The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

More documents

Recommendations

Info

- i�03{ [(NtVN) . (VNt x ifN) + (VNtN) . (Ntif x VN)] - (NtN)(VNt . if x VN) + (NtifN) · (VNt x VN)} 1 1 - - - 2 ( "2 C4 (8ik8jl + 8il8kj) + C58ij8kl ) x [(oiojNtakN) + (NtakOiOjN)] (NtaIN) - (�06 (8ik8jl + 8il8kj) + (05 - 67)8ij8kl) (NtakOiN)(ojNtaIN) 1 ( 1 - - - ) - 2 2 (C4 - C6) (8ik8jl + 8il8kj) + C78ij8kl x [(OiNtakOjN) + (OjNtakoiN)] (NtaIN) . 189 (F.21) The effective Lagrangian (F.21) is sufficient for the derivation of the short-range part of the twonucleon potential at NLO. To obtain the NNLO potential one needs the Lagrangian .c�� for contact interactions with four nucleon legs. Let us first discuss the terms with three derivatives acting on the nucleon field Hv (Hv) . • Terms with 0/-t, ov, 0p. It is, c1early, not possible to construct a Lorentz scalar of the form (F.22) if the operators f 1 ,2 contain only three derivatives and no velo city or spin operators (or, in general, if an odd number of the operators 0/-t' Vv and S(J enters f1 and f2) . • Terms with 0/-t, ov, op and S(J. The terms without insert ions of the totally anti-symmetrie tensor f et ß7 (J are not parity invariant. For the terms with f et ß7(J we first note that the derivatives must always act onto different fields Hv (Hv) beeause of the anti-symmetrie properties of the Levy-Civita tensor. The only two terms one ean build up are: f /-t VP (J { (Hv *a /-t 8 vHv )(Hv *a pS(JHv) + (Hv *a v 8 /-tHv)(Hv 8 pS(JHv)} , (F.23) if /-t VP (J { (Hv *a /-t 8 vHv)(Hv *a pS(JHv) - (Hv *a v 8 /-tHv)(Hv 8 pS(JHv)} . (F.24) The seeond term vanishes after applying partial integration. We will now show that the first term ean be eliminated from the Lagrangian using the EOM for the nucleon field and eqs. (F.2), (F.16) . Let us eonsider only the first term in eq. (F.23) to keep our notation more eompaet: f /-t Vp(J (Hv *a /-t 8 vHv)(Hv *a p S(JHv) = f /-t VP (J ( Hv *a /-t 8 vHv ) ( Hv *aet [vp Vet S(J - 2(Sp Set S(J + Set Sp S(J)] Hv) (F.25) = -2 f /-tv P (J ( Hv *a /-t 8 vHv) ( Hv *aet (i fpetß7 vß S7 S(J + Set {Sp, S(J}) Hv) + . .. . - � ---ct ( ) ( - �et = 2 z det(R) Hv a /-t (j vHv Hv 0 v ß S 7 S(JHv ) + ... ,
190 F. The complete set of the N N contact interactions with two derivatives where the matrix R is given by (F.26) We have used eq. (F.16) to obtain the seeond line of eq. (F.25). The term vp vQSrr in the squared braekets in the seeond line of this equation vanishes modulo higher order terms after applying the EOM for Hv . Sueh higher order terms are denoted in eq. (F.25) by the dots. To end up with the third line of eq. (F.25) we have used the last equality in eq. (F.2). The seeond term in the parenthesis (SQ {Sp, Srr}) vanishes beeause of the third equality in (F.2) and the anti-symmetrie property of the f/lyprr . Finally, the remaining term in the last line of eq. (F.25) involves the operator v . 1J or v . 71, as is obvious from eq. (F.26), and henee vanishes modulo higher order terms. • Terms with 8/l' 8y, 8p and Vrr. The terms involving the Levy-Civita tensor f /lyprr violate parity invarianee, whereas the remaining terms ean be eliminated using the EOM for the nucleon field. • Terms with 8/l' 8y, 8p and Vrr , VQ or Vrr, SQ or Srr, SQ are not Lorentz invariant. • Terms with 8/l' 8y, 8p, Srr, SQ and vß . Sueh terms involving f/l ' Y 'p'rr' are not parity invariant, whereas the remaining terms eontribute at higher orders . • All remaining terms with more than one insertion of the velo city operator ean be eliminated using the EOM for the nucleon field. Thus, there are no eontaet terms with three derivatives at order ßi = 3. Another possible kind of eontributions at this order within our power eounting seheme is from the terms with two derivatives suppressed by one power ofthe inverse nucleon mass (ljm-eorrections). Let us now take a closer look at sueh terms, whieh ean be written as (F.27) where the eonstants Bi are expressed in terms of the eoupling eonstants of the lower order Lagrangian. Note that sinee we do not eonsider the terms with six and more nucleon legs in the effeetive Lagrangian, whieh do not affeet the interaction between two nucleons, the only eonstants entering the B's are from LN and L�/v
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 25 and 26:
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
28 2. Low-momentum efIective theori
Page 41 and 42:
Page 43 and 44:
Page 45 and 46:
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 51 and 52:
Page 53 and 54:
42 -3 -5 -7 -9 2. Low-momentum effe
Page 55 and 56:
Page 57 and 58:
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 67 and 68:
Page 69 and 70:
Page 71 and 72:
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
66 3. The derivation o{ nuclear {or
Page 79 and 80:
Page 81 and 82:
Page 83 and 84:
Page 85 and 86:
74 3. The derivation o{ nuc1ear {or
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
84 .... . - - .... \ . • A .... .
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
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 117 and 118:
Page 119 and 120:
Page 121 and 122:
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: 138 4. The two-nuc1eon system: nume
Page 151 and 152: 140 4. The two-nucleon system: nume
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 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?