Event-shape engineering and heavy-flavour observables in relativistic heavy-ion collisions
PPrepared for submission to JHEP
Event-shape engineering and heavy-flavourobservables in relativistic heavy-ion collisions
Andrea Beraudo a Arturo De Pace a Marco Monteno a Marzia Nardi a Francesco Prino a a INFN, Sezione di Torino, via Pietro Giuria 1, I-10125 Torino
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
Traditionally, events collected at relativistic heavy-ion colliders are classifiedaccording to some centrality estimator (e.g. the number of produced charged particles)related to the initial energy density and volume of the system. In a naive picture the latterare directly related to the impact parameter of the two nuclei, which sets also the initialeccentricity of the system: zero in the case of the most central events and getting larger formore peripheral collisions. A more realistic modelling requires to take into account event-by-event fluctuations, in particular in the nucleon positions within the colliding nuclei:collisions belonging to the same centrality class can give rise to systems with differentinitial eccentricity and hence different flow harmonics for the final hadron distributions.This issue can be addressed by an event-shape-engineering analysis, consisting in selectingevents with the same centrality but different magnitude of the average bulk anisotropic flowand therefore of the initial-state eccentricity. In this paper we present the implementation ofthis analysis in the POWLANG transport model, providing predictions for the transverse-momentum and angular distributions of charm and beauty hadrons for event-shape selectedcollisions. In this way it is possible to get information on how the heavy quarks propagating(and hadronizing) in a hot environment respond both to its energy density and to itsgeometric asymmetry, breaking the perfect correlation between eccentricity and impactparameter which characterizes a modelling of the medium based on smooth average initialconditions. a r X i v : . [ phy s i c s . d a t a - a n ] D ec ontents Heavy flavour particles (
D/B mesons and Λ c/b baryons), arising from charm and beautyquarks produced in initial hard partonic scattering processes, have been always considereda probe of the deconfined medium one expects to form in relativistic heavy-ion collisions.The scope of first heavy-flavour measurements was simply to understand whether, in spiteof the large mass of the parent quarks, the distributions of final-state particles displayedthe same features observed in the case of light hadrons, i.e. a quenching of the spectraat high transverse momentum p T (with possible signatures of a mass and colour-chargedependence of parton energy loss) and a non-vanishing, positive elliptic-flow coefficient v .First data were limited to electrons from heavy-flavour decays, without the possibility todiscriminate between the charm and beauty contributions [1, 2]. It became then possibleto reconstruct D -mesons through some exclusive decay channels [3–8]. The message fromthese first measurements was that, although quantitatively a bit milder, the same quench-ing of the momentum spectra and elliptic (and triangular, as shown in Ref. [8]) asymmetryof the azimuthal distributions observed for light hadrons characterized also charm andbeauty particles. This entailed a quite strong coupling of the heavy quarks with the hotdeconfined plasma of quarks and gluons (QGP) supposed to be produced in the collisionof the two nuclei and possibly a non trivial modification of their hadronization due to thelarge density of light thermal partons nearby. From a comparison of the outcomes of trans-port calculations with experimental data it is in principle possible to extract informationon the value of the heavy-quark momentum-diffusion coefficient, a fundamental quantitywhich in the static limit in hot-QCD admits a rigorous definition in terms of Euclideancorrelators of chromo-electric fields [9, 10]. Recently, a systematic investigation based on aBayesian approach aiming at extracting the heavy-flavour diffusion coefficient from currentexperimental data has been carried out by some authors [11]. In this connection a compre-hensive study of the various theoretical uncertainties arising from the initial heavy-quarkspectrum, from Cold-Nuclear-Matter effects and from the modelling of the medium and of– 1 –adronization was carried out in Ref. [12]. This is somehow similar to what done for thecase of soft hadrons, where a comparison of hydrodynamics calculations with experimen-tal particle distributions allowed people to constrain within a quite narrow band anothertransport coefficient, the shear-viscosity to entropy-density ratio η/s , which turned out tobe close to the lower bound 1 / π postulated by the AdS/CFT correspondence [13].More recent measurements opened the possibility to get access to a richer information.Studies of D s and Λ c production in nuclear collisions have the potential to put the issueof medium-modification of heavy-flavour hadrochemistry on solid ground [14, 15], makingpossible to validate heavy-quark hadronization models based on the recombination withlight thermal partons. Experimental data on B meson production [16] allow one to studythe mass dependence of the heavy-quark medium interaction; if in the future these analysiswere extended to lower transverse momentum they would allow one to perform a theory-to-experiment comparison in a kinematic region in which transport calculations are underthe best control, reaching the goal of really measuring the heavy-quark diffusion coeffi-cient. Recently heavy-flavour studies have been extended to the case of proton-nucleuscollisions [17–20], with the aim of contributing to answer the still open question whetheralso in such small systems QGP droplets can be formed [21].Finally, the measurement of odd flow-harmonics of heavy-flavour hadrons can providea richer information on the initial conditions of the system formed after the collision of thetwo high-energy nuclei, like its tilted profile in the reaction plane (wounded nucleons tendingto deposit more energy along the direction of their motion) in the case of the directed flow v [22–24] or its event-by-event fluctuations (from the random nucleon positions) in the caseof the triangular flow v . The triangular flow v of D mesons in Pb-Pb collisions providedby transport calculations has been studied in some recent publications and theoreticalresults [25, 26] have been compared to experimental data from the CMS collaboration [8].A further possibility of accessing the response of the final particle distributions tothe initial asymmetries of the system, getting information both on the coupling of theheavy quarks with the medium and on its initial conditions, is given by the so-calledEvent-Shape-Engineering (ESE) studies. The basic idea is to select events belonging tothe same centrality class, but characterized by a different initial geometric (elliptic or tri-angular) asymmetry, getting subsamples of events with high/low eccentricity [27]. Such anapproach was proposed and adopted by the ALICE collaboration in the analysis of mo-mentum and azimuthal distributions of light hadrons [28], comparing the results obtainedin subsamples of collisions with large/small average elliptic-flow with the ones of an un-biased selection of events. Here, in the framework of a transport calculation, we wish toextend the approach to heavy flavour, studying how the different geometric asymmetry andthe resulting anisotropic flow of the medium affect the propagation of heavy quarks andleave their signatures in the final charm/beauty-hadron distributions [29]. Our findings willbe compared with recent experimental outcomes [30]. For independent phenomenologicalstudies of hard probes (heavy-flavour particles and jets) in heavy-ion collisions based onevent-shape-engineering see also Refs. [31–33].Our paper is organized as follows. In Sec. 2 we present our modelling of the backgroundmedium, focusing on the simulation of the initial conditions, on the selection – in the various– 2 –entrality classes – of the events belonging to different eccentricity subsamples and on theresulting light-hadron spectra decoupling from the fireball at the end of its hydrodynamicevolution. In Sec. 3 we briefly summarize our setup for the simulation of heavy-quarktransport and hadronization. In Sec. 4 we display the results of our transport calculationsperformed with the POWLANG model, which account both for the propagation of c and b quarks through the QGP and for their hadronization in the presence of a hot deconfinedmedium. Finally in Sec. 5 we discuss our results, suggesting possible future improvements. For the modelling of the medium produced in nucleus-nucleus collisions (in this paper weconsider Pb-Pb collisions at √ s NN = 5 .
02 TeV) we adopted the same approach describedin detail in Ref. [26], interfacing a Glauber Monte-Carlo (Glauber-MC) simulation of theinitial condition of the system to a hydrodynamic code (ECHO-QGP [34]) calculating thesubsequent evolution of the matter, under the assumption of longitudinal boost-invariance;the latter is a good approximation for observables around mid-rapidity and allows one tosolve a (2+1)-dimensional problem, reducing the computational time.In order to set the initial geometry we distribute nucleons within the two nuclei ac-cording to a Woods-Saxon distribution and we generate several thousands ( ∼ σ inNN = 70 mb was employed in the simulation. For agiven event each nucleon-nucleon collision is taken as a source of entropy production, witha Gaussian smearing σ . The initial entropy density in the transverse plane used to startthe hydrodynamic evolution of the system at the longitudinal proper time τ = 0 . s ( x, y ) = K πσ N coll (cid:88) i =1 exp (cid:20) − ( x − x i ) + ( y − y i ) σ (cid:21) . (2.1)The parameter K (with dimensions of an inverse length) sets the average entropy depositedby a single collision (so far we do not include fluctuations at the level of the individualnucleon-nucleon inelastic collisions). As in Ref. [26] for Pb-Pb collisions at √ s NN = 5 . Kτ = 6 .
37. For each event the above entropy density can be used as aweight to define complex eccentricities, which characterize the initial state (i.e. both theamount of anisotropy and its orientation in the transverse plane) and will be mapped intothe final hadron distributions by the subsequent hydrodynamic evolution [35]: (cid:15) m e im Ψ m ≡ − (cid:8) r ⊥ e imφ (cid:9) { r ⊥ } , with { ... } ≡ (cid:90) d r ⊥ s ( (cid:126)r ⊥ )( ... ) . (2.2)Modulus and orientation of the various azimuthal harmonics are given by: (cid:15) m = (cid:113) { r ⊥ cos( mφ ) } + { r ⊥ sin( mφ ) } { r ⊥ } (2.3)Ψ m = 1 m atan2 (cid:0) −{ r ⊥ sin( mφ ) } , −{ r ⊥ cos( mφ ) } (cid:1) (2.4)– 3 – ε / N e v ( d N e v / d ε ) ε / N e v ( d N e v / d ε ) Figure 1 . The elliptic (left panel) and triangular (right panel) eccentricity distribution of Pb-Pbcollisions at √ s NN = 5 .
02 TeV belonging to different centrality classes.
0 5 10 15 20 25 30 35 40 45 50
Centrality (%) ε Centrality (%) ε Figure 2 . Correlation between the elliptic (left panel) and triangular (right panel) eccentricity andthe centrality of the nucleus-nucleus collisions for our Glauber-MC sample of Pb-Pb events.
Using as an estimator the number of binary nucleon-nucleon collisions, we group thePb-Pb events in centrality classes (0-10%, 10-30% and 30-50%) and study within eachsample the distribution of initial elliptic and triangular eccentricity (cid:15) and (cid:15) . We will alsoconsider in some of the calculations a very peripheral class (60-80%). Results are shownin Fig. 1. Notice how, within a given centrality class, the eccentricity distribution is quitebroad, in particular for the case of (cid:15) whose large event-by-event fluctuations arise bothfrom the different impact parameter and from the random positions of the nucleons withinthe colliding nuclei. The strong dependence on the impact parameter is also evident fromthe sizable shift of the peak of the distribution towards larger values of (cid:15) going from centralto peripheral collisions. On the other hand, in the case of (cid:15) the eccentricity distributionsare narrower and the displacement of the peak when moving to a different centrality classis milder. This reflects the different origin of the triangular asymmetry, which (neglectingsub-nucleonic degrees of freedom) is entirely due to the event-by-event fluctuations in thepositions of the nucleons inside the colliding nuclei.In order to study how the initial asymmetry of the system is mapped by the hydrody-– 4 –amic/transport evolution into the azimuthal anisotropies of the final particle distributions(both light and heavy-flavour hadrons, the latter being the focus of this work) we select,for each centrality class, the 20% most eccentric and the 60% least eccentric events. Thiscorresponds to the selections adopted by the ALICE collaboration in the recent heavy-flavour analysis in Ref. [30]. We do this both for (cid:15) and (cid:15) . This, depending on the cases,amounts to subsamples of several hundreds/thousands of events. As evident from Fig. 2there is a strong anti-correlation between the elliptic eccentricity (cid:15) and the centralityof the collision (the effect is much milder for the case of (cid:15) ). In order to quantify theeffect it is useful to provide some typical numbers. For the 0-10% centrality class onehas, for an unbiased selection of events, (cid:104) N coll (cid:105) unbias .N coll = 1658 (in the average, events areweighted by N coll , since heavy-quark production scales with the number of binary colli-sions); applying cuts on elliptic and triangular eccentricity one gets (cid:104) N coll (cid:105) high − (cid:15) N coll = 1471, (cid:104) N coll (cid:105) low − (cid:15) N coll = 1730 and (cid:104) N coll (cid:105) high − (cid:15) N coll = 1604, (cid:104) N coll (cid:105) low − (cid:15) N coll = 1675, respectively. Hence, se-lecting events within a given centrality class of higher/lower eccentricity leads to a samplebiased toward lower/higher centrality. We have to bear in mind this observation in in-terpreting our numerical findings. Experimental analysis, profiting from a huge statistics,remove this bias performing their selection on eccentricity in very small bins of centrality. s(x,y) (fm -3 ) 10-30% Pb-Pb coll.-15 -10 -5 0 5 10 15x (fm)-15-10-5 0 5 10 15 y ( f m ) s(x,y) (fm -3 ) 10-30% Pb-Pb coll. (least ecc.)-15 -10 -5 0 5 10 15x (fm)-15-10-5 0 5 10 15 y ( f m ) s(x,y) (fm -3 ) 10-30% Pb-Pb coll. (most ecc.)-15 -10 -5 0 5 10 15x (fm)-15-10-5 0 5 10 15 y ( f m ) Figure 3 . The initial entropy-density for the 10-30% most central Pb-Pb collisions at √ s NN = 5 . (cid:15) (middle) and the 20% of events with the highest (cid:15) (right). The Glauber-MC modelling of the initial state can now be used as the initial conditionof the hydrodynamic evolution of the system, which we describe through the ECHO-QGPcode [34]. Its output provides the information on the background medium through whichthe propagation of the heavy quarks takes place. If we wished to perform fully realisticsimulations we should numerically solve the set of hydrodynamic equations for all the( ∼ (x,y) (fm -3 ) 10-30% Pb-Pb coll.-15 -10 -5 0 5 10 15x (fm)-15-10-5 0 5 10 15 y ( f m ) s(x,y) (fm -3 ) 10-30% Pb-Pb coll. (least ecc.)-15 -10 -5 0 5 10 15x (fm)-15-10-5 0 5 10 15 y ( f m ) s(x,y) (fm -3 ) 10-30% Pb-Pb coll. (most ecc.)-15 -10 -5 0 5 10 15x (fm)-15-10-5 0 5 10 15 y ( f m ) Figure 4 . The initial entropy-density for the 10-30% most central Pb-Pb collisions at √ s NN = 5 . (cid:15) (middle) and the 20% of events with the highest (cid:15) (right). eccentricity (i.e. low- (cid:15) n , high- (cid:15) n , unbiased if no cut is applied), we rotate each of them sothat its relevant symmetry plane ψ n is aligned along the x -axis and, starting from Eq. (2.1),we construct an average entropy-density distribution weighting each event by the numberof binary nucleon-nucleon collisions (since the QQ production scales with N coll , whichintroduces a bias towards more central events). In Figs. 3 and 4, referring to the 10-30%centrality class, we display the result of such a procedure for the study of the response toan elliptic and triangular deformation, respectively: the average initial conditions for theunbiased, low- (cid:15) n and high- (cid:15) n subsets of events are shown. p T (GeV) v π + ε π + ε π + ε π + ε π + ε π + ε Pb-Pb coll. @ 5.02 TeV p T (GeV) v π + ε π + ε π + ε π + ε π + ε π + ε Pb-Pb coll. @ 5.02 TeV
Figure 5 . The elliptic (left panel) and triangular (right panel) flow of charged pions in Pb-Pbcollisions at √ s NN = 5 .
02 TeV for different eccentricity selections within the same centrality class.
Having performed the hydrodynamic evolution of the medium we can check the effectof the eccentricity fluctuations on the light-hadron distributions, obtained from a stan-dard Cooper-Frye decoupling from a freeze-out hypersurface. In Fig. 5 we display theresulting elliptic and triangular flow coefficients v and v for charged pions, defined as v n = (cid:104) cos[ n ( φ − Ψ n )] (cid:105) . Notice how selecting events with high/low eccentricity producesa huge effect on the final angular distributions, comparable or even larger than changing– 6 – p T (GeV) v / v m b v /v /v /v /v /v /v p T (GeV) v / v m b v /v /v /v /v /v /v Figure 6 . Ratio of the charged-pion elliptic (left) and triangular (right) flow of high/low-eccentricity events over the minimum-bias ones within the same centrality class. Results of ourhydrodynamic modelling can be compared to ALICE data from Ref. [28]. centrality class, this in particular for the case of v . It is of interest to quantify the ef-fect by taking the ratio of the elliptic flow in the high/low- (cid:15) subsets over the one in theunbiased sample. Our results are displayed in Fig. 6. The effect of the event-shape selec-tion on the v coefficient appears in qualitative agreement with recent ALICE data [28],although obtained with different eccentricity cuts. Notice that our results for the v ineccentricity-selected events display some dependence on centrality. This probably reflectsour procedure of selection, which does not decouple completely eccentricity from centrality.On the other hand in the case of v , in which the initial geometric deformation arises en-tirely from fluctuations in the nucleon positions uncorrelated with the impact parameter,the results in the right panel of Fig. 6 do not show any dependence on centrality. s(x,y) (fm -3 ) 0-10% Pb-Pb coll. (0.3< ε <0.4)-15 -10 -5 0 5 10 15x (fm)-15-10-5 0 5 10 15 y ( f m ) -3 ) 10-30% Pb-Pb coll. (0.3< ε <0.4)-15 -10 -5 0 5 10 15x (fm)-15-10-5 0 5 10 15 y ( f m ) s(x,y) (fm -3 ) 30-50% Pb-Pb coll. (0.3< ε <0.4)-15 -10 -5 0 5 10 15x (fm)-15-10-5 0 5 10 15 y ( f m ) Figure 7 . The initial entropy-density for Pb-Pb collisions at √ s NN = 5 .
02 TeV of fixed ellipticeccentricity 0 . ≤ (cid:15) ≤ . Going back to Fig. 1 we note how the eccentricity distributions of different centralityclasses display a significant overlap: we can have events in which the system is initiallycharacterized by an equal degree of geometric deformation, but by very different dimen-sions and energy density. As a complementary study we select, in the different centrality– 7 – (x,y) (fm -3 ) 0-10% Pb-Pb coll. (0.2< ε <0.3)-15 -10 -5 0 5 10 15x (fm)-15-10-5 0 5 10 15 y ( f m ) s(x,y) (fm -3 ) 10-30% Pb-Pb coll. (0.2< ε <0.3)-15 -10 -5 0 5 10 15x (fm)-15-10-5 0 5 10 15 y ( f m ) s(x,y) (fm -3 ) 30-50% Pb-Pb coll. (0.2< ε <0.3)-15 -10 -5 0 5 10 15x (fm)-15-10-5 0 5 10 15 y ( f m ) Figure 8 . The initial entropy-density for Pb-Pb collisions at √ s NN = 5 .
02 TeV of fixed triangulareccentricity 0 . ≤ (cid:15) ≤ . p T (GeV) v π + <0.4 p <0.4π + <0.4 p <0.4π + <0.4 p <0.4 Pb-Pb coll. @ 5.02 TeV p T (GeV) v π + ε <0.3p 0-10%, 0.2< ε <0.3 π + ε <0.3p 10-30%, 0.2< ε <0.3 π + ε <0.3p 30-50%, 0.2< ε <0.3Pb-Pb coll. @ 5.02 TeV Figure 9 . The elliptic and triangular flow of pions and protons in Pb-Pb collisions at √ s NN = 5 . . ≤ (cid:15) ≤ . . ≤ (cid:15) ≤ .
3) butbelonging to different centrality classes. At low p T the results are mainly sensitive to the eccentricityof the initial condition rather than to the centrality class. classes, samples of events with an initial geometric asymmetry belonging to the same nar-row interval: we choose 0 . ≤ (cid:15) ≤ . . ≤ (cid:15) ≤ . v n curves for pions and protons in different central-ity classes display a strong overlap for a quite extended range of transverse momentum p T (with the partial exception of the triangular flow for the 30-50% class). It is of interest toperform the same study for the case of heavy-flavour particles, since their energy-loss anddegree of thermalization should be sensitive to the dimension and transport coefficients ofthe medium, both depending strongly on the centrality of the collision.– 8 – Heavy flavour transport and hadronization
After modelling the initial state with the Glauber-MC approach described in Sec. 2, ob-taining an average initial condition for the selected subsample of collisions (with cuts oncentrality and eccentricity), heavy quarks are distributed in the transverse plane accordingto the local density of binary collisions. Their propagation in the medium is then studiedthrough the relativistic Langevin equation∆ (cid:126)p/ ∆ t = − η D ( p ) (cid:126)p + (cid:126)ξ ( t ) . (3.1)containing a deterministic friction force quantified by the drag coefficient η D and a randomnoise term specified by its temporal correlator (cid:104) ξ i ( (cid:126)p t ) ξ j ( (cid:126)p t (cid:48) ) (cid:105) = b ij ( (cid:126)p t ) δ tt (cid:48) / ∆ t b ij ( (cid:126)p ) ≡ κ (cid:107) ( p )ˆ p i ˆ p j + κ ⊥ ( p )( δ ij − ˆ p i ˆ p j ) . (3.2)In the above, information on the background medium provided by hydrodynamics entersin two ways: first through its collective velocity flow u µ , whose knowledge is necessaryin order to perform the update of the heavy-quark momentum at each time-step in thelocal rest frame of the fluid; secondly through the temperature dependence of the transportcoefficients κ ⊥ / (cid:107) and η D , which quantify the coupling of the heavy quarks with the medium.Our simulations are performed adopting two independent choices for the above transportcoefficients, from weak-coupling (Hard-Thermal Loop [36]) and lattice-QCD calculations [9,37]. The differences in the final particle distributions obtained with these two sets oftransport coefficients provide an estimate of the current theoretical uncertainties and ofthe potential discriminating power of the experimental data. Lattice-QCD calculationsprovide in principle a non-perturbative result. However, they refer to the case of static,infinitely heavy, quarks; furthermore so far they are limited to the case of a pure gluonplasma and are affected by the unavoidable systematic uncertainties in extracting real-time information from simulations performed in an Euclidean spacetime. On the otherhand weak-coupling calculations can deal with the realistic case of finite-mass quarks withrelativistic momenta, but they are so far limited at the tree-level, with resummation ofmedium effects in the gluon propagators.Hadronization is modelled recombining at freeze-out the heavy quarks with light ther-mal partons from the same fluid-cell (an instantaneous decoupling with no further rescat-tering in the hadron-gas phase is assumed), forming Qq (or Qq ) strings which are thenfragmented according to the Lund model implemented in PYTHIA 6.4. This turns out tohave a major effect on the final charm and beauty hadron distributions. For deeper detailsabout the implementation of our transport calculations and our modelling of hadronizationwe refer the reader to our past publications [26, 38]. For a comprehensive review of trans-port calculations applied to the study of heavy-flavour observables in relativistic heavy-ioncollisions, emphasizing the various source of systematic uncertainties and the state of theart of the extraction of transport coefficients, see for instance Refs. [12, 39].– 9 – (GeV/c) T p AA R Charmed hadrons ε HTL, all ε HTL, small ε HTL, large ε lQCD, all ε lQCD, small ε lQCD, large POWLANG =5.02 TeV NN sPb Pb, Centrality 0 10% (GeV/c) T p ( unb i a s ed ) AA s e l e c t ed ) / R ε ( AA R (GeV/c) T p AA R Charmed hadrons HTL, varepsilon ε HTL, small ε HTL, large lQCD, varepsilon ε lQCD, small ε lQCD, large POWLANG =5.02 TeV NN sPb Pb, Centrality 10 30% (GeV/c) T p ( unb i a s ed ) AA s e l e c t ed ) / R ε ( AA R (GeV/c) T p AA R Charmed hadrons ε HTL, all ε HTL, small ε HTL, large ε lQCD, all ε lQCD, small ε lQCD, large POWLANG =5.02 TeV NN sPb Pb, Centrality 30 50% (GeV/c) T p ( unb i a s ed ) AA s e l e c t ed ) / R ε ( AA R Figure 10 . The nuclear modification factor R AA of charmed hadrons in the 0-10% (top), 10-30%(middle) and 30-50% (bottom) most central Pb-Pb collisions at √ s NN = 5 .
02 TeV. Results for the20% highest- (cid:15) and the 60% lowest- (cid:15) selection of events are compared to the unbiased case. Forboth choices of transport coefficients the results display only a mild and similar sensitivity to theinitial eccentricity. – 10 – Results
We start our event-shape-engineering study of heavy-flavour production in nucleus-nucleuscollisions considering the nuclear modification factor of charmed hadrons. We considerthe 0-10%, 10-30% and 30-50% centrality classes and we compare the results obtained forsubsamples of events corresponding – for a given centrality – to an unbiased selection, tothe 20% highest (cid:15) and to the 60% lowest (cid:15) . As one can see from Fig. 10, the effect of theeccentricity cuts is quite modest, at most of the order 10-20% when considering the high- (cid:15) sample, and consistent with the anticorrelation between eccentricity and centrality: onaverage, within a given centrality class, high- (cid:15) events correspond also to a larger impactparameter and hence to a lower initial size and density of the system. We remind that, inorder to remove this bias and get a cleaner decoupling between the density and the ellipticasymmetry of the medium, the ALICE collaboration performed the selection on eccentricityin very narrow centrality intervals, corresponding to bins of 1% of the total hadronic cross-section. This was possible taking advantage of the large available statistics: for eachcentrality class considered in the experimental analysis (10-30% and 30-50%) a number ofevents of order 10 was collected. For this first theroretical study on the subject we rely ona less demanding approach in terms of computing and storage resources, bearing in mindin interpreting our findings the not complete decoupling between the system density/sizeand its geometrical deformation in our selection of events. Notice that, even consideringvery narrow centrality bins as done by the ALICE collaboration in Ref. [28], a selection oncentrality can lead to an effect on the transverse-momentum distributions: Glauber-MCsimulations show a positive correlations between the initial density of the system and itseccentricity and this could explain the larger radial flow of light hadrons observed in eventswith larger average elliptic flow.We now move to the study of the elliptic flow. As one can see in Fig. 11, referring to the0-10%, 10-30% and 30-50% centrality classes, the effect of event-shape engineering is muchlarger in this case. If we focus on the ratio v high − (cid:15) /v unbiased2 , for all the three centralityclasses the 20% highest- (cid:15) events display an average v coefficient almost twice as large asthe one found in the unbiased sample. The size of the effect looks quite independent ofthe transport coefficients and the transverse momentum of the charmed hadron. Similarconsiderations hold for the ratio v low − (cid:15) /v unbiased2 , which looks quite flat around 0.7-0.8for a sufficiently broad range p T and independent of the coupling of the heavy quarkwith the medium (HTL vs lQCD curves in the figures). Actually, comparing the variouscentralities, the effect on v of selecting high-eccentricity events looks larger in the 0-10%class, in agreement with what already found for pions and displayed in the left panel ofFig. 6. To summarize, the charm-hadron v calculated with different transport coefficientsdisplays non-negligible differences, in particular for p T > ∼ v high − (cid:15) /v unbiased2 for charm quarks andhadrons in various classes of N coll . Here we include also a very peripheral (60-80%) sampleof events. As one can see in the left panel, curves at the quark level display a clear order-– 11 – (GeV/c) T p v Charmed hadrons ε HTL, all ε HTL, small ε HTL, large ε lQCD, all ε lQCD, small ε lQCD, large POWLANG =5.02 TeV NN sPb Pb, Centrality 0 10% (GeV/c) T p ( unb i a s ed ) s e l e c t ed ) / v ε ( v (GeV/c) T p v Charmed hadrons ε HTL, all ε HTL, lmall ε HTL, large ε lQCD, all ε lQCD, lmall ε lQCD, large POWLANG =5.02 TeV NN sPb Pb, Centrality 10 30% (GeV/c) T p ( unb i a s ed ) s e l e c t ed ) / v ε ( v (GeV/c) T p v Charmed hadrons ε HTL, all ε HTL, small ε HTL, large ε lQCD, all ε lQCD, small ε lQCD, large POWLANG =5.02 TeV NN sPb Pb, Centrality 30 50% (GeV/c) T p ( unb i a s ed ) s e l e c t ed ) / v ε ( v Figure 11 . The elliptic-flow coefficient v of charmed hadrons in the 0-10% (top), 10-30% (middle)and 30-50% (bottom) most central Pb-Pb collisions at √ s NN = 5 .
02 TeV. Results for the 20%highest- (cid:15) and to the 60% lowest- (cid:15) selection of events are compared to the unbiased case. For bothchoices of transport coefficients the results display a strong and similar sensitivity to the initialeccentricity. ing in centrality. The enhancement of the charm-quark v when selecting high-eccentricityevents gets larger moving from peripheral to central collisions: the denser and larger the– 12 – (GeV/c) T p ( unb i a s ed ) s e l e c t ed ) / v ε ( v POWLANG =5.02 TeV NN sPb Pb, Charm quarks, HTL ε Large (GeV/c) T p ( unb i a s ed ) s e l e c t ed ) / v ε ( v POWLANG =5.02 TeV NN sPb Pb, Charmed hadrons, HTL ε Large
Figure 12 . The ratio of the v coefficients for charm (quarks in the left panel, hadrons in the rightpanel) in high-eccentricity over unbiased events, for different centrality classes. medium, the stronger its effect on the propagation of charm quarks and hence the moreevident the signatures of its asymmetric geometry and flow in the final particle distribu-tions. Actually, varying the centrality seems to play a milder role at the hadron level:hadronization modelled via recombination with light partons from the medium probablywashes out part of the effect. Also experimental data by ALICE [30], although affectedby quite large error bars and dependent on the eccentricity estimator, seem to indicatethat there is not a big dependence on centrality, at least for the classes considered in theiranalysis (10-30% and 30-50%).Although not yet considered in the experimental analysis, it is of interest to performthe same event-shape-engineering study for the triangular flow v . Remember that in thiscase, for any selection on centrality and eccentricity (cid:15) , an average initial condition is builtafter rotating each event in the transverse plane by -Ψ , as described in Sec. 2. Our findingsare shown in Fig. 13 and look similar to the ones obtained for the elliptic flow. Consideringthe ratio v high − (cid:15) /v unbiased3 , for all the three centrality classes the 20% highest- (cid:15) eventsdisplay an average v coefficient almost twice as large as the one found in the unbiasedsample. The size of the effect looks quite independent of the transport coefficients andthe transverse momentum of the charmed hadron, although fluctuations look very largein p T regions in which the signal is small. The effect looks also pretty independent of thecentrality of the collision, as already found for pions and shown in the right panel of Fig. 6.Also in the case of the low- (cid:15) subsamples deviations from the unbiased results are of thesame order of what found for the elliptic flow.We decide now to adopt a different perspective and compare the results for the flow ofheavy-flavour particles in events characterized by the same initial eccentricity, but belong-ing to different centrality classes. Results for the elliptic and triangular flow are displayedin Figs. 14 and 15, referring to subsets of events with an initial asymmetry 0 . ≤ (cid:15) ≤ . . ≤ (cid:15) ≤ .
3, respectively. Our scope is to point out differences among the resultsin the different centrality classes which can be attributed to the energy-density and size– 13 – (GeV/c) T p v Charmed hadrons ε HTL, all ε HTL, small ε HTL, large ε LAT, all ε LAT, small ε LAT, large
POWLANG =5.02 TeV NN sPb Pb, Centrality 0 10% (GeV/c) T p ( unb i a s ed ) s e l e c t ed ) / v ε ( v (GeV/c) T p v Charmed hadrons ε HTL, all ε HTL, small ε HTL, large ε LAT, all ε LAT, small ε LAT, large
POWLANG =5.02 TeV NN sPb Pb, Centrality 10 30% (GeV/c) T p ( unb i a s ed ) s e l e c t ed ) / v ε ( v (GeV/c) T p v Charmed hadrons ε HTL, all ε HTL, small ε HTL, large ε lQCD, all ε lQCD, small ε lQCD, large POWLANG =5.02 TeV NN sPb Pb, Centrality 30 50% (GeV/c) T p ( unb i a s ed ) s e l e c t ed ) / v ε ( v Figure 13 . The triangular-flow coefficient v of charmed hadrons in the 0-10% (top), 10-30%(middle) and 30-50% (bottom) most central Pb-Pb collisions at √ s NN = 5 .
02 TeV. Results for the20% highest- (cid:15) and to the 60% lowest- (cid:15) selection of events are compared to the unbiased case.For both choices of transport coefficients the results display a strong and similar sensitivity to theinitial eccentricity. of the medium. In Sec. 2, in fact, we showed that the elliptic and triangular flow of lighthadrons (pions and protons) decoupling from the fireball depends essentially on the initial– 14 – (GeV/c) T p v POWLANG =5.02 TeV NN sPb Pb, Charm quarks, HTL ε ε ε
10 30%, all <0.4 ε
10 30%, 0.3< ε
30 50%, all <0.4 ε
30 50%, 0.3< (GeV/c) T p v POWLANG =5.02 TeV NN sPb Pb, Charmed hadrons, HTL ε ε ε
10 30%, all <0.4 ε
10 30%, 0.3< ε
30 50%, all <0.4 ε
30 50%, 0.3<
Figure 14 . The elliptic-flow coefficient v of charmed quarks (left panel) and hadrons (right panel)in Pb-Pb collisions at √ s NN = 5 .
02 TeV for various centrality classes. For each class we displayresults referring to the fixed eccentricity interval 0 . ≤ (cid:15) ≤ . (cid:15) but belonging to different centralityclasses display sizable differences, the response to the initial eccentricity being stronger for morecentral events. The difference is partially washed-out by hadronization via recombination. (GeV/c) T p v POWLANG =5.02 TeV NN sPb Pb, Charm quarks, HTL ε ε ε
10 30%, all <0.3 ε
10 30%, 0.2< ε
30 50%, all <0.3 ε
30 50%, 0.2< (GeV/c) T p v POWLANG =5.02 TeV NN sPb Pb, Charmed hadrons, HTL ε ε ε
10 30%, all <0.3 ε
10 30%, 0.2< ε
30 50%, all <0.3 ε
30 50%, 0.2<
Figure 15 . The triangular-flow coefficient v of charmed quarks (left panel) and hadrons (rightpanel) in Pb-Pb collisions at √ s NN = 5 .
02 TeV for various centrality classes. For each class wedisplay results referring to the fixed eccentricity interval 0 . ≤ (cid:15) ≤ . (cid:15) but belonging to differentcentrality classes display sizable differences, the response to the initial eccentricity being strongerfor more central events. The difference persists even after hadronization via recombination. – 15 –ccentricity of the medium and only marginally on the centrality class (see Fig. 5): in spiteof the very different temperature, size and lifetime of the medium, the final v and v ofsoft hadrons produced at hadronization look very similar. However the situation could bein principle different for heavy flavour particles, which do not come from the hadronizationof the bulk medium itself, but whose parents are the QQ pairs produced in hard scatter-ing processes occurring before the formation of a thermalized quark-gluon plasma. Theseheavy quarks, before decoupling, interact strongly with the fireball through which theypropagate and we expect that the different medium size, lifetime and temperature in thedifferent centrality classes should affect the final results. This is clearly visible in the leftpanels of Figs. 14 and 15: one gets very different results for the elliptic and triangular flowof charm quarks in events with the same initial eccentricity (orange curves) but belongingto different centrality classes, due to the different amount of energy-loss and diffusion suf-fered in the medium. Is the effect observable also in the final hadron distributions? As onecan see from the right panels of Figs. 14 and 15 at the level of charmed hadrons deviationsof the results among the different centrality classes are milder. This is particularly evidentin the case of the elliptic flow. The curves for the v corresponding to the unbiased selectionof events (grey curves) look very different going from central to more peripheral collisions;on the contrary if we focus on events of various centrality but corresponding to a very sim-ilar initial eccentricity (orange curves) the curves tend to merge, although this was not thecase at the quark level. This is clearly a consequence of hadronization, which in our modelproceeds via recombination of the heavy quarks with the light thermal partons from themedium, characterized by a very similar anisotropic flow in the different centrality classesif one consider events of comparable initial eccentricity. Notice that a difference amongevents with the same eccentricity but belonging to different centrality classes persists inthe case of the triangular flow of charm hadrons, as one can see in particular comparingthe 30-50% curve with the ones of the 0-10% and 10-30% centrality classes: this should notsurprise us too much, since the same different was present also in the case of light hadrons(see Fig. 9).We finally move to consider also beauty quarks and hadrons, focusing on the study oftheir elliptic flow and comparing the results to the ones found for lighter hadrons. In Fig. 16the v coefficients of beauty-hadron distributions obtained selecting the 20% highest- (cid:15) andthe 60% lowest- (cid:15) are shown and compared to the results referring to an unbiased selectionof events. The study is performed for the 0-10%, 10-30% and 30-50% centrality classes. Ourfindings are similar to what already obtained for charm: for all the centrality classes theratio v high / low − (cid:15) /v unbiased2 looks quite constant as a function of the transverse momentum p T and independent of the choice of the transport coefficients. In Fig. 17 the results forthe v of beauty hadrons with event-shape-engineering are compared to those for charmedhadrons and pions. The effect of the eccentricity selection is similar for particles withvery different masses and the largest deviations from unity of v ESE2 /v unbiased2 are observedin the 0-10% centrality class. Such a systematic comparison suggests that the quantity v ESE2 /v unbiased2 reflects essentially the initial geometric deformation of the system.– 16 – (GeV/c) T p v Beauty hadrons ε HTL, all ε HTL, small ε HTL, large ε lQCD, all ε lQCD, small ε lQCD, large POWLANG =5.02 TeV NN sPb Pb, Centrality 0 10% (GeV/c) T p ( unb i a s ed ) s e l e c t ed ) / v ε ( v (GeV/c) T p v Beauty hadrons ε HTL, all ε HTL, small ε HTL, large ε lQCD, all ε lQCD, small ε lQCD, large POWLANG =5.02 TeV NN sPb Pb, Centrality 10 30% (GeV/c) T p ( unb i a s ed ) s e l e c t ed ) / v ε ( v (GeV/c) T p v Beauty hadrons ε HTL, all ε HTL, small ε HTL, large ε lQCD, all ε lQCD, small ε lQCD, large POWLANG =5.02 TeV NN sPb Pb, Centrality 30 50% (GeV/c) T p ( unb i a s ed ) s e l e c t ed ) / v ε ( v Figure 16 . The elliptic-flow coefficient v of beauty hadrons in the 0-10% (top), 10-30% (middle)and 30-50% (bottom) most central Pb-Pb collisions at √ s NN = 5 .
02 TeV. Results for the 20%highest- (cid:15) and to the 60% lowest- (cid:15) selection of events are compared to the unbiased case. For bothchoices of transport coefficients the results display a strong and similar sensitivity to the initialeccentricity. – 17 – (GeV/c) T p v Hadrons ε , small π ε , large π , HTL ε Charm, small , HTL ε Charm, large , HTL ε Beauty, small , HTL ε Beauty, large
POWLANG =5.02 TeV NN sPb Pb, Centrality 0 10% (GeV/c) T p ( unb i a s ed ) s e l e c t ed ) / v ε ( v (GeV/c) T p v Hadrons ε , small π ε , large π , HTL ε Charm, small , HTL ε Charm, large , HTL ε Beauty, small , HTL ε Beauty, large
POWLANG =5.02 TeV NN sPb Pb, Centrality 10 30% (GeV/c) T p ( unb i a s ed ) s e l e c t ed ) / v ε ( v (GeV/c) T p v Hadrons ε , small π ε , large π , HTL ε Charm, small , HTL ε Charm, large , HTL ε Beauty, small , HTL ε Beauty, large
POWLANG =5.02 TeV NN sPb Pb, Centrality 30 50% (GeV/c) T p ( unb i a s ed ) s e l e c t ed ) / v ε ( v Figure 17 . Systematic comparison of the effect of the eccentricity selection on the elliptic flow oflight and heavy flavour hadrons in the 0-10% (top), 10-30% (middle) and 30-50% (bottom) mostcentral Pb-Pb collisions at √ s NN = 5 .
02 TeV. Deviations from the unbiased case look remarkablysimilar for pions, charmed and beauty hadrons.
Event-shape-engineering studies of particle p T -spectra and flow in relativistic heavy-ioncollisions, in which events are organized first in centrality classes and then in subsamples– 18 –f high/low eccentricity, have the potential to provide a richer information on the producedmedium, disentangling the effects of the size and density of the fireball from the ones relatedto its geometric asymmetry. In this paper we decided to focus on what one can learn inprinciple applying such a strategy to the study of heavy-flavour observables, showing resultsobtained with our POWLANG transport setup. In this case, in fact, one deals with externalprobes – the charm or beauty quarks – produced off-equilibrium in hard processes occurringbefore the formation of a thermalized Quark-Gluon Plasma. These heavy quarks then crossthe medium, interacting with its constituents, before hadronizing and being detected. Weexpect then that the initial density and size of the medium, beside its shape, affect thefinal momentum and angular distribution of charm and beauty hadrons.Notice that, at variance with the actual experimental situation in which an estimatorbased on the average flow measured in a different kinematic region is used as a a proxy of theinitial geometric asymmetry, in our simulations we can really select events on the basis oftheir initial elliptic o triangular eccentricity. On the other hand experimental analysis canrely on a huge statistics in each centrality class; performing an analogous theoretical studywith full event-by-event simulations would require huge computing and storage resources.Before starting a similar massive campaign it is important to get a solid estimate of thesize of the effect one can observe and of what one can learn on the medium and on itsinteraction with the external probes: this can be done within a simplified approach. Foreach of the considered subset of collisions we relied then on a one-shot hydrodynamicsimulation with a proper average initial condition. Of course, this prevented us fromdisentangling eccentricity and centrality as cleanly as in the experimental analysis and tostudy, for instance, correlations among radial, elliptic and triangular flow, but allowed usin any case to get a list of interesting results.We started our analysis with the nuclear modification factor of charm hadrons, findingthat, within a given centrality class, the selection of events with high/low initial eccentric-ity does not affect significantly the results. The small effect, at most of order 10-20%, lookscompatible with the positive correlation between eccentricity and impact parameter of thecollisions, which entails that more eccentric events are also on average more peripheral,hence leading to a milder quenching of the heavy-quark momentum. As already discussed,experimental analysis try to remove such an artificial correlation performing the selectionon eccentricity in very small centrality bins. The small size of the effect (deviations fromunity of the ratio of the heavy-flavour p T -distributions in high/low- (cid:15) events over the un-biased case being small), the current level of precision of the data and the slightly differentprocedure in performing the eccentricity selection do not allow to draw meaningful conclu-sions from a comparison with the present experimental data. However, in the near future,reducing the experimental uncertainties thanks to larger samples of data and performinga cleaner separation of eccentricity and centrality in theory calculations will allow one toextract a reacher information on the heavy-quark interaction with the medium.On the contrary, a selection based on the event-shape was found to lead to a majoreffect on the elliptic and triangular flow: we obtained results for the charmed hadron v and v in high-eccentricity events a factor 2 larger than in the unbiased case. Theratio v ESE n /v unbiased n looks quite constant as a function of p T . Interestingly, while results– 19 –or the v and v obtained with weak-coupling (HTL curves) or non-perturbative (lQCDcurves) transport coefficients display significant differences, the ratio between the high/low-eccentricity results and the unbiased case looks pretty independent of the modeling of theinteraction with the medium, suggesting that the effects depends mainly on the initialgeometry of the fireball. Also the dependence of v ESE n /v unbiased n on centrality is quite weak:for the triangular flow it is completely negligible; in the case of the elliptic flow, deviationsfrom unity of the ratio v high − (cid:15) /v unbiased2 tend to slightly decrease moving from centralto more peripheral collisions, the smallest effect being observed for charm quarks in the60-80% centrality class. This last observation suggests a limited interaction of the heavyquark in the case of a less thick and dense medium, which cannot leave the imprints of itsinitial geometry in the final angular distribution of charm quarks.We decided then to follow a complementary strategy, namely to select events of agiven initial eccentricity (cid:15) and (cid:15) and study how the results for the flow coefficients v and v change when considering different centrality classes. We started considering lighthadrons, coming from the hadronization of the bulk medium. We saw that in the case ofsoft hadrons decoupling from a freeze-out hypersurface, for a given initial eccentricity, theflow pattern looks essentially the same in the different centrality classes: anisotropies inthe particle distributions simply reflect the corresponding asymmetries in the fluid-velocityfield at freeze-out, arising from the hydrodynamic response of the medium to its initialgeometric deformation. In the case of heavy flavour distributions, however, things are morecomplicate, since we are not dealing with particles which are part of the bulk medium fromthe beginning of its evolution, but with hadrons arising from c and b quarks produced ininitial hard partonic processes, with momentum distributions described by perturbative-QCD. In this case we expect that the centrality of the collision plays an important role indetermining the response of the final particle distributions to the same initial geometricdeformation, since a medium of larger size, longer lifetime and higher density should affectmore strongly the propagation of the heavy quarks. This is what we actually observed atthe quark level, both for the v and the v : selecting events with the same (cid:15) / we founda larger elliptic/triangular flow of charm quarks in more central collisions. Hadronization,modeled in our scheme via recombination with light thermal partons following the flowof the medium, tends to wash out this difference, although some effect is still visible, inparticular in the case of v . We hope our observations can motivate future experimentalanalysis along this direction.Finally we moved to beauty, focusing on its elliptic flow, and our main finding is that,although the v of beauty hadrons is quite small, the effect of the eccentricity selection onthe azimuthal distributions, once normalized to the unbiased result, turns out to be of thesame size of the one of charmed and light hadrons.Our study presented in this paper must be considered just a first step in the directionof better constraining the heavy-quark interaction with the medium and the response to theevent-by-event fluctuations in the initial state of the latter. 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