Constraints on the chiral magnetic effect using charge-dependent azimuthal correlations in pPb and PbPb collisions at the CERN Large Hadron Collider

: Charge-dependent azimuthal correlations of same- and opposite-sign pairs with respect to the second- and third-order event planes have been measured in p Pb collisions at s NN = 8 . 16 TeV and PbPb collisions at 5.02 TeV with the CMS experiment at the LHC. The measurement is motivated by the search for the charge separation phenomenon predicted by the chiral magnetic effect (CME) in heavy ion collisions. Three- and two-particle azimuthal correlators are extracted as functions of the pseudorapidity difference, the transverse momentum ( pT ) difference, and the pT average of same- and opposite-charge pairs in various event multiplicity ranges. The data suggest that the charge-dependent three-particle correlators with respect to the second- and third-order event planes share a common origin, predominantly arising from charge-dependent two-particle azimuthal correlations coupled with an anisotropic flow. The CME is expected to lead to a v 2 -independent three-particle correlation when the magnetic field is fixed. Using an event shape engineering technique, upper limits on the v 2 -independent fraction of the three-particle correlator are estimated to be 13% for p Pb and 7% for PbPb collisions at 95% confidence level. The results of this analysis, both the dominance of two-particle correlations as a source of the three-particle results and the similarities seen between PbPb and p Pb, provide stringent constraints on the origin of charge-dependent three-particle azimuthal correlations and challenge their interpretation as arising from a chiral magnetic effect in heavy ion collisions. Charge-dependentazimuthalcorrelationsofsame-andopposite-signpairswithrespecttothesecond-andthird-order event planes have been measured in p Pb collisions at √ s NN = 8 . 16 TeV and PbPb collisions at 5.02 TeV with the CMS experiment at the LHC. The measurement is motivated by the search for the charge separation phenomenon predicted by the chiral magnetic effect (CME) in heavy ion collisions. Three- and two-particle azimuthal correlators are extracted as functions of the pseudorapidity difference, the transverse momentum ( p T ) difference, and the p T average of same- and opposite-charge pairs in various event multiplicity ranges. The data suggest that the charge-dependent three-particle correlators with respect to the second- and third-order event planes share a common origin, predominantly arising from charge-dependent two-particle azimuthal correlations coupled with an anisotropic ﬂow. The CME is expected to lead to a v 2 -independent three-particle correlation when the magnetic ﬁeld is ﬁxed. Using an event shape engineering technique, upper limits on the v 2 -independent fraction of the three-particle correlator are estimated to be 13% for p Pb and 7% for PbPb collisions at 95% conﬁdence level. The results of this analysis, both the dominance of two-particle correlations as a source of the three-particle results and the similarities seen between PbPb and p Pb, provide stringent constraints on the origin of charge-dependent three-particle azimuthal correlations and challenge their interpretation as arising from a chiral magnetic effect in heavy ion collisions.


I. INTRODUCTION
It has been suggested that in high-energy nucleus-nucleus (AA) collisions, metastable domains of gluon fields with nontrivial topological configurations may form [1][2][3][4]. These domains can carry an imbalance between left-and righthanded quarks arising from interactions of chiral quarks with topological gluon fields, leading to a local parity (P ) violation [3,4]. This chirality imbalance, in the presence of the extremely strong magnetic field, which can be produced in a noncentral AA collision, is expected to lead to an electric current perpendicular to the reaction plane, resulting in a final-state charge separation phenomenon known as the chiral magnetic effect (CME) [5][6][7]. Such macroscopic phenomena arising from quantum anomalies are a subject of interest for a wide range of physics communities. The chiral-anomalyinduced phenomena have been observed in magnetized relativistic matter in three-dimensional Dirac and Weyl materials [8][9][10]. The search for the charge separation from the CME in AA collisions was first carried out at RHIC at BNL [11][12][13][14][15] and later at the CERN LHC [16] at various centerof-mass energies. In these measurements, a charge-dependent azimuthal correlation with respect to the reaction plane was observed, which is qualitatively consistent with the expectation of charge separation from the CME. No strong collision energy dependence of the signal is observed going from RHIC to LHC energies, although some theoretical predictions suggested that the possible CME signal could be much smaller at the LHC than at RHIC because of a shorter lifetime of the magnetic field [17]. Nevertheless, theoretical estimates of the time evolution of the magnetic field have large uncertainties [17].
The experimental evidence for the CME in heavy ion collisions remains inconclusive because of several identified sources of background correlations that can account for part or all of the observed charge-dependent azimuthal correlations [18][19][20]. Moreover, the charge-dependent azimuthal correlation in high-multiplicity pPb collisions has been recently found to have a nearly identical value to that observed in PbPb collisions [21]. This is a strong indication that the observed effect in heavy ion collisions might predominantly result from background contributions. The CME-induced charge separation effect is predicted to be negligible in pPb collisions, as the angle between the magnetic field direction and the event plane is expected to be randomly distributed [21,22].
The charge separation can be characterized by the first Podd sine term (a 1 ) in a Fourier decomposition of the chargedparticle azimuthal distribution [23]: where φ − RP represents the particle azimuthal angle with respect to the reaction plane angle RP in heavy ion collisions (determined by the impact parameter and beam axis), and v n and a n denote the coefficients of P -even and P -odd Fourier terms, respectively. Although the reaction plane is not an experimental observable, it can be approximated in heavy ion collisions by the second-order event plane 2 , determined by the direction of the beam and the maximal particle density in the elliptic azimuthal anisotropy. The P -odd terms will vanish after averaging over events, because the sign of the chirality imbalance changes event by event. Therefore, the observation of such an effect is only possible through the measurement of particle azimuthal correlations. An azimuthal three-particle correlator γ 112 proposed to explore the first coefficient a 1 of the P -odd Fourier terms characterizing the charge separation [23] is Here, α and β denote particles with the same or opposite electric charge sign and the angle brackets reflect an averaging over particles and events. Assuming particles α and β are uncorrelated, except for their individual correlations with respect to the event plane, the first term on the right-hand side of Eq. (2) becomes v 1,α v 1,β , which is generally small and independent of the charge [12], while the second term is sensitive to the charge separation and can be expressed as a 1,α a 1,β . While the similarity of the pPb and PbPb data at 5.02 TeV analyzed by the CMS experiment pose a considerable challenge to the CME interpretation of the charge-dependent azimuthal correlations observed in AA collisions [21], important questions still remain to be addressed: is the correlation signal observed in pPb collisions entirely a consequence of background correlations? What is the underlying mechanism for those background correlations that are almost identical in pPb and PbPb collisions? Can the background contribution be quantitatively constrained with data and, if so, is there still evidence for a statistically significant CME signal?
In particular, among the proposed mechanisms for background correlations, one source is related to the chargedependent two-particle correlation from local charge conservation in decays of resonances or clusters (e.g., jets) [20]. By coupling with the anisotropic particle emission, an effect resembling charge separation with respect to the reaction plane can be generated. The observed characteristic range of the two-particle correlation in data is around one unit of rapidity, consistent with short-range cluster decays. In this mechanism of local charge conservation coupled with the elliptic flow, a background contribution to the three-particle correlator, γ 112 , is expected to be [24] γ bkg 112 = κ 2 cos(φ α − φ β ) cos 2(φ β − RP ) = κ 2 δ v 2 . (3) Here, δ ≡ cos(φ α − φ β ) represents the charge-dependent two-particle azimuthal correlator and κ 2 is a constant parameter, independent of v 2 , but mainly determined by the kinematics and acceptance of particle detection [24]. As both the charge conservation effect and anisotropic flow are known to be present in heavy ion collisions, the primary goal of this paper is to conduct a systematic investigation of how much of the observed charge-dependent correlations in the data can be accounted for by this mechanism.
Although the background contribution from local charge conservation is well defined in Eq. (3) and has been long recognized [17,20,24], it is still not known to what extent background contributions account for the observed γ 112 correlator. The main difficulty lies in determining the unknown value of κ 2 in a model-independent way. The other difficulty is to demonstrate directly the linear dependence on v 2 of γ bkg 112 , which is nontrivial as one has to ensure that the magnetic field, and thus the CME, does not change when selecting events with different v 2 values. Therefore, selecting events with a quantity that directly relates to the magnitude of v 2 is essential.
This paper aims to overcome the difficulties mentioned above and achieve a better understanding as to the contribution of the local charge conservation background to the chargedependent azimuthal correlation data. The results should serve as a new baseline for the search for the CME in heavy ion collisions. Two approaches are employed as outlined below.
(1) Higher-order harmonic three-particle correlator: in heavy ion collisions, the charge separation effect from the CME is only expected along the direction of the induced magnetic field normal to the reaction plane, approximated by the second-order event plane 2 . As the symmetry plane of the third-order Fourier term ("triangular flow" [25]) 3 is expected to have a weak correlation with 2 [26], the charge separation effect with respect to 3 is expected to be negligible. By constructing a charge-dependent correlator with respect to the third-order event plane, charge-dependent background effects unrelated to the CME can be explored. In particular, in the context of the local charge conservation mechanism, the γ 123 correlator is also expected to have a background contribution, with similar to that for the γ 112 correlator as given in Eq. (3). As the κ 2 and κ 3 parameters mainly depend on particle kinematics and detector acceptance effects, they are expected to be similar, largely independent of harmonic event plane orders. The relation in Eq. (5) can be generalized for all "higher-order harmonic" three-particle correlators, γ 1,n−1;n = κ n δ v n . Derivation of Eq. (5) as well as generalization to all higher-order harmonics can be found in Appendix A, which follows similar steps as for that of Eq. (3) given in Ref. [24]. One caveat here is that when averaging over a wide η and p T range, the κ n value may also depend on the η and p T dependence of the v n harmonic, which is similar, but not exactly identical, between the v 2 and v 3 coefficients [27,28]. By taking the difference of correlators between sameand opposite-sign pairs (denoted as γ 112 and γ 123 among three particles, and δ between two particles) to eliminate all charge-independent background sources, the following relation is expected to hold if the charge dependence of threeparticle correlators is dominated by the effect of local charge 044912-2 conservation coupled with the anisotropic flow: Therefore, an examination of Eq. (6) will quantify to what extent the proposed background from charge conservation contributes to the γ 112 correlator, and will be a critical test of the CME interpretation in heavy ion collisions.
(2) Event shape engineering (ESE): to establish directly a linear relationship between the γ correlators and v n coefficients, the ESE technique [29] is employed. In a narrow centrality or multiplicity range (so that the magnetic field does not change significantly), events are further classified based on the magnitude of the event-by-event Fourier harmonic related to the anisotropy measured in the forward rapidity region. Within each event class, the γ correlators and v n values are measured and compared to test the linear relationship. A nonzero intercept value of the γ correlators with a linear fit would reflect the strength of the CME.
With a higher luminosity pPb run at √ s NN = 8.16 TeV and using the high-multiplicity trigger in CMS, the pPb data sample gives access to multiplicities comparable to those in peripheral PbPb collisions, allowing for a detailed comparison and study of the two systems with very different expected CME contributions in the collisions [21]. Measurements of threeparticle correlators γ 112 and γ 123 and the two-particle correlator δ are presented in different charge combinations as functions of the pseudorapidity (η) difference (| η|), the transverse momentum (p T ) difference (| p T |), and the average p T of correlated particles (p T ). Integrated over η and p T , the event multiplicity dependence of three-and two-particle correlations is also presented in pPb and PbPb collisions. In pPb collisions, the particle correlations are explored separately with respect to the event planes that are obtained using particles with 4.4 < |η| < 5.0 from the pand Pb-going beam directions. The ESE analysis is performed for γ 112 as a function of v 2 in both pPb and PbPb collisions. This paper is organized as follows. After a brief description of the detector and data samples in Sec. II, the event and track selections are discussed in Sec. III, followed by the discussion of the analysis technique in Sec. IV. The results are presented in Sec. V, and the paper is summarized in Sec. VI.

II. DETECTOR AND DATA SAMPLES
The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diameter, providing a magnetic field of 3.8 T. Within the solenoid volume, there are four primary subdetectors, including a silicon pixel and strip tracker detector, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter (HCAL), each composed of a barrel and two endcap sections. The silicon tracker measures charged particles within the range |η| < 2.5. Iron and quartz-fiber Cherenkov hadron forward (HF) calorimeters cover the range 2.9 < |η| < 5.2. The HF calorimeters are constituted of towers, each of which is a twodimensional cell with a granularity of 0.5 units in η and 0.349 radians in φ. For charged particles with 1 < p T < 10 GeV and |η| < 1.4, the track resolutions are typically 1.5% in p T and 25-90 (45-150) μm in the transverse (longitudinal) impact parameter [30]. A detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref. [31].
The pPb data at √ s NN = 8.16 TeV used in this analysis were collected in 2016, and correspond to an integrated luminosity of 186 nb −1 . The beam energies are 6.5 TeV for the protons and 2.56 TeV per nucleon for the lead nuclei. The data were collected in two different run periods: one with the protons circulating in the clockwise direction in the LHC ring, and one with them circulating in the counterclockwise direction. By convention, the proton beam rapidity is taken to be positive when combining the data from the two run periods. A subset of PbPb data at √ s NN = 5.02 TeV collected in 2015 (30-80% centrality, where centrality is defined as the fraction of the total inelastic cross section, with 0% denoting the most central collisions) is used. The PbPb data were reprocessed using the same reconstruction algorithm as the pPb data, in order to compare directly the two colliding systems at similar final-state multiplicities. The three-particle correlator, γ 112 , data for pPb collisions at √ s NN = 8.16 TeV are compared to those previously published at √ s NN = 5.02 TeV [21] to examine any possible collision energy dependence. Because of statistical limitations, new analyses of higher-order harmonic three-particle correlator and event shape engineering introduced in this paper cannot be performed with the 5.02-TeV pPb data.

III. SELECTION OF EVENTS AND TRACKS
The event reconstruction, event selections, and the triggers, including the dedicated triggers to collect a large sample of high-multiplicity pPb events at √ s NN = 8.16 TeV, are similar to those used in previous CMS particle correlation measurements at lower energies [28,[32][33][34], as discussed below. For PbPb events, they are identical to those in Ref. [21]. Minimum bias pPb events at 8.16 TeV were selected by requiring energy deposits in at least one of the two HF calorimeters above a threshold of approximately 1 GeV and the presence of at least one track with p T > 0.4 GeV in the pixel tracker. In order to collect a large sample of high-multiplicity pPb collisions, a dedicated trigger was implemented using the CMS level-1 (L1) and high-level trigger (HLT) systems. At L1, the total number of towers of ECAL+HCAL above a threshold of 0.5 GeV in transverse energy (E T ) was required to be greater than a given threshold (120 and 150 towers), where a tower is defined by η× φ = 0.087 × 0.087 radians. Online track reconstruction for the HLT was based on the same offline iterative tracking algorithm to maximize the trigger efficiency. For each event, the vertex reconstructed with the greatest number of tracks was selected. The number of tracks with |η| < 2.4, p T > 0.4 GeV, and a distance of closest approach less than 0.12 cm to this vertex, was determined for each event and required to exceed a certain threshold (120,150,185,250).
In the offline analysis of pPb (PbPb) collisions, hadronic events are selected by requiring the presence of at least one (three) energy deposit(s) greater than 3 GeV in each of the two HF calorimeters. Events are also required to contain a primary vertex within 15 cm of the nominal interaction point along the 044912-3 beam axis and 0.15 cm in the transverse direction. In the pPb data sample, the average pileup (number of interactions per bunch crossing) varied between 0.1 to 0.25 pPb interactions per bunch crossing. A procedure similar to that described in Ref. [28] is used for identifying and rejecting pileup events. It is based on the number of tracks associated with each reconstructed vertex and the distance between multiple vertices. The pileup in PbPb data is negligible.
For track selections, the impact parameter significance of the track with respect to the primary vertex in the direction along the beam axis and in the transverse plane, d z /σ (d z ) and d T /σ (d T ), is required to be less than 3. The relative uncertainty in p T , σ (p T )/p T , must be less than 10%. Primary tracks, i.e., tracks that originate at the primary vertex and satisfy the high-purity criteria of Ref. [30], are used to define the event charged-particle multiplicity (N offline trk ). To perform correlation measurements, each track is also required to leave at least one hit in one of the three layers of the pixel tracker. Only tracks with |η| < 2.4 and p T > 0.3 GeV are used in this analysis to ensure high tracking efficiency.
The pPb and PbPb data are compared in classes of N offline trk , where primary tracks with |η| < 2.4 and p T > 0.4 GeV are counted. To compare with results from other experiments, the PbPb data are also analyzed based on centrality classes for the 30-80% centrality range.

IV. ANALYSIS TECHNIQUE
The analysis technique of three-particle correlations employed in this paper is based on that established in Ref. [21], with the extension of charge-dependent two-particle correlations, higher-order harmonic three-particle correlations, and correlation studies in different event shape classes (i.e., ESE analysis). The details are outlined below.

A. Calculations of two-and three-particle correlators
Without directly reconstructing the event plane, the expression given in Eq. (2) can be alternatively evaluated using a three-particle correlator with respect to a third particle [11,12], cos(φ α + φ β − 2φ c ) /v 2,c , where v 2,c is the elliptic flow anisotropy of particle c with inclusive charge sign. The three-particle correlator is measured via the scalar-product method of Q vectors. A complex Q vector for each event is defined as Q n ≡ M i=1 w i e inφ i /W , where φ i is the azimuthal angle of particle i, n is the Fourier harmonic order, M is the number of particles in the Q n calculation in each event, and w i is a weight assigned to each particle for efficiency correction, which is derived from a simulation using the HIJING event generator [35]. The W = M i=1 w i represents the weight of the Q vector. In this way, the three-particle correlator can be expressed in terms of the product of Q vectors, i.e., Q 1,α and Q 1,β , when particles α and β are chosen from different detector phase-space regions or carry different charge signs, where the angle brackets on the right-hand side denote an event average of the Q-vector products, weighted by the product of their respective total weights W . Here Q 2,trk is the charge inclusive Q 2 vector of all particles in the tracker region, and Q 2,HF± denotes the Q 2 -vector for particles c detected in the HF towers. When particles α and β are of the same sign and share the same phase space region (denoted as α = β), an extra term is needed to remove the contribution of a particle pairing with itself, so evaluation of the three-particle correlator is modified as where the Q 112 is defined as and the denominator of Eq. (9) is the respective event weight associated with Q 112 .
In the numerators of Eqs. (7) and (8), the particles α and β are identified in the tracker, with |η| < 2.4 and 0.3 < p T < 3 GeV, and are assigned a weight factor w i to correct for tracking inefficiency. The particle c is selected by using the tower energies and positions in the HF calorimeters with 4.4 < |η| < 5.0. This choice of η range for the HF towers imposes an η gap of at least two units with respect to particles α and β from the tracker, to minimize possible short-range correlations. To account for any occupancy effect of the HF detectors resulting from the large granularities in η and φ, each tower is assigned a weight factor w i corresponding to its E T value when calculating the Q vector. The denominator of the right-hand side of Eqs. (7) and (8) corresponds to the v 2,c using the scalar-product method [11,12], with Q 2,trk and Q 2,HF± denoting Q 2 vectors obtained from the tracker and the two HF detectors (positive and negative η side) with the same kinematic requirements as for the numerator. The three-particle correlator is evaluated for particles α and β carrying the same sign (SS) and opposite sign (OS). The SS combinations, (+,+) and (−,−), give consistent results and are therefore combined. For pPb collisions, the three-particle correlator is also measured with particle c from HF+ and HF−, corresponding to the pand Pb-going direction, respectively. For symmetric PbPb collisions, the results from HF+ and HF− are consistent with each other and thus combined.
The higher-order harmonic three-particle correlator, γ 123 , defined in Eq. (4), is evaluated in exactly the same way as the γ 112 correlator as follows when particles α and β do not overlap, with higher-order Q vectors for particles α and β of SS and OS. Similarly to Eq. (8) when particles α and β can overlap,  the γ 123 can be evaluated via where Q 123 is defined as and the respective event weight associated with Q 123 is the denominator of Eq. (12). Similarly, the charge-dependent two-particle correlator, δ ≡ cos(φ α − φ β ) , is also evaluated with Q vectors as δ = Q 1,α Q * 1,β when particles α and β are chosen from different detector phase-space regions or have opposite signs, or otherwise, and the respective event weight is the denominator of Eq. (13). The effect of the nonuniform detector acceptance is corrected by evaluating the cumulants of Q-vector products [36]. While the correction is found to be negligible for the γ 112 and δ correlators, there is a sizable effect of 5-10% correction to the γ 123 correlator.

B. Event shape engineering
In the ESE analysis, within each multiplicity range of pPb or centrality range of PbPb data, events are divided into different q 2 classes, where q 2 is defined as the magnitude of the Q 2 vector. In this analysis, the q 2 value is calculated from one side of the HF region within the range 3 < η < 5 for both pPb and PbPb collisions (weighted by the tower E T ), where in pPb collisions only the Pb-going side of HF is used because of the poor resolution from a relatively low charged-particle multiplicity on the proton-going side. In each q 2 class, the v 2 harmonic is measured with the scalar product method using a common resolution term (v 2,c ) as in the γ 112 correlator. Therefore, the v 2 from the tracker region can be expressed in terms of the Q-vectors as where particles from the HF are selected from the same region as particle c in the γ 112 correlator. In PbPb collisions, the particle c in the γ 112 correlator is taken from the HF detector that is at the opposite η side to the one used to calculate q 2 . However, the results are in good agreement with those where the particle c for γ 112 and q 2 is measured from the same side of the HF detector, which can be found in Appendix B. In pPb collisions, the particle c in the γ 112 correlator with respect to the Pb-and p-going sides is studied, when q 2 is measured only in the Pb-going side. The results are found to be independent of the side in which the particle c is detected. In Fig. 1, the HF q 2 distributions are shown for PbPb and pPb collisions in the multiplicity range 185 N offline trk < 250, where most of the high-multiplicity pPb events were recorded by the high-multiplicity trigger in this range. As indicated by the vertical dashed lines, the distribution is divided into several intervals with each corresponding to a fraction of the full distribution, where 0-1% represents the highest q 2 class. For each q 2 class, the three-particle γ 112 is calculated with the default kinematic regions for particles α, β, and c, and the v 2 harmonics from the tracker (|η| < 2.4) are also obtained by the scalar-product method [37]. The pPb and PbPb results are presented in Sec. V for both SS and OS pairs, as well as the differences found for the two charge combinations. In Fig. 2, the v 2 values for tracker particles as a function of the average q 2 in each HF q 2 class are shown. A proportionality close to linear is seen, indicating the two quantities are strongly correlated because of the initial-state geometry [38].

C. Systematic uncertainties
The absolute systematic uncertainties of the two-particle correlator δ, and three-particle correlators γ 112 and γ 123 , have been studied. Varying the d z /σ (d z ) and d T /σ (d T ) from less than 3 (default) to less than 2 and 5, and the σ (p T )/p T < 10% (default) to σ (p T )/p T < 5%, together yield the systematic uncertainties of ±1.0 × 10 −5 for the γ 112 , ±4.0 × 10 −5 for the γ 123 , and ±1.0 × 10 −4 for the δ correlator. The longitudinal primary vertex position (V z ) has been varied, using ranges |V z | < 3 cm and 3 < |V z | < 15 cm, where the differences with respect to the default range |V z | < 15 cm are ±1.0 × 10 −5 for the γ 112 , ±3.0 × 10 −5 for the γ 123 , and ±1.0 × 10 −4 for the δ correlator, taken as the systematic uncertainty. In the pPb collisions only, using the lower threshold of the high-multiplicity trigger with respect to the default trigger, a systematic uncertainty of ±3.0 × 10 −5 are yielded for all three correlators, which accounts for the possible trigger bias from the inefficiency of the default trigger around the threshold. In the pPb data sample, the average pileup can be as high as 0.25 and therefore the systematic effects from pileup have been evaluated. The full sample has been split into four different sets of events with different average pileup, according to their instantaneous luminosity during each run. The systematic TABLE I. Summary of systematic uncertainties in SS and OS three-particle correlators γ 112 and γ 123 , and two-particle correlator δ in pPb collisions at √ s NN = 8.16 TeV and PbPb collisions at 5.02 TeV.
A final test of the analysis procedures is done by comparing "known" charge-dependent signals based on the EPOS event generator [39] to those found after events are passed through a GEANT4 [40,41] simulation of the CMS detector response. Based on this test, a systematic uncertainty of ±2.5 × 10 −5 is assigned for the γ 112 , ±4.0 × 10 −5 for the γ 123 , and ±5.0 × 10 −4 for the δ correlators, by taking the difference in the correlators between the reconstructed and the generated level. Note that this uncertainty for the δ correlator is based on differential variables, where the uncertainty covers the maximum deviation from the closure test. For results that averaged over | η| < 1.6, the systematic uncertainty is found to be ±2.0 × 10 −4 when directly evaluating the average. The tracking efficiency and acceptance of positively and negatively charged particles have been evaluated separately, and the difference has been found to be negligible. All sources of systematic uncertainty are uncorrelated and added in quadrature to obtain the total absolute systematic uncertainty. No dependence of the systematic uncertainties on the sign combination, multiplicity, η, p T , or average-p T is found. The systematic uncertainties in our results are point-to-point correlated. In pPb collisions, the systematic uncertainty is also observed to be independent of particle c pointing to the Pb-or p-going direction, and thus it is quoted to be the same for these two situations. The systematic uncertainties are summarized in Table I.

A. Charge-dependent two-and three-particle correlators
Measurements of the charge-dependent three-particle (γ 112 , γ 123 ) and two-particle (δ) correlators are shown in Fig. 3 as functions of the pseudorapidity difference (| η| ≡ |η α − η β |) between SS and OS particles α and β, in the multiplicity range 185 N offline trk < 250 for pPb collisions at √ s NN = 8.16 TeV and PbPb collisions at 5.02 TeV. The SS and OS of δ correlators are shown with different markers to differentiate the twoparticle correlation from the three-particle correlation with a particle c in the forward rapidity. The pPb data are obtained with particle c in the Pb-and p-going sides separately. The The SS and OS three-particle correlators, γ 112 (upper) and γ 123 (middle), and two-particle correlator, δ (lower), as a function of | η| for 185 N offline multiplicity range 185 N offline trk < 250 for PbPb data roughly corresponds to the centrality range 60-65%.
Similar to the observation reported in Ref. [21], the threeparticle γ 112 [Figs. 3(a) and 3(b)] and γ 123 [Figs. 3(c) and 3(d)] correlators show a charge dependence for | η| up to about 1.6, in both pPb (5.02 [21] and 8.16 TeV) and PbPb (5.02 TeV) systems. Little collision energy dependence of the γ 112 data for pPb collisions is found from √ s NN = 5.02 TeV to 8.16 TeV within uncertainties (as will be shown later in Figs. 6 and 8 as a function of event multiplicity). For | η| > 1.6, the SS and OS correlators converge to a common value, which is weakly dependent on | η| out to about 4.8 units. In pPb collisions, the γ 112 correlator obtained with particle c from the p-going side is shifted toward more positive values than that from the Pb-going side by approximately the same amount for both the SS and OS pairs. This trend is reversed for the higher-order harmonic γ 123 correlator, where the Pb-going side data are more positive than the p-going side data. The Pb-going side results for the γ 112 correlator for the pPb collisions are of similar magnitude as the results for PbPb collisions, although a more pronounced peak structure at small | η| is observed in pPb collisions. The common shift of SS and OS correlators between the pand Pb-going side reference (c) particle may be related to sources of correlation that are charge independent, such as directed flow [the first-order azimuthal anisotropy in Eq. (1) FIG. 4. The SS and OS three-particle correlators, γ 112 (upper) and γ 123 (middle), and two-particle correlator, δ (lower), as a function of | p T | for 185 N offline momentum conservation effect, the latter being sensitive to the difference in multiplicity between pand Pb-going directions. The two-particle δ correlators [Figs. 3(e) and 3(f)] for both SS and OS pairs also show a decreasing trend as | η| increases and converge to the same values at | η| ≈ 1.6, similar to that for the three-particle correlators. The values of both OS and SS δ correlators are found to be larger in pPb than in PbPb collisions at similar multiplicities. As the δ correlator is sensitive to short-range jetlike correlations, reflected by the low-| η| region, this effect may be related to the higher-p T jets or clusters in pPb compared to PbPb collisions at similar multiplicities, as suggested in Ref. [28], because of short-range two-particle η-φ correlations.
To provide more detailed information on the particle p T dependence of the correlations, the γ 112 , γ 123 , and δ correlators are measured as functions of the p T difference (| p T | ≡ |p T,α − p T,β |) and average (p T ≡ (p T,α + p T,β )/2) of the SS and OS pairs in pPb and PbPb collisions, and shown in Figs. 4 and 5. The | p T |and p T -dependent results are averaged over the full |η| < 2.4 range. In particular, the charge-dependent correlations from the CME are expected to be strongest in the low-p T region [6].
For all correlators, similar behaviors between pPb and PbPb data are again observed. The trends in | p T | for γ 112 and γ 123 correlators seem to be opposite. The γ 112 correlator increases 044912-7 as a function of | p T |, while a decreasing trend is seen for the γ 123 correlator up to | p T | ≈ 2 GeV, where γ 123 becomes constant in | p T |. The opposite behavior observed between the γ 112 and γ 123 correlators is related to back-to-back jetlike correlations, which give a positive (negative) contribution to even-(odd-)order Fourier harmonics [42]. The δ correlators decrease monotonically as functions of | p T | for both SS and OS pairs in pPb and PbPb collisions. This trend of decreasing for δ is consistent with the expectation from either transverse momentum conservation or back-to-back jet correlations [19].
In terms of the p T dependence in Fig. 5, all three correlators for both SS and OS pairs show very similar behaviors in the low-p T region, which is likely a consequence of the same physical origin. However, an opposite trend starts emerging at p T ≈ 1.6 GeV, most evidently for γ 112 and δ. Within the 0.3 < p T < 3 GeV range, as p T increases toward 3 GeV, both particles of a pair tend to be selected with a high-p T value, while for low-p T or any | p T | values, the pair usually consists of at least one low-p T particle. This may be the reason for a different trend seen at high p T . The qualitative behavior of the data is captured by a multiphase transport model [43,44]. In Appendix C, all three correlators as functions of | η|, p T , and p T in different multiplicity and centrality ranges in pPb and PbPb collisions can be found. To explore the multiplicity or centrality dependence of the three-and two-particle correlators, an average of the data is taken over | η| < 1.6, corresponding to the region in Fig. 3 which exhibits charge dependence. The average over | η| < 1.6 is weighted by the density of particle pairs in | η|, and all further plots averaged over | η| < 1.6 are weighted similarly. The resulting | η|-averaged data of γ 112 , γ 123 , and δ are shown in Fig. 6 for both OS and SS pairs, as functions of N offline trk for pPb collisions at √ s NN = 8.16 TeV (particle c from the Pb-going side) and PbPb collisions at 5.02 TeV. Previously published pPb data at 5.02 TeV are also shown for comparison [21]. The centrality scale on the top of Fig. 6  results are measured in the same N offline trk ranges. The new pPb data at 8.16 TeV extend the multiplicity reach further than the previously published pPb data at 5.02 TeV (which stopped at N offline trk ≈ 300). Within the uncertainties, the SS and OS γ 112 correlators in pPb and PbPb collisions exhibit the same magnitude and trend as functions of event multiplicity. The pPb data are independent of collision energy from 5.02 to 8.16 TeV at similar multiplicities. This justifies the comparison of new pPb data and PbPb data at somewhat different energies. For both pPb and PbPb collisions, the OS correlator reaches a value close to zero for N offline trk > 200, while the SS correlator remains negative, but the magnitude gradually decreases as N offline trk increases. Part of the observed multiplicity (or centrality) dependence is understood as a dilution effect that falls with the inverse of event multiplicity [12]. The notably similar magnitude and multiplicity dependence of the three-particle correlator γ 112 observed in pPb collisions relative to that in PbPb collisions again indicates that the dominant contribution of the signal is not related to the CME. The results of SS and OS three-particle correlators as functions of centrality in PbPb collisions at √ s NN = 5.02 TeV are also found to be consistent with the results from lower energy AA collisions [12,16]. However, values of γ 123 correlators between pPb and PbPb are observed to be different, unlike those for γ 112 correlators. As the CME contribution to γ 123 is not expected, the data suggest different properties of backgrounds in pPb and PbPb systems. If the γ 112 correlator in pPb data is expected to be background dominated, as argued earlier, the similarity found to the PbPb data in γ 112 requires further understanding. The two-particle δ correlators show a similar trend in multiplicity between pPb and PbPb systems, but a larger splitting between OS and SS pairs is observed in pPb than in PbPb data.
To eliminate sources of correlations that are charge independent (e.g., directed flow, v 1 ) and to explore a possible charge separation effect generated by the CME or charge-dependent background correlations, the differences of three-particle correlators γ 112 and γ 123 and two-particle correlator δ between OS and SS are shown in Fig. 7 as functions of | η|, | p T |, and p T in the multiplicity range 185 N offline trk < 250 for pPb collisions at √ s NN = 8.16 TeV and PbPb collisions at 5.02 TeV. After taking the difference, the three-particle correlators γ 112 and γ 123 in pPb collisions with particle c from either the por Pb-going side and in PbPb collisions show nearly identical values, except in the high p T region. Note that for OS and SS correlators separately, this similarity between pPb and PbPb is only observed for the γ 112 correlator. As a function of  trk offline N δ Δ FIG. 8. The difference of the OS and SS three-particle correlators γ 112 (upper) and γ 123 (middle) and two-particle correlator δ (lower) averaged over | η| < 1.6 as a function of N offline trk in pPb collisions at √ s NN = 8.16 TeV and PbPb collisions at 5.02 TeV. The pPb results are obtained with particle c from Pb-and p-going sides separately. The δ correlator is denoted by a different marker for pPb collisions. The results of γ 112 for pPb collisions at 5.02 TeV from the CMS Collaboration [21] are also shown for comparison. Statistical and systematic uncertainties are indicated by the error bars and shaded regions, respectively.
| η|, the charge-dependent difference is largest at | η| ≈ 0 and drops to zero for | η| > 1.6 for both systems. The striking similarity in the observed charge-dependent azimuthal correlations between pPb and PbPb as functions of | η|, | p T |, and p T strongly suggests a common physical origin. As argued in Ref. [21], a strong charge separation signal from the CME is not expected in a very high-multiplicity pPb collisions, and not with respect to 3 (for the γ 123 correlator) in either the pPb or PbPb system. The similarity seen between high-multiplicity pPb and peripheral PbPb collisions for both γ 112 and γ 123 further challenges the attribution of the observed charge-dependent correlations to the CME. The two-particle correlator δ, on the other hand, is found to show a larger value in pPb than in PbPb collisions. The differences of three-particle correlators γ 112 and γ 123 and two-particle correlator δ between OS and SS are shown in Fig. 8 [21]. Similar to those shown in Fig. 7, the observed difference between OS and SS pairs in γ 112 and γ 123 is strikingly similar in pPb and PbPb collisions over the entire overlapping multiplicity range (and also independent of collision energy for γ 112 in pPb), while higher values of an OS-SS difference in δ are found for the pPb system.
To check if the mechanism of local charge conservation coupled with anisotropic flow can explain the observed charge dependence of the γ 112 and γ 123 correlators, the relation in Eq. (6) is used. The ratios of γ 112 and γ 123 to the product of δ and v n are shown in Fig. 9, averaged over | η| < 1.6, as functions of event multiplicity in pPb and PbPb collisions.
The v 2 and v 3 values for particles α or β are calculated with the scalar-product method with respect to the particle c. In pPb collisions, only results with the Pb-going direction are shown because the p-going direction data lack statistical 044912-10 precision, except for the multiplicity range 185 N offline trk < 250. The ratios shown in Fig. 9 for both systems are found to be similar between n = 2 and n = 3, on average with values slightly less than 2. This observation indicates that the measured charge dependence of three-particle correlators is consistent with mostly being dominated by charge-dependent two-particle correlations (e.g., from local charge conservation) coupled with the anisotropic flow v n . For a given n value, the ratios are also similar between pPb and PbPb collisions (and may reflect similar particle kinematics and acceptances), and approximately constant as functions of event multiplicity. Notably, the δ in Fig. 8 are different between the pPb and PbPb systems. However, the anisotropic flow harmonics v n are larger for PbPb collisions than for pPb collisions [28]. As a result, the product of δ and v n leads to similar values of γ 112 and γ 123 correlators between the pPb and PbPb systems, implying the κ 2 is similar to κ 3 .
The ratios of γ 112 and γ 123 to the product of δ and v n can also be studied as functions of | η|, p T , and p T in pPb and PbPb collisions, as shown in Fig. 10 for the multiplicity range of 185 N offline trk < 250. Here, the v n are calculated as the average v n of particles α and β, v n = (v n,α + v n,β )/2 [based on the relation derived in Eq. (A5) in Appendix A], and are weighted by the number of pairs of particles α and β in the given kinematic ranges when averaged over η or p T . The ratios involving γ 112 and γ 123 are again found to be similar differentially for all three variables in both pPb and PbPb collisions. This observation further supports a common origin of γ 112 and γ 123 from charge-dependent two-particle correlations coupled with the anisotropic flow.

B. Event shape engineering
To explore directly the background scenario in Eq. (3) in terms of a linear dependence on v 2 for the γ 112 correlator, results based on the ESE analysis are presented in this section.
The SS and OS three-particle correlators γ 112 averaged over | η| < 1.6, are shown as a function of v 2 (evaluated as the average v 2 value for each corresponding q 2 event class in Fig. 11) for the multiplicity range 185 N offline trk < 250 in pPb collisions at √ s NN = 8.16 TeV (upper) and PbPb collisions at 5.02 TeV (lower). The pPb results are obtained with particle c from the Pb-and p-going sides separately. Both SS and OS γ 112 correlators in both pPb (both beam directions for particle c) and PbPb collisions show a dependence on v 2 . A clear linear dependence on the v 2 value is not seen for any of the SS and OS correlators studied.
Similar to the analysis in Sec. V A, the difference between OS and SS correlators is taken in order to eliminate the chargeindependent sources of the correlators. The results, averaged over | η| < 1.6, are shown in Fig. 12 (upper), as a function of v 2 evaluated in each q 2 class, for the multiplicity range  11. The SS and OS three-particle correlators γ 112 averaged over | η| < 1.6 as a function of v 2 (evaluated as the average v 2 value for each corresponding q 2 event class) for the multiplicity range 185 N offline trk < 250 in pPb collisions at √ s NN = 8.16 TeV (upper) and PbPb collisions at 5.02 TeV (lower). The pPb results are obtained with particle c from Pb-and p-going sides separately. Statistical and systematic uncertainties are indicated by the error bars and shaded regions, respectively. data, where the first term corresponds to the v 2 -dependent background contribution with the slope parameter a equal to κ 2 δ [from Eq. (3)], which is assumed to be v 2 independent. The intercept parameter b denotes the v 2 -independent contribution (when linearly extrapolating to v 2 = 0) in the γ 112 correlator. In particular, as the CME contribution to the γ 112 is expected to be largely v 2 independent within narrow multiplicity (centrality) ranges, the b parameter may provide an indication to a possible observation of the CME, or set an upper limit on the CME contribution. As shown in Fig. 12, for both pPb and PbPb collisions in each multiplicity or centrality range, a clear linear dependence of the γ 112 correlator as a function of v 2 is observed. Fitted by a linear function, the intercept parameter b can be extracted. A one standard deviation uncertainty band is also shown for the linear fit. Taking the statistical uncertainties into account, the values of b are found to be nonzero for multiplicity range 185 N offline trk < 250 in pPb and 60-70% centrality in PbPb collisions. Observing a nonzero intercept b from Fig. 12 may or may not lead to a conclusion of a finite CME signal, as an assumption is made for the background contribution term, namely that δ is independent of v 2 . To check this assumption explicitly, the δ correlator is shown in Fig. 13 as a function of v 2 in different multiplicity and centrality ranges in pPb (upper) and PbPb (lower) collisions. It is observed that the value of δ remains largely constant as a function of v 2 in low-or intermediate-q 2 classes, but starts rising as v 2 increases in high-q 2 classes. The multiplicity, within a centrality or multiplicity range, decreases slightly with increasing q 2 , which qualitatively could contribute to the rising δ due to a multiplicity dilution effect. However, this is only found to be true for PbPb collisions, but not for pPb collisions. The other reason may be related to larger jetlike correlations selected by requiring large q 2 values. Events with higher multiplicities show a weaker dependence on v 2 than those with lower multiplicities, which is consistent with  13. The difference of the OS and SS two-particle correlators δ averaged over | η| < 1.6 as a function of v 2 evaluated in each q 2 class, for different multiplicity ranges in pPb collisions at √ s NN = 8.16 TeV (upper), and for different centrality classes in PbPb collisions at 5.02 TeV (lower). Statistical and systematic uncertainties are indicated by the error bars and shaded regions, respectively. the expectation that short-range jetlike correlations are stronger in peripheral events. Because of the possible bias towards larger jetlike correlations at higher q 2 from the ESE technique, the v 2 dependence of δ is hard to completely eliminate. This presents a challenge to the interpretation of the intercept values from the linear fits in Fig. 12.
In order to avoid the issue of δ being dependent on v 2 , the ratio γ 112 / δ as a function of v 2 is shown in Fig. 14 for different multiplicity ranges in pPb collisions at √ s NN = 8.16 TeV (upper) and for different centrality classes in PbPb collisions at 5.02 TeV (lower). Particularly in the scenario of a pure v 2 -dependent background, the ratio γ 112 / δ is expected to be proportional to v 2 . A linear function is fitted again using Here, comparing to the intercept parameter b in Eq. (15), the b norm parameter is equivalent to b scaled by the δ factor. The FIG. 14. The ratio between the difference of the OS and SS threeparticle correlators and the difference of OS and SS in δ correlators, γ 112 / δ, averaged over | η| < 1.6 as a function of v 2 evaluated in each q 2 class, for different multiplicity ranges in pPb collisions at √ s NN = 8.16 TeV (upper), and for different centrality classes in PbPb collisions at 5.02 TeV (lower). Statistical and systematic uncertainties are indicated by the error bars and shaded regions, respectively. A one standard deviation uncertainty from the fit is also shown.
fitted linear slope and intercept parameters, a norm and b norm , are summarized in Tables II and III   uncertainty is shown, which correspond to a 68% and 95% confidence level (C.L.), respectively. Within statistical and systematic uncertainties, no significant positive value for b norm is observed for most multiplicities in pPb or centralities in PbPb collisions. For multiplicity ranges 120 N offline trk < 150 and 150 N offline trk < 185 in pPb collisions, an indication of positive values with significances of more than two standard deviations is seen. However, results in these multiplicity ranges are likely to be highly sensitive to the very limited v 2 coverage using the ESE technique, as shown in the upper panel of Fig. 14. Overall, the result suggests that the v 2 -independent contribution to the γ 112 correlator is consistent with zero, and correlation data are consistent with the background-only scenario of charge-dependent two-particle correlations plus an anisotropic flow v n . This conclusion is consistent with that drawn from the study of higher-order harmonic three-particle correlators discussed earlier.
Based on the assumption of a non-negative CME signal, the upper limit of the v 2 -independent fraction in the γ 112 correlator is obtained from the Feldman-Cousins approach [45] with the measured statistical and systematic uncertainties. In Fig. 15 (lower), the upper limit of the fraction f norm , where f norm is the ratio of the b norm value to the value of γ 112 / δ , is presented at 95% C.L. as a function of event multiplicity. The v 2 -independent component of the γ 112 correlator is less than 8-15% for most of the multiplicity or centrality range. The combined limits from all presented multiplicities and centralities are also shown in pPb and PbPb collisions. An upper limit on the v 2 -independent fraction of the three-particle correlator, or possibly the CME signal contribution, is estimated to be 13% in pPb and 7% in PbPb collisions, at 95% C.L. Note that the conclusion here is based on the assumption of a CME signal independent of v 2 in a narrow multiplicity or centrality range. As pointed out in a study by the ALICE collaboration after this paper was submitted [46], the observed CME signal may be reduced as v 2 decreases for small v 2 values (e.g., <6%), due to a weaker correlation between magnetic field and event-plane orientations as a result of initial-state fluctuations. Depending on specific models of initial-state fluctuations, the upper limits obtained in this paper may increase relatively by about 20%, although still well within a few % level. On the other hand, covering a wide range of v 2 values in this analysis (6-15%), the v 2 dependence of the observed CME signal is minimized to the largest extent, especially for more central events. The data also rule out any significant nonlinear v 2 dependence of the observed CME signal, as suggested by Ref. [46]. Therefore, the high-precision data presented in this paper indicate that the charge-dependent three-particle azimuthal correlations in pPb and PbPb collisions are consistent with a v 2 -dependent background-only scenario, posing a significant challenge to the search for the CME in heavy ion collisions using three-particle azimuthal correlations.

VI. SUMMARY
Charge-dependent azimuthal correlations of same-and opposite-sign (SS and OS) pairs with respect to the second-and third-order event planes have been studied in pPb collisions at √ s NN = 8.16 TeV and PbPb collisions at 5.02 TeV by the CMS experiment at the LHC. The correlations are extracted via three-particle correlators as functions of pseudorapidity difference, transverse momentum difference, and p T average of SS and OS particle pairs, in various multiplicity or centrality ranges of the collisions. The differences in correlations between OS and SS particles with respect to both second-and third-order event planes as functions of η and multiplicity are found to agree for pPb and PbPb collisions, indicating a common underlying mechanism for the two systems. Dividing the OS and SS difference of the three-particle correlator by the product of the v n harmonic of the corresponding order  and the difference of the two-particle correlator, the ratios are found to be similar for the second-and third-order event planes, and show a weak dependence on event multiplicity. These observations support a scenario in which the chargedependent three-particle correlator is predominantly a consequence of charge-dependent two-particle correlations coupled to an anisotropic flow signal.
To establish the relation between the three-particle correlator and anisotropic flow harmonic in detail, an event shape engineering technique is applied. A linear relation for the ratio of three-to two-particle correlator difference as a function of v 2 is observed, which extrapolates to an intercept that is consistent with zero within uncertainties for most of multiplicities. An upper limit on the v 2 -independent fraction of the three-particle correlator, or the possible CME signal contribution (assumed independent of v 2 within the same narrow multiplicity or centrality range), is estimated to be 13% for pPb data and 7% for PbPb data at a 95% confidence level. The data presented in this paper provide new stringent constraints on the nature of the background contribution to the charge-dependent azimuthal correlations, and establish a new baseline for the search for the chiral magnetic effect in heavy ion collisions.

ACKNOWLEDGMENTS
We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centers and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: BMWFW and FWF (Austria);   FIG. 20. The SS and OS three-particle correlators γ 112 (upper) and γ 123 (middle) and two-particle correlator δ (lower) as a function of | η| for four multiplicity ranges in pPb collisions at √ s NN = 8.16 TeV (left) and PbPb collisions at 5.02 TeV (right). The pPb results obtained with particle c in Pb-going (solid markers) and p-going (open markers) sides are shown separately. The SS and OS two-particle correlators are denoted by different markers for both pPb and PbPb collisions. Statistical and systematic uncertainties are indicated by the error bars and shaded regions, respectively.

APPENDIX A: GENERAL RELATION OF v n HARMONICS AND TWO-AND THREE-PARTICLE AZIMUTHAL CORRELATIONS
In Sec. I, Eq. (5) can be derived in a way similar to Eq. (3), with details which can be found in Ref. [24]. Here, a general derivation of Eq. (5) for all higher-order-harmonic correlators is given.
Similar to Eq. (40) in Ref. [24], the general relation between the nth order anisotropy harmonic v n and the three-particle correlator with respect to the nth order event plane can be 044912-17 FIG. 21. The SS and OS three-particle correlators γ 112 (upper) and γ 123 (middle) and two-particle correlator δ (lower) as a function of | p T | for four multiplicity ranges in pPb collisions at √ s NN = 8.16 TeV (left) and PbPb collisions at 5.02 TeV (right) collisions. The pPb results obtained with particle c in Pb-going (solid markers) and p-going (open markers) sides are shown separately. The SS and OS two-particle correlators are denoted by different markers for both pPb and PbPb collisions. Statistical and systematic uncertainties are indicated by the error bars and shaded regions, respectively.
derived starting from γ 1,n−1;n ≡ cos[φ α +(n−1)φ β −n n ] where x denotes (p T ,η) and dx = p T dp T dη. ρ 2 is the twoparticle pair density distribution, which can be expressed 044912-18 The SS and OS three-particle correlators γ 112 (upper) and γ 123 (middle) and two-particle correlator δ (lower) as a function of p T for four multiplicity ranges in pPb collisions at √ s NN = 8.16 TeV (left) and PbPb collisions at √ s NN = 5.02 TeV (right). The pPb results obtained with particle c in Pb-going (solid markers) and p-going (open markers) sides are shown separately. The SS and OS two-particle correlators are denoted by different markers for both pPb and PbPb collisions. Statistical and systematic uncertainties are indicated by the error bars and shaded regions, respectively.
in terms of the single-particle density distribution and its underlying two-particle correlation function (see Sec. 2 in Ref. [24]), In the presence of collective anisotropy flow, the singleparticle azimuthal distribution can be expressed in terms of a Fourier series with respect to the event plane of the corresponding order, where ρ 0 (x) depends on p T and η only. The SS and OS three-particle correlators γ 112 (upper) and γ 123 (middle) and two-particle correlator δ (lower), as a function of | η| for five centrality classes in PbPb collisions at 5.02 TeV. The SS and OS two-particle correlators are denoted by different markers. Statistical and systematic uncertainties are indicated by the error bars and shaded regions, respectively.
The two-particle correlation function C describes intrinsic correlations that are insensitive to the event plane n , but only involve azimuthal angle difference φ = φ α − φ β . It can be also expanded in Fourier series [24], C( φ,x α ,x β ) = ∞ n=1 a n (x α ,x β ) cos (n φ), where a n (x α ,x β ) is the two-particle Fourier coefficient. By definition, a 1 (x α ,x β ) is equal to the two-particle correlator δ(x α ,x β ), introduced in Sec. I, as a function of x α and x β (i.e., p T and η of both particles). Therefore, we substitute Eqs. (A4) and (A2) into (A1) and obtain γ 1,n−1;n = 1 2N 2 ρ 0 (x α )ρ 0 (x β )a 1 (x α , The SS and OS three-particle correlators γ 112 (upper) and γ 123 (middle) and two-particle correlator δ (lower) as a function of | p T | for five centrality classes in PbPb collisions at √ s NN = 5.02 TeV. The SS and OS two-particle correlators are denoted by different markers. Statistical and systematic uncertainties are indicated by the error bars and shaded regions, respectively.
where N = ρ 0 (x)dx. This is the general equation explaining why a nonzero two-particle correlation δ(x α ,x β ) plus an anisotropy flow of nth order v n (x) contribute to the threeparticle correlator, γ 1,n−1;n . Therefore, this general form of γ 1,n−1;n can be applied to any order n and decomposed into the two-particle correlator δ and the nth order harmonic v n , where n = 2 and 3 are studied in detail in Sec. V A. The SS and OS three-particle correlators γ 112 (upper) and γ 123 (middle) and two-particle correlator δ (lower) as a function of p T for five centrality classes in PbPb collisions at √ s NN = 5.02 TeV. The SS and OS two-particle correlators are denoted by different markers.

044912-21
Statistical and systematic uncertainties are indicated by the error bars and shaded regions, respectively.

APPENDIX B: SUPPORTING RESULTS OF THE EVENT SHAPE ENGINEERING METHOD
As stated in Sec. IV B, the Q 2 vector is calculated using one side of the HF detector within the η range of 3-5 units. The default result in Sec. V B presents the γ 112 as a function of v 2 , where the particle c in the γ 112 correlator corresponds to the η range −5.0 to −4.4. However, the results are found to be independent of where the particle c is reconstructed, as it is shown in Fig. 16.
In Figs. 17 and 18, the denominators of Eq. (7), v 2,c , for different Q 2 classes with respect to HF+ and HF− in PbPb collisions at √ s NN = 5.02 TeV, and the Pb-going side of the HF in pPb collisions at 8.16 TeV, are shown as a function of v 2 in the tracker region. Here v 2,c is a measure of elliptic anisotropy of the transverse energy registered in the HF detectors without 044912-22 being corrected to the particle-level elliptic flow. It serves as the resolution correction factor when deriving the three-particle correlators or the v 2 values in the tracker region using the scalar-product method.
In Fig. 19, the average N offline trk is shown as a function of v 2 in different multiplicity and centrality ranges in pPb (upper) and PbPb collisions (lower), respectively. The average N offline trk is found to be weakly dependent on v 2 , but with a slight decreasing trend as v 2 increases. Similar to Fig. 13, the effect at low multiplicities is stronger than that at high multiplicities. Overall, this effect is negligible for the results shown in Sec. V B.

APPENDIX C: THREE-AND TWO-PARTICLE CORRELATOR AS FUNCTIONS OF DIFFERENTIAL VARIABLES IN DIFFERENT MULTIPLICITY AND CENTRALITY CLASSES
The figures in Appendix C show the γ 112 , γ 123 , and δ correlators as a function of | η|, | p T |, and p T in pPb collisions at √ s NN = 8.16 TeV and PbPb collisions at 5.02 TeV. In pPb and PbPb collisions, the results are shown for multiplicity ranges N offline trk = [120,150), [150,185), [185,250),and [250, In PbPb collisions, the results are also shown for five centrality classes from