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Out-of-equilibrium photon production and electric conductivity in a holographic Bjorken expanding plasma

We analytically compute the out-of-equilibrium direct photon production rate and electric conductivity, in a strongly coupled and expanding Bjorken plasma, from holography. Our results are valid at late times where the expanding plasma asymptotes Bjorken hydrodynamics. The out-of-equilibrium rates are substantially harder and larger, early on in the Bjorken expansion phase.

rate is given by [10,11] where k ≡ (k 0 , k) is a null 4-vector which we put on-shell k 0 = |k| and is the Wightman function for the electric current de-correlation. In thermal equilibrium, the Wightman function G < µν is related to the spectral density χ µν using the Bose-Einstein distribution n b (k 0 ) = 1/(e βk 0 − 1) Here G R µν (k) is the retarded electric current correlator in Fourier space.
Modelling the non-equilibrium dynamics of strongly coupled field theories from first principles is a notoriously difficult problem. In this context, holography proved to be a valuable framework to study the real-time evolution, and transport properties of certain strongly coupled field theories (see [12] for a discussion in the context of heavy-ion collisions). Within holography, the dynamics of the strongly coupled field theory is captured by general relativity in asymptotically Anti-de Sitter space. In this language, thermalization is described by the formation of a black hole [13], whose horizon is moving away from a boundary observer [14]. We will rely on the picture of the falling black hole, to extract the out-of-equilibrium photon production during the cooling process.
Bjorken [15] suggested a highly successful model for the central rapidity region of heavy-ion reactions, based on boost invariant hydrodynamics. The holographic dual gravity model which is based on the idea of the falling black hole in [14], was constructed in [16,17]. The main idea is to map the falling black hole onto a frame where the horizon is static and we can define (a time dependent) temperature [17]. The static frame provides a well defined framework for linear response theory. For example [18,19] used this idea to compute the diffusion of heavy quarks in these expanding backgrounds.
Within holography, the equilibrium: photon and dilepton production in N = 4 super Yang-Mills plasma was calculated in [11]. In holographic models for QCD (in the Veneziano limit) the photon production was derived in [20,21]. The authors in [22] extended the holographic discussion to anisotropic plasmas with magnetic fields. The authors in [23] computed the gradient corrections to the photon emission rate at strong coupling. Outof-equilibrium, the prompt photon and dilepton production was discussed in [24,25], using the holographic model of a falling shell. However, in the context of the falling shell the background metric is not explicitly time dependent, and connects smoothly to the equilibrium case which allows the authors to rely on Fourier transformations. In this work, we will compute the out-of-equilibrium photon production in a time dependent background corresponding to a strongly coupled, Bjorken expanding plasma.
The organization of the paper is as follows: In section II we briefly review the holographic set up for a falling black hole in bulk, dual to boost invariant Bjorken hydrodynamics on the boundary. We analyze the evolution of a U(1) vector gauge field, and derive the on-shell boundary action from which the pertinent retarded propagator on the boundary can be extracted. In section III, we use the holographic result to derive the photon emission rate in a Bjorken expanding and strongly coupled plasma. In section IV we derived a closed form result for the U(1) electric conductivity in out-of-equilibrium. Our conclusions are in V.

II. HOLOGRAPHIC SETUP
In the following, we study a strongly coupled SU(N c ) N = 4 super-Yang Mills theory at finite temperature and zero density. In order to study the photon production rate and conductivity, we couple a U (1) gauge field in terms of a Maxwell term to gravity where we assume the electromagnetic coupling to be small. To be more precise, the U (1) group is a subgroup of the global SU(4) R-symmetry.
The metric describing the asymptotic expanding fluid geometry is given by [16,17].
is the radial coordinate of AdS 5 , τ is the proper time, η the rapidity related to the longitudinal directions by x 0 = τ cosh η and x 3 = τ sinh η, where x 0 is the time coordinate. Moreover, we set the radius of AdS and the horizon to unity. The scaling variable v is given by where is the energy density and T 0 the initial temperature. Note that the horizon is located at v = 1 or z ∼ τ 1/3 . Following [18], we can transform the metric into a more canonical form, which resembles a static black hole.
At late times where the perfect fluid geometry is valid (i.e. τ → ∞ and v, u =const), the last two terms are suppressed, and can be ignored. Finally, rescaling the time coordinate by t/t 0 = 3/2 (τ /τ 0 ) 2/3 yields Using this background, we now consider the vec- where F = da. For now we will set the electromagnetic coupling e = 1 and recover it when we compute the transport quantities. At late times, there is no dependence on the transverse directions, and therefore we will only consider the dependence on the longitudinal direction. The equation of motion for the transverse fluctuations reads a x ⊥ ≡ a x ⊥ (t, η, u) The dependence on the longitudinal direction η is subleading at late times, and we can neglect it. Since the equation of motion explicitly depends on time we cannot use a simple Fourier transform, but have to perform a separation of variables by making the ansatz where we find that we can separate the time dependence with Here λ, c 1 , c 2 are independent of t, u, 0F1 is the regularized confluent hypergeometric function, and Y 0 refers to Bessel functions. Choosing the constants c 1 and c 2 accordingly, we eventually arrive at the expression which satisfies the in-going boundary condition at the horizon. We now define a Fourier-like transform where ψ ω (0) = 1. We will drop the subscript of ψ ≡ ψ ω in the following. With λ = ω and g 2 = ψ, we find for the spatial part which we will solve in the following.

A. Analytical solution
The solution to the equation of motion (14) should behave as an in-going wave at the horizon. Since the horizon is a regular singular point, we can expand the near horizon solution in a power series where α = ± i ω 4π T0 . We can recast eq. (14) formally as a Heun's differential equation which is solved by the hypergeometric functions The solution satisfying the in-going boundary condition at the horizon is the one proportional to c 1 , which implies c 2 = 0. Demanding that ψ ω (0) = 1 (since we are interested in the two point function) determines the second integration constant c 1 , and we find where 2F1 (a, b; c, d) is the regularized hypergeometric function 2F1 (a, b; c, d)/Γ(c).
In our ansatz, the on-shell action gives rise to the boundary term where we assumed which is valid for small ω. To first order in u the asymptotic expansion at the conformal boundary . To extract the expectation value, we have to subtract the logarithmic contribution by adding the appropriate counter term. The pre-factor of the logarithmic contribution enters the expectation value of the current as J x ⊥ = 2/3Rψ (l) 4π 2 ψ (s) T 2 0 − ψ (l) ω 2 . This contact term does not affect the photon production rate, or the real part of the conductivity, since they are related to the imaginary part of the retarded Green's function. It contributes, however, to the imaginary part of the conductivity. From eq. (20), we can then read off the renormalized retarded Green's function G R (ω) as where γ is the Euler number, H(n) gives the nth harmonic number H n , and ψ (n) (z) gives the nth derivative of the digamma function. In general, the symmetrized Wightman function G(ω) in momen-tum space is related to the imaginary part of retarded Green's function via G(ω) = − coth(ω/(2T 0 )) ImG R (ω) [26]. On the one hand, we find for the Wightman function at small frequencies ω T 0 On the other hand, for large frequencies ω T 0 , we find These results are obtained in the frame where the black hole is static, and are in general in agreement with the conductivity of the Schwarzschild AdS 5 black hole in [27] (see appendix A).

B. Connecting to the boosted frame
So far, we worked with the frequency ω with respect to the time t, in the frame where the black hole is static. In order to compute the photon production rate, we have to convert our result to frequencies with respect to the proper time in the Bjorken frame. We may define an inverse Fourier transform according to (25) If we introduce the relative and CM coordinates we find that for t 1 , t 2 1 The relative time is related to the proper time by With this in mind, we find where we introduced q 0 = ωτ 0 τ −1/3 , dω = τ 1/3 /τ 0 dq 0 and used T = τ −1/3 . The 4-momentum of the photon is given by where y is the photon rapidity in the Bjorken frame with rapidity η. The number of photons per unit volume, unit rapidity and mass in the Bjorken frame, is given by It is related to the frame where the black hole horizon is fixed by . (33)

III. OUT-OF-EQUILIBRIUM DIRECT PHOTON PRODUCTION RATE
In the following, we elaborate how our solution for the retarded Green's function of the transverse gauge field fluctuations is connected to the photon production rate (2). We can decompose the spectral function of the R current according to [11] with the transverse and longitudinal projector Taking the trace yields In general, the transverse and the longitudinal part contribute to the spectral function. However, we are interested in on-shell photons. For light-like momenta, the longitudinal part vanishes, and the photon-production rate is totally determined by the transverse part. Therefore, the rate of photon production per unit rapidity and mass, is given by integrating over the fluid spatial evolution where n b is the Bose-Einstein distribution discussed in eq. (4). In figure 1, we display the integrand which agrees with the scaling behavior of the equilibrium calculation [11].
In figure 2, we illustrate the dependence of the photon production on the proper time (and thus on the temperature for a given energy density). Starting from the green curve which corresponds to the smallest proper time τ = 2, the maximum shifts to lower momenta m T and decreases in magnitude. The red curve corresponds to τ = 10. This is in strong contrast to the result of [24] which considered a non-expanding plasma.

IV. OUT-OF-EQUILIBRIUM CONDUCTIVITY
The retarded current-current correlator also contains the information about the electrical conductivity σ of the expanding plasma, which is encoded in the zero frequency limit. More specifically, we have by [11] Thus, the real part of the conductivity is given by (38). In summary, the out-of-equilibrium conductivity is given by In figure 3, we illustrate the dependence of the real and imaginary part of the conductivity on the proper time (and thus on the temperature for a given energy density). Starting from the green curve which corresponds to the smallest proper time τ = 2, the value of the real part at m T = 0 increases with the proper time. Furthermore, the real part drops more rapidly as a function of the photon momentum m T for increasing proper time. The maximum in the imaginary part of the conductivity, moves towards lower frequencies and decreases in magnitude for increasing proper time from green to red. We also note that the peak is more pronounced for τ = 10.

V. CONCLUSIONS
In this work, we derived the out-of-equilibrium direct photon production rate and electrical conductivity, for an expanding Bjorken plasma. At late times, our results agree with the literature, however, by deriving the quantities in the time dependent background our results incorporate the history of the Bjorken expansion, and are dependent on the proper time and pseudo-rapidity. Since our metric is explicitly time dependent it is not possible to rely on Fourier transforms. However, in the Bjorken limit, we were able to re-cast the metric in the form of a static black hole, and factor out the time dependence with a Fourier-like transform based on Hankel functions. This trick, which is valid for moderate frequencies, allows us to compute the out-ofequilibrium transport quantities analytically.
We illustrated the dependence of the direct photon production rate on proper time, pseudo-rapidity, and photon momentum. At fixed pseudo-rapidity the peak in the production rate moves to lower momenta, for increasing proper time and is progressively suppressed. We observed a similar behavior for the imaginary part of the electrical conductivity. In contrast, the real part for small momenta strongly increases for larger proper times, but also falls off faster towards larger momenta.
Our results provide analytical insight into the outof-equilibrium transport of an expanding Bjorken plasma. It would be very interesting to extend our results to finite density along the lines of [28,29], and eventually strong background magnetic fields. Another interesting direction is to study metric fluctuations in order to compute transport quantities like the shear viscosity. Moreover, it would be interesting to consider non-Abelian symmetries to compute pion yields in heavy ion collisions with holographic techniques. We leave these tasks for future work.