The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

More documents

Recommendations

Info

4.3. Phase shifts 133 and a series of contact interactions. The obtained results are similar to ours at NLO. A furt her important observation made in this work is that the inclusion of the NNLO contact interactions alone does not allow to improve predictions for the phase shift relative to the NLO calculation. As is demonstrated in our work, only the inclusion of the complete NNLO corrections (4.3) to the potential given by the subleading TPE contribution and the contact terms allows to come closer to the Nijmegen PSA. Finally, let us briefly comment on the earlier work by Ord6nez et al. [76J. They performed the LO, NLO and NNLO calculations5 within the potential approach using time-ordered perturbation theory to obtain the N N force. The Cl, 3 ,4 couplings were not taken from 1f N scattering but instead considered as free parameters. Furthermore, the anti-symmetrization of the potential was not performed. As a consequence, many additional redundant parameters corresponding to contact interactions enter the expressions for the effective potential. A global fit in all lower partial waves has been performed to fix the 26 parameters at NNLO. This makes it difficult to separate the effects on the phase shifts of the individual contributions to the potential. The results for the 1 So and 3 SI phase shifts shown in figs. 6 and 7 in this paper are of the same quality as ours (our 1 So partial wave looks slightly better at intermediate energies). We can not compare the predictions for the effective range parameters, since no corresponding analysis has been done in ref. [78J. 4.3.2 P-waves In fig. 4.6 we show the corresponding partial waves together with the mixing parameter EI for the best global fit. In some cases, the differences between NLO and NNLO are modest, in 1 PI and 3 PI NLO is even somewhat bett er. That means that the chiral TPEP is too strong in these phases. Note also that in the 3 PI phase OPEP is dominant. Thus, the inclusion of the contact interaction does not lead to a visible change. In 3 P2, NNLO is still too strong but the prediction is considerably better than the NLO one. The energy dependence of EI is fairly precisely described at NLO and NNLO. These results are visibly better than the ones obtained in ref. [78J or in refs. [210J and [211], the latter being a NNLO calculation in the KSW scheme. This is shown in detail in fig. 4.5, where EI is plotted versus the cms nucleon moment um (p < 350 MeV) in comparison to the Nimegen PSA and the results from ref. [210J. The most important difference between our approach and the KSW one is, as already pointed out, that in the last one the pions are treated perturbatively. Thus, our results might be considered as an indication that in the two-nucleon system, pions have to be treated non-perturbatively (if one intends to describe data above p� 150 MeV).6 In the 1 PI and 3 Po channels the phase shift at leading order (iterated OPE without contact interactions) describes the data only at very low energies. Also, the phase shifts are sensitive to the choice of the cut-off. This is because no contact interactions appear in the potential at this order, that could compensate this cut-off dependence. Therefore, the LO approximation in these channels should not be taken seriously. In fig. 4.7 we show apart from the LO and NLO phase shifts also the curve, which results if one adds the contact term with two derivatives to the OPE potential. This is, certainly, not the complete NLO potential, since the leading TPE is missing. The results for such incomplete and complete NLO potentials are similar and very much improved relative to the LO calculation.7 Thus, we conclude that this improvement when going from LO 5 They also included the leading effects of intermediate D.-excitations. 6 This is, however, not the only difference between the two schemes. For example, in the KSW formalism the unitarity of the S-matrix is only given perturbatively. 7 A visible closeness of the phase shifts from the incomplete NLO and from NNLO is accidental.
134 10 8 4 2 o o • Nijmegen PSA NNLO NLO LO NNLO (KSW) NLO (KSW) 0.1 4. The two-nuc1eon system: numerical results / / / / " " 0.2 P [GeV] / / / / / / " " I I I I I / / / / / / / J //1 Figure 4.5: Predictions for the mixing parameter 1:1 for nucleon cms momenta p below 350 MeV. The short-dashed, long-dashed and solid curves represent our LO, NLO and NNLO results, in order. For comparison, the NLO [91J and NNLO [210J results in the KSW scheme are also shown. The filled squares depict the Nijmegen PSA results. to NLO is not due to inclusion of the leading TPE but mainly because the leading short range contact term with two derivatives is taken into account. The P-waves have also been recently calculated in the KSW scheme, see fig. 9 of ref. [211J. The leading non-vanishing contributions to the phase shifts come out in this approach at NL08 from the (non-iterated) OPE and the first corrections (NNLO) correspond to a single iteration of the OPE. The contact interactions with two and more derivatives are expected to contribute at higher orders. No free parameters appear in the calculation of the phase shifts in these channels. The results shown in ref. [211 J should, in principle, be compared to our LO calculations. The difference is that we iterate the OPE infinitely many times (and not just one time as in ref. [211 J). Comparing the NLO and NNLO phase shifts presented in [211 J with each other and with the phase shifts shown in fig. 4.6 we conclude, that the one-pion exchange becomes non-perturbative in spin triplet channels at momenta comparable with M7r • As already stressed before, one needs to include the leading short range effects to obtain a reasonable description of the phases at intermediate energies in the 1 P1 and 3 Po channels. This observation is also confirmed by the analysis of ref. [211], 8 At leading order these phase shifts are zero. 0.3 I
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: 82 3. The derivation of nuclear for
Page 95 and 96: 84 .... . - - .... \ . • A .... .
Page 101 and 102: 90 3. The derivation of nuc1ear for
Page 103 and 104: 92 3. The derivation of nuc1ear for
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 143: 132 4. The two-nuc1eon system: nume
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 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?