The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

60 3. The derivation of nuc1ear forces from chiral Lagrangians
Let us now make a few comments about eq. (3.74). This is, in fact, a crucial result for EFT,
since it says, that the interaction between the Goldstone bosons vanishes at low energies. Indeed,
it is seen in eq. (3.74) that one can always rewrite the Lagrangian in such a form, that only
derivative interactions between the Goldstone bosons are present. We will show below using the
power counting arguments that, as a consequence, the scattering amplitude for any process with
an arbitrary number of Goldstone particles is proportional to some positive power of external
momenta. We point out again that this statement concerning the scattering amplitude is independent
of any particular choice of the realization of the group G. In principle, we could equally
well use an arbitrary nonlinear realization (�, cjJ) to construct the Lagrangian, where cjJ denotes a
set of matter fields, their derivatives as well as derivatives of Goldstone fields. However, insisting
on the chiral invariance of the Lagrangian density would be a non-trivial problem because of the
complicated nonlinear transformation rules of the fields and their derivatives. Using the CCWZ
realization to construct the efIective Lagrangian via eq. (3.74) has the advantage that all building
blocks transform covariantly under G, Le. in the way shown in eq. (3.54), guaranteeing that any
H-scalar is also G-invariant.
To complete the discussion about the construction of a chiral invariant Lagrangian let us apply
now the chiral SU(Nf)v x SU(Nf)A group as a particular example of the general Lie algebra
(3.52). In that case all indices in eq. (3.52) vary from 1 to Ni - 1 and the structure constants G's
are represented in terms of the structure constants f's of the SU(Nf) group as follows:12
Gabe = 0, (3.75)
For example, for the SU(2)v x SU(2)A group we have fijk fijk, where fijk is the ordinary
three-dimensional totally antisymmetric tensor with f123 = 1. Note that one can represent the
chiral SU(Nf)V x SU(Nf)A group by SU(Nf)L x SU(Nf)R defining the corresponding generators
TL and TR as
(3.76)
where Vi and Ai are the generators of the SU(Nf)V and SU(Nf )A, respectively. Then SU(Nf)L
and SU(Nf)R build two independent subgroups of the whole chiral group with the Lie algebra
[(TL ) i , (TL)j]
[(TR) i , (TR)j]
[(TL ) i , (TR)j]
Let us now be more specific and concentrate on the chiral SU(2)v x
the following representation of the generators Vi and Ai fijk (TL h ,
fijk (TRh ,
o.
A - _ _ ilT i
l - 2 '
SU(2)A
(3.77)
group. We choose
where T i is a Pauli matrix and 1 is some flavor scalar matrix quantity with the properties:
12 = 1 , I t = I .
(3.78)
(3.79)
For example, 1 can be equal to the ')'5 matrix. Then, one has the same representation for the
SU(2)A generators as in the case of the QCD Lagrangian (3.1), as can be seen from eq. (3.9).
1 2 Here we use a definition of the chiral Lie algebra, which differs by the factor i from the one given in the last
section , eq. (3.36). This factar can be absorbed by the corresponding redefinition of the generators.