ELSEVIER

7 July 1994

Physics Letters B 331 (1994) 245-250

PHYSICS LETTERS B

Thermal photons and dileptons from non-equilibrium quark-gluon plasma

Michael Strickland Department of Physics, Duke University, Durham, NC 27708-0305, USA

Received I April 1994 Editor: G.E Bertsch

Abstract

The dependence on initial conditions of thermal production rates for photons and dileptons in the quark-gluon plasma is discussed. We also consider using these signals to experimentally determine plasma initial conditions. Based on the chemical rate equations of Bir6 et al., we show that later emissions in the thermal expansion conceal information about the initial conditions in the region of low Pr and M, making determination of initial conditions difficult.

1. Introduct ion

The formation and evolution of a quark-gluon plasma in central collisions of two massive nuclei has been the subject of considerable attention. Ex- periments are planned at the BNL Relativistic Heavy Ion Collider (RHIC) and the CERN Large Hadron Collider (LHC). In order to study the plasma it is necessary to have signatures that can be distinguished from the background processes. The most promising observable is the emission of high energy photons and dileptons [ 1 ]. These particles have large mean free paths due to the small cross section for electromag- netic interaction in the plasma. Since this is true over the entire range of expected plasma temperatures these emissions can be observed throughout the evolution.

There are, however, uncertainties in the calculation of these spectra. One of these has been the lack of detailed knowledge about the space-time evolution of the plasma. Recently T.S. Bir6 et al. have derived a set of rate equations describing the chemical equili- bration of quarks and gluons using the inside-outside

cascade model [2]. In this treatment they set the lo- cal momentum space isotropy time to be of the order 0.3 fm/c. From this time forward they use continuum dynamics to describe the chemical equilibration of the plasma, with the plasma temperature and the quark and gluon fugacities as parameters. For the plasma equilibration the dominant reaction mechanisms are: gg ~ ggg and gg .-~ qgt. To deal with the infrared divergences in these cross sections they use a simpli- fied version of the Braaten and Pisarksi resummation technique [ 3 ] by introducing momen tum- independen t screening masses into the propagators. Fig. 1 shows a plot of the temperature, quark fugacity, and gluon fu- gacity as a function of proper time at RHIC and LHC energies. Using this information we calculate the to- tal number of photons and dileptons produced as a function of transverse momentum and invariant mass respectively.

One of the outstanding questions in this area is that of plasma initial conditions. There is no consensus as to what the proper initial conditions should be. For one, arriving at a definite momentum space isotropy

0370-2693/94/$07.00 (g) 1994 Elsevier Science B.V. All rights reserved SSDI 0370-2693 (94) 00622-E

0.9

M. Strickland ~Physics Letters B 331 (1994) 245-250

(a)

~ 0 . 7

> 0.5

0.3

0.1

(b)

246

0.9

0.7

0.5

0.3

0.1 T

. , . , , , , , , , , . , , , , , , , , ~

0 2 4 6 8 10 12 0 2 4 6 8 10 12 x I m/c) x (fm/c)

Fig. 1. Time evolution of the temperature T and the fugacities ag and aq in the quark-gluon plasma created in Au + Au col- lisions. Initial conditions: (a) r0 = 0.3 fm/c, To = 0.5 GeV, dN~r/dy If= 1735, Jqo = ½ago, (b) r0 = 0.3 fm/c, To = 1.0 GeV,

dN~r/dy If= 5624, Aqo = Iago.

time is difficult, since this time depends strongly on the plasma volume considered. It is understood that this volume should be determined by the parton mean free paths, but whether we should use one mean free path or two is not understood. A simple change from one to two mean free paths changes the momentum space isotropy time also by a factor of two thus decreasing thermal emission rates.

Once the isotropy time has been set, the other ini- tial conditions are the plasma temperature, quark and gluon fugacities, and the final pion multiplicity. In our program we set the initial temperature and time and then calculate the initial fugacities needed to generate a given final pion multiplicity. We have not included soft pion production in our calculations. Additional sources of pions could lower the initial temperature and/or fugacities needed to achieve a given final mul- tiplicity and thus affect our spectra.

It should be kept in mind that the initial conditions used in our examples are merely educated guesses and should not be taken as gospel. In fact, our goal is to determine if we can use thermal electromagnetic emis- sions to experimentallydetermine plasma initial con- ditions. We find that, if background processes can be subtracted to only 3 GeV, determining initial condi- tions from these emissions is difficult. The difficulty stems from the fact that production of thermal dilep- tons and photons in this region (1 GeV < Pr, M < 3 GeV) depends strongly on the later stages of ther-

mal plasma development. Our analysis shows that in- formation about the initial state requires background subtraction to at least 5 GeV.

2. Dilepton production

There has been a good deal of research into dilep- ton production in the quark-gluon plasma. Whenever a new model for plasma evolution is proposed its im- pact on dilepton and photon emissions must be as- sessed. Recently, Geiger and Kapusta [4], Shuryak and Xiong [5], K~-npfer and Pavlenko [6], and Ka- pusta, McLerran, and Srivastava [7], have used the new understanding of plasma evolution to calculate dilepton yields.

The dominant reaction for thermal production of lepton pairs is the Drell-Yan mechanism: qgl --+ £+g- [8]. The reaction q(cl)g ~ q(cl) + g+g- could also play an important role in early stages of plasma evo- lution due to the abundance of gluons; however, we have not included it explicitly, but through a K-factor of 2. This is probably an underestimate, but should put us in the right range. To determine dilepton pro- duction rates, the cross section is combined with the thermal distribution functions and the thermal space- time evolution [9]:

dNe+ g- iV" / d4 x = ~ d3pld3p2f(Pl)f(P2)

X VrelOre+g- (M2), (1)

where f ( p ) is a thermal distribution function, Pl and P2 are the momenta, Vrel is the relative velocity, and the degeneracy factor .A/" = 20 (includes sum of up and down charges).

Since the energy of the incoming quarks is large we can factorize the JiJttner distribution function

f q'# - Aq ÷ eP o/T

O 0

= -- y ~ ( - - , ~ q ) k e - k p ° / T ,

k=l

(2)

where Aq is the quark fugacity. To perform the inte- gration we insert f ds 8(s - (p~ + p~')2), and after a little manipulation Eq. ( 1 ) becomes

1 0 "z

10 "s

' > 10 ~

c~ 10 .3

"O ~ 10 "6

Z "O

10 "r

10 "a

(a)

- - Thermal - - - Direct at 200 GeV

\ \ \

2 3 4 M ~GeV)

10 ¢

10 .3

10 ~

10 "s

10";'

10 a 5

M. Strickland / Physics Letters B 331 (1994) 245-250

(b)

_

\ \ - - Thermal - - - Direct at 7000 GeV

2 3 4 5

i (GeV)

Fig. 2. Thermal dilepton yield as a function of invariant mass for RHIC and LHC energies. Hard dilepton production is shown for comparison. Same initial conditions as in Fig. 1.

dNe+e- 5 - 2~7r 4 M2tre+e - (M 2) daxdM 2

(3O (3O

x f dpl fq(Pl) f dp2 fq(P2)° ( 3 )

0 M2/4pl

Performing this integration gives us the result

dNe+e-- = 5____~te2A2qTMg(z ) d4xdM 2 1 8~ "3

o o ( _ l ) k + e ~ _ M

g(z) -- Z KI(V '~ ). (4) kd=l

This calculation uses the Fermi distribution for the incoming particles and includes the effect of varying quark fugacity. The series converges very quickly, jus- tifying keeping only one term in k and g as was done in the past [ 8]; however, for our numerical calcula- tions we kept ten terms. To integrate over space and time, for longitudinal expansion, we change variables to d4x = 7rR2rdT-dy, where R is the smaller of the two nuclear radii. This gives

~-f dNe+e- 5ot2R 2

/ rdrA2q(r)T(~')g(z, ~'). (5) dydM 2 - 1 8 7 / . 2

1"0

The result of this integration is shown in Fig. 2 for RHIC and LHC energies. Also shown in the figure are the hard dilepton yields. These were calculated by combining the partonic subprocess qcl ~ g+g-

247

with nucleon structure functions (Duke-Owens) and multiplying by the geometric factor TAA. Associated or jet production of dileptons has not been included in Fig. 2 (b). Associated dilepton production has been shown to dominate hard Drell-Yan production at LHC energies [ 15].

3. Photon production

For large PT photon production there are two pro- cesses that must be considered: Compton-like scatter- ing gq( ~l) --~ Yq( (l) and annihilation q(l --+ Yg [ 10]. The thermal rate for these reactions is given by [ 11 ]

(E dN./ "~ O)i ~ \ d4xd3p ] i

_ .Af [ d 3 p l dap2 d3p3

2(27r) 8 J 2E1 2E2 2E3

x f l (El) fE(E2) ( 1 4- f3 (E3))

x (27r)4B(p~ +p~' - p ~ - P ~ ) I Mi 12, (6)

where f ( E ) is the thermal distribution function, Mi is the invariant amplitude for each contributing process, and the degeneracy factor .N" = 320/3 for Compton scattering and .N" = 20 for annihilation. The term ( 1 4- f3 (E3)) describes effects of quantum statistics in the final state.

3.1. Calculating the rate

Evaluation of this integral can be done as follows. First, we insert two delta functions which express energy-momentum conservation in the s and t chan- nels. We also make the substitution

2E3 d4p36(P~)®(P3°)" (7)

After these substitutions have been made we can re- express Eq. (6) as

N . / d s d t I Mi 12

o~i -- 16E(2~.) 7

x fdEtdEzfl (El)f2(E2) x [1 + f3(El +E2 - E) ]O(E1 +17,2 - E )

x (aE21 + bEl + c)-½, (8)

248 M. Strickland/Physics Letters B 331 (1994) 245-250

where 3.2. Screening the divergence

a = -(S + t>*,

b=2(s+t)(Es-E2t).

c = st(s + t) - (Es + E2t). (9)

If the incoming particles have large energies we can use the Boltzman distribution to good approximation, so that

fl(El)f2(E2) + e-(EI+EZ)‘T. (10)

The integrations over El and ET can be done using the change of variables x = El + E2 and y = E2 yielding

The next question that must be addressed is that of the cutoff k,. The divergence in the photon production rate mentioned above is caused by a diverging differ- ential cross section for small momentum transfer. In order to eliminate this divergence we must take many- body effects into account. In this treatment I plan to find the temperature and fugacity dependence of the quark effective mass using the methods of Weldon [ 121 which assume exact chiral invariance.

At zero temperature, chiral invariance implies that the fermion self-energy is of the form C = --a$ for a particle of momentum K”, where a is a functionof K*. Therefore, the fermion propagator is S = $/ ( 1 +a) K*.

The function ( 1 + a) only modifies the residue of the pole, the pole is still located at K* = 0.

Jv wi = 32E(2~)~ s

$dr] Mi(s,r) I2

Co

X J

dxe-X’T[1fj-3(x-E)]. (11)

S/4E

Now all that is left to do is to substitute the factorized Jtittner distribution for fs and the invariant amplitudes for the Compton and annhilation diagrams. However, we run into problems here since the total cross section is infinite. For now we introduce an integration cutoff which we will set later using many-body techniques. The new integration region for s and t is

--s + k,’ 5 t 5 -k;, 2k; < s 2 co. (12)

In the limit k,’ -+ 0 we get for the Compton process:

5 2aLY, w, = ~~A,A,T2e-EIT

(13)

and for annihilation:

5 2cm Wl=g d ~A,A,T2eCEIT

xf&$[In(k~;~l))-l-c]’

( 14)

where C = 0.57721 . . . is Euler’s constant.

At finite temperature this no longer holds since the heat bath defines a special Lorentz frame moving with four-velocity cP. With this addition the fermion self- energy is then of the form

C(K) = -a$ - b$, (15)

where a and b are Lorentz-invariant functions. With this the full fermion propagator becomes

S(K) = [(l+u)$+bjl-’ = [(l +u)$+bjlD-‘,

(16)

where D is the Lorentz-invariant function

D(k,w) = (1+u)*K2+2(1+a)bK~u+6*. (17)

In order to find the effective fermion mass, we have

to find the poles of the propagator S(K). Therefore, we need to know the functions a and b.The deter- mination of these functions follows directly from the calculations in Appendix A of Weldon’s paper [ 121 except where he uses f(p) = ( elP’j‘lIT f 1) -’ we in- sert the full Jtittner distribution. From this, it follows that if we replace the temperature by

T + KT,

where

K* = Q(Li*(A ) - Li2(--A 5-2 g

)) 4 ’

(18)

Liz(z) = F(-l)“(’ i21)‘, n=l

(19)

M. Strickland / Physics Letters B 331 (1994) 245-250 249

the rest of Weldon's arguments follow through un- changed giving

2 lg2K2T2" (20) mq =

Now that we have the effective mass, we can set the cutoff using the Braaten-Pisarski approach used in Ka- pusta et al [ 11 ]. They set k~ = 2m2q, giving

k2c = ½g2KZT2. (21)

Inserting this into Eqs. (13) and (14) gives us our final expression for the photon production rates: Compton:

5 25OtSAqAgT2e_E/T oJc- 9

° ° ( - - ~ q ) n [ l n ( 12E ) 1 ] X Z ( n + l ) z \ g Z K Z T ( n + l ) + ~ - C ,

n=0

(22)

Annhilation:

5 2~as rOa = -~ - - - ~ AqaqTZ e -F'/T

oo Ag [ In( 12E ) - 1 - C ] . x

( n + 1) 2 \g2K2T(n-+- 1) n=0

(23)

It should be kept in mind that this calculation is per- formed using a perturbative expansion in the coupling constant g, and that at temperatures expected within the plasma g2 ~ 5. This casts doubt on the validity of a perturbative approach, and further exploration of this question is needed.

Typical photon spectra are shown in Fig. 3. Again we show the hard photon yields, which were obtained following the method outlined in J.E Owens' review paper on direct photon production [ 13]. In addition to the direct production, Fig. 3 (b) contains the asso- ciated or jet production, which is the emission of pho- tons from jet fragmentation. Associated production of photons has been shown to dominate direct production for high energies and so must be included [ 14].

4. Discussion

Analyzing the photon and dilepton spectra at RHIC we see that the background processes can be sub-

10 0

10 "~ "7

G)

B- 10 .2 "O

Z "o 10"3

10 -4

(a)

Thermal Direct at 200 GeV

10 ~

10 o

10 -~

10 -z

10 ~

(b)

- - Thermal

\ 2 3 4 5 2 3 4 5

PT' 3eV) PT (GeV) Fig. 3. Thermal photon yield as a function of transverse momentum for RHIC and LHC energies. Hard photon productionis shown for comparison. Same initial conditions as in Fig. 1.

tracted for PT and M between approximately 1 to 3 GeV. Production from vector meson decays and hadronic collisions have not been included in the fig- ures. These further restrict the region for thermal mea- surements to the region between 2 and 3 GeV. Our analysis shows that emissions from the later stages of thermal development strongly affect photon and dilep- ton production for low pr and M. Fig. 4 shows a plot of the fractional yield of thermal dileptons and pho- tons as a function of M and Pr for different stages of plasma evolution. Each consecutive line from bot- tom to top represents approximately 0.6 fm/c elapsed. From this figure we see that dilepton and photon pro- duction for large pr and M is dominated by early time scales. Therefore, this region provides a record of the earliest stages of plasma evolution. On the other hand, for low energies production is spread evenly over all time scales thus concealing information about the ini- tial state.

There are more positive predictions by Geiger and Kapusta [4]. Using a parton cascade model they have studieddilepton radiation from the initial Drell-Yan re- actions all the way to equilibrium radiation. Accord- ing to their results, the secondary parton interactions dominate the primary interactions by at least a factor of 5 even at energies of 8 GeV. If this is true then the initial plasma temperature could be accurately de- termined using the logarithmic slope of the dilepton or photon yields. More research into this area is defi- nitely warranted.

250 M. Strickland / Physics Letters B 331 (1994) 245-250

1.0

018 ._ >- "~ 0.6

o

0.4

LL

0.2

0.0

(a)

i Ss

/

d ~ x t = 5.97 fm/c

x I = 0.87 fm/c

(b)

1.0

0.8

0,6 /

0.4

0.2 o o ~ = 5.97 fm/c o o ~ = 0.87 fm/c

0.0 2 3 4 1 2 3 4 5

M 3eV) PT (GeV)

Fig. 4. (a) Percent yield of dileptons as a function of invariant mass for different stages of thermal evolution. (b) Percent yield of photons. Both are at RHIC energies, but similar results hold at LHC.

The situation at LHC energies is similar. Associ- ated production of photons and dileptons restricts mea- surements to approximately 2 to 4 GeV. Once again, this region is not a good indicator of the initial con- ditions. I f experimentalists can find a way to sub- tract the background processes out to 5 or 6 GeV then accurate determination of the initial temperature is possible using the logari thmic slope of the thermal photon yields. Since associated photon and dilepton production (production from jet fragmentation) are the dominant background processes at LHC energies, the importance of medium effects on these processes should be studied. It is possible that secondary col- l isions could suppress this production by an order of magnitude. I f this were the case, then thermal photon yields could provide an excellent plasma thermometer at LHC energies.

In conclusion, determination of the initial temper- ature to within 0.25 GeV is possible using the loga- r i thmic slope of the photon production curves in the region between 2 and 3 GeV; however, accurate deter- mination is only possible if the hard production pro- cesses can be subtracted to approximately 5 GeV.

Acknowledgement

I would like to thank Berndt Mtiller for his guid- ance. Without his support this project would not have been completed. I would also like to thank Klaus Geiger, Xin-Nian Wang, and an anonymous referee for their comments. This work was supported in part by the U.S. Department of Energy under Grant No. DE-FG05- 90ER40592.

References

[1] E.V. Shuryak, Phys. Lett. B 78 (1978) 150; K. Kajantie and H.I. Miettinen, Z. Phys. C 9 (1981) 341; G. Domokos and J. Goldman, Phys. Rev. D 23 (1981) 203.

[2] T.S. Bir6, E. van Doom, B. MUller, M.H. Thoma and X.N. Wang, Phys. Rev. C 48 (1993) 1275.

[3] E. Braaten and R.D. Pisarski, Nucl. Phys. B 337 (1990) 569. [4] K. Geiger and J.I. Kapusta, Phys. Rev. Lett. 70 (1993) 1920. [5] E. Shuryak and L. Xiong, Phys. Rev. Lett. 70 (1993) 2241. [6] B. K~impfer and O.P. Pavlenko, Phys. Lett. B 289 (1992)

127. [7] J.I. Kapusta, L. MacLerran and D.K. Srivastava, Phys. Lett.

B 283 (1992) 145. [8] P.V. Ruuskanen in Quark Gluon Plasma, edited by R.C. Hwa

(World Scientific, Singapore, 1991). [9] C. Gale and J.I. Kapusta, Can. J. Phys. 67 (1989) 1200.

[10] M. Neubert, Z. Phys. C 42 (1989) 231. [ 11] J.l. Kapusta, P. Lichard and D. Seibert, Phys. Rev. D 44

(1991) 2774. [12] H. Weldon, Phys. Rev. D 26 (1982) 2789. [13] J.F. Owens, Rev. Mod. Phys. 59 No 2 (1987) 465. [14] S. Gupta, Phys. Lett. B 248 (1990) 453. [ 15] K.J. Eskola and X.N. Wang, Lawrence Berkeley Laboratory

preprint LBL-34409.