The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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4.3. Phase shifts 1 Elab [MeV] 11 NNLO* h 62.071 2 64.472 3 64.671 5* 63.659 10* 60.02 20 53.66 30 48.55 50* 40.49 100* 26.30 200 7.63 300 -6.41 NNLO 1 Nijm PSA I VPI PSA 1 Nijm93 1 AV18 CD-Bonn I 62.063 62.069 62.156 62.065 62.015 62.078 64.469 64.573 64.573 64.460 64.388 64.478 64.671 64.762 64.762 64.650 64.560 64.671 63.663 63.708 63.708 63.619 63.503 63.645 60.03 59.96 60.00 59.94 59.78 59.97 53.68 53.57 53.77 53.54 53.31 53.56 48.58 48.49 49.00 48.42 48.16 48.43 40.54 40.54 41.66 40.38 40.09 40.37 26.38 26.78 27.86 26.17 26.02 26.26 7.76 8.94 7.86 7.07 8.00 8.14 -6.24 -4.46 -5.55 -7.18 -4.54 -4.45 Table 4.3: ISO np phase shift for the best fit at NNLO (sharp cut-off, A = 875 MeV) compared to phase shift analyses and modern potentials. The parameters of the NNLO potential are fixed by fitting the Nijmegen PSA at six energies (E1ab = 1,5, 10, 25,50, 100 MeV). These energies are marked by the star. The parameters of the NNLO* potential are chosen to reproduce exactly the scattering length and the effective range as described in the text. I SO NLO -23.555 2.64 -0.58 5.4 -31 I SO NNLO -23.722 2.68 -0.61 5.1 -30 ISO NPSA -23.739 2.68 -0.48 4.0 -20 3S 1 NLO 5.434 1.711 0.075 0.77 -4.2 3S 1 NNLO 5.424 1.741 0.046 0.67 -3.9 3S 1 NPSA 5.420 1.753 0.040 0.67 -4.0 Table 4.4: Scattering lengths and range parameters for the S-waves at NLO and NNLO (global fits) compared to the Nijmegen PSA (NPSA). The values for V2, 3 , 4 in the 1 So channel are based on the np Nijm II potential and the values of the scattering length and the effective range are taken from the ref. [209]. The effective range parameters for the 3S 1 _3 D 1 channel are discussed in [102]. interactions without derivatives. The first corrections are given by dressing the one-pion exchange and the contact interactions with two derivatives by the leading order amplitude. This is the crucial difference to our power counting scheme, in which the OPE diagrams are of the same size as the leading contact interactions and thus should both be treated non-perturbatively. The NNLO corrections in the KSW scheme are given by various diagrams including contact interactions with 0, 2 and 4 derivatives as weH as pion exchange graphs. For more details see ref. [211]. Because of the perturbative treatment of the pion exchanges, the authors of ref. [211] could perform an analytic calculation of the amplitude including its renormalization. The resulting S-matrix satisfies the perturbative unitarity condition in the sense of the KSW power counting. Furthermore, at each 131
132 4. The two-nuc1eon system: numerical results I SO NLO* -23.739 2.68 -0.52 5.3 -30 ISO NNLO* -23.739 2.68 -0.61 5.1 -29.7 3S 1 NLO* 5.420 1.753 0.110 0.73 -3.9 3S 1 NNLO* 5.420 1.753 0.057 0.66 -3.8 Table 4.5: Scattering lengths and range parameters for the S-waves at NLO and NNLO. The corresponding LEes are obtained from the fit to a and r in the S-channel and from the first effective range parameter in the moment um expansion of E I , as described in text. order the amplitude is independent on the renormalization scale. Let us now comment on the results presented in ref. [211]. Imposing the constraints provided by the perturbative solution of the renormalization group equations (RGE's) and requiring that unphysical poles are removed from the amplitude at low momenta leave one free parameter at NLO and two parameters at NNLO for the 1 So and 3 SI channels, that were fixed by a fit to the phase shifts of the Nijmegen PSA. For the 1 So phase shift one observes a visible improvement when going from LO to NLO and from NLO to NNLO. At NNLO, the ISO phase shift goes very close to the one from the Nijmegen PSA up to cms momenta rv 300 MeV (E1ab rv 192 MeV), see fig. 3 in [211]. Such visible agreement of the phase shift with the data does, however, not yet allow to conclude about the quality of the results. For example, the authors of ref. [211] point out that the the 1 So phase shift at the cms momentum p = Mn is offits experimental value4 by 17% at NLO and by less then 1% at NNLO. On the other side, the effective range expansion (4.26) with the first two coefficients (a and r) yields at p = Mn the value for the phase shift, which differs from the experimental one by only rv 2%. Thus, this information is not enough to definitely conclude about improvement of the results obtained from the effective theory with explicit pions relative to the calculations within the pionless theory. A better testing ground for that is given by the shape parameters in the effective range expansion (4.26), which are expected to be sensitive to the pion physics, as discussed above, see also [100], [101]. At NLO, predictions for V2,3 , 4 totally disagree with the values derived from the Nijmegen PSA. The NNLO predictions for r and the v's, given in ref. [211] are r = 2.63 fm, V = 2 -1.2 fm3, V = 3 2.9 fm5 and V4 = -0.7 fm7, which still significantly differ from the experimental values shown in table. 4.4 and are considerably worse than our predictions indicated in the same table. Such dis agreement with the data is surprising since performing NNLO calculations without pions and choosing a finite cut-off of the order of Mn , see sec. 2.2, one expects to be able to reproduce the first four effective range parameters (a, r, V2 and V3 ) exactly. Therefore, no clear improvement relative to the pionless theory can be observed. For the 3 SI channel the situation turns out to be much worse than for the 1 So channel for the KSW scheme. Whereas the phase shift is described quite accurately at NLO, NNLO corrections are large and destroy the agreement with the data already at the cms momenta of the order of Mn [211]. This is shown in fig. 4 of that reference. The failure of EFT at NNLO is found to be due to large contributions resulting from the iteration of the pion exchanges, which is missed in the KSW approach. We would also like to comment on the recent work by Hyun et al. [106]. There, the 1 So channel is considered within an approach similar to ours. In particular, the authors of this reference consider the potential, which consists of the OPE and leading TPE contributions given in eqs. (4.1), (4.2) 4 As usual, we consider the values of the phase shifts from the Nijmegen PSA as experimental data.
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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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Page 107 and 108: 96 _ . . -.:;: " :,,,_ ._ . . _. 3.
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Page 125 and 126: 114 / / / / / / / 5 / / \ \ \ \ I I
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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
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Page 182 and 183: might be responsible for solving th
Page 184 and 185: derivative. Further helpful equalit
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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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