Partonic picture of nuclear shadowing at small x

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 637 (1998) 79-106

Partonic picture of nuclear shadowing at small x Zheng Huang, Hung Jung Lu, Ina Sarcevic Department of Physics, University of Arizona, Tucson, AZ 85721, USA Received 10 October 1997; revised 24 February 1998; accepted 15 April 1998

Abstract

We investigate the nuclear shadowing mechanism in the context of pcrturbative QCD and the Glauber-Gribov multiple scattering model. Using recent HERA data on nucleon structure function at small x, we put stringent constrains on the nucleon gluon density in the double-logarithm approximation. We suggest that the scaling violation of the nucleon structure function in the region of small-x and semihard scale Q2 can be reliably described by perturbative QCD which is a central key 1o the understanding of the scale dependence of the nuclear shadowing effect. Our results indicate that while the shadowing of the quark density arises from an interplay between "soft" and semihard QCD processes, while gluon shadowing is largely driven by a perturbative shadowing mechanism. We demonstrate that the gluon shadowing is a robust phenomenon at large Q2 and can be unambiguously predicted by perturbative QCD. @ 1998 Elsevier Science B.V. PACS: 24.85.+p, 25.75.-q, 12.38.Bx, 13.60.Hb Keywords: Nuclei; Partons

1. Introduction The theoretical studies of semihard processes play a increasingly important role in ultrarelativistic heavy-ion collisions at v,~ /> 200 GeV in describing the global collision features such as particle multiplicities and transverse energy distributions. These semihard processes can be reliably calculated in the framework of perturbative QCD and they are crucial for determining the initial condition for the possible formation of a quark-gluon plasma. Assuming the validity of the factorization theorem in perturbation theory, it is essential to know the parton distributions in nuclei in order to compute these processes. For example, to calculate the minijet production for Pr ~ 2 GeV or the charm quark production in the central rapidity region (y = 0), one needs to know the nuclear gluon structure function XgA(X, Q2) at Q ~ pr or mc and x = XBj = 2Pr/x/s or 2mc/x/~. For x/s/> 200 GeV, the corresponding value of x is x ~< 10 -2. Similarly, the nuclear quark density xfa(x, Q2) is needed for the Drell-Yan lepton-pair production process. 0375-9474/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PH S 0 3 7 5 - 9 4 7 4 ( 9 8 ) 0 0 2 1 0 - 3

80

z Huang et al./Nuclear Physics A 637 (1998) 79-106

The attenuation of quark density in a nucleus has been firmly established experimentally at CERN [ 1] and Fermilab [2] in the region of small values of the Bjorken variable x = Q 2 / 2 m v in deeply inelastic lepton scatterings (DIS) off nuclei, where Q2 is the four-momentum transfer squared, v is the energy loss of the lepton and m is the nucleon mass. The data, taken over a wide kinematic range, 10 -5 < x < 0.1 and 0.05 GeV 2 < Q2 < 100 GeV 2, show a systematic reduction of nuclear structure function F a ( x , Q 2 ) / A with respect to the free nucleon structure function F2u (x, Q2). There are also some indications of nuclear gluon shadowing from the analysis of J / ~ suppression in hadron-nucleus experiments [ 3]. Unfortunately, the extraction of nuclear gluon density is not unambiguous since it involves the evaluation of the initial parton energy loss and final state interactions [4]. These measurements present a tremendous challenge to the theoretical models of nuclear shadowing (for recent reviews, see Ref. [5] ). The partonic description of nuclear shadowing phenomenon has been extensively investigated in literatures in the context of a recombination model [6-8] and in a Glauber-Gribov multiple scattering theory [9-13]. These approaches predict different scale (Q2) dependence of the nuclear structure function than the conventional GribovLipatov-Altarelli-Parisi (GLAP) evolution equation [ 14]. Qualitatively, the scale evolution of the nuclear structure function is slower than the nucleon case, leading to a perturbative mechanism for the depletion of the parton density at large momentum transfer. The quantitative prediction of nuclear shadowing at large Q2 in general depends on the initial value of shadowing at a small momentum transfer where a perturbative calculation breaks down and some non-perturbative model or experimental input is needed. The so-called vector-meson-dominance (VMD) model [ 15] has been quite successful in predicting the quark shadowing at small Q2 where the experimental data is rich [ 1,2]. Presently there is no viable non-perturbative model for the gluon case and there is very little experimental information on the initial gluon shadowing value at some small Qo2. This, in principle, could make the prediction on gluon shadowing very uncertain. In this paper, however, we suggest that these exists a typical scale Qs2H~_ 3 GeV 2 in the perturbative evolution of the nuclear gluon density beyond which the nuclear gluon shadowing can be unambiguously predicted in the context of perturbative QCD. The so-called "semihard" scale Qs2H is determined by the strength of the scaling violation of the nucleon gluon density in the small-x region at which the anomalous dimension y - 3 l n x g N / I n Q2 is of O(1). We shall demonstrate that as Q2 approaches Q2H, the gluon shadowing ratio rapidly approaches a unique perturbative value determined by the Glauber-Gribov multiple scattering theory. At Q2 > QS2H' the nuclear gluon density, in contrast to the quark case, is almost independent of the initial distribution at Qo2 and its evolution is mainly governed by a GLAP equation. The suggested phenomenon is intimately related to the singular growth of the gluon density in the small-x region and the perturbative shadowing caused by the coherent parton multiple scattering. Here we present a quantitative study of the scale dependence of the nuclear shadowing phenomenon for both the quark and the gluon distributions based on the Glauber-Gribov diffractive approximation. A similar study has been performed by Eskola [ 16] in the recombination model. Our focus is the sensitivity of the nuclear shadowing at large QZ

Z Huang et al./Nuclear Physics A 637 (1998) 79-106

81

to the initial conditions. To determine the gluon content of the nucleon, we shall use a leading double logarithm approximation (DLA) where the scaling violation of the quark structure function F~ is entirely caused by the q --* qg splitting. The excellent agreement between our results based on the DLA and the measured values of F2u indicates that the perturbative evolution sets in as early as Q2 ~> 0.8 GeV 2 and the scaling violation of gluon structure function in the semihard region 0.8 GeV 2 ~> Q2 ~< Q~H can be reliably described by the DLA. This provide a basis for studying the Glauber-Gribov approach for both the quark and the gluon since it is derived in the same approximation where the separation of q~ or gg pair in the DLA is considered frozen during its passage in the nucleus. The reason for not using the available parton distributions for a nucleon such as those of MRS, CTEQ or GRV parameterizations [ 17] is that the nuclear structure function obtained in our approach does not contain a full GLAP kernel, as it will become clear below. Since the shadowing ratio depends on both the nuclear and the nucleon structure functions, they should be consistently calculated in the same approximation. We expect that the calculated ratio, however, has validity beyond the DLA. We shall demonstrate that the nuclear shadowing for the quark distribution is a leading twist effect due to the interplay between the "soft" and the "semihard" QCD processes, while for the gluon distribution, the shadowing is largely driven by the perturbative shadowing mechanism at small x. Both shadowings are shown to evolve slowly at large Q2 beyond the semihard scale. We discuss the x-dependence of the shadowing ratio and the non-saturating feature of the perturbative shadowing. As a by-product of the Glauber-Gribov approach, we also study the nuclear geometry dependence of the shadowing ratio, which cannot be addressed in the recombination model such as the GLR equation. Since the shadowing effect is intimately related to the effective number of scatterings, it should vary strongly as one changes the impact parameter. This leads to the breakdown of the conventional factorization of the nuclear parton density into the product of an impact parameter averaged parton distribution and the nuclear thickness function. The paper is organized as follows. In Section 2 we discuss the physical picture of the nuclear shadowing phenomenon. It can be regarded as a heuristic introduction to the central idea of our approach. In Section 3 we calculate the nucleon structure function FN(x, Q2) for large range of x and Q2 using gluon density obtained in the DLA. We determine the initial condition for FN(x, Q~) and for g(x, Q2) by fitting the Q2-dependence of F~ at x = 0.00008. The nuclear shadowing effect depends crucially on the gluon component of the nucleon. In Section 4 we give a brief introduction on the Glauber-Gribov multiple scattering model and discuss the dependence on the coherence length. The nuclear shadowing for Fza is calculated in Section 5 and the initial shadowing is parameterized in order to fit the available experimental data at the measured Q2 values. In Section 6, the gluon shadowing is computed using the GlauberGribov formula derived by Mueller [12] and Ayala F, Gay Ducati and Levin [13]. Section 7 is devoted to the discussion of the impact parameter dependence of the nuclear shadowing. We conclude our findings in Section 8.

82

Z. Huanget al./Nuclear PhysicsA 637 (1998) 79-106

2. Physical picture of nuclear shadowing At low Q2, Q2 < 1 GeV 2, in the deeply inelastic scattering, the interaction of the virtual photon with the nucleons in the rest frame of the target is most naturally described by a vector-meson-dominance (VMD) model [ 15 ], in which the virtual photon interacts with the nucleons via its hadronic fluctuations, namely the p, to and ~b mesons. The shadowing of the (sea) quark density is caused by the coherent multiple scatterings of the vector meson off the nucleons with a large hadronic cross section trVN, which can be most conveniently handled by the Glauber-Gribov diffractive approximation model [ 18]. The x-dependence of the shadowing ratio R(x,Q 2) = FA(x, Q2)/AF~(x, Q2) is explained by the coherent length of the virtual photon Ic ~- 1/2mx. For x < 10 -2, I¢ > RA = 4,~5 fm for large nuclei, the vector meson interacts coherently all nucleons inside the nucleus resulting in a uniform (in x) reduction of the total cross section, while for x > 10 -2, lc ~ RA the vector meson can only interact coherently with a fraction of all the nucleons and shadowing is reduced. A generalized VMD model describes the observed weak Q2 dependence reasonably well in the low Q2 region [19]. At Q2 > 1~2 GeV 2, the virtual photon can penetrate the nucleon and probe the partonic degrees of freedom where a partonic interpretation based on perturbative QCD is most relevant in the infinite momentum frame. It is well known that the Q2-dependence, the so-called scaling violation of the nucleon structure function F~(x, Q2) can be well accounted for by the GLAP evolution equation given some non-perturbative initial condition F~(x, Q2) [17]. In the small x region, the GLAP predicts a strong scaling violation of F~(x, Q2) due to the strong growth of the gluon density in the nucleon. In fact, a dominant perturbative evolution implies that the growth of F~(x, Q2) in ln Q 2 at fixed, small x is closely related to the growth in In 1/x at fixed Q2, i.e. the so-called double scaling violation observed in the DIS experiments for nucleon, most transparent in the Double Leading-Logarithm Approximation (DLA) of the gluon structure function. At extremely small x, the resummation of large In 1Ix is necessary leading to the BFKL evolution equation [20] which may also be derived in a Weiszacker-Williams classical field approximation [21]. It is not clear when the BFKL hard pomeron will become relevant since the GLAP evolution seems able to describe most of kinematical region that current experiments have probed so far. As we shall show, the main feature of the Q2-dependence of the nuclear shadowing is intimately related to this strong scaling violation in the small-x region. If the scale evolution of the nuclear structure function is described by GLAP then the shadowing ratio R(x, Q2), given its initial value, will quickly approach 1 because of the strong scaling violation FA(x, Q2) >> FA(x, Q2). This would imply that the shadowing is purely a non-perturbative effect and a higher twist effect. The key observation is that in the infinite momentum frame, because of the high nucleon and parton densities, the quarks and gluons that belong to different nucleons in the nucleus will recombine and annihilate, leading to the so-called recombination effect first suggested by Gribov, Levin and Ryskin [6] and later proven by Mueller and Qiu [7]. In the target rest frame, the virtual photon interacts with nucleons via its quark-antiquark pair (q(/) color-singlet

Z Huang et aL/Nuclear Physics A 637 (1998) 79-106

83

fluctuation [ 10]. If the coherence length of the virtual photon is larger than the radius of the nucleus, lc > RA, the q~ configuration interacts coherently with all nucleon with a cross section given by the color transparency mechanism for a point-like color-singlet configuration [22]. That is, the cross section is proportional to the transverse separation squared, r 2, of the q and ~. In the double logarithm approximation (DLA), the cross section can be expressed through the gluon structure function for a nucleon O'qgtN O( r~ OlsXt gDLA ( X t, l i r a ) ,

(1)

where x' = MZ/2mv and M is the invariant mass of qc7 pair. The total cross section O'qgtA can be calculated in a Glauber-Gribov eikonal approximation, O'qqA = f dZb 2{1

-

e-~qoNTa(b)/2},

(2)

where b is the impact parameter and TA (b) is the nuclear thickness function. The nuclear cross section is therefore reduced when compared to the simple addition of free nucleon cross sections. However, it is not clear that (1) and (2) will provide a satisfactory explanation of the non-vanishing shadowing effect at large Qz. Although for a given Q2, all q~ configurations with the separation size r~ > 1/Q 2 are possible, the virtual photon wavefunction [~Pz,'l ~ 1/r4 favors the smallest value of r 2 ~ 1/Q 2 where the cross section is small and no significant shadowing is expected. The solution to this problem comes from the fact that the nucleon structure function exhibits a strong scaling violation in the small x region at the moderate (semihard) momentum transfer 0.8 GeV 2 < Q2 < Qs2H where Qs2H_~ 3 GeV 2. As we shall show, the strong scaling violation in this region can be understood in the context of perturbative QCD. The gluon density in (1) grows rapidly with 1/r~ in the small x region, which can compensate the inherent "smallness" of the cross sections of "hard" QCD processes. As a consequence, the semihard processes can compete successfully with the "soft" processes as described in the VMD model and continue to provide the shadowing mechanism up to some moderately high value of Q2 ~ Qs2H. In terms of the anomalous dimension, y(Q2) = d ln xgu/d In Q2, the cross section in (1) can be written as (Q2)Z,

o'q4N cx

Q2

(3)

At large Q2, the anomalous dimension becomes greater than 1 and it delays the rapid decrease of the cross section due to the "smallness" of the configuration size, as first argued by Gribov, Levin and Ryskin in Ref. [6]. We refer to this as the perturbative shadowing mechanism as it can be computed in perturbative QCD. At even higher values of Q2, Q2 > Qs2n, the scaling violation becomes weak (logarithmic), and the perturbative shadowing mechanism ceases to be effective. In this case, both the nucleon and the nuclear structure function evolve slowly at the same rate. The shadowing ratio developed by both perturbative and non-perturbative mechanisms in the relatively low-Q 2 region persists in the high Q2 region but tends to 1 rather slowly as Q2 ___+c~.

84

z. Huang et al./Nuclear Physics A 637 (1998) 79-106

In contrast to the VMD model, the x-dependent cross section O-qONpredicts an additional x-dependence of the shadowing ratio apart from the coherence length dependence in the perturbative region. As a result, one does not in general expects a saturation of the shadowing ratio as x decreases below the coherence limit x < 10 -2. On the other hand, as Q2 decreases, the gluon distribution changes from a singular small-x behavior (hard pomeron regime) to the less singular one (soft pomeron regime). The shadowing ratio becomes flatter as Q2 decreases. The observed saturation of the shadowing ratio in the small-x region [2] is thus a non-perturbative, low-Q 2 feature. Similarly, the nuclear gluon density can be studied by considering a deeply inelastic scattering of a virtual colorless probe (e.g. the virtual "Higgs boson") with the nucleus where the interaction proceeds via a gluon pair gg component of the virtual probe. This has been considered by Mueller [ 12] and recently by Ayala F, Gay Ducati and Levin (AGL) [ 13]. In contrast to the quark distribution, the gluon structure functions for both a nucleon and a nucleus are poorly known, especially in the small-x regime though they are of extreme importance for the semihard QCD processes at high energies. Most of our knowledge on the gluon distribution comes from the theoretical expectations based on perturbative QCD constrained by the scaling violation of F~v and the prompt photon experiments [ 17]. The nuclear gluon density may also be studied in the Glauber-Gribov multiple scattering model where trq#N in (2) is to be replaced by trggN. Simple color group representation argument relates these two cross sections by O'ggN = 9O-qOU/4. However, what makes the gluon case very different from the quark case is the unknown initial (non-perturbative) gluon density in the nucleus. There is very little experimental information on the gluon shadowing at low Q2 and small x. The VMD model is not applicable here since the low-lying meson spectroscopy for gg-like states is poorly known. This would make this approach very uncertain in predicting the amount of gluon shadowing in the nucleus. However, as we shall show in this paper, the large O-ggN implies an even stronger scaling violation of gluon density than in the case of quarks making the perturbative shadowing a dominant mechanism. As a consequence, the shadowing ratio is largely determined by the perturbative calculation at Q2 = QZH. That is, the large anomalous dimension of gluon distribution function for a nucleon drives the shadowing ratio rapidly to approach the perturbative value at moderately large Q2, Q2 = Qsn, 2 and the initial shadowing (at low Q2) becomes relatively unimportant. For Q2 > QzH, the scaling violation becomes weaker and the shadowing ratio determined by the perturbative shadowing mechanism at the semihard scale persists and evolves slowly. The x-dependence of the shadowing ratio can be also predicted to be non-saturating, a generic feature of the perturbative shadowing. We therefore suggest that the nuclear gluon shadowing at Q2 > Q2H ~ 3 GeV 2 and at small x can be predicted in the context of perturbative QCD. The prediction is based on the scaling violation of the gluon structure function and the existence of the perturbative shadowing in the Glauber-Gribov multiple scattering model.

Z. Huang et al./Nuclear Physics A 637 (1998) 79-106

85

1-7

P

Fig. 1. The perturbative diagrams giving rise to the q~ pair fluctuation.

3. Lab frame description of nucleon structure function Since the nuclear shadowing ratio depends sensitively on the nucleon structure function, we shall develop a formalism to calculate both the nuclear and the nucleon structure functions simultaneously in order to reduce the uncertainty due to the nucleon structure function. This prevents us from using the well-known parametrizations such as MRS, CTEQ and GRV [ 17] which are based on the full GLAP equation because the nuclear structure function is not derived in the same formalism. We stress that the shadowing is only a few tens percentage effect, thus the uncertainty in the nucleon structure at the same level can cause a large effect on the shadowing ratio.

3.1. Quark distribution Consider a description of deeply inelastic lepton scattering off a nucleon in the laboratory frame. The inclusive cross section can be expressed via the virtual photonnucleon total cross section. The cross section for the absorption of a virtual photon in the small x region is dominated by the scattering off the sea-quark q~ pair. The generic perturbative QCD diagrams giving rise to the q~ fluctuation are shown in Fig. 1. The CM energy of the incoming virtual photon at small x is Q2

s = ( q ÷ p ) ~ ~_ 2p .q= -y- ,

(4)

where q and p are the four-momenta of the photon and the target nucleon. Thus a region of small x corresponds to a high energy scattering process at fixed Q2. The imaginary part of the amplitudes in Fig. 1 is related to the photoabsorption cross section which has been calculated by Nikolaev and Zakharov [ 11] assuming that the size of the qc] pair is frozen in the scattering process and that the one-gluon exchange dominates. This is nothing but the so-called double logarithm approximation (DLA) [6,23]. The transverse cross section can be cast into an impact parameter representation 1

o'(y*N) = f dz.ld2rl~(z,r)12O'q#N(r) , 0

(5)

86

Z. Huang et a l . / N u c l e a r Physics A 637 (1998) 79-106

where z is the Sudakov variable, defined to be the fraction of the qg/pair momentum carried by one of the (anti)quarks, r is the variable conjugate to the transverse momentum of (anti)quark of the pair, representing the transverse size of the pair. ~O(z, r) can be interpreted as the wave function of the qg/fluctuations of virtual photon. Thus, I¢(z,r)l 2 serves as the probability of finding a q~ pair with a separation r and a fractional momentum z. In an explicit form 6teem

Nf

eZiP(z)a2Kl ( a r ) 2 ,

I¢(z, r)l 2 = (2~r)Z ~

(6)

1

where a 2 = QZz(1 - z ) and Kl is the modified Bessel function. The factor P ( z ) = z 2 + (1 - z )2 is the universal splitting function of gauge boson into the fermion pair. Since Kl is a rapidly falling function, the integral in (5) has important contributions only in the region ar << 1. In this case, one has

I~(z'r)12

6tre m Nf 2 1 (2~)'2Zei-~' 1

~

(7)

where the expansion K l ( a r ) 2 --* 1/a2r 2 has been used. The photon wave function favors a q~ pair with small transverse separation. The cross section for the high energy interaction of a small-size q~ configuration with the nucleon can be unambiguously calculated in QCD in the region of small x by applying the QCD factorization theorem. In the DLA, the result is [25] 2

97"

2

t2

t

Crq¢s(r) = -~-r Ces(Q )xg~N

LA

t

(x,Q

/2

),

(8)

where xg pEA is the gluon distribution of the nucleon calculated in the DLA. i The fact that the cross section is proportional to the size of the point-like configuration is the consequence of the color transparency in QCD. Although the r 2 factor in (g) decreases fast as one goes to the short distance, the singular behavior of the wave function in (7) and the strong scaling violation of xgDNLA in the small-x region as r decreases can compensate the smallness of cross section due to the color transparency. As a result, the soft and hard physics compete in a wide range of Q2. The x / and Q,2 in (8) are related to z, x and r 2 as follows. The invariant mass squared of the q~ pair is M2 ~_

k~ z(1 - z )

(9) '

where kt2 ~ 1/r 2 is the transverse momentum squared of the (anti)quark. The virtuality of the gluon is then Q,2 = (kt - ( - k t ) ) 2 = 4 / r 2 while x' is given by J Since the gluon distribution, x g DLA is evolved, we do not include excitations of higher Fock states of the photon as studied in Ref. 124], because of the potential double counting,

Z Huang et al./Nuclear Physics A 637 (1998) 79-106 y t --

Me 2mtJ

k2 z(l - z)2mu

x aZr2 ,

87

(10)

where v is the energy loss of the lepton. The condition ar < 1 implies x ~ > x. Since r 2 > 4/Q 2, the integration region in z is limited by a2r 2 = QZr2z ( 1 - z) < 1. Because the integrand in (5) is symmetric under z ¢¢. l - z, one has f~ dz --* 2 f o d z where e is the solution to ar = 1 or x ~ = x. At small x, the free nucleon structure function can be written in terms of the virtual photon-nucleon cross section

F N ( x , Q 2) = Z

e2xfu(x'Q2) _ i

Q2

4,rr2ae~ o-( y* N) ,

( 11 )

where x f N ( x , Q 2) is the (sea) quark distribution function. Substituting (5) into the above equation and changing the integration variable z into x t in (5) dz = - x d x ' / (xaQ2r2), one obtains j

fN(X, Q2) = f N ( x , Q2) + ~

4/a~

~ 2 dx' f dr2 - ~ J --fi-~rqoU(X, r ) ,

3 X

(12)

4/Q2

where O'qgtUis given by (8). We have split the integration on r 2 into the region 4/Q 2 < r 2 < 4/Q2o where the perturbative calculation is valid and the region r 2 > 4/Q~ where the perturbation theory breaks down and the cross section is not calculable. The nonperturbative part is taken as initial condition f N ( x , Q 2) which should be determined by the experimental measurement. However, it is worthwhile to emphasize that because of the exponential cutoff of the large r 2 by Kj ( a r ) 2 in (5), the integration in r 2 is infrared safe. Even if one substitutes the small-r cross section into (12), the integral is only logarithmically divergent ~, In Q02, in contrast to the standard QCD infrared problem when Q2 ~ 0. This allows a smooth extrapolation of the "semihard" physics into the "soft" physics. Thus one can choose a rather small cutoff Q2 in order to integrate more perturbative contributions. Because of the infrared safety, the uncertainty due to the choice of Q02 is minimal. Substituting (8) into (12) and replacing 4/r 2 by Q~2, one recovers the familiar integrated form of the G L A P evolution equation at small x 1

Q2

1 I dx~ [ dQ~2 ,,-~2~.DLA X~ ~2 fN(X'O2)=fu(X'O2)+4--~a--~-J--~ast~ ,~N ( , e ). x

(13)

o;~

We note that (13) is not a full G L A P evolution equation since it does not involve the sea-quark contribution on the right-hand side. It is derived in the D L A where the small-x behavior is driven by the gluon contribution. To improve the larger x behavior, we insert a splitting function Pxq(Z) = z 2 + (1 - z) 2 where z = x / x ' in (13) and add the valence quark contributions to F~

Z. Huanget al./Nuclear PhysicsA 637 (1998) 79-106

88

FN(x,Q 2) : y ] e ~ [ x f N ( X , Q z) + xv,(x, Q2] , i

(14)

where xv's are the valence quark distributions, which are given by the GRV [17] parameterization (there is a very little difference among different parameterizations for valence quarks).

3.2. Scaling violation of F~ ( x, Q2) and gluon density The Q2 evolution or the scaling violation of the nucleon gluon density in the small-x region is quite different from that in the large-x region in the context of GLAP equation. In the region of small x, one has to sum up large asln 1/x contributions in addition to as In Q2 terms represented by the ladder diagrams with strong scale ordering in the leading logarithm approximation (LLA). To avoid summing up two large logarithms, one considers the DLA, i.e., the most leading diagrams involving (aslnQ21nl/x) n. Such a double logarithm appears only when vector particles (gluons) propagate in the t-channel of the ladder diagram as argued by Gribov, Levin and Ryskin [6]. Thus, the scale dependence of the nucleon quark distribution is directly related the small-x behavior of the gluon distribution. The DLA is valid in the kinematical region where as ln 1/xlnQ2/Q 2 ~ 1 while cesln l / x << 1 and aslnQ2/Q~ << 1 as well as as << 1. The precise validity range of DLA or of GLAP equation is currently still under debate. The general belief is that it describes a region up to a moderately large Q2, Q2 ~ 20 GeV 2, and a moderately small x, 10 -5 < x < 10 -2. In the small-x region the resummation of the gluon ladder diagrams has to be included [20]. For large-x, one needs to resum the leading-log as In Q2/Q~ terms as well, Recently it has been shown that the gluon density obtained in DLA with the resummed leading-log terms gives very good description of the double asymptotic scaling observed at HERA [23]. In the region of small-x and large Q2, the nucleon gluon density in DLA has the approximate solution [6],

xg DLA = XgN(X, QZ)lo(y),

(15)

where I0 is the Bessel function and y=2c

In

In

(') ~o

'

(16)

with t ~ In Q2/A2 and c = x/~6-/25, provided that input xgN(x, Q~) is not singular in x. The asymptotic behavior of lo(y) is that Io(y) ~ 1 as y --+ 0 and Io(y) ~ e Y / 2 x / ~ as y --~ cx~. That is, in this DLA, xg~ increases faster than any power of In 1Ix, but slower than a power of 1Ix. To improve the DLA at large Q2, Q2 > 20 GeV 2, the resummation of single logarithm, c~s ln Q2/Q~, is necessary. This results in additional Q2 dependence of the gluon distribution, a factor (t/to) -8 in (15) where 6 = 61/45 [23]. We find that this modification has very little numerical effect on the nuclear shadowing ratio in the region of interest.

Z. Huang et aL/Nuclear Physics A 637 (1998) 79-106

89

Motivated by a semiclassical expression xglv = x - ' ° ( Q Z ) L we define the exponent and ~o = OxgN/~ l n ( I / x ) . In the DLA

y = ~ ln xgu/~t

In 1/x

3/= tint/toOg.

(17)

The anomalous dimension can be large in the small-x and -t region. As t ~ to, w --~ 0 and y is finite. It is important to emphasize that the DLA form (15) is only valid when Q2 >> Q~. Our goal is to pin down the gluon density in the nucleon first. In our approach, we use the slope of F~, dF~/d In Q2, to determine the magnitude of the gluon distribution. We require that the gluon distribution in the DLA as given in (15) describes reasonably well the scaling violation of F2N for Q2 ranging from 0.85 to 100 GeV 2 at x = 0.00008. The Q2 dependence for all other values of x is then uniquely determined in the DLA. There are two unknown quantities in (15): Q~ and the initial condition XgN(X, Q2). Q~ determines where the perturbative calculation should begin to be valid and where the non-perturbative contribution can be separated into an initial condition xgN (x, Q~). We would like to have a reliable perturbative description of the structure function in the range from 1 to a few GeV which is extremely important in understanding the evolution of nuclear shadowing. This results in Q~ being rather small so that the DLA sets in at a relatively low Q2 = 1~2 GeV 2. Since xgN(x, Q2) has to be non-singular in the small x regime, we take xgN(X, Q~) = cgxg~vRV(x,Q~) at an initial scale Q~ since the initial gluon distribution in GRV is fairly flat while it has the proper behavior at x --+ 1. The initial condition for F~v is taken to be in the form motivated by a soft pomeron with a small intercept ce = 0.08

FN(x, Q~)

= c p x - ° ° 8 ( 1 - x) ~7,

(18)

where 7/ = 10 to account for the behavior as x ~ 1. We determine Q~, Cg and cp by requiring that the calculated Q2-dependence of F~v at x = 8 × 10 -5 fits the recent measurement from H1 Collaboration. We find the following unique solution Qo2 = 0.4,

Cg =

2.2,

cp = 0.2.

(19)

The Q2-dependence of F~ at other small values of x are then predicted using (13). In Fig. 2 we show our results for F u compared with the experimental data [26]. The gluon density in (15) describes the scale dependence of F~ very well even at relatively low Q2, Q2 ,,~ 0.85 GeV 2. The x-dependence of the nucleon structure function can also be predicted from (13). Fig. 3 shows the calculated FN(x,Q 2) from (13) as function of x at various Q2 as compared to H1 measured values [26]. The agreement is evident. Starting with a fairly flat distribution FzU(x,Q~), the nucleon structure function develops more and more singular shape in x from the perturbative evolution governed by the gluon distribution. Some deviation is visible for x > 10 -2, the region where we do not expect DLA to be valid.

90

Z. Huang et al./Nuclear Physics A 637 (1998) 79-106

10

[

I I I 1 ~

I

I 111111

./H

1~

X 0=1,9) 0.00008 - ~

I

I

I I Illll

H1 94&95

-- DLAgluon

-/

8 -o.ooo13

0.00020 - ~ -0.00032 - ~

o I

7

oo -I.

"0.00050

O

6 "0.00080

,

0.0013 LL

5 -0.0020 0.0080 4

0.013 I

30-1

I I IIIII1

100

I

I I Illlll

I

101 Q2(GeV2)

I I IIIIll

102

I

I I III

103

Fig. 2. The Q2-dependence of the nucleon structure function calculated using the DLA and compared with experiment data.

The results shown in Figs. 2 and 3 indicate that the gluon density in the DLA given in (15) faithfully captures the essential feature of Q2_ and x-evolution of nucleon structure function. The fact that we can describe F ( even at very low Q2, 0.85 < Q2 < 5 GeV 2 indicates that the perturbative approach is reliable in the semihard region.

3.3. Gluon distribution in the DLA Similarly to the quark case, one can consider the virtual gg pair interacting with the nucleons in the nuclear target. Using the well-known relationship between the gluon density and the deeply inelastic cross section [ 12], the gluon density can be written as [ 13] 1

x g u ( x , Q z) = - ~

dz

dZr[4,(z,r)lZo'~N(r) ,

(20)

o

where the wave function squared for a gluon pair is 10(z, r)12 - z(1 1 z~) [(aZKz(ar) - aK1 (ar)/r) 2 + aZKl (ar)2/r 2] ar<
(21)

and the cross section in the DLA is [25] ~r/-2

3 r2ot,t.41r2.xt DLA.x/ O'gu/v(r)=---~/ ) gN t , 4Jr2 / `).

(22)

Z Huang et al./Nuclear PhysicsA 637 (1998) 79-106

10

- I Illlllt I

-

I IIIIIII I

I

Q2=0.8 5 GeV 2 --

10

I Ilillll I

=_~- I

i ilillgj

I15-

1.2

--

I IIIIIlll I I ' i.*-IIm,q

==m=,l I IH.,,I ~ orm, r ~ '.'",r

I IIIIIIli r lll,,r

.

. %%

IIIIIIII I:::IIIII I '''''''l

.

I tlllllfi l irHliq

I ,H.,r

i ]~.l~

, I,I.

1.5

III I IIIIIIli ii~--llJi,, l

%%

,,,,,,i ,,,,,,J

,,,,,,,I

20 %%

I I[llltll

I II

,,,,,,,f

::

,,,,,,d

25

._

•

I t lifmli

8.5 %%

,,,,,,,I

%

I IIIIIlli I Illillll I II r lll~lll I Irlrlr, I r i ;

::

.

_.

15

I Illlllll

I IIIIIll I

::

5

::

3.5

%%,

=lml. I

DLA gluon

I IJm,I - f ]m,q

10

ll.I, I

91

%

%

-I Illlllll

I Iltlllil

I IIIIIIII

I i

I IIIlllll

I Ilfllll]

I IIIIIlll

I I

10 -5 10 -4 10 -3 10-210-5 10 -4 10 -3 10-210-5 10 ~ 10 -3 10 -2 X

X

X

Fig. 3. The x-dependence of the nucleon structure function calculated using the DLA and compared with the experiment data I26]. Note O'guN = 90"q~/4. Using (21) and (22) Eq. (20) can be written as 1

gN(X,Q 2) =gN(X,Q~) + ~

Q2

-Q,2 - c e ,~,~ ~t3'2)

--~ x

6

g~LA(x',Q'2)

(23)

og

a form in which the small-x limit splitting function in the DLA can be easily identified: Pe~,(z) ~ 6/z where z = x / x ~. Comparing (23) with (13), we note that due to the different quark and gluon splitting functions, as Qe increases the gluon density increases twelve times faster than the quark density, leading to a much stronger scaling violation. Similar to the quark case, (23) is not to be interpreted as the integral equation since the

92

Z. Huang et al./Nuclear Physics A 637 (1998) 79-106

right-hand side does not contain gN but g~LA. Namely, ~LA is the gluon density seen by a gg pair with a frozen transverse separation while gN is the gluon density seen by the probe, e.g., the virtual Higgs boson. However, since the integration kernel in (23) coincides with that of the DLA evolution equation, gN should be very close to gDLA. We confirm that numerically they differ by only few percent.

4. Glauber-Gribov multiple-scattering model The Glauber-Gribov multiple-scattering theory [ 18 ] treats nuclear collision as a succession of collisions of the probe with individual nucleons within nucleus. A partonic system (h), being a q~/ or a gg fluctuation, can scatter coherently from several or all nucleons during its passage through the target nucleus. The interference between the multiple scattering amplitudes causes a reduction of the hA cross section compared to the naive scaling result of A times the respective hN cross section, the origin of the nuclear shadowing. In the high-energy eikonal limit, the phase shift of the elastic hA scattering amplitude is just the sum of the individual hN scatterings. The total hA cross section is determined by the imaginary part of the forward elastic scattering amplitude as governed by the unitarity relation or the optical theorem. The unitarity of the total cross section is thus built-in within the Glauber-Gribov theory. In the high-energy eikonal limit where the projectile's momentum k is much larger than the inverse of the interaction radius, one describes the scattering amplitude by the Fourier transformation of the so-called profile function F(b) in the impact parameter space

f ( k , k') = ~ik f d2bei(k-k')bF(b),

(24)

where k' is the outgoing wave momentum. In the high energy scattering process, the collision is almost completely absorptive corresponding to an approximately real F(b) and pure imaginary amplitude. The total cross section is given by the optical theorem 4~'Im f ( 0 ) = 2 f d2bF(b). o'tot = ---if-

(25)

To extend these results to nuclei, one assumes that each collision with the nucleon modifies the piece of the incident wave passing through it by a factor 1 - FN(b - s) where s is the transverse position of the nucleon. The overall modification of the wave is thus given by A

H [ 1 - Flv(b - si) ]

.

(26)

i=1

Clearly, only those nucleons in the vicinity of impact parameter b can influence FA. The nucleus profile function is just the factor in (26) multiplied by the probability of finding nucleons in the positions rj . . . . . rA

Z. Huang et aL/Nuclear Physics A 637 (1998) 79-106

93

A

[~ (rl . . . . .

rA

)l 2 = ~

p(ri) ,

(27)

i=1 where p ( r i ) d 3 r i is the probability of finding a nucleon at ri. The nucleon number density is then p A ( r ) = ~ i P ( r i ) . For large A the nucleus profile function is given by A g

FA(b)

1 -- ] - I / d 3 r i p ( r i ) [ 1

- F N ( b - si)]

i=1 d A

P

= 1 -- H [ 1 - - / d 3 r i p ( r i ) F N ( b

= 1-

- si)]

d

i=1

1 - -~

d2sdZpA(s,z)Fu(b--s)

~-l-exp[-fdZsTA(S)F~(b-s)

(28)

1 ,

where we have used A >> 1. We shall use Gaussian parametrization of the nucleus thickness function defined by TA(b) -- f d z p a ( b , z )

_ eA A = _erR2

-~

,"a, h2/~2

(29)

where the mean nucleus radius R2A= 0.4R2ws and Rws = 1.3A U3 fro. Since the nucleon has a smaller transverse size than the typical nuclear dimension, one approximates the nucleon profile function in (28) by F N ( b - s ) ~_ O'hN~2(b - S ) / 2 and obtains the total h A cross section O'hA = /

(30)

d 2 b 2 [l -- e-tr"uTA[b)/2] .

For a Gaussian parametrization of TA(b), the integration on the impact parameter can be done exactly leading to 1 2

O'hA = 27rR2A Kh -- ~K h + . . . + (

_l)n-I

-- 27"rR~ [y + In Kh + El (Kh)] ,

~h nn! + . .

l (31)

where t
Z. Huang et al./Nuclear Physics A 637 (1998) 79-106

94

approached. The shadowing effect is determined by the strength of Kh, which is different for a h = qF/than for a h = gg case. So far we have sketched the necessary assumptions of applying the Glauber-Gribov theory for the elastic or the quasielastic scattering, that is, the longitudinal momentum remains the same during the passage of h through the nucleus. This condition can be relaxed to the forward production processes where there is a longitudinal momentum transfer appropriate to the q~ or the gg pair production. In this case, the scattering amplitude in (24) is modified to include the integration of the longitudinal position z due to the fact that an incident w a v e e ik~z for the photon 3, is changed to a final wave - F ( b ) e it~z for particle h. In general, k~ :~ k), and there is a phase difference between the 3' and h waves behind the target, i.e., e iak'~z where Akz = krz - k h. This implies that the coherence length cutoff is necessary for lc ~- l / A k z . If Ic > RA, the hadronic fluctuation h interacts coherently with all nucleons within the nucleus and the Glauber-Gribov theory is valid. On the other hand, if Ic < RA, h interacts coherently only with some of the nucleons and the interference effect is reduced. To estimate the coherence length l~, one calculates the change of the longitudinal momenta in the smali-x region

Ak z = ~

+ Qz _ V/~u2 _ M 2 ~_ Q2ZM+ 2v

= (x + x')m,

(32)

where M is the invariant mass of the q~ or the gg pair as defined in (10). Since the cross section is dominated by x' near x, the coherence length lc ~ 1 / ( 2 x m ) exceeds the radius of heavy nuclei RA ,-~ 4 fm as long as x < 2 × 10 -2 and the Glauber-Gribov multiple scattering model is adequate. For x > 2 × 10 -2, the Glauber-Gribov approximation breaks down and one should take into account the finite coherence length. Although we are mainly concerned with the small-x behavior of the nuclear shadowing, we would like to get some handle on the coherence length cutoff in the moderately large-x region. We propose to take a probability approach [ 13,27] and introduce a coherence length form factor ~-(Akz) = e x p [ - R ~ ( z a k z ) 2 / 4 ] [13] while preserving the nice structure of the Glauber-Gribov formula. The probability for a double scattering is reduced by ~', and that for a n-scattering is reduced by ~ - 1 so that one may make the following replacement in (30)

T

As r ~ 1, the original Glauber-Gribov formula is preserved, while as r ---+ 0, (33) gives o'ha = A O ' h N , i.e., there is no shadowing. In the kinematical region x < 10 -2, the coherence length cutoff does not play a role. Even for x > 10 -2, because the cross section becomes small, the effect is not very large.

Z Huang et al./Nuclear Physics A 637 (1998) 79-106

95

5. Nuclear quark distribution F A We now examine the effect of the gluon density in (15) on the shadowing of the nuclear structure function. Using (11) and with o-(y*A) calculated in the GlauberGribov multiple scattering model, 1

~r(~'*A)=fd2bfdz o

f d r IO(z, r)122( 1 -

e -~q°u(r'z)Ta(b)/2)

,

(34)

one obtains for (sea)quarks 1

Q2

x f a ( x , a 2) =xfA(x,a 2) + 3R 87r2axfax' j x,2 f x

dQ'2-~1 [ r + l n ( ~ : q r )

+E~(x~r)] ,

(35)

00~

where the effective number of scatterings x is given

2Art O~s(Qe)xgOvLA(x ' Q2) Kq(X, Q2) _ 3R~Q z

(36)

Expanding [ . . . ] in (35) and (42) and keeping only the first term which is proportional to xq, one recovers the result for the nucleon in the same DLA approximation. The nuclear structure function F2A(x,Q2) is defined through the total y*A cross section, similar to F ~ in (11). To improve the large-x behavior, we add to F A the valence quark contributions and assume that there is no shadowing for the valence quarks

F~(x,

e2[Xfa(X'

02) = Z

02) +

Axvi(x,

Q2].

(37)

i

In principle, the initial condition xfA(X,Q~) in (35) may be calculated in the vectormeson-dominance (VMD) model. Fortunately, there have been some experimental measurements on the nuclear quark shadowing in the low-Q 2 region so that the initial condition can be expressed in terms of the production of the initial nucleon structure function and the shadowing ratio Rq(x)

F~(x,Q 2) = ARq(x)FN(x,Q~),

(38)

where FN(x,Q~) is given in (18). Recently the E665 Collaboration has measured the nuclear shadowing effect in muon scatterings off 4°Ca and 2°8pb at low x [2]. In general, the small x data are measured at low Q2, e.g., for x = 6 × 1 0 - 4 the average (Q2) = 0.26 GeV 2 while x = 6 × 10 -3 corresponds t o ( Q 2 ) = 1.35 GeV 2. We parametrize the initial shadowing ratio Rq(x) at Q~ = 0.4 GeV 2 and use (37) to calculate the nuclear structure function at the measured (Q2) values at different x values. The nuclear shadowing ratio is defined by

Z. Huanget al./Nuclear PhysicsA 637 (1998) 79-106

96

2oepb 1.1 1.0

,~. I

~

~ ~ m

I

,

4°Ca

~ ~ H

I

[

III I

' ~I

I

I I II1111

I

I I IIIII I

I

I I' -q

/

_

Q2 (GeV 2)

0.9 0.8

0.7 -

0.6

1.1

IIIII

llll I

I

I

I Iltllll I IIIIII I

I I

I 1111111

I III1111

I

I

"7

I li lllll IIII

I I

I

I I

I I ILIII[ I I IIIll]

I I

I I IIII1[ I I IIIII I

I I

I 17 I I

1.0 0.9

-

Z

0.8 LL 0.7

E665 '95 2 Evolved to =p

[3

0.6

-

"3

0,5

lllll

10 -4

I

I I IIIIII

I

10 4

l I IIIII[

10-2 x

I

I I~

IIlI

104

I

I I IIIIII

I

10 4

I I Iltlll

I

I I

10.2 x

Fig. 4. The x-dependence of nuclear quark shadowing ratio Rq = FA/(AFN) at various Q2 for 2°spb and 4°Ca. The lower panels represent the parameterizations of initial shadowing ratios evolved to the experimental (Q2)cxp values. The non-saturation of Rq(x) as X decreases at large Q2 is the feature of the perturbative shadowing mechanism.

FA(x,Q 2) R°(x)F~(x,Q~) + AF:(x;Q2,Q 2) Rq(x,Q 2) = AF2~(x, Q2) = Fff(x, Q2) + AFff(x;Q2, Q2) ,

(39)

where zlF2 -- F2 - F2(Q 2) can be explicitly calculated by performing the double integration over x ~ and Q~2 in (35). The results for 4°Ca and 2°spb are shown in Fig. 4. The lower panels of Fig. 4 show our parametrizations of initial shadowing ratio at Qo2 evolved to the measured (Q2 / compared to the experimental data from E665. For (Q2} < Q2, we take it to be at the minimal scale Q2. As is evident from the comparison, our parametrizations faithfully represent the experimental data at small x. The shadowing ratio, Rq(x, a 2 ) , at high values of 02 are shown in the upper panels of Fig. 4. We note that at large Q2 the nuclear shadowing effect is reduced but does not diminish, i.e. it is not a higher twist effect. This can be understood as the interplay between the perturbative and the non-perturbative shadowing mechanisms. The nuclear quark density is initially shadowed by R°q at Q02 compared to the free nucleon case due to some non-

Z Huanget al./Nuclear PhysicsA 637 (1998) 79-106 1.0

.....

, -

-

.

GLAP

-

.

.

.

.

.

.

.

r

.

.

.

.

.

.

.

Evolution

Double Scattering --

.

97

Full G l a u b e r

I/-

/-

-

-

/

/

0.9 /

/,/"

/

_~ 0.8 oJ IJ.

/'

i/I

/'

ii I

0.7 . - J"

10 o

x=l 0 4

101

10 2

Q2 (GeV 2)

Fig. 5. The Q2-dependenceof nuclear quark shadowingratio Rq

=

FA/(AF2N) at x =

10 - 4

for 2°spb.

perturbative mechanism such as coherent scatterings of the p-meson as discussed in the VMD model [ 15]. As Q2 increases from Qo2, the nucleon structure function increases by zaFN(x; Q2, Q~) as governed by the GLAP evolution equation in the DLA. If the nuclear F z increases by the same amount (times A), i.e., we neglect the multiple scattering effect and evolve it according to the same GLAP equation, the initial shadowing effect in (39) decreases quickly due to the strong scaling violation at low Q2 (AFN(x; Q2, Q~) is not small compared to F~(x, Q2)). This is illustrated in Fig. 5 where the shadowing ratio is calculated by taking the first term in the expansion [ . . . ] in (35) and plotted as function of Q2 at a fixed x = 1 0 - 4 . The shadowing effect disappears quickly when both the nucleon and nuclear structure functions evolve according to the GLAP equation. However, because xg DLa increases rapidly as Q2 increases, the growth of the gluon density in a nucleon compensates for the suppression due to the smallness of the size of the qgl pair. As a result, Kq is not small in the semihard Q2 region. In fact, tCq increases as Q2 increases from Q~ to 2 GeV 2 and becomes greater than 1. The perturbative rescattering effect becomes important and needs to be taken into account in the Glauber-Gribov formalism. The effect of multiple scatterings slows down the Q2 evolution of F z as compared to F~, i.e.,

AF2A(x; Q2, Q~) < AFt(x; 0 2, Q~) ,

(40)

which provides a perturbative shadowing mechanism beyond Q02. This is illustrated in Fig. 5 where the Q2 evolution is indeed slower than the GLAP evolution case. As Q2 becomes even larger (Q2 > 5 GeV2), t
98

Z Huang et al./Nuclear PhysicsA 637 (1998) 79-106

shadowing ratio. Also plotted in Fig. 5 is the double scattering case where we take the first two terms in the expansion I 2 [Y + lnKq + EI(Kq)] ~-- Kq -- ~Kq.

(41)

AS it is evident from Fig. 5, the double scattering overestimates the amount of shadowing. Nevertheless, it accounts for most of the multiple scattering effect. The interplay between soft physics and the partonic picture of the nuclear shadowing is also evident in the x-dependence of Rq in Fig. 4. The apparent flatness of the shadowing ratio at low Q2 in the small-x region is altered by the perturbative evolution. This is due to the singular behavior of xg~vLA as x ~ 0 at large Q2 leading to the strong x-dependence of the effective number of scatterings, %(x). Thus, the saturation of the Rq as x decreases is a feature only at low Q2, Q2 < 2 GeV 2 where the gluon density is not so singular, and we do not expect the saturation to happen for higher values of Q2.

6. Predicting gluon shadowing In the previous section, we emphasized the importance of the interplay between the non-perturbative and perturbative shadowing mechanisms in order to predict the nuclear quark shadowing at large Q2. In principle, the prediction for the nuclear gluon shadowing could be uncertain because there is no experimental information on the gluon shadowing at low Q2. However, as we shall demonstrate below that this is not the case and the gluon shadowing can be unambiguously predicted for large Q2 in the context of perturbative QCD. The nuclear gluon distribution can be calculated in the Glauber-Gribov model similarly to the quarks [13] 1

Q2

xga(x'Q2) =xgA(X'Q2) + ~2R 2 / d x- '~ - jf dQ,21_[y+ln(Ke7-)+El(Kg7-)], 7"

(42)

The essential difference between the quark and gluon cases is different splitting functions and the effective number of scatterings, % = 9Kq/4 due to different color representations that the quark and the gluon belong to. These two effects combined result in the 12 times faster increase of the gluon density with Q2 than in the case of the sea quarks in the region of small x. The two important effects which make the gluon shadowing quite different from the quark shadowing are the stronger scaling violation in the semihard scale region and a larger perturbative shadowing effect. This can be seen by considering the shadowing ratio

Rg(x, QZ) = xgA(x, Q2) = xgu(x, Q~)R°(x) + AXgA(X; Q2, Q~) AXglV(X, QZ) xgu(x, Q2) + axgu(x; Q2, Q~) '

(43)

where R°g(X) is the initial shadowing ratio at Q0~ and Axg(x; Q2, Q2) is the change of the gluon distribution as the scale changes from Q02 to Q2. The strong scaling violation

Z Huang et al./Nuclear PhysicsA 637 (1998) 79-106

99

due to a larger Kg at small x causes dxgN(X;Q2,Q~) >> xglv(x,Q~) as Q2 is greater than 1~2 GeV 2. The dependence of Rg(x, Q2) on the initial condition R°(x) diminishes and the perturbative shadowing mechanism takes over, i.e.

Rg(x, Q2) __. ZIxgA(X;Q2, Q~) AXgN(X; Q2, Q~) ,

(44)

for Q2 > 2 GeV 2. The above statement can be made more rigorous by taking a partial derivative of (43) with respect to In Q2 at a fixed x ORg(x, Q2)

OlnQ 2

_y(QZ)[RgPT(x, Q 2 ) _ R g ( x , Q2)] ,

(45)

where the anomalous dimension y(Q2) and the so-called perturbative shadowing ratio RgPT are defined

y(QZ)

OlnxgN(X,Q 2) =

0 In Q2

PT= Oxga(x, Q2)/OlnQ z AcgxgN(x, Q2)/0 In Q2 "

Rg

'

(46)

Note that RPT is the ratio of the derivative of Xga to that of xgu, which is independent of the initial shadowing ratio and can be calculated perturbatively. In the small-x, R~ ~

T+lnKg+EI(Kg) Kg

.

1

.

t
u:~

--4"÷ . . 18

"'"

(47)

The interesting feature of (45) is the negative sign in front of Rg on the righthand side such that it is self-stabilizing. If Rg < R~ at Q2 and ORg/c9In Q2 > 0, the shadowing ratio increases with Q2; while for Rg < RT, the shadowing ratio decreases with Q2. As a result, the Q2-evolution of Rg is such that it is always in the process of approaching R~. Because RgPT also changes with Q2, the Q2-evolution of Rg never stops. The anomalous dimension y(Q2) determines how fast Re is approaching RgPT. Introduce an important scale Q~H = 3 GeV 2 ("SH" stands for "semihard') where y(QZH) ~-0(1). In the semihard region Qo2 < Q2 < Q~H, y(Q2) is large and Rg approaches RgPT(Q~H) fast regardless of what the initial shadowing one has to set at Q¢]. It is precisely in this region that Kg is large and (47) gives a large shadowing correction. As Q2 > QZH, the anomalous dimension .y(Q2) gets small and RgPT approaches 1 fast and the shadowing ratio Rg evolves slowly. Thus, the behavior of the QZ-evolution of the gluon shadowing can be understood by the interplay between the anomalous dimension of the nucleon gluon structure function and the perturbative shadowing mechanism. In Fig. 6 we show the Q2-behavior of T ( Q 2) and RgVr(Q2) at x = 10 - 4 for 2°gPb. The evolution of the shadowing ratio Ru with an initial condition R° = 0.5 is also plotted to illustrate the evolution governed by (45). To study the sensitivity of the shadowing ratio on the initial condition, one can solve (45) explicitly by making a transformation Ru = R'geXp[-fy(t')dt'] where t = lnQ 2. Then Re can be directly integrated out which yields

Z. Huang et al./Nuclear Physics A 637 (1998) 79-106 1.0 ..... I \ ........ i ........

100

0.9

\'\

0.8

Rg . .....................

'\

0.7

...............

/

"

O

\\

I

0.6 :.' ,..'

0.5

=

\ \ \ ~(Q2)

0.4 t

\'\'\-

0.3 0.2

100

10~ Q2 (GeV2)

102

Fig. 6. The Q2-dependenceof the anomalousdimension. y ( Q 2 ) and the perturbativeshadowingratio Rff (Q2) at x = 10 - 4 for 2°spb. Also plotted is the shadowing ratio Rg with an initial condition R~g= 0.5. t

Rg(t) =R~(to)e

+e

y( ff )efo Y(t")dt"RqPT ( tt) dff .

o

(48)

to

In the DLA, the integration of y ( t ) yields

xg LA(x, to)

1

- Xg~vLA(x,t) = Io(y) For large y, ~

(49)

~ 2 v f ~ e -y, 16, where y is defined in (16). This factor suppresses

the initial shadowing Rg(to) quickly for large t > tSH = In Q2H. The integration in (47) picks up most important contributions in the region to < t < tSH. Therefore, Rg(t) at large t is largely determined by the second term in (48), i.e., depends only on y and R ~ , especially in the small-x region. Fig. 7 demonstrates the stabilization of the gluon shadowing for any initial conditions as Q2 increases at various values of x for 2°spb. The shaded regions indicate the maximal uncertainty in predicting the gluon shadowing as they are bounded by two extreme initial conditions: R° = 1 and R°g = 0. One can almost uniquely determine the amount of the gluon shadowing for Q2 > 3 GeV 2. In general, the uncertainty gets bigger as x gets larger due to a weaker scaling violation, which is evident in Fig. 7. Neglecting the first term in (48) and approximating the perturbative shadowing by R~T = Ruvr (Qsn) 2 for Q2 < Q2 < QsH 2 and R ~ = 1 for Q2 > QsZn, one obtains from (48) the shadowing ratio for Q2 > Q2n

Z. Huang et al./Nuclear Physics A 637 (1998) 79-106

101

GLAP 0,8" z

~0.6×

n-"

0,4-

region

_ x = 1 0 -s

-

x=10 ~

z

,< X II _

X = 1 0 -3

10' 0 2 (GeV2)

_ x = 1 0 -2

I0" 10~

10' Q2 (GeV2)

Fig. 7. The Q2-dependence of nuclear quark shadowing ratio Rq = FA/(AF~) at x = 10 - 4 for 2°spb.

Rg(t)

= 1 - [1 - R~(tsH)] e -

fl. r't')at'

(50)

Comparing (50) with (48) we note that the solution (50) is the same as solution (48) with a new initial condition R° = RPT g r, ~,q2 s . J , Q2 replaced by Q2H and with R ~ = 1. The difference is that the evolution factor e x p [ - f ] no longer falls rapidly and Rg evolves rather slowly. Thus, the Q2-behavior of gluon shadowing at large Q2 can be roughly described by an initial condition at QszH followed by a GLAP evolution. The important point is that the initial condition at Q2 = Q2SH is pre-determined by the perturbative shadowing. Our main conclusion is that even though RgVr(Q2) approaches I quickly as Q2 increases beyond Q~. (a higher twist effect), the shadowing ratio at large Q2 is still governed by the perturbative shadowing at the semihard scale, namely R~(Q~H). The x-dependence of the gluon shadowing can also be predicted as long as Q2 > Qs2 where the influence of the initial condition is minimal. Fig. 8 shows the x-dependence of the gluon shadowing ratio at various Q2 values. The shaded region is bounded by the distributions calculated using two drastically different initial conditions Rg° = 0 and R°g(x) = 1. It is clear that the shape of the distribution is quite robust in the small-x

102

Z. Huang et al./Nuclear Physics A 637 (1998) 79-106

o,! ......

I'6"

.....

I'0"

x

J

...... I'

Fig. 8. The x-dependence of the gluon shadowing ratio at various Q2 values. The shaded region indicates the uncertainty due to the initial conditions.

region regardless of what initial conditions one may choose. Due to the perturbative nature of the shadowing, these distributions do not exhibit a saturation as x decreases. This is related to the singular behavior of the nucleon gluon distribution at large Q2 and small x. As long as the nucleon gluon density does not saturates at small x, the shadowing ratio does not saturate either. This is in accordance with the observation by Eskola, Qiu and Wang [ 8] in the context of the gluon recombination model. The rapid Q2-evolution of the gluon shadowing ratio in the low Q2 region found by Eskola [ 16] can be understood in our approach as the quick approach of the initial shadowing ratio to its perturbative value (see Fig. 7). In Eskola's approach some shadowing persists even at Q2 = 100 GeV 2 which can also be understood in our picture as the region of fully developed GLAP evolution. We note that multiple scatterings inside the nucleon are also possible in the region of very small x, where the eikonalization of the nucleon cross section is needed. However, in this region, the DLA breaks down and a BFKL resummation is necessary.

7. Giuon shadowing at a fixed impact parameter So far we have confined ourselves to the impact parameter averaged nuclear structure function. However, in the binary approximation of hard scattering processes in pA and AA collisions, one has to calculate the parton cross section at the nucleon-nucleon level at each impact parameter. In addition, to enhance the possible signal of a quark-gluon plasma in heavy-ion collisions, one triggers on central events where the impact parameter

Z. Huanget al./NuclearPhysicsA 637 (1998)79-106

103

is small (e.g. b < 2 fm). It is thus necessary to study the impact parameter dependence of the nuclear shadowing effect. Intuitively, one expects that the nuclear shadowing effect should depend on the local nuclear thickness at a given impact parameter. In the Glauber-Gribov approach, the effective number of scatterings is a product of the cross section and the nuclear thickness function. Since the shadowing is a non-linear effect in the effective number of scatterings, the impact parameter dependent shadowing ratio cannot be factorized into a product of an average shadowing ratio and the nuclear thickness function, as has been done in previous works [28]. The impact parameter dependent nuclear structure function X~A(X, Q2; b) is defined as

XgA(X, 02) = / d 2 b x ~ a ( x , Q2; b).

(51)

Comparing (51) with (30), one finds 1

4

Q2

Xga(x'a2;b) :xga(x'a2;b) + -~ S ~#x jf dat22 [1--e-~r~'NTa
(52)

where XDA(X, Q~; b) is the initial condition at a fixed impact parameter b. As shown in the previous section, the prediction for gluon shadowing ratio at Q2 > 3 GeV 2 does not depend on the initial condition. Thus, as an illustration we take the initial shadowing ratio to be 1 for all b's. The impact parameter dependent shadowing ratio is thus defined

Rg(x, Q2;b) = xga(x,Q 2;b) TA(b)XgN(X, Q2) •

(53)

In Fig. 9 we show the gluon shadowing ratio at various impact parameters. The upper panel is the shadowing ratio as function of b at a fixed x = 10 -4 and a fixed Q2 = 10 GeV 2. The parton density is more suppressed at a small impact parameter. The x-dependence of the impact parameter dependent shadowing ratio is also shown in the lower panel of Fig. 9.

8. Conclusions

We have studied the nuclear shadowing mechanism in the context of perturbative QCD and the Glauber-Gribov multiple scattering model. The nuclear shadowing phenomenon is a consequence of the parton coherent multiple scatterings. While the quark density shadowing arises from an interplay between "soft" physics and the semihard QCD process, the gluon shadowing is largely driven by a perturbative shadowing mechanism due to the strong scaling violation in the small-x region. The gluon shadowing is thus a robust phenomenon at large Q2 and can be unambiguously predicted by perturbative QCD.

104

Z. Huang et al./Nuclear Physics A 637 (1998) 79-106

1.0

'

I

'

I

'

I

'

I

'

'

I

'

I

'

~

2°sPb

0.9

I

.-9.0.8 Average //10_4 "o ~ Q2=10GeV2 - o.7 n" RA 0.6 0.5

I

0

~

....

,.,

°.

"o

0.8

I

~

I

,

I

,~

'"'1

........

I

'

rr 0.6 0.5 0.4

•

i

I

t

' ''""1

'..

~

"

-t ?bv6rfgeJ

J,-,"

, ,'

_

- - - b=3 fm - - - b=0fm

,

/'

/

/ ........

0 -s

I

.......'"""

~ ~

07

,

1 2 3 4 5 6 7 6 Impact Parameter b (fro)

~Pb

Q2=10 GeV 2 I

10"

........

I

10-3

........

I

10-2

......

10-1

Fig. 9. The impact parameter dependent gluon shadowing ratio calculated assuming R~g(b) = 1. The upper panel is the shadowing ratio as function of b at x = 10 - 4 and Q2 = 10 GeV 2. Tle lower panel shows the x-dependence of the shadowing ratio at Q2 = 10 GeV 2 and various values of b.

The scale dependence of the nucleon structure function was used to put a stringent constraint on the gluon density inside the nucleon in the double logarithm approximation. It is shown that the strong scaling violation of the nucleon structure function in the semihard momentum transfer region 0.8 < Q2 < 3 GeV 2 at small x can be reliably described by perturbative QCD and is a central key to the understanding of the scale dependence of the nuclear shadowing effect. We have also studied the impact parameter dependence of gluon shadowing and shown that it is a non-linear effect in the nuclear thickness function. It is important to correctly incorporate the impact parameter dependence of the nuclear structure function when one calculates the QCD processes in the central nucleus collisions.

Z. Huang et al./Nuclear Physics A 637 (1998) 79-106

105

Acknowledgements This work was supported through the U.S. Department of Energy under Contracts Nos. DE-FG03-93ER40792 and DE-FG03-85ER40213.

References 111 NMC Collaboration, P. Amaudruz et al., Z. Phys. C 51 (1991) 387; Z. Phys. C 53 (1992) 73; Phys. Lett. B 295 (1992) 195; NMC Collaboration, M. Ameodo et al., Nucl. Phys. B 441 (1995) 12. 121 E665 Collaboration, M.R. Adams et al., Phys. Rev. Lett. 68 (1992) 3266; Phys. Lett. B 287 (1992) 375; Z. Phys. C 67 (1995) 403. [3] D. Alde et al., Phys. Rev. Lett. 66 (1991) 133. 141 S. Gavin and J. Milana, Phys. Rev. Lett. 68 (1992) 1834; C.J. Benesh, J. Qiu and J.P. Vary, Phys. Rev. C 50 (1994) 1015. 151 B. Badelek and J. Kwiecifiski, Rev. Mod. Phys. 64 (1992) 927; M. Arneodo, Phys. Rep. 240 (1994) 301. 161 L.V. Gribov, E.M. Levin and M.G. Ryskin, Phys. Rep. 100 (1983) 1; E. Laenen and E. Levin, Nucl. Phys. B 451 (1995) 207. 171 A.H. Mueller and J. Qiu, Nucl. Phys. B 268 (1986) 427; J. Qiu, Nucl. Phys. B 291 (1987) 746. 181 K.J. Eskola, J. Qiu and X.-N. Wang, Phys. Rev. Lett. 72 (1994) 36. [91 L. Frankfurt and M. Strikman, Nucl. Phys. B 316 (1989) 340; L. Frankfurt, M. Strikman and S. Liuti, Phys. Rev. Lett. 65 (1990) 1725. 101 S.J. Brodsky and H.J. Lu, Phys. Rev. Lett. 64 (1990) 1342. I 1 I N.N. Nikolaev, Z. Phys. C 32 (1986) 537; N.N. Nikolaev and B.G. Zakharov, Phys. Lett. B 260 (1991) 414; Z. Phys. C 49 (1991) 607. 121 A.H. Mueller, Nucl. Phys. B 335 (1990) 115. 131 A.L. Ayala F, M.B. Gay Ducati and E.M. Levin, Nucl. Phys. B 493 (1997) 305. 141 V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 438; G. Altarelli and G. Parisi, Nucl. Phys. B 126 (1977) 298; Yu.L. Dokshitzer, Sov. Phys. JETP 46 (1977) 641. 151 see e.g., G. Piller, W. Ratzka and W. Weise, Z. Phys. A 352 (1995) 427; T.H. Bauer, R.D. Spital and D.R. Yennie and F.M. Pipkin, Rev. Mod. Phys. 50 (1978) 261. 161 K.J. Eskola, Nucl. Phys. B 400 (1993) 240. 171 A.D. Martin, R.G. Roberts and W.J. Stirling, Phys. Lett. B 387 (1996) 419; H.L. Lai, J. Huston, S. Kuhlmann, F. Olness, J. Owens, D. Soper, W.K. Tung and H. Weerts, Phys. Rev. D 55 (1997) 1280; M. Glueck, E. Reya and A. Vogt, Z. Phys. C 67 (1995) 433. 18] R.J. Glauber, in Lectures in Theoretical Physics, ed. W.E. Brittin et al. (Interscience Publishers, New York, 1959); R.J. Glauber and G. Matthiae, Nucl. Phys. B 21 (1970) 135; V.N. Gribov, Sov. Phys. JETP 29 (1969) 483. 191 G. Kerley and G. Shaw, Phys. Rev. D 56 (1997) 7291; D. Schildknecht, Acta Phys. Polonica B 28 (1997) 2453. 1201 E.A. Kuraev, L.N. Lipatov and V.S. Fadin, Sov. Phys. JETP 45 (1977) 199; Ya.Ya. Balitskii and L.V. Lipatov, Sov. J. Nucl. Phys. 28 (1978) 822; L.N. Lipatov, Sov. Phys. JETP 63 (1986) 904. [211 L. McLerran and R. Venugopalan, Phys. Rev. D 49 (1994) 2233; A. Kovner, L. McLerran and H. Weigert, Phys. Rev. D 52 (1995) 6231, 3809. 1221 A.J. Brodsky and A.H. Mueller, Phys. Lett. B 206 (1988) 685; G. Farrar, L. Frankfurt, M. Strikman and H. Liu, Phys. Rev. Lett. 64 (1990) 1996.

106

Z Huang et al./Nuclear Physics A 637 (1998) 79-106

1231 For recent studies, see R.K. Ellis, Z. Kunszt and E.M. Levin, Nucl. Phys. B 420 (1994) 517; R.D. Ball and S. Forte, Phys. Lett. B 335 (1994) 77. [241 N. N. Nikolaev and B.G. Zakharov, Z. Phys. C 64 (1994) 631. [25] V. Barone, M. Genovese, N.N. Nikolaev, E. Predazzi and B.G. Zakharov, Phys. Lett. B 326 (1994) 161; N.N. Nikolaev and B.G. Zakharov, Z. Phys. C 64 (1994) 631. [261 HI Collaboration, T. Ahmed et al., Nucl. Phys. B 470 (1996) 3; HI Collaboration, C. Adloff et al., DESY-97-042, e-Print Archive: hep-ex/9703012. [27] B. Kopeliovich and B. Povh, Phys. Lett. B 367 (1996) 329. 128] K.J. Eskola, Z. Phys. C 5l (1991) 633.