High $p_{T}$ azimuthal correlation functions for central events. Upper panel: Correlation function for $|\Delta\eta|$ < 0.5 (solid circles) and scaled correlation function for 0.5 < $|\Delta\eta|$ < 1.4 (open squares). Lower panel: Difference of the two correlation functions. Also shown are the fits to the data (described in the text).

$v_2$ of $\Lambda+\bar{\Lambda}$ as a function of $p_T$ for 11-45% centrality in Au+Au collisions at 130 GeV. Systematic errors of $\pm$0.005 for particle identification and background subtraction and $^{+0}_{-0.005}$ for nonflow effects.

$v_2$ of charged particles as a function of $p_T$ for minimum bias Au+Au collisions at 130 GeV.

$v_2$ of $K_s^0$ as a function of $p_T$ for minimum bias Au+Au collisions at 130 GeV. Systematic errors of $\pm$0.005 for particle identification and background subtraction and $^{+0}_{-0.005}$ for nonflow effects.

$v_2$ of $\Lambda+\bar{\Lambda}$ as a function of $p_T$ for minimum bias Au+Au collisions at 130 GeV. Systematic errors of $\pm$0.005 for particle identification and background subtraction and $^{+0}_{-0.005}$ for nonflow effects.

$p_T$-integrated $v_2$ of $h^-$, $K_s^0$, and $\Lambda+\bar{\Lambda}$ as a function of particle mass for minimum bias Au+Au collisions at 130 GeV.

Directed flow slope at mid-rapidity, $dv_{1}/dy|_{y=0}$, as a function of beam energy in 10-40\% Au+Au collisions. Dots represent deuterons. Open squares are the published results in~\cite{Adamczyk_2018} for protons. The band denotes the results for deuterons from AMPT transport model. Statistical uncertainties (bars) and systematic uncertainties (horizontal lines) are shown separately. For visibility, the data points are staggered horizontally.

The $p_{T}$ dependence of $v_1/A$ in $|y|<0.6$ for protons (open squares) in 10-40\% Au+Au collisions at $\sqrt{s_{NN}}$ = 7.7, 11.5, 14.5, 19.6, 27, and 39 GeV. Statistical uncertainties (bars) and systematic uncertainties (horizontal lines) are shown separately.

The $d_{T}$ dependence of $v_1/A$ in $|y|<0.6$ for deuterons (solid circles) in 10-40\% Au+Au collisions at $\sqrt{s_{NN}}$ = 7.7, 11.5, 14.5, 19.6, 27, and 39 GeV. Statistical uncertainties (bars) and systematic uncertainties (horizontal lines) are shown separately.

$z_{g}$ for Matched Trigger jets in pp$+$AuAu Data anti-kT R$=$0.4

$z_{g}$ for HardCore Recoil jets in AuAu Data anti-kT R$=$0.4

$z_{g}$ for HardCore Recoil jets in pp$+$AuAu Data anti-kT R$=$0.4

$z_{g}$ for Matched Recoil jets in AuAu Data anti-kT R$=$0.4

$z_{g}$ for Matched Recoil jets in pp$+$AuAu Data anti-kT R$=$0.4

$R_{g}$ for HardCore Trigger jets in AuAu Data anti-kT R$=$0.4

$R_{g}$ for HardCore Trigger jets in pp$+$AuAu Data anti-kT R$=$0.4

$R_{g}$ for Matched Trigger jets in AuAu Data anti-kT R$=$0.4

$R_{g}$ for Matched Trigger jets in pp$+$AuAu Data anti-kT R$=$0.4

$R_{g}$ for HardCore Recoil jets in AuAu Data anti-kT R$=$0.4

$R_{g}$ for HardCore Recoil jets in pp$+$AuAu Data anti-kT R$=$0.4

$R_{g}$ for Matched Recoil jets in AuAu Data anti-kT R$=$0.4

$R_{g}$ for Matched Recoil jets in pp$+$AuAu Data anti-kT R$=$0.4

Fully corrected two subjet $z_{sj}$ for $15 < p_{\rm{T, jet}} < 20$ GeV/c, R$=$0.4 anti-kT and R$=$0.1 subjets

Fully corrected two subjet $z_{sj}$ for $20 < p_{\rm{T, jet}} < 25$ GeV/c, R$=$0.4 anti-kT and R$=$0.1 subjets

Fully corrected two subjet $z_{sj}$ for $25 < p_{\rm{T, jet}} < 30$ GeV/c, R$=$0.4 anti-kT and R$=$0.1 subjets

Fully corrected two subjet $z_{sj}$ for $30 < p_{\rm{T, jet}} < 40$ GeV/c, R$=$0.4 anti-kT and R$=$0.1 subjets

Fully corrected two subjet $\theta_{sj}$ for $15 < p_{\rm{T, jet}} < 20$ GeV/c, R$=$0.4 anti-kT and R$=$0.1 subjets

Fully corrected two subjet $\theta_{sj}$ for $20 < p_{\rm{T, jet}} < 25$ GeV/c, R$=$0.4 anti-kT and R$=$0.1 subjets

Fully corrected two subjet $\theta_{sj}$ for $25 < p_{\rm{T, jet}} < 30$ GeV/c, R$=$0.4 anti-kT and R$=$0.1 subjets

Fully corrected two subjet $\theta_{sj}$ for $30 < p_{\rm{T, jet}} < 40$ GeV/c, R$=$0.4 anti-kT and R$=$0.1 subjets

$z_{sj}$ for HardCore Trigger jets in AuAu Data anti-kT R$=$0.4

$z_{sj}$ for HardCore Trigger jets in pp$+$AuAu Data anti-kT R$=$0.4

$z_{sj}$ for Matched Trigger jets in AuAu Data anti-kT R$=$0.4

$z_{sj}$ for Matched Trigger jets in pp$+$AuAu Data anti-kT R$=$0.4

$z_{sj}$ for HardCore Recoil jets in AuAu Data anti-kT R$=$0.4

$z_{sj}$ for HardCore Recoil jets in pp$+$AuAu Data anti-kT R$=$0.4

$z_{sj}$ for Matched Recoil jets in AuAu Data anti-kT R$=$0.4

$z_{sj}$ for Matched Recoil jets in pp$+$AuAu Data anti-kT R$=$0.4

$\theta_{sj}$ for HardCore Trigger jets in AuAu Data anti-kT R$=$0.4

$\theta_{sj}$ for HardCore Trigger jets in pp$+$AuAu Data anti-kT R$=$0.4

$\theta_{sj}$ for Matched Trigger jets in AuAu Data anti-kT R$=$0.4

$\theta_{sj}$ for Matched Trigger jets in pp$+$AuAu Data anti-kT R$=$0.4

$\theta_{sj}$ for HardCore Recoil jets in AuAu Data anti-kT R$=$0.4

$\theta_{sj}$ for HardCore Recoil jets in pp$+$AuAu Data anti-kT R$=$0.4

$\theta_{sj}$ for Matched Recoil jets in AuAu Data anti-kT R$=$0.4

$\theta_{sj}$ for Matched Recoil jets in pp$+$AuAu Data anti-kT R$=$0.4

$A_{J}$ for HardCore R=0.4 anti-kT jets with $\theta_{sj} > 0.1$ in Au$+$Au collisions

$A_{J}$ for HardCore R=0.4 anti-kT jets with $\theta_{sj} > 0.1$ in pp$+$AuAu reference

$A_{J}$ for HardCore R=0.4 anti-kT jets with $0.1 < \theta_{sj} < 0.2$ in Au$+$Au collisions

$A_{J}$ for HardCore R=0.4 anti-kT jets with $0.1 < \theta_{sj} < 0.2$ in pp$+$AuAu reference

$A_{J}$ for HardCore R=0.4 anti-kT jets with $0.2 < \theta_{sj} < 0.3$ in Au$+$Au collisions

$A_{J}$ for HardCore R=0.4 anti-kT jets with $0.2 < \theta_{sj} < 0.3$ in pp$+$AuAu reference

$A_{J}$ for Matched R=0.4 anti-kT jets with $\theta_{sj} > 0.1$ in Au$+$Au collisions

$A_{J}$ for Matched R=0.4 anti-kT jets with $\theta_{sj} > 0.1$ in pp$+$AuAu reference

$A_{J}$ for Matched R=0.4 anti-kT jets with $0.1 < \theta_{sj} < 0.2$ in Au$+$Au collisions

$A_{J}$ for Matched R=0.4 anti-kT jets with $0.1 < \theta_{sj} < 0.2$ in pp$+$AuAu reference

$A_{J}$ for Matched R=0.4 anti-kT jets with $0.2 < \theta_{sj} < 0.3$ in Au$+$Au collisions

$A_{J}$ for Matched R=0.4 anti-kT jets with $0.2 < \theta_{sj} < 0.3$ in pp$+$AuAu reference

tracking efficiency in pp collisions as a function of track momentum

tracking efficiency in AuAu collisions as a function of track momentum

Azimuthal distributions for Au+Au collisions compared to the expected distributions estimated by $D^{\rm model}$. All correlation functions require a trigger particle with $4<p_T^{\rm trig}<6$ GeV/$c$ and associated particles with $2<p_T<p_T^{\rm trig}$ GeV/$c$.

Azimuthal distributions for Au+Au collisions compared to the expected distributions estimated by $D^{\rm model}$. All correlation functions require a trigger particle with $4<p_T^{\rm trig}<6$ GeV/$c$ and associated particles with $2<p_T<p_T^{\rm trig}$ GeV/$c$.

Azimuthal distributions for Au+Au collisions compared to the expected distributions estimated by $D^{\rm model}$. All correlation functions require a trigger particle with $3<p_T^{\rm trig}<4$ GeV/$c$ and associated particles with $2<p_T<p_T^{\rm trig}$ GeV/$c$.

Azimuthal distributions for Au+Au collisions compared to the expected distributions estimated by $D^{\rm model}$. All correlation functions require a trigger particle with $3<p_T^{\rm trig}<4$ GeV/$c$ and associated particles with $2<p_T<p_T^{\rm trig}$ GeV/$c$.

Ratio of Au+Au and p+p for small-angle ($|\Delta\phi|$<0.75 rad) and back-to-back ($|\Delta\phi|$<2.24 rad) azimuthal regions versus number of participating nucleons ($N_{\rm part}$) for trigger particle intervals $4<p_T^{\rm trig}<6$ GeV/$c$ and $3<p_T^{\rm trig}<4$ GeV/$c$.

Systematic uncertainty on number of participating nucleons ($N_{\rm part}$) in Au+Au collisions.

Rapidity($y$) dependence of $v_1$ (top panels) and $v_2$ (bottom panels) of proton and $\Lambda$ baryons (left panels), pions (middle panels) and kaons (right panels) in 10-40% centrality for the $\sqrt{s_{NN}}$ = 3GeV Au+Au collisions. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points. The UrQMD and JAM results are shown as bands:golden, red and blue bands stand for JAM mean-field, UrQMD mean-field and UrQMD cascade mode, respectively. The value of the incompressibility $\kappa$ = 380 MeV is used in the mean-field option. More detailed model descriptions and data comparisons can be found in Supplemental Material.

Rapidity($y$) dependence of $v_1$ (top panels) and $v_2$ (bottom panels) of proton and $\Lambda$ baryons (left panels), pions (middle panels) and kaons (right panels) in 10-40% centrality for the $\sqrt{s_{NN}}$ = 3GeV Au+Au collisions. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points. The UrQMD and JAM results are shown as bands:golden, red and blue bands stand for JAM mean-field, UrQMD mean-field and UrQMD cascade mode, respectively. The value of the incompressibility $\kappa$ = 380 MeV is used in the mean-field option. More detailed model descriptions and data comparisons can be found in Supplemental Material.

Rapidity($y$) dependence of $v_1$ (top panels) and $v_2$ (bottom panels) of proton and $\Lambda$ baryons (left panels), pions (middle panels) and kaons (right panels) in 10-40% centrality for the $\sqrt{s_{NN}}$ = 3GeV Au+Au collisions. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points. The UrQMD and JAM results are shown as bands:golden, red and blue bands stand for JAM mean-field, UrQMD mean-field and UrQMD cascade mode, respectively. The value of the incompressibility $\kappa$ = 380 MeV is used in the mean-field option. More detailed model descriptions and data comparisons can be found in Supplemental Material.

$v_2$ scaled by the number of constituent quarks, $v_2/n_q$ , as a function of scaled transverse kinetic energy ($(m_T − m_0)/n_q$) for pions, kaons and protons from Au+Au collisions in 10-40% centrality at $\sqrt{s_{NN}}$ = 3, 27, and 54.4 GeV for positive charged particles (left panel) and negative charged particles (right panel). Colored dashed lines represent the scaling fit to data in 7.7, 14.5, 27, 54.4, and 200 GeV Au+Au collisions from STAR experiment at RHIC [43–45]. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points.

$v_2$ scaled by the number of constituent quarks, $v_2/n_q$ , as a function of scaled transverse kinetic energy ($(m_T − m_0)/n_q$) for pions, kaons and protons from Au+Au collisions in 10-40% centrality at $\sqrt{s_{NN}}$ = 3, 27, and 54.4 GeV for positive charged particles (left panel) and negative charged particles (right panel). Colored dashed lines represent the scaling fit to data in 7.7, 14.5, 27, 54.4, and 200 GeV Au+Au collisions from STAR experiment at RHIC [43–45]. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points.

$v_2$ scaled by the number of constituent quarks, $v_2/n_q$ , as a function of scaled transverse kinetic energy ($(m_T − m_0)/n_q$) for pions, kaons and protons from Au+Au collisions in 10-40% centrality at $\sqrt{s_{NN}}$ = 3, 27, and 54.4 GeV for positive charged particles (left panel) and negative charged particles (right panel). Colored dashed lines represent the scaling fit to data in 7.7, 14.5, 27, 54.4, and 200 GeV Au+Au collisions from STAR experiment at RHIC [43–45]. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points.

$v_2$ scaled by the number of constituent quarks, $v_2/n_q$ , as a function of scaled transverse kinetic energy ($(m_T − m_0)/n_q$) for pions, kaons and protons from Au+Au collisions in 10-40% centrality at $\sqrt{s_{NN}}$ = 3, 27, and 54.4 GeV for positive charged particles (left panel) and negative charged particles (right panel). Colored dashed lines represent the scaling fit to data in 7.7, 14.5, 27, 54.4, and 200 GeV Au+Au collisions from STAR experiment at RHIC [43–45]. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points.

$v_2$ scaled by the number of constituent quarks, $v_2/n_q$ , as a function of scaled transverse kinetic energy ($(m_T − m_0)/n_q$) for pions, kaons and protons from Au+Au collisions in 10-40% centrality at $\sqrt{s_{NN}}$ = 3, 27, and 54.4 GeV for positive charged particles (left panel) and negative charged particles (right panel). Colored dashed lines represent the scaling fit to data in 7.7, 14.5, 27, 54.4, and 200 GeV Au+Au collisions from STAR experiment at RHIC [43–45]. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points.

$v_2$ scaled by the number of constituent quarks, $v_2/n_q$ , as a function of scaled transverse kinetic energy ($(m_T − m_0)/n_q$) for pions, kaons and protons from Au+Au collisions in 10-40% centrality at $\sqrt{s_{NN}}$ = 3, 27, and 54.4 GeV for positive charged particles (left panel) and negative charged particles (right panel). Colored dashed lines represent the scaling fit to data in 7.7, 14.5, 27, 54.4, and 200 GeV Au+Au collisions from STAR experiment at RHIC [43–45]. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points.

$v_2$ scaled by the number of constituent quarks, $v_2/n_q$ , as a function of scaled transverse kinetic energy ($(m_T − m_0)/n_q$) for pions, kaons and protons from Au+Au collisions in 10-40% centrality at $\sqrt{s_{NN}}$ = 3, 27, and 54.4 GeV for positive charged particles (left panel) and negative charged particles (right panel). Colored dashed lines represent the scaling fit to data in 7.7, 14.5, 27, 54.4, and 200 GeV Au+Au collisions from STAR experiment at RHIC [43–45]. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points.

$v_2$ scaled by the number of constituent quarks, $v_2/n_q$ , as a function of scaled transverse kinetic energy ($(m_T − m_0)/n_q$) for pions, kaons and protons from Au+Au collisions in 10-40% centrality at $\sqrt{s_{NN}}$ = 3, 27, and 54.4 GeV for positive charged particles (left panel) and negative charged particles (right panel). Colored dashed lines represent the scaling fit to data in 7.7, 14.5, 27, 54.4, and 200 GeV Au+Au collisions from STAR experiment at RHIC [43–45]. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points.

$v_2$ scaled by the number of constituent quarks, $v_2/n_q$ , as a function of scaled transverse kinetic energy ($(m_T − m_0)/n_q$) for pions, kaons and protons from Au+Au collisions in 10-40% centrality at $\sqrt{s_{NN}}$ = 3, 27, and 54.4 GeV for positive charged particles (left panel) and negative charged particles (right panel). Colored dashed lines represent the scaling fit to data in 7.7, 14.5, 27, 54.4, and 200 GeV Au+Au collisions from STAR experiment at RHIC [43–45]. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points.

$v_2$ scaled by the number of constituent quarks, $v_2/n_q$ , as a function of scaled transverse kinetic energy ($(m_T − m_0)/n_q$) for pions, kaons and protons from Au+Au collisions in 10-40% centrality at $\sqrt{s_{NN}}$ = 3, 27, and 54.4 GeV for positive charged particles (left panel) and negative charged particles (right panel). Colored dashed lines represent the scaling fit to data in 7.7, 14.5, 27, 54.4, and 200 GeV Au+Au collisions from STAR experiment at RHIC [43–45]. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points.

$v_2$ scaled by the number of constituent quarks, $v_2/n_q$ , as a function of scaled transverse kinetic energy ($(m_T − m_0)/n_q$) for pions, kaons and protons from Au+Au collisions in 10-40% centrality at $\sqrt{s_{NN}}$ = 3, 27, and 54.4 GeV for positive charged particles (left panel) and negative charged particles (right panel). Colored dashed lines represent the scaling fit to data in 7.7, 14.5, 27, 54.4, and 200 GeV Au+Au collisions from STAR experiment at RHIC [43–45]. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points.

$v_2$ scaled by the number of constituent quarks, $v_2/n_q$ , as a function of scaled transverse kinetic energy ($(m_T − m_0)/n_q$) for pions, kaons and protons from Au+Au collisions in 10-40% centrality at $\sqrt{s_{NN}}$ = 3, 27, and 54.4 GeV for positive charged particles (left panel) and negative charged particles (right panel). Colored dashed lines represent the scaling fit to data in 7.7, 14.5, 27, 54.4, and 200 GeV Au+Au collisions from STAR experiment at RHIC [43–45]. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points.

$v_2$ scaled by the number of constituent quarks, $v_2/n_q$ , as a function of scaled transverse kinetic energy ($(m_T − m_0)/n_q$) for pions, kaons and protons from Au+Au collisions in 10-40% centrality at $\sqrt{s_{NN}}$ = 3, 27, and 54.4 GeV for positive charged particles (left panel) and negative charged particles (right panel). Colored dashed lines represent the scaling fit to data in 7.7, 14.5, 27, 54.4, and 200 GeV Au+Au collisions from STAR experiment at RHIC [43–45]. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points.

$v_2$ scaled by the number of constituent quarks, $v_2/n_q$ , as a function of scaled transverse kinetic energy ($(m_T − m_0)/n_q$) for pions, kaons and protons from Au+Au collisions in 10-40% centrality at $\sqrt{s_{NN}}$ = 3, 27, and 54.4 GeV for positive charged particles (left panel) and negative charged particles (right panel). Colored dashed lines represent the scaling fit to data in 7.7, 14.5, 27, 54.4, and 200 GeV Au+Au collisions from STAR experiment at RHIC [43–45]. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points.

$v_2$ scaled by the number of constituent quarks, $v_2/n_q$ , as a function of scaled transverse kinetic energy ($(m_T − m_0)/n_q$) for pions, kaons and protons from Au+Au collisions in 10-40% centrality at $\sqrt{s_{NN}}$ = 3, 27, and 54.4 GeV for positive charged particles (left panel) and negative charged particles (right panel). Colored dashed lines represent the scaling fit to data in 7.7, 14.5, 27, 54.4, and 200 GeV Au+Au collisions from STAR experiment at RHIC [43–45]. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points.

$v_2$ scaled by the number of constituent quarks, $v_2/n_q$ , as a function of scaled transverse kinetic energy ($(m_T − m_0)/n_q$) for pions, kaons and protons from Au+Au collisions in 10-40% centrality at $\sqrt{s_{NN}}$ = 3, 27, and 54.4 GeV for positive charged particles (left panel) and negative charged particles (right panel). Colored dashed lines represent the scaling fit to data in 7.7, 14.5, 27, 54.4, and 200 GeV Au+Au collisions from STAR experiment at RHIC [43–45]. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points.

$v_2$ scaled by the number of constituent quarks, $v_2/n_q$ , as a function of scaled transverse kinetic energy ($(m_T − m_0)/n_q$) for pions, kaons and protons from Au+Au collisions in 10-40% centrality at $\sqrt{s_{NN}}$ = 3, 27, and 54.4 GeV for positive charged particles (left panel) and negative charged particles (right panel). Colored dashed lines represent the scaling fit to data in 7.7, 14.5, 27, 54.4, and 200 GeV Au+Au collisions from STAR experiment at RHIC [43–45]. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points.

Collision energy dependence of (top panel) directed flow slope $dv_1/dy|_{y=0}$ for p, $\Lambda, \pi$(combined from $\pi^{\pm}$), K(combined from $K^{\pm}$ and $K_S^0$), $\phi$ and $\Xi^−$, and (bottom panel) elliptic flow $v_2$ for p, $\pi$(combined from $\pi^{\pm}$) in heavy- ion collisions. Collision centrality for all data from RHIC is 10-40%, except for 4.5 GeV where 10-30% is for $dv_1/dy|_{y=0}$ and 0-30% is for $v_2$. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points. The JAM and UrQMD results are shown as colored bands:golden, red and blue bands stand for JAM mean-field, UrQMD mean-field and UrQMD cascade mode, respectively. For clarity the x-axis value of the data points have been shifted.

Collision energy dependence of (top panel) directed flow slope $dv_1/dy|_{y=0}$ for p, $\Lambda, \pi$(combined from $\pi^{\pm}$), K(combined from $K^{\pm}$ and $K_S^0$), $\phi$ and $\Xi^−$, and (bottom panel) elliptic flow $v_2$ for p, $\pi$(combined from $\pi^{\pm}$) in heavy- ion collisions. Collision centrality for all data from RHIC is 10-40%, except for 4.5 GeV where 10-30% is for $dv_1/dy|_{y=0}$ and 0-30% is for $v_2$. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points. The JAM and UrQMD results are shown as colored bands:golden, red and blue bands stand for JAM mean-field, UrQMD mean-field and UrQMD cascade mode, respectively. For clarity the x-axis value of the data points have been shifted.

Collision energy dependence of (top panel) directed flow slope $dv_1/dy|_{y=0}$ for p, $\Lambda, \pi$(combined from $\pi^{\pm}$), K(combined from $K^{\pm}$ and $K_S^0$), $\phi$ and $\Xi^−$, and (bottom panel) elliptic flow $v_2$ for p, $\pi$(combined from $\pi^{\pm}$) in heavy- ion collisions. Collision centrality for all data from RHIC is 10-40%, except for 4.5 GeV where 10-30% is for $dv_1/dy|_{y=0}$ and 0-30% is for $v_2$. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points. The JAM and UrQMD results are shown as colored bands:golden, red and blue bands stand for JAM mean-field, UrQMD mean-field and UrQMD cascade mode, respectively. For clarity the x-axis value of the data points have been shifted.

Collision energy dependence of (top panel) directed flow slope $dv_1/dy|_{y=0}$ for p, $\Lambda, \pi$(combined from $\pi^{\pm}$), K(combined from $K^{\pm}$ and $K_S^0$), $\phi$ and $\Xi^−$, and (bottom panel) elliptic flow $v_2$ for p, $\pi$(combined from $\pi^{\pm}$) in heavy- ion collisions. Collision centrality for all data from RHIC is 10-40%, except for 4.5 GeV where 10-30% is for $dv_1/dy|_{y=0}$ and 0-30% is for $v_2$. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points. The JAM and UrQMD results are shown as colored bands:golden, red and blue bands stand for JAM mean-field, UrQMD mean-field and UrQMD cascade mode, respectively. For clarity the x-axis value of the data points have been shifted.

Collision energy dependence of (top panel) directed flow slope $dv_1/dy|_{y=0}$ for p, $\Lambda, \pi$(combined from $\pi^{\pm}$), K(combined from $K^{\pm}$ and $K_S^0$), $\phi$ and $\Xi^−$, and (bottom panel) elliptic flow $v_2$ for p, $\pi$(combined from $\pi^{\pm}$) in heavy- ion collisions. Collision centrality for all data from RHIC is 10-40%, except for 4.5 GeV where 10-30% is for $dv_1/dy|_{y=0}$ and 0-30% is for $v_2$. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points. The JAM and UrQMD results are shown as colored bands:golden, red and blue bands stand for JAM mean-field, UrQMD mean-field and UrQMD cascade mode, respectively. For clarity the x-axis value of the data points have been shifted.

Collision energy dependence of (top panel) directed flow slope $dv_1/dy|_{y=0}$ for p, $\Lambda, \pi$(combined from $\pi^{\pm}$), K(combined from $K^{\pm}$ and $K_S^0$), $\phi$ and $\Xi^−$, and (bottom panel) elliptic flow $v_2$ for p, $\pi$(combined from $\pi^{\pm}$) in heavy- ion collisions. Collision centrality for all data from RHIC is 10-40%, except for 4.5 GeV where 10-30% is for $dv_1/dy|_{y=0}$ and 0-30% is for $v_2$. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points. The JAM and UrQMD results are shown as colored bands:golden, red and blue bands stand for JAM mean-field, UrQMD mean-field and UrQMD cascade mode, respectively. For clarity the x-axis value of the data points have been shifted.

Collision energy dependence of (top panel) directed flow slope $dv_1/dy|_{y=0}$ for p, $\Lambda, \pi$(combined from $\pi^{\pm}$), K(combined from $K^{\pm}$ and $K_S^0$), $\phi$ and $\Xi^−$, and (bottom panel) elliptic flow $v_2$ for p, $\pi$(combined from $\pi^{\pm}$) in heavy- ion collisions. Collision centrality for all data from RHIC is 10-40%, except for 4.5 GeV where 10-30% is for $dv_1/dy|_{y=0}$ and 0-30% is for $v_2$. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points. The JAM and UrQMD results are shown as colored bands:golden, red and blue bands stand for JAM mean-field, UrQMD mean-field and UrQMD cascade mode, respectively. For clarity the x-axis value of the data points have been shifted.

Collision energy dependence of (top panel) directed flow slope $dv_1/dy|_{y=0}$ for p, $\Lambda, \pi$(combined from $\pi^{\pm}$), K(combined from $K^{\pm}$ and $K_S^0$), $\phi$ and $\Xi^−$, and (bottom panel) elliptic flow $v_2$ for p, $\pi$(combined from $\pi^{\pm}$) in heavy- ion collisions. Collision centrality for all data from RHIC is 10-40%, except for 4.5 GeV where 10-30% is for $dv_1/dy|_{y=0}$ and 0-30% is for $v_2$. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points. The JAM and UrQMD results are shown as colored bands:golden, red and blue bands stand for JAM mean-field, UrQMD mean-field and UrQMD cascade mode, respectively. For clarity the x-axis value of the data points have been shifted.

Collision energy dependence of (top panel) directed flow slope $dv_1/dy|_{y=0}$ for p, $\Lambda, \pi$(combined from $\pi^{\pm}$), K(combined from $K^{\pm}$ and $K_S^0$), $\phi$ and $\Xi^−$, and (bottom panel) elliptic flow $v_2$ for p, $\pi$(combined from $\pi^{\pm}$) in heavy- ion collisions. Collision centrality for all data from RHIC is 10-40%, except for 4.5 GeV where 10-30% is for $dv_1/dy|_{y=0}$ and 0-30% is for $v_2$. Statistical and systematic uncertainties are shown as bars and gray bands, respectively. Some uncertainties are smaller than the data points. The JAM and UrQMD results are shown as colored bands:golden, red and blue bands stand for JAM mean-field, UrQMD mean-field and UrQMD cascade mode, respectively. For clarity the x-axis value of the data points have been shifted.

The upper panel shows $v_{2}$ vs centrality using the conventional method, where the circles and triangles represent $v_{2}$ with and without $p_{t}$ weighting, respectively. The statistical error is smaller than the symbol size. The lower panel shows the statistical error on $v_{2}$ with $p_{t}$ weighting divided by the same without weighting.

The upper panel shows $v_{2}$ vs centrality using the conventional method, where the circles and triangles represent $v_{2}$ with and without $p_{t}$ weighting, respectively. The statistical error is smaller than the symbol size. The lower panel shows the statistical error on $v_{2}$ with $p_{t}$ weighting divided by the same without weighting.

The upper panel presents $v_{2}$ vs centrality from the scalar product method (triangles) and the conventional random subevent method (circles). All statistical errors are smaller than the symbol size. The statistical error for the scalar product method divided by that for the conventional method is shown in the lower panel.

The upper panel presents $v_{2}$ vs centrality from the scalar product method (triangles) and the conventional random subevent method (circles). All statistical errors are smaller than the symbol size. The statistical error for the scalar product method divided by that for the conventional method is shown in the lower panel.

Elliptic flow as determined from the fits to the $q$ distributions in different centrality bins. The circles are from the standard method with random subevents. For the squares, all the centralities were fit separately. For the triangles, centrality bins 2-8 were fit with the same value of the nonflow parameter.

Measured elliptic flow vs centrality for $Au+Au$ at $\sqrt{s_{NN}}=130$ GeV. The circles show the conventional $v_{2}$ with estimated systematic uncertainty due to nonflow [37], the stars show the fourth-order cumulant $v_{2}$ from the generating function, the crosses show the conventional $v_{2}$ from quarter events, and the squares show the fourth-order cumulant $v_{2}$ from the four-subevent method.

Reconstructed $v_{2}$ vs pseudorapidity from the conventional method (circles), from the second-order cumulant method (triangles), and from the fourth-order cumulant method (stars), in eight centrality bins. The upper left panel shows the most peripheral events, and the lower right the most central.

Reconstructed $v_{2}$ vs pseudorapidity from the conventional method (circles), from the second-order cumulant method (triangles), and from the fourth-order cumulant method (stars), in eight centrality bins. The upper left panel shows the most peripheral events, and the lower right the most central.

Reconstructed $v_{2}$ vs pseudorapidity from the conventional method (circles), from the second-order cumulant method (triangles), and from the fourth-order cumulant method (stars), in eight centrality bins. The upper left panel shows the most peripheral events, and the lower right the most central.

Reconstructed $v_{2}$ vs pseudorapidity from the conventional method (circles), from the second-order cumulant method (triangles), and from the fourth-order cumulant method (stars), in eight centrality bins. The upper left panel shows the most peripheral events, and the lower right the most central.

Reconstructed $v_{2}$ vs pseudorapidity from the conventional method (circles), from the second-order cumulant method (triangles), and from the fourth-order cumulant method (stars), in eight centrality bins. The upper left panel shows the most peripheral events, and the lower right the most central.

Reconstructed $v_{2}$ vs pseudorapidity from the conventional method (circles), from the second-order cumulant method (triangles), and from the fourth-order cumulant method (stars), in eight centrality bins. The upper left panel shows the most peripheral events, and the lower right the most central.

Reconstructed $v_{2}$ vs pseudorapidity from the conventional method (circles), from the second-order cumulant method (triangles), and from the fourth-order cumulant method (stars), in eight centrality bins. The upper left panel shows the most peripheral events, and the lower right the most central.

Reconstructed $v_{2}$ vs pseudorapidity from the conventional method (circles), from the second-order cumulant method (triangles), and from the fourth-order cumulant method (stars), in eight centrality bins. The upper left panel shows the most peripheral events, and the lower right the most central.

Reconstructed $v_{2}$ vs $p_{t}$ from the conventional method (circles), from the second-order cumulant method (triangles), and from the fourth-order cumulant method (stars), in eight centrality bins. The upper left panel shows the most peripheral events, and the lower right the most central.

Reconstructed $v_{2}$ vs $p_{t}$ from the conventional method (circles), from the second-order cumulant method (triangles), and from the fourth-order cumulant method (stars), in eight centrality bins. The upper left panel shows the most peripheral events, and the lower right the most central.

Reconstructed $v_{2}$ vs $p_{t}$ from the conventional method (circles), from the second-order cumulant method (triangles), and from the fourth-order cumulant method (stars), in eight centrality bins. The upper left panel shows the most peripheral events, and the lower right the most central.

Reconstructed $v_{2}$ vs $p_{t}$ from the conventional method (circles), from the second-order cumulant method (triangles), and from the fourth-order cumulant method (stars), in eight centrality bins. The upper left panel shows the most peripheral events, and the lower right the most central.

Reconstructed $v_{2}$ vs $p_{t}$ from the conventional method (circles), from the second-order cumulant method (triangles), and from the fourth-order cumulant method (stars), in eight centrality bins. The upper left panel shows the most peripheral events, and the lower right the most central.

Reconstructed $v_{2}$ vs $p_{t}$ from the conventional method (circles), from the second-order cumulant method (triangles), and from the fourth-order cumulant method (stars), in eight centrality bins. The upper left panel shows the most peripheral events, and the lower right the most central.

Reconstructed $v_{2}$ vs $p_{t}$ from the conventional method (circles), from the second-order cumulant method (triangles), and from the fourth-order cumulant method (stars), in eight centrality bins. The upper left panel shows the most peripheral events, and the lower right the most central.

Reconstructed $v_{2}$ vs $p_{t}$ from the conventional method (circles), from the second-order cumulant method (triangles), and from the fourth-order cumulant method (stars), in eight centrality bins. The upper left panel shows the most peripheral events, and the lower right the most central.

Elliptic flow vs pseudorapidity from the conventional method (circles), from the second-order cumulant method (triangles), from quarter events (crosses), and from the fourth-order cumulant method (stars), averaged over all centralities from bin 2-7, as defined in Figs. 14 and 15.

Elliptic flow vs transverse momentum from the conventional method (circles), from the second-order cumulant method (triangles), from quarter events (crosses), and from the fourth-order cumulant method (stars), averaged over all centralities from bin 2-7, as defined in Figs. 14 and 15.

The ratio of $v_{2}$ from the fourth-order cumulant divided by $v_{2}$, from the conventional method as a function of $p_{t}$, averaged over all centralities from bin 2-7, as defined in Figs. 14 and 15.

The ratio of $v_{2}$ from quarter events divided by the conventional $v_{2}$ as a function of $p_{t}$. In both cases, event planes were constructed from low-$p_{t}$ (< 0.5 GeV/c) particles. The data are averaged over all centralities from bin 2-7, as defined in Figs. 14 and 15.

$v_{2}/\varepsilon$ as a function of charged particle density in $Au+Au$ collisions. Data are from E877 at the AGS (squares), NA49 at the SPS (circles), and STAR at RHIC (stars). The AGS and SPS data have been obtained by conventional flow analysis. The STAR measurements are at $\sqrt{s_{NN}}=130$ GeV, and correspond to the final corrected elliptic flow based on fourth-order cumulants, and we assume $dN/dy=1.15dN/d\eta$. The horizontal shaded bands indicate the hydrodynamic limits for different beam energies [44].