The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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NNLO, this range is larger and extends from 650 to 1000 MeV. This can be understood from the chiral TPEP, which at NNLO includes 1f1f correlations. These introduce a new mass scale well above twice the pion mass. • We have shown that the NLO couplings can be combined in such a way that each combination feeds into one partial wave, see eqs. (4.16)-(4.23).1 More precisely, the nine couplings with four nucleon legs can be determined uniquely by a fit to the two Swaves, four P-waves and the mixing parameter E1 for nucleon laboratory energies below 100 MeV. This simplifies the fitting procedure enormously as compared to ref. [78], where a global fit in all low partial waves has been performed. As expected from the power counting underlying the EFT, the fits improve when going from LO to NLO to NNLO, compare fig. 4.l. • At NNLO, the resulting S-waves are of very high precision (for nucleon laboratory energies below 300 MeV), see e.g. tables 4.2,4.3 and fig. 4.4. The so-called range parameters collected in table 4.4 agree with what is found in the phase shift analysis. The P-waves are mostly well described, in particular the mixing parameter E1 is in good agreement with the phase shift analysis. We also note that above nucleon cms momenta of about 150 MeV, our NLO and NNLO results are far better than the ones obtained in the KSW scheme at NLO and NNLO. • All other partial waves are free of parameters. The D-waves, in particular 3 D1 and 3 D 3 are very well described. We have also discussed the cut-off sensitivity of these results. The NNLO TPEP is too strong in the trip let F-waves. For the peripheral waves, we recover the results of the Munich group [108], namely that in most cases OPE works well but chiral NNLO TPEP clearly improves the description of some partial waves like e.g. 3G5, 3 H5 or 3 h. • The deuteron properties are mostly well described, at NLO and NNLO, compare table 4.7. At NNLO, the deuteron wave functions show some interesting structure due to the appearance of two very deeply bound states. These are an artifact of the NNLO approximation. They have no influence on low energy properties and can be projected out completely from the theory. Our precise deuteron wavefunctions can be used for pion photoproduction, pion-deuteron scattering or Compton scattering off deuterium (still, the hybrid approach proposed by Weinberg [114] remains a useful tool). • We have also considered an approach with explicit D.. degrees of freedom in the TPEP. This NNLO-D.. approach leads to results very similar to the ones at NNLO in the theory without isobars, with the exception of the partial waves that are sensitive to pionic scalar-isoscalar correlations like e.g. 3 D 3 • We conclude that the inclusion of the D.. via resonance saturation of dimension two 1f N LECs capture the essential physics of the isobar in the two-nucleon system. We note, however, that a more systematic study of pion-nucleon scattering in an EFT including the D.. is needed to further quantify these statements. 4. Finally, we have considered electromagnetic and strong isospin violation in low-energy nucleon-nucleon scattering in the effective field theory formalism developed in ref. [91]. We now summarize the results of this investigation. • We first considered isospin violating parts of the effective Lagrangian. The terms related to the pion and the pion-nucleon system were discussed in [199], [222]-[224]. lThis was also noted by Kaplan et al. [91]. 169
170 5. Summary and outlook The isospin violating Lagrangian in the two-nucleon sector has been worked out by van Kolck in ref. [75]. We have reconsidered the corresponding terms using a different approach, namely the method of external fields . • After a brief introduction into the KSW formalism, we have discussed the leading isospin violating effects in the 1 So channel. In particular, the leading charge independence breaking effect is due to a combination of the neutral to charged pion mass difference in one-pion exchange diagrams together with an electromagnetic N N contact term. Its corresponding coupling constant scales as Q-2 but is numerically suppressed by the explicit appearance of the fine structure constant a rv 1/137. We have shown how the KSW power counting has to be modified in the presence of isospin violating operators . • We explicitely evaluated the 1 So phase shifts for the np, nn and Coulomb-subtracted pp systems at next-to-Ieading order. In addition, we have given a general classification of the various CIB and CSB corrections. This allows to understand so me phenomenologically found results. To the best of our knowledge the exact moment um space projection of the nucleon-nucleon interaction discussed and performed in sec. 2.3 has never been done before. In this thesis we have applied this formalism to some problems arising in the context of an effective field theory for the N N system. The method, however, might also provide new insights into many other interesting quest ions in nuclear physics. In particular, it opens the possibility of studying relativistic effects in a consistent and convergent manner, since the moment um components of the order or higher than the nucleon mass can be integrated out. Furthermore, it would be interesting to generalize the above formalism to three- and more-nucleon system. Further, the method of unitary transformation (projection formalism) has never been applied in the context of the chiral effective field theory. Previous calculations of the 2N interaction [73], [74], [76], [78] from chiral effective Lagrangians are based on time-ordered perturbation theory and lead to an energy-dependent potential. As already stressed above, the most important advantages of our formalism against time-ordered perturbation theory is energy-independence of the corresponding potential and the orthonormality of the related wave functions. Appropriate power counting rules, which allow to perform calculations within the projection formalism to any required order in the low-momentum expansion, have been worked out in sec. 3.6 and appendices A, B and C. Our results for nucleon-nucleon interactions derived from the most general chiral invariant Hamiltonian do not only show that the scheme originally proposed by Weinberg works qualitatively, it even works much better than it was expected, namely quantitatively. It extends the successful applications of effective field theory (chiral perturbation theory) in the pion and pion-nucleon sectors to systems with more than one nucleon. Clearly, one should now reconsider processes, which have been evaluated using Weinberg's hybrid approach [114] (?f - d scattering [114], [217], "(d -+ ?f o d [115], "(d -+ "(d [116]) and extend these considerations to systems with more than two nucleons. In addition, a fresh look at charge symmetry and charge independence breaking in the Hamiltonian formalism is called for (for earlier studies, see e.g. refs. [218], [99]). A very important and actual research field comprises the application of CHPT to 3N interactions. Whereas the first results in the KSW scheme and in the pionless effective theory were already presented for the three-body system, see e.g. refs. [119], [120], [121], [122], no calculations within the potential approach including the complete leading 3N force predicted by CHPT have yet been performed. It would be interesting to see whether the inclusion of the 3N forces of such type
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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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Page 176 and 177: Chapter 6 Summary and outlook In th
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Page 192 and 193: Appendix E Anti-symmetrization of t
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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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