Impact of magnetic-field fluctuations on measurements of the chiral magnetic effect in collisions of isobaric nuclei

We investigate the properties of electromagnetic fields in isobaric $_{44}^{96}\textrm{Ru}+\,_{44}^{96}\textrm{Ru}$ and $_{40}^{96}\textrm{Zr}+\,_{40}^{96}\textrm{Zr}$ collisions at $\sqrt{s}$ = 200 GeV by using a multiphase transport model, with special emphasis on the correlation between magnetic field direction and participant plane angle $\Psi_{2}$ (or spectator plane angle $\Psi_{2}^{\rm SP}$), i.e. $\langle{\rm cos}\ 2(\Psi_B - \Psi_{2})\rangle$ [or $\langle{\rm cos}\ 2(\Psi_B - \Psi_{2}^{\rm SP})\rangle$]. We confirm that the magnetic fields of $_{44}^{96}\textrm{Ru}+\,_{44}^{96}\textrm{Ru}$ collisions are stronger than those of $_{40}^{96}\textrm{Zr}+\,_{40}^{96}\textrm{Zr}$ collisions due to their larger proton fraction. We find that the deformation of nuclei has a non-negligible effect on $\langle{\rm cos}\ 2(\Psi_B - \Psi_{2})\rangle$ especially in peripheral events. Because the magnetic-field direction is more strongly correlated with $\Psi_{2}^{\rm SP}$ than with $\Psi_{2}$, the relative difference of the chiral magnetic effect observable with respect to $\Psi_{2}^{\rm SP}$ is expected to be able to reflect much cleaner information about the chiral magnetic effect with less influences of deformation.


I. INTRODUCTION
Lattice QCD calculations predicted that quarks and gluons are deconfined with their partonic degrees of freedom under the condition of high temperatures or the high baryon chemical potential, i.e., the formation of quark-gluon plasma (QGP). Relativistic heavy ion collisions are believed to be able to reach the condition of creating the QGP. On the other hand, a nonzero axial charge density of the QGP with a large magnetic-field B can lead to a dipole charge separation along the B direction, i.e., the so-called chiral magnetic effect (CME), which results in a generation of a vector current J [1][2][3][4][5], where σ 5 is the chiral magnetic conductivity and µ 5 is the chiral chemical potential arising from the nonzero axial charge density.
To measure the CME signal, people usually measure charge azimuthal correlation [6][7][8] between two particles α and β, which is defined as where φ α and φ β are the azimuthal angles of two charged particles and Ψ RP is the reaction plane angle, which is usually represented by the second order of the event participant plane Ψ 2 . From the CME expectation, the charge azimuthal correlation ∆γ = γ opp − γ same (the difference between opposite-pair and same-pair correlations) is expected to be proportional to B 2 and cos 2(Ψ B − Ψ 2 ) [9,10], i.e., However, the current main difficulty of measuring the CME signal is some backgrounds which we do not understand clearly [11][12][13][14]. For example, one of the difficulties of the CME observable interpretation is due to a large part of background contribution stemming from the coupling of resonance decay correlations and the flow v 2 arising from participant geometry [15][16][17]. To isolate the influence of those backgrounds, the isobar program at the Relativistic Heavy Ion Collider (RHIC) has been proposed and it collides 96 44 Ru + 96 44 Ru and 96 40 Zr + 96 40 Zr elements since they have a same nucleon number but the 10% difference in proton number. The same nucleon number indicates they should have similar bulk backgrounds (e.g., flow), however, the different proton number means they carry different magnitudes of magnetic fields. Therefore, the CME signal (due to the CME current J ) is expected to be different between the two isobaric collisions, as illustrated by Eq. (1). There has been some interesting research on isobaric collisions, see Refs. [10,[18][19][20][21][22][23][24][25].
If there are similar or even the same backgrounds in two isobaric collisions, the difference of the CME observable between two isobaric collisions is expected to be mainly due to the differences from the squared magnetic field and the correlation between magnetic-field direction Ψ B and participant plane Ψ 2 from Eq. (3). Meanwhile, because the magnetic field is mainly induced by spectator protons, people also proposed to replace the participant plane Ψ 2 with the spectator plane Ψ SP 2 , which is believed to be more strongly correlated with Ψ B [23,26]. In this paper, we focus on not only the magnetic field, but also the two correlations between magnetic-field direction Ψ B and participant plane angle Ψ 2 and between Ψ B and the spectator plane angle Ψ SP 2 . We systematically study 96 44 Ru + 96 44 Ru collisions and 96 40 Zr + 96 40 Zr collisions by a multiphase transport model (AMPT) model. Based on the above, the implications of our results to the future CME analysis in the isobaric experiment will be discussed.
The paper is organized as follows. In Sec. II, we provide a brief introduction to the AMPT model, our isobaric deformation settings, and the method to calculate magnetic fields. The numerical results for the properties of electromagnetic fields and some related correlations are presented and discussed in detail in Sec. III. Section IV contains our conclusions.

A. AMPT model
In this paper, we take advantage of a AMPT model [27] to investigate isobaric collisions. There are two versions of the AMPT model, the default version and the version with a string-melting mechanism. Both versions contain four important evolution stages of heavy ion collisions: initial state, parton cascade, hadronization, and hadron rescatterings. They both use the HIJING model [28,29] for generating the initial state of collisions. The main difference between the two versions is that in the string-melting version, strings and minijets are melted into partons so that there are more partons participating in the parton cascade than the default version. Therefore, the stringmelting version can better describe the cases when the QGP is produced, such as heavy ion collisions at the RHIC and Large Hadron Collider energies. The string-melting version currently only considers elastic collision processes between two partons [30], hadronization is simulated by a simple quark combination model, and hadron rescatterings are described by a hadron transport model [31]. In this paper, we choose the string-melting version to simulate In our convention, we choose the x axis along the direction of impact parameter b from the target center to the projectile center, the z axis along the beam direction, and the y axis perpendicular to the x and z directions.

B. Geometry configuration of isobaric collisions
For modeling 96 44 Ru and 96 40 Zr in the HIJING model, the spatial distribution of nucleons in their rest frame can be written in the Woods-Saxon form (in spherical coordinates), where the normal nuclear density ρ 0 = 0. 16 [33,34] and comprehensive model deductions [35]. For the first case (denoted as case 1 thereafter), 96 44 Ru is more deformed than 96 40 Zr, i.e., β Ru 2 =0.158 and β Zr 2 =0.08. However, the second case (denoted as case 2 thereafter) is the opposite, i.e., β Ru 2 = 0.053 and β Zr 2 = 0.217. As shown in Ref. [10], the systematic uncertainty has little influence on the multiplicity distribution. We focus on its impact on the CME signal of the correlator ∆γ. To cancel some theoretical uncertainties [10], we can take the ratio of the relative difference between the two collisions. The definition of the relative ratio in a quantity Q between 96 44 Ru + 96 44 Ru and 96 40 Zr + 96 40 Zr collisions is and Q can represent e|B|/m 2 π , cos 2(Ψ B − Ψ 2 ) , cos 2(Ψ B − Ψ SP 2 ) , (eB/m 2 π ) 2 cos 2(Ψ B − Ψ 2 ) and (eB/m 2 π ) 2 cos 2(Ψ B − Ψ SP 2 ) in our calculations. If R Q is close to zero, it implies a similarity between two isobaric systems, however, it implies a big difference if R Q is far away from zero. For relative differences of deformation, R β2 = 0.33 for case 1, but R β2 = −1.43 for case 2, which implies a larger deformation difference for case 2 than that for case 1.

C. Calculations of the electromagnetic field
Following Refs. [36][37][38][39], we use the same way to calculate the initial electromagnetic fields as where we use natural unit = c = 1, Z n is the charge number of the nth particle, for the proton it is one, R n = r − r n is the relative position of the field point r to the source point r n , and r n is the location of the nth particle with velocity v n at the retarded time t n = t−|r−r n |. The summations run over all charged protons in the system. We calculate the participant plane Ψ 2 by using the spatial distribution of partons from the string-melting mechanism before the parton cascade process starts. The participant plane can be given by where r p is the displacement of the participating partons from field point r = (0, 0, 0) and φ p is the azimuthal angle of the participating partons on the transverse plane [40,41]. Following Refs. [17,42,43], we calculate the spectator plane as where r s is the displacement of spectator neutrons only from one projectile from field point r = (0, 0, 0) and φ s is the azimuthal angle of spectator neutrons only from one projectile in the transverse plane. We check that our results change little even if we use spectator protons. We choose spectator neutrons because the zero-degree calorimeters at STAR Collaboration [44] only can measure neutrons. In the above two formulas, the bracket · · · mean taking the average over all participating partons or all spectator neutrons of projectile, respectively.

III. RESULTS AND DISCUSSIONS
A. Spatial distributions of electromagnetic fields in isobaric collisions Figure 1 shows the contour plots of B x,y,z , |B x,y,z | , E x,y,z and |E x,y,z | at t = 0 on the transverse plane in 96 44 Ru + 96 44 Ru collisions at √ s = 200 GeV for case 1 where the two upper panels are for b = 0 fm and the two lower panels are for b = 8 fm. We find that |B x | is far less than |B y | at r = 0, what is more, the maximum of the magnetic fields is in field point r = 0 for mid central collisions.  model, whereas centrality and N track are often used in experiments. We can easily find that the magnetic fields are almost zero in most central events and have the maximum at some peripheral events, which indicate we should search for the CME signals in peripheral collisions. Meanwhile, the average of the absolute value of electric fields gradually decreases as centrality increases. These results are similar to the results of electromagnetic fields for Au+Au collisions in shape, see Ref. [38]. Because the radius of the Ru nucleus is smaller than that of the Au nucleus, the maximum of the magnetic fields is found in about b = 9 fm for 96 44 Ru + 96 44 Ru collisions, and it is about b = 12 fm for Au+Au collisions. What is more, the Ru nucleus has less protons than the Au nucleus, so the magnitudes of electromagnetic fields for Ru+Ru collisions are smaller than those for Au+Au collisions. Figure 3 shows the electromagnetic fields in 96     Zr collisions for case 1 is about 4% in central events and increases to 6% in peripheral events. However, the relative difference for case 2 is about 4% in central events and gently decreases to 3% in mid-central events then increases to 11.5% in peripheral events. Note that our relative difference of B for case 1 is similar to Refs. [19,20]. It is easy to be understood that the electromagnetic fields of 96 40 Zr + 96 40 Zr collisions are smaller than 96 44 Ru + 96 44 Ru collisions because they have less protons. Unquestionably, the difference of magnetic fields is vital for measuring the CME and we indeed find differences in the magnetic fields between two isobaric collisions. Furthermore, we measure the CME signal as mentioned above by using the correlator ∆γ. Because ∆γ ∝ (eB/m 2 π ) 2 cos 2(Ψ B −Ψ 2 ) has similar flow due to the same atomic number, therefore it is key to check how different the cos 2(Ψ B − Ψ 2 ) are between two isobaric collisions, which will be discussed next. As the chiral anomalous effects always occur either along or perpendicular to the magnetic-field direction, it is important to find an experimental way to determine the direction of the magnetic field. With the help of finite correlation between Ψ B and Ψ 2 , ones fortunately are capable of accessing the magnetic field direction and then measuring the CME. In Figs. 8 and 9 we plot the accumulated histograms of Ψ B − Ψ 2 at b = 0, 4, 7, and 10 fm in   Zr collisions is about 5% in most central bins then decreases to −2% in most peripheral bins. For case 2, the relative ratio of cos 2(Ψ B − Ψ 2 ) is concave and about 7% in most central bins and then increases to about 27% in most peripheral bins. In peripheral bins, one can see that the relative differences of cos 2(Ψ B − Ψ 2 ) for case 1 and case 2 differ a lot, which is actually caused by the deformation, i.e., the larger deformation and the weaker correlation cos 2(Ψ B − Ψ 2 ) for case 2 as shown in Fig. 12. Figure 14 shows that the correlation (eB/m 2 π ) 2 cos 2(Ψ B − Ψ 2 ) in 96     and 12, we can see that it is caused by both the magnetic field and the correlation. Following the same way, the relative ratios of (eB/m 2 π ) 2 cos 2(Ψ B − Ψ 2 ) between 96 44 Ru+ 96 44 Ru collisions and 96 40 Zr+ 96 40 Zr collisions are presented in Fig. 15. From Fig. 15(b), we can clearly see for case 1, the relative ratio is flat near 10%. But for case 2, the relative ratio shows a clear increasing trend from central to peripheral events. By comparing the results from Figs. 7, 13, and 15, we find the relative ratio of (eB/m 2 π ) 2 cos 2(Ψ B − Ψ 2 ) is larger than the relative ratio of cos 2(Ψ B − Ψ 2 ) , which indicates that the magnetic field plays an important role on the CME observable. All the relative ratios between 96  Our results indicate that the deformation has almost no effect on cos 2(Ψ B − Ψ 2 ) and (eB/m 2 π ) 2 cos 2(Ψ B − Ψ 2 ) in central and mid central events, but can not be neglected in peripheral events. The previous subsection shows the results from the correlation between magnetic field and participant plane Ψ 2 , now we show the results from the correlation between magnetic field and spectator plane Ψ SP 2 .
b(fm)   2 ) is around two times larger than that between magnetic field and participant plane Ψ 2 , cos 2(Ψ B − Ψ 2 ) . In peripheral collisions, this correlation is much stronger and approaching one. In the same way, we also take the relative ratio between 96 44 Ru + 96 44 Ru collisions and 96 40 Zr + 96 40 Zr collisions for case 1 and case 2, as shown in Fig. 17. The relative ratios of cos 2(Ψ B − Ψ SP 2 ) for case 1 are gradually decreased from around 5% to 0. Compared with Fig. 13, the relative ratios of cos 2(Ψ B − Ψ SP 2 ) for both case 1 and case 2 are close to zero for noncentral collisions. This indicates that there is little difference in the terms of cos 2(Ψ B − Ψ SP 2 ) between the two isobaric collisions for both cases for non-central collisions, thanks to the strong correlation between Ψ B and Ψ SP 2 . It provides a natural advantage to detect the possible effects purely from the difference of magnetic fields, even with less influence of the deformation.   Zr collisions for case 1 and case 2 as functions of b, centrality, N part , and N track . Note that compared to Fig. 14, the magnetic field is the same, but cos 2(Ψ B − Ψ SP 2 ) makes a difference. Because the magnetic field has a stronger correlation with the spectator plane than the participant plane, (eB/m 2 π ) 2 cos 2(Ψ B − Ψ SP 2 ) is stronger than (eB/m 2 π ) 2 cos 2(Ψ B − Ψ 2 ) . The relative ratios of (eB/m 2 π ) 2 cos 2(Ψ B − Ψ SP 2 ) between 96 44 Ru + 96 44 Ru collisions and 96 40 Zr + 96 40 Zr collisions are presented in Fig. 19. For case 1, the ratio fluctuates near 15% which is similar to the relative ratio of (eB/m 2 π ) 2 cos 2(Ψ B − Ψ 2 ) . For case 2, the relative ratio increases from central to peripheral events which is similar to the trend of (eB/m 2 π ) 2 cos 2(Ψ B − Ψ 2 ) , but the magnitude is reduced from 40% to 20% for the peripheral collisions. Figure 20 gives a direct comparison between (eB/m 2 π ) 2 cos 2(Ψ B − Ψ 2 ) and (eB/m 2 π ) 2 cos 2(Ψ B − Ψ SP 2 ) as functions of the centrality bin for case 1 and case 2. We find that the relative ratios of (eB/m 2 π ) 2 cos 2(Ψ B − Ψ 2 ) and (eB/m 2 π ) 2 cos 2(Ψ B − Ψ SP 2 ) for case 1 are similar, because the deformation difference is relatively weak for case 1. However, we observe that the two methods present different results for case 2, i.e., the relative ratio for the participant plane is larger than the relative ratio for the spectator plane. Based on the above results, we have already known that the correlation with the spectator plane is stronger than that with the participant plane. Therefore, (eB/m 2 π ) 2 cos 2(Ψ B − Ψ SP 2 ) is mainly affected by the magnetic field, however, (eB/m 2 π ) 2 cos 2(Ψ B − Ψ 2 ) is affected by both magnetic field and cos 2(Ψ B − Ψ 2 ) . It suggests that we can observe a much cleaner magnetic field effect of CME with the correlation ∆γ with respect to the spectator plane than that with respect to the participant plane.

IV. CONCLUSIONS
To summarize, we have utilized the AMPT model to investigate the properties of electromagnetic fields in isobaric 96 44 Ru + 96 44 Ru collisions and 96 40 Zr + 96 40 Zr collisions at the RHIC energy of √ s=200 GeV. Meanwhile, the relative ratios of the magnetic fields are up to 10% for different centralities for case 1 and case 2. Furthermore, the correlations cos 2(Ψ B − Ψ SP 2 ) and (eB/m 2 π ) 2 cos 2(Ψ B − Ψ SP 2 ) are all much stronger than cos 2(Ψ B − Ψ 2 ) and (eB/m 2 π ) 2 cos 2(Ψ B − Ψ 2 ) for the two isobaric collisions. Moreover, deformation does affect the CME signals in isobaric collisions, especially for peripheral events in which the larger deformation leads to the weaker cos 2(Ψ B − Ψ 2 ) and (eB/m 2 π ) 2 cos 2(Ψ B − Ψ 2 ) . For case 1, the relative difference with respect to the spectator plane and that with respect to the participant plane look similar due to their small relative deformation difference. For case 2, the two relative differences look different due to their larger deformation difference. Since Ψ SP 2 has a much stronger correlation with Ψ B than Ψ 2 , the ∆γ correlator with respect to Ψ SP 2 is expected to reflect much cleaner information about the CME signal due to different magnitudes of magnetic fields between two isobaric collisions with less influences of deformation.