Charge-dependent correlations in heavy-ion collisions from stochastic hydrodynamics

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Gui-Rong Liang^,¹, Miao Li²¹ School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai, 519082, China
² Department of Physics, Southern University of Science and Technology, Shenzhen 518055, Guangdong, China

Received:2020-08-6Revised:2020-08-25Accepted:2020-08-26Online:2020-10-27

Abstract
Charge-dependent correlations from both background and charge separation contribute to experimental observables in heavy-ion collisions. In this paper, we use stochastic hydrodynamics to study background charge asymmetry due to fluctuations. Using the rapidity-dependent correlation and a simple ansatz for particle distributions, we find a fluctuation-induced correlation to provide a type of background F -correlation. Experimental data for Au+Au collisions at

$\sqrt{{s}_{\mathrm{NN}}}=200\,\mathrm{GeV}$

are compared. We also make predictions for F -correlations in isobar collisions. Combining this with our previous chiral magnetic effect results, we obtain δ -correlations for collisions in the three types of system. Computations from our model show an almost identical background with less than 2% difference for isobars, but roughly 10% difference for their charge separations. In combination with our earlier works, we provide a consistent method of calculating both the chiral magnetic effect and the charged background in the context of stochastic hydrodynamics.
Keywords： stochastic hydrodynamics;heavy-ion collisions;background charge fluctuations;isobar collisions;charge-dependent correlations

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Cite this article
Gui-Rong Liang, Miao Li. Charge-dependent correlations in heavy-ion collisions from stochastic hydrodynamics. Communications in Theoretical Physics[J], 2020, 72(11): 115304- doi:10.1088/1572-9494/abb7d9

1. Introduction

Heavy-ion collisions (HICs) provide a unique experimental approach to studying the creation of the quark gluon plasma (QGP) state in the early universe, and also to the production of the highest temperature and strongest magnetic field on earth. This facilitates the detection of the physical properties of extreme matter and novel quantum phenomena, such as the chiral magnetic effect (CME), the chiral separation effect (CSE) etc., some of which are macroscopic manifestations of a chiral anomaly. The search for evidence of anomaly-induced transport in HICs has garnered a great deal of attention in the past few decades [1 –3], and various frameworks have been developed to better understand the features of the signals generated by the evolution of axial charges, such as anomalous viscous fluid dynamics [4 –7], chiral kinetic theory [8 –10] and the multiphase transport model [11, 12]. However, experimental signals for the charge separation effect are overwhelmed by flow-related background contributions to charged hadron correlations. It is already known that this background noise originates from different effects, such as local charge conservation [13] and cluster particle correlations [14], among others [12, 15]. Various theoretical methods and experimental techniques have been employed to extract or exclude this significant background noise [16 –20]. However, a general framework for calculating both the background and the CME signal has not been fully explored to date. In [21, 22], we applied stochastic hydrodynamics to the calculation of the CME, taking into account fluctuation and dissipation effects. In this study, we employ the same procedure to discuss the evolution of the background vector charge, mainly focusing on its fluctuation origin. Assuming that non-equilibrium correlations over rapidity may be fixed by a freeze-out procedure and passed to final observables, we reproduce the F -type background correlations proposed in [17], thereby offering another possible origin of the background noise in addition to those proposed in [12 –15]. Together with our previous CME calculations, stochastic hydrodynamics is incorporated into the calculations of both background and CME correlations originating from fluctuation mechanisms.

Moreover, the isobar collision process would be a natural and superior playground for the examination of charge separation phenomena [23], particularly in light of the data soon to be released by the STAR collaboration. Isobars are those nuclei possessing the same number of nucleons, but different numbers of protons; here, we take Ru and Zr as our research objects. As only the protons carry charge and are therefore able to produce the initial magnetic field, it is believed that isobar collisions may generate the same background correlations, but with different CME signals, thereby contributing to a more convincing identification of CME. Our calculation results in terms of stochastic hydrodynamics show the following pattern: the background difference between isobars is strictly below 2%, and almost centrality-independent, as suggested by the difference in their colliding transverse overlap area, modeled via the Glauber Monte-Carlo method; in contrast, the charge separation difference is in the magnitude of 10%, and enhances with centrality, as indicated by the difference in their magnetic fields, as discussed in our most recent research [22]. Meanwhile, we find that background correlations scale in accordance with the system size, as the inverse of the atomic number, making the corresponding magnitude of isobars larger than that of Au. Given this system size-dependence, we assume that fluctuation-induced correlation takes up the same proportion in the final observables for different collision systems, and fit this proportionality with the datas for Au+Au collisions at

$\sqrt{{s}_{\mathrm{NN}}}=200\,\mathrm{GeV}$

; finally, we predict the δ -correlation in relation to isobar collisions, which is a direct measurable in the experiments.

The content of this study ranges from theoretical to phenomenological considerations. The general framework of stochastic hydrodynamics, incorporating both vector and axial charges, are introduced in section 2, in which we focus primarily on vector charge evolution, and determine the non-equilibrated electrical chemical potential correlation, together with its system size-dependence. In section 3, we take the Cooper-Frye freeze-out procedure and a particle distribution spectrum ansatz to pass the chemical potential correlations to the background observables, in order to match our model results to Au+Au collisions, and make predictions for Ru+Ru and Zr+Zr collisions; we then produce δ -correlations for the three types, and compare the background and CME signal difference ratio calculated in our model. Final conclusions and possible extensions are presented in section 4 .

2. Stochastic hydrodynamics for vector charges and chemical potential correlations

The full stochastic hydrodynamic equations given in [21, 22], incorporating both axial and vector charge evolutions and their interactions through the magnetic field, are summarized as follows:(1)

$ \begin{eqnarray}\left\{\begin{array}{l}{{\rm{\nabla }}}_{\mu }{J}_{A/V}^{\mu }=-2{q}_{A/V},\\ {J}_{A/V}^{\mu }={n}_{A/V}{u}^{\mu }-\sigma {{TP}}^{\mu \nu }{{\rm{\nabla }}}_{\nu }\left(\tfrac{{n}_{A/V}}{\chi T}\right)+{\xi }_{A/V}^{\mu },\\ {q}_{A}=\tfrac{{n}_{A}}{2{\tau }_{\mathrm{CS}}}+{\xi }_{q},\\ {q}_{V}=0,\end{array}\right.\end{eqnarray}$

where the subscript A /V denotes the corresponding axial or vector quantity. Here, q is the topological charge, which gives rise to the non-conservation of the axial charge, thus vector charge q_V =0;

${\tau }_{\mathrm{CS}}$

is the relaxation time of the axial charge. Both axial and vector conductivity take the numeric value

$\sigma \simeq 0.37{N}_{f}{T}_{0}$

, resulting from the lattice measurement as given in [24], and the axial and vector charge susceptibilities are given by free theory limits as

$\chi ={N}_{f}{N}_{c}{T}^{2}/3$

. We examine the results for three flavors in this paper. Along with the fluid velocity

${u}^{\mu }$

, the projection operator

${P}^{\mu \nu }={\eta }^{\mu \nu }+{u}^{\mu }{u}^{\nu }$

projects a quantity in 4 dimensions onto the 3-dimensional hyper-surface perpendicular to

${u}^{\mu }$

. The metric is taken to be

${\eta }_{\mu \nu }\,=\mathrm{diag}(-1,1,1,1)$

.

The noise average of the axial and vector thermal fluctuation,

${\xi }_{A/V}$

, and axial topological fluctuation, ξ_q, are set as follows, based on the standard fluctuation-dissipation theorem:(2)

$ \begin{eqnarray}\left\{\begin{array}{l}\left\langle {\xi }_{A/V}^{\mu }(x){\xi }_{A/V}^{\nu }(x^{\prime} )\right\rangle ={P}^{\mu \nu }2\sigma T\tfrac{\,{{\rm{d}}}^{4}(x-x^{\prime} )}{\sqrt{-g}},\\ \left\langle {\xi }_{q}(x){\xi }_{q}(x^{\prime} )\right\rangle =\left(\tfrac{\chi T}{{\tau }_{\mathrm{CS}}}\right)\tfrac{\,{{\rm{d}}}^{4}(x-x^{\prime} )}{\sqrt{-g}},\\ \left\langle {\xi }_{A}^{\mu }(x){\xi }_{V}^{\nu }(x^{\prime} )\right\rangle =0,\\ \left\langle {\xi }_{A/V}^{\mu }(x){\xi }_{q}(x^{\prime} )\right\rangle =0,\end{array}\right.\end{eqnarray}$

where we assume there are no cross-correlations between axial and vector fluctuations, nor between thermal and topological fluctuations, due to their independent origins.

In this paper, we focus on the evolution of the vector charge, n_V . The covariant form of the equation of motion for n_V from (1 ) without the CME effect is given by(3)

$ \begin{eqnarray}{{\rm{\nabla }}}_{\mu }\left({n}_{V}{u}^{\mu }\right)-{{\rm{\nabla }}}_{\mu }\left({\sigma }_{V}{{TP}}^{\mu \nu }{{\rm{\nabla }}}_{\nu }\left(\displaystyle \frac{{n}_{V}}{\chi T}\right)\right)=-{{\rm{\nabla }}}_{\mu }\left({P}^{\mu \nu }{\xi }_{\nu }\right).\end{eqnarray}$

This is similar to the dynamics of n_A, apart from the absence of a dissipative term concerning τ_CS on the left, and the topological fluctuation term concerning ξ_q on the right. Mathematically, we see that the finite relaxation time of axial charge

${\tau }_{\mathrm{CS}}$

leads to the axial dissipative term and topological fluctuations; if we take the

${\tau }_{\mathrm{CS}}\to \infty $

limit, these terms vanish, and what remains is in the same form as the dynamics of vector charges, n_V . Below, we will repeatedly use the

${\tau }_{\mathrm{CS}}\to \infty $

limit to check or extract calculation results for the vector charge dynamics of the axial counterparts.

Here, we work in Bjorken flow, with the local rest frame characterized by the Milne coordinates

$(\tau ,\eta ,{x}_{\perp })$

, and where the fluid velocity is given by

${u}^{\mu }=(1,0,{\bf{0}})$

. The dynamic equation can then be expanded:(4)

$ \begin{eqnarray}\left[\displaystyle \frac{\partial }{\partial \tau }-D\left({\partial }_{\perp }^{2}+\displaystyle \frac{{\partial }_{\eta }^{2}}{{\tau }^{2}}\right)\right]\left(\tau {n}_{V}\right)=-\tau {{\rm{\nabla }}}_{i}{\xi }^{i},\end{eqnarray}$

with D being the diffusion constant fixed by the Einsteinian relation

$D=\sigma /\chi $

. This is the governing equation for vector charge dynamics. As a physical check, we take a simple look at the total vector charge given by the integral of(5)

$ \begin{eqnarray}{N}_{V}=\int \,{\rm{d}}\eta \,{{\rm{d}}}^{2}{x}_{\perp }(\tau {n}_{V}),\end{eqnarray}$

where

$\tau \,{\rm{d}}\eta \,{{\rm{d}}}^{2}{x}_{\perp }$

serves as the volume element at the constant proper time,

$\tau $

. The dynamics of this total charge may be simply expressed as(6)

$ \begin{eqnarray}\displaystyle \frac{\,{\rm{d}}{N}_{V}}{\,{\rm{d}}\tau }=-\int \tau \,{\rm{d}}\eta \,{{\rm{d}}}^{2}{x}_{\perp }({{\rm{\nabla }}}_{i}{\xi }^{i})=0,\end{eqnarray}$

to reflect the conservation of the electric charge, which is in obvious contrast to the axial charge dynamics with a non-conservation term, induced by the topological property.

We have obtained the correlations for the case of a non-vanishing topological term from [21]; thus, we could simply take the limit

${\tau }_{\mathrm{CS}}\to \infty $

, thereby obtaining the local correlation for the conserved background electric charge as follows:(7)

$ \begin{eqnarray}\begin{array}{l}\displaystyle \int \,{{\rm{d}}}^{2}{x}_{\perp }\langle {\tau }_{f}{n}_{V}({\tau }_{f},\eta ,{x}_{\perp }){\tau }_{f}{n}_{V}({\tau }_{f},0)\rangle \\ \quad =\,{\chi }_{0}{T}_{0}{\tau }_{0}\left(\delta (\eta )-\displaystyle \frac{{{\rm{e}}}^{\tfrac{-{\eta }^{2}}{8B}}}{2\sqrt{2\pi B}}\right),\end{array}\end{eqnarray}$

where the factor B in the expression is given by(8)

$ \begin{eqnarray}B=\displaystyle \frac{3{D}_{0}}{2{\tau }_{0}^{1/3}}\left(\displaystyle \frac{1}{{\tau }_{0}^{2/3}}-\displaystyle \frac{1}{{\tau }_{f}^{2/3}}\right).\end{eqnarray}$

Here, we select

${\tau }_{0}$

and

${\tau }_{f}$

to be the thermolization and freeze-out times. The first positive delta-term localizes the correlation in a fluid cell, while the second negative term represents anti-correlation over rapidity.

To prepare for the particle distribution, the charge chemical potential is induced by

${\mu }_{V}={n}_{V}/{\chi }_{V}$

, with the susceptibility evolving over time, expressed as(9)

$ \begin{eqnarray}{\chi }_{V}={\chi }_{0}{\left(\displaystyle \frac{\tau }{{\tau }_{0}}\right)}^{-2/3}.\end{eqnarray}$

Based on equation (7 ), we see that the chemical potential correlation may be given by(10)

$ \begin{eqnarray}\begin{array}{l}\langle {\mu }_{V}({\tau }_{f},{\eta }_{1}){\mu }_{V}({\tau }_{f},{\eta }_{2})\rangle \\ \quad =\,\displaystyle \frac{1}{3{S}_{\perp }{T}_{0}\ {\tau }_{0}^{1/3}{\tau }_{f}^{2/3}}\left(\delta ({\eta }_{1}-{\eta }_{2})-\displaystyle \frac{{{\rm{e}}}^{\tfrac{-{\left({\eta }_{1}-{\eta }_{2}\right)}^{2}}{8B}}}{2\sqrt{2\pi B}}\right),\end{array}\end{eqnarray}$

where

${S}_{\perp }$

is the transverse overlap area, modeled by Glauber Monte-Carlo method listed in figure 1 . The system size-dependence of

${S}_{\perp }$

may be straightforwardly derived from the geometrical property in the Glauber analysis as

${S}_{\perp }\propto {A}^{2/3}$

, which is also indicated in the figure.

For Au+Au collisions at

$\sqrt{{s}_{\mathrm{NN}}}=200\,\mathrm{GeV}$

, the initial and final conditions comprise temperatures of

${T}_{0}=350\,\mathrm{MeV}$

and

${T}_{f}=154\,\mathrm{MeV}$

[25], and evolution times of

${\tau }_{0}=0.6\,\mathrm{fm}$

and

${\tau }_{f}=7\,\mathrm{fm}$

. For isobar collisions, the freeze-out time is taken to be the same as that for Au+Au collisions; the other three parameters are relevant to the saturation scale, Q_s, for which we adopt scaling with a system size of

${Q}_{s}\propto {A}^{\tfrac{1}{6}}$

, as given in [26]. The results are as follows:(11)

$ \begin{eqnarray}{T}_{0}\propto {Q}_{s}\propto {A}^{\tfrac{1}{6}},\qquad {\tau }_{0}\propto {Q}_{s}^{-1}\propto {A}^{-\tfrac{1}{6}},\qquad {\tau }_{f}\propto {A}^{\tfrac{1}{3}}.\end{eqnarray}$

Based on the above parameterization, and a very close numerical value of factor B, the resulting system dependence of the chemical potential correlation simply takes the form(12)

$ \begin{eqnarray}\langle {\mu }_{V}({\tau }_{f}){\mu }_{V}({\tau }_{f})\rangle \propto {A}^{-1},\end{eqnarray}$

which reflects the property of the fluctuation; i.e. it is suppressed by the system size. This scaling relation serves as the main source of the difference in background between Au and isobars, to be further examined below. The difference between isobars may be ascribed to slight deviations in the value of

${S}_{\perp }$

, which are less than 2%, and which we treat as virtually identical.

3. Cooper-Frye descriptions and particle distribution correlations

In order to establish the two particle distribution correlations measured in the experiment, we would have to employ the relativistic version of single particle distribution known as the Cooper-Frye formula [27]:(13)

$ \begin{eqnarray}E\displaystyle \frac{\,{\rm{d}}{N}_{i}}{\,{{\rm{d}}}^{3}p}=\displaystyle \frac{{g}_{i}}{{\left(2\pi \right)}^{3}}\int {p}^{\mu }\,{{\rm{d}}}^{3}{\sigma }_{\mu }{f}_{i}(x,p),\end{eqnarray}$

where i refers to a certain species of mesons produced in QGP, and the degeneracy g_i is taken to be 1 for each species. The phase-space distribution in a given frame at freeze-out time is approximated as

${f}_{i}(x,p)={{\rm{e}}}^{({p}_{\mu }{u}^{\mu }\pm {\mu }_{V}({\tau }_{f})+{\mu }_{i})/{T}_{f}}$

, where the plus or minus signs denote positive or negative charges. We consider only pions and kaons produced in heavy-ion collisions in our model, with their chemical potential given by

${\mu }_{\pi }\simeq 80\,\mathrm{MeV}$

and

${\mu }_{K}\simeq 180\,\mathrm{MeV}$

.

In the context of Bjorken flow, we express the particle 4-momentum and the fluid 4-velocity as(14)

$ \begin{eqnarray}\begin{array}{rcl}{p}^{\mu } & = & ({m}_{\perp }\cosh y,{{\bf{p}}}_{\perp },{m}_{\perp }\sinh y),\\ {u}^{\mu } & = & (\cosh \eta ,0,0,\sinh \eta ),\end{array}\end{eqnarray}$

where

${m}_{\perp }=\sqrt{{p}_{\perp }^{2}+{m}^{2}}$

is the transverse mass, and y the particle rapidity;

$\eta $

denotes spacetime rapidity in the Milne coordinates. We can therefore expand the Cooper-Frye distribution to obtain the azimuthal dependence as(15)

$ \begin{eqnarray}\displaystyle \frac{\,{\rm{d}}{N}_{\pm }^{i}}{\,{\rm{d}}\phi }=\displaystyle \frac{{S}_{\perp }}{{\left(2\pi \right)}^{3}}\int \,{\rm{d}}{m}_{\perp }{m}_{\perp }^{2}\int {\tau }_{f}\,{\rm{d}}y\,{\rm{d}}\eta \cosh (\eta -y){f}_{i}(x,p),\end{eqnarray}$

based on the fact that

${p}_{\perp }\,{\rm{d}}{p}_{\perp }={m}_{\perp }\,{\rm{d}}{m}_{\perp }$

. The lower bound of the

${m}_{\perp }$

integration takes the rest mass of the corresponding meson. The equation above describes a charge-neutral background, as indicated by its independence of the positive or negative sign. Since the charge asymmetry due to fluctuation is much smaller than the charge-neutral background,

${\mu }_{V}\ll {T}_{f}$

, the variation of the distribution is taken to the lowest order:(16)

$ \begin{eqnarray}\begin{array}{rcl}\delta \displaystyle \frac{\,{\rm{d}}{N}_{\pm }^{i}}{\,{\rm{d}}\phi } & = & \pm \displaystyle \frac{{S}_{\perp }}{{\left(2\pi \right)}^{3}}\displaystyle \int \,{\rm{d}}{m}_{\perp }{m}_{\perp }^{2}\displaystyle \int {\tau }_{f}\,{\rm{d}}y\,{\rm{d}}\eta \cosh (\eta -y)\\ & & \times {f}_{i}({\mu }_{V}=0){\mu }_{V}({\tau }_{f})/{T}_{f},\end{array}\end{eqnarray}$

where the plus or minus signs correspond to positive or negative charge. By a direct integration over the transverse mass, we introduce a shorthand notation as follows:(17)

$ \begin{eqnarray}\begin{array}{rcl}{h}_{i}(y-\eta ){T}_{f} & \equiv & \displaystyle \int \,{\rm{d}}{m}_{\perp }{m}_{\perp }^{2}\cosh (\eta -y)\\ & & \times \exp \left[-{m}_{\perp }\cosh (y-\eta )+{\mu }_{i})/{T}_{f}\right]\\ & = & {{\rm{e}}}^{\tfrac{-m\cosh (y-\eta )+{\mu }_{i}}{{T}_{f}}}{T}_{f}\left({m}^{2}+2m{\cosh }^{-1}(y-\eta ){T}_{f}\right.\\ & & \left.+2\cosh {\left(y-\eta \right)}^{-2}{T}_{f}^{2}\right),\end{array}\end{eqnarray}$

which enables us to express equations (15 ) and (16 ) in a neat form:(18)

$ \begin{eqnarray}\left\{\begin{array}{l}\tfrac{\,{\rm{d}}{N}_{\pm }^{i}}{\,{\rm{d}}\phi }=\,\tfrac{{S}_{\perp }{\tau }_{f}}{{\left(2\pi \right)}^{3}}\displaystyle \int {h}_{i}(y-\eta )\ \,{\rm{d}}y\,{\rm{d}}\eta \ {T}_{f},\\ \delta \tfrac{\,{\rm{d}}{N}_{\pm }^{i}}{\,{\rm{d}}\phi }=\pm \tfrac{{S}_{\perp }{\tau }_{f}}{{\left(2\pi \right)}^{3}}\displaystyle \int {h}_{i}(y-\eta )\ \,{\rm{d}}y\,{\rm{d}}\eta \ {\mu }_{V}({\tau }_{f}).\end{array}\right.\end{eqnarray}$

We define the total neutral background charge and the asymmetry caused by the fluctuation as(19)

$ \begin{eqnarray}{N}_{\pm }^{\mathrm{bg}}\equiv \displaystyle \sum _{i}{N}_{\pm }^{i},\qquad {{\rm{\Omega }}}_{\pm }\equiv \displaystyle \sum _{i}\delta {N}_{\pm }^{i},\end{eqnarray}$

note that

${{\rm{\Omega }}}_{\pm }$

is actually the fluctuation of the neutral

${N}_{\pm }^{\mathrm{bg}}$

and charge-dependent, so that(20)

$ \begin{eqnarray}\left\langle {{\rm{\Omega }}}_{\alpha }\right\rangle =0,\qquad \left\langle {{\rm{\Omega }}}_{\alpha }{{\rm{\Omega }}}_{\beta }\right\rangle \ne 0,\end{eqnarray}$

where

$\alpha /\beta $

denotes the

$+/-$

sign. Taking account of all of the above, we find the ratio of their correlations to be(21)

$ \begin{eqnarray}\begin{array}{l}\displaystyle \frac{\langle {{\rm{\Omega }}}_{\alpha }{{\rm{\Omega }}}_{\beta }\rangle }{\langle {N}_{\alpha }\rangle \langle {N}_{\beta }\rangle }=\displaystyle \frac{{\sigma }_{\alpha }{\sigma }_{\beta }}{{T}_{f}^{2}}\,\displaystyle \frac{\displaystyle \int \,{\rm{d}}{y}_{1}\,{\rm{d}}{\eta }_{1}\,{\rm{d}}{y}_{2}\,{\rm{d}}{\eta }_{2}\ H({y}_{1}-{\eta }_{1})H({y}_{2}-{\eta }_{2})\ \langle {\mu }_{V}({\tau }_{f},{\eta }_{1}){\mu }_{V}({\tau }_{f},{\eta }_{2})\rangle }{{\left[\displaystyle \int H(y-\eta ){\rm{d}}{y}{\rm{d}}\eta \right]}^{2}},\end{array}\end{eqnarray}$

where

$H(y-\eta )\equiv {\sum }_{i=\pi ,K}h(y-\eta )$

, and

${\sigma }_{\pm }=\pm 1$

.

In our previous works, since we used equilibrated results [21] or the total charge correlations [22], the integrals in the numerator and the denominator in the above ratio were canceled out, leaving only the chemical potential correlation. Here, however, we need to apply the rapidity-dependent correlation (10 ), and perform the full integral in order to calculate the results.

For the azimuthal distribution spectrum, we take the following ansatz:(22)

$ \begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\,{\rm{d}}\langle {N}_{\pm }\rangle }{\,{\rm{d}}\phi } & = & \displaystyle \frac{\langle {N}_{\pm }^{\mathrm{bg}}\rangle }{2\pi }\left[1+2\displaystyle \sum _{n=1}{v}_{n}\cos n(\phi -{{\rm{\Psi }}}_{n})\right]\\ & & +\displaystyle \frac{1}{4}{{\rm{\Omega }}}_{\pm }\cos (\phi -{{\rm{\Psi }}}_{\mathrm{RP}})+\displaystyle \frac{1}{4}{{\rm{\Delta }}}_{\pm }\sin (\phi -{{\rm{\Psi }}}_{\mathrm{RP}}),\end{array}\end{eqnarray}$

where the first part represents a Fourier expansion of the background neutral charge,

${N}_{\pm }^{\mathrm{bg}}$

, with v_n representing the n th harmonic flow, usually taken up to the second order. The second part relates to the background asymmetry,

${{\rm{\Omega }}}_{\pm }$

, due to fluctuation. Since this is the variation of the neutral background,

${N}_{\pm }^{\mathrm{bg}}$

, from the first part, it also takes the form of a cosine term, meaning that it is symmetric about the reaction plane; the third part relates to the asymmetry,

${{\rm{\Delta }}}_{\pm }$

, caused by CME, as the CME is induced by the magnetic field pointing perpendicular to the reaction plane, and thus a sine term is assumed. The normalization for the total charge asymmetry gives the factor of 1/4.

In the experiments, the reaction plane-dependent and reaction plane-independent correlations below are measured [16, 17]:(23)

$ \begin{eqnarray}\left\{\begin{array}{l}{\gamma }_{\alpha \beta }=\left\langle \cos ({\phi }_{1}^{\alpha }+{\phi }_{2}^{\beta }-2{{\rm{\Psi }}}_{\mathrm{RP}})\right\rangle ,\\ {\delta }_{\alpha \beta }=\left\langle \cos ({\phi }_{1}^{\alpha }-{\phi }_{2}^{\beta })\right\rangle ,\end{array}\right.\end{eqnarray}$

where the averages are taken over all particles in all events, and can be treated equivalently in analytical terms, as integrals over azimuthal angles

${\phi }_{1}^{\alpha }$

and

${\phi }_{2}^{\beta }$

. These correlations are known to have contributions from different origins. In [17], the following decomposition of the above charge-dependent correlations into the background F -correlation and CME signal H -correlation was proposed:(24)

$ \begin{eqnarray}\left\{\begin{array}{l}{\gamma }_{\alpha \beta }\simeq \kappa {v}_{2}{F}_{\alpha \beta }-{H}_{\alpha \beta },\\ {\delta }_{\alpha \beta }\simeq {F}_{\alpha \beta }+{H}_{\alpha \beta },\end{array}\right.\end{eqnarray}$

where κ is an undetermined factor ranging from 1 to 2, and numerically v₂ is much smaller than 1. This results in the extraction of the background being roughly expressed as

${F}_{\alpha \beta }\simeq {\gamma }_{\alpha \beta }+{\delta }_{\alpha \beta }$

. In our model, by substituting the background asymmetry contribution,

${{\rm{\Omega }}}_{\pm }$

, in the second part of equation (22 ), we find that it leads to(25)

$ \begin{eqnarray}{F}_{\alpha \beta }\simeq 2\cdot \displaystyle \frac{{\pi }^{2}}{16}\displaystyle \frac{\langle {{\rm{\Omega }}}_{\alpha }{{\rm{\Omega }}}_{\beta }\rangle }{\langle {N}_{\alpha }^{\mathrm{bg}}\rangle \langle {N}_{\beta }^{\mathrm{bg}}\rangle },\end{eqnarray}$

where the right-hand side can be computed via equation (21 ). The difference between the same charge correlation and the opposite charge correlation,

$({F}_{\mathrm{SS}}-{F}_{\mathrm{OS}})$

, gives another double to the above quantity of the same sign correlation.

The centrality-dependence of

${10}^{4}({F}_{\mathrm{SS}}-{F}_{\mathrm{OS}})$

for Au+Au collisions at

$200\,\mathrm{GeV}$

calculated from our model, and data extracted from STAR are shown in figure 1 . We note that the fluctuation-induced background present here is of the same type as the F-correlation in the experiment, apart from the inclusion of a proportion factor, fitted to be

${F}_{\mathrm{theo}}/{F}_{\exp }\simeq 0.69$

, indicating that fluctuation is also a source of the background correlation.

Figure 1.

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Figure 1.Centrality-dependence of the background F-correlation for Au+Au collisions at $\sqrt{{s}_{\mathrm{NN}}}=200\,\mathrm{GeV}$, based on our stochastic model (triangles joined by lines), the scaled results (dashed lines), and the STAR data from [28, 29], denoted by pentacles.

We assume the that proportion factor remains the same for different collision systems, and predict the F -correlations for Ru and Zr in figure 2 . Based on equations (12 ), (21 ), and (25 ), we find background signal scales with a system size of roughly

$F\propto {A}^{-1}$

, as confirmed by the numerical values. The predicted backgrounds for isobars are, as anticipated, almost identical.

Figure 2.

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Figure 2.Centrality-dependence of the modeled F -correlation (denoted by solid lines) and modeled δ -correlation (denoted by dashed lines) from our stochastic model for Au+Au, and isobaric collision at $\sqrt{{s}_{\mathrm{NN}}}=200\,\mathrm{GeV}$ . The plot legends are arranged in order from up to down, as shown in the graph. The yellow span represents the CME H -correlation calculated in our model in [22].

Furthermore, in our previous work [22], we used the same model to predict the CME signal H -correlations for both Au and isobars, using the decay time of the magnetic field

${\tau }_{B}\simeq 1.65\,\mathrm{fm}$

as the fitted parameter. Here, we were able to combine the background noise and the CME signal to obtain the charge-dependent δ -correlations measured in the experiment, based on the contribution from the sum of the two mechanisms,

$\delta \simeq F+H$

.

The centrality-dependence of the charge-dependent δ and F-correlations for the three types of system are shown in figure 2 . For clarity, we indicate the F backgrounds with solid lines, and the δ correlations with dashed lines. Plot legends are given on the right, from top to bottom, in the order of the correlation’s magnitude, in accordance with the curves shown on the left. Since the CME signals are positive, the δ -lines are above the F -lines, and the yellow shadow regions spanned by δ and the corresponding F represent the magnitudes of the CME.

However in figure 2, the relative H- /F- difference between the isobars are too small to distinguish. We therefore plot another diagram in figure 3, showing the following two quantities:(26)

$ \begin{eqnarray}\begin{array}{rcl}\left|\displaystyle \frac{{\rm{\Delta }}{H}_{\mathrm{differ}}}{{H}_{\mathrm{Zr}}}\right| & \equiv & \left|\displaystyle \frac{{H}_{\mathrm{Ru}}-{H}_{\mathrm{Zr}}}{{H}_{\mathrm{Zr}}}\right|,\\ \left|\displaystyle \frac{{\rm{\Delta }}{F}_{\mathrm{differ}}}{{F}_{\mathrm{Zr}}}\right| & \equiv & \left|\displaystyle \frac{{F}_{\mathrm{Ru}}-{F}_{\mathrm{Zr}}}{{F}_{\mathrm{Zr}}}\right|,\end{array}\end{eqnarray}$

where H and F, in this case, represent

$({H}_{\mathrm{SS}}-{H}_{\mathrm{OS}})$

and

$({F}_{\mathrm{SS}}-{F}_{\mathrm{OS}})$

. We can see that the background F -difference is primarily due to the slight difference in transverse overlap area, as indicated in equation (10 ), with a percentage of less than 2%, and a weak dependence on centrality, consistent with the data in table 1 . In contrast, the CME H -correlation difference is approximately 10%, mainly due to the different magnetic field or different proton numbers in isobars, as discussed in our previous paper [22], which is slightly enhanced, as the collisions tend to be more peripheral.

Figure 3.

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Figure 3.Centrality-dependence of the difference between isobar background F -correlations (blue dotted line) and CME H -correlation (red dotted line), based on our stochastic model for collisions at $\sqrt{{s}_{\mathrm{NN}}}=200\,\mathrm{GeV}$ . The F -difference is strictly below 2%, and has a very weak centrality-dependence, whereas the H -difference is around 10% and increases slightly with centrality.

Table 1.
Table 1.The centrality-dependence of transverse overlap area

${S}_{\perp }({\mathrm{fm}}^{2})$

, modeled via the Glauber Monte-Carlo method for Au, Ru, and Zr. Here,

${S}_{\perp }$

is taken to be the projection of the nucleon–nucleon cross-section,

${\sigma }_{\mathrm{NN}}$

, onto the transverse plane [30].

$10k$

events are run to generate the data. Averages are calculated using the impact parameter as the weight factor.

Centrality	0%–5%	5%–10%	10%–20%	20%–30%	30%–40%	40%–50%	50%–60%	60%–70%	70%–80%
Au+Au	147.9	128.9	106.1	83.0	64.8	49.7	36.6	25.5	16.2
Ru+Ru	92.5	81.6	67.8	53.8	42.3	32.8	24.7	17.6	12.1
Zr+Zr	91.3	80.5	67.0	53.1	41.8	32.5	24.4	17.4	11.9

New window|CSV

4. Conclusion

In this work, we have incorporated a stochastic hydrodynamic model into the calculations for background charge evolution in the context of heavy-ion collisions, finding that it provides the same type of F -correlation as that found in experimental data extracted from Au+Au collisions at

$\sqrt{{s}_{\mathrm{NN}}}=200\,\mathrm{GeV}$

by the STAR collaboration. We then predicted the background correlations for isobars using the same method. Combining this with the CME results obtained previously via our model, we calculate the δ -correlations, which are direct experimental observables, for Au+Au and isobar collisions. The significance of the results are twofold:

(1) the chiral magnetic effect signals and the charged background correlations due to axial or vector charge fluctuations can be calculated consistently in the context of stochastic hydrodynamics. While the background correlation originates from different effects, a considerable contribution can be traced back to the fluctuations of charges in the evolutions of QGP. The freeze-out process occurs long before the QGP reaches its electrical thermodynamic equilibrium, thus fixing the imbalance due to charge fluctuations and passing the stochastic results to the final observables.

(2) Similarly to CME, while the isobars have fewer colliding nucleons, as compared with Au, they have a larger magnitude of background contributions, due to the volume suppression effect, with the correlation scaling with the inverse of the atomic number. Regarding the background difference between each of the isobars, the deviations are less than 2%, owing to slight variations in the transverse overlap area between Ru and Zr.

We envisage two possible extensions of the stochastic hydrodynamic model for future study: we will attempt to generalize the model to calculate other experimental observables, such as the most state-of-the-art R -correlations for extracting pure CME signals relating to isobar collisions. It would also be interesting to apply these calculation procedures to other hydrodynamic models; for example, Gubser flow would be a good candidate for a clear comparison of the results with those obtained via Bjorken flow. We will continue our work along these lines, expecting to gain further insights.