8 November 2001

Physics Letters B 520 (2001) 124–130www.elsevier.com/locate/npe

A geometrical estimation of saturation of partonic densities

N. Armestoa, C.A. Salgadob

a Departamento de Física, Módulo C2, Planta baja, Campus de Rabanales, Universidad de Córdoba, E-14071 Córdoba, Spainb Laboratoire de Physique Théorique, Université de Paris XI, Bâtiment 210, F-91405 Orsay Cedex, France

Received 3 May 2001; received in revised form 30 July 2001; accepted 18 September 2001Editor: J.-P. Blaizot

Abstract

We propose a new criterium for saturation of the density of partons both in nucleons and nuclei. It is applicable to any multiplescattering model which would be used to compute the number of strings exchanged inep andeA collisions. The criterium isbased on percolation of strings, and the onset of percolation is estimated from expectations coming from the study of heavy-ioncollisions at high energies. We interpret this onset as an indication of saturation of the density of partons in the wave function ofthe hadron. In order to produce quantitative results, a particular model fitted to describe present HERA data and generalized tothe nuclear case is used. Nevertheless, with the number of scatterings controlled by the relation between inclusive and diffractiveprocesses, conclusions are weakly model-dependent as long as different models are tuned to describe the experimental data.This constitutes a new approach, based on the eikonal description of soft hadronic collisions, and different from others whichemploy either perturbative QCD ideas or semiclassical methods. It offers an alternative picture for saturation in the smallQ2

region. 2001 Elsevier Science B.V. All rights reserved.

PACS: 24.85.+p; 11.80.La; 13.60.Hb; 12.40.NnKeywords: Saturation; Partonic densities; Multiple scattering; Percolation

Much interest has recently been devoted to thesaturation of partonic densities [1], i.e., the changein the increase of the partonic densities from power-like to logarithmic or constant with decreasing partonmomentum fractionx, both in nucleons [2–4] and innuclei [5–7]. From the point of view of experimentaldata on lepton–hadron scattering, the most strikingfeature was the change in the logarithmic slope of theproton structure functiondF2/d ln (Q2) at x ∼ 10−4,the so-called Caldwell plot [8] (now known to bemainly due to aQ2 − x correlation), but the situationis not conclusive: nucleon data can be described notonly in approaches which consider saturation [2–4],

E-mail address: fa1arpen@uco.es (N. Armesto).

but also satisfactorily accommodated in the usualglobal fits [9] (also available for nuclei [10]) thatconsider the standard QCD evolution or resummation[11], starting from initial conditions at low photonvirtualities Q2 which do not include saturation (see[12] for an application to the Caldwell plot and [13]for a discussion on the present situation).

From a theoretical point of view, the saturationregime is a very interesting one characterized by asmall coupling constant and high occupation numbers,where a semiclassical description in terms of fieldshas been proposed [7]. Different models offer expla-nations based on multiple scattering (i.e., unitariza-tion) or gluon interaction, both in the case of nucleons[1–4] and nuclei [1,5–7]. These two approximationsto the problem are equivalent (see, e.g., [14]) in differ-

0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0370-2693(01)01153-4

N. Armesto, C.A. Salgado / Physics Letters B 520 (2001) 124–130 125

ent reference frames, but the models predict the onsetof saturation in different kinematical regions and thesaturation features are also diverse. In this short notewe will essay another approach to the problem, inher-ited from multiparticle production in nucleus–nucleus(AB) collisions at high energies and applicable to anymodel formulated in terms of multiple scatterings.

The concept of saturation, not of the density of par-tons in the hadronic wave function but of the num-ber of partons produced in the collision, was proposedsome time ago [15] inAB collisions at high energiesand has been reconsidered recently in the context ofthe search of the Quark Gluon Plasma (QGP); suchhigh partonic density should provide the initial con-dition for the possible thermalization of the createdsystem. Several related ideas have been used to com-pute the multiplicity of produced particles inAB colli-sions at the Relativistic Heavy Ion Collider (RHIC) atBNL and at the future Large Hadron Collider (LHC) atCERN [16,17]. For example, in [18] perturbative QCD(pQCD) is used to compute the initial number of glu-ons, quarks and antiquarks, which are limited accord-ing to the simple geometrical criterium that the num-ber of partons per unit of transverse space–times theirtransverse dimension (∝ 1/p2⊥) cannot be greater than1 (see [19] for other attempts in this direction). Be-sides, the semiclassical methods used in [7] have alsobeen employed to estimate the initial number of glu-ons in a heavy-ion collision [20].

On the other hand and in the framework of stringmodels for soft multiparticle production (see [21] andreferences therein), a simple geometrical criterium forsaturation has been proposed. In these models particleproduction comes from string breaking, strings whichare considered, in a first approximation, as formedand decayed independently. As the number of bi-nary nucleon–nucleon collisions (each one producing2 strings [21,22]) increases with increasing central-ity, energy or nuclear mass, this approximation shouldbreak down. The onset of this phenomenon can beestimated considering strings with a certain area inthe transverse space of the collision, and taking intoaccount the possibility of two-dimensional percola-tion of the strings when they overlap in this trans-verse space. Percolation is a second order phase tran-sition which takes place when clusters of overlappingstrings, with a size of the order of the total transversearea available, appear. This idea has been proposed in

AB collisions [23] and applied to signatures of QGP[24].

The purpose of this Letter is to use percolation ofstrings as an indication for the onset of saturationof the density of partons in nucleons and nuclei,a quantity which in our case is not directly relatedwith the partonic densities measured in DIS, as suchidentification [5,14] can only be done at highQ2, andour approach will be devoted to the lowQ2 regime.For this, we need a multiple exchange model forep

collisions which allows us to compute the numberof binary collisions, generalize it to the nuclear case,translate the number of collisions to a number ofstrings and estimate the density of strings to computewhether percolation takes place or not. The methodcan be applied to any multiple scattering model, andthe results in any of these models should be quite thesame (within the uncertainties due to the extrapolationof the model to nuclei and to higher energies orsmallerx) as long as the model is able to describe thefully inclusive and diffractive experimental data onepcollisions, see comments below.

Let us give a brief description of the model devel-oped in [4], which is the particular one we are goingto use to compute the number of binary collisions andthen of strings in order to give quantitative predictions.The goal of the model was the description of total anddiffractive data onep scattering at low and moderateQ2 and smallx. This region is where unitarity correc-tions are more important and where the transition fromnon-perturbative to perturbative QCD takes place.

In the proton rest frame, the virtual photon comingfrom the lepton fluctuates into aqq state. Then thishadronic state interacts with the proton. Unitaritycorrections are described by the multiple scattering ofthe qq fluctuation with the proton. In a quasieikonalapproach the total cross section is given by

σγ ∗ptot

(s,Q2)

(1)= 4g(Q2)∫

d2b

(1− exp{−Cχ(s,Q2, b)}

2C

),

whereg(Q2) is theγ ∗–(qq) coupling, 2χ(s,Q2, b) isthe elementary(qq)–p cross section at fixed impactparameterb, andC = 1.5 is a parameter taking intoaccount the diffractive dissociation of the proton. Forsmall sizesr of the qq pair, χ ∝ r2 from pQCDcalculations. Asr2 ∝ 1/Q2, for these small sizes

126 N. Armesto, C.A. Salgado / Physics Letters B 520 (2001) 124–130

χ ∝ 1/Q2. For large sizes of the fluctuation noQ2-dependence is expected. In [4] two components,corresponding to small (S) and large (L) sizes of theqq pair, were taken into account,χ = χL + χS . Thefact thatχS ∝ 1/Q2 while χL does not depend onQ2

makes the correction terms in Eq. (1) more importantfor theL part than for theS one. So, more scatteringsare present—in average—for theL than for theScomponent; for this reason and also due to the factthat we will consider smallQ2, only theL component,which is the dominant one [4] forQ2 � 2 (GeV/c)2,will be used in the actual computations. The energydependence of theseχ ’s is given by a single pomeronof intercept∆= 0.2,

(2)χL = CL

λLexp

{∆ξ − b2

4λL

}.

Here, ξ = ln s+Q2

s0+Q2 , λL = R2L + α′

P ξ with R2L =

3 GeV−2, and α′P = 0.25 GeV−2, the slope of the

pomeron trajectory, gives the lns behavior of thetotal cross section for very larges; besides,CL =0.56 GeV−2 and s0 = 0.79 GeV2. The variableξis chosen so thatχ ∝ x−∆ for large Q2 � s0 andχ ∝ (s/s0)

∆ for Q2 → 0; in this way, the model canbe used for photoproduction. For theS part, similarexpressions were used in [4] with an extrar2 factor inEq. (2).

The description of diffraction is a very importantingredient of the model. It is given by quadratic andhigher order terms inχ in the expansion of Eq. (1).Thus, the ratioσdiff/σtot controls the unitarity (mul-tiple scattering) corrections, i.e., the number of scat-terings and strings; this idea has been used to com-pute nuclear structure functions from a description ofdiffraction at HERA, see, e.g., [25]. So, any multiplescattering model able to reproduce the experimentaldata on this ratio should produce roughly the samenumber of scatterings (strings) and, consequently, thesame predictions for the onset of percolation and sat-uration. A triple pomeron term was introduced in [4]in order to reproduce large mass diffractive processes.This term is another source of shadowing correctionsto the total cross section. It will be used in the actualcomputations, see [4] for the full expressions and pa-rameters. The reggeon contribution which appears in[4] decreases with increasing energy and is negligible

at the energies under consideration, so it has been ig-nored.

The model in [4] has 9 free parameters that werefitted to experimental data on diffractive and totalep

cross sections for 0� Q2 � 10 (GeV/c)2 and 10−6 �x � 10−2. Once the parameters of the model are fitted,it is possible to know the mean number of collisions[22]:

n=∞∑n=1

n

∫d2b σn(s,Q

2, b)∑∞n=1

∫d2b′ σn(s,Q2, b′)

(3)=∫d2b2Cχ(s,Q2, b)∫

d2b′ [1− exp{−2Cχ(s,Q2, b′)}] ,where, forn� 1,

σn(s,Q2, b

)

(4)= g(Q2)

C

[2Cχ(s,Q2, b)]nn! exp(−2Cχ).

Notice that in these expressions,χ(s,Q2, b) containsthe triple pomeron contribution, so cuts in differentbranchings of one single fan diagram (i.e., one singletree of triple pomeron couplings) are included in thesameσn, which thus corresponds to the exchange ofn fan diagrams, each of them cut in one o more thanone of its branches. Thus Eq. (4) is a conservativeestimation, as these cuts could give rise to a largernumber of strings. Besides, all our expressions areasymptotic ones, not considering energy–momentumconservation (which could reduce slightly the numberof collisions at the lowest energies).

Neglecting isospin at high energies, the generaliza-tion of any multiple scattering model formulated forep to the case ofeA collisions is straightforward inthe Glauber–Gribov approach [26]: the number ofqq–nucleon collisions (the number of participating nucle-ons ofA) in this case, is given [27] in terms of theinelastic non-diffractive cross sections by

(5)〈npart〉 =Aσγ ∗pin

/σγ ∗Ain ∝A1/3,

with

σγ ∗Ain = g

(Q2)

(6)×∫

d2b

(1− exp

{−ATA(b)σγ ∗pin

g(Q2

)})

,

N. Armesto, C.A. Salgado / Physics Letters B 520 (2001) 124–130 127

TA(b)= ∫ ∞−∞ dzρA(z, �b) the profile function normal-

ized to 1 taken from [28] and

σγ ∗pin =

∞∑n=1

∫d2b σn

(s,Q2, b

)

(7)

= g(Q2)

C

∫d2b

[1− exp

{−2Cχ(s,Q2, b

)}].

So, the total number of collisions is given by〈ncoll〉 =〈npart〉n. As previously commented, in the actualcomputations we will only use theL component,Eq. (2), as it is the dominant one [4] forQ2 �2 (GeV/c)2 where our calculations will be done.

At this point, it could be argued that using the modelin [4] (or any other multiple scattering model) thereis a possibility to study saturation of the density ofpartons, both in nucleons and in nuclei, simply lookingat the point in which amplitudes in impact parameterspace become energy independent, or alternativelythe point in which cross sections reach a regimein which their energy behavior becomes identicalto that of the size of the target (expanding in thecase of a nucleon). Nevertheless, the generalizationof [4] to nuclei is not so obvious: ours is a verysimple one, but more rigorous generalizations [29]also rely in simplifications of the exact Gribov calculus[30] or Glauber–Gribov theory [26]. So we thinkthat an estimation as the one we perform, based ongeometrical criteria, is worthy, of simple and generalapplicability, and may provide, as in the case ofnucleus–nucleus collisions, an indication of the onsetof a high density, non-linear regime.

Let us establish now our criterium for saturationof the density of partons in the wave function ofthe target. As it was said, percolation is a non-thermal second order phase transition, which takesplace when clusters of overlapping objects acquire asize comparable to the total size available [31]. In ourcase the space is the transverse dimension available forthe collision, and the overlapping objects are strings.The parameter which controls the onset of percolationis the dimensionless string density

(8)η =Nt/T

(which may be related [17] with the dimensionlessdensity of gluons found in semiclassical models), with

(9)N = 2〈ncoll〉

the number of strings exchanged in the collision (eachcollision gives rise to two strings due to the pomerondominance at high energies, see [21,22]),t = πr2

0 isthe transverse dimension of the string,1 with r0 �0.20–0.25 fm as extracted from phenomenology [23,24], and T the total transverse area available forthe collision. This last quantity is not known andcould depend on the virtuality of the fluctuationQ2; however, for small and moderateQ2, it can beestimated to be the typical size of the vector mesonin which the virtual photon fluctuates (as this isthe smaller object in the interaction). So, we willuse T = 1 fm2 (a radius

√T/π � 0.56 fm). Also,

a size varying with the energy, in the spirit of anexpanding proton, could be explored; for example,a size increasing with increasing energy would slowthe corresponding increase of the density of stringsbut, for simplification, we will employ a fixed size.

The critical value forη where percolation takesplace, has been computed using different methods anddepends quite strongly on the profile of the nucleus(i.e., on the distribution of the overlapping objectsinside the available transverse space). For continuumtwo-dimensional percolation and from [31,32], wetake ηc � 1.12–1.50. Defining the string density asn = N/T and allowing for the different values ofr0andηc, we find a critical string density

(10)nc � 6–12 strings/fm2.

With this critical string density, it is tempting toestimate the behavior of theQ2 at which, for afixed s, saturation of the density of partons takesplace in this approach,Q2

sat [1–3,5–7,14]. However,in our case, beingχL almost independent onQ2, thedensity of strings is also almostQ2-independent forfixed s; besides, the model is only valid for smallQ2

and has not been designed forQ2-evolution. Moresignificant in our model is the value ofx or s where,for fixedQ2 andA, saturation takes place,xsat or ssat,respectively. Considering neither triple pomeron norreggeon contributions and approximating in Eq. (5)

σγ ∗Ain � g(Q2)πR2

0A2/3, with R0 � 1.2 fm, it is found

that xsat ∝ [Q2/(s0 + Q2)]A1/(3∆) and ssat ∝ (s0 +

1 In this approach this transverse dimension plays the rôle of anintrinsic scale of soft physics,Q2 → 0, where a description in termsof pQCD degrees of freedom becomes dubious.

128 N. Armesto, C.A. Salgado / Physics Letters B 520 (2001) 124–130

Fig. 1. String density inγ ∗–p, Be, Fe and Pb collisions versusxfor Q2 = 0.1 (GeV/c)2 (solid lines) andQ2 = 1 (GeV/c)2 (dashedlines). Dotted lines are the bounds on the critical string density forpercolation of strings.

Q2)A−1/(3∆), 1/(3∆) = 5/3. So, forQ2 → 0, xsatincreases linearly with increasingQ2 while ssat isroughly constant; these qualitative features will beobserved in the numerical results.

Let us turn to numerical evaluations. Using Eqs. (3)–(7), (9) and (10), we can now compute the string den-sity for different hadronic targets,Q2, and x or s.When this density becomes larger than the criticalvalue, Eq. (10), percolation will take place, which wewill interpret as a signal of the onset of saturation ofthe density of partons in the target. In Fig. 1 we presentresults for the string density inγ ∗–A collisions ver-susx for Q2 = 0.1 and 1 (GeV/c)2, with A = 1, 9,56 and 207 (corresponding top, Be, Fe and Pb, re-spectively). In Fig. 2 the same quantity is presentedversuss for Q2 = 0 and 2 (GeV/c)2. Some commentsare in order: first, from Fig. 1 it looks as if saturation(percolation) is favored by a higherQ2, apparently incontradiction to what is commonly expected. This isdue to the fact that, as we have said, the number ofrescatterings is hardly dependent onQ2—as can alsobe observed in Fig. 2, and thats is the variable whichcontrols this number (indeed, in Fig. 2 it can be seenthat ssat increases slightly with increasingQ2, as ex-

Fig. 2. String density inγ ∗–p, Be, Fe and Pb collisions versuss for photoproduction (solid lines) andQ2 = 2 (GeV/c)2 (dashedlines). Dotted lines are the bounds on the critical string density forpercolation of strings.

pected). So, for a fixeds at which percolation occurs,the higher theQ2 the higher thexsat (in agreementwith the naive expectations in the previous paragraph).Second, the dependence ofssat onA, parametrized asA−α , is found to be stronger in the numerical com-putations (α � 5/2) than the power 5/3 estimated inthe previous paragraph. This discrepancy is due to thetriple pomeron contribution included in the numeri-cal computations, which appear as a denominator inEq. (2), diminishing the ‘effective’∆ which appearsin ssat and thus makingα = 1/(3∆) larger.

To summarize, a criterium for saturation in the smallQ2 region applicable to any multiple scattering model,has been presented. To produce quantitative results,a multiple scattering model forγ ∗–p collisions in thisQ2 region [4] has been generalized to the nuclear case,and used to compute the number of exchanged strings.As multiple scattering (and thus the number of pro-duced strings) is controlled by the ratioσ diff/σ tot, thisnumber is related to experimental data on diffractionand the actual realization of the model is not crucial tocompute the string densities as long as it reproducesthe experimental data. Employing the ideas of perco-lation of strings taken from Heavy Ion Physics, the

N. Armesto, C.A. Salgado / Physics Letters B 520 (2001) 124–130 129

kinematical regions for the onset of percolation, whichhas been interpreted as saturation of the density of par-tons in the target,2 have been calculated. This consti-tutes a new approach, based on Regge phenomenologyand different from others which use either pQCD ideasor semiclassical methods; it offers an alternative pic-ture, based in hadronic degrees of freedom (strings),for saturation in the smallQ2 region. In view of theresults presented in the figures for the onset of per-colation, which for large nuclei may appear at not sosmallx, saturation could be observed in futureeA col-liders [33], and the effects of this second order phasetransition visible in correlations (as proposed inABcollisions [34]).

Acknowledgements

The authors express their gratitude to A. Capellafor a critical reading of the manuscript, to K.J. Eskolafor useful discussions on the model of saturation in[18], and to C. Pajares for his interest in this work andconstant encouragement. N.A. acknowledge financialsupport by CICYT of Spain under contract AEN99-0589-C02 and by Universidad de Córdoba, and C.A.S.a postdoctoral grant by Ministerio de Educación yCultura of Spain. Laboratoire de Physique Théoriqueis Unité Mixte de Recherche–CNRS–UMR No. 8627.

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