The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

140 4. The two-nucleon system: numerical results
More precisely, we have one independent parameter in each D-wave. Thus the cut-off independence
will be restored by the running of the corresponding LECs. To illustrate this, we show
in the lower panel of fig. 4.9 the partial N3LO results for the 1 D2 channel. The corresponding
potential consists of the NNLO terms plus one N3LO contact interaction. As expected, the cut-off
dependence of the phase shift is very much reduced compared to the NNLO result.9 Of course, this
illustrative example can not substitute for a complete N3LO calculation, but one should expect
very similar results. Note that according to our findings, the NNLO potential in all the D-wave
channels is not weak enough to be treated perturbatively, as it has been done in ref. [108]. The
potential has to be iterated to all orders in the LS equation. Only then one obtains a reasonable
description of the phase shifts in these partial waves. Concerning the F-waves, which are shown
in fig. 4.10, 1 F 3 and 1: 3 are well described, whereas the NNLO TPEP is visibly too strong in 3 F2,
3 F 3 , and 3 F4• This can be cured at higher orders by contact interactions. More precisely, a N3LO
calculation should be sufficient. Our phases in the F-waves look very similar to those shown in
ref. [108]. Consequently, a perturbative treatment of the potential in these channels is justified.
The leading non-vanishing contributions to the phases in the D-waves in the KSW sheme result
at NLO from the non-iterated OPE.1° The first corrections come out from a single iteration of
the OPE at NNLO. The results for the 1 D2 and 3 D2 phases shown in fig. 10 of [211] are rather
elose to our LO approximation. In the case of the 1 D2 channel, the OPE term projected onto the
corresponding partial wave is quite weak, so that iterating the potential in the LS equation does
not improve the result obtained with the Born approximation. For the 3 D2 partial wave, iteration
of the OPE is more important. The NNLO result of ref. [211] improves the Born approximation
significantly, see fig. 10 in [211], and is elose to our LO prediction. In the case of the 3 D 3 partial
wave, the authors of [211] observe a good agreement with the data at NNLO, i.e. with the Tmatrix
given by sum of the OPE and its single iteration. According to our analysis, we stress
that this agreement might be fortituous, because the terms resulting from furt her iterating the
OPE are of the same size as those ones ineluded in the NNLO calculation in [211]. In particular,
after summing up an infinite series of the iterated OPE we obtain the phase shift, see LO result
in fig. 4.8, which is totally different to the NNLO calculation of [211]. Furthermore, as already
pointed out above, the dominant effect in this channel is due to the correlated two-pion exchange,
which starts to contribute in the KSW scheme at higher orders and is not incorporated in the
NLO and NNLO calculations in [211].
Comparing our results to the ones obtained by Ord6iiez et al. [78] we note that our predictions for
the 1 D2 and 3 D 1 phase shifts and for the mixing parameter 1:2 are slightly better at intermediate
energies, whereas those one for 3 D2 are a somewhat worse. Unfortunately, the 3 D 3 partial wave
is not shown in ref. [78].
4.3.4 Peripheral waves
In figs. 4.11, 4.12 and 4.13 we show the G-, H- and I-waves together with the mixing parameters
1:4,5,6' These partial waves were first discussed in detail by the Munich group [108]. Their calculation
was perturbative and based on dimensional regularization of the TPE graphs. However,
for these partial waves the iteration becomes unimportant and our findings confirm their results.
The description of IG4, 3G 3 , 3G5, 3 H5, 3 H6, 3 h and 3 h is visibly improved by the NNLO TPEP.
Only in 116 the NLO result is better than the NNLO one. Of course, for the peripheral partial
9The situation here is very much similar to the one with the 1 H and 3 Po phases at leading order considered in
the preceding section.
10
As in the case of the P-waves, these phase shifts are zero at leading order.