Comparison of theoretical predictions to the measured total cross section for coherent $\rho^0$ production as a functionof $\sqrt{s_{NN}}$ . The measured cross section is based on the dataset collected with trigger B. The error bars show the sum of the statistical and systematic uncertainties. See text for details.

The correlation differences $\Delta\langle A^{2}\rangle=\delta\langle A^{2}_{UD}\rangle-\delta\langle A^{2}_{ LR}\rangle$ and $\Delta\langle A_{+}A_{-}\rangle=\delta\langle A_{+}A_{-}\rangle_{ UD}-\delta\langle A_{+}A_{-}\rangle_{ LR}$ as a function of $p_{T}$. The data are from 20-40% central (upper) and 0-20% central (lower) Au+Au collisions.

The charge asymmetry correlations $\delta\langle A^{2}\rangle$ (left panels) and $\delta\langle A_{+}A_{-}\rangle$ (center panels), and correlation differences $\Delta\langle A^{2}\rangle=\delta\langle A^{2}_{ UD}\rangle-\delta\langle A^{2}_{ LR}\rangle$ and $\Delta\langle A_{+}A_{-}\rangle=\delta\langle A_{+}A_{-}\rangle_{ UD}-\delta\langle A_{+}A_{-}\rangle_{ LR}$ (right panels), as a function of the azimuthal elliptic anisotropy of high-$p_{T}$ ($p_{T}>2$~GeV/$c$) particles (upper panels) and low-$p_{T}$ ($p_{T}<2$~GeV/$c$) particles (lower panels). Data are from 20-40% Au+Au collisions. The particle $p_{T}$ range of $0.15<p_{T}<2$~GeV/$c$ is used for both EP construction and asymmetry calculation.

The charge asymmetry correlations $\delta\langle A^{2}\rangle$ (left panels) and $\delta\langle A_{+}A_{-}\rangle$ (center panels), and correlation differences $\Delta\langle A^{2}\rangle=\delta\langle A^{2}_{ UD}\rangle-\delta\langle A^{2}_{ LR}\rangle$ and $\Delta\langle A_{+}A_{-}\rangle=\delta\langle A_{+}A_{-}\rangle_{ UD}-\delta\langle A_{+}A_{-}\rangle_{ LR}$ (right panels), as a function of the azimuthal elliptic anisotropy of high-$p_{T}$ ($p_{T}>2$~GeV/$c$) particles (upper panels) and low-$p_{T}$ ($p_{T}<2$~GeV/$c$) particles (lower panels). Data are from 20-40% Au+Au collisions. The particle $p_{T}$ range of $0.15<p_{T}<2$~GeV/$c$ is used for both EP construction and asymmetry calculation.

The wedge size dependences of charge multiplicity asymmetry correlations (upper panel), and their differences between out-of-plane and in-plane, $\Delta\langle A^{2}\rangle$ and $\Delta\langle A_{+}A_{-}\rangle$ (lower panel) for 20-40% central Au+Au collisions.

The wedge size dependences of charge multiplicity asymmetry correlations (upper panel), and their differences between out-of-plane and in-plane, $\Delta\langle A^{2}\rangle$ and $\Delta\langle A_{+}A_{-}\rangle$ (lower panel) for 20-40% central Au+Au collisions.

Charge multiplicity asymmetry correlations as a function of the wedge location, $\phi_{ w}$, in 20-40% central Au+Au collisions. The wedge size is $30^{\circ}$. The curves are the characteristic $\cos(2\phi_{ w}$) to guide the eye.

The wedge size dependence of the difference between the same-sign and opposite-sign, $\Delta\langle A^{2}\rangle-\Delta\langle A_{+}A_{-}\rangle$.

The values of $\Delta\langle A^{2}\rangle-\Delta\langle A_{+}A_{-}\rangle$, scaled by $N_{part}$, as a function of the measured average elliptic anisotropy $<v^{obs}_{2}}>$ in Au+Au collisions. The centrality bin number is labeled by each data point, 0 for 70-80% up to 8 for 0-5%.

$\Delta=\Delta\langle A^{2}\rangle-\Delta\langle A_{+}A_{-}\rangle$ as a function of $v^{obs}_{2}$, the event-by-event elliptical anisotropy of particle distributions relative to the second-harmonic event plane reconstructed from TPC tracks (left panel) and the first harmonic event plane reconstructed from the ZDC-SMD neutron signals (middle panel), in 20-40% central Au+Au collisions. Right panel: Average $\Delta$ for events with $|v^{obs}_{2}|<0.04$ relative to the TPC event plane as a function of centrality.

$\Delta=\Delta\langle A^{2}\rangle-\Delta\langle A_{+}A_{-}\rangle$ as a function of $v^{obs}_{2}$, the event-by-event elliptical anisotropy of particle distributions relative to the second-harmonic event plane reconstructed from TPC tracks (left panel) and the first harmonic event plane reconstructed from the ZDC-SMD neutron signals (middle panel), in 20-40% central Au+Au collisions. Right panel: Average $\Delta$ for events with $|v^{obs}_{2}|<0.04$ relative to the TPC event plane as a function of centrality.

$\Delta=\Delta\langle A^{2}\rangle-\Delta\langle A_{+}A_{-}\rangle$ as a function of $v^{obs}_{2}$, the event-by-event elliptical anisotropy of particle distributions relative to the second-harmonic event plane reconstructed from TPC tracks (left panel) and the first harmonic event plane reconstructed from the ZDC-SMD neutron signals (middle panel), in 20-40% central Au+Au collisions. Right panel: Average $\Delta$ for events with $|v^{obs}_{2}|<0.04$ relative to the TPC event plane as a function of centrality.

Left panel: $\Delta\langle A^{2}\rangle$ and $\Delta\langle A_{+}A_{-}\rangle$ as a function of $v^{obs}_{2}$ with a random EP in 20-40% Au+Au collisions. The lines depict the results with reconstructed EP lower right panel for comparison. Middle panel: The differences in $\Delta\langle A^{2}\rangle$ and $\Delta\langle A_{+}A_{-}\rangle$ between the reconstructed EP and random EP results, respectively. Right panel: $\Delta=\Delta\langle A^{2}\rangle-\Delta\langle A_{+}A_{-}\rangle$ as a function of $v^{obs}_{2}$ with a random EP (open triangles).

Left panel: $\Delta\langle A^{2}\rangle$ and $\Delta\langle A_{+}A_{-}\rangle$ as a function of $v^{obs}_{2}$ with a random EP in 20-40% Au+Au collisions. The lines depict the results with reconstructed EP lower right panel for comparison. Middle panel: The differences in $\Delta\langle A^{2}\rangle$ and $\Delta\langle A_{+}A_{-}\rangle$ between the reconstructed EP and random EP results, respectively. Right panel: $\Delta=\Delta\langle A^{2}\rangle-\Delta\langle A_{+}A_{-}\rangle$ as a function of $v^{obs}_{2}$ with a random EP (open triangles).

Left panel: $\Delta\langle A^{2}\rangle$ and $\Delta\langle A_{+}A_{-}\rangle$ as a function of $v^{obs}_{2}$ with a random EP in 20-40% Au+Au collisions. The lines depict the results with reconstructed EP lower right panel for comparison. Middle panel: The differences in $\Delta\langle A^{2}\rangle$ and $\Delta\langle A_{+}A_{-}\rangle$ between the reconstructed EP and random EP results, respectively. Right panel: $\Delta=\Delta\langle A^{2}\rangle-\Delta\langle A_{+}A_{-}\rangle$ as a function of $v^{obs}_{2}$ with a random EP (open triangles).

The correlation differences, $\langle A^{2}_{ LR}\rangle-\langle A^{2}_{ UD}\rangle$ and $\langle A_{+}A_{-}\rangle_{ LR}-\langle A_{+}A_{-}\rangle_{ UD}$, scaled by the number of participants $(\pi/4)^2 N_{part}$. Also shown are $\langle \cos(\phi_{\alpha}+\phi_{\beta}-2\phi_{c})\rangle/v_{2,c} \approx \langle \cos(\phi_{\alpha}+\phi_{\beta}-2\psi_{ RP}) \rangle$ of same- and opposite-sign particle pairs ($\alpha$ and $\beta$) calculated from particles used for the charge asymmetry correlations with particle $c$ from those used for EP construction. The upper panel shows the default results where particles are divided according to $\eta$; the lower panel shows results with particles divided randomly into two halves and compared to published correlators.

The correlation differences, $\langle A^{2}_{ LR}\rangle-\langle A^{2}_{ UD}\rangle$ and $\langle A_{+}A_{-}\rangle_{ LR}-\langle A_{+}A_{-}\rangle_{ UD}$, scaled by the number of participants $(\pi/4)^2 N_{part}$. Also shown are $\langle \cos(\phi_{\alpha}+\phi_{\beta}-2\phi_{c})\rangle/v_{2,c} \approx \langle \cos(\phi_{\alpha}+\phi_{\beta}-2\psi_{ RP}) \rangle$ of same- and opposite-sign particle pairs ($\alpha$ and $\beta$) calculated from particles used for the charge asymmetry correlations with particle $c$ from those used for EP construction. The upper panel shows the default results where particles are divided according to $\eta$; the lower panel shows results with particles divided randomly into two halves and compared to published correlators.

Upper panel: Positively charged particle multiplicity distributions versus the azimuthal angle, $\phi$, in 30-40% central Au+Au collisions. The data are shown separately for $\eta>0$ and $\eta<0$. The magnetic polarities of the STAR magnet have been summed and the $p_{T}$ is integrated over the range $0.15<p_{T}<2$~GeV/$c$. The inverse of these distributions, properly normalized, were used to correct for the track efficiency versus $\phi$. Lower panel: Reconstructed event-plane azimuthal angle distributions in 30-40% central Au+Au collisions. Particles within $0.15 < p_{T} < 2$~GeV/$c$ from $\eta<0$ and $\eta>0$ were used separately to reconstruct the EP.

Upper panel: Positively charged particle multiplicity distributions versus the azimuthal angle, $\phi$, in 30-40% central Au+Au collisions. The data are shown separately for $\eta>0$ and $\eta<0$. The magnetic polarities of the STAR magnet have been summed and the $p_{T}$ is integrated over the range $0.15<p_{T}<2$~GeV/$c$. The inverse of these distributions, properly normalized, were used to correct for the track efficiency versus $\phi$. Lower panel: Reconstructed event-plane azimuthal angle distributions in 30-40% central Au+Au collisions. Particles within $0.15 < p_{T} < 2$~GeV/$c$ from $\eta<0$ and $\eta>0$ were used separately to reconstruct the EP.

The relative charge asymmetry correlations, $\langle A_{+}A_{-}\rangle_{ UD}/\langle A_{+}A_{-}\rangle_{ LR}$, as a function of the number of participants, $N_{part}$, for four combinations of $\eta$ ranges used for EP reconstruction and asymmetry calculation.

The asymmetry correlations, $\langle A^{2}_{+,{ UD}}\rangle$ (left panel), $\langle A^{2}_{-,{ UD}}\rangle$ (middle panel), and $\langle A_{+}A_{-}\rangle_{ UD}$ (right panel), multiplied by the number of participants $N_{part}$, before and after the corrections for the $\phi$-dependent acceptance $\times$ efficiency. The colored curves are the corresponding statistical fluctuations and detector effects. The results are separated for $\eta>0$ and $\eta<0$. Only $\eta>0$ is shown for $\langle A^{2}_{+,{ UD}}\rangle$ and $\eta<0$ for $\langle A^{2}_{-,{ UD}}\rangle$ for clarity.

The asymmetry correlations, $\langle A^{2}_{+,{ UD}}\rangle$ (left panel), $\langle A^{2}_{-,{ UD}}\rangle$ (middle panel), and $\langle A_{+}A_{-}\rangle_{ UD}$ (right panel), multiplied by the number of participants $N_{part}$, before and after the corrections for the $\phi$-dependent acceptance $\times$ efficiency. The colored curves are the corresponding statistical fluctuations and detector effects. The results are separated for $\eta>0$ and $\eta<0$. Only $\eta>0$ is shown for $\langle A^{2}_{+,{ UD}}\rangle$ and $\eta<0$ for $\langle A^{2}_{-,{ UD}}\rangle$ for clarity.

The asymmetry correlations, $\langle A^{2}_{+,{ UD}}\rangle$ (left panel), $\langle A^{2}_{-,{ UD}}\rangle$ (middle panel), and $\langle A_{+}A_{-}\rangle_{ UD}$ (right panel), multiplied by the number of participants $N_{part}$, before and after the corrections for the $\phi$-dependent acceptance $\times$ efficiency. The colored curves are the corresponding statistical fluctuations and detector effects. The results are separated for $\eta>0$ and $\eta<0$. Only $\eta>0$ is shown for $\langle A^{2}_{+,{ UD}}\rangle$ and $\eta<0$ for $\langle A^{2}_{-,{ UD}}\rangle$ for clarity.

Left panel: Statistical fluctuation and detector effects in the charge asymmetry variance (multiplied by the number of participants $N_{part}$) from the $\eta<0$ region. The dotted curve shows the $1/N$ approximation, the dashed curve shows the statistical fluctuations $\langle A^{2}_{-,{ LR,stat}}\rangle$ obtained by the (50-50) method (see the text). The solid curve connecting the solid points shows the net effect of the statistical fluctuations and detector non-uniformities, $\langle A^{2}_{-,{ LR,stat+det}}\rangle$, obtained by the adding-$\pi$ method, and the crosses shows the same but obtained from the ``scramble'' method. See the text for details. Middle panel: The statistical fluctuation effects relative to the $1/N$ approximation (thin curves) and the effects due to the imperfect detector (thick curves). The dashed curves are for the region $\eta>0$ and the solid curves are for $\eta<0$. Right panel: The ratios of the statistical fluctuation and detector effects $\langle A^{2}_{+,{ LR,stat+det}}\rangle/\langle A^{2}_{-,{ LR,stat+det}}\rangle$ and $\langle A^{2}_{+,{ UD,stat+det}}\rangle/\langle A^{2}_{+,{ LR,stat+det}}\rangle$, separately for the $\eta>0$ and $\eta<0$ regions. The $\phi$-acceptance correction was applied for the results in this figure.

Left panel: Statistical fluctuation and detector effects in the charge asymmetry variance (multiplied by the number of participants $N_{part}$) from the $\eta<0$ region. The dotted curve shows the $1/N$ approximation, the dashed curve shows the statistical fluctuations $\langle A^{2}_{-,{ LR,stat}}\rangle$ obtained by the (50-50) method (see the text). The solid curve connecting the solid points shows the net effect of the statistical fluctuations and detector non-uniformities, $\langle A^{2}_{-,{ LR,stat+det}}\rangle$, obtained by the adding-$\pi$ method, and the crosses shows the same but obtained from the ``scramble'' method. See the text for details. Middle panel: The statistical fluctuation effects relative to the $1/N$ approximation (thin curves) and the effects due to the imperfect detector (thick curves). The dashed curves are for the region $\eta>0$ and the solid curves are for $\eta<0$. Right panel: The ratios of the statistical fluctuation and detector effects $\langle A^{2}_{+,{ LR,stat+det}}\rangle/\langle A^{2}_{-,{ LR,stat+det}}\rangle$ and $\langle A^{2}_{+,{ UD,stat+det}}\rangle/\langle A^{2}_{+,{ LR,stat+det}}\rangle$, separately for the $\eta>0$ and $\eta<0$ regions. The $\phi$-acceptance correction was applied for the results in this figure.

Left panel: Statistical fluctuation and detector effects in the charge asymmetry variance (multiplied by the number of participants $N_{part}$) from the $\eta<0$ region. The dotted curve shows the $1/N$ approximation, the dashed curve shows the statistical fluctuations $\langle A^{2}_{-,{ LR,stat}}\rangle$ obtained by the (50-50) method (see the text). The solid curve connecting the solid points shows the net effect of the statistical fluctuations and detector non-uniformities, $\langle A^{2}_{-,{ LR,stat+det}}\rangle$, obtained by the adding-$\pi$ method, and the crosses shows the same but obtained from the ``scramble'' method. See the text for details. Middle panel: The statistical fluctuation effects relative to the $1/N$ approximation (thin curves) and the effects due to the imperfect detector (thick curves). The dashed curves are for the region $\eta>0$ and the solid curves are for $\eta<0$. Right panel: The ratios of the statistical fluctuation and detector effects $\langle A^{2}_{+,{ LR,stat+det}}\rangle/\langle A^{2}_{-,{ LR,stat+det}}\rangle$ and $\langle A^{2}_{+,{ UD,stat+det}}\rangle/\langle A^{2}_{+,{ LR,stat+det}}\rangle$, separately for the $\eta>0$ and $\eta<0$ regions. The $\phi$-acceptance correction was applied for the results in this figure.

The asymmetry correlations, $\langle A^{2}_{+,{ UD}}\rangle$ (left panel), $\langle A^{2}_{-,{ UD}}\rangle$ (middle panel), and $\langle A_{+}A_{-}\rangle_{ UD}$ (right panel), multiplied by the number of participants $N_{part}$, separately for $\eta>0$ and $\eta<0$. The star and triangle depict the results from $d$+Au collisions. The net effects of the statistical fluctuations and detector non-uniformities are shown as the curves.

The asymmetry correlations, $\langle A^{2}_{+,{ UD}}\rangle$ (left panel), $\langle A^{2}_{-,{ UD}}\rangle$ (middle panel), and $\langle A_{+}A_{-}\rangle_{ UD}$ (right panel), multiplied by the number of participants $N_{part}$, separately for $\eta>0$ and $\eta<0$. The star and triangle depict the results from $d$+Au collisions. The net effects of the statistical fluctuations and detector non-uniformities are shown as the curves.

The asymmetry correlations, $\langle A^{2}_{+,{ UD}}\rangle$ (left panel), $\langle A^{2}_{-,{ UD}}\rangle$ (middle panel), and $\langle A_{+}A_{-}\rangle_{ UD}$ (right panel), multiplied by the number of participants $N_{part}$, separately for $\eta>0$ and $\eta<0$. The star and triangle depict the results from $d$+Au collisions. The net effects of the statistical fluctuations and detector non-uniformities are shown as the curves.

The statistical and detector effect subtracted asymmetry correlations, $\delta\langle A^{2}_{+,0^{\circ}\pm\Delta\phi_{w}}\rangle$ (left panel), $\delta\langle A^{2}_{-,0^{\circ}\pm\Delta\phi_{w}}\rangle$ (middle panel), and $\delta\langle A_{+}A_{-}\rangle_{ LR}$ (right panel), multiplied by the number of participants $N_{part}$, before and after the corrections for the $\phi$-dependent acceptance $\times$ efficiency.

The statistical and detector effect subtracted asymmetry correlations, $\delta\langle A^{2}_{+,0^{\circ}\pm\Delta\phi_{w}}\rangle$ (left panel), $\delta\langle A^{2}_{-,0^{\circ}\pm\Delta\phi_{w}}\rangle$ (middle panel), and $\delta\langle A_{+}A_{-}\rangle_{ LR}$ (right panel), multiplied by the number of participants $N_{part}$, before and after the corrections for the $\phi$-dependent acceptance $\times$ efficiency.

The statistical and detector effect subtracted asymmetry correlations, $\delta\langle A^{2}_{+,0^{\circ}\pm\Delta\phi_{w}}\rangle$ (left panel), $\delta\langle A^{2}_{-,0^{\circ}\pm\Delta\phi_{w}}\rangle$ (middle panel), and $\delta\langle A_{+}A_{-}\rangle_{ LR}$ (right panel), multiplied by the number of participants $N_{part}$, before and after the corrections for the $\phi$-dependent acceptance $\times$ efficiency.

Dynamical charge asymmetry variance after removing the statistical and detector effects, $\delta\langle A^{2}_{\pm,{ LR}}\rangle=\langle A^{2}_{\pm,{ LR}}\rangle-\langle A^{2}_{\pm,{ LR,stat+det}}\rangle$, scaled by the number of participants $N_{part}$. The results are consistent between positive and negative $\eta$ regions and between positive and negative charges.

The second-harmonic event-plane resolution of sub-events ($\eta<0$ and $\eta>0$) as a function of the number of participants.

Charge multiplicity asymmetry correlations, $\delta\langle A^{2}\rangle$ (left panel) and $\delta\langle A_{+}A_{-}\rangle$ (middle panel), and their differences between UD\ and LR\ (right panel) as a function of the event-plane resolution $\epsilon_{ EP}(f)=\sqrt{\mean{\cos2(\psi_{{ EP},\eta>0}(f)-\psi_{{ EP},\eta<0}(f))}}$ in 20-30% central Au+Au collisions. The solid lines are free linear fits to the data, while the dashed lines are linear fits with the intercept fixed to zero at $\epsilon_{ EP}(f)=0$.

Charge multiplicity asymmetry correlations, $\delta\langle A^{2}\rangle$ (left panel) and $\delta\langle A_{+}A_{-}\rangle$ (middle panel), and their differences between UD\ and LR\ (right panel) as a function of the event-plane resolution $\epsilon_{ EP}(f)=\sqrt{\mean{\cos2(\psi_{{ EP},\eta>0}(f)-\psi_{{ EP},\eta<0}(f))}}$ in 20-30% central Au+Au collisions. The solid lines are free linear fits to the data, while the dashed lines are linear fits with the intercept fixed to zero at $\epsilon_{ EP}(f)=0$.

TCharge multiplicity asymmetry correlations, $\delta\langle A^{2}\rangle$ (left panel) and $\delta\langle A_{+}A_{-}\rangle$ (middle panel), and their differences between UD\ and LR\ (right panel) as a function of the event-plane resolution $\epsilon_{ EP}(f)=\sqrt{\mean{\cos2(\psi_{{ EP},\eta>0}(f)-\psi_{{ EP},\eta<0}(f))}}$ in 20-30% central Au+Au collisions. The solid lines are free linear fits to the data, while the dashed lines are linear fits with the intercept fixed to zero at $\epsilon_{ EP}(f)=0$.

The TPC second-harmonic (upper left) and ZDC-SMD first harmonic (upper right) event-plane resolution as a function of $v^{obs}_{2}$ in 20-40% central Au+Au collisions. The TPC second-harmonic (lower left) and ZDC-SMD first harmonic (lower right) event-plane resolution for events with $|v^{obs}_{2}|<0.04$ as a function of centrality.

The TPC second-harmonic (upper left) and ZDC-SMD first harmonic (upper right) event-plane resolution as a function of $v^{obs}_{2}$ in 20-40% central Au+Au collisions. The TPC second-harmonic (lower left) and ZDC-SMD first harmonic (lower right) event-plane resolution for events with $|v^{obs}_{2}|<0.04$ as a function of centrality.

The TPC second-harmonic (upper left) and ZDC-SMD first harmonic (upper right) event-plane resolution as a function of $v^{obs}_{2}$ in 20-40% central Au+Au collisions. The TPC second-harmonic (lower left) and ZDC-SMD first harmonic (lower right) event-plane resolution for events with $|v^{obs}_{2}|<0.04$ as a function of centrality.

The TPC second-harmonic (upper left) and ZDC-SMD first harmonic (upper right) event-plane resolution as a function of $v^{obs}_{2}$ in 20-40% central Au+Au collisions. The TPC second-harmonic (lower left) and ZDC-SMD first harmonic (lower right) event-plane resolution for events with $|v^{obs}_{2}|<0.04$ as a function of centrality.