The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.4. Including nucleons 73
3.4 Including nucleons
In the previous section we have considered chiral perturbation theory for 1r7r scattering. Based on
the most general chiral invariant Lagrangian, one obtains the amplitude in form of an expansion
in powers of small external momenta. The spontaneously broken chiral symmetry guarantees
the weakness of the interactions between the Goldstone modes (pions). This allows to perform
perturbation theory for the S-matrix. At each order in the low momenta one has a finite number
of diagrams to be considered. More complicated graphs with more loops contribute, in general,
to higher orders in this expansion. This scheme can be naturally extended to include processes
with nucleons. The starting point is again the most general chiral invariant effective Lagrangian
for Goldstone bosons interacting with the matter fields, that can be constructed along the lines
introduced in the section 3.2.
The first systematic calculation of pion-nucleon scattering in the framework of chiral perturbation
theory have been performed by Gasser et al. in 1988 [68]. The nucleons have been treated in
this approach fully relativistically. In such a formulation one loses the one-to-one correspondence
between the number of loops and the power of small extern al momenta (pion four-momenta or
mass or nucleon three-momenta).30 The origin of this unwanted feature is that the inclusion of
nucleons unavoidably introduces a new mass scale in the problem, the nucleon mass. This scale is
not a soft one (like the pion mass). It cannot be regarded small and does not vanish in the chiral
limit.
An elegant solution of this problem has been proposed by Jenkins and Manohar in 1991 [69],
see also Bernard et al. in 1992 [70]. The idea is to use a simultaneous expansion in powers of
Q / Ax rv q / ( 41r f 1r) and Q / m, w he re m is the nucleon mass. Let us now briefly explain how such
an expansion may be performed. The four-momentum of the nucleon has the form
(3.157)
where vJl is the four velo city satisfying v 2 = 1. For a heavy nucleon it is convenient to parametrize
PJl as
(3.158)
where lJl is a small residual momentum v·l «m. The advantage of such a parametrization is that
the trivial kinematic dependence of PJl on the large term mVJl is now explicit. In the rest-frame
with vJl = (1,0,0,0) the three-momentum is entirely given by f and PJl = cj m2 + [2, T). Note
that in the limit m ---+ 00 the nucleon four-velocity becomes a conserved quantum number since
its change by a finite amount would lead to an infinite moment um transfer between the incoming
and outgoing nucleons [172]. To get rid of the nucleon mass term in the Lagrangian for the free
Dirac field
(3.159)
one can introduce the eigenstates H and h of the velo city operator p via
H =
p+ eimv.x'l!
v ,
(3.160)
where E� is the projector operator onto the eigenstates of the four-velocity operator v corresponding
to the eigenvalues ±1:
30 This is true for the conventional regularization schemes like, for example, dimensional regularization.
(3.161)