The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

62 3. The derivation of nuc1ear forces from chiral Lagrangians
where the quantity EJ.t is defined as
(3.88)
A more detailed discussion concerning the stereographical parametrization can be found in [161].
The use of the stereographical coordinates requires sometimes rat her tedious calculations. We will
now briefly discuss another more elegant way, which is commonly used in the analysis of processes
with one nucleon. Let us first introduce the standard notation and define the matrix u as
(3.89)
For the generators V and A we will again use the representation (3.78) with l = 1'5 , To establish
the transformation properties of the u we will apply eq. (3.55), which defines how the Cs transform
under go E SU(2)v X SU(2)A. An arbitrary group element go needs not necessarily to be
parametrized in terms of s and � via eq. (3.51). Equally weIl we can write
(3.90)
Here, eV,A are transformation parameters of the SU(2)V,A groups. We can now rewrite eq. (3.55)
using the parametrization (3.90) as
(3.91)
where s' is, as usual, a function of � and eV,A. Using the projectors PR and PL onto the states
with a definite chirality defined in eq. (3.5) we can express (ev . V + eA . A) as
The parameters e R,L are defined by
(3.92)
(3.93)
and VR,L == VPR,L are the generators of SU(2)R,L. Projecting eq. (3.91) onto states with adefinite
chirality we obtain the following equations:
PRi'R" V e�' v
PLih' V e-�' v
Here we have used the properties of the projectors PR,L:
and the following equalities:
P e·v Re e s'·V ,
P -e·v
Le e s'·V .
(3.94)
(3.95)
(3.96)
(3.97)
Since PR,L commute with the V, one can completely drop these projectors in eqs. (3.94), (3.95).
We can now easily read off the transformation properties of the u from eqs. (3.94), (3.95):
90 , h h- 1 h h- 1
U -=---7 U = RU = U L . (3.98)