The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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2.3. Going to higher energies: a toy model 41 numerically, we can determine the constants Ci by fitting eq. (2.116) to V'(q', q) - VIight (q', q). This is done numerically using the standard FORTRAN subroutines for polynomial fits to functions of one variable and taking the corresponding polynomials typically of order ten to eleven. Once this is done for the functions V(q, 0) and V(q, q) all constants Ci, C: in eq. (2.117) can be easily evaluated. For the potential parameters used so far and the choices of A = 400 MeV and A = 300 Me V, the resulting constants are given in table 2.1: Co C' 4 6 1 A = 400 MeV 11 11.23 1 -32.27 1 86.73 1 113.9 -913.6 I A = 300 -265.2 MeV 11 11.15 I -32.93 I 85.76 I 115.6 -232.6 C' -783.7 Table 2.1: The values of the coupling constants Ci, q in [GeV(-2-i )] for two choices of the cut-off A. Naturalness of the coupling constants Ci, q me ans that (2.118) C2n - = anAscale 2 , C 2n+2 where the an are numbers of order one. Indeed, as one can read off from table 2.1, such a common scale exists, namely Ascale = 600 MeV . (2.119) This is a reasonable value in the sense that it is very elose to the mass /1H of the heavy mesons, which is integrated out from the theory. Stated differently, the value for Ascale agrees with naive expectations. As it turns out, even for cut-off values like A = 300 MeV, there is only a small difference between the heavy meson mass and the ensuing natural mass scale. Another observation is that the values of the Ci, CI depend very weakly on the concrete choice of the cut-off A. Clearly, for such an expansion of the heavy mass exchange in terms of contact interactions to make sense, A has to be chosen below /1H. Furthermore, since we explicitly keep the light meson, A should not be smaller than the mass /1L. If one were to select such a value for the cut-off, one could also consider the possibility of expanding the light meson exchange in a string of contact terms. We do, however, not pursue this option in here. Therefore, we conelude that one should set A < Ascale but it is not possible to find a more precise relation. In fig. 2.10 MeV. we show how weIl the potential V' (q', q) is reproduced by the truncated expansion eq. (2.116) for A = 300 We have also calculated the two-body binding energy and the phase shifts using the form eq. (2.116) with the constants given in table 2.1. The corresponding results are shown in fig. 2.11 and in table 2.2. The agreement with the exact values is good. However, one sees that terms of rat her high order should be kept in the effective potential Vcontact in order to have the binding energy correct within a few percent. Note, however, that the value of the binding energy is unnaturally small compared to a typical hadronic scale like the pion mass or the scale of chiral symmetry breaking. The phase shifts are described fairly weIl for kinetic energies (in the lab) up to about 100 MeV if one retains the first three terms in the expansion eq. (2.116). For A = 400 MeV, one can not expect any reasonably fast convergence for the binding energy any more. This can be traced back to the fact that one is elose to the radius of convergence for momenta elose to the cut-off. More specificaIly, with q = q' = 400 Me V, the pertinent expansion parameter is (q,2 + q2)j A;cale ::::' 0.9. The ensuing very slow convergence is exhibited in table 2.2. Although we have found that Ascale rv 600 MeV and, therefore, the expansion of V' in terms of contact terms seems to converge for the chosen value of the cut-off A = 300 MeV, the ultimative
42 -3 -5 -7 -9 2. Low-momentum effective theories for two nuc1eons 0.3 0.8 0.6 004 0.2 0 -0.2 -004 q [GeV] 0.3 0 Figure 2.10: Left panel: effective two-nucleon potential V' (solid lines) in comparison with the truncated expansion (2.116) (dashed lines) for A = 300 MeV. Right panel: difference ßV between the effective potential V' and the truncated expansion as a function of momenta q, q ' . I E [MeVJ I E [MeVJ V( o ) V(O) + V(2) I V(O) + V(2) + V(4) I V( o ) + V(2) + V(4) + V(6) I V�ontact I 0.46 3.18 1.95 2.29 2.23 I 0.67 7.15 1.82 3.15 2.23 J Table 2.2: The values of the binding energy calculated with V�ontact' eq. (2.116), for A = 300 (second row) and A = 400 MeV (third row). 0.3 Me V test of convergence of the expansion (2.116) should be to calculate the quantum averages of operators V(O), V(2), V(4), V(6) , ... for the low-energy scattering and the bound states, as it was proposed in [81], [83J. There, the expectation values of various contact interactions in a pionless theory were estimated for the deuteron using the leading-order approximation (2.58) for the deuteron wave function. This allows only for a very rough estimate of the size of the operators in eq. (2.116). Furthermore, the value of the cut-off was chosen much below the scale, at which the effects of new physics appear. l9 With these assumptions it was found that all contact operators starting from the terms with two derivatives are of the same order. In our model we can perform numerically exact calculations of these quantities using not only the bound-state wave function but also the scattering wave functions without any additional and unnecessary assumptions. Using the relation eq. (2.93) one obtains for an arbitrary operator 0 (2.120) 19 In the pionless theory considered in [81], [83] this scale is associated with the pion mass m" . In our model this scale is given by the mass of the heavy meson.
Page 1 and 2: The Nucleon-Nucleon Interaction in
Page 3 and 4: 11 vom ursprünglichen wesentlich u
Page 5 and 6: IV Ordnung des Potentials besteht a
Page 7 and 8: VI sagen für die Schwerpunktsimpul
Page 9 and 10: Vlll betrachten, um eine komplette
Page 11 and 12: x 5 Isospin violation in the two-nu
Page 13 and 14: 2 1. Introduction nuclear force of
Page 15 and 16: 4 1. Introduction the Thcson-Melbou
Page 17 and 18: 6 1. Introduction pion-nucleon syst
Page 19 and 20: 8 1. Introduction completely domina
Page 21 and 22: 10 1. Introduction such interaction
Page 23 and 24: 12 2. Low-momentum effective theori
Page 39 and 40: 28 2. Low-momentum efIective theori
Page 47 and 48: 36 2 o -2 -4 -6 -8 -10 � -12 t-
Page 49 and 50: 38 (Eq - Ep )A(p, q) [GeV-2] 2 1.6
Page 51: 40 2. Low-momentum effective theori
Page 59 and 60: Chapter 3 The derivation of nuclear
Page 61 and 62: 50 3. The derivation of nuc1ear for
Page 63 and 64: 52 and for the SU(Nf h x SU(Nf)R [Q
Page 65 and 66: 54 3. The derivation of nuclear for
Page 77 and 78: 66 3. The derivation o{ nuclear {or
Page 85 and 86: 74 3. The derivation o{ nuc1ear {or
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92 3. The derivation of nuc1ear for
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94 , , , , , , , , , , , , , , , ,
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96 _ . . -.:;: " :,,,_ ._ . . _. 3.
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98 3. The derivation o{ nuc1ear {or
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100 3. The derivation o{ nuc1ear {o
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102 3. The derivation o{ nuclear {o
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104 3. The derivation of nuclear fo
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114 / / / / / / / 5 / / \ \ \ \ I I
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116 3. The derivation o{ nuc1ear {o
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120 4. The two-nuc1eon system: nume
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134 10 8 4 2 o o • Nijmegen PSA N
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136 4. The two-nucleon system: nume
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140 4. The two-nucleon system: nume
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146 --- .... . � - --- .... . ...
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148 � .... --- � � .... --- :
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4.5. Results for the NNLO-ß approa
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5.1 Isospin violating effective Lag
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5.1 Isospin violating effective Lag
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5.2 Brief introduction into the KSW
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5.3 The leading eIB and eSB effects
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5.3 The leading CIB and CSB effects
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5.5 Classification scheme 163 scatt
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Chapter 6 Summary and outlook In th
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2. Secondly, we considered the real
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NNLO, this range is larger and exte
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might be responsible for solving th
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derivative. Further helpful equalit
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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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