Pre-equilibrium photons from the early stages of heavy-ion collisions

We use QCD kinetic theory to compute photon production in the chemically equilibrating Quark-Gluon Plasma created in the early stages of high-energy heavy-ion collisions. We do a detailed comparison of pre-equilibrium photon rates to the thermal photon production. We show that the photon spectrum radiated from a hydrodynamic attractor evolution satisfies a simple scaling form in terms of the specific shear viscosity $\eta/s$ and entropy density $dS/d\zeta \sim {\scriptstyle \left(T\tau^{1/3}\right)^{3/2}}_\infty$. We confirm the analytical predictions with numerical kinetic theory simulations. We use the extracted scaling function to compute the pre-equilibrium photon contribution in $\sqrt{s_{NN}}=2.76\,\text{TeV}$ 0-20\% PbPb collisions. We demonstrate that our matching procedure allows for a smooth switching from pre-equilibrium kinetic to thermal hydrodynamic photon production. Finally, our publicly available implementation can be straightforwardly added to existing heavy ion models.

High-energy heavy-ion collisions produce an extremely hot and dense state of deconfined QCD matter.During the early stages of the collision, the QCD matter goes through a stage of kinetic and chemical equilibration, and the hydrodynamization of the Quark-Gluon Plasma (QGP) is swiftly achieved.Despite significant progress in the theoretical understanding of the early pre-equilibrium evolution [1,2], this stage is veiled from direct experimental observation using hadronic observables by the memory loss of the equilibrating medium and by the complicated nature of hadronization.However, electromagnetic probes, i.e., photons and dileptons, provide a unique tool to extract information about the pre-equilibrium epoch as they can escape the deconfined medium without rescattering [3,4].
Electromagnetic probes are produced during a heavyion collision through three main channels: hard scatterings in the first instants of the collision (prompt contribution), medium induced radiation (medium contribution), and the late-time hadronic decays (decay contribution).The sum of the first two channels (called the direct photons and dileptons) can be isolated by the subtraction of the decay products from the total yield.
By now it has been well established, that in order to describe the experimentally measured direct photon spectra in heavy ion collisions, it is essential to include the in-medium radiation.Specifically, the medium-induced photon radiation dominates over the prompt photon production at low and intermediate transverse momenta [5].Beyond providing an additional source of photon production, the anisotropic expansion of the QGP correlates the photons to the collective flow of hadrons.This results in the well-known measurement of a non-zero photon elliptic flow [6,7], i.e., the second Fourier component in the azimuthal angle.So far, the simultaneous description of photon yield and elliptic flow by theoretical models has proved to be challenging, as photon radiation from the more isotropic and hotter earlier stages competes with the more anisotropic photon emission in the later colder stages of the collision.These apparent complex interplay is what has been dubbed the direct photon puzzle in the literature [3,4].
So far, the photon production during the preequilibrium stages has been neglected in most theoretical studies [8,9], or described only in a simplified way [10][11][12][13], where pre-equilibrium photon emission was computed using the parametric behavior in the 'bottom-up' thermalization picture.Notable exceptions are provided by [14], where the pre-equilibrium photon production was computed in the QCD transport approach BAMPS [15].Additionally, Ref. [16] investigates the effects of momentum anisotropy on photon production.In Ref. [17] the suppression of photon emission in gluon-dominated initial stages was estimated from the chemical equilibration rate in QCD kinetic theory simulations.
The central objective of this paper is to simultaneously achieve a consistent theoretical description of the kinetic and chemical equilibration of QGP and the photon production during the early pre-equilibrium stage based on leading order QCD kinetic theory [18][19][20][21].Based on previous studies in strongly coupled holographic models of QGP and weakly coupled QCD kinetic theory, it has been observed that the early non-equilibrium evolution of QGP simplifies due to the phenomenon of hydrodynamic attractors (see Ref. [1,2,[22][23][24] and references therein).In the transversely homogeneous and longitudinally boost-invariant Bjorken expansion, the equilibration of plasmas with different energy densities and coupling strengths can be universally described in terms of a single scaled time variable w = τ T /(4πη/s).Here the specific shear viscosity η/s plays the role of coupling constant (stronger coupling-smaller viscosity).We will demonstrate that, under rather modest assumptions, the photon emission from the pre-equilibrium QGP also obeys a similar universal scaling relation.Specifically, the differential photon spectrum dN /d 2 x T d 2 p T dy can be expressed in terms of a universal function N γ of the scaled time variable w and scaled transverse momentum pT ∝ η/s p T /(sτ ) ∞ with (sτ ) ∞ being the entropy density per unit rapidity in the hydrodynamic phase, as where Cideal γ is a numerical constant describing the thermal photon production rate.Equation (1) along with the calculation of the scaling function N γ in leading order QCD kinetic theory are the main results of our paper 1 .
Based on Eq. ( 1) it becomes straightforward to compute the photon emission during the pre-equilibrium QGP stage, which smoothly matches to the thermal photon production in hydrodynamic simulations of equilibrated QGP.By performing this calculation explicitly, we find that although the pre-equilibrium photons make up only a small fraction of the total photon yield, their contribution at intermediate momenta p T ∼ 3 GeV can be dominant, such that a careful study of this momentum region may have the potential to differentiate between different QGP thermalization scenarios.
This paper is organized as follows: in Section II we present the relevant aspects of the pre-equilibrium photon production in the Effective Kinetic Theory framework.In Section III we derive a universal scaling function of the time-integrated photon spectrum.This formula is the most important result of this paper for phenomenological computations of photon production.Section IV is dedicated to the detailed comparison of non-equilibrium and thermal photon production.In particular, we verify the predicted scaling of the time-integrated photon spectrum.Using the scaling formula in Section V we add the pre-equilibrium photons to other sources of photon production in heavy-ion collisions and compare to the experimental data.We also study the sensitivity of the in-medium photon yield to the hydrodynamization time, and find it to be robust to the variation of this parameter.We conclude with a summary and outlook in Section VI.In Appendix A we document the collision process of quarks and gluons, while in Appendix B we detail the prompt and thermal photon computations for phenomenological comparisons.
We note that, throughout this manuscript, we denote four vectors as P or p µ , three vectors as ⃗ p, (transverse) two vectors as p T , their magnitude as p T and energy as E p = p 0 = |⃗ p| = p.

II. PHOTON PRODUCTION IN QCD KINETIC THEORY
We employ the high temperature effective kinetic theory of non-Abelian gauge theories (AMY EKT) [19] to describe the pre-equilibrium evolution of the QGP.The dynamics of color, flavor, spin, and polarization averaged single particle distribution function f a (X, ⃗ p) is governed by the Boltzmann equation within leading order QCD kinetic theory, we believe that it is also sensible to employ this expression for estimates of the preequilibrium photon production in different microscopic theories, e.g., by supplying the next-to-leading order photon rate [25] in the calculation of the coefficient Cideal γ to (partially) account for higher order corrections.
where the collision integral C 2↔2 a describes the elastic 2 ↔ 2 processes, C 1↔2 a encodes contributions from effective inelastic 1 ↔ 2 processes2 .The subscript a = g, q, γ indicates the particles species.In this work, f q represents a fermion distribution averaged over N f = 3 flavors, N c = 3 colors, and two spin states of massless quarks and anti-quarks (see Table I).
In the case of photons, we can relate the collision integrals to the local production rate dN γ /d 4 Xd 3 ⃗ p according to while the distribution function f γ is given as We will follow previous works in QCD kinetic theory [26,27], where transverse inhomogeneities and transverse expansion were neglected during the early pre-equilibrium phase.
We thus focus on homogeneous, boost-invariant systems for which the phasespace density only depends on local momenta P = (p T cosh(y − ζ), p T , p T sinh(y − ζ)) and Bjorken time τ = √ t 2 − z 2 .Here y = artanh(p z /p) is the momentumspace rapidity and ζ = artanh(z/t) the space-time rapidity.Since momentum and space-time rapidities only appear as a difference, we can trade the ζ integral for an integral over the longitudinal momentum p z = p T sinh(y − ζ) in the local rest frame and the Boltzmann equation simplifies to Our kinetic description includes all leading-order processes for quarks, gluons, and photons.The in-medium scattering matrix elements for the elastic collisions are evaluated using isotropic screening regulator [26][27][28][29][30].For the inelastic collisions, the in-medium splitting rates are found by solving an integral equation reproducing the Landau-Pomeranchuk-Migdal (LPM) effect.In the next section, we specify the relevant collision integrals for photon production: 2 ↔ 2 processes (Compton scattering, elastic pair annihilation) and effective collinear 1 ↔ 2 processes (Bremsstrahlung, inelastic pair annihilation).The analogous collision integrals for QCD processes have been documented in the previous works [27] and are given in Appendix A for completeness.
The kinetic processes would reach thermal equilibrium in an infinite system, including photon degrees of freedom.In realistic heavy ion collisions, the QGP fireball size ∼ 10 − 20 fm is much smaller than ∼ 300 − 500 fm needed for photons to re-interact [31].Therefore in photon collision integrals, we will not keep the detailed balance and will only consider the gain terms.For the same reason, we will neglect the effect of Bose enhancement on photons.

A. Elastic processes
The elastic processes included in our calculation are elastic pair annihilation (EPA) of quarks and antiquarks and Compton scattering (CS), where a quark/antiquark scatters with a gluon, see Fig. 1.The corresponding collision integrals are given in AMY formalism [19] as The matrix elements are summed over the degrees of freedom of incoming and outgoing particles and are given in Mandelstam variables by [32] M q q γg (⃗ where the quark degrees of freedom are d F = N c = 3, fundamental Casimir C F = (N 2 c − 1)/(2N c ) = 4/3, electromagnetic coupling constant e 2 ≈ 4π/137, strong coupling constant g 2 = 4πα s = λ/N c , and the sum of electric charges for N f = 3 flavours of quarks is s q 2 s = 2 3 .Because of the enhancement of soft t-and u-channel exchanges, the vacuum matrix elements Eqs.(7a) and (7b) would be divergent.Inside the medium, this is regulated by screenings effects through the insertions of Hard-Thermal loop (HTL) self-energies [19].Implementation of full HTL results is complicated [16,33], so instead, we use isotropic screening [26,[28][29][30], where the soft t-and u-channels are regulated by the screening mass . Namely, we substitute with ⃗ q u/t being the momentum of the exchange particle and ξ q = e/ √ 2 was determined in [26,34] to reproduce gluon to quark conversion gg → q q at leading order for isotropic distributions.

B. Inelastic processes
A full leading-order description of photon production includes inelastic contributions coming from inelastic pair annihilation (IPA) and Bremsstrahlung (BS) off quarks and antiquarks.The corresponding Feynman diagrams are presented in Fig. 2. The collision integrals are obtained from the AMY formalism [19].At leading order inelastic processes can be taken to be collinear.Therefore we follow [27] and rewrite the integrals as an effectively one-dimensional integral in terms of the collinear splitting (z) and joining (z = 1 − z) fractions The effective inelastic rate dΓ a bc dz is obtained by considering the overall probability of a single radiative emission/absorption over the course of multiple successive elastic interactions.The destructive interference of these scatterings leads to the suppression of radiative emission/absorption, known as the LPM effect.The splitting probability is reduced to an effective collinear rate by integrating over the parametrically small transverse momentum accumulated during the emission/absorption process.It can be expressed accordingly where α s = λ 4πNc = g 2 4π and P a bc is the leading-order QCD splitting function [35] The term Re 2p b • G(p b ) in Eq. ( 10) captures the relevant aspects of the current-current correlation function inside the medium to treat the LPM-effect via effective vertex resummation.It satisfies the integral equation where Here, m 2 ∞,a , m 2 ∞,b , m 2 ∞,c denotes the asymptotic masses of the different particles, namely, m 2 ∞,q = 2m 2 q for quarks and m 2 ∞,γ = 0 for photons.The thermal quark mass m q is given by where we used f q = f q for a system without net charges.The Casimir of the representation C R a , C R b , C R c are given by C R g = C A for gluons, C R q = C F for quarks and C γ = 0 for photons.Finally d Γel /d 2 q is the differential elastic scattering rate, which is stripped of its color factor and at leading order is given by [27] d Γel with the Debye mass The effective inelastic in-medium rates are sensitive to the density of elastic interaction partners due to the fact that inelastic interactions are induced by elastic interactions.The quantity g 2 T * in Eq. ( 12) characterizes then the rate of small angle scatterings in the plasma where, T * is defined such that in equilibrium it corresponds to the equilibrium temperature T * eq = T eq .Solutions to Eq. ( 12) are found by first performing a Fourier transform to the impact parameter space.One solves the resulting ordinary differential equation with specified boundary conditions [36,37].For details of numerical implementation, we refer to the previous work [27,37].

C. Comparison to equilibrium AMY rates
Before we proceed to the calculation of the preequilibrium photon production, we compare our numerical implementation of photon rates of Sections II A and II B to the equilibrium parametrizations of photon rates given in Ref. [20].In Fig. 3 we present the results for different coupling strengths λ = g 2 N c .In the left panel of Fig. 3 we compare different photon production channels.For small coupling, λ = 0.1, we find a perfect agreement between our and AMY results for all channels.We also note that the elastic processes are the dominant photon production channel for this coupling except for the low momentum region p < T .For larger coupling, λ = 10, the relative contribution of the elastic channel is greatly suppressed and becomes equal to the inelastic channel even for p > T .In this case, we see the difference in elastic processes between our and AMY calculations.We note that authors of [20] use different infrared regularisation in the elastic channel, so small differences are to be expected.The difference becomes more pronounced for larger values of the coupling where the subleading corrections to the infrared regulators become apparent.A similar trend is seen in the right panel of Fig. 3 where we show just the total photon rate for different couplings λ = 0.1, 1, 5, 10, 20.

III. SCALING LAWS FOR PHOTON SPECTRUM
The thermalization of QGP in QCD kinetic theory has been studied extensively [26][27][28][29][38][39][40][41][42][43][44].One of the key discoveries is the emergence of so-called hydrodynamic attractors.Namely, the key bulk characteristics of the QGP, like energy or pressure anisotropy, approximately follow a universal function, which is a function only of the scaled time w = τ T /(4πη/s).For example, the energy density e(τ ) in Bjorken-expansion follows where τ 4/3 e ∞ is the asymptotic equilibrium constant and E( w) is a universal function of scaled time w [45].This is the expected hydrodynamic behavior near equilibrium.The collapse of simulations with different initial conditions to the same curve E in deeply anisotropic regime 1) is called the hydrodynamic (energy) attractor.
The simplicity of non-equilibrium system following the attractor allowed analytical computation of entropy production in the non-equilibrium stages [45].Scaling with w has also been observed in the chemical equilibration of QGP [26,40], and recently been used to estimate the dilepton production during the pre-equilibrium stage [46,47].It is thus reasonable to expect that photon radiation off equilibrating fermions might also follow w-scaling.In this section, we will derive the analytical formulas for the photon spectra from expanding QGP.

A. Scaling for evolution along a hydrodynamic attractor
The Lorentz invariant photon spectrum radiated by the QGP is given by the volume integral where the phase-space density of photons is defined as and as before, the collision integral is determined from The integral Eq. ( 19) is similar to the Cooper-Frye integral for hadrons, but the surface integral is replaced by a volume integral.
As the momentum differential can be rewritten according to d 3 ⃗ p = d 2 p T dy, where y is the momentum spacerapidity, the transverse momentum spectrum is derived by inserting the Boltzmann equation into Eq.( 19) and is determined according to where we combined elastic and inelastic processes in a single integral C γ .Similar to the discussion around Eq. ( 5), for a boost-invariant expansion and transversely homogeneous expansion, the photon emission rate Now if we assume that the phase-space distributions of quarks and gluons obey w-scaling, such that in the local rest frame f a (X, P ) = f a ( w, ⃗ p/T eff (τ )), we can write the collision integral as a dimensionless function Cγ of w and scaled momenta where the effective temperature is defined by the Landaumatching condition with ν eff = ν g + 2N f 7 8 ν q .Using Eqs. ( 18) and ( 26) and the definition of w, we can write the effective temperature and Bjorken time in terms of w variable where By plugging Eqs. ( 25), (27a) and (27b) into Eq.( 24) we then obtain the scaling formula for photon spectrum emitted by the equilibrating QGP where we defined pT = (η/s) . We note that the photon production rate Cγ itself also explicitly contains the strong coupling constant (see Sections II A and II B).However, by looking at the ratio of the non-equilibrium rate to the thermal rate, this additional coupling dependence can be cancelled out.By following this logic and taking the limit wmin → 0 we define3 where is obtained as a moment of the equilibrium photon rate (see next section).We now see that as advertised in Section I the photon spectrum is naturally written in terms of the scaled transverse momentum pT = (4ν eff π 2 /90) ∞ and exhibits an overall (η/s) 2 scaling.

B. Scaling for ideal Bjorken expansion
Starting from Eq. ( 24) we can also derive the idealized thermal spectrum if we assume the system is in equilibrium for all times.In this case where the step function Θ depends on the starting and final temperatures of Bjorken expansion.
By interchanging the order of integration, we can find the integration limits on the rapidity integral.For convenience, we introduce a change of variables u = cosh ζ.
We split the z integral into three cases: z < p T /T (τ min ), p T /T (τ min ) < z < p T /T (τ max ) and z > p T /T (τ max ).Because u > 1, the first case does not satisfy z min < z < z max .The other two cases give The u-integrals can be done analytically and we get the final result for the photon spectra in ideal Bjorken expansion ∞ √ 4π wmin pT ) .
Importantly, in the limit of infinite expansion, the spectrum is exactly a power law where the normalization constant Cideal γ is given by Eq. ( 31).Together with other constants from our simulations, the numerical values of Cideal γ in leading order QCD kinetic theory for different couplings are listed in Table II s and initialization time of the simulation wmin for different coupling strengths λ.

IV. PHOTON PRODUCTION FROM NON-EQUILIBRIUM QGP EVOLUTION
We will now present our simulation results for the photon emission from kinetically and chemically equilibrating QGP.We start the kinetic evolution at τ 0 = 1/Q s with CGC-inspired gluon-dominated initial state.We use the same parametrization as in previous studies for the gluon distribution [26][27][28] and set the initial fermion distribution to zero where A = 5.34, Q 0 = 1.8Q s , ξ 0 = 10 and λ = g 2 N c = 4πN C α S .How such system approaches chemical and kinetic equilibrium has been studied in detail in previous works [26,27].

A. Instantaneous rates
In Figs.4a and 4b we show the quark and gluon distributions for different times as a function of momentum p/T eff (τ ) and longitudinal momentum fraction cos θ = p z /p for λ = 10 (α s = 0.27).The blue points are the non-equilibrium distribution functions and green lines correspond to the usual Fermi-Dirac distribution for quarks and Bose-Einstein distribution for gluons at temperature T = T eff (τ ), see Eq. (26).At very early times the f q is essentially zero, while f g is highly peaked at cos θ = 0 due to the initial momentum anisotropy.As the system evolves, quarks are produced from gluon fusion gg → q q and gluon splitting g → q q, and they also tend to peak around cos θ = 0. However as the longitudinal expansion becomes less prominent with time, the distributions become more isotropic (flatter in cos θ), but for w = 1, the time around which hydrodynamization starts, the gluons are still over-populated and quarks are under-populated with respect to their thermal distributions.Eventually, for large w > 2 (data not shown) both distribution functions smoothly approach the respective thermal distributions.In Figs.4c and 4d we show the photon production rate split up into elastic and inelastic channels according to Once again blue curves show the non-equilibrium rate, while the corresponding thermal production rates are shown in green. 4At early times all processes are es-sentially zero because there are no quarks in the plasma at the beginning.In the elastic channel shown in Fig. 4c, the Compton scattering and elastic pair annihilation make comparable contributions.The rate is significantly suppressed compared to thermal, because of suppressed quark distributions.We also see the slight peak at cos θ = 0, where the quark distribution is the highest.In Fig. 4d we show photon production in inelastic processes, where Bremsstrahlung dominates over the inelastic pair annihilation, because there are not many quarks to annihilate.We see a pronounced peak at cos θ = 0.For larger couplings in thermal equilibrium (see Fig. 3) the inelastic processes are more dominant than elastic photon production.We see that the same is true out of equilibrium for λ = 10.Evolving the system further in time shows that we start to recover the thermal production rates at times w ≳ 2 (data not shown).
For the comparison to experimental measurements, we are interested in the p T -spectrum.The corresponding production rate dN γ /(τ dτ d 2 x T d 2 p T dy) is given by the p z integral of the collision kernel in the fluid rest frame (see Eq. ( 24)).In Fig. 5 we show the results for λ = 10 as a function of p T /T eff (τ ) for different values of scaled time w.The non-equilibrium results are presented by blue solid lines, while the thermal rate is presented by green dashed lines.At early times the production rate is well below the thermal rate at low momentum.However the non-equilibrium spectrum is much harder.As we scale the momentum with T (τ ) = T eff (τ ), which is the largest at early times, we emphasize that the low p T /T (τ ) regime is produced earlier than hard regime.This is manifested in the fact that the thermal production rate is recovered in the soft regime first.At later times w > 1 the EKT  spectrum also approaches the equilibrium at higher momentum although it still slightly overshoots the thermal spectrum.
Finally, we look at the fully integrated photon production rate ∂n γ /∂τ and the energy loss rate ∂e γ /∂τ due to photon production, which can be obtained as ∞ .Different colors represent integration up to a specific time w as indicated by the color box.At the bottom the T eff (τ ) corresponding to the different times are indicated.Top axis is the transverse momentum in physical units for typical (T τ 1/3 ) 3/2 ∞ = 0.402 GeV (see discussion around Eq. ( 43)) and η/s = 0.08.Results are for λ = 10.
Fig. 6 shows the photon energy and number rates compared to the thermal rates as a function of w for different couplings λ = 5, 10, 20.We see that the relative modification with respect to the thermal rates is very large at early times w ≪ 1 and quite similar for different couplings, supporting our argument of a universal modification of thermal rates.It is also important to point out, that for w > 1 corresponding to the typical time scale for the onset of hydrodynamic behavior, the non-equilibrium photon production rate converges to the expected equilibrium photon rate up to small residual non-equilibrium correction.

B. Scaling of the time-integrated photon spectrum
The total contribution from pre-equilibrium photon production is given by the time-integrated spectrum Eq. (24).Our goal in this section is to verify that the photon spectrum from the full QCD kinetic simulations satisfies the scaling formula Eq. ( 29) derived in Section III A and extract the scaling curve.Therefore we will use the scaled transverse momentum variable pT = (η/s) 1/2 p T /(τ 1/3 T ) In QCD kinetic simulations we extract the specific shear viscosity η/s by fitting the late time behavior of the longitudinal pressure P L ≡ τ 2 T ζζ to the first order viscous hydrodynamics expectation for Bjorken expansion [29,44] Similarly, we extract the asymptotic constant (τ 1/3 T ) ∞ by fitting the late time behavior of the effective temperature T eff (τ ) to the exact result in first order viscous hydrodynamics We summarized the extracted values in Table II.
Of course, for comparisons with experimental data, it is important to estimate the typical value of τ 1/3 T ∞ in heavy ion collisions.This is done by considering the average entropy per unit rapidity dS/dζ = A ⊥ τ s(T ), where A ⊥ is the transverse area of the collision.At sufficiently late times after the collision, so that the system is close to local thermal equilibrium, but early enough that the transverse expansion can be neglected, the entropy density per rapidity is constant τ s(T ) = 4π 2 90 ν eff τ T 3 ∞ .Neglecting the entropy production during the transverse expansion, the same dS/dζ entropy per rapidity is converted to hadrons on the freeze-out surface.Using the known average entropy per charged particle S/N ch ≈ 7.5 [48] we relate the charged particle multiplicity and entropy per rapidity as dN ch dζ = N ch S dS dζ .We estimate the transverse area A ⊥ = 138 fm 2 .Using dN ch dζ ≈ 1600 [49] for central Pb-Pb collisions at LHC energies we then find the typical value of τ T 3 ∞ to be which corresponds to With this we can convert the scaled transverse momentum into physical units.The results of QCD kinetic theory simulations for timeintegrated production rate for λ = 10 are shown in Fig. 7.The different colors indicate up to which scaledtime w the photon spectrum is integrated.The arrows at the bottom indicate the effective temperature T eff at the corresponding time.On the upper x-axis we show the momentum in physical units for the typical value of (T τ 1/3 ) 3/2 ∞ , see Eq. (44).At early times the effective temperature is high and correspondingly the photon spectrum extends to much higher p T in absolute units.This regime is mainly produced up to times w ∼ 0.75.At lower momentum, we find a power law behavior.Although the slope stays the same for all times, the later time contributions are added at lower momentum.Therefore the spectrum at intermediate momenta gains a different power law behavior and approaches the ideal Bjorken evolution (see Eq. ( 37)).At later times (from w > 1) the spectrum is populated mostly at momentum below p T < 0.4 GeV.
In Fig. 8a we show the integrated spectra for different values of the coupling constant λ = 5, 10, 20 and for different final times w = 0.5, 1.0.We compare QCD kinetic theory results (solid lines) to an idealized Bjorken expansion, where the system is assumed to be in thermal equilibrium throughout the entire evolution, which is shown by dashed lines (see Section III B).In accordance with the discussion in Section III B, for an infinite expansion, this 'ideal' spectrum is a simple power-law (see Eq. ( 37)).As dotted lines we show the thermal production rates for finite w cut-off (see Eq. ( 36)), where we employed wmin = 0.For high p T they collapse on the appropriate 'ideal' curves, which is due to setting wmin = 0.At lower p T the curves with finite integration time deviate from the idealized curves as an effect of wmax < ∞.We see that a larger coupling corresponds to a larger ideal photon spectrum throughout the whole range of p T .Looking at the non-equilibrium contribution, we see that at high momentum, which is the regime produced the earliest in the evolution, more photons are produced the larger the coupling is.However, going down in momentum there is a point, where all three coupling coincide.For momentum below this point the hierarchy is reversed and for smaller couplings the production rate of photons is higher.As time evolves, the point were all three couplings agree, moves towards lower momentum and shifts from p T ≈ 0.8GeV for w = 0.51 towards p T ≈ 0.6GeV for w = 1.02.Overall the pre-equilibrium photon production is suppressed compared to the ideal thermal photon production, due to the suppression of quarks in the initial state.
In Fig. 8b we replot the same spectra using the predicted attractor scaling.
Namely, we use (η/s) 1/2 p T /(τ 1/3 T ) 3/2 ∞ as x-variable and divide the spectrum on the y-axis by (η/s) 2 Cideal γ .By construction, the ideal Bjorken expansion spectra collapse to the same curve (gray dashed) for all couplings.The same behav-ior is found for the finite wmax curve (gray dotted).The QCD kinetic theory results (solid lines) also become much closer to each other, especially at later times.This is the manifestation of the predicted attractor scaling in Eq. (29).Only in the high-p T sector, different couplings do not scale in the same way.This regime is produced at early times and is barely modified afterwards, such that a description in terms of an attractor solution is not appropriate at the time of production.
From Fig. 8b we can thus extract the universal scaling function for the integrated photon spectrum in nonequilibrium QGP expansion: Generally, the scaling curve for typical values of w = 1, the timescale on which the system hydrodynamizes, can be characterized as follows.At high regime, , it shows a steep fall-off as this regime is sensitive to the earliest evolution times, where only few quarks are present and thus few photons are produced.In the inter- ing curve for pre-equilibrium photon production shows the same power-law behavior as the ideal result, albeit with an overall suppression, which we attribute to the continued quark suppression during the pre-equilibrium evolution.Below ∞ ≲ 1 the scaling curve flattens and approaches a different power law behavior than in the intermediate region as the softer photons tend to be predominantly produced at later times w > 1.This is confirmed by the fact that the pre-equilibrium scaling curve shows essentially the same behavior as the ideal spectrum for a finite evolution time wmax , once again with a slight suppression due to the chemical non-equilibrium conditions.One of the remarkable consequences of the universality of the scaling result of Eq. ( 45) is the practical usefulness for phenomenology.Based on the scaling function in Eq. (45), it is directly possible to compute the preequilibrium photon production for a given initial condition for a hydrodynamic simulation by matching the viscosity η/s and the energy density at a matching time τ hydro .In Fig. 9 we show the photon spectra from the non-equilibrium rate that is integrated up to time τ hydro at which T eff (τ hydro ) = T hydro .The constant ( T τ 1/3 ) ∞ is obtained from Eq. ( 42) (assuming that entropy evolution is well described by 1st order hydrodynamic Bjorken expansion after τ hydro ) and the scaled time at τ hydro is simply whydro = τ hydro T hydro /(4πη/s).The two panels in Fig. 9 show the results for two different switching times τ hydro = 0.6 fm (left) and τ hydro = 1.0 fm (right), where for a fixed temperature T hydro a larger τ hydro corresponds to a larger whydro and larger ( T τ 1/3 ) ∞ .In each panel the photon spectra is shown for different temperatures  II).The different panels correspond to τ hydro = 0.6 fm (left) and τ hydro = 1.0 fm (right).
T hydro .Bands in Fig. 9 correspond to the results obtained for different values of η/s in the range of 0.08 − 0.16.Clearly, the first thing to notice is that for a fixed matching time τ hydro fluid cells with higher final temperature T hydro will produce more photons in the non-equilibrium evolution.Conversely, photon production from regions with smaller local temperatures is strongly suppressed.Secondly a later initialization time also results in a larger pre-equilibrium photon production yield, in particular at low p T as these photons tend to be produced later over the course of the evolution.By looking at the relative size of the bands in Fig. 9, one observes that the variation of the pre-equilibrium photon yield with changing η/s is rather small, indicating that uncertainties in the photon yield that stem from the microscopic dynamics of thermalization are not too dramatic.Although this might be surprising at first, it should be expected considering the momentum range sampled in Fig. 9.For the ideal spectrum in Eq. ( 37) we observe an exact cancellation of η/s by 1/p 4 T .The reader can observe that the physical momentum in Fig. 9, p T ∼ 1 − 5 GeV, corresponds to the universal momentum, p T (T τ 1/3 ) −3/2 ∼ 1 − 10.In such range the pre-equilibrium spectrum is close to the ideal one and we see an (almost exact) compensation and consequently only a small η/s dependence.Additionally, the spectrum does not depend on wmax , which would give another source of η/s dependence, as Fig. 7 shows that this momentum range is actually produced on timescales, which are well below the cut-off in w applied in our QCD kinetic simulations.

V. PHENOMENOLOGY OF THE PRE-EQUILIBRIUM PHOTONS
We will now describe how our results for the preequilibrium photon production can be employed in phenomenological calculations of the direct photon production.For phenomenological comparison to experimental photon data, the complete direct photon spectrum can be constructed as a sum of prompt and in-medium photons.The prompt photons are produced in hard scatterings at the initial instant of the collision, while the in-medium photons are a sum of the radiation coming from the deconfined medium (pre-equilibrium and equilibrium QGP) and photons produced by hadronic scatterings in an interacting hadron gas.A schematic timeline of different photon sources is summarized graphically in Fig. 10.Equilibrium QGP and hadronic radiation are relatively well-understood in-medium contributions.However, before this work the full computation of pre-equilibrium photons was lacking.In the next section we describe our procedure for computing pre-equilibrium photons and matching to the subsequent hydrodynamic evolution.For prompt, equilibrium QGP and hadron resonance gas contributions we follow Refs.[12,13] and reproduce the details of the computation in Appendix B.
A. Photons from the pre-equilibrium stage Pre-equilibrium photons are computed by matching the QCD kinetic theory evolution of the QGP to the initial conditions of the subsequent hydrodynamic evolution.To compute the pre-equilibrium spectrum for the photons, the two main ingredients are the scaling formula, Eq. ( 45), and the knowledge of the temperature profile at the start of the hydrodynamic evolution (see Fig. 10).Notice that the scaling formula readily provides the spectrum of all photons produced over the course of the pre-equilibirum evolution.Given an initial temperature profile T (τ hydro , x T ) at the initialization time τ hydro of a hydrodynamic evolution with η/s, one can then extract the local value of the scaling variable w(x T ), as well as the local energy scale (T τ 1/3 ) ∞ (x T ) for each fluid cell x T to obtain the pre-equilibrium photon spectrum.The total spectrum is given by the integral in the transverse plane The physical meaning of this is that we match the temperature in the fluid cell at hydrodynamic initialization time to the final state temperature in the kinetic theory evolution.By reading off the corresponding rescaled time w(x T ) it is possible to look up the spectra of photons produced by such a cell during the preceding non-equilibrium evolution. .The schematic timeline for photon production in heavy-ion collisions.The prompt photons are produced in hard scatterings at the instant of the collision.The pre-equilibrium photons (this work) are radiated during the first ∼ 1 fm of the evolution as the QGP is equilibrating.At τ hydro the pre-equilibrium production in QCD kinetic theory is smoothly matched (locally in the transverse plane and event-by-event) to equilibrium (AMY) photon rates in the hydrodynamic phase.The matching procedure is described in Section V A. Afterwards, thermal photons from the QGP are produced by folding thermal AMY rates with the hydrodynamic evolution.As the system goes through the cross-over at Tpc = 160 MeV, we switch to hadronic rates, which are the source of emission until the end of the collision evolution at Tmin = 120 − 140 MeV.
On a more pragmatic note, the η/s used in the matching is taken as an "external" hydro parameter, and not the η/s extracted from EKT for a given coupling constant λ.
Here we follow the phenomenological practice [26,39] of using the approximate independence of the coupling constant of the EKT evolution when expressed in scaled time w.The physical shear viscosity, which characterizes the realistic interaction strength of the QGP, is more reliably extracted from the hydrodynamic model to data comparisons [50,51] than perturbative computations due to large NLO corrections [52].For definiteness, we will use the value η/s = 0.08, which is on the lower side of the extracted values.Similarly, the normalization constant of thermal photon production Cideal γ in Eq. ( 46) can be taken as an external parameter.However, the NLO corrections to LO thermal photon production is small ∼ 20% at α s = 0.3 [25].Therefore in the following we will take Cideal γ = 0.573, which corresponds to AMY thermal rate for α s = 0.265, which we used for thermal productions during the hydrodynamic evolution.
The quantity (τ 1/3 T ) ∞ sets the physical scale, c.f. Eq. ( 43), and we will extract it using the exact formula at first viscous order The next step is to find the rescaled time, w which can be directly extracted as Notice that in the above expressions, both the evolution time w and the local energy scale (τ 1/3 T ) ∞ vary across the transverse plane, due to the x T dependence of the temperature profiles.We will use a constant η/s, but more generally transport coefficients in hydrodynamic simulations can be temperature dependent and therefore induce further x T dependence5 .Additionally, by varying wmin , one can explore the sensitivity to the initial time for the photon spectrum.In practice this is performed by subtracting from the spectra up to the desired minimum universal time.Namely, this is achieved by substituting the scaling function in (46) for In what follows, we apply this matching procedure to an event-by-event boost invariant (2+1D) hydrodynamical simulations.The realistic event-by-event temperature profiles in the transverse plane are taken from viscous hydrodynamic simulations tuned to experimental data.We use the same 200 Pb-Pb events at √ s NN = 2.76 TeV corresponding to the 0-20% centrality class as in Refs.[12,13].Hydrodynamics is initialized at τ 0 = 0.6 fm with the two-component Monte Carlo-Glauber model [53] and evolved with the VISHNU package [54,55] using the default model parameters.In particular, we use the lattice equation of state6 , a constant specific shear viscosity of η/s = 0.08 and zero bulk viscosity.See Refs.[12,13] for further details.
Finally, we note that the matching is done between transversely homogeneous EKT simulations and (2+1D) hydrodynamics at a constant τ hydro ≥ τ 0 .This matching time should be sufficiently small compared to the transverse system size ∼ 10 fm, such that we could neglect any transverse dynamics, e.g., transverse velocity profile.However, the matching time does not need to be constant, as in principle one only needs the initial hypersurface of the hydrodynamic simulation, which provides both temperature and proper time profiles.

B. Comparison to experimental data
Our main result for this section lies in Fig. 11, where we present our computation of the total photon yield in Pb-Pb collisions at 2.76 TeV, for the 0 − 20% centrality class.In Fig. 11, the multiple sources of photon production are presented such as the prompt (blue), the preequilibrium contribution (this work, green), and thermal photons from the QGP and the hadronic stage (yellow).We compare the summed total (red) to the measured ALICE photon yield [56,57] and find a good agreement.
When computing the pre-equilibrium contribution, we employ the photon yields for λ = 10, which corresponds to α s = 0.27, while the details of the calculation of prompt photon production, as well as the production of photons from the thermal QGP and hadronic sources are identical to the procedure of [12], but nevertheless included in Appendix B for completeness.Additionally, we have included a variation of minimum universal time wmin = 0.1 − 0.5 using the prescription described above.
Comparing the different sources, the thermal photon production is most prominent in the low momentum regime, whereas the EKT photons have the smallest contribution to the photon spectra in this region.Around p T ∼ 2.5 GeV, the thermal contribution becomes the smallest one and drops off rapidly with increasing transverse momentum.Between p T ∼ 2.5 GeV and p T ∼ 7 GeV the EKT photons are comparable to the spectra coming from prompt photons.Above p T ∼ 7 GeV the contribution from EKT falls off as well and the hard-momentum sector is mainly dominated by prompt photons.
When calculating the pre-equilibrium contribution in Fig. 11, we have chosen a matching time of τ hydro = 1.0 fm (see Section V A).Nevertheless, this choice is not unique, as the formalism in this paper allows for a smooth transition between the early out-of-equilibrium stage and the hydrodynamical evolution if w ≳ 1 [29].We quantify the validity for the matching times range we have chosen to explore, τ hydro = 0.6 − 1.0 fm, by looking at the spatially averaged scaling time variable for the average event at the extrema in this range.We find for our fixed value of η/s = 0.08, which is indeed in the regime where the non-equilibrium evolution of the QGP can be described hydrodynamically.
We explored the sensitivity of the resulting total yield to the variation of the matching time.In Fig. 12 we show the ratio of the thermal and pre-equilibrium contributions where the matching is once performed at τ hydro = 0.6 fm and once at τ hydro = 1 fm.The EKT curve is shown in green and the thermal contribution in yellow, while red shows the sum of EKT and thermal.Notably, the EKT and thermal ratios both deviate signficantly from each other.For the thermal yield, the ratio is insensible to the switching at low momentum, while for higher p T an earlier matching time means a larger yield.For the EKT photon yield this behavior is somewhat reversed.At low momentum a later switching time means a higher yield, while the EKT production is insensible to τ hydro at high p T .Strikingly, the sum of both contributions is independent of the switching time (< 5% variation).This allows for the matching to be smooth regardless of the matching time, as long as the relevant characteristic w is in the regime where the non-equilibrium evolution of the QGP can be described hydrodynamically.
Finally, when varying the initial time, we seem to get better agreement for larger wmin (lower end of the EKT band in Figure 11).The variation of the minimum time at which photons are produced is an interesting handle that needs to be explored further in future work.Nevertheless, we would like to note that, regarding the matching at large p T , more uncertainties may come into play when calculating the prompt contribution as they are commonly shown in the literature.For example, while error is computed in this figure by accounting for a change of the renormalization scale of the prompt processes (see Appendix B), the uncertainties of the parton distribution functions are not included in most of the literature.Additionally, the process is scaled using a simple MC-Glauber approach, which ignores the nuclear effects of the aforementioned distributions.These sources of uncertainties and novel parameterizations of nuclear effects need to be taken into account in the future to properly understand the photonic contributions at low and intermediate momenta.

VI. SUMMARY AND OUTLOOK
We computed the photon emission in chemically equilibrating QGP using leading order QCD kinetic theory.For phenomenological relevant coupling strengths (λ = 10, α s = 0.265), we showed that inelastic processes (Bremsstrahlung, inelastic pair annihilation) dominate the production over the elastic ones (Compton scattering, elastic pair annihilation).Especially Bremsstrahlung off quarks and antiquarks dominates all other four processes for the three couplings tested here.At early times the rates are suppressed significantly since the system is gluon dominated and few quarks are produced yet.p T (GeV)  [56,57].The solid red lines show the total computed photon spectra, which consist of prompt (blue), pre-equilibrium (green) and thermal rates in the hydrodynamic phase (yellow).
. The ratio of differential photon spectra in ki-netic+hydrodynamic evolution computed for two matching times τ hydro = 0.6, 1.0 fm.The yellow line is the ratio for thermal yields, green for EKT yields and red for the sum of EKT and thermal.The gray band correspond to changes of 5%.
Evolving in time the system cools down and the nonequilbrium rates smoothly approach their counterparts in thermal equilibrium.
In addition we showed that photons with high p T are predominantly produced at the earliest stages of heavyion collisions.In this regime we see a steep fall-off compared to photon production in an idealized expansion due to a strong quark suppression at this stage of the evolution.Conversely, the low-p T regime is produced at late times, where the photon spectra shows a characteristic power law behavior.The slope of the curve stays the same for all times although for later times the total contribution in this regime increases.In between an intermediate regime is established, which exhibits a different power law behavior.
Assuming that the pre-equilibrium evolution of the QGP can be described in terms of a single scaled time variable w, we showed that the photon spectra satisfies a simple scaling formula where the momentum is scaled by (η/s) 1/2 p T /(τ 1/3 T )

3/2
∞ with an additional overall normalization of 1/(η/s) 2 Cideal γ to the photon spectra.Here Cideal γ is a constant coming from the thermal photon spectra, if the system is assumed to be in equilibrium for all times.
The benefit of this universality is that our results can be applied to estimate the pre-equilibrium photon production in realistic simulations of heavy ion collisions.We applied our formula in event-by-event simulations and found a smooth matching to hydrodynamic photon yields.Compared to different sources of photons during a heavy-ion collision, the pre-equilibrium yield is small and is dominated by thermal contributions for low momenta and by prompt photons for the whole considered range of momenta.Nevertheless above p T ∼ 2.5GeV, the contribution from pre-equilibrium exceeds the thermal photon production in the hydrodynamic phase and is almost of the same order as the prompt contribution.
We implemented our formalism into the initial state framework KøMPøST [29,58].This version, called ShinyKøMPøST, allows the computation of preequilibrium photon spectra from the energy-momentum tensor profile at the hydrodynamic starting time τ hydro .Such profiles are naturally generated by KøMPøST propagation, such that our results can be used in future phenomenological studies.Similarly the study of different initial conditions is left for future works.Although the pre-equilibrium spectrum will be highly sensitive to the initial composition of quarks and gluons, this is beyond the scope of this paper.
Within this work, we computed the p T differential photon spectrum, which is arguably the simplest photon observable and is not directly sensitive to the anisotropy of photon production in early times.Nevertheless, it is conceivable that other observables, such as, e.g., photon v 2 or HBT could be more suited to identify the unique features of pre-equilibrium production [12].We leave such detailed investigations for future work.
Evidently, another logical extension of our formalism is the calculation of pre-equilibrium dilepton production in QCD kinetic theory.In this context, recent studies indicate that dileptons from the pre-equilibrium phase might dominate the production in an intermediate range of the invariant mass [46,47,59,60], making them an interesting candidate for further exploration of the preequilibrium phase of high-energy heavy-ion collisions.Preliminary results show that a similar universal scaling as for pre-equilibrium photon production can also be obtained for pre-equilibrium dilepton production, opening a new avenue for detailed phenomenological investigations.
Similarly, the emergence of such universal scaling during the pre-equilibrium phase is also interesting for other phenomenological applications, concerning, e.g., heavyflavor production, heavy-quark diffusion [61,62] or jet energy loss during the pre-equilibrium phase, which deserve further investigations.For the collision integrals we use the notation of Arnold, Moore and Yaffe (AMY) [19].Therefore the elastic collision integral for particle a with momentum p 1 involved in a scattering process a, b → c, d with momenta p 1 , p 2 ↔ p 3 , p 4 is given as where M ab cd is the corresponding matrix element summed over spin and color, F ab cd is the statistical factor, ν g = 2(N 2 c − 1) = 16 and ν q = 2N c = 6.The measure is determined as × (2π) 4 δ (4) (P 1 + P 2 − P 3 − P 4 ) . (A2) The statistical factor F ab cd is given as where the (+) takes into account Bose enhancement for gluons and the (−) Fermi blocking for quarks.

Inelastic collision integrals
The inelastic collision integral for particle a with momentum P 1 involved in a splitting process a → b, c with momenta P 1 ↔ P 2 , P 3 and in the inverse joining process a, b → c with momenta P 1 , P 2 ↔ P 3 has the form
In this work we do not want to concentrate on how to solve these equations in detail but rather refer to [27], where the authors do exactly this, which is the formalism we will also use.Nevertheless the dynamical evolution of distribution functions for quarks and gluons itself is important to our studies since they enter the photon production rates and is shown in Fig. 4.
To compute the thermal partonic radiation in this work, we used the well-known parametrization of the full leading order (LO) rate in thermal QGP, see Ref. [20].This rate contains the elastic contributions, which dominate at higher momenta, but also the in-medium collinear bremsstrahlung and inelastic pair annihilation.In particular, we include the suppression of the radiation due to interference when multiple scatterings happen, i.e., the LPM effect.The transition from the QGP to hadron resonance gas photon emission rate is done at T = 160 MeV.
The hadronic (thermal) stage photon emission rates are taken from Ref. [65], where two parametrizations are given: one for the bremsstrahlung originating from ππ scattering dominating at lower momenta and one for the in-medium ρ mesons scattering contribution.Additionally, we assume vanishing chemical potential, as we are investigating photon production at mid-rapidity in at LHC energies.The computation is performed by continuing the hydrodynamical evolution below the freezeout temperature and computing the contribution of these emission channels down to a minimum temperature.We have computed different spectra by shifting the minimum temperature between T = 120 − 140 MeV, and we have observed no change of photon spectrum in the relevant momentum range (p T > 0.5 GeV).
It is known that the computation of late-time photons from the hadronic stage requires a full microscopic afterburner treatment.However, as shown in Ref. [66], if one is interested in solely computing the emission at the level of the p T spectrum, hydrodynamical simulations have a qualitatively equal performance.

14 VI
of the time-integrated photon spectrum 9 V. Phenomenology of the pre-equilibrium photons 12 A. Photons from the pre-equilibrium stage 12 B. Comparison to experimental data

Figure 1 .
Figure 1.The t-channels of elastic processes contributing to photon production at leading order.Left: Compton scattering.Right: Elastic pair annihilation.The s-and u-channels are not shown but included in the calculations.

Figure 2 .
Figure 2. Inelastic processes contributing to photon production at leading order.Left: Bremsstrahlung.Right: Inelastic pair annihilation.
C γ is a function of the Bjorken time τ and local momentum P = (p T cosh(y − ζ), p T , p T sinh(y − ζ)) and we can express the ζ integral for an integral over the longitudinal momentum p z = p T sinh(y − ζ) in the local rest frame, such that

Figure 4 .
Figure 4. Thermalization of distribution functions for (a) gluons and (b) quarks in QCD kinetic theory at leading order and the corresponding leading order photon emission rates from (c) elastic and (d) inelastic processes.Blue points correspond to non-equilibrium distribution functions or rates, while green lines show expected thermal equilibrium distribution resp.rate.Each row corresponds to a different scaled time w = 0.2, 0.5, 1.0.Results are for λ = 10.

Figure 5 .Figure 6 .
Figure 5.The sum of all four contributions to the photon production rate at leading order as a function of pT /T eff (τ ) (subscript eff dropped for better readability).Blue solid curves correspond to non-equilibrium production, while green dashed lines represent the thermal contribution.Different panels correspond to different evolution times w = 0.25, 0.5, 1.0, 2.0.Results are for λ = 10.

3 / 2 ∞
to compare simulations with different values of the coupling constant λ corresponding to different values of the shear viscosity to entropy density ratio η/s.

Figure 10
Figure10.The schematic timeline for photon production in heavy-ion collisions.The prompt photons are produced in hard scatterings at the instant of the collision.The pre-equilibrium photons (this work) are radiated during the first ∼ 1 fm of the evolution as the QGP is equilibrating.At τ hydro the pre-equilibrium production in QCD kinetic theory is smoothly matched (locally in the transverse plane and event-by-event) to equilibrium (AMY) photon rates in the hydrodynamic phase.The matching procedure is described in Section V A. Afterwards, thermal photons from the QGP are produced by folding thermal AMY rates with the hydrodynamic evolution.As the system goes through the cross-over at Tpc = 160 MeV, we switch to hadronic rates, which are the source of emission until the end of the collision evolution at Tmin = 120 − 140 MeV.

Figure 11 .
Figure 11.The differential photon spectra for 0 − 20% centrality PbPb collisions at 2.76 TeV.The experimental ALICE results are shown as points[56,57].The solid red lines show the total computed photon spectra, which consist of prompt (blue), pre-equilibrium (green) and thermal rates in the hydrodynamic phase (yellow).

ACKNOWLEDGMENTS
We thank Jürgen Berges, Xiaojian Du, Charles Gale, Nicole Löher, Stephan Ochsenfeld, Jean-Francois Paquet and Klaus Reyers for their valuable discussions.OGM, PP and SS acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the CRC-TR 211 'Stronginteraction matter under extreme conditions'-project number 315477589 -TRR 211.AM acknowledges support by the DFG through Emmy Noether Programme (project number 496831614) and CRC 1225 ISOQUANT (project number 27381115).OGM and SS acknowledge also support by the German Bundesministerium für Bildung und Forschung (BMBF) through Grant No. 05P21PBCAA.The authors acknowledge computing time provided by the Paderborn Center for Parallel Computing (PC2) and the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

Appendix A: Quark and gluon collision integrals in QCD kinetic theory 1 .
Elastic collision integrals .