The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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4.1. Bound and scattering state equations 121 where J and E denote the quantities in the Blatt-Biedenharn parametrization. The bound state is obtained from the homogeneous part of eq. (4.5) and obeys where s = j = 1, [ = [' = 0,2 and Ed denotes the bound-state energy. ( 4.10) The potential (4.1)-(4.3) is only meaningful for momenta below a certain scale. This is obvious both from a conceptual and a practical point of view. Indeed, since we are working within the effective field theory approach, only the low-momentum matrix elements can be derived systematicaHy. Our perturbative expansion far the potential breaks down for momenta of the nucleons comparable with the scale Ax. Also from the practical point of view, the expressions (4.1)-(4.3) are, clearly, not acceptable for large values of the moment um transfer q. In particular, since the potential grows for large momenta, it leads to ultraviolet divergences in the LS equation (4.5). As it is appropriate in effective theory, we regularize the potential. This is done in the following way: (4.11) where fR(iJ) is a regulator function chosen in harmony with the underlying symmetries. In what follows, we work with two different regulator functions, ftarp(p) f �xpon (p) B(A2 _ p2) , exp( _p2n / A 2n) , ( 4.12) (4.13) with n = 2,3, ... , p = Ipl and p' = Ip'l. Such a regularization of the effective potential is also consistent with the discussion in chapter 2. We must introduce a cut-off in order to exclude the high-momentum intermediate states in the LS equation, that can not be described correctly within the low-energy effective theory. In eqs. (4.12), (4.13) we show the two different choices of the regulator function. The sharp cut-off is most appropriate for comparison with realistic phenomenological potentials. To enable such a comparison one first needs to integrate out the high-momentum components from the realistic potentials, since the effective one is defined only for momenta p, p' < A. Eliminating the highmoment um components can be performed via the unitary transformation, as described in section 2.3. We already performed that step but leave the discussion of the results to a furt her study. The sharp cut-off regularization scheme is suitable for deriving the phase shifts but leads to troubles when calculating some deuteron properties. This is because discontinuities in the moment um space are introduced at p, p' = A. In that case one should use a smooth regularization like the exponential one given in eq. (4.13). For very large integers n the exponential cut-off approximates the sharp one. Throughout, we work with n = 2. To the order we are working, the choice n = 1 has to be excluded since the terms of order p2,p /2 would be modified. For n = 2, the error we make is beyond the accuracy of the order we are calculating. We would like to point out that the low-energy observables should not be sensitive to the choice of the regulating function fR as weH as to an exact value of the cut-off. We will also confirm this statement by explicit calculations. The remaining (weak) dependence of the observables on the choice of fR and on the value of A can be systematically removed by including the higher order terms [141]. In the following seetions we will discuss these uncertainties in more details.
122 4. The two-nuc1eon system: numerical results 4.2 The fits In this section we, basicaIly, concentrate on the determination of the various coupling constants. For Mn:, in: and gA we use the following values: Mn: = 138.03 Me V , in: = 93 MeV , gA = 1.26 . ( 4.14) Furthermore, as already stated above, the values of C 1 ,3,4 are Cl = -0.81 GeV -1 , C3 = -4.70 GeV-1 , C4 = 3.40 GeV-1 . ( 4.15) Thus, all parameters entering the non-polynomial part of the potential are fixed. Consider now the polynomial part vcontact of the potential represented by contact interactions vcontact = V(O) + V(2). To pin down the nine parameters Cs, CT, Cl , ... ,C7 we do not perform global fits as done in ref. [78]. Rather we introduce the independent new parameters C2s+1l., J 02S+1l. via the following equations: J v contact (3 SI ) 41f (Cs - 3CT) + 1f (4C1 + C2 - 12C3 - 3C 4 - 4C6 - C7)(p2 + p'2) - 2 2 C1S0 + C1S0 (p + p' ) , (4.16) 41f (Cs + CT) + i (12C1 + 3C 2 + 12C3 + 3C4 + 4C6 + C7)(p2 + p'2) - 2 2 C3S1 +C3S 1 (p + p' ), (4.17) 2 ; (-4C1 + C2 + 12C3 - 3C4 + 4C6 - C7) (p p') = Cl PI (pp') , (4.18) 2 ; (-4C1 + C2 - 4C3 + C4 + 2C5 + 4C6 + C7) (p p') C3P1 (pp') , 2 ; (-4C1 + C2 - 4C3 + C 4 + 2C5) (pp') = C3P2 (pp') , 21f 3 (-4C1 - + C2 4C3 ) , + C4 + 4C5 + 12C6 - 3C7 (pp ) C1 Po (pp') , 2.j21f ( ,2 ,2 - 3 - 4C6 + C7) p = C3Dl-3S1 P , 2.j21f 2 2 - 3 - (4C6 + C7)p = C3Dl-3S1 P . (4.19) ( 4.20) (4.21) (4.22) (4.23) Here, Ves+1lj) denotes the matrix element (lsjlVllsj) and Ves+1lj _2s+1lj) is (lsjlVll'sj). The equations (4.16)-(4.23) can be obtained with the help ofthe formulae of appendix G for the partial wave decomposition. The main advantage of using the new parameters C2s+1[., 02s+1l. instead of J J the old ones Cs, CT and Cl , ... , C7 is that now we can fit each partial wave separately. This not only makes the fitting procedure extremely simple, but also leads to unique results. To leading order, the two S-waves are depending on one parameter each. At NLO, we have one additional parameter for 1 So and 3 SI as weIl as one parameter in each of the four P-waves and in EI' At NNLO (NNLO-Ll), l we have no new parameters, but must refit the various contact interactions due to the TPEP contribution in all partial waves. We have used two different methods to fix the 1We will denote by NNLO-ß the potential (4.24)
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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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Page 107 and 108: 96 _ . . -.:;: " :,,,_ ._ . . _. 3.
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Page 176 and 177: Chapter 6 Summary and outlook In th
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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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