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Jet (de)coherence in Pb-Pb collisions at the LHC Yacine Mehtar-Tani∗ Institut de Physique Théorique, CEA Saclay, F-91191 Gif-sur-Yvette, France Konrad Tywoniuk† Departament d’Estructura i Constituents de la Matèria and Institut de Ciències del Cosmos (IC-CUB), Universitat de Barcelona, Mart´ı i Franquès 1, 08028 Barcelona, Spain (Dated: July 15, 2015) We study the modifications of jets created in heavy-ion collisions at LHC energies. The inherent hierarchy of scales governing the jet evolution allows to distinguish a leading jet structure, which interacts coherently with the medium as a single color charge, from softer sub-structures that will be sensitive to effects of color decoherence. We argue how this separation comes about and show thatthispictureisconsistentwithexperimentaldataonreconstructedjetsattheLHC,providinga 4 1 quantitativedescriptionsimultaneouslyofthejetnuclearmodificationfactor,themissingenergyin 0 di-jeteventsandthemodificationofthefragmentationfunctions. Inparticular,wedemonstratethat 2 effectsduetocolordecoherencearemanifestintheexcessofsoftparticlesmeasuredinfragmentation functions in Pb-Pb compared to proton-proton collisions. n a J There is compelling evidence that a hot and dense tions due to partial color decoherence of its constituents 1 quark-gluonplasma(QGP)iscreatedinultra-relativistic [12]. The former component is responsible for radiative 3 heavy ion collisions [1]. Jets emerging from these colli- energy loss and, in particular, for transporting this en- ] sions are unique probes of the underlying dynamics [2]. ergyuptoverylargeangles. Thelatterenhancesthemul- h The features of a jet, such as its total energy p [19], de- tiplicity of soft particles inside the jet, and hence is as- p ⊥ rive from its constituents included within a reconstruc- sociated with radiation at relatively smaller angles. The - p tion radius Θ . The distribution of hadrons in the jet basicparametersofthetheoryarerelatedtothemedium jet e is sensitive to interactions with the dynamical medium properties via the transport coefficient qˆ, which interre- h [3]. Experimental results from Pb-Pb collisions at the latesthemomentumbroadeningandenergyloss,andthe [ LHC (√sNN = 2.76 TeV) have to date provided exten- in-medium mean free path λmfp. In addition, jet observ- 1 sive data of the modifications of fully reconstructed jets ables depend upon the geometry of the collision mainly v with respect to their proton-proton baseline. On the one through the mean path length L for jet propagation. 3 9 hand, a strong suppression of about 50 % of the in- Relying on the collimation property of high-energy ∼ 2 clusive jet yield is observed in central collisions across a QCDjets[13], wewillassumethroughoutthisworkthat 8 wide range in jet energy [4]. On the other hand, while themediumdoesnotresolvetheleadingsub-structuresof . 1 the angular correlation of di-jet events is consistent with thejet. Inthiscase,thejetinteractscoherentlywiththe 0 that in vacuum, the observed energy balance is strongly medium only via its total color charge and the ensuing 4 distorted and the missing energy of the leading jet is medium-induced branchings shift the emitted energy up 1 recovered only at very large angles with respect to the : toverylargeangleswithrespectthethejetaxis. Thisge- v di-jetaxis[5]. Thejetfragmentationstudiesprovidefur- ometrical (angular) separation between vacuum-like and i ther detail, confirming to a large extent that the hard X medium-induced branchings allows us to treat these two jet components escape the medium without large modi- r typesofprocessesindependentofeachotherandleadsto a fications while the soft components, typically occupying Eq. (6), to be discussed in further detail below. Devia- abroadangularrangewithinthejet,areenhanced[6–9]. tions from this simple picture are related to decoherence of vacuum-like radiation and corresponds to situations In this letter, we present a comprehensive analysis of when probing sub-structures of the jet that are being re- in-mediumjetmodificationsbasedonperturbativeQCD. solvedbythemedium[3],seeEq.(7). Wearguethatthis We argue that a consistent picture that accommodates correction is essential for explaining key features of soft thethreemaintrendsobservedinthedataemerges. This particles within the jet cone. The above considerations comes about due to the large separation of the intrinsic set out a strategy for enhancing the use jets to pin down jet scale Q Θ p and the characteristic momentum jet ⊥ ≡ medium effects. scale of the medium Q , to be defined below, such that s Q Q Λ , where Λ is the non-perturbative The inclusive spectrum of reconstructed jets in Pb- s QCD QCD (cid:29) (cid:29) scale ofQCD. Thisscale separationis also intimatelyre- Pb is suppressed compared to that in proton-proton lated to the interplay of two main mechanisms: induced [4]. At high p this is caused by energy loss due to ⊥ independent gluon radiation off the coherent jet [10, 11], mediuminducedradiation[10]. Presentlywewillassume whichstandsforthedominantmediumeffect,andcorrec- that these jets are mainly induced by primary quarks 2 and parameterize the vacuum spectrum by a power law, d2σjet/d2p p−n,withtheexponentn 5.6extracted ωc=60-100GeV,ωBH=1.5GeV p-p ⊥ ∝ ⊥ (cid:39) ωc=80GeV,ωBH=0.5-2.5GeV from experimental data [14]. The nuclear modification 1.2 CMSPrelim. factor is defined as d2Njet (p ) d2p 1 Rjet Pb-Pb ⊥ ⊥ , (1) AA ≡ T d2σjet(p ) d2p AA p-p ⊥(cid:14) ⊥ 0.8 tA where T is the nuclear overlap fu(cid:14)nction. The inclu- jeA AA R sive spectrum of jets after passing the medium can be 0.6 computed by convoluting the jet cross-section in vacuum, proportional to the quark cross-section, with the 0.4 distribution of quarks, Dmed after passing through the q medium, 0.2 d2Njet (p ) 1dx p d2σjet p⊥ Pb-Pb ⊥ Dmed x, ⊥,L p-p x (,2) 0 T d2p (cid:39) x q x d2p 50 100 150 200 250 300 350 400 AA ⊥ (cid:90)0 (cid:16) (cid:17) (cid:0)⊥ (cid:1) p [GeV] ⊥ where x is the fraction of the original quark energy carried by the quark after escaping the medium. For sim- FIG. 1. Calculation of the quenching factor with ω = 80 plicity, the geometry of the collision is accounted for on c GeV, Eq. (1), as a funtion of jet p for central Pb-Pb col- ⊥ averageintermsofaveragedvaluesforLandqˆ. Medium lisions. The dark (red) band includes the variation of ω BH effects due to induced radiation encoded in the distribu- around a central value of 1.5 GeV. The light (grey) band in- tion of quarks are found by solving the following kinetic cludes, in addition, a variation of ω ∈ [60, 100] GeV. The c rate equation [11, 15] experimental data are taken from [4]. ∂ 1 x Dmed(x,p ,L)= dz z, p ;L branchingtime,i.e.,t t +λ andreferto[11,17] ∂L i ⊥ Kij z ⊥ br → br mfp x (cid:90)0 (cid:16) (cid:17) for further details on the derivation of Eq. (3). Dmed ,p ,L zDmed(x,p ,L) , (3) The extracted distribution Dmed(x,p ,L) of quarks × j z ⊥ − j ⊥ q ⊥ originated from a quark is used to compare the results (cid:104) (cid:16) (cid:17) (cid:105) where the partonic distributions are xdNmed/dx fromEq.(1)withexperimentaldataonthenuclearmodi- Dmed(x,p ,L) with i = q,g [20]. The equatiion is gov≡- ficationfactoroffullyreconstructedjetsin0–10%central i ⊥ collisions from CMS [4]. We have allowed ω to vary erned by the branching rate, of parton j into parton i, BH between 0.5 and 2.5 GeV to gauge the uncertainty re- perunittime, (z,p ;t),whichcanbederiveddirectly ij ⊥ K lated to the infrared sector. This allows us to extract a from the one-gluon emission spectrum [10], value of ω = 80 GeV, see Fig. 1. For the purpose of c Ldt (z,p ;t)= αsP (z) ln cos(1+i)L , (4) illustration, we have also studied the sensitivity to ωc by Kij ⊥ 2π ij 2t allowing it vary around the central value, see Fig. 1. (cid:90)0 (cid:12)(cid:12) br (cid:12)(cid:12) In order to settle on a self-consistent set of parame- where αs is the strong coupling const(cid:12)(cid:12)ant (in this(cid:12)(cid:12)work, ters, we will from here on use a mean jet path length αs = 0.5 [16]), Pij(z) are the (unregularised) Altarelli- of L = 2.5 fm for 0–10% central Pb-Pb collisions. This Parisi splitting functions and t z(1 z)p /qˆ is choice is slightly reduced compared to the typical root br ⊥ eff ≡ − the branching time where z is the fraction of the energy mean square of the nuclear overlap in central Pb-Pb col- (cid:112) of parton j carried by parton i. Finally, the effective lisions motivated by the inherent surface bias of inclu- transportcoefficientprobedincourseofthebranchingis sive jet observables [18]. The value of L together with qˆ = 1 1+z2+[2C (j)/C 1](1 z)2 qˆ, where C (j) the extracted value of ω allows to relate all remaining eff 2 2 A− − 2 c is the color factor of the parton with label j, and qˆ is medium parameters. We notice further that all relevant (cid:0) (cid:1) consistently referring to the quenching parameter in the parameters vary only mildly within the range of relevant adjoint representation. L values and can therefore be expected to be well de- The form of the branching rate employed in this Let- scribed by their average values. For example, we extract ter, Eq. (4), is valid in the multiple scattering regime. the average transport coefficient qˆ= 5.1 GeV2/fm. It is characterized by the maximal gluon induced energy A crucial feature of the rate equation Eq. (3) is that it ω = qˆL2/2. The spectrum is regulated in the infrared describesquasi-democraticbranchingsofsoftgluonsand c when t is of the order of the mean free path λ , leads to turbulent flow of energy up to large angles [11]. br mfp which corresponds to the Bethe-Heitler (BH) frequency A particularly suited observable to study these effects is ω =qˆλ2 . We model this regime by regularizing the therefore the fraction of jet energy still remaining inside BH mfp 3 a cone defined by the jet reconstruction radius. We cal- to Λ by invoking the Local Parton-Hadron Duality QCD culate this quantity by hypothesis. The resulting parton spectrum can then be directly compared to hadron spectra by introducing an (θ <Θ ) 1dx Θjetdθ xdNimed , (5) energy independent scaling factor. E jet ≡(cid:90)0 (cid:90)0 i=q,g dθdx The collimation property of vacuum jets can be in- (cid:88) ferred directly from the fact that Dvac only depends on whichsumstheenergyofpartonsinsidethejetcone,i.e., the jet energy and cone angle in terms of Q, which is θ <Θjet. In terms oftransverse momenta this limitation the largest scale of the process. The separation of in- corresponds to k⊥ < xQ. On the other hand, the typi- trinsic jet and medium scales allow to find the modified caltransversemomentum ofaparton propagatinginthe fragmentation function directly via the jet calculus rule, plasma is given by the characteristic scale Q = √qˆL. s Hence, the angular condition can be turned into a con- 1 dz x Dcoh(x;Q,L)= Dvac ;Q Dmed(z,p ,L)(,6) dition on the parton energy, x > x0, where x0 Qs/Q. med z z q ⊥ Hence, we shall approximate, (θ <Θ ) (≡x>x ). (cid:90)x (cid:16) (cid:17) jet 0 E ≈ E In our case Qs = 3.6 GeV severely restricts the amount where Dqmed(x,p⊥,L) is the distribution of primary of soft induced radiation that is allowed within the cone. quarks [21]. Here we point out two crucial points con- The description of broadening will be discussed in more cerning Eq. (6). First and foremost, the subscript of the detail in a forthcoming work, see also [17]. resulting distribution refers to the coherent jet (color) We have computed the in-cone energy fraction for two structure that survives the medium interactions at this jet reconstruction angles using the previously extracted level of approximation. In other words, vacuum and medium parameters within the uncertainty due to the medium fragmentation take place independently of each variation in ωBH. We find that up to 14–19% of the other and are governed by separate evolution equations. energy flows out a cone of Θjet = 0.3 (x0 = 0.12). We Secondly, we have also neglected the variation of the in- scarcely recover more energy by opening the jet cone to trinsicjetscalewhichcomesaboutduetotheenergyloss Θjet =0.8 (x0 =0.045), in which case roughly 9–15% of atlargeanglesdiscussedabove. Asthiswasestimatedto the energy is still missing. This confirms that multiple contribute to a 20% variation to the jet scale, we will ∼ branching in the medium is an effective mechanism that allow for a certain variation of the jet energy scale of the transportsenergyfromhardtosoftquantaatlargeangles medium-modified jets. [11]. The results obtained here agree qualitatively with Remarkably, the simple picture incorporated in the estimates from the CMS collaboration on the out-of- Eq. (6), which has shown to be quite consistent up to coneenergyflowfordi-jetswhereitwasobservedthatthe now,breaksdowninthesoftsector(cf. greybandinFig. energyimbalancecouldberecoveredonlyatangleslarger 2). Thiscanbetracedbacktothetransversemomentum than0.8andwerecarriedbytrackswith0.5GeV<p⊥ < broadening of soft quanta, Eq. (5), which practically re- 4 GeV [5]. Moreover, the typical transverse momentum moves them from the cone. However, by comparing the broadening of the coherent jet due to scatterings in the minimal angle for induced radiation Θ = (qˆL3 12)−1/2 c plasmaisoftheorderofQs. Hence,onecanestimatethe [10,11],whichwithoursetofparameterscorrespondsto angular deviation of the sub-leading jet to be ∆Θjet 0.08,tothetypicaljetreconstructionradius,p(cid:14)resently Qs/p⊥ ∼ 0.036 for a jet p⊥ = 100 GeV. We note tha∼t, ∼considered to be Θjet = 0.3. This implies that sub- ∆Θjet Θjet, in agreement with the observation that leading structures of the jet are resolved by the medium (cid:28) most di-jets are back-to-back. [3, 12]. Postponing for the moment a more refined treat- Finally, we focus on the modifications of the frag- mentofjetenergyloss,wewillratheremphasizehowthis mentation functions of jets. Concretely we will con- breakdown of jet color coherence, initially studied in [3], centrate on the so called intra-jet energy distribution demandsamoresubtleandnoveltreatmentofsoftgluon of hadrons dNvac dln(1/x) Dvac(x;Q) which is typ- emission at relatively small angles. ≡ ically plotted in (cid:14)terms of the variable (cid:96) = ln(1 xh) Up to now, we have neglected the fact that the jet- where x = x2+(m /p )2 and x are ratios of the mediuminteractionsgiverisetoadditionalradiationthat h h ⊥ (cid:14) hadron and parton energies to the jet energy, respec- violates the strict AO of the vacuum shower [12]. Since (cid:112) tively. The Q dependence of Dvac is governed by the this component is geometrically separated from the AO Modified Leading-Logarithmic Approximation (MLLA) vacuum-like radiation and associated with large forma- evolutionequations[13]whichtakeintoaccountthedou- tion times, it is therefore not affected by the medium ble logarithmic contributions (DLA) as well as the full (e.g. by transverse momentum broadening). Note that set of single logarithmic corrections. One of the key fea- since this contribution also is subleading in DLA, it is tures of this evolution is the angular ordering (AO) of enoughtoincludetheeffectfromthefirstnontrivialsplit- subsequent emissions which is a manifestation of color ting. Thisallowsustoaddthiscontributionincoherently coherence. The evolution takes place between the jet to the full, medium-modified intrajet distribution. The scale Q and the hadronization scale Q which can be set intrajet distribution in heavy-ion collisions can thus be 0 4 Fig. 2,whilethelowerpaneldetailstheratioofthelatter 5 Coh to the former. We compare to experimental data from Coh+Decoh 4 Vacuum CMS for jets with p⊥ > 100 GeV [8]. First, the vacuum CMSPrelim.: medium,0-10% baseline data are reproduced by the MLLA equation by CMSPrelim.: vacuum dℓ 3 adjusting the relevant parameters (Q0 = 0.4 GeV, mh = (cid:30) 1.1 GeV and K = 1.6) to optimize the description, de- N d 2 picted by a sold (blue) line in the upper part of Fig. 2. Due to the energy loss in the medium, we have allowed 1 the jet scale of the medium-modified jets to vary within 0 E [100,125] GeV (we plot the results for the extreme Coh ∈ cases). Inwhatfollows,thevariationoftheBHfrequency 2 Coh+Decoh CMSPreliminary,0-10% was found to be negligible and the central value ω = BH o 1.5 1.5 GeV was used. The result of using only Eq. (6), ti a depicted by the dashed (grey) lines, which assume co- R 1 herent radiation, yields a suppression of the distribution at all (cid:96) as compared to that in vacuum. This reflects 0.5 the energy loss via soft gluon radiation at large angles 0 1 2 3 4 5 off the total charge of the jet and is in agreement with ℓ the suppression of the nuclear modification factor. How- ever, the data indicates that the suppression turns into an enhancement when (cid:96) (cid:38) 3 in the most central col- FIG.2. Upperpanel: thelongitudinalfragmentationfunction plotted as a function of (cid:96)=ln1(cid:14)x. Lower panel: the ratio of lisions [8]. Accounting for color decoherence as given in medium-modified and vacuum fragmentation functions. The Eq.(7)wedescribetheexcessofsoftparticlesinthemea- experimental data are taken from [8]. See text for further sured medium-modified fragmentation function, see the details. thin-solid (red) curves in Fig. 2. The resulting ratio of medium-to-vacuum distributions show the characteristic dip and enhancement behavior with increasing (cid:96) around written as the sum of two components, the humpbacked plateau. Note that the MLLA equation Djet (x;Q,L)=Dcoh(x;Q,L)+∆Ddecoh(x;Q,L),(7) isvalidatintermediatevaluesof(cid:96)andthattheregionof med med med small (cid:96)(cid:46)1 is sensitive to energy conservation and hence where Dcoh is the coherent modified jet spectrum found med should be discarded. On the other hand, for (cid:96) 4.5 the fromEq.(6)andthedecoherenceofin-conevacuumradi- ∼ distribution in reaching the limits of phase space and is ationiscontainedin∆Ddecoh. Wecomputetherealcon- med verysensitivetonon-perturbativephysicsandtheprecise tribution at two successive emissions at DLA accuracy jet energy scale. with the inclusion of running coupling effects, yielding Tosummarize,wehaveinvestigatedseveraljetobserv- E dω(cid:48) Θjet dθ(cid:48) ablesthathaverecentlybeenmeasuredattheLHC.Our ∆Ddecoh(x;Q,qˆ,L)= model based on the QCD limit of color coherence is con- med ω(cid:48) θ(cid:48) (cid:90)ω (cid:90)Q0/ω sistentwiththedifferentfeaturesseenindataandweare θmax dθ able to pin down departures from this picture in the soft ∆ (θ(cid:48))α (ω(cid:48)θ(cid:48)) α (ωθ), (8) × med s θ s sector of fragmentation functions, which we argue is an (cid:90)θ(cid:48) evidence for partial decoherence. Our approach further where the decoherence parameter reads ∆ (θ(cid:48)) = 1 med shows how jets produced in these collisions can be used − exp[ θ(cid:48)qˆL3/12] [12] and θ = min(Θ ,Q /ω). In max jet med asapowerfultooltoextractinformationabouttheQGP − this context, Q = max (θ(cid:48)L)−1,Q is the hardest med s and color coherence. scale of the splitting. To test the sensitivity of the re- (cid:2) (cid:3) sulting distribution to this parameter we have varied Q while keeping Q at the central value such that med s 0.8 < lnQ Q < 3.2. As a further refinement, we med 0 will also demand that the first splitting occurs inside (cid:14) ACKNOWLEDGMENTS the medium. This puts a constraint on the formation time of the first gluon, i.e., t(ω(cid:48)) (ω(cid:48)θ(cid:48)2)−1 < L. For f (cid:39) consistency, we will also count the traversed path length K. T. is supported by a Juan de la Cierva fellowship from the production point by shifting L L t(ω(cid:48)) in andbytheresearchgrantsFPA2010-20807,2009SGR502 f → − ∆ (θ(cid:48)). and by the Consolider CPAN project. Y. M. -T. is sup- med The resulting vacuum and medium distributions for portedbytheEuropeanResearchCouncilundertheAd- jets with Q = 30 GeV are shown in the upper panel of vanced Investigator Grant ERC-AD-267258. 5 [11] J. -P. Blaizot, E. Iancu and Y. Mehtar-Tani, Phys. Rev. Lett. 111, 052001 (2013) [12] Y. Mehtar-Tani, C. A. Salgado and K. Tywoniuk, Phys. ∗ [email protected] Rev. Lett. 106, 122002 (2011) † [email protected] Y. Mehtar-Tani, C. A. Salgado and K. Tywoniuk, Phys. [1] D. d’Enterria, arXiv:0902.2011 [nucl-ex]. Lett. B 707, 156 (2012) [2] Y.Mehtar-Tani,J.G.MilhanoandK.Tywoniuk,Int.J. [13] Y. L. Dokshitzer, V. A. Khoze, A. H. Mueller and Mod. Phys. A 28, 1340013 (2013) S. I. Troian, “Basics of perturbative QCD,” Gif-sur- [3] J. Casalderrey-Solana, Y. Mehtar-Tani, C. A. Salgado Yvette, France: Ed. Frontieres (1991) and K. Tywoniuk, Phys. Lett. B 725, 357 (2013) V.A.KhozeandW.Ochs,Int.J.Mod.Phys.A12,2949 [4] G.Aadetal.[ATLASCollaboration],Phys.Lett.B719, (1997) 220 (2013) [14] A. 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