The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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3.2. Effective Lagrangians 55 would observe parity doublets Ih) and Q'Alh) in the hadron speetrum. No sueh parity doubling is observed in the real world. Another argument in favor of spontaneously breaking of the chiral symmetry is the presenee ofvery light pseudoscalar mesons (pions in the case of Nf = 2), which ean be associated with the Goldstone bosons. In fact, spontaneous breaking of the chiral symmetry is also evident for other reasons like, for instance, large differenees in vector and axial-vector spectral functions, results of lattice ealculations and so on [158]. Clearly, only SU(Nf)A is spontaneously broken.7 Otherwise one would also observe scalar Goldstone bosons. A more evident reason is the presenee of SU(Nf)V multiplets in the hadron spectrum. For instanee, SU(Nf = 2)v corresponds to an ordinary isospin transformation group. The high quality of the isospin symmetry of the strong interaction may be demonstrated by comparing the proton and neutron masses mp = 938.27 MeV and mn = 939.57 MeV, which are indeed very elose to eaeh other. In faet, the ehiral symmetry is broken not only spontaneously but also explicitly due to the presenee of the quark mass term .c�CD in the QCD Lagrangian (3.1). Let us now be more eonerete and regard Nf = 2. Furthermore, we will assume that the matrix M is already in diagonal form: M ( � u �d ) �(mu + md) (� �) + �(mu -md) ( � �1 ) 1 1 2(mu + md)I + 2(mu -md)T3 , (3.50) where I is the unit matrix in 2 dimensions and T3 a Pauli matrix. Whereas the term proportional to the quark mass difference breaks both SU(2)v and SU(2)A symmetries (or, equivalently, SU(2)L and SU(2)R), the one proportional to mu + md remains SU(2)v invariant. It turns out that the quark masses mu rv 3 --;-7 MeV and md rv 7 --;- 15 MeV are mueh smaller than the typieal hadronie mass seale of the order of rv as a small perturbation. 3.2 Effective Lagrangians 1 Ge V. Thus, the quark mass term .c�CD indeed ean be considered In the last seetion we have discussed the general symmetry aspeets of QCD. Now we would like to eoneentrate on the two fiavor case. Let us furt her switeh from quark and gluon to hadron degrees of freedom. At present, one is not able to direetly determine the strueture and dynamics ofhadrons from QCD. Therefore, it is useful to apply the powerful method of effective field theory (EFT) technique to analyze the interactions between the hadrons at low energies. Some examples of such a teehnique were already given in the previous ehapter. Clearly, one would like to extraet all possible information from QCD in order to minimize the number of free parameters in the effective Lagrangian. One expects that the ehiral symmetry of the QCD Lagrangian is very important for EFT eonsiderations, sinee the lightest hadronic degrees of freedom, the pions, are elosely related to its spontaneous breaking. We will see below that the spontaneously broken ehiral symmetry indeed provides strong eonstraints on the strueture of pion-pion and pion-baryon interaetions. The starting point in the EFT program is the construction of the effective Lagrangian. Apart from the usual requirements like loeality, invarianee und er Lorentz transformations, parity, timereversal invarianee and hermeticity the effective Lagrangian should also be chirally symmetrie. 7 In fact, Vafa and Witten have proven that the global SU(Nf)v symmetry of QCD can not be spontaneously broken [159). This holds more generally for any vector-like (gauge) theory.
56 3. The derivation of nuc1ear forces from chiral Lagrangians The procedure of implementing the broken symmetry transformation on Goldstone fields was first discussed by Weinberg [160] in 1968 in the context of the chiral SU(2)v x SU(2)A group broken to SU(2)v and generalized one year later by Coleman, Callan, Wess und Zumino (CCWZ realization) [67] to the case of a general compact group G broken to an arbitrary subgroup H. Here we will basically follow this last work. An arbitrary element 9 of the group G can be parametrized ass where {A} and {V} are the sets of generators of the group G satisfying the Lie algebra [Vi, Vj] [Aa, Ab] [Vi, Aa] CijkVk, CabcAc + Cabk Vk , CiabAb· (3.51) (3.52) Here, the C's are the antisymmetric structure constants of the group G. Clearly, the chiral Lie algebra (3.36) is a particular case of eqs. (3.52). Note that the generators Vi form a closed Lie algebra. Therefore, the transformation h = eS 'v corresponds to the subgroup H of G, which is required to be unbroken. The CCWZ realization is defined by the mappings where ( and s ' have to satisfy � � ( = ( (�, go) ,
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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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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
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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
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
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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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Page 105 and 106: 94 , , , , , , , , , , , , , , , ,
Page 107 and 108: 96 _ . . -.:;: " :,,,_ ._ . . _. 3.
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Page 111 and 112: 100 3. The derivation o{ nuc1ear {o
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106 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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