Medium–induced jet evolution: wave turbulence and energy loss

# Medium–induced jet evolution: wave turbulence and energy loss

Leonard Fister    and Edmond Iancu
###### Abstract

We study the gluon cascade generated via successive medium-induced branchings by an energetic parton propagating through a dense QCD medium. We focus on the high-energy regime where the energy of the leading particle is much larger than the characteristic medium scale , with the jet quenching parameter and the distance travelled through the medium. In this regime the leading particle loses only a small fraction of its energy and can be treated as a steady source of radiation for gluons with energies . For this effective problem with a source, we obtain exact analytic solutions for the gluon spectrum and the energy flux. These solutions exhibit wave turbulence: the basic physical process is a continuing fragmentation which is ‘quasi-democratic’ (i.e. quasi-local in energy) and which provides an energy transfer from the source to the medium at a rate (the energy flux ) which is quasi-independent of . The locality of the branching process implies a spectrum of the Kolmogorov-Obukhov type, i.e. a power-law spectrum which is a fixed point of the branching process and whose strength is proportional to the energy flux: for . Via this turbulent flow, the gluon cascade loses towards the medium an energy , which is independent of the initial energy of the leading particle and of the details of the thermalization mechanism at the low-energy end of the cascade. This energy is carried away by very soft gluons, which propagate at very large angles with respect to the jet axis. Our predictions for the value of and for its angular distribution appear to agree quite well, qualitatively and even semi-quantitatively, with the phenomenology of di-jet asymmetry in nucleus-nucleus collisions at the LHC.

###### Keywords:
Perturbative QCD. Heavy Ion Collisions. Jet quenching. Wave turbulence
institutetext: Institut de Physique Théorique de Saclay, F-91191 Gif-sur-Yvette, France

## 1 Introduction

The experimental observation of the phenomenon known as ‘di–jet asymmetry’ in Pb+Pb collisions at the LHC Aad:2010bu (); Chatrchyan:2011sx (); Chatrchyan:2012nia (); Aad:2012vca (); Chatrchyan:2013kwa (); Chatrchyan:2014ava (); Aad:2014wha (); Gulhan:2014 () has triggered intense theoretical efforts MehtarTani:2010ma (); MehtarTani:2011tz (); CasalderreySolana:2010eh (); Qin:2010mn (); CasalderreySolana:2011rz (); Blaizot:2012fh (); CasalderreySolana:2012ef (); Blaizot:2013hx (); Blaizot:2013vha (); Iancu:2014aza (); Kurkela:2014tla (); Blaizot:2014ula (); Apolinario:2014csa () aiming at understanding the evolution of an energetic jet propagating through a dense QCD medium, such as a quark–gluon plasma. The crucial observation is that the part of the jet fragmentation which is triggered by interactions inside the medium is controlled by relatively soft gluon emissions, with energies well below the characteristic medium scale and a formation time much smaller than . (Here, is the jet quenching parameter, is the distance travelled by the ‘leading particle’ — the energetic parton which has initiated the jet — through the medium, and the ‘formation time’ is the typical duration of the branching process.) This observation has far reaching consequences:

The soft gluons can be easily deviated towards large angles by rescattering in the medium, so their abundant production via jet fragmentation may explain the significant transport of energy at large angles with respect to the jet axis — the hallmark of di–jet asymmetry. Also, the subsequent emissions of soft gluons can be viewed as independent from each other and hence described as a classical, probabilistic, branching process. Indeed, the quantum coherence effects and the associated interference phenomena are efficiently washed out by rescattering in the medium MehtarTani:2010ma (); MehtarTani:2011tz (); CasalderreySolana:2010eh (): the loss of color coherence occurs on a time scale comparable to that of the branching process, so that gluons that emerge from a splitting propagate independently of each other Blaizot:2012fh ().

Based on such considerations, one has been able to derive a classical effective theory for the gluon cascade generated via successive medium–induced gluon branchings Blaizot:2013hx (); Blaizot:2013vha () (see also Refs. Baier:2000sb (); Jeon:2003gi () for earlier, related, studies). This is a stochastic theory for a Markovien process in which the branching rate is given by the BDMPSZ spectrum Baier:1996kr (); Baier:1996sk (); Zakharov:1996fv (); Zakharov:1997uu (); Baier:1998kq () for a single, medium–induced, gluon emission. The branching probability corresponding to a distance is parametrically of order , with . This probability becomes of order one (meaning that the branching dynamics becomes non–perturbative) when . As we shall see, this ‘soft’ scale is truly semi–hard (in the ballpark of a few GeV), meaning that there is a significant region in phase–space where perturbation theory breaks down. The effective theory put forward in Refs. Blaizot:2013hx (); Blaizot:2013vha () allows one to deal with such non–perturbative aspects, by resuming soft multiple branchings to all orders.

The original analysis in Blaizot:2013hx () demonstrated that the non–perturbative dynamics associated with multiple branchings has a remarkable consequence: it leads to wave turbulence KST (); Nazarenko (). The leading particle, whose initial energy is typically much larger than the non–perturbative scale , promptly and abundantly radiates soft gluons with energies and thus loses an amount of energy of order event by event (that is, with probability of order one). After being emitted, these soft primary gluons keep on branching into even softer gluons, and their subsequent branchings are quasi–democratic : the two daughter gluons produced by a typical splitting have comparable energies111It is interesting to notice that a similar branching process occurs in a different physical context, namely the thermalization of the quark–gluon plasma produced in the intermediate stages of a ultrarelativistic heavy ion collision: during the late stages of the ‘bottom–up’ scenario Baier:2000sb (), the hard particles lose energy towards the surrounding thermal bath via soft radiation giving rise to quasi–democratic cascades Kurkela:2011ti ().. The locality of the branchings in is the key ingredient for turbulence. It leads to a power–law spectrum , which emerges as the Kolmogorov–Zakharov (KZ) fixed point KST (); Nazarenko () of the branching process (this KZ spectrum is formally similar to the BDMPSZ spectrum), and to an energy flux which is independent of — the turbulent flow. An energy flux which is uniform in means that the energy flows from the high–energy end to the low–energy end of the cascade, without accumulating at any intermediate value of . For an ideal cascade, where the branching law remains unchanged down to arbitrary small values of , the energy carried by the flow accumulates into a condensate at . In practice, we expect the branching process to be modified when the gluon energies become comparable to the medium ‘temperature’ (the typical energy of the medium constituents): the soft gluons with ‘thermalize’, meaning that they transfer their energy towards the medium. Assuming the medium to act as a perfect sink at , we conclude that the rate for energy loss is fixed by the turbulent flow and thus independent of the details of the thermalization mechanism (‘universality’).

An essential property of the turbulent flow is the fact that it allows for the transfer of a significant fraction of the total energy towards arbitrarily soft quanta. To better appreciate how non–trivial this situation is, let us compare it with a more traditional parton cascade in perturbative QCD: the DGLAP cascade, as driven by bremsstrahlung in the vacuum. In that case, the typical splittings are very asymmetric, due to the ‘infrared’ () singularity of bremsstrahlung, and lead to a rapid rise in the number of gluons with small values of the energy fraction . Yet, the total energy carried by these ‘wee’ gluons with is very small: the energy fraction contained in the region of the spectrum at vanishes as a power of when . Most of the original energy remains in the few partons with larger values of . This is due to the fact that, after a very asymmetric splitting, the parent parton preserves most of its original energy.

By contrast, for the medium–induced cascade, the energy contained in the bins of the spectrum at is only a part of the total energy associated with modes softer than . The other part is the energy carried by the turbulent flow, which ends up at arbitrarily low values of (at least, for an ideal cascade) and hence is independent of . Depending upon the size of the medium, this flow energy can be as large as the original energy of the leading particle (see the discussion in Sect. 2). In the presence of a thermalization mechanism at , the above argument remains valid so long as , with . In practice, this ‘thermal’ value is quite small, so most of the energy lost by the gluon cascade towards the medium is associated with the turbulent flow, and not with the (BDMPSZ–like) gluon spectrum222Incidentally, this explains why earlier studies of the energy distribution based on the BDMPSZ spectrum alone, which have not included the effects of multiple branchings, concluded that there should be very little energy in the gluon cascade at small and large angles Salgado:2003rv (), and thus failed to predict the phenomenon of di–jet asymmetry.. Without this flow, there would be no significant energy transfer towards very small .

Soft gluons propagate at large angles with respect to the jet axis: , where is the typical transverse momentum acquired by the gluon via rescattering in the medium, and is at most weakly dependent upon . So, the ability of the medium–induced cascade to abundantly produce soft gluons provides a natural explanation for the main feature of di–jet asymmetry: the fact that the energy difference between the trigger jet and the away jet is carried by many soft ( GeV) hadrons propagating at large angles () with respect to the axis of the away jet Chatrchyan:2011sx (). This qualitative explanation has been originally proposed in Blaizot:2013hx () and further developed in Refs. Blaizot:2013vha (); Iancu:2014aza (); Kurkela:2014tla (); Blaizot:2014ula (). However, these previous studies were not fully conclusive, as they did not explicitly consider the kinematical regime which is pertinent for di–jet asymmetry. Namely, they focused on the ‘low–energy’ regime where the energy of the leading particle (LP) is smaller than the medium scale . Albeit the value of is not precisely known from first principles, its current phenomenological estimates are well below the energy  GeV of the trigger jet in the experimental measurements of di–jet asymmetry (see the discussion in Sect. 2). It is our main objective in this paper to provide a thorough analysis of the high–energy regime at , including its implications for the phenomenology.

In order to describe our results below, it is useful to recall the physical meaning of the medium scale : this is the highest possible energy of a medium–induced emission by a parton with energy which crosses the medium over a distance . The emission of a gluon with energy has a formation time and hence a small probability : this is a rare event. Still, such rare but hard emissions dominate the average energy loss by the LP, estimated as Baier:1996kr (); Baier:1996sk (); Zakharov:1996fv (); Zakharov:1997uu (). Hence, a very energetic particle with loses only a small fraction of its original energy and thus emerges from the medium with an energy , which is much larger than the maximal energy of its radiation. Accordingly, the spectrum shows a gap between a peak at , which represents the LP, and a continuum at , which describes the radiation. The detailed structure of the peak is irrelevant for studies of the di–jet asymmetry: the energy carried by the LP is very closely collimated around the jet axis, within a small angle333Here, is the transverse momentum broadening acquired by the LP while crossing the medium over a distance . Some typical values are  GeV,  GeV, and hence . , which is much smaller than the angular opening of the experimental ‘jet’. This is in agreement with the experimental observation Aad:2010bu (); Chatrchyan:2011sx () that the azimuthal distribution of di–jets in Pb+Pb collisions is as narrowly peaked at as the corresponding distribution in p+p collisions.

In view of the above, our subsequent analysis will focus on the radiation part of the spectrum at , where and . This part includes the essential physics of multiple branching leading to energy loss via many soft particles propagating at large angles. For the purposes of this analysis, the LP can be treated as a steady source of radiation for gluons with energy fractions . For this effective problem with a source, we will be able to construct exact solutions for the gluon spectrum at any time , and also for the energy flux (the rate for energy transfer through the cascade; see Sect. 3 for a precise definition). This energy flux, and more precisely its ‘flow’ limit , is the most interesting quantity in the present context, since it controls the energy transfer by the gluon cascade to the medium.

A non–zero ‘flow’ component in the energy flux is the main signature of turbulence KST (); Nazarenko () (e.g., there is no such a component for the DGLAP cascade). An important property of turbulence, which follows from the locality of the branchings, is the fact that, within the ‘inertial range’ deeply between the ‘source’ and the ‘sink’, the spectrum is fully determined by the energy flux together with the KZ scaling law. For the standard turbulence in 3+1 dimensions, this relation is known as the ‘Kolmogorov–Obukhov spectrum’. For our present problem in 1+1 dimensions (energy and time), the ‘source’ is the leading particle, the ‘sink’ is the thermal bath, and the ‘inertial range’ correspond to . A priori, our problem differs from the familiar turbulence set–up via its explicit time–dependence: the source acts only up to a finite time , which moreover is quite small, in the sense that . Notwithstanding, we shall demonstrate that a time–dependent generalization of the Kolmogorov–Obukhov relation holds for the problem at hand: the gluon spectrum at is fully determined by the flow component of the energy flux, together with the characteristic scaling behavior of the BDMPSZ spectrum (the KZ scaling for the present problem). Namely, we shall find , where the proportionality constant is under control.

The energy transferred by the gluon cascade to the medium can be identified with the energy carried away by the flow, i.e. the time integral of between and . For the high–energy regime under consideration, this quantity turns out to be independent of the original energy of the LP and to have a transparent physical interpretation444This estimate for holds to leading order in  ; see Eq. (74) and the plots in Sect. 5 for more accurate results.: , where and is a pure number which can be interpreted as the average number of soft primary emissions with energies . Such soft gluons are radiated by the LP with probability of order one and they subsequently transfer their energy towards the medium via successive, quasi–democratic, branchings. A typical gluon cascade is illustrated in Fig. 1. Using phenomenologically motivated values for and , we find  GeV (see Sect. 5). Since carried by very soft gluons, with energies , this energy propagates at very large angles with respect to the jet axis, at least as large as . ( is the typical propagation angle of the soft primary gluons, and its above estimate will be discussed in Sect. 5.) By progressively increasing the jet opening angle within a rather wide range, say from up to , we can recover part of the missing energy, but only very slowly : most of this energy lies at even larger angles, (see Fig. 1 and Sect. 5 for details). The above predictions — the numerical estimate for the energy loss at large angles and its extremely weak dependence upon the jet opening angle — are in good agreement, qualitative and even semi–quantitative, with the phenomenology of di–jet asymmetry at the LHC Chatrchyan:2011sx (); Aad:2012vca (); Chatrchyan:2013kwa (); Gulhan:2014 (). Vice versa, we believe that these particular LHC data could not be understood in a scenario which neglects multiple branchings, nor in one which uses a vacuum–like model for the in–medium gluon fragmentation, that is, a model which ignores the quasi–democratic nature of the soft branchings and the associated turbulent flow.

Our paper is organized as follows. In Sect. 2 we shall introduce, via qualitative considerations and parametric estimates, the main physical scales which control the medium–induced gluon branching and allow one to separate between various physical regimes. In Sect. 3, we shall consider the low–energy regime at as a warm–up. Besides a succinct review of the main results obtained in Ref. Blaizot:2013hx (), this section will also contain some new material, like the explicit calculation of the energy flux and a first discussion of the Kolmogorov–Obukhov relation. Sects. 4 and 5 will be devoted to the main new problem of interest for us here: the high–energy regime at . Sect. 4 will present the main theoretical developments: the justification of the effective problem with a source, the exact, analytic and numerical, solutions for the radiation spectrum at and for the turbulent flow, the democratic nature of the branchings and its physical implications, and the proof of the (time–dependent version of the) Kolmogorov–Obukhov relation for the branching dynamics at hand. Finally, in Sect. 5 we shall discuss some phenomenological consequences of this dynamics for the energy lost by the jet via soft gluons propagating at large angles.

## 2 Typical scales and physical regimes

We would like to study the gluon cascade generated via successive medium–induced gluon branchings by an original gluon — the ‘leading particle’ (LP) — with energy which propagates through a dense QCD medium along a distance . For the present purposes, the medium is solely characterized by a transport coefficient , known as the ‘jet quenching parameter’, which measures the dispersion in transverse momentum acquired by a parton propagating through this medium per unit length (or time). Depending upon its energy, the leading particle can either escape the medium, or disappear inside it (in the sense of not being distinguishable from its products of fragmentation). The actual scenario depends upon the ratio between and a characteristic medium scale , which is the maximal energy of a gluon whose emission can be triggered by multiple scattering in the medium: gluons with an energy have a formation time of order and an emission probability of order . Another energy scale that will play an important role in what follows is the soft scale  : gluons with have a relatively short formation time and an emission probability of order 1. This scale is ‘soft’ since at weak coupling and since one generally has in the applications to phenomenology (see below).

More generally, the elementary probability for a gluon with energy to be radiated (via the BDMPSZ mechanism) during a time interval can be parametrically estimated as

 ΔP∼¯αΔttbr(ω)∼¯α√^q2ωΔt, (1)

where is the ‘gluon formation time’ — more precisely, the typical duration of a branching process in which the softest of the two daughter gluons has an energy . Eq. (1) holds so long as . When , the multiple branchings become important and the evolution of the gluon cascade becomes non–perturbative (in the sense that the effects of multiple branchings must be resumed to all orders). As clear from Eq. (1), for any , there exists a sufficiently soft sector where the branching dynamics is non–perturbative: this occurs at . In particular, for , this yields back the ‘soft’ scale aforementioned: .

The above discussion in particular implies that the quantity sets the scale for the energy lost by the LP in a typical event : with a probability of , the LP particle emits primary gluons with energies of , and thus loses an energy . Accordingly, the typical energy loss, as measured event–by–event, is sensitive to multiple branchings. On the other hand, the average energy loss is dominated by rare but hard emissions, with energies , for which the effects of multiple branchings are negligible. One finds indeed

 ⟨ΔE⟩≃∫ωmaxdωωdNdω≃¯α∫ωmaxdω√ωcω∼¯α√ωcωmax, (2)

where the gluon spectrum is essentially the elementary probability for a single branching, Eq. (1), evaluated for and the upper limit is typically much larger than . The integral in Eq. (2) is dominated by its upper limit, i.e. by energies .

The global features of the medium–induced gluon cascade depend upon the relative values of these three scales , , and . Namely, for a given medium scale , one can distinguish between three interesting physical regimes, depending upon the energy of the leading particle : (i) high energy , (ii) intermediate energy , and (iii) low energy . Recalling that , we see that the ‘high energy’ regime can also be viewed as the limit where the in–medium path is relatively small, whereas the ‘low energy’ case corresponds to relatively large values of .

In case (i), both the average energy loss and its typical value are much smaller than , and the probability to find the LP outside the (already narrow) energy interval is negligibly small. Accordingly, in this case there is a gap in the spectrum between a ‘peak’ at representing the leading particle and a ‘continuum’ at representing the radiated gluons.

In case (ii), the typical energy loss is still much smaller than , so the leading particle survives in most of the events, yet there is a sizable fraction of the events, of or larger, where both fragmentation products carry similar energies. Accordingly, the LP peak is visible in the spectrum, but there is no gap anymore. The average energy loss is still smaller than the original energy , but it represents a relatively large fraction of it, of order .

In case (iii), both the typical and the average energy loss are of order , meaning that the LP undergoes strong fragmentation and ‘disappears’ in most, if not all, of the events. Of course, this should be also the faith of the very soft ) gluons produced via radiation in cases (i) and (ii). So, in this third case, the spectrum contains no peak or other structure suggestive of the LP.

To summarize, the first two cases have in common the fact that the LP survives after crossing the medium, but they differ in the actual shape of the spectrum (with or without a gap). The last two cases are both characterized by the absence of a gap, but they differ in the fact that the LP peak is still visible in case (ii), whereas it is totally washed out in case (iii).

To make contact with the phenomenology, we chose  GeV/fm (a reasonable estimate for a weakly coupled quark–gluon plasma Baier:1996kr () which moreover appears to be consistent with recent analyses of data Burke:2013yra ()), , and let vary from 2 to 6 fm. For the three particular values  fm, we deduce  GeV and  GeV. Hence, when one is interested in the phenomenology of high–energy jets with  GeV, as in the studies of di–jet asymmetry at the LHC, one should mainly consider the case (i) above. On the other hand, for studies of the nuclear modification factor , where the energies of the measured hadrons vary from 1 GeV to about 20 GeV, one is mostly in the situations covered by cases (ii) and (iii). These last two cases have been thoroughly discussed in the recent literature, in particular in relation with the disappearance of the leading particle and the energy transport at large angles Blaizot:2013hx (); Blaizot:2013vha (); Iancu:2014aza (); Kurkela:2014tla (); Blaizot:2014ula (), but to our knowledge the first case has not been studied in detail so far. From the previous discussion, is should be clear that this is the most relevant case for a study of di–jet asymmetry in Pb+Pb collisions at the LHC. This is the main problem that we would like to address in what follows.

## 3 The low–energy regime

In preparation for the discussion of the high–energy regime at , it is useful to first review some known results concerning the low and intermediate regimes at Blaizot:2013hx (); Blaizot:2013vha () (see also Refs. Baier:2000sb (); Jeon:2003gi (); Schenke:2009gb () for earlier, related, studies). These two regimes can be simultaneously discussed, as they refer to different limits of a same theoretical description.

### 3.1 The rate equation

Throughout this paper we shall focus on the gluon spectrum integrated over transverse momenta, i.e.

 D(ω,t)≡ωdNdω=∫d2k ωdNdωd2k, (3)

where and denote the energy and respectively transverse momentum of a gluon in the cascade, is the number of gluons, and it is understood that the evolution time obeys . The function describes the energy distribution within the cascade and its evolution with time. For sufficiently soft gluons at least, namely so long as , and to leading order555A class of particularly large radiative corrections, which are enhanced by the double–logarithm , can be effectively resummed into the effective dynamics by replacing the ‘bare’ value of the jet quenching parameter by its renormalized value, as recently computed in Refs. Liou:2013qya (); Iancu:2014kga (); Blaizot:2014bha (); Iancu:2014sha (); Wu:2014nca (). in , this evolution can be described as a classical stochastic branching process Blaizot:2012fh (); Blaizot:2013hx (); Blaizot:2013vha (), with the elementary splitting rate determined by the BDMPSZ spectrum Baier:1996kr (); Baier:1996sk (); Zakharov:1996fv (); Zakharov:1997uu (); Baier:1998kq (). Specifically, the differential probability per unit time and per unit for a gluon with energy to split into two gluons with energy fractions respectively and is

 d2Pbrdzdt=αs2πPg→g(z)tbr(z,ω), (4)

where , with , is the leading order gluon–gluon splitting function, is the number of colors, and is the typical duration of the branching process:

 tbr(z,ω)≡√z(1−z)ω^qeff(z),^qeff(z)≡^q[1−z(1−z)]. (5)

Note that this branching time depends upon both the energy of the parent gluon and the splitting fraction , and that it is much smaller than whenever at least one of the two daughter particles, with energies and respectively , is soft compared to .

The elementary splitting rate (4) together with the requirement of probability conservation completely specifies the structure of the stochastic branching process and, in particular, the evolution equation obeyed by the gluon spectrum. So long as , this equation reads

 ∂D(x,τ)∂τ=¯α∫dzK(z)[√zxD(xz,τ)−z√xD(x,τ)], (6)

in convenient notations where , is the energy fraction with respect to the leading particle, and

 τ≡√^qEt=√2xctL,xc≡ωcE, (7)

is the reduced time (the evolution time in dimensionless units). Notice that for the physical problems discussed in this section. The splitting kernel is defined as

 K(z)≡f(z)[z(1−z)]3/2=K(1−z),f(z)≡[1−z(1−z)]5/2. (8)

It depends only upon the splitting fraction since the corresponding dependence upon the energy (fraction) of the leading particle, cf. Eq. (4), has been explicitly factored out in writing Eq. (6).

We shall refer to the r.h.s. of Eq. (6) as the ‘branching term’ and denote it as . This is the sum of two terms, which can be recognized as the familiar ‘gain’ and ‘loss’ terms characteristic of a branching process. The first term, which is positive and nonlocal in , is the gain term : it describes the rise in the number of gluons at due to emissions from gluons at larger . The respective integral over is restricted to by the support of . The second, negative, term, which is local in , represents the loss term and describes the reduction in the number of gluons at due to their decay into gluons with smaller . Taken separately, the gain term and the loss term in Eq. (6) have endpoint singularities at , but these singularities exactly cancel between the two terms and the overall equation is well defined.

As anticipated, Eq. (6) encompasses the two regimes at ‘low’ and ‘intermediate’ energies introduced in Sect. 2. In fact, there is no fundamental difference between the dynamics in these two regimes, rather they differ only in the maximal value for the reduced time which is allowed in practice. This maximal value, namely , increases with the medium size , but decreases with the energy of the leading particle. In the ‘intermediate energy’ regime, the evolution is limited to relatively small times, , whereas in the ‘low energy’ one, it can extend up to much larger values: . This explains the qualitative differences between the two regimes that were anticipated in Sect. 2 and will be now demonstrated via explicit solutions to Eq. (6).

### 3.2 The spectrum and the flow energy

To study the effects of multiple branchings, one needs a non–perturbative solution to Eq. (6). Whereas it is straightforward to solve this equation via numerical methods, for the purpose of demonstrating subtle physical phenomena, it is much more convenient to dispose of an analytic solution. Such a solution has been obtained in Ref. Blaizot:2013hx (), but for the simplified kernel , which is obtained from Eq. (8) after replacing the slowly varying factor in the numerator by 1. This simplified kernel has the same singularities at and as the original kernel , hence it is expected to have similar physical implications, at least qualitatively. (This will also be checked via numerical simulations later on; see e.g. Fig. 4.)

For the simplified kernel and the initial condition , corresponding to a single gluon (the ‘leading particle’ ) carrying all the energy at , the exact solution reads Blaizot:2013hx ()

 D(x,τ)=¯ατ√x(1−x)3/2 exp{−π¯α2τ21−x}. (9)

This is recognized as the product between the BDMPSZ spectrum Baier:1996kr (); Baier:1996sk (); Zakharov:1996fv (); Zakharov:1997uu (); Baier:1998kq () (which is the same as the result of the first iteration of Eq. (6)),

 D0(x,τ)=¯ατ√x(1−x)3/2, (10)

and a Gaussian factor describing, at early times, the broadening of the peak associated with the LP Baier:2001yt () and, at late times, the suppression of the spectrum as whole.

To be more specific, consider increasing the time from up to the maximal value , where we recall that . When , the r.h.s. of Eq. (9) approaches , as it should. So long as is small enough for , the spectrum exhibits a pronounced peak in the vicinity of , which describes the leading particle: the maximum of this peak lies at with and its width around is of order . The fact that the peak gets displaced below 1 is a consequence of the Gaussian factor in Eq. (9), which strongly suppresses the spectrum for close to 1, within a window

 1−x≲π¯α2τ2≪1. (11)

The physical origin of this suppression should be clear in view of the discussion in Sect. 2: for close to 1, the quantity is the energy lost by the leading particle via radiation. Eq. (11) shows that the typical value of this energy is , with the non–perturbative scale for the onset of multiple branching, as introduced in Sect. 2. That is, the LP copiously radiates very soft gluons, for which the emission probability is of , and thus loses an energy of order . Interestingly, this energy loss is enhanced by the relatively large numerical factor , which can be interpreted as the average number of gluons with energy that are emitted by the LP during a time interval . This interpretation will be supported by other findings below.

Let us now increase towards larger values . This is of course possible only in the ‘low energy’ regime where . Then the Gaussian suppression extends to all values of , the LP peak gets washed out — it broadens, it moves towards smaller values of , and its height is decreasing — and eventually disappears from the spectrum. One can say that a LP with energy has a finite ‘lifetime’ inside the medium, of order or, in physical units (cf. Eq. (7)),

 Δt∼1¯α√E^q . (12)

More precisely, this means that the LP has fragmented into gluons which carry a sizable fraction of its original energy . Via successive branchings, the energy gets degraded to lower and lower values of , and it is interesting to understand this evolution in more detail. A priori, one might expect this energy to accumulate in the small– part of the spectrum, and notably at (corresponding to ), but Eq. (9) shows that this is actually not the case: for , Eq. (9) reduces to

 D(x,τ)≃¯ατ√x e−π¯α2τ2, (13)

which has exactly the same shape in as the small– limit of the BDMPSZ spectrum, Eq. (10). In fact, Eq. (13) formally looks like the BDMPSZ spectrum produced via a single emission by the LP, times a Gaussian factor describing the decay of the LP with increasing time. This interpretation seems to imply that multiple branchings are not important at small , but from the discussion in Sect. 2 we know that this cannot be true: after a time , the single–branching probability becomes of order one (meaning that multiple branching becomes important) for all the soft modes obeying . This last condition can also be inferred from Eq. (10): when , the BDMPSZ spectrum becomes of .

We are thus facing an apparent paradox — in spite of the importance of multiple branching, the energy does not get accumulated in the bins of the spectrum at small — which finds its solution in the phenomenon of wave turbulence Blaizot:2013hx (). The BDMPSZ spectrum at small is not modified by the fragmentation because this represents a fixed point of the rate equation (6) at small : the branching term vanishes (meaning that the ‘gain’ and ‘loss’ terms compensate each other) when evaluated with the ‘scaling’ spectrum . This can be recognized as the Kolmogorov–Zakharov (KZ) spectrum KST (); Nazarenko () for the branching process at hand. In turn, the existence of this fixed point implies that, via successive branchings, the energy gets transmitted from large to small , without accumulating at any intermediate value of : it rather flows throughout the spectrum and accumulates into a condensate at .

This is illustrated in Fig. 2, where the exact solution (9) is represented as a function of for several values of , up to relatively large values, such that . The early–time set of curves, at , where the LP peak is still visible, is representative for the ‘intermediate energy’ regime, where the late–time curves, from which the LP has disappeared and where the spectrum is seen to be suppressed as a whole, correspond to the ‘low energy’ case.

The energy flow can also be studied analytically, on the basis of Eq. (9). To that aim, consider the energy balance between spectrum and flow. The energy fraction contained in the spectrum after a time is computed as Blaizot:2013hx ()

 E(τ)=∫10dxD(x,τ)=e−π¯α2τ2, (14)

and decreases with time. The difference

 Eflow(τ)≡1−E(τ)=1−e−π¯α2τ2, (15)

is the energy fraction carried by the flow, i.e. by the multiple branchings, and which formally ends up in a condensate at . For sufficiently large times (corresponding to the low–energy regime), this can be as large as the total initial energy of the LP.

It is also interesting to consider the small time limit of Eq. (15), that is

 Eflow(τ)≃π¯α2τ2=2πxs(τ)forπ¯α2τ2≪1. (16)

This result can be interpreted as follows: is the average number of primary gluons with energies of the order of that are emitted by the leading particle during a time . This number is independent of or , since such gluon emissions occur with probability of order one. Stated differently, the typical time interval between two successive such emissions is of order . [This interval can be estimated from the condition that , with given by Eq. (1) with  ; this implies .] After being emitted, these soft primary gluons rapidly cascade into even softer gluons and thus eventually transmit (after a time estimated as above) their whole energy to the arbitrarily soft quanta which compose the flow. This argument also shows that the gluons with energies not only are emitted with a probability of during an interval of order , but also have a ‘lifetime’ before they branch again with probability of .

### 3.3 Energy flux, turbulence, and thermalization

The physical interpretation of Eq. (15) in terms of multiple branchings and, in particular, its relation to turbulent flow become more transparent if one studies a more differential quantity, the energy flux . This is defined as the rate for energy transfer from the region to the region . Since the energy in the region is decreasing with time, via branchings, it is natural to define the flux as the following, positive, quantity

 F(x0,τ) ≡−∂E>(x0,τ)∂τ=∂E<(x0,τ)∂τ, (17)

where is the energy fraction contained in the bins of the spectrum with , that is,

 E>(x0,τ)=∫1x0dxD(x,τ), (18)

whereas the complementary quantity is the energy fraction carried by the modes with . In turn, is the sum of two contributions : the flow energy (15) and the energy contained in the bins of the spectrum at  ; that is,

 E<(x0,τ) =1−E>(x0,τ)=Eflow(τ)+∫x00dxD(x,τ). (19)

Using the above definitions together with Eq. (9) for , it is straightforward to numerically compute the energy flux , with the results displayed in Fig. 3. For a physical discussion, it is convenient to focus on the behavior at small . In that region, one can use Eq. (19) together with the small– approximation to the spectrum, Eq. (13), to deduce the analytic estimate

 F(x0,τ) ≃[2π¯α2τ+2¯α√x0(1−2π¯α2τ2)]e−π¯α2τ2. (20)

The first term within the square brackets, which is independent of , is the flow contribution,

 Fflow(τ)≡∂Eflow(τ)∂τ=2π¯α2τe−π¯α2τ2, (21)

while the second term, proportional to , is the rate at which the energy changes in the region of the spectrum at . Clearly, the flow component in Eq. (21) dominates over the non–flow one at sufficiently small values of , such that . This is also visible in Fig. 3, where the various curves become indeed flat at sufficiently small .

We thus see that the small– behavior of the flux, and unlike the corresponding behavior of the spectrum, does reveal the non–perturbative nature of the multiple branchings and of the associated scale : in the soft region at , the flux is controlled by its ‘flow’ component and is quasi–independent of . A uniform energy flux is the distinguished signature of (wave) turbulence KST (); Nazarenko (). It physically means that the energy flows through the spectrum without accumulating at intermediate values of . To see this, let us compute the rate of change for — the energy fraction contained within the interval :

 E(x1,x2,τ)=∫x2x1dxD(x,τ)  ⟹  ∂E(x1,x2,τ)∂τ=F(x2,τ)−F(x1,τ). (22)

This rate vanishes if the flux is independent of . In our case, the flux is not strictly uniform, not even at very small values of (see Eq. (20)). Yet, the energy flux which crosses a bin at is much larger than the rate for energy change in that bin: the energy flows through the bin, without accumulating there.

It is intuitively clear that a quasi–uniform flux requires the branchings to be quasi–local in (or ‘quasi–democratic’). Since, if the typical branchings were strongly asymmetric, then after each branching most of the energy would remain in the parent gluon and the energy would accumulate in the bins at large . It is also quite clear, in view of the general arguments in Sect. 2, that in the non–perturbative region at the branchings are indeed quasi–local: a gluon with energy splits with probability of during a time interval irrespective of the value of the splitting fraction. Hence, there is no reason why special values like or should be favored. A more elaborate argument in favor of democratic branchings will be presented in Sect. 4.4.

The locality of the interactions is a fundamental property of turbulence KST (); Nazarenko (). In the traditional turbulence problem, where the energy is injected by a time–independent source which is localized in energy and produces a steady spectrum, this property ensures that the energy spectrum in the ‘inertial range’ (i.e. sufficiently far away from the source) can be expressed in terms of the (steady) flux and a special power–like spectrum, the ‘Kolmogorov–Zakharov spectrum’, which is a fixed–point of the ‘collision term’. In the case of hydrodynamic turbulence in 3+1 dimensions, this relation between the energy spectrum and the flux is known as the ‘Kolmogorov–Obukhov spectrum’.

For the problem at hand, where the ‘source’ is the leading particle originally localized at , the ‘inertial region’ corresponds to , the ‘collision term’ term is the branching term , and the fixed–point solution is the scaling spectrum . But unlike for the more conventional set–up, our current problem is clearly not stationary: the ‘source’ (the LP) loses energy and can even disappear at large times, so both the spectrum and the energy flux have non–trivial time dependencies. Notwithstanding, it turns out that the fundamental relation alluded to above, between the energy spectrum and the flux, also holds for the time–dependent physical problem at hand. Namely, by inspection of Eqs. (13) and (21), it is clear than one can write

 D(x,τ)≃12π¯αFflow(τ)√xforx≪1. (23)

This relation can be recognized as a version of the celebrated Kolmogorov–Obukhov scaling adapted to the current problem and generalized to a time–dependent situation. Note that Eq. (23) involves only the flow contribution to the flux, albeit this relation holds for any and not only in the ‘non–perturbative’ sector at . At this level, the relation (23) might look fortuitous, but in Sect. 4.4 we shall present a general argument showing that it has a deep physical motivation.

So far, we have implicitly assumed that the branching dynamics as described by Eq. (6) extends all the way down to , that is, it includes arbitrarily soft gluons. In reality, the dynamics should change at sufficiently low energies, for various reasons. First, when the gluons in the cascade become as soft as the medium constituents — that is, their energies become comparable to the temperature — they rapidly thermalize via collisions in the medium and thus ‘disappear’ from the cascade. Second, the BDMPSZ branching law (4) assumes the dominance of multiple soft scattering and hence it ceases to be valid when the branching time becomes as low as the mean free path between successive collisions in the medium. This condition restricts the gluon energies to values . For a weakly coupled quark–gluon plasma, the ‘Beithe–Heitler’ scale is comparable to the temperature . (Indeed, in this case, one has and to parametric accuracy.) With this example in mind, we shall not distinguish between these two scales anymore, but simply assume that the dynamics described by Eq. (6) applies for all the energies , i.e., for all . In all the interesting problems, the thermal scale is small enough to allow for multiple branchings: . For instance, in the case of a weakly coupled plasma, the above condition is tantamount to , which is indeed satisfied since the interesting values for are much larger than the typical relaxation time of the plasma.

Notice that we implicitly assume here that the thermalization mechanism acts as a ‘perfect sink’ at . (A similar assumption was made e.g. in the ‘bottom–up’ scenario for thermalization Baier:2000sb ().) That is, the surrounding medium absorbs the energy from the cascade at a rate equal to the relevant flux , without modifying the branching dynamics at higher values . This is a rather standard assumption in the context of turbulence and is well motivated for the problem at hand, as we argue now. To that aim, one should compare the relaxation time aforementioned, which represents the characteristic thermalization time at weak coupling, with the lifetime of a gluon generation (the time interval between two successive branchings) for gluons with energy , which is the characteristic time scale for the turbulent flow. This can be estimated as explained at the end of Sect. 3.2, and reads (to parametric accuracy)

 Δt(ω)∼1¯αtbr(ω)∼1¯α√ω^q. (24)

Using and the perturbative estimate , one deduces . We thus conclude that the physics of thermalization is as efficient in dissipating the energy as the turbulent flow. This implies that there should be no energy pile–up towards the low–energy end of the cascade.

Under these assumptions, it is interesting to compute the total energy lost by the cascade towards the medium, i.e. ‘the energy which thermalizes’. This is the same as the energy which has the crossed the bin during the overall time , namely (cf. Eq. (19))

 Eth≡E<(xth,τL)≃1−e−π¯α2τ2L+2¯ατL√xthe−π¯α2τ2L, (25)

where the approximate equality holds since . Eq. (25) is recognized as the sum of the flow energy, Eq. (15), and of the energy that would be contained in the spectrum at , cf. Eq. (13). Using and , it is easy to check that the flow component dominates over the spectrum piece, and hence . This implies that the energy lost by the gluon cascade towards the medium is independent of the details of the thermalization process, like the precise value of . This universality too is a well known feature of a turbulent process KST (); Nazarenko ().

## 4 The high–energy regime

With this section, we begin the study of the main physical problem of interest for us in this paper, namely the gluon cascade produced in the medium by a very energetic leading particle, with original energy . The main new ingredient as compared to the previous discussion is a kinematical restriction on the primary gluon emissions that can be triggered by interactions in the medium: the energy of the gluons emitted by the LP cannot exceed a value in order for the respective formation times to remain smaller than . When , this restriction has important consequences: it implies that the LP loses only a small fraction of its total energy, of order . Our main focus in what follows will not be on this average energy lost by the LP (this is well understood within the original BDMPSZ formalism, including multiple soft emissions of primary gluons Baier:2001yt ()), but rather on the further evolution of this radiation via multiple branchings and the associated flow of energy towards small values of and large angles.

### 4.1 The coupled rate equations

Since the radiation is restricted to relatively low energies , or , it is clear that the part of the spectrum at higher energies has to be associated with the LP. This makes it natural to decompose the overall spectrum as

In reality, the LP piece is a rather narrow peak located in the vicinity of (see below), so there is a large gap between the two components of the spectrum.

The evolution of the radiation via successive branchings involves no special constraint, so the respective rate equation can be obtained simply by replacing according to Eq. (26) in the r.h.s. of the general equation Eq. (6) (restricted to , of course). This yields

where the source is the energy per unit time and per unit radiated by the LP:

 S(x,τ)≡¯α∫dzK(z)√zxDLP(xz,τ). (28)

It is here implicitly understood that this source has support at and that it acts over a limited interval in time, at , which is moreover small, , in the high–energy regime of interest. The integral over in the gain term of Eq. (27) is restricted to , where the lower limit is introduced by the support of the function .

In the rate equation for the leading particle, one needs to enforce the condition that the radiated gluons have energy fractions smaller than . The ensuing equation reads (with )

 ∂DLP(x,τ)∂τ =¯α∫dzK(z){Θ(z−xx+xc)√zxDLP(xz,τ) −z√xDLP(x,τ)[Θ(z−1+xcx)+Θ(xcx−z)]} (29)

where the various –functions enforce the kinematical constraint: In the gain term, one requires that the unmeasured gluon emitted (with splitting fraction ) by the LP (with initial energy fraction ) be softer than : . In the loss term, one requires that one of the daughter gluons be soft: either , or .

As it should be clear from the previous discussion, the functions and at any time also depend upon , hence upon the overall size of the medium, via the kinematical constraints on the gluon emissions. This shows that the dynamics in this high energy regime is non–local in time ; e.g., the branching rate in Eq. (4.1) ‘knows’ about the maximal time via the various –functions, which involve . This property reflects a true non–locality of the underlying quantum dynamics: it takes some time to emit a gluon and this time cannot be larger than . Accordingly, at any , one should only initiate emissions whose energies are smaller than  : gluon fluctuations with higher energies would have no time to become on–shell. The kinematical constraint reflects only in a crude way the actual non–locality of the quantum emissions. The classical description at hand, as based on rate equations, is truly appropriate only for the sufficiently soft emissions with small formation times . Fortunately, these are the most important emissions for the physics problems that we shall here address.

In the zeroth order approximation, which is strictly valid as , one can use , and then the source in Eq. (28) reduces to the BDMPSZ spectrum, as expected:

 S0(x)≡¯αxK(x)≃¯α√x. (30)

In writing the second, approximate, equality we have used the fact that is small, , to simplify the expression of the splitting kernel (cf. Eq. (8)): for .

We shall now argue that the expression (30), which is time–independent, remains a good approximation for all the times of interest. Of course, the spectrum of the LP changes quite fast with increasing , notably due to the prompt radiation of very soft quanta with energy fractions . This leads to a broadening of the LP peak on the scale , similar to that exhibited by Eq. (9) at small times. Yet, the probability to emit a relatively hard gluon with is very small, of . Accordingly, the support of the function remains limited to a narrow band at , which is well separated from the radiation spectrum at . Hence, the integration over in Eq. (28) is effectively restricted to a narrow range close to , namely , and the integral can be approximated as

 S(x,τ)≃¯αxK(x)∫dx′DLP(x′,τ)≃¯α√x[1+O(x,¯αxc)]. (31)

Here we have used the fact that the overall strength of the function , i.e. the energy fraction carried by the LP after a time , can be estimated as

 ELP(τ)≡∫dxDLP(x,τ)≃1−2¯ατ√xc, (32)

that is, the initial energy minus the energy lost via radiation of soft gluons with , cf. Eq. (30).

To summarize, after an evolution time , the energetic LP loses only a small fraction of its total energy and its spectral density remains peaked near . Accordingly, it can be effectively treated as a steady source for the soft radiation at . This is verified in the plots in Fig. 4, where we perform two types of comparisons: (i) between the evolution with the exact kernel in Eq. (8) and that with the simplified kernel , and (ii) between the solution to the coupled system of equations (27) and (4.1) and that to the effective equation with a source, i.e. Eq. (27) with . As one can see in this plot, the two choices for the kernel lead indeed to results which are qualitatively similar and numerically very close to each other. Furthermore, the radiation spectrum at produced by the ‘model’ equation with a source is indeed close to the respective prediction of the coupled rate equations. (In fact, for the exact kernel , this similarity looks even more striking — the respective curves almost overlap with each other at sufficiently small — but in our opinion this is merely a coincidence.) In the next subsection, we shall construct an exact analytic solution for the equation with the source, for the case of the simplified kernel .

In the remaining part of this section, we shall concentrate on the solution to the following equation

which, as above argued, offers a good approximation for the dynamics of the medium–induced radiation by a leading particle with high energy . This is an inhomogeneous equation with vanishing initial condition and can be solved with the help of the respective Green’s function: