Probes of the quark-gluon plasma and plasma instabilities

Penetrating probes in heavy-ion collisions, like jets and photons, are sensitive to the transport coefficients of the produced quark-gluon plasma, such as shear and bulk viscosity. Quantifying this sensitivity requires a detailed understanding of photon emission and jet-medium interaction in a non-equilibrium plasma. Up to now, such an understanding has been hindered by plasma instabilities which arise out of equilibrium and lead to spurious divergences when evaluating the rate of interaction of hard probes with the plasma. In this paper, we show that taking into account the time evolution of an unstable plasma cures these divergences. We calculate the time evolution of gluon two-point correlators in a setup with small initial momentum anisotropy and show that the gluon occupation density grows exponentially at early times. Based on this calculation, we argue for a phenomenological prescription where instability poles are subtracted. Finally, we show that in the Abelian case instability fields do not affect medium-induced photon emission to our order of approximation.


I. INTRODUCTION
In heavy-ion collisions at ultra-relativistic energies a dense medium of quarks and gluons is formed: the quarkgluon plasma (QGP) [1]. The medium expands and cools until the quarks and gluons coalesce into soft hadrons which rescatter and fly to the detectors. Remarkably, this time evolution of the QGP is captured by hydrodynamics making the QGP a relativistic fluid [2]. A major goal of heavy-ion collision experiments is to characterize this QGP using transport coefficients, such as shear and bulk viscosity, which quantify its response to weak perturbations and are fundamental properties of QCD. Hydrodynamic studies have shown that the ratio of shear viscosity to entropy density of QGP is the lowest of any known material [2], but arguably the precise value is only known within a factor of two or so. Another major goal of these experiments is to understand how the QGP is formed in the first place. Specifically, it needs to be understood how an initial collision of two heavy nuclei at high energies gives rise to a macroscopic fluid within a time frame of a fm/c or even less.
Explaining equilibration and transport coefficients in heavy-ion collisions relies on knowledge of the nonequilibrium physics of the quark-gluon plasma. Up until now transport coefficients of the QGP have mostly been extracted by fitting hydrodynamic studies to experimental results of the yield and angular distribution of soft hadrons [2]. An alternative way is offered by hard probes of the QGP, such as photons, jets and heavy quarks. As an example, jets broaden and lose energy as they interact with the QGP medium [3]. The rate of interaction and its dependence on the energy of a jet particle depends in detail on the makeup of the fluid. Thus the interaction with a thermally equilibrated fluid and a fluid with shear flow will be different, meaning that the energy loss of jets is sensitive to the QGP's shear viscosity [4]. To use jets or photons to get the QGP's shear viscosity requires thorough understanding of hard probes in non-equilibrium QGP.
Many issues arise when calculating interaction of hard probes with a non-equilibrium plasma. An important challenge comes from instabilities intrinsic to weakly coupled plasmas. These Weibel instabilities [5,6] come about when quasiparticles that are anisotropically distributed in momentum space radiate soft gluons, the density of which grows exponentially with time. The system is thus intrinsically time dependent. Microscopic calculations of hard probes in non-equilibrium plasma have so far not taken this time dependence into account leading to spurious divergences, such as in the rate of jet particles splitting while interacting with the QGP [7][8][9][10].
This paper is organized as follows: In Sec. II we explain how instabilities in weakly coupled QGP lead to spurious divergences when studying hard probes in a nonequilibrium plasma. In Secs. III and IV we calculate the time evolution of gluon correlators in a non-equilibrium plasma with slight initial momentum anisotropy. We argue that for phenomenological applications the contribution of instabilities should be subtracted. Finally, in Secs. V and VI we show that in the Abelian case instability fields do not affect medium-induced photon emission to our order of approximation. Details of calculations are relocated to appendices.

II. BACKGROUND
Instabilities in a weakly coupled non-equilibrium quark-gluon plasma lead to spurious divergences when calculating e.g. the rate of photon emission from the plasma or the rate of jet-medium interaction. Understanding the origin of these divergences requires some background on weakly coupled plasmas and quantumfield theoretical calculations of photon emission.
The ultimate goal when calculating photon emission in a non-equilibrium plasma, is to learn about the QGP formed in experiments by using photons. This necessitates a flexible approach where rates of photon production can be combined with hydrodynamic simulations of heayv-ion collisions. Specifically, we have two conditions: 1. The rate of photon production should only depend on the properties of the medium in that instant, and not on the medium's history. This requires t process t medium (1) where t process is the time it takes to emit a photon and t medium is the time scale over which the medium changes substantially.
2. The rate of photon production should depend solely on macroscopic variables, like pressure and shear flow, that can be obtained from hydrodynamic calculations. This is achieved by describing the medium by a quasiparticle momentum distribution f (p) that corresponds to the macroscopic variables 1 .
These two conditions have immediate consequences for quantum field theory calculations of photon production. The first condition says that the medium is effectively static during the emission of a photon. We thus want to specify a quasiparticle distribution f (p) at an initial time t 0 → −∞ which will appear in bare propagators. Assuming that f (p) remains the same during photon emission, we can use the same bare propagators at all times. Since time ranges from −∞ to ∞ we can do Fourier transforms and work in frequency space which provides huge simplification. Naively, we expect the results for the rate of photon production to have the same form as in thermal equilibrium, with equilibrium distributions replaced by a more general distribution f (p) 2 .
Unfortunately, this simple picture does not work in general. As explained in greater detail below, one generally gets a non-sensical, infinite rate of photon production when assuming a static medium characterized by a momentum distribution f (p). The culprit are instabilities in the plasma which give rise to rapid exponential growth in the density of soft gluons, violating the assumption of a static medium. These instabilities arise for any momentum distribution that is anisotropic, i.e. f (p) = f (p). (In the case of thermal equilibrium or other isotropic distributions the instabilities are not present and one can assume a static medium.) In fact, the same problem of divergent rates is present when calculating e.g. the rate of jet-medium interaction [7,8], heavy-quark potential [9] and even the rate of interaction among the quasiparticles comprising the medium [10].
Understanding this problem better requires a detailed discussion of weakly coupled QCD plasmas that are sufficiently close to equilibrium. Such plasmas are characterized by two energy scales. Firstly, there are quasiparticles -quarks and gluons -which are localized and propagate freely, apart from occasionally interacting with each other. Their phase space behaviour can be described by kinetic theory [10], and their distribution functions obey a Boltzmann equation where the distribution f (t, x; p) changes because of external forces F and collisions between quasiparticles, as described by C. Here colour indices have been suppressed for simplicity. The quasiparticles radiate gluon fields with energy gΛ where g 1 is the coupling constant. These longwavelength, soft gluons have high occupancy and can thus be described using classical field theory. Specifically, they obey the classical equations of motion for a gluon field A µ , where D µ is a covariant derivative, F µν is the chromoelectromagnetic tensor, and j µ is a current which comes from the quark and gluon quasiparticles. These two coupled equations, Eqs. (2) and (3), tell us that quasiparticles source gluon fields which deflect the quasiparticles in turn. They can be solved simultaneously, giving rise to an effective field theory for the longwavelength gluons called Hard Thermal Loops (HTL) [13]. We write the quasiparticle momentum distribution as where δf is a small fluctuation around the distribution f 0 specified at the initial time t 0 → −∞. Dropping the subleading collision kernel, Eq. (2) then becomes where an external force F[A µ ] due to an applied gauge field A µ sources fluctuation δf . Solving for the fluctuation gives a current j µ [A µ ] ∼ d 3 p p µ p δf which linear response theory tells us is related to the applied field A µ through j(P ) = Π ret (P )A(P ). We thus get the retarded self-energy for soft gluons [14] Π µν ret (Q) ∼ g 2 which depends explicitly on the initial momentum distribution f 0 . Here P and Q are four-momenta while p = |p| is the three-momentum.
FIG. 1: Different processes in a weakly coupled quark-gluon plasma: (a) photon production through two-to-two scattering, (b) two-to-two scattering with gluon exchange, (c) photon emission triggered by in-medium interactions, (d) gluon emission triggered by in-medium interactions Eq. (6) contains a wealth of information on how soft gluons propagate in the medium. Continuing with the assumption of a static medium, the retarded propagator in momentum space where Π µν ret is given by Eq. (6). A pole of the retarded propagator, ω = E(p) − iΓ(p), contributes dω 2πi in the time domain. This shows that ω = E(p) is the dispersion relation of the excitation and Γ(p) is the decay width. Whenever the initial momentum distribution f 0 is anistropic, instabilities are present in the system. In [10,15] it was shown that a new pole, ω = iγ, appears in the retarded gluon propagator in the upper half complex plane. It corresponds to exponential growth e γt in soft gluon density in the time domain. This happens as energy is transferred from quasiparticles to the soft chromomagnetic field as it deflect the quasiparticles which source an even stronger field [6]. This instability in soft gluon density has been studied extensively numerically, see [6] and references therein.
The presence of instabilities invalidates the assumption of a static medium. This can for instance be seen when evaluating photon production from the medium. At leading order in the strong coupling constant g, photons are produced through two distinct channels. The first channel is two-to-two scattering with a photon in the final stage, Fig. 1a, which is unaffected by instabil-ities in gluon density since the mediator is a quark. 3 Its rate has been calculated in a non-equilibrium plasma for various momentum distributions [16][17][18]. The second channel is medium-induced bremsstrahlung of a collinear photon, see Fig. 1c. A quark is brought slightly off-shell by kicks from the medium's soft gluons, which allows it to emit a photon. The probability for a kick to give the quark transverse momentum q ⊥ is whereK µ = K µ /k is the direction of the quark [10,11].
Here the crucial ingredient is the correlator G µν rr (x, y) = 1 2 {A µ , A ν ]} which describes the density of soft excitations. In a static medium with initial time t 0 = −∞ it is given by with G adv = G * ret and Π aa denotes the probability to create the excitation. During emission of a collinear photon, the quark can receive arbitrarily many kicks from the soft gluons. The kicks act coherently and tend to reduce the rate of emission; this is the Landau-Pomeranchuk-Migdal effect [12,19,20]. Thus in a static medium the rate of photon production through bremsstrahlung has a complicated dependence on C(q ⊥ ) which can be seen in Eqs. (20) and (21) [11].
We can now finally see how instabilities invalidate the assumption of a static medium when calculating photon production through medium-induced bremsstrahlung. Roughly speaking the rr propagator for the instability mode can be approximated as Substituting this into Eq. (9) and ignoring q z dependence gives We thus see that for slowly growing modes γ → 0 at finite q ⊥ the probability for interacting with soft gluons diverges. This is a sign that our handling of instabilities in a static medium is incorrect. The function C(q ⊥ ) in Eq. (9) not only appears in photon production but also when calculating the rate of jet-medium interaction [19,21], as well as interaction of quasiparticles [10]. All of these processes thus suffer from the same divergence in a naive calculation in a static medium. Furthermore, a similar problem arises when calculating the imaginary part of the heavy quark potential in a non-equilibrium medium [9].

III. OVERVIEW OF RESULTS AND IMPLICATIONS FOR PHENOMENOLOGY
To calculate photon production through bremsstrahlung in a non-equilibrium QGP we must go beyond the assumption of a static medium. Otherwise, we get non-sensical results because of instabilities in soft gluon density. However, including the time evolution of the medium in general is a complicated task, especially since we can no longer do Fourier transforms which are essential to get simple equations for the LPM effect. To be able to handle this task, we consider the simplest setup imaginable for the medium and draw lessons from it for more realistic settings. This furthermore gives a rare opportunity to do analytic calculation in non-equilibrium plasma.
In our setup the plasma is initially comprised of hard quasiparticles with energy Λ while soft gluons with energy gΛ are absent. The initial condition at t 0 = 0 is given by a slightly anisotropic quasiparticle distribution f 0 (p). The anisotropy is defined by where p z , p ⊥ are the momentum distribution's typical momenta. We assume that ξ g. This guarantees that the growth of instabilities is slow enough for us to have handle on the calculation. We furthermore only consider times shortly after the initial time. This ensures that the density of the soft gluons does not become so high that the HTL approximation is invalidated.
In Sec. IV we calculate the propagators that describe soft gluons in this setup. The retarded correlator becomes The propagator is written in the time domain where t x , t y > 0 are the times of the two fields. Since we assume an infinite spatial extension we can define a threemomentum p by Fourier transform. The function is the same as in Eq. (7). It generally has poles in the upper half complex plane which correspond to instabilities [15]. Crucially, we must choose a contour α that goes above all poles in the upper half complex plane, as in Fig. 2(a). An instability pole p 0 = iγ then gives G ret ∼ e γ(tx−ty) for t x > t y which grows exponentially, showing that the system is unstable to perturbations. Choosing the contour in this way, also guarantees that The important ingredient when calculating photon emission is the rr correlator of soft gluons which describes the soft gluon density. To find an expression for it we must separate between two scales, namely the soft scale gΛ and the instability growth rate γ ∼ ξgΛ 4 . As an example we write the retarded correlator in Eq. (15) as where G ret only has poles and branch cuts of order gΛ while γ i are all poles of order ξgΛ, including instability poles. Using a number of controlled approximations, explained in Sec. IV, we then get the rr correlator at early times when the gluon occupation density is not so high that the HTL approximation is invalidated. It is and G adv = G * ret . The rr propagator in Eq. (17) has a clear physical interpretation. The first term has no information about the initial time. It is of the same form as the rr correlator in a static medium, Eq. (9), except that all instability poles have been subtracted. The second term shows exponential growth due to instabilities at scale γ ∼ ξgΛ. It vanishes at the initial time t x = t y = 0 when the instability modes are not occupied. It is furthermore finite for slow growth rate γ → 0.
At very early times the instability part in Eq. (17) can be neglected. The probability of a quark to get a transverse kick q ⊥ is then Here all instability poles have been subtracted in G rr so the probability is finite. As time goes on the occupation density of the soft gluons increases due to instabilities and the second term of Eq. (17) must be included. This complicated task is discussed in Sec. V. As even further time passes the HTL approximation used to derive Eq.
(17) breaks down and numerical calculations are needed to evaluate the evolution of instability modes.
Instability modes have the potential of violating our assumptions for photon production in heavy-ion collisions. The fluctuating soft gluon cloud in Eq. (19) is sourced by the hard quasiparticles at each instant so that its effect on photon production only depends on the instantaneous, macroscopic properties of the medium. Conversely, the instability contribution depends on the whole history of the medium and can only be included in phenomenological calculations with great difficulty.
Fortunately, detailed classical-statistical simulations suggest that plasma instabilities only play a role in the very early stages of heavy-ion collisions [23,24]. These calculations describe a weakly coupled, highly occupied classical system with fluctuating initial conditions coming from the color-glass condensate. There the instabilities are important in the approach to a universal, nonequilibrium attractor but once the attractor is reached, detailed information on the initial stages is forgotten and the dynamics is dominated by a turbulent cascade towards higher energies until thermalization is reached.
For the phenomenology of photon production in a nonequilibrium QGP, it is therefore reasonable to neglect the contribution of instabilities and use the function C in Eq. (19). This function is non-trivial and has not been calculated fully for a given non-equilibrium distribution. Using it guarantees a finite rate which still includes the essential non-equilibrium information, both from the nonequilibrium quasiparticle distribution f , as well as the soft gluon cloud sourced at each instant by the quasiparticles. The same procedure works for medium-induced jet splitting which also depends on the function C. We will report on photon production in a non-equilibrium QGP, using this procedure [25,26].
Applying this prescription, the rate of emitting photons with momentum k through bremsstrahlung is when the photon is emitted in the z direction [11]. Here Re f (p) can be thought of as the probability for the quark to gain transverse momentum p because of medium kicks. It is solved by the Boltzmann-like equation where C comes from Eq. (19).

IV. CORRELATORS FOR UNSTABLE FIELDS
We now turn to derive the correlators in Eqs. (14) and (17) for a weakly anisotropic plasma shortly into its evolution. The retarded propagator is defined by where Π ret is the retarded self-energy and G 0 ret is the bare retarded propagator (see e.g. [27] and Sec. 3 of [28]). In a static system, such as thermal equilibrium, this equation can be solved by Fourier transforming to the frequency domain, thanks to translational invariance which guarantees that G ret (x, y) = G ret (x − y). We must take a different route to solve Eq. (22) since time translational invariance is broken by instabilities. We assume that our system has infinite spatial extension so that the spatial dependence can be described in Fourier space. We start our system at initial time t 0 = 0. The time integrals in Eq. (22) range over all times greater than the initial time. Using the properties of retarded functions, we write 5 where the dependence on three-momentum is omitted. We have Π ret (z, w) = Π ret (z − w) in the HTL approximation, valid at the early times we consider when the soft gluon density is not too high. This equation has the same form as in equilibrium because the initial time does not appear explicitly. Furthermore, G ret (x + τ, y + τ ) is a solution of Eq. (23) for any τ . This suggests that we can write G ret (x, y) = G ret (x − y). We will therefore try to find a solution 6 for some function G ret (k). It's enough to find one such solution because the solution of Eq. (23) is unique. The contour α goes along the real line and above all instability poles that G ret (k) might have in the upper half complex plane, see Fig. 2(a). This ensures that G ret (x, y) = 0 for y > x.
We will now evaluate the last term in Eq. (23)  and Π ret we write that term as The time integrals can be done explicitly. This would not be possible if the time integrals were written for all z, w ≥ 0 since the integral with e −ik3(w−y) would not converge with k 3 in the upper half complex plane. In the end we get Some tricks are needed to evalute Eq. (26). We notice that the function f (k 1 , k 2 , k 3 ) has no poles in its variables. Thus, we can include a principal value for each term in the function by substituting Here 1 , 2 , 3 > 0 are set to zero in the end. The result must be independent of the order in which they are set to zero. As is shown in App. A we can then evaluate the momentum integrals in Eq. (26) using the residue given by Eq. (24). However, we prefer avoiding formal integrals which do not converge. In the end, the time domain is the only physical domain in a non-equilibrium system and Gret(k) is just some function that gives the correct retarded function when substituted in Eq. (24). theorem. Doing so requires continuing the integration contours to the correct half plane which only contains poles of the function f . The final result is From this we immediately see that confirming our expression for G ret in Eq. (14). We note that the advanced correlator can easily be shown to be where G adv (k) = G ret (k) * and the integration contour is α =α * which goes below all poles of G adv (k).
We have found the retarded and advanced correlators.
The other two-point correlator is G rr = 1 2 {A µ (x), A ν (y)} which gives the occupation density of gluonic modes in the medium. It is (32) where the integration limits have been rewritten using properties of the retarded and advanced functions, as well as the initial time t 0 = 0 [28]. In general there is an additional term corresponding to correlation with the initial state. Assuming that there are no soft gluons in the initial state, we can omit that term but it could easily be included in our calculations. The integrals depend explicitly on the initial time so we expect that G rr (x, y) = G rr (x − y). Substituing the Fourier transform of the HTL Π aa as well as Eqs. (24) and (31) gives after doing the time integrals.
In order to evaluate the remaining integrals in Eq. (33) we must think about the scales of the problem. The retarded correlator is at two momentum scales: Here, G ret only has poles and branch cuts of order gΛ which are all in the lower half complex plane while γ i are all poles of order ξgΛ, with ξ g the initial anisotropy of the system. We have split G ret in a fluctuating part G ret that is continually sourced by quasiparticles and an instability part that describes includes time evolution. Similarly, we write where G adv = G * ret . The self-energy Π aa only has poles and branch cuts of order gΛ.
We need to be careful when writing the retarded correlator as in Eq. (34). The correlator has a branch cut from ω = −|k| − i to ω = |k| − i which corresponds to Landau damping. The branch cut is most often chosen to lie just below the real axis but then it will be partially at the scale ξgΛ which spoils the separation of scales in Eq. (34). The remedy is to choose a branch cut that avoids the ξgΛ region, see Fig. 4. This results in new decaying modes on the second Riemann sheet [29]. Ultimately, the retarded propagator only exists in the time domain where it is independent of the branch cut we choose.
We will use controlled approximations to evalute the rr correlator in Eq. (33). Firstly, we assume that x, y 1/g 2 Λ so that sufficient time has passed since the system was initialized. This allows us to drop any term with e −iax where Im a < 0 and Im a ∼ gΛ, as correlations with the initial condition are damped when sufficient time has passed. Secondly, we can assume that x−y ∼ 1/g 2 Λ since this is the time that medium-induced emission takes. This allows us to drop any term with e −iax where Re a ∼ gΛ, as it oscillates very rapidly during emission and cancels out. The terms we drop would also be present in a thermally equilibrated system started at an initial time t 0 = 0. They tell us little about the non-equilibrium physics we are interested in.
These approximations allow us to to evaluate the rr propagator at early times. Using the same calculational tricks as before, a lengthy calculation given in App. A shows that where the terms correspond to fluctuating contributions k ∼ gΛ, instability contributions k ∼ ξgΛ or their cross terms.
Eq. (36) has a simple interpretation. Schematically, a mode e −iEt−γt of the retarded function contributes to the rr correlator in a system in thermal equilibrium with initial condition at t → −∞. This expression has a pole at k 0 = E − iγ. However, in a non-equilibrium system with initial time at t = 0 the corresponding integral is (38) which has no pole. In a similar fashion, there should strictly speaking be no poles in Eq. (36): for a pole b ∼ gΛ of G ret we should have Nevertheless, in using our approximations we have dropped all terms ∼ e −ibx since sufficient time has passed to eliminate all traces of an initial time t 0 = 0. Conversely, we must retain the analogous factors e γx for instability modes since they grow exponentially in time.
It is instructive to rewrite Eq. (36). We can drop crossterms between instability and fluctuating modes since the decay or oscillations of fluctuating modes dominates over the slow growth rate of instability terms. 8 The last term in Eq. (36) has no poles because of the exponentials and can thus be written with a different contour Here β is a contour that goes along the real line and above all instability poles of G ret in the upper half plane and below all instability poles of G adv in the lower half plane, see Fig. 3. Doing the contour integrals then gives where we can ignore poles 9 of Π aa (k) and write Π aa (a i ) ≈ Π aa (a * j ) ≈ Π aa (0). Using that a(x − y) ∼ ξ/g 1 gives our final expression for the full rr correlator which reproduces Eq. (17): where γ i and A i are functions of the three-momentum k.
where d ∼ gΛ gives oscillations and c ∼ ξgΛ gives exponential growth. Averaging over the time of interaction in mediuminduced emission corresponds to introducing an initial time t 0 which varies over scale 1/g 2 Λ. This can e.g. be done by integrating e (id+c)(t−t 0 ) e −t 2 0 /2σ 2 over the intial time t 0 where the Gaussian with width σ ∼ 1/g 2 Λ corresponds to averaging the time of emission over the time a typical emission takes. Integrating over t 0 then gives a factor e − 1 2 σ 2 (d 2 −c 2 +2icd) which is heavily suppressed since d c and σd 1. A full field theoretical calculation gives the same exponential suppression. 9 The fact that poles of Πaa can be ignored can be seen as follows: Let's write Πaa as A/(k −B) where B ∼ gΛ is a pole and A is the residue. Upon performing the contour integral, the pole B will contribute A/(B − a i )(B − a * j ) ∼ A/(g 2 Λ) while an instability pole will contribute A/(a i −B)(a i −a * j ) ∼ A/ξg 2 Λ which is much bigger.

FIG. 6:
A diagram for medium-induced photon production in the presence of a background field. The red gluons denote medium kicks at energy gΛ which are time ordered. The blue lines denote kicks from the background field at energy ξgΛ. They are not time ordered.

V. QUARK PROPAGATORS IN LONG-WAVELENGTH ABELIAN BACKGROUND FIELDS
We have argued that for phenomenological applications the time-dependent instability field in Eq. (17) should simply be subtracted, leaving a simple expression for photon production in a non-equilibrium plasma. It is nevertheless interesting to explore the effect of the long-wavelength instability field, both from a theoretical point of view, as well as as a first step towards including classical fields at the early stages of heavy-ion collisions.
We will now calculate how the long-wavelength background fields modify photon emission, focusing on the case of an Abelian plasma. In particular we consider how the background fields modify medium-induced bremsstrahlung as seen in Fig. 1c which suffers from spurious divergences when one assumes a static, nonequilibrium plasma. Our setup is fairly general: The medium can be described by the rr correlator in Eq. (17) but also by any other rr correlator which has two different scales, fluctuating time-independent excitations with energy gΛ and a time-dependent background field with energy ξgΛ, ξ g. This calculation also extends easily to jet-medium interaction and quasiparticle splitting as seen in Fig. 1d. Our goal is to sum up non-perturbative effects of the background field at a given order in l∆t 1 where l ∼ ξgΛ is the small momentum of the background field and ∆t ∼ 1/g 2 Λ is the time photon emission takes.
The two energy scales, i.e. the fluctuating field at gΛ and the background field at ξgΛ, affect photon emission in very different ways. The time for collinear bremsstrahlung of photons is ∼ 1/g 2 Λ which is very long compared to the time 1/gΛ for a typical medium kick. Thus the medium-kicks are ordered in time and diagrams with crossed rungs like in Fig. 5 are suppressed. On the other hand the long-wavelength background field has wavelength ∼ 1/ξgΛ which is much longer than the time for photon emission. Thus we must evaluate diagrams like in Fig. 5 for the background field. These diagrams are complicated because of the color factors and can only realistically be summed up in the case of an Abelian background field or a non-Abelian background field in the large N c limit where only planar diagrams contribute. We focus on the Abelian case here. Our goal is to do a calculation that includes both medium kicks and the background field as can be seen in Fig. 6.
We make a few assumptions about the scales of the problem. Firstly, we assume that the momentum l of the long-wavelength background field satisfies where ∆t is the time the emission of a photon takes. In our case ∆t ∼ 1/g 2 Λ so for instability fields ξ g. Furthermore, we assume that γ∆t 1 where 1/γ is the time over which the background fields change appreciably. We also assume 1 Λ∆t (44) where Λ is the hard scale of the medium and ∆t is the time an emission takes. Medium-induced emission of photons or gluons takes time ∼ 1/g 2 Λ which is long enough to fulfill the condition. In general Eq. (44) is satisfied for off-shell photon emission with virtuality Q 2 Λ 2 . We finally assume that the wavelength of the background fields cannot be so long that it correlates two subsequent gluon emissions. In other words where 1/Λ (∆t) 2 ∼ Λ/g 4 , the mean free path for gluon emission.
Quark propagators are modified in the presence of background fields. The bare retarded propagator is where ∆t = t x − t y and P = (1, p) and P = (−1, p) denote different polarizations. Adding one background field insertion, Fig. 7, gives We can take the background field A µ out of the time integral since it changes slowly. This gives Here we have expanded in ∆t l || 1 with l || = p · l. Terms with ∆t in Eq. (48) denote a potential phase rotation in the background field. The subleading term with l || (∆t) 2 , gives the first derivative of the background field A µ and thus denotes the effect of electromagnetic fields on photon emission. Higher order terms are not amenable to evaluation using our methods. 10 The retarded quark propagator with an arbitrary number of background field insertions is This simple form is achieved by summing over all the different permutations of attaching n background field insertions. The analogous expression for the advanced propagator has an overall minus sign and θ(−∆t) instead of θ(∆t). Eq. (49) can be derived by noting that the dependence on background field momentum is for n ordered instability insertions. Performing the integral and expanding in l i || =p · l i gives a complicated expression. It is hugely simplified by summing over all permutations of attaching n background field insertions, and using that permute {l1,...,lj } 1 (l 1 + · · · + l j )(l 2 + · · · + l j ) . . . l j = 1 l 1 . . . l j (51) 10 The omitted terms in Eqs. Finally, we must evaluate how the rr propagator is modified in the presence of a long-wavelength background field. The bare rr propagator in the time domain is describes the momentum distribution for incoming quarks and outgoing antiquarks, respectively. There are many ways to add background field insertions in the ra basis. As an example Fig. 9 shows the three possible ways of including two background field insertions. To find them we have used that a background field insertion has index r, that each vertex has an odd number of a indices and that bare aa propagators vanish [12]. Assuming that the momentum distributions in each propagator are the same to our order of approximation, f q (k) ≈ f q (k+l 1 ) ≈ f q (k+l 1 +l 2 ) 11 , and using Eq. (52), most of the terms cancel [11]. We end up with adv .
A similar cancellation takes place for any number of background field insertions so that in the end S (n) rr (t x , t y ; p; {l 1 , . . . , l n }) = 1 2 − F (k) S (55)

VI. MEDIUM-INDUCED PHOTON EMISSION IN ABELIAN BACKGROUND FIELDS
We turn to evaluating photon emission in an Abelian background field. For simplicity, we begin by only considering the long wavelength background field with momentum ξgΛ, considering medium kicks with momentum gΛ below. 11 By making the approximation fq(k + l) ≈ fq(k) we ignore how quarks are rotated in the background field during emission. This correction is of order l · ∇f (p) ∼ l Λ f (p). Such terms have a combination of retarded and advanced propagator with no simple time ordering. The time integral at the vertex with momentum contribution l will thus give T , the time that has passed since the initial conditions that specified the momentum distribution f (p). Choosing T ∆t so that the momentum distribution describes the quarks just before they emit the photon, it is easy to see that the correction is subleading to Eq. (49).
On-shell photon emission from on-shell quarks is kinematically suppressed in the absence of kicks from a background field or a medium. The rate of on-shell photon emission is given by Π γ 12 which goes like the four-point quark correlator S 1122 , see Fig. 10. We show in Appendix B that where we have gone to the ra basis in the closed-time path formalism [27] defined by The momentum factors in Eq. (56) describe different channels. As an example with p 0 > 0 we get f q (p+k) (1 − f q (p)) which denotes a quark with momentum p + k emitting a photon with momentum k through bremsstrahlung. The rate of emitting an on-shell photon with momentum k through quark bremsstrahlung then goes like as can be seen in Fig. 11. The frequency integral gives δ(k − |p + k| + p) which vanishes under integration over p for an on-shell photon. This is simply because the emission is kinematically suppressed. We now turn on the background field and see whether on-shell photon emission becomes possible. Since we have assumed that A = 0 but G rr = 1 2 {A, A} = 0 we must pair up the background field insertions into rr two-point functions to account for fluctuations in the background field. An example of a contributing can be seen in Fig.  5.
The upper quark rail with momentum p + k becomes after summing over all possible permutations of n 1 background field insertions. Similarly, the lower quark rail with momentum k becomes with n 2 instability insertions. The extra factor of e −i i k·li(ty−tx) arises because he momentum flow into r r r r a r a r + r r r a r r a r + r r r a r a r r the advanced propagator is p + k + l i where l i come from the retarded propagator. When pairing up background insertions we must integrate over the momenta in rr propagators, l and l. Pairing up two background fields insertions on the upper quark rail gives a factor where we used that the momentum flow is l in one insertion, and −l in the other insertion. The first term, (∆t) 2 , describes a phase shift and the second term describes how the dispersion relation changes because of fluctuating background fields. Pairing up two background field insertions on the lower quark rail gives the same factor D. Finally, pairing up an insertion from the upper rail and an insertion from the lower rail gives which has the value −D. We must now sum over all possible ways of attaching background field insertions to the two quark rails. A typical diagram can be seen in Fig. 5. Fortunately, we have already summed over all ways of ordering field insertions on each quark rail in Eqs. (59) and (60). Thus we only need to sum over the number of insertions on each rail and the different ways of joining them in rr propagators. Assuming that there are m 1 rr propagators where both ends are on the upper quark rail, m 2 propagators with both ends on the lower quark rail and m 3 pairs that join the two propagator, the time dependence becomes There is a total of 2n background field insertions. The combinatorial factors account for that the diagram remains the same after interchanging different propagators between the same rails or interchanging the ends of a propagator. (We do not divide by 2 m3 since we have not permuted instability insertions between the rails.) The combinatorial sum in Eq. (63) gives Thus the total contribution of the background field cancels out in the Abelian case. The same cancellation takes place in the other channels, namely for an antiquark emitting a photon and in quark-antiquark pair annihilation. The effect of a long-wavelength background field on photon emission is vanishing to our order of approximation. This is true in the absence of a medium but it turns out to be equally true when there is a medium kicking the quarks back and forth in the transverse plane. To show this, we need retarded propagators including both medium kicks and the effect of a long-wavelength back-ground field. With a medium kick at time t and k background field insertions before the kick and n−k insertions after the kick, see Fig. 12, the quark propagator becomes S (k) ret (t x , t; p; {l 1 , . . . , l k }) I(t; q) × S (n−k) ret (t, t y ; p + q + l 1 + · · · + l k ; {l k+1 , . . . , l n }) where I is the vertex factor for the medium kick and q is the momentum flow in the kick. We have defined for background field insertions left of the kick and for insertions right of the kick. We have furthermore used that p+q ≈ p to avoid corrections of order gl(∆t) 2 to Eq. (65).
We now sum over all ways of attaching the n background field insertions, either before or after the medium kick. This gives The effect of the instabilities is in a factor which does not depend on the time of the medium kick. This argument can clearly be extended for any number of medium kicks. Thus, the effect of medium kicks and n background field insertions factorizes and the dependence on the background field strength is exactly the same as in the case without a medium. The same argument that lead to Eq. (64) then shows that the background field does not affect photon emission at leading and next-to-leading order in l∆t and the rate is given by Eq. (20).

VII. CONCLUSIONS
Non-equilibrium QCD plasmas at weak coupling contain instabilities which lead to exponential growth in soft gluon density with time. This makes the plasma inherently time-dependent. We thus argue that quantum field theory calculations that assume a static non-equilibrium plasma do not work. In particular, the rate of mediuminduced jet splitting or photon production cannot be evaluated in a static plasma out of equilibrium.
To resolve this affair, we have calculated the time dependence of soft gluon correlators in a simple setup with a slightly anisotropic initial momentum distribution of quasiparticles. Using tools of non-equilibrium quantum field theory, we have derived the retarded correlator, Eq. (14), and rr correlator, Eq. (17), at early times. As expected, the rr correlator shows exponential growth in the soft gluon density with time because of instabilities. Using this correct correlator leads to finite and well-behaved rates for medium-induced photon production and jet splitting. The rates depend on the density of gluons in a fluctuating cloud that is sourced at each instant by quasiparticles, as well as the density in instability modes which changes with time. Including the effect of the time-dependent instability modes is difficult in general, but we show that in an Abelian plasma the instabilities' effect on photon production vanishes up to next-to-leading order in l∆t 1 where l is the energy scale of the instabilities and ∆t is the time needed to emit a photon.
In phenomenological applications, numerical work using classical-statistical field theory suggests that instability modes are no longer highly occupied once the hydrodynamical stage is reached [23,24]. Thus we suggest that one can simply subtract the instability contribution, when calculating photon production or jet-medium interaction in the non-equilibrium plasma produced in heavyion collisions. This gives well-behaved rates that only depend on the instantaneous properties of the medium, see Eq. (20). They contain both non-equilibrium momentum distributions and the non-equilibrium, fluctuating soft gluon cloud. The rate equations can then be solved numerically whenever a momentum distributions of quarks and gluons is specified, see [25,26]. Thus it becomes possible to calculate the effect of shear viscous flow on jet evolution in the plasma, as well as photon production, through all leading-order channels. Combined with a hydrodynamical model of the QGP fluid, this might make it possible to constrain the viscosity of the QGP using jet physics and photons.

ACKNOWLEDGMENTS
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. S. H. gratefully acknowledges a scholarship from the Fonds de recherche du Québec -Nature et technologies. (28). The first term becomes − dk 1 2π We can continue the k 3 integral by adding a semicircle in the upper half plane with a large radius, see Fig. 2b. Using the residue theorem we avoid all poles of G ret so only the poles in the square bracket contribute. We similarly continue the k 2 to the upper half plane. This gives where the integration contour is simply the real line. In a similar fashion, the second term in Eq. (26) becomes The third term in Eq. (26) is slightly trickier to evalute. We write the k 3 integration contour as α = R + i γi where γ i go around the instability poles in the upper half complex plane, see Fig. 2c. The part with the real line integration gives after doing the k 1 and k 2 integrals. The part with integration over the γ i contour can be done explicitly giving We are finally in a position to find the retarded propagator. Adding up the contributions of Eqs. (A2), (A3), (A4) and (A5) and using identities of θ-functions we get which leads directly to Eq. (29) and thus to Eq. (14) as we wanted to show. We next evaluate the rr correlator. Specifically, we will show how Eq. (36) follows from Eq. (33) using the approximations described in Chapter IV. Just like for the evaluation of G ret there are no poles when k 1 = k 2 or k 2 = k 3 which allows us to insert principal values. This gives that where we have substituted the scale sepaparation of Eqs.
(34) and (35). The evaluation of Eq. (A7) depends on the scale one is working at. We begin by evaluating terms at the scale gΛ, i.e. terms with G ret and G adv . We do this one exponential at a time. The first exponential term (i.e. all terms with e −ik2(x−y) ) can be evaluated exactly by con-tinuing the k 1 integral to the upper half complex plane, the k 3 to the lower half complex plane and applying the residue theorem. Then all poles of G ret and G adv are avoided and one gets In the second exponential term (i.e. all terms with −e −ik1x e ik2y ) in Eq. (A7) we continue the k 3 integral to the lower half plane giving exactly. In order to evalute the k 1 integral we need to use our approximations. Because of the exponential we must continue the contour to the lower half complex plane.
Applying the residue theorem we get a contribution from all poles and branch cuts of G ret but they all contain a factor e −ibx with b ∼ gΛ and can thus be dropped according to our approximations. Thus only poles with k 2 = k 1 contribute, giving dk 2 2π G ret (k 2 ) Π aa (k 2 ) G adv (k 2 ) e −ik2(x−y) In the same way, the third exponential term in Eq. (A7) is and the fourth exponential is Adding up the different terms in Eqs. (A8), (A10), (A11), (A12) and using identities for θ-functions, we get that the contribution to the rr propagator at the scale gΛ is We next turn to evaluating terms in Eq. (A7) at the scale ξgΛ, i.e. the contribution of instability poles in the retarded and advanced propagators. As before the contribution of the first exponential is In the second exponential in Eq. (A7) we continue the k 3 integral to the lower half plane to get Now when we continue the k 1 integral to the lower half plane we get a contribution from k 1 = k 2 as well as a contribution from k 1 = a i leading to i,j dk 2 2π Similarly, the third exponential in Eq. (A7) is and the fourth one is Adding up the contributions in Eqs. (A14), (A16), (A17), (A18) then gives that the contribution to G rr at the scale ξgΛ is The calculation for mixed terms with, say, contribution at scale gΛ from the retarded correlator and contribution at scale ξgΛ from the advanced correlator proceeds analogously. The final results is precisely Eq. (36).
We finally note how the momentum factors work out when there is a medium as well as background fields. Adding n background field insertions to the bare rr propagator gives S (n) rr (t x , t y ; p; {l 1 , . . . , l n }) for the P part and similarly for the P part. The same argument as in Chapter VI then allows us to factor out the effect of background fields for any combination of rr, retarded and advanced propagator and shows that the effect of the background field vanishes.