Charge-dependent ﬂow and the search for the chiral magnetic wave in Pb-Pb collisions at

We report on measurements of a charge-dependent ﬂow using a novel three-particle correlator with ALICE in Pb-Pb collisions at the CERN Large Hadron Collider (LHC), and discuss the implications for observation of local parity violation and the chiral magnetic wave (CMW) in heavy-ion collisions. Charge-dependent ﬂow is reported for different collision centralities as a function of the event charge asymmetry. While our results are in qualitative agreement with expectations based on the CMW, the nonzero signal observed in higher harmonics correlations indicates a possible signiﬁcant background contribution. We also present results on a differential correlator, where the ﬂow of positive and negative charges is reported as a function of the mean charge of the particles and their pseudorapidity separation. We argue that this differential correlator is better suited to distinguish the differences in positive and negative charges expected due to the CMW and the background effects, such as local charge conservation coupled with strong radial and anisotropic ﬂow.


Introduction
Parity (P) is a major symmetry of classical physics, being present in rigid-body dynamics, classical electrodynamics, and gravity.In the development of quantum mechanics, parity conservation was assumed.It was not until the 1950s [1] that the possibility of parity violation was considered, and soon after it was definitively demonstrated experimentally in nuclear decays [2,3].In the modern picture, P-and CP-violation in weak interactions are widely established experimentally and well understood theoretically.In strong interactions there is very little or no global P-violation, as determined by measurements of the neutron electric dipole moment [4,5].However, there is no first-principles reason why P-and CP-violation should not exist in strong interactions.P-and CP-violation as a general feature of quantum field theories was first explored in the 1970s [6,7], and a proposal to use heavy-ion collisions as a tool for studying P-and CP-violation first appeared as early as the 1980s [8].Specific proposals for a search for local P-violating effects in heavy-ion collisions appeared in the last decade [9][10][11][12][13][14].
Collisions of heavy nuclei at ultra-relativistic energies create a hot, dense medium that appears to have partonic degrees of freedom and evolve hydrodynamically.In non-central collisions the initial overlap region is non-isotropic, which, due to particle interactions, leads to a momentum-space anisotropy of the produced particles.This anisotropy can be described using a Fourier expansion of the azimuthal distribution of particles [15].Non-central collisions are also characterized by large orbital momentum and, what is important to this study, very large magnetic fields.Numerical estimates [16,17] indicate that at LHC energies the field strength can be as large as B ≈ m 2 π /e ≈ 10 14 T. In a vacuum, the magnetic field induced by the spectators decays in time quadratically (B ∝ t −2 ) and the lifetime of the magnetic field at LHC energies is extremely short, decreasing 6 orders of magnitude over the course of 0.5 fm/c.However the presence of electrical charges (such as the quarks in the QGP) means there is finite electrical conductivity.By Lenz's law the change in magnetic field is opposed by the charge carriers in the conductor, so the temporal decay of the magnetic field is significantly slowed [18,19].
The Chiral Magnetic Effect (CME) [20] is a process of charge separation with respect to the reaction plane.In the QCD vacuum there can exist gluonic configurations with nonzero topological charge, which can be an instanton or a sphaleron.At high temperatures, the sphaleron rate is expected to be dominant.The presence of such gluonic configurations with topological charge is what drives the Pviolation process.For example, in a region with negative topological charge, left-handed quarks will become right-handed, and right-handed quarks will remain right-handed.The strong magnetic field created in heavy-ion collisions interacts with the magnetic moment of the quarks and orients the spins of of quarks with positive (negative) electric charge to be parallel (anti-parallel) to the field direction.Under the assumption of massless quarks, right-handed quarks have their spins and momenta aligned.This will cause positive (negative) quarks to move parallel (anti-parallel) to the magnetic field, leading to a positive electric current and thus a positive electric charge dipole.Due to the chiral symmetry restoration, u and d quarks have only their bare Higgs mass, which is of the order of a few MeV/c 2 .This is sufficiently small to regard the quarks as effectively massless.Based on simple geometrical arguments, the magnetic field direction is always normal to the reaction plane, and therefore straightforwardly accessible to experiment.The Chiral Separation Effect (CSE) [21] is a similar effect in which the presence of a vector charge, e.g.electric charge, causes a separation of chiralities.For example, the presence of a net positive electric charge will induce a positive axial current along the direction of the magnetic field, i.e. right (left) handed quarks moving parallel (anti-parallel) to the magnetic field.
The CME can be summarized in a relatively simple equation: where J V is the vector current (electric charge current in this case), N c is the number of colors (3 in QCD but other numbers of colors, i.e.SU (N c ) gauge fields, are of interest in theory), µ A is the axial chemical potential (which encodes the anomaly-induced chiral imbalance), and B is the magnetic field.
The CSE can similarly be summarized as: where J A is the axial current (flow of axial charges, i.e. chiralities) and µ V is the vector (electric) chemical potential.The coupling between these two phenomena leads to a wave propagation of the electric charge, resulting in an electric charge quadrupole moment of the system.This is called the Chiral Magnetic Wave (CMW) [22][23][24].Importantly, for a given (net) charge state of the system, the quadrupole moment always has the same sign and is therefore present in an average over events with the same vector charge state, meaning it may lead to a signal strong enough to be observed directly in experiment.
As mentioned above, the azimuthal distribution of particles can be written as a Fourier expansion: where ϕ is the azimuthal angle of the particle, v n is the Fourier coefficient, and ψ n is the symmetry plane, which in principle can be different for each harmonic number n.
Taking into account the well-known modulation of particle emission due to elliptic flow parameterized with the Fourier coefficient v 2 (see e.g.[25]), one can write the azimuthal distribution of charged particles due to the CMW [24] as where the charge asymmetry A = (N + − N − )/(N + + N − ) is determined in some kinematic region (for example in the experimental acceptance), and the parameter r encodes the strength of the electric quadrupole due to the CMW.Bjorken flow [26] relates the pseudorapidity of a particle to its longitudinal production point.Due to space-momentum correlations, the charge asymmetry A, determined in the experimental acceptance, corresponds to the local charge asymmetry in a certain region of the fireball.
ALICE measurements of the charge dependent correlations in search for the CME have been published in [27].This paper presents the ALICE results on the charge dependent elliptic flow as a function of the event charge asymmetry.As the event charge asymmetry A strongly depends on the experimental acceptance and tracking efficiency, we also present related results on a differential three particle correlator that allows much more detailed study of the underlying physics mechanisms.We also present the corresponding measurements for higher harmonics flow that should be largely insensitive to the CMW but sensitive to the possible background effects.
2 Analysis methodology 2.1 v n as a function of A and integral three-point correlators In the theoretical work on the CMW [22-24, 28, 29] as well as the analysis published by STAR [30] of Au-Au collisions at √ s NN = 200 GeV, the observable has been the charge-dependent flow coefficient v ± n as a function of the charge asymmetry A. Experimentally, the charge asymmetry defined in a specified kinematic region must be corrected for detector efficiency, as discussed in [30][31][32].The effect of the correction is to increase the slope of positive or negative particle v ± n vs. A.This correction to the slope, though absolutely necessary, has the undesired feature of introducing additional sources of systematic uncertainty.
In addition to measuring v ± n as a function of A for a specified event-selection criterion (e.g. a centrality interval), one can measure the covariance of v ± n and A, i.e. v ± n A − A v ± n , as a function of some eventlevel variable (e.g.centrality) [33].The harmonic coefficient is, by definition, v n = cos(n(ϕ 1 − ψ n )) , where ϕ 1 is the azimuthal angle of a particle in the event and ψ n is the n-th harmonic symmetry plane.This makes v ± n A − A v ± n a three-point correlator.The first point is the flow particle, the second point is the event plane (which is an estimator for the true symmetry plane), and the third point is the event charge asymmetry.In cases where the event plane is determined with a second particle, as is the case with the two-particle cumulant method, this correlator can also be called a three-particle correlator.
The strength of the covariance of any two variables is independent of the sample size as long as the correlation is statistically significant, so no correction for efficiency is needed for the three-point correlator.For that reason, the correlation strength is identical for the full set of particles in some collection of events and for some randomly selected subset of particles in the same collection of events [34].
From Eq. 4, it follows that v ± 2 ≈ v 2 ∓ rA/2, which in turn allows one to write By measuring ∆v 2 as a function of A, it is possible to extract r directly in experiment.One can also substitute these terms into the three particle correlator, giving Clearly, either of these approaches can be generalized to arbitrary harmonic v n , yielding in principle a different r for each harmonic.Because of the symmetry of the CMW effect, the expectation is that r, and therefore the three-particle correlator, is significant only for the second harmonic and strongly suppressed for higher harmonics.Since A is efficiency dependent, it is necessary to scale down the observed width to account for the natural broadening due to the binomial sampling.To calculate the scale factor, we compared the widths of A in a Monte Carlo simulation before and after reconstruction effects, where the full detector response was implemented within the GEANT3 [35] framework.Since the ALICE tracking efficiency is independent of centrality, this scale factor is also independent of centrality.This correction to σ 2 A introduces an additional source of systematic uncertainty, in essentially the same way as the correction to the slope for v ± n vs A. In the present analysis, this uncertainty represents a roughly 6% normalization uncertainty on the points.Both the correction and the associated uncertainty vary depending on detector acceptance and efficiency, analysis cuts, etc.A key advantage of measuring the three-particle correlator, rather than r (directly or indirectly), is that it is efficiency independent and does not require any correction.This reduces the overall systematic uncertainty on the measurement, which better facilitates comparisons across experiments as well as with theory calculations.
In this analysis, two-particle cumulants are always used to calculate v n (which ignores correlations not related to anisotropic flow (non-flow), as well as flow fluctuations).Then, the integral correlator is: For this equation the inner average represents an average over all particles in a single event, and the outer average represents an average over all events.The first particle has the selected charge and the second particle is of both charges.For the integral correlator reported below, the same particles are used to calculate v n and A. The slopes extracted from the three-particle correlator were checked against the slopes extracted directly from ∆v 2 vs.A and found to be perfectly consistent with each other.

Differential three-point correlators
A key advantage of the novel three-point correlator is that it permits more differential studies and as such has more discriminating power.The charge asymmetry in the event can be generalized to the charge of a particle in the event, which we will call q 3 .The average of all charges in the event is equal to the charge asymmetry, i.e. q 3 event ≡ A. Under this generalization the correlator (Eq.6) becomes v n q 3 − q 3 v n .
We also use the additional notation of q 3 1 for denoting the mean of q 3 evaluated when selecting on the charge of the first particle q 1 .This is important because, by construction, the correlator v n q 3 − q 3 v n contains reducible correlations, i.e. correlations that can be expressed in terms of lower order correlations [33,34].These reducible correlations are removed by the construction v n q 3 − q 3 1 v n , which is therefore a three-point cumulant.
Using these relations, we estimate the differential correlator in the following way: The evaluation of a differential correlator is a very important feature of this study.Rather than measuring only event quantities, one can also measure the relationship between the flow at a particular kinematic coordinate and the charge of the third particle at another particular coordinate.This means the effect can be measured as a function of the separation in pseudorapidity of particles 1 and 3, for example.This differential nature allows for a much more detailed study of the origin of the correlation and provides stronger experimental constraints on the theoretical modeling of such effects.
Throughout this paper we use the subscript notation used above for the first, second, and third particles.The charge, azimuthal angle, and pseudorapidity of the first particle are q 1 , ϕ 1 , and η 1 , respectively.Similarly, the azimuthal angle of the second particle is ϕ 2 , and the charge and pseudorapidity of the third particle are q 3 and η 3 , respectively.
3 Experimental apparatus and data analysis ALICE [36,37] is a dedicated heavy-ion experiment located at the Large Hadron Collider at CERN.It is composed of a wide array of detector subsystems.Those used in the present analysis are the V0 detectors, the Inner Tracking System (ITS), and the Time Projection Chamber (TPC).The V0 detectors consist of scintillator arrays and are used for triggering and centrality determination.There are two V0 detectors, V0A and V0C.The V0A is located 340 cm from the nominal interaction point and the V0C is installed at 90 cm distance in the opposite direction.The V0A covers 2.8 < η < 5.1 in pseudorapidity and the C-side spans -3.7 < η < -1.7.The ITS is used for both tracking and vertex determination.The ITS is composed of three subsystems, each having two cylindrical layers of silicon detectors.Each of the layers covers at least |η| < 0.9 in pseudorapidity to match the TPC acceptance.The TPC is the primary tracking detector at midrapidity.The TPC is a large gas volume detector separated into two regions by a central electrode, positioned in a solenoidal magnetic field of 0.5 T. The gas volume is contained in a cylindrical electric field cage with an inner radius of 85 cm and an outer radius of 2.5 m, spanning the full azimuth 0 < ϕ < 2π.It extends 5.0 m in the z-direction, providing coverage of the full radial track length for pseudorapidity |η| < 0.9.
The present manuscript reports an analysis of Pb-Pb collisions at is an online trigger with a threshold on the multiplicity in the V0 detectors such that it corresponds to the 10% most central events.The semi-central trigger is defined similarly such that it corresponds to the 50% most central events.The centrality is estimated using the mean multiplicity in the V0 detectors, and the centrality is required to be within 5% (absolute) of the centrality estimate using the TPC multiplicity to avoid multiplicity fluctuations in the central region.The longitudinal position of the primary vertex is required to be within 10 cm of the nominal center of the ALICE coordinate system in order to ensure uniform detector acceptance.
Tracks are selected in the kinematic region |η| < 0.8 and 0.2 GeV/c < p T < 5.0 GeV/c.They are required to have at least 70 TPC clusters, and the percentage of registered hits to crossed TPC pad rows to be at least 80%.The track fit is required to have χ 2 per cluster (2 degrees of freedom) less than 4.0.Additional tracking information from the ITS is used when it is available, i.e. when the track trajectory in the TPC points to an active area of the ITS.The distance of closest approach to the reconstructed vertex is required to be within 3.2 cm in the z-direction and within 2.4 cm in xy-plane.Due to the excellent azimuthal uniformity of the TPC response, no correction for azimuthal acceptance is needed, nor is one applied.The results are corrected for the p T dependence of the tracking efficiency, which is about 80% at low p T and about 90% at high p T .The correction procedure is to randomly exclude tracks in such a way that the effective efficiency is made to be uniform in p T .The result is a 2-3% reduction in v n .
To assess systematic uncertainties, the analysis is repeated for different operational conditions (i.e. the two orientations of the experimental magnetic field), different event selection criteria, different track selection cuts, and different track reconstruction methods.The uncertainties associated with each specific selection or condition are observed to be uncorrelated and thus added in quadrature to determine the overall systematic uncertainty.All aforementioned sources were found to generally contribute with similar magnitude.Many observables reported in this manuscript have values very close to zero, so that reporting systematic uncertainties as a percentage of these values obscures their true meaning.For numerical stability, systematic uncertainties are evaluated as a percentage of v n A or v n q 3 .This quantity alone is not necessarily physically meaningful, because it contains detector specific effects.It does, however, set a natural scale for the uncertainties.Once uncertainties are assessed, their absolute value is determined and then plotted together with the data points.

v 2 vs. A
Figure 1 shows v + 2 and v − 2 as a function of the observed (uncorrected) event charge asymmetry A in the 30-40% centrality class.Clearly visible is an increase in v − 2 with increasing A, along with a corresponding decrease in v + 2 .This is qualitatively consistent with expectations from the CMW [22-24, 28, 29] as well as with the STAR results [30].Visually the relationship does not appear exactly linear for either charge.For a given centrality selection, there is some range of multiplicities.Since A is a combination of numbers of particles, it is a combination of different negative binomial distributions, which are broader for lower numbers of particles.Therefore, for larger values of |A|, one is sampling events with lower multiplicities, which can affect the value of v 2 .
Figure 2 shows ∆v 2 = v − 2 − v + 2 as a function of the observed A in the left panel and of A corrected for efficiency in the right panel.To obtain the corrected A, we analyzed HIJING [38] simulations propagated through a detector description implemented in the GEANT3 [35] framework to determine the true (generated particle level) A as a function of the observed (reconstructed track level) A. It can be seen that the effect of the correction is a modest increase in the slope.Again, these results are qualitatively consistent with CMW expectations and with the STAR data [30].

Integral correlator results as a function of centrality
Considering the observed increase (decrease) of v − 2 (v + 2 ) with increasing A, discussed in the preceding section, we expect a positive (negative) covariance of v − 2 (v + 2 ) with A, and indeed this is exactly what is seen in the integral correlator.Additionally, it enables a convenient study of the evolution of the correlation as a function of event-level observables.
Figure 3 shows the integral correlator of the second harmonic as a function of centrality.A substantial increase in the correlation strength is seen as the collisions become more peripheral.This can be caused by a combination of several factors.The magnetic field strength increases as the impact parameter increases since there are more spectators and thus the current gets stronger.This would cause the correlations due to the CMW to get stronger.Additionally, local charge conservation (LCC) effects could play a role [33].It is important to note that neither of these necessarily comes at the expense of the other; in principle the observable could have contributions from both of these and/or additional contributions from as yet unknown sources of correlation.
Local charge conservation is the production of charged pairs at the same spacetime point.For N particles, there are N/2 correlated pairs, and N (N − 1) combinatoric pairs, meaning the correlation strength is proportional to N/(N (N − 1)), or approximately 1/N .Figure 4 shows the difference between the  charges for the second harmonic correlator multiplied by dN ch /dη [39], where dN ch /dη is used as a proxy for the total number of particles, to examine the role of the dilution of LCC correlations on the correlator.Considerable centrality dependence remains.
The three-particle correlator is studied using other harmonics as well.This provides important additional constraints because P-violating effects are expected to occur with respect to the reaction plane, therefore higher harmonics should have very little or no correlations.The three-particle correlator for the third harmonic is shown in Fig. 5 and the fourth harmonic is shown in Fig. 6.In both cases the left panel shows the correlator for positive and negative charges separately, and the right panel shows the charge difference of the correlator multiplied by dN ch /dη .In both of these cases, the centrality dependence of the charge dependence is flat, in contrast to the second harmonic.This may suggest a different nature of the correlation.It could also reflect a weaker centrality dependence of v 3 compared to that of elliptic flow.

Slopes of ∆v 2 vs. A
Figure 7 shows a comparison between slope parameters r estimated in this analysis and from the STAR analysis [30] of Au-Au collisions at √ s NN = 200 GeV.For the STAR data, the v 2 is evaluated for  charged pions with 0.15 GeV/c < p T < 0.5 GeV/c, in contrast with the present results which are for unidentified hadrons with 0.2 GeV/c < p T < 5.0 GeV/c.Overall, the slopes are surprisingly similar when considering the different collision energies and multiplicities, as well as the different kinematic acceptance (in addition to the different p T selection, the STAR results correspond to the pseudorapidity range |η| < 1.0).The STAR data exhibit a somewhat stronger centrality dependence than the ALICE data.Moreover, the STAR data exhibit a stronger centrality dependence than predicted by the theoretical models invoking the CMW for Au-Au at 200 GeV [24].Additionally, hydrodynamical models have been developed to attempt to explain the STAR results without invoking the CMW [40,41].However, no theoretical modeling or calculations at all, regardless of mechanism, are available for Pb-Pb collisions at √ s NN = 2.76 TeV.

Differential correlator results as a function of ∆η
As discussed above, the definition of the three particle differential correlator includes q 3 1 -the mean charge of the third particle when evaluated with a selection on q 1 .The quantity q 3 1 − q 3 is shown as a function of ∆η = η 1 − η 3 in Fig. 8.The measurements are performed as a function of |∆η| and  shown as a function of ∆η with the points reflected about ∆η = 0.This conditional mean of q 3 depends significantly on ∆η and has the opposite sign when q 1 is flipped.The effect is most pronounced for ∆η ≈ 0, and the weakest when ∆η is large.When the first particle is negative, the third particle has a slightly positive mean charge, and when the first particle is positive, the third particle has a slightly negative mean charge.Note that the quantity q 3 1 − q 3 is proportional to the charge balance function [33] and as such reflects the charge correlation length.
Figure 9 shows the three-particle correlator for the second harmonic as a function of ∆η.The correlator exhibits a rather non-trivial dependence on ∆η: a peak with a "typical hadronic width" of about 0.5-1 units of rapidity and a possible change of the sign at about ∆η ≈ 1 (note however these points are consistent with zero within the systematic uncertainties).Both of those features qualitatively agree with possible background contribution from local charge conservation combined with strong radial and elliptic flow [33].Unfortunately there exist no predictions for this observable from the CMW.
The three-particle correlator for the third and fourth harmonics as a function of ∆η is shown in Fig. 10.The strength of the correlations is significantly reduced, by a factor about 3 in the case of the third harmonic and at least a factor of 5 for the fourth harmonic.The fourth harmonic correlator is consistent with zero within errors.Neglecting flow fluctuations, the CMW expectations for higher harmonics correlators     would be zero; unfortunately there are no reliable calculations of the effect of flow fluctuations.The (background) contribution due to the local charge conservation should roughly scale with the magnitude of the flow [33] and is qualitatively consistent with the experimental results.More detailed calculations in both scenarios, as well as more precise data, are obviously needed for a more definitive conclusion.

Summary and Outlook
Novel three-particle correlators have been employed in an experimental search for the CMW.Results have been shown for the second, third, and fourth harmonic for the integrated correlator of the chargedependent flow as a function of centrality and the differential correlator as a function of pseudorapidity separation.A clear dependence of the positive and negative particle anisotropic flow on the event charge asymmetry is presented for different centralities in Pb-Pb collisions.The slopes of this dependence, determined by two different methods, are consistent and qualitatively agree with the expectations for CMW, as well as similar to those measured by the STAR Collaboration at the top RHIC energy.The observed nonzero signal in higher harmonics correlations indicates a possible strong background contribution, likely from LCC in combination with strong radial and anisotropic flow.We also have presented results on the differential correlator, which is more sensitive to the detail of the underlying physics and help to discriminate between the CMW scenario and the background effects.The second harmonic results show a fairly large correlation, and the strength of the correlation strongly decreases with increasing harmonic number.Further input from theory is needed to give detailed constraints on the magnitude and range of background vs. CMW correlations.LHC Run-2 will include Pb-Pb collisions at √ s NN = 5.02 TeV and will offer substantially higher integrated luminosity, which will largely improve statistical precision of these measurements, and may also help reduce some of the systematic uncertainties.One of the chief benefits of increased statistical precision would be the possibility to evaluate the three-particle correlator with identified particles.The species of both particle 1 and 3 is of potential interest.The different collision energy affects both the peak strength (which increases) and the lifetime (which decreases) of the magnetic field induced in the collision, which provides additional information.For that reason, an analysis of this correlator at lower collisions energies, for example at RHIC, would provide important additional insights.
76 TeV, collected by ALICE during the 2010 and 2011 years of LHC operations.In the early part of the 2010 operation, the Pb-Pb minimum bias (MB) trigger was a 2-out-of-3 coincidence of a) signals in two pixel chips in the outer layer of the SPD, b) a signal in the V0A, c) a signal in the V0C.In the later part of the 2010 operation and for all of the 2011 operation, the Pb-Pb MB trigger required a coincidence of both V0 detectors.The data sample used in this analysis comprises approximately 1.7×10 7 MB triggered events in the 2010 data set.In the 2011 set, we use a mix of the central, semi-central, and MB triggers.The central trigger

Fig. 1 :
Fig. 1: (Color online) Harmonic coefficients v + 2 (red squares) and v − 2 (blue circles) as a function of the observed event charge asymmetry A in the 30-40% centrality class.Statistical uncertainties only.

Fig. 3 :
Fig. 3: (Color online) Three-particle correlator for positive (red squares) and negative (blue circles) particles for the second harmonic as a function of centrality.Statistical (systematic) uncertainties are indicated by vertical bars (shaded boxes).

Fig. 4 :
Fig.4: Charge difference of the three-particle correlator for the second harmonic as a function of centrality, multiplied by dN ch /dη[39].Statistical (systematic) uncertainties are indicated by vertical bars (shaded boxes).

Fig. 5 :Fig. 6 :
Fig. 5: (Color online) Three-particle correlator for the third harmonic (left panel) for positive (red squares) and negative (blue circles) particles, and the charge difference multiplied by dN ch /dη (right panel).Statistical (systematic) uncertainties are indicated by vertical bars (shaded boxes).

Fig. 7 :Fig. 8 :
Fig. 7: (Color online) Slope parameter r as a function of centrality, including points from STAR [30].Statistical (systematic) uncertainties are indicated by vertical bars (shaded boxes).Not shown is the 6% systematic normalization uncertainty due the MC correction for σ 2 A . η

Fig. 10 :
Fig. 10: (Color online) Three-particle correlator for the third (left panel) and fourth (right panel) harmonics, for positive (red squares) and negative (blue circles) particles.Statistical (systematic) uncertainties are indicated by vertical bars (shaded boxes).