The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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Consequently, only four of the eight contact terms in eq. (E.9) are independent. Choosing those terms to be (E.12) and requiring that the anti-symmetrized contributions of eqs. (E.9) and (E.12) are the same, we end up with the following result using eqs. (E.10), (E.ll): 3"(8 "(7 "(1 - "(3 - 4 - 4 ' -4"(3 - 12"(4 + "(5 - "(7 , "(7 "(8 "(2 - "(4 - 4 + 4 ' -4"(3 + 4"(4 + "(6 - "(8 . 183 (E.13)
Appendix F The complete set of N N contact interactions with two derivatives The most general effective Lagrangian for nucleons given in refs. [76], [78] contains 14 contact interactions with two derivatives, which are multiplied by the coefficients C 1 , ... 14. In this appendix we would like to show that 7 of these 14 terms are not independent from the others and have fixed coefficients. Moreover, considering the couplings accompanying these terms as apriori unknown and non-vanishing quantities would violate general symmetry principles of the theory.1 The situation he re is similar to the familiar case of the nucleon kinetic energy term Nt\l 2 j(2m)N in the Lagrangian corresponding to the Hamiltonian (3.199). The Lorentz invariance of the theory would be broken if one allows an arbitrary coefficient for the term Nt\l 2 N. As explained in the text, the nucleons in the effective theory can be described by velocity dependent fields Hv and hv, see eq. (3.160). 2 The total momentum Pp. of the nucleon is then parametrized by a pair (Vp., Ip.) via eq. (3.158). The small component field hv can be eliminated from the theory, which leads to 1jm-corrections in the effective Lagrangian. Clearly, such correction terms have fixed coefficients, defined by the structure of the original relativistic Lagrangian. Explicitly integrating out the small component field hv has not yet been worked out for nucleon interactions with four and more legs. Instead of explicitly integrating out hv, it is easier in such a case to write down all possible terms consisting of Hv , vp. and Sp., which is defined by and of covariant derivatives of the nucleon and pion fields.3 Note that the spin operator Sp. obeys the following relations (in four dimensions): 2 3 1 S· v = 0, S = -4' {Sp., Sv} = 2 (vP.vv - 9p.v) , [Sp., Sv] = ifp.vpuV P S u , (F.2) where we have used the convention f0 1 2 3 = 1. One can require the reparametrization invariance [182] of the effective Lagrangian to fix the coefficients for the 1jm-corrections and to eliminate the terms violating Lorentz symmetry. The effective Lagrangian expressed in terms of velo city dependent fields is reparametrization invariant if it preserves its form under the reparametrization (F.1) (Vp., Ip.) +-t (Wp., kp.) = (Vp. + qp. , Ip. - q p.) , (F.3) m 1 In particular, the Galilean invarianee would be broken. 2 In sec. 3.4 we have suppressed the label v for those fields. 3 AH Dirae bilinears fIv rHv (r = {I, 'Y5 , 'YI" 'Y5'YI" O"l' v }) ean be expressed in terms of (fIvHv), (fIvSI' Hv ), v!' and of the anti-symmetrie Levi-Civita tensor El'vpu , see e.g. [72]. 184
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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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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 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 202 and 203: Consequently, it is not possible to
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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