Search for the Chiral Magnetic Effect in Relativistic Heavy-Ion Collisions

Search for the Chiral Magnetic Effect in Relativistic Heavy-Ion Collisions

Jie Zhao
Abstract

Relativistic heavy-ion collisions provide an ideal environment to study the emergent phenomena in quantum chromodynamics (QCD). The chiral magnetic effect (CME) is one of the most interesting, arising from the topological charge fluctuations of QCD vacua, immersed in a strong magnetic field. Since the first measurement nearly a decade ago of the possibly CME-induced charge correlation, extensive studies have been devoted to background contributions to those measurements. Many new ideas and techniques have been developed to reduce or eliminate the backgrounds. This article reviews these developments and the overall progress in the search for the CME.

Keywords: Chiral magnetic effect; topological charge; heavy-ion collisions; QGP; QCD

Received Day Month Year

Revised Day Month Year

PACS numbers: 25.75.-q, 25.75.Gz, 25.75.Ld, 25.75.Ag

1 Introduction

Quark interactions with topological gluon configurations can induce chirality imbalance and local parity violation in quantum chromodynamics (QCD). In relativistic heavy-ion collisions, this can lead to observable electric charge separation along the direction of the strong magnetic field produced by spectator protons. This phenomenon is called the chiral magnetic effect (CME). An observation of the CME-induced charge separation would confirm several fundamental properties of QCD, namely, approximate chiral symmetry restoration, topological charge fluctuations, and local parity violation. Extensive theoretical efforts have been devoted to characterize the CME, and intensive experimental efforts have been invested to search for the CME in heavy-ion collisions at BNL’s Relativistic Heavy Ion Collider (RHIC) and CERN’s Large Hadron Collider (LHC).

Transitions between gluonic configurations of QCD vacua can be described by instantons/sphelarons and characterized by the Chern-Simons topological charge number. Quark interactions with such topological gluonic configurations can change their chirality, leading to an imbalance in left- and right-handed quarks (nonzero axial chemical potential ); , is the number of light quark flavors and is the topological charge of the gluonic configuration. Thus, gluonic field configurations with nonzero topological charges induce local parity violation. It was suggested that in relativistic heavy-ion collisions, where the deconfinement phase transition and an extremely strong magnetic field are present, The chirality imbalance of quarks in the local metastable domains will generate an electromagnetic current, , along the direction of the magnetic field. Quarks hadronize into charged hadrons, leading to an experimentally observable charge separation. The measurements of this charge separation provide a means to study the non-trivial QCD topological structures in relativistic heavy-ion collisions.

In heavy-ion collisions, particle azimuthal angle distribution in momentum space is often described with a Fourier decomposition:

(1)

where , and is the reaction-plane direction, defined to be the direction of the impact parameter vector and expected to be perpendicular to the magnetic field direction on average. The parameters and account for the directed flow and elliptic flow. The parameters can be used to describe the charge separation effects. Usually only the first harmonic coefficient is considered. Positively and negatively charged particles have opposite values, , and are proportional to . However, they average to zero because of the random topological charge fluctuations from event to event, making a direct observation of this parity violation effect impossible. Indeed, the measured of both positive and negative charges are less than at the 95% confidence level in Au+Au collisions at = 200 GeV. The observation of this parity violation effect is possible only via correlations, e.g. measuring with the average taken over all events in a given event sample. The correlator is designed for this propose:

(2)

and are the reaction plane dependent backgrounds in in-plane and out-plane directions, which are assumed to largely cancel out in their difference, while there are still residual background contributions (e.g. momentum conservation effect ). At mid-rapidity, the is averaged to zero, and the contribution () is expected to be small. Moreover, the background is expected to be charge independent. By taking the opposite-sign (OS) and same-sign (SS) difference those charge independent backgrounds can be further cancelled out. Thus, usually the correlator is used:

(3)

where OS and SS describe the charge sign combinations between the and particle.

The correlator can be calculated by the three-particle correlation method without an explicit determination of the reaction plane; instead, the role of the reaction plane is played by the third particle, . Under the assumption that particle is correlated with particles and only via common correlation to the reaction plane, we have:

(4)

where is the elliptic flow parameter of the particle , and , and are the azimuthal angles of particle , and , respectively.

2 Challenges and Strategies

A significant has indeed been observed in heavy-ion collisions at RHIC and LHC. The first measurement was made by the STAR collaboration at RHIC in 2009 . Fig. 1 shows their correlator as a function of the collision centrality in Au+Au and Cu+Cu collisions at = 200 GeV. Charge dependent signal of the same-sign and opposite-sign charge correlators have been observed. Similarly, Fig. 2 shows the and correlator as a function of the collision centrality in Au+Au collisions at = 7.7-200 GeV from STAR and in Pb+Pb collisions at 2.76 TeV from ALICE. At high collision energies, charge dependent signals are observed, and is larger than . The difference between and , , decreases with increasing centrality, which would be consistent with expectation of the magnetic field strength to decrease with increasing centrality. At the low collision energy of =7.7 GeV, the difference between the and disappears, which could be consistent with the disappearance of the CME in the hadronic dominant stage at this energy. Thus, these results are qualitatively consistent with the CME expectation.

Fig. 1: (Color online) correlator in Au+Au and Cu+Cu collisions at = 200 GeV. Shaded bands represent uncertainty from the measurement of . The thick solid (Au+Au) and dashed (Cu+Cu) lines represent HIJING calculations of the contributions from three-particle correlations. Collision centrality increases from left to right.
Fig. 2: (Color online) correlator as a function of centrality for Au+Au collisions at 7.7-200 GeV, and for Pb+Pb collisions at 2.76 TeV. Grey curves are the charge independent results from MEVSIM calculations.

There are, however, mundane physics that could generate the same effect as the CME in the variable, which contribute to the background in the measurements. An example is the resonance or cluster decay (coupled with ) background; the variable is ambiguous between a back-to-back OS pair from the CME perpendicular to and an OS pair from a resonance decay along . Calculations with local charge conservation and momentum conservation effects can almost fully account for the measured signal at RHIC. A Multi-Phase Transport (AMPT) model simulations can also largely account for the measured signal. In general, these backgrounds are generated by two particle correlations coupled with elliptic flow ():

(5)

Thus, a two particle correlation of from resonance (cluster) decays, coupled with the of the resonance (cluster), will lead to a signal.

Experimentally, various proposals and attempts have been put forward to reduce or eliminate backgrounds, exploiting their dependences on and two particle correlations. (1) Using the event shape selection, by varying the event-by-event exploiting statistical (event-by-event methods) and dynamical fluctuations (event shape engineering method), it is expected that the independent contribution to the can be extracted. (2) Isobaric collisions and Uranium+Uranium collisions have been proposed to take advantage of the different nuclear properties (such as proton number, shape). (3) Control experiments of small system p+A or d+A collisions are used to study the background behavior, where backgrounds and possible CME signals are expected to be uncorrelated because the participant plane and the magnetic field direction are uncorrelated due to geometry fluctuations in these small system collisions. (4) A new idea of differential measurements with respect to reaction plane and participant plane are proposed, which takes advantage of the geometry fluctuation effects on the participant plane and the magnetic field direction in A+A collisions. (5) A new method exploiting the invariant mass dependence of the measurements is devised, which identifies and removes the resonance decay backgrounds, to enhance the sensitivity of CME measurement. (6) New correlator is designed to detect the CME-driven charge separation. In the following sections we will review these proposals and attempts in more detail.

3 Event-by-event selection methods

The main background sources of the measurements are from the elliptic flow () induced effects. These backgrounds are expected to be proportional to the . One possible way to eliminate or suppress these induced backgrounds is to select “spherical” events with exploiting the statistical and dynamical fluctuations of the event-by-event . Due to finite multiplicity fluctuations, one can easily vary the shape of the final particle momentum space, which is directly related to the backgrounds.

By using the event-by-event , STAR has carried out the first attempt to remove the backgrounds. Fig. 3 shows the charge multiplicity asymmetry correlator () as a function of the event-by-event . The event-by-event () can be measured by the vector method:

(6)

where sums over particles (used for the correlator) in each event; is the event plane (EP) azimuthal angle, reconstructed from final-state particles, as a proxy for participant plane () that is not experimentally accessible. To avoid self-correlation, particles used for the EP calculations are exclusive to the particles used for and correlator. The results show strong correlation between the correlator and the . By selecting the events with , the correlator is largely reduced. The correlator shows similar correlation with from the preliminary STAR data.

Fig. 3: (Color online) charge multiplicity asymmetry correlations () as a function of from Au+Au collisions at = 200 GeV.

A similar method selecting events with the (see Eq. 4) variable has been proposed recently. To suppress the related background, a tighter cut, , is proposed to extract signal. The cut is tighter because corresponds to a zero harmonic to any plane, while corresponds to zero harmonic with respect to the reconstructed EP in the event.

These methods assume the background to be linear in of the final-state particles. However, the background arises from the correlated pairs (resonance/cluster decay) coupled with the of the parent sources, not the final-state particles. In case of resonance decays: , where depends on the resonance decay kinematics, and is the of the resonances, not the decay particles’. It is difficult, if not at all impossible, to ensure the of all the background sources to be zero. Thus, it is challenging to completely remove flow background by using the event-by-event or methods.

4 Event shape engineering

Because of dynamical fluctuations of the event-by-event , one could possibly select events with different initial participant geometries (participant eccentricities) even with the same impact parameter. By restricting to a narrow centrality, while varying event-by-event , one is presumably still fixing the magnetic filed (mainly determined by the initial distribution of the spectator protons). This provides a way to decouple the magnetic field and the , and thus a possible way to disentangle background contributions from potential CME signals. This is usually called the event shape engineering (ESE) method.

In ESE, instead of selecting on , one use the flow vector to possibly access the initial participant geometry, selecting different event shapes by making use of the dynamical fluctuations of . The ESE method is performed based on the magnitude of the second-order reduced flow vector, , defined as:

(7)

where is the magnitude of the second order harmonic flow vector and M is the multiplicity. The sum runs over all particles/hits, is the azimuthal angle of the -th particle/hit, and is the weight (usually taken to be the of the particle or energy deposition of the hit).

Fig. 4: (Color online) Schematic comparison of the event-by-event selection (A) and the ESE (B) methods. Different boxes represent different phase spaces, usually displaced in . The green color for in (A) reflects that the is calculated with respect to the event-plane. The correlator is usually calculated from the correlation between the particle of interest (POI, here the POI refers to the and particles in Eq. 1) and the event-plane.

Figure 4 is a schematic comparison of the event-by-event selection and the ESE methods. Basically, the most important difference between these two groups of methods lies in which phase space to calculate the or variables for event selection. In the event-by-event selection methods, the same phase space of the particle of interest (POI) is used for event selections, thus these methods take advantage of statistical as well as dynamical fluctuations of the POI. In the ESE method, a different phase space is used (often displaced in ), so that the event selection is dominated by the dynamical fluctuations, because statistical fluctuations of POI and event selection are independent. The dynamical fluctuations stem out of the common origin of the initial participant geometries. Thus a zero should correspond to an average zero of the background sources of the POI. However, a zero is unlikely accessible directly from data, so extrapolation is often involved.

Fig. 5: (Color online) (Left) The distributions in multiplicity range in Pb+Pb collisions. Red dash line represents the selection used to divide the events into multiple classes. (Right) The correlation between and in p+Pb and Pb+Pb collisions based on the selections of the events.

Figure 5(left) shows the distribution in Pb+Pb collisions from CMS. The events of a narrow multiplicity bin are divided into several classes with each corresponding to a fraction of the full distribution, where 0-1% represents the events with the largest value, and 95-100% corresponds to the events with smallest value, and so on. Fig. 5(right) shows that the is closely proportional to , suggesting those two quantities are strongly correlated because of the common initial-state geometry. One could thus use the to select events with different , and study the dependence of the correlator. In a similar way, the correlator is also calculated in each class.

Fig. 6(upper left) shows correlator as a function of in different centralities in Pb+Pb collisions from ALICE. To compensate for the dilution effect, correlator was multiplied by the charged-particle density in a given centrality bin () in the lower left panel. The results show strong dependence on , and the scaled correlator falls approximately onto the same linear trend for different centralities. This is qualitatively consistent with the expectation from background effects, such as resonance decay coupled with . Therefore, the observed dependence on indicates a large background contribution to correlator.

By restricting to a given narrow centrality, the event shape selection is expected to be less affected by the magnetic field. The different dependences of the CME signal and background on () could possibly be used to disentangle the CME signal from background. Fig. 6(right) shows the dependence of the from Monte Carlo Glauber calculation. The CME signal is assumed to be proportional to , where and are the magnitude and azimuthal direction of the magnetic field. The calculation shows that the CME signal weakly depends on within each given centrality (Fig. 6 right panel) and approximately linear. To extract the contribution of the possible CME signal to the current measurements, a linear function is fit to the data:

(8)

Here accounts for a overall scale. is the normalised slope, reflecting the dependence on . In a pure background scenario, the correlator is linearly proportional to and the parameter is equal to unity, Eq. 4 is reduced to . On the other hand, a significant CME contribution would result in non-zero intercepts at = 0 of the linear functional fits shown in Fig. 6(top left).

Fig. 6: (Color online) (Left top) The correlator and (Left bottom) the charged-particle density scaled correlator as a function of for shape selected events with for various centrality classes. Error bars (shaded boxes) represent the statistical (systematic) uncertainties. (Right) The expected dependence of the CME signal on for various centrality classes from a MC-Glauber simulation. The solid lines depict linear fits based on the variation observed within each centrality interval.

In a naive two components model with signal and background, a measured observable () can be expressed as:

(9)

and are the values of the observable from signal and background, represents the fraction of signal contribution in the measurement. The from the fit to the measured data is thus a combination of CME signal slope () and the background slope ():

(10)

where represents the CME fraction to the correlator from the measurements, and is the slope parameter from the MC calculations in Fig. 6 right panel. Figure 7(left) shows the centrality dependence of from fits to data and to the signal expectations based on MC-Glauber, MC-KLN CGC and EKRT models. Fig. 7(right) presents the estimated from the three models. The extracted for central (0-10%) and peripheral (50-60%) collisions have currently large uncertainties. Combining the points from 10-50% neglecting a possible centrality dependence gives , and for the MC-Glauber, MC-KLN CGC and EKRT models inputs of , respectively. These results are consistent with zero CME fraction and correspond to upper limits on of 33%, 26% and 29%, respectively, at 95% confidence level for the 10-50% centrality interval.

Fig. 7: (Color online) (Left) Centrality dependence of the parameter from a linear fit to the correlator in Pb+Pb collisions from ALICE and from linear fits to the CME signal expectations from MC-Glauber, MC-KLN CGC and EKRT models. (Right) Centrality dependence of the CME fraction extracted from the slope parameter of fits to data and different models. Points from MC simulations are slightly shifted along the horizontal axis for better visibility. Only statistical uncertainties are shown.

The above analysis is model-dependent, relying on precise modeling of the magnetic field with a given centrality. The CMS collaboration took a different approach, cutting on very narrow centrality bins and assuming the magnetic field to be constant within each centrality bin. The background contribution to the correlator is approximated to be:

(11)

Here, represents the charge-dependent two-particle azimuthal correlator and is a constant parameter, independent of , but mainly determined by the kinematics and acceptance of particle detection. The , and are experimental measured observables. With event shape engineering to select event with different , the above Eq.4 can be tested. The charge-independent background sources are eliminated by taking the difference of the correlators () between same- and opposite-sign pairs. In the background scenario, the is expected to be:

(12)

A linear function was used to extract the -independent fraction of the correlator:

(13)

where could be possibly the contribution from CME signal.

Fig. 8: (Color online) The ratio between () and correlators, , averaged over as a function of evaluated in each class, for different multiplicity ranges in p+Pb (left) and Pb+Pb (right) collisions.
Fig. 9: (Color online) (Left) Extracted intercept parameter and (Right) corresponding upper limit of the fraction of -independent correlator component, averaged over , as a function of in p+Pb and Pb+Pb collisions from CMS.

Figure 8 shows the ratio of as function of for different multiplicity ranges in p+Pb (left) and Pb+Pb (right) collisions. The values of the intercept parameter are shown as a function of event multiplicity in Fig. 9(left). Within statistical and systematic uncertainties, no significant positive value for is observed. Result suggests that the -independent contribution to the correlator is consistent with zero, and the results are consistent with the background-only scenario of charge-dependent two-particle correlations. Based on the assumption of a nonnegative CME signal, the upper limit of the -independent fraction in the correlator is obtained from the Feldman-Cousins approach with the measured statistical and systematic uncertainties. Fig. 9(right) shows the upper limit of the fraction , the ratio of the value to the value of , at 95% CL as a function of event multiplicity. The -independent component of the 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 p+Pb and Pb+Pb collisions. An upper limit on the -independent fraction of the three-particle correlator, or possibly the CME signal contribution, is estimated to be 13% in p+Pb and 7% in Pb+Pb collisions, at 95% CL. The results are consistent with a -dependent background-only scenario, posing a significant challenge to the search for the CME in heavy ion collisions using three-particle azimuthal correlations.

Fig. 10: (Color online) The (left top), (left middle), (left bottom), averaged over as a function of in p+Pb and Pb+Pb collisions. The p+Pb results are obtained with particle from Pb- and p-going sides separately. The ratio of and to the product of and in p+Pb collisions for the Pb-going direction (right top) and Pb+Pb collisions (right bottom). Statistical and systematic uncertainties are indicated by the error bars and shaded regions, respectively.

The CME-driven charge separation are expected along the magnetic field direction normal to the reaction plane, estimated by the second-order event plane (). The third-order event plane () are expected to be weakly correlated with , thus the CME-driven charge separation effect with respect to is expected to be negligible. In light of -dependent background-only scenario, where background can be expressed as Eq 4. A similar correlator () with respect to third-order event plane () are constructed to study the background effects:

(14)

In the flow-dependent background-only scenario, the and mainly depend on particle kinematics and detector acceptance effects, and are expected to be similar, largely independent of harmonic event plane order. Fig. 10(left) shows the (), , correlator as a function of multiplicity in p+Pb and Pb+Pb collisions. Fig. 10(right) shows the ratio of the () and to the product of and . The results show that the ratio is similar for =2 and 3, and also similar between p+Pb and Pb+Pb collisions, indicating that the is similar to . These results are consistent with the flow-dependent background-only scenario.

The event shape selection provides a very useful tool to study the background behavior of the . All current experimental results at LHC suggest that the are strongly dependent on the and consistent with the flow-background only scenario. In summary, the independent contribution are estimated by different methods from STAR, ALICE and CMS, and current results indicate that a large contribution of the correlator is from the related background.

5 Isobaric collisions

The CME is related to the magnetic field while the background is produced by -induced correlations. In order to gauge differently the magnetic field relative to the , isobaric collisions and Uranium+Uranium collisions have been proposed. The isobaric collisions are proposed to study the two systems with similar but different magnetic field strength, such as and , which have the same mass number, but differ by charge (proton) number. Thus one would expected very similar at mid-rapidity in and collisions, but the magnetic field, proportional to the nuclei electric charge, could vary by 10%. If the measured is dominated by the CME-driven charge separation, then the variation of the magnetic field strength between and collision provides an ideal way to disentangle the signal of the chiral magnetic effect from related background, as the related backgrounds are expected to be very similar between these two systems.

To test the idea of the isobaric collisions, Monte Carlo Glauber calculations of the spatial eccentricity () and the magnetic field strength form and collisions have been carried out. The Woods-Saxon spatial distribution is used:

(15)

where is the normal nuclear density, is the charge radius of the nucleus, represent the surface diffuseness parameter. is the spherical harmonic. The parameter is almost identical for and : fm. fm and 5.020 fm are used for and , and are used for both the proton and nucleon densities. The deformity quadrupole parameter has large uncertainties; there are two contradicting sets of values from current knowledge, and (referred to as case 1) vis a vis and (referred to as case 2). These would yield less than 2% difference in , hence a less than 2% residual background, between  and  collisions in the 20-60% centrality range. In that centrality range, the mid-rapidity particle multiplicities are almost identical for  and  collisions at the same energy.

The magnetic field strengths in and collisions are calculated by using Lienard-Wiechert potentials alone with HIJING model taking into account the event-by-event azimuthal fluctuation of the magnetic field orientation. HIJING model with the above two sets (case 1 and 2) of Woods-Saxon densities are simulated. Fig. 11(a) shows the calculation of the event-averaged initial magnetic field squared with correction from the event-by-event azimuthal fluctuation of the magnetic field orientation,

(16)

for the two collision systems at 200 GeV. Fig. 11(b) shows that the relative difference in , defined as , between and collisions is approaching 15% (case 1) or 18% (case 2) for peripheral events, and reduces to about 13% (cases 1 and 2) for central events. Fig. 11(b) also shows the relative difference in the initial eccentricity (), obtained from the Monte-Carlo Glauber calculation. The relative difference in is practically zero for peripheral events, and goes above (below) 0 for the parameter set of case 1 (case 2) in central collisions. The relative difference in from  and  collisions is expected to closely follow that in eccentricity, indicating the -related backgrounds are almost the same (different within 2%) for and collisions in 20-60% centrality range.

Fig. 11: (Color online) (a) Event-averaged initial magnetic field squared at the center of mass of the overlapping region, with correction from event-by-event fluctuation of magnetic field azimuthal orientation, for and collisions at 200 GeV, and (b) their relative difference versus centrality. Also shown in (b) is the relative difference in the initial eccentricity. The solid (dashed) curves correspond to the parameter set of case 1 (case 2).

Based on the available experimental measurements in Au+Au collisions at 200 GeV and the calculated magnetic field strength and eccentricity difference between and collisions, expected signals from the isobar collisions are estimated:

(17)

where represents the scaled correlator ( account for the dilution effect). The is the related background fraction of the correlator. Fig. 12(left) shows the with events for each of the two collisions types, assuming of the comes from the related background, and compared with . Fig. 12(right) shows the magnitude and significance of the projected relative difference between and collisions as a function of the background level. With the given event statistics and assumed background level, the isobar collisions will give 5 significance.

Fig. 12: (Color online) (Left) Estimated relative difference in and in the initial eccentricity for and collisions at 200 GeV. (Right) Magnitude and significance of the relative difference in the CME signal between and , as a function of the background level.

The above estimates assume Woods-Saxon densities, identical for proton and neutron distributions. Using the energy density functional (EDF) method with the well-known SLy4 mean field  including pairing correlations (Hartree-Fock-Bogoliubov, HFB approach), assumed spherical, the ground-state density distributions for  and are calculated. The results are shown in Fig. 13(left). They show that protons in Zr are more concentrated in the core, while protons in Ru, 10% more than in Zr, are pushed more toward outer regions. The neutrons in Zr, four more than in Ru, are more concentrated in the core but also more populated on the nuclear skin. Fig. 13(right) shows the relative differences between and collisions as functions of centrality in and with respect to and from AMPT simulation with the densities calculated by the EDF method. Results suggest that with respect to , the relative difference in and are as large as 3%. With respect to , the difference in and becomes even larger (10%), and the difference in is only 0-15%. These studies suggest that the premise of isobaric sollisions for the CME search may not be as good as originally anticipated, and could provide additional important guidance to the experimental isobaric collision program.

Fig. 13: (Color online) (Left) Proton and neutron density distributions of the and nuclei, assumed spherical, calculated by the EDF method. (Right) Relative differences between and collisions as functions of centrality in and with respect to and from AMPT simulation with the densities from the left plot.

6 Uranium+Uranium collisions

Isobaric collisions produce different magnetic field but similar . One may produce on average different but same magnetic fields, this may be achieved by uranium+uranium collisions. Unlike the nearly spherical nuclei of gold (Au), uranium (U) nuclei have a highly ellipsoidal shape. By colliding two uranium nuclei, there would be various collision geometries, such as the tip-tip or body-body collisions. In very central collisions, due to the particular ellipsoidal shape of the uranium nuclei, the overlap region would still be ellipsoidal in the body-body U+U collisions. This ellipsoidal shape of the overlap region would generate a finite elliptic flow, giving rise to the background in the measurements. On the other hand, the magnetic field are expected to vanish in the overlap region in those central body-body collisions. Thus in general the magnetic field driven CME signal will vanish in these very central collisions. By comparing central Au+Au collisions of different configurations, it may be possible to disentangle CME and background correlations contributing to the experimental measured signal. In 2012 RHIC ran U+U collisions. Preliminary experimental results in central U+U have been compared with the results from central Au+Au. However, the geometry of the overlap region is much more complicated than initially anticipated, and the experimental systemic uncertainties are under further detailed investigation. So far there is no clear conclusion in term of the disentangle of the CME and related background from the preliminary experimental data yet.

7 Small system p+A or d+A collisions

The small system p+A or d+A collisions provides a control experiment, where the CME signal can be “turned off”, but the related backgrounds still persist. In non-central heavy-ion collisions, the , although fluctuating, is generally aligned with the reaction plane, thus generally perpendicular to . The measurement is thus entangled by the two contributions: the possible CME and the -induced background. In small-system p+A or d+A collisions, however, the is determined purely by geometry fluctuations, uncorrelated to the impact parameter or the direction. As a result, any CME signal would average to zero in the measurements with respect to the . Background sources, on the other hand, contribute to small-system p+A or d+A collisions similarly as to heavy-ion collisions. Comparing the small system p+A or d+A collisions to A + A collisions could thus further our understanding of the background issue in the measurements.

Fig. 14: (Color online) Single-event display from a Monte-Carlo Glauber event of a peripheral Pb+Pb (a) and a central p+Pb (b) collision at 5.02 TeV. The open gray [solid green (light gray)] circles indicate spectator nucleons (participating protons) traveling in the positive z direction, and the open gray [solid red (dark gray)] circles with crosses indicate spectator nucleons (participating protons) traveling in the negative z direction. In each panel, the calculated magnetic field vector is shown as a solid magenta line and the long axis of the participant eccentricity is shown as a solid black line.

Figure 14 shows a single-event display from a Monte Carlo-Glauber event of a peripheral Pb+Pb (a) and a central p+Pb (b) collision at 5.02 TeV. In A+A collisions, due to the geometry of the overlap region, the eccentricity long axis are highly correlated with the impact parameter direction. Meanwhile the magnetic field direction is mainly determined by the positions of the protons in the two colliding nucleus, which is also generally perpendicular to the impact parameter direction. Thus in A+A collisions, these two direction are highly correlated with each other. Consequently, the measurements are entangled with the background and possible CME signal. While in p+A (Fig. 14, b) due to fluctuations in the positions of the nucleons, the eccentricity long axis and magnetic field direction are no longer correlated with each other. So the measurements in p+A collisions with respected to the eccentricity long axis (estimated by ) will lead to zero CME signal on average, and similarly for d+A collisions.

The recent measurements in small system p+Pb collisions from CMS have triggered a wave of discussions about the interpretation of the CME in heavy-ion collisions. The correlator signal from p+Pb is comparable to the signal from Pb+Pb collisions at similar multiplicities, which indicates significant background contributions in Pb+Pb collisions at LHC energy.

Fig. 15: (Color online) The opposite-sign and same-sign three-particle correlator averaged over as a function of in p+Pb and Pb+Pb collisions at = 5.02 TeV from CMS collaboration. Statistical and systematic uncertainties are indicated by the error bars and shaded regions, respectively.
Fig. 16: (Color online) The difference of the opposite-sign and same-sign three-particle correlators (a) as a function of for and (b) as a function of , averaged over , in p+Pb and Pb+Pb collisions at = 5.02 TeV from CMS collaboration. The p-Pb results are obtained with particle from Pb- and p-going sides separately. Statistical and systematic uncertainties are indicated by the error bars and shaded regions, respectively.

Figure 15 shows the first measurements in small system p+A collisions from CMS, by using p+Pb collisions at 5.02 TeV compared with Pb+Pb at same energy. The results are plotted as a function of event charged-particle multiplicity (). The p+Pb and Pb+Pb results are measured in the same ranges up to 300. The p+Pb results obtained with particle c in Pb-going forward direction. Within uncertainties, the SS and OS correlators in p+Pb and Pb+Pb collisions exhibit the same magnitude and trend as a function of event multiplicity. By taking the difference between SS and OS correlators, Fig 16 shows the and multiplicity dependence of correlator. The p+Pb and Pb+Pb data show similar dependence, decreasing with increasing . The distributions show a traditional short range correlation structure, indicating the correlations may come from the hadonic stage of the collisions, while the CME is expected to be a long range correlation arising from the early stage. The multiplicity dependence of correlator are also similar between p+Pb and Pb+Pb, decreasing as a function of , which could be understood as a dilution effect that falls with the inverse of event multiplicity. There is a hint that slopes of the dependence in p+Pb and Pb+Pb are slightly different in Fig. 16(b), which might be worth further investigation. The similarity seen between high-multiplicity p+Pb and peripheral Pb+Pb collisions strongly suggests a common physical origin, challenges the attribution of the observed charge-dependent correlations to the CME.

It is predicted that the CME would decrease with the collision energy due to the more rapidly decaying at higher energies. Hence, the similarity between small-system and heavy-ion collisions at the LHC may be expected, and the situation at RHIC could be different. Similar control experiments using p+Au and d+Au collisions are also performed at RHIC.

Fig. 17: (Color online) The preliminary , (Left panel) and (Right panel) correlators in p+Au and d+Au collisions as a function of multiplicity, compared to those in Au+Au collisions at = 200 GeV from STAR collaboration. Particles , and are from the TPC pseudorapidity coverage of with no gap applied. The is obtained by two-particle cumulant with gap of . Statistical uncertainties are shown by the vertical bars and systematic uncertainties are shown by the caps.

Fig. 17(left) shows the and results as functions of particle multiplicity () in p+A and d+A collisions at  GeV. Here is taken as the geometric mean of the multiplicities of particle and . The corresponding Au+Au results are also shown for comparison. The trends of the correlator magnitudes are similar, decreasing with increasing . The results seem to follow a smooth trend in over all systems. The results are less so; the small system data appear to differ somewhat from the heavy-ion data over the range in which they overlap in . Similar to LHC, the small system results at RHIC are found to be comparable to Au+Au results at similar multiplicities (Fig. 17, right). While in the overlapping range between p(d)+Au and Au+Au collisions, the data differ by 20-50%. This seems different from the LHC results where the p+Pb and Pb+Pb data are found to be highly consistent with each other in the overlapping range. However, the CMS p+Pb data are from high multiplicity collisions, overlapping with Pb+Pb data in the 30-50% centrality range, whereas the RHIC p(d)+Au data are from minimum bias collisions, overlapping with Au+Au data only in peripheral centrality bins. Since the decreasing rate of with is larger in p(d)+Au than in Au+Au collisions, the p(d)+Au data could be quantitatively consistent with the Au+Au data at large in the range of the 30-50% centrality. It is interesting to note that this is similar to the observed difference in the slope of the dependence in p+Pb and Pb+Pb by CMS  as mentioned previously. Considering these observations, the similarities in the RHIC and LHC data regarding the comparisons between small-system and heavy-ion collisions are astonishing.

Since the p+A and d+A data are all backgrounds, the should be approximately proportional to the averaged of the background sources, and in turn, the of final-state particles. It should also be proportional to the number of background sources, and, because is a pair-wise average, inversely proportional to the total number of pairs as the dilution effect. The number of background sources likely scales with multiplicity, so the . Therefore, to gain more insight, the was scaled by :

(18)

Fig. 18 shows the scaled as a function of in p+A and d+A collisions, and compares that to in Au+Au collisions. AMPT simulation results for d+Au and Au+Au are also plotted for comparison. The AMPT simulations can account for about of the STAR data, and are approximately constant over . The in p+A and d+A collisions are compatible or even larger than that in Au+Au collisions. Since in p+A and d+A collisions only the background is present, the data suggest that the peripheral Au+Au measurement may be largely, if not entirely, background. For both small-system and heavy-ion collisions, the is approximately constant over . It may not be strictly constant because the correlations caused by decays (), depends on the which is determined by the parent kinematics and can be somewhat -dependent. Given that the background is large, suggested by the p+A and d+A data, the approximate -independent in Au+Au collisions is consistent with the background scenario.

Fig. 18: (Color online) The scaled three-particle correlator difference in p+Au and d+Au collisions as a function of , compared to those in Au+Au collisions at = 200 GeV from the preliminary STAR data. AMPT simulation results for d+Au and Au+Au are also plotted for comparison.

Due to the decorrelation of the and the magnetic field direction in small system p(d)+A collisions, the comparable measurements (with respect to the ) in small system p(d)+A collisions and in A+A collisions at the same energy from LHC/RHIC suggests that there is significant background contribution in the measurements in A+A collisions, where the measurements (with respect to the ) in small system p(d)+A collisions are all backgrounds. While, by considering the fluctuating proton size, Monte Carlo Glauber model calculation shows that there could be significant correlation between the magnetic field direction and direction in high multiplicity p+A collisions, even though the magnitude of the correlation is still much smaller than in A+A collisions. Those calculations may indicate possibilities of studying the chiral magnetic effect in small systems.

The decorrelation of the and the magnetic field direction in small system p(d)+A collisions provides not only a way to “turn off” the CME signal, but also a way to “turn off” the -related background. The background contribution to the measurement with respect to the magnetic field direction would average to zero due to this decorrelation effect in system p(d)+A collisions. So the key question is weather we could measure a direction that possibly accesses the magnetic field direction. The magnetic field is mainly generated by spectator protons and therefore experimentally best measured by the 1st-order harmonic plane () using the spectator neutrons.

Fig. 19: (Color online) The preliminary correlator in p+Au collisions with respect to of spectator neutrons measured by the ZDC-SMD, compared to the measured with respected to in p(d)+Au and Au+Au collisions at = 200 GeV from STAR.

Fig. 19 shows the preliminary measurement in p+Au collisions with respect to of spectator neutrons measured by the shower maximum detectors of zero-degree calorimeters (ZDC-SMD) from STAR. The measurement is currently consistent with zero with large uncertainty. In the future with improved experimental precision, this could possibly provide an excellent way to search for CME in small systems.

8 Measurement with respect to reaction plane

Again, one important point is that the CME-driven charge separation is along the magnetic field direction (), different from the participant plane (). The major background to the CME is related to the elliptic flow anisotropy (), determined by the participant geometry, therefore the largest with respect to the . The and in general correlate with the , the impact parameter direction, therefore correlate to each other. While the magnetic field is mainly produced by spectator protons, their positions fluctuate, thus is not always perpendicular to the . The position fluctuations of participant nucleons and spectator protons are independent, thus and fluctuate independently about . Fig. 20 depicts the display from a single Monte-Carlo Glauber event in mid-central Au+Au collision at 200 GeV.

Fig. 20: (Color online) Single-event display from a Monte-Carlo Glauber event of a mid-central Au+Au collision at 200 GeV. The gray markers indicate participating nucleons, and the red (green) markers indicate the spectator nucleons traveling in positive (negative) z direction The blue arrow indicates the magnetic field direction. The long axis of the participant eccentricity is shown as the black arrow. The magenta arrow shows the direction determined by spectator nucleons.

The eccentricity of the transverse overlap geometry is by definition . The overlap geometry averaged over many events is an ellipse with its short axis being along the ; its eccentricity is and

(19)

The magnetic field strength with respect to a direction is: . And

(20)

The relative difference of the eccentricity () or magnetic field strength () with respect to and are defined below:

(21)

where

(22)

The and are not experimentally measured. Usually the event plane () reconstructed from final-state particles is used as a proxy for . can be used as a proxy for :

(23)

Although a theoretical concept, the may be assessed by Zero-Degree Calorimeters (ZDC) measuring spectator neutrons. Similar to Eq.8,8,8,8, these relations hold by replacing the with . For example,

(24)
Fig. 21: (Color online) Relative differences , from Monte Carlo Glauber model (upper panel) and , from AMPT (lower panel) for (a,f) Au+Au, (b,g) Cu+Cu, (c,h) Ru+Ru, and (d,i) Zr+Zr at RHIC, and (e,j) Pb+Pb at the LHC. Both the Woods-Saxon and EDF-calculated densities are shown for the Monte Carlo Glauber calculations, while the used density profiles are noted for the AMPT results.

Figure 21(upper panel) shows and calculated by a Monte Carlo Glauber model for Au+Au, Cu+Cu, Ru+Ru, Zr+Zr collisions at RHIC and Pb+Pb collisions at the LHC. The results are compared to the corresponding . These numbers agree with each other, indicating good approximations used in Eq. 8,8. Fig. 21(lower panel) shows and calculated from AMPT simulation