The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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