Pseudo-rapidity distributions of charged hadrons in pp and p

s=0.9,2.36,7TeV,thenormalizationfactorisobtainedandthetheoretical

resultsareingoodagreementwiththeexperimentaldata.Then,consideringthein uenceofnucleonhardpartonstransversedistributiononthenumberofparticipantsinpAcollisionsbyaGlauberMonteCarlomethod,wealso

√

givethepredictiveresultsforthemultiplicitydistributionsinpPbcollisionsat

s=4.4TeVwillbe

alsogiven.ThedataattheLHCwillallowtoprobethenucleargluondistributionsatverysmallBjorkenxdo-main(10 4–10 6).Thus,thesemeasurementsareveryimportantfortestingthenatureoftheCGC.Inthispa-per,basingontheCGCformalism,wewillinvestigatethechargehadronmultiplicitydistributionsinppcolli-sionsandgivethepredictiveresultsforpPbcollisions.IntheCGCformalism,thecrosssectionforinclusivehadronproductioncanbegivenbytheconvolutionof

theunintegratedgluondistribution(UGD)ofthepro-ton(ornucleus)fromtheprojectileandthetarget[1,7].Forthechargehadronpseudo-rapiditydistributions,thecrosssectioncanbeobtainedbyintegratingtheinclusiveproductionoverptandaJacobiantransformation[7,8].Correspondingly,theUGDfunctionoftheprotoncanbeobtainedfromthedipole-protonscatteringamplitudebyaFouriertransform.Inthispaper,thesimpleUGDfunc-tionfromtheGolec-BiernatandW¨ustho (GBW)model[9],whichhassuccessfullydescribedboththeHERAandRHICdata,isusedforppcollisions.Throughaχ2anal-ysisoftheCMSdataforppcollisions,thenormaliza-tionfactorKthatdescribestheconversionofpartonstohadronscanbeobtained.

InpAcollisions,thenumberofparticipatingnucle-onsinthecollisionsmustbeconsidered,andthesimpleandappropriatemethodtocalculatethisnumberistheGlauberMonte-Carlo(GMC)approach[10].Atsuperhighenergydomain,thecontributionfromsmall-xglu-onsdominatesthemechanismofinelastichadroniccolli-sionsandthein uenceofthetransversespatialdistribu-tionofhardpartonsinthenucleonmustbeconsideredintheGMCapproach[11–13].Thesimplyandpopularlyusedmethodtoconsiderthehardpartonstransversedis-tributionisconsideredthenucleonasahardsphere.Inthispaper,inordertogiveaaccuratepredictiveresults

d3

p

=K

2p

2s p1(x1,kt) p2

t

pt

dk22tα×(x2,(p k)2t),

(1)

whereCF=(Nc2

1)/(2πNc),x1,2=(pt/

√sisthecenterofmassenergy. pistheunintegratedgluondistributionofaproton.ThenormalizationfactorKcanbedeterminedbyaglobal ttoppdataatvariousenergies.

ThemultiplicitydistributionperunitrapiditycanbegivenbyintegratingEq.(1)dN

S

overpt

d2pdσtEdp2t

dy

=

KCF

p2t

4π2αs(Qs,p)

Q2s,p(x)

,(3)

wherethesaturationscaleistakenas[15]Q2s,p(y)=Q0 x√0

Qexp(±y) λ2

¯,(4)

withand¯theparametersσ0=23mb,Q0=0.6GeV,x0=0.01

λ

=0.205.Therunningcouplingconstantαs,isas-sumedtofreezeatαmax=0.52[16]

α(Q2)=min 12πs,αmax Λ2

,

(5)

whereΛ=0.226.Inordertoaccountforlarge-xe ectsinthegluondistribution,thedistributionfunctionisalwaysmultipliedby(1 x)4.

Tocalculatethedistributionversepseudo-rapidity,oneshouldexpressrapidityyintermsofpseudo-rapidityη

2y(η)=

1

cosh2η+µ+sinhηcosh2

η+µ2

sinhη

,(6)

andtheJacobiancanbeobtainedby

J(η)=

y

s)=0.24/(0.13+0.32

√

sexpressedinunitsofTeV[15].

ForpAcollisions,thesaturationscaleofthenucleuscanbegivenas

Q2y)=Npart,AQ2

s,A(s,p(y),

(8)

whereNpart,Aisthenumberofparticipatingnucleonsin

thecollisions.IntheGMCapproach,thenumberofpar-ticipantscanbegivenby[10]

Npart,A( b)= P(| b rii=1,2...A

|),(9)

where bistheimpactparameterofthepAcollisions,and

theset ri,whichcorrespondstothecoordinatesofthenucleonsinthetarget,canbepickedrandomlyaccordingtoaWoods-Saxondistribution[10,17]

ρ(r)=ρ10

a ,

whereρ0correspondstothenucleondensityinthecenterofthenucleus,Rcorrespondstothenuclearradiusandacorrespondstothe“skindepth”.IntheGMCframe-work,thenucleonsisalwayssimplyconsideredas“hardsphere”(HS),andthefunction

PHS(| b ri|)=Θ(| b ri| dmax),

(10)

where

ds)

max=

s)=52,60,65.75,70.45mbat

√

datadN

dηpp,j

dN data dN

pp,j

dη

dηpp,j

theexperiment.Withχ2min/(degreeoffreedom)=0.2597,we ndthenormalizationfactork(=K/Spp)isequalto√0.098.Thetheoreticalresultsinppcollisionsat

err

denotesthesystematicerrorsin

s=0.9TeV(a),2.36TeV(b)and7TeV

(c).ThesolidanddashedcurvesaretheresultsoftheGBWmodelandtheKLNmodel,respectively.ThedatacomefromCMS[5,6].

dη

pA

= b2

b1

b2

b1

db2πb

dN

,

ppinA

(13)

(b)]A}db2πb{1 [1 σT

(b)isthethicknessfunction[13],andthepseudo-whereT

dN

rapiditydistributionsinpAcollisions,

s=4.4TeVfordi erentcentrality:0–100%

(solidcurve),0–50%(dashedcurve),50%–100%(dottedcurve).Theresultswillbeveri edbytheLHCexperi-mentsinthenearfuture.

Insummary,accordingtotheCGCformalism,wehavecalculatedthehadronpseudo-rapiditydistributionattheLHC.Withtheχ2analysisoftheexperimentaldatainppcollisions,thenormalizationfactorisobtainedandthetheoreticalresultsareingoodagreementwiththeexperimentaldata.Inordertogiveaaccuratepredic-tiveresultsforpAcollisions,thehadronpartonstrans-versedistributioninnucleonisalsoconsideredintheGMCapproach,andthepredictiveresultswillbevali-datedbythefutureLHC

experiments.

Fig.3.Pseudo-rapiditydistributionofcharged

√

particlesinminimumbiaspPbcollisionsat

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