String and Parton Percolation

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by  C. Pajares

EPJ manuscript No.
(will be inserted by the editor)
STRING and PARTON PERCOLATION
C. Pajares
Departamento deF´ısica de Part´ıculas Elementais e InstitutoGalego de F´ısica deAltas Enerx´ıas
5 Universidadede Santiago de Compostela, 15782 Santiago deCompostela, Spain
0
0
Received: date/ Revised version: date
2

n Abstract. A brief review to string and parton percolation is presented. After a short introduction, the
a
main consequences of percolation of color sources on the following observables in A-A collisions: J/ψ
J
suppression, saturation of the multiplicity, dependence on the centrality of the transverse momentum
4
fluctuations, Cronin effect and transverse momentum distributions, strength of the two and three body
1
Bose-Einstein correlations and forward-backward multiplicity correlations, are presented. The behaviour

of all of them can be naturally explained by the clustering of color sources and the dependence of the
1
fluctuations of the numberof theseclusters on the density.
v
5
PACS. 2 5.75.-q, 12.38.Mh, 24.85.+p
2
1
1
0 1 Introduction 2 Local parton percolation and J/ψ
5 suppression
0 Whatconditionsarenecessaryinthepre-equilibriumstage
/ to achieve deconfinement and perhaps subsequent quark-
h
gluon plasma formation? This question on the occurence Hardprobes,suchasquarkonia,probethemediumlocally,
p
- ofcolordeconfinementinnuclearcollisionswithoutassum- and thus test only if it has reached the percolation point
p ing prior equilibration has been addressedon the basic of and the resulting geometric deconfinement at their loca-
e two closely related concepts, string or parton percolation tion.Itisthusnecessarytodefineamorelocalpercolation
h
[1]-[2] and parton saturation [3]-[4]-[5].In this paper we criterium [7].
:
v will study the first subject. As we mentioned before, at the percolating critical
Xi Consider a flat two dimensional surface S (the trans- density,1/3ofthesurfaceremainsempty.Hencediscden-
verse nuclear area), on which N small disc of radius r sity in the percolating cluster must be greaterthan 3 ηc .
ar (the transverse partonic or string size) are randomly dis0- In fact numerical studies show that percolation se2tsπri02n
tributed, allowing overlapping. With increasing density when the density of partons in the largest cluster reaches
n N/πR2 (wetakehereS =πR2),clustersofincreasing thecriticalvalue1.72/πr2,slightlylargerthan 31.13.This
siz≡eappear.The crucialfeatureis thatthis clusterforma- 0 2 πr02
result provides the required local test: if the parton den-
tion shows critical behaviour: in the limit N and
→ ∞ sity at a certain point in the transverse nuclear collision
R withnfinite,the clustersizedivergesatacertain
→∞ plane has reached this level, the medium there belongs
critical density. The percolation threshold is given by
to a percolating cluster and hence to a deconfined parton
N condensate. In Fig. 1 the percolation probability and its
ηc =πr02πR2 (1) derivative as a function of η are shown.
LetusapplytheaboveideatoJ/ψsuppressioninA-A
andits value 1.13is determinedby numericalstudies.For
collision [8]. We denote by n (A) the density of nucleons
finiteN andR,percolationsetsinwhenthelargestcluster s
in the transverse plane and by dN (x,Q2)/dy the parton
spans the entire surface from the center to the edge. Be- q
distributionfunctions.(Atcentralrapidityy =0,wehave
cause of overlap, a considerable fraction of the surface is
x = k /√s, where k denotes the transverse momentum
still empty at the percolationpoint in fact, at the thresh- T T
ofthe partons andthus k Q). The localpartonperco-
old, only 1 exp( ηc) 2/3 of the surface is covered by T ≃
− − ≃ lation condition is
discs.
In high energynuclearcollision,the strings orpartons
are originated from the nucleons within the colliding nu- n (A) dNq(x,Q2c) = 1.72 (2)
clei, therefore their distribution on the transverse area of s (cid:18) dy (cid:19) π/Q2
x=QC/√s c
the collision is highly non uniform with more nucleons
and hence more strings or partons in the center than in For a given A-A collision at a fixed centrality and en-
the edge. In this case the value of η becomes higher [6]. ergy, the relation (2) determines Q .
c C
2 C. Pajares: STRINGand PARTONPERCOLATION
1 qq¯pairs in this field. With growing energy and/or atomic
0.9 numberofcolliding particles,the number ofstringsgrows
and they start to overlap, forming clusters. At a critical
y 0.8
bilit 0.7 densityamacroscopicclusterappearsthatmarkstheper-
ba colation phase transition.
o 0.6
Pr The percolation theory governs the geometrical pat-
on 0.5 tern of the string clustering. Its observable implications,
ati 0.4 however, require introduction of some dynamics to de-
Percol 00..23 η sfocrrmibeedsbtryinsgevienrtaelroavcteirolanp,pi.ien,gtshterinbgesh.aviour of a cluster
0.1 c
It is assumed that a cluster behaves as a single string
0
1 1.2 1.4 1.6 1.8 2 2.2 2.4 withahighercolorfieldQ→n correspondingtothevectorial
η sum of the color charge of each individual Q→ string. The
1
resulting color field covers the area S of the cluster. As
n
Fig.1.Percolationprobabilityanditsderivativeasafunction
Q→= n Q→, and the individual string colors may be
of η n i 1
orientePdinanarbitrarymannerrespectivetooneanother,
2 2
the averageQ→ Q→ is zero, and Q→ =nQ→ .
1i 1j n 1
ForPb Pbcollisionat√s=17.4GeV,Q 0.7GeV.
c
The scales−of the charmonium states χc and≃ψ′, as de- Knowing the charge color Q→n, one can compute the
termined by the inverse of their radii calculated in po- particle spectra produced by a single cluster ofsuch color
tential theory, are around 0.6 GeV and 0.5 GeV respec- charge and area Sn using the Schwinger formula. For the
tively, therefore the parton condensate can thus resolve multiplicity µn and average p2T of particles, < p2T >n,
thesestatesandallχ andψ statesformedinside the per- produced by a cluster of n strings one finds [11]-[12]
′
colating cluster disappear. The location is determined by
the collision density. The first onset of J/ψ suppression
in Pb Pb collision at SPS should occur at Npart 125, µ = nSnµ ; <p2 > = nS1 <p2 > (3)
where−theJ/ψ sduetofeed-downfromχ andψ st≃atesin n r S 1 T n r S T 1
′ c ′ 1 n
the percolating cluster are eliminated. Directly produced
J/ψ′s survive because of their smaller radii (leading to a whereµ1and<p2T >1arethemeanmultiplicityandmean
scales of 0.9–1.0 GeV) and its dissociation requires more p2 of particles produced by a single string with a trans-
T
centralcollisions,whichlead to a better resolution,i.e. to verse area S1 = πr02. For strings just touching each other
anincreaseofQc.ForQc =1.0GeV weneedNpart 320. Sn =nS1 and hence µn =nµ1;<p2T >n=<p2T >1 as ex-
TheresolutionscaleofthedirectJ/ψcannotbereac≃hedin pected (simple fragmentation of n independent strings).
S-Ucollisions,thereforeonlyonestoppatternsuppression In the opposite case of maximum overlapping, Sn = S1
is obtained for this case. and therefore µn = √nµ1,< p2T >n= √n < p2T >1, so
ForAu-AucollisionatRHIC,theincreasepartonden- that the multiplicity results maximally supressed. Notice
sity shift the onset of percolation to a higher resolution that a certain conservation rule holds
scale,sothatfromthethresholdon,allcharmoniumstates
µ
including J/ψ s are supressed,starting atN 90,i.e. n <p2 > =µ <p2 > (4)
′ part ≃ n T n 1 T 1
a single step suppression pattern occurs.
ForthecaseIn-IncollisionsatSPSenergies,thethres- and also the scaling low
hold for directly produced J/ψ s is not reached even for
′
the most central collisions, and again a single step sup- <p2 > /µ S =<p2 > /µ S (5)
T n n n T 1 1 1
pression patterm is expected.
In the limit of high density
Adetaileddiscussionandcomparisonwithexperimen-
tal data can be found in references [7] and [9].
η 1
<nS /S >= ,η =N πr2/πR2
1 n 1 exp( η) ≡ F2(η) S 0
3 String percolation − −
(6)
Thus
Multiparticle production is currently described in terms
ofcolorstringsstretchedbetweenpartonsoftheprojectile µ=N F(η)µ ; <p2 >=<p2 > /F(η) (7)
S 1 T T 1
andtarget,whichdecayintonewstringsthroughq q¯pro-
−
duction and subsequently hadronize to produce observed The universalscalinglaw(5)is validfor allprojectiles
hadrons. Color strings may be viewed as small discs in and targets, different energies and centralities, being in
the transverse space, πr2, r = 0.2 0.25 fm, filled with reasonableagreementwithexperimentaldata[13].Asim-
0 0 −
the color field created by the colliding partons. Particles ilarscalingisfoundinthecolorglasscondensateapproach
are produced by the Schwinger mechanisms [10] emitting [14].
C. Pajares: STRINGand PARTO NPERCOLATION 3
Notice that N N N4/3 at central rapidity (at <pT>1 <pT>2 <p >
S ∼ coll ∼ A <pT>1 T 1
the fragmentation region N N N ). As F(η)
S part A
∼ ∼ ∼
N1/3,µ N .
A ∼ A
Therefore,themultiplicityperparticipantdoesnotde-
<p > <p > <p >
pend on the number of participants, there is saturation. T 1 T 3 T n
Numerical studies show a good agreement with SPS and
RHIC data [12]. The prediction for central Pb Pb colli- No fluctuations No fluctuations
sions(N =400)atLHC(√s=5.5GeV)isµ/0−.5N = Fig. 2. Clusterformation forlow, intermediateand high den-
A part
8.6andthetotalchargedmultiplicityperunitrapidity(at sity respectively.
centralrapidity)is1800.Thisnumbersareverysimilarto
theonesobtainedfromHeradata,assumingscaling,using
PQCD and the BK equation [16]-[17].
T
p 6
F
4 Transverse momentum fluctuations 5
4
The behaviour of the transverse momentum fluctuations
3
canbeunderstoodasfollows:Atlowdensitiesmostofthe
particlesareproducedbyindividualstringswiththesame 2
< p > , so the fluctuations must be small. Similarly, at
T 1
large density, above the percolation critical point, there 1
is essentially one cluster formed by almost all the strings
created in the collision and therefore fluctuations are not 0
0 50 100 150 200 250 300 350 400
expected either. Indeed, the fluctuations are expected to Number of participants
bemaximalwhenthenumberofdifferentclustersbecomes
larger,justbelowthe percolationcriticaldensity(seeFig.
2). In this case in addition to the normal fluctuations Fig. 3. FpT(0/0)vsthenumberofparticipants.Experimental
aroundthe meantransversemomentumofa singlestring, datafrom PHENIX at √s=200 GeV. Solid line our results.
there are more fluctuations due to the different average
transverse momentum of each cluster.
5 Universal transverse momentum
Experimentally, it has been measured the quantity
distributions
In order to know the transverse momentum distributions
ω ω <p2 > <p >2
Fpt ≡ Datωar−andormandom ω = p T<p−T > T colnuestnere,edasndthtehefrmageamnensqtautairoendfturnacntsivoenrsfe(mx,opmTe)ntfourmedaicsh-
(8)
tribution of the clusters, W(x), which is related to the
whereω denotesthecorrespondingnormalizedfluc-
random cluster size distribution through Eq. (3). For f(x,p ) we
T
tuations in the case of statistically independent particle assume the Schwinger formula, f(x,p ) = exp( p2x),
emission. T − T
used also for the fragmentation of a Lund string [21], at
InFig.3ourresult[18]comparedwiththeexperimen- first approximation x is related to the string tension or
tal data is shown. A reasonable agreement is obtained. equivalently to the inverse of the mean transverse mo-
There is an alternative explanation based on the occur- mentumsquared.FortheweightfunctionW(x)wechoose
renceatRHIC ofminijets whichwillenhance the pT fluc- the gamma distribution
tuations.Athighcentrality,itiswellestablished,thesup-
γ
pression of high pT particles at RHIC what can explain W(x)= (γx)k−1exp( γx). (9)
the suppression of fluctuations seen at lower centralities. Γ(k) −
According to this picture, at SPS where the production The reason of this choice is the following: In periph-
of minijets is negligible, this behaviour is not expected, eral heavy ion collisions there is almost not overlapping
contrary to the expectations of percolation of strings. between the formed strings and therefore the cluster size
Insteadofp fluctuations,theNA49Collaboration[19] distributionispeakedaroundlowvalues.Mostoftheclus-
T
has measured (< n2 > < n >2)/ < n >) at SPS ters are made of one single string. As the centrality in-
for Pb-Pb collisions−, as a−functio−n of the cen−trality of the creases the number of strings grows, so there are more
collision,showinga maximumat lowcentrality andbeing and more overlapping among the strings and the cluster
1athighcentrality.Thisbehaviourhasnothingtodowith size distribution is strongly modified, according to Fig.
minijets. On the contrary, it is explained naturally in our 4where three cluster distributions correspondingto three
approach [20]. differentcentralitiesofthecollisionareshown.Eachcurve
4 C. Pajares: STRINGand PARTONPERCOLATION
and the normalized p distribution is
T
dN (k 1)F(η) 1
f(p ,y)= − (13)
T dy k <p2T >1 1+ F(η)p2T k
(cid:16) k<p2T>1(cid:17)
The equation (12) can be seen as a superposition of
chaotic color sources (clusters) where 1/k fixes the trans-
verse momentum fluctuations. At small density η << 1,
the strings are isolated and there are not fluctuations,
k . When the density increases, there will be some
→ ∞
overlappingofstringsformingclusters,thefluctuationsin-
creaseandk decreases.Theminimumofk willbereached
when the fluctuations in the number of strings per clus-
ter reach its maximum. Above this point, increasing η,
these fluctuations decrease and k increases. In the limit,
when only one cluster of all strings is formed, there are
not fluctuations and again k .
→∞
The obtained power-like behaviour (p2) k, with an
T −
exponent k related to some intrinsic fluctuations, is com-
mon to many apparently different systems, as sociologi-
cal, biological or informatic ones. Distributions like the
Fig. 4. Schematic representation of the numberof clusters as citations of scientific works, or other complex networks
a function of the number of strings of each cluster at three [25][26] where the probability P(m) of having a given
different centralities (the solid line corresponds to the most nodewithmlinksisdescribedbythefreescalepowerlaw
peripheral one and the pointed line to themost central one). P(m) (m)−k with k related to the fluctuations in the
∼
numberoflinksobeythesamebehaviour.Also,ithasbeen
shown[27]-[28]thatmaximizationofthenonextensivein-
in Fig. 4 can be reproduced by gamma distributions with formation Tsallis entropy leads to the same distribution
different k values. (12).
Moreover, the increase of centrality can be seen as a The universal behaviour indicates the importance of
transformation of the cluster size distribution of the type thecommonfeaturespresentinthosephenomena,namely,
the cluster structure and the fluctuations in the number
xP(x) xkP(x)
of objets per cluster.
P(x) ... ... (10)
→ <x> → → <xk > → ¿From (13) one can calculate
This kind of transformation were studied long time
ago by Jona-Lasinio in connection to the renormalization dlnf 2F(η) p2 >
= − T , (14)
gthroeucpenintrparliotbieasbiilsisetiqcutivhaeloernyt[2to2].aAtcrtaunaslfloyrmanatiinocnrewasheicohf dlnpT (cid:16)1+ F(kη)<pp2T2T>i(cid:17)<p2T >1i
changes cells (single strings) by blocks (clusters) and the
where “i” refers to the different particle especies.
caonrdre<spopn2d>ing.vTarhiaesbeletsrµan1safnordm<atpio2Tn>s1ofofththeetcyeplelsobfytµhne As p2T →0 this reduces to −2F(η)p2T/<p2T >i.
T n This behaviour has been confirmed by the PHOBOS
chain (10) have been used also to study the probability Collaboration.As <p2 > <p2 > <p2 > the
associatedtoeventswhichsatisfysomerequirements[23]. T 1p ≥ T 1k ≥ T 1π
absolute value is larger for pions than for kaons and than
Theγ andkparametersofthegammadistributionare
for protons.
related to the mean x and dispersion of the distribution
Now,letusdiscusstheinterplaybetweenlowandhigh
through
p . One defines the ration R (p ) between central and
T CP T
peripheral collisions as
k <x2 > <x>2 1
<x>= − = (11)
γ <x>2 k f (p ,y =0)/N
R (p )= ′ T c′oll (15)
CP T
We use Eq (7) to take into account the effect of over- f(p ,y =0)/N
T coll
lappingofstrings,andhencef(x,m )=exp( p2xF(η)).
T − T where the distribution in the numerator corresponds to
Therefore we obtain
higher densities η > η. In the p 0 limit, taking into
′ T
1 = ∞dxexp( p2xF(η)) account that 32 ≤ k−k1 ≤ 1 and th→at F(η′) < F(η), we
1+ F(η)p2T k Z0 − T obtain
(cid:16) k<p2T>1(cid:17) 2
γ F(η )
(γx)k 1exp( γx) (12) R (0) ′ <1 (16)
− CP
Γ(k) − ≃(cid:18)F(η)(cid:19)
C. Pajares: STRINGand PARTONPERCOLATION 5
approximatelyindependentofk andk .Asη /η increases,
′ ′
theratioR (0)decreases,inagreementwithexperimen- λ 0.8
CP
tal data. 0.7
As p increases we have
T 0.6
R (p ) 1+F(η)p2T/<p2T >1i (17) 0.5
CP T ∼ 1+F(η′)p2T/<p2T >1i 0.4
and R (p ) increases with p (again, F(η)>F(η )) 0.3
CP T T ′
At large p ,
T 0.2
R (p ) F(η) k′(p2)k k′ (18) 0.1
CP T ∼ F(η ) k T − 0 0.2 0.4 0.6 0.8 1 1η.2
′
Fig. 5. λasafunctionofη.Experimentalpointsareforsemi-
whichmeansthatifweareinthelowdensityregimek >k
′
centralS-Pbcollisions(filledtriangle),18%centralPb-Pbcolli-
and R (p ) > 1, and we reproduce the Cronin effect.
CP T sions(nonfilledcircle)and10%centralPb-Pbcollisions(filled
As we increase the energy, the density increases and on
box) at SPS.
reaches the high density regime where k < k and sup-
′
pressionofp occurs.TheCronineffectdisappearsathigh
T
energiesand/ordensities.Thecriticaldensityatwhichthe
Cronin effect disappears is the same at which the trans-
verse momentum fluctuations presents a maximum.
ω 1
R (p )fortwodifferentparticles,forinstancepand
CP T
π, becomes, at intermediate p ,
T 0
RCpP(pT) <p2T >1p k′−k (19) -1
Rπ (p ) ≃(cid:18)<p2 > (cid:19)
CP T T 1π
-2
As < p2 > >< p2 > , in the high density limit
T 1p T 1π
(Au-Au collisions k′ >k) we expect a ratio larger than 1, -3
as the experimental data show.
As far as we approach the low density limit, the ratio -4
decreases, becoming closer to 1 or even lower. 0 0.2 0.4 0.6 0.8 1 1.2 η
Amoredetailedcomparisonwithexperimentaldataon
Fig. 6. ω asafunctionofη.ExperimentalpointsareforS-Pb
Au-Au, d-Au collisions discussion of the forward rapidity
semicentral collisions and 9% central Pb-Pb collisions at SPS.
regioncanbefoundinreference[24].Anoverallreasonable
agreement is obtained. It is very remarkable that such
agreement is based on the universal behaviour of the p
T
distribution given by Equation (13).
a chaotic source λ = 1, and the production of particles
from several clusters can be seen as the superposition of
6 Bose-Einstein Correlations chaotic sources. In this schemce, λ = ns/n, being ns the
number of pairs produced in the same cluster and n the
T
totalnumberofpairs.Inthisway,λisproportionaltothe
MostofthestudiesoftwobodyBose-Einstein(B.E)corre-
inverse of the number of independent sources (clusters),
lationshavepaidattentionto the parametersR ,R ,
side out
therefore it decreases with the density up to the critical
R and not to the strength of the correlation, defined by
L
percolation value. ¿From this critical value, it increases
the chaoticity parameter
with density. This behaviour is shown in Fig. 5.
C (0,0)=λ (20)
2 Similar considerations can be done concerning the s-
Experimentally, due to Coulomb interference and to trength of the three body B-E correlations,ω. NA44 Col-
the necessaryextrapolationsthere aremanyuncertainties laboration has obtained for S-Pb collisions, ω = 0.20
±
in its evaluation, however, some trend of the dependence 0.02 0.19 and for central Pb-Pb collisions ω = 0.85
± ±
ofλonthemultiplicitycanbeestablished.First,SPSmin- 0.02 0.21. STAR collaboration obtains for central Au-
±
imum bias data for O-C, O-Cu, O-Ag and O-Au collision Au collisions values of ω close to 1. This sharp variation
show that λ decreases as the size of the target increases, from ω = 0.2 to ω = 0.8 1 in a small range of η is eas-
−
from λ=0.92 up to λ=0.16. However for S-Pb and Pb- ily explained in the framework of percolation of strings.
Pb central collisions, where the values of η are larger λ is Nowω becomesproportionaltotheinverseofthesquared
also larger, λ 0.5 0.7. of the number of independent sources (clusters) what can
≃ −
This behaviour can easily explained in the percola- acomodatestrongervariationcomparedtothecaseoftwo
tion framework [29]. Each cluster can be considered as body. In Fig. 6 our result is shown [30].
6 C. Pajares: STRINGand PARTONPERCOLATION
7 Forward-Backward Correlations Acknowledgements. I thank the organizers for such a
nice meeting and E.G. Ferreiro for a critical reading of
An useful observableto check the percolationapproachis the manuscript.This workhasbeen doneunder contracts
the forward-backwardcorrelation measured by the quan- FPA2002-01161ofCICyTofSpain,andPGIDIT03PXIC-
tity 20612PNfrom Galicia.
D =<n n > <n ><n > (21)
FW F B F B
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