The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

More documents

Recommendations

Info

5.1 Isospin violating effective Lagrangian 153 unnaturally large S-wave scattering lengths. This can be dealt with in different manners. One is the approach proposed by Weinberg, in which chiral perturbation theory is applied to the kernel of the Lippmann-Schwinger equation (effective potential). This approach is very similar to the Hamiltonian formalism used in this work and explained in detail in chapter3. This route was followed by van Kolck et al. , see refs [75], [74], [76], [78]. A different fashion to deal with low-energy nucleon-nucleon scattering is the approach recently proposed by Kaplan, Savage and Wise (KSW) [91].2 Essentially, one resums the lowest order local four-nucleon contact terms rv Co (Nt N)2 (in the S-waves) to generate the large scattering lengths and treats the remaining effects perturbatively, in particular also pion exchange. This means that most low-energy observables are dominated by contact interactions. The chiral expansion for NN scattering entails a new scale ANN of the order of 300 MeV, so that one can systematically treat external momenta up to the size of the pion mass. There have been suggestions that the radius of convergence can be somewhat enlarged [88], but in any case ANN is considerably sm aller than the typical scale of about 1 Ge V appearing in the pion-nucleon sector . For recent calculations of various properties of the two-nucleon system within this formalism see e.g. refs. [87], [88], [92], [95], [98], [210], [211]. In this context, it appears to be particularly interesting to study CIB (or in general isospin violation) which is believed to be dominated by long range pion effects. That is done here.3 First, we write down the leading strong and electromagnetic four-nucleon contact terms. It is important to note that in contrast to the pion or pion-nucleon sector, one can not easily lump the expansion in small momenta and the electromagnetic coupling into one expansion but rat her has to treat them separately.4 Then we consider in detail CIB. The leading effect starts out at order aQ-2, where Q is the generic expansion parameter in the KSW approach. It sterns from OPE plus a contact term of order a with a coefficient aE� l ) of natural size that scales as Q-2. Similarly, the leading CSB effect results from contact terms with four nucleon legs of order a and order E == mu - md, which also scale as Q-2. While in the case of E�l) this scaling property is enforced by a cancellation of a divergence, the situation is a priori different for CSB. However, for a consistent counting of all isospin breaking effects related to strong or electromagnetic (ern) insertions, one should count the quark mass difference and virtual photon effects similarly. Note, however, that these CIB and eSB terms are numerically much smaller than the leading strong contributions which scale as Q-l because a« 1 and (mu - md)/Ax «1. The corresponding constants, which we call E�1,2), together with the strong parameters (as given in the work of KSW) can be determined by fitting the three scattering lengths a pp , ann, an p and the np effective range.5 That allows to predict the moment um dependence of the np and the nn 1 So phase shifts. Based on these observations, we can in addition give a general classification for the relevant operators contributing to eIB and eSB in this scheme. Additional work related to long-range Coulomb photon exchange is necessary in the proton-proton system. We do not deal with this issue here but refer to recent work using EFT approaches in refs. [219] , [220], [221]. 5.1 Isospin violating effective Lagrangian Let us first discuss the various parts of the effective Lagrangian underlying the analysis of isospin violation in the two-nucleon system. To include virtual photons in the pion and the pion-nucleon 2 There exist by now modifications of this approach and it has been argued that it is equivalent to cut-off schemes. We do not want to enter this discussion here but rather stick to its original version. 3For a first look at these effects in an EFT framework (based on the Weinberg power counting) , see the work of van Kolck [75]. Electromagnetic corrections to the one-pion exchange potential have been considered in [218]. 4 This is because of the different power counting schemes in these cases. 5Whenever we talk of the pp system, we assume that the Coulomb effects have been subtracted.
154 5. Isospin violation in the two-nuc1eon system system is by now a standard procedure [199], [222]-[227]. The lowest order (dimension two) pion Lagrangian takes the form (5.3) with f7r = 92.4MeV the pion decay constant, V' ft the (pion) covariant derivative containing the virtual photons, ( ) denotes the trace in flavor space. Further, X contains the light quark mass matrix M. Speaking more precisely, X can be expressed in the most general case in terms of the scalar and pseudoscalar external sour ces s and p,6 respectively, as follows: X(x) = 2B(s(x) + ip(x)), (5.4) where the constant B is defined in eq. (3.134). Clearly, one has to set p = 0, s = M to end up with the usual QCD Lagrangian. Further, the last term in eq. (5.3) contains the nucleon charge matrix Q=ediag(I,0),7 and leads to the charged to neutral pion mass difference, 8M2 = M;± - M;o, via 8M2 = 87fcxC/ f;, i.e. C = 5.9· 10-5 GeV4. Note that to this order the quark mass difference mu - md does not appear in the meson Lagrangian (due to G-parity). That is chiefly the reason why the pion mass difference is almost entirely an electromagnetic (ern) effect. The equivalent pion-nucleon Lagrangian to second order8 takes the form Nt (iDo - g; a . ü) N + Nt { �� + Cl (X+) + (C2 - :�) U6 + C3UftUft C str 7rN + l (q + 4�) [O"i,O"j]UiUj + C5 (X+ - �(X+)) +., .} N , (5.5) which is the standard heavy baryon effective Lagrangian in the rest-frame vft = (1,0,0,0). Further, m is the nucleon mass and Uft the chiral viel-bein introduced in eq. (3.102), uft '" -ÖftrP/ f7r + ... , The quantities X± are defined via (5.6) and transform covariantly under chiral rotation, i.e. via 90 I h h-l X± -"--'- -t X± = X± , (5.7) where the compensator field h is defined in eq. (3.99). The contact interactions to be discussed below do not modify the form of this Lagrangian (for a general discussion, see e.g. ref. [71]). Strong isospin breaking is due to the operator '" C5. Electromagnetic terms to second order are given by [226]:9 C;� = f;Nt {h (Q� - Q�) + hQ+(Q+) + h(Q� + Q�) + f4(Q+)2} N , (5.8) 6The extern al fields s and p are introduced in the QCD Lagrangian via the term -q(s - ip/5)q. The seal ar sour ce s has already been discussed in sec. 3.2. The transformation properties of the p under charge conjugation, parity transformation and chiral rotation can be obtained requiring invariance of the above term in the same way as it has been done for s in sec. 3.2. In particular, one obtains: p ---+ pe = pT, P ---+ pP = -p and p � pi = hLphj/ = hRph"[,l, where hR,L are the same matrices as in in eq. (3.98). Note further that the field p is hermitian: pt = p. 70r equivalently, one can use the quark charge matrix e( � + T3 ) /2. 8 Here and in what follows we count the nucleon mass in the same way as the scale Ax ' Note furt her that this Lagrangian was used in the context of the NNLO two-nucleon potential in sec. 3.8 as weil as in the discussion of the three-nucleon interactions in sec. 3.9. gOne of these four terms can be eliminated using the matrix relation given in ref. [227].
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 125 and 126: 114 / / / / / / / 5 / / \ \ \ \ I I
Page 127 and 128: 116 3. The derivation o{ nuc1ear {o
Page 131 and 132: 120 4. The two-nuc1eon system: nume
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 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 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

Create successful ePaper yourself

Delete template?

Save as template?