The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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3.8. Two-nucleon potential Here, g� (1 - g�) ( ) . 641fmf� Tl ' T2 Z (Tl + (T2 . P x P 1f + q q . ( � � ) ( � I �) (4M 2 2)A( ) A(q) = 1 q 2q 2M1f - arctan -- . Furt her , P(k, if) in eq. (3.309) is a polynomial in momenta of at most second degree and has the same structure as the expressions in eq. (3.270).44 Thus, its explicit form is of no relevance here, since it only contributes to the renormalization of the couplings Cs , CT , Ci . Again, no explicit dependence on the renormalization scale appears in the expression (3.310) für the non-polynomial terms. In ref. [108] it is also pointed out that all NNLO one-pion exchange diagrams containing vertex corrections do not yield any pion-nucleon form factor and only lead to renormalization of the 1fN coupling gA (i.e. they have the same form as the proper one-pion exchange). Clearly, the same holds true for all NNLO graphs representing vertex corrections to short range interactions. For the LECs Cl,3,4 entering eq. (3.310) we should take the values obtained from fitting 1fN phases in the threshold region, see e.g. ref. [71] or, alternatively, from fitting the invariant amplitudes inside the Mandelstarn triangle, i.e. in the unphysical region [195]. The so determined parameters are only slightly different, but these small differences will play a role later on. For example, the LECs CI,3,4 from fit 1 of ref. [71] are Cl = -1.23 GeV-l, C3 = -5.94 GeV-l and C4 = 3.47 GeV-l. A recent investigation of the subthreshold amplitudes [195] leads to slightly different values, Cl = -0.81 GeV-l, C3 = -4.70 GeV-l and C4 = 3.40 GeV-l. It is this latter set we will use in the following. These values are also consistent with the re cent determination from the proton-proton interaction based on the chiral two-pion exchange potential [198]. Kaiser et al. also included l/m corrections into the expression (3.310) for the two-pion exchange potential an NNLO. This is consistent with the power counting scheme in the one-nucleon sector. In our power counting scheme with Q/m rv Q2 / A�, where Q corresponds to the low moment um scale, such l/m terms appear first one order higher. Nevertheless, we have decided to keep these corrections, since otherwise one cannot directly use the values of the LECs Cl,3,4 as determined from the 1fN sector in the presence of the l/m terms. By doing so we only include some sm aller terms of higher order in the potential. It can easily be checked, that those terms are indeed small as compared to the Cl,3,ccontributions. For example, the constant C4 enters eq. (3.310) only in the combination C4 + 1/ ( 4m). Correspondingly, the pertinent C4 contribution is numerically more than 10 times larger than the l/m correction. To end this section we would like to point out once more that eqs. (3.269), (3.300), (3.270) and (3.310) define the unique expressions for the effective potential up to NNLO, i.e. the results do not depend on the regularization scheme, as discussed above. 105 (3.310) (3.311) 3.8.3 Phenomenological interpretation of some of the 1f N LEes and the role of the ß(1232) It is well known that the values of some of the LECs can be understood phenomenologically in terms of the resonance saturation hypotheses. This is discussed in detail in the reference [199] for the purely meson sector and in ref. [200] for pion-nucleon interactions. The idea of such a phenomenological interpretation is very simple. Consider the most general chiral invariant Lagrangian for pions, nucleons and their excitations. The resonances can be integrated out from 44 More precisely, after performing the partial wave decomposition, fCk, if) leads in each partial wave to polynomials in p and pi of at most second degree.
106 3. The derivation of nuclear forces from chiral Lagrangians the Lagrangian if one regards their excitation energies to be very large (or, equivalently, if one considers 7r N scattering at energies much smaller than the excitation energies). This generates a I I + , / , / , / , / , / , / .. Figure 3.16: Ll-resonance saturation of the 7rN LECs C3 and C4. Double solid line represents Ll-isobar. For remaining notations see figs. 3.6, 3.14. series of local pion-nucleon operators of increasing dimension with the coupling constants fixed by details of the pion- (nucleon-) resonance interactions. Certainly, one expects that the resonances with the lowest excitation energy give the dominant contributions to the values of the LECs. In particular, the Ll(1232) with the excitation energy of only about two pion masses is expected to be quite important for the low-energy pion-nucleon dynamics. This expectation is confirmed by a phenomenological analyses [200]. It turns out that the value of C3 is dominated by the Ll contributions. The Ll-isobar also leads to a sizable contribution to C4, whereas the value of Cl is mainly saturated by the scalar-isoscalar meson contributions.45 In many approaches to the N N interactions the effects of the Ll-excitations in the intermediate states are explicitly included. Because of the relative small value of the LlN-mass splitting (293 MeV) such effects might be quite important for the low-energy N N dynamics. Already in their pioneering work [78], Ord6iiez et al. incorporated the leading effects due to the intermediate Llexcitations treating the LlN-mass splitting as a small quantity on the same footing as the pion mass. Although the LlN-mass splitting does not vanish in the chiral limit, such a phenomenological extension of chiral perturbation theory is quite useful in many situations and may be formulated in a systematic fashion (the so-called "small scale expansion") [201]. In this approach one starts from the most general chiral invariant Lagrangian for relativistic pions, nucleons and deltas. To describe the spin- and isospin-3/2 field corresponding to the Ll, one typically uses the Rarita-Schwinger formalism [202], i.e. one introduces a vector-spinor field W jl (x) that satisfies the equation of motion (3.312) with the subsidiary condition (3.313) Furthermore, the Rarita-Schwinger spinors for the spin 3/2 field are constructed by coupling spin- 1 vector to spin-1/2 Dirac spinor fields via Clebsch-Gordon coefficients. One then introduces a complete set of the projection operators (p3/2)jll/ and (PiY\w with i, j = 1,2 onto the spin-3/2 and spin-1/2 components that satisfy the algebra [201] ( P 3/2) jll/ + ( Pn 1/2 ) jll/ + ( P22 1/2 ) jll/ = g jll/ , (3.314) (3.315) 45 In the one-boson exchange models of the nuclear interaction one typically intro duces the effective (J meson with a mass of ab out 600 MeV to parametrize the strong pionic correlations observed in this channel.
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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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Page 107 and 108: 96 _ . . -.:;: " :,,,_ ._ . . _. 3.
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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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