Different approaches to gluon shadowing at small x are reviewed. Some
available results relevant for RHIC and LHC are compared.Comment: 6 pages, LaTeX2e, uses enclosed cernrep.cls, one eps figure enclosed
  using graphicx, contribution to the Yellow Report on Hard Probes in Heavy Ion
  Collisions at the LH

Armesto, N.

Salgado, C. A.

English

arXiv

Different approaches to gluon shadowing at small x are reviewed. Some available results relevant for RHIC and LHC are compared.Different approaches to gluon shadowing at small x are reviewed. Some available results relevant for RHIC and LHC are compared

Salgado, C.A.

CERN Document Server

Gluon distributions in nuclei at small x: guidance from different modelsN. Armestoa,b and C. A. Salgadoba Departamento de Fı´sica, Universidad de Co´rdoba, Spainb Theory Division, CERN, SwitzerlandThe difference between the structure functions measured in nucleons and nuclei [1] is a veryimportant and well known feature of the study of nuclear structure and nuclear collisions. At smallvalues of the Bjorken variable x (< 0:01, shadowing region), the structure function F2 per nucleonturns out to be smaller in nuclei than in a free nucleon. This shadowing corresponds to a shadowing ofparton densities in nuclei. While at small x valence quarks are of little importance and the behaviour ofthe sea is expected to follow that of FA2 , the gluon distribution, which is not an observable quantity, isbadly determined and represents one of the largest uncertainties in computation of cross sections bothfor moderate and large scales Q2 in collinear factorization [2]. For example, the uncertainty in thedetermination of the glue for Pb at Q2 ∼ 5 GeV2 at LHC for y = 0, x ∼ mT =√s ∼ 10−4 ÷ 10−3, is afactor ∼ 3 (see Fig. 1), which for the corresponding cross section in PbPb collisions results in a factor∼ 9.In this situation and while waiting for new experimental data to come from lepton-ion [3, 4, 5] orpA colliders, the guidance from different theoretical models is of uttermost importance to perform safeextrapolations from the region where experimental data exist to those interesting for LHC studies. Twodifferent approaches to the problem have been essayed: On the one hand, there exist models which try toexplain the origin of shadowing, usually in terms of multiple scattering (in the frame where the nucleus isat rest) or parton interactions in the nucleus wave function (in the frame in which the nucleus is movingfast). On the other hand, other models parameterize parton densities inside the nucleus at some scale Q20large enough for perturbative QCD to be applied reliably, and then evolve these parton densities usingthe DGLAP [6, 7, 8] evolution equations; in this way, the origin of the differences of partons densitiesin nucleons with respect to nuclei is not addressed, but contained in the parameterization at Q20 which isobtained from a fit to experimental data.1. Multiple scattering and saturation modelsThe nature of shadowing is well understood qualitatively: In the rest frame of the nucleus, the incomingphoton splits, at small enough x, into a qq¯ pair long before reaching the nucleus, with a coherence lengthlc ∝ 1=(mNx) with mN the nucleon mass, which at small enough x becomes of the order of or greaterthan the nuclear size. Thus this qq¯ pair interacts coherently with the nucleus with typical hadronic crosssections, which results in absorption [9, 10, 11, 12, 13] (see [14] for a simple geometrical approach in thisframework). In this way nuclear shadowing is a consequence of multiple scattering and is thus relatedwith diffraction (see e.g. [15, 16]).Multiple scattering is usually formulated in the dipole model [13, 17], which is equivalent to kT -factorization [18] at leading order. In this framework the γ-nucleus cross section is expressed throughthe convolution of the probability of the transversal or longitudinal γ to split into a qq¯ pair of transversedimension r times the cross section for scattering of such dipole with the nucleus. It is this dipole-nucleus cross section at fixed impact parameter which saturates (i.e. gets a maximum value allowed byunitarity), most frequently by multiple scattering in the Glauber-Gribov approach [19, 20, 21, 16]. Thisdipole-target cross section is related through a Bessel-Fourier transform to the so-called unintegratedgluon distribution ’A(x; kT ) in kT -factorization [22, 19, 20], which in turn can be related to the usualcollinear gluon density throughxGA(x;Q2) =∫ Q2Λ2dk2T ’A(x; kT ) (1)(with Λ2 some infrared cut-off, if required) although this identification is only true for large Q2 Q2s [23, 24], with Q2s the saturation momentum corresponding to a transverse length scale where thesaturation of the dipole-target cross section occurs.Other formulations of multiple scattering do not use the dipole formulation but relate shadowingwith diffraction [15, 16] by Gribov theory. In this way diffraction in lepton scattering on nucleons isrelated to nuclear shadowing.An equivalent explanation to multiple scattering in the frame in which the nucleus is movingfast, is that gluon recombination due to the overlap of the gluon clouds from different nucleons, makesgluon density in nucleus with mass number A smaller than A times that in a free nucleon [25, 26]:at small x the interaction develops over longitudinal distances z ∼ 1=(mNx) which become of theorder of or larger than the nuclear size, leading to the overlap of gluon clouds from different nucleonslocated in a transverse area ∼ 1=Q2. These studies have received great theoretical impulse with thedevelopment of semiclassical ideas in QCD and the appearance of non-linear equations for evolutionin x in this framework (see [27, 28] and references therein), although saturation appears to be differentfrom shadowing [23, 24] (i.e. the reduction in the number of gluons as defined in DIS is not apparent).In the semiclassical framework the gluon field at saturation reaches a maximum value and becomesproportional to the inverse QCD coupling constant, gluon correlations are absent and a form for thedipole-target cross section appears which leads to a geometrical scaling of γ-target cross section; suchscaling has been found in small x nucleon data [29], and also in analytical and numerical solutions ofthe non-linear equations [30, 31, 32, 33], but apparently the region where this scaling should be seenin lepton-nucleus collisions has not been reached yet in available experimental data [34]. Apart fromeA colliders [3, 4, 5], LHC will be the place to look for non-linear effects. However, the consequencesof such high density configurations may be masked in AB collisions by other effects, as those due tofinal state interactions. pA collisions would be, thus, essential for this type of studies (see e.g. [35]).Moreover, they would be required to fix the baseline for other studies in AB collisions as QGP searchand characterisation. As a final comment, let us indicate that the non-linear terms in [25, 26] are of ahigher-twist nature and in this case the low density limit recovers the DGLAP equations [6, 7, 8], whilethe non-linear equations for evolution in x [27, 28] do not correspond to any definite twist and their linearlimit correspond to the BFKL equation [36, 37].Let us comment a little more on the difference between shadowing for gluons and saturation. Insaturation models [27, 28] saturation, defined by a maximum value of the gluon field or by the scatteringbecoming black, and shadowing for gluons, defined by the ratio xGA=(xGN ) being < 1, are apparentlydifferent phenomena, i.e. saturation does not necessarily lead to shadowing for gluons [23, 24]. Indeed,in the framework of numerical studies of the non-linear equations for small x evolution, the unintegratedgluon distribution turns out to be a universal function of just one variable  = k2T =Q2s [30, 31, 32] (in [33]this universality is analytically shown to be fulfilled up to k2T much larger than Q2s), vanishing quicklyfor k2T > Q2s; this scaling appears also in the analysis of DIS experimental data on nucleons [29] and hasbeen searched for in nuclear data [34]. In nuclei, Q2s increases with increasing nuclear size, centralityand energy. So, through the relation with the collinear gluon given by Eq. (1), this scaling implies thatthe integral gives the same value (up to logarithmic corrections if a perturbative tail ∝ 1=k2T exists forlarge k2T  Q2s), provided that the upper limit of the integration domain Q2  Q2s. Thus the ratioxGA=(xGN ) keeps equal to 1 (or approaching 1 as a ratio of logs if the perturbative tail exits) with xfor Q2  Q2s. So saturation or non-linear evolution do not automatically guarantee the existence ofshadowing for gluons. As shadowing is in many models described in terms of multiple scattering, thisresult [23, 24] looks somewhat surprising.2. DGLAP evolution modelsOn the other hand, a different approach is taken in [38, 39, 40, 41]: parton densities inside the nucleusare parameterized at some scale Q20 ∼ 1÷5 GeV2 and then evolved using the DGLAP [6, 7, 8] evolutionequations. In this way, all nuclear effects on parton densities are included in the parameterization atQ20, which is obtained from a fit to experimental data. The differences in the approaches come mainlyfrom the sets of data used (e.g. the use of Drell-Yan data or not) to constrain the parton distributions. Inthese calculations the lack of experimental data makes the gluon to be badly constrained at very smallx; experimental data on the evolution of FA2 with log Q2 give direct constrains [42] through DGLAPevolution but only for x ≥ 0:01. Please refer to [42] and references therein for more discussions on thiskind of models.3. Comparison between different approachesThe results from different models usually depend on additional semiphenomenological assumptions andoften contradict each other. For example, concerning the Q2-dependence of the effect, in [9, 10, 11,12, 13] it is argued that qq¯ configurations of a large dimension give the dominant contribution to theabsorption, which results essentially independent of Q2. On the other hand, in the gluon recombinationapproach of [25, 26] the absorption is obtained as a clear higher-twist effect dying out at large Q2. Finally,in the models [38, 39, 40, 41] which use DGLAP, all Q2-dependence comes from QCD evolution andis thus of a logarithmic, leading-twist nature. Predictions (particularly for the gluon density) on thex-evolution towards small x turn out to be very different.Let us compare between different approaches whose numerical results are available (when con-sidering an approach based on the dipole model as in [19], one should keep in mind the difficulties toidentify at small and moderate Q2 the integral of the unintegrated gluon distribution with the ordinarygluon density, see Eq. (1) and comments below it). A comparison at Q2 = 5 GeV2 for the ratio ofgluon densities in Pb over proton, can be found in Fig. 1. There it can be seen that the results of[19, 38, 39, 40, 21] at x ' 10−2 (relevant for RHIC) roughly coincide, while they are higher than thoseof [16, 43]; at x ' 10−4 ÷ 10−5 (accessible at LHC) the results of [19] become smaller than those of[38, 39, 40, 21], get close to those of [16] and are still larger than those of [43]. Apart from the constraintscoming from existing DIS experimental data on nuclei which are very loose for the glue at small x, in[38, 39, 40] the saturation of gluon shadowing comes mainly from the initial condition for DGLAP evo-lution where this saturation has been imposed, while [19, 16, 21] are multiple scattering models. In [43]the behaviour of the glue, in the form of a Q2-independent parameterization, has been fixed at x ∼ 10−2in order to reproduce charged particle multiplicities in AuAu collisions at RHIC. In this approach thestrongest gluon shadowing is obtained. However, it seems to be in disagreement with existing DIS data[42]. Additional caution has to be taken to compare results from multiple scattering models with thosecoming from DGLAP analysis [38, 39, 40]: the ratios for the glue at some moderate, fixed Q2 and verysmall x may result smaller (e.g. in [19]) than the ratios for F2 at the same x, Q2, which might lead toproblems with leading-twist DGLAP evolution, see [42].ACKNOWLEDGEMENTSN. A. thanks CERN Theory Division, Departamento de Fı´sica de Partı´culas at Universidade de Santiagode Compostela, Department of Physics at University of Jyva¨skyla¨, Helsinki Institute of Physics andPhysics Department at BNL, for kind hospitality during stays in which parts of this work have beendeveloped; he also acknowledges financial support by CICYT of Spain under contract AEN99-0589-C02 and by Universidad de Co´rdoba. C. A. S. is supported by a Marie Curie fellowship of the EuropeanCommunity programme TMR (Training and Mobility of Researchers) under the contract number HPMF-CT-2000-01025.Fig. 1: Ratios of gluon distribution functions from different models at Q2=5 GeV2; HKM refers to the results from [40],Sarcevic to those from [21], EKS98 to those from [38, 39], Frankfurt to those from [16], Armesto to those from [19, 20] andnew HIJING to those from [43]. The bands represent the ranges of x relevant for processes with a scale Q2 = 5 GeV2 at RHICand LHC.References[1] M. Arneodo, Phys. Rept. 240 (1994) 301.[2] J. C. Collins, D. E. Soper and G. Sterman, Adv. Ser. Direct. High Energy Phys. 5 (1988) 1.[3] M. Arneodo, A. Bialas, M. W. Krasny, T. Sloan and M. Strikman, arXiv:hep-ph/9610423.[4] H. Abramowicz et al. [TESLA-N Study Group Collaboration], DESY-01-011.[5] EIC White Paper, preprint BNL-68933, Eds. A. Deshpande, R. Milner and R. Venugopalan.[6] Y. L. Dokshitzer, Sov. Phys. JETP 46 (1977) 641 [Zh. Eksp. 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Gluon distributions in nuclei at small x: guidance from different models

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