The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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3.8. Two-nucleon potential ,, ' \ 1 , , , , , , , , , , , \ , I " , , 2 5 6 \ 1 1 , \ 1 1 1 1 \ \ \ 1 Figure 3.10: First corrections to the NN potential: irreducible self-energy and vertex correction graphs. For notations see fig. 3.6. , , , , 1 I , \ 'I 1 \ I, ' , , , , , \ , , 1 3 7 , , , , , \ \ , , , 1 1 , , 2 3 4 Figure 3.11: First corrections to the NN potential: reducible self-energy graphs. For notations see fig. 3.6. following equations: Oij + iEijkTk Oij + iEijkO"k 4 \ \ 95 (3.266) (3.267) The initial (final) moment um of the nucleons in the c. m. system is denoted by p (p'). The transferred and average momenta are given by � -I -+ q = p -p, 2 (3.268) � p+ p' k= --
96 _ . . -.:;: " :,,,_ ._ . . _. 3. The derivation of nuc1ear forces from chiral Lagrangians _. . _ .:: . .,. :",_. _. . _ . _ . . _._. __ ._ . . _. _ . . _._."._._ . . _. 2 - . . _._. __ ._ . . -. _ . . _._.04._._ . . _. _. __ . :: ,., 1""_ ._ . . _. Figure 3.12: Reducible graphs. In (1), a truly reducible diagram is shown. In the projection formalism, one has graphs like (2) and (3). These correspond to diagram 9 in fig. 3.9. The horizontal dot-dashed lines count the free energies of the particles cut. respectively. The leading order potential can then be written as: 2 3 4 5 6 Figure 3.13: NLÜ one-loop corrections to the four-nucleon contact interactions in the projection formalism. For notations see fig. 3.6. 3 (3.269) The first corrections to this result appear at order two. We now enumerate all such corrections. Since we want to compare both schemes, we will first discuss the results obtained within oldfashioned perturbation theory and then present the potential derived using the method of unitary transformation. We start with the tree-diagram contribution given by the graph of fig. 3.8 with various contact interactions from eq. (3.237). This leads to the potential (2) VNN,tree C �2 C k2 (C �2 � � C k2 )( 1 q + 2 + 3 q + 4 0"1 • 0"2 + Z 5 . x q) 2 + C6 (if· o\) (if· i'h) + C7 (k . ih)(k . (2 ) , (3.270) ) ' C 0\ + ih (k ;;'I
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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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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
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Page 127 and 128: 116 3. The derivation o{ nuc1ear {o
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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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