The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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4.3. Phase shifts 0.45 .---.,.- -- -,- -- -- -r- -.,- --r- --, 0.4 0.35 0.3 0.25 0.2 0.15 0. 1 0.05 .l. .. .- -- --'- -- -- -'- --'- -- --'- -- -' O ��... o 0.05 0.1 0.15 0.2 0.25 0.3 2.5 r--- -,- -_,- -.- -- -,- -- -,- -- -, 2 1.5 0.5 0 �4=::. o 0.05 0.1 0.15 0.2 0.25 0.3 0 -0. 1 -0.2 -0.3 -0.4 -0.5 -0.6 0 0.05 0.1 .. .L. .- -- -l.- --'- -- --'- -- -'- -- -- -' E6 [deg] -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 143 o ��=---.-�- -�-�- -, -1 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35 o 0.05 0.1 0.15 0.2 0.25 0.3 O ��-_r-_,--.- -._- -. -0.4 "--_ _'___-'-_ ___'_ _ __'_ __ L----I o 0.05 0.1 0.15 0.2 0.25 0.3 0.15 0.2 0.25 0.3 Figure 4.13: Predictions for the I-waves and the mixing parameter E6 (in degrees) for nucleon laboratory energies E1ab below 300 MeV. For notations, see fig. 4.4
144 4. The two-nuc1eon system: numerical results waves OPE does already a fairly good job, but the improvement in some of these phases due to the NNLO TPEP clearly underlines the importance of chiral symmetry in a precise description of low-energy nuclear physics. The most interesting situation in the peripheral partial waves appears for the 3 G5 phase,u In these channel the OPE and the leading TPE are not sufficient to describe the phase shift correctly at energies higher than 100 MeV and the largest discrepancy of our NLO predictions with the data is observed. Adding the subleading TPE exchange (NNLO potential) that incorporates the leading effects of the heavier meson exchanges and intermediate ..6.-excitations, which is hidden in the values of the couplings Cl,3,4 allows to improve the NLO result considerably and to obtain an excellent parameter free and cut-off independent description of the phase shift. Performing the calculations with the NNLO-Ll potential we were even able to separate the leading effects of the ..6.'s. We will comment on that later on. It is comforting to have such a clear indication that the values for Cl,3,4 fixed from 7f N scattering are consistent with N N calculations. 4.4 Deuteron properties We now turn to the bound state properties. At NNLO (NLO), we consider the exponential regulator (4.13) with n = 2 and A = 1.05 (0.60) GeV, which reproduces the deuteron binding energy within an accuracy of about one third of a permille (2.5 percent).We make no attempt to reproduce this number with better precision.12 The results for the phase shifts, which correspond to these values of the exponential regulator, are very similar to those obtained with the sharp cutoff A = 0.875 (0.50) GeV. For completeness, we list in table 4.6 the values of the coupling constants in the 3 SI - 3 Dl channel corresponding to the exponential regulator. NLO -0.0363 0.186 -0.190 NNLO -14.497 15.588 -4.358 NNLO-..6. -8.637 7.264 -0.447 Table 4.6: The values of the LECs as determined from the 3 SI _ 3 Dl channel. We use an exponential cut-off with A = 0.6 GeV and 1.05 GeV at NLO and NNLO (NNLO-..6.), respectively. The LEC 0 3 5 1 is in 104 GeV-2 , while the others are in 104 GeV-4 . The parameters of the NLO, NNLO and NNLO-..6. potentials are obtained from fitting to the Nijmegen PSA. In table 4.7 we collect the deuteron properties in comparison to the data and two realistic potential model predictions (the pertinent formulae are given in app. H). We give the results for NLO and NNLO. We note that the deviation of our prediction for the quadrupole moment compared to the empirical value is slightly larger than for the realistic potentials. The asymptotic D / S ratio, called ry, and the strength of the asymptotic wave function, As, are weIl described. The D-state probability, which is not an observable, is most sensitive to small variations in the cut-off. At NLO, it is comparable and at NNLO somewhat larger than the one obtained in the CD-Bonn or ll Note that the 3Zj=I+I-phases show, in general, the largest effects when going from NLO to NNLO and thus are most sensitive to the short range physics. The three extreme cases are the 1 P 2 -, 3 D 3 - and 3G5-waves. We assurne some sort of cancelation in the potential in these channels, wh ich leads to a suppression of the OPE and the leading TPE contributions. 12Note that the deuteron binding energy is not used to fit the free parameters in the potential.
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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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74 3. The derivation o{ nuc1ear {or
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84 .... . - - .... \ . • A .... .
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Page 107 and 108: 96 _ . . -.:;: " :,,,_ ._ . . _. 3.
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Page 113 and 114: 102 3. The derivation o{ nuclear {o
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Page 125 and 126: 114 / / / / / / / 5 / / \ \ \ \ I I
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Page 145 and 146: 134 10 8 4 2 o o • Nijmegen PSA N
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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
Page 180 and 181: NNLO, this range is larger and exte
Page 182 and 183: might be responsible for solving th
Page 184 and 185: derivative. Further helpful equalit
Page 186 and 187: Appendix B Operators contributing t
Page 188 and 189: Appendix C Small scale expansion wi
Page 190 and 191: 179 (C.16) l�r-2. (C.17) h + l2 =
Page 192 and 193: Appendix E Anti-symmetrization of t
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Page 196 and 197: where qJ-! is chosen to satisfy (v
Page 198 and 199: To keep the presentation self-conta
Page 200 and 201: - i�03{ [(NtVN) . (VNt x ifN) + (
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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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