frno till SPIN EFFECTS IH LOW ENERGY PPOTON-ANTIPROTON FORWARD ELASTIC SCATTERING M. Lacombe, B. Loiseau, B. Moussallam, R. Vinh Hau Division de Physique Théorique, Institut de Physique Nucléaire, 91406 Orsay and LPTHE, Université Pierre et Marie Curie, 75230 Paris Cedex OS (France) Abstract Predictions of the pp forward elastic cross section and of the real to imaginary ratio p given by the NN interaction we proposed recently are reported. Our cross sections compare very well with recent measurements at 400<p-<730MeV/c and 700 MeV/c. It is also shown that spin effects cannot be neglected in the determination of p as usually assumed. IPSO/ TH 82-16 Laboratoire associé au C.N.R.S.
In a recent letter, new measurements of the pp forward elastic cross sections at low energies (400 < p t < 730 MeV/c) were reported. These 2) data complete earlier bubble chamber results obtained at 700 MeV/c. In both letters, the data were used in conjunction with pp total cross sections measurements to deduce the real to imaginary ratio Re [SjCfO) + «3 (t=0)] p - Im [GjCt-O) + * 3 (t=0)] of the forward spin independent nuclear amplitude via the following approximation : - An (ohc/bt) 2 F(t) 2 *l/hxo,_ / «tic) 2 (1 + p 2 ) exp(-bt) (1) tot + (aa tqt I 6t) F(t) exp(- j bt) <p cos S - sin «) where -t is the four momentum transfer, a the fine structure constant, 8 the velocity in the laboratory of the incident p, b the slope parameter - 3) of the pp diffraction peak, and the *.*s are the usual helicity amplitudes. F(t)» CI + t/0.71)~ is the Coulomb form factor of the proton and 6 - [tn 9.5t + 0.5772] a6~ is the phase of Coulomb amplitude as used in 2 references 1 and 2. In these two last expressions t is to be expressed in (GeV) Formula (1) is, however, written with the assumption that only the spin independent amplitude is dominant in the forward direction. If spin dependence is not neglected., and if one assumes the same slope for all amplitudes, equation (1) should be replaced by - 4n <anc/<st) 2 F(t) 2 +<l/ïï)(o tot / 4nc) 2 (1 + P 2 ) (1 + n 2 ) exp(-bt) (2) * (a0 tot ' S t ) F U ) e x p ( " 2" b t ) < P C O S 6 " B i n 6 )
2 2 *,(t=0) 2 + # 3(t=0) - «j(t=0) 2 where ti «*,(t-0) + * 3(t=0) 2 Formula (1) was used in references O and 2) to deduce p as a function of energy. Comparing the results thus obtained with those given by a dispersion relation analysis, they concluded that it was necessary to include an effective pole term in the unphysical region near the pp threshold (at m = 1884 ± 9 MeV with a residue = 0.97 ± 0.08). The purpose of this note is twofold. Firstly, we wish to report predictions of the pp forward elastic cross section and of the ratio p given by 4) the NN interaction we proposed recently. Secondly, we want to show that the widely used formula (1) with the neglect of spin dependence can lead to incorrect results for p, and consequently, the comparison with theoretical models as done In references I) and 2) can be misleading. This remark applies not only to the analysis of earlier works but should be borne In mind also in new projects currently under investigation, for example at CERN '. In reference 4), the existing data for observables on pp scattering were used to constrain the phenomenological parameters of the NN optical model. However, in this fit, the forward elastic cross sections of references 1) and 2) were not included. The results presented here are therefore predictions of the model. The agreement of our results with the data of references 1) and 2) is very good, the x 2 /data being of 1.06 and 1.18 respectively. Some samples of these results are shown In figures 1 to 4. Along with the excellent fit to the elastic cross sections at medium and backward angles, as well as 4) to the total cross sections, this result confirms the ability of our model to describe accurately both pp elastic scattering and annihilation. The values of p, calculated in our model, are shown in Table I. Also shown there are the calculated values of the spin dependence parameter n. It can be seen that the values of the parameter p are small whereas those of the parameter n are significant, and tl therefore cannot be neglected. These results suggest that the conclusions drawn from earlier analyses of the parameter p which neglect the n term are invalid, and we would like to urge experimentalists to use formulat2) rather than formula (1) in the analysis of their
3. forward scattering data. In this case however, some additional theoretical constraints may have to be imposed on the various parameters P, I) and b in a search for the best fit. Otherwise, the solution might not be unique. Actually, formula(2) by itself is a relationship between do/dft, O, pand n in the approximation that in the forward region the angular dependence is essentially given by the slope parameter b. He have checked that, with the quantities p, n and o t given by our model, it is a good approximation to the exact differential cross section predicted by the same model. For 0 < 0_. < 40, we find for b the values shown in Table I. These values of b are slightly higher than those of reference 2) but agree with more recent values '. In conclusion, we suggest that, in future analyzes for determining the ratio p of the real to imaginary amplitudes, i) care should be taken on spin effects, ii) one should use experimental data of the forward differential and the total cross sections in combination with theoretical values of n and b such as those given in Table I. He would like to thank Drs. K. Nakamura and T. Kamae for sending us the numerical values of their data.
REFERENCES [1] H. Iwasaki, H. Aihara, J. Chiba, H. Fujil, T. Kamae, K. Nakaraura, T. Sutniyoshi, Y. Takada, T. Takeda and M. Yaroauchi, Phys. Lett. 103B (1981) 247. [2] H. Kaseno, R. Hamatsu, K. Kawano, M. Klmura, M. Takanaka, T. YamagaCa, I. Kita, K. Takahashi, K. Tanahashi, H. Kohno, and S. MatsumoCo, Phys. Lett. 61B (1976) 203 ; 68B (1977) 487 (E). [3] H. Hoshizaki, Suppl. Progr. Theor. Phys. 42 (1968) 107. [4] J. Côté, M. Lacombe, B. Loiseau, B. Moussallam, R. Vinh Hau, Phys. Rev. Lett. 48 (1982) 1319 [S] Th. Walcher, private communication. [6] S. Sakamoto, T. Hashimoto, F. Sai and S.S. Yamamoto, Nucl. Phys. B195 (1981) l.
r i T Lab (MeV) 50.0 96.3 138.4 192.4 233.0 250.0 300.0 350.0 P -0.181-0.0972-0.0464-0.0073 +0.0098 +0.0143 +0.0201 +0.0193 n 0.325 0.309 0.299 0.284 0.272 0.268 0.257 0.247 _ 2 (GeV) 34.4 25.7 22.4 19.6 18.4 17.9 17.0 16.2 Table I The values of p and p from our model and of the slope parameter b to be used in formula 2. L._
FIGURE CAPTIONS Fig.1 : Differential cross section of pp forward elastic scattering at T.. - 96.3MeV. The data points (f) are from reference 1. The solid curve is the prediction of our NN model and the dashed curve the contribution of our nuclear amplitude. As in Fig.I but for T. - 192.4 MeV. As in Fig.l but for T. «138.4 MeV. As in Fig.l but for T,. «233 MeV. Also shown the data points (4) of reference 2 at t.» 230.5 MeV. Lao
1 A d fmb/sr) dn V / ' 100 I, =96.3 MeV < lab (f ab =436 MeV/c,9cm.(dgg) Fig. 1 I
r i f de; Cmb/sr) 100 Tj ab =138.4 MeV 75 1^=528 MeV/c 50 25 0 10 20 30,6cm.(deg) Fig. 2
r ~i T hb =192.4 MeV P b =631 MeVè Fig. 3
r i t ^(^b/sr) dfl Tf Qb = 233.0 MeV 100, b = 697 MeV/c Fig. 4 I