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Formation of bremsstrahlung in an absorptive QED/QCD medium

The radiative energy loss of a relativistic charge in a dense, absorptive medium can be affected significantly by damping phenomena. The effect is more pronounced for large energies of the charge and/or large damping of the radiation. This can be understood in terms of a competition between the formation time of bremsstrahlung and a damping time scale. We discuss this competition in detail for the absorptive QED and QCD medium, focusing on the case in which the mass of the charge is large compared to the in-medium mass of the radiation quanta. We identify the regions in energy and parameter space, in which either coherence or damping effects are of major importance for the radiative energy loss spectrum. We show that damping phenomena can lead to a stronger suppression of the spectrum than coherence effects.


I. INTRODUCTION
The quenching of jets associated with the strong suppression of the yields of high transverse momentum hadrons in comparison with proton-proton collisions has been observed experimentally in relativistic heavyion collisions at the Relativistic Heavy Ion Collider at BNL [1,2] and at the Large Hadron Collider at CERN, cf. [3] for a recent review. Originally, such a signal was predicted by Bjorken [4] as an evidence for the formation of a deconfined plasma state of QCD matter, in which propagating partons suffer from an enhanced in-medium energy loss.
Considered as dominant contribution, the radiative energy loss of relativistic partons due to medium-induced gluon radiation was, for instance, studied in [5][6][7][8][9][10][11][12][13]. Important for the discussion of bremsstrahlung in a dense medium are possible coherence effects. These lead to a suppression of the radiation spectrum compared to the spectrum due to incoherent scatterings. This was first realized by Landau, Pomeranchuk [14] and Migdal [15] who studied in QED the possibility of a destructive interference of radiation amplitudes, which stem from multiple scatterings of an electric charge traversing dense matter, within the formation time of radiation (LPM effect). This effect was later generalized to QCD in [7,8].
Likewise important are modifications of the radiation spectrum due to the dielectric polarization of the medium, which is known as the Ter-Mikaelian (TM) effect [16]. The medium polarization results in a change of the dispersion relation of the radiated quanta. In this way, the spectrum becomes regulated in the soft and collinear regions. In QCD, the TM effect on the radiation spectrum was investigated by considering bremsstrahlung gluons with a finite in-medium mass [17, * Electronic address: mbluhm@ncsu.edu 18].
The concept of a formation time (or length) of radiation turned out to be extremely fruitful for the discussion of radiative energy losses [7]. It allows for a semiquantitative understanding of the pattern of the radiation spectra. The spectrum reduction due to the LPM effect, for example, was qualitatively analyzed in this way for QED in [19,20] and for QCD in the review [21]: If the radiation formation length, which depends on the energy ω of the emitted quantum and on the properties of the dense medium (e.g. the transport coefficientq), is large compared to the mean free path in the medium, multiple scatterings will contribute coherently to the emission of a single radiation quantum. This represents the LPM regime of coherent radiation. In the opposite case, radiation quanta will be formed independently at each individual scattering. This represents the Bethe-Heitler (BH) regime of incoherent radiation.
In this work, we extend this analysis in order to include damping phenomena within an absorptive medium into the discussion. The latter are responsible for a reduction of the radiative energy loss of the charge and the associated radiative energy loss spectrum as was recently studied in [22]. We stress that this behaviour is not a simple consequence of the rather trivial reduction of any radiation spectrum in the far distance due to the absorption of already formed radiation but a result of the effect which damping has on the creation of the radiation itself. Just as the TM effect alters the radiation probability in the soft part of the spectrum [23], damping mechanisms lead also to a modification of the probability for emitting bremsstrahlung. This can be understood by viewing the formation of bremsstrahlung to be hampered by damping effects, in particular, when formation times become large.
The article is organized as follows: In Sec. II, our results [22] for the radiative energy loss spectrum per unit length of a relativistic charge in an absorptive, electromagnetic plasma are reviewed. We highlight that for a semi-quantitative discussion of this spectrum both the radiation formation time, as discussed in Sec. III A, and the time scale associated with damping effects are important. Their competition is analyzed in detail in Sec. III B. This allows for an identification of the regions in energy and parameter space, in which damping phenomena are of importance. A physical discussion in terms of the spectra is given in Sec. III C. In Sec. IV, we apply the methods of Sec. III in order to study phenomenologically the influence of damping effects on the gluon bremsstrahlung spectrum in the hot QCD plasma and advocate some possible physical consequences in the conclusions in Sec. V, where our results are summarized. Throughout this work natural units are used, i. e. = c = 1 with c ≃ 0.197 GeV·fm.

II. RADIATIVE ENERGY LOSS SPECTRUM IN AN ABSORPTIVE PLASMA
The radiative energy loss spectrum per unit length of an asymptotic, relativistic charge q with energy E and mass M (Lorentz-factor γ = E/M ) traversing a polarizable and absorptive plasma was found in [22] to be substantially reduced by both medium polarization and damping effects. This study was performed for an electro-magnetic plasma, where small fractional photon energies ω/E were considered. In [22], the dense medium was modelled by a complex squared index of refraction, with m and Γ accounting for the in-medium mass and damping rate of the radiation quanta, respectively. Considering an infinite medium with permeability µ(ω) = 1, one finds for the radiative energy loss spectrum for positive ω in linear response theory [22]. Here, α = q 2 /(4π) is the coupling, β = 1 − 1/γ 2 ≃ 1 − 1/(2γ 2 ), n r is the real part of n(ω) = n r (ω) + in i (ω) andq denotes the mean accumulated transverse momentum squared of the deflected charge per unit time. We note that in [22]q denoted only one-half of the mean accumulated transverse momentum squared of the deflected charge per unit time, see also [24]. The factor F (t ) in Eq. (2) enters the spectrum for every charge trajectory defining ∆ r ≡ r(t) − r(t ′ ) with t = t−t ′ and reads F ≡ exp[−ω|n i |∆r], cf. [22,24]. Since ∆r ≥ 0, it introduces a genuine damping factor in the spectrum Eq. (2) for an absorptive medium which is related to the absolute value of the imaginary part of n(ω). A specific form for F (t ) ≃ exp[−ω|n i |βt (1 −qt/(12E 2 ))] was used in [22]. This form is a consequence of the particular trajectory considered in [14] yielding ∆r ≃ βt [1 −qt/(12E 2 )], which takes explicitly into account the effect of multiple scatterings in the approximation of small deflection angles accumulated within the timedurationt. The made approximation is, however, strictly valid only fort ≪ t diff = 6E 2 /q [24]. For larger times a different dependence than the one used in [22] is to be expected since, physically, F (t ) should decrease monotonically. Since fort ≪ t diff the term quadratic int, being proportional tot/t diff , represents only a minor correction to ω|n i |βt in F (t ) and since t diff is indeed a very large time scale for relativistic charges, in the following it suffices to consider only the linear time-dependence in the exponential damping factor, i.e.
where we define as damping time scale.
In the limit n r = 1 and n i = 0, Eq. (2) becomes which agrees with the result for the radiation spectrum reported in [14] ifq is properly identified with the parameters used therein. Substituting in Eq. (5)t by where A =qE 2 /(3ωM 4 ). The latter integral may be evaluated analytically in terms of the known Fresnelintegrals S(y) and C(y) resulting in with y −1 = √ 2πA. For small A, one finds formally from Eq.
The ω-independent result in Eq. (8) provides a particularly suitable reference point for an analysis of in-medium effects on the radiative energy loss spectrum.
In [22], it has been shown that for non-zero n i the differential spectrum in Eq. (2) is reduced with increasing E (for fixed Γ) or with increasing Γ (for fixed E). This is a consequence of the increasing influence of F (t ) in Eq. (2) with increasing Γ, but also with decreasing ω/E, as was discussed in [25]. The behaviour of the spectrum can be understood by analyzing the competition of two different time scales: The formation time of radiation for negligible damping t f , and the damping time, t d , at which the radiation amplitude in Eq. (2) is essentially suppressed due to F (t ).
The formation time of radiation in the limit Γ ≪ m ≪ ω, i.e. in the case of negligible damping effects, can be found, following [14], from a condition for the phase of the oscillating function in Eq. (2) reading with k(ω) = √ ω 2 − m 2 . The first term, which is linear in t f , is specific for a single scattering process, while the second term, which is quadratic in t f , is genuine for multiple scatterings [14]. Considering in Eq. (9) either the linear or the quadratic term in t f as the dominant contribution, t f may be estimated by t f ≃ min{t With these time scales, it is possible to rewrite Eq. (2) normalized with respect to the BH-result in Eq. (8) as with t BH ≃ 2γ 2 /ω. The integral in Eq. (12) can be evaluated exactly [27], leading to instead. This particular limit comprises the LPM effect as discussed in [19]. Accordingly, we define Moreover, it turns out that Eq. (14) is also valid if t d is large compared to either t In case t d is small with respect to t and evaluate the simple integrals realizing that higherorder terms int give rise to subdominant contributions. In the regime, in which t d ≪ t (s) f , in which case Eq. (17) is also found, and In order to highlight the influence of an absorptive medium on coherence effects in that medium, one may normalize the full spectrum in Eq. (2) rather with respect to the spectrum expression (−d 2 W LP M ) relevant in the regime, in which coherence effects dominate (t  and (18) account fairly well for the behaviour of the radiative energy loss spectrum in an absorptive medium in the different physical regimes.
The above scaling laws show impressively the importance of the parameter-sensitive time scales t (s) and t d : From their interplay, the structure of the radiative energy loss spectrum in comparison with the BH-result as reference spectrum can be obtained semi-quantitatively. In the next section, we will study these time scales and their competition in detail and discuss the meaning of our findings for the spectra.  , respectively, as defined in Eqs. (19) and (20). The solid curve depicts our estimate for t f given by the minimum of t . In addition, long-dashed horizontal lines show the damping time t d ≃ 1/Γ for increasing Γ from top to bottom. Accordingly, for fixed γ damping mechanisms play an increasing role with increasing Γ, cf. text for details. energy x = ω/E. Then, Eqs. (10) and (11) become The method of approximating t f by the minimum of t while from Eq. (20) follows. The function in Eq. (21) exhibits a maximum at Fig. 2). Moreover, it approaches 2/m for γ ≫ 1 when x approaches the minimal allowed value x * /γ which arises from the absence of radiation in the plasma for xE < m, cf. [22]. We note that the exact expression for t In the following, we restrict the discussion to the case m ≪ M such that x * ≪ 1. Then, one finds that t In this case, the formation time will be given by t For γ > γ (1) c , instead, one finds two values of x as shown in Fig. 2, for which t f (x). These intersection points are given by is independent of γ and represents the minimal amount of time necessary to radiate a photon through a multiple scattering process. To obtain the intersection points and the corresponding formation times, we have approximated t [17], or by t f (x) and therefore t (m) f determines the formation time of radiation, whereas outside of this region radiation is still dominated by single scatterings, in particular for x < x 1 , cf. also [19]. The behaviour observed for m = 0 is in striking contrast to the m = 0 case. In this case, the expressions for the formation times become t , which are equivalent to the known results [14] for bremsstrahlung photons from incoherent (single) and coherent (multiple) scatterings, respectively. In the region 0 ≤ x x 2 one finds t [14]. The difference stems from the evident reduction of t (s) f compared to t BH in the small-x region due to polarization effects when m = 0.
The lower bound x 1 decreases with γ −1/3 , while the upper bound x 2 increases linearly with γ. Consequently, the region in which multiple scattering processes become effective increases with increasing γ (or increasing energy E for fixed M ) as evident from Fig. 3. For γ → γ (2) c , x 2 tends to 1. Thus, for γ > γ  recovered, cf. [25]. However, with increasing x towards 1 corrections, which are not present in Eq. (9), have to be taken into account. These are discussed later in Sec. IV A in the case of QCD. Generically, such corrections lead to a reduction of both t becomes stronger reduced. This implies that the region in which multiple scattering processes dominate effectively shrinks and the Bethe-Heitler bremsstrahlung spectrum is necessarily recovered for x ∼ 1. As long as m ≪ M and γ < γ (2) c , one may refrain from including corrections for x close to 1. Otherwise, these corrections would become mandatory, which is, however, beyond the scope of this section.

B. Competition between photon formation and damping
In an absorptive medium, damping of radiation effects can modify the aforementioned picture significantly. Considering m ≪ xE and Γ ≪ xE, the exponential damping factor in Eq. (3) becomes F (t ) ≃ e −Γt . This gives rise to a damping time, t d ≃ 1/Γ (cf. Eq. (4)), as highlighted in Fig. 2 by long-dashed horizontal lines, at which, compared to the undamped case, the amplitude in Eq. (2) is reduced by a factor 1/e.
In case t d is large compared to the formation time of radiation, damping mechanisms will not significantly affect the spectrum, as evident from Fig. 1. This situation is also illustrated by the upper long-dashed horizontal line in Fig. 2 gion in x, then damping will be of significance in that region. In particular, it implies that any coherent interference of radiation amplitudes happens only during t d rather than t f , because the formation process of the radiation quantum will be hampered by damping mechanisms. Strictly speaking, any finite value of Γ enters |n r | and thus modifies Eq. (9). The above parametric discussion of t f , however, is still valid qualitatively as long as we demand Γ < m.
The relevant scales for discussing the influence of damping mechanisms are t d and t onset f ≃ M 2 /q or, equivalently, Γ andq/M 2 . In case t d > t onset f or Γ <q/M 2 , one expects to find a region in γ-x-space, in which coherence effects are of importance. In the opposite case, however, i.e. if t d < t onset f or Γ >q/M 2 , damping effects will be significant and coherence effects will play no role. This stems from the definition of t onset f , see the comment below Eq. (23). Again, we do not include corrections necessary for x close to 1 in this discussion.
The lower bound x 3 ∝ γ −1 decreases faster than x 1 with increasing γ but tends to x 1 as γ → γ (1) d , while the upper bound x 4 ∝ γ increases slower than x 2 with increasing γ and tends to 1 as γ → γ (2) d ∼q/(Γ 2 M ). This leads to the situation exhibited in Fig. 4 (panel (a)). For γ > γ (2) d , damping mechanisms are significant for the whole x-range above x 3 , while for γ (2) d > γ > γ (1) d a region for x > x 4 emerges, in which undamped multiple scattering processes become relevant. In case γ (2) c > γ > γ (1) d , this picture will be modified to the extent that, in addition, a region for x > x 2 (> x 4 ) exists, in which undamped single scatterings dominate the formation of radiation again.
In the special case Γ →q/M 2 , one finds that γ This implies, in particular, that for any γ > γ (1) c coherence effects are superseded by damping mechanisms and, thus, play no significant role for the formation of radiation anymore.
For even larger damping rates Γ >q/M 2 (case sketched in panel (b) of Fig. 4), damping effects will be influential already for γ < γ (1) c : As t f is given by t      x 5 > x 2 (as also illustrated by the lower long-dashed horizontal line in Fig. 2). Therefore, damping mechanisms hamper the formation of radiation in a large region of γ-x-space for Γ >q/M 2 . We note that x 5 tends to 1 as

C. Physical implications
The above analysis discussed the competition of the time scales t  and t d . In this section, we want to interpret our findings in the context of the spectrum. In case of negligible damping, i.e. t d large compared to the other time scales, the spectrum behaves in line with Eq. (14). In case of a non-negligible damping rate, however, we identified three different regimes, in which, depending on x and γ, damping effects can be important: Among these regimes, regime (b) is particularly interesting. From our discussion of the competition of the different time scales in Secs. III A and III B, one could understand the physical process taking place as the hampering of the formation of radiation in a multiple scattering process, because without damping in the medium radiation would be formed coherently (t   Here, x cr represents a γ-independent critical x-value determined from assuming t d = t which, in case of Γ <q/M 2 , allows for a discrimination of the nature of the physical process taking place in the γ-x-space regions, in which damping plays a role (i.e. in region (I) of Fig. 4 panel (a)). With increasing Γ →q/M 2 , one finds x cr → ∞.
Corresponding to this interpretation, the physical picture simplifies tremendously: For Γ <q/M 2 (see panel (a) of Fig. 4) and any fixed x < x cr , damping effects are strong enough to influence the radiation formation significantly already after a single scattering of the charge (shaded region (SD) between x 3 and x cr ), while for fixed x > x cr damping effects influence the formation of radiation in multiple scattering processes only (shaded region (MD) between x cr and x 4 ), provided γ is large enough. For x < x cr , one passes therefore with increasing γ from single undamped (region (SU) below the x 3 -curve) to single damped (shaded region (SD)) scattering processes, which determine the spectrum. For x > x cr , instead, one passes from single undamped (region (SU) below the x 1x 2 -curve) to multiple undamped (region (MU) in between the x 4 -curve and the x 1 -x 2 -curve) to multiple damped (shaded region (MD)) scatterings with increasing γ.
For Γ ≥q/M 2 (see panel (b) of Fig. 4), damping effects are so strong that effectively the radiation formation is hampered for any x after a single scattering of the charge (shaded region (SD) between x 3 and x 5 ) as long as γ is large enough. Otherwise, single undamped scatterings determine the spectrum (region (SU) below the x 3 -x 5curve), and there is no room left for coherence effects.

IV. APPLICATION TO QCD PHENOMENOLOGY
We now extend our study to the consideration of damping phenomena in a strongly interacting medium. To our best knowledge, the influence of the damping of radiation has not yet been investigated in the context of radiative energy loss in QCD matter. Given the universality of the possible physical processes, it is reasonable to assume that dissipative effects lead equally to a damping of the radiation spectrum in QCD matter, which may formally be written down similar to Eq. (12) up to a different normalization factor and essential differences in the entering time scales due to the non-Abelian character of QCD. In Sec. IV A, we start our discussion, as in Sec. III A, by analyzing the functional behaviour and the parametric dependence of the formation time t f of gluon bremsstrahlung by ignoring possible damping mechanisms. In this case, estimates for t f exist, cf. e.g. [28]. Then, in Sec. IV B, we include damping phenomena into the discussion in the same way as done in Sec. III B by introducing a competing time scale t d . The phenomenological consequences for the gluon bremsstrahlung spectrum are analyzed in Sec. IV C in analogy to Sec. III C and a discussion of possible physical mechanisms leading to the damping of the gluon radiation is found in Sec. IV D.
A. Analysis of the gluon bremsstrahlung formation time in the absence of damping The formation time of gluon bremsstrahlung in the QCD plasma can be estimated, following [28], from the inverse expectation value of the energy imbalance between the final state (a relativistic on-mass-shell parton with momentum p plus an on-mass-shell bremsstrahlung gluon with momentum k) and a relativistic on-mass-shell single parton state carrying the same total momentum P = p + k. This leads to the condition [28] where m s and m g denote the masses of the emitting color charge and of the emitted bremsstrahlung gluon, respectively, while p B is defined as p B = (p k ⊥ − k p ⊥ )/P and E ≃ P . In order to make a connection with Sec. III A, we consider in Eq. (24) first the special case in which k ⊥ vanishes, while the emitting charge experiences small deflections through soft, elastic scatterings in the medium. Accordingly, p 2 B ≃ x 2 t fqs , whereq s is the mean squared transverse momentum per unit time picked up by the charge, and Eq. (24) renders into It is interesting to note that in the small x ≪ 1 limit this condition equation will be equivalent to the condition Eq. (9) discussed in detail above, apart from a factor q = 6q s , if in Eq. (9), now written in terms of x, the function k(x) is expanded for m ≪ xE, only terms of O(1/γ) are kept, and M and m are identified with m s and m g , respectively.
The dominant contribution to the radiative energy loss of a relativistic parton in the hot QCD plasma stems, however, from the rescatterings of the radiated gluon in the medium [7]. Thus, we consider in Eq. (24) rather the case in which the bremsstrahlung gluon undergoes soft, elastic scatterings during its formation, while any deflection of the emitting color charge is assumed to be of minor importance. This leads to p 2 B ≃ (1−x) 2 k 2 ⊥ ≃ (1−x) 2 t fqg , whereq g is the mean squared transverse momentum per unit time picked up by the radiated gluon. Then, the condition for the gluon formation time Eq. (24) becomes We note that the above equation is strictly applicable only in the limit of highly energetic gluons, i.e. when k ≃ xE. For gluon energies of the order of the gluon mass, in contrast, it has to be generalized. This can be achieved conveniently by considering rather the imbalance of parallel momenta in the above mentioned scattering process, while the energy is conserved. The resulting condition equation for the gluon formation length (the formation length translates easily into a formation time) looks similar to Eq. (26), where one has to replace only m 2 g /(2xE) by m 2 g /(ω + k ) and (1 − x)q g by [2ω/(ω + k ) − x]q g . As a self-consistency condition, the gluon energy must satisfy These modifications introduce, however, merely numerical changes of order O(1) in the prefactors so that Eq. (26) is completely sufficient for the parametric discussion we aim at in the following. As in Sec. III A, the functional behaviour of the formation time for gluon bremsstrahlung t f with x depending on γ = E/m s and on the parameter values for m s , m g andq g can be analyzed conveniently by estimating t f by the minimum of t which both increase with γ, t represents the formation time of bremsstrahlung gluons in a coherent (multiple) scattering process. This procedure for determining t f is shown in Fig. 5. We stress here that the above expression for t (s) f is only applicable for finite m g , while in the limit of a vanishing in-medium gluon mass it has to be modified as is discussed below. The results of the following analysis are summarized graphically in Figs. 5, 6 and 7, while the definitions of the appearing variables are listed in Tab. II.
Compared to Eqs. (22) and (21), corrections for x close to 1 are now taken into account in both expressions for  , respectively, as defined in Eqs. (28) and (29), while the solid curve depicts our estimate for t f given by the minimum of t In the parametric analysis, one has to distinguish between two distinct cases, namely m 3 g >q g and m 3 g <q g . In perturbative QCD (pQCD), m 3 g >q g because the in-medium gluon mass m g ∼ gT [17,18,29], whilê q g ∼ g 4 T 3 [30] for small QCD running coupling g ≪ 1 at large temperatures T . We start our discussion with this case. Nonetheless, we do not want to restrict the analysis of the parameter space to the case m 3 g >q g only, because the opposite case could be satisfied in the nonperturbative (strong coupling) regime.  f (x 4 ) for x 4 ≫ x * , respectively. The lower bound x 3 decreases with γ −1 (but slower than x 1 ) and will approach x 1 only if Γ →q g /m 2 g , for which also γ (1) c . The upper bound x 4 , instead, increases with γ (but faster than x 2 ). Thus, the x-region in which damping mechanisms are of importance increases with increasing γ and/or increasing Γ.
In contrast, for larger Γ >q g /m 2 g (case sketched in panel (b) of Fig. 6) one finds t d < t onset f (see also in Fig. 5 the lower long-dashed horizontal line). Then, damping mechanisms become important already for γ > γ c , in a region between x 5 ∼ m 2 g /(Γγm s ) and x 4 (region (II)). The lower bound We note that x 5 decreases faster than x 1 with increasing γ. Thus, we find for the case Γ >q g /m 2 g that damping mechanisms influence the formation of gluon bremsstrahlung in a large region of γ-x-space, implying a negligible role of coherence effects on the radiative energy loss spectrum.
Considering now the case m 3 g <q g , i.e. when t at x 1,S , we find the situation depicted in Fig. 7: For fixed damping rates Γ <q 1/3 g , damping effects are of importance for γ > γ (1) d in a region between x 3 and x 4 (region (I)) as in the above case m 3 g >q g (cf. panel (a) of Fig. 6). We conclude that, also in the m 3 g <q g case, damping effects become more important for the formation of gluon bremsstrahlung with increasing γ and/or Γ. As Γ →q 1/3 g , one would find that x 3 < x 1,S for any γ and, thus, damping effects would be of significance in almost the entire γ-x-space. However, here we refrain from considering damping rates Γ as large as Γ →q 1/3 g or even larger. This is because for Γ →q 1/3 g one finds Γ > m g such that the underlying assumption, that the inclusion of damping phenomena does not qualitatively influence the above Γ-independent discussion of t f , would become invalid. In any case, we expect the effect of damping to be more pronounced with increasing Γ in this regime, too.
C. Impact on the spectra The above formation and damping time analysis allows us to discuss qualitatively the influence of different in-medium effects such as damping on the radiation spectrum in QCD. Given the generic structure of the spectrum discussed in Sec. II, we make use of the scaling laws determined there. However, here we want to analyze the behaviour of the radiative power spectrum dI/dx relative to the soft, i.e. x ≪ x * , Gunion-Bertsch (GB) power spectrum limit of gluon radiation off massive partons dI GB /dx from incoherent scatterings. The latter was determined within scalar QCD in [31] as an extension of the result by Gunion and Bertsch [32] for massless partons. With this normalization, the analogon of the scaling law in Eq. (14) reads as with t f ≃ min{t Considering first the case m 3 g >q g , the spectra ratio is dI/dI GB = 1 for small x as seen in the small-x regions (SU) in Fig. 8, because t f is given by t (s) f for such x (cf. also Fig. 5). With increasing x, the ratio dI/dI GB is reduced, where the reduction is proportional to a specific power of the gluon fractional energy in line with the appropriate scaling law.
For negligible damping rates Γ, as illustrated by solid curves in Fig. 8 panels (a) and ( f , which holds for x 1 < x < x 2 . This includes, in particular, the known BDMPS-Z suppression of the power spectrum [7][8][9] f , then dI/dI GB ∝ (1 − x)/x 2 is found. This behaviour is a consequence of the finite parton mass m s in Eq. (28) and, thus, a feature specific to heavy quarks, cf. [26].
For non-negligible damping rates Γ, one finds at most four different physical regions and sizeable parts of the spectrum can become additionally reduced. For Γ <  Fig. 5). This highlights the increasing influence of damping effects on the radiation spectrum with increasing Γ. q g /m 2 g (see dashed curve in Fig. 8 panel (a)), damping effects influence the regions at intermediate x, that were dominated by multiple undamped (MU) and single undamped (SU) scatterings in the absence of damping. For x cr < x < x 2 , damping effects are so strong that effectively the formation of radiation is hampered already after a single scattering process, even though t f , and follows, assuming x cr > x * , as x cr ∼ 1/(1 + m s Γ/q g ). Passing from small x-values towards x = 1, thus, the spectrum is dominated first by single undamped scatterings, then by multiple undamped scattering processes with the known BDMPS-Z suppression, followed by multiple damped (MD) processes, where the spectrum suppression is ∝ For x cr < x < x 2 , single damped (SD) scatterings dominate the spectrum with a modification factor and for x > x 2 a region (SU) follows with dI/dI GB ∝ (1 − x)/x 2 . The spectrum suppression in the damped regions (MD) and (SD) is evidently stronger than the suppression ∝ x −1/2 in the BDMPS-Z regime.
With increasing Γ, the regions dominated by damping mechanisms increase. For Γ >q g /m 2 g , the physical situation changes significantly (cf. panel (b) in Fig. 8). In a large x-region between x 5 and x 4 (cf. also Fig. 6 panel (b)) single damped scatterings determine the power spectrum according to Eq. (34), and there is no room left for coherence effects. It is noteworthy, however, that for larger x the term ∝ 1/(1 − x) dominates in Eq. (34) such that the effect of a power spectrum suppression due to damping becomes reduced.
The opposite case m 3 g <q g is shown in Fig. 9. For small x, multiple undamped scattering processes dominate the power spectrum and dI/dI GB < 1 already in this region. For negligible Γ, the physical situation changes only for x > x 2 , where single undamped scatterings determine the spectrum. For non-negligible Γ <q 1/3 g , damping effects become influential at intermediate x, where for x < x cr a region of multiple damped scatterings and for x > x cr a region of single damped scatterings emerges. Again, the spectrum suppression in the regions (MD) and (SD) is stronger than in the regime (MU), in which coherence effects dominate the radiation spectrum.

D. Discussion of possible damping mechanisms
Damping phenomena specific to hot QCD matter become important for the radiative energy loss of an energetic parton, when they start to influence the formation process of the bremsstrahlung gluons. The gluon damping rate is related to the imaginary part of the poles in the corresponding in-medium propagator [33]. In perturbative QCD approaches, it has been calculated both for collective gluon modes in the plasma [34,35] as well as for hard gluons [36]. For collective gluons with momenta and energies of O(gT ) both elastic and inelastic processes contribute equally to a damping rate Γ ∼ g 2 T . In case of a hard gluon with either momentum or energy of O(T ) or larger only elastic processes have been considered in the evaluation of the gluon damping rate in [36], yielding Γ ∼ g 2 T ln(1/g) for g ≪ 1. However, elastic rescatterings of a bremsstrahlung gluon during its formation are already taken into account byq g in our considerations. The nature of the damping mechanisms we have in mind here is different: As possible inelastic processes leading to a damping of gluon radiation, one might consider either quark-anti-quark pair production or secondary bremsstrahlung creation from a gluon during its formation process.
For estimating the parametric dependence of the latter process, one might view the emission of the secondary bremsstrahlung gluon in the dense medium as a BDMPSmechanism [8]. In doing so, a preformed gluon with fractional energy x > x LP M , which is, however, small compared to 1, emits a secondary bremsstrahlung gluon of fractional energy x ′ in an inelastic scattering process. In order to be of influence on the formation of the preformed gluon, the formation time of the secondary gluon must be small compared to the formation time of its emitter, which can be realized if x ′ ≪ x, cf. Fig. 5. The production rate for the secondary gluon as a measure for the damping rate of the preformed emitter gluon is obtained by integrating the weighted BDMPS-radiation spectrum [8]. Neglecting the part in the integral, which is suppressed with x −3/2 , one finds a gluon damping rate Γ ∼ g 4 T up to corrections of order ln(1/g). We note that a similar parametric dependence was found in [37] for the gluon production rate, which is of interest for the chemical equilibration of the hot QCD plasma. For such a damping rate, the case Γ <q g /m 2 g would be realized in pQCD, leading to the situation depicted in panel (a) of Fig. 6.

V. CONCLUSION
Damping phenomena in a dense, absorptive plasma can influence substantially the radiative energy loss of an energetic charge traversing this medium. They manifest themselves in a non-trivial reduction of the associated radiation spectrum off that charge as compared to the spectrum from incoherent scatterings in non-absorptive matter. The effect is more pronounced for large energies E (or equivalently large Lorentz-factors γ) of the charge and/or large damping rates Γ of the radiated quanta in the medium. This behaviour can be understood semiquantitatively by making use of the concept of a formation time for radiation: Damping mechanisms reduce the radiative energy loss spectrum if the typical time scale for the damping of radiation quanta with fractional energy x = ω/E is small compared to their formation time such that these effects influence already the creation of the radiation.
In this article, we analyzed systematically the interplay between these competing time scales for an absorptive QED and QCD medium. We started with the case of a polarizable and absorptive, infinite electro-magnetic plasma, for which analytical results for the radiative energy loss spectrum per unit length have been derived in [22,24]. Then, we extended the phenomenological discussion to QCD matter. This is the first time that the consequences of damping are studied in QCD as in perturbative QCD approaches damping phenomena were so far considered as negligible, higher-order effects.
In both cases, i.e. QED and QCD, we identified parametrically the regions in γ-x-space, in which either coherence or damping effects significantly influence the associated radiation spectrum, cf. Figs. 4, 6 and 7. We showed that, generically, damping effects become important in an intermediate-x regime, which grows with increasing γ and/or Γ. Any suppression of the spectrum in this regime has to be attributed to damping phenomena rather than coherence effects. We showed that the radiation spectrum is stronger suppressed through damping effects than it is through coherence effects. Restricting our analysis for QCD matter to the case of heavy quarks, we visualized this feature in Figs. 8 and 9 by using typical values for the entering parameters. The utilized concept should, however, be generalizable to the study of light partons as well.
As damping effects become pronounced for large γ, the study of hadronic correlations at high transverse momenta p T might open the avenue for experimentally measuring the absorptive properties of hot QCD mat-ter. Moreover, the effect could be a key ingredient for the understanding of the observed heavy meson spectra: If t d becomes the dominant scale, the radiation spectra turn out to be mass-independent, which would constitute a step towards solving the non-photonic single-electron puzzle [38]. Corresponding phenomenological investigations are presented in [39], where the quark-mass independence of the heavy-flavor meson quenching at large p T is quantified and a comparison with the available experimental data from the ALICE collaboration on the D meson quenching suggests a rather strong damping rate Γ/T ≃ 0.75. Also, we expect our results to be of relevance in heavy-quark tagged jet physics: On the one hand, either the finite in-medium gluon mass m g or the parameterq 1/3 g represent a natural lower energy cutoff for bremsstrahlung gluons in the soft region of the spectrum. Damping effects, on the other hand, hamper the formation of hard or intermediate hard gluons, providing effectively an upper limit for the radiative energy loss of an energetic projectile, similar to coherence effects but with a stronger impact on the power spectra.
In this work, we neglected the consideration of any dependence on a finite path length L. Nevertheless, whether the energetic charge is created in the remote past or originating from the interior bulk of a thick medium, only quanta with a formation length smaller than the distance L travelled by the charge can contribute to the medium-induced radiative energy loss, cf. [26]. This implies that in the regions of γ-x-space, in which damping effects are relevant, finite path length effects play no additional role for the radiation spectrum off these charges. Following the heuristic arguments presented in [8], we expect that the averaged radiative energy loss of such partons in the hot QCD plasma is proportional to L/Γ in case 1/Γ < L. Only for partons stemming from the outer crust of the medium of thickness 1/Γ finite path length effects are important and the averaged radiative energy loss is proportional to L 2 . We considered, moreover, x-independent damping rates in our analysis. In general, however, Γ should depend on the energy of the radiation quanta. We leave such studies for future investigations.