Nuclearand ParticlePhysics Proceedings NuclearandParticlePhysicsProceedings00(2017)1–7 Parton energy loss in QCD matter KonradTywoniuk TheoreticalPhysicsDepartment,CERN,Geneva,Switzerland 7 1 0 2 Abstract n QCDjets,producedcopiouslyinheavy-ioncollisionsatLHCandalsoatRHIC,serveasprobesofthedynamics a J of the quark-gluon plasma (QGP). Jet fragmentation in the medium is interesting in its own right and, in order to extractpertinentinformationabouttheQGP,ithastobewellunderstood. Wepresentabriefoverviewofthephysics 1 2 involved and argue that jet substructure observables provide new opportunities for understanding the nature of the modifications. ] h Keywords: QCDJets,JetQuenching p - p e 1. Introduction tomultiplechargesaccountingforpossibleinterference h effects. WewillreviewtheformeraspectsinSec.2and [ The study of perturbative probes of the quark-gluon the latter in Sec. 3. We will also discuss the applica- 1 plasma, and QCD jets in particular, is currently in its tionofthemediummodificationsonthelevelofjetsub- v golden age with the development of jet reconstruction structuremeasurementsinSec.4. Thisdiscussionisin 3 techniques for heavy-ion collisions at LHC and RHIC, nowaymeanttobeexhaustivebutwillimmediatelyil- 7 seee.g. [1,2,3]andtheseproceedings. Thesemeasure- lustrate the importance of whether sub-jets are treated 0 ments provide in many ways a more rigorous connec- as independent or coherent. Jet substructure provides 6 0 tionbetweenexperimentalmeasurementsandtheoryor therefore a new handle on the dynamics that can help . Monte-Carlo studies because of the implicit resumma- pinpointthemicroscopicprocessesunderlyingthemea- 1 0 tion of collinear divergences. On the other hand, suc- suredmodifications. 7 cessfuljetreconstructionintheextremeenvironmentof The choice of focus here is of course a biased se- 1 heavy-ioncollisionsischallengingandcomparisonsbe- lection, and not all recent progress in the field can be : v tweenmodelsanddatashouldbedonewithcare[4,5]. covered. A very interesting topic which deserves fur- i Until recently most studies, both experimental and ther study is the back-reaction of medium dynamics X phenomenological,dealtwithjetanddi-jetratesaswell to the propagation of the jet, see e.g. [48, 8, 9, 10]. r a as inclusive properties of jets, fragmentation functions While these aspects certainly are important for quanti- andjetshapes,andmeasurementsoflarge-angleenergy tative comparisons to experimental data, we currently flow around jets. However, novel measurements of jet have not much to say about their qualitative features. substructuresinnuclearcollisions[6]haverecentlyin- WesummarisebrieflyinSec.5. vigorated the discussion and opened new possibilities formeasuringandunderstandingmediummodifications 2. Radiativepartonenergyloss ofjets. The alleys of recent progress can predominantly be A single hard parton traversing a coloured medium categorisedaccordingtotwochiefaspectsofin-medium undergo successive elastic interactions which modify jet physics. Firstly, the propagation of a single colour their kinematics, mainly leading to the transverse mo- chargeinthemediumand,secondly,thegeneralisation mentum broadening (cid:104)k2(cid:105) = qˆL, characterised by the ⊥ /NuclearandParticlePhysicsProceedings00(2017)1–7 2 parameter qˆ in a medium of length L. The most ef- thefirstlineofEq.(1),thisdistributionbecomes ficient energy degradation mechanism is therefore re- (cid:114)ω (cid:20) πω (cid:21) alisedthroughanenhancedrateofsplitting. Assuming D ((cid:15))= s exp − s , (2) multiplesoftscattering,thespectrumofinducedquanta QW (cid:15)3 (cid:15) with energy ω radiated off a hard gluon is strongly where a more realistic form can be tabulated [25]. It cut-off at a characteristic energy ω ≡ qˆL/2 and reads c canrelatethejetspectruminthepresenceofamedium [11,12,13,14] tothatinvacuum,dN /dp2,as jet(0) T ωdNdBωDMPS =α¯ (cid:113)1 (cid:16)2ωωωcc(cid:17)2 ωω<>ωωc , (1) ddNpj2Tet =(cid:90)0∞d(cid:15)DQW((cid:15))dNjet(0d)(pp2TT +(cid:15)). (3) 12 ω c This allows to calculate the quenching factor Q (p ) QW T where α¯ ≡ 2αsNc/π. For further details and refine- astheratioofmediumtovacuumspectra. ments, see e.g. [15, 16]. The behaviour in the soft While the physics of transverse momentum broad- sector is characteristic of the Landau-Pomeranchuk- ening and radiative energy loss has been known for a Migdal (LPM) interference between scattering centres while, recently progress has been made toward under- and arises because the formation time of the gluon standing their respective radiative corrections [27, 28, (cid:112) scales as t = ω/qˆ. One can also find a compact an- 29]. Usually, one assumes that the interactions with f alytical expression for uncorrelated scatterings, the so- the medium are quasi-instantaneous (with respect to called“firstorderinopacity”spectrum[17,18];inthis the relevant timescales). However, allowing for short- case,theLPMeffectsuppressesthehardsector. lived, and thus soft, fluctuations one finds corrections The parameter ω determines the energy of gluons whichcanmostnaturallyberecastascorrectionstothe c that have been broadened along the whole medium medium parameter qˆ. For instance, the first double- length and are emitted at the minimal angle (qˆL3)−1/2. logarithmiccorrectionreads It is also controls the mean energy loss (cid:104)∆E(cid:105) ∼ qˆL2. α N L These emissions are rare O(α ), though. However, the ∆qˆ (cid:39) s cqˆln2 , (4) s energy scale ω = α¯2ω determines the regime when 2π l0 s c we have to take into account multiple branchings, i.e. where the shortest timescale l is some cut-off scale. (cid:82) 0 dωdN /dω > 1. Since their formation times is The inclusion of these fluctuations to all orders leads ωs BDMPS shorter than the medium length, a cascading process toarenormalisationequationthataccountsforatower takes place which transports these gluons to large an- offluctuations,orderedinformationtime,andtakesone gles,θ > α¯−2(qˆL3)−1/2. Whenwereconstructtheenergy fromthevalueofqˆ(l0),i.e. describingthemicroscopic oftheleadingpartoninacone,thiseffectisresponsible propertiesofthemediumatscalel ,toqˆ(L),whichin- 0 forsizeableenergyleakage[19,20,21,22]. cludes the contribution from additional fluctuations in To get a clearer picture, let us put some numbers themedium. Foralargemed√ium,qˆ(L) ∝ Lγ wherethe on these equations. For L = 4 fm, qˆ = 1 GeV2/fm anomalous dimension γ = 2 α¯ [28]. This novel rela- and α¯ = 0.3, we find ω = 80 GeV and ω = 7 tionaffectshowboththeaveragetransversemomentum c s Gev. For this energy range, the corresponding range broadening and energy loss scale with the size of the of emission angles, estimated from momentum broad- medium. (cid:112) ening as θ ∼ qˆL/ω, yields 0.025 < θ < 0.28. BDMPS ForajetreconstructedinaconeofR=0.3,thistypical 3. Interferenceinmulti-gluonprocesses choiceofmediumparametersindicatethatrareandhard BDMPSemissionspopulatethein-conejetdistribution The“running”ofqˆisanexampleofaresummationof whilemultiplebranchingtransportenergyout-of-cone. fluctuations in the medium that overlap. In this partic- Thedetailsofthissoftcascadehasbeenstudiedinquite ularsituation, thefluctuationsarestronglyorderedand somedetailanditsconnectionwiththephysicsofther- caneasilyberesummed.However,onecouldworrythat malisation has been highlighted [23]. We will come in other situations, multiple fluctuations that interfere backtothisinsightinSec.4. which each other would arise and thus ruin the proba- As a reminder, we note that the soft emissions can bilistic picture of independent emissions that underlies be resummed into a probability distribution, called the muchofthediscussionintheprevioussection. Besides, quenching weight (QW), of losing a finite amount of asknownfromjetphysicsinvacuum,thesecorrections energy[24,25,26]. Takentheformofthespectrumin givecrucialinputtoMonte-Carloshowergeneratorsof /NuclearandParticlePhysicsProceedings00(2017)1–7 3 thefragmentationprocessandwouldserveforthesame colour sources will ultimately be resolved and permit- purpose for dedicated generators of jets in heavy-ion ted to radiate. After this particular time one therefore collisions. findsanadditionalcontributiontothespectrum,namely Thetwo-gluonrateinadensemediumwascalculated thatofanemissionspectrumoffanon-shellcolourcur- in a series of noteworthy works [30, 31, 32, 33]. They rent (Gunion-Bertsch spectrum). The timescale where provided an independent confirmation of the double- thepositivecontributiontotheratesetsinissimplythe logarithmic contributions discussed above. For most formation time of the hard gluon. Hence, the decoher- configurations the corrections to the probabilistic pic- ence time is equal to the formation time or, in other ture were small except whenever the gluon energies words, the hard gluon gets resolved immediately after were strongly separated, i.e. one gluon being much emission. softer than the other. Strikingly, in this case the found In the second limit, see Fig. 1b, the formation time correctionswerenegativeimplyingareducedrate. This ordering is reversed. This happens whenever the en- canbeinterpretedasaninterferenceeffectowingtothe ergy of the soft gluon is small. In this case, the phys- fact that, from the viewpoint of the shortest-lived fluc- icalpictureisquiteintuitive: theparentpartonandthe tuation,theparentpartonandtheother,relativelylong- hardgluonformadipolethatinteractandradiateinthe livedfluctuationcannotberesolved[31].Physicallythis medium.Infact,onerecoversexactlythespectrumoffa means that the shortest fluctuation can only be emitted colourcharged“antenna”thatwasinitiallycalculatedat off the total colour charge and not by each of the legs firstorderinopacity[36,37]andgeneralisedtomultiple independently. scatteringin[38,39,40]. Inthelatter,generalsituation This striking result connects the physics of multiple the interference effects are controlled by the so-called medium-induced emission to the physics of jet frag- decoherenceparameter mentation and modification in the medium. However, there are several subtle differences between the two ∆ =1−e−(L/td)3, (5) decoh cases. Firstly,splittingsinducedbythemediumarenot collineardivergentincontrasttovacuumradiation.Sec- where we identify the decoherence time td = ondly, their formation is similar to their decoherence [12/(qˆθ2)]1/3,whereθ istheemissionangleofthehard H H time, i.e. the time when a typical medium fluctuation gluon. For long decoherence times, td > L, the dipole can resolve it from the parent, see for a discussion on is not resolved by the medium and radiates medium- thispoint. Thesetimescalescanpossiblydifferalotfor inducedradiationcoherentlyasthetotalcolourcharge. vacuum radiation and we will come back to two cases Additionally, it can radiate (fragment) vacuum-like ac- below. cording to the rules of angular ordering. In the oppo- Inordertoshedmorelightontheseissues,oneshould sitecase, td (cid:28) L, thedipolede-coheres, i.e. bothcon- consider the full two-gluon spectrum, differential in stituentsbecomeindependentofoneanother. Notethat bothenergiesandangles. Whilethefullsplittingfunc- inbothcasesthedecoherencetimeismuchlargerthan tionwasfirstcalculatedin[34]atfirstorderinopacity, theformationtime,tf (cid:28)td. two limits of the spectrum, relevant for jet fragmenta- Further work is need to understand intermediate tion in medium, were meticulously analysed [35]. For regimes. Nevertheless, to summarise this section, the simplification,oneofthegluonswastreatedas“hard”, effectsofcolourcoherencehavebeenfirmlyestablished i.e. its transverse momentum is much bigger than the by several calculations. This points to a simple organ- medium kick, while the other not. Let us spend some isingprincipleputforwardin[41]. Rewritingthedeco- timeexplainingtheselimitsseparately. Theseareillus- herence parameter (5) to highlight a characteristic de- (cid:112) tratedinFig.1. coherence angle θ = 12/(qˆL3), one argues that the d In the first limit, see Fig. 1a, the formation time of medium only can modify jet substructures at large an- the hard gluon is much longer than the formation time gles θ > θ . The resolved substructures, in particular d of the soft one. This is denoted the “collinear limit” the jet core, fragment internally as in the vacuum and sincethehardgluonisemittedverycloseinangletothe lose energy independently of one another. A signifi- parent parton. In fact, due to angular ordering the soft cantfractionoftypicaljetsinheavy-ioncollisionscould gluonisformallyonlyradiatedofftheparentpartonin remaincompletelyunresolvedbythemediumhowever the vacuum.1 Nevertheless, in a large medium the two theyarestillaffectedbyenergylosseffectsduetotheto- tal(quark/gluon)colourchargeofthejet.Correctionsto 1Softemissionscanonlybeemittedwithinaconedeterminedby this picture also can also account for the gradual erad- theemitter.Inthecollinearlimit,thisconeshrinkstozero. ication of angular ordering of the jet constituents and /NuclearandParticlePhysicsProceedings00(2017)1–7 4 Θ ~ Θ Θ ≫Θ H S S H Θ S ΘH Θ ΘS H τH ≫ τS τH ≪ τS (a) (b) Figure1: Sketchofthetwokinematiclimitsofthedoubleemissionratecalculatedin[35]. Inbothpanelsthehard gluonisbluewhilethesoftgluonisredandtheblobrepresentsallpossibleplacementsofthein-mediumexchange. (a)Thecollinearlimit,leftpanel: theangleofemissionofthehardgluonisverysmallanditsformationtimeislong comparedtothesoftgluonformationtime. (b)Thesoftlimit,rightpanel: inthislimittheformationtimeofthehard gluonisveryshortcomparedtothesoftgluononeandtheanglesofemissionofbothgluonsarecomparable. Figures takenfrom[35]. lead to an enhancement of soft gluons radiated within P(z) is the relevant Altarelli-Parisi splitting function thejetcone[42]. Nevertheless,acompleteunderstand- (stripped of its colour factor). However, given it’s ingofhowjetsformandinteractinthemediumisstill collinear divergence ∼ α¯ln(cid:0)R/R (cid:1) (β = 0) we have 0 missing. to resum multiple emissions into the relevant Sudakov formfactor. Physically, thismeanstakingintoaccount allthegroomedemissionsforR(cid:29)R . Wecanthen,for 4. Jetsubstructureinmedium 0 instance, define the probability to split to two sub-jets Inordertogainfurtherinsightintothemechanismsat withmomentumfractionzgas play,andalsoencouragedbyrecentexperimentalmea- (cid:90) R surements, it is natural to consider jet substructure ob- p(z )= dθ∆(θ)Pvac(z ,θ)Θ (z ,θ), (7) g g cut g servables. Aparticularlyclearprocedure,called“Soft- 0 Drop”2 [44,45], selectsapairofsubjets, startingfrom whereR →0andthestep-functionin(7)embodiesthe 0 amaximalangularseparationatthejetconesizeR,that condition in Eq. (6), for details see [44, 45, 46]. The satisfiesthecriterion relevantSudakovreads z>zcutθβ, (6) (cid:34) (cid:90) R (cid:90) 1 (cid:35) ∆(θ)=exp − dθ(cid:48) dzPvac(z,θ(cid:48))Θ (z,θ(cid:48)) , (8) cut wherez ≡ min(pT1,pT2)/(pT1+ pT2), pT1(2) isthesub- θ 0 jet energy and θ their angular separation. Candidates andisequivalenttothe1-jetrate,i.e. itistheprobabil- that do not satisfy the condition (6) are discarded or ityofnosplittingsbetweenthemaximalangleRandθ. “groomed”. This procedure therefore corresponds to GivenaresolutionangleR ,theprobabilityoffindinga 0 clustering all jet constituents into an angular ordered pair that satisfies the SoftDrop condition, aka the two- treeandlookforthefirst“hard”branching,accordingto pronged probability, is therefore P = 1 − ∆(R ) 2prong 0 (6). Itisalsoworthkeepinginmindthattheprocedure [46]. Strikingly, after the resummation the splitting can be made to terminate at some minimal resolution probability becomes independent of α¯, thus not on the angle R . Typical values chosen for the experimental 0 valueofα northecolourorflavourofthesplitting,and analysesarez =0.1,β=0andR=0.4,R =01. s cut 0 exhibits the universal 1/z-behaviour at small-z for the In vacuum, the “hard” branching is inherently sen- β=0case[45]. sitive to the fundamental splitting function, which for When considering the medium modifications of this gluon-gluon splitting reads Pvac(z,θ) = α¯P(z)/θ where observable, we are guided by the insight found in the previoussectionsthatimplyanapproximateseparation 2Wheneverβ = 0,SoftDropisequivalenttothemodifiedMass- oftwotypesofradiation: multiple,softonlarge-angles Dropprocedure[43]. and rare, hard emission in the jet cone [46], see also /NuclearandParticlePhysicsProceedings00(2017)1–7 5 [47]. Hence, having to deal with two sub-jets we have withthefinal-statejetspectrum,andreads(β=0) to decide, according to some criterium, whether they loseenergycoherentlyorindependently. Secondly, for dN R (cid:90) ∞ (cid:90) (cid:15) p(z )=α¯ln d(cid:15) d(cid:15)(cid:48)D ((cid:15)−(cid:15)(cid:48))D ((cid:15)(cid:48)) jets with pT = 100−200 GeV our back of the enve- dp2T g R0 0 0 QW QW lope estimate shows that hard BDMPS radiation could p (cid:32)z p +(cid:15)(cid:48)(cid:33)dN (p +(cid:15)) beidentifiedbytheSoftDropasactualjetsubstructures. × T P g T (0) T Θ(z −z ), (9) The effect should be small O(α ) and care should be pT +(cid:15) pT +(cid:15) dp2T g cut s taken when aiming for a quantitative a description of forz <1/2. Thistimethesplittingfunctionitselfisdi- thedata. Nevertheless,letuscomebacktothisexciting g rectlyaffectedbythefactthatenergylossoftheoutgo- pointlaterandcurrentlyfocusonthefirstaspect,sub-jet inglegsisindependent. Thiscanbeseenbyexpanding coherence. the Altarelli-Parisi splitting function for (cid:15),(cid:15)(cid:48) (cid:28) p in T Duetoenergylosseffectstheprobabilityofthesplit- thesmall-zgregionwhereitreads tingisintimatelyrelatedtothesuppressionofthespec- trum itself. In order to simplify the discussion, let us (cid:32)zgpT +(cid:15)(cid:48)(cid:33) 1 (cid:32) (cid:15)(cid:48) (cid:33) P (cid:39) 1− . (10) considertwoclearlydefinedscenariosandreviewtheir p +(cid:15) z z p T g g T consequences,formoredetailssee[46]. Inthefirstsce- nariothewholejet,andthereforeallitssub-jets,isun- Thecharacteristicenergy-splittingvariablecanbeseen resolvedbythemedium. Inthesecondscenarioallsub- toshiftaszg → zg+zloss,wherezloss ∼ ωs/pT fromdi- jets are resolved, thus independent. In order to study mensionalarguments,resultinginaflatteningofthezg- thesescenarioswewillmakeuseofaprobabilisticsetup distribution. Furthermore,Eq.(9)containstwoquench- whereenergyloss(whetherelasticorradiative)canaf- ing weights in contrast to only one in the “coherent” fectanyresolvedsub-jet. scenario.Thissignalsforthefirsttimethestrongeffects ofenergylosswhenappliedtoincoherentsubstructures In the former, “coherent” case none of the inter-jet withintheoriginaljet. splittings are modified but the spectrum is overall sup- In order to understand how to disentangle the split- pressedbecauseofenergyloss,asgivenbyEq.(3).This ting probability in Eq. (9), imagine a situation where impliesthat,intheabsenceofanyothersourceofradia- mostofthequenchingthejetasawholeistakenbythe tion,Eq.(7)holds. Theproperwayofaddinganewra- mostenergeticleg(carryingmomentumfraction1−z , g diativemechanism,namelyin-coneBDMPSemissions, for z < 1/2). The jet spectrum on the left-hand side g is on the level of probabilities. Hence, we have to re- ofEq.(9)isagaingivenby(3). Theremainingquench- duce the vacuum probability in order to obtain a prop- ingaffectsonlythesoftlegandcannowberesummed erlynormalisedtotalprobabilityofradiation. Aftertak- into a modified Sudakov form factor that accounts for ing appropriate care of the angular restrictions (for in- energy loss. This resummed quenching effect strongly stance, the introduction of a minimal resolution angle suppressestheprobabilityoftwo-prongedobjectscom- shouldfurthersuppressthecontributionofvacuumradi- paredtothevacuum. ation)weshouldexpectanenhancementofthesplitting ItbecomesclearthataddingtheBDMPSspectrumon probability at small-z because of the medium-induced the level of probabilities complicates the situation fur- bremsstrahlung that scales as z−3/2. This enhancement theranditisnotourgoalheretopresentadefinitean- diesrapidlyoffwithenergy∼ p−T1/2, seeEq.(1). Inef- swer. Wecouldarguethatthestrongeffectsofincoher- fect, the two-pronged probability P should be en- 2prong ent energy loss strongly distorts the vacuum spectrum, hancedcomparedtothevacuum. thusbeinghardtoreconcilewiththetrendsobservedin experimental data. A more realistic calculation should Taken at face value, this scenario illustrates that the provide an interpolation between the two extreme sce- SoftDrop procedure presents a unique possibility to nariosdiscussedsofar. Besides, effectsofasoftback- measure directly medium-induced quanta rather than groundcorrelatedwiththejet, e.g. generatedbyback- simplybeingsensitivetoitsgeneralconsequences,such reaction,couldinfluencetheinterpretationoftheresult. asenergyloss,etc. Nevertheless, the potentially unique prospect of Thesecondscenariosketchedaboveismorecompli- a (semi-)direct measurement of the medium-induced cated. 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