Commun. Theor. Phys. (Beijing, China) 38 (2002) pp. 59{64 c International Academic Publishers Vol. 38, No. 1, July 15, 2002 Nuclear Eect Study Under K

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1 Commun. Theor. Phys. (Beijing, China) 38 (2002) pp. 59{64 c International cademic Publishers Vol. 38, No. 1, July 15, 2002 Nuclear Eect Study Under K Factor's Nonconstancy HOU Zhao-Yu, 1 2 WU Wen-Wang, 3 HE Ming-Zhong, 4 and ZHNG Ben-i 2 1 Department of Physics, Shijiazhuang Normal College, Shijiazhuang , China 2 Graduate School, the Chinese cademy of Engineering Physics, Beijing , China 3 Department of Mathematics and Physics, Shijiazhuang Railway College, Shijiazhuang , China 4 Department of Electronics, Hebei Normal University, Shijiazhuang , China (Received March 6, 2002) bstract consistent approach to estimating nuclear eect functions Rv (x 2 ) and Rs (x 2 ) based on numerical iteration technique is presented in the quark-parton model when taking into account the nonconstancy of quantum chromodynamics correction factor K. Rv (x 2 ) and Rs (x 2 ) correspond respectively to the valence quark distributions for one bound nucleon within the nucleus and to the sea quark ones. Related numerical analysis is given for nuclei 12 6 C, 40 20Ca, and 56 26Fe. s the basis, it adopts both experimental data of the high energy proton-nucleus Drell{Yan process and of the high energy lepton-nucleus deep inelastic scattering. PCS numbers: Hb, Fb, Ep, eV Key words: K-factor, Drell{Yan process, deep inelastic scattering, nuclear parton distribution 1 Introduction In the wake of the development of quark-parton model study, Drell{Yan (DY) process, [1] like deep inelastic scattering, has been widely used to investigate nucleon structure and certain important nuclear features such as quark distributions. For more than ten years, the quantum chromodynamics (QCD) correction K factor [2;5] has been adopted to deal with the order- s (Q 2 ) corrections to the DY cross section so as to simplify the actual calculation of perturbative QCD. This factor would be taken as a constant value due to the approximation of O( s ) QCD corrections, as examined by a number of experiments concerned with the cases of free nucleon as target. However, further experimental analyses on p{p, p{, and { collisions reveal its nonconstancy. [6;9] s an essential quantity for DY phenomenology, the K factor has certain correlations with nuclear medium eect, for instance, the QCD corrections from the gluon Compton scattering and other terms have been proven to be dependent upon the parton distributions in the nucleus. Therefore we should carefully treat each contribution of QCD-correction terms rather than simply using the K factor of constancy to replace the QCD corrections. Especially our eort should be aimed at extracting nuclear eect from available data of experimental information. The present approach under K nonconstancy could be regarded as a try. This paper provides a consistent method to obtain nuclear medium eect and gives related computed results of ratio functions Rv (x 2 ) and Rs (x 2 ) for 12 6 C, 40 20Ca, and 56 26Fe as target nuclei. In the calculation we have adopted the parton momentum distributions of free nucleon given by M. Gluck et al. [10] 2 Drell Yan Cross Section Ratio Following the quark-parton model the Drell{Yan process of high energy h{ collision h + ;! + ; + X (1) can be explained as that a quark (anti-quark) from the incident hadron and an anti-quark (quark) from the target nucleus annihilate to a virtual photon which subsequently materialize into a pair of + and ; (Fig. 1). Fig. 1 Hadrons h and nuclei collision into + ; pair generation process. On the consideration of the contribution due to the distribution functions of partons in nucleus, the corresponding dierential cross section may be expressed [11 12] as d 2 = K h (x 1 x 2 Q 2 ) 42 H h (x dx 1 dx 2 9Sx 2 1 x 2 Q 2 ) (2) 1 x2 2 where x 1 (x 2 ) represents the Bjorken variable for a beam The project supported by the Chinese Engineering Physics Institute for Research Doctor Subsidizes (2001)

2 60 HOU Zhao-Yu, WU Wen-Wang, HE Ming-Zhong, and ZHNG Ben-i Vol. 38 (target) particle, p S is the center-of-mass energy and the ne structure constant. Q 2 Sx 1 x 2. K is the QCD correction factor, and H h is called as H-function dependent on parton distributions with the denition H h (x 1 x 2 Q 2 ) = X f e 2 f[x 1 q h f (x 1 Q 2 )x 2 q f (x 2 Q 2 ) + x 1 q h f (x 1 Q 2 )x 2 q f (x 2 Q 2 )] : (3) In Eq. (3) e f is the charge number of the f avour quark, and qf h and q f are the momentum distribution functions of f avour quark in the incident hadron and in the target nucleus respectively. When the incident hadron h is the proton, the H function reads In the above expression [13] where index N denotes the free nucleon. It is easy to nd H p (x 1 x 2 Q 2 ) = H v (x 1 x 2 Q 2 )R v (x 2 ) + H s (x 1 x 2 Q 2 )R s (x 2 ) (4) R v (x 2 ) = q fv(x 2 Q 2 )=q N fv(x 2 Q 2 ) R s (x 2 ) = q fs(x 2 Q 2 )=q N fs(x 2 Q 2 ) (5) H v (x 1 x 2 Q 2 ) = 1 9 x 1 n Z 4u 1 s 2h x u2 v + 1 ; Z i d 2 v + Z ds 2h 1 x d2 v + 1 ; Z io u 2 v (6) H s (x 1 x 2 Q 2 ) = 1 9 x 1f4(u 1 v + u 1 s)x 2 u 2 s + (d 1 v + d 1 s)x 2 d 2 s + s 1 s x 2s 2 s + s 1 s x 2s 2 s4u 1 s x 2u 2 s + d 1 s x 2d 2 sg (7) where Z and are the proton number and the mass number of the target nucleus respectively. When the incident hadron is proton and the target is deuteron, we suppose R D s = R D v = 1 and hence obtain H pd (x 1 x 2 Q 2 ) = 1 18 x 1[4u 1 s + d 1 s )x 2 (u 2 v + d 2 v + 8u 1 s x 2u 2 s + 2 d 1 s x 2d 2 s] s for the QCD correction factor K, its denition can be written as x 1[4(u 1 v + u 1 s)x 2 u 2 s + (d 1 v + d 1 s)x 2 d 2 s + s 1 s x 2s 2 s + s 1 s x 2s 2 s] : (8) K(x 1 x 2 Q 2 ) = X i where i DY, C (Compton), nn (annihilation), d i dx 1 dx 2 = Z 1 d i dx 1 dx 2. d DY dx 1 dx 2 (9) Z 1 h d^i (t 1 t 2 ) dt 1 dt 2 Q i (t 1 t 2 ) + d^i (t 2 t 1 ) Q ~ i (t 1 t 2 ) (10) x 1 x 2 dx 1 dx 2 dx 1 dx 2 here, d^ i (t 1 t 2 )=dx 1 dx 2 (i = DY, C, nn) is the dierential cross section for the process with index i whose formula given in Ref. [2]. In Eq. (10), we can easily obtain the following expressions [14] Q DY (t 1 t 2 ) = Q nn (t 1 t 2 ) = 1 9 [4(u1 v + u 1 s)u 2 s + (d 1 v + d 1 s) d s 2 + s 1 ss 2 s]rs (t 2 ) (11) ~Q DY (t 1 t 2 ) = Q ~ nn (t 1 t 2 ) = 1 9 [2u1 s(u 2 v + d 2 v) + d 1 s(d 2 v + u 2 v)]rv (t 2 ) [4u1 s u2 s + d s 1 d2 s + s 1 s s2 s]rs (t 2 ) (12) Q C (t 1 t 2 ) = 1 9 g1 h 5 2 (u2 v + d 2 v) i R v (t 2 ) g1 [4u 2 s + d 2 s + s 2 s + u 2 s + d 2 s + s 2 s]r s (t 2 ) (13) ~Q C (t 1 t 2 ) = 1 9 [4(u1 v + u 1 s + u 1 s) + d 1 v + d 1 s + d 1 s + s 1 s + s 1 s]g 2 (14) where g 1 and g 2 are the gluon distribution functions in the free nucleon and in the bound nucleon of target nucleus respectively. t last the DY cross section ratio reads where T =D (x 2 ) = Z d 2 p (x 2 ).Z d 2 pd (x 2 ) dx 1 dx 1 = M (k) (x 2 )Rv (x 2 ) + N (k) (x 2 )Rs (x 2 ) (15) dx 1 dx 2 dx 1 dx 2!(x 2 ) Z K p (x M (k) 1 x 2 Q 2 ) (x 2 ) = H v (x 1 x 2 Q 2 )dx 1 (16) x 2 1 x2 2 i

3 No. 1 Nuclear Eect Study Under K Factor's Nonconstancy 61 Z K p (x N (k) 1 x 2 Q 2 ) (x 2 ) = H x 2 1 x2 s (x 1 x 2 Q 2 )dx 1 (17) 2 Z K pd (x 1 x 2 Q 2 )!(x 2 ) = H pd (x 1 x 2 Q 2 )dx 1 : (18) x 2 1 x2 2 The experimental data of T =D (x 2 ) from Drell{Yan experiments of h{ collision at FNL E772 have been provided in Ref. [15]. From these data we have reproduced the corresponding tted result as shown in Fig. 2. Fig. 2 T =D of the Drell{Yan data for C/ 2 H, Ca/ 2 H, and Fe/ 2 H. 3 The Nucleon Structure Function Ratio In the quark-parton model, the nucleons are described as those composed of point-like and quasi-free quarks. The nucleon structure function is dened as F 2 (x 2 Q 2 ) = X f For the bound nucleon in nucleus, the average nuclear structure function is F 2 (x 2 Q 2 ) = X f e 2 f x 2q f (x 2 Q 2 ) : (19) e 2 f x 2[R v (x 2 )q N fv(x 2 Q 2 ) + R s (x 2 )q N fs(x 2 Q 2 )] = ~ XR v (x 2 ) + ~ Y R s (x 2 ) : (20) where ~X 1 Z 2h 9 x (4u2 v + d 2 v) + 1 ; Z i (4d 2 v + u 2 v) (21) ~Y 1 9 x 2(4u 2 s + 4u 2 s + d 2 s + d 2 s + s 2 s + s 2 s) : (22) For the conventional nucleus we account for the neutron excess eect and have F 2 (x 2 Q 2 ) = F 2 (x 2 Q 2 ) ; 1 2 (N ; Z)[F n 2 (x 2 Q 2 ) ; F p (x 2 2 Q 2 )] (23) where F n 2 and F p 2 are free nucleon and free proton structure functions respectively, F n 2 (x 2 Q 2 ) = 1 9 x 2[4(d 2 v + d 2 s + d 2 s ) + u 2 v + u 2 s + u 2 s + s 2 s + s 2 s] (24) F p 2 (x 2 Q 2 ) = 1 9 x 2[4(u 2 v + u 2 s + u 2 s) + d 2 v + d 2 s + d 2 s + s 2 s + s 2 s] : (25) If we neglect the weak nuclear eect in deuteron, its structure function can be written as F D 2 (x 2 Q 2 ) = X f e 2 f x 2q N f (x 2 Q 2 ) = 5 18 x 2(u 2 v + d 2 v) x 2(4u 2 s + 4u 2 s + d 2 s + d 2 s + s 2 s + s 2 s) : (26) Hence we may dene the nucleon structure function ratio in the following, [16] R =D (x 2 ) F 2 (x 2 Q 2 )=F D 2 (x 2 Q 2 ) : (27)

4 62 HOU Zhao-Yu, WU Wen-Wang, HE Ming-Zhong, and ZHNG Ben-i Vol. 38 The experimental data on R =D are available from the l- DIS experiment at CERN. [17;21] Based on these original data we have produced the corresponding tted result by making use of numerical tting technique as plotted in Fig. 3. Fig. 3 R =D of the DIS data for C/D, Ca/D, and Fe/D. 4 Method of nalysis and Related Results where The ratio of the structure function can be expressed in the simple form ~R(x 2 ) = ~ XR v (x 2 ) + ~ Y R s (x 2 ) (28) ~R(x 2 ) = R =D (x 2 )F D 2 (x 2 Q 2 ) + (x 2 Q 2 ) (29) (x 2 Q 2 ) = 1 2 (N ; Z)[F n 2 (x 2 Q 2 ) ; F p 2 (x 2 Q 2 )] : (30) In a similar way we can also express the cross section ratio of nuclear Drell{Yan process as T (x 2 ) = M (k) R v (x 2 ) + N (k) R s (x 2 ) (31) where T (x 2 ) =!(x 2 )T =D (x 2 ). So equations (28) and (31) become a set of nonlinear equations. Rv and Rs can be in principle determined. We have the solution in iterative form as follows: R (k+1) s (x 2 ) = R (k+1) v (x 2 ) = where k is the iterative number, for k = 0 we may take R (0) s k = 1 Rs (1) and Rv (1) solutions on Rv and Rs. R ~ Y ~ X ~ ~ Y T N (k) M (k) N (k) R ~ Y ~ X ~ ~ Y M (k) T M (k) N (k) (32) (33) = Rv (0) = 1 in the expressions of M (0), N (0), hence for can be deduced. Through the iterative procedure we can obtain a set of suciently approximate Numerical results of Rv Fe (x 2 ), Rs C (x 2 ), Rs Ca (x 2 ), and Rs Fe (x 2 ) have been obtained as shown in Fig. 4. From Fig. 4 it appears to show that the nuclear eect of valence quark distribution and that of sea quark are quite dierent. In the whole calculation region, Rv (x 2 ) decreases as x 2 increases. t the position x 2 = 0:05 the value of Rv (x 2 ) is very large, that is, valence quark distribution in the bound nucleon of nucleus is much greater than that for the case of free nucleon. With the increase of x 2 Rv (x 2 ) decreases gradually, and until x 2 = 0:3, Rv (x 2 ) approaches to 1, namely, the valence quark distribution in the bound nucleon of nucleus changes into the same as that in free nucleon. s for Rs (x 2 ) it is always less than 1 when x 2 0:3 and approaches to zero at the position x 2 = 0:05, that is, in the region x 2 0:3 the sea quark distributions in bound nucleon of nucleus are always below the sea quark distribution in free nucleon. The variation of Rv (x 2 ) for the three nuclei is that Fe > Ca > C, which shows that quark distribution induced nuclear

5 No. 1 Nuclear Eect Study Under K Factor's Nonconstancy 63 eect in C, Ca, and Fe are slightly dierent. Fig C, 40 20Ca, 56 26Fe nuclear eects on the valence quark and sea quark distributions. Fig C, 40 20Ca, 56 26Fe on the valence quark and sea quark distributions probability at statistical error.

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