The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
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140 4. <strong>The</strong> two-nucleon system: numerical results<br />
More precisely, we have one <strong>in</strong>dependent parameter <strong>in</strong> each D-wave. Thus the cut-off <strong>in</strong>dependence<br />
will be restored by the runn<strong>in</strong>g of the correspond<strong>in</strong>g LECs. To illustrate this, we show<br />
<strong>in</strong> the lower panel of fig. 4.9 the partial N3LO results for the 1 D2 channel. <strong>The</strong> correspond<strong>in</strong>g<br />
potential consists of the NNLO terms plus one N3LO contact <strong>in</strong>teraction. As expected, the cut-off<br />
dependence of the phase shift is very much reduced compared to the NNLO result.9 Of course, this<br />
illustrative example can not substitute for a complete N3LO calculation, but one should expect<br />
very similar results. Note that accord<strong>in</strong>g to our f<strong>in</strong>d<strong>in</strong>gs, the NNLO potential <strong>in</strong> all the D-wave<br />
channels is not weak enough to be treated perturbatively, as it has been done <strong>in</strong> ref. [108]. <strong>The</strong><br />
potential has to be iterated to all orders <strong>in</strong> the LS equation. Only then one obta<strong>in</strong>s a reasonable<br />
description of the phase shifts <strong>in</strong> these partial waves. Concern<strong>in</strong>g the F-waves, which are shown<br />
<strong>in</strong> fig. 4.10, 1 F 3 and 1: 3 are well described, whereas the NNLO TPEP is visibly too strong <strong>in</strong> 3 F2,<br />
3 F 3 , and 3 F4• This can be cured at higher orders by contact <strong>in</strong>teractions. More precisely, a N3LO<br />
calculation should be sufficient. Our phases <strong>in</strong> the F-waves look very similar to those shown <strong>in</strong><br />
ref. [108]. Consequently, a perturbative treatment of the potential <strong>in</strong> these channels is justified.<br />
<strong>The</strong> lead<strong>in</strong>g non-vanish<strong>in</strong>g contributions to the phases <strong>in</strong> the D-waves <strong>in</strong> the KSW sheme result<br />
at NLO from the non-iterated OPE.1° <strong>The</strong> first corrections come out from a s<strong>in</strong>gle iteration of<br />
the OPE at NNLO. <strong>The</strong> results for the 1 D2 and 3 D2 phases shown <strong>in</strong> fig. 10 of [211] are rather<br />
elose to our LO approximation. In the case of the 1 D2 channel, the OPE term projected onto the<br />
correspond<strong>in</strong>g partial wave is quite weak, so that iterat<strong>in</strong>g the potential <strong>in</strong> the LS equation does<br />
not improve the result obta<strong>in</strong>ed with the Born approximation. For the 3 D2 partial wave, iteration<br />
of the OPE is more important. <strong>The</strong> NNLO result of ref. [211] improves the Born approximation<br />
significantly, see fig. 10 <strong>in</strong> [211], and is elose to our LO prediction. In the case of the 3 D 3 partial<br />
wave, the authors of [211] observe a good agreement with the data at NNLO, i.e. with the Tmatrix<br />
given by sum of the OPE and its s<strong>in</strong>gle iteration. Accord<strong>in</strong>g to our analysis, we stress<br />
that this agreement might be fortituous, because the terms result<strong>in</strong>g from furt her iterat<strong>in</strong>g the<br />
OPE are of the same size as those ones <strong>in</strong>eluded <strong>in</strong> the NNLO calculation <strong>in</strong> [211]. In particular,<br />
after summ<strong>in</strong>g up an <strong>in</strong>f<strong>in</strong>ite series of the iterated OPE we obta<strong>in</strong> the phase shift, see LO result<br />
<strong>in</strong> fig. 4.8, which is totally different to the NNLO calculation of [211]. Furthermore, as already<br />
po<strong>in</strong>ted out above, the dom<strong>in</strong>ant effect <strong>in</strong> this channel is due to the correlated two-pion exchange,<br />
which starts to contribute <strong>in</strong> the KSW scheme at higher orders and is not <strong>in</strong>corporated <strong>in</strong> the<br />
NLO and NNLO calculations <strong>in</strong> [211].<br />
Compar<strong>in</strong>g our results to the ones obta<strong>in</strong>ed by Ord6iiez et al. [78] we note that our predictions for<br />
the 1 D2 and 3 D 1 phase shifts and for the mix<strong>in</strong>g parameter 1:2 are slightly better at <strong>in</strong>termediate<br />
energies, whereas those one for 3 D2 are a somewhat worse. Unfortunately, the 3 D 3 partial wave<br />
is not shown <strong>in</strong> ref. [78].<br />
4.3.4 Peripheral waves<br />
In figs. 4.11, 4.12 and 4.13 we show the G-, H- and I-waves together with the mix<strong>in</strong>g parameters<br />
1:4,5,6' <strong>The</strong>se partial waves were first discussed <strong>in</strong> detail by the Munich group [108]. <strong>The</strong>ir calculation<br />
was perturbative and based on dimensional regularization of the TPE graphs. However,<br />
for these partial waves the iteration becomes unimportant and our f<strong>in</strong>d<strong>in</strong>gs confirm their results.<br />
<strong>The</strong> description of IG4, 3G 3 , 3G5, 3 H5, 3 H6, 3 h and 3 h is visibly improved by the NNLO TPEP.<br />
Only <strong>in</strong> 116 the NLO result is better than the NNLO one. Of course, for the peripheral partial<br />
9<strong>The</strong> situation here is very much similar to the one with the 1 H and 3 Po phases at lead<strong>in</strong>g order considered <strong>in</strong><br />
the preced<strong>in</strong>g section.<br />
10<br />
As <strong>in</strong> the case of the P-waves, these phase shifts are zero at lead<strong>in</strong>g order.