The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

128 4. The two-nuc1eon system: numerical results
differs typically by factors of 2 ... 10. In what follows, we will only discuss the best solution at
NNLO.
4.3 Phase shifts
4.3.1 S-waves
In fig. 4.4 we show the two S-waves at LO, NLO and NNLO for the cut-off values given above.
Clearly, the lowest order OPEP plus non-derivative contact terms is insufficient to describe the 1 So
phase (as it is weIl-known from effective range theory and previous studies in EFT approaches).
The much more smooth 3 SI phase is already fairly weIl described at leading order. For energies
above 100 MeV, the improvement by going from NLO to NNLO is clearly visible. The corresponding
values of the S-wave phase shifts at certain energies are given in tables 4.2,4.3. For
comparison, we also give the results of the Nijmegen and VPI PSA [206] and of three modern
potentials (Nijmegen 93 [33], Argonne V18 [37] and CD-Bonn [29]). Our NNLO result for 1 So is
visibly better than the one obtained in ref. [207].
It is also of interest to consider the scattering lengths and effective range parameters as it was
done in the cases of the pionless theory and in the model calculations in chapter 2. The effective
range expansion takes the form (written here for a genuine partial wave)
( 4.26)
with p the nucleon cms momentum, a the scattering length, r the effective range and V2, 3,4 the
shape parameters. It has been stressed in ref. [208] that the shape parameters are a good testing
ground far the range of applicability of the underlying EFT since a fit to say the scattering length
and the effective range at NLO leads to predictions for the Vi. In table 4.4, we present our results
for the S-waves in comparison to the ones obtained from the Nijmegen PSA. Note that in the
coupled channel we have used the Blatt and Biedenharn parametrization of the S-matrix in order
to be able to compare our findings with those of ref. [102]. For both S-waves we observe a good
agreement with the data (apart from the last coefficient in the ISO channel). In general, the results
for the 1 So partial wave are not as precise as for the 3 SI chan ne 1. This has to be expected because
of the unnaturally large value of the scattering length in the first case. At this point we would
like to remind the reader of the similar analysis for the effective range coefficients performed for
the pionless theory in sec. 2.2 and for our model in sec. 2.3. In the first case one can not obtain
any predictions for the effective range coefficients, that are not used to fix the LECs. This is
because one describes a purely short-range physics in such a theory. In the second example we
included the known long-range force into the effective Hamiltonian. It is natural to assurne that
the values of the effective range coefficients are governed by the lowest mass scale associated with
the longest range part of the interaction. In the model considered in sec. 2.3 we had, however, no
large separation between the scales related to the long- and short-range parts of the underlying
force. Therefore, only the first effective range (shape) parameter not used in the fit could be
predicted accurately. The fairly precise description of the shape parameters shown in table 4.4
may be considered as a clear indication of large separations between the relevant scales in the case
of the realistic N N interactions. Furthermore, we would like to point out a clear improvement for
all quantities with exception of V2 for the 1 So partial wave when going from NLO to NNLO. Note
that in both cases we had 2 (3) free parameters in the 1 So e Sl-3 Dd channel (and the cut-off).
In our opinion, this is an important argument in favor of a good convergence of our expansion for
the effective potential (which survives the iteration).