Higher-order corrections to heavy-quark jet quenching
HHigher-order corrections to heavy-quark jet quenching
Boris Blok ∗ Department of Physics, Technion - Israel Institute of Technology, Haifa, Israel
Konrad Tywoniuk † Department of Physics and Technology,University of Bergen, 5020 Bergen, Norway (Dated: January 24, 2019)
Abstract
We calculate higher-order corrections to the quenching factor of heavy-quark jets due to hard,in-medium splittings in the framework of the BDMPS-Z formalism. These corrections turn outto be sensitive to a single mass-scale m ∗ = (ˆ qL ) / , where ˆ q is the medium transport coefficientand L the path length, and allow to draw a distinction between the way light, with m < m ∗ (in contrast to massless m = 0), and genuinely heavy, with m > m ∗ , quark jets are quenchedin the medium. We show that the corrections to the quenching factor at high energies aredouble-logarithmic and qualitatively of the same order as for the massless quark jet. ∗ [email protected] † [email protected] a r X i v : . [ h e p - ph ] J a n . INTRODUCTION Jets are formed in the process of soft and collinear QCD radiation that results ina spray of collimated hadrons and energy deposition in the detector [1]. In heavy-ioncollisions, partons traverse a hot and dense nuclear medium that leaves an imprint on thesubsequent jet formation, for reviews see [2–4]. Currently it is widely accepted that theBDMPS-Z formalism of radiative energy loss [5–7] describes the propagation and multiplescattering of quark and gluon jets in the nuclear QCD medium that is produced in heavyion collisions at LHC, for a review see Ref. [8].Jet quenching is a multi-scale problem. Even for massless partons, there is an convo-luted interplay between the intrinsic jet scales, such as the mass of the jet, and the scalesof the medium, including typically the medium transport coefficient and the medium size.For quarks, the non-zero mass introduces another scale. It is well known that the collineardivergence is regulated by the characteristic dead-cone angleΘ = mE . (1)where m is the mass of the heavy quark and E ≡ p T is the jet energy. As a consequenceof the strong suppression of gluon radiation inside the dead-cone angle, heavy-quark jetsfragment differently from jets originating from their massless counterparts or from gluons[1, 9–11].Radiative energy loss was calculated by BDMPS-Z for massless partons [5–7]. It wasfirst pointed out in Ref. [12] that the quenching of massive quarks would be differentfrom massless ones because of the dead-cone effect. The resulting restriction of the phasespace for radiation, and hence energy loss, leads to a systematically smaller suppressionof single-inclusive hadron spectra the larger the mass of the constituent quarks. This wasfollowed up by a more thorough analysis in [13, 14], where it was shown that the heavyquark quenching factors get further corrections when the correct phase space constraintsare taken into account. For results within the limit of dilute media, see also [15, 16].In summary, based on radiative processes alone one expects a smaller rate of emissionsoff massive quarks compared to massless ones, that brings about a mass-hierarchy of thesuppression. In contrast, low- p T heavy mesons have a similar modification as the pions[17–19]. This has prompted many investigations of additional elastic energy loss processes,for a review see [20]. It is however worth keeping in mind that the final suppression ofheavy mesons and heavy flavor jets depends also on the details of the partonic cross2ections and the problem is still an open one.While most of the previous contributions has focussed on small p T , where the crosssections are the largest, we will mainly focus on the genuinely high- p T regime whereperturbative corrections play a crucial role. This regime is within the reach of the exper-iments at LHC, see e.g. [19, 21] . Recently higher-order corrections to the quenching ofindependent, massless quark/gluon jets were calculated [22, 23]. The results demonstratehow these contribution lead to the enhanced quenching of massless quark/gluon jets ascompared to single partons. The role of in-medium jet splittings and their color coherenceproperties has also been emphasized in other contexts, see e.g. [24, 25]. Consequently, itwill be of interest to extend the previous efforts to include mass effects. In this work, weconsider higher-order corrections to the quenching of a heavy-quark jet, i.e. a jet formedas a result of the fragmentation of a leading massive quark.Our main result is that higher-order corrections lead to an enhanced suppression forheavy-quark jets relative to the leading BDMPS-Z result, corresponding to the quenchingof a single parton. The magnitude is determined by the phase space available for theradiation of hard gluons within the jet, and is of similar magnitude as for quark/gluonjets in general. However, due to the restricted phase space determined by the dead-cone(1), the mass sets the scale where significant deviations between massive and masslessjets can be observed. We identify a critical mass scale that permits to observe suchdiscriminating features in the high- p T regime.The paper is organized in the following way. In Sec. II, we introduce the generalizedquenching weight and discuss its expansion in terms of the strong-coupling constant.We calculate the radiative energy loss due to multiple, soft BDMPS-Z radiation off asingle heavy quark and a heavy quark-gluon dipole in Sec. III, and obtain the evolutionequations and expressions for related quenching factors. The details of the calculations ofthe associated spectra and rates are given in Appendices A and B, respectively, where thebasic formulae for the interference contributions to antenna radiation are derived in detail.In Sec. IV, we finally map out the logarithmic phase space for higher-order correctionsand present explicit expressions for the collimator function of heavy-quark jet togetherwith numerical results. We summarize our results and give a brief outlook in Sec. V.3 I. GENERALIZED QUENCHING WEIGHT
Assuming small energy losses in the medium, (cid:15) (cid:28) p T , and accounting for a steeplyfalling hard spectrum, the spectrum of heavy-quark jets in heavy-ion collisions can bewritten as d σ d p T = (cid:90) ∞ d (cid:15) P ( (cid:15), L | m ) d σ d q T (cid:12)(cid:12)(cid:12)(cid:12) q T = p T + (cid:15) (cid:39) d σ d p T Q ( p T ) . (2)where d σ / d p T is the Born-level jet production cross section, P ( (cid:15), L | m ) is an energy-lossprobability distribution associated with a massive particle and L is the medium length(below we shall suppress the arguments L and m , unless it is unclear from the context).The jet suppression factor Q ( p T ), introduced in the second step, is the Laplace transformof the energy loss distribution P ( (cid:15) ), i.e. Q ( p T ) ≡ ˜ P ( n/p T ) = (cid:82) ∞ d (cid:15) e − n(cid:15)/p T P ( (cid:15) ) [26], wherethe effective power of the steeply falling spectrum is n = dd ln p T ln d σ d p T [12, 27]. The jetsuppression factor permits an expansion in the strong-coupling constant that accounts forthe energy loss of in-medium jet splittings, Q ( p T ) = Q (0) ( p T ) + Q (1) ( p T ) + O ( α s ) . (3)The first term in the expansion is the quenching of the jet total charge which, for a heavy-quark initiator, is given by Q (0) ( p T ) = ˜ P q ( n/p T ). This distribution is dominated by softgluon radiation that transfers energy from the jet axis to large angles.The resummation of higher-order terms leads to an additional suppression factor whichwas referred to as the “collimator” function in Ref. [23]. These corrections correspond tothe energy-loss of composite, partonic systems created inside the medium during the jetformation. Hard splittings in the jet cone can be described by vacuum splitting functions.Hence the next-to-leading correction to the jet quenching factor, that involves the (realand virtual) emission and subsequent quenching of an additional gluon [23], takes theform Q (1) ( p T ) = (cid:90) R θ d θ ( θ + Θ ) (cid:90) d z α s π P gq ( z )Θ( t f (cid:28) L ) (cid:2) Q gq ( θ, p T | m ) − Q q ( p T | m ) (cid:3) , (4)where P gq ( z ) is the Altarelli-Parisi splitting function and Q gq ( θ, p T ) is the quenching factorof a composite quark-gluon system propagating in the medium [22]. This equation holdswhenever the splitting takes place early in the medium. This enforces the formation time, t f = 2 / [ z (1 − z ) Eθ + z − z E Θ ], to be short compared to that of any process in the medium,in particular the medium length L . As will be discussed in more detail later, an important4ime-scale is the so called decoherence time t d which corresponds to the time when a dipoleof size x ⊥ ∼ θt , characterized by its opening angle θ , is resolved by medium fluctuations.The characteristic wave-length of the latter can be estimated via diffusive broadening as λ ⊥ ∼ (ˆ qt ) − / . The two length-scales become of the same order at t d ∼ (ˆ qθ ) − / . Inthis limit, t f (cid:28) t d (cid:28) L , the splitting process completely factorizes out on the level of thecross section [28] and effectively forms a color-charged antenna. This composite systemundergoes further medium-induced radiation in the medium that turns out to be sensitiveto its opening angle [22]. However, there can also be strong cancellations between the twoquenching factors in the squared brackets in (4) for small-angle emissions, when t d > L or θ < θ c , where θ c ∼ (ˆ qL ) − / , due to interference effects.Higher-order corrections naturally follow a similar logic, becoming sensitive to morecomplicated radiation patterns. In the large- N c limit the picture is simplified further,since a jet in this case can be decomposed into a set of mutually independent color-singletdipoles, whose radiation is added to that of the total charge radiation [29]. The quenchingof the total charge can therefore be factorized out, and the total quenching factor becomes, Q ( p T | m ) = Q q ( p T | m ) C ( p T , R | m ) , (5)where C ( p T , R | m ) is the collimator function that accounts for the quenching of higher-order(real and virtual) jet emissions. The resummation of such emissions takes, in the generalcase, the form of a non-linear evolution equation for the collimator function but, in thelimit of strong quenching, one can neglect all real emissions and resum the virtual terms,i.e. the second term in the squared brackets of Eq. (4). We will discuss the collimator inmore detail in Sec. IV. In the remaining part of the paper, we will describe the radiativequenching of a heavy-quark system and identify the relevant time-scales that play a rolein this problem in order to compute and resum these corrections. We focus on the high- p T regime and (relatively) large quark masses, where elastic energy losses, see e.g. [20], canbe neglected. Our results at low- p T are therefore not completely realistic. However, weemphasize that for genuinely heavy quarks the high- p T regime (meaning p T (cid:38) − III. COMPUTING THE QUENCHING FACTORS
In this section we compute the quenching weights, that is energy loss probability dis-tributions that resum multiple soft, gluon radiation responsible for transporting energy5rom the leading particle to large angles. As mentioned in the Introduction, it will beconvenient to work directly in Laplace space, defined as P ( (cid:15), L | m ) = (cid:90) C d ν πi ˜ P ( ν, L | m )e ν(cid:15) , (6)where the contour C runs parallel to the imaginary axis in the complex- ν plane, Re ν =const., to the right of any singularity of ˜ P ( ν, L | m ). To recap, P ( (cid:15), L | m ) acts as a prob-ability distribution for radiating gluons that in total carry an energy (cid:15) off a particlewith mass m after propagating through a medium of length L , and the quenching factor Q ( p T ) = ˜ P ( n/p T , L | m ). It will be convenient to define a “regularized” splitting rate, γ ij ( ν, t ) = (cid:90) ∞ d ω (cid:0) e − νω − (cid:1) Γ ij ( ω, t ) , (7)where γ i ( ν, t ) ≡ γ ii ( ν, t ) and we have already anticipated the possibility of interferencecontributions between two different particles labeled “ i ” and “ j ” that refer to quarks,antiquarks or gluons. Here Γ ij ( ω, t ) is the rate of (interference) emissions in the medium,where the soft gluon is emitted by a parton i and absorbed by the parton j in the complexconjugate amplitude. We derive the generic interference spectrum off a color-chargedantenna in Appendix A, and derive concrete expressions for the direct and interferencerates in Appendix B within the multiple-soft scattering approximation. A. Quenching of a single parton
Let us start by considering a single propagating particle. Medium interactions canenhance the probability of gluon emissions. In Laplace space, the resummation of soft,medium-induced gluons takes the form of a rate equation, ∂ t ˜ P i ( ν, t ) = γ i ( ν, t ) ˜ P i ( ν, t ) , (8)with initial condition ˜ P ( ν,
0) = 1, whose solution is simply given by ˜ P i ( ν, L ) = e (cid:82) L d t γ i ( ν,t ) .For a massive quark, the emission rate of soft gluons was derived in Eq. (B11), and readsΓ q ( ω, t | m ) = ¯ α Θ (cid:20) − Im ψ (cid:18) − i ζ / (cid:19) − ζ − / − π (cid:21) , (9)where ψ ( x ) is the digamma function, ¯ α ≡ α s C F /π and the expression in the squaredbrackets is a function of the scaling variable ζ ≡ ω/ω DC , where ω DC = (cid:0) ˆ q (cid:14) Θ (cid:1) / = (cid:0) ˆ q p T /m (cid:1) / . (10)6he expression (9) is valid only for ζ <
1. It turns out the spectrum is strongly suppressedat ω > ω DC , where we also observe negative contributions owing to the treatment of thehigh-energy behavior which goes beyond the leading-logarithmic accuracy of our calcu-lation [30]. In order to avoid these unphysical contributions, and retain the informationabout the physical scales, we approximate the rate byΓ q ( ω, t | m ) ≈ ¯ α (cid:112) ˆ q/ω Θ( ω DC − ω ) . (11)In Laplace space, this becomes γ q ( ν, t | m ) = 2 ¯ α (cid:114) ˆ qω DC (cid:2) − e − νω DC − √ πνω DC erf ( √ νω DC ) (cid:3) , (12) ≈ − α (cid:112) ˆ q (cid:0) √ πν − √ ω DC (cid:1) for ν − (cid:46) ω DC , (13)where erf( x ) is the error function. This is qualitatively similar to what is obtained inRef. [12], although the precise form of the cut-off at ω DC determines a numerical constantin front of the second term in the brackets (according to the authors of Ref. [12], thisfactor is ∼ . m → C F → N c , e.g.Γ q ( ω, t | m = 0) = ¯ α (cid:112) ˆ q/ω and γ g ( ν, t | m = 0) = − α (cid:112) π ˆ qν , (14)for massless quarks. Neglecting corrections O (1 /N c ), which shortly will be further moti-vated, we find that the gluon rates by Γ g = 2Γ q and γ g = 2 γ q .While these rates are time-independent in the limit of soft gluon emissions, this isviolated at large energies. This approximation breaks down for emissions with formationtimes of the order of the medium length, corresponding to a critical energy ω c ∼ ˆ qL that brings about a power-like cut-off of the spectrum and, therefore, the rate as well.The constraint from the dead-cone angle is stronger than this absolute limit whenever ω DC < ω c which, in turn, implies that Θ > θ c . This marks the regime where the massof the quark should start affecting the general properties of radiative energy loss that isdominated by LPM interference effects. B. Quenching of a two-parton system (color-charged dipole)
Let us now turn to the higher-order corrections to this picture, that arise from a quark-gluon antenna propagating in dense QCD media. Considering for the moment the energyloss of a quark-gluon dipole that is formed quasi-instantaneously after the hard vertex,7n Laplace space the joint energy loss distribution factorizes in the large- N c limit into theproduct of energy loss off a total charge (triplet) and a color-singlet dipole,˜ P gq ( ν, t | m ) = ˜ P q ( ν, t ) ˜ P q q ( ν, t | m ) , (15)where ˜ P q ( ν, t ) ≡ ˜ P q ( ν, t | m = 0) and we have decomposed the gluon into a quark-antiquark pair g = ( q , q ) (where q is an antiquark). Recall that only the quark thatforms part of the dipole q = q is massive. Note that the quenching of the total (quark)charge is not sensitive to the mass of the initial particle, since it is inherited from the radi-ated gluon. Instead, the mass controls the energy loss of the additional irreducible singlet˜ P q q ( ν, t | m ) ≡ ˜ P sing ( ν, t | m ). We will confirm below that, in the completely decoherentlimit, the mass will be associated with the total color charge, as expected.The two factors in (15) satisfy two separate evolution equations. First, the single-particle quenching is given by Eq. (8), where the splitting rate γ q is explicitly given by(14), keeping in mind that this fictitious quark is massless. The singlet, dipole quenchingweight is determined by solving the differential equation [22] ∂ t ˜ P sing ( ν, t | m ) = γ dir ( ν, t ) ˜ P ( ν, t ) ˜ P ( ν, t | m ) + γ int ( ν, t ) S ( t ) . (16)The initial condition at t = 0 (corresponding to the time when the antenna was formedin the medium) is again trivial, ˜ P sing ( ν, | m ) = 1. The direct and interference rates werederived in Appendix B, and are given by γ dir ( ν, t ) = γ ( ν, t ) + γ ( ν, t ) , (17) γ int ( ν, t ) = γ ( ν, t ) + γ ( ν, t ) . (18)The two direct terms correspond to emissions off the two legs, and similarly the inter-ference terms correspond to emitting a gluon from one leg and “absorbing” it (in thecomplex conjugate amplitude) on the other (see Fig. 4 for details). For the singlet dipolewe have γ dir ( ν, t ) = 2 γ q ( ν, t ), since γ ( ν, t ) = γ ( ν, t ) ≡ γ q ( ν, t ).Note that Eq. (16) contains a inhomogeneous term arising from the possibility of in-terferences between the dipole consituents. The interference spectra have a more complexstructure since they involve both color and quantum decoherence processes [31–33]. Theinterference spectrum associated with a massive dipole is given explicitly in Eq. (A9), seealso [34], and evaluated in the multiple-soft scattering approximation in Eqs. (B20) and(B21). As discussed further in Appendix B, color decoherence is related to the survival8robability of a color-singlet dipole and is explicitly factorized out in the so-called deco-herence parameter S ( t ) in (16). This factor is responsible for the previously introducedtime scale t d ∼ (ˆ qθ ) − / . At late times, t > t d , the dipole decoheres and the particlescan radiate independently. Conversely, a coherent splitting corresponds to the situationwhen t d > L and the pair remains coherent during the passage through the medium. Thisapplies to the regime of small angles, θ < θ c where θ c ∼ (ˆ qL ) − / .In our formulation, the medium-induced interference rates (in energy-space) are them-selves suppressed at a time-scale t quant ∼ ( θ ω ) − . This scale can nevertheless be neglectedby noting that t quant ∼ ( θ br ( ω ) /θ ) / t d , where θ br ( ω ) = (ˆ q/ω ) / is the branching angle ofmedium-induced gluons. Since energy loss is governed by soft gluons, ω ∼ ¯ α ω c , that para-metrically go to large angles, in particular out of the jet cone θ br ( ω ) ∼ ¯ α − / θ c > R > θ ,this implies that t d < t quant [22, 32]. Since the multiplicity of hard gluons is small, N ( ω > ¯ α ω c ) ∼ O ( α s ) and correspondingly their contribution to energy-loss is small [27]we will neglect such quantum effects in the following. Hence, for our purposes, i.e. attimes t < t d < t quant , the interference rate is approximated as γ int ( ν, t ) ≈ − γ dir ( ν, t ), seeEqs. (B22) and (B23). This property is independent of the mass.The solution to the rate equation can be written symbolically as˜ P sing ( ν, L | m ) = ˜ P ( ν, L ) ˜ P ( ν, L | m )+ (cid:90) L d t ˜ P ( ν, L − t ) ˜ P ( ν, L − t | m ) γ int ( ν, t ) S ( t ) . (19)The extension of the time-integral of the second term is limited by the shortest time-scale where interferences are suppressed. In the leading-logarithmic approximation it issufficient to consider only large-angle radiation where, parametrically, the energy radiatedvia medium-induced gluons leave the jet cone. In this case the integral is limited by t < t d ,as discussed above, and the singlet distribution can then be approximated by˜ P sing ( ν, L | m ) ≈ ˜ P q ( ν, L − t d ) ˜ P q ( ν, L − t d | m ) , (20)where we have reinstated the mass dependence. Hence, the decoherence time acts asa “delay” for when energy loss processes start affecting the irreducible dipole and, inthe limit t d (cid:28) L , the dipole constituents decohere early in the medium and lose energyindependently along the whole medium length.To summarize, the delay effect is strictly associated with the color dynamics of thedipole and, since this involves the shortest relevant time-scale, does not depend on themass of the constituents. It might, at first look, seem strange that the mass-effect on9uenching is delayed although it is intimately linked with the quark-initiator and, hence,the total charge. For instance, considering long decoherence times, t d ≥ L , applying tosmall-angle emissions θ ≤ θ c , the color-charged antenna is quenched as a massless quark,rather than a massive one. This effect gives rise to a mismatch between real and virtualemissions at small angles. This turns nevertheless out to be a sub-leading effect, seeEq. (29). IV. HEAVY-QUARK COLLIMATOR FUNCTIONA. Quenching of total charge
The first term in the expansion in (3) corresponds to the quenching of a single, massivequark. After implementing the result in (13), we find that Q (0) ( p T | m ) ≡ Q q ( p T | m ) = exp (cid:34) − αL (cid:18) π ˆ qnp T (cid:19) / (cid:35) exp (cid:34) αL (cid:18) ˆ qm p T (cid:19) / (cid:35) , (21)where the first term corresponds to the quenching of a massless color parton, while thesecond is a mass-dependent enhancement factor. The mass-independent term implies thatthe regime of strong quenching of massless quarks, − ln Q (0) ∼ O (1), is given by p T < nω s , (22)up to numerical factors, where ω s ∼ ¯ α ˆ qL is a soft scale corresponding to large multi-plicity of medium-induced emissions, N ( ω < ω s ) > O (1). We can rewrite the quenchingfactor as Q (0) ( p T | m = 0) ≈ exp (cid:2) − √ πN ( p T /n ) (cid:3) [27], which is interpreted as a Sudakovsuppression factor for medium-induced gluons with energies ω > p T /n .The regime with an additional strong enhancement of heavy compared to masslessquarks arises for p T < mθ − s ∼ ¯ α / m (ˆ qL ) / , (23)where θ s ∼ θ br ( ω s ) ∼ ¯ α − / θ c . This condition is equivalent to demanding that θ s < Θ ,which implies that the regime of multiple, soft gluon emissions is cut off by the dead-cone angle. So there exists a regime of strong massless-quark quenching with additionalstrong effects from heavy-quark quenching whenever for masses m < n ¯ α / Q s where Q s ∼ (ˆ qL ) / . To limit the scope of our qualitative analysis, we will assume that theindex of the steeply falling spectrum n , combined with the medium parameters ˆ q and L ,is large enough to work in the regime of strong quenching effects.10 . Scale analysis Let us now turn to the next-to-leading correction (4), coming from the emission ofan additional hard gluon off the initiating heavy quark early in the medium. We willcurrently focus on the leading-logarithmic contributions, leaving an analysis of sub-leadinglogarithmic contributions for the future. In this context we only consider strong orderingof scales and we will therefore not typically keep track of numerical factors that are anywaybeyond the precision of this analysis. We will also work in the large- N c limit, where wecan exploit the factorization of the color-charged dipoles, as in Eq. (15).
1. Massless quarks
Before turning to effects related to the mass of the jet particles, let us summarize thescale analysis for massless partons. In terms of angles, we have several characteristicscales: the jet radius R , the minimal medium resolution angle θ c ≡ θ br ( ω c ) and thetypical angle for soft gluon emissions θ s ≡ θ br ( ω s ). Note that the two medium scales areparametrically separated by the smallness of the coupling constant, θ s ∼ ¯ α − / θ c .If R > θ s the energy loss of jet is small since all radiated BDMPS gluons remain insidethe jet. On the other hand, if θ c > R , i.e. the jet angle is less than decoherence angle,the propagation of the jet is not influenced by subjet structure, and it is equivalent topropagation of a total color charge, i.e. one parton through the medium. The typicalordering that interests us, where the quenching could be substantial and where higher-order effects are non-trivial, is therefore θ s > R > θ c . (24)In what remains, we will assume that this hierarchy holds and, besides, that it is also thephenomenologically most relevant one.However, note that the minimal angle θ c only is relevant for high- p T jets, p T > ω c .Conversely, for p T < ω c the decoherence time is necessarily always shorter than themedium length, t d < L . In this case there is still the possibility for a regime of shortformation times, t f < t d , but in this case this condition implies that θ > θ d , where θ d ∼ (cid:18) ˆ qp T (cid:19) / . (25)For massless quarks, this regime is double-logarithmic in the jet scale [23], see below, butthe p T -range is automatically limited by ω c . The window for a regime of short formation11 uark mass Distinctive angle Critical jet p T Critical parton p T m < m ∗ (ˆ q/p T ) / m / ˆ q ¯ α / m (ˆ qL ) / m > m ∗ (ˆ qL ) − / m (ˆ qL ) / ¯ α / m (ˆ qL ) / TABLE I: Summary of the scales for light and heavy quarks.times closes whenever θ d = R , or p T ∼ (ˆ q/R ) / . In the following, we will thereforedistinguish between high- p T (with p T > ω c ) and low- p T (with p T < ω c ) jets.
2. Massive quarks
For massive quarks, the dead-cone angle (energy, etc.) introduces another physical scaleto the problem. For a finite dead-cone, QCD radiation is no longer genuinely collinearlyenhanced which necessitates a scale-dependent scheme to properly include mass-effectsfor resummed observables, see e.g. [1]. We will only stick to the leading-logarithmicapproximation and only consider emissions θ > Θ , i.e. θ / ( θ + Θ ) ≈ θ − Θ( θ − Θ ) in(4). Hence, for R ∼ Θ the heavy-quark jet only contains a single quark.Comparing the mass scale to other relevant medium scales, in particular comparingΘ and θ c , can become involved because of the p T -dependence of the former. In order toorganize the discussion, it will be useful to introduce a critical value of the mass, namely m ∗ ≡ (ˆ qL ) / . (26)From now on we will call quarks with m > m ∗ genuinely heavy , and quarks with m < m ∗ for light (in contrast to massless, m = 0). We have also summarized the discussion aboutthe relevant scales in Table I.For heavy quarks, the dead-cone angle becomes comparable to the coherence angle,Θ = θ c , at large- p T , i.e. p T > ω c . Rewriting the same condition, this happens at acritical energy p T = mθ − c ∼ m (ˆ qL ) / . Hence, we expect the heavy-quark jet quenchingto deviate from the light-quark jet quenching at a scale that is parametrically larger, bya factor ¯ α − / , than the soft scale identified for the quenching of the total charge, cf.Eq. (23). In other words, while the quenching of a single heavy quark starts deviatingfrom the massless one at relatively low p T due to the enhancement factor in (21), a jetinitiated by a heavy-quark should start deviating from the behavior of a massless quark orgluon jet already in the high- p T regime, since by definition m (ˆ qL ) / > ω c . Considering12igh- p T single-inclusive mesons to be proxies of single-parton dynamics, see e.g. [35],this analysis therefore predicts a different behavior of heavy-quark jets and heavy-quarkmesons over a large range in p T . We will come back to a possible experimental signaturefor this effect in Sec. IV D.For light quarks, Θ (cid:54) = θ c for p T ≥ ω c and we have instead to consider the low- p T regime, i.e. p T < ω c . In this regime, the condition t f < t d implies that θ > θ d whichis estimated in (25). Therefore Θ = θ d when p T ∼ m / ˆ q , which can be considered arelatively soft energy-scale (comparing it to the soft scale in (23) gives m ∼ ¯ α / (ˆ qL ) / which is compatible with the prior assumption about the smallness of the mass). Weconclude therefore that the light-quark jets behave similarly to massless jets, as far as thehigher-order corrections go, and start deviating from this behavior only when Θ ∼ θ s ,where the quenching of the total charge gets suppressed. This follows very closely thetrend of single-parton, or single-inclusive meson, quenching.The corresponding kinematical Lund planes for high- and low- p T heavy-quark jets areillustrated in Fig. 1, where we have spanned the plane in the logarithmic variables 1 /z and 1 /θ . At fixed coupling, the plane is equally filled with splittings with probability2 ¯ α , up to a color factor. The two diagonal lines, with slopes − / −
2, delineate theconditions t f = t d and t f = L , respectively. The area between the two lines correspondsto in-medium radiation with t d (cid:46) t f < L , or k ⊥ (cid:46) √ ˆ qω , which is strongly influenced bymedium interactions and broadening.The high- p T regime, p T > ω c , is plotted on the left side of Fig. 1, where we have markedthe location of the critical angle θ c with a (red) dotted line. Similarly, the dead-cone angleis marked, and corresponds to an energy scale ω DC at t f = t d . The low- p T regime, p T < ω c is conversely plotted on the left in Fig. 1. One observes immediately that the critical angle θ c is replaced by θ d . In both figures we have assumed that the dead-cone is appreciable,i.e. Θ > θ c for p T > ω c and Θ > θ d for p T < ω c , and marked out the phase space availablefor hard, in-medium splittings of the heavy-quark. C. Higher-order contributions to quenching
The analysis in the preceding section allows us to calculate the higher-order contribu-tions to jet quenching. Isolating the quenching of the initiating parton, that correspondsto the total color charge of the jet, into an overall pre-factor, see Eq. (5), these contribu-tions are collected into the collimator function. Using Eq. (4) and the definition in (5),13 n p T ω c ln p T ω DC ln p T R / ˆ q / ln p T R L ln R − ln Θ − ln θ − c ln( p T L ) p T > ω c ln 1 /z ln 1 /θ ln p T ω DC ln p T R / ˆ q / ln p T R L ln R − ln Θ − ln θ − d ln( p T L ) p T < ω c ln 1 /z ln 1 /θ FIG. 1: Illustration of the DLA phase space for higher-order quenching effects formassive particles, marked by the black, lined area. The two lines correspond to t f = L (upper line with slope −
2) and t f = t d (lower line with slope − / p T > ω c and R > Θ > θ c (left) and p T < ω c and R > Θ > θ d (right).we see that the first-order correction the collimator function is C (1) ( p T | m ) ≈ α (cid:90) R Θ d θθ (cid:90) p T min((ˆ q/θ ) / , ( θ L ) − ) d ωω × (cid:20) Q q ( p T ) Q q ( p T | m ) Q q ( p T , L − t d ) Q q ( p T , L − t d | m ) − (cid:21) , (27)where we have treated the splitting vertex in the leading-logarithmic approximation andadopted the notations of the previous section.It is worth pointing out two limits of this equation. For t d (cid:28) L , we can neglect thedecoherence times in the real term, i.e. the first term in the squared brackets, to obtain C (1) ( p T | m ) (cid:12)(cid:12) t d (cid:28) L ≈ α (cid:90) R max(Θ , θ c , θ d ) d θθ (cid:90) p T (ˆ q/θ ) / d ωω (cid:2) Q q ( p T ) − (cid:3) , (28)which, when taking m →
0, is equal to the contribution of massless quark quenching.When Θ > max( θ c , θ d ) the angular phase space is more restricted for heavy-quark jets,and therefore we expect a relatively smaller impact of the collimator function than inthe massless case. Here it is worth pointing out that the quenching factor on the right-hand side of (28) arises due to the quenching of the additional (massless) gluon since, atlarge- N c , Q q ( p T ) = Q g ( p T ), which is a generic property of Sudakov suppression factors.Before continuing, we point out a new contribution in the small-angle limit in theregime that is unique to massive-quark jets. It appears for t d > L , or θ < θ c , relevant for14 T > ω c , where Q ( p T , L − t d | m ) = 1. We are left with C (1) ( p T ) (cid:12)(cid:12) t d >L,p T >ω c ≈ α (cid:90) θ c Θ d θθ (cid:90) p T ( θ L ) − d ωω (cid:20) Q q ( p T ) Q q ( p T | m ) − (cid:21) . (29)However, Θ < θ c for p T > mθ − c , which leaves the factor Q ( p T ) / Q ( p T | m ) − (cid:38) ¯ α O (1),and therefore the contribution in this regime is sub-leading ∼ O ( ¯ α ). We will thereforealtogether neglect this regime when working in the leading-logarithmic approximation.Let us now evaluate the next-to-leading contributions for massless, light and heavyquarks. For completeness, we repeat here the resulting collimator function for masslessquarks at first order in α s , that reads [23] C (1) ( p T | m = 0)[ Q q ( p T ) −
1] = α ln Rθ c (cid:16) ln p T ω c + ln Rθ c (cid:17) for p T > ω c , α ln p T R / ˆ q / for (ˆ q/R ) / < p T < ω c . (30)Turning now to the new results, for light quarks we obtain C (1) ( p T | m < m ∗ )[ Q q ( p T ) −
1] = α ln Rθ c (cid:16) ln p T ω c + ln Rθ c (cid:17) for p T > ω c , α ln p T R / ˆ q / for m / ˆ q < p T < ω c , α ln p T Rm (cid:16) ln p T Rm + ln m (ˆ qp T ) / (cid:17) for (ˆ q/R ) / < p T < m / ˆ q , (31)and for heavy quarks we get instead, C (1) ( p T | m > m ∗ )[ Q q ( p T ) −
1] = α ln Rθ c (cid:16) ln p T ω c + ln Rθ c (cid:17) for p T > m (ˆ qL ) / , α ln p T Rm (cid:16) ln p T Rm + ln m (ˆ qp T ) / (cid:17) for (ˆ q/R ) / < p T < m (ˆ qL ) / . (32)Equations (30)–(32) are written with logarithmic accuracy, i.e. we neglected all O (1)numerical factors that enter the arguments of the logarithms. The inclusion of thesefactors change the scales in the arguments of the logarithms of the order of 1 −
2, butdoes not change any qualitative conclusions we make.Let us briefly comment on further contributions to the collimator at higher-order (next-to-next-to-leading, and higher). Examining the structure of Eq. (15), one realizes thatthe dipole that “contains” the heavy-quark is distinct from further dipoles in the sensethat it is massive while further dipoles, originating from other gluon emissions, are mass-less. However, as discussed in detail above, this distinction gives rise only to sub-leadingcorrections and for our purposes, having separated out the specific quenching factor ofthe originating parton (that also carried the total color charge), it is adequate to treat alldipoles on equal footing. 15he problem then reduces to the massless case with a modified phase space, as detailedabove. The resummation of higher-order contributions to the collimator involves solvinga non-linear evolution equation and was derived in Ref. [23]. It goes beyond the scope ofour investigation to solve this equation here for the massive case. Furthermore, since weare interested in a relatively modest p T range in order to be sensitive to the dead-cone,the phase space is limited and the first, non-trivial term should provide a good estimateof the effects. Using the same arguments as in [23], we therefore expect that the fullcollimator function is well approximated by C ( p T ) ≈ exp (cid:2) C (1) ( p T ) (cid:3) . (33)In particular, the strong quenching limit, Q ( p T ) (cid:28)
1, returns the correct exponentiationof the virtual terms and also the fixed point at p T → ∞ , where Q ( p T ) → C ( p T ) →
1, is reproduced.
D. Numerics
To emphasize the effects of higher-order contributions we propose the following phe-nomenological quantity, J AA ( p T , R | m ) = R jet AA ( p T , R | m ) R meson AA ( p T | m ) , (34)which is a ratio of nuclear modification factors of heavy-quark jets to heavy-quarks.Within our approximations, this ratio is simply the collimator function for massless andmassive quarks J AA ( p T , R | m ) ≈ C ( p T , R | m ), where we have utilized that R jet AA ( p T , R | m ) (cid:39) Q ( p T , R | m ) and R meson AA (cid:39) Q q ( p T | m ). The latter can be justified in the sense that, althoughthe quenching factor of a heavy-quark is not a direct measurable, it is closely related tothe quenching factor of the corresponding heavy-meson [35]. The underlying idea buildson the assumption of a similar path-length dependence for the two observables.Before we then present our numerical calculations, it is worth emphasizing our ap-proximations in computing the single-quark quenching factors based on the soft-gluonapproximation to the full BDMPS-Z spectrum. It was already pointed out in Ref. [27],that the sub-leading logarithmic and numerical factors play an important role for comput-ing the right order of quenching effects and this was also adopted in [12]. For the momentwe focus only on the effect of massless quark quenching, which enters the dynamics of thecollimator functions, cf. (4). Keeping these corrections, the massless quenching factor in16 � �� ��� ��������������������� � � [ ��� ] � ( � � ) FIG. 2: The quenching factor for massless quark Q ( p T | m = 0) that enters the calculationof the collimator function. The full (lower) line corresponds to the quenching factor withthe leading BDMPS-Z soft-gluon spectrum, cf. first term in (21), while the dashed(upper) line contains sub-leading corrections, cf. ( ?? ). The shaded area between thecurves corresponds to the uncertainty in modeling radiative energy loss.Eq. (21) should read Q ( p T | m = 0) = exp[ − α ( (cid:112) π ˆ qL n/p T − ln(2) ln(ˆ qL n/ (2 p T )) − . . (35)The sub-leading terms result in a faster approach of the quenching factor to unity. Wehave plotted the quenching factors for massless quarks that enter the calculation of thecollimator function in Fig. 2, where the parameters were chose as described below and theuncertainty arising from modeling radiative energy loss is marked with the shaded region.As we have done throughout, we will assume the medium to be static and describedby averaged parameters ˆ q and L . We have chosen ˆ q = 1 GeV / fm and L = 2 . α = 0 .
15 [36]. These are are qualitatively in the same range as the values obtainedin more sophisticated extractions from comparisons to experimental data. Furthermore,we compute the collimator function R = 0 . n = 5, that was extracted from a fit of the jet data [37]. Withthese choices m ∗ (cid:39) . m = 1 . m = 5 GeV) is heavy.We have adopted the approximations and the two ways of estimating the associated,massless quenching factors as discussed above in computing J AA ( p T ) in Fig. 3. As a result,the ratio is simply given by the collimator functions that have been explicitly computed to17 � �� ��� ��������������������� � � [ ��� ] � �� ( � � ) �� �� ��� ��������������������� � � [ ��� ] � �� ( � � ) FIG. 3: The ratio Eq. (34) as a function of energy for a light quark (charm, m = 1 . m = 5 GeV) (dashed, orange line).On the left we have used only the leading term of soft-gluon approximation of theBDMPS-Z spectrum; on the right side, we included numerical corrections to thequenching, as in Refs. [12, 27].next-to-leading order in Eqs. (30)–(32), with the associated quenching factors computedin Fig. 2, and whose full resummation is given in Eq. (33). The solid, black (dashed,organe) curves represent the collimator function for charm (bottom) quark jets. On theleft side, we used the massless quenching factor associated with the first leading term ofthe BDMPS-Z soft-gluon spectrum while, on the right side, we included the sub-leadinglogarithmic and numerical factors, as was done in Refs. [12, 27]. It is clear from Fig. 3that the order of magnitude of the effect does depend on the inclusion of these sub-leadingBDMPS-Z corrections since the full expression (35) approaches unity much faster thanthe leading term alone. However, the characteristic p T where the collimator function ofcharm and bottom starts deviating on the level of ∼
10% is roughly the same in the twocases (for our choice of parameters, this corresponds to roughly p T ∼
50 GeV).
V. CONCLUSIONS
We have calculated a subset of higher-order corrections for massive-quark jet propa-gating in the quark-gluon plasma that are enhanced by logarithms of the jet energy. Weconsidered the corrections due to rapid split of the leading particle into hard dipoles wellwithin the medium. This contribution also plays an important role in the context of jetsubstructure [24]. We have shown that these corrections lead to the enhancement of jet18nergy loss, and consequently to the decrease of the jet quenching factor, like it was foundfor the case of massless quark/gluon jets previously.The additional suppression can be factorized into a collimator function, that enhancesthe quenching factor associated with the leading particle and is here evaluated in theleading-logarithmic asymptotics. We have demonstrated that these corrections are essen-tially determined by phase space restrictions available for dipole creation — on the onehand related to the criterium of early splitting, in particular with formation times t f < t d ,and, on the other hand, at angles larger than the associated dead-cone angle. Thesesemi-quantitive estimates show that these corrections are of the same order of magnitudeas for massless quark at high- p T , contrary to leading order results [12], where the substan-tial difference between quenching factors of massless and heavy quark jets was found atrelatively low p T . The reason is that in significant part of the parameter space the heavyand massless quark corrections are just the same, and in the remaining region they onlydiffer by the argument of the logarithm.Our main results are given by Eqs. (31) and (32), where we demonstrate that, from thepoint of view of higher corrections, we can divide the quarks into light , with m < ( qL ) / and heavy , m > ( qL ) / . For light quarks the corrections are exactly the same as for themassless quark, see Eq. (30), and start to be weakly mass dependent only for rather smallenergies of order p T ≤ m / ˆ q . For heavy quarks, given in Eq. (32), already at large p T > ω c the dead-cone effect start to play role.Our work shows that antenna corrections are essentially the same for massless andheavy quarks, leading to correct from the experimental point of view decrease of quenchingfactors of heavy quark jets relative to the Dokshitzer-Kharzeev result [12]. Consequentlythey had little influence on the problem first raised in [12], namely that, contrary tocalculation focussing on radiative energy loss, there is not much difference between heavyand massless quark quenching factors experimentally [17, 18] (although the situation canimprove in the future [19]). Nevertheless, since the BDMPS-Z calculation is the leadingmechanism available for jet energy loss at high- p T it is of great interest to test it fordifferent parts of the parameter space.We have also derived the spectra and rates related to a color-charged, massive dipoleformed early in the medium, see Eq. (A9) and (A10), and Eqs. (B10), (B11), (B12) and(B16).We did not include the restrictions on the phase space that were found to be important19or calculation of massless and massive quark quenching weights, see [38] and [13, 14]. Wedo not expect that these restrictions will make qualitative influence on our results. Thereason is that the corrections that we calculated were dominated by small frequenciesregime outside the dead-cone while the corrections discussed in [13, 14] are mostly impor-tant for frequencies in, or close to, the dead-cone region. Besides, the analysis of theseconstraints needs detailed numerical investigation.Our results are valid in the leading-logarithmic approximation at high energies. Goingto higher logarithmic precision and for applications at very high energies we expect thecorrections to satisfy nonlinear integral equations similar to the one that was suggestedin [23]. In this context, the effects of the dead-cone suppression in secondary heavy-quarkproduction, i.e. from the splitting of a gluon into a massive q ¯ q pair, are still largelyunexplored. For a clearer interpretation, these contributions could perhaps be suppressedusing state-of-the-art grooming techniques, employed in [39].The results worked out here demonstrate the factorization between the leading termin the quenching weight, calculated in [12] and higher order corrections, and introducesa new way to address jet observables involving massive quark, e.g. the “leading particleeffect” [40–42], that we plan to address in forthcoming works. From a phenomenologicalpoint of view, it will be interesting to study the ratio of quenching factors of heavy quarkjet and heavy mesons as a check of the current approach, cf. Eq. (34). ACKNOWLEDGMENTS
KT is supported by a Starting Grant from Bergen Research Foundation (“Thermalizingjets: novel aspects of non-equilibrium processes at colliders”) and the University of Bergen.BB was supported by the Israel Science Foundation (ISF) grant no. 2025311. The authorsthank the CERN Theory Department for hospitality, where most of the work was done.
Appendix A: Derivation of the interference spectrum
Let us derive the rates of gluon emissions of massless and massive particles. Consider aparton splitting at very short time-scales that quasi-instantaneously forms a dipole insidethe medium. Using conventional arguments in perturbation theory, one can thereafterdefine a subsequent emission spectrum off this system. For further applications, it willbe sufficient to calculate the interference spectrum between the two constituents of the20ipole since direct emissions can be recovered by setting the dipole opening angle to zero.For concreteness, and in order to clarify the color structure of the process, considera color-charged dipole originating from a q ( (cid:126)p ) → q ( (cid:126)p ) + g ( (cid:126)p ) splitting, where we haveidentified the momentum flow of the splitting [43]. We assume, for the time being, thatthe gluon is massive; this allows us to easily generalize the formula for arbitrary dipoles.The amplitude describing the emission of a soft gluon inside a medium of size L is the sumof two terms that correspond to the possibility of being radiated by dipole constituents.They are given by M ∼ gω (cid:90) L d t e i ω ( n +Θ ) t ( ∂ x + iω n ) · (cid:15) ∗ λ G ab ( k , L ; x , t ) (cid:12)(cid:12) x = n t × (cid:2) V n ( L, t ) t b V n ( t, (cid:3) ij U dc n ( L, t cjk J k ( (cid:126)p ) , (A1) M ∼ gω (cid:90) L d t e i ω ( n +Θ ) t ( ∂ x + iω n ) · (cid:15) ∗ λ G ab ( k , L ; x , t ) (cid:12)(cid:12) x = n t × V n ( L, ij (cid:2) U n ( L, T b U n ( t, (cid:3) dc t cjk J k ( (cid:126)p ) , (A2)up to factors that cancel in the cross section and where the fundamental color matrix t cjk accounts for the color conservation of the system (for completeness, we recall that[ T b ] ac ≡ if abc ). The color factor related to the dipole splitting will be factored in thefinal expression of the emission spectrum. The dead-cone angles are denoted Θ i ≡ m i /E i and n i ≡ p i /E i determine the trajectories of the dipole constituents. Finally, J k ( (cid:126)p )represents the initial quark current.Medium interactions can be encapsulated into color rotation matrices. To keep explicittrack of the color we denote by V n ( t,
0) = P exp[ ig (cid:82) t d s t · A ( s, n s )] a (path-ordered)Wilson line in the fundamental representation that resums interactions with the mediumthat is modeled by a background field A ( t, x ). Similarly, U n ( t,
0) denotes a Wilson line inthe adjoint representation, with the substitution t b → T b . In Eqs. (A1) and (A2) theseobjects describe the propagation of the dipole constituents through the medium along fixedtrajectories. In contrast, a soft gluon emission in the medium can experience momentumbroadening due to transverse momentum exchanges with medium constituents. Theseinteractions are encapsulated in the dressed propagator G ab , that is given by G ab ( x , t ; x , t ) = (cid:90) r ( t )= x r ( t )= x D r exp (cid:20) i E (cid:90) t t d t ˙ r ( t ) (cid:21) U ab r ( t ) ( t , t ) , (A3)where the trajectory of the gluon r is explicitly time-dependent.We proceed now with the calculation of the interference spectrum. It was first com-puted for a color-charged dipole involving a massive quark in [34] but an explicit expression21 t ¯ t a) ∞ t ¯ t b) FIG. 4: Two contributions to the interference spectrum of a heavy-quark–gluon dipole,where the (dressed) propagators of the hard quark (gluon) are represented by adouble-line (spiral) while the soft, medium-induced gluon emission is represented by awavy line. We draw the diagrams for ¯ t > t .is not available in the literature. It is therefore meaningful to rederive the spectrum inmore generality here. The spectrum involves in total four terms, corresponding to the dif-ferent time orderings of emission (absorption) in the amplitude and the complex-conjugateamplitude. We depict the two main possibilities in Fig. 4. The process in Fig. 4a depictsan emission from the heavy-quark (in the amplitude) and later absorption by a gluon (inthe complex conjugate), while the second diagram, Fig. 4b, describes an emission fromgluon and subsequent absorption by the heavy-quark. The remaining contributions canbe found by adding the complex conjugate of these terms to the final answer, and will beautomatically included in the expressions below.We now have for the diagram Fig. 4a where gluon is emitted at time t and absorbed ata later time ¯ t , i.e. ¯ t > t , by a heavy quark. Then, the double-differential spectrum readsd I int dΩ k (cid:12)(cid:12)(cid:12)(cid:12) a = 1 N c C F g ω (cid:90) L d t (cid:90) Lt d¯ t e − i ω ( n +Θ )¯ t + i ω ( n +Θ ) t ( ∂ ¯ x − iω n ) · ( ∂ x + iω n ) × TrΣ (cid:90) u , y , y e − i k · ( y − y ) N c − (cid:104) Tr G † ( ¯ x , ¯ t ; y , L ) G ( y , L ; u , ¯ t ) (cid:105)× N c − (cid:104) Tr U † n ( t, ¯ t ) G ( u , ¯ t ; x , t ) (cid:105) N c − (cid:104) Tr U † n (0 , t ) U n ( t, (cid:105) (cid:12)(cid:12) x = n t ¯ x = n ¯ t , (A4)where dΩ k = d ω d k / [(2 π ) ω ] is the Lorentz-invariant phase-space element, TrΣ ≡ if abc tr( t a t b t c ) = − N c ( N c − / (cid:82) u ≡ (cid:82) d u . The normalizationfactor ( N c C F ) − takes care of the averaging over the colors of the initial quark and divides22ut the color structure related to the dipole splitting. The in-medium two-point correla-tors appearing in (A4) are known from the literature and, after some simplifications, werewrite (A4) asd I int dΩ k (cid:12)(cid:12)(cid:12)(cid:12) a = − N c g ω (cid:90) L d t (cid:90) Lt d¯ t (cid:90) u e − i k · u − Ncn ( L − ¯ t ) σ ( u ) e − i ω ( n +Θ )¯ t + i ω ( n +Θ ) t S ( t ) × ( ∂ ¯ x − iω n ) · ( ∂ x + iω n ) K ( ¯ x , x )e − i ω n (¯ t − t )+ iω n · (¯ x − x ) (cid:12)(cid:12) x = n t, ¯ x = u + n ¯ t , (A5)where S ( t ) = exp (cid:34) − N c n (cid:90) ¯ tt d s σ ( n s ) (cid:35) (A6)and K ( ¯ x , x ) = (cid:90) r (¯ t )=¯ xr ( t )= x D r exp (cid:34) iω (cid:90) ¯ tt d s ˙ r − N c n (cid:90) ¯ tt d s σ ( r ) (cid:35) , (A7)where we have assumed that the medium is described by a static density n . The overallcolor factor in (A5), − N c /
2, corresponds to the correct (negative) interference charge.After performing the derivatives and shifting the coordinates, we finally obtaind I int dΩ k (cid:12)(cid:12)(cid:12)(cid:12) a = − N c g ω (cid:90) L d t (cid:90) Lt d¯ t e − i k · u − Ncn ( L − ¯ t ) σ ( u ) e i ω n t e i ω (Θ t − Θ ¯ t ) S ( t ) × ( ∂ x + iω n ) · ∂ ¯ x K ( ¯ x , x ) (cid:12)(cid:12) ¯ x = u , x = n t . (A8)The calculation of the second diagram in Fig. 4b is completely analogous. From sym-metry, it turns out that d I/ d ω | b = d I/ d ω | a (Θ ↔ Θ ). Hence, summing up all fourcontributions, our final result for the emission spectrum therefore reads,d I int dΩ k = − N c g ω (cid:90) ∞ d t (cid:90) ∞ t d¯ t (cid:90) d u e − i k · u − Ncn ( L − ¯ t ) σ ( u ) × e i ω n ij t (cid:104) e i ω (Θ i t − Θ j ¯ t ) + e i ω (Θ j t − Θ i ¯ t ) (cid:105) S ( t ) × ( ∂ x + iω n ij ) · ∂ ¯ x K ( ¯ x , x ) (cid:12)(cid:12) ¯ x = u , x = n ij t . (A9)We have checked that this formula reproduces the double-differential interference spec-trum in vacuum. It was first discussed in Ref. [34]. Note, however, that our formuladiffer from the related Eqs. (4.1) and (4.2) in Ref. [34] by the second term in the squarebrackets.The in-medium energy spectrum is found by integrating out the transverse momentum,in which case we obtaind I int d ω = − N c α s ω (cid:90) ∞ d t (cid:90) ∞ t d¯ t e i ω n t (cid:104) e i ω (Θ t − Θ ¯ t ) + e i ω (Θ t − Θ ¯ t ) (cid:105) S ( t ) × ( ∂ x + iω n ) · ∂ ¯ x K ( ¯ x , x ) (cid:12)(cid:12) ¯ x =0 , x = n t , (A10)23hich is the main formula for analyzing the direct and interference rates in this work. Appendix B: Direct and interference radiation rates in the multiple-soft scatter-ing approximation
The formulas in Eqs. (A9) and (A10) can easily be generalized to any splitting processinvolving a gluon emission (for the time being let us disregard photon splitting into quark-antiquark). The general formula for the energy spectrum, cf. Eq. (A10), reads,d I ij d ω = α s ω Q ij (cid:90) t L d t (cid:90) t L t d¯ t e i ω n ij t (cid:104) e i ω (Θ i t − Θ j ¯ t ) + e i ω (Θ j t − Θ i ¯ t ) (cid:105) S ( t ) × ( ∂ y + iω n ij ) · ∂ x K ( x , y ) (cid:12)(cid:12) x = , y = n ij t , (B1)where Q ij ≡ Q i · Q j = ( Q − Q − Q ) / Q i corresponds to the color charge-vectorof the emitter (e.g. for a quark Q q = C F and for a gluon Q g = N c ). The particle i isassociated to the dead-cone angle Θ i and direction n i = p i /E i (and similarly for particle j ). The two terms describe two possible emission processes, namely one that is initiatedby particle i and one initiated by particle j . The case under study, the Q → Q + g splittingthat contributes to the fragmentation of a heavy-quark jet, corresponds to Θ j = 0 and Q ij = − N c /
2, as derived explicitly in (A9).In (B1), S ( t ) is a two-point function describing the decoherence of the antenna beforethe splitting occurs and is often referred to as the decoherence parameter [31, 32, 44, 45].It depends explicitly on the opening angle of the pair, in particular S ( t ) = 1 for n = 0.The formula in (B1) generalizes the result previously obtained in [34]. It is worth pointingout that, due to the Galilean symmetry of the problem, the spectrum does not dependon the directions of the emitters apart from the dipole opening angle n .Working in the multiple-soft scattering limit of medium interactions, the n -point func-tions can be found exactly. It corresponds to the following approximation on the interac-tion term, N c nσ ( r ) ≈ ˆ q r /
2, where ˆ q is the jet quenching parameter. For the decoherenceparameter, we immediately obtain S ( t ) = exp[ − ˆ q n t / K ( x , y ) = ω Ω2 πi sinh Ω τ exp (cid:26) iω Ω4 (cid:20) tanh Ω τ x + y ) + coth Ω τ x − y ) (cid:21)(cid:27) , (B2)where τ ≡ ¯ t − t and Ω ≡ (1 + i ) (cid:112) ˆ q/ω/
2. The propagator in (B2) constraints the extentof the time difference between gluon emission and absorption τ (cid:46) (cid:112) ˆ q/ω ≡ t br . Hence, in24he limit t br (cid:28) L which holds for soft gluon emissions ω (cid:28) ˆ qL that are most importantfor energy loss at not too high energies [27], we can approximate the time integration overthe time-difference of the emissions in the amplitude and in the complex-conjugate as (cid:90) Lt d¯ t = (cid:90) L − t d τ ≈ (cid:90) ∞ d τ , (B3)see, e.g., [46]. Formally, this allows to treat multiple radiation as independent with aconstant rate.Focussing again on a q → q + g splitting, we therefore obtain the formulas for thedirect and interference rates. In order to separate the dynamics of the dipole before and during the emission-time of medium-induced gluon, we will explicitly define the emissionrate Γ ij ( ω, t ) as d I ij d ω d t = S ( t ) × ˜Γ ij ( ω, t ) , (B4)where the decoherence parameter is responsible for the long-distance color decoherenceprocesses that are happening prior to the emission. Setting the dipole opening angle tozero, n →
0, we find the independent heavy-quark and gluon spectra.˜Γ q ( ω, t ) = α s C F ω (cid:90) ∞ d τ e − i ω Θ τ ∂ x · ∂ y K ( x , y ) (cid:12)(cid:12)(cid:12) x = y = , (B5)˜Γ g ( ω, t ) = α s N c ω (cid:90) ∞ d τ ∂ x · ∂ y K ( x , y ) (cid:12)(cid:12)(cid:12) x = y = , (B6)where we have restored the correct mass and color factors, and introduced the definitionof the dead-cone angle Θ ≡ θ q used throughout. The interference spectrum contains twoterms, given by˜Γ ( ω, t ) = − α s N c ω i ω ( n +Θ ) t (cid:90) ∞ d τ ( ∂ y + iω n ) · ∂ x K ( x , y ) (cid:12)(cid:12) x =0 , y = x , (B7)˜Γ ( ω, t ) = − α s N c ω i ω ( n − Θ ) t (cid:90) ∞ d τ e − i ω Θ τ ( ∂ y + iω n ) · ∂ x K ( x , y ) (cid:12)(cid:12) x =0 , y = x . (B8)Furthermore, we can altogether neglect the phases involving the terms Θ t since, in thedouble-logarithmic approximation the hard gluon splitting takes place at angles muchlarger than the dead-cone angle.Note that this expression, at finite n (cid:54) = 0, diverges both in the limit of small t and τ .However, this contribution is completely independent of medium parameters, ∼ ω / [ πtτ ],and can therefore be regularized by subtracting the vacuum contribution. This is incontrast to the regularization of the direct terms, which only exhibit a divergence at25mall τ , ∼ ω / [ πτ ], but is cured in the same way. It follows that all medium-inducedcontributions vanish in the ˆ q → ij ( ω, t ) ≡ ˜Γ ij ( ω, t ) − ˜Γ ij ( ω, t ) (cid:12)(cid:12) ˆ q → . (B9)After applying this regularization, we immediately find the rate of direct emissions by amassless gluon, Γ g ( ω, t ) = α s N c π (cid:114) ˆ qω , (B10)recovering the well-known LPM rate valid for soft-gluon emission. For the correspondingrate off a massive quark, we findΓ q ( ω, t ) = α s C F π Θ (cid:34) − Im ψ (cid:32) − i (cid:115) Θ ω ˆ q (cid:33) − (cid:114) ˆ q Θ ω − π (cid:35) , (B11)where ψ ( x ) is the digamma function.Let us continue with the interference spectra. Considering first (B7), we define α (cid:48) = ωt (Θ + n ) / ≈ ωt n / α = α (cid:48) + ω x / (4 t f ), where t f ≡ (cid:112) ω/ ˆ q , the first interfer-ence term reads simplyΓ ( ω, t ) = − α s N c πω t e − ω t f x (cid:20) ω t f x cos α (cid:48) − α + tt f (cos α − sin α ) (cid:21) . (B12)Here, putting Θ → α and α (cid:48) factors, we recover immediately interference spec-trum for a massless antenna.The second interference term in Eq. (B8) is complicated due to the additional phasefactor related to the finite quark mass. We define the following integral, I ≡ (cid:90) ∞ d τ sinh Ω τ e a coth Ω τ + bτ = 2Ω e a Γ (cid:18) − b (cid:19) U (cid:18) − b , , − a (cid:19) , (B13)which will become useful later. We have made sure that Re b/ (2Ω) > −
1, which allowedus to deform the integration contour to lie along the real axis. We will also need thefollowing integral (cid:90) ∞ d τ sinh Ω τ coth Ω τ e a coth Ω τ + bτ = ∂∂a I = I + I , (B14)where I ≡
4Ω e a (cid:18) − b (cid:19) Γ (cid:18) − b (cid:19) U (cid:18) − b , , − a (cid:19) . (B15)26he second interference term can then be written asΓ ( ω, t ) = α s N c ω i ω ( n − Θ ) t ( ω Ω) π (cid:2)(cid:0) − i n tω (cid:1) I + i n t ω Ω ( I + I ) (cid:3) − ˜Γ ( ω, t ) (cid:12)(cid:12) ˆ q → , (B16)with parameters a = iω Ω n t / b = − iωθ q /
2, and where the vacuum subtractionterm is simply ˜Γ ( ω, t ) (cid:12)(cid:12) ˆ q → = α s N c ω i ω ( n − Θ ) t ω πt . (B17)We have explicitly checked that taking the massless limit, Θ →
0, inside of the squarebrackets of Eq. (B16) reproduces the first term of the interference contribution given byEq. (B12).Although the formulas derived above can be evaluated numerically, we aim at under-standing the problem through the fundamental scales appearing in these expressions. Asa final step, let us therefore spend some time on discussing the different time-scales ap-pearing, starting with the interferences. Neglecting the dead-cone angle compared to thedipole opening angle, we find that one of the exponentials in Eqs. (B12) and (B16) startoscillating at times t ∼ t = 1 n ω . (B18)This scale is related to quantum coherence [32, 45], imposing that the wavelength ofemitted quanta resolve the dipole x ⊥ ( t ) > λ ⊥ ( t ), where λ ⊥ ∼ ω/t . In the vacuum t ∼ t f , which leads immediately to the angular ordering condition. The second time-scaleappearing in the exponentials is t = (ˆ q n ω ) − / . (B19)This scale is not an independent scale since we can rewrite it as t = ( t t ) / , where t d isthe color decoherence scale introduced above. This only allows for two possible orderings, t d < t < t and t < t < t d , and in neither of the cases does t constitute the shortest,and therefore most relevant, time-scale.It is also possible to show that t /t ∼ θ br ( ω ) /θ , where θ ≡ | n | and θ br ( ω ) =(ˆ q/ω ) / is the typical emission angle for medium-induced gluons. Hence, since we mainlyare interested in large-angle gluon emissions that contribute to energy-loss, θ br ( ω ) > R >θ , t > t and the first ordering is actually realized. This means that the decoherencetime, which resides in the function S ( t ) sets the shortest time-scale where the interferences27ill be suppressed. For our purposes, it is therefore possible to show that we can altogetherneglect the phases involving these time-scales in the interference terms, leaving us withΓ ( ω, t ) (cid:39) − Γ g ( ω, t )Θ (cid:0) ω − ω (cid:1) , (B20)Γ ( ω, t ) (cid:39) − Γ q ( ω, t )Θ (cid:0) ω − ω (cid:1) , (B21)where ω ≡ (ˆ q/θ ) / . In Laplace space this becomes, γ ( ν, t ) (cid:39) − α (cid:114) ˆ qω (cid:2) − e − νω − √ πνω erf( √ νω ) (cid:3) , (B22) γ ( ω, t ) (cid:39) − α (cid:114) ˆ qω min (cid:2) − e − νω min − √ πνω min erf( √ νω min ) (cid:3) , (B23)where ω min ≡ min( ω DC , ω ) which is simply ω min = ω in the leading-logarithmic approx-imation ( θ > Θ ). Hence, γ (cid:39) γ ≈ − α √ ˆ q ( √ πν − √ ω ). [1] Y. L. Dokshitzer, V. A. Khoze, A. H. Mueller, and S. I. Troian, Basics of perturbativeQCD (1991).[2] D. d’Enterria, (2009), arXiv:0902.2011 [nucl-ex].[3] Y. Mehtar-Tani, J. G. Milhano, and K. Tywoniuk, Int. J. Mod. Phys.
A28 , 1340013 (2013),arXiv:1302.2579 [hep-ph].[4] J.-P. Blaizot and Y. Mehtar-Tani, Int. J. Mod. Phys.
E24 , 1530012 (2015), arXiv:1503.05958[hep-ph].[5] R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne, and D. Schiff, Nucl.Phys.
B483 , 291(1997).[6] R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne, and D. Schiff, Nucl.Phys.
B484 , 265(1997).[7] B. Zakharov, JETP Lett. , 615 (1997).[8] R. Baier, D. Schiff, and B. G. Zakharov, Ann. Rev. Nucl. Part. Sci. , 37 (2000), arXiv:hep-ph/0002198 [hep-ph].[9] Y. L. Dokshitzer, V. S. Fadin, and V. A. Khoze, Z. Phys. C15 , 325 (1982).[10] Y. L. Dokshitzer, V. A. Khoze, and S. I. Troian,
Jet Studies Workshop at LEP and HERADurham, England, December 9-15, 1990 , J. Phys.
G17 , 1602 (1991).[11] Y. L. Dokshitzer, V. A. Khoze, and S. I. Troian, Phys. Rev.
D53 , 89 (1996), arXiv:hep-ph/9506425 [hep-ph].
12] Y. L. Dokshitzer and D. E. Kharzeev, Phys. Lett.
B519 , 199 (2001), arXiv:hep-ph/0106202[hep-ph].[13] N. Armesto, C. A. Salgado, and U. A. Wiedemann, Phys. Rev.
D69 , 114003 (2004),arXiv:hep-ph/0312106 [hep-ph].[14] N. Armesto, A. Dainese, C. A. Salgado, and U. A. Wiedemann, Phys. Rev.
D71 , 054027(2005), arXiv:hep-ph/0501225 [hep-ph].[15] M. Djordjevic and M. Gyulassy, Nucl. Phys.
A733 , 265 (2004), arXiv:nucl-th/0310076[nucl-th].[16] L. Zhang, D.-F. Hou, and G.-Y. Qin, (2018), arXiv:1812.11048 [hep-ph].[17] D. Sharma (PHENIX),
Quark matter. Proceedings, 22nd International Conference on Ultra-Relativistic Nucleus-Nucleus Collisions, Quark Matter 2011, Annecy, France, May 23-28,2011 , J. Phys.
G38 , 124082 (2011).[18] T.-W. Wang (CMS),
Proceedings, 26th International Conference on Ultra-relativisticNucleus-Nucleus Collisions (Quark Matter 2017): Chicago, Illinois, USA, February 5-11,2017 , Nucl. Phys.
A967 , 656 (2017).[19] S. Chatrchyan et al. (CMS), “Projected Heavy Ion Physics Performance at the High Lumi-nosity LHC Era with the CMS Detector,” (2017), CMS-PAS-FTR-17-002.[20] G.-Y. Qin and X.-N. Wang, Int. J. Mod. Phys.
E24 , 1530014 (2015), [,309(2016)],arXiv:1511.00790 [hep-ph].[21] S. Chatrchyan et al. (CMS), Phys. Rev. Lett. , 132301 (2014), [Erratum: Phys. Rev.Lett.115,no.2,029903(2015)], arXiv:1312.4198 [nucl-ex].[22] Y. Mehtar-Tani and K. Tywoniuk, Nucl. Phys.
A979 , 165 (2018), arXiv:1706.06047 [hep-ph].[23] Y. Mehtar-Tani and K. Tywoniuk, Phys. Rev.
D98 , 051501 (2018), arXiv:1707.07361 [hep-ph].[24] J. Casalderrey-Solana, Y. Mehtar-Tani, C. A. Salgado, and K. Tywoniuk,
Proceedings, 26thInternational Conference on Ultra-relativistic Nucleus-Nucleus Collisions (Quark Matter2017): Chicago, Illinois, USA, February 5-11, 2017 , Nucl. Phys.
A967 , 564 (2017).[25] P. Caucal, E. Iancu, A. H. Mueller, and G. Soyez, Phys. Rev. Lett. , 232001 (2018),arXiv:1801.09703 [hep-ph].[26] Assuming that d σ we have only accounted for the first term in the expansion (1 + x ) − n ≈ e − nx (1 + nx / . . . ).
27] R. Baier, Y. L. Dokshitzer, A. H. Mueller, and D. Schiff, JHEP , 033 (2001), arXiv:hep-ph/0106347 [hep-ph].[28] J. Casalderrey-Solana, D. Pablos, and K. Tywoniuk, JHEP , 174 (2016),arXiv:1512.07561 [hep-ph].[29] A. Bassetto, M. Ciafaloni, and G. Marchesini, Phys. Rept. , 201 (1983).[30] All results for the medium-induced spectra and rates hold in the soft limit where we haveexplicitly subtracted the vacuum component. Negative contributions in these quantitiestherefore indicate an over-subtraction at larger gluon energies.[31] Y. Mehtar-Tani, C. A. Salgado, and K. Tywoniuk, Phys.Lett. B707 , 156 (2012).[32] J. Casalderrey-Solana and E. Iancu, JHEP , 015 (2011).[33] Y. Mehtar-Tani, C. A. Salgado, and K. Tywoniuk, JHEP , 064 (2012).[34] M. R. Calvo, M. R. Moldes, and C. A. Salgado, Phys. Lett.
B738 , 448 (2014),arXiv:1403.4892 [hep-ph].[35] F. Arleo, Phys. Rev. Lett. , 062302 (2017), arXiv:1703.10852 [hep-ph].[36] Note also that the value of the strong-coupling constant α s can, in principle, be differentin the quenching factor and in the collimator function or, more precisely, the running takesplace at different scales: in the quenching factor with the typical medium transverse scaleand in the collimator with typical jet k ⊥ D68 , 014008 (2003), arXiv:hep-ph/0302184 [hep-ph].[39] L. Cunqueiro and M. Ploskon, (2018), arXiv:1812.00102 [hep-ph].[40] J. D. Bjorken, Phys. Rev.
D17 , 171 (1978).[41] Y. I. Azimov, Y. L. Dokshitzer, and V. A. Khoze, Sov. J. Nucl. Phys. , 878 (1982), [Yad.Fiz.36,1510(1982)].[42] Y. I. Azimov, Y. L. Dokshitzer, V. A. Khoze, and S. I. Troian, Yad. Fiz. , 777 (1984),[Sov. J. Nucl. Phys.40,498(1984)].[43] The notation (cid:126)p = ( E, p ) involves the light-cone energy variable E ≡ ( p + p ) / p = ( p , p ). Similarly, all time coordinates (and lengths) refer tothe light-cone variable t ≡ ( x + x ) / , 122002 (2011).[45] Y. Mehtar-Tani, C. A. Salgado, and K. Tywoniuk, JHEP , 197 (2012).
46] J.-P. Blaizot, F. Dominguez, E. Iancu, and Y. Mehtar-Tani, JHEP , 143 (2013),arXiv:1209.4585 [hep-ph]., 143 (2013),arXiv:1209.4585 [hep-ph].