Electric Fields and Chiral Magnetic Effect in Cu + Au Collisions

The non-central Cu + Au collisions can create strong out-of-plane magnetic fields and in-plane electric fields. By using the HIJING model, we study the general properties of the electromagnetic fields in Cu + Au collisions at 200 GeV and their impacts on the charge-dependent two-particle correlator $\gamma_{q_1q_2}=<\cos(\phi_1+\phi_2-2\psi_{RP})>$ (see main text for definition) which was used for the detection of the chiral magnetic effect (CME). Compared with Au + Au collisions, we find that the in-plane electric fields in Cu + Au collisions can strongly suppress the two-particle correlator or even reverse its sign if the lifetime of the electric fields is long. Combining with the expectation that if $\gamma_{q_1q_2}$ is induced by elliptic-flow driven effects we would not see such strong suppression or reversion, our results suggest to use Cu + Au collisions to test CME and understand the mechanisms that underlie $\gamma_{q_1q_2}$.


I. INTRODUCTION
Over the past few years, there has been increasing interest in the quantum-anomaly-related transport phenomena and their experimental signals in heavy-ion collisions. Such anomalous transports include chiral magnetic effect (CME) [1][2][3], chiral vortical effect (CVE) [4][5][6], chiral separation effect (CSE) [7,8], chiral electric separation effect (CESE) [9,10], etc. All these effects involve a net chiral imbalance in the quark-gluon plasma (QGP) and their occurrence, if confirmed, provides us hitherto unique evidence for the local P and CP violation in QGP. For review on the anomalous transports in heavy-ion collisions, see for example Refs [11][12][13].
The CME in QGP can be neatly expressed as where the summation is over all light quark flavors, N c is the number of colors, µ 5 is the chiral chemical potential, q f is the electric charge of quark flavor f , and B is a magnetic field. Equation (1) represents an electric current induced by a magnetic field in QGP with net chirality (characterized by µ 5 ). In real heavy-ion collisions, the magnetic fields are generated mostly in the direction perpendicular to the reaction plane (For discussions about the event-by-event fluctuation of the magnetic field orientation, see Refs. [14]), therefore the CME is expected to induce a charge separation with respect to the reaction plane which may be detected, as proposed by Voloshin [15], via the two-particle correlator where q 1 and q 2 are charges of particles 1 and 2, φ 1 , φ 2 , and ψ RP are the azimuthal angles for the particles 1, 2, and the reaction plane, respectively, and the average in Eq. (2) is taken over events. Negative same-sign (SS) correlator γ SS ≡ (γ ++ + γ −− )/2 and positive opposite-sign (OS) correlator γ OS ≡ γ +− could constitute a strong evidence for the occurrence of CME. The measurements have been carried out by STAR Collaboration [16][17][18][19] at RHIC for Au + Au and Cu + Cu collisions and ALICE Collaboration [20] at LHC for Pb + Pb collisions. The experimental results of γ q 1 q 2 showed consistent behavior with the expectation of CME. However, there still remain debates [21][22][23][24][25][26] on the CME interpretation of the data because γ q 1 q 2 may have contributions from other effects, potentially the elliptic-flow (v 2 ) driven ones, e.g., the transverse momentum conservation (TMC) [23,25] and local charge conservation (LCC) [26]. (If one measures the difference between the SS and OS correlators, the TMC contribution can be subtracted as it is charge independent, but the LCC remains to contribute.) It was proposed [27] to use the central U + U collisions which are expected to have sizable elliptic flows but no magnetic fields to disentangle the CME and v 2 driven contributions. Preliminary results for γ q 1 q 2 in U + U collisions have been reported by STAR collaboration [18], but there are still uncertainties, see discussions in Ref. [28]. The purpose of this paper is to propose to use another colliding system, that is, the Cu + Au collisions at RHIC to test CME-induced and the v 2 -driven contributions to γ q 1 q 2 . The idea is the following. If γ q 1 q 2 is dominated by v 2 -driven effects, we expect that γ q 1 q 2 (more precisely, ∆γ ≡ γ OS − γ SS , because the directed flow in Cu + Au collisions may contribute to both γ OS and γ SS . This contribution does not exist in Au + Au collisions.) would not change too much from Au + Au collisions to Cu + Au collisions. (A plausible guess would be that ∆γ as a function of centrality in Cu + Au collisions lie between that in Cu + Cu and Au + Au collisions). However, on the other hand, if γ q 1 q 2 is driven by CME, then the correlator in Cu + Au collisions will be very different from that in Au + Au collisions. This is because that, in noncentral Cu + Au collisions, due to the asymmetric collision geometry, there can generate electric fields pointing from Au to Cu nuclei which can induce an in-plane charge separation in addition to the out-of-plane charge separation due to CME. Thus the resulting charge dipolar distribution will deviate from the out-of-plane direction, and as a consequence, both the SS and OS correlators will descend, see Fig. 1 for an illustration. If the in-plane electric fields are so strong (or somehow equivalently, if their In-plane electric field is negligible. The dipole is driven by CME, and thus is along the magnetic field direction. (Middle): Weak electric field. It will drive the dipole to skew from the magnetic field direction, and thus reduce the magnitudes of both γ OS and γ SS . (Right): Strong electric field. The charge dipole will be driven by the electric field to nearly the in-plane direction, and thus the signs of γ OS and γ SS will be reversed. lifetime is long. We will not distinguish these two situations.) that the resulting charge dipole is mainly along the in-plane direction, one should observe a positive γ SS and a negative γ OS as opposite to that in Au + Au collisions, see the most right panel in Fig. 1. Thus if the measurement in Cu + Au collisions shows a strong reduction or even sign-reversion for γ OS and γ SS comparing to that in Au + Au collisions, it will be a hint for the EM-field (i.e., CME plus electric field) interpretation of γ q 1 q 2 . Otherwise, it will indicate a v 2 -driven interpretation of γ q 1 q 2 .
The paper is organized as following. In Sec. II we will report our numerical simulations for some general properties of the EM fields in Cu + Au collisions. In Sec. III we will then use these results to study the correlator γ q 1 q 2 in Cu + Au collisions. Finally we summarize and discuss in Sec. IV. Throughout this paper, we use the natural units = k B = c = 1.

A. Setups
The aim of this section is to give an event-by-event calculation of the electromagnetic fields in Cu + Au collisions. The EM fields in symmetric collisions like Au + Au at RHIC and Pb + Pb at LHC have been investigated in Refs. [29][30][31][32]. We focus on the fields at the initial time t = 0 which we define as the moment when the two colliding nuclei are maximally overlapped. We use the Liénard-Wiechert potentials to calcu-late the fields: (4) where e > 0 is the proton charge, R n = r − r n (t) is the relative position of the field point r to the nth proton at time t, r n (t), and v n is the velocity of the nth proton. The summations run over all protons in the projectile (we set it as Cu nucleus) and target (Au nucleus). The Liénard-Wiechert potentials are singular at R n = 0. In order to regulate the singularities we treat protons as uniformly charged spheres with radius R p . This is reflected in the charge number factor Z n (R n ) in Eqs. (3) and (4): when the field point locates outside the nth proton (in the rest frame of the proton) we set Z n = 1, otherwise we cut the proton into two spherical domes at R n and sum the contributions of these two domes. The in-medium charge radius R p of proton is unknown, we use the recent measurement of the rms charge radius of proton in vacuum, R p = 0.8775 fm [33]. Varying R p from 0.7 fm to 0.9 fm brings about 10% shift to the numerical results but no qualitative conclusion is altered. We set the projectile beam direction as z axis and the x-axis is along the impact parameter vector (pointing from Au nucleus to Cu nucleus) so that the reaction plane is the x-z plane. Finally, the positions of nucleons in the rest frame of a nucleus are sampled according to the Woods-Saxon distribution.
In real collisions, not only the strengths of the EM fields, but also their orientations with respect to the matter geometry (which we refer to as the participant planes) vary from event to event. We define the nth participant plane angle ψ n and eccentricity ǫ n as (see e.g. Refs. [34,35]): where ρ(r ⊥ ) is the participant density projected onto the transverse plane. For each event, we use the HIJING model [36][37][38] to describe the collision dynamics and determine the centrality, the density, the angles ψ n , etc, and apply Eqs. (3)-(4) to calculate the EM fields. Fig. 2 we show the impact parameter and centrality dependence of the event averaged EM fields at the mass center of the collision region, where the center of mass is defined through r cm ⊥ = d 2 r ⊥ ρ(r ⊥ )r ⊥ . First, the strength of the event averaged magnetic field (which is along the −y direction) in Cu + Au collisions at 200 GeV is comparable to that in Au + Au collisions [32]. Second, as a consequence of the asymmetry of the collision geometry, we find a strong event averaged electric field pointing from the Au nucleus to Cu nucleus. The strength of this electric field is smaller than the magnetic field, but still very large. The effect of this strong electric field on the two-particle correlator γ q 1 q 2 will be studied in Sec. III. In Fig. 3, we show B 2 and E 2 in Cu + Au collisions as functions of centrality. They reflect the event-by-event fluctuation of the strengths of the EM fields. This is particularly clear in central collisions where although the event-averaged EM fields vanish, the event-averaged squares of them are finite.  2. Spacial distribution.-In Fig. 4 and Fig. 5 we show the spacial distributions (on the transverse plane) of the event averaged magnetic and electric fields for Cu + Au collisions at 200 GeV. Obviously, the fields are very inhomogeneous in the overlapping region. Comparing to the Au + Au collisions [32], the asymmetric geometry of the Cu + Au collisions brings asymmetries to the spacial distributions of the fields. This is most evident for E x at b = 10 fm: E x is positive everywhere in the overlapping region in Cu + Au collisions. 10 10 x fm 3.

2.
3. which measure how strongly the azimuthal direction of the electric field (or magnetic field) correlate to the nth harmonic angle ψ n of the participants. See Refs. [14,28] and Sec. III for the reasoning of studying these correlations. Similar with what has been observed in Au + Au collisions [14], the correlations between ψ B and the odd harmonics, ψ 1 , ψ 3 , are practically zero, while the correlations between ψ B and the even harmonics, ψ 2 , ψ 4 , are nonzero for moderate centralities but get suppressed compared with the case in the hardsphere limit without event-by-event fluctuation (where, e.g., cos [2(ψ B − ψ 2 )] = −1). For the electric field case, we observe that ψ E is practically not correlated to ψ 3 and ψ 4 , but there is a strong back-to-back correlation between ψ E and ψ 1 (the same as that in Au + Au collisions [14]). Differently from the Au + Au case, in noncentral Cu + Au collisions there is a clear positive correlation between ψ E and ψ 2 signaling a persistent in-plane electric field. We note that at centrality 0.9, the correlations cos [2(ψ B − ψ 2 )] and cos [2(ψ E − ψ 2 )] change signs. This is because at that centrality region the most probably collisions involve only two participants distributed in the in-plane direction, and thus lead ψ 2 to be mostly perpendicular to the reaction plane.

A. Formalism
Both the electric field and the magnetic field can drive charges to flow via the normal Ohm's law and the CME, respectively, and can build certain charge distribution patterns. In general, we can write down a single-particle distribution per rapidity as where q = ± is the charge of the particle, N q is the number of hadrons with charge q per rapidity, v 0 1 is the directed flow parameter at E = B = 0, d E and d B characterize the strength of the dipole induced by E and B, respectively, (We will estimate d E and d B latter), and χ = ± is a random variable accounting for the fact that in a given event there may be a sphaleron or an anti-sphaleron transition resulting in charge separation parallel or anti-parallel to the B-field via CME. The event average of χ is zero. Equation (5) accounts for three kinds of "flows": the directed flow along the azimuthal angle ψ 1 , the CME along ψ B , and the normal electric conduction along ψ E . From the single-particle distribution (5), we first observe that the directed flow of charge q reads where · · · denotes event average. The second equality is due to the fact that the event average of χ is zero. Thus in Cu + Au collisions, the measured directed flow is likely to be charge dependent. This may provide us a possibility to extract the electric conductivity of the produced matter by measuring the difference between the directed flows of positive and negative hadrons because d E is proportional to the electric conductivity σ [See Eq. (13)]. This idea has been exploited in Ref. [39] and we will not discuss it further.
Our main concern will be the two-particle correlator (2). The single-particle distribution (5) results in a two-particle distribution in the form 1 from which one can derive We see that the directed flow appears in the first two terms. For Au + Au collisions these two terms vanish because in that case the directed flow appears owing to fluctuation and does not correlate to the reaction plane. But for Cu + Au collisions, these two terms can be finite. However, if we study the difference between the opposite-sign and same-sign correlators, the directed flow terms are subtracted and we obtain Note that this correlator also subtracts other charge insensitive contributions (e.g.,the TMC) and has been extensively discussed in, for example, Refs. [11,26]. Let us estimate the dipole strengths d E and d B . Definē Because the electric field is roughly in-plane as we have simulated (i.e. ψ E ≈ ψ RP ), it transports charges across the y − z plane during a proper time interval 0 to τ E by an amount or where τ E is the lifetime of the electric field, η is the spacetime rapidity, Y is a characteristic length scale of the overlapping region in y direction, and σ is electric conductivity. We have assumed that σ and E are constant in time and homogeneous in space. Equation (11) should be equal to the integration of f 1 (+, φ) −f 1 (−, φ) from ψ RP − π/2 to ψ RP + π/2, that is (To simplify the notation, we set ψ RP = 0), where N = N + + N − is the total multiplicity in a unit rapidity. Note that if there is a net charge, δN = N + − N − 0, the directed flow could also transport charges, that is why we usē f 1 : it subtracts such non-EM-field induced charge transport. From Eq. (11) and Eq. (12), we obtain In a similar way, we can estimate d B as with σ 5 the CME conductivity, τ B the lifetime of the magnetic fields, and X a characteristic length scale of the overlapping region in x direction. By adopting a simple hard-sphere model for the nucleus, we have (15) where b is the impact parameter and R Cu and R Au are the radii of Cu and Au nuclei.

B. Results
In the above modeling, the unknown parameters are the electric conductivity σ, the CME conductivity σ 5 , and the lifetimes of fields τ E,B . For σ, we will use the recent lattice QCD result [40] with C EM = f q 2 f = (2/3)e 2 if we consider u, d and s quarks. We will take T = 350 MeV in our numerical simulation. The CME conductivity is given by σ 5 = 3C EM µ 5 /(2π 2 ). As it is σ 2 5 appearing in ∆γ, thus instead of determining µ 5 itself we will determine µ 2 5 which measures the fluctuation of the chiral chemical potential because µ 5 = 0. For a given τ B , we determine µ 2 5 (as a function of T 2 ) by fitting ∆γ, Eq. (9), to the experimental result for Au + Au collisions reported by STAR collaboration [18]. The results are shown in Fig. 7 where we choose τ B = 5 fm. (The EM fields generated by the spectators decay very fast [32]. It is likely that the presence of a conducting medium may increase the lifetime of the magnetic fields, but there still remain debates, see discussions in Refs. [41][42][43]. In general, the lifetime of the electric field is expected to be shorter than magnetic fields.) At small centralities, the fitting is surprisingly good; at centrality larger than 60% our model does not apply because the two nuclei do not overlap and Eqs. (15) (with R Cu replaced by R Au ) fails.
With µ 2 5 obtained, we apply Eq. (9) to Cu + Au collisions and obtain ∆γ for Cu + Au collisions. The results are shown in Fig. 8. We see that, as expected, the in-plane electric fields give a negative contribution to ∆γ while the out-of-plane magnetic fields give a positive contribution (via the CME). For lifetime τ E = 2 fm (panel (a) of Fig. 8), we observe a suppression of ∆γ owing to the in-plane electric fields. This suppression becomes clearer for a longer lifetime τ E = 3 fm (panel (b) of Fig. 8). For an even longer lifetime, τ E = 4 fm, the electric fields dominate ∆γ and ∆γ becomes negative ( panel (c) of Fig. 8). Based on these simulations, we hypothesize that the correlator ∆γ in Cu + Au may be much smaller or even reversed compared with that in Au + Au collisions.  [18]. The line is our fitting curve by using Eq. (9). The blue, red, and black curves represent the contributions from magnetic field, electric field, and the total, respectively.

IV. SUMMARY AND DISCUSSIONS
In summary, by using HIJING model, we simulated the generation of the EM fields in Cu + Au collisions at 200 GeV and studied the general properties of the EM fields, see Sec. II. We found that a strong electric field along the impact parameter direction exists in Cu + Au collisions. This in-plane electric field gives a negative contribution to the two-particle correlator ∆γ (defined in Eq. (9)). By using a one-component model (i.e., by assuming ∆γ is driven by EM-field induced effects) we studied how the in-plane electric field influences ∆γ in Cu + Au collisions and found that it may strongly suppress or even change the sign of ∆γ compared with that in Au + Au collisions.
Some discussions are in order: (1) The correlator ∆γ may receive other non-EM-field induced contributions (contained in the irreducible component, C(q 1 , φ 1 ; q 2 , φ 2 ), of the two-particle distribution), potentially the LCC, which are not taken into account in the analysis in Sec. III. However, inclusion of the non-EM-field induced contributions will not affect the role played by the in-plane electric field in Cu + Au collisions. The in-plane electric field always tends to drive an in-plane component of the charge dipole and thus tends to suppress ∆γ.
(2) However, there is the possibility that ∆γ is dominated by non-EM-field induced effects and the CME and electric field play minor roles. In this case, one does not expect a strong suppression or reversion of ∆γ in Cu + Au collisions compared with that in Au + Au collisions. This provides us a feasible way to clarify how important the EM fields can be: If the experimental data in Cu + Au collisions really show a strong suppression or reversion of ∆γ, it will strongly indicate that the EM fields play dominant roles in ∆γ (it will also indicate a large electric conductivity of the quark-gluon matter); otherwise it will be a hint for the v 2 -driven interpretation of ∆γ.
(3) The modeling of ∆γ in Sec. III is illustrating, it can be improved by adopting a more realistic model, e.g., the twocomponent model [11,28] for ∆γ (i.e., to include the contributions driven by v 2 ), by taking into account more realistic spacetime dependence of the EM fields, and by including the final state interactions [44,45]. These will be our future tasks and the results will be presented in future publications.