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Chiral magnetic effect in isobar collisions from stochastic hydrodynamics

  • We studied the chiral magnetic effect in AuAu, RuRu, and ZrZr collisions at sNN=200GeV. The axial charge evolution was modeled with stochastic hydrodynamics, and geometrical quantities were calculated with the Monte Carlo Glauber model. By adjusting the relaxation time of the magnetic field, we found our results are in good agreement with background subtracted data for AuAu collisions at the same energy. We also made predictions for RuRu and ZrZr collisions. We found a weak centrality dependence on initial chiral imbalance, which implies that the centrality dependence of chiral magnetic effect signals results mainly from the effects of the magnetic field and volume factor. Furthermore, our results show an unexpected dependence on system size. While the AuAu system has larger chiral imbalance and magnetic field, it was observed to have a smaller signal for the chiral magnetic effect due to the larger volume suppression factor.
  • The anomalous transport of chiral magnetic effect (CME) has gained significant attention over the past few years [1, 2]. If the local parity odd domain is present in the quark-gluon plasma produced in heavy ion collisions, CME leads to charge separation along the magnetic field generated in off-central collisions:

    je=fNcq2f2π2μAB,

    (1)

    where the chiral imbalance, μA, characterizes local parity violation. This offers the possibility of detecting local parity violation in quantum chromodynamics (QCD). Charge separation has been actively sought experimentally [3-5]. However, we are still far from consensus on the status of CME, largely due to the difficulty in determining CME both experimentally and theoretically; see [6-9] for recent reviews. Experimentally, charge separation needs to be measured through charged hadron correlation on an event-by-event basis. Unfortunately, charged hadron correlation is dominated by flow-related background with various possible origins [10-12]. Various observables and experimental techniques have been proposed and implemented to exclude flow-related background [13-16]. In addition, STAR collaboration proposes to search for CME in isobar collisions [17]. Since isobars have the same atomic number but different proton numbers, the corresponding collisions are supposed to generate the same flow background, but different magnetic field, and thus, different charge separation. This can unambiguously distinguish the CME contribution.

    Theoretically describing CME is also difficult. Both μA and B contain large uncertainties. Their peak values are known to be set by the axial charge production in glasma phase [18, 19] and the moving charge of spectators [20], respectively. However, their further evolution is model dependent. Different theoretical frameworks, such as anomalous viscous fluid dynamics (AVFD) [21-24], chiral kinetic theory [25-27], and the multiphase transport model [28, 29], have been employed to study the time evolution of axial/vector charges. All these frameworks treat axial charge as an approximately conserved quantity in the absence of parallel electric and magnetic fields. However, axial charges are not conserved, due to gluon dynamics. In fact, it is the same origin for the initial axial charge. In [30], the authors incorporated fluctuation and dissipation of axial charge in the framework of stochastic hydrodynamics. We found that independent of the initial conditions, the variance of axial charge always approaches the thermodynamic limit after sufficient time, due to the interplay of fluctuation and dissipation. In [30], we used the thermodynamic limit for the axial charge to model CME. While being model-independent, the study missed an important fact: most charge separation occurs at early stages of quark-gluon plasma formation, when μAand B have not decayed appreciably. This study aims to incorporate the initial axial charge and investigate the coupled dynamics of axial and vector charge. In particular, we make predictions for the CME contribution to isobar collisions.

    This paper is organized as follows. In Sec. 2, we generalize the stochastic hydrodynamics framework to include axial and vector charge, which are coupled through anomalous effects in the presence of the magnetic field. Subsequently, we will justify the claim that for phenomenologically relevant magnetic fields, the back-reaction of vector charge to axial charge is negligible. In Sec. 3, we derive axial charge evolution with a non-vanishing initial value. The obtained axial charge is used to calculate charge separation. We make predictions for CME in isobar collisions, using AuAu collisions as a reference. We conclude and discuss directions for future work in Sec. 4.

    The stochastic hydrodynamic equations for axial charge in the absence of magnetic field are given in [30]. In a magnetic field, axial charge couples to vector charge through the chiral magnetic effect and chiral separation effect (CSE). The full stochastic hydrodynamic equations for axial and vector charges are

    {μJμA=nAτCS2ξqJμA=nAuμ+λnVeBμσTPμνν(nAχAT)+ξμA,

    (2)

    and

    {μJμV=0JμV=nVuμ+λnAeBμσTPμνν(nVχVT)+ξμV.

    (3)

    nA and nV are axial and vector charge density, respectively. The axial current is not conserved, due to the topological configuration of gluons. This leads to the dissipative term, nAτCS, and fluctuating noise term, ξq. The constitutive equations for axial and vector current consist of a co-moving term, an anomalous mixing term, a diffusive term, and a thermal noise term. uμ is the fluid velocity, which defines the projection operator, Pμν=gμν+uμuν and the magnetic field in the fluid cell, Bμ=12ϵμναβgFαβuν. ξA, ξV, and ξq are considered as Gaussian white noises:

    ξμA(x)ξνA(x)=Pμν2σATd4(xx)g,ξμV(x)ξνV(x)=Pμν2σVTd4(xx)g,ξq(x)ξνq(x)=ΓCSd4(xx)g,ξμA(x)ξq(x)=ξνV(x)ξq(x)=0,

    (4)

    with ΓCS being the Chern-Simon diffusion constant characterizing the magnitude of topological fluctuation.

    For application to CME in heavy ion collisions, we fix the parameters as follows. We use the free theory limit for axial and vector charge susceptibilities, χA=χV=χ=NfNcT2/3. The coefficient of the mixing term, λ, is determined by the chiral magnetic/separation effect as λ=1χNc2π2. The quark mass effect on CSE can be neglected [31]. For three flavours, we have χ=3T2 and λ=12π2T2. ΓCS is the Chern-Simon diffusion constant, obtained from the extrapolated weak coupling results: ΓCS=30α4sT4 [32] with αs=0.3. The relaxation time of the axial charge, τCS, is fixed by the Einstein relation: τCS=χT2ΓCS. σA and σV are the conductivities for the axial and vector currents, and are not required in our analysis.

    The axial/vector charge is treated as a perturbation in the background hydrodynamic flow. We consider heavy ion collisions at the top RHIC collision energy, sNN=200GeV, using Bjorken flow as the background. In the Milne coordinates (τ,η,x,y), the fluid velocity is uμ=(1,0,0,0). We can show that the total axial charge is conserved up to a mixing term and the topological fluctuation-induced terms. To do so, we substitute the constitutive equation into the conservation equation in (2) and integrate over the volume, τdηd2x=gdηd2x. The identity, μVμ=1gμ(gVμ), dropping the boundary terms, yields

    dηd2x(τ(τnA)τ(σATPτνν(nAχT))+τ(τξτA))=dηd2x(τnAτCS2τξq).

    (5)

    Note that Pμνuν=0 and ξμAξνAPμν. Thus, Pτν=0 and ξτA=0. We then arrive at

    τNA=NAτCSdηd2x2τξq,

    (6)

    with NA=τdηd2x. The absence of the diffusive term, thermal noise term, and mixing term is consistent with the fact that these terms only lead to the redistribution of axial charge. The counterpart for the vector charge is simpler: τNV=0, because vector charge is strictly conserved.

    We assume that the initial axial charge created by the chromo flux tube is homogeneous in the transverse plane. The boost-invariant Bjorken expansion maintains a homogeneous distribution in the longitudinal direction. The homogeneous axial charge leads to charge separation via CME. This simplified picture is modified by three effects: diffusion, thermal noise, and CSE. The thermal noise and diffusion correspond to the fluctuation and dissipation of charge, which bring the charge to equilibrium. The CSE is not balanced by other effects. We will now show that its effect is sub-leading.

    Let us compare the axial charge, nA, and CSE modification, λBnV. Since χA=χV, it is equivalent to comparing μA and λBμV. Since B drops quickly with time, the CSE effect is maximized at initial time. We estimate the initial nA following [21] as

    nA(τ0)2Q4s(πρ2tubeτ0)Ncoll16π2S,

    (7)

    where Qs is the saturation scale and ρtube1fm is the width of the flux tube. τ0 is the initial proper time. For AuAu collisions, we employ Qs1GeV and τ0=0.6fm. The number of binary collisions, Ncoll, and transverse overlap area, S, are calculated using a Monte Carlo Glauber model [33-36] with the centrality dependence listed in Table 1.

    Table 1

    Table 1.  Geometrical quantities from the MC-Glauber model for Au, Ru and Zr. Ncoll, S, and L are the number of binary collisions, transverse overlap area, and width of the participants' region along the cross-line between the transverse and reaction planes, respectively. S is considered to be the projection of the nucleon-nucleon cross-section σNN onto the transverse plane [37], and L is calculated through the same algorithm as S. 10k events are run to generate the data. Averages are found using the impact parameter b as the weight factor.
    centrality 0-5% 5%-10% 10%-20% 20%-30% 30%-40% 40%-50% 50%-60% 60%-70% 70%-80%
    Au
    Ncoll 1049.8 843.9 594.8 369.1 217.4 121.6 62.2 29.2 12.7
    S/fm2 147.9 128.9 106.1 83.0 64.8 49.7 36.6 25.5 16.2
    L/fm 13.2 11.9 10.3 8.6 7.3 6.1 5.0 4.1 3.1
    Ru
    Ncoll 387.5 316.3 228.9 146.6 90.9 53.6 30.0 15.8 8.1
    S/fm2 92.5 81.6 67.8 53.8 42.3 32.8 24.7 17.6 12.1
    L/fm 10.5 9.5 8.3 7.0 6.0 5.1 4.3 3.5 2.9
    Zr
    Ncoll 395.6 322.5 232.1 149.0 91.8 54.0 30.1 15.7 8.0
    S/fm2 91.3 80.5 67.0 53.1 41.8 32.5 24.4 17.4 11.9
    L/fm 10.4 9.4 8.2 7.0 6.0 5.0 4.2 3.5 2.9
    DownLoad: CSV
    Show Table

    The initial temperature is taken as T0=350MeV. These combined give μA36MeV with weak centrality dependence. In contrast, μV is estimated from [38]

    μB(s)a1+s/b,

    (8)

    with a1.27GeV and b4.3GeV at sNN=200GeV, μB27MeV, corresponding to μV9MeV. Taking the peak value of B10m2π, we determine that λeBμV/μA3%. Since the magnetic field decays rapidly with time, a more realistic estimation method for the back-reaction is to use the time-averaged magentic field. Assuming the following functional form of magnetic field [39, 40],

    eB(τ)=eB01+(τ/τB)2,

    (9)

    and averaging between initial time τ0=0.6fm and freeze-out time τ=7fm, we obtain λeBavgμV/μA1% for τB=2fm and λeBavgμV/μA0.4% for τB=1fm. Therefore, we can safely neglect the CSE effect on axial charge redistribution. Similar analysis shows that the same is true for the isobar collisions.

    Since the back-reaction from vector charge is negligible, we can trace the evolution of the total axial charge and use it to determine the average μA for CME phenomenology. In [30], we derived the hydrodynamic evolution of the total axial charge with an initial value. It is given by

    NA(τ)2=NA(τ0)2e3(1(ττ0)2/3)(τ0τCS0)+dηd2x2Γ0τ0τCS0(1e3(1(ττ0)2/3)(τ0τCS0)).

    (10)

    The initial conditions for the AuAu collisions at sNN=200GeV has been discussed in the previous subsection. The counterpart for the isobars scales accordingly. We adopt the scaling of Qs with the system size from [41] and the initial time for Bjorken hydrodynamics from [42]. The freeze-out time is determined by the same freeze-out temperature, Tf=154MeV. We list the scalings as follows:

    QsA16,T0QsA16,τ01/QsA16,τfA13.

    (11)

    The axial chemical potential is calculated using the average axial charge

    μA(τ)=nA(τ)2χ(τ)=NA(τ)2V(τ) χ(τ),

    (12)

    with V(τ)=SτΔη being the total volume. The rapidity span is taken to be |η|<2 with Δη=4. Consequently, the axial chemical potential is determined as

    μA(τ)=μA0(ττ0)13e3(1(ττ0)2/3)(τ0τCS0)+3T30τ0 Δη S n2A0[1e3(1(ττ0)2/3)(τ0τCS0)],

    (13)

    where the square root factor is a modification to the simple τ1/3 dependence when relaxation of axial charge is ignored. The initial axial chemical potential is determined by the initial axial charge density nA0 given in (7) via μA0=nA0χ0=nA03T20.

    Then, we determine the scalings of the initial axial charge density and chemical potential. From the empirical scaling for the AuAu collisions [33, 37] in the Glauber model,

    SN23part,NcollN43part,

    (14)

    where Npart is the number of participant nucleons, we have

    SNcoll.

    (15)

    Thus, from (7) , nA0 has only weak centrality dependence. The system size dependence of nA0 and μA0 can be easily obtained using (11):

    nA0A12,μA0A16.

    (16)

    The centrality dependence of initial chemical potential μA0 for Au and isobars are listed in Table 2. A weak centrality dependence is observed for AuAu and a slightly enhanced dependence is observed for Ru and Zu, due to the deviation from the empirical scaling (14). The system size dependence (16) is approximately consistent with Table 2.

    Table 2

    Table 2.  Centrality dependence of μA0(MeV).
    centrality 0-5% 5%-10% 10%-20% 20%-30% 30%-40% 40%-50% 50%-60% 60%-70% 70%-80%
    Au 36.11 37.15 37.90 38.14 37.53 36.56 35.49 34.97 36.19
    Ru 31.13 31.89 32.63 32.93 32.99 32.63 32.45 33.06 34.55
    Zr 31.85 32.62 33.29 33.62 33.51 33.08 32.89 33.35 34.84
    DownLoad: CSV
    Show Table

    Now we can calculate the chiral magnetic current using (1), whose time integral gives the total charge separation

    Qe=τfτ0dτ τdηL CeμA eB=CeΔη Lτfτ0dτ τμA(τ) eB(τ),

    (17)

    where Ce=fq2feNc2π2 and L is the width of the participants' region along the cross-line between the transverse plane and reaction plane, sampled from the MC-Glauber Model, as seen in Table 1. Hence, τdηL represents the area that the CME current penetrates in the reaction plane. We integrate it from initial thermalization time τ0 to freeze-out time τf, whose values are determined in (11). The effective electric chemical potential is then induced by the total electric charge asymmetry as,

    μe(τf)=QeVf χe(τf)=3Lπ2 eSτf T2fτfτ0dττμA(τ) eB(τ),

    (18)

    where Vf=SτfΔη/2 and χe(τf)=13fq2fNcTf2 denote the volume of QGP above the reaction plane and the electric charge susceptibility at freeze-out time, respectively.

    The magnetic field in the lab frame is calculated from the Liénard-Wiechert potentials as

    eB(t,r)=e24πdr3ρZ(r) 1v2[R2(R×v)2]3/2v×R ,

    (19)

    where R=rr(t) is the vector pointing from the proton position r(t) at time t to the position r of the field point. v is the velocity of the protons, chosen to be v2=1(2mN/sNN)2, where sNN/2 is the energy for each nucleon in the center-of-mass frame and mN is the mass of the nucleon. The impact parameter vector is set to be along the x-axis, so that the xz plane serves as the reaction plane and xy as the transverse plane. We sample the positions of protons in a nucleus in the rest frame by the Woods-Saxon distribution,

    ρZ(r)11+exp(rR0a),

    (20)

    where R0=6.38fm and a=0.535fm for Au, and R0=5.085fm and 5.020fm for Ru and Zr, respectively, with a=0.46fm for both isobars. The homogeneity and boost-invariance of the magnetic field is assumed, and the power-decaying form follows from (9) with the peak value eB0 set by (19) at t=r=0 along the y-axis. The dependence on nucleus shape discussed in [43] is not included in our analysis. As a result, the centrality dependence of eB0 for Au, Ru, and Zr are shown in Fig. 1. We see that the magnitude of the magnetic field is suggested by the proton numbers of the corresponding nucleus, and that the difference between isobars is indicated as 10%.

    Figure 1

    Figure 1.  (color online) Centrality dependence of the event-averaged magnetic field oriented out of the reaction plane, with triangles for Au, squares for Ru, and circles for Zr.

    The characteristic decay time of the magnetic field τB has a large uncertainty in various models [44-46]. We treat it as a fitting parameter, fixing it by matching the CME signal for AuAu collisions calculated in our model to the flow-excluded charge separation measured by the STAR collaboration at sNN=200GeV [5], as discussed in Section 3.3. Thus, τB=1.65fm. We will assume the same τB for isobars at the same collision energy, and use our model to make predictions for CME signals for Ru and Zr.

    Finally, we obtain eμe for different centralities in Fig. 2. Despite the system of AuAu having larger μA0 and eB, it gives smaller eμe than the systems of Ru and Zr. This is due to the larger volume factor in (18). We will obtain the scaling in the following subsection.

    Figure 2

    Figure 2.  (color online) Centrality dependence of the event-averaged electric chemical potentials induced by the chiral magnetic effect, with triangles for Au, squares for Ru, and circles for Zr.

    To determine the scalings of the magnitude of the electric chemical potential for different heavy ions, we substitute (9) and (13) into (18), and sort it into several blocks as

    μe(τf)=3π2 T2f LB0S 1τf τfτ0dτ τ1+(τ/τB)2 μA0(ττ0)13e3(1(ττ0)2/3)(τ0τCS0)+3T30τ0 Δη S n2A0[1e3(1(ττ0)2/3)(τ0τCS0)].

    (21)

    The first block, 3π2 T2f, holds identically for the three types of nucleus. The second block, LB0S, is determined entirely from the geometry of the nuclei, that is, the distribution of nucleons. The third block, 1τf τfτ0 , accounting for the integral average, scales as τfτ0τf1. The fourth block, μA0(ττ0)13, is determined by the initial condition from the glasma, which we already discussed in Section 2. The square root factor accounts for the damping and fluctuation in our stochastic model.

    We first determine the scaling of the geometrical term LB0S. Throughout the following analysis, the empirical proportionality relationship R0A1/3 is implied. For L and S, the geometrical property from the Glauber model is straightforward,

    SR20A2/3,LR0A1/3,

    (22)

    which is also in agreement with (14), if we assume that the number of participants scales with the volume NpartR3A.

    To analyze the magnetic field, we have to know its dependence on the centrality. Note that (19) is the dependence on the impact parameter, but at a given centrality, the averaged impact parameter is different for each type of nuclei. Since we are comparing the signal in each fixed centralities, we have to know how the averaged impact parameter scales for different nuclei in each centrality.

    Following from [33], the distribution of the total cross section σtot holds well for b<2R0,

    dσtotdb2πb,

    (23)

    thus, the total cross section scales is given as,

    σtotR0bdbR20A2/3,

    (24)

    which is a reasonable scaling in term of dimensions. From [47], the following geometric relation between centrality c and impact parameter b also holds to a very high precision for b<2R0,

    b(c)cσtotπ.

    (25)

    Thus, for a given centrality c, the average impact parameter for different nucleus scales with

    b(c)σ1/2totA1/3.

    (26)

    To proceed to determine the scaling of the magnetic field, we take the multiple-pole expansion of (19) and treat the monopole as our scaling of the magnetic field for different nucleus at a given centrality c, thus it is given by

    B0(c)Z/b(c)2ZA2/3.

    (27)

    Therefore, the geometrical combination block scales is given as

    LB0SZA.

    (28)

    Next, we look at the chemical potential block, without damping and fluctuation effects. The scaling of the initial chemical potential is already discussed in Section 2, and is μA0A1/6, but considering the volume expansion, which contains τ0, it scales as

    μA0(ττ0)13A1/9.

    (29)

    Lastly, the most ambiguous block is the square root factor accounting for the damping and fluctuation effect. From the above analysis, the scaling of the fluctuation is set by

    3T30τ0 Δη S n2A0A1.

    (30)

    However, fluctuation is generally small compared to initial contribution from the glasma. If we neglect it, the square root factor simply scales with 1. Incorporating the contributions from both of them, we may write the scaling of the square root factor as Aζ, with 0<ζ<12.

    Combining all of the above terms together, we have the scaling of the electric chemical potential as

    μe(τf)(ZA)A19Aζ=ZA(ζ+89),

    (31)

    with 0<ζ<12. When we consider only the CME from the initial condition, ζ=0; when we consider only the fluctuation effect ζ=12. Otherwise, ζ lies between them. Our numerical data for Au and isobars suggest a rough value of ζ14. However, there are deviations in each centrality, mainly due to our simplified scaling of the magnetic field using the monopole.

    To proceed, we will first employ the Cooper-Frye freeze-out procedure [48] to obtain the spectrum of the single particle distribution,

    EdNd3p=g(2π)3pμd3σμf(x,p),

    (32)

    where g is the degeneracy factor, taken to be 1 for each species of mesons (K±, π±) produced in QGP. The 4-momentum of the particle and Bjorken spacetime 4-velocity are given by

    pμ=(mcoshy, p,msinhy),uμ=(coshη,0,0,sinhη),

    (33)

    with m=p2+m2. Note that y is the particle rapidity and η is the spacetime rapidity. Thus, we could expand the Cooper-Frye formula to

    dNdϕdypdp=g(2π)3τfdηd2x mcosh(ηy)f(x,p).

    (34)

    The phase-space distribution of the i-th particle species at freeze-out time is given in Boltzmann approximation as,

    fi(x,p)=e(pμuμ±eμe(τf)+μi)/Tf,

    (35)

    where ±μe(τf) is the positive or negative electric chemical potential at freeze-out time caused by CME, as in Fig. 2, which is much smaller than the freeze-out temperature Tf154MeV [49], and μi is the chemical potential for the i-th species. Here, we consider only pions and kaons in our calculations with respect to heavy ion collisions, with μπ80MeV for pions and μK180MeV for kaons. Thus, we can approximate the distribution to the lowest order in μe as

    δfi(x,p)=fi(μe=0) ±eμe(τf)Tf,

    (36)

    which leads to the azimuthal distribution of the ith positive or negative charged particle Ni± created from CME being

    δdNi±dϕ=gi S(2π)3dmm2τfdydη cosh(ηy)×fi(μe=0) ±eμe(τf)Tf.

    (37)

    Here, we used the fact that pdp=mdm. The lower bound of m integration is the rest mass of the corresponding meson. The integration domain for particle rapidity should be taken according to the experiments as |y|<1, and the spacetime rapidity as |η|<2. Note the sign difference on the right hand side of the above equation; the charge asymmetry of the particle distribution is due to CME. Since the magnetic field points to the upper half of the QGP region from the lower half across the reaction plane, positive charge accumulates in the upper region and negative charge in the lower one. Therefore, μe changes sign across the reaction plane. Similarly, the multiplicity of charged particles from the background is obtained consistently from (34) as

    dNi±dϕ=gi S(2π)3dmm2τfdydηcosh(ηy)fi(μe=0),

    (38)

    where there is no sign difference between positive and negative charges, indicating that the background is electrically neutral.

    To acquire the total charged particle multiplicity from CME Δ± and from the neutral background Nbg±, index i should be summed over different species. Thus, we define

    Δ±iδNi±,Nbg±iNi±,

    (39)

    where again ± denotes positive or negative charge. Note that since we assume the whole QGP to be electrically neutral, the fluctuation of the electric chemical potential is averaged to be zero, μe(τf)=0, but the two-point correlation is taken to be the square of the electric chemical potential itself, μe(τf)2μe(τf)2. Further, note that our electric chemical potential μe calculated in Section 2 is an effective quantity; it is not η-dependent and decouples in the integrals. Then, from (37), (38) and (39), denoting α,β=± and σ±=±1, we have the following average and proportionality relations:

    Δα=0,ΔαΔβNbgαNbgβσασβ(eμe(τf))2T2f.

    (40)

    The average relation on the left is interpreted straightforwardly as the conservation of electric charge. The proportionality relation on the right is a measurement of the asymmetry. The CME induced term Δ± is treated as a perturbation to the electrically neutral background as heat bath with temperature Tf.

    Subsequently, we analyze the background angular distribution dN±/dϕ, which reflects the charge-independent evolution of the medium determined by the event-by-event fluctuating initial state. Pursuant to this, we take the Fourier expansion of the background angular distribution as

    dNbg±dϕ=Nbg±2π[1+2n=1vncosn(ϕΨn)],

    (41)

    where Ψn indicates the participant plane angle of order n. Note that we have dropped the sine term in the Fourier decomposition because the distribution is symmetric about the participant plane. Coefficient vn is defined as the nth order harmonic flow. Typically, the directed flow v1 is generally chosen to be 0 if the distribution is measured in a symmetric rapidity region [13, 50]. Therefore, in the following calculation, we only retained the next leading term from the elliptic flow v2.

    Next, we assume the following ansatz [1] for the total generated charged single-particle spectrum originating from both the background and the CME:

    dN±dϕ=dNbg±dϕ+14Δ±sin(ϕΨRP),

    (42)

    where the form of the CME-induced term is proportional to sin(ϕΨRP) owing to the symmetry of the distribution about the magnetic field, which is perpendicular to the reaction plane, and where the factor 1/4 is consistent with our definition (39).

    In contrast to our previous work [30], we choose our correlated two-particle spectrum not only as a product of the single spectrum, but also to include an underlying correlation term proposed in [13] as

    ρ(ϕ1,ϕ2)=dNαdϕα1dNβdϕβ2[1+n=0ancosn(ϕ1ϕ2)],

    (43)

    with α,β=±. The cosine correlation term is reaction-plane-insensitive. Here, we only consider the leading term a1 (with normalization leading to a0=0).

    By employing all these values, two types of two particle correlations, γ and δ, which are measured in the heavy-ion collision experiments, are given as

    {γαβ=cos(ϕα1+ϕβ22ΨRP)δαβ=cos(ϕα1ϕβ2),

    (44)

    where the average cosφ of the angle φ=(ϕα1+ ϕβ22ΨRP) or (ϕα1ϕβ2) is taken over events, that is, integrated over ϕ1 and ϕ2 as

    cosφ=ρ(ϕ1,ϕ2) cosφ dϕα1dϕβ2ρ(ϕ1,ϕ2) dϕα1dϕβ2.

    (45)

    This will result in

    {γαβ=v2a1cos2(Ψ2ΨRP)π216ΔαΔβNbgαNbgβδαβ=a12(1+v22)+π216ΔαΔβNbgαNbgβ.

    (46)

    These forms of γ and δ correlators are consistent with the proposal discussed in [5, 13]:

    {γαβ=κv2FαβHαβ,δαβ=Fαβ+Hαβ,

    (47)

    with Fαβ denoting the background and Hαβ denoting the CME contribution, and κ being an undetermined factor ranging from 1 to 2. Therefore, by matching the above sets of equations and using (40), we claim that the CME signal takes the following form:

    Hαβ=π216ΔαΔβNbgαNbgβσασβ π216(eμe(τf))2T2f.

    (48)

    The difference between the same charge correlation HSS and opposite charge correlation HOS is thus expressed as

    (HSSHOS)2π216(eμe(τf))2T2f.

    (49)

    The centrality dependence of 104(HSSHOS) for Au and isobars is shown in Fig. 3. We also plot the signal for AuAu collision at 200GeV with data extracted from STAR, by solving (47) as

    Figure 3

    Figure 3.  (color online) Centrality dependence of the CME signal from our stochastic model for AuAu and isobaric collision at sNN=200GeV, with triangles for Au, squares for Ru, and circles for Zr. We also list the data for AuAu collisions at sNN=200GeV, extracted from STAR [51, 52], with pentacles, for comparison.

    Hαβ=κv2δαβγαβ1+κv2,

    (50)

    where κ is taken to be 1, numerical values of γ and δ are taken from [51], and values of v2 are taken from [52]. Notably, by adjusting the τB parameter, the CME signal from our model is in a good agreement with that from the experiments. Moreover, with the same τB(1.65fm), we predict the signals for Ru and Zr, which are larger than that of Au, due to the square of the scaling of μe(τf) as Z2A2(ζ+89), with roughly ζ14, as we discussed in Section 3.2.

    We have calculated the axial charge evolution using the stochastic hydrodynamics model, and used it to derive the chiral magnetic effect in off-central collisions of AuAu, RuRu, and ZrZr. By matching the results from our model with the background subtracted experimental data, we have fixed the relaxation time for the magnetic field. We use the same relaxation time to make predictions for the CME signal for collisions of RuRu and ZrZr. Two significant results have been obtained in our analysis.

    First, while the axial and vector charges are coupled through the chiral magnetic effect and chiral separation effect, we found that the influence of vector charge on axial charge is negligible at top RHIC collision energy. This allows us to decouple the evolution of axial charge from the vector charge.

    Secondly, we study the centrality and system size dependencies of the CME signal. The initial chiral imbalance μA0 is found to have only weak centrality dependence. The centrality dependence of the CME signal mainly results from the magnetic field and QGP volume factor. As for the system size dependence, although larger systems provide enhanced magnetic field and chiral imbalance, the electric charge asymmetry characterized by eμe is suppressed due to the larger volume factor. Consequently, we found larger absolute charged particle correlation in isobar collisions than in AuAu collisions.

    The present study readily generalizes to collisions of large nuclei at higher energies, where we expect that Bjorken flow approximation will still apply. It would be interesting to see if the energy dependence matches with current experimental data at different energies. At lower energies, the Bjorken flow approximation becomes inaccurate. One possible approach to address this issue is to implement stochastic noise numerically in the existing AVFD model. We will report studies along these lines in the future.

    We are grateful to Huanzhong Huang, Guoliang Ma, Dirk Rischke, and Gang Wang for discussions. G.R.L also acknowledges the Institute for Theoretical Physics at Frankfurt University for the warm hospitality where part of this work has been done.

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  • [1] D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, Nucl. Phys. A, 803: 227-253 (2008) doi: 10.1016/j.nuclphysa.2008.02.298
    [2] K. Fukushima, D. E. Kharzeev, and H. J. Warringa, Phys. Rev. D, 78: 074033 (2008) doi: 10.1103/PhysRevD.78.074033
    [3] L. Adamczyk et al., Phys. Rev. C, 88(6): 064911 (2013) doi: 10.1103/PhysRevC.88.064911
    [4] B. Abelev et al., Phys. Rev. Lett., 110(1): 012301 (2013) doi: 10.1103/PhysRevLett.110.012301
    [5] L. Adamczyk et al., Phys. Rev. Lett., 113: 052302 (2014) doi: 10.1103/PhysRevLett.113.052302
    [6] D. Kharzeev, J. Liao, S. Voloshin et al., Prog. Part. Nucl. Phys., 88: 1-28 (2016) doi: 10.1016/j.ppnp.2016.01.001
    [7] J. Zhao and F. Wang, Prog. Part. Nucl. Phys., 107: 200-236 (2019) doi: 10.1016/j.ppnp.2019.05.001
    [8] X.-G. Huang, Rept. Prog. Phys., 79(7): 076302 (2016) doi: 10.1088/0034-4885/79/7/076302
    [9] J. Liao, Pramana, 84(5): 901-926 (2015) doi: 10.1007/s12043-015-0984-x
    [10] S. Schlichting and S. Pratt, Phys. Rev. C, 83: 014913 (2011)
    [11] A. Bzdak, V. Koch, and J. Liao, Phys. Rev. C, 83: 014905 (2011) doi: 10.1103/PhysRevC.83.014905
    [12] F. Wang, Phys. Rev. C, 81: 064902 (2010) doi: 10.1103/PhysRevC.81.064902
    [13] A. Bzdak, V. Koch, and J. Liao, in Strongly Interacting Matter in Magnetic Fields, edited by D. Kharzeev, K. Landsteiner, A. Schmitt et al., (Springer-Verlag Berlin Heidelberg, 2013), pp. 503-536
    [14] F. Wen, J. Bryon, L. Wen et al., Chin. Phys. C, 42(1): 014001 (2018)
    [15] H.-j. Xu, J. Zhao, X. Wang et al., Chin. Phys. C, 42(8): 084103 (2018) doi: 10.1088/1674-1137/42/8/084103
    [16] N. Magdy, S. Shi, J. Liao et al., Phys. Rev. C, 97(6): 061901 (2018) doi: 10.1103/PhysRevC.97.061901
    [17] V. Koch, S. Schlichting, V. Skokov et al., Chin. Phys. C, 41(7): 072001 (2017)
    [18] K. Fukushima, D. E. Kharzeev, and H. J. Warringa, Phys. Rev. Lett., 104: 212001 (2010) doi: 10.1103/PhysRevLett.104.212001
    [19] M. Mace, S. Schlichting, and R. Venugopalan, Phys. Rev. D, 93(7): 074036 (2016) doi: 10.1103/PhysRevD.93.074036
    [20] V. Skokov, A. Illarionov, and V. Toneev, Int. J. Mod. Phys. A, 24: 5925-5932 (2009) doi: 10.1142/S0217751X09047570
    [21] Y. Jiang, S. Shi, Y. Yin et al., Chin. Phys. C, 42(1): 011001 (2018) doi: 10.1088/1674-1137/42/1/011001
    [22] S. Shi, Y. Jiang, E. Lilleskov et al., Annals Phys., 394: 50-72 (2018) doi: 10.1016/j.aop.2018.04.026
    [23] Y. Hirono, T. Hirano, and D. E. Kharzeev, arXiv: hepph/1412.0311
    [24] S. Shi, H. Zhang, D. Hou et al., arXiv: nucl-th/1910.14010
    [25] A. Huang, Y. Jiang, S. Shi et al., Phys. Lett. B, 777: 177-183 (2018) doi: 10.1016/j.physletb.2017.12.025
    [26] Y. Sun, C. M. Ko, and F. Li, Phys. Rev. C, 94(4): 045204 (2016)
    [27] Y. Sun and C. M. Ko, Phys. Rev. C, 95(3): 034909 (2017) doi: 10.1103/PhysRevC.95.034909
    [28] W.-T. Deng, X.-G. Huang, G.-L. Ma et al., Phys. Rev. C, 94: 041901 (2016) doi: 10.1103/PhysRevC.94.041901
    [29] G.-L. Ma and B. Zhang, Phys. Lett. B, 700: 39-43 (2011) doi: 10.1016/j.physletb.2011.04.057
    [30] S. Lin, L. Yan, and G.-R. Liang, Phys. Rev. C, 98(1): 014903 (2018) doi: 10.1103/PhysRevC.98.014903
    [31] S. Lin and L. Yang, Phys. Rev. D, 98(11): 114022 (2018) doi: 10.1103/PhysRevD.98.114022
    [32] G. D. Moore and M. Tassler, JHEP, 02: 105 (2011)
    [33] M. L. Miller, K. Reygers, S. J. Sanders et al., Ann. Rev. Nucl. Part. Sci., 57: 205-243 (2007) doi: 10.1146/annurev.nucl.57.090506.123020
    [34] B. Alver, M. Baker, C. Loizides et al., arXiv: nucl-ex/0805.4411
    [35] C. Loizides, J. Nagle, and P. Steinberg, SoftwareX, 1-2: 13-18 (2015)
    [36] C. Loizides, J. Kamin, and D. d’Enterria, Phys. Rev. C, 97(5): 054910 (2018) doi: 10.1103/PhysRevC.97.054910
    [37] B. Abelev et al., Phys. Rev. C, 79: 034909 (2009) doi: 10.1103/PhysRevC.79.034909
    [38] P. Braun-Munzinger, J. Cleymans, H. Oeschler et al., Nucl. Phys. A, 697: 902-912 (2002)
    [39] Y. Yin and J. Liao, Phys. Lett. B, 756: 42-46 (2016) doi: 10.1016/j.physletb.2016.02.065
    [40] H.-U. Yee and Y. Yin, Phys. Rev. C, 89(4): 044909 (2014) doi: 10.1103/PhysRevC.89.044909
    [41] D. Kharzeev and M. Nardi, Phys. Lett. B, 507: 121-128 (2001) doi: 10.1016/S0370-2693(01)00457-9
    [42] G. Başar and D. Teaney, Phys. Rev. C, 90(5): 054903 (2014) doi: 10.1103/PhysRevC.90.054903
    [43] H.-J. Xu, X. Wang, H. Li et al., Phys. Rev. Lett., 121(2): 022301 (2018) doi: 10.1103/PhysRevLett.121.022301
    [44] L. McLerran and V. Skokov, Nucl. Phys. A, 929: 184-190 (2014) doi: 10.1016/j.nuclphysa.2014.05.008
    [45] U. Gursoy, D. Kharzeev, and K. Rajagopal, Phys. Rev. C, 89(5): 054905 (2014) doi: 10.1103/PhysRevC.89.054905
    [46] K. Tuchin, Phys. Rev. C, 93(1): 014905 (2016) doi: 10.1103/PhysRevC.93.014905
    [47] W. Broniowski and W. Florkowski, Phys. Rev. C, 65: 024905 (2002) doi: 10.1103/PhysRevC.65.024905
    [48] F. Cooper and G. Frye, Phys. Rev. D, 10: 186 (1974)
    [49] D. Teaney, arXiv: nucl-th/0204023
    [50] S. A. Voloshin, Phys. Rev. C, 70: 057901 (2004) doi: 10.1103/PhysRevC.70.057901
    [51] B. Abelev et al., Phys. Rev. C, 81: 054908 (2010) doi: 10.1103/PhysRevC.81.054908
    [52] G. Agakishiev et al., Phys. Rev. C, 86: 014904 (2012) doi: 10.1103/PhysRevC.86.014904
  • 加载中

Cited by

1. Kharzeev, D.E., Liao, J., Tribedy, P. Chiral magnetic e®ect in heavy ion collisions: The present and future[J]. International Journal of Modern Physics E, 2024, 33(9): 2430007. doi: 10.1142/S0218301324300078
2. Grieninger, S., Morales-Tejera, S. Real-time dynamics of axial charge and chiral magnetic current in a non-Abelian expanding plasma[J]. Physical Review D, 2023, 108(12): 126010. doi: 10.1103/PhysRevD.108.126010
3. Liu, H., Chu, P.-C. Elliptic flow splitting of charged pions in relativistic heavy-ion collisions | [相对论重离子碰撞中 π 介子椭圆流劈裂][J]. Wuli Xuebao/Acta Physica Sinica, 2023, 72(13): 132101. doi: 10.7498/aps.72.20230454
4. Zhao, X.-L., Ma, G.-L., Ma, Y.-G. Electromagnetic field effects and anomalous chiral phenomena in heavy-ion collisions at intermediate and high energy | [中高能重离子碰撞中的电磁场效应和手征反常现象][J]. Wuli Xuebao/Acta Physica Sinica, 2023, 72(11): 20230245. doi: 10.7498/aps.72.20230245
5. Huang, A., Shi, S., Lin, S. et al. Accessing topological fluctuations of gauge fields with the chiral magnetic effect[J]. Physical Review D, 2023, 107(3): 034012. doi: 10.1103/PhysRevD.107.034012
6. An, X., Bluhm, M., Du, L. et al. The BEST framework for the search for the QCD critical point and the chiral magnetic effect[J]. Nuclear Physics A, 2022. doi: 10.1016/j.nuclphysa.2021.122343
7. Becattini, F., Buzzegoli, M., Palermo, A. et al. Polarization as a signature of local parity violation in hot QCD matter[J]. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 2021. doi: 10.1016/j.physletb.2021.136706
8. Fang, R.-H., Dong, R.-D., Hou, D.-F. et al. Thermodynamics of the System of Massive Dirac Fermions in a Uniform Magnetic Field[J]. Chinese Physics Letters, 2021, 38(9): 091201. doi: 10.1088/0256-307X/38/9/091201
9. Buividovich, P.V., Smith, D., von Smekal, L. Numerical study of the chiral separation effect in two-color QCD at finite density[J]. Physical Review D, 2021, 104(1): 014511. doi: 10.1103/PhysRevD.104.014511
10. Bu, Y., Demircik, T., Lublinsky, M. All order effective action for charge diffusion from Schwinger-Keldysh holography[J]. Journal of High Energy Physics, 2021, 2021(5): 187. doi: 10.1007/JHEP05(2021)187
11. Giacalone, G.. Turning up and down strong magnetic fields in relativistic nuclear collisions[J]. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 2020. doi: 10.1016/j.physletb.2020.135915
12. Liang, G.-R., Li, M. Charge-dependent correlations in heavy-ion collisions from stochastic hydrodynamics[J]. Communications in Theoretical Physics, 2020, 72(11): 115304. doi: 10.1088/1572-9494/abb7d9

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Gui-Rong Liang, Jinfeng Liao, Shu Lin, Li Yan and Miao Li. Chiral Magnetic Effect in Isobar Collisions from Stochastic Hydrodynamics[J]. Chinese Physics C. doi: 10.1088/1674-1137/44/9/094103
Gui-Rong Liang, Jinfeng Liao, Shu Lin, Li Yan and Miao Li. Chiral Magnetic Effect in Isobar Collisions from Stochastic Hydrodynamics[J]. Chinese Physics C.  doi: 10.1088/1674-1137/44/9/094103 shu
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Chiral magnetic effect in isobar collisions from stochastic hydrodynamics

  • 1. School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai 519082, China
  • 2. Physics Department and Center for Exploration of Energy and Matter, Indiana University, 2401 N Milo B. Sampson Lane, Bloomington, Indiana 47408, USA
  • 3. Key laboratory of Nuclear Physics and Ion-beam Application (MOE) & Institute of Modern Physics, Fudan University, Shanghai 200433, China
  • 4. Department of Physics, Southern University of Science and Technology, Shenzhen 518055, Guangdong, China

Abstract: We studied the chiral magnetic effect in AuAu, RuRu, and ZrZr collisions at sNN=200GeV. The axial charge evolution was modeled with stochastic hydrodynamics, and geometrical quantities were calculated with the Monte Carlo Glauber model. By adjusting the relaxation time of the magnetic field, we found our results are in good agreement with background subtracted data for AuAu collisions at the same energy. We also made predictions for RuRu and ZrZr collisions. We found a weak centrality dependence on initial chiral imbalance, which implies that the centrality dependence of chiral magnetic effect signals results mainly from the effects of the magnetic field and volume factor. Furthermore, our results show an unexpected dependence on system size. While the AuAu system has larger chiral imbalance and magnetic field, it was observed to have a smaller signal for the chiral magnetic effect due to the larger volume suppression factor.

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    1.   Introduction
    • The anomalous transport of chiral magnetic effect (CME) has gained significant attention over the past few years [1, 2]. If the local parity odd domain is present in the quark-gluon plasma produced in heavy ion collisions, CME leads to charge separation along the magnetic field generated in off-central collisions:

      je=fNcq2f2π2μAB,

      (1)

      where the chiral imbalance, μA, characterizes local parity violation. This offers the possibility of detecting local parity violation in quantum chromodynamics (QCD). Charge separation has been actively sought experimentally [3-5]. However, we are still far from consensus on the status of CME, largely due to the difficulty in determining CME both experimentally and theoretically; see [6-9] for recent reviews. Experimentally, charge separation needs to be measured through charged hadron correlation on an event-by-event basis. Unfortunately, charged hadron correlation is dominated by flow-related background with various possible origins [10-12]. Various observables and experimental techniques have been proposed and implemented to exclude flow-related background [13-16]. In addition, STAR collaboration proposes to search for CME in isobar collisions [17]. Since isobars have the same atomic number but different proton numbers, the corresponding collisions are supposed to generate the same flow background, but different magnetic field, and thus, different charge separation. This can unambiguously distinguish the CME contribution.

      Theoretically describing CME is also difficult. Both μA and B contain large uncertainties. Their peak values are known to be set by the axial charge production in glasma phase [18, 19] and the moving charge of spectators [20], respectively. However, their further evolution is model dependent. Different theoretical frameworks, such as anomalous viscous fluid dynamics (AVFD) [21-24], chiral kinetic theory [25-27], and the multiphase transport model [28, 29], have been employed to study the time evolution of axial/vector charges. All these frameworks treat axial charge as an approximately conserved quantity in the absence of parallel electric and magnetic fields. However, axial charges are not conserved, due to gluon dynamics. In fact, it is the same origin for the initial axial charge. In [30], the authors incorporated fluctuation and dissipation of axial charge in the framework of stochastic hydrodynamics. We found that independent of the initial conditions, the variance of axial charge always approaches the thermodynamic limit after sufficient time, due to the interplay of fluctuation and dissipation. In [30], we used the thermodynamic limit for the axial charge to model CME. While being model-independent, the study missed an important fact: most charge separation occurs at early stages of quark-gluon plasma formation, when μAand B have not decayed appreciably. This study aims to incorporate the initial axial charge and investigate the coupled dynamics of axial and vector charge. In particular, we make predictions for the CME contribution to isobar collisions.

      This paper is organized as follows. In Sec. 2, we generalize the stochastic hydrodynamics framework to include axial and vector charge, which are coupled through anomalous effects in the presence of the magnetic field. Subsequently, we will justify the claim that for phenomenologically relevant magnetic fields, the back-reaction of vector charge to axial charge is negligible. In Sec. 3, we derive axial charge evolution with a non-vanishing initial value. The obtained axial charge is used to calculate charge separation. We make predictions for CME in isobar collisions, using AuAu collisions as a reference. We conclude and discuss directions for future work in Sec. 4.

    2.   Stochastic hydrodynamics for axial and vector charges
    • The stochastic hydrodynamic equations for axial charge in the absence of magnetic field are given in [30]. In a magnetic field, axial charge couples to vector charge through the chiral magnetic effect and chiral separation effect (CSE). The full stochastic hydrodynamic equations for axial and vector charges are

      {μJμA=nAτCS2ξqJμA=nAuμ+λnVeBμσTPμνν(nAχAT)+ξμA,

      (2)

      and

      {μJμV=0JμV=nVuμ+λnAeBμσTPμνν(nVχVT)+ξμV.

      (3)

      nA and nV are axial and vector charge density, respectively. The axial current is not conserved, due to the topological configuration of gluons. This leads to the dissipative term, nAτCS, and fluctuating noise term, ξq. The constitutive equations for axial and vector current consist of a co-moving term, an anomalous mixing term, a diffusive term, and a thermal noise term. uμ is the fluid velocity, which defines the projection operator, Pμν=gμν+uμuν and the magnetic field in the fluid cell, Bμ=12ϵμναβgFαβuν. ξA, ξV, and ξq are considered as Gaussian white noises:

      ξμA(x)ξνA(x)=Pμν2σATd4(xx)g,ξμV(x)ξνV(x)=Pμν2σVTd4(xx)g,ξq(x)ξνq(x)=ΓCSd4(xx)g,ξμA(x)ξq(x)=ξνV(x)ξq(x)=0,

      (4)

      with ΓCS being the Chern-Simon diffusion constant characterizing the magnitude of topological fluctuation.

      For application to CME in heavy ion collisions, we fix the parameters as follows. We use the free theory limit for axial and vector charge susceptibilities, χA=χV=χ=NfNcT2/3. The coefficient of the mixing term, λ, is determined by the chiral magnetic/separation effect as λ=1χNc2π2. The quark mass effect on CSE can be neglected [31]. For three flavours, we have χ=3T2 and λ=12π2T2. ΓCS is the Chern-Simon diffusion constant, obtained from the extrapolated weak coupling results: ΓCS=30α4sT4 [32] with αs=0.3. The relaxation time of the axial charge, τCS, is fixed by the Einstein relation: τCS=χT2ΓCS. σA and σV are the conductivities for the axial and vector currents, and are not required in our analysis.

      The axial/vector charge is treated as a perturbation in the background hydrodynamic flow. We consider heavy ion collisions at the top RHIC collision energy, sNN=200GeV, using Bjorken flow as the background. In the Milne coordinates (τ,η,x,y), the fluid velocity is uμ=(1,0,0,0). We can show that the total axial charge is conserved up to a mixing term and the topological fluctuation-induced terms. To do so, we substitute the constitutive equation into the conservation equation in (2) and integrate over the volume, τdηd2x=gdηd2x. The identity, μVμ=1gμ(gVμ), dropping the boundary terms, yields

      dηd2x(τ(τnA)τ(σATPτνν(nAχT))+τ(τξτA))=dηd2x(τnAτCS2τξq).

      (5)

      Note that Pμνuν=0 and ξμAξνAPμν. Thus, Pτν=0 and ξτA=0. We then arrive at

      τNA=NAτCSdηd2x2τξq,

      (6)

      with NA=τdηd2x. The absence of the diffusive term, thermal noise term, and mixing term is consistent with the fact that these terms only lead to the redistribution of axial charge. The counterpart for the vector charge is simpler: τNV=0, because vector charge is strictly conserved.

    • 2.1.   Back-reaction from the vector current

    • We assume that the initial axial charge created by the chromo flux tube is homogeneous in the transverse plane. The boost-invariant Bjorken expansion maintains a homogeneous distribution in the longitudinal direction. The homogeneous axial charge leads to charge separation via CME. This simplified picture is modified by three effects: diffusion, thermal noise, and CSE. The thermal noise and diffusion correspond to the fluctuation and dissipation of charge, which bring the charge to equilibrium. The CSE is not balanced by other effects. We will now show that its effect is sub-leading.

      Let us compare the axial charge, nA, and CSE modification, λBnV. Since χA=χV, it is equivalent to comparing μA and λBμV. Since B drops quickly with time, the CSE effect is maximized at initial time. We estimate the initial nA following [21] as

      nA(τ0)2Q4s(πρ2tubeτ0)Ncoll16π2S,

      (7)

      where Qs is the saturation scale and ρtube1fm is the width of the flux tube. τ0 is the initial proper time. For AuAu collisions, we employ Qs1GeV and τ0=0.6fm. The number of binary collisions, Ncoll, and transverse overlap area, S, are calculated using a Monte Carlo Glauber model [33-36] with the centrality dependence listed in Table 1.

      centrality 0-5% 5%-10% 10%-20% 20%-30% 30%-40% 40%-50% 50%-60% 60%-70% 70%-80%
      Au
      Ncoll 1049.8 843.9 594.8 369.1 217.4 121.6 62.2 29.2 12.7
      S/fm2 147.9 128.9 106.1 83.0 64.8 49.7 36.6 25.5 16.2
      L/fm 13.2 11.9 10.3 8.6 7.3 6.1 5.0 4.1 3.1
      Ru
      Ncoll 387.5 316.3 228.9 146.6 90.9 53.6 30.0 15.8 8.1
      S/fm2 92.5 81.6 67.8 53.8 42.3 32.8 24.7 17.6 12.1
      L/fm 10.5 9.5 8.3 7.0 6.0 5.1 4.3 3.5 2.9
      Zr
      Ncoll 395.6 322.5 232.1 149.0 91.8 54.0 30.1 15.7 8.0
      S/fm2 91.3 80.5 67.0 53.1 41.8 32.5 24.4 17.4 11.9
      L/fm 10.4 9.4 8.2 7.0 6.0 5.0 4.2 3.5 2.9

      Table 1.  Geometrical quantities from the MC-Glauber model for Au, Ru and Zr. Ncoll, S, and L are the number of binary collisions, transverse overlap area, and width of the participants' region along the cross-line between the transverse and reaction planes, respectively. S is considered to be the projection of the nucleon-nucleon cross-section σNN onto the transverse plane [37], and L is calculated through the same algorithm as S. 10k events are run to generate the data. Averages are found using the impact parameter b as the weight factor.

      The initial temperature is taken as T0=350MeV. These combined give μA36MeV with weak centrality dependence. In contrast, μV is estimated from [38]

      μB(s)a1+s/b,

      (8)

      with a1.27GeV and b4.3GeV at sNN=200GeV, μB27MeV, corresponding to μV9MeV. Taking the peak value of B10m2π, we determine that λeBμV/μA3%. Since the magnetic field decays rapidly with time, a more realistic estimation method for the back-reaction is to use the time-averaged magentic field. Assuming the following functional form of magnetic field [39, 40],

      eB(τ)=eB01+(τ/τB)2,

      (9)

      and averaging between initial time τ0=0.6fm and freeze-out time τ=7fm, we obtain λeBavgμV/μA1% for τB=2fm and λeBavgμV/μA0.4% for τB=1fm. Therefore, we can safely neglect the CSE effect on axial charge redistribution. Similar analysis shows that the same is true for the isobar collisions.

    • 2.2.   Evolution of the axial chemical potential

    • Since the back-reaction from vector charge is negligible, we can trace the evolution of the total axial charge and use it to determine the average μA for CME phenomenology. In [30], we derived the hydrodynamic evolution of the total axial charge with an initial value. It is given by

      NA(τ)2=NA(τ0)2e3(1(ττ0)2/3)(τ0τCS0)+dηd2x2Γ0τ0τCS0(1e3(1(ττ0)2/3)(τ0τCS0)).

      (10)

      The initial conditions for the AuAu collisions at sNN=200GeV has been discussed in the previous subsection. The counterpart for the isobars scales accordingly. We adopt the scaling of Qs with the system size from [41] and the initial time for Bjorken hydrodynamics from [42]. The freeze-out time is determined by the same freeze-out temperature, Tf=154MeV. We list the scalings as follows:

      QsA16,T0QsA16,τ01/QsA16,τfA13.

      (11)

      The axial chemical potential is calculated using the average axial charge

      μA(τ)=nA(τ)2χ(τ)=NA(τ)2V(τ) χ(τ),

      (12)

      with V(τ)=SτΔη being the total volume. The rapidity span is taken to be |η|<2 with Δη=4. Consequently, the axial chemical potential is determined as

      μA(τ)=μA0(ττ0)13e3(1(ττ0)2/3)(τ0τCS0)+3T30τ0 Δη S n2A0[1e3(1(ττ0)2/3)(τ0τCS0)],

      (13)

      where the square root factor is a modification to the simple τ1/3 dependence when relaxation of axial charge is ignored. The initial axial chemical potential is determined by the initial axial charge density nA0 given in (7) via μA0=nA0χ0=nA03T20.

      Then, we determine the scalings of the initial axial charge density and chemical potential. From the empirical scaling for the AuAu collisions [33, 37] in the Glauber model,

      SN23part,NcollN43part,

      (14)

      where Npart is the number of participant nucleons, we have

      SNcoll.

      (15)

      Thus, from (7) , nA0 has only weak centrality dependence. The system size dependence of nA0 and μA0 can be easily obtained using (11):

      nA0A12,μA0A16.

      (16)

      The centrality dependence of initial chemical potential μA0 for Au and isobars are listed in Table 2. A weak centrality dependence is observed for AuAu and a slightly enhanced dependence is observed for Ru and Zu, due to the deviation from the empirical scaling (14). The system size dependence (16) is approximately consistent with Table 2.

      centrality 0-5% 5%-10% 10%-20% 20%-30% 30%-40% 40%-50% 50%-60% 60%-70% 70%-80%
      Au 36.11 37.15 37.90 38.14 37.53 36.56 35.49 34.97 36.19
      Ru 31.13 31.89 32.63 32.93 32.99 32.63 32.45 33.06 34.55
      Zr 31.85 32.62 33.29 33.62 33.51 33.08 32.89 33.35 34.84

      Table 2.  Centrality dependence of μA0(MeV).

    3.   Chiral magnetic effect in isobar collisions

      3.1.   Effective electrical chemical potential for isobars

    • Now we can calculate the chiral magnetic current using (1), whose time integral gives the total charge separation

      Qe=τfτ0dτ τdηL CeμA eB=CeΔη Lτfτ0dτ τμA(τ) eB(τ),

      (17)

      where Ce=fq2feNc2π2 and L is the width of the participants' region along the cross-line between the transverse plane and reaction plane, sampled from the MC-Glauber Model, as seen in Table 1. Hence, τdηL represents the area that the CME current penetrates in the reaction plane. We integrate it from initial thermalization time τ0 to freeze-out time τf, whose values are determined in (11). The effective electric chemical potential is then induced by the total electric charge asymmetry as,

      μe(τf)=QeVf χe(τf)=3Lπ2 eSτf T2fτfτ0dττμA(τ) eB(τ),

      (18)

      where Vf=SτfΔη/2 and χe(τf)=13fq2fNcTf2 denote the volume of QGP above the reaction plane and the electric charge susceptibility at freeze-out time, respectively.

      The magnetic field in the lab frame is calculated from the Liénard-Wiechert potentials as

      eB(t,r)=e24πdr3ρZ(r) 1v2[R2(R×v)2]3/2v×R ,

      (19)

      where R=rr(t) is the vector pointing from the proton position r(t) at time t to the position r of the field point. v is the velocity of the protons, chosen to be v2=1(2mN/sNN)2, where sNN/2 is the energy for each nucleon in the center-of-mass frame and mN is the mass of the nucleon. The impact parameter vector is set to be along the x-axis, so that the xz plane serves as the reaction plane and xy as the transverse plane. We sample the positions of protons in a nucleus in the rest frame by the Woods-Saxon distribution,

      ρZ(r)11+exp(rR0a),

      (20)

      where R0=6.38fm and a=0.535fm for Au, and R0=5.085fm and 5.020fm for Ru and Zr, respectively, with a=0.46fm for both isobars. The homogeneity and boost-invariance of the magnetic field is assumed, and the power-decaying form follows from (9) with the peak value eB0 set by (19) at t=r=0 along the y-axis. The dependence on nucleus shape discussed in [43] is not included in our analysis. As a result, the centrality dependence of eB0 for Au, Ru, and Zr are shown in Fig. 1. We see that the magnitude of the magnetic field is suggested by the proton numbers of the corresponding nucleus, and that the difference between isobars is indicated as 10%.

      Figure 1.  (color online) Centrality dependence of the event-averaged magnetic field oriented out of the reaction plane, with triangles for Au, squares for Ru, and circles for Zr.

      The characteristic decay time of the magnetic field τB has a large uncertainty in various models [44-46]. We treat it as a fitting parameter, fixing it by matching the CME signal for AuAu collisions calculated in our model to the flow-excluded charge separation measured by the STAR collaboration at sNN=200GeV [5], as discussed in Section 3.3. Thus, τB=1.65fm. We will assume the same τB for isobars at the same collision energy, and use our model to make predictions for CME signals for Ru and Zr.

      Finally, we obtain eμe for different centralities in Fig. 2. Despite the system of AuAu having larger μA0 and eB, it gives smaller eμe than the systems of Ru and Zr. This is due to the larger volume factor in (18). We will obtain the scaling in the following subsection.

      Figure 2.  (color online) Centrality dependence of the event-averaged electric chemical potentials induced by the chiral magnetic effect, with triangles for Au, squares for Ru, and circles for Zr.

    • 3.2.   Scaling relationship of the electrical chemical potential for different heavy ions

    • To determine the scalings of the magnitude of the electric chemical potential for different heavy ions, we substitute (9) and (13) into (18), and sort it into several blocks as

      μe(τf)=3π2 T2f LB0S 1τf τfτ0dτ τ1+(τ/τB)2 μA0(ττ0)13e3(1(ττ0)2/3)(τ0τCS0)+3T30τ0 Δη S n2A0[1e3(1(ττ0)2/3)(τ0τCS0)].

      (21)

      The first block, 3π2 T2f, holds identically for the three types of nucleus. The second block, LB0S, is determined entirely from the geometry of the nuclei, that is, the distribution of nucleons. The third block, 1τf τfτ0 , accounting for the integral average, scales as τfτ0τf1. The fourth block, μA0(ττ0)13, is determined by the initial condition from the glasma, which we already discussed in Section 2. The square root factor accounts for the damping and fluctuation in our stochastic model.

      We first determine the scaling of the geometrical term LB0S. Throughout the following analysis, the empirical proportionality relationship R0A1/3 is implied. For L and S, the geometrical property from the Glauber model is straightforward,

      SR20A2/3,LR0A1/3,

      (22)

      which is also in agreement with (14), if we assume that the number of participants scales with the volume NpartR3A.

      To analyze the magnetic field, we have to know its dependence on the centrality. Note that (19) is the dependence on the impact parameter, but at a given centrality, the averaged impact parameter is different for each type of nuclei. Since we are comparing the signal in each fixed centralities, we have to know how the averaged impact parameter scales for different nuclei in each centrality.

      Following from [33], the distribution of the total cross section σtot holds well for b<2R0,

      dσtotdb2πb,

      (23)

      thus, the total cross section scales is given as,

      σtotR0bdbR20A2/3,

      (24)

      which is a reasonable scaling in term of dimensions. From [47], the following geometric relation between centrality c and impact parameter b also holds to a very high precision for b<2R0,

      b(c)cσtotπ.

      (25)

      Thus, for a given centrality c, the average impact parameter for different nucleus scales with

      b(c)σ1/2totA1/3.

      (26)

      To proceed to determine the scaling of the magnetic field, we take the multiple-pole expansion of (19) and treat the monopole as our scaling of the magnetic field for different nucleus at a given centrality c, thus it is given by

      B0(c)Z/b(c)2ZA2/3.

      (27)

      Therefore, the geometrical combination block scales is given as

      LB0SZA.

      (28)

      Next, we look at the chemical potential block, without damping and fluctuation effects. The scaling of the initial chemical potential is already discussed in Section 2, and is μA0A1/6, but considering the volume expansion, which contains τ0, it scales as

      μA0(ττ0)13A1/9.

      (29)

      Lastly, the most ambiguous block is the square root factor accounting for the damping and fluctuation effect. From the above analysis, the scaling of the fluctuation is set by

      3T30τ0 Δη S n2A0A1.

      (30)

      However, fluctuation is generally small compared to initial contribution from the glasma. If we neglect it, the square root factor simply scales with 1. Incorporating the contributions from both of them, we may write the scaling of the square root factor as Aζ, with 0<ζ<12.

      Combining all of the above terms together, we have the scaling of the electric chemical potential as

      μe(τf)(ZA)A19Aζ=ZA(ζ+89),

      (31)

      with 0<ζ<12. When we consider only the CME from the initial condition, ζ=0; when we consider only the fluctuation effect ζ=12. Otherwise, ζ lies between them. Our numerical data for Au and isobars suggest a rough value of ζ14. However, there are deviations in each centrality, mainly due to our simplified scaling of the magnetic field using the monopole.

    • 3.3.   CME signal to be compared in experiments

    • To proceed, we will first employ the Cooper-Frye freeze-out procedure [48] to obtain the spectrum of the single particle distribution,

      EdNd3p=g(2π)3pμd3σμf(x,p),

      (32)

      where g is the degeneracy factor, taken to be 1 for each species of mesons (K±, π±) produced in QGP. The 4-momentum of the particle and Bjorken spacetime 4-velocity are given by

      pμ=(mcoshy, p,msinhy),uμ=(coshη,0,0,sinhη),

      (33)

      with m=p2+m2. Note that y is the particle rapidity and η is the spacetime rapidity. Thus, we could expand the Cooper-Frye formula to

      dNdϕdypdp=g(2π)3τfdηd2x mcosh(ηy)f(x,p).

      (34)

      The phase-space distribution of the i-th particle species at freeze-out time is given in Boltzmann approximation as,

      fi(x,p)=e(pμuμ±eμe(τf)+μi)/Tf,

      (35)

      where ±μe(τf) is the positive or negative electric chemical potential at freeze-out time caused by CME, as in Fig. 2, which is much smaller than the freeze-out temperature Tf154MeV [49], and μi is the chemical potential for the i-th species. Here, we consider only pions and kaons in our calculations with respect to heavy ion collisions, with μπ80MeV for pions and μK180MeV for kaons. Thus, we can approximate the distribution to the lowest order in μe as

      δfi(x,p)=fi(μe=0) ±eμe(τf)Tf,

      (36)

      which leads to the azimuthal distribution of the ith positive or negative charged particle Ni± created from CME being

      δdNi±dϕ=gi S(2π)3dmm2τfdydη cosh(ηy)×fi(μe=0) ±eμe(τf)Tf.

      (37)

      Here, we used the fact that pdp=mdm. The lower bound of m integration is the rest mass of the corresponding meson. The integration domain for particle rapidity should be taken according to the experiments as |y|<1, and the spacetime rapidity as |η|<2. Note the sign difference on the right hand side of the above equation; the charge asymmetry of the particle distribution is due to CME. Since the magnetic field points to the upper half of the QGP region from the lower half across the reaction plane, positive charge accumulates in the upper region and negative charge in the lower one. Therefore, μe changes sign across the reaction plane. Similarly, the multiplicity of charged particles from the background is obtained consistently from (34) as

      dNi±dϕ=gi S(2π)3dmm2τfdydηcosh(ηy)fi(μe=0),

      (38)

      where there is no sign difference between positive and negative charges, indicating that the background is electrically neutral.

      To acquire the total charged particle multiplicity from CME Δ± and from the neutral background Nbg±, index i should be summed over different species. Thus, we define

      Δ±iδNi±,Nbg±iNi±,

      (39)

      where again ± denotes positive or negative charge. Note that since we assume the whole QGP to be electrically neutral, the fluctuation of the electric chemical potential is averaged to be zero, μe(τf)=0, but the two-point correlation is taken to be the square of the electric chemical potential itself, μe(τf)2μe(τf)2. Further, note that our electric chemical potential μe calculated in Section 2 is an effective quantity; it is not η-dependent and decouples in the integrals. Then, from (37), (38) and (39), denoting α,β=± and σ±=±1, we have the following average and proportionality relations:

      Δα=0,ΔαΔβNbgαNbgβσασβ(eμe(τf))2T2f.

      (40)

      The average relation on the left is interpreted straightforwardly as the conservation of electric charge. The proportionality relation on the right is a measurement of the asymmetry. The CME induced term Δ± is treated as a perturbation to the electrically neutral background as heat bath with temperature Tf.

      Subsequently, we analyze the background angular distribution dN±/dϕ, which reflects the charge-independent evolution of the medium determined by the event-by-event fluctuating initial state. Pursuant to this, we take the Fourier expansion of the background angular distribution as

      dNbg±dϕ=Nbg±2π[1+2n=1vncosn(ϕΨn)],

      (41)

      where Ψn indicates the participant plane angle of order n. Note that we have dropped the sine term in the Fourier decomposition because the distribution is symmetric about the participant plane. Coefficient vn is defined as the nth order harmonic flow. Typically, the directed flow v1 is generally chosen to be 0 if the distribution is measured in a symmetric rapidity region [13, 50]. Therefore, in the following calculation, we only retained the next leading term from the elliptic flow v2.

      Next, we assume the following ansatz [1] for the total generated charged single-particle spectrum originating from both the background and the CME:

      dN±dϕ=dNbg±dϕ+14Δ±sin(ϕΨRP),

      (42)

      where the form of the CME-induced term is proportional to sin(ϕΨRP) owing to the symmetry of the distribution about the magnetic field, which is perpendicular to the reaction plane, and where the factor 1/4 is consistent with our definition (39).

      In contrast to our previous work [30], we choose our correlated two-particle spectrum not only as a product of the single spectrum, but also to include an underlying correlation term proposed in [13] as

      ρ(ϕ1,ϕ2)=dNαdϕα1dNβdϕβ2[1+n=0ancosn(ϕ1ϕ2)],

      (43)

      with α,β=±. The cosine correlation term is reaction-plane-insensitive. Here, we only consider the leading term a1 (with normalization leading to a0=0).

      By employing all these values, two types of two particle correlations, γ and δ, which are measured in the heavy-ion collision experiments, are given as

      {γαβ=cos(ϕα1+ϕβ22ΨRP)δαβ=cos(ϕα1ϕβ2),

      (44)

      where the average cosφ of the angle φ=(ϕα1+ ϕβ22ΨRP) or (ϕα1ϕβ2) is taken over events, that is, integrated over ϕ1 and ϕ2 as

      cosφ=ρ(ϕ1,ϕ2) cosφ dϕα1dϕβ2ρ(ϕ1,ϕ2) dϕα1dϕβ2.

      (45)

      This will result in

      {γαβ=v2a1cos2(Ψ2ΨRP)π216ΔαΔβNbgαNbgβδαβ=a12(1+v22)+π216ΔαΔβNbgαNbgβ.

      (46)

      These forms of γ and δ correlators are consistent with the proposal discussed in [5, 13]:

      {γαβ=κv2FαβHαβ,δαβ=Fαβ+Hαβ,

      (47)

      with Fαβ denoting the background and Hαβ denoting the CME contribution, and κ being an undetermined factor ranging from 1 to 2. Therefore, by matching the above sets of equations and using (40), we claim that the CME signal takes the following form:

      Hαβ=π216ΔαΔβNbgαNbgβσασβ π216(eμe(τf))2T2f.

      (48)

      The difference between the same charge correlation HSS and opposite charge correlation HOS is thus expressed as

      (HSSHOS)2π216(eμe(τf))2T2f.

      (49)

      The centrality dependence of 104(HSSHOS) for Au and isobars is shown in Fig. 3. We also plot the signal for AuAu collision at 200GeV with data extracted from STAR, by solving (47) as

      Figure 3.  (color online) Centrality dependence of the CME signal from our stochastic model for AuAu and isobaric collision at sNN=200GeV, with triangles for Au, squares for Ru, and circles for Zr. We also list the data for AuAu collisions at sNN=200GeV, extracted from STAR [51, 52], with pentacles, for comparison.

      Hαβ=κv2δαβγαβ1+κv2,

      (50)

      where κ is taken to be 1, numerical values of γ and δ are taken from [51], and values of v2 are taken from [52]. Notably, by adjusting the τB parameter, the CME signal from our model is in a good agreement with that from the experiments. Moreover, with the same τB(1.65fm), we predict the signals for Ru and Zr, which are larger than that of Au, due to the square of the scaling of μe(τf) as Z2A2(ζ+89), with roughly ζ14, as we discussed in Section 3.2.

    4.   Conclusion
    • We have calculated the axial charge evolution using the stochastic hydrodynamics model, and used it to derive the chiral magnetic effect in off-central collisions of AuAu, RuRu, and ZrZr. By matching the results from our model with the background subtracted experimental data, we have fixed the relaxation time for the magnetic field. We use the same relaxation time to make predictions for the CME signal for collisions of RuRu and ZrZr. Two significant results have been obtained in our analysis.

      First, while the axial and vector charges are coupled through the chiral magnetic effect and chiral separation effect, we found that the influence of vector charge on axial charge is negligible at top RHIC collision energy. This allows us to decouple the evolution of axial charge from the vector charge.

      Secondly, we study the centrality and system size dependencies of the CME signal. The initial chiral imbalance μA0 is found to have only weak centrality dependence. The centrality dependence of the CME signal mainly results from the magnetic field and QGP volume factor. As for the system size dependence, although larger systems provide enhanced magnetic field and chiral imbalance, the electric charge asymmetry characterized by eμe is suppressed due to the larger volume factor. Consequently, we found larger absolute charged particle correlation in isobar collisions than in AuAu collisions.

      The present study readily generalizes to collisions of large nuclei at higher energies, where we expect that Bjorken flow approximation will still apply. It would be interesting to see if the energy dependence matches with current experimental data at different energies. At lower energies, the Bjorken flow approximation becomes inaccurate. One possible approach to address this issue is to implement stochastic noise numerically in the existing AVFD model. We will report studies along these lines in the future.

      We are grateful to Huanzhong Huang, Guoliang Ma, Dirk Rischke, and Gang Wang for discussions. G.R.L also acknowledges the Institute for Theoretical Physics at Frankfurt University for the warm hospitality where part of this work has been done.

Reference (52)

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