The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

More documents

Recommendations

Info

3.2. Effective Lagrangians 59 where e and s' are given by eq. (3.55). Therefore, the new set (�, �)* indeed defines the CCWZ realization. It is now clear how to construct the effective Lagrangian from the es,
60 3. The derivation of nuc1ear forces from chiral Lagrangians Let us now make a few comments about eq. (3.74). This is, in fact, a crucial result for EFT, since it says, that the interaction between the Goldstone bosons vanishes at low energies. Indeed, it is seen in eq. (3.74) that one can always rewrite the Lagrangian in such a form, that only derivative interactions between the Goldstone bosons are present. We will show below using the power counting arguments that, as a consequence, the scattering amplitude for any process with an arbitrary number of Goldstone particles is proportional to some positive power of external momenta. We point out again that this statement concerning the scattering amplitude is independent of any particular choice of the realization of the group G. In principle, we could equally well use an arbitrary nonlinear realization (�, cjJ) to construct the Lagrangian, where cjJ denotes a set of matter fields, their derivatives as well as derivatives of Goldstone fields. However, insisting on the chiral invariance of the Lagrangian density would be a non-trivial problem because of the complicated nonlinear transformation rules of the fields and their derivatives. Using the CCWZ realization to construct the efIective Lagrangian via eq. (3.74) has the advantage that all building blocks transform covariantly under G, Le. in the way shown in eq. (3.54), guaranteeing that any H-scalar is also G-invariant. To complete the discussion about the construction of a chiral invariant Lagrangian let us apply now the chiral SU(Nf)v x SU(Nf)A group as a particular example of the general Lie algebra (3.52). In that case all indices in eq. (3.52) vary from 1 to Ni - 1 and the structure constants G's are represented in terms of the structure constants f's of the SU(Nf) group as follows:12 Gabe = 0, (3.75) For example, for the SU(2)v x SU(2)A group we have fijk fijk, where fijk is the ordinary three-dimensional totally antisymmetric tensor with f123 = 1. Note that one can represent the chiral SU(Nf)V x SU(Nf)A group by SU(Nf)L x SU(Nf)R defining the corresponding generators TL and TR as (3.76) where Vi and Ai are the generators of the SU(Nf)V and SU(Nf )A, respectively. Then SU(Nf)L and SU(Nf)R build two independent subgroups of the whole chiral group with the Lie algebra [(TL ) i , (TL)j] [(TR) i , (TR)j] [(TL ) i , (TR)j] Let us now be more specific and concentrate on the chiral SU(2)v x the following representation of the generators Vi and Ai fijk (TL h , fijk (TRh , o. A - _ _ ilT i l - 2 ' SU(2)A (3.77) group. We choose where T i is a Pauli matrix and 1 is some flavor scalar matrix quantity with the properties: 12 = 1 , I t = I . (3.78) (3.79) For example, 1 can be equal to the ')'5 matrix. Then, one has the same representation for the SU(2)A generators as in the case of the QCD Lagrangian (3.1), as can be seen from eq. (3.9). 1 2 Here we use a definition of the chiral Lie algebra, which differs by the factor i from the one given in the last section , eq. (3.36). This factar can be absorbed by the corresponding redefinition of the generators.
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 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
Page 69: 58 3. The derivation of nuclear for
Page 77 and 78: 66 3. The derivation o{ nuclear {or
Page 85 and 86: 74 3. The derivation o{ nuc1ear {or
Page 95 and 96: 84 .... . - - .... \ . • A .... .
Page 105 and 106: 94 , , , , , , , , , , , , , , , ,
Page 107 and 108: 96 _ . . -.:;: " :,,,_ ._ . . _. 3.
Page 109 and 110: 98 3. The derivation o{ nuc1ear {or
Page 111 and 112: 100 3. The derivation o{ nuc1ear {o
Page 113 and 114: 102 3. The derivation o{ nuclear {o
Page 115 and 116: 104 3. The derivation of nuclear fo
Page 121 and 122:
110 3. The derivation of nuclear fo
Page 123 and 124:
Page 125 and 126:
114 / / / / / / / 5 / / \ \ \ \ I I
Page 127 and 128:
116 3. The derivation o{ nuc1ear {o
Page 129 and 130:
Page 131 and 132:
120 4. The two-nuc1eon system: nume
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Page 143 and 144:
Page 145 and 146:
134 10 8 4 2 o o • Nijmegen PSA N
Page 147 and 148:
136 4. The two-nucleon system: nume
Page 149 and 150:
Page 151 and 152:
140 4. The two-nucleon system: nume
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
146 --- .... . � - --- .... . ...
Page 159 and 160:
148 � .... --- � � .... --- :
Page 162 and 163:
4.5. Results for the NNLO-ß approa
Page 164 and 165:
5.1 Isospin violating effective Lag
Page 166 and 167:
5.1 Isospin violating effective Lag
Page 168 and 169:
5.2 Brief introduction into the KSW
Page 170 and 171:
5.3 The leading eIB and eSB effects
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
Page 178 and 179:
2. Secondly, we considered the real
Page 180 and 181:
NNLO, this range is larger and exte
Page 182 and 183:
might be responsible for solving th
Page 184 and 185:
derivative. Further helpful equalit
Page 186 and 187:
Appendix B Operators contributing t
Page 188 and 189:
Appendix C Small scale expansion wi
Page 190 and 191:
179 (C.16) l�r-2. (C.17) h + l2 =
Page 192 and 193:
Appendix E Anti-symmetrization of t
Page 194 and 195:
Consequently, only four of the eigh
Page 196 and 197:
where qJ-! is chosen to satisfy (v
Page 198 and 199:
To keep the presentation self-conta
Page 200 and 201:
- i�03{ [(NtVN) . (VNt x ifN) + (
Page 202 and 203:
Consequently, it is not possible to
Page 204 and 205:
(j ± 1, 1jIVIJ =f 1, 1j) Here, Pj
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
show all

The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?