The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.2. Eifective Lagrangians 63
Here hR,L == iJR,L'V denote the SU(2)v-like15 transformations16 given by the matrices e(}R,L'V
and the so-called compensator field h is defined via
(3.99)
Let us repeat once more the logic to make this presentation more dear. An arbitrary group
element go E SU(2)v X SU(2)A can be parametrized via eq. (3.51) in terms of s and � or by a pair
eA, ev as in eq. (3.90) (or, equivalently, by eR and eL given in eq. (3.93)). The quantities s' and,
therefore, also the matrix h can be determined from eq. (3.55). To obtain the transformed fs (or
u' defined in eq. (3.89)), one can again make use of eq. (3.55) or, equivalently, apply eq. (3.98).
The second way is more preferable for our furt her consideration. Note that the matrix h depends,
in general, on space-time due to its explicit dependence on the fs whereas the hR,L in eq. (3.98)
do not.
Let us introduce another useful matrix U as
From eq. (3.98) we see that it transforms as
(3.100)
(3.101)
Consider now the quantity
uJ1, == iut (8J1,U) ut = i (ut 8J1,u - u8J1,ut) = -i ((8J1,ut)u + u8ttut) = -iu (8J1,Ut) u = u1 . (3.102)
Note that because of the definition (3.78) for the SU(2)v generators one has ut = u-1. In the
second and third equalities we have used the relation
(3.103)
One verifies that utt corresponds to the matrix representation of the covariant derivative of �. For
that one recalls that covariant derivatives are characterized by their transformation properties
under the chiral group. Speaking more precisely, the covariant derivative Dtt� should transform in
the same way as � under the subgroup H, which is in our case SU(2)v. The transformation of Dtt�
under the whole group SU(2)v x SU(2)A should have the form of the SU(2) v-transformation.
The only difference is that now the transformation parameters (the s" s entering h) are functions
of the group element and of the fs, given by eq. (3.55). Since we use the matrix form u defined in
eq. (3.89) for the fs, we have to consider the transformation properties of the u and not of the fs
under SU(2)v. As we already know from eq. (3.59), for a SU(2)v-transformation h = e(}v'
v
EH
one has h = h. Furthermore, as follows from eq. (3.93), eR = eL = ev since eA = O. Therefore,
hL = hR = h. Consequently, the matrix u transforms under h E H = SU(2)v as
u ---+ u' = huh-1 • (3.104)
One can easily check using eqs. (3.104) and (3.33) that the �'s are the basis of the adjoint representat
ion of SU(2)v as they should be, see eq. (3.60). For the covariant derivative of � in the matrix
15It should be kept in mind that the (h,L are, in general, not scalars with respect to parity operation as the
transformation parameters of an ordinary SU(2)v rotation.
lßWe refrain here from introducing the commonly used notation, in which the matrices hR,L are denoted by 9R,L.
In our opinion, this leads to confusion, since hR,L do not belong to the corresponding representation of the groups
SU(2)R,L.