The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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4.2. The nts 60 0; 40 CD � 0 20 Cf) � I 0 0 -l z -20 -40 0 0.1 70 0; 50 CD � 0 Cf) 30 • 0.2 0.3 I - - - - - - - - - - 0 -l (\J z 10 -1 0 0 0.1 0.2 0.3 E1ab [GeV] Figure 4.3: Phase shifts for the two solutions in the 1 So-wave as dicussed in the text. At NLO (upper panel), these are indistinguishable. At NNLO (lower panel), one of the solutions (dashed line) shows an unacceptable behaviour at higher momenta and is discarded. 127
128 4. The two-nuc1eon system: numerical results differs typically by factors of 2 ... 10. In what follows, we will only discuss the best solution at NNLO. 4.3 Phase shifts 4.3.1 S-waves In fig. 4.4 we show the two S-waves at LO, NLO and NNLO for the cut-off values given above. Clearly, the lowest order OPEP plus non-derivative contact terms is insufficient to describe the 1 So phase (as it is weIl-known from effective range theory and previous studies in EFT approaches). The much more smooth 3 SI phase is already fairly weIl described at leading order. For energies above 100 MeV, the improvement by going from NLO to NNLO is clearly visible. The corresponding values of the S-wave phase shifts at certain energies are given in tables 4.2,4.3. For comparison, we also give the results of the Nijmegen and VPI PSA [206] and of three modern potentials (Nijmegen 93 [33], Argonne V18 [37] and CD-Bonn [29]). Our NNLO result for 1 So is visibly better than the one obtained in ref. [207]. It is also of interest to consider the scattering lengths and effective range parameters as it was done in the cases of the pionless theory and in the model calculations in chapter 2. The effective range expansion takes the form (written here for a genuine partial wave) ( 4.26) with p the nucleon cms momentum, a the scattering length, r the effective range and V2, 3,4 the shape parameters. It has been stressed in ref. [208] that the shape parameters are a good testing ground far the range of applicability of the underlying EFT since a fit to say the scattering length and the effective range at NLO leads to predictions for the Vi. In table 4.4, we present our results for the S-waves in comparison to the ones obtained from the Nijmegen PSA. Note that in the coupled channel we have used the Blatt and Biedenharn parametrization of the S-matrix in order to be able to compare our findings with those of ref. [102]. For both S-waves we observe a good agreement with the data (apart from the last coefficient in the ISO channel). In general, the results for the 1 So partial wave are not as precise as for the 3 SI chan ne 1. This has to be expected because of the unnaturally large value of the scattering length in the first case. At this point we would like to remind the reader of the similar analysis for the effective range coefficients performed for the pionless theory in sec. 2.2 and for our model in sec. 2.3. In the first case one can not obtain any predictions for the effective range coefficients, that are not used to fix the LECs. This is because one describes a purely short-range physics in such a theory. In the second example we included the known long-range force into the effective Hamiltonian. It is natural to assurne that the values of the effective range coefficients are governed by the lowest mass scale associated with the longest range part of the interaction. In the model considered in sec. 2.3 we had, however, no large separation between the scales related to the long- and short-range parts of the underlying force. Therefore, only the first effective range (shape) parameter not used in the fit could be predicted accurately. The fairly precise description of the shape parameters shown in table 4.4 may be considered as a clear indication of large separations between the relevant scales in the case of the realistic N N interactions. Furthermore, we would like to point out a clear improvement for all quantities with exception of V2 for the 1 So partial wave when going from NLO to NNLO. Note that in both cases we had 2 (3) free parameters in the 1 So e Sl-3 Dd channel (and the cut-off). In our opinion, this is an important argument in favor of a good convergence of our expansion for the effective potential (which survives the iteration).
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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 105 and 106: 94 , , , , , , , , , , , , , , , ,
Page 107 and 108: 96 _ . . -.:;: " :,,,_ ._ . . _. 3.
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Page 145 and 146: 134 10 8 4 2 o o • Nijmegen PSA N
Page 147 and 148: 136 4. The two-nucleon system: nume
Page 151 and 152: 140 4. The two-nucleon system: nume
Page 157 and 158: 146 --- .... . � - --- .... . ...
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Page 166 and 167: 5.1 Isospin violating effective Lag
Page 168 and 169: 5.2 Brief introduction into the KSW
Page 170 and 171: 5.3 The leading eIB and eSB effects
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
Page 178 and 179: 2. Secondly, we considered the real
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Page 184 and 185: derivative. Further helpful equalit
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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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