The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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)