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 ${}_{44}^{96}\mathrm{Ru}+{}_{44}^{96}\mathrm{Ru}$ and ${}_{40}^{96}\mathrm{Zr}+{}_{40}^{96}\mathrm{Zr}$ collisions at $\sqrt{s}=200\mathrm{GeV}$ by using a multiphase transport model with special emphasis on the correlation between magnetic-field direction and participant plane angle ${\mathrm{\Psi }}_2$ (or spectator plane angle ${\mathrm{\Psi }}_2^{\mathrm{SP}}$), i.e., $\langle cos2({\mathrm{\Psi }}_B-{\mathrm{\Psi }}_2)\rangle$ [or $\langle cos2({\mathrm{\Psi }}_B-{\mathrm{\Psi }}_2^{\mathrm{SP}})\rangle$]. We confirm that the magnetic fields of ${}_{44}^{96}\mathrm{Ru}+{}_{44}^{96}\mathrm{Ru}$ collisions are stronger than those of ${}_{40}^{96}\mathrm{Zr}+{}_{40}^{96}\mathrm{Zr}$ collisions due to their larger proton fraction. We find that the deformation of nuclei has a non-negligible effect on $\langle cos2({\mathrm{\Psi }}_B-{\mathrm{\Psi }}_2)\rangle$ especially in peripheral events. Because the magnetic-field direction is more strongly correlated with ${\mathrm{\Psi }}_2^{\mathrm{SP}}$ than with ${\mathrm{\Psi }}_2$, the relative difference of the chiral magnetic effect observable with respect to ${\mathrm{\Psi }}_2^{\mathrm{SP}}$ is expected to be able to reflect much cleaner information about the chiral magnetic effect with less influences of deformation.

Figure 1

The spatial distributions of the electromagnetic fields on the transverse plane at $t=0$ for $b=0$ (upper panels) and $b=8\mathrm{fm}$ (lower panels) in ${}_{44}^{96}\mathrm{Ru}+{}_{44}^{96}\mathrm{Ru}$ collisions at $\sqrt{s}=200\mathrm{GeV}$ for case 1 where the unit is $m_{\pi }^2$. The black solid circles indicate the two colliding nuclei.

Figure 2

The electromagnetic fields at $t=0$ and $\mathbf{r}=\mathbf{0}$ as functions of (a) impact parameter $b$, (b) $N_{\mathrm{part}}$, and (c) $N_{\mathrm{track}}$ in ${}_{44}^{96}\mathrm{Ru}+{}_{44}^{96}\mathrm{Ru}$ collisions at $\sqrt{s}=200\mathrm{GeV}$ for case 1.

Figure 3

The electromagnetic fields at $t=0$ and $\mathbf{r}=\mathbf{0}$ as functions of (a) impact parameter $b$, (b) $N_{\mathrm{part}}$, and (c) $N_{\mathrm{track}}$ in ${}_{44}^{96}\mathrm{Ru}+{}_{44}^{96}\mathrm{Ru}$ collisions at $\sqrt{s}=200\mathrm{GeV}$ for case 2.

Figure 4

The electromagnetic fields at $t=0$ and $\mathbf{r}=\mathbf{0}$ as functions of (a) impact parameter $b$, (b) $N_{\mathrm{part}}$, and (c) $N_{\mathrm{track}}$ in ${}_{40}^{96}\mathrm{Zr}+{}_{40}^{96}\mathrm{Zr}$ collisions at $\sqrt{s}=200\mathrm{GeV}$ for case 1.

Figure 5

The electromagnetic fields at $t=0$ and $\mathbf{r}=\mathbf{0}$ as functions of (a) impact parameter $b$, (b) $N_{\mathrm{part}}$, and (c) $N_{\mathrm{track}}$ in ${}_{40}^{96}\mathrm{Zr}+{}_{40}^{96}\mathrm{Zr}$ collisions at $\sqrt{s}=200\mathrm{GeV}$ for case 2.

Figure 6

The magnetic field at $t=0$ and $\mathbf{r}=\mathbf{0}$ as functions of (a) impact parameter $b$, (b) centrality, (c) $N_{\mathrm{part}}$, and (d) $N_{\mathrm{track}}$ in ${}_{44}^{96}\mathrm{Ru}+{}_{44}^{96}\mathrm{Ru}$ collisions and ${}_{40}^{96}\mathrm{Zr}+{}_{40}^{96}\mathrm{Zr}$ collisions for case 1 and case 2 at $\sqrt{s}=200\mathrm{GeV}$.

Figure 7

The relative ratio of the magnetic field as functions of (a) impact parameter $b$, (b) centrality, (c) $N_{\mathrm{part}}$, and (d) $N_{\mathrm{track}}$ in isobaric collisions at $\sqrt{s}=200\mathrm{GeV}$ for case 1 and case 2.

Figure 8

The event-by-event histograms of ${\mathrm{\Psi }}_B-{\mathrm{\Psi }}_2$ at impact parameters $b=0,4$, 7, and 10 fm in ${}_{44}^{96}\mathrm{Ru}+{}_{44}^{96}\mathrm{Ru}$ collisions at $\sqrt{s}=200\mathrm{GeV}$ for case 1. Here, ${\mathrm{\Psi }}_B$ is the azimuthal direction of the $\mathbf{B}$ field (at $t=0$ and $\mathbf{r}=\mathbf{0}$), and ${\mathrm{\Psi }}_2$ is the second-harmonic participant plane.

Figure 9

The event-by-event histograms of ${\mathrm{\Psi }}_B-{\mathrm{\Psi }}_2$ at impact parameters $b=0$, 4, 7, and 10 fm in ${}_{40}^{96}\mathrm{Zr}+{}_{40}^{96}\mathrm{Zr}$ collisions at $\sqrt{s}=200\mathrm{GeV}$ for case 1. Here, ${\mathrm{\Psi }}_B$ is the azimuthal direction of the $\mathbf{B}$ field (at $t=0$ and $\mathbf{r}=\mathbf{0}$), and ${\mathrm{\Psi }}_2$ is the second-harmonic participant plane.

Figure 10

The scatter plots on the ${\mathrm{\Psi }}_B-{\mathrm{\Psi }}_2$ plane at impact parameters $b=0$, 4, 7, and 10 fm in ${}_{44}^{96}\mathrm{Ru}+{}_{44}^{96}\mathrm{Ru}$ collisions at $\sqrt{s}=200\mathrm{GeV}$ for case 1, where ${\mathrm{\Psi }}_B$ is the azimuthal direction of the $\mathbf{B}$ field (at $t=0$ and $\mathbf{r}=\mathbf{0}$), and ${\mathrm{\Psi }}_2$ is the second-harmonic participant plane.

Figure 11

The scatter plots on the ${\mathrm{\Psi }}_B-{\mathrm{\Psi }}_2$ plane at impact parameters $b=0$, 4, 7, and 10 fm in ${}_{40}^{96}\mathrm{Zr}+{}_{40}^{96}\mathrm{Zr}$ collisions at $\sqrt{s}=200\mathrm{GeV}$ for case 1, where ${\mathrm{\Psi }}_B$ is the azimuthal direction of the $\mathbf{B}$ field (at $t=0$ and $\mathbf{r}=\mathbf{0}$), and ${\mathrm{\Psi }}_2$ is the second-harmonic participant plane.

Figure 12

The correlation $\langle cos2({\mathrm{\Psi }}_B-{\mathrm{\Psi }}_2)\rangle$ as functions of (a) impact parameter $b$, (b) centrality, (c) $N_{\mathrm{part}}$, and (d) $N_{\mathrm{track}}$ in ${}_{44}^{96}\mathrm{Ru}+{}_{44}^{96}\mathrm{Ru}$ collisions and ${}_{40}^{96}\mathrm{Zr}+{}_{40}^{96}\mathrm{Zr}$ collisions at $\sqrt{s}=200\mathrm{GeV}$ for case 1 and case 2.

Figure 13

The relative ratio of $\langle cos2({\mathrm{\Psi }}_B-{\mathrm{\Psi }}_2)\rangle$ as functions of (a) impact parameter $b$, (b) centrality, (c) $N_{\mathrm{part}}$, and (d) $N_{\mathrm{track}}$ in isobaric collisions at $\sqrt{s}=200\mathrm{GeV}$ for case 1 and case 2.

Figure 14

The correlation $\langle {(eB/m_{\pi }^2)}^{2}cos2({\mathrm{\Psi }}_B-{\mathrm{\Psi }}_2)\rangle$ as functions of (a) impact parameter $b$, (b) centrality, (c) $N_{\mathrm{part}}$, and (d) $N_{\mathrm{track}}$ in ${}_{44}^{96}\mathrm{Ru}+{}_{44}^{96}\mathrm{Ru}$ collisions and ${}_{40}^{96}\mathrm{Zr}+{}_{40}^{96}\mathrm{Zr}$ collisions at $\sqrt{s}=200\mathrm{GeV}$ for case 1 and case 2.

Figure 15

The relative ratio of $\langle {(eB/m_{\pi }^2)}^{2}cos2({\mathrm{\Psi }}_B-{\mathrm{\Psi }}_2)\rangle$ as functions of (a) impact parameter $b$, (b) centrality, (c) $N_{\mathrm{part}}$, and (d) $N_{\mathrm{track}}$ in isobaric collisions at $\sqrt{s}=200\mathrm{GeV}$ for case 1 and case 2.

Figure 16

The correlation $\langle cos2({\mathrm{\Psi }}_B-{\mathrm{\Psi }}_2^{\mathrm{SP}})\rangle$ as functions of (a) impact parameter $b$, (b) centrality, (c) $N_{\mathrm{part}}$, and (d) $N_{\mathrm{track}}$ in ${}_{44}^{96}\mathrm{Ru}+{}_{44}^{96}\mathrm{Ru}$ collisions and ${}_{40}^{96}\mathrm{Zr}+{}_{40}^{96}\mathrm{Zr}$ collisions at $\sqrt{s}=200\mathrm{GeV}$ for case 1 and case 2.

Figure 17

The relative ratio of $\langle cos2({\mathrm{\Psi }}_B-{\mathrm{\Psi }}_2^{\mathrm{SP}})\rangle$ as functions of (a) impact parameter $b$, (b) centrality, (c) $N_{\mathrm{part}}$, and (d) $N_{\mathrm{track}}$ in isobaric collisions at $\sqrt{s}=200\mathrm{GeV}$ for case 1 and case 2.

Figure 18

The correlation $\langle {(eB/m_{\pi }^2)}^{2}cos2({\mathrm{\Psi }}_B-{\mathrm{\Psi }}_2^{\mathrm{SP}})\rangle$ as functions of (a) impact parameter $b$, (b) centrality, (c) $N_{\mathrm{part}}$, and (d) $N_{\mathrm{track}}$ in ${}_{44}^{96}\mathrm{Ru}+{}_{44}^{96}\mathrm{Ru}$ collisions and ${}_{40}^{96}\mathrm{Zr}+{}_{40}^{96}\mathrm{Zr}$ collisions at $\sqrt{s}=200\mathrm{GeV}$ for case 1 and case 2.

Figure 19

The relative ratio of $\langle {(eB/m_{\pi }^2)}^{2}cos2({\mathrm{\Psi }}_B-{\mathrm{\Psi }}_2^{\mathrm{SP}})\rangle$ as functions of (a) impact parameter $b$, (b) centrality, (c) $N_{\mathrm{part}}$, and (d) $N_{\mathrm{track}}$ in isobaric collisions at $\sqrt{s}=200\mathrm{GeV}$ for case 1 and case 2.

Figure 20

The relative ratios of (a) $\langle {(eB/m_{\pi }^2)}^{2}cos2({\mathrm{\Psi }}_B-{\mathrm{\Psi }}_2)\rangle$ and (b) $\langle {(eB/m_{\pi }^2)}^{2}cos2({\mathrm{\Psi }}_B-{\mathrm{\Psi }}_2^{\mathrm{SP}})\rangle$ as functions of the centrality bin in isobaric collisions at $\sqrt{s}=200\mathrm{GeV}$ for case 1 and case 2.