The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

where qJ-! is chosen to satisfy (v + q/m)2 = 1, and, consequently, v . q = O(q2/m). As pointed
out in section 3.4, it is more convenient to work with the fields Hv defined in eq. (3.170) 4 instead
of Hv , since the first ones transform covariantly, via eq. (3.172), und er the reparametrization
transformation (F.3), see ref. [182]:
Hw = eiq.x Hv . (F.4)
At order l/m the two fields Hv and Hv are connected via:
- ( i Hv = 1 +
2m {J ) Hv · (F.5)
In what follows, we will not be interested in the explicit calculation of the l/m-corrections to
the contact interactions. This is because in our power counting scheme one has: Q / m '" Q2 / A�.
Consequently, all l/m-terms are suppressed against the corresponding ones, whose coefficients
scale by inverse powers ofAx' As will be shown below, such l/m-corrections to contact interactions
with four and more nucleon legs contribute at higher order and are irrelevant for our calculations.
Therefore, we can set
Further, we will need various properties for the building blocks of the effective Lagrangian, which
are listed in table F.1.
I Operator 11 H C x --+ x ' = Ax
vp + - AJ-!v V v
Hv Hv + + HvHv
HvSJ-!Hv + + det(A)AJ-!v HvSvHv
Table F.l: Transformation properties for the building blocks of the effective Lagrangian
under hermitian (H) and charge (C) conjugations and homogeneous Lorentz transformation
(x --+ x ' = Ax).
Consider first the contact interactions without derivatives and pion fields. Here and in what
follows, we will not consider the contact interactions with insert ions of the isospin matrices, since
those ones can be eliminated via Fierz reshuffiing or anti-symmetrization of the corresponding
N N potential, as detailed in appendix E. The two possible terms are
where CS,T are some constants. All other terms can be reduced to those two using the first
equality in (F.2) and the fact that v2 = 1. The terms in eq. (F.7) are, clearly, reparametrization
invariant.5
4 In the general case of nucleons interacting with pions and/or external fields one has to replace the derivative
8" in eq. (3.170) by the covariant one. In this appendix we will, however, not consider such a general situation.
5 As is obvious from eq. (F.4), only terms with derivatives of Hv can break the reparametrization invariance if
no l/m corrections are considered.
185
(F.6)
(F.7)