CChiral Magnetic Effects in
Nuclear Collisions
W. Li and G. Wang Department of Physics and Astronomy, Rice University, Houston (TX), USA,77005; email: [email protected] Department of Physics and Astronomy, University of California, Los Angeles(CA), USA, 90095; email: [email protected]. Xxx. Xxx. Xxx. YYYY. AA:1–28https://doi.org/10.1146/((please addarticle doi))Copyright c (cid:13)
YYYY by Annual Reviews.All rights reserved
Keywords chiral magnetic effect, chiral magnetic wave, chiral vortical effect,heavy-ion collisions, quark-gluon plasma
Abstract
The interplay of quantum anomalies with strong magnetic field and vor-ticity in chiral systems could lead to novel transport phenomena, suchas the chiral magnetic effect (CME), the chiral magnetic wave (CMW)and the chiral vortical effect (CVE). In high-energy nuclear collisions,these chiral effects may survive the expansion of a quark-gluon plasmafireball and be detected in experiments. The experimental searches forthe CME, the CMW and the CVE, have aroused extensive interest overthe past couple of decades. The main goal of this article is to reviewlatest experimental progress in search for these novel chiral transportphenomena at Relativistic Heavy Ion Collider at BNL and the LargeHadron Collider at CERN. Future programs to help reduce uncertain-ties and facilitate the interpretation of the data are also discussed. a r X i v : . [ nu c l - e x ] F e b ontents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Magnetic field and vorticity in nuclear collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43. CME searches in nuclear collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1. Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2. Results in large A+A systems: evidence for the CME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3. Approaches to disentangle the signal vs. background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4. Outlook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164. CMW searches in nuclear collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.1. Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2. Results in A+A collisions: evidence for the CMW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3. Approaches to disentangle the signal vs. background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.4. Outlook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235. CVE searches in nuclear collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1. Introduction
An object or system is chiral if it is not invariant under mirror imaging. A chiral systemcarrying an imbalance of right- and left-handed particles can be characterized by an axialchemical potential ( µ ). In a system of charged chiral fermions with a finite µ value, anelectric current can be induced when a strong external magnetic field ( −→ B ) is applied, −→ J e ∝ µ −→ B . (3), Na Bi (4), TaAs (5) and TaP (6)).In this article, we review the progress of decades-long efforts in searching for the CME inhigh-energy nuclear collisions.In ultra-relativistic heavy-ion (e.g., lead or gold) collisions, a new phase of hot and densenuclear matter is created with a temperature above several trillion kelvins, consisting ofdeconfined quarks and gluons, dubbed the quark-gluon plasma (QGP) (112, 113, 114, 115).The chiral symmetry restoration realized in a QGP renders nearly massless or chiral quarks.It has been suggested that local chiral domains with finite µ values may be formed in theinitial stage of a QGP, via topological charge fluctuations (related to chiral anomaly) inthe vacuum of quantum chromodynamics (QCD) (1, 2, 7, 8, 9, 10). Within each domain,there is an imbalance of right- and left-handed chiral quarks (although the global chiralimbalance vanishes after averaging over infinite number of domains). In non-central heavy-ion collisions, extremely strong magnetic fields ( B ∼ T) can be formed in the QGP,mostly by energetic spectator protons (1, 7). Therefore, the two preconditions (finite µ and −→ B field) for the CME may be realized in heavy-ion collisions, leading to an observableeffect such as an electric current along the −→ B . The direction of −→ B field is approximatelyperpendicular to the reaction plane (Ψ RP ) that contains the impact parameter and the beammomenta of a collision. As a result, the CME in nuclear collisions will manifest an electriccharge transport phenomenon across the reaction plane. The experimental observation of he CME in high-energy nuclear collisions will have far-reaching impacts in many frontiersof fundamental physics: the evolution of the strongest magnetic field ever created, thetopological phases of QCD, and the chiral symmetry restoration of strong interactions.This prompted a community-wide effort to search for the CME at the Relativistic HeavyIon Collider (RHIC) in the Brookhaven National Lab (BNL) and the Large Hadron Collider(LHC) in CERN over the past two decades.Motivated by the CME search and other modes of collective motions of a QGP, theazimuthal distribution of particles of given transverse momentum ( p T ) and pseudorapidity( η ) in an event is typically decomposed by a Fourier series: dN α dφ ∝ v ,α cos(∆ φ ) + 2 v ,α cos(2∆ φ ) + ... + 2 a ,α sin(∆ φ ) + ..., φ is the azimuthal angle of a particle, and ∆ φ = φ − Ψ RP . The subscript α (+ or − )denotes the charge sign of a particle. Conventionally, coefficients v , v and v of parity-eventerms are called “directed flow”, “elliptic flow”, and “triangular flow”, respectively. Theyreflect the hydrodynamics response of the QGP medium to the initial collision geometryand to its fluctuations, respectively (116). The coefficient a (with a , − = − a , + ) of theparity-odd term quantifies the electric charge separation with respect to the reaction plane,e.g., due to the CME.Another transport phenomenon complementary to the CME is the chiral separationeffect (CSE) (11, 12), in which a current of chiral charges along the −→ B field is induced bya finite chemical potential of vector charges (e.g., electric charges): −→ J ∝ µ v −→ B . v ). Observation of the CMW does not necessarily depend on the observation of theCME, as the latter requires an initial µ from the QCD chiral anomaly (which may besmall), while the former only needs a local net electric charge density.Finally, in analogy to the −→ B -field induced anomalous chiral transport effects, similarphenomena can also take place when a chiral system carries a global angular momentumunder rotation. The fluid rotation can be quantified by vorticity, −→ ω = −→(cid:53) × −→ v , where −→ v is the flow velocity field. Given a large vorticity, the chiral vortical effect (CVE) (15) caninduce a vector current: −→ J v ∝ µ µ v −→ ω . −→ B , the CVE is driven by µ v −→ ω of the QGP. Here the subscript“v” denotes “vector”, which can be, e.g., “ B ” (baryonic charge) or “ e ” (electric charge).The µ B is not affected by the presence of −→ B field, making it a better tool to search forthe CVE through the baryonic-charge separation in heavy-ion collisions. In the case of theCVE search, the subscript α in Eq. 2 would represent baryon or anti-baryon numbers.There are more chiral magnetic/vortical effects proposed, such as the chiral electricseparation effect (CESE) (16, 17) and the chiral vortical wave (CVW) (18) (see Ref (90)for a review on these effects). • Chiral Magnetic Effects in Nuclear Collisions 3 his article focuses on the key experimental results over the past couple of decades insearch for the chiral magnetic/vortical effects in high-energy nuclear collisions: the probe ofthe initial magnetic field and vorticity in Section 2, experimental searches for the electric-(baryonic-)charge separation in Section 3 (Section 5), and the electric quadrupole momentin Section 4. An outlook for future developments is discussed in Section 6.
2. Magnetic field and vorticity in nuclear collisions
Intuitively, one may regard the magnetic field (vorticity) as the driving force of the CME(CVE), the chirality imbalance as the initial condition, and the electric-(baryonic-)chargeseparation as the manifestation. The existence and magnitude of the magnetic field (vor-ticity) could be independently constrained by other experimental observable.The initial magnetic field is roughly estimated to be, eB ∼ γα EM Z/b , α EM (cid:39) / b is the impact parameter, and γ is the Lorentz factor. The largecharge number, Z , and small impact parameter, b , lead to an extremely strong magneticfield. A typical noncentral Au+Au collision at √ s NN = 200 GeV produces an initial eB ∼ / (1fm ) ∼ m π (or 10 T). Many model calculations have attempted to quantify theelectromagnetic field in detail on the event-by-event basis, in terms of its spatial distribution,the fluctuation of its orientation as well as the dependence on colliding nuclei, centralityand beam energy (see Refs. (20, 21, 22) for examples). A major uncertainty in calculationsof the magnetic field −→ B is its lifetime during the evolution of the QGP created in heavy-ion collisions (see Refs. (23, 24, 25, 26) for examples). The time dependence of −→ B afterthe initial impact crucially depends on whether/when/how a conducting medium may beformed. In the vacuum, the magnetic field will quickly decay as two ions pass by, whereasif the QGP medium carries an electric conductivity, a longer duration may be sustained.The electric conductivity (27) and the time evolution of quark densities (28) can bestudied via charge-dependent directed flow of final-state hadrons in asymmetric A+A colli-sions, such as Cu+Au. The difference in the number of protons between Au and Cu createsa strong electric field in the initial stage of the collision, pointing from Au to Cu. Thelifetime of the electric field might be very short (e.g. t ∼ .
25 fm/ c depending on conduc-tivity from Refs. (27, 28)). If quarks and antiquarks are produced sufficiently early in thecollision, they would experience a Coulomb force, and the degeneracy in v is lifted betweenpositively and negatively charged particles (21, 27): v ± = v ± d E (cid:104) cos(Ψ RP − Ψ E ) (cid:105) , E denotes the azimuthal angle of the electric field, and the coefficient d E charac-terizes the strength of dipole deformation induced by the electric field, proportional to theelectric conductivity of the medium. Here v represents the rapidity-even component of di-rected flow that dominates over the rapidity-odd one in asymmetric collisions. In symmetriccollisions, v often denotes the latter.The STAR collaboration has measured charge-dependent v even1 and the difference ∆ v even1 as functions of p T in both Cu+Au and Au+Au collisions at 200 GeV (29, 30). For p T < c , the ∆ v even1 seems to increase with p T in Cu+Au collisions, while it is consistentwith zero in Au+Au collisions. The parton-hadron-string-dynamics (PHSD) model (28) is a ynamical transport approach in the partonic and hadronic phases, and has calculated the∆ v even1 driven by the initial electric field. The model assumes that all electric charges areaffected by the electric field, which results in a large separation of v even1 between positive andnegative particles. After scaling down the calculated ∆ v even1 by a factor of 10, the modeldescribes rather well the p T dependence of experimental data for p T < c . Thisqualitative observation of the strong initial electric field in asymmetric collisions providesan indirect evidence for the strong initial magnetic field in heavy-ion collisions, which sharesthe same conducting medium with the electric field, and hence could also leave an imprinton the final-stage particles.Heavy-flavor quarks, such as charm (c) and beauty (b) , are produced much earlier thanlight-flavor quarks in a collision because of their large masses. As a result, heavy quarkshave a better chance to witness the strong electric and magnetic fields (117, 118). Insymmetric A+A collisions, it has been predicted that a rapidity-dependent splitting of D and ¯ D meson v and v (from the magnetic field), and v (from the electric field) will begenerated. The same phenomenon is expected to occur for light hadrons but at a muchreduced magnitude because of the much later production time. Experimental efforts are onongoing to accumulate high-precision data sets to explore these effects.Vorticity results from the interplay of global rotation and shear viscosity of the QGP inheavy-ion collisions. In a noncentral collision, the majority of the global angular momentum, −→ L , is carried away by spectator nucleons. However, about 10 −
20% of −→ L could remain inthe QGP and be approximately conserved over time (31, 32). This implies that the CVEcan be developed over a relatively long duration. The angular momentum is largely alignedwith the magnetic-field direction, both perpendicular to the reaction plane, so the CME andCVE are very much alike in terms of their experimental observables. Attempts to computelocal vorticity −→ ω and its space-time distribution have also been made extensively (31, 32,33, 34, 35, 36).Experimentally, the global polarization of hyperons such as the Λ baryon has beenused to probe both the QGP vorticity and the magnetic field. The local vortical effectscan generate a positive spin polarization for both Λ and ¯Λ, whereas the coupling of thehadronic magnetic dipole moment to the magnetic field will produce a negative (positive)contribution for Λ (¯Λ). Therefore, observing a splitting between Λ and ¯Λ polarization willbe a direct evidence for the magnetic field. Note that Λ is typically produced in a laterstage of the collision, so its sensitivity to the initial magnetic field could be limited by thelifetime of the magnetic field. The first observation of global Λ and ¯Λ polarization in heavy-ion collisions has been reported by the STAR collaboration (37). At √ s NN <
100 GeV,the signal is on the order of a few percent, and displays some hint of a weak beam-energydependence. Current statistical precision of data is insufficient to study polarization for Λand ¯Λ separately. This question will be addressed in the second beam energy scan (BES-II)program at RHIC (38) to search for evidence of the magnetic field.Further searches for the initial magnetic field (vorticity) have been proposed throughphoton (vector meson) polarization measurements (39). The initial magnetic helicity ( −→ E ·−→ B )of the collision system can be quite large, with opposite signs in the upper and lower hemi-spheres. Because of the chiral anomaly, helicity can be transferred back and forth betweenthe magnetic flux and fermions, so that the magnetic helicity could last long enough toyield photons with opposite circular polarizations in the hemispheres above and below thereaction plane (40, 41, 42, 43). The initial global quark polarization could effectively lead toa polarization of photons (41), and hence cause an asymmetry in photon polarization (44). • Chiral Magnetic Effects in Nuclear Collisions 5 his local imbalance of photon circular polarization could be investigated via the polar-ization preference with respect to the reaction plane for photons that convert into e + e − pairs (39). Similarly, vector mesons that decay into two daughters can also have theirpolarization preferences measured with the scheme outlined in Ref. (39), and the helicityseparation in this case originates from vorticity (44, 45, 46).Another proposal aiming at measuring the imprint left by the initial magnetic fieldfocuses on pairs of oppositely charged particles in the whole evolution of heavy-ion col-lisions (47). The pertinent mechanism is the distortion of the relative angle betweenpositively- and negatively-charged particles inside a pair. Two observables are adopted todetect this effect: one based on the same framework as that measuring global Λ polarization,and the other based on a slightly modified balance function. The knowledge documentedin Ref (47) will facilitate the experimental efforts to quantify the strong magnetic field inhigh-energy nuclear collisions.
3. CME searches in nuclear collisions
In this section, methodologies employed to search for the CME (as well as other anoma-lous chiral effects) in nuclear collisions are reviewed, followed by experimental results atRHIC (48, 49, 50, 51, 52, 58, 59, 60, 53, 54) and the LHC (55, 56, 57). These include datain large A+A collision systems like Au+Au, U+U and Pb+Pb, and also small systems suchas p+Au, d+Au and p+Pb. In particular, direct comparison of large- and small-systemdata in recently years are proven to provide a powerful tool in better understanding back-ground contributions. Approaches developed for quantifying the background contributionsand extracting true CME signals are also discussed. In the outlook, prospects of futureprograms at RHIC for isobaric collisions and the Beam Energy Scan (BES-II), and at thehigh-luminosity LHC are reviewed.
Measurements of the CME-induced charge separation across the reaction plane are primarilyexplored by the so-called three-point γ correlator , first proposed in Ref. (61).It is tempting to directly measure the event-averaged a , ± coefficient from the single-particle azimuthal distribution in Eq. 2. However, since the sign of the µ value fluctuatesbetween positive and negative on an event-by-event basis with equal probability (global par-ity for QCD should be conserved), the event-averaged a , ± values are zero by construction.The γ correlator (later often referred to as γ ) is designed to observe the fluctuations ofcharge separations or a , ± coefficients with respect to the reaction plane (61), γ ≡ (cid:104)(cid:104) cos( φ α + φ β − RP ) (cid:105)(cid:105) = (cid:104)(cid:104) cos(∆ φ α ) cos(∆ φ β ) − sin(∆ φ α ) sin(∆ φ β ) (cid:105)(cid:105) = ( (cid:104) v ,α v ,β (cid:105) + B IN ) − ( (cid:104) a ,α a ,β (cid:105) + B OUT ) , , α and β in an event andover all events. The expansion of the γ correlator reveals the difference between in-plane and out-of-plane projections of azimuthal correlations. The third term of Eq. 7, (cid:104) a ,α a ,β (cid:105) ,represents a measurement of the variance (or fluctuations) of a , ± coefficients, which is themain target for the CME search. There are other terms that are presumably unrelated tothe CME. The first term, (cid:104) v ,α v ,β (cid:105) , is related to the directed flow that is expected to be harge independent and unrelated to the magnetic field in symmetric A+A systems. The B IN and B OUT terms represent other possible background correlations (as will be discussedin detail later) in and out-of the reaction plane, respectively. By taking a difference betweenopposite-sign and same-sign γ correlators,∆ γ ≡ γ OS − γ SS , (cid:104) v ,α v ,β (cid:105) terms cancel out, as well as a large portion of the background terms ( B IN and B OUT ) that are reaction-plane independent. There may still be a residual reaction-plane dependent background in ( B IN − B OUT ), at a level proportional to the magnitudeof elliptic flow coefficient ( v ). This is the major unknown source of backgrounds in ∆ γ measurements. In practice, the reaction plane is approximated with the “event plane”(Ψ EP ) reconstructed with measured particles, and then the measurement is corrected forthe finite event plane resolution. The main advantages of the γ correlator lie in its directconnection to the a coefficient and a relative straight forward procedure for correcting theevent plane resolution.Several alternative methods to the γ correlator were also proposed, with the goal ofproviding complementary sensitivity to the CME signal and backgrounds. These includethe modulate sign correlator (MSC) (50), the charge multiplicity asymmetry correlator(CMAC) (51), the multi-particle charge-sensitive correlator ( R Ψ m (∆ S )) (53, 54, 62), andthe signed balance functions (63). It is not a surprise that these methods provide largelyoverlapping information with the γ correlator, as they all make use of the same inputs ofparticle azimuthal correlations. These alternative methods are mostly in the process ofbeing applied to experimental data. Therefore, we will focus on reviewing experimentalresults of the γ correlator and its derivatives in the following discussion. Measurements of the three-point γ correlator have been performed extensively in Au+Aucollisions over a wide range of RHIC energies and in Pb+Pb collisions at top LHC energies.Figure 1 shows the result of opposite-sign and same-sign correlators ( γ × N part ), respec-tively, in Au+Au collisions at 200 GeV, measured by the STAR collaboration (48, 49, 50).Here, the number of participating nucleons, N part , is used as a multiplicative factor tocompensate for the expected dilution of signals with increasing number of domains (withrandom signs of µ ) toward more central Au+Au collisions. In this way, results become lesscentrality dependent.An apparent charge dependence of the three-point correlator is observed, where valuesof γ OS112 ( ∼
0) are generally higher than those of γ SS112 ( < γ correlations, γ OS112 and γ SS112 should have the same magnitude but opposite signs, symmetricaround zero. This is obviously not the case in data, indicating that charge-independentbackgrounds must be present (e.g., momentum conservation and collective flow effects).Following the first observation at 200 GeV, the charge-dependent γ correlator hasbeen measured over a wide range of collision energies at RHIC and the LHC, shown in Fig. 2,as a function of centrality for Pb+Pb collisions at 2.76 TeV (55), and for Au+Au collisionsat 200, 62.4, 39, 27, 19.6, 11.5 and 7.7 GeV (52). The charge-independent backgrounds • Chiral Magnetic Effects in Nuclear Collisions 7
Most central p a r t N × γ − − − − × opposite charge, Y7 Ψ , Y7 Ψ , Y4 Ψ same charge, Y7 Ψ , Y7 Ψ , Y4 Ψ Au+Au 200 GeV
Figure 1
Charge-dependent γ × N part measured with the 1st-order spectator plane (Ψ ) and the2nd-order participant plane (Ψ ) versus centrality for Au+Au collisions at 200 GeV (48, 49, 50).Y4 and Y7 represent STAR results from the RHIC runs 2004 and 2007, respectively. tend to be more prominent in lower beam energies, where the multiplicity is lower. TheMEVSIM model calculation (65) with momentum conservation effects but no CME canqualitatively capture this feature of experimental data. The difference between γ OS112 and γ SS112 persists up to the LHC energies and down to the RHIC BES energies, while there isa hint of diminishing at 7.7 GeV. To focus on the charge-dependent correlation signals andisolate background effects that are most relevant to CME searches, the difference between γ OS112 and γ SS112 is also often studied. opposite chargesame charge
27 GeV Au+Au (e) − (a) (f)
200 GeV Au+Au (b) (g)
MEVSIM (c) (h)
39 GeV Au+Au (d) % Most central × ] 〉 ) PP ψ β φ + α φ c o s ( 〈 ≡ γ [ Figure 2
Charge-dependent γ measured with the the 2nd-order participant plane (Ψ PP ) versuscentrality for Pb+Pb collisions at 2.76 TeV (55), and for Au+Au collisions at 200, 62.4, 39, 27,19.6, 11.5 and 7.7 GeV (52). .3. Approaches to disentangle the signal vs. background Extraordinary claims require extraordinary evidence.
While the A+A data have providedevidence in line with the observation of the CME, no definitive conclusion can be drawnyet because of the presence of several charge-dependent background contributions. A classof main charge-dependent backgrounds can be generalized into the local charge conserva-tion (LCC) or ordering effect in decays of resonance and/or cluster-like intermediate states.These resonances and clusters develop a correlation with the reaction plane during theanisotropic hydrodynamic expansion of the QGP, seeded by an asymmetric initial geome-try. Precision theoretical calculations of those background contributions are not available,because of their nonperturbative nature. Data-driven approaches and phenomenologicalmodels have been developed to quantitatively constrain the backgrounds and examine if itis necessary to invoke the CME mechanism to explain the experimental data. For exam-ple, a model calculation incorporating the LCC effect and anisotropy flow (68) was able tocapture the STAR data without introducing any CME signal.A generic scenario of anisotropic cluster emission (also called “flowing clusters”) wereoriginally considered in Ref. (61) to investigate the background terms B IN and B OUT in the γ correlator: B IN − B OUT B IN + B OUT ≈ v , cl (cid:104) cos( φ α + φ β − φ cl ) (cid:105)(cid:104) cos( φ α − φ β (cid:105) ) , φ cl is the cluster emission azimuthal angle, and φ α and φ β are the azimuthal an-gles of two decay products. The v , cl of clusters contains both flow and so-called nonflowcontributions (e.g., short-range correlations within a cluster). The flowing cluster modelcan be generalized to a larger portion of or even the full event, through the mechanismsof transverse momentum conservation (TMC) (66, 67) and/or local charge conservation(LCC) (68). Ideally, the two-particle correlator, δ ≡ (cid:104) cos( φ α − φ β ) (cid:105) , should be proportionalto (cid:104) a ,α a ,β (cid:105) , but in reality it is strongly dominated by short-range two-particle correlationbackgrounds. For example, the TMC effect leads to the following pertinent correlationterms in ∆ δ and ∆ γ (67):∆ δ TMC → − N (cid:104) p T (cid:105) (cid:104) p T (cid:105) F v , Ω ) − v , F ¯ v , Ω − (¯¯ v , F ) , γ TMC112 → − N (cid:104) p T (cid:105) (cid:104) p T (cid:105) F v , Ω − ¯¯ v , F − ¯¯ v , F (¯ v , Ω ) − (¯¯ v , F ) ≈ κ TMC112 · v , Ω · ∆ δ TMC , κ TMC112 = (2¯ v , Ω − ¯¯ v , F ) /v , Ω , and ¯ v and ¯¯ v represent the p T - and p T -weightedmoments of v , respectively. The subscript “F” denotes an average of all produced particlesin the full phase space; the actual measurements will be only in a fraction of the fullspace, denoted by “Ω”. The background contribution due to the LCC effect has a similarcharacteristic structure as the above (66, 68). This motivates a normalization of ∆ γ by v and ∆ δ : κ ≡ ∆ γ v · ∆ δ . κ is larger than κ TMC112 , may a CME signal be present. In Au+Au collisionsat 200 GeV, the κ TMC112 values estimated in the model using PHOBOS v data (69, 70) arearound 1 . κ values from background-only simulations of A Multi-Phase Transport (AMPT) model (71, 72, 73) show a seemingly • Chiral Magnetic Effects in Nuclear Collisions 9 onstant of 1 . −
80% centrality range (74). On the other hand, the STAR data forsuch collisions typically bear κ ≥ γ correlationcould arise from the CME. However, the AMPT model may not capture all backgroundcontributions, and thus cannot be relied on to quantify the CME signal contribution. (GeV) NN s κ
10 60%
BES II error projection
Au+Au Pb+Pb
Figure 3 κ measured with the participant plane versus beam energy for 10-60% Pb+Pb collisions at 2.76TeV (55), and for 10-60% Au+Au collisions at 200, 62.4, 39, 27, 19.6, 11.5 and 7.7 GeV (52). Theerror projection for the RHIC BES II is also presented. Figure 3 shows the beam-energy dependence of κ for 10-60% Pb+Pb collisions at2.76 TeV (55), and for 10-60% Au+Au collisions at 200, 62.4, 39, 27, 19.6, 11.5 and 7.7GeV (52). The data display a rise-and-fall trend, with a peak around 27 GeV Au+Aucollisions and a drop approaching the background level at 7.7 GeV. The disappearance ofthe CME is expected at low collision energies where the partonic interactions are dominatedby the hadronic ones, and quarks are not massless any more. At LHC energies, althoughthe initial magnetic field has a much stronger peak magnitude than at RHIC, it dropsmore rapidly, possibly vanishing before the formation of the QGP. Without any electricconductivity in the QGP, by the time scale of 0.1 fm/ c , the remaining magnetic field atLHC is lower than that at RHIC typically by two orders of magnitude. Therefore, a smallerCME coud be anticipated at LHC than at RHIC energies.Data-driven approaches to constrain the background contributions are discussed below,which have the advantage of being model independent. These are general strategies: (1)to vary the signal while keeping the background fixed, such as using small systems andhigher-order γ correlators; (2) to vary the background while keeping the signal fixed, suchas the event shape engineering. In a non-central A+A collisions, the participantplane of the lenticular overlap region is, although fluctuating, generally strongly correlatedwith the reaction plane, or perpendicular to the magnetic field, as illustrated in Fig. 4 (left)for a Pb+Pb collision. Conversely, in a p+Au(Pb) collision, the overlapping geometry is en-tirely determined by fluctuations, and the participant plane is essentially uncorrelated withthe reaction plane or the magnetic field direction, as illustrated in Fig. 4 (right). Therefore,even if the magnetic field may still be comparable in magnitude to A+A systems in suchsmall-system collisions, its decoupling from the participant plane will greatly suppress thepossible CME contribution in γ . Meanwhile, it has been observed in recent years thatsmall systems exhibit similar bulk properties to large A+A systems (119, 120). All these
10 W. Li and G. Wang ake the small system an ideal data-driven testing ground for turning off the possible CMEsignal and understanding the pure-background contributions.
Figure 4
Cartoons for demonstrating the correlation (and the decorrelation) between the participant plane(black arrows) and the reaction plane (red arrows) in A+A (left) and p+A (right) collisions.
The ∆ γ correlator in p+Pb collisions has been measured by CMS (56), shown inFigure 5 (left) as a function of multiplicity, together with Pb+Pb data at 5.02 TeV (56). Atthe same multiplicity, the p+Pb and Pb+Pb data are nearly identical over a wide range ofmultiplicities. This observation indicates that the charge separation signal observed in A+Acollisions is likely to be dominated by, if not entirely, background correlations unrelated tothe CME. In this analysis, a large η gap of at least 2 units is required between particle α , β and the event plane, largely eliminating backgrounds directly from short-range corre-lations. Long-range nonflow correlations such as di-jets are also present and their effectstend to diminish as event multiplicity increases or from peripheral to most central A+Acollisions. Although small-system data strongly indicate the dominance of backgrounds upto semi-peripheral A+A events, caution should be taken when extrapolating to more centralcollisions, where contributions of different physics processes may vary.STAR has also measured the ∆ γ correlator in small systems of p+Au and d+Aucollisions at 200 GeV (58). The results of ∆ γ scaled by dN ch /dη/v are shown in Fig. 5.An interesting ordering of ∆ γ pAu > ∆ γ dAu > ∆ γ AuAu is seen if compared at the samemultiplicity. Similar to the CMS observation, this seems to indicate that the previouslyobserved ∆ γ in peripheral Au+Au collisions are entirely dominated by backgrounds.However, note that a key difference between the CMS and STAR analyses is the η gapimposed. The STAR analysis implements a much smaller η gap between particle α , β andthe event plane, and thus likely includes additional contribution of short-range correlations.Indeed, when different η gaps are introduced, the results could vary by a factor of 2 inp+Au and d+Au (58). The multiplicity range covered by STAR and CMS analyses is alsoquite different, which may result in different long-range nonflow correlation backgrounds.Future program with small systems and upgraded detectors at RHIC and the LHC will helpprovide a more meaningful and consistent comparison between difference energies. The γ measurements are usu-ally implemented with the second-order event plane, which is approximately perpendicularto the magnetic field direction. A new correlator, γ (57), with respect to the third-order • Chiral Magnetic Effects in Nuclear Collisions 11 /d ch dN { } / v h / d c h d N ´ g D = 200 GeV NN s p+Au d+Au Au+Au (Y2004)Au+Au (Y2007) p+Au d+Au : 0{2} in v hD : 0.5{2} in v hD : 1.0{2} in v hD : 1.4{2} in v hD Figure 5
The opposite-sign (SS) and same-sign (OS) difference ∆ γ correlator as a function of eventmultiplicity in p+Pb and Pb+Pb collisions at 5.02 TeV from CMS (56) (left) and p+Au, d+Auand Au+Au collisions at 200 GeV from STAR (58) (right). event plane, was motivated for the background study, γ ≡ (cid:104)(cid:104) cos( φ α + 2 φ β − ) (cid:105)(cid:105) . γ correlator captures pure background contributions thatare proportional to v · ∆ δ (following a similar derivation to Eq. 11). Similar to Eq. 12, anormalized quantity can be defined (57): κ ≡ ∆ γ v · ∆ δ , κ as a data-driven background estimate for κ . Indeed, bothCMS data of Pb+Pb collisions at 5.02 TeV (57) (illustrated in the lower panel of Fig. 6)and STAR preliminary data of Au+Au collisions at 200 GeV (75) show that κ and κ are close to each other in most centrality intervals studied. This indicates little fraction, ifany, of the CME contribution to the measured κ . The expectation of similar κ and κ values in a pure background environment is further validated by the CMS p+Pb data(entirely dominated by backgrounds) as shown in the upper panel of Fig. 6, again suggestinglittle room for any CME contribution in Pb+Pb data at LHC energies.In model calculations without any CME signal, AMPT suggests that κ and κ arenot identical in Au+Au collisions at 200 GeV, with the latter higher than the former byabout 50% (or a ratio of 3 /
2) (74). This indicates that higher-harmonic anisotropy is more“damped” in the system evolution, and κ is at best a qualitative estimate for κ , whichmay depend heavily on properties of system evolution. This may not be surprising, basedon previous derivations, where κ and κ are not exactly identical. Looking closely at
12 W. Li and G. Wang rkoffline N dD n / v , n - ; n gD (Pb-going) c f n = 2, (Pb-going) c f n = 3, CMS | < 1.6 hD |pPb 8.16 TeV PbPb centrality(%)55 45 3565 trkoffline N dD n / v , n - ; n gD n = 2n = 3 PbPb 5.02 TeV| < 1.6 hD | Figure 6 κ and κ measured with the participant plane in p+Pb collisions for the Pb-going directionat 8.16 TeV (upper) and Pb+Pb collisions at 5.02 TeV (lower) (56, 57). the CMS data in both Pb+Pb and p+Pb in Fig. 6, κ is indeed generally above κ .More work is still needed to better understand and/or redefine the κ to draw quantitativeconclusion on the possible CME contribution to κ .Another derivative of the γ correlator and its normalized quantity are (74) γ ≡ (cid:104)(cid:104) cos( φ α − φ β + 2Ψ ) (cid:105)(cid:105) κ ≡ ∆ γ v · ∆ δ . γ uses the second-order event plane, it is similar to γ in the sense that itis linked to the CME only via a v · ∆ δ term. The AMPT simulations (74) show that κ is close to unity in most cases, providing another data-driven gauge of the backgroundbaseline. The event-shape engineering technique was proposed toquantitatively remove backgrounds related to long-range collective (or flow) correlations ina data-driven way (80, 81). The idea is based on the expectation that the CME signal isindependent of v , while the dominant background is proportional to v , as supported by κ and κ data in p+Pb:∆ γ = κ BKG112 · v · ∆ δ + ∆ γ CME112 . v = 0, it will then arrive at ∆ γ | v =0 = ∆ γ CME112 .Note that some dependence of the CME signal on v is expected when v becomes very • Chiral Magnetic Effects in Nuclear Collisions 13 mall and event plane resolution is poor. This effect is studied by Monte Carlo models, aswill be mentioned later.The “standard” procedure of event-shape engineering is to keep the following threetypes of particles independent of each other in an event: (A) the particles that are usedto engineer the event shape, (B) the particles of interest ( α, β ), and (C) the particles thatreconstruct the event plane (Ψ EP ). In other words, they should come from three differentsub-events. In practice, the flow vector of sub-event A, −→ q = ( q A x , q A y ), controls the eventshape: q A x = 1 √ N N (cid:88) i cos(2 φ A i ) 17. q A y = 1 √ N N (cid:88) i sin(2 φ A i ) , −→ q is equivalent to v but contains effects of statistical fluctuations.For different q A bins, v B2 and ∆ γ B112 are calculated for particles in sub-event B, with theevent plane estimated from sub-event C. Then ∆ γ B112 is plotted as a function of v B2 , andthe extrapolation of ∆ γ B112 to the value at v B2 = 0 yields the true CME contribution withno flow-related backgrounds. Although q A and q B are linearly correlated on average, thereis a spread between them on an event-by-event basis, arising from statistical fluctuations.Therefore, even the lowest q A bin close to zero could correspond to a sizable v B2 . Systematicuncertainties and model dependence could be introduced when ∆ γ B112 is extrapolated overa wide unmeasured v B2 region.Figure 7 demonstrates the application of this event-shape engineering approach by theCMS collaboration in Pb+Pb collisions at 5.02 TeV for various centrality ranges (57). Aratio of ∆ γ to ∆ δ is taken and plotted as a function of v to eliminate a weak dependenceof ∆ δ on v (especially for peripheral events). A linear fit to data and extrapolation to v = 0results in a vertical intercept well consistent with zero, indicating little or no CME signal.Under the assumption of linear extrapolation, an upper limit on the fraction of possibleCME signals in ∆ γ can be extracted, which is less than 7% at 95% confidence level(C.L.) in Pb+Pb collisions from the CMS data, shown in Fig. 7 (top right). As mentionedearlier, a large fraction of v region towards zero v is lack of data points and relies entirelyon extrapolation. The ALICE collaboration took into account possible v dependence ofthe CME signal as v approaches to zero based on several Monte Carlo initial-state models,and also obtained fractions of residual CME signals in ∆ γ , shown in Fig. 7 (bottomright). The conclusion of CMS and ALICE measurements are consistently holding thatat the LHC energies, the observed ∆ γ correlator is consistent with 100% flow-drivenbackground contributions. The CME signal contribution to the ∆ γ correlator, if indeedpresent, has an upper limit of only a few %.One way to avoid the long extrapolation in v is to sacrifice the independence betweensub-events A and B. When particles of interest are used to define q , the lowest q bin naturallycorresponds to a v value very close to zero, and the extrapolation is technically much morereliable. The caveat on this approach is that only the “apparent” flow (including statisticalfluctuations) is under control. It is still possible that a resonance parent has a finite v , andits decay daughters have zero contribution to q . In this case, even at zero q or v , there existsa fake CME signal in ∆ γ . This approach has been tested with a background-only AMPTmodel. The disappearance of background is demonstrated when ∆ γ is extrapolated to
14 W. Li and G. Wang ero q (82). Another caveat on the event-shape engineering in general is that in realitythe CME signal and flow magnitude could have an intrinsic correlation, e.g., related to thecentrality or impact parameter dependence. In that case, the projection to v = 0 couldlead to an over-subtraction of the background. Model studies on the centrality dependenceof the magnetic field and initial-state eccentricity will help understand this effect. Centrality (%) C M E f − MC-GlauberMC-KLN CGCEKRT = 2.76 TeV NN s Pb − ALICE Pb | < 0.8 η | c < 5.0 GeV/ T p trkoffline N no r m f
95% CL Interval PbPb 5.02 TeV95% CL Interval pPb 8.16 TeV,(Pb-going) c φ CMS
PbPb centrality(%)55 45 3565 | < 1.6 η∆ | Combinedlimits pPb PbPb
Figure 7
Left: ratios of ∆ γ to ∆ δ as a function of v for different centrality classes in Pb+Pb collisionsat 5.02 TeV. Right: upper limits on fractions of possible CME signals in ∆ γ . Derivatives of the γ observable with respect to differenttypes of event planes, i.e., the participant plane (Ψ PP ) and the spectator plane (Ψ SP ), werealso proposed. Here, the idea is that the CME signal is most correlated with Ψ SP , while theflow-driven backgrounds are largely correlated with Ψ PP . Making use of the decorrelationbetween Ψ SP and Ψ PP , ∆ γ can be decomposed into two components: ∆ γ flow112 and ∆ γ CME112 ,following the relations below (83):∆ γ { PP } = ∆ γ flow112 { PP } + ∆ γ CME112 { PP } , γ { SP } = ∆ γ flow112 { SP } + ∆ γ CME112 { SP } , a · ∆ γ flow112 { PP } + ∆ γ CME112 { PP } / a , a = (cid:104) cos(2Ψ PP − SP ) (cid:105) , denoting the decorrelation between the two event planes.By eliminating ∆ γ flow112 { PP } from the equations, the CME signal can be extracted,∆ γ CME112 { PP } = (∆ γ { SP } − a · ∆ γ { PP } ) / (1 / a − a ) . γ correlator data with respect to Ψ PP and Ψ SP are already presented in Fig. 1(corresponding to Ψ and Ψ ), but higher statistics are needed for a firm conclusion.There are also other method developments involving differential measurements of the γ correlators, such as those as functions of the particle pair’s relative pseudorapidity (∆ η ) (60) • Chiral Magnetic Effects in Nuclear Collisions 15 nd invariant mass ( m inv ) (84). The former approach assumes that the CME-inducedcorrelations should not be of short ranges in rapidity, and hence the decomposition of γ (∆ η )into several Gaussian distributions could separate the contributions of different physicsorigins, or at least exclude the short-range correlations. However, the typical correlationlength of the CME-induced correlations is still elusive on the theoretical side, and for a clearinterpretation, the γ (∆ η ) results need to be compared with other correlators that aredominated by backgrounds, such as γ (∆ η ) and γ (∆ η ). Data from CMS in p+Pb andPb+Pb collisions do not indicate any obvious difference in ∆ η dependence of γ and γ correlators. The latter approach focuses on the backgrounds due to resonance decays, andattempts to remove such contributions by rejecting particle pairs with small m inv . Again,the theoretical guidance is needed on the m inv dependence of the CME signal, the lack ofwhich hinders a definite conclusion. Along the line of disentangling the possible CME signal and flow-driven backgrounds viadata-driven approaches, comparison between U+U and Au+Au collisions as a function ofcentrality is another promising way of obtaining new insights. A uranium nucleus have13 more protons than a gold nucleus, which in turn causes stronger magnetic fields inU+U than Au+Au collisions at the same number of participating nucleons ( N part ). Thedifference in the magnetic field is compensated by the difference in ellipticity at lower N part ,but becomes overwhelming towards very central events with higher N part . This direction isbeing pursued at RHIC.Following this direction of fixing v -driven backgrounds, collisions of isobaric nuclei,such as Ru and
Zr, have been proposed (77). The Ru+Ru and Zr+Zr collisions atthe same beam energy are almost identical in terms of hadronic particle production, buttheir initial magnetic fields differ because of the charge difference between the Ru and Zrnuclei. Figure 8 shows that the relative difference in magnetic field magnitudes betweenRu+Ru and Zr+Zr collisions is around 13% for central events, and increases to 15% − β ) of Ru and Zr nuclei (86, 87, 88). Thereis a little relative difference in eccentricity (that drives v ), but much smaller than thatin the magnitude field. AMPT simulations have shown that the relative difference in theCME signal embedded in the model between the two isobars is robust and can survive thefinal-state interactions (89).The isobaric program was carried out at RHIC in 2018, where about 3 billion minimumbias events for each of the two isobaric systems have been collected by the STAR experiment.The data are being analyzed. In a simple scenario of identical v between Ru+Ru and Zr+Zrsystems, there are three signatures that would indicate a CME signal (90): (1) ∆ γ Ru+Ru112 > ∆ γ Zr+Zr112 ; (2) ∆ δ Ru+Ru112 < ∆ δ Zr+Zr112 ; (3) (∆ γ Ru+Ru112 − ∆ γ Zr+Zr112 ) / (∆ δ Ru+Ru112 − ∆ δ Zr+Zr112 ) = (cid:104) cos[2(Ψ RP − Ψ EP )] (cid:105) . If the CME contributes to more than 20% of the ∆ γ signal inthe 20–60% centrality range, a more than 7 σ difference between Ru+Ru and Zr+Zr data isexpected. Taking into account possible difference in v between the two isobaric systems,the κ and κ observable can be invoked to mitigate the effect for a fair comparison. Ifno difference is seen between Ru+Ru and Zr+Zr within experimental uncertainties, upperlimits on the CME contribution to ∆ γ and other CME-motivated observable can beobtained. However, it should be also noted that there are other effects that may shadow
16 W. Li and G. Wang
Most central )] > R P Ψ B Ψ c o s [ ( ) π < ( e B / m ≡ s q B case 1case 2Ru+Ru Zr+Zr = 200 GeV NN s(a) % Most central R e l a t i ve d i ff e r e n ce − case 1case 2 sq B R ∈ R = 200 GeV NN s(b) Figure 8
Theoretical calculation of the initial magnetic field squared with correction from azimuthalfluctuation (a) and their relative difference (b) versus centrality for Ru+Ru and Zr+Zr collisionsat 200 GeV (85). Also shown is the relative difference in initial eccentricity. The solid (dashed)lines correspond to the deformity parameter set of case 1 (case 2). the charge difference between the two isobars. For example, if Zr has a thicker neutron skinthan Ru, the initial magnetic field in Zr+Zr is then stronger than originally estimated, andthe difference in the magnetic field between Zr+Zr and Ru+Ru will diminish (91, 92).The RHIC BES-II program has been ongoing from 2018 to 2022, with the main goal ofsearching for a critical endpoint of the QCD phase diagram. It will significantly expand thedata sets collected during the BES-I phase for a variety of collision energy. The projectedstatistical uncertainties on the κ measurement is displayed with the shaded band in Fig. 3,which are nearly invisible. These future results and in comparison with realistic modelcalculations, such as AMPT and anomalous-viscous fluid dynamics (AVFD) model (93),will draw a more definitive conclusion on the possible disappearance of CME signal inAu+Au collisions at low beam energies. The newly installed Event Plane Detector (EPD)in the STAR experiment introduces a sizable η gap between the particles of interest andthe event plane, and will help suppress the nonflow effects, which are particularly sizablefor low multiplicity events at lower beam energies. For some beam energies, the η coverageof the EPD could span into the beam rapidity to capture spectator nucleons. This willfacilitate the comparison between the ∆ γ { SP } and ∆ γ { PP } correlators.
4. CMW searches in nuclear collisions
In this section, searches for another anomalous chiral effect, chiral magnetic wave (CMW),in nuclear collisions is reviewed. Methodologies employed to search for the CMW signal areoutlined, followed by reviews of experimental results at RHIC (94) and the LHC (95, 96)in both large and small systems. Background contributions to the CMW searches arediscussed. The section ends with a future outlook.
In heavy-ion collisions, the CMW-induced electric quadrupole evolves with hydrodynamicexpansion of the QGP, and results in a charge-dependent elliptic flow ( v ). Taking pions • Chiral Magnetic Effects in Nuclear Collisions 17 s an example, the v values for π + and π − are expected to be identical from hydrody-namic flow alone. On top of this baseline v base2 ( π ± ), the CMW will introduce an additionalcontribution (13) v ( π ± ) = v base2 ( π ± ) ∓ ( q e ¯ ρ e ) A ch , q e , ¯ ρ e and A ch = ( N + − N − ) / ( N + + N − ) are the quadrupole moment, the netcharge density and the final-state charge asymmetry of a collision event, respectively. With (cid:104) A ch (cid:105) always positive, the A ch -integrated v of π − ( π + ) should be above (below) the baselinebecause of the CMW, leading to a splitting between v ( π + ) and v ( π − ). For A ch -integrated v , it has been proposed that the baseline v for π + and π − may be modified by otherphysics mechanisms (97, 98) unrelated to the CMW. Therefore, the most unambiguous wayto search for the CMW signal in nuclear collisions is to measure the full A ch dependence ofpion v , or the slope of ∆ v ( v difference between π − and π + ) as a function of A ch . In thefollowing discussions, the r parameter is used to represent the slope of ∆ v ( A ch ), where apositive r indicates a possible CMW signal.In the method above, a correction to the observed A ch for the finite detector efficiencyis required. An alternative approach of a three-particle correlator that is less dependent onthe efficiency correction was proposed (99), (cid:104)(cid:104) cos[ n ( φ − φ )] q (cid:105)(cid:105) = (cid:104) cos[ n ( φ − φ )] q (cid:105) − (cid:104) cos[ n ( φ − φ )] (cid:105)(cid:104) q (cid:105) . φ and φ are the azimuthal angles of particles 1 and 2 ( π ± ), and q is the charge( ±
1) of particle 3. The single brackets represent the average over particles and events,and the double bracket denotes the cumulant. Without charge-dependent correlations, thiscorrelator should be equal to zero. A positive difference in this correlator between π − and π + will signify the CMW contribution.Similar to searches for the CME, more alternative methods have been proposed that mayprovide different sensitivity to the CMW signal and backgrounds. For example, in analogyto the multiparticle charge-dependent correlator for the CME search (53, 54, 62), a novel cor-relator has been recently proposed in search of the CMW-induced electric quadrupole (100).This new method does not require A ch either. Using AMPT model calculations, this ap-proach has shown a potential in improving the methodology of the CMW search.In reviewing experimental results below, we focus on the slope parameter r of ∆ v ( A ch )and its derivatives, which is the most widely studied observable. First evidence for the CMW has been reported by STAR in the measurement of charge-dependent pion v as a function of A ch in Au+Au collisions at 200 GeV, shown in Fig. 9(left) (94). A significant splitting between π + and π − v data is observed, which has anapproximately linear dependence on A ch . The ∆ v ( A ch ) result is also shown in Fig. 9(right), where the slope parameter r is extracted by a linear fit. These observations areconsistent with the expectation of the CMW.The measured slope parameter ( r ) is presented in Fig. 10 as a function of centralityfor Pb+Pb collisions at 2.76 TeV (95) and for Au+Au collisions from 7.7 to 200 GeV (94).A universal rise-and-fall trend is observed in the centrality dependence of r for most beamenergies except for 11.5 and 7.7 GeV, where the r slopes are consistent with zero, althoughstatistical uncertainties are still large. This trend is in line with the CMW expectation
18 W. Li and G. Wang h Observed A - ( % ) v p + p Au+Au 200 GeV: 30-40% < 0.5 GeV/c T (a) ch A - - ) ( % ) + p ( ) - v - p ( v - (b) 0.2903 – r = 3.1985 Figure 9
Left: pion v as a function of observed charge asymmetry. Right: v difference between π − and π + as a function of charge asymmetry after corrected for the detector efficiency, for 30–40%centrality Au+Au collisions at 200 GeV. (94) as it approximately follows how the magnitude of the magnetic field is expected to evolvewith centrality. The comparison between the STAR data for 200 GeV Au+Au and theALICE data for 2.76 TeV Pb+Pb reveals a striking similarity, especially considering themany differences between the two measurements such as collision energies, multiplicities,and kinematic acceptance: particles of interest in the STAR data are charged pions with0 . < p T < . c and | η | <
1, while those in the ALICE data are unidentified hadronswith 0 . < p T < c and | η | < .
8. Note that in most central and peripheral events,the r data at 200 GeV are consistent with zero within uncertainties or even negative,whereas it still remains significantly positive at 2.76 TeV. This may indicate an interplayof different physics mechanisms between RHIC and LHC energies.The three-particle correlator (as defined in Eq. 24) for the 2 nd and 3 rd harmonics hasbeen measured by ALICE as a function of centrality in Pb+Pb collisions at 2.76 TeV (95).The ordering between negative and positive particles of interest also supports the CMEpicture in noncentral collisions. The correlation strength substantially increases in moreperipheral collisions, which may go beyond the pure CMW interpretation. Not surprisingly,background sources (e.g., the LCC effect) could also contribute to the r measurement, asfor the γ correlator in the CME search. There are two major types of physical backgrounds proposed that may contribute to the r measurement: isospin chemical potential ( µ I ) (107), namely imbalance between numbers ofup and down quarks, and local charge conservation (LCC) effect (108), similar to that forthe CME search.A hydrodynamic study (107) incorporating finite initial µ I suggests that a simple viscoustransport of charges, combined with certain initial conditions, will lead to a sizable v splitting for charged pions. According to analytical calculations of the anisotropic Gubserflow, the ∆ v for pions is proportional to both the shear viscosity and µ I , and µ I is linearlyrelated to A ch . This model further predicts a negative r for charged kaons with largermagnitudes than the pion r , since µ I and strangeness chemical potential ( µ S ) are differentfor kaons and pions. This idea does not seem to be supported by preliminary STAR data • Chiral Magnetic Effects in Nuclear Collisions 19
20 40 60 − Pb + Pb 2760 GeV − Au + Au 27 GeV
Au + Au 200 GeV
Au + Au 19.6 GeV
Au + Au 62.4 GeV
Au+Au 11.5 GeV
Au + Au 39 GeV
Au+Au 7.7 GeV % Most central ( % ) S l ope pa r a m e t e r r Figure 10
The slope parameter ( r ) as a function of centrality for Pb+Pb collisions at 2.76 TeV (95) and forAu+Au collisions at 7.7-200 GeV (94). The grey bands represent systematic uncertainties. Incomparison with data of 200 GeV Au+Au, UrQMD calculations (101) (open square) and a CMWmodel calculation with a magnetic field duration time of 5 fm/ c (102) (dashed-dotted line) arealso shown. showing that the kaon r is consistent with the pion r within uncertainties for variouscollision energies at RHIC (105, 106).The LCC mechanism is also able to qualitatively explain the positive r observed fromdata, when convoluted with the characteristic dependence of v on η and p T in a limiteddetector acceptance (108). Similar to the CME background study, this effect to the CMWis demonstrated with a model of “flowing” clusters that locally conserve charges whendecaying into final-state particles, e.g., a pair of particles with opposite charges. Such apair could cause a non-zero A ch , if one of the particles falls outside the detector acceptance.If this process preferentially occurs in a phase space with smaller v , such as a lower- p T or higher- η region, a positive r can then be induced, no matter the missing particle ispositive- or negative-charged. A first estimate of the LCC contribution within a simplifiedmodel, however, appears to underestimate the r observed in the STAR measurements inAu+Au at 200 GeV by an order of magnitude (94). The LCC scenario also predicts thata narrower η coverage may lead to a stronger r , since it is more likely that one particlefrom a charge-conserved cluster escapes the detector. In a preliminary STAR result wherethe η coverage is reduced to half, r does not seem to display a significant variation withinexperimental uncertainties (106). Another prediction by the LCC scenario is that the r slope is proportional to the baseline v . This motivated a new noramlized r observable, r norm2 = r /v , by the CMS collaboration (96).Transport Monte Carlo models have been used to study contributions from backgroundsor conventional physics. The slope parameters extracted from UrQMD calculations forAu+Au collisions at 200 GeV (101), shown in Fig. 10, are consistent with zero for the 10–70% centrality range, as opposed to the positive signal observed in the real data. On theother hand, the simplified CMW calculations with a magnetic field duration time of 5 fm/ c demonstrate a centrality dependence of r similar to the data (102). AMPT calculations
20 W. Li and G. Wang lso show that a positive r caused by the electric quadrupole can survive the final-stateinteractions (100). A quantitative comparison between data and theoretical calculationsstill requires further work. In particular, a model that incorporates both the CMW signaland background contributions ( such as the AVFD model (93)) will help facilitate thecomparison with data.The transported-quark model (97) argues that at lower beam energies, the A ch -integrated v difference between particles and anti-particles can be explained by the ef-fect of quark transport from projectile nucleons to mid-rapidity. The assumption is thatquark coalescence mechanism still holds at low energies, and the v of transported quarksis larger than that of produced quarks. The model, however, suggests a negative slope for v ( π − ) − v ( π + ) as a function of A ch (103), which is opposite to the data for most centralityintervals.Data-driven approaches to constrain the CMW background contributions to the r measurement are discussed below, following the same strategy as for the CME search tovary the signal (background) while keeping the background (signal) fixed, using small-system collisions and higher-order flow coefficients. Following the same idea as for the CME search, measure-ments of charge-dependent v and slope parameter r in small p+A systems can providea baseline of pure background contributions, because of decorrelation between the eventplane and the magnetic field direction. Figure 11 displays the CMS measurements of nor-malized r for charged hadrons in p+Pb and Pb+Pb collisions at 5.02 TeV (96). The r norm2 values are comparable between p+Pb and Pb+Pb collisions (with p+Pb data even larger),with little dependence on event multiplicity or centrality. Similar to the conclusion drawnfor the CME, these results again suggest that the r slopes observed in peripheral Pb+Pbcollisions at 5.02 TeV are likely to be dominated by background contributions unrelated tothe CMW.Preliminary STAR results show that in minimum bias p+Au and d+Au collisions at200 GeV, the r value is consistent with zero within current uncertainties. This wouldsupport the observation of a CMW signal in Au+Au collisions at 200 GeV (105, 106). Asdiscussed earlier for the CME, an energy dependence of the CMW signal is possible dueto much shorter lifetime of magnetic field at higher energies when two ions pass by muchfaster. However, higher precision small-system data at RHIC, especially covering a widerange of multiplicity as for the LHC data, are still needed to draw a definitive conclusion.One technical detail relevant in the r measurement arises from the A ch dependence ofthe mean p T for particles of interest. Such an effect is not expected from the CMW signalbut may occur in certain background scenario (e.g., the LCC effect). Since v monotonicallyincreases with p T up to p T ∼ p T with A ch will naturallylead to a A ch -dependent v after integrated over the full p T range. As shown in the leftpanel of Fig. 11, with a proper normalization, a significant fraction of the v ( A ch ) slopecould be explained by the (cid:104) p T (cid:105) ( A ch ) slope in the CMW data of p+Pb and Pb+Pb collisionsat 5.02 TeV from CMS (96). As the relation between v and (cid:104) p T (cid:105) is not necessarily linear,a detailed simulation is needed to estimate the exact fraction of the (cid:104) p T (cid:105) contribution inthe CMS r results before extracting any upper limit on the possible residual CMW signalin the data. On the other hand, the STAR measurements have focused on a narrow rangeof low- p T pions (0 . < p T < . c ) to minimize the effect of (cid:104) p T (cid:105) contributions totheir results (94, 106). It should be noted that in the LCC scenario, even for final-state • Chiral Magnetic Effects in Nuclear Collisions 21 rkoffline N æ T p Æ no r m , r r m r CMS r pPb 5.02 TeV PbPb 5.02 TeV æ T p Æ norm r < 3.0 GeV/c T Centrality (%) nno r m r æ trkoffline N Æ
26 81 197 403 716 1139
CMS
PbPb 5.02 TeV < 3.0 GeV/c T v v Figure 11
Left: the linear slope parameters, r norm , for v (filled symbols) and mean p T (open symbols) asfunctions of event multiplicity in p+Pb and Pb+Pb collisions at 5.02 TeV. Right: r norm2 and r norm3 as functions of the centrality class in Pb+Pb collisions at 5.02 TeV. Average N offlinetrk valuesfor each centrality class are indicated on the top axis. particles selected with a fixed p T value, they could still originate from clusters over a wide p T range. As a function of A ch , (cid:104) p T (cid:105) of clusters will still vary, leading to a finite r slope.In this sense, the effect of (cid:104) p T (cid:105) cannot be eliminated, but is a generic LCC effect. In the LCC mecha-nism, a finite slope parameter for the third-order Fourier coefficient ( v ), r , should alsoarise (also true for all higher-order v n ) that is proportional to the baseline v magnitude.As both v and v have an approximately linear dependence on p T at low p T , a scalingrelation between the normalized r and r slopes is expected, (cid:104) r norm2 ≡ r v base2 (cid:105) ∼ (cid:104) r norm3 ≡ r v base3 (cid:105) κ as a data-driven background estimate for κ . TheCMW is not expected to generate a finite r slope, as the third-order event plane has nearlyno correlation with the reaction plane and/or the magnetic field direction.The test of this prediction has been carried out by CMS, as shown in Fig. 11 (right) for r norm2 and r norm3 as a function of centrality in Pb+Pb collisions at 5.02 TeV (96). With highprecision, the normalized r and r slopes agree well with each other, indicating the datacan be explained entirely by the LCC scenario without any CMW contribution. Futuretests in a pure-background scenario, such as the p+A data as a data-driven test groundand realistic model calculations, will help further validate this scaling relation.In the measurement of r and r slopes, cautions should be taken to the possible ex-istence of so-called nonflow effects (e.g., short-range correlations), which can give rise to a“trivial” contribution to r (109). The quantitative influence of this effect to r n depends ondetails but it is generally more significant for low multiplicity events, such as low-energy,peripheral A+A and p+A collisions. The sign of its contribution to r and r is opposite.This effect can be partially mitigated by imposing a large η gap in measuring v and v coefficients as implemented in the CMS measurement.
22 W. Li and G. Wang .4. Outlook
The isobaric collisions at RHIC could benefit the search of the CMW in a similar wayas that of the CME, except that the observed difference between Ru+Ru and Zr+Zr isexpected to be less significant in the CMW observable than the CME one. The reason isthat r is proportional to the magnetic field, whereas γ is proportional to the magneticfield squared, which amplifies the difference between Ru+Ru and Zr+Zr. In spite of that,with the current statistics taken by the STAR experiment in the RHIC run 2018, therecould still be a 3 σ difference in r between the two isobaric systems, if dominated by theCMW signal contribution.The RHIC BES-II program will greatly reduce statistical uncertainties of r measure-ments in Au+Au collisions at lower energies, and it is of great interest to see if r reallygoes to zero at 11.5 or 7.7 GeV. With a large η gap, the newly-installed EPD will helpsuppress nonflow contribution, which could fake a sizable r at lower energies.The CMW consists of two chiral gapless modes traveling at the same speed (13): theright-handed (left-handed) wave transports the right-handed (left-handed) density and cur-rent in the direction parallel (antiparallel) to the −→ B direction. A more general theoreticalanalysis (110) studied various possible collective modes based on a non-neutral-backgroundQGP (i.e. with nonzero µ and/or µ ) in external electric and/or magnetic fields, and founda new type of collective motion, the chiral electric wave (CEW), arising from CESE andpropagating in parallel/antiparallel to the −→ E field. In symmetric collisions there should beno net electric field on average, but asymmetric collisions like Cu+Au could provide a testground for the CEW measurements.
5. CVE searches in nuclear collisions
The experimental manifestation of the CVE is similar to that of the CME, except thatthe baryonic charge separation, instead of the electric charge separation, is induced withrespect to the reaction plane. As a result, the γ correlators developed for the CME searchare also applicable to search for the CVE, if replacing different electric charge combinationsby combinations of baryon and anti-baryon numbers.A natural choice is to study correlations between protons and anti-protons. However,there is an ambiguity that (anti)protons also carry electric charges so the CME may alsoarise, which is indistinguishable from the CVE. Neutral baryons, such as Λ, will provide acleaner probe to search for the baryonic charge separation effect.Although the Λ baryon is electrically neutral, it carries a strange quark, which has alarger mass and thus could be less “chiral”. Therefore, strange quarks may behave differ-ently from up/down quarks in the chiral dynamics in heavy-ion collisions. As a first stepto validate the behavior of Λ, preliminary STAR γ measurements have been performedon Λ- h + (¯Λ- h − ) and Λ- h − (¯Λ- h + ) as functions of centrality in Au+Au collisions at 200GeV (111). In this analysis, (anti)protons have been excluded from h ± in the correlatorto avoid any possible CVE contribution. No charge dependence is observed in this mea-surement, which assures that the Λ baryon manifests no electric charge effect in the γ correlation. Furthermore, the Λ- h correlation provides a baseline for the Λ- p correlationin search of the CVE. Any possible signal observed in the latter should not arise from theCME contribution.Preliminary STAR data have been obtained on γ of Λ- p (¯Λ-¯ p ) and Λ-¯ p (¯Λ- p ) asfunctions of centrality in Au+Au collisions at 200 GeV (111). The same-baryonic-charge • Chiral Magnetic Effects in Nuclear Collisions 23 orrelation is below the opposite-baryonic-charge correlation from mid-central to peripheralcollisions. This baryonic-charge separation with respect to the event plane is consistentwith the presence of a CVE signal.Just like in the CME search, comprehensive investigations of background contributionsare necessary. In fact, similar background effects in the CME search could also come intoplay in the CVE study. For example, in analogy to the local charge conservation, the localbaryonic-charge conservation could play a similar role as LCC when coupled to the ellipticflow ( v ). Most of ideas and tools developed to constrain background contributions in theCME search can be easily migrated to the CVE search. The relation between κ , κ and κ of Λ- p correlations in both small and large systems shall shed light on the nature ofbackground contributions. The event-shape engineering technique is applicable to eliminatethe v dependent background and determine the true CVE signal (or set an upper limit).As Λ baryons are less abundantly produced than pions, higher luminosity data samplesare generally required to achieve good precision for the CVE search in future programs,such as isobaric runs and BES-II at RHIC. No difference in the Λ- p correlations is expectedin the data of Ru+Ru and Zr+Zr collisions, since vorticity is supposed to be the same forthese isobaric systems. The CVE analyses will benefit from the BES-II program, becausevorticity increases at lower beam energies, and hence the CVE may yield stronger signalsto be observed.
6. Summary
In chiral systems, the interplay of quantum anomalies with a strong magnetic field or vor-ticity is predicted to result in a variety of novel transport phenomena such as the chiralmagnetic effect (CME), the chiral magnetic wave (CMW) and the chiral vortical effect(CVE). These phenomena probe topological properties of the QCD vacuum, and can be ex-plored experimentally in high-energy nuclear collisions via the charge-dependent azimuthalcorrelations of produced hadrons from the QGP medium. In this article, we have reviewedthe latest progress of experimental searches for these chiral transport effects at RHIC, BNLand the LHC, CERN over the past couple of decades.The three-particle charge-dependent γ correlator has been introduced as the mainworkhorse in the search for the CME. Clear differences between the opposite- and same-sign γ correlations are observed, which persist up to the LHC energies and down to the RHICBES energies, with a hint of diminishing at 7.7 GeV. This is in line with expectation of theCME picture. However, possible background contributions such as resonance decays, localcharge conservation and transverse momentum conservation are identified, which could ex-plain a sizable fraction of, if not all, the observed charge separation. Their contributionshave to be carefully quantified before any claim of the discovery of the CME can be made.Intensive efforts have been made to constrain the CME backgrounds with phenomenologicalmodels and data-driven approaches.We reviewed several data-driven background studies including the small-system (p+A,d+A) data, the γ correlator with respect to higher-order event planes, where no CMEsignals are expected; and the event-shape engineering technique, with the goal of varying thesignal and background contributions in a controlled way. Present results suggest that little,if any, contribution of the CME signal is present in the experimental observable of Pb+Pbcollisions at LHC energies, with an upper limit less than only a few %. However, futurework of applying these studies to lower RHIC energies at BES-II and isobaric programs is
24 W. Li and G. Wang till promising to identify a possible CME signal that is expected to be larger in that energyregime.The status of experimental searches for the CMW is reviewed, with the main focus onthe measurement of charge-dependent elliptic flow as a function of the observed event chargeasymmetry. Positive signals have been observed at both RHIC and LHC energies, support-ing the CMW picture. Similar to the CME search, efforts have been made to understandpossible contributions of background sources using small system data and higher-order flowharmonics, where no CMW signals are expected. Striking similarity between the p+Pband Pb+Pb data at LHC energies suggest the dominance of backgrounds, while p+Au andd+Au results at RHIC energies seem to be consistent with zero, indicating a possible CMWsignal in Au+Au collisions. Future programs of isobaric collisions and BES-II at RHIC willhelp clarify if a CMW signal indeed exists and its potential beam energy dependence.Experimental searches for the CVE signal has been performed with the same γ correlatorbut replacing charged particles by (anti)Λ and (anti)proton in Au+Au collisions at 200 GeV.A difference between the opposite- and same-baryonic-charge correlations is observed fromperipheral to mid-central collisions, consistent with the CVE expectation. However, just likein the CME search, extensive work to quantitatively understand background contributionsis still needed to draw a definitive conclusion on the observation of the CVE signal. DISCLOSURE STATEMENT
The authors are not aware of any affiliations, memberships, funding, or financial holdingsthat might be perceived as affecting the objectivity of this review.
ACKNOWLEDGMENTS
Gang Wang is supported by the U.S. Department of Energy, Office of Nuclear Physics, underthe Grant DE-FG02-88ER40424. Wei Li is supported by the U.S. Department of Energy,Office of Nuclear Physics, under the Grant de-sc0005131 and the Welch Foundation (GrantNo. C-1845).
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