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HERA combined reduced cross sections $\sigma_{r,\rm NC}^{+}$ for NC $e^{+}p$ scattering at $\sqrt{s} = 225$ GeV; $\delta_{\rm stat}$, $\delta_{\rm uncor}$ and $\delta_{\rm cor}$ represent the statistical, uncorrelated systematic and correlated systematic uncertainties, respectively; $\delta_{\rm rel}$, $\delta_{\gamma p}$, $\delta_{\rm had}$ and $\delta_{1}$ to $\delta_{4}$ are the correlated sources of uncertainties arising from the combination procedure. The uncertainties are quoted in percent relative to $\sigma_{r,\rm NC}^{+}$.

HERA combined reduced cross sections $\sigma_{r,\rm NC}^{-}$ for NC $e^{-}p$ scattering at $\sqrt{s} = 318$ GeV; $\delta_{\rm stat}$, $\delta_{\rm uncor}$ and $\delta_{\rm cor}$ represent the statistical, uncorrelated systematic and correlated systematic uncertainties, respectively; $\delta_{\rm rel}$, $\delta_{\gamma p}$, $\delta_{\rm had}$ and $\delta_{1}$ to $\delta_{4}$ are the correlated sources of uncertainties arising from the combination procedure. The uncertainties are quoted in percent relative to $\sigma_{r,\rm NC}^{-}$.

Structure function $xF_3^{\gamma Z}$ for different values of $Q^2$ and $x_{\rm Bj}$.

Structure function $xF_3^{\gamma Z}$ averaged over $Q^2 \ge 1000$ GeV$^2$ at the scale 1000 GeV$^2$.

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.