Antiproton-Nucleus Interaction and Coulomb Effect at High Energies

Commun. Theor. Phys. (Beijing, China 43 (2005 pp. 699 703 c International Academic Publishers Vol. 43, No. 4, April 15, 2005 Antiproton-Nucleus Interaction and Coulomb Effect at High Energies ZHOU Li-Juan, 1 WU Qing, 2 GU Yun-Ting, 3 MA Wei-Xing, 1,4 TAN Zhen-Qiang, 3 and HU Zhao-Hui 1 1 Department of Information and Computing Science, Guangxi University of Technology, Liuzhou 545006, China 2 Department of Physics, Qingdao University, Qingdao 266071, China 3 Department of Physics, Guangxi University, Nanning 530004, China 4 Institute of High Energy Physics, the Chinese Academy of Sciences, P.O. Box 918, Beijing 100049, China (Received May 9, 2004 Abstract The Coulomb effect in high energy antiproton-nucleus elastic and inelastic scattering from 12 C and 16 O is studied in the framework of Glauber multiple scattering theory for five kinetic energies ranged from 0.23 to 1.83 GeV. A microscopic shell-model nuclear wave functions, Woods Saxon single-particle wave functions, and experimental pn amplitudes are used in the calculations. The results show that the Coulomb effect is of paramount importance for filling up the dips of differential cross sections. We claim that the present result for inelastic scattering of antiproton- 12 C is sufficiently reliable to be a guide for measurements in the very near future. We also believe that antiproton nucleus elastic and inelastic scattering may produce new information on both the nuclear structure and the antinucleon-nucleon interaction, in particular the p-neutron interaction. PACS numbers: 13.75.Cs, 21.10.Sf Key words: p-nucleus scattering, Glauber scattering theory, nuclear wave functions 1 Introduction With the construction and subsequent operation of the Low Energy Antiproton Ring (LEAR, beams of low energy antiproton with previously unobtainable intensity and quality are possible. Elastic and inelastic scattering experiments were performed on several nuclei in both the p and sd shell [1 5] as well as on targets of heavier mass. In the very near future experiments with antiprotons having momentum of up to 2 GeV/c will be possible. In our previous paper [6] we studied anti-proton elastic and inelastic scattering with a variety of pn interactions deduced from different experiments. The calculations reproduced data both for differential cross sections and polarizations. However, we could not get a good understanding of the depth of the minima of differential cross sections. Now, we study the contribution from the Coulomb effect to the dips in this paper and try to improve our previous theoretical descriptions. It is the purpose of this article to report our new results of Coulomb effect on p-nucleus scattering for energies that span this new energy region and for which the elementary pn amplitudes are known. Another purpose is studying 12 C structure and predict antiproton- 12 C interaction for experimentally measuring the excitation cross section of the 2 + state of 12 C. The Glauber model [7] has proven capable of providing an excellent description of the low-energy p-nucleus scattering. [8,9] Unlike the case for proton-nucleus scattering, the Glauber model appears to provide a reasonably accurate description of antiproton-nucleus scattering even at 47.9 MeV. [10] The results from Glauber model calculations at 180 MeV agree astonishingly well with experiments. [11,12] A plausible reason for the validity of the Glauber model is the presence of several partial waves that contribute to the reaction already at low energies, thus producing a very forward-peaked cross section near threshold. This earlier work using the Glauber model has been summarized at the 1988 PANIC meeting by Dalkarov and Karmonov. [10] Subsequently, the Glauber model was used to study the effects of annihilation and spin-flip on p-nucleus reactions. [12 14] It was shown that the spin effect is negligible. In this work we shall use the Glauber model, which is not only quite accurate, but requires a minimum of input data for the calculations. This is essential when no p-nucleus reaction data exist. Only the pn elementary amplitude, which is characterized by the total pn cross section and the ratio of the real to imaginary pn forward amplitude, the value of the diffraction slope parameter is required. These have been measured by Kaseno et al. [15] at 700 MeV/c and at several energies above 700 MeV/c by Jenni et al. [16] Our version of the Glauber model uses microscopic wave functions determined from shell-model calculations. With the use of the experimentally determined pn interaction, we produce a zero-parameter calculations of Coulomb effect in the elastic and inelastic scattering that includes both Pauli correlations and nucleon-nucleon correlations that are included via the shell model. The project supported in part by National Natural Science Foundation of China under Grant Nos. 10247004, 10075081 and the Natural Science Foundation of Guangxi Province of China under Grant No. 0339008

700 ZHOU Li-Juan, WU Qing, GU Yun-Ting, MA Wei-Xing, TAN Zhen-Qiang, and HU Zhao-Hui Vol. 43 The paper is organized as follows. Section 2 briefly introduces our formalism and microscopic nuclear wave functions. Our numerical results are given in Sec. 3. Finally, section 4 is reserved for concluding remarks. 2 Glauber Formalism and Microscopic Nuclear Wave Functions 2.1 Glauber Formalism We start by recalling the formalism of Glauber multiple scattering theory and microscopic nuclear wave functions. For hadron-nucleus scattering with Coulomb effect taken into account, the relevant scattering amplitude is given by F fi = F c ( q δ fi + H c.m. (q ik 2π e i q b Γ c fi( b d b, (1 where F c ( q is the Coulomb scattering amplitude for a point charge target and is given by Ref. [17] F c ( q = 2ξ k q 2 e iφc, (2 ( q φ c = 2ξ ln + 2η, (3 2k ξ = Zα m k. (4 Here Z is the charge number of the target nucleus, m is the nucleon mass, α = 1/137 and k denotes the initial wave number in the center of mass system. The quantity η is defined by η = arg Γ(1 + iξ = ξψ(x = 1 + + n=0 ( 1 + ξ 1 + n arctan ξ, (5 1 + n where ψ(x is the digamma function and ψ(x = 1 = 0.577 21. The Γ(1 + iξ in Eq. (5 is Euler s Gamma function. The notation arg stands for the principal argument of a complex number Γ(1 + iξ. H c.m. (q = exp[q 2 /4Aα ] is the center-of-mass correction factor [18] with α being harmonic oscillator parameter of the target nucleus. The Coulomb corrected profile function Γ c fi ( b is determined by Γ c fi( b = e ixc( b δ fi e i xc( b δ fi Γ fi ( b, (6 where x c ( b and x c ( b are Coulomb phase shift of a pointlike charge and an extended charge distribution, respectively. They are given by with x c (b = 2ξ ln(kb, (7 x c (b = x c (b + x c (b (8 1 x c (b = 2ξ4πb 3 dx 1 ( b x 4 ρ c x 0 [ ln ( 1 + 1 x 2 x x 2 ], (9 x where ρ c (b/x is the charge distribution of the target nucleus. For instance, three parameter Fermi distribution 1 + wr 2 /c 2 ρ c (r = ρ 0, (10 1 + e (r c/z where the parameters c, Z, w varying as change of nucleus and their values are given by Ref. [17]. Given ρ c (r, the x c (b can be easily obtained by carrying out the integration in Eq. (9 numerically. Γ fi ( b in Eq. (6 is nuclear profile function given by Γ fi ( A b = ψ f 1 [1 Γ j ( b s j ] ψ i, (11 j=i where ψ f and ψ i are the final and initial nuclear wave functions, respectively. Γ j ( b s j is the two-dimensional Fourier transform of elementary two-body scattering amplitude f j ( q, Γ j ( b s j = 1 e i q ( b s j f j ( q d q, (12 2πik f j ( q is of spin-isospin dependence, and can be written as [19] f j ( q = f A ( q + f B ( q ( σ p ˆn( σ t ˆn + f C ( q ( σ p + σ t ˆn + f D ( q ( σ p ˆm( σ t ˆm + f E ( q ( σ p l ( σ t l, (13 where σ p and σ t are spin operators of the projectile (p and target nucleon (t, respectively. We define ˆn = k i k f k i k f = q k i, (14 q = k i k f, (15 where k i and k f are the initial and final momentums of projectile particle, while ˆm = k i k f k i = ˆq, (16 k f ˆl = ki + k f k i + k f. (17 It is obvious that ˆn, ˆm, and ˆl are unit vectors. Of course, f l ( q are also operators in isospin space, f l ( q = f s l ( q + f v l ( q τ p τ t (18 with l = A, B, C, D, and E. The τ p and τ t are isospin operators of projectile and target nucleon, respectively. f s (q is isoscalar part of amplitude f l (q, while fl v (q is the isovector part of f l (q. In Eq. (13 the first term f A ( q denotes nospin-flip amplitude, the term containing f C ( q is spin-flip amplitude, while the term containing f B (q, f D (q, and f E (q are double spin-flip amplitudes. Inspecting Eqs. (11 and (12 shows that there are only two ingredients entering Glauber amplitude. One is nuclear structure represented by the initial and final nuclear

No. 4 Antiproton-Nucleus Interaction and Coulomb Effect at High Energies 701 wave functions, ψ i,f, and another is two-body elementary interaction indicated by f j (q. For our present purpose here, we firstly study the Coulomb effect on differential cross section using the above formalism, and neglecting the spin-flip parts of two-body amplitude in Eq. (13. We also parameterize the central part f A (q in the form f A ( q = ikσ (1 iρ 2 q 2 e β /2, (19 4π where the σ, ρ, and β are respectively total cross section, ratio of the real to imaginary part of forward amplitude, and slope parameter. They are of energy dependence, and are determined by experimental data. 2.2 Microscopic Nuclear Wave Functions The initial and final nuclear wave functions that enter Eq. (11 are calculated as a sum of Slater determinants using a version of the Glasgow shell-model code. [20] The nuclei 16 O and 40 Ca are treated as closed shell nuclei for which there is a single determinant. The 12 C is treated within the complete p-shell basis for which there are 51 determinants. The wave functions for mass twelve were obtained using the matrix elements of Cohen and Kurath. [21] Because the wave functions are expressed in terms of Slater determinants, the many-body matrix elements of the Glauber operator, Eq. (11, may be expressed as the sum of A A determinants. [22] No problems of timeordering occur. Our method of evaluating the scattering amplitude has the virtue that the effect of antisymmetry, or Pauli correlations, are explicitly included. No free parameters enter the calculations. The single-particle wave functions are calculated by summing a Woods Saxon central potential and binding the nucleons at their experimental energies. The single particle functions are then expanded in terms of threedimensional Hermite polynomials, for which the matrix elements of the profile functions are evaluated as described in Refs. [17] and [18]. Analytic expressions have been obtained for the single-particle profile function in the case of harmonic-oscillator functions. [23] However, in the case of using an effective charge β c 1, the quadrupole component of the matrix element of the single-particle profile function is enhanced by a factor of β c and the integral is performed numerically. 3 Numerical Results and Discussions The values of the total pn cross section σ, the ratio of the real-to-imaginary pn forward amplitude ρ, and the value of the diffraction slope parameter β used in our calculations are given in Table 1. Table 1 The values of the total pn cross section σ, the ratio ρ of the real to imaginary pn forward amplitude, and the value of the diffraction slope parameter β used in the calculations. The values for p = 0.779 GeV/c are from Kaseno et al. [15] The parameters for the remaining four energies are from Jenni et al. [16] p (GeV/c T (GeV σ (fm 2 ρ β (GeV 2 0.779 0.232 13.2 0.26 15.9 1.174 0.565 10.6 0.22 14.9 1.412 0.757 10.0 0.24 14.2 1.776 1.070 9.2 0.14 13.2 2.607 1.833 8.1 0.04 13.1 One may expect the pn interaction to be renormalized inside the nucleus. In Ref. [10] it was found that a better reproduction of 180 MeV p scattering data could be achieved by increasing the magnitude of σ by about 5 to 10 %. There is a precedent for this: in studying pion and kaon induced reactions, Mizoguchi and Toki [23] found a similar effect. However, in this paper we shall not take this into account; if experience at lower energies is a guide, our calculations will be slightly lower than experiment. The experimental values have uncertainties of a few percent. The calculated results are essentially independent of small variations in the value of β. Changing ρ will essentially only affect the depth of the minima. However, it has been determined by experiments and we have no freedom to change. The Coulomb effect is very important for filling up the depth of the minima. According to our calculations σ is also of essential sensitivity to the theoretical predictions. The errors on the σ are relatively small. The principal unknowns are the values of the three parameters for pn interaction. We have taken them to be equal to the pp interaction, an approximation that has worked quite well at lower energies. Figure 1 shows the result for elastic scattering of antiprotons on 12 C, 16 O, and 40 Ca for five kinetic energies from 0.23 to 1.07 GeV. The forward angle cross section increases as one increases both the target mass and the incident energy. The maximum energy of antiproton with the upgraded LEAR should be intermediate between 1.07 and 1.83 GeV. Thus, even at the maximum energies, one should anticipate being able to measure the angular distribution for scattering from 16 O out to the second minimum and perhaps the subsequent maximum. It is in this region that effects due to the single particle wave function and the value of the diffraction slope parameter β pn of the pn interaction will become more pronounced. Measurement of the angular distribution for 16 O or several of the Ca isotopes should allow one to more reliably extract the pn interaction from experiment.

702 ZHOU Li-Juan, WU Qing, GU Yun-Ting, MA Wei-Xing, TAN Zhen-Qiang, and HU Zhao-Hui Vol. 43 Figure 2 shows the Coulomb effect of the depth of minima of differential cross section. As is seen from Fig. 2, the Coulomb effect is of paramount importance for filling up the dips. Therefore, for any accurate calculation one must include the Coulomb effect. Figure 3 shows the result at the five energies of inelastic excitation of the 2 + of 12 C. Although the complete p-shell basis is used in the calculation, it is known from electromagnetic excitations that one must use an effective charge in order to reproduce the strength of the 2 + 0 + transitions. [24] We have used an effective charge of β c = 1.5, a value consistent with electromagnetic transitions and other hadron induced reactions. A value of β c = 1 would have resulted in angular distributions of essentially identical shape but having a magnitude of approximately two smaller. We emphasize that unlike earlier work, our approach includes one-body through A-body scattering (always with the eikonal restriction that no nucleon is struck more than once. The p strongly interacts with the nucleus and a single scattering approximation could be quite inaccurate. Fig. 2 Coulomb effect on differential cross section of antiproton- 16 O elastic scattering at the energy of 180 MeV. The dashed line is the result given by Coulomb effect, while solid line denotes the result without Coulomb effect taken into consideration. Fig. 1 Differential cross section in laboratory system for p-elastic scattering on 12 C, 16 O, and 40 Ca. The solid lines are results for 0.232 GeV (Fig. 1(a, 0.757 GeV (Fig. 1(b, and the dashed lines for 0.565 GeV (Fig. 1(a, 1.070 GeV (Fig. 1(b. Fig. 3 Differential cross section in laboratory system for p-inelastic scattering to the 2 + of 12 C at five energies. Note that the 232 and 565 MeV curves extend to 45 degrees whereas the remaining curves extend only to 30 degrees. An effective charge of β = 1.5 is used. Although no experimental data are now available for comparison with the theoretical predictions in this paper, the present results are sufficiently reliable to be a guide

No. 4 Antiproton-Nucleus Interaction and Coulomb Effect at High Energies 703 for measurements in the very near future. We believe that antiproton-induced elastic and inelastic scattering on nuclei at intermediate energies are particularly interesting. Some new and possibly unexpected phenomena may occur owing to the unique features of the many-body system. It may also produce new information on both the nuclear structure and the antiproton nucleon interaction, in particular the p-neutron interaction. 4 Concluding Remarks Within the Glauber multiple scattering formalism, we study Coulomb effect in antiproton-nucleus elastic scattering. A calculation for inelastic scattering of p 12 C is also performed without any free parameters. There are two ingredients in the Glauber formalism that determine the Glauber amplitude: one is nuclear wave function and the other is two-body pn interaction. We use a realistic nuclear wave function, which has been verified by many investigations. Our input of two-body amplitude is determined by experiments. Therefore, the calculations indeed have no free parameters. The results are reliable, which is very good for examining the Coulomb effect. Figure 1 shows our predictions for elastic scattering on the 12 C, 16 O, and 40 Ca. The results will be served as theoretical guide for experimental measurements in the future. Speaking of Fig. 2, as it is seen from the figure, Coulomb effect is evidently paramount importance for filling up the dips of the differential cross section. Namely, Coulomb effect is very sensitive to the depths of the minima of differential cross section. We got a good improvement to fit the experimental data compared with our previous calculations. By use of the same approximation, we also calculated the inelastic excitation of the 2 + of 12 C by antiproton. Although no experiments data are now available for comparison with the theoretical prediction in this paper, the present results are sufficiently reliable to be a guide for measurements in the very near future. We believe that antiproton induced elastic and inelastic scattering on nuclei at intermediate energies are particularly interesting. Some new and possibly unexpected phenomena may occur due to the unique features of the many-body system. It may also produce new information on both nuclear structure and the antinucleon-nucleon interaction, in particular the p-neutron interaction. In conclusion, we would like to emphasize again that the depths of minima of differential cross section is very sensitive to Coulomb effect, which changes the depths of the minima of differential cross section in a remarkable way and reproduce experimental data successfully. Therefore, in any accurate calculation, the Coulomb effect has to be taken into consideration. References [1] D. Garreta, et al., Phys. Lett. B135 (1984 266; B139 (1984 464. [2] D. Garreta, et al., Phys. Lett. B149 (1984 649; B151 (1985 473. [3] V. Ashford, et al., Phys. Rev. C30 (19984 1080. [4] W. Brueckner, et al., Phys. Lett. B158 (1986 180. [5] M.C. Lemaire, et al., Nucl. Phys. A456 (1986 557. [6] Gu Yun-Ting, et al., High Energy Physics and Nuclear Physics, 28 (2004 72. [7] R.J. Glauber, Lecture in Theoretical Phyiscs, Interscience, New York (1959 Vol. 1, pp. 315 414. [8] O.D. Dalkarov and V.A. Karmonov, Nucl. Phys. A478 (1988 635c. [9] O.D. Dalkarov and V.A. Karmonov, Phys. Lett. B147 (1984 1. [10] O.D. Dalkarov, V.A. Karmonov, and A.V. Trukhov, Yad. Fiz. 45 (1987 688 [Sov. J. Nucl. Phys. 45 (1987 430]. [11] Ma Wei-Hsing and D.D. Strottman, Phys. Rev. C44 (1991 615. [12] Tan Zhen-Qiang and Gu Yun-Ting, J. Phys. G15 (1989 1699. [13] Tan Zhen-Qiang and Ma Wei-Xing, Nuovo Cimento A103 (1990 185. [14] Tan Zhen-Qiang, Ma Wei-Hsing, and Gu Yun-Ting, Nuovo Cimento A103 (1990 1347. [15] H. Kaseno, et al., Phys. Lett. B61 (1976 203; B68 (1977 487. [16] P. Jenni, et al., Nucl. Phys. B94 (1975 1. [17] E. Oset and D.D. Strottman, Nucl. Phys. A355 (1981 437; A377 (1982 297. [18] V. Franco and R.J. Glauber, Phys. Rev. 142 (1966 1195. [19] R.H. Bassel and C. Wilkin, Phys. Rev. 174 (1968 1179. [20] R. Whitehead, Nucl. Phys. A182 (1972 290; R.R. Whitehead, A. Watt, B.J. Cole, and I. Morrison; in Advances in Nuclear Physics, eds. Y.M. Baranger and E. Vogt, Plenum, New York (1978 Vol. 9. [21] S. Cohen and D. Kurath, Nucl. Phys. 73 (1965 1. [22] E. Oset and D. Strottman, At. Data, Nucl. Data Tables 28 (1983 531. [23] M. Mizoguchi and H. Toki, Nucl. Phys. A513 (1990 685.