Lepton pairs from a detonating quark-gluon plasma

Volume 168B, number 4

LEPTON PAIRS FROM

PHYSICS LETTERS

A DETONATING

13 March 1986

QUARK-GLUON PLASMA

J. C L E Y M A N S and J. F I N G B E R G Department of Theoretical Physics, University of Bielefeld, D-4800 Bielefeld 1, Fed. Rep. Germany Received 25 November 1985; revised manuscript received 18 December 1985

The situation is considered where a quark-gluon plasma is produced in high energy heavy ion collisions. Because of its rapid expansion this plasma may become supercooled and transform into hadrons via a detonation wave. the cross section for lepton pair production is calculated in this case and compared with earlier calculations where the plasma follows thermodynamic equilibrium. It is concluded that with the detonating mechanism most lepton pairs are producezl inside the hadronic gas resulting from the detonation. To distinguish experimentally between different scenarios it is necessary to measure the angular distribution of the leptons which is predicted to be sin20 in the rest frame of the lepton pair for a detonating quark-gluon plasma.

It is widely agreed upon that lepton pairs form one of the most interesting signals of a quark-gluon plasma in high energy heavy ion collisions [1]. Several experiments are presently being planned focusing on this. Recent theoretical work [2,3] on the subject considers thermal production of lepton pairs in a quark-gluon plasma undergoing rapid longitudinal expansion. During the expansion the temperature changes according to the hydrodynamic equations following from energymomentum conservation. In refs. [2,3] the plasma cools off in an isentropic way and follows a path in the e (energy density) and T (temperature) plane with the system always being in thermal equilibrium. In this letter we show the results following from a very different scenario. We consider the case where the transition from plasma phase to hadron phase proceeds via a detonation wave. There are two reasons why this is worthwhile investigating. First of all, it was recently shown [4] that a detonation is the most efficient mechanism to release the energy density of the plasma phase into hadrons, estimates lead to an energy flux which is several orders of magnitude larger than results obtained for any other known mechanism [5-9]. Second, a detonation wave propagates rapidly through the plasma [4], the velocity being near light velocity. For these reasons, detonations are an interesting mechanism for the transition from quark-gluon plasma to hadrons. The allowed regions in the eq (quark energy 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

density) and eh (hadron energy density) plane have been calculated for the equation of state of the bag model [ 10] and are shown in fig. 1. The curly line corresponds to an isentropic transition, the region below corresponds to the transition we are considering while above the inverse transition occurs. To have a realistic situation we concentrate on that part of the detona-

Physico[ /

B

Forbidden

4.26 ~'~'~l a\ \g\ rDef ation\ 4

\\

Detonation

1.26 1 !

0.26

,

2

I

3.26

~hlB

Fig. 1. Allowed and forbidden regions in the (eq, e h) plane. Indicated are the detonation and deflagration regions o f interest. B is the bag constant. The curly line corresponds to the AS = 0

transition. 405

Volume 168B, number 4

PHYSICS LETTERS

13 March 1986

For the hadronic phase we consider a gas of pions in thermodynamic equilibrium. In this case the corresponding rate is given by

12

B

dNTr/d4x d4q = (ot2/487r4) O(M2 - 4m 2) IF~r(M2)l2

8

X (1 - m2/M2)(1 - 4,,,rr/m ~ 2 / ~ . t 2 ~)3 / 2 (1 + 2m2/M 2) 4

//

/.o

X exp [ - (Mi/r) cosh (y - 0)], 0

o~ - - I |[

[ /1/I Jr----- - - - - ~ - - |

0.53

1.0

15

[ 187

T/Tc

Fig. 2. Path followed by a detonation wave in the (e, T) plane.

where F~r(M2) is the electromagnetic form factor of the pion parametrized with a simple Breit-Wigner form as Fn(M 2)

tion region which requires the smallest amount of supercooling for the quark-gluon plasma. To emphasize this, we reproduce it in the (e, T) plane as shown in fig. 2. The path chosen is indicated by the dashed line. The quark-gluon plasma supercools to a given temperature and through a detonation transforms into a superheated hadronic gas. The temperatures at which this process occurs are determined as, e.g., in refs. [4, 10] from the conservation laws for the energy-momentum tensor

aUTuv = 0,

(1)

and from the local form of the first and second laws of thermodynamics de = T ds +/a dn.

(2)

In what follows we consider the case where the chemical potential,/a, is zero. It is well known that in the Boltzmann approximation the rate for lepton pair production is independent of the chemical potential. To zeroth order in QCD the rate in the quark-gluon plasma is given by dNq/d4x d4q = (or2/12~ -4) O(M2 - 4m 2)

2UM2 - mp2 + i mprp), no.,__

X exp[-

(M±/T) c o s h ( y - 0 ) ] ,

(3)

where q is the four-momentum of the lepton pair, M the invariant mass, 0 the rapidity of the fluid element, y the rapidity of the lepton pair, and the Boltzmann approximation was used * 1. * 1 Our no tation follows that o f ref. [ 2].

406

(5)

with mp and Up being the mass and the width of the p meson. The spacetime evolution of the plasma enters when integrating (3) and (4)over these variables. It is useful to introduce at this point the temperature profile function [1i,12] d~(T, 0) = f d4x 6(T - T(x)) 6(0 - O(x)),

(6)

where T(x) and O(x) specify how the temperature and the fluid rapidity change as a function of space and time. It has been argued that the presence of a plateau structure in the central region of rapidity space of high energy collisions implies boost invariance along the beam direction [13]. For this to be fulfilled it is sufficient to have 0 = Y,

(7)

which implies in the absence of dissipative terms in the energy-momentum tensor the following relations

T = T O(r 0 [r)C2s,

s7 = constant,

(8, 9)

where c 2 is the speed of sound and Y is the spacetime rapidity: Y -_2 1 In [(t +

X (1 + 2m2/M2)(1 - 4 , ~, , #2,/,n, 1 2 a) l / 2

(4)

z)/(t-

z)],

(10)

and r is the eigentime of the fluid element. For a detonation the temperature profile function is shown in fig. 3. It is obvious from this figure that the dominant contribution to the lepton pair production rate comes from the final hadronic gas and not from the quark-gluon plasma. The reason for this result is very simple to understand. Because of the entropy law the transition from quarks and gluons to hadrons is inhibited by the large number of degrees of freedom in

Volume 168B, number 4

PHYSICS LETTERS

~

103 -

13 March 1986

re Profile Function

I02 - ~ Q u a r k s

10 I I

1

0.53

I

0.75

I I I I I

Tfo/ Tc

I

1.0

I

1.5

To/ Tc

1.87 T/T¢

Fig. 3. Temperature prof'de function for a detonation.

the first phase as compared to the second one. In a detonation scenario the quark-gluon plasma becomes substantially supercooled, the temperature therefore becomes too low to produce many heavy lepton paks since the kinetic energy of the quarks is not very high on the average. The hadronic gas, however, is superheated after the detonation and therefore easily produces a large number of heavy lepton pairs. It is not clear what the contribution from the transition region is since this is a situation where one is far from thermodynamic equilibrium. If one argues that this transition is a very rapid one it is probably safe to neglect this. We also neglect deviations from eqs. (3), (4) due to non-equilibrium production of lepton pairs. The results of our calculations are shown in fig. 4. We now compare this to the smooth transition scenario considered in refs. [2,3]. In this case the temperature profile function contains a delta-function contribution from the mixed phase. We have redone the calculation of ref. [3] by taking into account the mixed phase and also production from the ffmal hadronic gas. The lifetime of the mixed phase can be long. An estimate can be obtained if one assumes the transition to proceed as given by eq. (9) SQ TQ = s H TH

102 N• ~b ¢-

I

i

,

I

I

,

i

I

I

1

0 JO

E lO_Z "0 ~E 10-4 "10 L~ "10 10.6 0

I

1

2

3

M [GeV]

Fig. 4. Cross section for the production of muon pairs. Q indicates the contribution from the quark-gluon plasma while H indicates the contribution from the hadronic gas resulting from the detonation.

leading to ~'H = ~ rQ, where TQ is the proper time at the end of the q u a r k gluon plasma phase (a few fermi's) and ~'H is the proper time at the beginning of the hadronic phase. The fac407

Volume 168B, number 4

i

E

PHYSICS LETTERS

10-2

5

6

10_4

F

1

I 2

0 0

-

1 1

o

3

4 ~

"1

~ 2

3

M [GeV] Fig. 5. Cross section for the production of muon pairs calculated for the equilibrium transition. The individual contributions are: (1) pure quark-gluon plasma phase, (2) quark-gluon plasma in the mixed phase, (3) pion gas in the mixed phase, (4) pion gas, (5) result from ref. [3], (6) sum of (1) -(4).

13 March 1986

nario the rate is dominated everywhere b y the contribution from the pion gas. The two models can nevertheless be distinguished experimentally by measuring the angular distribution o f one lepton in the rest frame of the lepton pair with respect to the beam axis. for • the smooth transition it will be given by 1 + cos~0 if the invariant mass is above 2 GeV since it results from the annihilation o f spin 1/2 objects while in the detonation transition it will be 1 - cos20 everywhere as it results from the ejected hot pion gas where spin zero objects create the lepton pairs. This argument is correct provided the expansion is purely in the longitudinal direction. The transverse expansion will tend to make the distribution more isotropic. This could be taken into account in a way similar to the one proposed by Collins and Soper [ 14] for the D r e l l - Y a n production mechanism of lepton pairs• We acknowledge useful discussions with T. Matsui.

References tor 37/3 comes from counting the degrees o f freedom in each phase. The lifetime o f the intermediate phase could thus indeed be very long. Figs. 4 and 5 were calculated for the case o f an o x y g e n - p l a t i n u m collision with a beam energy of 225 GeV. The fact that not all collisions will be central has been taken into account by a geometric averaging over all possible configurations. We consider two spherical objects having different radii which penetrate each other and the q u a r k - g l u o n plasma is produced only in those parts where they overlap• Wherever necessary the calculations for the rates were done numerically using two different algorithms to check the accuracy. A t no stage did we approximate integrands b y delta-functions as, e.g., in ref. [2]. Comparing the rates for each scenario, i.e. figs. 4 and 5 one notices that there is actually very little difference in the overall order of magnitude even though the individual contributions are very different• For example, in the smooth transition scenario the contribution of the q u a r k - g l u o n plasma dominates above invariant masses o f 2 GeV while in the detonation sce-

408.

[1] L. McLerran, Quark Matter 1984, Proc. Fourth Intern~ Conf. on Ultra-relativistic nucleus-nucleus collisions (Helsinki, Finland), ed. K. Kajantie, Lecture Notes in Physics, Vol. 221 (Springer, Berlin, 1985). [2] L. MeLerran and T. Toimela, Phys. Rev. D31 (1985) 545. [3] R.C. Hwa and K. Kajantie, Phys. Rev. D32 (1985) 1109. [4] J. Cleymans, E. Nyk/inen and E. Suhonen, University of Bielefeld preprint BI-TP 85/23. [5] L. Van Hove, Z. Phys. C27 (1985) 135. [6] M• Danos and J. Rafelski, Phys. Rev. D27 (1983) 671; University of Cape Town preprint UCT-TP 7/84. [7] B. Banerjee, N.K. Glendenning and T. Matsui, Phys. Lett. 127B (1983) 453. [8] B. Miller and J.M. Eisenberg, Nuel. Phys. A435 (1985) 791. [9] B. Miller, The physics of the quark-gluon plasma, Leeture Notes in Physics, Vol. 225 (Springer, Berlin, 1985). [10] M. Gyulassy, K. Kajantie, H. Kurki-Suonio and L. McLerran, Nucl. Phys. B237 (1984) 447. [ 11 ] E.V. Shuryak, Phy~ Lett. 78B (1978) 150; Soy. J. Nuel. Phys. 28 (1978) 408. [12] L. Van Hove, CERN preprint TH 4204/85. [13] J.D. Bjorken, Phys. Rev. D27 (1983) 140. [14] J.C. Collins and D.E. Soper, Phys. Rev. D16 (1977) 2219.