Fluctuations of the transverse energy in Pb+Pb collisions and Jpsi suppression

1、Fluctuations of the transverse energy in Pb+Pb collisions and Jpsi suppressiona r X i v :n u c l -t h /0012003v 2 15 O c t 2001Fluctuations of the transverse energy in Pb +P b collisions and J/suppressionJo rg Hu fner a,b ,Boris Z.Kopeliovich b,c and Alberto Polleri a 1a Institut fu r Theoretische P

2、hysik der Universita t,Philosophenweg 19,D-69120Heidelberg,Germany.b Max Planck Institut fu r Kernphysik,Postfach 103980,D-69029Heidelberg,Germany.c Joint Institute for Nuclear Research,Dubna,141980Moscow Region,Russia.February 8,2008Abstract The observed J/suppression in P b +P b collisions shows a

3、 drop at those large values of the transverse energy E T which arise from ?uctuations.The validity of existing models for J/suppression can be extended into this domain of E T by introducing an ad hoc factor proportional to E T .We propose a formalism in which the in?uence of E T ?uctuations on the

4、J/suppression can be calculated and discuss the conditions under which the ad hoc factor is obtained.The observed dependence on transverse energy of J/production in P b +P b collisions at 158A GeV is the ?rst unambiguous signal for an anomalous mechanism for charmonium suppression,i.e.one which goes

5、 beyond what is already observed in proton-nucleus collisions and in reactions with light ions.In order to identify the detailed nature of the anomalous mechanism,it is important to investigate each of its features using a minimum of adjustable parameters.The past experience has shown that models su

6、cceed to reproduce the data only after several parameters are adjusted.Three years ago,the NA50collaboration has re-measured the J/suppression in P b +P b collisions in the region of large transverse energy E T and has corrected its earlier result 1.Rather than being ?atas a function of E T the new

7、data show a drop at E T 100GeV.For an overview of the ?eld,the data and their interpretation we refer to the reviews 2,3and the proceedings of the Quark Matter 99conference 4.Within the picture of the quark-gluon plasma the newly observed drop is interpreted as the expected onset of J/melting 5.Two

8、other groups,Capella et al.6and Blaizot et al.7,modify their previously successfully proposed expressions for the observed J/suppression by introducing the factor ?(E T )=E T 1Present address :Physik Department,TU Mu nchen,James-Franck-Stra?e,D-85747Garching,Germany.e-mail:polleriphysik.tu-muenchen.

9、deimpact parameter b ,but rather by ?uctuations in E T around the mean value ET (b ),which is deter-mined by the collision /doc/c02bd8727fd5360cba1adb8b.htmlrger values of E T lead to a correspondingly larger suppression.For the comover description by Capella et al.the modi?cation

10、 by the factor ?has a small e?ect and fails to describe thedata2,while the same modi?cation to the cut-o?model by Blaizot et al.quantitatively describes the newly observed break.These partial and full successes,respectively,call for a derivation of the ad hoc formula.This is what is attempted in thi

11、s paper.The expression for the charmonium ()production cross section in A +B collisions is usually written asdABabs N T A 1?exp?abs N T B2In a recent paper Capella et al.8argue,that the drop shown in the data is misleading,since the data are evaluated with respect to minimum bias events,whose E T di

12、stribution di?ers from the E T distribution of events in which also a is observed.The“critical participant density”n c p,a parameter adjusted to n c p=3.7fm?2,represents the density above whichabsorption is100%e?ective.The QGP phase transition may(but must not)be the mechanism for the critical trans

13、ition.In ref.7the-function in eq.(5)is smeared at the expense of a further parameter,which is obtained from a?t to the data:1+tanh(n c p?n p( b, s)S cut F SI( b, s)=3After the?rst version of our paper had appeared on the web,Chaudhuri9posted a paper addressing similar issues.we deal with partons or

14、hadrons.We consider the6-dimensional phase-space for the particles and divide it into cells numbered as i=0,.,k,where the index0is reserved for the cell in whichis found together with its comoving particles.We denote by m i the number of particles in the i-th cell and by p i(m i)their probability di

15、stribution.It will be characterized by a mean valuem i and a variance2i.The probability to?nd m0particles in cell0in an event with a total of M produced particles is then?(M,m0)= m1,.,m k p0(m0).p k(m k)(M?k i=0m k).(8) Note that we do not sum over m0.We also introduce the probability distributionT(

16、E T,M)=e T2?2ETexp?(E T?e T M)2/2?2ET,(9)which describes the correlation between the observed transverse energy E T and the total number M of particles,e T being the mean transverse energy which each particle contributes to the total E T.The correlation between the observed transverse energy E T and

17、 the number m0of particles,which suppressis then given byP(E T,m0)= MT(E T,M)?(M,m0).(10) If Gaussian distributions are used for the correlation functionsT and?and if the sums over M and m0are replaced by integrals,eq.(10)can be evaluated exactly,but a rather cumbersome expression results.For the ca

18、se of many cells,k?1,the result,can be factorized asP(E T,m0)=P T(E T,b)P c(m0,m0,E T).(11)Here the distribution of the?uctuations in E T isP T(E T,b)=122ETexp?(E T?E T(b)2/22ET,(12)whereE T is the mean value of the produced transverse energy for a given value of the impact parameter.In eq.(12)P T d

19、oes not depend on the number m0of particles in cell0because they contribute a negligible amount.However,the distribution function P c of particle number m0does depend on E T viaP S(m0,m0,E T)=122c(13)exp?(m0?m0( b, s)?(E T)2/220,where the dependence on transverse energy is contained in the expressio

20、n?(E T)=1+0(E T?E T(b)m0e TE TET(16)as assumed in refs.6,7.A value for the second factor e TE T/2ETin0can be deduced from a?t of calculated E T distributions(using eq.(12)to the measured one.One?nds values between1and0.7 6,7.Although we have no direct information on the ratio20/m0,the multiplicity d

21、istribution of produced particles in pp collisions are known to be negative binomials10for which20/m0=1+m0a NN collision,a hadron with rapidity y a can arise from each string,provided it covers the rapidity y a.Therefore the probability to?nd a hadron with rapidity y a in the event shown in Fig.1isP

22、(y a)= dy1dy2w1(y1)w2(y2)(y2?y a)+(y a?y1),(17) where each-function refers to the contribution of one string.Furthermore,the probability to observe two hadrons,one at rapidity y a and another of y b isP(y a,y b)= dy1dy2w1(y1)w2(y2)(18)(y2?y a)(y2?y b)+(y a?y1)(y b?y1).Let us identify y a with the ra

23、pidity of the charmonium yand y b with the rapidity y ETof a hadron contributing to the observed E T.The probabilityx for cross talk(x1)is then given by the ratiox(y,y ET )=P(y,y ET)2iandw1(y1)=c e?12(Y?y2),(20) where the normalization drops in the ratio eq.(18).In the experiment under consideration

24、,we areinterested in the cross talk between the comoving hadrons(3y4)and the hadrons in transverseenergy E T(1y ET2.3).Instead of integratingx over these intervals,we evaluate eq.(19)at themean values y=3.5,y ET=1.65.A straightforward calculation then leads tox(3.5,1.65)=0.82.(21) The high value of

25、the coe?cient for cross talk(x=1corresponds to complete cross talk)re?ects the property of the probability distributions w i in that they are largest for those strings which span the full rapidity range.In our calculation,we have neglected the strings involving sea quarks.Since these strings are sho

26、rter,their inclusion would reduce the value ofx.With this result,the factor?(E T)from eq.(14)has to be modi?ed to?(E T)=1+0x E T?E T(b)with0from eq.(15)andx from eq.(19).Eq.(22)is the?nal result of our paper.We now look at the experimental J/suppression as a function of E T for P b+P b collisions at

27、 158A GeV and compare with it the results of several calculations in order to show the importance of the e?ects discussed in this paper(Fig.2).We have repeated the calculation of7with the form given in eq.(7)for the anomalous suppression.However,the expression?=E T/E T in front of n p( b, s) is repl

28、aced by the expression?derived in eq.(22)with=0x which describes the in?uence of the shape of the probability distributions and of the incomplete cross talk.Then=0means that no?uctuations are taken into account while=1gives the maximal in?uence of?uctuations as in 6,7.A reasonable estimate may be=0.

29、8,but we also give a curve for=0.4.While=1and =0.8are compatible with the data,=0.4and smaller values are de?nitely ruled out,provided the e?ect discussed in ref.8is not too important.In this paper we have derived and discussed the phenomenological prescriptions by Capella et al.6and Blaizot et al.7

30、to calculate J/suppression in the region of large values of E T where the transverse energy?uctuates.The derivation has shown which physical parameters determine the strength with which?uctuations in(E T?E T)/E T in?uence J/suppression.These are the width parameters for the multiplicity distribution

31、s of comovers and for the E T distributions and the correlation function for incomplete cross talk.We conclude that the ad hoc factor,eq.(1),holds to a good approximation.Acknowledegement:This work has been supported in part by the German federal ministry BMBF under contract number06HD642.References

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