The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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.