Cone size dependence of jet suppression in heavy-ion collisions
CCone size dependence of jet suppression in heavy-ion collisions
Yacine Mehtar-Tani ∗ RIKEN BNL Research Center and Physics Department,Brookhaven National Laboratory, Upton, NY 11973, USA
Daniel Pablos † and Konrad Tywoniuk ‡ Department of Physics and Technology, University of Bergen, 5007 Bergen, Norway (Dated: January 7, 2021)The strong suppression of high- p T jets in heavy ion collisions is a result of elastic and inelasticenergy loss suffered by the jet multi-prong collection of color charges that are resolved by mediuminteractions. Hence, quenching effects depend on the fluctuations of the jet substructure that areprobed by the cone size dependence of the spectrum. In this letter, we present the first complete,analytic calculation of the inclusive R -dependent jet spectrum in PbPb collisions at LHC energies,including resummation of energy loss effects from hard, vacuum-like emissions occurring in themedium and modeling of soft energy flow and recovery at the jet cone. Both the geometry of thecollision and the local medium properties, such as the temperature and fluid velocity, are givenby a hydrodynamic evolution of the medium, leaving only the coupling constant in the medium asa free parameter. The calculation yields a good description of the centrality and p T dependenceof jet suppression for R = 0 . PACS numbers: 12.38.-t,24.85.+p,25.75.-q
Introduction.
Jets are collimated sprays of energeticparticles produced in collider experiments that act asproxies of accelerated quark and gluon degrees of freedomoriginating from elementary large momentum-transferprocesses or decays of massive bosons. In this context,precision computations of QCD jet events play a crucialrole in a wide range of fundamental measurements at col-liders [1, 2], including measurements of the Higgs bosonproperties [3] and searches beyond the Standard Model.In contrast, jet physics in heavy ion collisions probesthe discovery frontier to potentially reveal and detail newemergent QCD phenomena in dense partonic systems.The creation of a short-lived, hot and dense state of de-confined matter, also known as the quark-gluon plasma(QGP), leaves a strong imprint on high- p T probes [4, 5].This phenomenon, commonly referred to as “jet quench-ing”, was observed for the first time at RHIC [6–8] andlater at the LHC [9–14]. Currently, the exact mechanismsresponsible for jet modifications, including details of theenergy transport from high-energy to low-energy modesand color/quantum decoherence of multi-partonic states,are under intense investigation.The basic mechanisms of parton energy loss were un-derstood and formalized in the 90’s and implemented forRHIC phenomenology [15–19], where a number of ap-proximations, in particular for the medium induced ra- ∗ [email protected] † [email protected] ‡ [email protected] diative spectrum, were then necessary to allow for ana-lytic computations. This introduced a theoretical biason model calculations, absent from full numerical ap-proaches [20–22], that could be alleviated by incorpo-rating the two main scattering regimes: the Rutherfordscattering regime, dominated by a single hard momentumtransfer, and the low momentum regime where multiplescatterings contribute with order one probability. Fur-thermore, with the measurements of fully reconstructedjets at the LHC and RHIC, it was soon recognized thathigher order corrections accounted for by parton cas-cades are not negligible. The need to address these ef-fects spurred the rapid development of Monte Carlo (MC)event generators [23–33] which, to some extent, rely onmodeling of the quantum nature of jet evolution. In par-allel with this computational effort, tremendous concep-tual progress has been made in addressing these questionsby analyzing the interference structure of two successivesplittings within the medium [34–38]. To leading loga-rithmic accuracy, it has been shown that the in-mediumjet evolution is characterized by an early vacuum par-ton cascade whose constituents either get resolved by themedium due to color decoherence, whereas unresolvedsplittings factorize from the in-medium evolution, losingenergy coherently as a single color charge [39, 40]. Thiswork aims to address these two challenges within a first-principle analytic framework.In addition to the modification of the hard compo-nents of the jet and their interactions with the plasmaconstituents, there are non-universal contributions to jetobservables pertaining to how soft jet constituents ther-malize in the plasma. In analogy with hadronization ef- a r X i v : . [ h e p - ph ] J a n fects, these non-perturbative contributions are bound tobe modeled. This leads us to one of the most impor-tant questions in jet quenching physics: what is the rel-ative magnitude of the uncertainties associated with de-scribing the hard, perturbative structures—that are sys-tematically improvable—and the soft, infrared featuresof medium-modified jets as a function of their kinemat-ics? Providing a quantitative answer to this question iscrucial if one aims to establish the predictive power ofweak coupling techniques in jet quenching phenomenol-ogy and probe the transport properties of the QGP. Withthis work, we aim to provide an answer to this fundamen-tal question.In this letter, we report a first-principle calculation ofthe single-inclusive jet spectrum and its cone size depen-dence in heavy ion collisions where high density effectsare resummed to all orders. Even though jets with alarger cone do retain a larger fraction of the lost energy,a priori reducing jet suppression compared to a smallerone, we show that resumming the additional energy lossinduced by the cone-size dependent jet substructure fluc-tuations yields a final jet suppression that is very mildlydependent on R . The well established connection be-tween energy loss dynamics and coherence effects, whichdetermine the actual resolved phase space of the jet inthe medium, allows us to confront our results with high-statistics experimental data merely by constraining thestrength of the QCD coupling in the medium. Additionalfluctuations on the path and medium density exploredby the jet, which vary event by event, are taken care ofby embedding our framework into a realistic heavy-ionenvironment in which the medium is described by theexplosion of a liquid droplet of deconfined QCD matter. Theoretical formalism.
The spectrum of jets withcone size R in proton-proton collisions is given by the con-volution of the initial hard parton spectra with the cor-responding fragmentation function. The latter describesthe energy remaining within the jet at different angularresolutions R , starting from a large value R ∼
1. For asteeply falling initial spectrum, it can be written as [41] σ pp ( p T , R ) = (cid:88) k = q,g f ( n − /k ( R | p T , R ) ˆ σ k ( p T , R ) , (1)where n ≡ n k ( p T , R ) is the power-index of the cross-section of the hard parton with flavor k . This is cal-culated at leading order (LO) at the factorization scale Q , such that ˆ σ k = f i/A ⊗ f j/A ⊗ ˆ σ ij → k ( l ) , and involvesa convolution of parton distribution functions (PDFs) f i/A ( x, Q ) with the 2 → σ ij → kl . The moment of the fragmentation function of aninitial hard parton with flavor k , i.e. f ( n )jet /k ( R | p T , R ) = (cid:82) d x x n f jet /k ( R | x, R ), gets both quark and gluon con-tributions, f ( n )jet /k = (cid:80) i = q,g f ( n ) i/k , due to flavor conversionduring the DGLAP evolution [41–44].Correspondingly, the cross section in nucleus-nucleuscollisions (AA) are convolved with a probability distri-bution P ( (cid:15) ) describing medium-induced energy loss out of the jet cone, and reads σ AA ( p T , R ) = (cid:88) i = q,g (cid:90) ∞ d (cid:15) P i ( (cid:15) )˜ σ ppi ( p T + (cid:15), R ) , (2)where ˜ σ ppi corresponds to the quark/gluon contributionto the total cross section in Eq. (1) (the tilde serves asa reminder that the proton PDFs are replaced by nu-clear PDFs). Finally, the flavor dependent resummedquenching factors (QF) Q i ( p T , R ) ≡ (cid:82) ∞ d (cid:15)P i ( (cid:15) )˜ σ ppi ( p T + (cid:15) ) / ˜ σ ppi ( p T ) account for the energy loss by a jet with mo-mentum p T and size R during the passage of a back-ground medium [45]. In the limit of large power index n , we we invoke the asymptotic expansion ˜ σ pp ∝ ( p T + (cid:15) ) − n ∼ p − nT e − n(cid:15)/p T (cid:0) O ( n(cid:15) /p T ) (cid:1) , which to leadingorder allows us to identify the QF with the Laplace trans-form (LT) of the energy loss probability, i.e., Q ( p T ) = (cid:82) ∞ d (cid:15)P ( (cid:15) )e − ν(cid:15) | ν = n/p T , where we have omited the flavorsubscript for clarity.The factorization (2) reduces trivially to the jet pro-duction cross section in the absence of final-state interac-tions, Eq. (1), by setting the quenching factors to unity, Q i →
1, and replacing the nuclear PDFs by standardproton PDFs. It is justified by the fact that out of conevacuum evolution takes place at much shorter times thanenergy loss and was used as a basis for the extraction ofthe quenching weights from the data [46, 47].A novel ingredient of our setup are the quenching fac-tors Q i ( p T , R ) that resum contributions to the total en-ergy loss of a jet consisting of many color charges thatinteract with the medium. Every splitting that occursat short time scales within the medium, gives rise toan additional color current that can scatter with theplasma constituents and source further medium-inducedenergy loss. The magnitude of this effect can be gaugedby comparing the formation time of a splitting, t f =2 / [ z (1 − z ) p T θ ], to the characteristic time scale themedium needs to resolve the product of the splitting,namely t d = [ˆ q θ / − / [34–37]. Here, ˆ q ≡ d (cid:104) k ⊥ (cid:105) / d t is the transport coefficient that encodes medium proper-ties, the so-called jet quenching parameter [16]. Hence,jet splittings occurring at time scales much shorter thanthe related medium time scale, that is if t f (cid:28) t d (cid:28) L ,are unaffected by the medium and obey the same prop-erties as vacuum splittings [39, 40]. The latter inequalityimplies that a splitting with θ < θ c , where the criticalangle is θ c = (ˆ q L / − / , will not be resolved by themedium. With these considerations in mind, one canshow that in the large n limit, owing to the fact that theconvolution of energy loss probability distributions re-duces to a direct product of quenching factors (in Laplacespace), the evolution equation for the resummed quench-ing weight is [39] ∂Q i ( p, θ ) ∂ ln θ = (cid:90) d z α s ( k ⊥ )2 π p ( k ) ji ( z )Θ res ( z, θ ) × [ Q j ( zp, θ ) Q k ((1 − z ) p, θ ) − Q i ( p, θ )] , (3)where k ⊥ = z (1 − z ) pθ , p ( k ) ji ( z ) are the un-regularizedAltarelli-Parisi splitting functions and the phase spaceconstraint is given by Θ res ( p, R ) = Θ( t f < t d < L ).Above, it is understood that p is evaluated at p ≡ p T .This distinction is necessary when solving Eq. (3) sincethe initial condition also depends on p T . The non-linearevolution equations account for the energy loss of themulti-prong jet substructures that are generated by earlycollinear splittings.The initial conditions for the resummed quenching fac-tors Q i ( p, R ) at R = 0 are the bare quenching factorsfor single partons. In this work, we have Q i ( p,
0) = Q (0)rad ,i ( p T ) Q (0)el ,i ( p T ), where the two bare quenching factorsare the LT of the corresponding probability distributionsthat describe radiative and elastic energy loss [45, 48],For their precise definitions, see Eqs. (8) and (9) below.The radiative and elastic energy loss are driven by thetransport coefficients ˆ q and ˆ e [49], respectively, which arerelated by the fluctuation-dissipation relation ˆ e = ˆ q/ (4 T )in a weakly-coupled plasma [50] (where ˆ e g = ˆ e for glu-ons, and ˆ e q = C F N c ˆ e g for quarks). The quenching factordue to radiative energy loss off a single parton is simplythe exponential of the LT for a single inclusive gluon ra-diative spectrum [45, 48]. For our purposes, we shouldrather consider how single partons contribute to the en-ergy loss of the jet by accounting for the energy that istransported outside of the jet reconstruction cone. Tothis aim, we exploit the wide parametric angular separa-tion between the regime of soft emissions that undergoa rapid turbulent cascade responsible for transportingenergy from the jet scale to the medium temperaturewhere dissipation forces take over, and the regime of colli-mated semi-hard emissions, which experience broadeningthrough collisions with the medium constituents [51–55].The medium-induced gluon radiation spectrum hasbeen computed up to next-to-leading order (NLO) withinthe improved opacity expansion (IOE) in the soft limit[56–58] and unifies both the BDMPS approach with theGLV/higher-twist formalism [59, 60], which has proven tobe an important ingredient for phenomenological stud-ies [61]. The IOE was also shown to be very accuratewhen compared to exact numerical solutions [62]. It isexpressed as d I NLO / d ω = d I (0) / d ω + d I (1) / d ω , withd I (0) d ω = 2 α s C R πω ln | cos Ω L | , (4)d I (1) d ω = α s C R ˆ q π Re (cid:90) L d s − k ( s ) ln − k ( s ) Q e − γ E , (5)where Ω = (1 − i ) (cid:112) ˆ q/ (4 ω ), k ( s ) = i ω Ω2 [cot Ω s − tan Ω( L − s )], and the strong coupling constant runs withthe typical transverse momentum of the emission, i.e. α s = α s (cid:0) (ˆ qω ) / (cid:1) [63]. In this expansion, the effective transport coefficient ˆ q differs from the bare ˆ q by a fac-tor that reflects the full leading logarithmic contribution,i.e. ˆ q = ˆ q ln Q µ ∗ , (6) . . . . R = 0 . . . . . √ s = 5 .
02 ATeV R AA − − − − R AA Jet p T [GeV] 10 − − − FIG. 1. Calculation of inclusive jet R AA in PbPb collisions at √ s = 5 .
02 ATeV, compared to ATLAS data [65], for differentcentralities. where ˆ q = g N c m D T / (4 π ) for a thermal medium inthe Hard Thermal Loop (HTL) theory and the lower cut-off scale is µ ∗ = m D exp[ − γ E ] / m D computed at LO in a thermal medium reads m D = 3 g T / Q depends itself on the amount ofrescattering in the medium and can be found by solvingthe transcendental equation Q = ˆ q ω ln Q /µ ∗ [64]. Inour framework, the medium coupling g med , the only freeparameter that determines energy loss, is to be extractedfrom the comparison to experimental data.We first consider semi-hard gluons that are emit-ted within the range ω s < ω (cid:46) ω c , where ω c ≡ ˆ q ln(ˆ q L/µ ∗ ) L / ω s ≡ ( g N c / (2 π ) ) π ˆ q L is the energy scale at whichemission probability is of order one, determining the on-set of turbulent energy loss [66]. Their broadening dis-tribution reflects the typical transverse momentum kicksreceived in the plasma. The fact that the two termsentering the full NLO spectrum are dominated by differ-ent kinds of processes has to be reflected in the typicalbehavior of the respective broadening distribution. Inthis way, the softer gluons from Eq. (4), with ω (cid:28) ω c and small transverse kicks k ⊥ ∼ ˆ qL , will experienceGaussian broadening, while the harder emissions fromEq. (5), with ω (cid:29) ω c and typically large transversemomenta k ⊥ > ω/L (cid:29) ˆ qL , where the first inequalityarises from demanding that t f = ω/k ⊥ < L , are governedby a power-law behavior, ∼ ˆ q L/k ⊥ . We assume thatthe effect of broadening appears as a multiplicative fac-tor B (cid:0) ωR ; Q (cid:1) = (d I/ d ω ) − (cid:82) ∞ ( ωR ) d k ⊥ d I/ (d ω d k ⊥ ),representing the probability for the emitted gluon to betransported out to an angle larger than the jet cone, θ > R , where Q denotes a characteristic broaden-ing scale. This is concretely realized by integrating thebroadening probability distribution, derived in [67], forangles larger than the jet cone. This distribution andthe proposed factorized form correctly interpolate be-tween the multiple-scattering and higher-twist regimes.The full out-of-cone spectrum consists of two terms, fromthe IOE expansion up to NLO with their correspondingbroadening factors [52], such thatd I > d ω = B (cid:0) ωR ; Q s / (cid:1) d I (0) d ω + B (cid:0) ωR ; max (cid:2) Q s , ω/ ( π L ) (cid:3)(cid:1) d I (1) d ω . (7)Since emissions can take place anywhere along the in-medium path, one also has to average over the radi-ation time. This is approximated by simply setting Q = ˆ qL/ ω and k ⊥ regime up to a logarithmic factor that we neglect. Inbrief, d I/ (d ω d k ⊥ ) (cid:39) α ˆ q Lπ k ⊥ ( . . . ) ≈ ω/ ( π Lk ⊥ ) × d I (1) d ω ,where the ellipses represent the logarithmic contributionsand d I (1) / d ω ∼ π ¯ αq L /ω is the limiting behavior ofEq. (5) for ω (cid:29) ω c and k ⊥ (cid:29) ˆ qL . The B distributionwas used in the second term to insure proper normaliza-tion. For more details see the supplemental material.Soft gluons, with T < ω < ω s , cascade quasi-instantaneously to the thermal scale [66] and should ef-fectively be treated within hydrodynamics. Their emis-sion rate is therefore not affected by transverse momen-tum broadening. Assuming that their distribution be-comes approximately uniform in the solid angle aroundthe jet, we account for the possibility that a fraction ofthis energy ends up back in the jet cone by modifying ω → ω (1 − ( R/R rec ) ), where the recovery angle R rec isa free parameter [68]. An analogous modification is ap-plied to the elastic quenching factor. Emissions at ω < T belong to the Bethe-Heitler regime, and are not relevantfor our present phenomenological application [62, 69].Putting all the pieces together, the final expression forthe radiative bare quenching factor reads Q (0)rad ( p T ) = exp (cid:34) − (cid:90) ∞ ω s d ω d I > d ω (cid:0) − e − νω (cid:1) − (cid:90) ω s T d ω d I (0) d ω (cid:16) − e − νω (1 − ( RR rec ) ) (cid:17) (cid:35) , (8) where the parton flavor index is implicit and ν ≡ n/p T .We have approximated d I NLO / d ω (cid:39) d I (0) / d ω in the softregime. The bare quenching factor for elastic energy lossis Q (0)el ( p T ) = exp (cid:34) − ˆ eLν (cid:32) − (cid:18) RR rec (cid:19) (cid:33)(cid:35) , (9)also with implicit parton flavor dependence. It resultsfrom taking the LT of δ ( (cid:15) − ˆ eL (1 − R /R )). We shall seethat our results at small cone sizes are not very sensitiveto the above modeling of energy recovery. Numerical results.
Using the bare quenching factorsfor the radiative (8) and elastic (9) contributions to en-ergy loss, we numerically solve the coupled evolutionequations in Eq. (3). The cone-size dependence of thebare quenching factors, through the broadening effectsencoded in d I > / d ω and resulting in more energy loss forsmaller R , are to a large extent washed away by the evo-lution. This is because wider jets have a larger resolvedphase space and hence comprise more radiating chargesthan the narrower jets, effectively hampering energy re-covery.In our numerical computations, we fix the values of thetwo free parameters of our setup, g med and R rec . The en-ergy recovery parameter R rec has been varied between R rec = π/ R rec = (5 / π/
2, which was estimatedfrom a linearized approach to model the QGP wake for[70, 71]. To constrain g med we have compared our re-sults for the widely used nuclear suppression factor forjet production, also known as R AA , for jets with R = 0 . p T ∼
100 GeV against high-statistics experimen-tal data from ATLAS for the 0–10% centrality class ofPbPb collisions at √ s = 5 .
02 ATeV [65]. In order to com-pute R AA , we have taken the ratio between the nuclearand the vacuum spectra, both defined through Eqs. (1)and (2), which comprise a weighted sum of the quark andgluon jet contributions to the full spectrum [72]. Event-by-event in-medium path fluctuations of a jet throughthe QGP have been taken into account by embedding ourframework into a realistic heavy ion environment as simu-lated in the VISHNU hydrodynamical model [73], see thesupplemental material for further details. The value of g med is thus constrained by the experimental data to bewithin the range g med ∈ { . , . } . We emphasize thatthe magnitude of quenching is predominantly driven bythe emission of copious soft gluons [45], see also Tab. Ibelow. The extracted parameters yield an average value (cid:104) ˆ q (cid:105) (cid:39) .
41 GeV /fm in 0-10% central PbPb collisionsthat is well within the perturbative regime, see the sup-plemental material for more information on the centralitydependence of key parameters. However, the logarithmiccorrections to the bare medium parameters, resulting in Q = 14 . for the factorization scale and ˆ q = 2 . /fm, produce a relatively large maximal mediumenergy scale ω c ≈
65 GeV. These effects only become ap-parent when carefully treating the dominant scatteringregimes of the full spectrum, as achieved with Eqs. (4)and (5). . . . . √ s = 5 .
02 ATeV0-10% R AA ( R ) / R AA ( R = . ) Jet p T [GeV] R = 0 . R = 0 . R = 0 . R = 0 . FIG. 2. Double ratio of inclusive jet R AA for different jetradius R over R AA for R = 1 . √ s = 5 . θ c [ θ c / , θ c ] (cid:46) ∼ n ± ∼ R rec [1 , ∞ ] (cid:46) ω s [ ω s / , ω s ] (cid:46) R AA from varying key parameters for cone sizes R = 0 . − .
6, seetext for further details.
We show results for R AA as a function of jet p T andcentrality in Fig. 1, confronted against data from AT-LAS [65] for R = 0 . R AA around p T ∼ p T dependence and centrality evolution of jet sup-pression. We note that the downturn of R AA at thehighest jet p T is due to the nuclear PDF modificationsimprinted in the initial hard parton spectra of PbPb col-lisions. Finally, we quantify the R (in)dependence of jetsuppression by taking double ratios of the full resultsfor R AA as in Fig. 2, with the largest size R = 1 inthe denominator. Such notably mild dependence of jetsuppression with R is in agreement with ALICE resultsat low- p T [74] and with recent experimental preliminarydata from CMS at high- p T [75]. Summary and discussion.
We have provided a first,analytical description of the cone-size dependent jet spec-trum in heavy ion collisions at the LHC implemented in arealistic event-by-event setup including nuclear geometryand hydrodynamic expansion of the quark-gluon plasmaand accounting for multiple scattering effects. By ad-equately introducing the notion of single-parton energyloss within the context of a multi-parton object such asa jet, in which the phase space is determined not only bythe jet p T and size R but also on whether splittings areresolved by the medium as determined by color coher- ence effects, we have shown that the resummation of thequenching factors yield results for jet suppression thatare very mildly dependent on R . Our procedure to em-bed our formalism into a realistic heavy ion environmenthas proven successful, given in particular our good de-scription of the centrality evolution of high-statistics ex-perimental data. The various tools and formalism intro-duced in this work can be systematically improved andwill be applied to other jet substructure observables inheavy ion collisions in the future.While the error bands provided in the plots above stemmainly from the constraining power of the experimentaldata at R = 0 . g med , we wouldcurrently like to discuss the sensitivity of our results onthe various assumptions made in the setup in order toidentify the main sources of uncertainty. Our main find-ings are summarized in Tab. I for moderate cone sizes0 . ≤ R ≤ .
6, see also the supplemental material for afull scan of parameters and their centrality dependence.First, the inclusion of the higher-twist radiative spectrumin the IOE is of mild importance for this observable, sincesuch emissions typically occur at small angles, but it im-proves the description at high- p T . Furthermore, as ex-pected, notable bias effects can be identified through thestrong sensitivity to the power of the steeply falling spec-trum n , which point to the importance of higher orderterms in the large n expansion that can be calculatedsystematically. More importantly, comparing the effectof changing the hard phase space (through θ c ) and theparameters governing the behavior and recovery of softgluons (through ω s and R rec ), we note that an increasedprecision in the perturbative sector is still needed beforethe sensitivity to non-perturbative effects start to dom-inate. For instance, going beyond leading logarithmicaccuracy to compute θ c will be important to rigorouslystudy the interesting marked centrality dependence ofthis critical angle. The importance of the recovery pa-rameter R rec has been gauged between two limiting sce-narios of R rec = 1 corresponding to almost complete en-ergy recovery for large- R jets and R rec = ∞ correspond-ing to no energy recovery. Surprisingly, we find relativelylittle sensitivity to this parameter at these moderate conesizes. Conversely, the sensitivity becomes the dominantsource of uncertainty only at large- R , i.e. R ≈
1, jets.In conclusion, we have demonstrated how the cone-size dependent jet spectrum and R AA is largely governedby the energy loss off hard, resolved jet splittings in themedium through copious, soft gluon radiation and theirsubsequent broadening out of the jet cone and elasticdrag. The results obtained by analyzing the various com-ponents of our setup emphasize the importance of analyt-ical tools to guide more sophisticated numerical models,such as MC parton showers. Acknowledgements.
Y. M.-T. was supported by theU.S. Department of Energy, Office of Science, Office ofNuclear Physics, under contract No. DE- SC0012704, byLaboratory Directed Research and Development (LDRD)funds from Brookhaven Science Associates and by theRHIC Physics Fellow Program of the RIKEN BNLResearch Center. K. T. and D. P. are supported by a Starting Grant from Trond Mohn Foundation(BFS2018REK01) and the University of Bergen. [1] G. P. Salam, Eur.Phys.J.
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The broadening distribution is related to the dipolescattering amplitude via a Fourier transform. In a staticmedium it reads P ( k ) = (cid:90) d x e − i x · k − Q s x log x µ ∗ , (10)within the leading logarithmic approximation. Here, µ ∗ represents a model-dependent, infrared scale, see belowEq. (6) for its precise definition in the HTL model ofmedium interactions. Its asymptotic behavior are as fol-lows P ( k ) (cid:39) (cid:40) πQ s e − k /Q s k ⊥ (cid:28) Q πQ s k k ⊥ (cid:29) Q . (11)In the latter case, we have neglected additional logarith-mic terms. In the regime of multiple scattering, partonsplitting take place as a two-stage process where, first,the splitting products separate and decohere and, sec-ond, they broaden independently [37, 82]. Therefore, wecan easily propose a similar ansatz for the integrated,out-of-cone spectrum ω d I > d ω = (cid:90) ∞ ( ωR ) d k ⊥ ω d I d ω d k ⊥ , (cid:39) B (cid:0) ωR ; Q (cid:1) × ω d I d ω , (12)where B (cid:0) ωR ; Q (cid:1) = Q π (cid:82) ∞ y d x P ( x ) with y =( ωR ) /Q and Q broad is the characteristic broadeningscale.The gluon emission regime where ω < ω c is dominatedby multiple scatterings. For ω > ω c , the dominant con-tribution to the spectrum is provided by a single, hardscattering with the medium. In this case, the uninte-grated GLV ( N = 1) spectrum reads ω d I d ω d k = 8¯ α ˆ q (cid:90) d s (cid:90) d q (2 π ) k · qk ( k − q ) ( q + µ ) × (cid:20) − cos (cid:18) ( k − q ) ω s (cid:19)(cid:21) , (13)where k = ( k x , k y ) is the transverse momentum vectorand k ⊥ ≡ | k | . At large energy and transverse momen-tum, i.e. k ⊥ (cid:29) ω/L and ω (cid:29) µ L , the spectrum behavesas ω d I d ω d k ⊥ = 2¯ α ˆ q Lπ k ⊥ (cid:20) log k ⊥ L ω + log k ⊥ µ − γ E (cid:21) . (14)Given that the integrated spectrum in the limit ω (cid:29) ω c becomes ω d I d ω = ¯ α π q L ω , we can thus express the above unintegrated spectrum as a function of the integrated one ω d I d ω d k ⊥ ≈ ω d I d ω × ωπ L k ⊥ (cid:2) . . . (cid:3) , (15)where the ellipses correspond to the logarithmic correc-tions in Eq. (14). For the purpose of our applications, inthe following we will neglect such corrections.The two discussed limits can be accounted for by ex-panding the full potential around the harmonic oscilla-tor [67], where the x -dependent logarithm is neglectedin Eq. (10). The formula for broadening up to next-to-leading order in the improved opacity expansion reads P ( x ) = 4 πQ s e − x × (cid:8) − λ (cid:2) e x − − x ) (cid:0) Ei( x ) − log(4 x ) (cid:1)(cid:3)(cid:9) , (16)where we have defined x ≡ k /Q s , and the expansionparameter is λ = (ln Q s /µ ∗ ) − with Q s = ˆ q L ln Q s /µ ∗ found implicitly. For our purposes, λ is restricted to bewithin 0 and 1. After integrating out the angle, we get B ( y ) = − λ + e − y (cid:8) λ (cid:2) y (cid:0) Ei( y ) − log 4 y (cid:1)(cid:3)(cid:9) (17)with y ≡ ( ωR ) /Q s . In addition, we should also takean average of the production point. A good estimate, atleast for the leading order part, is to consider Q s → Q s / N = 1 spectrum at high ω is welldescribed by considering instead max (cid:0) Q s , ω/ ( π L ) (cid:1) . Bare and resummed quenching factors
We plot the bare and resummed quenching factors forquarks and gluons in Fig. 3 as a function of p T for valuesof the medium parameters corresponding to 0–5% cen-trality PbPb collisions at √ s = 5 .
02 ATeV. As discussedabove, these include both the effect of radiative and elas-tic energy loss, incorporated in the bare quenching factorsas Q (0) i ( p T ) = Q (0)rad ,i ( p T ) Q (0)el ,i ( p T ) , (18)where i = q, g . The p T dependence appears mainlythrough the ratio n i ( p T ) /p T , where n i ( p T ) is the power-law indices of the spectra of quark and gluon initiatedjets. These weight factors also constitute the initial con-ditions for the non-linear evolution in Eq. (3) for thefully resummed weights Q i ( p T , R ). The error bands cor-respond to varying the medium coupling g med ∈ [2 . , . R rec ∈ [5 π/ , π/ R of thebare quenching factors (dashed), with more energy lossfor smaller R , is no longer present in the resummed case(solid). The energy loss is even enhanced when com-paring R = 0 . R = 0 .
4. Both for quark initiatedjets (top) and gluon initiated jets (bottom), resummedquenching factors show only slight differences among thedifferent R . . . . .
81 0-10%00 . . . . √ s = 5 .
02 ATeV Q q u a r k R=0.2R=0.4R=0.6R=0.8R=1 Q g l u o n p T (GeV) BareResummed
FIG. 3. Bare and resummed quenching weights for quarks(upper panel) and gluons (lower panel) for central collisionsat √ s = 5 .
02 ATeV.
Embedding into a heavy ion environment
In order to account for the fluctuations in jet energyloss due to the different paths that the jet can explorethrough the expanding QGP, we need to embed our the-oretical framework into a realistic heavy ion background.The necessary steps are the following: • Sample the production point in the transverseplane ( x, y ) using the overlap of the thickness func-tions of the two nuclei, T AB ( x, y ; b ) = T A ( x − b/ , y ) T B ( x + b/ , y ), where b is the impact pa-rameter of the nuclear collision and the thicknessfunction is the transverse density of nucleons ofthe Lorentz contracted nuclei, distributed accord-ing to the Woods-Saxon density function (see [83]for more details on Glauber modeling). • Assign a random orientation in the transverse planeand a random value of rapidity within the range − ≤ y ≤ • Following the path of the jet, compute the inte-grated values of the necessary physical variables,which in general depend on the local temperature T and fluid velocity u , until the jet exits the QGPphase at T c = 145 MeV (possible energy loss effectsduring the hadron gas phase, which could be morerelevant for the softer particles [84–87], have been ignored for the moment and will be addressed inthe future). The values of T and u are read fromevent averaged hydrodynamic profiles for the evo-lution of an expanding droplet of liquid QGP [73]in PbPb collisions at √ s = 5 .
02 ATeV for differentcentrality classes.Given that quantities like the fluid temperature T aregiven in the local fluid rest frame, we need to considerthe actual distance traversed by the jet at each time stepwithin such reference frame [88]: dx F = d t (cid:113) (cid:126)v + γ F (cid:0) (cid:126)u − (cid:126)u · (cid:126)v + ( (cid:126)u · (cid:126)v ) (cid:1) , (19)where (cid:126)v ≡ (cid:126)p/E is the jet axis and (cid:126)u and γ F are thelocal fluid velocity and Lorentz factor, respectively. Upto numerical factors, the set of physical variables that weneed are found by integrating along the path Γ( t ) of a jetsuch as, for example: L = (cid:90) Γ( t ) dx F , (20)ˆ q ∝ L (cid:90) Γ( t ) dx F T ( x ) (cid:18) p · u ( x ) p (cid:19) , (21)and analogously for the rest of variables: T , m D , θ c , ˆ e and w c . Note that, due to the presence of a flowing medium,transport coefficients get a dilution factor ( p · u ) /p , where p is the four-momentum of the jet and u the fluid four-velocity [89].In this way, we obtain a set of representative historiesfor the jet in-medium path so we can use the event-by-event values of the path integrated physical variables tocompute jet energy loss jet-by-jet. Our final results areobtained by averaging over all configurations.To provide the reader some guidance on the typicalmagnitude of these quantities, we show in Table IIthe average values for several of the relevant physicalvariables that enter our calculation, as a function ofcentrality, where we set g med = 2 .
25. We point outthe interesting correlation between the presence ofdecoherence effects, for θ c < R , and complete coherence, θ c ∼ R , as a function of centrality. Sensitivity of results to systematic uncertainties
The calculation presented in this work is based on aperturbative description of hard, vacuum-like and semi-hard, medium-induced emissions. In addition, the regimeof soft gluon emissions where thermalization plays an im-portant role are modelled using parametric estimates. Inorder to systematically quantify the theoretical uncer-tainties imbued in our modelling, we have performed ascan varying most of the parameters that go into thecomputation. This is presented as a relative change in R AA at p T (cid:39)
110 GeV for several jet radius R for the0 Centrality ˆ q [GeV /fm] L [fm] θ c w c [GeV] w s [GeV] ˆ e [GeV/fm] m D [GeV] T [GeV]0–5% 0.46 5.9 0.11 80.4 21.2 1.35 0.74 0.2485–10% 0.43 5.6 0.13 63.2 17.2 1.25 0.73 0.24610–20% 0.41 5.0 0.15 49.3 13.9 1.17 0.72 0.24220–30% 0.38 4.4 0.18 35.4 10.4 1.08 0.70 0.23830–40% 0.34 3.9 0.23 23.7 7.4 0.95 0.68 0.23140–50% 0.29 3.3 0.28 15.7 5.2 0.82 0.65 0.2250–60% 0.25 2.8 0.36 9.7 3.5 0.69 0.61 0.2160–70% 0.20 2.2 0.47 5.4 2.2 0.54 0.57 0.20 TABLE II. Average values of the physical quantities entering our calculation. The strength of the QCD coupling constant hasbeen set to g med = 2 . R n + 1 n − w s / w s θ c / θ c g − . g + 0 . R rec = 1 R rec = ∞ w/o NLO0 . . . . . . . . . TABLE III. Relative change in R AA around p T (cid:39)
110 GeV, compared to the central results that use g = 2 .
25 and R rec = π/ √ s = 5 . g med = 2 .
25 and R rec = π/
2, as shown in Ta-ble III. Clearly, R AA is sensitive to the index n of thehard, steeply falling spectrum. This is a reflection of thebias effect related to jet selection. Next, a large sensitiv-ity is reported in the variation of θ c in the phase spaceof the resummed quenching factors. This points to theimportance of pushing beyond the leading logarithmic precision presently implemented. We also report littlesensitivity to the precise value of the transition betweenthe semi-hard and soft gluon regimes, encoded in the pa-rameter ω s . Finally, we observe that even though we vary R rec between two extreme values such as R rec = 1 (fullrecovery of energy at R = 1) and R rec = ∞ (no recoveryat all, for any R ), these non-perturbative effects remainmoderate ( (cid:46) R = 0 ..