The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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Consequently, it is not possible to construct Bi in terms of CS,T and of Ci 1 0 and, therefore, there are no l/m-corrections to contact terms with two derivativesY Finally, we conclude that there are no terms in L�N=3). IOIf we would not eliminate the terms in eq. (F.8) using the EOM for the nucleon field, then we would have l/m corrections to the contact interactions with two derivatives. llNote that one can have l/m 2 -corrections with two derivatives, which, however, would contribute at order ßi = 4. 191
Appendix G Partial wave decomposition of the N N potential In this appendix we would like to deseribe the partial wave deeomposition of the two-nucleon potential. For that we first rewrite the potential V in the form V = Vc + Va- ih · ih + VSL i �(o\ + 5 2 ) · Cf x ifJ + Va-L 51 · (ifx k) 5 2 · (if x k) + Va-q (51 · ifJ (5 2 · ifJ + Va-k (51 . k) (5 2 · k) (G.l) with six functions Vc (p,p',z), ... , Va-k (P,P',z) depending on p == Ipl, p' == Ip'l and the eosine of the angle between the two momenta is ealled z. These functions may depend on the isospin matriees T as weIl. To perform the partial wave deeomposition of V, i. e. to express it in the standard lsj representation, we have followed the steps of ref. [193]. In particular, we start from the helicity state representation Iß A1A 2 ), where ß = vip and Al and A 2 are the helicity quantum numbers eorresponding to nucleons 1 and 2, respeetively. We then expressed the potential in the IjmA1A 2 ) representation using the transformation matrix (ß A1A2IjmA1A2 ), given in ref. [193]. The final step is to switeh to the Ilsj) representation. The eorresponding transformation matrix (lsjmlJmA1A 2 ) is given in refs. [194], [193]. For j > 0, we obtained the following expressions for the non-vanishing matrix elements in the Ilsj) representation: (jOj I V IjOj) (jljlVljlj) 27r [ 1 1 dz {Vc - 3Va- + p,2 p2(z2 - 1)Va-L - q2Va-q - k2Va- k} Pj (z) , 27r [ 1 1 dz {[Vc + Va- + 2P'PZVSL - p,2 p2(1 + 3z2)Va-L + 4k2Va-q + lq2Va-k] X Pj (z) + [-p'p VSL + 2p, 2 p2zVa-L - 2p'p (Va-q - l Va- k)] x (Pj-1(z) + Pj +1(Z)) } , (j ± 1, IjlVlj ± 1, Ij) 27r [ 1 1 dz {p'p [ -VSL ± 2j � 1 ( -p'PZVa-L + Va-q - l Va-k)] x Pj(z) + [Vc + Va- + p'pzVsL + p, 2 p2(1 _ z2)Va-L (G.2) 192
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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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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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Page 157 and 158: 146 --- .... . � - --- .... . ...
Page 159 and 160: 148 � .... --- � � .... --- :
Page 162 and 163: 4.5. Results for the NNLO-ß approa
Page 164 and 165: 5.1 Isospin violating effective Lag
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
Page 194 and 195: Consequently, only four of the eigh
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) + (
Page 204 and 205: (j ± 1, 1jIVIJ =f 1, 1j) Here, Pj
Page 206 and 207: Bibliography [1] E. Rutherford, Phi
Page 208 and 209: [62] J. Goldstone, A. Salam and S.
Page 210 and 211: [137] V. Bernard, N. Kaiser and Ulf
Page 212: [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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