NNuclear Physics A 00 (2018) 1–6
NuclearPhysics A
Di–jet asymmetry and wave turbulence
Edmond Iancu
Institut de Physique Th´eorique de Saclay, F-91191 Gif-sur-Yvette, France
Abstract
We describe a new physical picture for the fragmentation of an energetic jet propagating through a dense QCD medium, whichemerges from perturbative QCD and has the potential to explain the di–jet asymmetry observed in Pb–Pb collisions at the LHC.The central ingredient in this picture is the phenomenon of wave turbulence, which provides a very e ffi cient mechanism for thetransport of energy towards the medium, via many soft particles which propagate at large angles with respect to the jet axis. Keywords:
Perturbative QCD, Heavy Ion Collisions, Jet Quenching, Wave Turbulence
1. Introduction
One of the most interesting discoveries of the heavy ion program at the LHC is the phenomenon known as di–jetasymmetry — a strong imbalance between the energies of two energetic back–to–back jets produced in an ultrarel-ativistic nucleus-nucleus collision. This is attributed to the e ff ect of the interactions of one of the two jets with thedense QCD matter that it traverses, while the other leaves the system una ff ected. Originally identified [1] as missingtransverse energy within a conventionally defined ‘jet’ with small angular opening (the same as for the trigger jet),this phenomenon has been subsequently shown, via more detailed studies [2, 3], to consist in the transport of a partof the jet energy towards large angles and by soft particles. The total amount of energy thus transferred from small tolarge angles (about 10 to 20 GeV) is considerably larger than the typical transverse momentum, ∼ ff ect is large and potentially non–perturbative.Yet, there exists a mechanism within perturbative QCD which naturally leads to energy loss at large angles:the BDMPSZ mechanism for medium–induced gluon radiation [4, 5]. Most previous studies within this approachhave focused on the energy lost by the leading particle, as controlled by relatively hard emissions at small angles.More recently, in the wake of the LHC data, the attention has been shifted towards softer emissions (which occur atlarge angles) and, more generally, towards a global understanding of the in–medium jet evolution. This raised thedi ffi culty of including the e ff ects of multiple gluon branchings, which become important for the soft emissions. Afterfirst studies of interference phenomena, which exhibited the role of medium rescattering in destroying the colourcoherence between partonic sources [6, 7], we have recently demonstrated [8] that the in–medium jet evolution canbe reformulated (to the perturbative accuracy of interest) as a classical stochastic process. This allows for systematicnumerical studies via Monte Carlo methods, like for jets fragmenting in the vacuum. It also allows for analyticstudies, at least for particular problems, like the recent study of the energy flow throughout the cascade in Ref. [9].This study revealed a remarkable phenomenon, which is new in the context of QCD and which, besides its conceptualinterest, has also the potential to explain the LHC data for di-jet asymmetry: the wave turbulence . The developmentsin Refs. [8, 9] will be briefly reviewed in what follows, with emphasis on the physical picture of wave turbulence. Email address: [email protected] (Edmond Iancu) a r X i v : . [ h e p - ph ] A p r . Iancu / Nuclear Physics A 00 (2018) 1–6
2. Medium–induced radiation `a la BDMPSZ
The BDMPSZ mechanism relates the radiative energy loss by an energetic parton propagating through a denseQCD medium (‘quark–gluon plasma’) to the transverse momentum broadening via scattering o ff the medium con-stituents. A central concept is the formation time τ br ( ω ) — the typical times it takes a gluon with energy ω (cid:28) E to beemitted. ( E is the energy of the original parton, a.k.a. the ‘leading particle’.) The gluon starts as a virtual fluctuationwhich moves away from its parent parton via quantum di ff usion: the transverse separation b ⊥ grows with time as b ⊥ ∼ ∆ t /ω . The gluon can be considered as ‘formed’ when it loses coherence w.r.t to its source, meaning that b ⊥ is at least as large as the gluon transverse wavelength λ ⊥ = / k ⊥ . But the gluon transverse momentum k ⊥ is itselfincreasing with time, via collisions which add random kicks ∆ k ⊥ at a rate given by the jet quenching parameter ˆ q : ∆ k ⊥ ∼ ˆ q ∆ t . The ‘formation’ condition, b ⊥ > ∼ / ∆ k ⊥ for ∆ t > ∼ τ br , implies τ br ( ω ) (cid:39) (cid:115) ω ˆ q , k br = ˆ q τ br ( ω ) (cid:39) (2 ω ˆ q ) / , θ br (cid:39) k br ω (cid:39) (cid:32) q ω (cid:33) / , (1)where k br and θ br are the typical values of the gluon transverse momentum and its emission angle at the time offormation. Eq. (1) applies as long as (cid:96) (cid:28) τ br ( ω ) < L , where L is the length of the medium and (cid:96) is the mean freepath between successive collisions. The second inequality implies an upper limit on the energy of a gluon that can beemitted via this mechanism, and hence a lower limit on the emission angle: ω < ∼ ω c ≡ ˆ qL / θ br > ∼ θ c ≡ / ( ˆ qL ) / .The BDMPSZ regime corresponds to ˆ qL (cid:29) θ c (cid:28)
1. Choosing ˆ q = / fm (the weak couplingestimate [4] for a QGP with temperature T =
250 MeV) and L = ω c (cid:39)
40 GeV and θ c (cid:39) . ω (cid:28) ω c ): this is the product of the standard bremsstrahlung spectrum for the emission of asingle gluon times the average number of emissions which can occur within the plasma, that is L /τ br : ω d N d ω (cid:39) α s N c π L τ br ( ω ) = ¯ α (cid:114) ω c ω , (2)with ¯ α ≡ α s N c /π . Note that the number of emissions L /τ br is much smaller than the number of collisions L /(cid:96) ,since several successive collisions can coherently contribute to a single emission; this is known as the LPM e ff ect(Landau, Pomeranchuk, Migdal) and leads to the characteristic ∼ / √ ω dependence of the BDMPSZ spectrum (2).By integrating this spectrum over all the energies ω ≤ ω c , one estimates to the total energy loss by the leading particle: ∆ E tot = (cid:90) ω c d ω ω d N d ω ∼ ¯ αω c ∼ ¯ α ˆ qL . (3)The above integral is dominated by its upper limit: the total energy loss is controlled by the hardest possible emissions,those with energies ω ∼ ω c . Such hard emissions, however, propagate at small angles θ ∼ θ c w.r.t. to the jet axis,so they remain a part of the conventionally defined ‘jet’ and thus cannot contribute to the di-jet asymmetry. On theother hand, the soft gluons with ω (cid:28) ω c are emitted directly at large angles θ (cid:29) θ c and, moreover, these angles arefurther enhanced after emission via rescattering in the medium: a gluon which crosses the medium over a distance ∼ L acquires a transverse momentum broadening k ⊥ ∼ ˆ qL ≡ Q s , which for ω (cid:28) ω c is in fact larger than therespective momentum acquired during formation: Q s (cid:29) k br ( ω ). Accordingly, a soft gluon emerges at a typical angle θ ( ω ) ∼ Q s /ω which is even larger than θ br ( ω ) — and of course much larger than θ c . It is interesting to try and estimatethe typical energy which would be transported in this way at angles larger than a given value θ , with θ (cid:29) θ c : ∆ E ( θ > θ ) = (cid:90) ω d ω ω d N d ω ∼ ¯ α √ ω c ω ∝ √ θ with ω ≡ Q s θ . (4)This is only a small fraction ( θ c /θ ) / of the total energy loss (3), but it is lost at large angles, so it counts for theenergy loss by the jet . Yet, Eq. (4) does not show the right trend to explain the LHC data: this estimate decreases quite The ‘transverse directions’ refer to the 2–dimensional plane orthogonal to the 3–momentum of the leading particle (the ‘longitudinal axis’). . Iancu / Nuclear Physics A 00 (2018) 1–6 c L0 Figure 1. A medium–induced cascade: the leading particle emits one hard gluon with ω ∼ ω c (at a small angle θ ∼ θ c ) together with a myriad ofsoft gluons with ω < ∼ α s ω c , which in turn generate gluon cascades (at relatively large angles) via successive, quasi–democratic, branchings. fast with increasing θ , thus predicting that most of the energy loss should lie just outside the jet cone (and thus beeasily recovered when gradually increasing the jet angular opening). This contradicts the results of a detailed analysisby CMS [2], which show that most of the ‘missing’ energy is deposited at very large angles θ > . ω ∼ ω c , this probability if of O ( ¯ α ), showing thathard emissions are relatively rare events. But when ω ∼ ¯ α ω c , this probability becomes of O (1), meaning that thesoft emissions with ω < ∼ ¯ α ω c can occur abundantly, event-by-event. For such small energies, the result (2) must becorrected to account for multiple emissions and, especially, multiple branchings of the soft emitted gluons [8, 9].
3. Democratic branchings and wave turbulence
Multiple soft emissions by the leading particle have already been discussed by BDMPS [10]: they change theenergy distribution of the leading hadron, but not the inclusive spectrum (2) for the soft radiation, nor its (unrealistic)prediction for the angular distribution of the radiation, Eq. (4). What is more important for the present purposes, isthe fate of the soft gluons after being emitted. The probability for a gluon with energy ω to split into two daughtergluons with energy fractions x and 1 − x is obtained by replacing ω → x (1 − x ) ω in Eq. (2). Hence, when ω ∼ ¯ α ω c ,this probability is of O (1) for generic values of x : the soft parent gluon is certain to split and its branching is quasi–democratic (i.e. unbiased towards the endpoints at x = x = x ) [9]. For even softerenergies, ω (cid:28) ¯ α ω c , the lifetime ∆ τ of a gluon generation, i.e. the time interval between two successive branchings,is considerably smaller than the medium size : ∆ τ ∼ (1 / ¯ α ) τ br ( ω ) (cid:28) L . Hence, such soft gluons undergo successivebranchings leading to gluon cascades. Being quasi–democratic, these branchings e ffi ciently degrade the energy tosmaller and smaller values of x . And since the gluons produced by these branchings are softer and softer, they geteasily deviated by the collisions in the medium to larger and larger angles (see Fig. 1). Thus, the quasi–democraticand quasi–deterministic cascade provides a very e ffi cient mechanism for transporting energy at large angles. Thismechanism is a manifestation of a phenomenon well known in other fields of physics : the wave turbulence [11, 12].Before we characterize this new phenomenon in mode detail, let us describe the formalism which allows us totreat multiple branching [8]. In principle, one can construct a parton cascade by iterating the 1 → classical branching process. It turns out to that this is also the right procedure for the quantum problem at hand, but in thiscase such a procedure is highly non-trivial, as it could be invalidated by interference phenomena . Recall e.g. theevolution of a jet via successive parton branching in the vacuum : the daughter partons produced by one splittingremain ‘color-coherent’ with each other (their total color charge is fixed to be equal to the respective charge of theparent parton) until the next splitting of any of them. This coherence implies interferece e ff ects between the emissionsby the two daughter partons, which in that context are well known to be important: they lead to the angular ordering of successive emissions, which ultimately favors jet collimation [13]. This estimate for ∆ τ follows from the condition that the emission probability P ( ω ) (cid:39) ¯ α ( ∆ τ/τ br ( ω )) become of O (1). . Iancu / Nuclear Physics A 00 (2018) 1–6 =E x’ = x/z x d x (1 (cid:239) z)xzx Figure 2. The change in the gluon spectrum D ( x , τ ) ≡ x (d N / d x ) due to one additional branching g → gg . Remarkably, the situation in that respect appears to be simpler for parton branching in the medium [6–8]: thedaughter partons e ffi ciently randomize their color charges via rescattering in the medium and thus lose their mutualcolor coherence already during the formation process [8]. Accordingly, the interference e ff ects are suppressed (ascompared to the independent branchings) by a phase–space factor τ br ( ω ) / L , which is small whenever ω (cid:28) ω c . Thisimplies that the successive medium–induced emissions can be e ff ectively treated as independent of each other andtaken into account via a probabilistic branching process , in which the BDMPSZ spectrum plays the role of a branchingrate. Such a process has already been used in applications to phenomenology, albeit on a heuristic basis [10, 14, 15].The general branching process is a Markovian process in 3 + ω ) and transverse momentum ( k ⊥ ), and its evolution when increasing the medium size L (see Ref. [8] fordetails). This process is well suited for numerical studies via Monte-Carlo simulations. But analytic results have beenobtained too [9], for a simplified process in 1 + gluon spectrum D ( x , τ ) ≡ x (d N / d x ), where x ≡ ω/ E is the energyfraction carried by a gluon from the jet and the ‘evolution time’ τ is the medium size in dimensionless units, as definedin the equation below. The quantity D ( x , τ ) obeys a ‘rate’ equation [9, 14, 15], which reads, schematically, ∂ D ( x , τ ) ∂τ = I [ D ]( x , τ ) ≡ Gain[ D ] − Loss[ D ] , with τ ≡ ¯ α (cid:114) ˆ qE L . (5)where the ‘collision term’ I [ D ] (a linear functional of D ( x , τ )) is the di ff erence between a ‘gain’ term and a ‘loss’term, as illustrated in Fig. 2. The ‘gain’ term describes the increase in the number of gluons with a given x viaradiation from gluons with a larger x (cid:48) = x / z , with any x < z <
1. The ‘loss’ term expresses the decrease in the numberof gluons at x via their decay x → zx , (1 − z ) x , with any 0 < z <
1. By construction, the first iteration of this equationcoincides with the BDMPSZ spectrum (2), which in our new notations reads (for relatively soft gluons with x (cid:28) D (1) ( x (cid:28) , τ ) (cid:39) τ √ x . (6)This approximation breaks down when D (1) ( x , τ ) ∼ O (1), meaning for x < ∼ τ (the familiar condition ω < ∼ ¯ α ω c inthese new notations). In this non–perturbative regime at small x , one needs an exact result which resums multiplebranchings to all orders. Such a solution has been presented in [9] and reads (for x (cid:28) D ( x (cid:28) , τ ) (cid:39) τ √ x e − πτ . (7)Formally, one can read Eq. (7) as ‘BDMPSZ spectrum by the leading particle × survival probability for the latter’.However, unlike Eq. (6), the spectrum (7) also includes the e ff ects of multiple branchings. That is, the energy in agiven bin with x (cid:28) x (cid:48) > x , through successive splittings. The persistence of the scaling spectrum D s ≡ / √ x under thisevolution demonstrates that this spectrum is a fixed point of the collision kernel: I [ D s ]( x ) = x (cid:28)
1. In turn, thismeans that the rate for energy transfer from one parton generation to the next one is independent of the generation(i.e. of x ). This property is the distinguished signature of wave turbulence [11, 12]: via successive splittings, theenergy flows from large x to small x without accumulating at any intermediate value of x . It rather accumulates intoa condensate at x =
0. Since there is only a finite amount of energy available (the energy E of the leading particle),4 . Iancu / Nuclear Physics A 00 (2018) 1–6 x √ x D ( x , ) τ Figure 3. Plot (in Log-Log scale) of √ xD ( x , τ ) as a function of x for various values of τ (full lines from bottom to top: τ = . , . , . , . , . τ = . , . it follows that the total energy which is contained in the spectrum (i.e. in the bins at 0 < x ≤
1) must decrease withtime. Indeed, a direct calculation yields [9] (cid:90) d x D ( x , τ ) = e − πτ = ⇒ E flow ( τ ) ≡ − (cid:90) d xD ( x , τ ) = − e − πτ . (8)The quantity E flow ( τ ) is the energy fraction carried away by the flow and which formally ends up in the condensate.As we shall shortly discuss, this energy is in fact transferred to the medium, at very large angles.These considerations are illustrated in Fig. 3 which shows the spectrum for various values of τ . At small τ (cid:28) / √ π , the small– x part of the spectrum rises linearly with τ , as shown by Eq. (6) (see the full lines in Fig. 3). At thesame time, the leading–particle peak, which originally was a δ –function at x =
1, moves at 1 − x (cid:39) πτ and becomesbroader. For larger times τ > ∼ / √ π , the source disappears and the spectrum is globally suppressed by the Gaussianfactor in (7); yet, the scaling behavior D ∝ / √ x is still visible at small x (see the dotted lines in Fig. 3).
4. Energy loss at large angles
The emergence of a flow component E flow ( τ ) in the energy transport down the cascade explains one of the maincharacteristics of (wave) turbulence: this is a very e ffi cient mechanism for transferring energy between two widelyseparated scales — here, from x = x =
0. To see this, let us compute the energy transferred after time τ below a given value x (cid:28)
1. This includes two components: the energy which is contained in the spectrum, in thebins at 0 < x < x , and the flow energy, which is independent of x (since accumulated at x = E ( x ≤ x , τ ) = τ √ x e − πτ + (1 − e − πτ ) (cid:39) τ √ x + πτ , (9)where the second, approximate, equality holds for πτ (cid:28)
1. Note that, even for small times, the flow component dom-inates over the non–flow one provided x < τ , that is, in the non–perturbative regime at small x where the multiplebranching becomes important. For larger times τ > ∼ / √ π , the flow piece dominates for any x and approaches unity,meaning that the whole energy can be lost towards arbitrarily soft quanta, which propagate at arbitrarily large angles.To understand how remarkable this situation is, let us compare it with the more familiar example of the DGLAPevolution (say, for a jet in the vacuum), where there is no flow . (The DGLAP equation too can be viewed as a ‘rateequation’, cf. Eq. (5), with the logarithm of the virtuality playing the role of the ‘evolution time’.) In that case, thesplittings are typically asymmetric ( x → x → x .Yet most of the energy remains in the few partons with relatively large values of x . Indeed, for the DGLAP cascade,the energy is fully contained within the spectrum (no flow) and the energy sum-rule (cid:82) d xD ( x , τ ) = x in the support of the function D ( x , τ ) at time τ . Conversely, one can show that a necessarycondition for the emergence of (turbulent) flow is quasi–democratic branching [11].5 . Iancu / Nuclear Physics A 00 (2018) 1–6 So far, we have assumed that the evolution remains unchanged down to x =
0, but physically this is not the case:when the gluon energies become as low as the typical energy scale in the medium — the ‘temperature’ T ∼ x → x th ≡ T / E in Eq. (9). This energy loss is independent of the detailsof the thermalization mechanism and even of the medium temperature (since dominated by the flow component, aswe shall shortly see). This universality is the hallmark of turbulence: the rate for energy transfer at the lower end ofthe cascade is fixed by the turbulent flow alone, and thus is independent of the specific mechanism for dissipation.To make contact with the phenomenology, we notice that for a jet with E =
100 GeV ≈ ω c , Eq. (5) implies τ ≡ ¯ α √ ω c / E (cid:39) ¯ α (cid:39) .
3, which is quite small. The flow piece in Eq. (9), which is independent of x , dominatesover the non–flow piece for any x < τ (cid:39) .
1, a value much larger than the thermalization scale x th (cid:39) .
01. Thus, inevaluating the energy loss via thermalization, one can keep only the flow component in the small– τ version of Eq. (9),as anticipated. Returning to physical units, one finds ∆ E th (cid:39) E E flow (cid:39) υ ¯ α ω c , (10)where υ would be equal to 2 π according to Eq. (9), but a more precise calculation yields υ (cid:39) .
96 [9]. This is formallysuppressed by an additional power of ¯ α as compared to the total energy loss by the leading particle, Eq. (3), whichwe recall is controlled by hard gluon emissions ( ω ∼ ω c ) at small angles ( θ ∼ θ c ). However, the flow contribution inEq. (10) is numerically quite large (because υ is a reasonably large number) and moreover this is associated with softemissions at large angles. It thus has the potential to explain the LHC data for di–jet asymmetry.With ω c =
40 GeV, Eq. (10) predicts ∆ E th (cid:39)
20 GeV, a value that compares well with the experimental observa-tions. This energy is carried by the relatively soft quanta at the lower end of the cascade ( x ∼ x th ), that is, by particleswhose energies are comparable to the ‘temperature’ T of the medium. Precisely because they are so soft, these par-ticles propagate at very large angles with respect to the jet axis. To obtain a parametric estimate for these angles,we recall that a gluon with energy ω < ∼ ¯ α ω c has a lifetime ∆ τ ∼ (1 / ¯ α ) τ br ( ω ) < ∼ L , during which it accumulates atransverse momentum broadening k ⊥ ∼ ˆ q ∆ τ = (1 / ¯ α ) k br , via collisions in the medium. Accordingly, this gluon shouldemerge at an angle (compare to Eq. (1)) θ ( ω ) ∼ √ ¯ α θ br ( ω ) ∼ √ ¯ α (cid:32) q ω (cid:33) / . (11)A lower limit on this angle is obtained by choosing ω ∼ ¯ α ω c ∼ θ (cid:39) .
5. But for a typical gluon with ω ∼ T ∼ ÷ θ ∼ O (1). This is in qualitative and even quantitative agreement with the detailed analyses of the databy CMS [2] and ATLAS [3], which show that most of the ‘missing’ energy lies at very large angles θ > ∼ . Acknowledgements
This research is supported by the European Research Council under the Advanced Investi-gator Grant ERC-AD-267258.
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