Constraining the magnitude of the Chiral Magnetic Effect with Event Shape Engineering in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76$ TeV

In ultrarelativistic heavy-ion collisions, the event-by-event variation of the elliptic flow $v_2$ reflects fluctuations in the shape of the initial state of the system. This allows to select events with the same centrality but different initial geometry. This selection technique, Event Shape Engineering, has been used in the analysis of charge-dependent two- and three-particle correlations in Pb-Pb collisions at $\sqrt{s_{_{\rm NN}}} =2.76$ TeV. The two-particle correlator $\langle \cos(\varphi_\alpha - \varphi_\beta) \rangle$, calculated for different combinations of charges $\alpha$ and $\beta$, is almost independent of $v_2$ (for a given centrality), while the three-particle correlator $\langle \cos(\varphi_\alpha + \varphi_\beta - 2\Psi_2) \rangle$ scales almost linearly both with the event $v_2$ and charged-particle pseudorapidity density. The charge dependence of the three-particle correlator is often interpreted as evidence for the Chiral Magnetic Effect (CME), a parity violating effect of the strong interaction. However, its measured dependence on $v_2$ points to a large non-CME contribution to the correlator. Comparing the results with Monte Carlo calculations including a magnetic field due to the spectators, the upper limit of the CME signal contribution to the three-particle correlator in the 10-50% centrality interval is found to be 26-33% at 95% confidence level.

Parity symmetry is conserved in electromagnetism and is maximally violated in weak interactions. In strong interactions, global parity violation is not observed even though it is allowed by quantum chromodynamics. Local parity violation in strong interactions might occur in microscopic domains under conditions of finite temperature [1][2][3][4] due to the existence of the topologically non-trivial configurations of the gluonic field, instantons and sphalerons. The interactions between quarks and gluonic fields with non-zero topological charge [5] change the quark chirality. A local imbalance of chirality, coupled with the strong magnetic field produced in heavy-ion collisions (B ∼ 10 15 T) [6][7][8], would lead to charge separation along the direction of the magnetic field, which is on average perpendicular to the reaction plane (the plane of symmetry defined by the impact parameter vector and the beam direction), a phenomenon called Chiral Magnetic Effect (CME) [9][10][11][12]. Since the sign of the topological charge is equally probable to be positive or negative, the charge separation averaged over many events is zero. This makes the observation of the CME experimentally difficult and possible only via correlation techniques.
Azimuthal anisotropies in particle production relative to the reaction plane, often referred to as anisotropic flow, are an important observable to study the system created in heavy-ion collisions [13,14]. Anisotropic flow arises from the asymmetry in the initial geometry of the collision. Its magnitude is quantified via the coefficients v n in a Fourier decomposition of the charged particle azimuthal distribution [15,16]. Local parity violation would result in an additional sine term [17] dN d∆ϕ α ∼ 1 + 2v 1,α cos(∆ϕ α ) + 2a 1,α sin(∆ϕ α ) + 2v 2,α cos(2∆ϕ α ) + ..., where ∆ϕ α = ϕ α − Ψ RP , ϕ α is the azimuthal angle of the particle of charge α (+, −) and Ψ RP is the reaction-plane angle. The first (v 1,α ) and the second (v 2,α ) coefficients are called directed and elliptic flow, respectively. The a 1,α coefficient quantifies the effects from local parity violation. Since the average a 1,α = 0 over many events, one can only measure a 2 1,α or a 1,+ a 1,− . The charge-dependent twoparticle correlator is not convenient for such a study, because along with the signal a 1,α a 1,β (β denotes the charge) there is a much stronger contribution from correlations unrelated to the azimuthal asymmetry in the initial geometry ("non-flow"). These correlations largely come from the inter-jet correlations and resonance decays. To increase the CME contribution it was proposed to use the following correlator [17] γ αβ ≡ cos(ϕ α + ϕ β − 2Ψ RP ) = cos(∆ϕ α ) cos(∆ϕ β ) − sin(∆ϕ α ) sin(∆ϕ β ) that measures the difference between the correlation projected onto the reaction plane and perpendicular to it. In practice, the reaction-plane angle is estimated by constructing the event plane angle Ψ 2 using azimuthal particle distributions, which is why this correlator is often described as a three-particle correlator. This correlator suppresses background contributions at the level of v 2 , the difference between the particle production in-plane and out-of-plane. Examples of such background sources are the local charge conservation (LCC) coupled with elliptic flow [18,19], momentum conservation [19][20][21], and directed-flow fluctuations [22]. The most significant background source for CME measurements is the LCC.
The measurements of charge-dependent azimuthal correlations performed at the Relativistic Heavy Ion Collider (RHIC) [23][24][25][26] are in qualitative agreement with the expectations for the CME. The ALICE measurements at the Large Hadron Collider (LHC) [27, 28] also do not allow for a firm conclusion about the CME signal at the LHC energies because the interpretation of the results is complicated due to possible background contributions. The Event Shape Engineering (ESE) technique was proposed to disentangle background contributions from the potential CME signal [29]. This method makes it possible to select events with eccentricity values significantly larger or smaller than the average in a given centrality class [30,31] since v 2 scales approximately linearly with eccentricity [32]. The CME contribution is expected to mainly scale with the magnetic field strength and to not have a strong dependence on the eccentricity [33], while the background varies significantly. Therefore ESE provides a unique tool to separate the CME signal from the background for the three-particle correlator.
The CMS Collaboration has recently reported the measurement of the three-particle correlator γ αβ in p-Pb collisions at √ s NN = 5.02 TeV [34], where the direction of the magnetic field is expected to be uncorrelated to the reaction plane [35]. The magnitude of the correlator in p-Pb and Pb-Pb collisions is comparable for similar final-state charged-particle multiplicities. This measurement indicates that the contribution of the CME to this observable in this multiplicity range is small.
In this paper we report the measurements of the two-particle correlator δ αβ , the three-particle correlator γ αβ , and the elliptic flow v 2 of unidentified charged particles. These measurements are performed for shape selected and unbiased events in Pb-Pb collisions at √ s NN = 2.76 TeV. An upper limit on the CME contribution is deduced from comparisons of the observed dependence of the correlations on the event v 2 to that estimated using Monte Carlo (MC) simulations of the magnetic field of spectators with different initial conditions. While this paper was in preparation, a paper employing a similar approach to estimate the fraction of the CME signal in the three-particle correlator was submitted by the CMS Collaboration [36].
The data sample recorded by ALICE during the 2010 LHC Pb-Pb run at √ s NN = 2.76 TeV is used for this analysis. General information on the ALICE detector and its performance can be found in [37,38]. The Time Projection Chamber (TPC) [37,39] and Inner Tracking System (ITS) [37,40] are used to reconstruct charged-particle tracks and measure their momenta with a track-momentum resolution better than 2% for the transverse momentum interval 0.2 < p T < 5.0 GeV/c [38]. The two innermost layers of the ITS, the Silicon Pixel Detector (SPD), are employed for triggering and event selection. Two scintillator arrays (V0) [37,41], which cover the pseudorapidity ranges −3.7 < η < −1.7 (V0C) and 2.8 < η < 5.1 (V0A), are used for triggering, event selection, and the determination of centrality [42] and Ψ 2 . The trigger conditions and the event selection criteria are described in [38]. An offline event selection is applied to remove beam induced background and pileup events. Approximately 9.8 · 10 6 minimumbias Pb-Pb events with a reconstructed primary vertex within ±10 cm from the nominal interaction point in the beam direction are in 0-60% centrality interval used for this analysis.
Charged particles reconstructed using the combined information from the ITS and TPC in |η| < 0.8 and 0.2 < p T < 5.0 GeV/c are selected with full azimuthal coverage. Additional quality cuts are applied to reduce the contamination from secondary charged particles (i.e. particles originating from weak decays, conversions and secondary hadronic interactions in the detector material) and fake tracks (with random associations of space points). Only tracks with at least 70 space points in the TPC (out of a maximum of 159) with an average χ 2 per degree-of-freedom for the track fit lower than 2, a distance of closest approach (DCA) to the reconstructed event vertex smaller than 2.4 cm in the transverse plane (xy) and 3.2 cm in the longitudinal direction (z) are accepted. The charged particle track reconstruction efficiency was estimated from HIJING simulations [43,44] combined with a GEANT3 [45] detector model, and found to be independent of the collision centrality. The reconstruction efficiency, which may bias the determination of the p T averaged charge-dependent correlations and flow, increases from 70% at p T = 0.2 GeV/c to 85% at p T ∼ 1.5 GeV/c where it has a maximum. It then gradually decreases and is flat at 80% for p T > 3.0 GeV/c. The systematic uncertainty of the efficiency is about 5%.
The event shape selection is performed as in [30] based on the magnitude of the second-order reduced flow vector, q 2 [46], defined as where |Q 2 | = Q 2 2,x + Q 2 2,y is the magnitude of the second order harmonic flow vector and M is the mul- Table 1: Summary of absolute systematic uncertainties. The uncertainties depend on centrality and shape selection, whose minimum and maximum values are listed here. Fig. 1: (Colour online) Unidentified charged particle v 2 for shape selected and unbiased events as a function of collision centrality. The event selection is based on q 2 determined in the V0C with the lowest (highest) value corresponding to 0-10% (90-100%) q 2 . Points are slightly shifted along the horizontal axis for better visibility. Error bars (shaded boxes) represent the statistical (systematic) uncertainties. tiplicity. The vector Q 2 is calculated from the azimuthal distribution of the energy deposition measured in the V0C. Its x and y components and the multiplicity are given by where the sum runs over all channels i of the V0C detector (i = 1 − 32), ϕ i is the azimuthal angle of channel i and w i is the amplitude measured in channel i. The large gap in pseudorapidity (|∆η| > 0.9) between the charged particles in the TPC used to determine v 2 , δ αβ and γ αβ and those in the V0C suppresses non-flow effects. Ten event-shape classes with the lowest (highest) q 2 value corresponding to 0-10% (90-100%) q 2 are investigated for each centrality interval.
The flow coefficient v 2 is measured using the event plane method [16]. The orientation of the event plane Ψ 2,EP is estimated from the azimuthal distribution of the energy deposition measured by the V0A detector. The event plane resolution is calculated from correlations between the event planes determined in the TPC and the two V0 detectors separately [16]. The non-flow contributions to the v 2 coefficient and charge-dependent azimuthal correlations are greatly suppressed by the large rapidity separation between the TPC and the V0A (|∆η| > 2.0).
The absolute systematic uncertainties are evaluated from the variation of the results with different selection criteria on the reconstructed collision vertex, different magnetic field polarities, as well as by estimating the centrality from multiplicities measured by the TPC or the SPD rather than the V0 detector. Changes of the results due to variations of the track-selection criteria (e.g. changing the DCA xy and z ranges, number of the TPC space points, using tracks reconstructed by the TPC only) are considered as part of the systematic uncertainties. The effect of reconstruction efficiency on the measurements  is checked by randomly rejecting tracks to ensure a flat acceptance in p T . The detector response is studied using HIJING and AMPT [47] simulations, where the v 2 coefficients and the charge-dependent azimuthal correlations obtained directly from the models are compared with those from reconstructed tracks. The largest contribution to the systematic uncertainties is given by the detector response. The checks related to the reconstruction efficiency, magnetic field polarity and track-selection criteria also yield significant deviations from the nominal values for v 2 , γ αβ and δ αβ , respectively. The contributions from all sources are added in quadrature as an estimate of the total systematic uncertainty. The resulting systematic uncertainties are summarized in Table 1. Figure 1 presents the unidentified charged particle v 2 averaged over 0.2 < p T < 5.0 GeV/c for shape selected and unbiased samples as a function of collision centrality. The measured v 2 for the shape selected events differs from the average by up to 25%, which demonstrates that events with the desired initial spatial anisotropy can be experimentally selected. Sensitivity of the event shape selection deteriorates for peripheral collisions due to the low multiplicity and for central collisions due to the reduced magnitude of flow [30].
The centrality dependence of γ αβ for pairs of particles with same and opposite charge for shape selected and unbiased events is shown in the top panel of Fig. 2 Fig. 3: (Colour online) Top: Difference between opposite and same charge pair correlations for γ αβ as a function of v 2 for shape selected events together with a linear fit (dashed lines) for various centrality classes. Bottom: Difference between opposite and same charge pair correlations for γ αβ multiplied by the charged-particle density [48] as a function of v 2 for shape selected events for various centrality classes. The event selection is based on q 2 determined in the V0C with the lowest (highest) value corresponding to 0-10% (90-100%) q 2 . Error bars (shaded boxes) represent the statistical (systematic) uncertainties. between pairs of particles with only positive and only negative charges since the two combinations are found to be consistent within statistical uncertainties. The correlation of pairs with the same charge is stronger than the correlation for pairs of opposite charge for both shape selected and unbiased events. The ordering of the correlations of pairs with same and opposite charge indicates a charge separation with respect to the reaction plane. The magnitude of the same and opposite charge pair correlations depends weakly on the event shape selection (q 2 , i.e. v 2 ) in a given centrality bin.
The bottom panel of Fig. 2 shows the centrality dependence of δ αβ for pairs of particles with same and opposite charge for shape selected and unbiased samples. As reported in [27], the magnitude of the correlation for the same charge pairs is smaller than for the opposite charge combinations. This is in contrast to the CME expectation, indicating that background dominates the correlations. The same and opposite charge pair correlations are insensitive to the event-shape selection in a given centrality bin.
The difference between opposite and same charge pair correlations for γ αβ can be used to study the charge separation effect. This difference is presented as a function of v 2 for various centrality classes in the top panel of Fig. 3. The difference is positive for all centralities and its magnitude decreases for more central collisions and with decreasing v 2 (in a given centrality bin). At least two effects could be   Fig. 4: (Colour online) The expected dependence of the CME signal on v 2 for various centrality classes from a MC-Glauber simulation [50] (see text for details). No event shape selection is performed in the model, and therefore a large range in v 2 is covered. The solid lines depict linear fits based on the v 2 variation observed within each centrality interval. responsible for the centrality dependence: the reduction of the magnetic field with decreasing centrality and the dilution of the correlation due to the increase in the number of particles [24] in more central collisions. The difference between opposite and same charge pair correlations multiplied by the chargedparticle density in a given centrality bin, dN ch /dη (taken from [48]), to compensate for the dilution effect, is presented as a function of v 2 in the bottom panel of Fig. 3. All the data points fall approximately onto the same line. This is qualitatively consistent with expectations from LCC where an increase in v 2 , which modulates the correlation between balancing charges with respect to the reaction plane [49], results in a strong effect. Therefore, the observed dependence on v 2 points to a large background contribution to γ αβ .
The expected dependence of the CME signal on v 2 was evaluated with the help of a Monte Carlo Glauber [50] calculation including a magnetic field. In this simulation, the centrality classes are determined from the multiplicity of charged particles in the acceptance of the V0 detector following the method presented in [42]. The multiplicity is generated according to a negative binomial distribution with parameters taken from [42] based on the number of participant nucleons and binary collisions. The elliptic flow is assumed to be proportional to the eccentricity of the participant nucleons and approximately reproduces the measured p T -integrated v 2 values [51]. The magnetic field is evaluated at the geometrical center of the overlap region from the number of spectator nucleons following Eq. (A.6) from [11] with the proper time τ = 0.1 fm/c. The magnetic field is calculated in 1% centrality classes and averaged into the centrality intervals used for data analysis. It is assumed that the CME signal is proportional to |B| 2 cos(2(Ψ B − Ψ 2 )) , where |B| and Ψ B are the magnitude and direction of the magnetic field, respectively. Figure 4 presents the expected dependence of the CME signal on v 2 for various centrality classes. Similar results are found using MC-KLN CGC [52,53] and EKRT [54] initial conditions. The MC-KLN CGC simulation was performed using version 32 of the Monte Carlo k T -factorization code (mckt) available at [55], while the TRENTO model [56] was employed for EKRT initial conditions.
To disentangle the potential CME signal from background, the dependence on v 2 of the difference between opposite and same charge pair correlations for γ αβ and the CME signal expectations are fitted with a linear function (see lines in Figs. 3 (top panel) and 4, respectively): where p 0 accounts for the overall scale, which cannot be fixed in the MC calculations, and p 1 reflects the slope in the signal normalised to unity at v 2 = v 2 . In a pure background scenario, the correlator is directly proportional to v 2 and the p 1 parameter is equal to unity. The presence of a significant CME contribution, on the other hand, would result in non-zero intercepts at v 2 = 0 of the linear functions shown in Fig. 3. The ranges used in these fits are based on the v 2 variation observed within each centrality interval. The centrality dependence of p 1 from fits to data and to the signal expectations based on MC-Glauber, MC-KLN CGC and EKRT models is reported in Fig. 5. In this case, p 1 from data and MC models can be related according to where f CME denotes the CME fraction to the charge dependence of γ αβ and is given by Figure 6 presents f CME for the three models used in this study. The CME fraction cannot be extracted for central (0-10%) and peripheral (50-60%) collisions due to the large statistical uncertainties on p 1 extracted from data. The negative values for the CME fraction obtained for the 40-50% centrality range (deviating from zero by one σ ), if confirmed, would indicate that our expectations for the background contribution to be linearly proportional to v 2 are not accurate. Combining the points from 10-50% neglecting a possible centrality dependence gives f CME = 0.10 ± 0.13, f CME = 0.08 ± 0.10 and f CME = 0.08 ± 0.11 for the MC-Glauber, MC-KLN CGC and EKRT models, respectively. These results are consistent with zero CME fraction and correspond to upper limits on f CME of 33%, 26% and 29%, respectively, at 95% confidence level for the 10-50% centrality interval. The CME fraction agrees with the observations in [36] where the centrality intervals overlap.
In summary, the Event Shape Engineering technique has been applied to measure the dependence on v 2 of the charge-dependent two-and three-particle correlators δ αβ and γ αβ in Pb-Pb collisions at √ s NN = 2.76 TeV. While for δ αβ we observe no significant v 2 dependence in a given centrality bin, γ αβ is found to be almost linearly dependent on v 2 . When multiplied by the corresponding charged-particle density,  Fig. 6: (Colour online) Centrality dependence of the CME fraction extracted from the slope parameter of fits to data and MC-Glauber [50], MC-KLN CGC [52,53] and EKRT [54] models, respectively (see text for details). The dashed lines indicate the physical parameter space of the CME fraction. Points are slightly shifted along the horizontal axis for better visibility. Only statistical uncertainties are shown.
to compensate for the dilution effect, a linear dependence is observed consistently across all centrality classes. Using a Monte Carlo simulation with different initial-state models, we have found that the CME signal is expected to exhibit a weak dependence on v 2 . These observations imply that the dominant contribution to γ αβ is due to non-CME effects. In order to get a quantitative estimate of the signal and background contributions to the measurements, we fit both γ αβ and the expected signal dependence on v 2 with a first order polynomial. This allows the resulting fraction of the CME signal to be estimated in the centrality range 10-50%, but not for the most central (0-10%) and peripheral (50-60%) collisions due to large statistical uncertainties. Averaging over the centrality range 10-50% gives an upper limit of 26% to 33% (depending on the initial-state model) at 95% confidence level for the CME contribution to the difference between opposite and same charge pair correlations for γ αβ .