The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

58 3. The derivation of nuclear forces from chiral Lagrangians
Thus, the most general chiral invariant effective Lagrangian for �, cjJ can be constructed solely
from the cjJ's. This simplifies the procedure enormously. Indeed, according to eq. (3.54), the cjJ's
transform "covariantly" under G: the only difference between the G- and H- transformations is
that in the first case the parameters s are complicated nonlinear functions of the ts and of the
group element. Therefore, any H-scalar built up from the cjJ's is also G-invariant.
Since we would like to describe the interaction between those fields � and cjJ within the EFT approach,
we have also to include derivatives of �, cjJ into the Lagrangian. Although these derivatives
transform linearly under the subgroup H, as it follows from (3.60) and (3.63), it is clear that the
field derivatives, in general, do not belong to the CCWZ realization (3.53)-(3.55). The number of
the ts is equal to the number of the A's and, therefore, the field derivatives cannot belong to the
set of the ts. In addition, they do not belong to the set of the cjJ's. For example, the derivative
0/lcjJ transforms as
(3.66)
The first term in this equation does not vanish in general, since the parameters s ' depend on
�. Therefore, eq. (3.66) violates eq. (3.54). In particular, eq. (3.64) is not valid any more for
derivatives. Although in that case s' = 0,
the derivative of s ' is different from zero. Consequently,
we can not proceed in the above way to construct the effective Lagrangian including the field
derivatives. To generalize the formalism introduced above to include the field derivatives, let us
first consider the following problem. In addition to the set of the ts we introduce fields \)I such
that the group G is realized (nonlinearly) on (�, \)I) in the following manner: the �'s form the
CCWZ realization of G and transform via eq. (3.53) and the \)I's transform linearly under H and
in some arbitrary way under the whole group G:
go (�, \)I) = (t, \)I'), (3.67)
where e are given by eq. (3.55) and \)I' are (nonlinear) functions of go, � and \)I. How to construct
the CCWZ realization for both � and \)I? We first note that for the specific group transformation
go = e-f A eq. (3.67) takes the form
(3.68)
where the �'s are the particular choice of \)I' corresponding to the transformation go = e-�·A .
Following ref. [67], we define a pair (�, �)* by
- _ _ -
(�, \)I) * = (� ,\)I) - e �·A (0, \)I) . (3.69)
We will now show that (�, �)* belongs to the CCWZ realization. For an arbitrary group element
go E G we have:
(3.70)
Now we have to apply the transformation eS'.v on both quantities in the bracket in the last
equality of eq. (3.70). We first note that the point � = 0 is invariant und er H-transformations.
This follows from eq. (3.55) and from the uniqueness of the parametrization (3.51). Further, the
�'s transform linearly under H in the same way as the \)I's, for which a linear transformation
property under H has been required. Thus, we obtain for eq. (3.70):
(3.71)