Inhomogeneous Shadowing Effect in High-Energy p-a Drell Yan Process

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1 Commun. Theor. Phys. (Beijing, China) 50 (2008) pp c Chinese Physical Society Vol. 50, No. 1, July 15, 2008 Inhomogeneous Shadowing Effect in High-Energy p- Drell Yan Process WNG Hong-Min, 1, SUN Xian-Jing, 2 and ZHNG Ben-i 3 1 Physics Department, cademy of rmored Forces Engineering of PL, Beijing , China 2 Institute of High Energy Physics, the Chinese cademy of Sciences, Beijing , China 3 Institute of pplied Physics and Computational Mathematics, Beijing , China (Received September 11, 2007) bstract Having studied the initial state energy loss versus nuclear shadowing for the Drell Yan dimuon pair production in the color string model, the inhomogeneous shadowing effect is considered in this paper. We find that the inhomogeneous shadowing effect does amend the rate of energy loss per unit path length, de/dz. Finally, the theoretical results for the Drell Yan differential cross-section ratios are compared with the E772 and E866 data. It is found that the theoretical results are in good agreement with the experimental data. PCS numbers: Qk, h, p Key words: inhomogeneous shadowing effect, color string model, Drell Yan process 1 Introduction The inhomogeneous shadowing effect, which means shadowing should depend on the spatial position of the interacting parton within the nucleus, is another important initial-state nuclear effect being found recently. lthough deep inelastic scattering (DIS) [1] experiments are typically insensitive to this position dependence, some spatial inhomogeneity has been observed in νn scattering. [2] Thus the effect should be sensitive to the impact parameter, b, at which the collision occurs so that the results depend on the collision centrality. Central collisions with low impact parameter should exhibit stronger shadowing effects than collisions in the nuclear periphery. The initial state energy loss effect in nuclear matter, which is another nuclear effect apart from the nuclear shadowing effect, has been a matter of considerable theoretical and phenomenological interest over the past years. [3 9] Because the energy loss and shadowing effects are too similar to be disentangled, there is not agreed rate of energy loss per unit path length, de/dz (GeV/fm), in the cold nuclear matter. First, a set of shadowing parameter without corrections for energy loss should be given to disentangle the two initial-state nuclear effects. Fortunately, in 2001, Hirai, Kumano, and Miyama (HKM) [10] proposed a set of nuclear parton distribution functions, which were obtained by a χ 2 global analysis of experimental data on nuclear structure functions without including the corrections for energy loss. In our present analysis we use the HKM nuclear shadowing parametrization. Second, the correct path length over which the energy loss takes place should be given. Recently, a more elaborate treatment, which describes the space-time pattern for DY pair production off a nucleus, is given in the color string model. [8,11 12] In this space-time pattern, as the inhomogeneous shadowing effect, the energy loss effect is also dependent on the impact parameter b, which makes inhomogeneous shadowing more important. Having studied the inhomogeneous shadowing effect in the Glauber model, [6,9] the inhomogeneous shadowing effect is studied in the color string model. By a χ 2 analysis of the experimental data given by Fermi National ccelerator Laboratory (FNL) E772 [13] and E866 [7] the rates of quark energy loss per unit path length are given. Finally, the calculated results for the ratios of the Drell Yan differential cross-section are compared with the experimental data. 2 Method ccording to the inhomogeneous shadowing theory, the leading order (LO) hadronic Drell Yan [14] differential cross section at impact parameter b is [9,15] d 2 σ dx 1 dm ( b) = K 8πα2 1 e 2 9M x 1 s f[q p f (x 1, M 2 ) f Q f (x 2, M 2, b ) + q p f (x 1, M 2 )Q f (x 2, M 2, b )], (1) where K is the high-order QCD correction and α is the fine structure constant. The nuclear parton density, Q f ( Q f )(x, M2, b ), is the product of the nucleon parton density qf N( qn f )(x, M2 ) and a shadowing function S f P,S (, x, M2, b ). The first subscript P refers to the choice of shadowing parametrization, while the second S, refers to the spatial dependence. Taking into account the proton and neutron numbers of both nuclei, the Q f ( Q f )(x, M2, b ) can be given as [9,15] Q f ( Q f )(x, M 2, b )=S f P,S (, x, M2, b)q N f ( q N f )(x, M 2 ). (2) The project supported by the Innovation Foundation of the cademy of rmored Forces Engineering of PL under Grant 20062L10 whmw@sina.com.cn

2 176 WNG Hong-Min, SUN Xian-Jing, and ZHNG Ben-i Vol. 50 There are two forms of the spatial dependence, one is proportional to the local nuclear density S f P,WS, and the other is proportional to the parton path through the nucleus, S f P,ρ. In this paper, we show only results for Sf P,ρ. The S f P,ρ parametrization is, in general, S f P,ρ (, x, M2, b ) = 1 + N ρ (S f P (, x, M2 ) 1) In Fig. 1, we compare the SūP,ρ (b) of Be, C, Ca, Fe, and W at the similar values of the homogeneous shadowing ratios. [17] It is shown that the inhomogeneous shadowing effect of heavy nucleus is stronger than those of light nucleus. dzρ ( b, z) dzρ (0, z), (3) where N ρ is chosen so that 1 d 2 bdzρ ( b, z)s f P,ρ (, x, M2, b, z) = S f P (, x, M2 ). ρ ( b, z) is the nucleon density in the nucleus, depending on the impact parameter b and the longitudinal coordinate z with (1/) d 2 bdzρ ( b, z) = 1. The nucleon densities of the heavy nucleus are assumed to be Woods Saxon distributions based on measurements of the nuclear charge distributions. [16] The nucleon density distributions of the light nucleus can be given by harmonic-oscillator model. [16] Fig. 2 The space-time pattern for Drell Yan dimuon pair production. The lower example illustrates a case where the dimuon pair is produced in the first inelastic interaction without energy loss effect, and the upper one illustrates a case when the beam hadron experiences a soft inelastic interaction prior to the hard interaction, in which the l l is produced. Fig. 1 The SūP,ρ(b) of Be, C, Ca, Fe, and W at the similar values of the homogeneous shadowing ratios. Now we will take account of the initial-state energy loss effect. ccording to the space-time pattern given in the color string model, [8,11,12] there are two cases for the dimuon production in the Drell Yan process. s shown at the bottom in Fig. 2, the first is that the dimuon pair may be produced in the first inelastic interaction without energy loss effect. The second, which is illustrated at the top in Fig. 2, is that the beam hadron may experience its first soft inelastic interaction at the point z 1. Then the leading projectile quark in the debris of the beam hadron will pass through the nuclear matter with energy loss and reach the point z where the dimuon pair is produced. So the energy E of the projectile hadron in the second case will decrease by E = de/dz z with z = z z 1. Thus the lepton pair production cross section in the Drell Yan process can be written as [11,12] d 2 σ = 1 d 2 b dx 1 dm 1 { d 2 σ pp z dzρ ( b, z) exp [ σ in dz 1 ρ ( ] b, z 1 ) dx 1 dm ( b)[1 δ( b, z)] + 1 d 2 σ pn 2 dx 1 dm ( b)[1 + δ( } b, z)] z d 2 b dzρ ( b, z) dz 1 σ in ρ ( z1 b, z 1 ) exp [ σ in dz 2 ρ ( ] b, z 2 ) { d 2 σ pp dx 1 dm ( b)[1 δ( b, z)] + 1 d 2 σ pn 2 dx 1 dm ( b)[1 + δ( } b, z)], (4) where the first term corresponds to the first case for the dimuon pair production, and the exponential factor in the second term, which corresponds to the second case, requires that there is not inelastic interaction of the beam hadron prior to the point z 1. In Eq. (4), σ in ( 30 mb) is the nucleon-nucleon inelastic cross section. δ(b, z) is the relative difference of the neutron and proton densities, δ(b, z) = (ρ n ρ p )/ρ, and should be equal to 1 2Z/ when ρ and ρ p are taken the same form

3 No. 1 Inhomogeneous Shadowing Effect in High-Energy p- Drell Yan Process 177 given by Ref. [16]. Thus the lepton pair production cross section in the Drell Yan process can be rewritten as d 2 σ = 1 z d 2 b dzρ ( dx 1 dm b, z) exp [ σ in dz 1 ρ ( ] d 2 σ b, z 1 ) dx 1 dm ( b ) + 1 z d 2 b dzρ ( b, z) dz 1 σ in ρ ( z1 b, z 1 ) exp [ σ in dz 2 ρ ( ] d 2 σ b, z 2 ) dx 1 dm ( b), (5) where (d 2 σ/dx 1 dm)( b) in the first term is given by Eq. (1) and (d 2 σ /dx 1 dm)( b) in the second term should be rewritten as d 2 σ dx 1 dm ( b) = K 8πα2 1 9M x 1 s e 2 f[q p f (x 1, M 2 ) Q f (x 2, M 2, b ) + q p f (x 1, M 2 )Q f (x 2, M 2, b )], (6) f with the re-scaled quantity x 1 = x 1 + x 1 = x 1 + de dz z E. For comparing with the experimental data from the E772 [13] and E866 [7] collaborations, we introduce the nuclear Drell Yan differential cross-section ratios: dm d 2 σ p 1 /dx 1 dm R 1 / 2 (x 1 ) = dm d2 σ p 2 /dx1 dm. (7) In our theoretical analysis, χ 2 is calculated with the nuclear Drell Yan ratios R 1 / 2 as χ 2 = j (R data 1 / 2,j Rtheo 1 / 2,j )2 (R err 1 / 2,j )2, (8) where R data 1 / 2,j (Rtheo 1 / 2,j) indicates the experimental data (theoretical values) for the ratio R 1 / 2, and R err 1 / 2,j denotes the systematic errors in the experiment. 3 Results and Discussions By employing the HKM for nuclear parton distribution functions together with Martin, Roberts, Stirling, and Thorne (MRST) [18] parton distribution functions in a proton, we have adjusted de/dz (free parameter) to fit the entire set of ratios C/D, Ca/D, Fe/D, W/D, Fe/Be and W/Be from E772 [13] and E866 [7] with Eq. (8). Considering inhomogeneous shadowing effect, with χ 2 min / (degree of freedom) = 1.116, we find the rate of energy loss: de/dz = 1.79 GeV/fm, which is less than the resust of homogeneous shadowing, de/dz = 2.06 GeV/fm, calculated in Ref. [12]. From Fig. 1, we can see that the nuclear shadowing effect of central collisions, where the projectile quark also has a longer path length before the dimuon pair production, is stronger than that of the peripheral collisions. This will enhance the energy loss effect and induce a small de/dz in the inhomogeneous shadowing theoretical result. Fig. 3 The nuclear Drell Yan cross section ratios R 1 / 2 on Ca to D for various M intervals. The solid and dashed curves correspond to the results without and with energy loss, respectively. The experimental data are taken from E772. [13]

4 178 WNG Hong-Min, SUN Xian-Jing, and ZHNG Ben-i Vol. 50 Fig. 4 The nuclear Drell Yan cross section ratios R 1 / 2 on W to D for various M intervals. The comments are the same as in Fig. 3 and the experimental data are from E772. [13] Fig. 5 The nuclear Drell Yan cross section ratios R 1 / 2 on W to Be for various M intervals. The comments are the same as in Fig. 3 and the experimental data are from E866. [7] For comparing with the experimental data, the calculated results, taking account of the inhomogeneous shadowing effect, are shown in Figs. 3 5 for the Drell Yan ratios R 1 / 2. Figure 3 shows the ratios of the cross section per nucleon for p-ca to p-d and the experimental data are from E772. [13] The solid and dashed curves correspond to the results without and with energy loss. Theoretically, being taken account of the energy loss effect, the sea quark

5 No. 1 Inhomogeneous Shadowing Effect in High-Energy p- Drell Yan Process 179 in the projectile proton will decrease for x 1 > x 1, so the cross section decreases. Figures 4 and 5 show the cross section per nucleon for p-w to p-d and p-w to p-be, respectively. From comparison with the experimental data, it is found that our theoretical results considered energy loss effect are in good agreement with the experimental data. In summary, by a χ 2 analysis of the experimental data given by FNL E772 and E866, the rate of quark energy loss per unit path length for inhomogeneous shadowing effect is given. Comparing with the experimental data, it is shown that the theoretical results with energy loss are in good agreement with the experimental data. The inhomogeneous shadowing effect can occur in both lepton pair production and J/ψ formation. Since J/ψ production is an important topic of study for nuclear collisions, this research should also further the understanding in cold nuclear matter to set a proper baseline for quarkonium suppression in collisions. References [1] J.J. ubert, et al., Nucl. Phys. B (1987); M. rneodo, Phys. Rep (1994). [2] T. Kitagaki, et al., Phys. Lett. B (1988). [3] S. Gavin and J. Milana, Phys. Rev. Lett (1992). [4] S.J. Brodsky and P. Hoyer, Phys. Lett. B (1993). [5] R. Baier, et al., Nucl. Phys. B (1997). [6] Jian-Jun Yang and Guang-Lie Li, Eur. Phys. J. C 5 (1998) 719; C.G. Duan, L.H. Song, L.J. Huo, and G.L. Li, Eur. Phys. J. C 29 (2003) 557. [7] M.. Vasilier et al., Phys. Rev. Lett. 83 (1999) [8] M.B. Johnson et al., Phys. Rev. Lett. 86 (2001) [9] Hong-Min Wang, Xian-Jing Sun, and Ben-i Zhang, Phys. Scr. 75 (2007) 651. [10] M. Hirai, S. Kumano, and M. Miyama (HKM), Phys. Rev. D 64 (2001) [11] B.Z. Kopeliovich and F. Niedermayer. JINR Report No. JINRE , 1984, (a scanned version in the KEK library: index ). [12] Wang Hong-Min, Sun Xian-Jing, and Hou Zhao-Yu, High Energy Physics and Nuclear Physics 31 (2007) [13] D.M. lde et al., Phys. Rev. Lett. 64 (1990) [14] S. Drell and T.M. Yan, Phys. Rev. Lett. 25 (1970) 316. [15] S.R. Klein and R. Vogt, Phys. Rev. Lett. 91 (2003) [16] C.W. dejager, H. devries, and C. devries, tomic Data and Nuclear Data Tables 14 (1974) 485. [17] K.J. Eskola, V.J. Kolinen, and C.. Salgado (EKS), Eur. Phys. J. C 9 (1999) 61. [18].D. Martin, R.G. Roberts, W.J. Stirling, and R.S. Thorne, (MRST), Eur. Phys. J. C 4 (1998) 463.