleon field Hv are redundant and
can be dropped in the effective Lagrangian. Further, contracting Öv with the second term in the
parenthesis in eq. (F.16) yields only the spatial derivative, since SO = ° in the rest-frame system.
Finally, we point out that the large (small) component field Hv (hv) defined in eq. (3.160), which
is, in general, a four component Dirac spinor, turns into the two component Pauli spinor N (N):
(F.17)
With these rules it is easy to rewrite the Lagrangian (F.11) in the same notation as in eq. (F.13)
and to establish the connection between the constants CL.,14 and CY1, ... ,18' Concretely, one obtains:
7 The zeroth component of SJl. vanishes because of its definition (F.l) and eq. (F.14).
8 We use here Latin letters to denote the components of various three-vectors. Clearly, we do not distinguish
between co- and contravariant quantities in such cases.

188 F. The complete set of the N N contact interactions with two derivatives
For the terms without derivatives entering eqs. (F.7) and (F.12) one obtains:
G�O = -la12 '
(F.18)
Gs = -2Gs, (F.19)
Let us now take a closer look at the terms in eq. (F.ll). All interactions entering this equation
can be divided into three different classes: the terms al, 2 , 3 ,4 do not contain Sft, the terms a5,6,7,8
involve just one and the terms ag, ... ,18 two insertions of the spin-operator. One can make use of
partial integration for each of these classes separately, in order to eliminate redundant terms. For
example, the acterm can be expressed as a linear combination of the al, 2 , 3-interactions and, thus,
can be completely omitted. It is a matter of choice which terms are required as basis of independent
operators. For instance, to end up with the Lagrangian (F.13) one has to drop the a4,8,17,18-terms,
see eq. (F.18). We, however, point out that not all remaining interactions with two insertions of
the spin-operator (ag, ... ,16-terms in eq. (F.ll) or G7, ... ,lcterms in eq. (F.ll)) are independent.
Only 7 and not 8 such interactions are independent from each other, since one can express e.g. the
terms a9 ,lO,ll as linear combinations of the other a12 , ... ,16-couplings. To keep only independent
contact interactions in the effective Lagrangian we will use an operator basis, which is different
from eq. (F.13). More precisely, we choose the a = {I, 2, 3, 5, 6, 7, 12, 13, 14, 15, 16, 17, 18} -terms
(altogether 13 interactions instead of 14 entering eq. (F.13)).
Let us now work out the consequences of the requirement of these terms to be reparametrization
invariant. Performing the (infinitesimal) transformation (F.4) and replacing w by v at the end, g
we obtain the following constraints on the ai 's:
al - a2 + 2a3 1
'2a5 + a6
1
-a5 - a7 2
a14 + a16 - a18
1 1
a12 + -a15 - -a17
2 2
1 1
a1 3 + -a15 - -a17
2 2
0,
0,
0, (F.20)
0,
Consequently, only seven independent coupling constants enter the corresponding Lagrangian.
Denoting such couplings by Cl, ... ,7 and switching to the nucleon rest-frame system, our final
result for the reparametrization invariant set of independent N N contact interactions at order
L:1i = 2 reads:
.c��
- 1Cl [(NtVN)2 + (VNtN)2] - (Cl + C2)(NtVN) . (VNtN)
- 1C2(NtN) [NtV2N + V2NtN]
9 The complete effect of the reparametrization transformation (F.3) is given by the shift (F.4) of the nucleon
field, if no I/rn corrections are taken into account. Therefore, reparametrization invariance in that case turns into
Galilean invariance.
0,
0.

- i�03{ [(NtVN) . (VNt x ifN) + (VNtN) . (Ntif x VN)]
- (NtN)(VNt . if x VN) + (NtifN) · (VNt x VN)}
1 1 - -
- 2 ( "2 C4 (8ik8jl + 8il8kj) + C58ij8kl )
x [(oiojNtakN) + (NtakOiOjN)] (NtaIN)
- (�06 (8ik8jl + 8il8kj) + (05 - 67)8ij8kl) (NtakOiN)(ojNtaIN)
1 ( 1 - - - )
- 2 2 (C4 - C6) (8ik8jl + 8il8kj) + C78ij8kl
x [(OiNtakOjN) + (OjNtakoiN)] (NtaIN) .
189
(F.21)
The effective Lagrangian (F.21) is sufficient for the derivation of the short-range part of the twonucleon
potential at NLO. To obtain the NNLO potential one needs the Lagrangian .c�� for
contact interactions with four nucleon legs. Let us first discuss the terms with three derivatives
acting on the nucleon field Hv (Hv) .
• Terms with 0/-t, ov, 0p.
It is, c1early, not possible to construct a Lorentz scalar of the form
(F.22)
if the operators f 1 ,2 contain only three derivatives and no velo city or spin operators (or, in
general, if an odd number of the operators 0/-t' Vv and S(J enters f1 and f2) .
• Terms with 0/-t, ov, op and S(J.
The terms without insert ions of the totally anti-symmetrie tensor f et ß7 (J are not parity
invariant. For the terms with f et ß7(J we first note that the derivatives must always act onto
different fields Hv (Hv) beeause of the anti-symmetrie properties of the Levy-Civita tensor.
The only two terms one ean build up are:
f /-t VP (J { (Hv *a /-t 8 vHv )(Hv *a pS(JHv) + (Hv *a v 8 /-tHv)(Hv 8 pS(JHv)} , (F.23)
if /-t VP (J { (Hv *a /-t 8 vHv)(Hv *a pS(JHv) - (Hv *a v 8 /-tHv)(Hv 8 pS(JHv)} . (F.24)
The seeond term vanishes after applying partial integration. We will now show that the
first term ean be eliminated from the Lagrangian using the EOM for the nucleon field and
eqs. (F.2), (F.16) . Let us eonsider only the first term in eq. (F.23) to keep our notation
more eompaet:
f /-t Vp(J (Hv *a /-t 8 vHv)(Hv *a p S(JHv)
= f /-t VP (J ( Hv *a /-t 8 vHv ) ( Hv *aet [vp Vet S(J - 2(Sp Set S(J + Set Sp S(J)] Hv)
(F.25)
= -2 f /-tv
P (J ( Hv *a /-t 8 vHv) ( Hv *aet (i fpetß7 vß S7 S(J + Set {Sp, S(J}) Hv) + . ..
. - � ---ct ( ) ( - �et = 2 z det(R) Hv a /-t (j vHv Hv 0 v ß S 7 S(JHv ) + ... ,

190 F. The complete set of the N N contact interactions with two derivatives
where the matrix R is given by
(F.26)
We have used eq. (F.16) to obtain the seeond line of eq. (F.25). The term vp vQSrr in the
squared braekets in the seeond line of this equation vanishes modulo higher order terms after
applying the EOM for Hv . Sueh higher order terms are denoted in eq. (F.25) by the dots.
To end up with the third line of eq. (F.25) we have used the last equality in eq. (F.2). The
seeond term in the parenthesis (SQ {Sp, Srr}) vanishes beeause of the third equality in (F.2)
and the anti-symmetrie property of the f/lyprr . Finally, the remaining term in the last line
of eq. (F.25) involves the operator v . 1J or v . 71, as is obvious from eq. (F.26), and henee
vanishes modulo higher order terms.
• Terms with 8/l' 8y, 8p and Vrr.
The terms involving the Levy-Civita tensor f /lyprr violate parity invarianee, whereas the
remaining terms ean be eliminated using the EOM for the nucleon field.
• Terms with 8/l' 8y, 8p and Vrr , VQ or Vrr, SQ or Srr, SQ are not Lorentz invariant.
• Terms with 8/l' 8y, 8p, Srr, SQ and vß .
Sueh terms involving f/l ' Y 'p'rr' are not parity invariant, whereas the remaining terms eontribute
at higher orders .
• All remaining terms with more than one insertion of the velo city operator ean be eliminated
using the EOM for the nucleon field.
Thus, there are no eontaet terms with three derivatives at order ßi = 3.
Another possible kind of eontributions at this order within our power eounting seheme is from the
terms with two derivatives suppressed by one power ofthe inverse nucleon mass (ljm-eorrections).
Let us now take a closer look at sueh terms, whieh ean be written as
(F.27)
where the eonstants Bi are expressed in terms of the eoupling eonstants of the lower order Lagrangian.
Note that sinee we do not eonsider the terms with six and more nucleon legs in the
effeetive Lagrangian, whieh do not affeet the interaction between two nucleons, the only eonstants
entering the B's are from LN and L�/v

Consequently, it is not possible to construct Bi in terms of CS,T and of Ci 1 0 and, therefore, there
are no l/m-corrections to contact terms with two derivativesY Finally, we conclude that there
are no terms in L�N=3).
IOIf we would not eliminate the terms in eq. (F.8) using the EOM for the nucleon field, then we would have l/m
corrections to the contact interactions with two derivatives.
llNote that one can have l/m 2 -corrections with two derivatives, which, however, would contribute at order
ßi = 4.
191

Appendix G
Partial wave decomposition of the
N N potential
In this appendix we would like to deseribe the partial wave deeomposition of the two-nucleon
potential. For that we first rewrite the potential V in the form
V =
Vc
+ Va- ih · ih + VSL i �(o\ + 5 2 ) · Cf x ifJ + Va-L 51 · (ifx k) 5 2 · (if x k)
+ Va-q (51 · ifJ (5 2 · ifJ + Va-k (51 . k) (5 2 · k) (G.l)
with six functions Vc (p,p',z), ... , Va-k (P,P',z) depending on p == Ipl, p' == Ip'l and the eosine
of the angle between the two momenta is ealled z. These functions may depend on the isospin
matriees T as weIl. To perform the partial wave deeomposition of V, i. e. to express it in the
standard lsj representation, we have followed the steps of ref. [193]. In particular, we start from
the helicity state representation Iß A1A
2 ), where ß = vip and Al and A
2 are the helicity quantum
numbers eorresponding to nucleons 1 and 2, respeetively. We then expressed the potential in
the IjmA1A
2 ) representation using the transformation matrix (ß A1A2IjmA1A2 ), given in ref. [193].
The final step is to switeh to the Ilsj) representation. The eorresponding transformation matrix
(lsjmlJmA1A
2 ) is given in refs. [194], [193].
For j > 0, we obtained the following expressions for the non-vanishing matrix elements in the
Ilsj) representation:
(jOj I V IjOj)
(jljlVljlj)
27r [ 1
1 dz {Vc - 3Va- + p,2 p2(z2 - 1)Va-L - q2Va-q - k2Va- k} Pj (z) ,
27r [ 1
1 dz {[Vc + Va- + 2P'PZVSL - p,2 p2(1 + 3z2)Va-L + 4k2Va-q + lq2Va-k]
X Pj (z) + [-p'p VSL + 2p, 2 p2zVa-L - 2p'p (Va-q - l Va- k)]
x (Pj-1(z) + Pj +1(Z)) } ,
(j ± 1, IjlVlj ± 1, Ij) 27r [ 1
1 dz {p'p [ -VSL ± 2j � 1 ( -p'PZVa-L + Va-q - l Va-k)]
x Pj(z) + [Vc + Va- + p'pzVsL + p, 2 p2(1 _ z2)Va-L (G.2)
192

(j ± 1, 1jIVIJ =f 1, 1j)
Here, Pj (z) are the conventional Legendre polynomials. For j = 0 the two non�vanishing matrix
elements are
(OOOIV IOOO)
(1l01V 11l0)
211" [ 1 1 dz {Vc - 3Va + p ,2 p 2 (z 2 - l)VaL
211" [ 1 1 dz { ZVC + zVa + p ' p(z 2 - l)VSL
- ((p ,2 + p 2 )z _ 2p ' p) Vaq -
�
- q 2 Vaq - k
2 Va k} ,
+ p ,2 2
p z(1 - z2)VaL
((p ,2 + p 2 )z + 2p ' p) Va k} .
193
(G.3)
Note that sometimes another notation is used in which an additional overall "-" sign enters the
expressions for the off�diagonal matrix elements with l = j + 1, l' = j - 1 and l = j - 1, l' = j + 1.
Our results (G.2), (G.3) agree with the corresponding ones of ref. [193] apart from the off�diagonal
matrix elements (j ± 1, 1jIVaL a1 . (if x k) a2 . (if x k)1J =f 1, 1j) and with those ones for the on�
energy�shell matrix elements given in ref. [108] (up to an overall factor).
By deriving the effective N N potential we assume exact isospin invariance. In that case one can
express the operators Vcn a = {C, (T, SL, (TL, (Tq, (Tk} as
Therefore, the contribution to a state with total isospin I = 0, 1 is given by
(G.4)
(G.5)
The expressions (G.2), (G.3) can be brought into a different but equivalent form using the following
recurrence relation for the Legendre polynomials:
(G.6)

Appendix H
Formulae for the deuteron properties
Here, we collect the formulae needed to calculate the deuteron properties. We denote by u( r)
and w(r) the S- and D-wave coordinate space wave functions, further, the momentum space
representations of u(r)/r and w(r)/r by ü(p) and w(p), respectively. We have
N ormalization
D - state probability
Quadrupole moment
Asymptotic S - state
Asymptotic D /S - ratio T}
RMS (matter) radius :
1000 dpp2 [ü(p)2 + w(p)2] = 1000
1000 dpp2 w(p)2 = 1000 dr w(r)2 ,
Qd = � (oo drr2 w(r) [VSu(r) -w(r)]
20 Jo
dr [u(r)2 + w(r)2] = 1 , (H.1)
= _� (oo dP{VS[p2dÜ(P) dw(p) + 3pw(p) dü(P)]
20 Jo dp dp dp
(H.2)
+p2 (d��)) 2 + 6w(p)2 } , (H.3)
u(r) ----7 As e - ,,( T for r ----7 00 , (H.4)
w(r) ----7 T} As (1 + ;r + h : )2) e- ,,(T for r ----7 00 , (H.5)
rd = l [1000 drr2 [u(r)2 + w(r)2] f/2 , (H.6)
with r = Jm IEdl = 45.7MeV (using m = (mp + mn)/2). Note that the momentum-space representation
of Qd given in eq. (H.3) shows why one cannot use a sharp momentum-space regulator
to calculate this quantity. We also remark that the D-state prob ability is not an observable.
Meson-exchange current corrections to Qd are not given.
194

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