The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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4.3. Phase shifts 80 60 40 20 o -20 -40 o 0.05 0.1 160 140 120 100 80 60 � � - - - _ 40 20 o -20 -40 o 0.05 0.1 ISO [deg] 0.15 0.2 0.25 0.3 3S 1 [deg] 0.15 0.2 0.25 0.3 Figure 4.4: Predictions for the S-waves (in degrees) for nucleon laboratory energies Elab below 300 MeV (0.3 GeV). The dotted, dashed and solid curves represent LO, NLO and NNLO results, in order. The squares depict the Nijmegen PSA results. In the upper and the lower panel, 1 So and 3 SI, respectively, are shown. 129
130 4. The two-nuc1eon system: numerical results 1 E1ab [MeV] 11 NNLO* NNLO 1 Nijm PSA 1 VPI PSA 1 Nijm93 1 AV18 CD-Bonn h 147.735 147.727 147.747 147.781 147.768 147.749 147.748 2 136.447 136.450 136.463 136.488 136.495 136.465 136.463 3 128.763 128.781 128.784 128.788 128.826 128.786 128.783 5* 118.150 118.196 118.178 118.129 118.240 118.182 118.175 10* 102.56 102.67 102.61 102.41 102.72 102.62 102.60 20 85.99 86.21 86.12 85.67 86.35 86.16 86.09 30 75.84 76.14 76.06 75.46 76.40 76.12 75.99 50* 62.35 62.79 62.77 62.12 63.36 62.89 62.63 100* 42.33 43.06 43.23 42.98 44.33 43.18 42.93 200 19.54 20.68 21.22 20.88 22.82 21.31 20.88 300 4.15 5.58 6.60 5.08 8.44 7.55 6.70 Table 4.2: 3 Sl np phase shift for the global fit at NNLO (sharp cut-off, A = 875 MeV) compared to phase shift analyses and modern potentials. The parameters of the NNLO potential are fixed by fitting the Nijmegen PSA at six energies (E1ab = 1,5, 10,25, 50(, 100) MeV). These energies are marked by the star. The parameters of the NNLO* potential are chosen to reproduce exactly the scattering length and the effective range as described in the text. To show that our results are stable and do not depend on the fitting procedure we present in table 4.5 the NLO* and NNLO* predictions. Here, the LECs are fixed directly from the first effective range parameters, as discussed above. Overall, the agreement between the numbers in tables 4.4 and 4.5 is quite satisfactory. The NLO results turn out to be slightly more sensitive to the fitting procedure. As is pointed out in ref. [208], one also can perform the moment um expansion for E1 in a similar way as for the S-waves. The authors of the reference [208] show the values of the corresponding parameters extracted by themselves from the Nijmegen PSA. They use, presumably, the different (Stapp) parametrization of the S-matrix in that case. We have not found any original reference of the Nijmegen group concerning the moment um expansion for E1 . We thus refrain from discussing this issue any further.3 For completeness, we collect the experimental values for the S-wave scattering lengths and effective ranges: as = (-23.758 ± 0.010) fm , Ts = (2.75 ± 0.05) fm , at = (5.424 ± 0.004) fm , Tt = (1.759 ± 0.005) fm . ( 4.27) ( 4.28) Quite recently the results for the NNLO calculation of the phase shifts in the S, P and D channels within the KSW scheme [91] have been published [211]. Here we would like to compare our findings with those from ref. [211]. The leading and non-perturbative contribution to the N N Smatrix in S channels results in the KSW approach from summing up an infinite series of the contact 3Note, however, that in order to fix the LEes in the 3S1 _3 D1 channel we have used the leading coefficient 91 = 1.66 fm3 in the momentum expansion of the EI . This value agrees with the one given in ref. [208]. In principle, we could take apart from the 3 SI scattering length and effective range the deuteron binding energy as the third quantity to fix the three free parameters in the potential. However we refrain from doing that because of a strong correlation between the deuteron binding energy and the 3 SI effective range parameters.
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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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74 3. The derivation o{ nuc1ear {or
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Page 107 and 108: 96 _ . . -.:;: " :,,,_ ._ . . _. 3.
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Page 168 and 169: 5.2 Brief introduction into the KSW
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Page 172 and 173: 5.3 The leading CIB and CSB effects
Page 174 and 175: 5.5 Classification scheme 163 scatt
Page 176 and 177: Chapter 6 Summary and outlook In th
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Page 184 and 185: derivative. Further helpful equalit
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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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