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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3.3. <strong>Chiral</strong> perturbation theory with pions 69<br />
where the superscript (2) stands for the number of derivatives and the quark mass matrix <strong>in</strong>sertions<br />
as expla<strong>in</strong>ed above. Proceed<strong>in</strong>g <strong>in</strong> the same way as we did for [)2) one can construct the nextto-Iead<strong>in</strong>g<br />
order Lagrangian C(4) , which conta<strong>in</strong>s 8 <strong>in</strong>dependent terms25 [63], [64]:<br />
c(4) LI (OJ1ut oJ1U)2 + L 2 (0J1UOvut) (oJ1Uo v ut) + L 3(0J1U0J1ut OvUo v ut)<br />
+ L4(0J1U0J1ut) (M(U + ut)) + L5 (OJ1UoJ1ut (MU + UtM)) (3.141)<br />
+ L6(M(U + Ut))2 + L7((Ut - U)M)2 + L8((U M)2 + (ut M)2) ,<br />
where LI,. " ,L8 are the so-called low energy constants (LEC's), that have to be fixed from<br />
experiment.26<br />
3.3 <strong>Chiral</strong> perturbation theory with pions<br />
In this section we will outl<strong>in</strong>e the basic ideas of chiral perturbation theory on an example, which is<br />
the elastic pion-pion scatter<strong>in</strong>g. <strong>The</strong> start<strong>in</strong>g po<strong>in</strong>t is the most general Lagrangian constructed as<br />
described <strong>in</strong> the last section. Apart from the chiral symmetrie part that can be classified <strong>in</strong> powers<br />
of derivatives act<strong>in</strong>g on the pion fields, various symmetry break<strong>in</strong>g terms with <strong>in</strong>sertions of the<br />
quark mass matrix and conta<strong>in</strong><strong>in</strong>g any number of field derivatives enter the efIective Lagrangian.<br />
Based on that Lagrangian, one can calculate the tree diagrams for arbitrary processes with any<br />
number of pions. Furthermore, the lead<strong>in</strong>g approximation to the scatter<strong>in</strong>g amplitude can be<br />
obta<strong>in</strong>ed us<strong>in</strong>g the <strong>in</strong>teractions from c(2), s<strong>in</strong>ce <strong>in</strong>sert ions of vertices from higher order Lagrangians<br />
lead to a suppression <strong>in</strong> powers of momenta or quark masses. Such an expansion clearly makes<br />
sense only if the momenta of the pions are small.<br />
<strong>The</strong> efIective field theory technique allows to go beyond the tree approximation and to systematically<br />
calculate the quantum corrections represented by loop graphs. All ultraviolet divergences<br />
can be absorbed <strong>in</strong> a redef<strong>in</strong>ition of the coupl<strong>in</strong>g constants accompany<strong>in</strong>g the <strong>in</strong>teractions <strong>in</strong> the<br />
Lagrangian order by order <strong>in</strong> powers of low external momenta and quark masses. To see how this<br />
works <strong>in</strong> practice, we need to establish the power count<strong>in</strong>g rules [60] for an arbitrary scatter<strong>in</strong>g<br />
process. <strong>The</strong> on-shell S-matrix element for a reaction with N external pions can be expressed<br />
as27<br />
(3.142)<br />
where PI , P2 , ... , PN are momenta of external pions and TI is a product of phase space factors correspond<strong>in</strong>g<br />
to the <strong>in</strong>com<strong>in</strong>g and outgo<strong>in</strong>g particles. <strong>The</strong> transition amplitude M can be rewritten<br />
<strong>in</strong> the form<br />
(3.143)<br />
M == M(Q,g,fL) = QV f( Q ,g) .<br />
Here, Q denotes a generic external momentum, g corresponds to a comb<strong>in</strong>ation of the pert<strong>in</strong>ent<br />
coupl<strong>in</strong>g constants and fL is the renormalization scale. To calculate 1/ we have to count all derivatives,<br />
pion propagators, moment um <strong>in</strong>tegrations and 6-functions for a specific diagram, s<strong>in</strong>ce these<br />
are the only dimensionsfull elements apart from the coupl<strong>in</strong>g constants. For a process with I <strong>in</strong>ner<br />
pion l<strong>in</strong>es we have:<br />
1/ = L Vi di - 21 + 4L , (3.144)<br />
25This is the most general Lagrangian for SU(3). For the two-fiavor case, not all of these terms are <strong>in</strong>dependent<br />
from each other.<br />
26 In order to be able to perform renormalization <strong>in</strong> the presence of external sources one also needs to <strong>in</strong>clude <strong>in</strong><br />
eq. (3.141) furt her terms, which do not conta<strong>in</strong> the pion fields. For more details see e.g. refs. [63], [64].<br />
27Here we will only consider connected diagrams.<br />
fL