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The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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4.3. Phase shifts 133<br />

and a series of contact <strong>in</strong>teractions. <strong>The</strong> obta<strong>in</strong>ed results are similar to ours at NLO. A furt her<br />

important observation made <strong>in</strong> this work is that the <strong>in</strong>clusion of the NNLO contact <strong>in</strong>teractions<br />

alone does not allow to improve predictions for the phase shift relative to the NLO calculation.<br />

As is demonstrated <strong>in</strong> our work, only the <strong>in</strong>clusion of the complete NNLO corrections (4.3) to the<br />

potential given by the sublead<strong>in</strong>g TPE contribution and the contact terms allows to come closer<br />

to the Nijmegen PSA.<br />

F<strong>in</strong>ally, let us briefly comment on the earlier work by Ord6nez et al. [76J. <strong>The</strong>y performed the<br />

LO, NLO and NNLO calculations5 with<strong>in</strong> the potential approach us<strong>in</strong>g time-ordered perturbation<br />

theory to obta<strong>in</strong> the N N force. <strong>The</strong> Cl, 3 ,4 coupl<strong>in</strong>gs were not taken from 1f N scatter<strong>in</strong>g but <strong>in</strong>stead<br />

considered as free parameters. Furthermore, the anti-symmetrization of the potential was not<br />

performed. As a consequence, many additional redundant parameters correspond<strong>in</strong>g to contact<br />

<strong>in</strong>teractions enter the expressions for the effective potential. A global fit <strong>in</strong> all lower partial waves<br />

has been performed to fix the 26 parameters at NNLO. This makes it difficult to separate the<br />

effects on the phase shifts of the <strong>in</strong>dividual contributions to the potential. <strong>The</strong> results for the 1 So<br />

and 3 SI phase shifts shown <strong>in</strong> figs. 6 and 7 <strong>in</strong> this paper are of the same quality as ours (our 1 So<br />

partial wave looks slightly better at <strong>in</strong>termediate energies). We can not compare the predictions<br />

for the effective range parameters, s<strong>in</strong>ce no correspond<strong>in</strong>g analysis has been done <strong>in</strong> ref. [78J.<br />

4.3.2 P-waves<br />

In fig. 4.6 we show the correspond<strong>in</strong>g partial waves together with the mix<strong>in</strong>g parameter EI for<br />

the best global fit. In some cases, the differences between NLO and NNLO are modest, <strong>in</strong> 1 PI<br />

and 3 PI NLO is even somewhat bett er. That means that the chiral TPEP is too strong <strong>in</strong> these<br />

phases. Note also that <strong>in</strong> the 3 PI phase OPEP is dom<strong>in</strong>ant. Thus, the <strong>in</strong>clusion of the contact<br />

<strong>in</strong>teraction does not lead to a visible change. In 3 P2, NNLO is still too strong but the prediction is<br />

considerably better than the NLO one. <strong>The</strong> energy dependence of EI is fairly precisely described<br />

at NLO and NNLO. <strong>The</strong>se results are visibly better than the ones obta<strong>in</strong>ed <strong>in</strong> ref. [78J or <strong>in</strong><br />

refs. [210J and [211], the latter be<strong>in</strong>g a NNLO calculation <strong>in</strong> the KSW scheme. This is shown<br />

<strong>in</strong> detail <strong>in</strong> fig. 4.5, where EI is plotted versus the cms nucleon moment um (p < 350 MeV) <strong>in</strong><br />

comparison to the Nimegen PSA and the results from ref. [210J. <strong>The</strong> most important difference<br />

between our approach and the KSW one is, as already po<strong>in</strong>ted out, that <strong>in</strong> the last one the pions<br />

are treated perturbatively. Thus, our results might be considered as an <strong>in</strong>dication that <strong>in</strong> the<br />

two-nucleon system, pions have to be treated non-perturbatively (if one <strong>in</strong>tends to describe data<br />

above p� 150 MeV).6<br />

In the 1 PI and 3 Po channels the phase shift at lead<strong>in</strong>g order (iterated OPE without contact<br />

<strong>in</strong>teractions) describes the data only at very low energies. Also, the phase shifts are sensitive to<br />

the choice of the cut-off. This is because no contact <strong>in</strong>teractions appear <strong>in</strong> the potential at this<br />

order, that could compensate this cut-off dependence. <strong>The</strong>refore, the LO approximation <strong>in</strong> these<br />

channels should not be taken seriously. In fig. 4.7 we show apart from the LO and NLO phase<br />

shifts also the curve, which results if one adds the contact term with two derivatives to the OPE<br />

potential. This is, certa<strong>in</strong>ly, not the complete NLO potential, s<strong>in</strong>ce the lead<strong>in</strong>g TPE is miss<strong>in</strong>g.<br />

<strong>The</strong> results for such <strong>in</strong>complete and complete NLO potentials are similar and very much improved<br />

relative to the LO calculation.7 Thus, we conclude that this improvement when go<strong>in</strong>g from LO<br />

5 <strong>The</strong>y also <strong>in</strong>cluded the lead<strong>in</strong>g effects of <strong>in</strong>termediate D.-excitations.<br />

6 This is, however, not the only difference between the two schemes. For example, <strong>in</strong> the KSW formalism the<br />

unitarity of the S-matrix is only given perturbatively.<br />

7 A visible closeness of the phase shifts from the <strong>in</strong>complete NLO and from NNLO is accidental.

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