The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

118 3. The derivation of nuclear forces from chiral Lagrangians
citations. The situation here is similar to the NNLO two-nucleon potential. In that case one
has three independent 1f1fNN vertices, two of which, the C3- and q-couplings, are governed by
intermediate Ll-excitations. An explicit inclusion of the Ll's leads to NLO corrections to the 2N
potential, if the delta-nucleon mass splitting is treated as the small quantity, on the same footing
with M1f' We will now consider the situation with the 3N force if the Ll's are included. Then,
many additional contributions to the leading 3N force at v = -1 arise from eq. (3.323), that
correspond to diagrams with intermediate Ll-excitation. The first term in the second line and
the last term in this equation refer not only to the TPE graphs of fig. 3.23 but also to the TPE
3N force with one intermediate Ll shown in fig. 3.26 (1). The second term in the first li ne of
eq. (3.323) together with the second and third terms in the last line lead to OPE diagram with
virtual Ll-excitation shown in fig. 3.26 (2). Finally, the first term in the last line of this expression
refers to diagram 3 in fig. 3.26. Note that in all cases one gets the same result from the projection
formalism and time-ordered perturbation theory, since reducible topologies are impossible
for diagrams shown in fig. 3.26. Furthermore, one verifies that if the mass of the Ll is regarded
to be large, the graphs of fig. 3.26 go into the diagrams shown in fig. 3.25. The saturation of the
corresponding vertices due to intermediate Ll-excitations is now transparent.
One should stress that not all these 3N forces are new. For example, the TPE 3N interaction
with one intermediate delta is included in many phenomenological models, like for instance, the
Fujita-Miyazawa [42] or the Urbana [197] force. The remaining 3N forces 2 and 3 in fig. 3.26 that
include contact interactions are, certainly, less common, since most of the "realistic" 3N forces
are based on meson-exchange models. However, implicitly, such short range contact interactions
are also present in form of the heavy meson-exchange.
Finally, we would like to discuss another kind of forces, which may aIso appear in the threebody
calculations: two-body interactions, that depend on the total moment um P of a twonucleon
subsystem. Such three-body-like two-nucleon forces do not affect the pure two-nucleon
calculations in the cms, where P = 0, but become important for processes including additional
particles (for example, pion production on the deuteron) and for three and more nucleons, where
the total moment um P of the two-nucleon subsystem is not conserved any more. The Lagrangian
eq. (F.13) given in refs. [77], [78] for the contact interactions with two derivatives will necessarily
lead to the forces of this kind. Moreover, fitting the N N phase shifts will only fix seven parameters
of the fourteen entering eq. (F.13), see e.g. [74], [76], [78], [127]. Thus, the P-dependent forces
appear to have unknown coefficients. Such a situation is, clearly, not satisfactory, since Lorentz
invariance would be violated, see appendix F. In this appendix we have analyzed all possible
contact interactions with four nucleon legs up to order Lli = 3, requiring their reparametrization
invariance. It turns out that only seven independent coupling constants enter the Lagrangian
L�;.,= 2), which all can be fixed from nucleon-nucleon scattering. Furthermore, the N N potential
at NLO does not depend on the total momentum P (as it should because of Galilean invariance).
We have also found that no 1/m-corrections to the contact terms with two derivatives appear in
the effective Lagrangian and that no terms enter L�;.,=3). Consequently, also our NNLO potential
does not depend on P. The P-dependent N N forces may only appear at order N3LO and will
not introduce new unknown coefficients beyond the ones needed in the 2N cms.