Geometrical scaling and the dependence of the average transverse momentum on the multiplicity and energy for the ALICE experiment

Larry McLerran

^a,b,c

, Michal Praszalowicz

^d,∗

aPhysicsDept.,Bldg. 510A,BrookhavenNationalLaboratory,Upton,NY11973,USA

bRIKENBNLResearchCenter,Bldg.510A,BrookhavenNationalLaboratory,Upton,NY11973,USA cPhysicsDepartment,ChinaCentralNormalUniversity,Wuhan430079,China

dM.SmoluchowskiInstituteofPhysics,JagiellonianUniversity,S.Lojasiewicza11,30-348Krakow,Poland

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received31July2014

Receivedinrevisedform19November2014 Accepted21December2014

Availableonline24December2014 Editor:J.-P.Blaizot

We review therecent ALICE data oncharged particlemultiplicity inp–p collisions,and show that it exhibits GeometricalScaling(GS)withenergydependencegivenwithcharacteristicexponentλ=⁰.22.

Next,startingfromtheGShypothesisandusingresultsoftheColorGlassCondensateeffectivetheory, wecalculate^pT ^asafunctionNch includingdependenceonthescatteringenergy W .Weshow that

^pT^{bothinp–pandp–Pbcollisionsscalesintermsofscalingvariable}(W/W0)^λ/(2+λ)√_N

ch/S_⊥ where S_⊥ ismultiplicity-dependent interactionareainthetransverseplane.Furthermore,wediscusshowthe behavioroftheinteractionradiusR atlargemultiplicitiesaffectsthemeanpTdependenceonNch,and makeapredictionthat^pT^athighmultiplicityshouldreachanenergy-independent limit.

©^{2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

Recently, the ALICE Collaboration has published charged par- ticle spectra from p–p collisions at three LHC energies 0.9, 2.76 and7TeV.Thisdata,togetherwithp–PbandPb–Pbdatahasbeen subsequentlyusedtoconstructotherquantities,inparticulartotal multiplicity and mean pT asa function of charged particle mul- tiplicity Nch [1,2]. Thisdependence has beennext testedagainst hypothesisofGeometricalScaling(GS)proposedinRef.[3]witha conclusionthat“ALICEp–pandp–Pbdataatlowandintermediate multiplicitiesarecompatiblewiththeproposedscaling”withsub- stantialdeparture from scalingat largermultiplicities [2].In this Letter weshow that GSworks infact muchbetter, provided one takesintoaccount energydependenceoftheinteraction radiusat fixedmultiplicity.

The paper is organized asfollows. First we introduce the ba- sicconcepts ofGeometrical Scalingfor pT spectraandshow that the ALICEdataexhibit GS over a limitedrageof pT withenergy dependence determined by the characteristic exponent λ=⁰.22.

Thisvalue isslightlylower thanthe one foundfromthe analysis [4]ofsinglenon-diffractiveCMSdata[5],andtheonefound[6]in DeepInelasticScattering(DIS)[7].Next,wederivetheformulafor

*

Correspondingauthor.

E-mailaddresses:mclerran@me.com(L. McLerran),michal@if.uj.edu.pl (M. Praszalowicz).

meantransverse momentumanddiscussmultiplicity dependence oftheinteractionradius.Weuseresultsofthecalculations[8]per- formed within the Color GlassCondensate (CGC)effective theory [9]. Finally we discussthe energy dependence of the interaction radiusandshowthatverygoodscalingofmean pT isseeninthe ALICEdata.Wealsoarguethatthecharacterofthe energydepen- dencechangesforlargemultiplicitieswheretheinteractionradius should reachenergy-independent value.Suchbehaviorhastestable phenomenologicalconsequences.Wefinishwithconclusions.

Multi-particleproductionatlow andmoderatetransversemo- mentaprobesthenonperturbativeregimeofQuantumChromody- namics (QCD). Yet at high energies an overall picture drastically simplifies due to the existence of an intermediate energy scale, thesaturationmomentum Qs(x).Particle productionproceedsvia gluon scattering whose distribution is determined by the ratio pT/Qs(x) where Bjorken x corresponds to the longitudinal gluon momentum. Therefore,on dimensionalgrounds, thegluon multi- plicitydistributionisgivenintermsoftheuniversalfunctionF(

τ

) [11–13]

dN_g

dyd²p_T

=

^S⊥F

( τ )

(1)

where

τ ₌

^p

2 T

Q_s²

(

x

)

⁽²⁾

http://dx.doi.org/10.1016/j.physletb.2014.12.046

0370-2693/©^{2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

Fig. 1. Multiplicitydistributionofchargedparticlesinp–pcollisionsat0.9TeV(blueup-triangles),2.76TeV(reddown-triangles)and7TeV(blackfullcircles)plottedas functionsofpT(left)andasfunctionsofthescalingvariable√

τforλ=0.22 (right).(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferred tothewebversionofthisarticle.)

isthescalingvariable.Here

Q_s²

(

x

) =

Q₀²



x0

x



λ

(3)

isthe saturation momentum, Q0 is an arbitraryscale parameter forwhichwetake1 GeV/c,andforx₀ wetake10⁻³.Ouranalysis ofGSpresentedinthepresentpaperisnotsensitivetotheactual value ofx0 and/or Q0,butonly to thevalue of λ.Bjorken x’sof colliding partons for mid rapidity production take the following form

x

=

^pT

W

.

(4)

HereW=√

s isthescatteringenergyandS_⊥ isatransversearea whichwill be specified later. Eq. (1)exemplifies the property of theparticlespectra knownasGeometrical Scaling(GS)where an observablethatinprincipledependsontwokinematicalvariables, suchasp_TandW (orBjorkenx),dependsinpracticeonaspecific combinationof them through the scaling variable.In Eq. (1) we havesuppressed strong couplingconstant

α

s whose dependence onpT isexpectedtointroduceweakGSviolation.

Eq.(1)appliestothescatteringofsymmetricsystems(pp,AA).

For pA scattering, which we will also discuss here, there are in principletwo saturationscales:theoneoftheproton Q_s^(p)=^Qp, andthat of thenucleus, Q_s^(A)=^QA [14,15]. Thisissue hasbeen discussedinRef.[3]withaconclusionthat forhighenoughmul- tiplicitiesandforcentralrapiditiesthetwoscalesshouldhavethe sameenergydependence,meaningthat Qp/QA=^{const.Thiscon-} ditionisenoughforGSoftheform(1)toholdforpAcollisionsas well.Atestofthisassumptionwillbeprovidedbytheforthcoming pAdataatadifferentLHCenergy.

Geometricalscaling[16]hasbeenintroduced inthecontextof DISatHERAandlaterextendedtoparticleproductioninhadronic collisions[4,17,18].Thesaturationscaleappearsduetothenonlin- eareffectsinpartonevolutionwithgrowingenergy.Thisevolution is in general described by the JIMWLK equation [19] which for large Nc reducesto the Balitsky–Kovchegovequation [20]. These equationspossesstravelingwavesolutionswhichexplicitlyexhibit GS[21].

Agooddescriptionoflargeenergyscattering,orequivalentlyof smallBjorkenx’s,istheeffectivefieldtheory[9]oftheColorGlass Condensate(CGC) (for an introduction andreview seeRef. [10]).

Inthe theory ofthe CGC,hadronsafter acollision stretch inthe longitudinal direction strong gluonic fields that are coherent in

thetransverse planeover theradius 1/Qs.Multi-particleproduc- tion proceeds by the decay of these flux tubes, and it has been shownthatthedominantcontributioncomesfromtheproduction ofgluonswithpT≤^Qs.Thismechanismisabletoexplaindiffer- ent features of highenergy p–p collisions including e.g. negative binomial distribution[22] orridge correlationsin highmultiplic- ityevents[23].In thispaperwe shallusepredictionsofthe CGC effectivetheoryfortheinteractionradiiasfunctionsofgluonmul- tiplicityinp–pandp–PbcollisionsdiscussedinRef.[8].

AnimmediateconsequenceofEq.(1)isthat pT spectraatdif- ferent energies fall on one universal curve ifplottedin terms of thescalingvariable

τ

.¹ ThequalityofGSdependsonthevalueof theexponentλ enteringthedefinitionofthesaturationscale(3).

Inordertodetermineλinamodel-independent wayweemploya methodofratioswhereweconstruct

Rik

( τ ) =

^dN

(

Wi

, τ )

dyd²p_T



_dN

(

Wk

, τ )

dyd²p_T (5)

which,accordingto(1),shouldbe equaltounityifGSispresent.

Inpractice R_ik≈^{1 in awindow}

τ

min<

τ

<

τ

max.Forparticles of small pT (i.e. small

τ

),comparableto ΛQCD and/orpionmass,we donotexpecttheargumentsthatleadtoGStobeapplicable,and forlarge pT weenterintoadomainoflargeBjorken x’s(4)where GSisexplicitlyviolatedandperturbativeQCDtakesover.InEq.(5) wehaveassumedthatthenumberofchargedparticlesispropor- tional to the number of produced gluonsand the proportionally factor does not depend on energy (so-called parton–hadron du- ality). We have checked by explicit calculations of mean square deviations of Rik’s from unity that the best value of λ for the ALICE datathat givesthesmallest

χ

² over thelargestinterval in

τ

is equal to0.22 [24]. This isillustrated in Fig. 1where in the leftpanel weplotdN/dyd²pT asfunctionsof pT andasfunctions of √

τ

for λ=⁰.22 (right panel). We see that spectra at differ- entenergiesoverlapwithinawindowupto√

τ

∼^4.Inordernot to bebiasedby the logarithmic scaleof Fig. 1,we constructtwo ratios R12 andR13 corresponding to the LHC energies W1=^7, W₂=².76 andW₃=⁰.9 TeV,respectively.Theseratiosareplotted inFig. 2whereagainweplotthemasfunctionsof pT (leftpanel) andasfunctionsof√

τ

(rightpanel).Weseerelativelygoodscal- ing where the weakrise Rik’s with√

τ

can be attributed to the residualdependenceofλupon p²_T [17].

The behaviorof ratiosRik shownin Fig. 2 isalmost identical asinthe caseofthe CMSdataanalyzed inRef. [4]. Howeverwe

1 Inwhatfollowsweshalluse√

τwhichforλ=^{0 reducestop}T/Q0.

Fig. 2. RatiosR12(i.e. 7TeVto2.76TeV–reddown-triangles)andR13(i.e. 7TeV to0.9TeV–blueup-triangles)plottedasfunctionsofpT (left)andasfunctions ofscalingvariable√τ forλ=0.22 (right).(Forinterpretationofthereferencesto colorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

havefoundin[4]that λCMS=⁰.27 ratherthan0.22 andthatthe GSwindow extendsto slightlyhigher

τ

.This maybe duetothe differenteventselection(single non-diffractiveatCMSvs. inelas- ticin ALICE)anddifferent pseudorapidity coverage (|

η

|<2.4 at CMS vs. |

η

_|<0.3 at ALICE). In a recent study of GS in prompt photon production [25] the optimalrange ofλ turned out to be 0.22–0.28.These differencesinλ maybe infact dueto some ad- ditionalweak energydependence of the multiplicity distribution (1),like the one of

α

s or some energy dependence ofthe unin- tegratedglue.Moreoverone shouldbareinmindthatthereisno factorizationtheoremformultiparticleproductionink_T-dependent gluon densityformalism. StudyingALICE datawe have found for example[24],thatbetterGSqualityisachievedforthedifferential cross-section,ratherthan forthemultiplicity,withλ∼⁰.32.Fur- therdiscussionoftheseissueswillbepresentedelsewhere[24].

HavingestablishedtheexistenceofGSintheALICEdatawecan nowproceedtotheanalysisoftotalmultiplicityandpT.Inorder tocalculateintegralsoverp_TweneedaJacobian:

p_T

= ¯

^Qs

(

W

) τ

^1/(2+λ)

,

dp²_T

=

²

2

+ λ

Q

¯

_s²

(

W

) τ

^−λ/(2+λ)d

τ ,

(6) wherewehaveintroducedanaverage saturationscale

Q

¯

_s

(

W

) =

^Q0



W

Q₀



λ/(2+λ)

,

(7)

whichcanbethoughtofasasolutionoftheequation

Q_s²

( ¯

Q_s

/

W

) = ¯

^Q_s²

.

(8) Notethatduetoourchoice ofx0=^{10−3 inEq.(3) p}T and Q0 in Eq.(7)areinGeVandW isinTeV.

Itfollowsthat dNg

dy

=

^{A S}_⊥Q

¯

_s²

(

W

)

(9)

where A isanintegralovertheuniversalfunctionF(

τ

) A

=

²

π

2

+ λ



τ

^−2+λλ F

( τ )

d

τ

(10)

andassuch isenergy-independent.Theconstant A can, however, dependonparticlespeciesproducedinthecollision.

TheuniversalfunctionF(

τ

)isnotknownfromfirstprinciples, however,in mostphenomenologicalapplications wheregood de- scriptionofthepT spectraisgivenintermsofTsallisparametriza- tion,onecanshowthat[26]

which is energy-dependent. Here again pT and Q0 in Eq. (12) are in GeV and W is in TeV. It follows that for the ALICE data

τ

max≈^{700, 1300 and 1700 for scattering energies W} = 0.9,2.76 and 7 TeV, respectively. Since n is here numerically the highest power, the contributions of the unphysical tail, i.e. from

τ

>

τ

max,are dumped approximatelyas(1/

τ

max)^n/2−1 multiplied by a small coefficient (n

κ

)ⁿ. Therefore in what follows we shall neglect finite energy effects on the integrals of F(

τ

). We have checkednumericallythatthecontributiontothemeantransverse momentumcomingfromthiseffectisatthepermillelevel.

Eq.(9)canbethoughofasadefinitionofthesaturationscale, whichisessentiallygivenasgluonnumberdensitypertransverse area. One should howeverkeep in mindthat the transverse area itself doesdepend onmultiplicityaswell, sinceitcorresponds to the overlaparea oftwo collidingsystemsatfixed impactparam- eter b. The smaller b the larger S_⊥. This dependence will be of importance in the following, where we discuss mean pT depen- denceonmultiplicity.

ItfollowsfromEq.(9)thatparticlemultiplicityinmidrapidity growslikeapowerofthescatteringenergy,whichisinremarkable agreement withthe LHC data [27]. The power of this growth is solelygivenbytheenergyriseoftheaveragesaturationscale Q¯_s². Numerically for λ=⁰.22, we findthat λ/(2+ λ)⁰.099, which again is in agreement with experimental results [27]. From this simple analysiswe conclude that S_⊥ is energy-independent over theLHCenergyrange(orveryweaklydependentforlargerenergy span).

Whenthisformulaisappliedtominimumbiashadron–hadron collisions, we are basically fixing the averagehadron radius. This average radius seems to be a slowly varying function of energy.

However, and this will be of primary importance in the follow- ing, if we fix dNg/dy and then change energy, then S_⊥ has to change with energy as well in agreement with Eq. (9). This is becausedifferentradiiaresampledatthedifferentimpactparam- eters. Ifwe varythe densityofparticleper unit area, by varying thesaturationmomentum,andthenrequirefixedmultiplicity,we necessarilywillsampledifferentimpactparameterscorresponding todifferentareas.

Inheavyioncollisions S_⊥isequaltothegeometricaltransverse size of the overlap of the colliding nuclei. As such it is related to the centrality of the event and, in consequence, to the event multiplicity.ItislessclearwhatisgeometricalinterpretationofS_⊥ inp–pcollisions.Inamodelwithanimpactparameterdependent saturationscale Q_s^b(b_⊥)[29]wehave:

Q

¯

_s²S_⊥

=



d²r_⊥Q_s^b

(

r_⊥

)

²

,

(13)

where the integral extends over the overlap area S_⊥ of collid- ing protonsata givenimpactparameterb.Itisthereforeobvious that alsoin thecaseofp–p scatteringthereshould bea relation between S_⊥ andmultiplicityinagivenevent.Indeed,thisdepen- dence hasbeen calculated within the CGC framework [8], which

Fig. 3. EnergydependenceofmeanpT. CompilationofdatafromRef.[28].Solid linecorrespondstothepowerlawbehaviorofEq.(14):0.227×^{W0.099,whilethe} dashedlinecorrespondstotheCMSlogarithmicfitaimedat describingalsolow energydata:0.413−0.0171ln s+0.00143ln²s,withW²=s.

predicts that S_⊥ dependson N_ch^2/3 linearly, andthen saturatesat some constant value. This behavior has a simple geometrical in- terpretation: number ofparticles produced in hadronic collisions is proportional to the active overlap volume. Once the maximal volumeisreached, furthergrowth ofmultiplicityis duesolelyto fluctuations.

AveragepTcanbeeasilycalculatedusingEqs.(1)and(6)giving



^pT

 =



pT dNg dyd²p_Td²pT



_dNg

dyd²pTd²pT

= ¯

^Qs

(

W

)

^B

A (14)

where B

=

²

π

2

+ λ



d

τ τ

^12−λ+λF

( τ ).

(15)

Eq.(14)hastwoimportantconsequences.First,itgivesrightaway theenergydependenceof^pT^whichisillustratedinFig. 3where goodagreementwiththedatatakenfromRef. [28]canbeeseen.

Second, at some fixed energy W0 one can express Q¯s(W0) in termsofthegluonmultiplicity(9),whichgives



^pT

|

W0

=

^B A



dN_g

/

dy A S_⊥

(

dN_g

/

dy

) |

W0

.

(16)

Note that for fixed dNg/dy interaction size S_⊥ in Eq. (16) de- pends,asexplained above, onthe referenceenergy W0 andalso ondN_g/dy itself,whichisrelatedtothenumberofchargedparti- clesN_chinthekinematicalrangeofagivenexperiment:

N_ch

=

¹

γ 

y



y

dNg

dy dy

.

(17)

Herethecoefficient

γ

relatesgluonmultiplicitytothemultiplicity ofobservedchargedhadronswithin therapidity intervaly.For ALICEdataused inthispapery=⁰.6.Theinteraction radius R characterizingthe volumefromwhich theparticles are produced andwhichis relatedto S_⊥=

π

R²,dependsin a naturalwayon the third root ofdNg/dy, i.e. R=^R(³

dNg/dy)=^R(√³

γ

N_ch) [8]

andonthecollisionenergyW0.

In the following we shall use a slightly modified formulafor

^pT^{,whichtakesintoaccount}nonperturbativeeffectsandcontri- butionsfromtheparticlemassesencodedinaconstant

α

:



^pT

|

W0

= α + β

√

Nch

R

( √

³

γ

N_ch

)|

W₀

.

(18)

Fig. 4. R(N¹g^/3)for threedifferentenergies.Thelowestsolid(black)curvecorre- spondstotheparametrization(19)atW0=^{7 TeV,whereastwouppersolidcurves} correspondto(19)multipliedbytheenergy-dependent factor(W0/W)^λ/(2+λ)for W=2.76 TeV (red)and0.9 TeV(blue).Highmultiplicitysaturationisschematically depictedbydashedlines.Verticalthindashedlinescorrespondtothehighestmul- tiplicitiesanalyzedbytheALICECollaborationat0.9 TeV(mostleftblue),2.76 TeV (middlered)and7 TeV(mostrightblack).(Forinterpretationofthereferencesto colorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Here

α

andβareconstantsthatdonotdependonenergy.Formula (18)hasbeenproventoworkvery wellinp–pat7TeVandalso inp–Pbcollisionsat5.02TeVattheLHC[3].

The interaction radius R(³

dN_g/dy) has been calculated in Ref.[8].HereweshallusetheparametrizationofRef.[3]:

Rpp

(

x

)

=

⎧ ⎨

⎩

0

.

387

+

⁰

.

0335x

+

⁰

.

274 x²

−

⁰

.

0542 x³ if x

<

3

.

4

1

.

538 if x

≥

³

.

4

[

^fm

],

⁽¹⁹⁾

forp–pcollisionsat7TeVand R_pPb

(

x

)

=

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

0

.

21

+

0

.

47 x if x

<

3

.

5 1

.

184

−

⁰

.

483 x

+

⁰

.

305 x²

−

⁰

.

032 x³ if 3

.

5

≤

^x

<

5

2

.

394 if x

≥

⁵

[

^fm

],

⁽²⁰⁾

forp–Pbcollisionsat5.02TeV.

In our analysis of the 7 TeV ALICE data, it turns out we are only sensitive to radii where we are in the first interval for the dependence of R on multiplicity. This is the region where im- pactparameterisvarying.Intheveryhighmultiplicityregion,the radiussaturatesandatlowerenergiesithappensforlowermulti- plicities.

The energydependenceof(18)followsfromthe generalform givenbyEq.(14).Inordertofindanexplicitformulaformean p_T atanyscatteringenergyW ,i.e. for^pT|W,onehastorecompute R(√³

γ

Nch),but–asaconsequenceofEq.(14)–oneshouldobtain that



pT

|

W

= α + β

√

N_ch R

( √

³

γ

N_ch

) |

W

= α + β



W

W₀



λ/(2+λ)

√

Nch

R

( √

³

γ

N_ch

)|

W₀

(21)

whereW₀ correspondstotheenergyforwhichtheinteractionra- diushasbeencomputed. Forp–p W0=^{7 TeV ifweuse R}pp from Eq. (19), and for p–Pb W0=⁵.02 TeV corresponding to RpPb of Eq.(20).

Eq. (21) implies that the effective interaction radius at fixed multiplicityvarieswithenergyas(W₀/W)^λ/(2+λ)R|W0.Thisisde- pictedinFig. 4whereweplotR(N¹_g^/3)forthreedifferentenergies.

Fig. 5. MeanpTinp–pcollisionsforthreedifferentLHCenergies7TeV(fullblack circles),2.76TeV(fullreddown-triangles)and0.9TeV(fullbluedown-triangles) togetherwith theoretical parametrizationsofEq. (21).Solid linescorrespondto theinteractionradiiR showninFig. 4asthesolidlinesaswell,whereasdashed linesshowthechangeinthemultiplicitydependencecausedbysharpR saturation showninFig. 4asdashedlines.(Forinterpretationofthereferencestocolorinthis figurelegend,thereaderisreferredtothewebversionofthisarticle.)

Thelowestsolid (black)curve correspondsto theparametrization (19)atW0=^{7 TeV,whereastwouppersolidcurvescorrespondto} (19)multipliedbytheenergy-dependent factor(W0/W)^λ/(2+λ)for W=².76 TeV (red)and0.9 TeV(blue).

Itishoweverclearthatthepowerlawincreaseofeffectivein- teractionradius atlow energieshastotamed atsome point.Itis not possible ina model-independent way to find how thisactu- allyhappens.Therefore wehaveassumedasimplistic modelthat radiiatallenergiessaturateatabout1.5 fm,avaluecorresponding totheCGCpredictionat7 TeV.ThisisshowninFig. 4bydashed linescorrespondingtothesharpcut-offoftheinteractionradii.Of course,the sharpcut-off is a verynaive assumption andin real- itytheapproachtoradiussaturationiscertainlymorecomplicated assuggestedbythedisagreementofthe0.9TeVwithfixedradius saturationhypothesis.Thisissuecan bealsoaddressedbyrevert- ingthelogicandbyextractingtheinteractionradiusfromthedata.

AnattemptinthisdirectionisbrieflydiscussedinRef.[30].

Wecannowchecktheseideas againstexperimentusingALICE dataonp–p,p–PbandPb–Pbscattering[1,2].Letusfirstconsider the case whereinteraction radii at different energies saturate at differentvaluescorrespondingtothesolidlinesinFig. 4.

Wehaveusedthep–pdataat7TeVasthereferencefittingto itformula(18)withthefollowingresult:

α =

⁰

.

268 GeV

, β =

⁰

.

153 GeV fm

, γ =

¹

.

138

.

(22) Onewouldnaivelyexpect

γ 

³ 2

1



y

whichforALICEpseudo-rapidity interval|

η

|<0.3 wouldgive 2.5 rather than 1.138. Predictions for other two LHC energies follow from Eq. (21) with no other parameters. The result is plotted in Fig. 5 and one can see good but not perfect agreement with the data. One can observe that the 2.76 TeV points (red down- triangles) at higher multiplicities tend towards the 7 TeV curve, andthattwolast0.9 TeVpoints(blueup-triangles)seemtoshow thesimilartendency. Thisbehaviorcanbe attributedto thefixed valuesaturationofR asdepictedinFig. 4bythedashedlines.The effectoffixed R isshownbythe dashed linesinFig. 5.Onecan seethat2.76 TeVdatafollowquitecloselythefixedsaturationra- dius prediction starting from Nch∼^{17, whereas the 0.9 TeV are} wellbelowthedashedline.

It is important to note at this point that the fit leading to Eq. (22) is performed over the multiplicities measured at 7 TeV that correspond to the mostright (black dashed) vertical linein

Fig. 6. Mean^pT^{inp–pcollisionsat7TeV(fullblackcircles),inp–Pbcollisions} at5.02TeV(fullbrowndiamonds)andinPb–Pbcollisionsat2.76TeV(fullpurple squares)togetherwithparametrizationsofEqs.(18)and(24).(Forinterpretation ofthereferencestocolorinthisfigurelegend,thereaderisreferredtotheweb versionofthisarticle.)

Fig. 4.Onecanseethatthefitisdriventotallybythealmostlinear rise ofR with N¹_g^/3 withsome sensitivitytothecurvaturebefore saturation,anddoesnotdependonthevalueofthesaturationra- dius.Onthecontrary,lowerenergydatainfixedsaturationradius scenario,aremoresensitivebothtothecurvatureandthevalueof the saturation radius forwhich,however, we do not havemodel calculation. Therefore our analysiscan be only qualitative atthis point. With moredata athigher multiplicities one could make a globalfittodisentanglethefunctionaldependenceofR onmulti- plicityinamodel-independent way.

Analogouslywecancalculate^pT^{forp–Pbcollisionsusingthe} samevaluesofparameters(22)withR=^RpPbofEq.(20).There- sultisplottedinFig. 6.ForcomparisonwealsoplotinFig. 6^pT forp–pcollisionsat7TeV.

Finallywewouldliketocheckifmean p_TinPb–Pbcanbealso describedbyformula(18).Unfortunatelythereisnocalculationof the interactionradiusdependenceondNg/dy forheavy ioncolli- sions.Makingtheplausibleassumptionthat

R_PbPb

=

^const

· 

³

N_ch (23)

whichsimplystatesthatthesaturationradiuswhereformula(23) should flatten is much larger than in the case of p–p and p–Pb collisions, andshouldnot play anyrole intheregion wheredata for thelatter reactions are available. We haveperformeda fit to thePb–Pbdatausingthefollowingformula



^pT



PbPb

= α

PbPb

+ β

PbPb

√

N_ch

√

3

N_ch (24)

obtaining

α

PbPb

=

⁰

.

43 GeV,

β

PbPb

=

⁰

.

11 GeV

.

(25) ThedataandthefitarealsoplottedinFig. 6.

Yet another illustration of the mean p_T scaling is shown in Fig. 7 where we plot ^pT ^{as a function of the scaling variable} (W/W0)^λ/(2+λ)√

N_ch/S_⊥bothforp–pandp–Pbcollisions.Wesee quitesatisfactoryscalingincontrarytotheclaimofRef.[2]where the scaling variable has not been rescaled by the energy factor (W/W0)^λ/(2+λ). We cannot superimpose the Pb–Pb data on the plot in Fig. 7 because we do not know the absolute normaliza- tionofRPbPb (23),whichcanbefoundonlybyexplicitcalculation withintheCGCeffectivetheory.

From thissimpleexercise wemayconcludethat mean pT de- pendenceonchargedparticlemultiplicitycanbewelldescribedin anapproachbasedontheColorGlassCondensateandGeometrical Scaling. More understandingis certainly requiredas farasheavy

Fig. 7. MeanpTinp–pcollisionsat7TeV(fullblackcircles),2.76TeV(fullred down-triangles),0.9TeV(fullblueup-triangles)andinp–Pbcollisionsat5.02TeV (fullbrowndiamonds)plottedintermsofscalingvariable(W/W0)^λ/(2+λ)√

Nch/S_⊥. Forp–pW0=^{7 TeV andforp–PbW}0=⁵.02 TeV.(Forinterpretationoftherefer- encestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthis article.)

iondataareconcerned, although anonset ofGSinnuclear colli- sionshasbeenalreadyreported[18,25].Wehaveestablishedthat ALICEdataonchargedparticlemultiplicityinp–pcollisionsexhibit GeometricalScalingwithinareasonablewindowinscalingvariable

τ

withexponentλ=⁰.22.Therearesomedifferencesinthevalue ofλextractedfromdifferentexperimentsanddifferentreactions, however,allresultsfallwithinawindow0.22–0.32.Furtherstudies tounderstandthesefineeffectsareclearlyneeded.Themainfind- ingof thepresentwork concernsthe energydependenceof^pT whichisgivenbytheenergydependenceoftheaveragesaturation scale Q¯s(W).Ourfinal plot, Fig. 7,demonstrates very goodscal- ingof^pT^{bothinp–pandp–Pb}collisions.Newresultsathigher energies,especiallyinthecaseofp–Pb,willprovideanimportant testoftheseideas.

Wehavealsoarguedthattheinteractionradiiatdifferentener- giesshould forlarge multiplicities converge tosome fixed value.

Such tendency is clearly seen at 2.76 TeV and presumably at 0.9 TeV at multiplicities above 20. Our simplistic sharp cut-off modelfailstodescribe0.9 TeVdata,butthatcouldbepresumably curednotaffectingtheotherenergies,byallowingforasomewhat largervalue ofthe saturationradius andcareful modeling ofthe curvaturebeforesaturation.Wefinditquiteunexpectedthatsuch simpleobservable as^pT^{canprovidesuchnontrivial}information ontheenergyandmultiplicitybehavioroftheinteractionradius.

Acknowledgements

This research of M.P. has been supported by the Polish NCN grant2011/01/B/ST2/00492.Theresearch ofL.M.issupported un- derDOEContractNo.DE-AC02-98CH10886.

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