PHYSICAL REVIEW D VOLUME 38, NUMBER 9 1 NOVEMBER 1988

Muon pairs and photon pairs as signals for the hadronization phase transition at finite baryon density

G. Janhavi and P. R . Subramanian Department of Nuclear Physics, University of Madras, Madras 600 025, India

(Received 4 March 1988; revised manuscript received 8 July 1988)

In an isentropic expansion of a baryon-rich quark-gluon plasma (QGP) formed in heavy-ion col- lisions and undergoing a first-order phase transition into a hadron-resonance gas, it is found that an analysis of the production cross sections of dimuons in the region where the invariant mass M is greater than 2.25 GeV is enough for the detection of a QGP, whereas an analysis in the region with M Z 7.5 GeV is required for photon pairs. While the production cross section of dimuons is found to be almost independent of the equation of state used for the QGP, that for photon pairs depends on the equation of state of the QGP. We conclude that for the detection of the QGP muon pairs are better signals than photon pairs, whereas for a study of the equation of state of the QGP photon pairs seem to be more suitable than muon pairs.

I. INTRODUCTION

There is a s~ecula t ion that normal nuclear matter un- dergoes a phase transition to a quark-gluon plasma (QGP) at very high densities and temperatures, and it is expected to occur in ultrarelativistic heavy-ion collisions. Although the order of the phase transition is not yet clearly known, one can find it to be a first-order phase transition if chemical equilibrium between the different hadronic species in the hadron-resonance-gas (HRG) phase and quarks in the Q G P phase is a s ~ u m e d . ' , ~

Some of the signals suggested for the existence of a QGP phase are the following: (i) pf p- and y y produc- ti0x-1,~~~ (ii) production of antimatter clusters,* (iii) strangeness production,5 (iv) charm production,6 (v) J / 4 suppression,' (vi) the slope parameters of K + and K - en- ergy spectra from relativistic heavy-ion collision^,^ (vii) multiplicity fluctuations in high-energy heavy-ion col- l i s i o n ~ , ~ etc. We are interested in the first one: namely, the dilepton (muon pairs) and photon-pair productions. Since the photons and leptons do not have strong interac- tions with the constituents of the QGP, they are easily emitted outside the Q G P once they are formed inside and hence may serve as better signals.

Recently, the invariant-mass distribution of the pro- duction rates of lepton pairs'0p1' and photon have been calculated for a baryonless plasma which is likely to be formed in the midrapidity region of collisions with El,, > 100 GeV/nucleon. However, a baryon-rich plasma which is more realistic may be attained at much lower energies ( - 10 GeV/nucleon). The purpose of this paper is to study the invariant-mass distributions of dimuon- and photon-pair production rates in the QCD phase transition at finite baryon chemical potentials. We calculate these rates for two equations of state for the Q G P phase as in Ref. 1.

Our calculation differs from those of the o t h e r s " ~ ' ~ in two ways. First, we consider the baryon-rich region (p,# 0 ) of the plasma which is formed at much lower en-

ergies than that needed to produce a baryonless plasma (pq =0) ; second, the time evolution of the temperature is different from that given in Refs. 10 and 11. Whereas the temperature remains constant throughout the expansion in the two-phase coexistence region in the above refer- ences, in our case' the temperature rises in the coex- istence region. This rise in temperature is due to the emission of latent heat which occurs in a first-order phase transition from the QGP phase to the H R G phase. Moreover, one of the assumptions in the dynamics of the QCD phase transition is the conservation of entropy and baryon number. At any point ( p c , T c ) on the phase- transition curve, the number of degrees of freedom and hence the specific entropy in the Q G P phase are larger than those in the H R G phase. Therefore, to conserve simultaneously both the total entropy and baryon num- ber, the temperature should increase during the hadroni- zation process. Heinz, Lee, and Rhoades-Brown also have considered signatures for deconfinement at finite baryon density.' Our discussions are applicable to the nuclear fragmentation regions and will be relevant to the future experiments at the Brookhaven Alternating Gra- dient Synchrotron (Au + Au collisions at 15 GeV/nucleon) and the CERN Super Proton Synchrotron.

The paper is organized as follows. In Sec. I1 we con- sider briefly the dynamics of the hadronization phase transition. We present the expressions for the dimuon production rates including the chemical potential of quarks in Sec. 111. Section IV provides expressions for photon-pair production rates incorporating the finite baryon density. Our conclusions are presented in Sec. V.

11. DYNAMICS OF THE QCD PHASE TRANSITION AT FINITE BARYON DENSITY

In this section we discuss briefly the dynamics of the hadronization phase transition following Ref. 1. We con- sider a quark-gluon plasma formed in relativistic heavy-

2808 @ 1988 The American Physical Society

38 - MUON PAIRS AND PHOTON PAIRS AS SIGNALS FOR THE. . . 2809

ion collisions which undergoes a first-order phase transi- V( t )= V(t ,) t / to , (2.1) tion into a hadron-resonance gas assuming the conserva- where to is the time at which the expansion of the blob of tion of total entropy and total baryon number during the Q G p starts. expansion process. We also assume Bjorken's scenario of The pressure of the QGP phase is determined by the an expanding system in which volume is directly propor- equation of state for the Q G P which is derived from the tional to time: thermodynamic potential f iQGP:

where p,- =pu =pd =p9 is the quark chemical potential, B is the bag constant, a, (p9 , T ) is the running coupling con- stant, and V is the volume of the blob of QGP. The first term on the right-hand side of Eq. (2.2) is the gluon contribu- tion to the pressure of the QGP. The baryonic chemical potential pb =3pq. The baryon density, entropy density, and the energy density are obtained from Eq. (2.2):

When pq =0, our expressions for P Q G p , EQGpI and sQGp reduce to those given in Refs. 10 and 11.

The H R G phase, which is assumed to be a mixture of mesons, baryons, and their resonances of mass below 2 GeV, is described by the appropriate relativistic Bose- Einstein and Fermi-Dirac distributions. T-he calculations also include the Hagedorn's pressure-ensemble correction for the finite size of hadrons.

If h, and hh denote the probabilities for the system to exist in the pure QGP phase and the pure H R G phase, respectively, then the following equations are satisfied:

and

Using the above equations, one can calculate the time evolution of the temperature T ( t ) , the quark chemical potential p q ( t ) , the probabilities h , ( t ) , baryon densities p, ( t 1, and the energy densities e i ( t ) , where i =q , h , in the pure Q G P phase, mixed phase, and the pure H R G phase. These quantities will be used in the calculation of the pro- duction rates for dimuon and photon pairs which are studied in the following sections.

111. DIMUON PRODUCTION

The most dominant channel for dimuon production in the QGP phase is supposed to be qq-p+p-+X. Here

q = u , d only. In the pure H R G phase the predominant channel for dimuon production is considered to be .rr+.rr--p+pFL- +X. For convenience, let us define the following differential operators:

a),,=d/d4x d M 2 ,

D,, = d /dy d M ,

and the quantities

NQ: + =number of muon pairs produced 99 P P

in the Q G P phase through

the process qq-p+p- +X ,

N"_ + _ =number of muon pairs produced 99 P P

in the mixed phase through

the process qq-p+p- + X , (3.3b)

N~~~ + =number of muon pairs produced Ti+"---p p

in the H R G phase through

the process . r r + r P + p + p - + X , ( 3 . 3 ~ )

~ r n i x e d + - +P+P- =number of muon pairs produced Ti Ti

in the mixed phase through

the process . r r+ . r r - -+p+pp+X . (3.3d)

Now, the number of muon pairs in a space-time volume d4x and of invariant mass M is given by3

2810 G . JANHAVI AND P. R. SUBRAMANIAN - 3 8

x , u ~ P + P = [ 5 T ' ~ ' o / 3 ( 2 r ) ~ ] section for the annihilation of qg pairs at temperature T into lepton pairs and is given by

x Jocdx dy[exp(x -r)+ 11-I o=(4.rra2/M2)( 1 +2m?/M2)( 1 - 4 m ? / ~ ~ ) ' / ~ . (3.5)

x [exp(y + z ) + l]- '8(xy -u2 /4 ) , Here a is the fine-structure constant and m,=0.106

(3'4) GeV=m,, the mass of the muon. After the y integra-

where z = p q / T and u =M/T. o is the elementary cross tion, Eq. (3.4) reduces to

Similarly, the number of dimuons in a space-time volume d4x and of invariant mass M in the pure H R G phase is given by3

where

(The chemical potential of pions is zero.) The pion form factor F, is given by

/ F,(M~) 1 2 = m ~ / [ ( r n ~ - ~ 2 ) 2 + m ~ r 2 ] , (3.9)

mp=0.776 GeV, and T=0.155 GeV . Since the Bjorken volume element is

where t is the proper time, y is the space-time rapidity, and x , are the transverse coordinates, we get the differential cross sectionlo

where f can be any one of the quantities given in Eqs. (3.3a)-(3.3d). We now get the following differential cross sections for the production of dimuons.

(i) Pure QGP phase:

~ B ~ , N r ~ ~ + ~ _ ( T ( t ) , p ~ ( t ) ) , (3.12)

where to is the proper time at which the hydrodynamic expansion of the blob of QGP starts and t, is the proper time at which the system leaves the pure Q G P phase.

(ii) Coexistence region (QGP):

where th is the proper time at which the system leaves the coexistence phase and enters the pure H R G phase. Note

that the integrand contains a weight factor A, ( t ) . (iii) Coexistence region (HRG):

where A h ( t ) = l - h q ( t ) . (iv) Pure H R G phase:

where tf if the proper time at which the system reaches the freezeout temperature.

As in Refs. 10-12, to is assumed to be 1 fm/c. Other proper times depend on the equation of state and the dy- namics of the hadronization phase transition. Their values are given below for the two equations of state con- sidered: (i) t, = 2 fm/c, th = 6.2 fm/c, and t 19.6 fm/c S" for B =400 ~ e ~ / f m ~ , e ( t0 )=4 .2 GeV/fm , p/po=6.5, and S / A =20 (stiffer equation of state); (ii) t, -2 fm/c, th = 5.4 fm/c, and t -24.8 fm/c for B = 250 ~ e ~ / f m ~ , l - 4 t 0 ) = 2 . 6 GeV/fm , p/po=7.0, and S / A = 1 1 (softer equation of state). The S / A values considered by us are likely to be attained in the forthcoming experiments at Brookhaven National Laboratory and CERN. More- over, we have

where R , is the nuclear radius. In our calculations we consider Au + Au collisions.

We calculate the cross sections given in Eqs.

MUON PAIRS AND PHOTON PAIRS AS SIGNALS FOR T H E . . . 2811

- - - - Q G P - - - - - , H R G

M ( G e V ) 0 1 2 3 4

FIG. 1. Dimuon production cross sections as a function of the invariant mass M of the muon pairs for B =400 MeV/fm3 and S/A =20. The dashed-dotted line corresponds to the con- tribution from the process qq+pfp-+X. The dashed line gives the contribution from the . r r + ~ annihilation and the solid line denotes the sum of the quark and hadron contributions.

0

FIG. 2. As in Fig. 1, but for B=250 ~ e ~ / f m ' and S/A=11.

-

, \ ,' \ - - QGP

-----. H R G \

(3.12)-(3.15) numerically. The total contribution to the D total total + - =DyMN -- yM P P 49 P+P-

dimuon production cross section from q?j annihilation is +D total obtained by adding Eqs. (3.12) and (3.13) and is given by YM ,+T-+P+P- ' (3.19)

D total mixed yM 44-P-P- = D y M N ~ ~ p + p - + D y ~ N q q + P + P - .

(3.17)

By adding Eqs. (3.14) and (3.15), we obtain the total muon-pair production cross section from rf r- annihila- tlon:

The total lepton-pair production cross section is

The contributions arising from different regions to the di- mensionless quantity 1n(M2~, ,N) are shown in Figs. 1 and 2 for the two equations of state considered.

IV. PRODUCTION RATES FOR THE PHOTON PAIRS

The photon-pair production in the pure QGP phase is mainly through the elementary reaction qq-+2y. In the pure HRG phase, the photon-pair production proceeds mainly via r + r - - 2 y . The number of photon pairs in a space-time volume d4x and of invariant mass M in the QGP and HRG phases, respectively, are given

where

h = m q / T , W=M2/2m;-1, P = ( W ~ - - ~ ) ' / * , and z = p q / T .

The annihilation cross section for the qtj-2y process is4

I a 49-*Y - ( ~ * ) = [ 1 3 6 ~ a ~ / 2 7 ( ~ * - 4 r n ~ ) ] ( 1 + 4 r n ~ / ~ ~ - 8 r n ~ / ~ ~ ) l n -4rni /M2) ' /*] -1

2812 G. JANHAVI AND P. R. SUBRAMANIAN

where

h=m, /T , w=M2/2rn:--1, and P=( ~ ~ - 1 ) " ~ .

For the 7~+a-+2y process, the annihilation cross section is

Using Eq. (3.1 1) we get the following differential cross when the system expands isentropically. This implies sections for the photon-pair production in the QGP that to produce high-mass muon pairs, high temperatures phase, mixed phase, and the H R G phase. are necessary which are available in the initial QGP

(i) Pure QGP phase: phase. At low temperatures (HRG phase) many low- mass muon pairs are produced. In the region 1 < M < 2.25 GeV both the phases coexist and contribute

D Y M NQGP qq-2y =Jt:t d t d 2 x l to the production rates. We have found that almost the same range of M is obtained for the cases (i) S / A =20,

x 2 ) x M N z 4 2 y ( T(t)!pq(f)) . (4.7) B =400 Mev/fm3 and (ii) S / A = 1 1, B =250 Mev/fm3. This means that the production of dimuons is not affected much by the equation of state for the QGP.

(ii) Coexistence region (QGP): The behavior of photon-pair production rates (Figs. 3 and 4) is different from that of the dimuon production

D ~ r n i x e d th Y~ qa+2y = Jlq t d t d2x, h q ( t ) . rates. Here, only for M > 7.5 GeV ( M > 8.5 GeV) for

B =400 ~ e ~ / f m ~ ( B = 250 Mev/fm3) the Q G P contribu- X ~ , , ~ ~ ~ 2 y ( ~ ( t ) , p q ( t ) ) . (4.8) tion tries to dominate and its behavior beyond 10 GeV

(9.25 GeV) is not clear. For most of the time only the H R G phase contributes. Although the trend of the

(iii) Coexistence region (HRG): figures is the same for both the values of B, the photon- pair production seems to be sensitive to the equation of

(iv) Pure H R G phase:

where the proper times are already defined in Sec. 111. The notation used here is the same as before. For the two equations of state mentioned in Sec. 111, Figs. 3 and 4 show, respectively, the contributions arising from different regions to the invariant-mass distributions of the dimensionless quantity ln( M2DYMN).

statefor the QGP. We may conclude that to detect the production of a

. -. . . - . - QGP+ HRG

r. ,. . ._ - . . 9 - - , \

% - - - \ ,/ -, , .-__

V. CONCLUSIONS FIG. 3 . The photon-pair production cross sections vs the in- variant mass of the photon pairs for B=400 MeV/fm3 and

We observe from Figs. 1 and 2 that for M < 1 GeV, the S / A =20. Dashed-dotted line, quark contribution; dashed line, H R G contribution to the dimuon production dominates hadron contribution; solid line, sum of the contributions from while above 2.25 GeV, the QGP contribution is more quarks and hadrons.

MUON PAIRS AND PHOTON PAIRS AS SIGNALS FOR T H E . . . 2813

---. QGP HRG

- QGP+ HRG

pairs as convenient signals for the detection of the forma- tion of a baryon-rich plasma. However, for a study of the equation of state of the QGP, photon pairs may be better than the muon pairs. Work is in progress to include the contribution to the photon-pair productions from other channels such as the decay of T O , 11, etc. Although the isentropic expansion is mainly (1 + 1 )-dimensional in the QGP phase and in the mixed phase, it becomes a full (3 + 1)-dimensional one in the H R G phase.I0 We have to take this into account in our future work. As suggested by ~ c ~ e r r a n , ' ) it will be interesting to compare our dimuon production rates with those from the Drell-Yan mechanism14 and study the production rates for photon pairs at very high temperatures of the order of 500 MeV.

FIG. 4. As in Fig. 3, but for B=250 MeV/fm3 and S / A = l l .

baryon-rich Q G P in heavy-ion collisions, using the dimu- on signals, one should analyze the invariant-mass distri- bution of dimuon production rates in the region where M > 2.25 GeV. However, the photon-pair production as a signature requires an analysis of the region where M Z 8 GeV. Hence the dimuons are better than the photon

ACKNOWLEDGMENTS

It is a pleasure for us to thank Dr. M. Rajasekaran, Dr. Bikash Sinha, and Professor V. Devanathan for fruitful discussions, the Department of Atomic Energy, Govern- ment of India, and the University Grants Commission, New Delhi, for support, the latter through its Committee on Strengthening the Infrastructure of Science and Tech- nology (COSIST) program. One of us (G.J.) is grateful to the University Grants Commission for financial support.

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