Elliptic flow of electrons from beautyhadron decays extracted from Pb–Pb collision data at \(\sqrt{s_\mathrm{NN}} = 2.76\ \hbox {TeV}\)
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Abstract
We present a calculation of the elliptic flow of electrons from beautyhadron decays in semicentral Pb–Pb collisions at centreofmass energy per colliding nucleon pair, represented as \(\sqrt{s_\mathrm{NN}}\), of 2.76 TeV. The result is obtained by the subtraction of the charmquark contribution in the elliptic flow of electrons from heavyflavour hadron decays in semicentral Pb–Pb collisions at \(\sqrt{s_\mathrm{NN}} = 2.76\ \hbox {TeV}\) recently made publicly available by the ALICE collaboration.
1 Introduction
In view of the experimental results on the elliptic flow of heavyflavour particles, an important question that remains open is whether beauty quarks take part in the collective motion in the medium. The first measurement of the \(v_\mathrm{2}\) of nonprompt \(\hbox {J}/\psi \) mesons from \(\mathrm {B}\)hadron decays is compatible with zero within uncertainties in two kinematic regions, \(6.5< p_\mathrm{T} < 30\ \hbox {GeV}/\mathrm{c}\) and \(y < 2.4\), and \(3< p_\mathrm{T} < 6.5\ \hbox {GeV}/\mathrm{c}\) and \(1.6< y < 2.4\), in 10–60% Pb–Pb collisions at \(\sqrt{s_\mathrm{NN}} = 2.76\ \hbox {TeV}\) at the LHC [27]. In this paper, we present a method to subtract the contribution of charm quarks in the published measurement of the elliptic flow of electrons from heavyflavour hadron decays in semicentral Pb–Pb collisions at \(\sqrt{s_\mathrm{NN}} = 2.76\ \hbox {TeV}\) performed by the ALICE collaboration. The calculation uses as input the \(v_\mathrm{2}\) coefficients of prompt D mesons and electrons from heavyflavour hadron decays measured by the ALICE collaboration [32, 36] and three different results for the relative contribution of electrons from beautyhadron decays to the yield of electrons from heavyflavour hadron decays [38, 39, 40, 41].
2 Methodology
In the following, we present the currently published measurements and, in case there is no available measurement, our calculations of the three observables required to obtain the elliptic flow of electrons from beautyhadron decays. Based on available results on open heavy flavours at RHIC and LHC, the most suitable system for this analysis is the Pb–Pb collision system at \(\sqrt{s_\mathrm{NN}} = 2.76\ \hbox {TeV}\) in the 20–40% centrality class, which corresponds to the centrality range where the measured \(v_\mathrm{2}\) of electrons from heavyflavour hadron decays is observed to be positive with a maximum significance [36] and thus a possible elliptic flow of electrons from beautyhadron decays is expected to be more significant. In this analysis, the \(v_\mathrm{2}\) and \(R_\mathrm{\mathrm{AA}}\) of heavyflavour particles are assumed to be the same at slightly different midrapidity ranges (\(y < 0.5\), 0.7 and 0.8) in which the measurements needed in the calculation are available. Indeed, no dependence on rapidity was observed in recent ALICE results on those observables for electrons from heavyflavour hadron decays at midrapidity (\(y < 0.7\) for \(v_\mathrm{2}\) and \(y < 0.6\) for \(R_\mathrm{\mathrm{AA}}\) measurements) and muons from heavyflavour hadron decays at forward rapidity (\(2.5< y < 4\)) [24, 36].
2.1 Elliptic flow of electrons from heavyflavour hadron decays
The result on the elliptic flow of electrons from heavyflavour hadron decays (\(v_{2}^{e \leftarrow c+b}\)) at midrapidity (\(y < 0.7\)) in 20–40% Pb–Pb collisions at \(\sqrt{s_\mathrm{NN}} = 2.76\ \hbox {TeV}\) published by the ALICE collaboration [36] is used in this analysis. The \(v_{2}^{e \leftarrow c+b}\) is measured in the interval \(0.5< p_\mathrm{T} < 13\ \hbox {GeV}/\mathrm{c}\) with the event plane method [30]. A positive value is observed in the interval \(2< p_\mathrm{T} < 2.5\ \hbox {GeV}/\mathrm{c}\) with significance of \(5.9\sigma \) [36].
2.2 Relative contribution of electrons from beautyhadron decays to the yield of electrons from heavyflavour hadron decays
The measurement of the relative contribution of electrons from beautyhadron decays to the yield of electrons from heavyflavour hadron decays (R) has been published by the ALICE collaboration only in pp collisions at \(\sqrt{s} = 2.76\ \hbox {TeV}\) [38, 39]. The factor R is measured using the track impact parameter and electronhadron azimuthal correlation methods. Results obtained with both techniques are compatible within uncertainties. The coefficient R measured in pp collisions with the electronhadron azimuthal correlation method is used in the analysis with the caveat that initial and finalstate effects modify the yield of electrons from heavyflavour hadron decays in heavyion collisions. In particular, the coefficient R at high \(p_\mathrm{T}\) is expected to be higher in Pb–Pb collisions compared to pp collisions, since the inmedium energy loss of charm quarks is predicted to be larger than the one of beauty quarks [23]. Therefore, the factor R at high \(p_\mathrm{T}\) in Pb–Pb collisions at \(\sqrt{s_\mathrm{NN}} = 2.76\ \hbox {TeV}\) is expected to have an exclusive value between the measured factor R in pp collisions at \(\sqrt{s} = 2.76\ \hbox {TeV}\) and unity. Consequently, according to Eq. 3, the minimum value of the \(v_\mathrm{2}\) of electrons from beautyhadron decays can be computed with the R measured in pp collisions.
The result on the factor R from the BAMPS heavyflavour transport model [40, 41], which includes collisional and radiative inmedium energy loss of heavy quarks, is also employed in the analysis to obtain the \(v_\mathrm{2}\) of electrons from beautyhadron decays. The choice of the BAMPS model is justified by the good agreement of the predictions for the \(R_\mathrm{\mathrm{AA}}\) of electrons from beauty and heavyflavour hadron decays for \(p_\mathrm{T} > 3\,\hbox {GeV}/\mathrm{c}\) in central Pb–Pb collisions at \(\sqrt{s_\mathrm{NN}}= 2.76\ \hbox {TeV}\) with what measured by the ALICE collaboration [23, 24].
In this analysis, the R coefficient measured in pp collisions using the electronhadron azimuthal correlation technique and the ones obtained with \(\hbox {POWHEG}+\hbox {PYTHIA}\) (EPS09NLO) and with the BAMPS model are used to estimate the elliptic flow of electrons from beautyhadron decays.
2.3 Elliptic flow of electrons from charmhadron decays

the \(p_\mathrm{T}\)differential yield, which is used as a probability distribution for finding a \(\hbox {D}^{0}\) meson with a certain \(p_\mathrm{T}\);

the \(p_\mathrm{T}\)differential \(v_{2}\), which is used to obtain the \(\varphi _\mathrm{{D}^{0}}  \varPsi _{2}\) probability distribution with Eq. 2.
The \(v_\mathrm{2}\) of prompt \(\hbox {D}^{0}\) mesons in 20–40% Pb–Pb collisions at \(\sqrt{s_\mathrm{NN}} = 2.76\ \hbox {TeV}\) (bottom panel of Fig. 2) is obtained by the arithmetic average of the measured prompt \(\hbox {D}^0\)meson \(v_\mathrm{2}\) in Pb–Pb collisions at the same collision energy in the 10–30 and 30–50% centrality classes [32]. Indeed, experimental results show that the \(v_\mathrm{2}\) of heavyflavour particles increases with the centrality class [17, 32, 36, 37], which is consistent with the qualitative expectation of increasing of the elliptic anisotropy from central to peripheral nucleusnucleus collisions. The statistical and systematic uncertainties are propagated considering the prompt \(\hbox {D}^0\)meson \(v_\mathrm{2}\) in the 10–30 and 30–50% centrality classes as uncorrelated as a conservative estimation. In the \(\hbox {D}^0\)meson \(v_\mathrm{2}\) measurement by the ALICE collaboration, the central value was obtained by assuming that the \(v_\mathrm{2}\) coefficients of prompt D mesons and D mesons from Bmeson decays are the same [32]. However, the systematic uncertainty related to this assumption, referred to as systematic uncertainty from the B feeddown subtraction, was evaluated by the ALICE collaboration. It was assumed that the \(v_\mathrm{2}\) of prompt D mesons from Bmeson decays should be between zero and \(v_\mathrm{2}\) of prompt D mesons, resulting in the upper and lower limits of the systematic uncertainty, respectively. Therefore, the B feeddown contribution decreases the absolute value of the \(\hbox {D}^0\)meson \(v_\mathrm{2}\) and thus the systematic uncertainty is restricted to the upper (lower) limit when the \(v_\mathrm{2}\) is positive (negative). Since the measured \(\hbox {D}^0\)meson \(v_\mathrm{2}\) coefficients are negative in the \(8< p_\mathrm{T} < 12\) and 12 \(< p_\mathrm{T}<\) 16 GeV/c intervals in the 10–30 and 30–50% centrality classes, respectively, the resulting propagated systematic uncertainty from the B feeddown subtraction contains lower and upper limits.
Finally, the estimated \(p_\mathrm{T}\) and \(v_\mathrm{2}\) distributions of \(\hbox {D}^{0}\) mesons in 20–40% Pb–Pb collisions at \(\sqrt{s_\mathrm{NN}} = 2.76\ \hbox {TeV}\) are used to obtain the \(v_{2}^{e \leftarrow c} = \langle \cos \left[ 2 \left( \varphi _{e}  \varPsi _{2} \right) \right] \rangle \) in the same collision system using the PYTHIA event generator. The azimuthal angle of electrons (\(\varphi _{e}\)) takes into account the angular separation between electrons and their parent \(\hbox {D}^{0}\) mesons.
2.3.1 Statistical uncertainty
The statistical uncertainty of the \(\hbox {D}^{0}\)meson \(v_\mathrm{2}\) is used as input for the MC simulation to obtain the statistical uncertainty of the \(v_{2}^{e \leftarrow c}\). The statistical uncertainties of the measurements used to obtain the \(p_\mathrm{T}\)differential yield of \(\hbox {D}^{0}\) mesons are considered in the fit of the \(\hbox {D}^{0}\)meson probability distribution. Further variations are considered as systematic uncertainties.
2.3.2 Systematic uncertainty
The systematic uncertainties from data and from the B feeddown subtraction of the \(\hbox {D}^{0}\)meson \(v_\mathrm{2}\) (bottom panel of Fig. 2) are used as input for the MC simulation to obtain the systematic uncertainty of the \(v_{2}^{e \leftarrow c}\). The following is a discussion on other sources of systematic uncertainty that can influence the \(v_{2}^{e \leftarrow c}\) estimation.
In order to validate the Eq. 4, the \(p_\mathrm{T}\)differential yield and \(R_\mathrm{\mathrm{AA}}\) of prompt \(\hbox {D}^{0}\) mesons in 40–80% Pb–Pb collisions at \(\sqrt{s_\mathrm{NN}} = 2.76\ \hbox {TeV}\) [19] are also used as reference to obtain the \(p_\mathrm{T}\)differential yield of \(\hbox {D}^{0}\) mesons in the 20–40% centrality class. The result is the same as the one obtained with the 0–20% centrality class (top panel of Fig. 2).
The ALICE result on the \(\hbox {D}^{0}\)meson \(R_\mathrm{AA}\) in the 30–50% centrality class [32] is used as an alternative for the \(\hbox {D}^{0}\)meson \(R_\mathrm{\mathrm{AA}}\) estimation in the 20–40% centrality class. No significant difference is observed in the resulting \(v_{2}^{e \leftarrow c}\) with respect to the one obtained with the \(R_\mathrm{AA}\) estimated by the average of the \(\hbox {D}^{0}\)meson \(R_\mathrm{AA}\) measurements in the 0–20 and 40–80% centrality classes weighted by the corresponding yield of \(\hbox {D}^{0}\) mesons in each centrality class.
The systematic uncertainties of the measurements of the \(p_\mathrm{T}\)differential yield and \(R_\mathrm{\mathrm{AA}}\) of prompt \(\hbox {D}^{0}\) mesons are considered in the fit of the \(\hbox {D}^{0}\)meson \(p_\mathrm{T}\) distribution in 20–40% Pb–Pb collisions. No significant difference is observed in the resulting \(v_{2}^{e \leftarrow c}\) with respect to the one considering only the statistical uncertainty in the fit. For further investigation, The BAMPS result on the \(p_\mathrm{T}\) distribution of D mesons at \(y < 0.8\) in 20–40% Pb–Pb collisions at \(\sqrt{s_\mathrm{NN}} = 2.76\ \hbox {TeV}\) [40, 41] is also used to compute the \(v_{2}^{e \leftarrow c}\). The relative difference of the obtained \(v_{2}^{e \leftarrow c}\) using the estimated \(p_\mathrm{T}\) distribution of \(\hbox {D}^{0}\) mesons and the BAMPS result, which increases from 1 to 20% in the interval \(2< p_\mathrm{T} < 8\ \hbox {GeV}/\mathrm{c}\), is included in the systematic uncertainty.
The effect of the \(\hbox {D}^{0}\)meson \(v_\mathrm{2}\) estimation in 20–40% Pb–Pb collisions using the arithmetic average of the \(\hbox {D}^{0}\)meson \(v_\mathrm{2}\) measurements in the 10–30 and 30–50% centrality classes is investigated in this analysis. For this purpose, the trend of the unidentified charged particle \(v_\mathrm{2}\) as a function of the average number of binary collisions (\(\langle N_{coll }\rangle \)) [47] is assumed to be the same as the one for \(\hbox {D}^{0}\) mesons. The \(v_\mathrm{2}\) as a function of \(\langle N_{coll }\rangle \) is obtained from a parametrisation of the centralitydependent \(v_\mathrm{2}\) measurement of unidentified charged particles integrated over the interval \(0.2< p_\mathrm{T} < 5 \hbox {GeV}/\mathrm{c}\) [48]. The corresponding result exhibits a linear dependence between \(v_\mathrm{2}\) and \(\langle N_{coll }\rangle \) with a negative slope for \(\langle N_{coll }\rangle > 220\). For comparison, the parametrisation is also obtained from the centralitydependent \(v_\mathrm{2}\) measurement of unidentified charged particles integrated over the interval \(10< p_\mathrm{T} < 20\ \hbox {GeV}/\mathrm{c}\) [49]. The linear dependence between \(v_\mathrm{2}\) and \(\langle N_{coll }\rangle \) is the same as the one obtained for particles in a lower \(p_\mathrm{T}\) interval. The \(\hbox {D}^{0}\)meson \(v_\mathrm{2}\) is then obtained by the average of the \(\hbox {D}^{0}\)meson \(v_\mathrm{2}\) in the 10–30 and 30–50% centrality classes weighted by the \(v_\mathrm{2}\) coefficients of the corresponding \(\langle N_{coll }\rangle \) values [47]. The relative difference of the obtained \(\hbox {D}^{0}\)meson \(v_\mathrm{2}\) with respect to the one obtained with the arithmetic average is negligible for \(p_\mathrm{T} < 8\ \hbox {GeV}/\mathrm{c}\) and its average is 19% for \(p_\mathrm{T} > 8\ \hbox {GeV}/\mathrm{c}\), which is still compatible within uncertainties. The \(v_{2}^{e \leftarrow c}\) coefficients obtained with the two approaches show a relative difference of 2% in the range \(2< p_\mathrm{T} < 8\ \hbox {GeV}/\mathrm{c}\). This deviation is considered as a consequence of statistical fluctuations in the \(\hbox {D}^{0}\)meson \(v_\mathrm{2}\) measurement for \(p_\mathrm{T} > 8\ \hbox {GeV}/\mathrm{c}\) and thus no systematic uncertainty is assigned for this effect.
In order to investigate the impact of the assumption of the particle mass ordering of the elliptic flow [50] used to determine the systematic uncertainty from the B feeddown subtraction in the \(\hbox {D}^{0}\)meson \(v_\mathrm{2}\) measurement, one can assume that the \(v_\mathrm{2}\) of prompt D mesons from Bmeson decays should be between zero and the unidentified charged particle \(v_{2}\). The unidentified charged particle \(v_\mathrm{2}\) in 20–40% Pb–Pb collisions is obtained by the average of the \(v_\mathrm{2}\) measurements in the 20–30 and 30–40% centrality classes [49] weighted by the corresponding \(\langle N_{coll }\rangle \) values. The \(v_\mathrm{2}\) coefficients of prompt \(\hbox {D}^{0}\) mesons and unidentified charged particles are compatible within uncertainties as well as the \(v_{2}^{e \leftarrow c}\) obtained with these two results. Therefore, the lower limit of the systematic uncertainty from the B feeddown subtraction can be positioned at the central values of the prompt Dmeson \(v_\mathrm{2}\) and \(v_{2}^{e \leftarrow c}\) without strictly considering that the Bmeson \(v_\mathrm{2}\) is expected to be lower than the Dmeson \(v_{2}\).
As a consequence of the \(p_\mathrm{T}\) interval (\(2< p_\mathrm{T} < 16\ \hbox {GeV}/\mathrm{c}\)) of the \(\hbox {D}^{0}\)meson \(v_\mathrm{2}\) and \(p_\mathrm{T}\)differential yield measurements, the \(v_{2}^{e \leftarrow c}\) is obtained in the range \(2< p_\mathrm{T} < 8\ \hbox {GeV}/\mathrm{c}\). The fraction of electrons with \(p_\mathrm{T} > 2\ \hbox {GeV}/\mathrm{c}\) that come from \(\hbox {D}^{0}\) mesons with \(p_\mathrm{T} < 2\ \hbox {GeV}/\mathrm{c}\) is negligible according to PYTHIA simulations. The effect of the \(p_\mathrm{T}\) upper limit of the \(\hbox {D}^{0}\)meson measurements is studied by evaluating the \(v_{2}^{e \leftarrow c}\) with extrapolation of the \(p_\mathrm{T}\) and \(v_\mathrm{2}\) distributions of \(\hbox {D}^{0}\) mesons up to \(26\ \hbox {GeV}/\mathrm{c}\). The transverse momentum extrapolation is obtained from the powerlaw fit function shown in the top panel of Fig. 2, while the impact of the \(v_\mathrm{2}\) of \(\hbox {D}^{0}\) mesons is estimated by explicitly setting its value, in the interval \(16< p_\mathrm{T} < 26\ \hbox {GeV}/\mathrm{c}\), to either zero, or constant at high \(p_\mathrm{T}\), or maximum value of the prompt \(\hbox {D}^{0}\)meson \(v_\mathrm{2}\) (shown in Fig. 2). The highest relative difference in these three scenarios, which increases from 0.3 to 40% in the interval \(2< p_\mathrm{T} < 8\ \hbox {GeV}/\mathrm{c}\), is assigned as a conservative systematic uncertainty.
The effect of the midrapidity range of \(\hbox {D}^{0}\) mesons is investigated by obtaining the \(v_{2}^{e \leftarrow c}\) using the \(\hbox {D}^{0}\)meson \(p_\mathrm{T}\) distribution in the rapidity range \(y < 1.6\) as input for the simulation. The \(\hbox {D}^{0}\)meson \(v_\mathrm{2}\) is considered to be the same in this rapidity range, because no dependence on rapidity was observed in ALICE results on leptons from heavyflavour hadron decays [24, 36] as discussed previously. The relative difference of the obtained \(v_{2}^{e \leftarrow c}\) with respect to the one using the \(\hbox {D}^{0}\)meson \(p_\mathrm{T}\) distribution in the rapidity range \(y < 0.8\) is negligible and thus no additional systematic uncertainty is considered due to the rapidity effect.
In this analysis, the \(v_\mathrm{2}\) and shape of the \(p_\mathrm{T}\)differential yields of charm hadrons are assumed to be the same as the ones measured for \(\hbox {D}^{0}\) mesons. This is justified by the fact that the \(v_\mathrm{2}\) coefficients of \(\hbox {D}^{0}\), \(\hbox {D}^{+}\) and \(\hbox {D}^{*+}\) mesons are compatible within uncertainties in 30–50% Pb–Pb collisions at \(\sqrt{s_\mathrm{NN}} = 2.76\ \hbox {TeV}\) [31], also the prompt \(\hbox {D}_{s}^{+} v_\mathrm{2}\) is compatible within uncertainties with the prompt nonstrange D meson \(v_\mathrm{2}\) in 30–50% Pb–Pb collisions at \(\sqrt{s_\mathrm{NN}} = 5.02\ \hbox {TeV}\) [34]. In addition, the ratios of the yields of \(\hbox {D}^{+}/\hbox {D}^{0}\) and \(\hbox {D}^{*+}/\hbox {D}^{0}\) were observed to be constant within uncertainties in pp collisions at \(\sqrt{s} = 7\ \hbox {TeV}\) and no modification of the ratios was observed within uncertainties in central and semicentral Pb–Pb collisions at \(\sqrt{s_\mathrm{NN}} = 2.76\ \hbox {TeV}\) [51]. A possible hint for an enhancement of the \(\hbox {D}_{s}^{+}/\hbox {D}^{0}\) ratio is observed in 0–10% Pb–Pb collisions at \(\sqrt{s_\mathrm{NN}} = 2.76\ \hbox {TeV}\) [52], but the current uncertainties do not allow for a conclusion. The effect of different decay kinematics of charm particles is estimated by simulating the \(v_\mathrm{2}\) of electrons from combined D, \(\hbox {D}^{*}\), \(\hbox {D}_{s}\), and \(\varLambda _{c}\) particle decays taking into account the fraction of charm quarks that hadronise into these particles [53] and using the same simulation input as used in the analysis (\(p_\mathrm{T}\)differential yield and \(v_\mathrm{2}\) of \(\hbox {D}^{0}\) mesons). The obtained \(v_{2}^{e \leftarrow c}\) is compatible with the one using \(\hbox {D}^{0}\)meson decay and thus no systematic uncertainty is considered due to this effect. In order to exemplify the impact of a possible production enhancement of \(\hbox {D}_ {s}^{+}\) and \(\varLambda _{c}\) particles in Pb–Pb collisions with respect to pp collisions, their fragmentation fractions are increased by a factor 2 and 5, respectively, in the simulation of the combined charm meson \(v_{2}\). The relative difference of the obtained \(v_{2}^{e \leftarrow c}\) and the one using \(\hbox {D}^{0}\)meson decay is negligible for \(p_\mathrm{T} < 3\ \hbox {GeV}/\mathrm{c}\) and its average is 5% for \(p_\mathrm{T} > 3\ \hbox {GeV}/\mathrm{c}\).
Finally, the \(\hbox {D}^{0}\)meson \(v_\mathrm{2}\) systematic uncertainty from data is summed in quadrature with other sources of systematic uncertainty that affect significantly the \(v_{2}^{e \leftarrow c}\) estimation, which are the \(p_\mathrm{T}\) distribution of \(\hbox {D}^{0}\) mesons and the limited \(p_\mathrm{T}\) interval of the \(\hbox {D}^{0}\)meson measurements. They are considered as uncorrelated since the effect from the \(p_\mathrm{T}\) distribution of \(\hbox {D}^{0}\) mesons is obtained with the BAMPS result and the effect from the limited \(p_\mathrm{T}\) interval of the \(\hbox {D}^{0}\)meson measurements is obtained by extrapolations. The term “from data” is maintained later in this paper to distinguish all sources of systematic uncertainty from the systematic uncertainty related to the B feeddown subtraction of the \(\hbox {D}^{0}\)meson \(v_\mathrm{2}\) measurement, which is shown separately.
2.4 Elliptic flow of electrons from beautyhadron decays
The \(v_\mathrm{2}\) of electrons from beautyhadron decays (\(v_{2}^{e \leftarrow b}\)) is obtained from Eq. 3 using the R, \(v_{2}^{e \leftarrow c+b}\) and \(v_{2}^{e \leftarrow c}\) results presented in their respective sections.
The three results are considered as statistically independent. First, the factor R was measured in a different collision system (pp collisions) or obtained with calculations. Second, the \(v_{2}^{e \leftarrow c}\) is obtained with a simulation using measurements of \(\hbox {D}^{0}\) mesons reconstructed via the hadronic decay channel \(\mathrm {D}^{0} \rightarrow \mathrm {K}^{} \pi ^{+}\) in a different centrality class than in the \(v_{2}^{e \leftarrow c+b}\) measurement.
Even though the systematic uncertainties of the R, \(v_{2}^{e \leftarrow c+b}\) and \(v_{2}^{e \leftarrow c}\) results might be partially correlated, especially concerning the particle identification selection criteria, the limited public information prevents a more accurate treatment of these uncertainties. Therefore, they are assumed to be uncorrelated as a conservative estimation. As an example of the effect of a possible overestimation, if the systematic uncertainties of the \(v_{2}^{e \leftarrow c}\) and \(v_{2}^{e \leftarrow c+b}\) results decrease by 30% in the interval 2 \(< p_\mathrm{T} < 8\ \hbox {GeV}/\mathrm{c}\), the systematic uncertainty from data of the \(v_{2}^{e \leftarrow b}\) result is expected to decrease by approximately 24%.
Therefore, the statistical and systematic uncertainties of the R, \(v_{2}^{e \leftarrow c+b}\) and \(v_{2}^{e \leftarrow c}\) results are propagated as independent variables. The \(v_{2}^{e \leftarrow b}\) systematic uncertainties from data and from the B feeddown subtraction are asymmetric as a consequence of the systematic uncertainty asymmetry of the measurements used in this analysis. The systematic uncertainty from data is evaluated according to the method described in [54], where the positive and negative deviations are obtained separately and their average is added in quadrature. For verification, the alternative approach presented in [55] is also applied in this analysis. No significant difference between these methods is observed. Since the asymmetry of the systematic uncertainty from the B feeddown subtraction only comes from the \(v_{2}^{e \leftarrow c}\) result, the limits of the \(v_{2}^{e \leftarrow b}\) systematic uncertainty are the deviations resulting from the upper and lower limits of the \(v_{2}^{e \leftarrow c}\) systematic uncertainty.
3 Results
The \(v_\mathrm{2}\) coefficients of electrons from beautyhadron decays in 20–40% Pb–Pb collisions at \(\sqrt{s_\mathrm{NN}} = 2.76\ \hbox {TeV}\) obtained with different approaches of the factor R (top panel of Fig. 3) are shown in Fig. 4. The result computed with the coefficient R in pp collisions is an estimation of the minimum value, as discussed previously. The \(v_\mathrm{2}\) of electrons from beautyhadron decays in 20–40% Pb–Pb collisions at \(\sqrt{s_\mathrm{NN}} = 2.76\ \hbox {TeV}\) is compatible with zero within approximately \(1 \sigma \) of the total uncertainty, obtained by summing in quadrature the different uncertainty contributions, in all \(p_\mathrm{T}\) intervals and different R coefficients. However, the large statistical and systematic uncertainties prevent a definite conclusion. The result is consistent with the measured \(v_\mathrm{2}\) of nonprompt \(\hbox {J}/\psi \) mesons from \(\mathrm {B}\)hadron decays in 10–60% Pb–Pb collisions at \(\sqrt{s_\mathrm{NN}} = 2.76\ \hbox {TeV}\) [27], which is also compatible with zero within uncertainties.
4 Conclusions
Average of the \(v_\mathrm{2}\) coefficients of electrons from charm and beautyhadron decays obtained in the transverse momentum interval \(2< p_\mathrm{T} < 8\ \hbox {GeV}/\mathrm{c}\) in 20–40% Pb–Pb collisions at \(\sqrt{s_\mathrm{NN}} = 2.76\ \hbox {TeV}\). The reported errors are the combined statistical and systematic uncertainties. See text for more details
Result  R approach  Average \(v_\mathrm{2}\) 

\(e \leftarrow c\)  –  \(\mathrm 0.150_\mathrm{ 0.028}^\mathrm{+ 0.034}\) 
\(e \leftarrow b\)  ALICE  \(\mathrm 0.014 _\mathrm{ 0.042}^\mathrm{+ 0.039}\) 
\(e \leftarrow b\)  POWHEG+PYTHIA(EPS09NLO)  \(\mathrm 0.010 _\mathrm{ 0.052}^\mathrm{+ 0.047}\) 
\(e \leftarrow b\)  BAMPS  \(\mathrm 0.032_\mathrm{ 0.030}^\mathrm{+ 0.028}\) 
5 Outlook
In the presented method, the elliptic flow of electrons from beautyhadron decays can be determined by using three observables that have largely been measured at the LHC and RHIC. Based on available results of these observables, the procedure was applied using measurements performed by the ALICE collaboration. The method demonstrated to be effective; however, the current statistical and systematic uncertainties of the ALICE results prevent a definite conclusion whether the collective motion of the medium constituents influences beauty quarks. A better accuracy of the results on heavyflavour particles has been achieved in measurements in Pb–Pb collisions at \(\sqrt{s_\mathrm{NN}} = 5.02\ \hbox {TeV}\) [34] and it is expected to be further improved with the ALICE upgrade, which is foreseen to start in 2019.
In particular, the upgrade of the Inner Tracking System (ITS) detector will improve the determination of the distance of closest approach to the primary vertex, momentum resolution and readout rate capabilities [57]. These improvements will allow for more precise measurements of D mesons down to low transverse momenta and for reducing the systematic uncertainties from data and from the B feeddown subtraction. The latter will be possible with the direct measurement of the fraction of prompt D mesons and D mesons from Bmeson decays, which is expected to be accessible with relative statistical and systematic uncertainties smaller than 1 and 5% [57], respectively, for prompt \(\hbox {D}^{0}\) mesons. In addition, the ITS upgrade will enable the tracking of electrons down to approximately \(0.05\ \hbox {GeV}/\mathrm{c}\) and enhance the capability to separate prompt from displaced electrons [57], improving the reconstruction of electrons that do not originate from heavyflavour hadron decays needed for the background subtraction. Moreover, the systematic uncertainty of the elliptic flow of electrons from beautyhadron decays can be further improved by taking into account correlations among different contributions.
The capability of the heavyflavour measurements will also enhance with the increase of luminosity. For instance, the current relative statistical uncertainty of the Dmeson \(v_\mathrm{2}\) measurement in Pb–Pb collisions is 10% for an integrated luminosity of \(0.1\ \hbox {nb}^{1}\), while it is expected to be 0.2% for a scenario with an integrated luminosity of \(10\ \hbox {nb}^{1}\) [57]. Also the elliptic flow coefficients of \(\hbox {D}_{s}\) and \(\varLambda _{c}\) particles are expected to be achievable with a relative statistical uncertainty of 8 and 20% [57], respectively.
Therefore, the presented method can be used to extract the elliptic flow of electrons from beautyhadron decays with better precision with future measurements of the three needed observables.
Notes
Acknowledgements
We would like to thank Carsten Greiner and Florian Senzel for providing the BAMPS results, as well as Francesco Prino for fruitful discussions. We are grateful for the support of the Deutsche Forschungsgemeinschaft (DFG) through the Research Training Group “GRK 2149: Strong and Weak Interactions—from Hadrons to Dark Matter”; Bundesministerium für Bildung und Forschung (BMBF) under the project number 05P15PMCA1; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP).
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