- Email: [email protected]
Thermal photons and dileptons from non-equilibrium quark-gluon plasma
7 July 1994 PHYSICS LETTERS B
ELSEVIER
PhysicsLettersB 331 (1994) 245-250
Thermal photons and dileptons from non-equilibriumquark-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. Introduction
The formation and evolution of a quark-gluon plasma in central collisions of two massive nuclei has been the subject of considerable attention. Experiments 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 electromagnetic 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 equilibration of quarks and gluons using the inside-outside
cascade model [2]. In this treatment they set the local 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 simplified version of the Braaten and Pisarksi resummation technique [ 3 ] by introducing m o m e n t u m - i n d e p e n d e n t screening masses into the propagators. Fig. 1 shows a plot of the temperature, quark fugacity, and gluon fugacity as a function of proper time at RHIC and LHC energies. Using this information we calculate the total 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
M. Strickland ~Physics Letters B 331 (1994) 245-250
246
(a)
mal plasma development. Our analysis shows that information about the initial state requires background subtraction to at least 5 GeV.
(b) 0.9
0.9 ~0.7
0.7
> 0.5
0.5
0.3
0.3
0.1
0.1
2. Dilepton production
T .
0
,
2
.
,
4
,
,
6
,
,
8
,
,
,
10 12
x I m/c)
.
0
,
2
,
,
4
,
,
6
,
,
8
,
~
10 12
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 collisions. 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 initial 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 multiplicity 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 emissions to experimentallydetermine plasma initial conditions. We find that, if background processes can be subtracted to only 3 GeV, determining initial conditions from these emissions is difficult. The difficulty stems from the fact that production of thermal dileptons and photons in this region (1 GeV < Pr, M < 3 GeV) depends strongly on the later stages of ther-
There has been a good deal of research into dilepton production in the quark-gluon plasma. Whenever a new model for plasma evolution is proposed its impact on dilepton and photon emissions must be assessed. Recently, Geiger and Kapusta [4], Shuryak and Xiong [5], K~-npfer and Pavlenko [6], and Kapusta, 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 evolution 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 production rates, the cross section is combined with the thermal distribution functions and the thermal spacetime evolution [9]:
dNe+gd4x
=~
iV" /
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 ÷ ePo/T O0
= -- y ~ ( - - , ~ q ) k e - k p ° / T ,
(2)
k=l
where Aq is the quark fugacity. To perform the integration we insert f ds 8(s - (p~ + p~')2), and after a little manipulation Eq. ( 1 ) becomes
M. Strickland/ PhysicsLettersB 331 (1994)245-250 (a)
(b)
---
Thermal -
Direct at 200 GeV 10 .3
10 "s
'>
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].
10 ¢
1 0 "z
10 ~
10 ~
10 .3
10 "s
c~ "O
~
247
3. Photon production 10 "6
_
Z "O 10";'
10 "r
\\\ 10 "a
2
3
\\ - - Thermal - - - Directat 7000 GeV
10 a
4
2
5
3
4
5
i (GeV)
M ~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.
daxdM 2
- 2~7r4 M2tre+e- (M 2)
(3O
(3O (3)
5____~te2A2qTMg(z)
18~"3 oo
( _ el )~k_+
g(z) -- Z
M
KI(V'~
).
dydM 2
-
187/. 2
[
d3pl dap2 d3p3
x (27r)4B(p~ +p~' - p ~ - P ~ ) I
Mi 12,
(6)
3.1. Calculating the rate
This calculation uses the Fermi distribution for the incoming particles and includes the effect of varying quark fugacity. The series converges very quickly, justifying keeping only one term in k and g as was done in the past [ 8]; however, for our numerical calculations 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 5ot2R 2
.Af
(4)
kd=l
dNe+e-
d4xd3p ] i
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 4f3 (E3)) describes effects of quantum statistics in the final state.
Performing this integration gives us the result
dNe+e-- =
\
2(27r) 8 J 2E1 2E2 2E3 x f l (El) fE(E2) ( 1 4- f3 (E3))
x f dpl fq(Pl) f dp2 fq(P2)° 0 M2/4pl
d4xdM 2
(E dN./ "~ O)i ~ _
5
dNe+e-
For large PT photon production there are two processes that must be considered: Compton-like scattering gq( ~l) --~ Yq( (l) and annihilation q(l --+ Yg [ 10]. The thermal rate for these reactions is given by [ 11 ]
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 channels. We also make the substitution
d4p36(P~)®(P3°)"
2E3
(7)
After these substitutions have been made we can reexpress Eq. (6) as
~-f
/ rdrA2q(r)T(~')g(z, ~').
(5)
N
o~i -- 16E(2~.) 7 . / d s d t I Mi
12
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-
x fdEtdEzfl (El)f2(E2) x [1 + f3(El + E 2 - E ) ] O ( E 1 +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>*,
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 differential cross section for small momentum transfer. In order to eliminate this divergence we must take manybody 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. 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 selfenergy is then of the form
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
Jv wi = 32E(2~)~
$dr]
Mi(s,r)
I2
s
Co
X
dxe-X’T[1fj-3(x-E)]. J
(11)
S/4E
Now all that is left to do is to substitute the factorized Jtittner distribution for fs and the invariant amplitudes for the Compton and annhilation diagrams. However, we run into problems here since the total cross section is infinite. For now we introduce an integration cutoff which we will set later using many-body techniques. The new integration region for s and t is
C(K)
= -a$
- b$,
(15)
where a and b are Lorentz-invariant functions. this the full fermion propagator becomes S(K)
= [(l+u)$+bjl-’
= [(l +u)$+bjlD-‘, (16)
where D is the Lorentz-invariant --s + k,’ 5 t 5 -k;,
2k; < s 2 co.
function
(12)
In the limit k,’ -+ 0 we get for the Compton process: 5 2aLY, w, = ~~A,A,T2e-EIT
(13) and for annihilation: Wl=g
With
D(k,w)
(17)
In order to find the effective fermion mass, we have to find the poles of the propagator S(K). Therefore, we need to know the functions a and b.The determination of these functions follows directly from the calculations in Appendix A of Weldon’s paper [ 121 except where he uses f(p) = ( elP’j‘lIT f 1) -’ we insert the full Jtittner distribution. From this, it follows that if we replace the temperature by T +
5 2cm ~A,A,T2eCEIT d
= (1+u)*K2+2(1+a)bK~u+6*.
KT,
(18)
where K*
xf&$[In(k~;~l))-l-c]’ ( 14) where C = 0.57721 . . . is Euler’s constant.
=
Q(Li*(A 5-2
Liz(z)
g
) - Li2(--A 4 )) ’
= F(-l)“(’ n=l
i21)‘,
(19)
M. Strickland/ PhysicsLettersB 331 (1994)245-250 (a)
the rest of Weldon's arguments follow through unchanged giving
mq2 = lg2K2T2"
k2c ½g2KZT2.
(21)
=
Inserting this into Eqs. (13) and (14) gives us our final expression for the photon production rates:
Compton:
10 ~ Thermal Direct at 200 GeV
10 "~
10 o
10.2
1 0 -~
Z "o 10"3
10 -z
B-
"O
\ 10 ~
°°(--~q)n[ln( n=0
(n+l) z
12E
)
\gZKZT(n+l)
1 +~-C
] , (22)
Annhilation: 5 -2~as rOa = -~ - - ~ AqaqTZ e -F'/T x
oo n=0
Ag (n+
1) 2
3
PT' 3eV)
oJc- 9
[In( 12E \g2K2T(n-+-
)-1-C]. 1)
(23) It should be kept in mind that this calculation is performed 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 associated or jet production, which is the emission of photons 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-
Thermal
G)
1 0 -4
5 25OtSAqAgT2e_E/T
--
"7
2
XZ
(b)
10 0
(20)
Now that we have the effective mass, we can set the cutoff using the Braaten-Pisarski approach used in Kapusta et al [ 11 ]. They set k~ = 2m2q, giving
249
4
5
2
3
4
5
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 figures. These further restrict the region for thermal measurements to the region between 2 and 3 GeV. Our analysis shows that emissions from the later stages of thermal development strongly affect photon and dilepton production for low pr and M. Fig. 4 shows a plot of the fractional yield of thermal dileptons and photons as a function of M and Pr for different stages of plasma evolution. Each consecutive line from bottom to top represents approximately 0.6 fm/c elapsed. From this figure we see that dilepton and photon production 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 initial 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 reactions all the way to equilibrium radiation. According 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 determined using the logarithmic slope of the dilepton or photon yields. More research into this area is definitely warranted.
250
M. Strickland / Physics Letters B 331 (1994) 245-250
(a) 1.0 018
._ >"~
0.6
o 0.4
LL
Ss
i
/
0.0
I would like to thank Berndt Mtiller for his guidance. 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 o f Energy under Grant No. DE-FG05- 90ER40592.
1.0
0.8 0,6
/
0.4 0.2
0.2 d
Acknowledgement
(b)
~
2
x t = 5.97 f m / c
o
o ~ = 5.97 fm/c
x I = 0.87 f m / c
o
o ~ = 0.87 fm/c
3
M
3eV)
4
0.0
1
2
3
4
References 5
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 L H C energies is similar. Associated production o f photons and dileptons restricts measurements to approximately 2 to 4 GeV. Once again, this region is not a good indicator of the initial conditions. I f experimentalists can find a way to subtract the background processes out to 5 or 6 GeV then accurate determination o f the initial temperature is possible using the logarithmic slope o f the thermal photon yields. Since associated photon and dilepton production (production from jet fragmentation) are the dominant background processes at LHC energies, the importance o f medium effects on these processes should be studied. It is possible that secondary collisions could suppress this production by an order of magnitude. If this were the case, then thermal photon yields could provide an excellent plasma thermometer at LHC energies. In conclusion, determination of the initial temperature to within 0.25 GeV is possible using the logarithmic slope o f the photon production curves in the region between 2 and 3 GeV; however, accurate determination is only possible if the hard production processes can be subtracted to approximately 5 GeV.
[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~impferand 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.
ELSEVIER
PhysicsLettersB 331 (1994) 245-250
Thermal photons and dileptons from non-equilibriumquark-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. Introduction
The formation and evolution of a quark-gluon plasma in central collisions of two massive nuclei has been the subject of considerable attention. Experiments 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 electromagnetic 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 equilibration of quarks and gluons using the inside-outside
cascade model [2]. In this treatment they set the local 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 simplified version of the Braaten and Pisarksi resummation technique [ 3 ] by introducing m o m e n t u m - i n d e p e n d e n t screening masses into the propagators. Fig. 1 shows a plot of the temperature, quark fugacity, and gluon fugacity as a function of proper time at RHIC and LHC energies. Using this information we calculate the total 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
M. Strickland ~Physics Letters B 331 (1994) 245-250
246
(a)
mal plasma development. Our analysis shows that information about the initial state requires background subtraction to at least 5 GeV.
(b) 0.9
0.9 ~0.7
0.7
> 0.5
0.5
0.3
0.3
0.1
0.1
2. Dilepton production
T .
0
,
2
.
,
4
,
,
6
,
,
8
,
,
,
10 12
x I m/c)
.
0
,
2
,
,
4
,
,
6
,
,
8
,
~
10 12
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 collisions. 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 initial 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 multiplicity 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 emissions to experimentallydetermine plasma initial conditions. We find that, if background processes can be subtracted to only 3 GeV, determining initial conditions from these emissions is difficult. The difficulty stems from the fact that production of thermal dileptons and photons in this region (1 GeV < Pr, M < 3 GeV) depends strongly on the later stages of ther-
There has been a good deal of research into dilepton production in the quark-gluon plasma. Whenever a new model for plasma evolution is proposed its impact on dilepton and photon emissions must be assessed. Recently, Geiger and Kapusta [4], Shuryak and Xiong [5], K~-npfer and Pavlenko [6], and Kapusta, 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 evolution 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 production rates, the cross section is combined with the thermal distribution functions and the thermal spacetime evolution [9]:
dNe+gd4x
=~
iV" /
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 ÷ ePo/T O0
= -- y ~ ( - - , ~ q ) k e - k p ° / T ,
(2)
k=l
where Aq is the quark fugacity. To perform the integration we insert f ds 8(s - (p~ + p~')2), and after a little manipulation Eq. ( 1 ) becomes
M. Strickland/ PhysicsLettersB 331 (1994)245-250 (a)
(b)
---
Thermal -
Direct at 200 GeV 10 .3
10 "s
'>
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].
10 ¢
1 0 "z
10 ~
10 ~
10 .3
10 "s
c~ "O
~
247
3. Photon production 10 "6
_
Z "O 10";'
10 "r
\\\ 10 "a
2
3
\\ - - Thermal - - - Directat 7000 GeV
10 a
4
2
5
3
4
5
i (GeV)
M ~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.
daxdM 2
- 2~7r4 M2tre+e- (M 2)
(3O
(3O (3)
5____~te2A2qTMg(z)
18~"3 oo
( _ el )~k_+
g(z) -- Z
M
KI(V'~
).
dydM 2
-
187/. 2
[
d3pl dap2 d3p3
x (27r)4B(p~ +p~' - p ~ - P ~ ) I
Mi 12,
(6)
3.1. Calculating the rate
This calculation uses the Fermi distribution for the incoming particles and includes the effect of varying quark fugacity. The series converges very quickly, justifying keeping only one term in k and g as was done in the past [ 8]; however, for our numerical calculations 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 5ot2R 2
.Af
(4)
kd=l
dNe+e-
d4xd3p ] i
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 4f3 (E3)) describes effects of quantum statistics in the final state.
Performing this integration gives us the result
dNe+e-- =
\
2(27r) 8 J 2E1 2E2 2E3 x f l (El) fE(E2) ( 1 4- f3 (E3))
x f dpl fq(Pl) f dp2 fq(P2)° 0 M2/4pl
d4xdM 2
(E dN./ "~ O)i ~ _
5
dNe+e-
For large PT photon production there are two processes that must be considered: Compton-like scattering gq( ~l) --~ Yq( (l) and annihilation q(l --+ Yg [ 10]. The thermal rate for these reactions is given by [ 11 ]
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 channels. We also make the substitution
d4p36(P~)®(P3°)"
2E3
(7)
After these substitutions have been made we can reexpress Eq. (6) as
~-f
/ rdrA2q(r)T(~')g(z, ~').
(5)
N
o~i -- 16E(2~.) 7 . / d s d t I Mi
12
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-
x fdEtdEzfl (El)f2(E2) x [1 + f3(El + E 2 - E ) ] O ( E 1 +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>*,
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 differential cross section for small momentum transfer. In order to eliminate this divergence we must take manybody 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. 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 selfenergy is then of the form
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
Jv wi = 32E(2~)~
$dr]
Mi(s,r)
I2
s
Co
X
dxe-X’T[1fj-3(x-E)]. J
(11)
S/4E
Now all that is left to do is to substitute the factorized Jtittner distribution for fs and the invariant amplitudes for the Compton and annhilation diagrams. However, we run into problems here since the total cross section is infinite. For now we introduce an integration cutoff which we will set later using many-body techniques. The new integration region for s and t is
C(K)
= -a$
- b$,
(15)
where a and b are Lorentz-invariant functions. this the full fermion propagator becomes S(K)
= [(l+u)$+bjl-’
= [(l +u)$+bjlD-‘, (16)
where D is the Lorentz-invariant --s + k,’ 5 t 5 -k;,
2k; < s 2 co.
function
(12)
In the limit k,’ -+ 0 we get for the Compton process: 5 2aLY, w, = ~~A,A,T2e-EIT
(13) and for annihilation: Wl=g
With
D(k,w)
(17)
In order to find the effective fermion mass, we have to find the poles of the propagator S(K). Therefore, we need to know the functions a and b.The determination of these functions follows directly from the calculations in Appendix A of Weldon’s paper [ 121 except where he uses f(p) = ( elP’j‘lIT f 1) -’ we insert the full Jtittner distribution. From this, it follows that if we replace the temperature by T +
5 2cm ~A,A,T2eCEIT d
= (1+u)*K2+2(1+a)bK~u+6*.
KT,
(18)
where K*
xf&$[In(k~;~l))-l-c]’ ( 14) where C = 0.57721 . . . is Euler’s constant.
=
Q(Li*(A 5-2
Liz(z)
g
) - Li2(--A 4 )) ’
= F(-l)“(’ n=l
i21)‘,
(19)
M. Strickland/ PhysicsLettersB 331 (1994)245-250 (a)
the rest of Weldon's arguments follow through unchanged giving
mq2 = lg2K2T2"
k2c ½g2KZT2.
(21)
=
Inserting this into Eqs. (13) and (14) gives us our final expression for the photon production rates:
Compton:
10 ~ Thermal Direct at 200 GeV
10 "~
10 o
10.2
1 0 -~
Z "o 10"3
10 -z
B-
"O
\ 10 ~
°°(--~q)n[ln( n=0
(n+l) z
12E
)
\gZKZT(n+l)
1 +~-C
] , (22)
Annhilation: 5 -2~as rOa = -~ - - ~ AqaqTZ e -F'/T x
oo n=0
Ag (n+
1) 2
3
PT' 3eV)
oJc- 9
[In( 12E \g2K2T(n-+-
)-1-C]. 1)
(23) It should be kept in mind that this calculation is performed 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 associated or jet production, which is the emission of photons 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-
Thermal
G)
1 0 -4
5 25OtSAqAgT2e_E/T
--
"7
2
XZ
(b)
10 0
(20)
Now that we have the effective mass, we can set the cutoff using the Braaten-Pisarski approach used in Kapusta et al [ 11 ]. They set k~ = 2m2q, giving
249
4
5
2
3
4
5
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 figures. These further restrict the region for thermal measurements to the region between 2 and 3 GeV. Our analysis shows that emissions from the later stages of thermal development strongly affect photon and dilepton production for low pr and M. Fig. 4 shows a plot of the fractional yield of thermal dileptons and photons as a function of M and Pr for different stages of plasma evolution. Each consecutive line from bottom to top represents approximately 0.6 fm/c elapsed. From this figure we see that dilepton and photon production 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 initial 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 reactions all the way to equilibrium radiation. According 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 determined using the logarithmic slope of the dilepton or photon yields. More research into this area is definitely warranted.
250
M. Strickland / Physics Letters B 331 (1994) 245-250
(a) 1.0 018
._ >"~
0.6
o 0.4
LL
Ss
i
/
0.0
I would like to thank Berndt Mtiller for his guidance. 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 o f Energy under Grant No. DE-FG05- 90ER40592.
1.0
0.8 0,6
/
0.4 0.2
0.2 d
Acknowledgement
(b)
~
2
x t = 5.97 f m / c
o
o ~ = 5.97 fm/c
x I = 0.87 f m / c
o
o ~ = 0.87 fm/c
3
M
3eV)
4
0.0
1
2
3
4
References 5
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 L H C energies is similar. Associated production o f photons and dileptons restricts measurements to approximately 2 to 4 GeV. Once again, this region is not a good indicator of the initial conditions. I f experimentalists can find a way to subtract the background processes out to 5 or 6 GeV then accurate determination o f the initial temperature is possible using the logarithmic slope o f the thermal photon yields. Since associated photon and dilepton production (production from jet fragmentation) are the dominant background processes at LHC energies, the importance o f medium effects on these processes should be studied. It is possible that secondary collisions could suppress this production by an order of magnitude. If this were the case, then thermal photon yields could provide an excellent plasma thermometer at LHC energies. In conclusion, determination of the initial temperature to within 0.25 GeV is possible using the logarithmic slope o f the photon production curves in the region between 2 and 3 GeV; however, accurate determination is only possible if the hard production processes can be subtracted to approximately 5 GeV.
[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~impferand 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.