The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
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where qJ-! is chosen to satisfy (v + q/m)2 = 1, and, consequently, v . q = O(q2/m). As po<strong>in</strong>ted<br />
out <strong>in</strong> section 3.4, it is more convenient to work with the fields Hv def<strong>in</strong>ed <strong>in</strong> eq. (3.170) 4 <strong>in</strong>stead<br />
of Hv , s<strong>in</strong>ce the first ones transform covariantly, via eq. (3.172), und er the reparametrization<br />
transformation (F.3), see ref. [182]:<br />
Hw = eiq.x Hv . (F.4)<br />
At order l/m the two fields Hv and Hv are connected via:<br />
- ( i Hv = 1 +<br />
2m {J ) Hv · (F.5)<br />
In what follows, we will not be <strong>in</strong>terested <strong>in</strong> the explicit calculation of the l/m-corrections to<br />
the contact <strong>in</strong>teractions. This is because <strong>in</strong> our power count<strong>in</strong>g scheme one has: Q / m '" Q2 / A�.<br />
Consequently, all l/m-terms are suppressed aga<strong>in</strong>st the correspond<strong>in</strong>g ones, whose coefficients<br />
scale by <strong>in</strong>verse powers ofAx' As will be shown below, such l/m-corrections to contact <strong>in</strong>teractions<br />
with four and more nucleon legs contribute at higher order and are irrelevant for our calculations.<br />
<strong>The</strong>refore, we can set<br />
Further, we will need various properties for the build<strong>in</strong>g blocks of the effective Lagrangian, which<br />
are listed <strong>in</strong> table F.1.<br />
I Operator 11 H C x --+ x ' = Ax<br />
vp + - AJ-!v V v<br />
Hv Hv + + HvHv<br />
HvSJ-!Hv + + det(A)AJ-!v HvSvHv<br />
Table F.l: Transformation properties for the build<strong>in</strong>g blocks of the effective Lagrangian<br />
under hermitian (H) and charge (C) conjugations and homogeneous Lorentz transformation<br />
(x --+ x ' = Ax).<br />
Consider first the contact <strong>in</strong>teractions without derivatives and pion fields. Here and <strong>in</strong> what<br />
follows, we will not consider the contact <strong>in</strong>teractions with <strong>in</strong>sert ions of the isosp<strong>in</strong> matrices, s<strong>in</strong>ce<br />
those ones can be elim<strong>in</strong>ated via Fierz reshuffi<strong>in</strong>g or anti-symmetrization of the correspond<strong>in</strong>g<br />
N N potential, as detailed <strong>in</strong> appendix E. <strong>The</strong> two possible terms are<br />
where CS,T are some constants. All other terms can be reduced to those two us<strong>in</strong>g the first<br />
equality <strong>in</strong> (F.2) and the fact that v2 = 1. <strong>The</strong> terms <strong>in</strong> eq. (F.7) are, clearly, reparametrization<br />
<strong>in</strong>variant.5<br />
4 In the general case of nucleons <strong>in</strong>teract<strong>in</strong>g with pions and/or external fields one has to replace the derivative<br />
8" <strong>in</strong> eq. (3.170) by the covariant one. In this appendix we will, however, not consider such a general situation.<br />
5 As is obvious from eq. (F.4), only terms with derivatives of Hv can break the reparametrization <strong>in</strong>variance if<br />
no l/m corrections are considered.<br />
185<br />
(F.6)<br />
(F.7)