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Abstract

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The production of single photons in relativistic heavy ion collisions at CERN SPS, BNL RHIC and CERN LHC energies is re-examined in view of the recent studies of Aurenche et al which show that the rate of photon production from quark gluon plasma, evaluated at the order of two loops far exceeds the rates evaluated at one-loop level which have formed the basis of all the estimates of photons so far. We find that the production of photons from quark matter could easily out-shine those from the hadronic matter in certain ideal conditions.

Single photons can be counted among the first signatures [@first] which were proposed to verify the formation of deconfined strongly interacting matter- namely the quark gluon plasma (QGP). Along with dileptons- which will have similar origins, they constitute electro-magnetic probes which are believed to reveal the history of evolution of the plasma, through a (likely) mixed phase and the hadronic phase, as they do not re-scatter once produced and their production cross section is a strongly increasing function of temperature. During the QGP phase, the single photons are believed to originate from Compton ($q\,(\overline{q})\,g\,\rightarrow\,q\,(\overline{q})\,\gamma$) and annihilation ($q\,\overline{q}\,\rightarrow\,g\,\gamma$) processes [@joe; @rolf] as well as bremsstrahlung processes ($q\,q\,(g)\,\rightarrow\,q\,q\,(g)\,\gamma$). Recently in the first evaluation of single photons within a parton cascade model [@pcm], it was shown [@pcmphot] that the fragmentation of time-like quarks ($q\,\rightarrow\,q\,\gamma$) produced in (semi)hard multiple scatterings during the pre-equilibrium phase of the collision leads to a substantial production of photons (flash of photons!), whose $p_T$ is decided by the $Q^2$ of the scatterings and not the temperature as in the above mentioned calculations.

The upper limit for production of single photons in $S+Au$ collisions at SPS energies [@wa80] has been used by several authors to rule out simple hadronic equations of states [@prl] and the final results for the $Pb+Pb$ collisions at SPS energies are eagerly awaited.

In a significant development Aurenche et al [@pat] have recently evaluated the production of photons in a QGP up to two loops and shown that the bremsstrahlung process gives a contribution which is similar in magnitude to the Compton and annihilation contributions evaluated up to the order of one loop earlier [@joe; @rolf]. This is in contrast to the ‘expectations’ that the bremsstrahlung contributions drop rapidly with energy (see Ref. [@kryz; @dipali] for estimates within a soft photon approximation). They also reported an entirely new mechanism for the production of hard photons through the annihilation of an off-mass shell quark and an anti-quark, where the off-mass shell quark is a product of scattering with another quark or gluon and which completely dominates the emission of hard photons. This process is similar to the annihilation of quarks in the presence of the chromo-electric field which may develop when two nuclei pass through each other due to colour exchange, and which can absorb the unbalanced energy and momentum to ensure the feasibility of the process which is absent in vacuum [@avijit].

If confirmed, this has far reaching consequences for the search of single photons from the relativistic heavy ion collisions.

The rate for the production of hard photons evaluated to one loop order using the effective theory based on resummation of hard thermal loops is given by [@joe; @rolf]: $$E\frac{dN}{d^4x\,d^3k}=\frac{1}{2\pi^2}\,\alpha\alpha_s\,
                          \left(\sum_f e_f^2\right)\, T^2\,
                       e^{-E/T}\,\ln(\frac {cE}{\alpha_s T})$$ where the constant $c\approx$ 0.23. The summation runs over the the flavours of the quarks and $e_f$ is the electric charge of the quarks in units of charge of the electron. The rate of production of photons due to the bremsstrahlung processes evaluated by Aurenche et al is given by: $$E\frac{dN}{d^4x\,d^3k}=\frac{8}{\pi^5}\,\alpha\alpha_s\,
                          \left(\sum_f e_f^2\right)\, 
                        \frac{T^4}{E^2}\,
                       e^{-E/T}\,(J_T-J_L)\,I(E,T)$$ where $J_T\approx$ 4.45 and $J_L\approx - $4.26 for 2 flavours and 3 colour of quarks. For 3 flavour of quarks, $J_T\approx$ 4.80 and $J_L\approx - $4.52. $I(E,T)$ stands for; $$\begin{aligned}
I(E,T)&=&\left[ 3\zeta(3)+\frac{\pi^2}{6}\frac{E}{T}+
        \left(\frac{E}{T}\right)^2\ln(2)\right.\nonumber\\
        & &+4\,Li_3(-e^{-|E|/T})+2\,Li_2(-e^{-|E|/T})\nonumber\\
        & &\left. -\left(\frac{E}{T}\right)^2\,\ln(1+e^{-|E|/T})\right]~,\end{aligned}$$ and the poly-logarith functions $Li$ are given by; $$Li_a(z)=\sum_{n=1}^{+\infty}\frac{z^n}{n^a}~~.$$

And finally the contribution of the $q\overline{q}$ annihilation with scattering obtained by them is given by: $$E\frac{dN}{d^4x\,d^3k}=\frac{8}{3\pi^5}\,\alpha\alpha_s\,
                          \left(\sum_f e_f^2\right)\, ET \,
                          e^{-E/T}\,(J_T-J_L)$$

We plot these rates of emission of photons from a QGP at $T=$ 250 MeV (Fig. 1) for an easy comparison. The dashed curve gives the contribution of the Compton and annihilation processes evaluated to the order of one loop by Kapusta et al [@joe], the dot-dashed curve gives the bremsstrahlung contribution evaluated to two-loops by Aurenche et al [@pat] while the solid curve gives the results for the annihilation with scattering evaluated by the same authors. The dotted curve gives the results for the bremsstrahlung contribution evaluated within a soft-photon approximation (and using thermal mass for quarks and gluons) obtained by Pal et al [@dipali]. We see that at larger energies the annihilation of quarks with scattering really dominates over the rest of the contributions by more than a order of magnitude.

How much of this dominance does survive when we integrate the radiation of photons over the history of evolution of the system, specially as the QGP phase occurring in the early stages of the evolution necessarily occupies smaller four-volume compared the hadronic matter, which is known to have an emission rate similar to the quark matter at a given temperature [@joe] at least when only the Compton and the annihilation terms are used?

In order to ascertain this we consider central collision of lead nuclei at SPS, RHIC and LHC energies. We assume that a chemically and thermally equilibrated quark-gluon plasma is formed at $\tau_0=$ 1 fm/$c$ at SPS and at 0.5 fm/$c$ at RHIC and LHC energies. While there are indications that the plasma produced at the energies under consideration may indeed attain thermal equilibrium at around $\tau_0$ chosen here [@pcm; @therm], it is not quite definite that it may be chemically equilibrated. It may be recalled that the parton cascade model which properly accounts for multiple scatterings uses a cut-off in momentum transfer and virtuality to regulate the divergences in the scattering and the branching amplitudes for partons. This could underestimate the extent of chemical equilibration, by a cessation of interactions when the energy of the partons is still large which would not be the case if the screening of the partonic interactions could be accounted for. The self-screened parton cascade [@sspc] on the other hand attempts to remove these cut-offs by estimating the screening offered by the partons which have larger $p_T$ (and hence materialize earlier) to the partons which have smaller $p_T$ (and hence materialize later). However it does not explicitly account for multiple scattering except for what is contained in the Glauber approximation utilized there.

In these exploratory calculations we assume a chemical equilibration at the time $\tau_0$ such that the initial temperature is obtained from the Bjorken condition [@bj]; $$\frac{2\pi^4}{45\zeta(3)}\,\frac{1}{\pi R_T^2}\frac{dN}{dy}=4 a
T_0^3\tau_0$$ where we have chosen the particle rapidity densities as 825, 1734, and 5625 respectively at SPS, RHIC, and LHC energies for central collision of lead nuclei [@kms] and taken $a=47.5\pi^2/90$ for a plasma of mass-less quarks (u, d, and s) and gluons.

We assume the phase transition to take place at $T=$ 160 MeV, and the freeze-out to take place at 100 MeV. We use a hadronic equation of state consisting of all the hadrons and resonances from the particle data table which have a mass less then 2.5 GeV [@jean]. The rates for the hadronic matter have been obtained [@joe] from a two loop approximation of the photon self energy using a model where $\pi-\rho$ interactions have been included. The contribution of the $A_1$ resonance is also included according to the suggestions of Xiong et al [@li]. The relevant hydrodynamic equations are solved using the procedure [@hydro] discussed earlier and a integration over history of evolution is performed [@jean].

In Fig. 2 we show our results for central collision of lead nuclei at energies which are reached at CERN SPS. We give the contribution of the quark matter (from the QGP phase and the mixed phase) labeled as QM and that of the hadronic matter (from the mixed phase and the hadronic phase) separately. We see that if we use the rates obtained earlier by Kapusta et al, there is no window when the radiations from the quark-matter could shine above the contributions from the hadronic matter. However, once the newly obtained rates are used we see that the quark matter may indeed out-shine the hadronic matter up to $p_T=$ 2 GeV, from these contributions alone. Note that by tracking the history from $\tau_0$= 1 fm/$c$ onward, we have not included the pre-equilibrium contributions [@pcmphot] which will make a large contribution at higher momenta. The contribution of hard QCD photons [@qcd] obtained by scaling the results for $pp$ collisions by the nuclear thickness.

The results for RHIC energies (Fig. 3) are quite interesting as now the window over which the quark matter out-shines the hadronic contributions stretches to almost 3 GeV. Once again the addition of the pre-equilibrium contributions at larger $p_T$ would substantially widen this window.

At LHC energies this window extends to beyond 4 GeV, and considering that perhaps the local thermalization at LHC (and also at RHIC) could be attained earlier than what is definitely a very conservative value here, these results provide the exciting possibility that if these conditions are met the quark matter may emit photons which may be almost an order of magnitude larger than those coming from the hadronic matter over a fairly wide window. As mentioned earlier, the pre-equilibrium contribution (due to the very larger initial energy) should be much larger here and we may have the exciting possibility that the quark matter may out-shine the hadronic matter over a very large window indeed.

How will the results change if the QGP is not in chemical equilibrium? While it is not easy to perform the estimates similar to the one done by Aurenche et al for a chemically non-equilibrated plasma, it is reasonable to assume that the rates will fall simply because then the number of quarks and gluons will be smaller. Some of this short-fall will be off-set by the much larger temperatures which the parton cascade models predict. If one considers a chemically equilibrating plasma [@smm] then the quark and gluon fugacities will increase with time and at least the contributions from the latter stages will not be strongly suppressed. It is still felt that the loss of production of high $p_T$ (from early times) photons due to chemical non-equilibration would be more than off-set by the increased temperature and the pre-equibrium contribution, which can be quite large.

We conclude that the newly obtained rates for emission of photons from QGP (evaluated to the order of two loops) suggest that if chemically equilibrated plasma is produced then there will exist a fairly wide window where the photons from quark matter may outshine the photons from hadronic matter. Even in the absence of chemical equilibration these results indicate an enhanced radiation from the quark matter which is of considerable interest.

Acknowledgments {#acknowledgments .unnumbered}
===============

The author gratefully acknowledges the hospitality of University of Bielefeld where part of this work was done. He would also like to acknowledge useful discussions with Jean Cleymans and Francois Gelis. He is especially grateful to Haitham Zaraket for suggesting that the exact expression for the bremsstrahlung contribution be used.

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