The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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2.2. Two nucleons at very low energies 19 For large A the term a2rm7fh in the expression (2.29) becomes dominant. Obviously, for positive effective range r > 0 an upper bound for the values of A exists, above which no real solutions for Co and C2 are available. This statement is quite general but not new: it can be derived from Wigner's theorem [84], which is based on such general principles like causality and unitarity. The theorem says that the rate dJ(k)jdk by which the phase shift can change with energy is bounded from below by some function of the range R of the potential, the moment um k and the phase shift J. Thus, since the effective range in the 1 So and 3 SI channels is positive, one cannot take the cut-off A to infinity. At first sight, this might look like a failure of the cut-off regularization. Indeed, the regularization scale (cut-off) is usually taken to infinity if the standard renormalization technique is used for renormalizable field theories. However, in an effective theory the cut-off becomes physically relevant. Introducing a cut-off in an effective theory provides finiteness of the amplitude. At the same time, it leads to suppression of all high-moment um virtual states, for which no information is available. That is why in an effective theory one should take the cut-off below the scale, at which new physics is expected to appear. In the low-energy nucleonnucleon effective theory such new physics is associated with pions. We cannot expect that our derivative expansion of the potential will correctly represent effects of pion exchanges at energies rv Mn . Thus, if we would take the cut-off A much larger than the pion mass Mn , we would simply incorporate wrang physics. This explains why the cut-off should remain finite and of the order of Mn · Up to now we have only considered a general scattering problem in the S-channel. We have neither restricted ourselves to a concrete regularization scheme nor to a particular partial wave. Let us now consider the 1 So channel and compare different regularization schemes for that case. We will consider cut-off and dimensional regularizations. Let us start with the cut-off theory and specify the function JA (p2). For simplicity, we will take This leads to (k) = -i7fm m 1 A + k I 2k + 2k n A - k . Further, one finds for the particular choice (2.34) of the function JA(p2) that Ii = 0, ti = (i + l )Ai+l 1 . For the constants Co and C2 one obtains with Co 9D + m(7f - 2aA) (m(97f - 8aA) ± 18VD) 5DAm - 3 ( m (7f - 2aA)) mA3 1 ± VD ' D = �m2 (37f2 - aA (127f + aA(7frA - 16))) . For the inverse of the T-matrix we get from eq. (2.16) a quite simple expression: 1 = Am (1 + (7f - 2aA)2 ) _ To n (k) 2aA(7f - 2aA) - a2(4 - 7frA)k2 I(k)k2 . (2.34) (2.35) (2.36) (2.37) (2.38) (2.39) (2.40)
20 2. Low-momentum effective theories for two nucleons One can immediately check that the first two terms in the effective range expansion (2.21) are correctly reproduced. Let us now take a look at the experimental data in the 1 S o channel. All information we need to this order of ca1culation is contained in the scattering length a and the effective range r. One could first try to estimate their values from simple dimensional analysis: since the longest range part of the nuclear force is given by the OPE, it is natural to assume that the scale entering the values of the coefficients in the effective range expansion is of the order of the pion mass M'/[ . The experimental values of this quantities are given in eq. (1.1). Our naive estimation works fairly well for the effective range r: 2 r = (2.75 ± 0.05) fm '" - , M'/[ but it fails to describe the scattering length, for which we have 17 a = (-23.758 ± 0.010) fm '" - M'/[ . (2.41 ) (2.42) The scattering length takes an unnatural large value in the 1 So channel. As already stressed before, we have to take the cut-off A of the order of the pion mass. That is why we can make use of the relation aA ' " aM'/[ » 1 , (2.43) to simplify the express ions (2.37)-(2.40). From eq. (2.39) we obtain The coupling constants Co and C2 take the values Co '" 1 ( 192 - 97frA 36� ) mA 80 - 57frA ± 5V16 - 7frA ' __ 3 ( ) 2� _ 1± mA3 V16 - 7frA . (2.44) (2.45) (2.46) As already discussed above, the inverse of the on-shell T -matrix can be represented at very low energies in terms of the effective range expansion (2.21). This follows from analytic properties of the amplitude. From the expression (2.40) we see that the effective range expansion is reproduced provided that 2A(7f - 2aA) k2 < ' " 4A2 . (2.47) a (4 - 7fr A) 7fr A - 4 Since the cut-off A as well as the inverse of the effective range are of the order of the pion mass, one obtains from the inequality (2.47) (2.48) In these estimations we do not care about the numerical factors like 4/7f. The condition (2.47) shows that the range of applicability of our effective range theory is in accordance with our expectation: we reproduce physics correctly at momenta smaller than the cut-off A. This holds also for a --+ 00.
Page 1 and 2: The Nucleon-Nucleon Interaction in
Page 3 and 4: 11 vom ursprünglichen wesentlich u
Page 5 and 6: IV Ordnung des Potentials besteht a
Page 7 and 8: VI sagen für die Schwerpunktsimpul
Page 9 and 10: Vlll betrachten, um eine komplette
Page 11 and 12: x 5 Isospin violation in the two-nu
Page 13 and 14: 2 1. Introduction nuclear force of
Page 15 and 16: 4 1. Introduction the Thcson-Melbou
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
Page 29: 18 2. Low-momentum effective theori
Page 39 and 40: 28 2. Low-momentum efIective theori
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
Page 65 and 66: 54 3. The derivation of nuclear for
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70 3. The derivation of nuclear for
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72 3. The derivation o{ nuclear {or
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74 3. The derivation o{ nuc1ear {or
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78 3. The derivation o{ nuclear {or
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84 .... . - - .... \ . • A .... .
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90 3. The derivation of nuc1ear for
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92 3. The derivation of nuc1ear for
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94 , , , , , , , , , , , , , , , ,
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96 _ . . -.:;: " :,,,_ ._ . . _. 3.
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98 3. The derivation o{ nuc1ear {or
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100 3. The derivation o{ nuc1ear {o
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102 3. The derivation o{ nuclear {o
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104 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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