The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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Contents 1 Introduction 2 Low-momentum effective theories for two nucleons 11 2.1 What is effective? . . . . . . . . . . . . 11 2.2 Two nucleons at very low energies . . . . 15 2.3 Going to higher energies: a toy model . . 23 2.3.1 Method of unitary transformation 24 2.3.2 Regularization of the decoupling equation 29 2.3.3 Numerical realization of the projection formalism . 32 2.3.4 Unitary transformation for a potential of the Malfliet-Tjon type 33 2.3.5 Implication for effective theories 37 2.3.6 Coordinate space representation . . . . . . . . . . . 46 3 The derivation of nuclear forces from chiral Lagrangians 48 3.1 Chiral symmetry . . . . . . . . . . . . 48 3.2 Effective Lagrangians . . . . . . . . . . 55 3.3 Chiral perturbation theory with pions 69 3.4 Including nucleons . . . . . . . . . . . 73 3.5 Bloch-Horowitz scheme and the method of unitary transformation 78 3.6 Application to chiral invariant Hamiltonians . . . . . . . . . 81 3.7 Nuclear forces using the method of unitary transformation . 87 3.8 Two-nucleon potential . . . . . . . . . . . . . . . . . . 91 3.8.1 Expressions and discussion . . . . . . . . . . . . . . 91 3.8.2 Renormalization of the N N potential at NLO . . . . 3.8.3 Phenomenological interpretation of some of the 7fN LECs and the role of 100 the ß(1232) . . . 105 3.9 Three-nucleon potential . . . . . . . . . . . . 113 4 The two-nucleon system: numerical results 4.1 Bound and scattering state equations . 4.2 The fits . . . . 4.3 Phase shifts . . 4.3.1 S-waves 4.3.2 P-waves 4.3.3 D- and F-waves 4.3.4 Peripheral waves 4.4 Deuteron properties . . 4.5 Results for the NNLO-ß approach IX 1 119 119 122 128 128 133 136 140 144 149
x 5 Isospin violation in the two-nucleon system 5.1 Isospin violating effective Lagrangian . . . . . . . . . . . . 5.2 Brief introduction into the KSW approach . . . . . ... . 5.3 The leading CIB and CSB effects in the N N 1 So channel 5.4 Numerical results . . 5.5 Classification scheme 6 Summary and outlook A Dimensional analysis of the projected decoupling equation (3.206) B Operators contributing to eq. (3.206) at order r C Small scale expansion within the projection formalism D Divergent loop integrals E Anti-symmetrization of the contact interactions F The complete set of N N contact interactions with two derivatives G Partial wave decomposition of the N N potential H Formulae for the deuteron properties 152 153 156 159 162 163 165 172 175 177 180 181 184 192 194
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: Vlll betrachten, um eine komplette
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 53 and 54: 42 -3 -5 -7 -9 2. Low-momentum effe
Page 59 and 60: 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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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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