Correlation between mean transverse momentum and multiplicity of charged particles in $pp$ and $p\bar{p}$ collisions: from ISR to LHC

0.28 MB·

by  E. O. Bodnya

Correlation between mean transverse momentum and
multiplicity of charged particles in pp and pp collisions:
from ISR to LHC
E.O.Bodnya , V.N.Kovalenko†, A.M.Puchkov† and G.A.Feofilov†
∗
4
1 UniversityofCalifornia,Berkeley,USA
∗
0 †SaintPetersburgStateUniversity,Russia
2

n Abstract. We present our analysis of the available experimental data on correlation between mean transverse momentum
a and charged particles multiplicity ( pT -Nch) at central rapidity in pp and pp collisions at √s from 17 GeV to 7 TeV. A
J multi-pomeronexchangemodelbashedoinRegge-Gribovapproachprovidesquantitativedescriptionof p -N correlation
h Ti ch
9 dataandtheirenergydependence.Resultsarefoundtobeinagreementwithstringfusionmodelhypothesis.
2
Keywords: ppandppcollisions,chargedmultiplicity,transversemomentum,correlations,quark-gluonstrings,pomerons,stringinteraction

PACS: 13.85.-t
]
h
p
- INTRODUCTION
p
e
h It came first from the observations in the cosmic rays experiments [1, 2] and it was confirmed later in numerous
[ colliderstudiesthatthemajorityofparticles(pions)producedinhighenergyppandppcollisionshavetheirtransverse

1 momentumpT assmallasafewhundredsofMeV/c.Theexistenceofcorrelationofchargedparticlesmeantransverse
momentum( p )withtheeventmultiplicity(N )wasestablishedinawideenergyrange,changingfromthenegative
v T ch
h i
4 (atlowcollisionenergiesof√s=17GeV)topositiveone(above40GeV),seeacompilationofresultsin[3].Besides
3 this, a peculiardependenceof pT spectra on the eventcomplexity– a tendencyof flatteningof the mean transverse
5 momentum of charged particles with the multiplicity – was established with the growth of collision energy. It was
7 arguedbyVanHove[4]thatthisanomalousbehavior(aplateau-likestructure)ofthemeantransversemomentumas
.
1 afunctionofhadronmultiplicitycouldbeasignalforthedeconfinementtransitionofhadronicmatterin ppand pp
0 highenergycollisions.
4 Recently, the new set of detailed experimentaldata on the mean transverse momentum p versus the charged-
T
1 h i
particlemultiplicityN appeared,showingdramaticdifferenceofcorrelationfunctionsfordifferentcollidingsystems
ch
:
v [5].Inppandp–PbcollisionsatLHCenergies,astrongincreaseof pT withNchisobserved,whichismuchstronger
h i
i thanthatmeasuredinPb–Pbcollisions.Inthelastcaseastrikingflatbehaviourof p -N correlationwasfoundfor
X h Ti ch
thefirsttimeinawideregionofcollisioncentralities.Itisdiscussedin[5]thatforthe ppcollisions,thebehaviorof
r
a pT -NchcorrelationwithNchcouldbeattributed,withinamodelofhadronizingstrings,tomultiple-partoninteractions
h i
andtoafinal-statecolorreconnectionmechanism.Atthesametimeitisnotedin[5]thatthedatain p–PbandPb–Pb
collisionscannotbedescribedbyanincoherentsuperpositionofnucleon-nucleoncollisionschallengingmostofthe
eventgenerators.
However,quantitativeandconsistentdescriptionofbehaviorof p -N correlationinnucleon-nucleoncollisions
T ch
h i
and, in particular,its energy dependence,still remains an open basic problem.It is evidentthat the last one should
be completely understood, on a single approach base, before any use of event-generator calculations for the more
complicatedcases of p–Pb or Pb–Pb collisions analysis. Overviewof differentmodelstrying to explain the effects
observedin p -N correlationsinhighenergyppand ppcollisionscouldbefoundin[3].
T ch
h i
In the present study we continue the analysis [6, 7] of p -N correlation for chargedparticles using available
T ch
h i
data in a wide energy range of pp and pp collisions and basing on the Regge-Gribov approach. It is assumed [8],
[9]thatlow-p multiparticleproductionisaresultafew(n)pomeronexchange.Afterthisexchangehadronsbecome
T
joinedbynpairsofstrings.Fissionofthestringsduringtheseparationofhadronsresultsinproductionofn-pomeron
showers, so that the multiplicity exceeds by a factor n the one-pomeron shower multiplicity. In case of identical
stringsthe p distributionsofchargedparticleshavenodependenceonmultiplicity.However,whenthereispossible
T
interactionbetween strings [10] one may expectthe p distributionsof chargedparticles to be differentfrom those
T
produced by non-interacting strings. It is in the case of interaction between quark-gluon strings in a form of color
strings fusion [11] that new types of particle-emitting sources (strings) might be formed, characterized by a higher
energydensity(describedbystringtensionparametert).Asaresult,thehighervaluesofmeantransversemomentaof
chargedparticlescouldbeexpectedinhadronizationofsuchtypesofstrings.
We are using a modified variant [6, 7] of multi-pomeronmodel [3] with collectivity effects. Similar to [3], both
multiplicity of charged particles and their transverse momenta are treated within the same model where the multi-
pomeronexchangeiscombinedwithSchwingermechanism[12]ofparticleproduction.Thusthecollectivityeffects
like string fusion are included into the model [3] in an effective way by a single parameter b , and the p -N
T ch
h i
correlationappearstobetheintrinsicpropertyofthemodel.
Itisshownbelowthatthisconceptallowstoprovidegoodquantitativedescriptionof p -N correlationpattern
T ch
h i
andvariousobservablesin ppand ppcollisionsandtheirenergydependenceinawideenergyrange.Itisalsoshown
thatthemodelhasthepredictivepowerthatallowsextrapolationtothehighercollisionenergies.Besides,themodel
couldbeextendedtothelong-range p and N correlationswhichmightbestudiedbyusingevent-by-eventdata
T ch
h i h i
inseparatedpseudorapidityintervals.
The paper is organizedas follows: in the next section we give a very brief descriptionof the modified variantof
multi-pomeronexchangemodelwithcollectivityusedinouranalysis.InSection3wedescribehowthevaluesofthe
parametersin the modelare extractedby fits to experimentaldata. In Section 4 we discuss the results obtainedand
showsomepredictions.Finally,wepresentourconclusions.
MODIFIEDMULTI-POMERON EXCHANGEMODEL
Inthepresentstudyweusethemodificationsofthemulti-pomeronexchangemodelwithstringfusion[3]asdescribed
in [6, 7]. We consider the multiplicity and the transverse momenta of charged particles produced in high energy
nucleon-nucleoncollisionsintoagivenpseudo(rapidity)intervald .
Basing on[9], [13] and following[3], the function f(N ,p ;z;k,b ,t) is introducedto describe bothmultiplicity
ch T
andtransversemomentumdistributionsinsoftmulti-particleproductioninhadroncollisions:
f(N ,p ;z;k,b ,t)=C (cid:229)¥ 1 1 exp( z)n(cid:229)−1zl exp( 2nkd )(2nkd )Nch 1 exp p p2T . (1)
ch T wn=1zn − − l=0 l!!· − Nch! ·nb t (cid:18)− nb t (cid:19)
HereC isanormalizationfactorwhichisintroducedinordertoenforcethenormalizationcondition:
w
¥
¥
2p (cid:229) f(N ,p ;z;k,b ,t)p dp =1. (2)
ch T T T
Nch=0 Z0
Thefunction f (seeeq.(1))representscombinedprobabilitydistributionofthenumberofchargedparticlesN and
ch
theirtransversemomentumspectra(p ).Thefirstfactorineq.(1)givesa probabilityofnpomeronsproductionina
T
singleevent[9,13].Parameterzisresponsibleforthedependenceofthisdistributiononcollisionenergy√s.Weuse
thefollowingvaluesoftheparameters[14]:
2Cg sD
z= R2+a log(s), D =0.139, a ′=0.21GeV−2, g =1.77GeV−2, R20=3.18GeV−2, C=1.5. (3)
′
The second multiplier in eq.(1) represents the probability of N particles production in n-pomeron showers.
ch
The Poisson distribution is used with mean being proportional to the number of pomerons n and the width of
pseudo(rapidity)intervald .Parameterkis,accordingly,themeannumberofchargedparticlesperrapidityunitfrom
onestring.
Thelastfactorwasintroducedineq.(1)in[3].ItreflectsSchwingermechanismofparticleproduction[12],which
prescriptsthetransversespectrumofchargedparticlesfromonestringtobeofaGaussianform:
d2N p (p2+m2)
ch exp − t , (4)
dp2 ∼ t
T (cid:18) (cid:19)
wheret correspondstothestringtension.
Asimplifiedassumptionisusedinourwork:wedonotdiscriminatetheparticlespecies,consideringonlyproduction
ofpions.Thustheparticlemassdependenceintheexpressioneq.(1)iseliminated.
Inthecurrentmodel,similarto[3],thestringinteractionisintroducedinaneffectivewaythroughasingleparameter
b responsibleforthecollectivityeffects,thatgivesusaneffectivestringtensionnb t.
p -N correlationfunctioninthepresentmodeliscalculatedas
T ch
h i
¥
f(N ,p ;z;k,b ,t)p2dp
ch T T T
hpTiNch = R0¥ f(N ,p ;z;k,b ,t)p dp , (5)
ch T T T
0
R
or
¥
¥
hpTiNch = ¥ f(Nch,pT;z1;k,b ,t)pTdpT n(cid:229)=1Z0 fn(Nch,pT;z;k,b ,t)p2TdpT, (6)
0
R
where f (N ,p ;z;k,b ,t)p2dp hereisdefinedsimilartoeq.(1),butwithoutthesumsymbol.
n ch T T T
Note,thatitispossibletointegrateoutp dependencefromeq.(1)andobtainthechargedparticlesdistributionand
T
meanmultiplicity:
¥
P(N )=2p f(N ,p ;z;k,b ,t)p dp , (7)
ch ch T T T
Z
0
¥
N (s)= (cid:229) N P(N )=2 n(s) k(s)d . (8)
ch ch ch
h i h i
Nch=0
Thedependenceofthemeantransversemomentumofchargedparticlesonenergycanbecalculatedusing:
¥ ¥
(cid:229) Nch f(Nch,pT;z;k,b ,t)p2TdpT ¥ ¥
hpTi(√s)= Nc(cid:229)h¥ =0NchR0¥ f(Nch,pT;z;k,b ,t)pTdpT =2p Nc(cid:229)h=0hNNcchhiZ0 f(Nch,pT;z;k,b ,t)p2TdpT. (9)
Nch=0 0
R
Oneshouldmention,thatthepicturewheretransversemomentumofchargedparticlesincreaseswithmultiplicity
and collision energyis naturalfor a string fusion model[15, 16]. Accordingto this model, with the increase of the
collisionenergyonemayexpectageneralgrowthinthenumberofmulti-pomeronsinexchangeinnucleon-nucleon
collision. Thusthe total numberof stringswill increase andthey couldstart to overlap,formingclusters. In case of
stringfusiononemayexpecttheformationofnewsourceswith higherstringtensionand,therefore,the increaseof
bothmeanmultiplicityandmeanp ofchargedparticlesemittedfromsuchtypefusedstrings.Itcouldcausenon-zero
T
correlationbetweentransversemomentumandmultiplicitywithallowancetobothpositive[14,17,18]andnegative
[19,20] p -N correlations.
T ch
h i
Thisincreaseinmean p isrelevanttothedegreeofstringsoverlapintransversespacethatischaracterizedbythe
T
stringdensityparameter[15,16]whichisdenotedhereash .Accordingtostringfusionmodel,themeanmultiplicity
t
m andmeanvalueofp2 emittedfromaclusteroffusedstringsaremodifiedcomparedtomultiplicitym andtransverse
T 0
momentumsquared p2ofasinglestring:
0
m =m √h , p2 =p2√h .
0 t T 0 t
Herem and p2aretreatedassomeuniversal,independentonenergyparameters,astheycorrespondtotheproperties
0 0
of single (none-fused)string. Thusone can obtain the followingratio, which in case of the string fusion we should
expectenergyindependent:
m m
0
= (10)
p2 p2
T 0
In this connection,we have to note that it is possible to obtain this ratio with two basic quantities of our model:
k (mean number of particles produced per unit rapidity by one emitter) and the mean value of squared transverse
FIGURE 1. Left: Experimental data on charged particle pseudorapidity density at midrapidity vs. collision energy (see com-
pilation of pp and pp collision data in [21] and the modified multi-pomeron exchange model – solid line (see text). Right:
Energydependence of parameter k obtained inthepresent model usingdataat different energies. Straightline–approximation
byk=0.255+0.0653log√s.
b
momentum n t.Sothebothquantities,asineq.(10),couldbedefinedusingthe p -N correlationfunctionat
T ch
∼h i h i
thegiven√s.Therefore,theconditioneq.(10)couldbecheckedintheframeworkofthepresentmodelbyplottingthe
values
k
R= (11)
b
n t
h i
asafunctionofcollisionenergyusing2parametersextractedfromtheexperimentaldataatthegiven√s.
DETERMINATION OFMODELPARAMETERS
Determination ofthe energy dependence ofthe parameter k
In ourmodifiedvariant[6, 7],contraryto [3], k – themean numberofparticlesproducedperunitofrapidityper
string–isnotconsideredasafreemodelparameterbutassumedto betheenergydependent:k=k(√s).Thisenergy
dependencewasfixedinthemodelbyfittingasetofavailableexperimentaldataonchargedparticlesyieldsvs.√s.
We used the experimentaldata on the energydependenceof the chargedparticle multiplicity density in pp and pp
collisions(seecompilationin[21]).Theparameterk wasdeterminedbyfittingthedataintheenergyrangefrom20
GeVto7TeV,theresultoffittingisshownintheFig.1(left)bysolidline.
ChargedparticlespseudorapiditydensityintheleftplotFig.1wascalculatedbasingontheeq.8,wherethemean
numberofpomerons n(s) wasestimatedatagiven√susingRegge-Gribovapproach,andthepseudorapiditywidth
d wasselectedtomathchthierelevantexperimentalone.Thustheparameterkwasdefinedbyfittingexperimentaldata
(seetheleftplotFig.1).WehavetonoteherethattheISRdata(diamondsintheleftplotFig.1)werenotusedinour
fittingprocedure,sothattheresultsofextrapolationofourmodelareshowninthisregion.
As a result of this fitting parameter k was defined. The values of the parameter k are plotted as a function of √s
in Fig.1 (right). A smooth logarithmic growth of multiplicity density from one string with energy was obtained as:
k=0.255+0.0653log√s.Thisresultisinagreementwithstringfusionmodelpredictions[15,16],wherethegrowth
ofstringtensionwiththecollisionenergyisexpectedforfusedstringsduetothegrowingstringdensity.
FIGURE2. pT -Nchcorrelationfunctionat√s=17GeVinawindowof1.5rapidityunits.Datafrom[22].Curve–resultsoffit
inmodifiedmuhlti-piomeronexchangemodel,usingeq.(5),todefineparametersb andt.Contributionsofseveralfirstterms(n=1,
2,3...)ofprocessesgoingviatheexchangeofnsoftpomeronsarealsoshown.
FIGURE3. ...thesameasinFig.2for√s=31GeV.Data-from[23], y <2.
| |
Determination oftheenergy dependence ofthe parameterst and b
Twofreeparametersofthemodifiedmodel,b -thattakesaccountofstringfusion,andtheaveragestringtension
parameter t, are extracted in our work from the available experimental data on p -N correlation in proton-
T ch
h i
(anti)protoncollisionsatwideenergyregionfrom17GeVto7TeV.Theexpressionforthecorrelationfunctioneq.5
wasusedinourfittingprocedure.Someresultsoffittingof p -N correlationin ppand ppcollisionsatdifferent
T ch
h i
energies,startingfrom17GeVandto7TeV,areshowninFig.2–5(afulldatasetisavailablein[6]).
Parametersb andtwereextractedbyfitting p -N correlationdata,therelevantvaluesareshownasfunctionon
T ch
h i
energyattheFig.6.Bothparametersareshowingsmoothbehaviorwiththecollisionenergy.
For the parametert two subsets are obtained:t = 0.566GeV2 andt = 0.428GeV2 which are veryclose to those
obtainedin[3].Asitwasexplainedpreviously[3],thisfactpointstothesystematiceffectsofthedifferentexperimental
estimatesofthemeanp valuesforlow-multiplicityevents(i.e.forasinglepomeronexchangeregion).Theuppervalue
T
ofstringtensionisusedheret =0.566GeV2astheoneensuringthevaluesofeventmean p valuescorrespondingto
T
FIGURE4. ...thesameasinFig.2for√s=540GeV.Datafrom[24], h <2.4
| |
FIGURE5. ...thesameasinFig.2for√s=7TeV.DatafromCMS[25].Midrapidity,pseudorapidityintervalis h <2.4.
| |
theexperiment(seeFig.9below).
In case of b A smooth behavior of parameter b with energy is obtained and approximated at fig. 6 by b =
b ((1 log√s b 2)b 1).Herewehaveb =1.16s´0.39,b =0.19 0.08,b =2.52 0.03.
0 0 1 2
− − ± ±
RESULTS AND DISCUSSION
Multi-pomeronexchangecontributionsvs.√s
Fig.7containsresultsofourcalculationsofmeannumberofpomerons(solidline)andtheirvariancevs.collision
energy.Onemayseearathersmoothgrowthofthemeannumberofpomeronswithenergy,reachingthevalueabout3
attheLHCregion.Thisincreaseiscombinedwithamuchfastergrowthofvarianceofthemeannumberofpomerons.
Contributionstothecorrelationfunctionofseveralfirstterms(n=1,2,3etc.–seeformula(6))areshownatvarious
energiesintheFigures2-5.Onemayseethatthedominantcontributionofsinglepomeronexchangeisconcentrated
FIGURE6. Energydependenceofparametersb andtextractedbyfittingexperimentaldataon pT -Nchcorrelation.
h i
FIGURE7. Resultsofcalculationsofthemeannumberofpomerons(solidline)andofvarianceofthemeannumberofpomerons
vs. ppcollisionenergy.
in the low multiplicity region, bringingthere the valuesof p from 0.32 GeV/c to 0.4GeV/c (forthe relevant
T
h i ∼
collisionenergiesof17-7000GeV,seeFigures2and5).
With the growing energy the role of the multi-pomeron exchange increases. The number of pomerons and fluc-
tuations in the number of pomerons are growing with energy leading to the observed flattening of the p - N
T ch
h i
correlationfunctions.
We have to note here that the growth of fluctuations in the number of pomerons exchanged in the collision of
protonsattheLHCenergiesbringsrelevantlargefluctuationsinthenumberofparticle-emittingstringsand,therefore,
hasanaturalpredictionoftheexistenceofalong-range p -N correlations.Thelastonesmightbeobservedinthe
T ch
h i
event-by-eventstudiesofobservablesmeasuredinseparatedpseudorapidityintervals.
Charged-particlemultiplicitydistribution
The multiplicity distributions of charged particles formed as the result of processes going via the exchange of n
soft pomeronsin pp collisions at the given energy,calculated in the modified multi-pomeronexchange model, are
comparedtotherelevantexperimentaldataintheFigure8.Calculationsweredoneusingequation(7).Examplesare
FIGURE8. Chargedparticlesmultiplicitydistributionsin ppcollisionsatvariousenergies√s(200GeV,900GeV)[25]and
in ppcollisions(2360 GeV,7TeV)[26]inpseudorapidity interval h <0.5(dots).Theresultsofthemodifiedmulti-pomeron
exchangemodel(lines)arecalculatedwiththemodelparametersb an|d|tasdefinedinFig.1,6.
shownforthecollisionenergiesof√s=200,900,2360and7000GeV.
Wemayconcludeherethattheoveralltendenciesofmultiplicitydistributionsarewellreproducedinwideenergy
range. The model predictions slightly deviate from experimental data, that is observed in the region of the high-
multiplicitytails,howeverthiscouldberelatedbothtosomeassumptionsusedinthepresentcalculationsandtothe
stillpossiblecontributionsofhardprocesses.
Energydependenceof p values
T
h i
Results of calculations in the modified multi-pomeronmodel with collectivity of the mean p values vs. √s are
T
comparedtotheexperimentaldataintheFigure9.Calculationsweredoneusingequation(9).Theparametersofthe
modelarethesameasweredefinedabove(seeFig.1,6).
Energydependenceofstringfusionmodelratiok/p2
T
We also made checks of the string fusion model condition (10). Results are shown in the Fig. 10 approximated
by a straightline. Thus we can concludehere that the collectivity effects, relevantto quark-gluonstring interaction
phenomenon(string fusion),could be consideredas importantin pp and pp¯ collisions and in a wide energyrange,
muchwiderthanitwaspreviouslyexpected.
FIGURE 9. Energy dependence of mean p values calculated in the modified multi-pomeron model with collectivity and
T
comparisontotheexperiment.Data–(blackpoints) wereborrowedfrom[27].Redpoints,approximated byacurve–modified
multi-pomeronexchangemodelwithparametersk,b andtasinFig.1,6.
Ænkæbt, GeV-2
1.4
1.2
1
0.8
0.6
0.4
0.2
0
102 103
s, GeV
FIGURE 10. Relation to string fusion hypothesis: ratio of k- mean multiplicity of charged particles from one string to char-
acteristicstring tension at different ppand pp collision energies. Dots– experimental values from our analysis. Straight line –
approximation.
k
=0.87 0.08. (12)
n b t ±
h i
CONCLUSIONS
Allfeaturesof p -N correlationsexperimentallyobservedinthecentralrapidityregionin ppand ppcollisions,in
T ch
h i
awideenergyrangefromtheISRtoTevatronandLHC,aresuccessfullydescribedintheframeworkofanewvariant
ofthemulti-pomeronexchangemodelinwhichstringcollectivityhasbeenincludedinaneffectiveway.Thecrucial
featureofnewmodelisthesmoothlogarithmicgrowthofmeanmultiplicityperstring(theparameterk)withenergy,
whereasinpreviousworksitwassupposedtobeconstant.Thusthenumberofthemodelparametersisreducedfrom
threetotwo:b parameter,responsibleforthecollectivepropertiesandt stringtension.Bothb andt areshowing
− −
smoothenergydependencewhenextractedfromtheavailableexperimentaldataon p -N correlationsin ppand
T ch
h i
ppcollisions.
The transition fromnegativeto positive p -N correlationpattern and tendencyfor flattening of the p with
T ch T
h i h i
multiplicity and with increase of energy √s from 17 GeV to 7000 GeV in pp and pp collisions are quantitatively
described within this single approach. The experimental data on multiplicity distributions in the whole interval of
collisionenergiesandthe p vs.energydependencearealsowellreproducednumerically.Finally,thepredictionsof
T
thestringfusionmodelarehalsiocheckedfortheenergydependenceoftwomodelquantities-kandb .Alltheseresults
on p -N correlationanalysisin ppand ppcollisionsat√sfrom17GeVto7TeVarefoundtobeinagreement
T ch
h i
withstringfusionmodelhypothesis.
ACKNOWLEDGMENTS
Authors are grateful to D.A.Derkach and V.V.Vechernin for useful discussions and permanent interest. This work
was partially supportedby the SPbSU grant11.38.66.2012.V.N.Kovalenkohas also benefited from Special SPbSU
Rector’sScholarship.
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