Effect of magnetic field on the photon radiation from quark-gluon plasma in heavy ion collisions

# Effect of magnetic field on the photon radiation from quark-gluon plasma in heavy ion collisions

B.G. Zakharov L.D. Landau Institute for Theoretical Physics, GSP-1, 117940, Kosygina Str. 2, 117334 Moscow, Russia
August 1, 2019
###### Abstract

We develop a formalism for the photon emission from the quark-gluon plasma with an external electromagnetic field. We then use it to investigate the effect of magnetic field on the photon emission from the quark-gluon plasma created in collisions. We find that even for very optimistic assumption on the magnitude of the magnetic field generated in collisions its effect on the photon emission rate is practically negligible. For this reason the magnetic field cannot generate a significant azimuthal asymmetry in the photon spectrum.

## I Introduction

There is now a variety of experimental data on hadronic observables in collisions at RHIC and LHC that show that hadron production in high energy collisions goes via formation of a hot quark-gluon plasma (QGP) fireball. The major arguments in favor of the QGP formation at RHIC and LHC are the observation of a strong suppression of high- particle spectra (the so-called jet quenching phenomenon) and the success of the hydrodynamical models in describing the flow effects in hadron production in collisions. The results of the jet quenching UW_JQMC (); RAA12 (); JETC1 (); CUJET () and hydrodynamical Heinz_hydro1 () analyses support the production time of the QGP fm. However, this is only a qualitative estimate, because the value of is not well constrained by the data on the jet quenching and the flow effects. For jet quenching it is due to a strong reduction of the radiative parton energy loss in the initial stage of the QGP evolution by the finite size effects Z_OA (); Gale-Caron (). For this reason jet quenching is not very sensitive to the first fm/c of the matter evolution. And for the flow effects it is due to the low transverse velocities in the initial stage of the fireball evolution and the correlations of with the viscosity of the QGP in the hydrodynamical fits Heinz_hydro2 (); Heinz_tau ().

It is believed that the photon spectrum in the low and intermediate region may be more sensitive to the initial stage of the QGP evolution than the hadronic observables. Because the thermal photons radiated from the QGP leave the fireball without attenuation and the photon emission rate is largest in the initial hottest stage of the QGP evolution Shuryak (). The measurements of the photon spectrum in collisions performed at RHIC PHENIX_ph1 (); PHENIX_ph_v2 (); PHENIX_ph_PR () and LHC ALICE_ph () show that there is some excess of the photon yield (above the photons from hadron decays and from the hard perturbative mechanism) at GeV. It is widely believed that it is related to the photon emission from the QGP. However, the results of pQCD calculations of the thermal contribution to the photon spectrum are only in a qualitative agreement with the data obtained at RHIC and LHC (see Shen1 () and references therein). Say, the theoretical predictions obtained in recent analysis Gale_best () using a sophisticated viscous hydrodynamical model of the fireball evolution underestimate the photon spectrum by a factor of . It was observed that the thermal photons exhibit a significant azimuthal asymmetry (elliptic flow) comparable to that for hadrons. It is difficult to reconcile this fact with the expectation that the thermal photons should be mostly radiated from the hottest initial stage of the QGP where the flow effects should be small (this is often called the direct photon puzzle). It was suggested Dusling () that in the standard pQCD scenario of the thermal photon emission the flow effect for photons may be related to the viscous effects in the QGP that lead to a deviation of the parton distribution functions in the QGP from the equilibrium ones. The numerical results of Gale_best () show that the viscosity of the QGP may be an important source of the photon momentum anisotropy. However, in the analysis Gale_best () the viscous effects have been accounted for only for the LO pQCD processes (Compton) and (annihilation), and have not been included for the higher order collinear processes and AMY1 ().

The direct photon puzzle stimulated searches for novel mechanisms of the photon production in collisions that could generate a significant azimuthal asymmetry. In Ref. Kharzeev1 () it was suggested that the large photon azimuthal anisotropy may be related to a novel photon production mechanism stemming from the conformal anomaly and a strong magnetic field in noncentral collisions. However, the contribution of this mechanism becomes important only for a sufficiently large magnitude of the magnetic field, which is not supported by calculations for realistic evolution of the plasma fireball Z_maxw (). In Ref. Snigirev () it was argued that the observed photon asymmetry may be due to an intensive bremsstrahlung like synchrotron radiation resulting from the interaction of escaping quarks with the collective confining color field at the surface of the QGP. For this mechanism the asymmetry arises due to bigger surface emission from the almond-shaped QGP fireball along the direction of the impact parameter vector (as shown in Fig. 1). In Ref. T1 () it was suggested that the significant photon can be related to the real synchrotron emission from the thermal quarks in a strong magnetic field generated in noncentral collisions. Since the magnetic field in the noncentral collisions is mostly perpendicular to the reaction plane (this direction corresponds to axis, if axis is directed along the impact parameter of the collision as shown in Fig. 1) the synchrotron radiation rate is largest in the direction along of the impact parameter vector.

For this reason the synchrotron mechanism leads naturally to a strong azimuthal asymmetry of the photon emission. This explanation works only if the contribution of the synchrotron mechanism to the photon emission rate is significant. The analysis of Ref. T1 () shows that in the central rapidity region at GeV the contribution of the synchrotron mechanism may be comparable with the observed photon yield in Au+Au collisions at TeV. However, the calculations of Ref. T1 () are of a qualitative nature. In T1 () the calculations are performed for purely synchrotron radiation. But in the QGP each quark undergoes multiple scattering due to interaction with other thermal quarks and gluons. One can expect that it will lead to a reduction of the coherence/formation length of the photon emission, and to suppression of the synchrotron emission. In reality for the QGP with magnetic field one simply cannot distinguish between the synchrotron radiation and the bremsstrahlung due to multiple scattering, and one has to treat both the mechanisms on an even footing. In this case the effect of magnetic field on the photon emission can only be defined as the difference between the photon emission rate from the QGP with and without magnetic field. Also, in T1 () the comparison with the experimental photon spectrum has been performed by integrating over the QGP four volume neglecting the longitudinal and transverse expansion of the QGP. The neglect of the longitudinal expansion of the QGP may be too crude approximation. For a QGP with zero velocity the energy of a quark radiating a photon with a given momentum is smaller than that in the comoving frame for the QGP with the longitudinal expansion. Since the quark (anti-quark) thermal distribution decreases exponentially with quark energy, the approximation of zero QGP velocity can overestimate considerably the photon spectrum. Another issue that can result in overestimation of the synchrotron contribution is the use in T1 () of the current quark masses. In the QGP quarks acquire a thermal quasiparticle mass , that appears after the Hard Thermal Loop (HTL) resummation (which is very important already for the LO processes Baier_ph ()). Since the synchrotron spectrum reduces with the charged particle mass, the accounting of the quark quasiparticle mass, that is much bigger than the current quark masses, should suppress considerably the effect of magnetic field.

Besides the photon bremsstrahlung addressed in T1 () the magnetic field can affect the photon production via the annihilation mechanism . The analysis of the collinear processes and for the QGP without magnetic field shows that the annihilation contribution is even more important than bremsstrahlung at the photon momenta AMY1 (). The purpose of the present work is to address the effect of magnetic field on both the processes and (below we will call the magnetic field modification for both these processes as the synchrotron contribution). We develop a formalism which treats on an even footing the effect of multiple scattering and curvature of the quark trajectories in the collective magnetic field in the QGP. Our analysis is based on the light cone path integral (LCPI) formalism LCPI (), which was previously successfully used AZ () for a very simple derivation of the photon emission rate from the higher order collinear processes and obtained earlier by Arnold, Moore and Yaffe (AMY) AMY1 () using methods from thermal field theory with the HTL resummation. It is known that the higher order diagrams corresponding to these processes contribute to leading order AGZ2000 (), and turn out to be as important as the LO processes and . Contrary to the collinear processes the LO processes should not be affected by the presence of an external magnetic field. Our results differ drastically from that of T1 (). We find that even for very optimistic magnitude of the magnetic field for RHIC and LHC conditions its effect on the photon emission from the QGP is very small.

The plan of the paper is as follows. In Sec. 2 we first discuss the physical picture of the processes and . We show that for the magnitude of the magnetic field of interest for collisions these process remain in the collinear regime. Then we develop a formalism for evaluation of their contribution to the photon emission from the QGP with magnetic field in the medium rest frame. In Sec. 3 we discuss how to compute the photon spectrum from the plasma fireball in collisions. We discuss the model of the fireball and the possible magnitude of the magnetic field for the most optimistic scenario for the synchrotron photon emission. In Sec. 4 we present our numerical results. Sec. 5 summarizes our work. Some of our results concerning the photon emission rate from the QGP at rest have been reported in an earlier short communication Z_JETPL ().

## Ii Bremsstrahlung and pair annihilation in the QGP with magnetic field

In this section we discuss the photon emission rate per unit time and volume in the equilibrium QGP with magnetic field in the QGP rest frame. Similarly to the analyses AMY1 (); AZ () of the processes and for zero magnetic field we treat quarks and photons as relativistic quasiparticles with energies much larger than their quasiparticle masses and 111We assume that the photons emitted in the QGP adiabatically become massless after escaping from the plasma fireball.. For the weakly coupled QGP with flavors and read AMY1 ()

 mq=gT/√3, (1)
 mγ=eT3√(3+Nf)/2, (2)

where is the QCD coupling constant, is the electron charge. In numerical calculations we take to account for qualitatively the suppression of strange quarks at moderate temperatures. Since the effect of the nonzero photon mass is very small, and our results are close to that for massless photon.

### ii.1 Physical picture of photon emission and photon formation length

The physical picture behind the derivation of the photon emission rate in the QGP without magnetic field from the processes and given in AMY1 (); AZ () is the fact that in the weakly coupled QGP the hard partons with energy undergo typically only small angle multiple scattering due to interaction with the random soft gluon fields at the momentum scale . And the large angle scattering with the momentum transfer is a very rare process. The typical quark scattering angle at the longitudinal scale about the photon coherence/formation length, , is small AZ (). Due to this fact the processes and are dominated by the collinear configurations, when the photon is emitted practically in the direction of the initial quark for (and in the direction of the momentum of the pair for ). For a QGP with magnetic field this picture will remain valid if

 LfRL≪1, (3)

where is the quark Larmor radius in the magnetic field ( is the quark electric charge in units of ). Let us demonstrate that the condition (3) is satisfied for the fields with that are of interest for collisions. Making use the formulas of the LCPI approach for the bremsstrahlung due to multiple scattering LCPI () and for the synchrotron emission Z_sync_QCD () one can obtain qualitative estimate

 Lf∼min(L1,L2), (4)

 L1∼2Eq(1−x)SLPMm2qx, (5)
 L2∼(24Eqx(1−x)f2)1/3. (6)

Here is the suppression factor due to the Landau-Pomeranchuk-Migdal (LPM) effect LP (); Migdal (), is the photon fractional longitudinal momentum, . For (4) gives simply the formation length for the synchrotron emission in vacuum Z_sync_QCD (). The LPM suppression factor can be easily estimated in the oscillator approximation corresponding to the description of multiple scattering in terms of the transport coefficient in the BDMPS BDMPS () approach to the induced gluon emission. In the oscillator approximation (see below (51)) LCPI (), where (we take here ) A qualitative pQCD estimate gives (see below). From the point of view of the photon emission from the QGP the interesting -region is . Making use of (5), (6) one can obtain for at for quark ()

 L1∼1T√Eq/T, (7)
 L2∼4(Eqc2m4π)1/3. (8)

From (7) and (8) one can see that for the QGP temperatures (here MeV is the deconfinement temperature EoS ()) we have in the energy region of interest GeV, i.e. we have . Then we obtain

 LfRL∼zqc(mπT)3/2(mπEq)1/2 (9)

From (9) for we obtain that at GeV for quark . Thus the condition (3) is reasonably satisfied even at . The contribution of the annihilation may be expressed via the spectrum of the transition (see below). By repeating the above estimates for one can show that for this case the condition (3) is also satisfied.

### ii.2 Basic formulas

The above analysis shows that, similarly to the QGP without magnetic field AMY1 (); AZ (), for the QGP produced in collisions in the presence of magnetic field we can treat the processes and as the collinear ones. And the contribution of these processes to the photon emission rate per unit time and volume in the plasma rest frame can be written as AZ (); AMY1 ()

 dNdtdVd{\bf k}=dNbrdtdVd{\bf k}+dNandtdVd{\bf k}, (10)

where the first term corresponds to and the second one to . The bremsstrahlung contribution reads AZ ()

 dNbrdtdVd{\bf k}=dbrk2(2π)3∑s∫∞0dpp2nF(p)[1−nF(p−k)]θ(p−k)dPsq→γq({\bf p},{\bf k})dkdL, (11)

where is the number of the quark and antiquark states,

 nF(p)=1exp(p/T)+1 (12)

is the thermal Fermi distribution, and is the probability distribution of the photon emission in the QGP per unit length from a fast quark of type . Since we work in the small angle approximation, we can take the vectors p and k parallel. The quantity should be evaluated accounting for the quark interaction with the random soft gluon field generated by the thermal partons and with the smooth external electromagnetic field.

The annihilation contribution can be expressed via the probability distribution for the photon absorption with the help of the detailed balance principle which gives AZ ()

 dNandtdVd{\bf k}=[1+nB(k)]−1dNabsdtdVd{\bf k}, (13)

where is the Bose distribution. The photon absorption rate on the right-hand side of (13) can be written as

 dNabsdtdVd{\bf k}=dannB(k)(2π)3∑s∫∞0dp[1−nF(p)][1−nF(k−p)]θ(k−p)dPsγ→q¯q({\bf k},{\bf p})dpdL, (14)

where is the number of the photon helicities, is the probability distribution per unit length for the transition ( is the quark momentum and is the antiquark momentum, and similarly to we can take the vectors p and k parallel). Using the relation

 nB(k)1+nB(k)[1−nF(p)][1−nF(k−p)]=nF(p)nF(k−p) (15)

from (13), (14) one obtains AZ ()

 dNandtdVd{\bf k}=dan(2π)3∑s∫∞0dpnF(p)nF(k−p)θ(k−p)dPsγ→q¯q({\bf k},{\bf p})dpdL. (16)

Let us consider first calculation of the bremsstrahlung contribution. In the LCPI formalism LCPI () the probability of the transition (for a quark with charge ) per unit length can be written in the form (we use here the fractional photon momentum instead of )

 dPq→γqdxdL=2Re∞∫0dzexp(−izλf)^g(x)[K({\boldmathρ}2,z|{\boldmathρ}1,0)−Kvac({\boldmathρ}2,z|{\boldmathρ}1,0)]∣∣∣{\boldmathρ}1,2=0, (17)

where with , (in general for transition ), is the vertex operator, given by

 ^g(x)=V(x)M2(x)∂∂{\boldmathρ% }1⋅∂∂{\boldmathρ}2 (18)

with

 V(x)=z2qαem(1−x+x2/2)/x, (19)

the fine-structure constant. in (17) is the retarded Green function of a two dimensional Schrödinger equation, in which the longitudinal coordinates (along the initial quark momentum) plays the role of time, with the Hamiltonian

 ^H=−12M(x)(∂∂{\boldmathρ})2+v({\boldmathρ}), (20)

and

 Kvac({\boldmathρ}2,z|{\boldmathρ}1,0)=M(x)2πizexp[iM(x)({\boldmathρ}2−% {\boldmathρ}1)22z] (21)

is the Green function for . The potential can be written as

 v=vf+vm, (22)

where is due to the fluctuating gluon fields of the QGP, and is related to the mean electromagnetic field. The mean field component of the potential reads

 vm=−{\bf f}{\boldmathρ}, (23)

where , F is transverse component (to the parton momentum) of the Lorentz force for a particle with charge . The effect of the longitudinal Lorentz force (which exists for nonzero electric field) is small for the relativistic partons, and we neglect it. The component reads

 vf=−iP(xρ). (24)

Here the function can be written as

 P({\boldmathρ})=g2CF∞∫−∞dz[G(z,0⊥z)−G(z,{\boldmathρ},z)], (25)

where is the QCD coupling, is the quark Casimir, is the gluon correlator (the color indexes are omitted)

 G(x−y)=uμuν⟨⟨Aμ(x)Aν(y)⟩⟩. (26)

Here is the light-like four vector along the axis. The gluon correlator may be expressed via the HTL gluon polarization operator. Making use of an elegant sum rule for the transverse and longitudinal HTL gluon self-energies derived in PA_C () the function may be written as AZ ()

 P({\boldmathρ})=g2CFT(2π)2∫d{\bf q}⊥[1−exp(i{\boldmathρ}{\bf q}⊥)]C({\bf q% }⊥), (27)
 C({\bf q}⊥)=m2D{\bf q}2⊥({% \bf q}2⊥+m2D), (28)

where is the Debye mass. In AZ () it was demonstrated that for the case without external field calculation of the spectrum given by (17) within the LCPI formalism with the use of (27), (28) is equivalent to solving the integral equation obtained in the AMY analysis AMY1 () in the momentum representation. And the formulas (10), (11), (16) reproduce exactly the AMY photon emission rate.

In the approximation of static color Debye-screened scattering centers (in the sense of quark multiple scattering in the QGP) GW () the function reads

 P({\boldmathρ})=nσq¯q(ρ)2, (29)

where is the number density of the color centers, and

 σq¯q(ρ)=CTCFα2s∫d{\bf q}⊥[1−exp(i{\bf q}⊥{\boldmathρ})]({\bf q% }2⊥+m2D)2 (30)

is the well known dipole cross section NZ12 () with being the color center Casimir.

Both for the HTL scheme (27), (28) and the static approximation (29), (30) at approximately . At the function in the static model differs from that in the HTL scheme just by the normalization factor (for ). The replacement of the factor in the dipole cross section in the static model by in the HTL scheme leads to unlimited growth of at large (due to zero magnetic mass in the HTL approximation), while for static model flattens at . However, this difference is not very important from the point of view of the photon emission, because the contribution of the region is relatively small (in the sense of the path integral representation of the Green function entering to (17)).

We will work in the oscillator approximation

 P(ρ)=Cpρ2, (31)

which is widely used in jet quenching analyses JQ_OA1 (); JQ_OA2 (); JQ_OA3 (); JQ_OA4 (); JQ_OA5 (); JQ_OA6 (). The can be expressed via the transport coefficient BDMPS (), describing gluon transverse momentum broadening in the QGP, as . In numerical calculations we use and set GeV at MeV. This value is supported by estimate of within the static model via the magnitude of the dipole cross section at that allows to describe well the data on jet quenching in collisions within the LCPI scheme RAA12 (). It also agrees with the qualitative pQCD calculations of Ref. Baier_qhat () that give with the QGP energy density) (it gives ). Note that the estimate obtained in Baier_qhat () agrees with the relation between and the ratio of the shear viscosity to the entropy density

 ^q∼1.25T3s/η (32)

obtained in MMW_qhat () if one takes the quantum limit value Son ().

### ii.3 Photon spectrum in the oscillator approximation

For the quadratic the Hamiltonian (20) takes the oscillator form (we omit arguments of functions for brevity, where possible)

 ^H=−12M(∂∂{\boldmathρ})2+MΩ2{\boldmathρ}22−{\bf f% }{\boldmathρ} (33)

with

 Ω=√−iCpx2/M. (34)

The Green function for the Hamiltonian (33) is known explicitly (see, for example, FH ())

 K({\boldmathρ}2,z2|{\boldmathρ}1,z1)=MΩ2πisin(Ωz)exp[iScl({\boldmathρ}2,z2|{\boldmathρ}1,z1)], (35)

where , and is the classical action. The action can be written as a sum with

 Sosc({\boldmathρ}2,z2|{\boldmathρ}1,z1)=MΩ2sin(Ωz)[cos(Ωz)({% \boldmathρ}21+{\boldmathρ}22)−2{\boldmathρ}1{\boldmathρ}2], (36)
 Sf({\boldmathρ}2,z2|{\boldmathρ}1,z1)=MΩ2sin(Ωz)[{\bf P}({\boldmathρ}1+{\boldmathρ}2)−W], (37)

where

 {\bf P}=2{\bf f}[1−cos(Ωz)]MΩ2, (38)
 W=2{\bf f}2M2Ω4[1−cos(Ωz)−Ωzsin(Ωz)2]. (39)

Then, after including the vacuum term in (17), a simple calculation gives

 dPdxdL=2V(x)(Iosc+ΔI). (40)

Here corresponds to the pure oscillator case (). It reads

 Iosc=1πRe∫∞0dz⎡⎣1z2−(Ωsin(Ωz))2⎤⎦exp(−izλf). (41)

And gives the synchrotron correction. It can be written as a sum with

 I1=1πRe∫∞0dz(Ωsin(Ωz))2[1−exp(−U)]exp(−izλf), (42)
 I2=1πRe∫∞0dziMΩ38sin3(Ωz){\bf P}2exp(−U−izλf), (43)

where

 U=iMΩW2sin(Ωz). (44)

For numerical calculations it is convenient to introduce the dimensionless integrals

 ¯Iosc,1,2=π|Ω|Iosc,1,2, (45)

and to use the dimensionless integration variable . Then we obtain for

 ¯Iosc(κ)=Re∫∞0dτexp(iπ/4)τ2(1−τ2sinh2τ)exp(−(1+i)τ√2κ), (46)
 ¯I1(κ,ϕ)=Re∫∞0dτexp(iπ/4)sinh2τ[1−exp(−U)]exp(−(1+i)τ√2κ), (47)
 ¯I2(κ,ϕ)=ϕ2Re∫∞0dτ(1−coshτ)2sinh3τexp(−(1+i)τ√2κ−U), (48)

where now

 U=(1−i)ϕ2√2[τ−2tanh(τ/2)], (49)

and the dimensionless parameters and read , .

In the low density limit () . The higher order terms in describe the LPM effect. The ratio of to the leading order term gives the LPM suppression factor

 SLPM=3¯Iosc/κ. (50)

From (46), (50) one can obtain for two limiting cases of strong () and weak () LPM effect LCPI ():

 SLPM≈3κ√2(κ≫1),SLPM≈1−16κ421(κ≪1). (51)

In the limit and the integrals (42), (43) take the form (we denote them )

 Is1=1πRe∫∞0dzz2exp(−izλf)[1−exp(−i{\bf f}2z324M)], (52)
 Is2=1πRe∫∞0dzi{\bf f}2z8Mexp(−izλf−i{\bf f}2z324M). (53)

Similarly to the case of (42), (43) it is convenient to go from (52), (53) to dimensionless integrals. Now we define them as

 ¯Is1,2=πλfIs1,2. (54)

Using the dimensionless integration variable from (42), (43) taking the limit we obtain

 ¯Is1(ϕs)=Re∫∞0idττ[exp(−(1−i)ϕsτ3√2)−1]exp(−(1+i)τ√2), (55)
 (56)

where . Functions (55), (56) may be expressed via the Airy function (here is the Bessel function)

 ¯Is1(ϕs)=−π∫∞zdtAi(t), (57)
 ¯Is2(ϕs)=−2πzAi′(z), (58)

where . Our probability of photon emission in the limit is reduced to the well known quasiclassical formula for the synchrotron spectrum BK (); LL4 () in QED.

For one can obtain similar formulas. But now ( is the quark fractional momentum) , , and

 V(x)=z2qαemNc[x2+(1−x)2]/2, (59)
 Ω=√−iCp/M. (60)

The factor in (59) accounts for summing over the quark color indices for process. For it does not appear in (19) since the sum over the quark color states is included in the factor in (11).

Note that for the contribution of multiple scattering alone the oscillator approximation is equivalent to Migdal’s calculations in QED within the Fokker-Planck approximation Migdal (). The oscillator approximation can lead to large errors in description of the gluon/photon emission from fast partons produced in hard reactions in the regime when the formation length is much bigger than the QGP size Z_OA (); Z_phot (). In this regime the oscillator approximation underestimates strongly the gluon/photon spectrum. However, this problem does not arise for the photon emission by the thermal quarks. In this case we have a situation similar to that for the photon emission from a quark propagating in an infinite medium. In this regime the errors of the oscillator approximation should not be large.

## Iii Photon spectrum in Aa collisions

### iii.1 Integration over space-time coordinates

For the collision at a given impact parameter the thermal contribution to the photon spectrum (we will consider the central rapidity region ) can be written as

 dNdyd{\bf k}T=∫dtdVω′dN(T′,F′,k′)dt′dV′d{\bf k}′, (61)

where primed quantities correspond to the comoving frame, and (here we consider a photon as a massless particle). In (61) we write explicitly the arguments of the photon emission rate in the comoving frame. The argument is the absolute value of the transverse (to the direction of the emitted photon) Lorentz force acting on a particle with electric charge . Note that the photon emission rate in the comoving frame does not depend directly on the azimuthal direction of the photon momentum, and angular dependence of the left hand side of (61) stems solely from the dependence of the photon emission rate on the right hand side on the photon momentum and on the Lorentz force . The value of may be written via the photon four momentum in the c.m. frame of the collision as

 ω′=uμkμ, (62)

where

 uμ=⎛⎜ ⎜⎝1√1−{\bf v}2,{\bf v}√1−{\bf v}2⎞⎟ ⎟⎠ (63)

is the four velocity of the QGP cell. The value of also can be expressed via the photon four momentum and the four velocity of the QGP cell. In the matter comoving frame

 F′=e∣∣{\bf E}′⊥+[{\bf n}′×{\bf B}′]∣∣, (64)

where is the unit vector in the direction of the photon momentum, is tranverse (to the vector ) component of the electric field. In terms of the electromagnetic field tensor in the c.m. frame of the collisions (64) can be written as

 F′=e√−LμLμ, (65)

where

 Lμ=Fμνkμuδkδ. (66)

As usual we write the four volume integration in (61) changing the integration variables , to the proper time and rapidity

 τ=√t2−z2,Y=12ln(t+zt−z). (67)

In these coordinates

 dNdyd{\bf k}T=∫τdτdYd{\boldmathρ}ω′dN(T′,F′,k′)dt′dV′d{\bf k}′. (68)

The use of the formulas (62), (65), (66) allows one to avoid the Lorentz transformations from the quantities in the c.m. frame of collisions to the ones in the comoving frame of the QGP. It makes the calculations for an expanding QGP as simple as for a QGP at rest.

Note that from (62) it is clear that the -integration in (68) is dominated by the region . Because the photon emission rate in the QGP rest frame in the integrand in (68) falls rapidly with , and from (62) one obtains (we neglect the transverse expansion). Since the dominating contribution in the -integration in (68) comes from fm, the effective -volume for the integration over and is fm. It is by a factor of smaller than that of T1 (), where the - and -integrations have been performed for , and (which gives ) over the region fm.

### iii.2 Model of the fireball

It is widely believed that the plasma fireball is produced in collisions after thermalization of the glasma color tubes created in interaction of the Lorentz-contracted nuclei term_gl (). The typical time of evolution of the glasma color fields is about several units of , where ( GeV for RHIC and LHC conditions Lappi_qs ()) is the saturation scale of the nuclear parton distributions. It means that even for a very fast thermalization of the glasma color fields one can apply the formulas obtained for the equilibrium QGP only at fm. The thermalization time fm means practically instantaneous process of the glasma thermalization at , and does not seem to be realistic. Nevertheless, in some analyses of the photon production Sinha (); Mitra () the authors use and fm for RHIC and LHC energies,respectively. But such small values do not have a theoretical justification. In the present analysis we use a more realistic value of fm used in the analysis Gale_best (). To account for qualitatively the fact that the process of the QGP production is not instantaneous we take the entropy density in the interval . However, the contribution of this region is relatively small (due to the factor in the integrand in (68)).

We describe the plasma fireball in the thermalized stage at in the Bjorken model Bjorken () without the transverse expansion that gives the entropy density . For the ideal gas model with it gives in the plasma phase. However, the lattice calculations show EoS () that for the temperature range of interest MeV the entropy density exhibits a significant deviation from the dependence. For this reason it seems reasonable Gale_HG () to determine the plasma temperature from the temperature dependence of the entropy density predicted by lattice calculations. In our analysis we determined from the entropy density obtained in EoS (). At it gives the temperature greater than that for the ideal gas dependence by %. This relatively small increase in may be important for the photon emission rate, because its -dependence comes mostly from the exponential factor (stemming from the Fermi distribution (12)), which at is sensitive even to a small variation of .

In Bjorken’s model the entropy density of the QGP at a given impact parameter vector b of the collision can be written as

 s(τ,{\boldmathρ},Y,{\bf b})=1τdS(%\boldmath$ρ$,Y,{\bf b})d{\boldmathρ}dY, (69)

where is the distribution of the entropy in the impact parameter plane and rapidity. For simplicity we take a Gaussian distribution of the entropy in the rapidity

 dS({\boldmathρ},Y,{\bf b})d{\boldmathρ}dY=dS({\boldmathρ},Y=0,{\bf b})d{\boldmathρ}dYexp(−Y2/2σ2Y). (70)

For Au+Au collisions at TeV we take for the width in which allows to reproduce qualitatively the experimental pseudorapidity distribution of the charged particles . However, the results are not sensitive to the exact choice of , because the dominating contribution to the -integral in (68) comes from .

We calculate the initial density profile in the impact parameter plane of the entropy at the proper time assuming that it is proportional to the charged particle pseudorapidity density at calculated in the two component wounded nucleon Glauber model KN ()

 dNch({\boldmathρ},{\bf b})dηd{% \boldmathρ}=dNppchdη[(1−α)2dNpart({\boldmathρ},{\bf b})d{\boldmathρ}+αdNcoll({\boldmathρ},{\bf b})d{% \boldmathρ}], (71)

where is the pseudorapidity multiplicity density for collisions, and

 dNpart({\boldmathρ},{\bf b})d{\boldmathρ}=TA(|{\boldmathρ}−{\bf b}/2|)[1−exp(−σppTA(|{\boldmathρ}+{\bf b}/2|))]+TA(|{\boldmathρ}+{\bf b}/2|)[1−exp(−σppTA(|{\boldmathρ}−{\bf b}/2|))], (72)
 dNcoll({\boldmathρ},{\bf b})d{\boldmathρ}=σppTA(|{\boldmathρ}−{\bf b}/2|)TA(|{\boldmathρ}+{\bf b}/2|). (73)

Here is the nuclear profile function calculated with the Woods-Saxon nuclear distribution

 nA(r)=N1+exp[(r−RA)/a], (74)

where is the normalization constant,