The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

2.3. Going to higher energies: a toy model 41
numerically, we can determine the constants Ci by fitting eq. (2.116) to V'(q', q) - VIight (q', q). This
is done numerically using the standard FORTRAN subroutines for polynomial fits to functions of
one variable and taking the corresponding polynomials typically of order ten to eleven. Once this is
done for the functions V(q, 0) and V(q, q) all constants Ci, C: in eq. (2.117) can be easily evaluated.
For the potential parameters used so far and the choices of A = 400 MeV and A = 300 Me V, the
resulting constants are given in table 2.1:
Co C' 4
6
1 A = 400 MeV 11 11.23 1 -32.27 1 86.73 1 113.9 -913.6
I A = 300
-265.2
MeV 11 11.15 I -32.93 I 85.76 I 115.6 -232.6
C'
-783.7
Table 2.1: The values of the coupling constants Ci, q in [GeV(-2-i )] for two choices of the cut-off
A.
Naturalness of the coupling constants Ci, q me ans that
(2.118)
C2n
- = anAscale 2 ,
C 2n+2
where the an are numbers of order one. Indeed, as one can read off from table 2.1, such a common
scale exists, namely
Ascale =
600
MeV . (2.119)
This is a reasonable value in the sense that it is very elose to the mass /1H of the heavy mesons,
which is integrated out from the theory. Stated differently, the value for Ascale agrees with naive
expectations. As it turns out, even for cut-off values like A = 300 MeV, there is only a small
difference between the heavy meson mass and the ensuing natural mass scale.
Another observation is that the values of the Ci, CI depend very weakly on the concrete choice
of the cut-off A. Clearly, for such an expansion of the heavy mass exchange in terms of contact
interactions to make sense, A has to be chosen below /1H. Furthermore, since we explicitly keep
the light meson, A should not be smaller than the mass /1L. If one were to select such a value
for the cut-off, one could also consider the possibility of expanding the light meson exchange in a
string of contact terms. We do, however, not pursue this option in here. Therefore, we conelude
that one should set A < Ascale but it is not possible to find a more precise relation. In fig. 2.10
MeV.
we show how weIl the potential V' (q', q) is reproduced by the truncated expansion eq. (2.116) for
A = 300
We have also calculated the two-body binding energy and the phase shifts using the form
eq. (2.116) with the constants given in table 2.1. The corresponding results are shown in fig. 2.11
and in table 2.2. The agreement with the exact values is good. However, one sees that terms
of rat her high order should be kept in the effective potential Vcontact in order to have the binding
energy correct within a few percent. Note, however, that the value of the binding energy is
unnaturally small compared to a typical hadronic scale like the pion mass or the scale of chiral
symmetry breaking. The phase shifts are described fairly weIl for kinetic energies (in the lab) up to
about 100 MeV if one retains the first three terms in the expansion eq. (2.116). For A = 400 MeV,
one can not expect any reasonably fast convergence for the binding energy any more. This can
be traced back to the fact that one is elose to the radius of convergence for momenta elose to
the cut-off. More specificaIly, with q = q' = 400 Me V, the pertinent expansion parameter is
(q,2 + q2)j A;cale ::::' 0.9. The ensuing very slow convergence is exhibited in table 2.2.
Although we have found that Ascale rv 600 MeV and, therefore, the expansion of V' in terms of
contact terms seems to converge for the chosen value of the cut-off A = 300 MeV, the ultimative