The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

168 5. Summary and outlook
• In sec. 3.8.1 we have also discussed the structure of the NNLO potential in the method
of unitary transformation. It turns out that in that case one obtains the same result
in both the projection formalism and time-ordered perturbation theory. The NNLO
potential has been calculated within a different (but leading to the same result) scheme
by the Munich group [108].
• In sec. 3.8.3 we have generalized the projection formalism to include effects of the
virtual Ll-isobar excitation within the "small scale expansion" and discussed the leading
contribution to the N N potential from the intermediate Ll 'so In that case, both the
method of unitary transformation and time-ordered perturbation theory lead to the
same result.
• Furthermore, we have considered the leading contributions to the three-nucleon potential.
As in time-ordered perturbation theory, we find that these sum up to zero.
However, the mechanism for the cancelation between so me of the graphs is distinctively
different in the projection formalism since it is not related to an intricate cancelation
of terms generated by the iteration of the energy-dependent two-nucleon potential in
old-fashioned time-ordered perturbation theory. Rather, in the approach here, these
cancelations can be traced back to the appearance of "reducible" graphs whose precise
meaning is explained in section 3.8.1. These diagrams are in fact responsible for the
orthonormality of the wave functions and are thus sometimes called "wave function
re-orthonormalization" graphs.
• We have also constructed the most general reparametrization invariant effective Lagrangian
for the contact interactions with four nucleon legs up to order Lli = 3, L}J;;5�3)
(reparametrization invariance is a consequence of Lorentz invariance of the underlying
theory, see ref. [182]). The Lagrangian used in previous calculations, see e.g. [74], [76],
[78], [127], contains fourteen free parameters Ci,...,14 and leads to two-nucleon forces,
which depend on the total moment um P of two nucleons. Whereas such P-dependent
forces do not affect calculations of the two-nucleon system in the c.m.s., they would be
very important for processes including other particles and for three and more nucleons.
We have shown that requiring the reparametrization invariance for the contact terms
in the effective Lagrangian yields several constraints on the parameters Ci. Only seven
of these fourteen parameters are really independent. The resulting NLO and NNLO
two-nucleon potentials do not depend on the total moment um P.
3. Thirdly, in chapter 4 we have calculated nuclear forces and properties of the two-nucleon
system based on a chiral effective field theory and the projection formalism. The results of
this investigation can be summarized as follows:
• We considered the two-nucleon potential at NNLO resulting from the projection formalism.
It consists of one- and two-pion exchange diagrams, including dimension two
(Lli = 2) insertions from the pion-nucleon Hamiltonian. The corresponding LEes have
been taken from an investigation of rrN scattering [195]. In addition, there are two and
seven contact interactions without and with two derivatives, respectively. The coupling
constants of these terms must be fixed by a fit to data.
• For large momenta, the potential becomes unphysical and has to be regularized. We
performed this regularization on the level of the Lippmann-Schwinger equation, as
explained in sec. 4.1 using either a sharp or an exponential regulator function. At
NLO, physics does not depend on the cut-off in the range between 400 and 650 Me V. At