The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

166 5. Summary and outlook
• We have also considered an alternative way of determining the operator A from the
half-shell two-nucleon T-matrix. This leads to the linear equation (2.94), which has
been solved numericaIly. For all selected values of the cut-off A the solutions of the
linear and nonlinear equations for A agree perfectly with each other.
• We have examined a low-momentum expansion of the potential. For that we have
expanded the heavy meson exchange term in a string of local operators with increasing
dimension but kept the light meson exchange unchanged, see eqs. (2.115), (2.116).
The corresponding coupling constants accompanying these local operators, which are
monomials of even power in the momenta, can be determined precisely from the exact
solution. For A =300 (400) MeV their values are given in table 2.1. We have
shown that they are of "natural" size, i.e. of order one, with respect to the mass scale
Ascale = 600 MeV. We have also discussed the relation of this scale to the mass of the
heavy meson, which is integrated out, and the convergence properties of such type of
expansion. In particular, to recover the binding energy within a few percent, one has
to retain terms of rather high order in this expansion, cf. eq. (2.116). This is to be
expected due to the unnatural smallness of this energy on any hadronic mass scale. The
3 SI scattering phase shift can be weIl reproduced up to kinetic energies llab ':::' 120 Me V
with the first three terms in the contact term expansion.
• Based on the expanded heavy meson exchange term, we have also determined the
constants Ci directly from a fit to the phase shifts. This is equivalent to the procedure
performed in an effective field theory approach. We could show that as long as one does
not include polynomials of order six (or higher), the resulting values of these constants
are close to their exact ones. Furthermore, the binding energy is reproduced within
2%. Including dimension six terms, the fits become unstable. This can be traced back
to the fact that the contribution of such terms to the phase shifts are very small (at
low and moderate energies) and thus can not really be pinned down.
• We have also studied the quantum averages of the expanded potential in the bound and
scattering states. For A = 300 MeV, the expansion parameter is of the order of 1/2 and
we find fast convergence for the bound and the low-Iying scattering states, as shown
in table (2.3). As expected, for scattering states with higher energy, the convergence
becomes slower.
• We have determined the effective range parameters using the expansion (2.116) for
the potential and demonstrated the predictive power of such an effective theory. In
particular, having fixed the free parameters in the potential from the first n terms in
the effective range expansion one obtains a prediction for the next coefficient, which has
not been used in the fit. This is different from the pionless effective theory considered
in the seetion 2.2, where no predictions are possible.
• In the model space of small momenta only, one can also study the non-Iocalities in
the coordinate space representation. We have shown that for typical cut-off values, the
effective potential V(x, x') is highly non-Iocal and looks very different from the original
one. For very large values of the cut-off, one recovers the original local potential.
While this thesis was being written down, a similar work done by Bogner et al. [214] has
been reported, showing a considerable research activity in this field. In this work, effective
low-momentum potentials defined below a given moment um space cut-off A are obtained
from the Bonn-A and Paris potentials using the folded-diagram method of Kuo, Lee and
Ratcliff [215]. The method leads to non-hermitian potentials and preserves the half-shell
NN T-matrix.