The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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