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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132 4. <strong>The</strong> two-nuc1eon system: numerical results<br />
I SO NLO* -23.739 2.68 -0.52 5.3 -30<br />
ISO NNLO* -23.739 2.68 -0.61 5.1 -29.7<br />
3S 1 NLO* 5.420 1.753 0.110 0.73 -3.9<br />
3S 1 NNLO* 5.420 1.753 0.057 0.66 -3.8<br />
Table 4.5: Scatter<strong>in</strong>g lengths and range parameters for the S-waves at NLO and NNLO. <strong>The</strong><br />
correspond<strong>in</strong>g LEes are obta<strong>in</strong>ed from the fit to a and r <strong>in</strong> the S-channel and from the first<br />
effective range parameter <strong>in</strong> the moment um expansion of E I , as described <strong>in</strong> text.<br />
order the amplitude is <strong>in</strong>dependent on the renormalization scale. Let us now comment on the<br />
results presented <strong>in</strong> ref. [211]. Impos<strong>in</strong>g the constra<strong>in</strong>ts provided by the perturbative solution of<br />
the renormalization group equations (RGE's) and requir<strong>in</strong>g that unphysical poles are removed<br />
from the amplitude at low momenta leave one free parameter at NLO and two parameters at<br />
NNLO for the 1 So and 3 SI channels, that were fixed by a fit to the phase shifts of the Nijmegen<br />
PSA. For the 1 So phase shift one observes a visible improvement when go<strong>in</strong>g from LO to NLO and<br />
from NLO to NNLO. At NNLO, the ISO phase shift goes very close to the one from the Nijmegen<br />
PSA up to cms momenta rv 300 MeV (E1ab rv 192 MeV), see fig. 3 <strong>in</strong> [211]. Such visible agreement<br />
of the phase shift with the data does, however, not yet allow to conclude about the quality of the<br />
results. For example, the authors of ref. [211] po<strong>in</strong>t out that the the 1 So phase shift at the cms<br />
momentum p = Mn is offits experimental value4 by 17% at NLO and by less then 1% at NNLO. On<br />
the other side, the effective range expansion (4.26) with the first two coefficients (a and r) yields at<br />
p = Mn the value for the phase shift, which differs from the experimental one by only rv 2%. Thus,<br />
this <strong>in</strong>formation is not enough to def<strong>in</strong>itely conclude about improvement of the results obta<strong>in</strong>ed<br />
from the effective theory with explicit pions relative to the calculations with<strong>in</strong> the pionless theory.<br />
A better test<strong>in</strong>g ground for that is given by the shape parameters <strong>in</strong> the effective range expansion<br />
(4.26), which are expected to be sensitive to the pion physics, as discussed above, see also [100],<br />
[101]. At NLO, predictions for V2,3 , 4 totally disagree with the values derived from the Nijmegen<br />
PSA. <strong>The</strong> NNLO predictions for r and the v's, given <strong>in</strong> ref. [211] are r = 2.63 fm, V = 2 -1.2<br />
fm3, V = 3 2.9 fm5 and V4 = -0.7 fm7, which still significantly differ from the experimental values<br />
shown <strong>in</strong> table. 4.4 and are considerably worse than our predictions <strong>in</strong>dicated <strong>in</strong> the same table.<br />
Such dis agreement with the data is surpris<strong>in</strong>g s<strong>in</strong>ce perform<strong>in</strong>g NNLO calculations without pions<br />
and choos<strong>in</strong>g a f<strong>in</strong>ite cut-off of the order of Mn , see sec. 2.2, one expects to be able to reproduce<br />
the first four effective range parameters (a, r, V2 and V3 ) exactly. <strong>The</strong>refore, no clear improvement<br />
relative to the pionless theory can be observed.<br />
For the 3 SI channel the situation turns out to be much worse than for the 1 So channel for the<br />
KSW scheme. Whereas the phase shift is described quite accurately at NLO, NNLO corrections<br />
are large and destroy the agreement with the data already at the cms momenta of the order of<br />
Mn [211]. This is shown <strong>in</strong> fig. 4 of that reference. <strong>The</strong> failure of EFT at NNLO is found to be<br />
due to large contributions result<strong>in</strong>g from the iteration of the pion exchanges, which is missed <strong>in</strong><br />
the KSW approach.<br />
We would also like to comment on the recent work by Hyun et al. [106]. <strong>The</strong>re, the 1 So channel is<br />
considered with<strong>in</strong> an approach similar to ours. In particular, the authors of this reference consider<br />
the potential, which consists of the OPE and lead<strong>in</strong>g TPE contributions given <strong>in</strong> eqs. (4.1), (4.2)<br />
4 As usual, we consider the values of the phase shifts from the Nijmegen PSA as experimental data.