Color screening in relativistic heavy ion collisions

Physics Letters B 283 ( 1992 ) 171-173 North-Holland

PHYSICS LETTERS B

Color screening in relativistic heavy ion collisions Tam,is S. Bir6 lnstitut.Fdr Theorettsche Phj'sik, Justus-Lwbtg-Universitdt, 14:6300 Giessen, FRG

Berndt Miiller and Xin-Nian Wang Physics Department, Duke University, Durham, A'C27706, US,,t

Received 20 Fcbruar)' 1992:revised manuscript received 4 March 1992

We calculate the color screening length in a non-equilibrated gluon gas formed by interacting minijets in relativistic heavy-ion collisions. We show that that the screening length is too short at CERN LHC collider energyto permit the formation of independent flux-tubes or strings. The prediction for RHIC energies is somewhat ambiguous.

The physical processes occurring at very early times (r<< 1 fm/c) in violent collisions between heavy nuclei are probably best described in the framework of perturbative quantum chromodynamics (PQCD). It is an interesting question whether those processes, which can be adequately treated by PQCD, can lead to the formation of a locally thermalized system of quarks and gluons, the quark-gluon plasma [1,2]. Such a thermalization scenario, which has been discussed in the context of the QCD parton model [ 35 ], would be in contrast to the often advocated model of independent flux-tubes or strings [6]. It has been argued [7] that the success of independent string fragmentation models [ 8,9 ] in describing the low energy data of hadron-nucleus and nucleus-nucleus collisions is due to the weak interaction between two transversely overlapping strings. However, at extremely high energies, a huge number ofgluons with transverse momenta of a few GeV/c will be produced through semihard parton scatterings. It is apparently questionable whether independent strings can be formed in the presence of such a dense, though not yet equilibrated system of strongly interacting gluons. In order to explore, whether there can exist a preequilibrium stage in string formation governed by non-perturbative physics, we calculate here the color screening length in a dense medium of quarks and gluons produced by semi-hard QCD interactions among partons in colliding nuclei. Our analysis is

based on the HIJING Monte Carlo code [ 10], which models the earliest stage of the nuclear collision as Glauber-type superposition of independent parton interactions determined by PQCD. Because gluon-gluon scattering is by far the dominant semihard process in hadronic interactions, most of the perturbativcly scattered and radiated partons are gluons. We may, therefore, in first approximation initially neglect the influence of quarks on the screening of color forces between scattered partons. Following the standard calculation (in Coulomb gauge) of screening in the time-like gluon propagator in a medium of gluonic excitations [ 1 ], we obtain the following expression for the screening mass: m -~-

~

lira iq~

,0

d3k

q'Vkf{k)

,

( 1)

°

where o~s is the strong coupling constant a n d f ( k ) is the phase space density ofgluons. The Debye screening length of the force between static color charges is given by 2 = 1/m. In the case of the ideal equilibrium gluon gas, f (k) is simply the Bose-Einstein distribution and we can recover the well-known screening mass in an ideal gluon gas m2=4no~T 2. We assume that the above calculation is still valid for a system of non-equilibrated gluons produced in the early stage of heavy ion interactions. The phase space density is then related to the gluon distribution g(k.r, y) = d-'A~ / dk~-dy which can be calculated from PQCD,

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Volume 283, number 3,4

.f(k)-

PHYSICS LETTERS B

2 ( 2;,t)-" 1 g ( k , , )') . g(; I' Ikl

(2)

where kv and y are the transverse m o m e n t u m and rapidity of the initially produced gluons, g o = 16 is the degeneracy factor for gluons, and I" is the volume of the fireball. Because the gluon distribution will initially be anisotropic with respect to the beam axis. the screening mass of a propagating gluon depends on its direction with respect to that axis. At colliding energies corresponding to the future R H I C and C E R N - L H C heavy-ion colliders (w."'s= 0.2 and 6 TeV/nucleon, respectively) P Q C D calculations predict the existence of a central plateau in the rapidity distribution of initially scattered gluons, extending between rapidities _* Y. For simplicity, we assume 1

g(kr,y)=~-~g(kr)[O(y+Y)-O(y-Y)],

(3)

and g(k-r)=d,~.'G/dk 2] is the transverse m o m e n t u m distribution integrated over the whole rapidity range. After partial integration, we then obtain from eq. ( 1 ) for the longitudinal and transverse screening mass. respectively, in central AA collisions, ,

48a~

m ] - = r,R'~g~; -,

m2=mr

s i n - ' ( t a n h Y) f J Y

( l+sinh

,

dkTg(kT)

Ysin-L(tanh Y)

)'

(4)

(5)

where we have set I ' = r,~zRe~ [ 1 I ]. Here r, characterizes the formation time for the initially produced gluons and R~ is the nuclear radius of the colliding nuclei. Using the uncertainty principle argument, we take the formation time to be r. = 1/ (k-r) and ( k v ) is t h e average transverse m o m e n t u m of the produced gluons. Since the total number of produced partons is proportional to the number of binary nucleon-nucleon interactions, g(k-r) has roughly a nuclear dependence of A -'. Therefore, the screening mass grows as .,t -'/~. Notice that for large rapidity range Yat high energies, m~ is very, close to m~. From the above equations, we see that all we need now to calculate the screening mass is the transverse m o m e n t u m distribution which can be obtained from PQCD. To have a rough estimate, we first assume an exponential distribution, g(kl ) = N,,exp( - kr/ko). 172

11 June 1992

with an average transverse m o m e n t u m ( k r ) = 2 k . and the total number of produced gluons : V , = 2N.k{~. Substituting these into eq. (4), we obtain m~ ~ 3zta~ N~;

(6)

R~ e r Therefore, the pre-equilibrium screening mass is only directly related to the rapidity density of the initially produced gluons. At low energies (x.,s < 50 G e V / n u cleon ), where the minijet cross section is very,, small, we can estimate ,'\'c~~ 2aj~,, (po) T , , (b), where T.,, ( b ) is the nuclear overlap function at an impact parameter b and Po is a low Pl cutoff for minijet production [10]. At CERN-SPS energy, x.."s=20 GeV/nucleon. a~.,(p.=2 G e V ) ~ 0 . 2 mb in a rapidity range Y=2. For central A u + A u collisions, T ( 0 ) ~ 30/mb. we have i\'¢;- 12 with p 1 > 2 GeV, leading to l / m ~ ~ 2 fm. Therefore, the effect of semihard production of gluons is much too small to invalidate the independent string or flux-tube picture at CERN-SPS energies. At high energies, the cross section of semihard scatterings becomes large. We have to consider other nuclear effects and the initial and final state radiation. Shown in fig. 1 as histograms are the results of the calculation using the H I J I N G [ 10] Monte Carlo model for central A u + A u collisions. In this model, binary approximation for the independent parton scatterings is assumed and an effective nuclear dependent structure function is also included to take into account thc parton shadowing effect. Initial and final state radiation is treated implicitely via PQCD. Due to the initial and final state radiation, the transverse m o m e n t u m distribution of produced gluons at R H I C energy is approximately falling exponentially [ 5 ]. We want to emphasize that this is not in contradiction with the power law behavior of the jet differential cross section whose transverse energy is the sum of all the produced partons in a finite phase space. However. at C E R N - L H C energy, even the total gluon distribution has an apparent power law tail at high kr. To simplify our calculation here, we parametrize the H I J I N G calculation at the R H I C and LHC energies respectively as given in table I and shown in fig. 1 as dashed lines. Listed in table 1 also are the rapidity range Y. total number of produced gluons Nc~ and their average transverse m o m e n t u m (,k-r>. We note that both the total number of initially produced gluons

Volume 283. number 3,4

104

I

IOs

I

PHYSICS LETTERS B

1

1

I

I

"~.

,.~

,

lO-a

2, = 0 . 1 3 f m .

I

LHC

tO ~

~o-,

I

~\

%', I

1

RHIC

t~:%~

Au+Au (b=0) 1

l

I

I

l

1

I

I

1

l

2

3

4

5

6

7

8

9

10

kT (GeV/e) Fig. I. Transverse momentum distributions ofinitiall)i produced gluons at RHIC ( vS = 0.2 TeV/nucleon ) and LHC ( \ s= 6 TeV/ nucleon ) energies as calculated from HIJING Monte Carlo model (histograms). The dashed lines are the parametrizations as given in table 1. Table 1 The rapidity range Y, parametrization of transverse momentum distribution g(kr ), total number of produced gluons NG, and the average transverse momentum
Y

(TeV/nucleon)

g(kl)

A~

(GeV- 2)

0.2

2.5

6.0

5.0


500 exp ( - kT/0.9 ) kl +0.3 9× 10~' (kT+2.9) 64

2~=0.4fm

).-r=0.13fm(LHC),

(8)

where we h a v e used the value o~,=0.3. T h e values p r e d i c t e d for L H C energy are small due to the very high density o f initially scattered gluons at this energy. The,,' indicate that one c a n n o t expect the form a t i o n o f e x t e n d e d c o h e r e n t color field configurations, such as color flux-tubes with a radius o f a b o u t 0.5 fm [ 12]. in such an e n v i r o n m e n t . C o l o r forces will be screened before n o n p e r t u r b a t i v c infrared aspects o f Q C D set in. T h e s i t u a t i o n at R H I C energies is s o m e w h a t a m b i g u o u s since the screening length is not m u c h s m a l l e r than the radius o f an i n d e p e n d e n t flux-tube. H o w e v e r , as c o m p a r e d to the e s t i m a t e o f the screening mass in t h e r m a l Q C D . 1 / m , , ~ 0.36 fm at 7 = 2 0 0 MeV and ( ~ = 0 . 6 , it is still not unlikely that a q u a r k - g l u o n plasma could be created directly f r o m the rescattering o f the gluons initially p r o d u c e d in s e m i h a r d collisions. T h i s work has been s u p p o r t e d in part by the U S D e p a r t m e n t o f Energy ( G r a n t D E - F G 0 5 9 0 E R 4 0 5 9 2 ) , by the N o r t h C a r o l i n a S u p e r c o m p u t ing Center, the B u n d e s m i n i s t e r i u m f'tir F o r s c h u n g und T e c h n o l o g i e , and by the Gesellschafl f'tir S c h w e r i o n e n forschung.

References

(GeV) 570

1.12

8113

1.76

and t h e i r a v e r a g e t r a n s v e r s e m o m e n t u m at L H C energy are m u c h larger t h a n at R H I C . W i t h the a b o v e input o f the H I J I N G p r e d i c t i o n s o f initially p r o d u c e d gluon d i s t r i b u t i o n we o b t a i n the following values o f the screening length for central Au + Au collisions: 2.=0.38fm,

11 June 1992

(RHIC),

(7)

[ 1 ] B. Mtiller, The physics of the quark-gluon plasma, Lecture Notes in Physics ( Springer, Berlin. 1985 ). [2] L. McLerran, Re',. Mod. Phys. 58 (1986) 1021. [3] R. Hwa and K. Kajantie, Phys. Rev. Lett. 56 (1986) 695. [4] JP. Blai:,ot and A.H. Mueller, Nucl. Phys. B 289 (1987) 847. [5] K. (ieiger and B. Miiller, Nucl. Phys. B 369 (1992) 600. [6] See e.g.T. Matsut, Nucl. Ph?s. A 461 (1987) 49c. [71M. Grabiak, J.A. Casado, and M. Gyulassy, Z. Phys. (" 49 ( 1991 ) 283. [81 B. Andcrsson, G. Gustafson. G. lngclman and T. Sj6strand, Phys. Rep. 97 (1983) 31. [9] A. Capella, U. Sukhatmc and J. Tran Thanh Van. Z. Ph),s. C 3 (1980) 329. [ I 0 ] X.-N. Wang and M. Gyulassy, Phys. Rev. D 44 ( 1991 ) 3501. [ l 1 ] J.D. Bjorkcn. Phys. Re'.'. D 27 (1983) 140. [ 12] K. Sailer et al.. Ph),s. Lett. B 240 (1990) 381.

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