The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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4.2. The fits 125 "*" if the LECs were fixed from the effective range parameters. We have found that the observables (phase shifts) depend rather weekly on what fitting procedure is used. The values of the LECs do not change much except for some C's at NLO, where significant variations were observed. Note furt her that in the S-waves, especially in 1 So, isospin breaking effects like e.g. the charged to neutral pion mass difference, are known to be important. We do not consider such effects in this work and thus take an average value for the pion mass. The actual values of the S-wave LECs see m to be rat her sensitive to the choice of the pion mass. For the phases 1 So, 3 Sl and E1 we use the errors as given in ref. [33], for all other partial waves we assign an absolute error of 3%. This number is arbitrary, but taking any other value would not change the fit results, only the total X2• To perform the fits, we have to specify the value of the cut-off A in the regulator functions as defined in eqs. (4.12), (4.13). At NLO, for any choice between 380 MeV and 600 MeV, we get very similar fits (a very shallow X2 distribution in each partial wave). At NNLO, this shallow distribution turns into a plateau, which shifts to higher values of the cut-off. We can now use values between 600 MeV and 1 GeV. The results obtained using the sharp and the exponential cut-off are very similar. For illustration, we consider the sharp cut-off with A = 500 Me V at NLO and A = 875 MeV at NNLO. We show in fig. 4.1 the absolute quadratic deviations of the fit to the Nijmegen phase shift analysis (PSA), defined by (8fit - 8PSA)2. Consider first the S-waves. Both in 1 So and 3 Sl, we observe a clear improvement when going from LO to NLO to NNLO (NNLO-Ll). A similar pattern holds for the P-waves and E1 , although the differences between NLO and NNLO are somewhat less pronounced. Note that the 3 Po wave is very sensitive to the value of the pion mass, therefore the slightly better NLO fit should not be considered problematic. The corresponding parameters of the coupling constants are collected in table 4.1. Cl So Cl So C 3S1 C3S1 Cl PI C3P1 C3Po C3P2 C3Dl-3S1 NLO 1 NLO* 1 NNLO 1 NNLO* 1 NNLO-Ll 11 -0.1337 -0.09276 -4.249 -4.246 -5.731 1.822 2.125 11.95 11.84 13.82 -0.1304 -0.1022 -6.508 -6.655 -0.9767 -0.3933 0.02432 11.29 11.28 3.446 0.3442 0.3442 -2.045 -2.045 -2.494 -0.3941 -0.3941 -7.061 -7.061 -8.188 1.335 1.335 -2.832 -2.832 -2.993 -0.1907 -0.1907 -8.056 -8.056 -8.431 -0.03170 -0.03568 -3.424 -3.516 -0.5623 Table 4.1: The values of the LECs as determined from the low partial waves. We use the sharp cut-off with A = 500 MeV and 875 MeV at NLO and NNLO, respectively. The Ci are in 104 GeV-2 while the others are in 104 GeV-4. The parameters of the NLO, NNLO and NNLO Ll potentials are obtained from fitting to the Nijmegen PSA. The LECs of the NLO*, NNLO* potentials in the 1S0 and 3S1 _3 D1 channels are fixed to reproduce exactly the effective range parameters. To illustrate the dependence on the cut-off, we show in fig. 4.2 the running of the two (three) couplings in the 1S0 eSd channels at NNLO (note that the third parameter in 3S1 comes in via the mixing with the 3 D1 wave). We notice that the variation of these LECs over a wide range of cut-offs is rat her modest. We also mention that using the 7rN parameters from refs. [200, 205, 71]
126 4. The two-nuc1eon system: numerical results leads to a considerably worse X 2 in the fits. We take that as an indication that the determination of the Ci based on the method employed in ref. [195] is more reliable than fitting to 7rN phase shifts (as long as one works to third order in the chiral expansion). We remark that using the parameters of ref. [195], the deuteron binding energy Ed comes out as NLO Ed = -2.175 MeV , NNLO Ed = -2.208 MeV , (4.25) Le. the NNLO result is already within 7.5 permille of the experimental number. Fine tuning in the parameters in the deuteron channel would allow to get the binding energy at the exact value of -2.224575(9) MeV without leading to any noticeable change in the phase shifts. We later consider the deuteron channel separately with an exponential regulator. This will lead to an improved binding energy but no attempt is made to match the exact value in all digits. At this point we would like to comment on the increase in the cut-off values when going from NLO to NNLO (NNLO-b.). Consider first the leading order result. Lepage [141] has pointed out that the inclusion of the one-pion exchange does not lead to a remarkable increase in the cut-off values compared to a pionless theory. In order to fit the phase shifts in the S-waves one should choose the cut-off below 500-600 MeV even if the contact interactions with two derivatives are taken into account (within our power counting scheme such contact interactions contribute first at next-to-Ieading order and are of the same size as the lading two-pion exchange terms). Lepage assumed that such a low value of the cut-off is due to the missed physics associated with the two-pion exchange. So, naively, one would expect that the inclusion of the leading two-pion exchange contributions at NLO would allow to take larger cut-off values. However, that does not happen. This was also pointed out in refs. [105], [106]. According to our analysis, only at NNLO, after the subleading two-pion exchange contributions are taken into account, one can increase the cut-off up to 800 to 1000 MeV. The inclusion of the dimension two operators of the pion-nudeon interaction at NNLO encodes some information about heavy meson exchange as weIl as virtual isobar excitations, as discussed in detail in ref. [200]. In the present work we were able to separate the leading effects of the b.-resonance (NNLO-b.). The clear increase in the cut-off values when going from NLO to NNLO-b. indicates the importance of physics associated with heavier mass states like e.g. the b.-resonance. Our NNLO (NNLO-b.) TPEP is sensitive to moment um scales sizably larger than twice the pion mass (as it would be the case for uncorrelated TPE) and deltanucleon mass splitting. Consequently, the cut-off has to be chosen safely above these scales, say above 500 MeV (with the sharp regulator). The upper limit of about 1 GeV is related to the cancelations in the S-waves (fine-tuning), since for too large values of A it is no longer possible to keep this intricate balance. It is, however, comforting to see that including more physics in the potential leads indeed to a wider range of applicability of the EFT. FinaIly, we need to discuss one furt her topic. Performing the fits, we have found two minima in both the ISO and the 3S1 channel. This is not unexpected. Indeed, the same happens in the case of the pionless theory considered in the section 2.2. In particular, the requirement of reproducing exactly the scattering length and the effective range within the effective theory with the sharp cut-off leads to the equations (2.37), (2.38) for the constants Co and C2. Thus, one observes two equivalent solutions for Co and C2. Such a situation with two solutions appears also in the NLO and NNLO theory with pions. At NLO, we find very similar predictions for the phase shifts and observables as weIl as a very similar quality of the fits in the ISO and 3S 1 _3 D1 channels for both solutions, see the upper panel in fig. 4.3. So, there is no real criterion to prefer one of these solutions. However, at NNLO, the behavior of the phase shifts at higher energies differs quite remarkably as it is illustrated in the lower panel of fig. 4.3. Also, the X 2 for these two solutions
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The Nucleon-Nucleon Interaction in
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11 vom ursprünglichen wesentlich u
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IV Ordnung des Potentials besteht a
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VI sagen für die Schwerpunktsimpul
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Vlll betrachten, um eine komplette
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x 5 Isospin violation in the two-nu
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2 1. Introduction nuclear force of
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4 1. Introduction the Thcson-Melbou
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6 1. Introduction pion-nucleon syst
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8 1. Introduction completely domina
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10 1. Introduction such interaction
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12 2. Low-momentum effective theori
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28 2. Low-momentum efIective theori
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36 2 o -2 -4 -6 -8 -10 � -12 t-
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38 (Eq - Ep )A(p, q) [GeV-2] 2 1.6
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42 -3 -5 -7 -9 2. Low-momentum effe
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Chapter 3 The derivation of nuclear
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50 3. The derivation of nuc1ear for
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52 and for the SU(Nf h x SU(Nf)R [Q
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54 3. The derivation of nuclear for
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66 3. The derivation o{ nuclear {or
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72 3. The derivation o{ nuclear {or
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Page 107 and 108: 96 _ . . -.:;: " :,,,_ ._ . . _. 3.
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Page 176 and 177: Chapter 6 Summary and outlook In th
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Appendix B Operators contributing t
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Appendix C Small scale expansion wi
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179 (C.16) l�r-2. (C.17) h + l2 =
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Appendix E Anti-symmetrization of t
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Consequently, only four of the eigh
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where qJ-! is chosen to satisfy (v
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To keep the presentation self-conta
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- i�03{ [(NtVN) . (VNt x ifN) + (
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Consequently, it is not possible to
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(j ± 1, 1jIVIJ =f 1, 1j) Here, Pj
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Bibliography [1] E. Rutherford, Phi
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[62] J. Goldstone, A. Salam and S.
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[137] V. Bernard, N. Kaiser and Ulf
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[204] J.M. Blatt and L.C. Biedenhar
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The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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