The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

Appendix F
The complete set of N N contact
interactions with two derivatives
The most general effective Lagrangian for nucleons given in refs. [76], [78] contains 14 contact
interactions with two derivatives, which are multiplied by the coefficients C 1 , ... 14. In this appendix
we would like to show that 7 of these 14 terms are not independent from the others and have fixed
coefficients. Moreover, considering the couplings accompanying these terms as apriori unknown
and non-vanishing quantities would violate general symmetry principles of the theory.1 The
situation he re is similar to the familiar case of the nucleon kinetic energy term Nt\l 2 j(2m)N in
the Lagrangian corresponding to the Hamiltonian (3.199). The Lorentz invariance of the theory
would be broken if one allows an arbitrary coefficient for the term Nt\l 2 N.
As explained in the text, the nucleons in the effective theory can be described by velocity dependent
fields Hv and hv, see eq. (3.160). 2 The total momentum Pp. of the nucleon is then parametrized
by a pair (Vp., Ip.) via eq. (3.158). The small component field hv can be eliminated from the
theory, which leads to 1jm-corrections in the effective Lagrangian. Clearly, such correction terms
have fixed coefficients, defined by the structure of the original relativistic Lagrangian. Explicitly
integrating out the small component field hv has not yet been worked out for nucleon interactions
with four and more legs. Instead of explicitly integrating out hv, it is easier in such a case to write
down all possible terms consisting of Hv , vp. and Sp., which is defined by
and of covariant derivatives of the nucleon and pion fields.3 Note that the spin operator Sp. obeys
the following relations (in four dimensions):
2 3 1
S· v = 0, S = -4' {Sp., Sv} = 2 (vP.vv - 9p.v) , [Sp., Sv] = ifp.vpuV P S u , (F.2)
where we have used the convention f0 1 2 3 = 1. One can require the reparametrization invariance
[182] of the effective Lagrangian to fix the coefficients for the 1jm-corrections and to eliminate
the terms violating Lorentz symmetry. The effective Lagrangian expressed in terms of velo city
dependent fields is reparametrization invariant if it preserves its form under the reparametrization
(F.1)
(Vp., Ip.) +-t (Wp., kp.) = (Vp. + qp. , Ip. - q p.) , (F.3)
m
1 In particular, the Galilean invarianee would be broken.
2 In sec. 3.4 we have suppressed the label v for those fields.
3 AH Dirae bilinears fIv rHv (r = {I, 'Y5 , 'YI" 'Y5'YI" O"l' v }) ean be expressed in terms of (fIvHv), (fIvSI' Hv ), v!'
and of the anti-symmetrie Levi-Civita tensor El'vpu , see e.g. [72].
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