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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126 4. <strong>The</strong> two-nuc1eon system: numerical results<br />
leads to a considerably worse X 2 <strong>in</strong> the fits. We take that as an <strong>in</strong>dication that the determ<strong>in</strong>ation<br />
of the Ci based on the method employed <strong>in</strong> ref. [195] is more reliable than fitt<strong>in</strong>g to 7rN phase<br />
shifts (as long as one works to third order <strong>in</strong> the chiral expansion). We remark that us<strong>in</strong>g the<br />
parameters of ref. [195], the deuteron b<strong>in</strong>d<strong>in</strong>g energy Ed comes out as<br />
NLO Ed = -2.175 MeV ,<br />
NNLO Ed = -2.208 MeV , (4.25)<br />
Le. the NNLO result is already with<strong>in</strong> 7.5 permille of the experimental number. F<strong>in</strong>e tun<strong>in</strong>g<br />
<strong>in</strong> the parameters <strong>in</strong> the deuteron channel would allow to get the b<strong>in</strong>d<strong>in</strong>g energy at the exact<br />
value of -2.224575(9) MeV without lead<strong>in</strong>g to any noticeable change <strong>in</strong> the phase shifts. We<br />
later consider the deuteron channel separately with an exponential regulator. This will lead to an<br />
improved b<strong>in</strong>d<strong>in</strong>g energy but no attempt is made to match the exact value <strong>in</strong> all digits.<br />
At this po<strong>in</strong>t we would like to comment on the <strong>in</strong>crease <strong>in</strong> the cut-off values when go<strong>in</strong>g from<br />
NLO to NNLO (NNLO-b.). Consider first the lead<strong>in</strong>g order result. Lepage [141] has po<strong>in</strong>ted<br />
out that the <strong>in</strong>clusion of the one-pion exchange does not lead to a remarkable <strong>in</strong>crease <strong>in</strong> the<br />
cut-off values compared to a pionless theory. In order to fit the phase shifts <strong>in</strong> the S-waves one<br />
should choose the cut-off below 500-600 MeV even if the contact <strong>in</strong>teractions with two derivatives<br />
are taken <strong>in</strong>to account (with<strong>in</strong> our power count<strong>in</strong>g scheme such contact <strong>in</strong>teractions contribute<br />
first at next-to-Iead<strong>in</strong>g order and are of the same size as the lad<strong>in</strong>g two-pion exchange terms).<br />
Lepage assumed that such a low value of the cut-off is due to the missed physics associated with<br />
the two-pion exchange. So, naively, one would expect that the <strong>in</strong>clusion of the lead<strong>in</strong>g two-pion<br />
exchange contributions at NLO would allow to take larger cut-off values. However, that does not<br />
happen. This was also po<strong>in</strong>ted out <strong>in</strong> refs. [105], [106]. Accord<strong>in</strong>g to our analysis, only at NNLO,<br />
after the sublead<strong>in</strong>g two-pion exchange contributions are taken <strong>in</strong>to account, one can <strong>in</strong>crease the<br />
cut-off up to 800 to 1000 MeV. <strong>The</strong> <strong>in</strong>clusion of the dimension two operators of the pion-nudeon<br />
<strong>in</strong>teraction at NNLO encodes some <strong>in</strong>formation about heavy meson exchange as weIl as virtual<br />
isobar excitations, as discussed <strong>in</strong> detail <strong>in</strong> ref. [200]. In the present work we were able to separate<br />
the lead<strong>in</strong>g effects of the b.-resonance (NNLO-b.). <strong>The</strong> clear <strong>in</strong>crease <strong>in</strong> the cut-off values when<br />
go<strong>in</strong>g from NLO to NNLO-b. <strong>in</strong>dicates the importance of physics associated with heavier mass<br />
states like e.g. the b.-resonance. Our NNLO (NNLO-b.) TPEP is sensitive to moment um scales<br />
sizably larger than twice the pion mass (as it would be the case for uncorrelated TPE) and deltanucleon<br />
mass splitt<strong>in</strong>g. Consequently, the cut-off has to be chosen safely above these scales, say<br />
above 500 MeV (with the sharp regulator). <strong>The</strong> upper limit of about 1 GeV is related to the<br />
cancelations <strong>in</strong> the S-waves (f<strong>in</strong>e-tun<strong>in</strong>g), s<strong>in</strong>ce for too large values of A it is no longer possible<br />
to keep this <strong>in</strong>tricate balance. It is, however, comfort<strong>in</strong>g to see that <strong>in</strong>clud<strong>in</strong>g more physics <strong>in</strong> the<br />
potential leads <strong>in</strong>deed to a wider range of applicability of the EFT.<br />
F<strong>in</strong>aIly, we need to discuss one furt her topic. Perform<strong>in</strong>g the fits, we have found two m<strong>in</strong>ima <strong>in</strong><br />
both the ISO and the 3S1 channel. This is not unexpected. Indeed, the same happens <strong>in</strong> the case<br />
of the pionless theory considered <strong>in</strong> the section 2.2. In particular, the requirement of reproduc<strong>in</strong>g<br />
exactly the scatter<strong>in</strong>g length and the effective range with<strong>in</strong> the effective theory with the sharp<br />
cut-off leads to the equations (2.37), (2.38) for the constants Co and C2. Thus, one observes two<br />
equivalent solutions for Co and C2. Such a situation with two solutions appears also <strong>in</strong> the NLO<br />
and NNLO theory with pions. At NLO, we f<strong>in</strong>d very similar predictions for the phase shifts and<br />
observables as weIl as a very similar quality of the fits <strong>in</strong> the ISO and 3S 1 _3 D1 channels for<br />
both solutions, see the upper panel <strong>in</strong> fig. 4.3. So, there is no real criterion to prefer one of these<br />
solutions. However, at NNLO, the behavior of the phase shifts at higher energies differs quite<br />
remarkably as it is illustrated <strong>in</strong> the lower panel of fig. 4.3. Also, the X 2 for these two solutions