The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.4. Inc1uding nuc1eons 75
to eliminate derivatives of the nucleon field in higher order Lagrangians. In eq. (3.167) the dots
correspond to terms of order Q and higher. Using this equation we can rewrite derivatives acting
on the nucleon field in interactions like � ... 82n fJk'J! in terms of other operators of the same
and higher orders. Such terms are anyway present in the effective Lagrangian, which contains all
possible chiral invariant interactions. Thus, eliminating the nucleon time derivatives only modifies
the values of couplings accompanying the interactions in L. With other words, one can simply
drop time derivatives acting on the nucleon field in all interaction terms, leaving a single time
derivative in the free Dirac Lagrangian. Another more elegant way to eliminate the derivatives of
the nucleon field is to use (nonlinear ) field transformations [133]. For a recent discussion on that
approach as well as for concrete examples see the reference [71]. For various problems related to
the inclusion of higher derivative operators in the Lagrangian see refs. [176], [177], [178], [179].
As soon as one has solved the problem with the time derivatives acting on the nucleon fields, the
nonrelativistic reduction can be performed in a similar way as for free nucleons. The only difference
is that the equations of motion for the large and small component fields contain corrections from
higher order terms. Eliminating h from the Lagrangian yields then coefficients of 1/ m corrections.
Equivalently, the effective Lagrangian can from the beginning be derived in terms of nonrelativistic
nucleon fields. In such a case not all of the coupling constants are arbitrary: so me of them have
definite coefficients proportional to inverse powers of the nucleon mass. Those coefficients are
fixed by the Lorentz invariance of the original relativistic Lagrangian. The simplest example of
such terms is the nonrelativistic kinetic energy Nt\!2/ (2m)N. Technically, one can obtain those
fixed coefficients in the easiest manner requiring the so-called reparametrization invariance of the
Lagrangian [182]. Consider the heavy baryon effective field theory discussed at the beginning of
this section. Instead of the usual fields for nucleons it is convenient to introduce the large and
small component fields according to eq. (3.160). If the nucleon mass is very large, the four velo city
becomes a conserved quantum number. As can be seen from eq. (3.160), the fields H and h are
velocity dependent and should be, in principle, denoted by the index v.32 The effective Lagrangian
is then given by
Leff = L Lv (Hv, vf.t, ... ) , (3.168)
v
where dots correspond to other ingredients like, for instance, covariant derivatives of the nucleon
and pion fields. Reparametrization invariance says that we could equally well chose another
parametrization of the four moment um to describe the same physics:
(3.169)
where qf.t has to satisfy (vf.t + qf.t/m)2 = 1. Luke and Manohar [182] proposed to construct the
effective Lagrangian in terms of Hv defined as33
where
- 1 + #
Hv = Hv ,
J2(1 +w·v)
(3.170)
(3.171)
32We did not do that in the above discussion since the velo city is anyway conserved.
33In a more general case when one has to incorporate external fields and/or pions the derivative of the nucleon
field in eqs. (3.170), (3.171) should be replaced by a covariant one.