Wounded quarks in A+A, p+A, and p+p collisions

Piotr Bozek,1, ∗ Wojciech Broniowski,2, 3, † and Maciej Rybczynski3, ‡

1AGH University of Science and Technology, Faculty of Physics andApplied Computer Science, al. Mickiewicza 30, 30-059 Cracow, Poland

2The H. Niewodniczanski Institute of Nuclear Physics,Polish Academy of Sciences, 31-342 Cracow, Poland

3Institute of Physics, Jan Kochanowski University, 25-406 Kielce, Poland(Dated: ver. 2, 3 May 2016)

We explore predictions of the wounded quark model for particle production and properties of theinitial state formed in ultrarelativistic heavy-ion collisions. The approach is applied uniformly toA+A collisions in a wide collision energy range, as well as for p+A and p+p collisions at the CERNLarge Hadron Collider (LHC). We find that generically the predictions from wounded quarks forsuch features as eccentricities or initial sizes are close (within 15%) to predictions of the woundednucleon model with an amended binary component. A larger difference is found for the size in p+Pbsystem, where the wounded quark model yields a smaller (more compact) initial fireball than thestandard wounded nucleon model. The inclusion of subnucleonic degrees of freedom allows us toanalyze p+p collisions in an analogous way, with predictions that can be used in further collectiveevolution. The approximate linear dependence of particle production in A+A collisions on the num-ber of wounded quarks, as found in previous studies, makes the approach based on wounded quarksnatural. Importantly, at the LHC energies we find approximate uniformity in particle productionfrom wounded quarks, where at a given collision energy per nucleon pair similar production of ini-tial entropy per source is needed to explain the particle production from p+p collisions up to A+Acollisions. We also discuss the sensitivity of the wounded quark model predictions to distributionof quarks in nucleons, distribution of nucleons in nuclei, and to the quark-quark inelasticity profilein the impact parameter. In our procedure, the quark-quark inelasticity profile is chosen in such away that the experiment-based parametrization of the proton-proton inelasticity profile is properlyreproduced. The parameters of the overlaid multiplicity distribution is fixed from p+p and p+Pbdata.

PACS numbers: 25.75.-q, 25.75Gz, 25.75.Ld

I. INTRODUCTION

The idea of wounded quarks [1, 2] was proposed shortlyafter the successful concept of wounded nucleons [3], inthe quest of understanding in a natural way particle pro-duction in high-energy nuclear collisions. In particular, inthese early applications one looked for appropriate scal-ing of particle production with the number of partici-pants, and the idea of wounding, i.e., production froma participant independent on the number of its collisionwith participants from the other nucleus, proved veryuseful. It is based on the Glauber approach [4] adaptedto inelastic collisions [5], and the soft particle productionmay be justified in terms of the Landau-Pomeranchukeffect (for review see [6]).

With the advent of the BNL Relativistic Heavy-IonCollider (RHIC), the idea of wounded nucleons was re-vived by the PHOBOS Collaboration [7]. However, toexplain the multiplicity distribution in Au+Au collisionsit was necessary to include a component proportional tothe binary collisions [8], i.e., depending nonlinearly onthe number of the wounded nucleons. Then the average

∗ Piotr.Bozek@fis.agh.edu.pl† Wojciech.Broniowski@ifj.edu.pl‡ Maciej.Rybczynski@ujk.edu.pl

number of particles produced at mid-rapidity is

dNch

dη∼ 1− α

2NW + αNbin, (1)

where NW denotes the number of wounded nucleons andNbin the number of binary collisions, with the first terminterpreted as soft, and the second as hard production [8].

In heavy-ion phenomenology, the entropy density pro-file in the initial state is the most important source ofuncertainty in hydrodynamic models. Besides the MonteCarlo Glauber initial condition, IP-Glasma and KLN ini-tial conditions are frequently used [9, 10]. One shouldnote that differences in the parameterizations of the ini-tial entropy deposition give different eccentricities of thefireball [11]. Within the Monte Carlo Glauber approach,an understanding in more microscopic terms of the phe-nomenological binary collisions component in Eq. (1) isdesired, whereby uncertainties in the initial entropy de-position in the Glauber model should be reduced.

Eremin and Voloshin [12] noticed that the RHICmultiplicity data can be naturally reproduced within awounded quark model. With a larger number of con-stituents (three with quarks) and a reduced cross sectionof quarks compared to nucleons one can obtain an ap-proximately linear increase in particle production withthe number of wounded quarks, denoted as QW in this

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paper, namely:

dNch

dη∼ QW. (2)

The quark scaling has also been reported for the SPS en-ergies [13]. It has been vigorously promoted the PHENIXCollaboration [14, 15]. Experimental data on particleproduction has been compared to the number of woundednucleons or wounded quarks. The number of woundedquarks or nucleons in a collisions is obtained from aGlauber Monte Carlo code [16, 17], where for the modelwith quark constituents additionally three quarks are dis-tributed in each nucleon and collisions occur betweenpairs of quarks. An approximate, uniform scaling of themultiplicity and of the transverse energy with the num-ber of wounded quarks is claimed [14, 15], also includingthe LHC data [18].

The basic effect of the subnucleonic degrees of freedomin particle production is a stronger combinatorics, whichaccomplishes the approximately linear scaling of produc-tion with the number of constituents. An intermediatecombinatorics is realized in the quark-diquark model [19],which led to proper description of the RHIC data as wellas the proton-proton scattering amplitude (including thedifferential elastic cross section) at the CERN Intersect-ing Storage Rings (ISR) energies.

The purpose of this paper is to explore in detail pre-dictions of the wounded quark approach to particle pro-duction and properties of the initial state, as well as toinvestigate sensitivity to the assumptions concerning thedistribution of quarks and the quark-quark collision pro-file. While this work was nearing completion, a similarstudy for A+A collisions by Zheng and Yin [20] appeared.In Ref. [18] the scaling of particle production with thenumber of quarks is discussed, whereas in Ref. [21] dif-ferent ways of generating quark positions in the nucleonare compared. In addition, Loizides [22] presented a de-tailed study of the initial state in heavy-ion collisions formodels involving a number of partons in a nucleon dif-ferent than three.

The difference in this work compared to other stud-ies is that our methodology fixes the proton shape andthe quark-quark inelasticity profile to reproduce accu-rately the inelasticity profile in proton-proton collisions.It is important, since the details of the model of par-tons in nuclei, such as the effective size of the nucleon,the quark-quark cross section, and the shape of the in-elasticity profile in quark-quark scattering turn out tobe important for the results. With the fitted parametersof the quark distribution in nucleons and quark scatter-ing, we calculate the basic characteristics (eccentricities,sizes, and their event-by-event fluctuations) of the initialstate in A+A collisions. We also apply the approach top+A and p+p collisions, not analyzed in other works.This extension is important, as it checks consistency ofthe constituent quark scaling, as well as probes the limitof collectivity in small systems.

The outline of our paper is as follows: In Sec. II A

we present our method of fixing the parameters of themodel in terms of the proton-proton scattering ampli-tude, with further details given in Appendix A. Next,we discuss the overlaying of fluctuations on the Glaubersources, with the gamma distribution in the initial stageand the Poisson distribution at hadronization, convolv-ing to the popularly used negative binomial distribution.These extra fluctuations are necessary to describe thep+A and p+p collisions, or the A+A collisions at low-est centralities; they also contribute to eccentricities orevent-by-event fluctuation measures. In Sec. III we lookat the A+A collisions at a wide energy range, testingthe wounded quark scaling of hadron production and itssensitivity to model assumptions. We then pass to thestudy of eccentricities (including the hypercentral U+Ucollisions) and size fluctuations. Section IV presents theresults for p+A collisions, where we use the measuredmultiplicity distributions of produced hadrons to fix theparameters of the overlaid distribution. We then obtainthe eccentricities and sizes of the initial fireball. We findthat in the case of wounded quarks the initial size is sig-nificantly more compact than in the standard woundednucleon model. In Sec. V we analyze in an analogousway the p+p collisions, which is possible with subnucle-onic degrees of freedom. The obtained initial fireballs forhighest-multiplicity p+p collisions could be used in stud-ies of subsequent collective evolution of the system. Thepossibility of including more wounded constituents is ex-plored in Sec. VI. Finally, Sec. VII contains our summary,with the main conclusion that the concept of woundedquarks works uniformly from p+p to Pb+Pb collisionsat the LHC energies, providing approximately linear scal-ing of hadron production with the number of sources, andresulting in properties of the initial fireball close to thepredictions with wounded nucleons amended with binarycollisions. At RHIC, we observe dependence of the scal-ing on the colliding nuclei, and substantial difference withthe p+p channel (about 30%), which indicates the needfor improvement of the theoretical description at lowercollision energies.

II. GENERAL FORMULATION

A. Quarks inside nucleons

Constituent quarks are localized within nucleons,which results in their clustering as opposed to free, un-confined distribution in the nucleus. For that reason weattempt to model their distributions in a realistic way.Firstly, we randomly place the centers of quarks in thenucleons according to the radial distribution

ρ(r) =r2

r30

e−r/r0 , (3)

with a shift to the center of mass of the nucleon aftergenerating the positions of the three quarks. Positionsof nucleons in a nucleus are distributed according to a

3

standard Woods-Saxon radial density profile with appro-priate parameters [23, 24], or are taken from outside mi-croscopic calculations, such as [25]. Nuclear deforma-tion [26–29] is incorporated for deformed nuclei, such as63Cu, 197Au, or 238U, used at RHIC.

The quark-quark wounding profile (inelasticity profilein the impact parameter b) is for simplicity assumed tohave the Gaussian shape

pinqq(b) = 2πbe−πb

2/σqq , (4)

where σqq is the quark-quark inelastic cross section.The choice of this profile, together with the

parametrization (3), is important, as it determines theprobability of a collision at the transverse separation b,which influences the wounded quark scaling and the prop-erties of the formed initial state. The numbers r0 andσqq are treated as adjustable parameters and, at eachcollision energy, are fitted to the nucleon-nucleon totalinelastic cross section and to the nucleon-nucleon inelas-ticity profile. The latter can be obtained straightfor-wardly from working parameterizations of the NN scat-tering data, which describe both inelastic and elastic col-lision amplitudes. Here we take the parametrization ofRef. [30] based on the Barger-Phillips model [31]. Thedetails of our procedure are given in Appendix A. Thebottom line of this procedure is that we reproduce withsufficient accuracy the experimental inelasticity profile ofthe NN collisions. The values of our parameters are listedin Table I in Appendix A.

In the wounded quark Glauber model of the PHENIXCollaboration [14, 15], the size of the proton is energyindependent, whereas the quark-quark cross section in-creases with the collision energy. In a different approach,the increase of the nucleon-nucleon cross section with en-ergy is achieved with the increase of the nucleon size [32].When fitting the inelastic profile at different energies inour approach we find that the effective size of the nucleonincreases very weakly with the energy, while the quark-quark cross section increases sizeably with the energy.The corresponding change in the quark-quark inelastic-ity profile is determined by σqq, cf. Eq. (4).

B. Overlaid distribution

The number of particles produced in p+p collisionsfluctuates. In the wounded quark model only a part ofthese fluctuations can be accounted for by fluctuationsof the number of wounded quarks in a collisions. Todescribe the experimentally measured distributions, thecharged hadron distribution should be written as a con-volution of the numbers of hadrons nk produced fromeach wounded quark,

P (n) =∑i

Pwq(i)× (5)∑n1,n2,...,ni

Phq(n1)Phq(n2) . . . Phq(ni)δn,n1+n2+...+ni ,

where Pwq is the distribution of the number i of woundedquarks and Phq is the overlaid distribution of the num-ber of hadrons from an individual wounded quark. Theabove formula should be used uniformly for p+p, p+A,and A+A collisions if the production mechanism is to beuniversal. In the following we use the negative binomialdistribution to parametrize Phq, namely

PNB(n|n, κ) =Γ(n+ κ)nnκκ

Γ(κ)n!(n+ κ)n+κ, (6)

where Γ(z) is the Euler Gamma function, the average isgiven by n, and κ = n2/(var(n) − n) (larger κ meanssmaller fluctuations). This form has been widely usedto describe the multiplicity distributions in p+p colli-sions [33]. The parameters n and κ of the negative bi-nomial distribution can be adjusted to reproduce theobserved multiplicity distribution in p+p (Sec. V) orp+A collisions (Sec. IV A). In collisions of heavy nucleithe multiplicity distribution, or the mean multiplicity ineach centrality bin, is almost independent on the over-laid distribution, except for very central collisions whereit should be taken into account. However, effects of theoverlaid distribution show up also in A+A collisions invarious event-by-event fluctuation quantities.

When dividing the dynamics of the collision into twostages, namely 1) the formation of the initial fireballand 2) its subsequent evolution and decay into hadrons,one should also separate the multiplicity fluctuations intotwo corresponding parts. The first part comes from thefluctuations in the initial entropy deposition in the fire-ball, whereas the other part comes from entropy produc-tion during the expansion of the fireball and subsequenthadronization. Note that a smooth, linear increase of en-tropy in viscous hydrodynamics is effectively included inthe normalization coefficient. We neglect possible fluc-tuations in the hydrodynamic phase [34, 35] (which areexpected not to be very large [36]) and from correlatedparticle emissions at freeze-out. We follow the usual as-sumption that the particle emission is given by a Poissonprocess, with the mean proportional to the entropy inthe fluid element.

As the negative binomial distribution is a convolutionof the gamma and the Poisson distributions, the entropydistribution P (s) in the fireball is given by a convolutionof a gamma distribution,

PΓ(s|n, κ) =sκ−1κκ

Γ(κ)nκe−κs/n, (7)

with the distribution of the number of wounded quarks,

P (s) =∑i

Pwq(i)PΓ(s|in, iκ). (8)

Microscopically, it means that each wounded quark de-posits a random entropy s taken from the gamma distri-bution (7), with same parameters as the parameters ofthe negative binomial distribution adjusted to the mul-tiplicity distributions. The fluctuations in entropy de-position increase the fluctuations and deformations of

4

the initial fireball. The effect is especially important insmall systems or for central A+A collisions [29, 37–39].Throughout this work, according to the above discussion,we overlay a negative binomial distribution on top of thewounded quark distribution when calculating the hadronmultiplicity distribution in p+p and p+A collisions, andwe overlay a gamma distribution when calculating theentropy profile of the initial fireball and its properties,such as eccentricities or size.

C. Smearing of sources

On physical grounds, the Glauber sources must pos-sess a certain transverse size; smoothness of the initialcondition is also required by hydrodynamics. For thatreason one needs to smooth the initial distribution. Wefollow the Gaussian prescription, where the entropy den-sity from a single source centered at a transverse point(x0, y0) is given with the profile

g(x, y) =1

2πσ2exp

(− (x− x0)2 + (y − y0)2

2σ2

). (9)

Unless otherwise stated, we use arbitrarily σ = 0.2 fm inthe wounded quark model and σ = 0.4 fm in the woundednucleon model. One should note that in small systemsthe fireball eccentricities or sizes depend sensitively onthe value of σ.

III. A+A COLLISIONS

A. Multiplicity distribution

We begin the presentation of our results by showing theoutcome of the wounded quark model for the A+A colli-sions. The results, obviously, depend on the centrality ofthe collision, which is defined experimentally via the re-sponse of detectors, hence its detailed modeling may becomplicated. Assuming that the centrality is obtainedfrom the multiplicity of the collision, we may evaluateit as percentiles of the event-by-event distribution of thenumber of sources, which is accurate enough except forthe very central and very peripheral events [40]. A bet-ter way is to obtain centrality from the number of sourceswith weights from an overlaid distribution (see Sec. II B).However, in presenting the results in the field it is cus-tomary to use the number of wounded nucleons NW, asobtained from the Glauber simulations with the modelof Eq. (1). We follow this convention, thus NW serves asa label the centrality classes, even when these are deter-mined by the wounded quarks.

The basic test of the wounded quark model is its verydefinition of Eq. (2), whereby the production of hadronsshould be proportional to the number of wounded quarksQW. In particular, the ratio of the multiplicity of chargedhadrons at mid-rapidity dNch/dη divided by the num-ber of wounded QW, should not depend on centrality

0.0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

1 10 102

♣♣♣♣♣♣♣♣

♣

dNch/dη/QW

〈NW 〉

UA5, p + p,√s = 19.6 GeV

UA5, p + p,√s = 200 GeV

ALICE, p + p,√s = 2760 GeV

PHENIX, d + Au,√sNN = 200 GeV

PHENIX, 3He + Au,√sNN = 200 GeV

PHENIX, Cu + Au,√sNN = 200 GeV

PHENIX, Au + Au,√sNN = 19.6 GeV

PHENIX, Au + Au,√sNN = 200 GeV

PHENIX, U + U,√sNN = 193 GeV

ALICE, Pb + Pb,√sNN = 2760 GeV

FIG. 1. Experimental multiplicity of charged hadrons perunit of pseudorapidity (at mid-rapidity) divided by the num-ber of wounded quarks, dNch/dη/QW, plotted as function ofcentrality expressed via the number of wounded nucleons.We also show the results for the p+p collisions (filled sym-bols at 〈NW〉 = 2) discussed in Sec. V. The data are fromRefs. [15, 41–43].

or on colliding nuclei. To find QW corresponding to agiven centrality we have carried out simulations in theGlauber model with the help of appropriately modifiedGLISSANDO code [23, 37]. In the standard calculation,the nuclear profiles of heavy nuclei are obtained fromthe Woods-Saxon form with the nucleon-nucleon expul-sion radius of 0.9 fm [44]. For the deuteron we use theHulthen wave function, and for 3He we take the distribu-tions from the method of Ref. [45] as provided in Ref. [17].The quarks are then generated in nucleons according toEq. (3), whereas the inelasticity profile for the quark-quark collisions is of the Gaussian form (4).

The results are shown in Fig. 1. We note that, in qual-itative agreement with the earlier studies [12–15], the de-pendence of dNch/dη/QW on centrality is approximatelyflat, and increases with the collision energy. We note thatsome deviation from a linear scaling between the initialentropy and the final particle density in pseudorapidity ispossible due to different mean transverse momentum ordifferent entropy production at different centralities [46]

5

0.9

1.2

1.5

1.8

2.1

0 100 200 300 400

dN

ch/dη/Q

W

〈NW 〉

ALICE,√sNN = 2760 GeV

standardPb from Alvioli et al.hard-sphere qq profile

FIG. 2. Comparison of dNch/dη/QW in Pb+Pb collisions at√sNN = 2.76 TeV [43], obtained with the standard GLIS-

SANDO simulations with the nuclear Woods-Saxon distribu-tion (squares), compared to the case of correlated nuclear dis-tributions of Alvioli et al. from Ref. [25]. We also show thecalculation with quarks implemented as in the Glauber MonteCarlo of Refs. [15, 22] (diamonds), i.e., with the nucleon pro-file from Ref. [15] and a hard-sphere quark-quark inelasticityprofile.

during the evolution of the system. In contrast, the flat-ness is not the case of the wounded nucleon model ofEq. (1), where, as is well known, the ratio dNch/dη/NW

increases considerably with Nw. Moreover, the value ofdNch/dη/QW (i.e., the average number of charge hadronsper unit of rapidity coming from a single wounded quark),is at a given energy roughly similar for various consid-ered reactions. For the LHC energies it is also consistentwith the p+p collisions, where the data are taken fromRef. [42] and QW is obtained in Sec. V. At RHIC energiesof√sNN = 200 and 19.6 GeV the p+p point, obtained

with the data from Ref. [41], is noticeably higher (about30%) than the band for the A+A collisions. Also, thedependence on the colliding nuclei at RHIC indicates asystematic uncertainly of the approach.

The above discussed universality of the hadron produc-tion in p+p vs A+A, holding fairly well for the LHC andto a lesser degree for the lower energies, is an importantcheck of the idea of the independent production from thewounded quark sources. We note that at lower energiesit is not very accurate. Also, as the plots in Fig. 1 arenot really flat, the wounded quark scaling is approximate.This observation may suggest that the effective numberof subnucleonic degrees of freedom at RHIC energies issmaller that three, as implied, e.g., by the quark-diquarkmodel [19], and at higher energies may be larger thanthree (cf. Sec. VI).

To assess the sensitivity of the approximate woundedquark scaling for the particle production, we check thedependence on the nucleon distributions in nuclei, as well

0.2

0.4

0.6

0.8

1.0

50 100 150 200 250 300 350 400

〈ε2〉

〈NW 〉

Pb+Pb,√sNN = 2760 GeV, wounded quarks

Pb+Pb,√sNN = 2760 GeV, wounded nucleons

Au+Au,√sNN = 200 GeV, wounded quarks

Au+Au,√sNN = 200 GeV, wounded nucleons

FIG. 3. Average ellipticity vs centrality for two selected reac-tions, evaluated in the wounded quark and wounded nucleonmodels.

as on the treatment of quarks. First, in Fig. 2 we com-pare our standard GLISSANDO calculation described inSec. II A with Woods-Saxon distributions (squares) toan analogous simulation with distributions obtained fromRef. [25], where central two-body nucleon-nucleon corre-lations are incorporated (circles). We note that the lattercase yields even more flat result as a function of central-ity. The reason for this behavior is a longer tail in theone-body nucleon distributions of Ref. [25] as comparedto the Woods-Saxon profile [23].

Second, we compare our standard results to the calcu-lation made as in the quark Glauber Monte Carlo codedescribed in Ref. [15, 22] (diamonds), i.e., with a modifieddistribution of quarks in the nucleon and a hard-spherequark-quark inelasticity profile (the resulting nucleon-nucleon inelasticity profile in this case is different fromthe experimental parametrization used in this paper, cf.Appendix A). We notice a substantial difference from ourstandard result, with a larger breaking of the woundedquark scaling. The conclusion emerging from Fig. 2 isthat the modeling of the subnucleonic structure shouldbe done as accurately and realistically as possible. Thesame concerns the nuclear distributions, the distributionsof quarks inside nucleons, or the quark-quark inelasticityprofiles, as such details influence the bulk properties inA+A collisions. Still some systematic errors, stemmingfrom the model assumptions on the nuclear and subnu-cleonic structure of the nuclei, are unavoidable.

B. Eccentricities

A basic feature of phenomenology of relativistic heavy-ion collisions is the development of harmonic flow due togeometry [48] and to event-by-event fluctuations [49–54].In this section we compare predictions of the wounded

6

0.2

0.4

0.6

0.8

1.0

50 100 150 200 250 300 350 400

∆ε 2/〈ε 2〉

〈NW 〉

Pb+Pb,√sNN = 2760 GeV, wounded quarks

Pb+Pb,√sNN = 2760 GeV, wounded nucleons

Au+Au,√sNN = 200 GeV, wounded quarks

Au+Au,√sNN = 200 GeV, wounded nucleons

FIG. 4. The fluctuation measure ∆ε2/〈ε2〉 vs centrality, eval-uated in the wounded quark and wounded nucleon modelsand compared to the data from ATLAS Collaboration [47](circles).

nucleon and wounded quark models for the eccentricitiesof the initial state. For the wounded nucleon case we usethe mixing parameter α = 0.145 for

√sNN = 200 GeV

and α = 0.15 for√sNN = 2.76 TeV. The Gaussian

smearing parameter is σ = 0.4 fm for wounded nucle-ons and σ = 0.2 fm for wounded quarks (the value ofσ has tiny effects in the A+A collisions). We note fromFig. 3 that the wounded quarks lead to larger ellipticitycompared to the wounded nucleon case, with the effectreaching about 15% at the peripheral collisions.

We have checked that for the most central collisionsthe ratio ε3/ε2 approaches the limit of the independentsource model (cf. Eq. (7.8) of Ref. [55], see also the dis-cussion in Ref. [56]), namely

ε3

ε2=〈ρ2〉〈ρ3〉

√〈ρ6〉〈ρ4〉 , (10)

where 〈ρn〉 denote the moments of the initial density forcollisions at at zero impact parameter in the transverse

radius ρ =√x2 + y2. Numerically, the ratio of Eq. (10)

is very similar for the wounded nucleon and woundedquark models and is ∼ 1.1. As is known, the proximityof the most central values of ε2 and ε3 leads to problemsin the description of v2 and v3 in viscous hydrodynamics,where triangular flow is quenched more strongly than theelliptic flow [57]. As a result, the predicted values ofv3 are significantly smaller than v2, in contrast to theexperiment [58].

When switching from nucleon to quark participantstwo effects influence the entropy distribution in the fire-ball, with opposite effect on the eccentricities. Firstly, ad-ditional fluctuations at subnucleonic scales appear fromindividual entropy deposition from wounded quarks. Sec-ondly, the entropy is spread around the nucleon-nucleon

10-3

10-2

10-1

1

0.0 0.5 1.0 1.5 2.0 2.5 3.0

P(ε2/〈ε

2〉)

ε2/〈ε2〉

c=20%-25%wounded quarkswounded nucleons

FIG. 5. Distribution of ε2/〈ε2〉 for centrality 20-25%, com-pared to the data for v2/〈v2〉 from ATLAS Collaboration [47](circles).

collision point among the different positions of individualwounded quarks in the colliding nucleons. The second ef-fect can be estimated from the rms size of the fireball inp+p collisions (Sect. V); it is of the same order (0.4 fm)as the width of the Gaussian centered at the positionsof the wounded nucleons. The second effect comes ina similar way in both models and gives a reduction ofthe eccentricities. On the other, hand the first effect ap-pears only when using subnucleonic degrees of freedomand leads to larger eccentricities in the wounded quarkmodel. Correspondingly, as presented in Fig. 4, scaledevent-by-event fluctuations of ellipticity are reduced withwounded quarks, which brings the results closer to thedata in semi-central collisions. The horizontal line inFig. 4 corresponds to the limit of

√4/π − 1 reached in

the most central events [55].Results analogous to Figs. 3 and 4 for the triangularity

are very similar for the wounded quark and the woundednucleon models, except for 〈ε3〉 at peripheral collisions,where wounded quarks give higher values than woundednucleons, up to about 15%.

In Figs. 5 and 6 we compare the distributions ofεn/〈εn〉 for ellipticity (n = 2) and triangularity (n = 3)for the two considered approaches. We note that theresults are close to each other, with somewhat smallertails for the case of the wounded quarks (solid lines) forn = 2, which moves the model a bit further from the datacompared the the case of the wounded nucleons (dashedlines).

A long-standing problem for the Glauber-based sim-ulations were the results for the 238U+ 238U collisions,measured at RHIC at

√sNN = 200 GeV [59]. It had

been expected that due to the intrinsic prolate deforma-tion of the 238U nucleus, the results of the most centralcollisions for the ellipticity should exhibit a knee struc-ture [60]. It was later argued [29] that this structuremay be washed out with large fluctuations of the over-laid distribution. Also, as shown recently, in a Glauber

7

10-3

10-2

10-1

1

0.0 0.5 1.0 1.5 2.0 2.5 3.0

P(ε3/〈ε

3〉)

ε3/〈ε3〉

c=20%-25%wounded quarkswounded nucleons

FIG. 6. Same as in Fig. 5 but for the triangularity ε3.

0.0

0.05

0.1

0.15

0.2

10-3

10-2

10-1

1

〈ε2〉

centrality [%]

wounded quarkswounded nucleons

U+U,√sNN = 200 GeV

FIG. 7. Ellipticity ε2 for the 238U+238U collisions at√sNN =

200 GeV for the wounded quark (solid line) and woundednucleon (dashed line) models, plotted for centralities lessthan 1%.

approach with shadowing [61] the model predictions com-pare well to the data. Here we compare the predictionsof the wounded quark and wounded nucleon models. Itis evident from Fig. 7 that at hyper-central collision thewounded nucleon model leads to fall-of of ellipticity withdecreasing centrality c, whereas wounded quarks give aflattening, i.e., no knee behavior, in qualitative accor-dance to the data and the wounded quark model studiesreported in [59].

C. Size fluctuations

In Ref. [62] it was proposed that transverse size fluc-tuations of the initial state lead to transverse momen-tum fluctuations. The mechanism is based on the sim-ple fact that more compressed matter, as may happen

from statistical fluctuations in the initial state, leads tomore rapid radial hydrodynamic expansion, which thenprovides more momentum to hadrons. The mechanismwas later tested with 3+1 dimensional viscous event-by-event hydrodynamics [63], with a somewhat surpris-ing result that the effect leads to even larger (by about30%) fluctuations than needed to explain the experimen-tal data [64, 65], in particular for most central collisions.The basic formula for the mechanism of Ref. [62] is thatthe event-by-event scaled standard deviation of the av-erage transverse momentum is proportional to the cor-responding quantity for the transverse size, 〈r〉, of theinitial state:

∆〈r〉〈〈r〉〉 = β

∆〈pT 〉〈〈pT 〉〉

, (11)

where 〈〈.〉〉 denotes the event-by-event average of thequantity averaged in each event, ∆ denotes the standarddeviation, and the measure of the transverse size is

〈r〉2 =

∫dx dy s(x, y)(x2 + y2)∫

dx dy s(x, y). (12)

The function s(x, y) is the entropy density in the trans-verse coordinates (x, y). The constant β ∼ 0.3 dependson the hydrodynamic response, but not on the centralityof the collision, which allows for predictions. We intro-duce the variable

〈r〉 =√〈r〉2, (13)

which is analyzed event by event, in particular 〈〈r〉〉 isthe event-by-event average of the size of Eq. (13).

The results for the event-by-event scaled standard de-viation of the size are presented in Fig. 8. We com-pare the wounded nucleon (1) and the wounded quark(2) models with an overlaid gamma distribution. Wenote that at low centralities the size fluctuations in bothmodels are very close for the central collisions, whereasfor peripheral collisions the size fluctuations are larger inthe wounded quark model. The results for the two sam-ple reactions, Au+Au at

√sNN = 200 GeV and Pb+Pb

at√sNN = 2.76 TeV, are virtually indistinguishable, as

the corresponding curves lie on top of each other.

IV. p+A COLLISIONS

Ultrarelativistic p+A collisions are an important test-ing ground for theoretical approaches, due to the ex-pected onset of collectivity [38, 66–68]. It is thus im-portant to address this issue with the model based onwounded quarks, as we search a uniform description ofparticle production. In this section we present the resultsfor hadron multiplicity, the fireball size, and eccentrici-ties.

8

Au+Au 200GeV

Pb+Pb 2.76TeV

wounded quarks

wounded nucleons

0 100 200 300 4000.00

0.05

0.10

0.15

0.20

<NW>

Δ<r>/<<r>>

FIG. 8. Scaled size fluctuations of the initial fireball, plottedas a function of centrality evaluated in the wounded quarkand nucleon models with an overlaid gamma distribution.The results for Au+Au at

√sNN = 200 GeV and Pb+Pb

at√sNN = 2.76 TeV overlap.

A. Multiplicity distribution

To reproduce the experimental hadron multiplicity dis-tribution in p+A collisions, in particular the tail at veryhigh multiplicities, one needs to overlay an additionaldistribution over the Glauber sources, as explained inSec. II B. The physical meaning of this procedure is thatsources deposit entropy at a varying strength. The mech-anism was already proposed in the original woundednucleon model [1]. Experimental data are well repro-duced with an overlaid negative binomial distribution(Sec. II B). In Fig. 9 we compare the wounded-nucleonand the wounded-quark model predictions, compared tothe CMS data [69]. The high-multiplicity tail is properlyreproduced when the κ parameter of the negative bino-mial distribution is about 0.5. In Fig. 9 we use κ = 0.54for the wounded quark model, whereas κ = 0.9 for thewounded nucleon model (α = 0 in Eq. (1)). Here wefollow the convention that the parameters of the nega-tive binomial distribution correspond to overlaying overthe nucleons and not the nucleon pairs, as implied byEq. (1). The parameter n is adjusted such that meanexperimental multiplicities are reproduced at a given ex-perimental acceptance and efficiency, namely n = 3.9 forthe wounded quark and n = 6.2 for wounded nucleon.

We note a fair description of the tail of the distribu-tions in Fig. 9, whereas, admittedly, there are departuresat low values of Nch.

In Fig. 9 the coordinate Nch corresponds to the numberof tracks in the CMS detector. Unfortunately, no hadronmultiplicity data corrected for acceptance and efficiencyare available, such that at present we cannot overlay thep+Pb point on other results in Fig. 1.

chN20 40 60 80 100 120 140 160 180 200 220

)

chP

(N

5−10

4−10

3−10

2−10

wounded quarks + NBwounded nucleons + NBCMS

p+Pb 5.02TeV

FIG. 9. Multiplicity of charged hadrons in p+Pb collisionsfrom the wounded quark and wounded nucleon models, com-pared to the preliminary CMS data [69]. Appropriate nega-tive binomial distribution is overlaid over the distribution ofsources (see text).

0 50 100 1500

0.5

1

1.5

chN

<<r>

> [

fm]

wounded quarkswounded nucleonswounded nucleons(compact source)

p+Pb 5.02 TeV

FIG. 10. Initial size of the fireball in p+Pb collisions at√sNN = 5.02 TeV evaluated from Eq. (13) and plotted

against the multiplicity of produced charged hadrons. Thesolid line corresponds to the wounded quark model with thesmearing parameter σ = 0.2 fm, and is compared to woundednucleons with σ = 0.4 fm in the standard form (dashed line)and the compact source version (dot-dashed line).

B. Fireball in p+Pb

The size and the shape of the fireball are importantcharacteristics of the initial state. The size of the fire-ball determines the transverse push and interferometrycorrelations [70], whereas the eccentricities generate har-monic flow coefficients in the collective expansion. In thehydrodynamic model the best description of the data isobtained when the entropy is deposited in between thetwo colliding nucleons, in the so called compact sourcescenario [38, 71], where the entropy is deposited in the

9

0 50 100 1500

0.1

0.2

0.3

0.4

0.5

0.6

chN

2ε

wounded quarkswounded nucleonswounded nucleons(compact source)

a)p+Pb 5.02 TeV

0 50 100 1500

0.1

0.2

0.3

0.4

0.5

0.6

chN

3ε

wounded quarkswounded nucleonswounded nucleons(compact source)

b)p+Pb 5.02 TeV

FIG. 11. Ellipticity (panel a) and triangularity (panel b) ofthe initial fireball in p+Pb collisions at

√sNN = 5.02 TeV

from the wounded quark model with smearing parameter σ =0.2 fm, plotted as functions of the multiplicity of producedcharged hadrons.

middle between the two colliding nucleons. The prescrip-tion corresponds to Eq. (1) with α = 1. The effectivenumber of sources is NW−1 in this case, and the param-eters of the overlaid binomial distribution fitted to theexperiment as in Fig. 9 are n = 6.6 and κ = 1.0.

In Fig. 10 we show the size measure of Eq. (13) plottedas a function of the number of produced charged hadrons.We note that the results from the wounded quark modelare closer to the compact source variant of the woundednucleon model than to its standard version. Thereforethe wounded quark model gives a microscopic motiva-tion for the prescription used in the compact source sce-nario [38].

In Fig. 11 we present the ellipticity and the triangu-larity in the wounded quark model. The deformation ofthe initial fireball is similar as in the standard woundednucleon model [38]. We expect that the shape and sizeof the fireball obtained in the wounded quark model is agood initial condition for the hydrodynamic descriptionof measurements made in p+Pb collisions.

p+p ÈΗÈ<1 ALICE param.wounded quarks+NB

wounded quarks+NB+string fluct.

7TeV2.76TeV´0.01

0 50 100 150 20010-15

10-12

10-9

10-6

0.001

1

Nch

PHN ch

L

FIG. 12. Charged particle multiplicity distribution for |η| < 1.Solid lines denote a parametrization of experimental datafrom the ALICE Collaboration [42] for

√s = 7 TeV (up-

per curves) and for√s = 2.76 TeV (lower curves, multiplied

by 0.01). The multiplicity distribution from the woundedquark model convoluted with a negative binomial distribu-tion is denoted with dotted lines, whereas dashed lines repre-sent the calculation where both string fluctuations in rapidityand a negative binomial distribution are convoluted for eachwounded quark.

V. p+p COLLISIONS

With subnucleonic degrees of freedom we have an op-portunity to analyze the p+p collisions. The proton-proton inelastic collision profile in the wounded quarkmodel is described using quark-quark collisions (Sec. II Aand Appendix A). The total inelastic cross section at eachenergy is reproduced with an energy dependent quark-quark cross section.

A. Multiplicity distributions

The mean number of wounded quarks in p+p collisionsincreases mildly with the energy, as we find QW = 2.6,2.75, and 2.81 at

√s = 200 GeV, 2.76 TeV, and 7 TeV

respectively. The multiplicity of charged hadrons comesfrom a convolution of the multiplicity distributions ofparticles produced from each wounded quark and the dis-tribution of the number of wounded quarks (Sec. II B).

In Fig. 12 we show the multiplicity distribution ofcharged hadrons (for the acceptance window |η| < 1) ininelastic p+p collisions at two energies: 2.76 and 7 TeV.The experimental results are represented by a numericalparametrization using a sum of two negative binomialdistributions [42]. These curves for the multiplicity dis-tributions are reproduced in the wounded quark model byadjusting the parameters of the negative binomial distri-bution convoluted with the wounded quark distribution.We find n = 3.3 and κ = 0.52 at 7 TeV, and n = 2.8and κ = 0.55 at 2.76 TeV (dotted lines in Fig. 12). Theagreement of our model with the ALICE phenomenolog-ical fit, while not perfect, holds approximately over 12

10

0 50 1000

0.1

0.2

0.3

0.4

0.5

0.6

chN

nε=0.2fmσ 2ε

=0.2fmσ 3ε=0.4fmσ 2ε

=0.4fmσ 3ε

p+p 7 TeV

FIG. 13. Ellipticity ε2 (solid lines) and triangularity ε3 (dot-ted lines) of the fireball in p+p collisions at

√s = 7 TeV

with the smoothing width σ = 0.2 fm (thick lines) and 0.4 fm(thin lines), as a function of the mean charged multiplicity for|η| < 2.4.

orders of magnitude.The scenario discussed above assumes that all the

wounded quarks contribute to particle production in theconsidered pseudorapidity interval. Alternative scenar-ios are possible, where each wounded quark contributesto particle production in a limited pseudorapidity inter-val. An example of such a case is the flux tube model,where strings decay into particles in a rapidity intervallimited by the rapidity of the leading charges [72–74].Without entering into details of a particular model ofthe random deposition of entropy in rapidity, we makea simple estimate using an extreme scenario. Namely,a wounded quark contributes (or not) with probability1/2 to hadron production in the central rapidity inter-val. It is a scenario assuming largest possible fluctuationsfrom a flux-tube mechanism. The goal of this exercise isto show that in both extreme scenarios, equal, smoothin rapidity contribution from each wounded quark andstrongly fluctuating contribution from each quark, theobserved hadron multiplicity distribution can be repro-duced. In the model with fluctuating contribution fromeach wounded quark, the parameters of the negative bi-nomial distribution are n = 7.9 and κ = 1.7 at 7 TeV,and n = 7.4 and κ = 2.2 at 2.76 TeV (dashed lines inFig. 12). These estimates show that in the woundedquark model there is a possibility to accommodate fora mechanism involving additional fluctuations in the en-ergy deposition in a fixed rapidity interval, with a properadjustment of the parameters of the convoluted negativebinomial distribution.

B. Fireball in p+p

In an inelastic p+p collisions two or more woundedquarks take part in the collision. The distribution of the

chN10 20 30 40 50 60 70 80

<<

r>>

[fm

]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

=0.4fmσ7 TeV, =0.2fmσ7 TeV, =0.2fmσ2.76 TeV,

p+p

FIG. 14. Initial size of the fireball in p+p collisions at theLHC energies evaluated with different Gaussian smearing pa-rameters and plotted against the multiplicity of producedcharged hadrons.

wounded quarks in the transverse plane generates trans-verse shape deformations of the initial fireball. In such asmall system it is essential to check the sensitivity to therange of the deposition of entropy from each quark. Wetake a Gaussian profile (9) of width of 0.2 or 0.4 fm. Thedistribution in the transverse plane takes into account thefluctuations in the entropy deposition from each woundedquark, with an overlaid gamma distribution with parame-ters adjusted to the multiplicity distribution as explainedabove. The total entropy in the fireball is rescaled to themean multiplicity for |η| < 2.4. The eccentricities showa very weak dependence on multiplicity (Fig. 13). Wenote that the triangularity is much smaller than the el-lipticity. In a scenario with a collective expansion in p+pcollisions this would give v3 much smaller than v2. Thesmoothing parameter of the Gaussian in the initial fire-ball has a decisive role in determining the magnitude ofthe eccentricity.

In Fig. 14 we display the average size of the systemformed in p+p collisions at two LHC energies and fortwo values of the Gaussian smearing parameters. Thisobservable reflects the size of the proton, the quark-quarkinelasticity profile, and the Gaussian smearing parame-ter. We note that the size, especially for the case ofσ = 0.4 fm, is not much smaller than in the p+Pb sys-tem shown in Fig. 10. This opens the opportunity ofusing the fireball from the wounded quark model in hy-drodynamic calculations for p+p collisions [75, 76].

VI. MORE PARTONS

The number of effective subnucleonic degrees of free-dom in the nucleon could be different from three, wherequark constituents are assumed. For instance, at the ISR

11

energies the proton-proton scattering amplitude can bevery well described using a quark-diquark model of thenucleon [77, 78]. In the preceding sections we have as-sumed that the proton is composed of three quarks, asused in many other recent studies [14, 18, 20, 21]. Ifprotons are composed of numerous Np partons, proton-proton collisions can be described in the Glauber op-tical model [79]. In the optical limit, the parameterdefining the inelastic collisions is N2

pσpp, where σpp isthe parton-parton cross section. Similarly, in the MonteCarlo Glauber model, when the number of partons inthe proton increases, the parton-parton cross section de-creases. A model with different numbers of partons inthe proton was recently studied by Loizides [22], with ablack-disc prescription for the parton-parton scattering.The cross section σpp can then be adjusted to reproduceσNN for each Np. It was found that with a large num-ber of partons in a nucleon the increase of the number ofwounded partons when going from peripheral to centralevents is stronger.

In our study we adjust the distribution profile of par-tons in the nucleon, as well as σpp and the parametersof the parton-parton inelastic scattering profile to repro-duce the inelasticity profile in p+p scattering, in the sameway as for Np = 3, as presented in Appendix A. We findσpp = 22.96, 11.93, 7.32, and 4.96 mb for Np = 2, 3, 4,and 5, respectively. The relation σpp ∝ 1/N2

p holds onlyapproximately for the considered small number of par-tons in the nucleon.

The number of produced hadrons in A+A collisionsis taken as proportional to the number of wounded par-tons. The reduction of the cross section with the num-ber of partons in the nucleon means that the number ofwounded partons approaches the binary scaling in theGlauber model. It means that with decreasing parton-parton cross section the number of produced hadrons as afunction of centrality interpolates between the woundednucleon scaling and the binary scaling. This dependencecan be used to estimate the effective number of par-tons involved in inelastic nucleon-nucleon collisions. Thecharged particle density in Pb+Pb collisions divided bythe number of partons is shown in Fig. 15. We notice thatthe number that best describes the scaling of the particlemultiplicity is 3 or slightly above, where the curve is flatas a function of centrality. This argument gives supportto results for collisions at the LHC energies presented inthe preceding sections, where Np = 3 was used.

VII. CONCLUSIONS

We have explored in detail the Glauber Monte Carloimplementation of the wounded quark model, applied toparticle production and for calculating the characteristicsof the initial state in ultrarelativistic heavy-ion collisions.We have applied the model to p+p, p+A, d+A, He+A,Cu+Au, Au+Au, U+U, and Pb+Pb collisions at RHICand the LHC energies, confirming an approximate linear

0.6

0.9

1.2

1.5

1.8

2.1

2.4

0 100 200 300 400

dNch/dη/PW

〈NW 〉

ALICE,√sNN = 2760 GeV

Np = 2Np = 3Np = 4Np = 5

FIG. 15. The average multiplicity per unit of pseudorapidity,dNch/dη, divided by the number of wounded partons PW forthe Pb+Pb collisions at

√sNN = 2.76 TeV (empty symbols),

evaluated with different numbers of constituent quarks in thenucleon, NW. The filled symbols at 〈NW〉 = 2 correspond top+p collisions.

scaling of the charged hadron density at midrapidity withthe number of wounded quarks.

To constrain the parameters of the effective subnucle-onic degrees of freedom in the nucleon, we reproduce theproton-proton inelasticity profile in the impact parame-ter by adjusting the distribution of quarks in the nucleonand the quark-quark inelasticity profile. We find thatthe effective size of the proton increases weakly with thecollision energy, whereas the growth of the quark-quarkcross section is substantial and is responsible for the in-crease of the inelastic nucleon-nucleon cross section. Themultiplicity distributions in p+p and p+Pb collisions areused to constraint the overlaid entropy fluctuations, heretaken as an additional gamma distribution superposedover the distribution of the initial Glauber sources.

Our conclusions are as follows:

1. The production of particles at mid-rapidity fol-lows the wounded quark scaling, with the quantitydNch/dη/QW displaying approximately a flat be-havior with centrality (cf. Fig. 1). This confirmsthe earlier reports [14, 15, 18, 20] of the woundedquark scaling, but with phenomenologically moti-vated parameters of the quark-quark interaction.At the LHC collision energy of

√sNN = 2.76 TeV

the production per wounded quark in Pb+Pb colli-sions is compatible with the analogous quantity inp+p collisions, which is an essential feature for theconsistency of the approach, displaying the univer-

12

sality of the particle production based on superpo-sition of individual collisions.

2. At lower collision energies, such as√sNN =

200 GeV, the universality is far from perfect andthe obtained scaling is approximate, exhibitingsome dependence on the reaction. Moreover, wenote in Fig. 1 that the corresponding p+p point ishigher by about 30% from the band of other re-actions. This indicates that at lower energies thereare corrections to the independent production fromthe wounded quarks. Also, this may suggest asmaller number of effective subnucleonic degrees offreedom than three at these energies.

3. Average eccentricities and their event-by-eventfluctuations show a similar dependence on central-ity in the wounded quark model and in the woundednucleon model amended with binary collisions ofEq. (1). The average eccentricities are somewhatlarger in the wounded quark model compared tothe wounded nucleon case, reflecting the presenceof additional fluctuations at subnucleonic scales.

4. For collisions of the deformed U+U nuclei at RHIC(√sNN = 200 GeV), we find that the average ellip-

ticity in the wounded quark model flattens at verycentral collisions, as opposed to the wounded nu-cleon case. This is qualitatively in agreement withthe absence of the knee structure in v2 in experi-mental data.

5. Event-by-event fluctuations of the initial size, re-sponsible for the transverse-momentum fluctua-tions, are very similar in the wounded quark modeland in the wounded nucleon model amended withbinary collisions, with wounded quarks yieldingsomewhat larger values for peripheral collisions.

6. The analysis of the distribution of particles at highmultiplicity in p+Pb and p+p collisions at the LHCenergies shows, that the needed overlaid distribu-tion, taken in the negative binomial form, receivesthe same parameters. This is needed for the con-sistency of the approach.

7. For p+Pb collisions, we find that the size and theeccentricity of the fireball is similar as in the com-pact source implementation of the wounded nucleonmodel that best describes the data after the hy-drodynamic evolution. In that way, we reduce theuncertainty in the initial conditions in small sys-tems and obtain a more microscopic motivation formechanism of the entropy deposition in the initialstage.

8. For p+p collisions we generate initial fireballs whichlater may be used in hydrodynamic studies. It isexpected that the resulting triangular flow shouldbe much smaller from the elliptic flow.

9. We have also tested the hypothesis that the effec-tive number of partons in the nucleon is smalleror larger than three. However, particle produc-tion data at the LHC are fairly well describedusing three partons, identified with constituent(wounded) quarks.

ACKNOWLEDGMENTS

We thank Andrzej Bia las, Adam Bzdak, ConstantinLoizides and Jurgen Schukraft for useful discussions. Re-search supported by the Polish Ministry of Science andHigher Education, (MNiSW), by the National ScienceCentre, Poland grants DEC-2015/17/B/ST2/00101 andDEC-2012/06/A/ST2/00390.

Appendix A: Quark-quark wounding profile

In this appendix we present in detail our procedureof fixing the inelasticity profile of the quark-quark col-lisions, based on the experimental NN scattering data.The inelasticity profile in NN collisions is defined via theNN scattering amplitude

pinNN(b) = 4p(Imh(b)− p|h(b)|2), (A1)

where b is the impact parameter, p is the CM mo-mentum of the nucleon, and in the eikonal approxima-tion bp = l + 1/2 +O(s−1) the scattering amplitude ish(b) = fl(p) +O(s−1). Following the lines of Ref. [80],we take a working parametrization for h(b) from Ref. [30],whose functional form is based on the Barger-Phillipsmodel [31].

When the nucleon is composed of three constituentquarks, its inelasticity is a folding of the inelasticity of

0.0

0.4

0.8

1.2

1.6

2.0

0.0 0.4 0.8 1.2 1.6 2.0

2πbpin NN(b)/σin NN

b [fm]

√s = 7000 GeV√s = 200 GeV

FIG. 16. Inelasticity profiles 2πbpinNN obtained from foldingthree quarks (thick dashed line for

√s = 200 GeV and thick

solid line for√s = 7 TeV), compared to the parametrization

of Ref. [31] (thin solid lines), plotted as functions of the impactparameter b.

13

the quark and the distribution of quarks in the nucleon.Here we proceed in an approximate way, using the ex-pressions (3,4). The shape of the resulting NN inelastic-ity profile is determined be dimensionless ratio σqq/r

20,

whereas the normalization, i.e., the NN inelastic crosssection σin

NN =∫

2πbdb pinNN(b), is controlled with σqq. A

sample result of the fit for√sNN is shown in Fig. 16.

Richer functional forms of Eqs. (3,4) would allow for aneven better agreement, but for the present applicationthe accuracy is by far sufficient.

In Table I we give the obtained values of model param-eters for various collision energies. As the experimentalpin

NN(b) may be very well approximated with the Γ profileof Ref. [81],

pinNN(b) = AΓ

(1/ω, πAb2/(ωσin

NN))/Γ(1/ω), (A2)

with Γ(a, x) denoting the incomplete Euler Γ function,we also list parameters A and ω in Table I, as well as theinelastic NN cross section σin

NN.We note that the size parameter r0 in Table I corre-

sponds to the rms radius of the nucleon (including the

CM corrections as obtained from the Monte Carlo sim-ulation) equal from 0.70 fm at

√sNN = 17.3 GeV to

0.84 fm at√sNN = 7 TeV. This range is comparable to

estimates for the size of the nucleon.

TABLE I. Parameters of the quark distribution in the nucleonof Eq. (3), r0, quark-quark inelastic cross section of Eq. (4),σqq, as well as parameters A and ω from Eq. (A2) for variousNN collision energies. The last column lists the total inelasticnucleon-nucleon cross section σin

NN.

√sNN [GeV] r0 [fm] σqq [mb] A ω σin

NN [mb]

17.3 0.25 5.1 0.94 0.83 31.5

62.4 0.27 5.8 0.92 0.79 35.6

130 0.27 6.5 0.97 0.80 38.8

200 0.27 7.0 0.97 0.76 41.3

2760 0.29 11.9 0.99 0.57 64.1

5020 0.30 13.6 1.0 0.53 70.9

7000 0.30 14.3 1.0 0.51 74.4

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