Elliptic flow of muons from heavy-flavour hadron decays at forward rapidity in Pb-Pb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV

The elliptic flow, $v_{2}$, of muons from heavy-flavour hadron decays at forward rapidity ($2.5<y<4$) is measured in Pb--Pb collisions at $\sqrt{s_{\rm NN}}$~=~2.76 TeV with the ALICE detector at the LHC. The scalar product, two- and four-particle $Q$ cumulants and Lee-Yang zeros methods are used. The dependence of the $v_2$ of muons from heavy-flavour hadron decays on the collision centrality, in the range 0--40\%, and on transverse momentum, $p_{\rm T}$, is studied in the interval $3<p_{\rm T}<10$~GeV/$c$. A positive $v_2$ is observed with the scalar product and two-particle $Q$ cumulants in semi-central collisions (10--20\% and 20--40\% centrality classes) for the $p_{\rm T}$ interval from 3 to about 5 GeV/$c$. The $v_2$ magnitude tends to decrease towards more central collisions and with increasing $p_{\rm T}$. It becomes compatible with zero in the interval $6<p_{\rm T}<10~{\rm GeV/}c$. The results are compared to models describing the interaction of heavy quarks and open heavy-flavour hadrons with the high-density medium formed in high-energy heavy-ion collisions. The model calculations describe the measured $v_2$ within uncertainties.


Introduction
Experiments with ultra-relativistic heavy-ion collisions aim at investigating the properties of stronglyinteracting matter at very high temperatures and energy densities. Quantum Chromodynamics (QCD) calculations on the lattice predict, under these conditions, the formation of a Quark-Gluon Plasma (QGP), where colour confinement vanishes and chiral symmetry is partially restored [1][2][3][4][5]. Heavy quarks (charm and beauty) are created in initial hard-scattering processes on a time scale shorter than the QGP formation time. Subsequently, they interact with the medium constituents via inelastic [6,7] and elastic [8][9][10] processes. Therefore, heavy quarks are regarded as effective probes of the QGP properties.
Heavy-quark energy loss due to in-medium interactions can be studied by means of the nuclear modification factor R AA , defined as the ratio of the yield of heavy-flavour particles measured in nucleus-nucleus (AA) collisions to that observed in proton-proton (pp) collisions scaled by the number of binary nucleonnucleon collisions. The PHENIX and STAR Collaborations measured, in central Au-Au collisions at √ s NN = 200 GeV, a strong suppression corresponding to a R AA of about 0.2-0.3 for heavy-flavour decay electrons at mid-rapidity (y) and transverse momentum p T > 5 GeV/c [11][12][13][14][15][16][17]. A similar suppression was also measured by the STAR Collaboration for mid-rapidity D 0 mesons [18]. A significant suppression was also observed by the PHENIX Collaboration at forward rapidity for muons from heavy-flavour hadron decays in central Cu-Cu collisions at √ s NN = 200 GeV [19]. At the LHC, the ALICE Collaboration reported a similar effect in central Pb-Pb collisions at √ s NN = 2.76 TeV for D mesons at mid-rapidity [20] and muons from heavy-flavour hadron decays at forward rapidity [21] in the interval 2 < p T < 16 GeV/c and 4 < p T < 10 GeV/c, respectively. The CMS Collaboration measured a significant suppression of non-prompt J/ψ from beauty-hadron decays in the interval 6.5 < p T < 30 GeV/c (3 < p T < 30 GeV/c) and |y| < 2.4 (1.6 < |y| < 2.4) [22,23]. A first measurement of non-prompt J/ψ by the ALICE Collaboration at mid-rapidity (|y| < 0.8) and in the interval 4.5 < p T < 10 GeV/c has been recently published [24].
Further insights into the QGP evolution and the in-medium interactions can be gained from the study of the azimuthal anisotropy of particles carrying heavy quarks which, in contrast to light quarks, have experienced the full system evolution. The study of azimuthal anisotropy is a field of intense experimental and theoretical investigations (see [25] and references therein). In non-central collisions, the initial spatial anisotropy of the overlap region, elongated in the direction perpendicular to the reaction plane, defined by the beam axis and the impact parameter of the collision, is converted into an anisotropy in momentum space through rescatterings [26]. Experimentally, the study of the particle azimuthal anisotropy is based on a Fourier expansion of azimuthal distributions given by: where ϕ and p T are the particle azimuthal angle and transverse momentum, respectively. The Fourier coefficients, v n , characterize the anisotropy of produced particles and Ψ n is the azimuthal angle of the initial-state symmetry plane for the n th harmonic, introduced to account for the event-by-event fluctuations of the initial nucleon density profile. The second Fourier coefficient, v 2 , which can also be expressed as v 2 = cos[2(ϕ − Ψ 2 )] , is named elliptic flow.
The v 2 of heavy-flavour hadrons is expected to provide information on the collective expansion and degree of thermalization of heavy quarks in the medium at low p T (p T < 2 − 3 GeV/c). The participation of heavy quarks in the collective expansion is expected to give a positive v 2 [26]. Moving towards intermediate p T (3 < p T < 6 GeV/c), the v 2 Fourier coefficient is also expected to be sensitive to the presence of recombination processes in the hadronization of heavy quarks [27,28]. At high p T (p T > 6 GeV/c), the v 2 measurement can constrain the path-length dependence of the in-medium parton energy loss, which becomes the dominant contribution to the azimuthal anisotropy and is also predicted to give a positive v 2 [29,30], thus complementing the R AA measurement.
The PHENIX Collaboration reported a positive v 2 of heavy-flavour decay electrons at mid-rapidity in Au-Au collisions at √ s NN = 200 GeV, reaching a maximum value of about 0.15 at p T = 1.5 GeV/c in semi-central collisions [14,15,31]. A similar behaviour was also observed by the STAR Collaboration [32]. Recently, a v 2 value significantly larger than zero was measured for D mesons at mid-rapidity in Pb-Pb collisions at √ s NN = 2.76 TeV [33,34]. A complementary measurement at the same energy, provided by the heavy-flavour decay muon elliptic flow at forward rapidity (2.5 < y < 4), is of great interest in order to provide new constraints for models that implement the heavy-quark interactions with the medium. Finally, the measurement is also important for the interpretation of the J/ψ elliptic flow results at forward rapidity [35] in terms of a regeneration production from deconfined charm quarks in the medium.
In this Letter, we present the measurement of the elliptic flow of muons from heavy-flavour hadron decays at forward rapidity (2.5 < y < 4) in Pb-Pb collisions at √ s NN = 2.76 TeV recorded with the ALICE detector. The elliptic flow is measured using different methods: scalar product [36], two-and fourparticle Q cumulants [37,38] and Lee-Yang zeros [39][40][41]. These methods exhibit different sensitivities to flow fluctuations and correlations not related to the azimuthal asymmetry in the initial geometry (nonflow effects). The v 2 coefficient is measured as a function of p T in the interval 3 < p T < 10 GeV/c and in three centrality classes in the range 0-40%. The centrality dependence of v 2 is presented in the interval 3 < p T < 10 GeV/c. The Letter is organized as follows. The ALICE detector, with an emphasis on the muon spectrometer, and the data sample are presented in Section 2. The analysis details, the methods for the v 2 measurement, the inclusive muon v 2 determination, the procedure for the subtraction of the background of muons from decays of light-flavour hadrons and the study of systematic uncertainties, are described in Section 3.
The v 2 results for muons from heavy-flavour decays are presented in Section 4 and compared to model calculations in Section 5. Finally, conclusions are given in Section 6.

ALICE experiment and data sample
The ALICE detector is described in detail in [42,43]. The apparatus is composed of a set of central barrel detectors (pseudo-rapidity coverage |η| < 0.9) located inside a solenoid magnet that generates a field of 0.5 T parallel to the beam direction, a muon spectrometer (−4 < η < −2.5 1 ) and a set of detectors for event characterization and triggering located in the forward and backward η regions. The muon spectrometer consists of a passive front absorber made of carbon, concrete and steel, a beam shield, a 3 T·m dipole magnet, tracking chambers, a muon filter (iron wall) and trigger chambers. The muon tracking system is composed of five stations, each including two planes of cathod pad chambers, with the third station inside the dipole magnet. The muon tracking system is completed by four trigger planes of resistive plate chambers downstream of the iron wall, which absorbs hadrons that punch through the front absorber, as well as secondary particles produced inside it and low momentum muons (p < 4 GeV/c). Two scintillator arrays (V0) covering the pseudo-rapidity intervals −3.7 < η < −1.7 and 2.8 < η < 5.1 are used for triggering, for collision centrality determination and for beam-induced background rejection. The Zero Degree Calorimeters (ZDC), located at 114 m from the centre of the detector on both sides, can detect spectator protons and neutrons and are also used for the offline rejection of beam-induced background and electromagnetic interactions. The Silicon Pixel Detector (SPD), that composes the two innermost layers of the Inner Tracking System (ITS), is used for the interaction vertex reconstruction. The Time Projection Chamber (TPC), which measures charged-particle tracks with full azimuthal coverage in |η| < 0.9, is used in this analysis for the measurement of the reference particles (Section 3.1).
The results presented in this Letter are obtained from the data sample recorded with ALICE during the 2011 Pb-Pb run. The data were collected with a minimum-bias trigger requiring the coincidence of signals in the two V0 arrays in synchronization with the passage of two crossing bunches. In addition, the recorded event sample was enriched with central and semi-central Pb-Pb collisions by applying thresholds, at the trigger level, on the V0 signal amplitude. The beam-induced background (beam-gas interactions) was reduced by using the timing information from the V0 and ZDC detectors. Furthermore, a minimal energy deposit in the ZDC was required to reject the contribution from electromagnetic Pb-Pb interactions. Only events with a reconstructed primary vertex within ± 10 cm from the nominal position of the interaction vertex along the beam direction are analyzed. The Pb-Pb collisions are classified according to their degree of centrality by means of the sum of the amplitudes of the signals in the V0 detector and the centrality classes are defined as percentiles of the total hadronic Pb-Pb cross section [44]. The analysis is carried out in three centrality classes: 0-10%, 10-20% and 20-40%. The analysed data sample corresponds to an integrated luminosity of 11.3 µb −1 in the 0-10% centrality class and of 3.5 µb −1 in the other two centrality classes.

Data analysis
The elliptic flow of muons from heavy-flavour hadron decays, v where f µ←π,K is the muon background fraction, defined as the ratio of the yield of muons from primary charged pion and kaon decays to that of inclusive muons. The measurement of the v µ←HF 2 coefficient is carried out in the interval 3 < p T < 10 GeV/c in order to limit the systematic uncertainty on the subtraction of the muon background contribution.

Track selection
The selection criteria for particles of interest, muon tracks, are similar to those used in the previous analyses of pp collisions at √ s = 2.76 TeV and 7 TeV and Pb-Pb collisions at √ s NN = 2.76 TeV [21,45]. The tracks are required to be within the geometrical acceptance of the muon spectrometer, with −4 < η < −2.5 and 170 • < θ abs < 178 • , where θ abs is the polar angle measured at the end of the absorber. In order to improve the muon identification, a reconstructed track in the tracking chambers is required to match a track segment in the trigger chambers. This leads to a very efficient rejection of the background produced by charged hadrons, which are absorbed in the iron wall. Furthermore, a cut on the product p ·DCA of the track momentum p and distance of closest approach (DCA) to the primary vertex is applied to remove the beam-induced background tracks and fake tracks coming from the superposition of several particles crossing the muon spectrometer. This cut is set to 6σ , where σ is extracted from a Gaussian fit to the p · DCA distribution measured in two intervals of θ abs , corresponding to different materials in the front absorber. After the cuts are applied, in the region p T > 3 GeV/c the residual background to heavy-flavour decay muons consists of muons from decays of primary charged pions and kaons 2 and it amounts to 5-15%, depending on p T and on collision centrality (Section 3.4).
The mid-rapidity charged-particle tracks used to determine the flow vector Q n or the generating function (Section 3.2) are called in the following reference particles. They are defined as tracks measured in the TPC in |η| < 0.8. These are required to have at least 70 associated space points out of the maximum of 159, a χ 2 per degree of freedom (ndf) for the momentum fit in the range χ 2 /ndf < 2 and a transverse momentum value in the interval 0.2 < p T < 5 GeV/c. Additionally, tracks are rejected if their distance of closest approach to the primary vertex is larger than 3 cm in the plane transverse to the beam direction or in the longitudinal direction.

Flow analysis methods
The elliptic flow measurement is carried out using various methods that have different sensitivities to flow fluctuations and non-flow effects [46]. Flow fluctuations are mainly due to event-by-event fluctuations of the initial density profile, while non-flow effects correspond to correlations not related to the azimuthal anisotropy in the initial state, such as resonance decays, jets and Bose-Einstein correlations between identical particles. It is worth mentioning that, in the present analysis, most of these non-flow effects are strongly suppressed by introducing an η gap between reference particles and particles of interest [47]. In this analysis, the scalar product [36], two-and four-particle Q cumulants [37,38] and Lee-Yang zeros [39][40][41] methods are employed. The description of these methods will be limited to the features specific to the present analysis. The  {LYZ − Sum} are obtained using Lee-Yang zeros with product and sum generating functions. The superscripts µ and µ ← HF refer to inclusive muons and muons from heavy-flavour hadron decays, respectively. It is worth mentioning that these methods are more accurate than the standard event plane method, which yields a measurement lying between the event-averaged mean value and the rootmean-square value in the presence of flow fluctuations [48,49]. Moreover, the multi-particle correlation methods (four-particle Q cumulants and Lee-Yang zeros) are less affected by non-flow correlations than two-particle correlation methods, but they cannot be used reliably when the muon flow magnitude is small and when the number of muons is small in the selected phase-space region e.g. in central and peripheral collisions, respectively [37,39]. Under these conditions, the scalar product and two-particle cumulant methods provide a v 2 measurement in a wider centrality range.
The scalar product method [36, 48], derived from the standard event plane technique [48], is based on the measurement of the flow vector Q n [36] computed from reference particles. In order to determine the elliptic flow, the Q 2 vector in a given event is expressed as: where ϕ j is the particle azimuthal angle and N is the multiplicity of reference particles.
With this method the 2 nd harmonic coefficient is given by: where the brackets in the numerator indicate the average over muons at forward rapidity, in all events. The vector Q 2 is calculated from Eq. (3) and the vector u 2,i = (cos2ϕ i , sin2ϕ i ) is the unit vector of the i th muon. In the denominator, each sample of reference particles used to compute Q 2 is divided into two sub-samples of same multiplicity in symmetrical η intervals, −0.8 < η < −0.5 and 0.5 < η < 0.8, separated by a η gap of one unit of pseudo-rapidity, labeled with the superscripts A and B and the brackets correspond to the average over events.
The cumulant technique [37,38] is based on a cumulant expansion of multi-particle azimuthal correlations. Different order cumulants have different sensitivities to flow fluctuations. In the present analysis, two-and four-particle cumulants are used to extract the muon elliptic flow. The results presented in the following are obtained from a direct calculation of multi-particle cumulants performed by using the Q-cumulant technique [38], which is based on the moments of the magnitude of the flow vector Q 2 . It is worth mentioning that in this approach the cumulants are not biased by the interferences between various harmonics. The reference elliptic flow values V 2 evaluated from the 2 nd order cumulant c 2 {2} and 4 th order cumulant c 2 {4} with reference particles are given by respectively. Once the reference elliptic flow is estimated, the muon elliptic flow with respect to the reference elliptic flow is obtained from the 2 nd and 4 th order cumulants according to: where d 2 {2} and d 2 {4} are the 2 nd and 4 th order cumulants of selected muons [38].
The Lee-Yang zeros method [39][40][41] relies on correlations involving all particles in the event. This is the limit of cumulants when the order of cumulants goes to infinity. The method is based on the location of the zeros in the complex plane, of a generating function of azimuthal correlations, which relates the position of the first minimum of the generating function to the magnitude of the reference elliptic flow V 2 defined as: where M is the multiplicity of reference particles and the average is taken over all events. For this purpose, the following complex-valued generating function is evaluated as a function of a positive real variable r and few, typically five, equally spaced reference angles ϑ (LYZ − Prod method): The first positive minimum of |G ϑ (ir)|, denoted as r ϑ 0 , allows one to estimate V ϑ 2 , which can be written as V ϑ 2 = j 01 /r ϑ 0 , where j 01 2.405 is the first root of the Bessel function. Once the first minimum r ϑ 0 is determined, the differential muon elliptic flow is estimated with respect to the reference flow V ϑ 2 as detailed in [41]. Finally, the result is averaged over all ϑ angles. An alternative form of the generating function provided with the LYZ − Sum method is: The version of the method involving a product for the construction of the generation function (Eq. (7)) was designed to disentangle interferences between different harmonics, which is not the case with the generating function using a sum of the individual reference particle contributions. Both generating functions are used in this analysis.
Note that, for all methods, autocorrelation effects are avoided because the particles (muons) used in the determination of the flow are not included in the estimation of the reference flow.

Inclusive muon elliptic flow
The elliptic flow of inclusive muons, v µ 2 , is studied with two-particle correlation methods (scalar product and two-particle Q cumulants) in the centrality intervals 0-10%, 10-20% and 20-40%. In the 20-40% centrality interval, the multi-particle correlation methods (four-particle Q cumulants and Lee-Yang zeros) are also used.
Several sources of systematic uncertainty affecting the muon elliptic flow measurement are considered. These take into account the changes due to the variations of the reference particle selection criteria as in [33,34,50], to allow us to check the robustness of the v µ 2 measurement. Since the collision impact parameter distribution could slightly depend on the observable used for the centrality determination, a systematic uncertainty is estimated by repeating the analysis using the number of clusters in the outermost layer of the SPD and the number of tracks in the TPC as centrality estimators, instead of the V0 signal amplitude. The systematic uncertainty due to the effect of TPC tracks from different Pb-Pb collisions piled-up in the same recorded event is estimated by applying a tighter cut to remove outliers in the multiplicity distribution of reference particles. This is done by requiring that the centrality values determined using the V0 signal amplitude and the number of TPC tracks do not differ by more than 5%. An additional systematic uncertainty specific to the scalar product is evaluated by varying the η gap between the two sub-events from 1 to 0.8 η-units (see Eq. (4) and [36]). The various systematic uncertainties are added in quadrature. They tend to increase with increasing p T (see Fig. 1). A summary of the systematic uncertainties, in the interval 3 < p T < 4.5 GeV/c, is presented in Table 1.   Table 1: Systematic uncertainty sources affecting the inclusive muon elliptic flow measurement in the 0-10%, 10-20% and 20-40% centrality classes for the interval 3 < p T < 4.5 GeV/c. They are given as a percentage of the v 2 value. Figure 1 shows the p T -differential muon elliptic flow (v µ 2 ) in the 0-10%, 10-20% and 20-40% centrality classes as obtained using the various methods. The values of v µ 2 slightly increase from central to semi-central collisions and this effect is more pronounced in the p T interval 2 < p T < 4 GeV/c. The two-particle correlation methods (scalar product and two-particle Q cumulants) give consistent results over the whole p T range, indicating that these methods have a similar sensitivity to non-flow effects 3 and in particular to flow fluctuations. A similar agreement is found when comparing the multi-particle correlation methods (four-particle Q cumulants and Lee-Yang zeros) to each other. No significant difference between the v µ 2 results extracted with Lee-Yang zeros using either the sum or product generating function is seen, hence indicating that interferences between harmonics are negligible [51]. The fourparticle Q cumulants and Lee-Yang zeros are expected to be less affected by non-flow effects than scalar  Fig. 1: p T -differential inclusive muon v 2 in 2.5 < y < 4 and various centrality intervals, in Pb-Pb collisions at √ s NN = 2.76 TeV. The symbols are placed at the centre of the p T interval and, for visibility, the points from twoparticle Q cumulants and Lee-Yang zeros with product generating function are shifted horizontally. The vertical error bars represent the statistical uncertainty, the horizontal error bars correspond to the width of the bin (not shown for the shifted data points) and the open boxes are the systematic uncertainties. The p T intervals used with the Lee-Yang zeros method are different with respect to the other methods. Upper panels: results from twoparticle correlation flow methods (scalar product and two-particle Q cumulants) in the 0-10% (left) and 10-20% (right) centrality intervals. Lower panels: results in the 20-40% centrality interval from two-particle correlation flow methods (scalar product and two-particle Q cumulants) and from four-particle Q cumulants (left), and from four-particle Q cumulants and Lee-Yang zeros (right). product or two-particle Q cumulants [52]. Moreover, since four-particle Q cumulants give comparable results as Lee-Yang zeros, one can conclude that non-flow correlations are almost completely removed at the 4 th order. Finally, the central values of v µ 2 obtained with four-particle Q cumulants or Lee-Yang zeros are systematically smaller than with two-particle correlation methods, although compatible within uncertainties. Such differences may indicate that initial fluctuations play a role in the development of the final momentum-space anisotropy.

Muon background subtraction
The subtraction of the muon background contribution to the measured v The p T -and η-differential elliptic flow of charged particles in |η| < 2.5, v ch 2 , is extrapolated to forward rapidity using: v where v ch 2 (p T , 2 < |η| < 2.5) is the measured charged-particle elliptic flow in 2 < |η| < 2.5 with the event plane method. Since the v ch 2 (p T ) measured by the ATLAS Collaboration is affected by statistical fluctuations, it is assumed that in the interval 10 < p T < 20 GeV/c, needed to simulate the decay muons up to p T = 10 GeV/c, v ch 2 remains constant with a value given by the one measured in the interval 10 < p T < 12 GeV/c. The extrapolation factor F(η) is calculated by parameterizing the η-differential v ch 2 measured by the ATLAS Collaboration in various p T intervals with a second order polynomial. In the interval 7 < p T < 20 GeV/c, the ATLAS v ch 2 does not show a dependence on η in |η| < 2.5. Therefore, for p T > 7 GeV/c, F(η) is computed as the average between a flat extrapolation function and the extrapolation factor obtained with the parabolic parameterization in 4 < p T < 7 GeV/c.
The mid-rapidity charged pion and kaon p T distributions measured in Pb-Pb collisions are extrapolated to forward rapidity using the same strategy as in [21] and summarized in the following. Assuming that the nuclear modification factor R π,K AA of charged pions and kaons in Pb-Pb collisions does not depend on rapidity up to y = 4 [21, 56], the p T distributions of charged pions and kaons at forward rapidity can be expressed as: where T AA is the average nuclear overlap function in centrality classes under study, estimated as described in [57]. The systematic uncertainty introduced by the assumption on R π,K AA will be discussed later. The rapidity extrapolation of the mid-rapidity pion and kaon p T -differential cross sections measured in pp collisions [21, 58] is done according to: d 2 σ π,K pp dp T dy = d 2 σ π,K pp dp T dy y=0 · exp −y 2 2σ 2 y , σ y being estimated from Monte-Carlo event generators (see [21] for details).
The elliptic flow of muons from charged pion and kaon decays, v µ←π,K 2 , in 2.5 < y < 4 and in various centrality classes 4 , is obtained by means of fast simulations using v ch 2 (η, p T ) given by Eq. (9) and charged pion and kaon p T distributions as obtained from Eq. (10)- (11). The absorber effect is accounted for by rejecting the pions and kaons that do not decay within a distance corresponding to one interaction length from the beginning of the absorber. The simulation was repeated twice, considering that charged particles are either all pions or all kaons. The estimated v µ←π,K 2 decreases with increasing p T from about 0.085 at p T = 3 GeV/c to 0.035 at p T = 10 GeV/c in the 10% most central collisions. In the 20-40% centrality interval, the v µ←π,K 2 values vary between 0.19 (p T = 3 GeV/c) and 0.08 (p T = 10 GeV/c) The background fraction, f µ←π,K , is calculated as the ratio of the p T -differential yield of muons from charged pion and kaon decays in 2.5 < y < 4 obtained in the simulation to the measured p T -differential yield of inclusive muons. It decreases as p T increases, from about 12% (15%) at p T = 3 GeV/c to 5% (7%) at p T = 10 GeV/c in the 0-10% (20-40%) centrality class.
The systematic uncertainties affecting the estimated v µ←π,K 2 are summarized in Table 2. They originate from i) the method used to measure the charged-particle v ch 2 in ATLAS, ii) the η and p T extrapolation of 4 The v µ←π,K 2 of muons from charged pion and kaon decays in the 20-40% centrality class is then obtained from the mean of the charged-particle v 2 in 20-30% and 30-40% centrality classes, with an additional systematic uncertainty provided by the difference with respect to the results in these two centrality classes. v ch 2 and iii) the treatment of the charged-particle v ch 2 in the fast simulation procedure. As the event plane method was used for the v ch 2 measurement in ATLAS, the results range between the mean ( v ch 2 ) and R.M.S. ( (v ch 2 ) 2 ) of the true v ch 2 values due to fluctuations, depending on the event plane resolution which varies with the collision centrality [49]. According to a Monte-Carlo Glauber model [49], the ratio v 2 2 / v 2 is expected to vary from about 1.06 to 1.15. Consequently, a conservative systematic uncertainty of 15% is applied to account for this bias and is propagated to v µ←π,K 2 . The systematic uncertainty due to the η extrapolation of v ch 2 is evaluated using several fit functions (first and third order polynomials, and Gaussian function) in the region p T < 7 GeV/c, and for larger p T values an additional systematic uncertainty due to the extrapolation procedure is considered. The latter is determined by comparing the results obtained with the two extrapolation functions used in the interval p T > 7 GeV/c. The systematic uncertainty due to the assumption on v ch 2 in the region p T > 10 GeV/c is estimated by varying v ch 2 between 0 and the value in 10 < p T < 12 GeV/c in the fast simulations. Such uncertainty affects mainly the high p T region (p T > 7 GeV/c). Finally, the systematic uncertainty obtained by treating charged particles separately as pions and kaons is found to be negligible. The various systematic uncertainty sources are propagated to the estimated v µ←π,K 2 and added in quadrature.

Source
Systematic uncertainty (%) Input v ch 2 bias 9 v ch 2 η extrapolation 9-12 v ch 2 high p T extrapolation 13-15 π and K in fast simulations < 1 for the interval 3 < p T < 10 GeV/c. They are stated as a percentage of the v 2 value. The given range reflects the dependence on the collision centrality.
The systematic uncertainty on f µ←π,K , detailed in [21], includes the uncertainty on the generated p T distributions of muons from charged pion and kaon decays, and the uncertainty on the measured inclusive muon p T distributions. The former originates from the input charged pion and kaon distributions, the rapidity extrapolation and the absorber effect. The systematic uncertainty on the measured inclusive muon yields contains the systematic uncertainty on detector response, residual mis-alignment and centrality dependence of the efficiency. This gives a total systematic uncertainty on f µ←π,K of about 21% in the interval 3 < p T < 4.5 GeV/c with almost no dependence on the collision centrality. Finally, as done for the measurement of the heavy-flavour decay muon R AA [21], the systematic uncertainty due to the unknown suppression of charged particles at forward rapidity is calculated by varying f µ←π,K from 0 to two times the estimated value. This corresponds to a variation of R π,K AA (p T ) at forward rapidity from 0 up to two times [R π,K AA (p T )] y=0 . This systematic uncertainty amounts to 10-30% in the interval 3 < p T < 4.5 GeV/c, depending on the collision centrality and the flow analysis method.  is obtained by adding in quadrature the two contributions. It amounts to about 12%-36% in the interval 3 < p T < 4.5 GeV/c, depending on the collision centrality and the flow analysis method.   , in 2.5 < y < 4 and various centrality intervals, in Pb-Pb collisions at √ s NN = 2.76 TeV. The symbols are placed at the centre of the p T interval and, for visibility, the points from two-particle Q cumulants and Lee-Yang zeros with product generating function are shifted horizontally. The meaning of the symbols is the same as in Fig. 1. The horizontal error bars are not plotted for shifted data points. The p T intervals used with the Lee-Yang zeros method are different with respect to the other methods. Upper panels: results from two-particle correlation flow methods (scalar product and two-particle Q cumulants) in the 0-10% (left) and 10-20% (right) centrality intervals. Lower panels: results in the 20-40% centrality interval from two-particle correlation flow methods (scalar product and two-particle Q cumulants) and from four-particle Q cumulants (left), and from four-particle Q cumulants and Lee-Yang zeros (right). See the text for details. muon elliptic flow (Fig. 1). When comparing the results to those obtained for inclusive muons (Fig. 1), one can notice that v µ←HF 2 and v µ 2 are similar due to the small background fraction (5% to 15%) in the p T interval 3-10 GeV/c. The differences between the various methods are similar to those discussed for the measurement of the inclusive muon v µ 2 i.e. i) scalar product and two-particle Q cumulants give compatible results, ii) consistent results are also found with four-particle Q cumulants and Lee-Yang zeros, and iii) the v µ←HF 2 values extracted from these multi-particle correlation methods are smaller, although still compatible within uncertainties, than the ones obtained with two-particle correlation methods. As mentioned in Section 3.3, such differences are expected if initial-state fluctuations play a role in the development of the final momentum-space anisotropy.

Results
A positive v µ←HF 2 is observed at intermediate p T for the 20-40% and 10-20% centrality classes with a significance larger than 3σ when combining statistical and systematic uncertainties. In the 20-40% centrality class, the values of the significance in the interval 3 < p T < 4 GeV/c (4 < p T < 5.5 GeV/c) are 4 σ ( 3.2 σ ) and 4.3 σ (3.8 σ ) with scalar product and two-particle Q cumulants, respectively. In the 10-20% centrality class and in the interval 3 < p T < 4.5 GeV/c, the values of the significance correspond to 4.4σ both with scalar product and two-particle Q cumulants. This behaviour results from the interplay between the significant interaction of heavy quarks with the expanding medium and the path-length  of muons from heavy-flavour hadron decays decreases with increasing p T and becomes compatible with zero in the high p T region. Figure 3 shows the centrality dependence of the p T -integrated (3 < p T < 10 GeV/c) elliptic flow of muons from heavy-flavour hadron decays. It is investigated with scalar product and two-particle Q cumulants, which can be applied in a wider event-multiplicity (i.e. centrality) interval compared to multi-particle correlation methods. A significant decrease of the v 2 magnitude towards central collisions is observed. This is expected from the decrease of the initial spatial anisotropy from semi-central to central collisions. ALICE has measured the elliptic flow of prompt D mesons in |y| < 0.8 in three centrality classes in the interval 0-50% with various two-particle correlation methods [33,34]. Similar trends as those reported here for muons from heavy-flavour decays are observed, although in different p T and rapidity intervals. In particular, a positive v 2 was observed for D mesons in semi-central collisions in 2 < p T < 6 GeV/c.
The positive elliptic flow of muons from heavy-flavour hadron decays has been observed in a p T interval from 3 to about 5 GeV/c where the charm contribution is expected to be dominant with respect to the beauty component according to perturbative QCD calculations [21]. This measurement supports the interpretation of the J/ψ positive v 2 at forward rapidity [35] in terms of a significant contribution to J/ψ production from recombination of flowing charm quarks in the deconfined medium.

Comparison with models
The results presented in this publication may constrain models describing the interactions of heavy quarks with the medium via elastic (collisional) and inelastic (radiative) processes, and in particular the parton energy loss dependence on the path-length within the medium.  [63][64][65] is a partonic transport model based on the Boltzmann approach to multi-parton scatterings. It includes collisional processes with a running strong coupling constant. The lack of radiative contributions is accounted for by scaling the binary cross section with a correction factor, tuned to describe the nuclear modification factor and elliptic flow results at RHIC energies. Vacuum fragmentation functions are used for the hadronization. Figure 4 (left) shows that the p T -differential elliptic flow of muons from heavy-flavour hadron decays in the 20-40% centrality class is described reasonably well by the three models. However, the BAMPS model tends to slightly underestimate the R AA of muons from heavy-flavour decays in the 10% most central collisions, while the MC@sHQ+EPOS model tends to overestimate it. The TAMU model describes the R AA measurement over the entire p T interval within uncertainties and tends to slightly underestimates the v

Conclusions
In summary, we have reported on a measurement of the elliptic flow of muons from heavy-flavour hadron decays at forward rapidity in central and semi-central Pb-Pb collisions at √ s NN = 2.76 TeV with the ALICE detector at the LHC.
Measurements have been carried out using several methods which exhibit different sensitivity to initial-state fluctuations and non-flow correlations. The systematic comparison of scalar product, two-and four-particle Q cumulants and Lee-Yang zeros helps in understanding the processes that build up the observed differences between two-particle correlation methods and multi-particle correlation methods and suggests that flow fluctuations are significant.
The magnitude of the elliptic flow of muons from heavy-flavour hadron decays increases from central to semi-central collisions and decreases with increasing p T , becoming compatible with zero at high p T The results indicate a positive elliptic flow with the scalar product and two-particle Q cumulants in semicentral collisions (10-20% and 20-40% centrality classes) for the p T interval from 3 to about 5 GeV/c with a significance larger than 3σ The data are described by transport model calculations within uncertainties, although a simultaneous description of R AA and v 2 remains a challenge. The results reported in this Letter in various centrality classes may provide further important constraints to the models.

Acknowledgements
The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The       [24] ALICE Collaboration, J. Adam et al., "Inclusive, prompt and non-prompt J/ψ production at mid-rapidity in        [56] ATLAS Collaboration, A. Milov, "Centrality dependence of charged particle spectra and R CP in Pb+Pb collisions at √ s NN = 2.76 TeV with the ATLAS detector at the LHC," J. Phys. G38 (2011) 124113, arXiv:1107.0460 [nucl-ex].