# Event-shape engineering and heavy-flavour observables in relativistic heavy-ion collisions

## Abstract

Traditionally, events collected at relativistic heavy-ion colliders are classified according 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 latter are directly related to the impact parameter of the two nuclei, which sets also the initial eccentricity of the system: zero in the case of the most central events and getting larger for more 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 different initial eccentricity and hence different flow harmonics for the final hadron distributions. This issue can be addressed by an event-shape-engineering analysis, consisting in selecting events with the same centrality but different magnitude of the average bulk anisotropic flow and therefore of the initial-state eccentricity. In this paper we present the implementation of this analysis in the POWLANG transport model, providing predictions for the transverse-momentum and angular distributions of charm and beauty hadrons for event-shape selected collisions. 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 its geometric asymmetry, breaking the perfect correlation between eccentricity and impact parameter which characterizes a modelling of the medium based on smooth average initial conditions.

## 1 Introduction

Heavy flavour particles (*D* / *B* mesons and \(\varLambda _{c/b}\) baryons), arising from charm and beauty quarks produced in initial hard partonic scattering processes, have been always considered a 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 spite of the large mass of the parent quarks, the distributions of final-state particles displayed the same features observed in the case of light hadrons, i.e. a quenching of the spectra at high transverse momentum \(p_T\) (with possible signatures of a mass and colour-charge dependence of parton energy loss) and a non-vanishing, positive elliptic-flow coefficient \(v_2\). First data were limited to electrons from heavy-flavour decays, without the possibility to discriminate between the charm and beauty contributions [1, 2]. It became then possible to reconstruct *D*-mesons through some exclusive decay channels [3, 4, 5, 6, 7, 8]. The message from these first measurements was that, although quantitatively a bit milder, the same quenching of the momentum spectra and elliptic (and triangular, as shown in Ref. [8]) asymmetry of the azimuthal distributions observed for light hadrons characterized also charm and beauty particles. This entailed a quite strong coupling of the heavy quarks with the hot deconfined plasma of quarks and gluons (QGP) supposed to be produced in the collision of the two nuclei and possibly a non trivial modification of their hadronization due to the large density of light thermal partons nearby. From a comparison of the outcomes of transport calculations with experimental data it is in principle possible to extract information on the value of the heavy-quark momentum-diffusion coefficient, a fundamental quantity which in the static limit in hot-QCD admits a rigorous definition in terms of Euclidean correlators of chromo-electric fields [9, 10]. Recently, a systematic investigation based on a Bayesian approach aiming at extracting the heavy-flavour diffusion coefficient from current experimental data has been carried out by some authors [11]. In this connection a comprehensive study of the various theoretical uncertainties arising from the initial heavy-quark spectrum, from Cold-Nuclear-Matter effects and from the modelling of the medium and of hadronization was carried out in Ref. [12]. This is somehow similar to what done for the case of soft hadrons, where a comparison of hydrodynamics calculations with experimental particle distributions allowed people to constrain within a quite narrow band another transport coefficient, the shear-viscosity to entropy-density ratio \(\eta /s\), which turned out to be close to the lower bound \(1/4\pi \) 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 \(\varLambda _c\) production in nuclear collisions have the potential to put the issue of medium-modification of heavy-flavour hadrochemistry on solid ground [14, 15], making possible to validate heavy-quark hadronization models based on the recombination with light thermal partons. Experimental data on *B* meson production [16] allow one to study the mass dependence of the heavy-quark medium interaction; if in the future these analysis were 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 under the best control, reaching the goal of really *measuring* the heavy-quark diffusion coefficient. Recently heavy-flavour studies have been extended to the case of proton-nucleus collisions [17, 18, 19, 20], with the aim of contributing to answer the still open question whether also in such small systems QGP droplets can be formed [21].

Finally, the measurement of odd flow-harmonics of heavy-flavour hadrons can provide a richer information on the initial conditions of the system formed after the collision of the two high-energy nuclei, like its tilted profile in the reaction plane (wounded nucleons tending to deposit more energy along the direction of their motion) in the case of the directed flow \(v_1\) [22, 23, 24] or its event-by-event fluctuations (from the random nucleon positions) in the case of the triangular flow \(v_3\). The triangular flow \(v_3\) of *D* mesons in Pb–Pb collisions provided by transport calculations has been studied in some recent publications and theoretical results [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 to the initial asymmetries of the system, getting information both on the coupling of the heavy quarks with the medium and on its initial conditions, is given by the so-called Event-Shape-Engineering (ESE) studies. The basic idea is to select events belonging to the same centrality class, but characterized by a different initial geometric (elliptic or triangular) asymmetry, getting subsamples of events with high/low eccentricity [27]. Such an approach was proposed and adopted by the ALICE collaboration in the analysis of momentum and azimuthal distributions of light hadrons [28], comparing the results obtained in subsamples of collisions with large/small average elliptic-flow with the ones of an unbiased selection of events. Here, in the framework of a transport calculation, we wish to extend the approach to heavy flavour, studying how the different geometric asymmetry and the resulting anisotropic flow of the medium affect the propagation of heavy quarks and leave their signatures in the final charm/beauty-hadron distributions [29]. Our findings will be compared with recent experimental outcomes [30]. For independent phenomenological studies of hard probes (heavy-flavour particles and jets) in heavy-ion collisions based on event-shape-engineering see also Refs. [31, 32, 33]. Notice however that Refs. [31, 32], although representing pioneering work in the field, are mainly focused on the high transverse-momentum region of particle distributions and are based on a rather schematic modeling of the interaction of the hard probe with the medium. Our ambition here is to perform a full transport calculation interfaced with a realistic hydrodynamic simulation of the evolution of the background medium.

*c*and

*b*quarks through the QGP and for their hadronization in the presence of a hot deconfined medium. Finally in Sect. 5 we discuss our results, suggesting possible future improvements.

## 2 Modelling of the background medium

For the modelling of the medium produced in nucleus–nucleus collisions (in this paper we consider Pb–Pb collisions at \(\sqrt{s_{\mathrm{NN}}}=5.02\) TeV) we adopted the same approach described in detail in Ref. [26], interfacing a Glauber Monte-Carlo (Glauber-MC) simulation of the initial condition of the system to a hydrodynamic code (ECHO-QGP [34]) calculating the subsequent 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 to solve a (2+1)-dimensional problem, reducing the computational time.

*K*(with dimensions of an inverse length) sets the average entropy deposited by a single collision (so far we do not include fluctuations at the level of the individual nucleon-nucleon inelastic collisions). As in Ref. [26] for Pb–Pb collisions at \(\sqrt{s_{\mathrm{NN}}}=5.02\) TeV we choose \(K\tau _0=6.37\). For each event the above entropy density can be used as a weight to define complex eccentricities, which characterize the initial state (i.e. both the amount of anisotropy and its orientation in the transverse plane) and will be mapped into the final hadron distributions by the subsequent hydrodynamic evolution [35]:

Using as an estimator the number of binary nucleon-nucleon collisions, we group the Pb–Pb events in centrality classes (0–10%, 10–30% and 30–50%) and study within each sample the distribution of initial elliptic and triangular eccentricity \(\epsilon _2\) and \(\epsilon _3\). We will also consider in some of the calculations a very peripheral class (60–80%). Results are shown in Fig. 1. Notice how, within a given centrality class, the eccentricity distribution is quite broad, in particular for the case of \(\epsilon _2\) whose large event-by-event fluctuations arise both from the different impact parameter and from the random positions of the nucleons within the colliding nuclei. The strong dependence on the impact parameter is also evident from the sizable shift of the peak of the distribution towards larger values of \(\epsilon _2\) going from central to peripheral collisions. On the other hand, in the case of \(\epsilon _3\) the eccentricity distributions are narrower and the displacement of the peak when moving to a different centrality class is milder. This reflects the different origin of the triangular asymmetry, which (neglecting sub-nucleonic degrees of freedom) is entirely due to the event-by-event fluctuations in the positions of the nucleons inside the colliding nuclei.

*x*-axis and, starting from Eq. (1), we construct an average entropy-density distribution weighting each event by the number of binary nucleon-nucleon collisions (since the \(Q\overline{Q}\) production scales with \(N_{\mathrm{coll}}\), which introduces a bias towards more central events). In order to verify that our modeling of the initial state leads to a realistic density of the medium we integrate Eq. (1) over the transverse plane, getting the entropy per unit space-time rapidity \(dS/d\eta _s=\tau _0\int d\mathbf {x}_\perp s(\mathbf {x}_\perp )\). For the 0–10% centrality class one gets \(dS/d\eta _s\approx 10{,}560\) which, for an adiabatic expansion, should be directly related to the rapidity density of charged particles detected in the final state. For a gas of relativistic bosons one has \(s/n\approx 3.6\); taking into account that one detects only charged hadrons one gets \(dN^\mathrm{ch}/dy\approx (2/3)\cdot 10{,}560/3.6\approx 1955\), in nice agreement with the particle multiplicity measured in Pb–Pb collisions at \(\sqrt{s_{\mathrm{NN}}}=5.02\) TeV. Our initial density in central collisions is also in overall agreement with the values obtained by other groups performing heavy-flavour transport calculations and summarized in Table 1 of Ref. [12]. In Figs. 3 and 4, referring to the 10–30% centrality class, we display the result of the above modeling of the initial condition for the case of an elliptic and triangular deformation, respectively: the average initial conditions for the unbiased, low-\(\epsilon _n\) and high-\(\epsilon _n\) subsets of events are shown.

The ratios \(\langle \epsilon _n\rangle ^\mathrm{selected}/\langle \epsilon _n\rangle ^{\mathrm{unbiased}}\) obtained selecting, in the various centrality classes, the 0–20% and 0–60% most/least eccentric events

Centrality | \(\langle \epsilon _2\rangle ^{\mathrm{high}}/\langle \epsilon _2\rangle ^{\mathrm{mb}}\) | \(\langle \epsilon _2\rangle ^{\mathrm{low}}/\langle \epsilon _2\rangle ^{\mathrm{mb}}\) | \(\langle \epsilon _3\rangle ^{\mathrm{high}}/\langle \epsilon _3\rangle ^{\mathrm{mb}}\) | \(\langle \epsilon _3\rangle ^{\mathrm{low}}/\langle \epsilon _3\rangle ^{\mathrm{mb}}\) |
---|---|---|---|---|

0–10% | 1.845 | 0.646 | 1.868 | 0.667 |

10–30% | 1.561 | 0.765 | 1.879 | 0.652 |

30–50% | 1.469 | 0.784 | 1.801 | 0.675 |

Going back to Fig. 1 we note how the eccentricity distributions of different centrality classes display a significant overlap: we can have events in which the system is initially characterized by an equal degree of geometric deformation, but by very different dimensions and energy density. As a complementary study we select, in the different centrality classes, samples of events with an initial geometric asymmetry belonging to the same narrow interval: we choose \(0.3\le \epsilon _2\le 0.4\) and \(0.2\le \epsilon _3\le 0.3\) for the elliptic and triangular deformation. The resulting average initial conditions of the systems are displayed in Figs. 7 and 8 for the 0–10%, 10–30% and 30–50% centrality classes. Starting from such initial state it is then of interest to study the flow of light hadrons decoupling from the medium at the end of its hydrodynamic evolution, in order to check whether the angular distributions of final-state particles respond only to the initial geometry of the system or whether its very different energy density plays a role. As one can see from Fig. 9 the major role is played by the initial geometric deformation: the \(v_n\) curves for pions and protons in different centrality 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 to perform the same study for the case of heavy-flavour particles, since their energy-loss and degree of thermalization should be sensitive to the dimension and transport coefficients of the medium, both depending strongly on the centrality of the collision.

## 3 Heavy flavour transport and hadronization

## 4 Results

We finally move to consider also beauty quarks and hadrons, focusing on the study of their elliptic flow and comparing the results to the ones found for lighter hadrons. In Fig. 16 the \(v_2\) coefficients of beauty-hadron distributions obtained selecting the 20% highest-\(\epsilon _2\) and the 60% lowest-\(\epsilon _2\) are shown and compared to the results referring to an unbiased selection of events. The study is performed for the 0–10%, 10–30% and 30–50% centrality classes. Our findings are similar to what already obtained for charm: for all the centrality classes the ratio \(v_2^{\mathrm{high/low}-\epsilon _2}/v_2^\mathrm{unbiased}\) 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 for the \(v_2\) of beauty hadrons with event-shape-engineering are compared to those for charmed hadrons and pions. The effect of the eccentricity selection is similar for particles with very different masses and the largest deviations from unity of \(v_2^{\mathrm{ESE}}/v_2^\mathrm{unbiased}\) are observed in the 0–10% centrality class. Such a systematic comparison suggests that the quantity \(v_2^{\mathrm{ESE}}/v_2^{\mathrm{unbiased}}\) reflects essentially the initial geometric deformation of the system.

## 5 Discussion and perspectives

Event-shape-engineering studies of particle \(p_T\)-spectra and flow in relativistic heavy-ion collisions, in which events are organized first in centrality classes and then in subsamples of high/low eccentricity, have the potential to provide a richer information on the produced medium, disentangling the effects of the size and density of the fireball from the ones related to its geometric asymmetry. In this paper we decided to focus on what one can learn in principle applying such a strategy to the study of heavy-flavour observables, showing results obtained with our POWLANG transport setup. In this case, in fact, one deals with external probes – the charm or beauty quarks – produced off-equilibrium in hard processes occurring before the formation of a thermalized Quark-Gluon Plasma. These heavy quarks then cross the medium, interacting with its constituents, before hadronizing and being detected. We expect then that the initial density and size of the medium, beside its shape, affect the final momentum and angular distribution of charm and beauty hadrons.

Notice that, at variance with the actual experimental situation in which an estimator based on the average flow measured in a different kinematic region is used as a a proxy of the initial geometric asymmetry, in our simulations we can really select events on the basis of their initial elliptic o triangular eccentricity. On the other hand experimental analysis can rely on a huge statistics in each centrality class; performing an analogous theoretical study with 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 the size of the effect one can observe and of what one can learn on the medium and on its interaction with the external probes: this can be done within a simplified approach. For each of the considered subset of collisions we relied then on a one-shot hydrodynamic simulation with a proper average initial condition. Of course, this prevented us from disentangling eccentricity and centrality as cleanly as in the experimental analysis and to study, for instance, correlations among radial, elliptic and triangular flow, but allowed us in any case to get a list of interesting results.

We started our analysis with the nuclear modification factor of charm hadrons, finding that, within a given centrality class, the selection of events with high/low initial eccentricity does not affect significantly the results. The small effect, at most of order 10–20%, looks compatible with the positive correlation between eccentricity and impact parameter of the collisions, 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 selection on eccentricity in very small centrality bins. The small size of the effect (deviations from unity of the ratio of the heavy-flavour \(p_T\)-distributions in high/low-\(\epsilon _2\) events over the unbiased case being small), the current level of precision of the data and the slightly different procedure in performing the eccentricity selection do not allow to draw meaningful conclusions from a comparison with the present experimental data. However, in the near future, reducing the experimental uncertainties thanks to larger samples of data and performing a cleaner separation of eccentricity and centrality in theory calculations will allow one to extract 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 major effect on the elliptic and triangular flow: we obtained results for the charmed hadron \(v_2\) and \(v_3\) in high-eccentricity events a factor 2 larger than in the unbiased case. The ratio \(v_n^{\mathrm{ESE}}/v_n^{\mathrm{unbiased}}\) looks quite constant as a function of \(p_T\). Interestingly, while results for the \(v_2\) and \(v_3\) obtained with weak-coupling (HTL curves) or non-perturbative (lQCD curves) transport coefficients display significant differences, the ratio between the high/low-eccentricity results and the unbiased case looks pretty independent of the modeling of the interaction with the medium, suggesting that the effects depends mainly on the initial geometry of the fireball. Also the dependence of \(v_n^{\mathrm{ESE}}/v_n^{\mathrm{unbiased}}\) on centrality is quite weak: for the triangular flow it is completely negligible; in the case of the elliptic flow, deviations from unity of the ratio \(v_2^{\mathrm{high}-\epsilon _2}/v_2^{\mathrm{unbiased}}\) tend to slightly decrease moving from central to more peripheral collisions, the smallest effect being observed for charm quarks in the 60–80% centrality class. This last observation suggests a limited interaction of the heavy quark in the case of a less thick and dense medium, which cannot leave the imprints of its initial geometry in the final angular distribution of charm quarks.

We decided then to follow a complementary strategy, namely to select events of a given initial eccentricity \(\epsilon _2\) and \(\epsilon _3\) and study how the results for the flow coefficients \(v_2\) and \(v_3\) change when considering different centrality classes. We started considering light hadrons, coming from the hadronization of the bulk medium. We saw that in the case of soft hadrons decoupling from a freeze-out hypersurface, for a given initial eccentricity, the flow pattern looks essentially the same in the different centrality classes: anisotropies in the particle distributions simply reflect the corresponding asymmetries in the fluid-velocity field at freeze-out, arising from the hydrodynamic response of the medium to its initial geometric deformation. In the case of heavy flavour distributions, however, things are more complicate, since we are not dealing with particles which are part of the bulk medium from the beginning of its evolution, but with hadrons arising from *c* and *b* quarks produced in initial 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 in determining the response of the final particle distributions to the same initial geometric deformation, since a medium of larger size, longer lifetime and higher density should affect more strongly the propagation of the heavy quarks. This is what we actually observed at the quark level, both for the \(v_2\) and the \(v_3\): selecting events with the same \(\epsilon _{2/3}\) we found a larger elliptic/triangular flow of charm quarks in more central collisions. Hadronization, modeled in our scheme via recombination with light thermal partons following the flow of the medium, tends to wash out this difference, although some effect is still visible, in particular in the case of \(v_3\). We hope our observations can motivate future experimental analysis along this direction.

Finally we moved to beauty, focusing on its elliptic flow, and our main finding is that, although the \(v_2\) of beauty hadrons is quite small, the effect of the eccentricity selection on the azimuthal distributions, once normalized to the unbiased result, turns out to be of the same size of the one of charmed and light hadrons.

Our study presented in this paper must be considered just a first step in the direction of better constraining the heavy-quark interaction with the medium and the response to the event-by-event fluctuations in the initial state of the latter. In the future we can certainly improve our results, employing an event-by-event approach allowing a study of all possible correlations of the various kind of flow (radial, elliptic and triangular) among themselves and with the fluctuations of the initial geometry of the medium. This, however, will be a very demanding task from the point of view of computing time and storage resources. We believe that this first cheaper exploratory study has already been able to provide some interesting indications motivating future, more refined, ESE-analysis addressing for instance the triangular flow and more peripheral centrality classes. It also permits a first comparison with the experimental data, which in the following years – with increasing statistics – will become more precise.

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