Example of combinatorial background subtracted invariant mass distributions and the extracted yields as a function of $\cos \theta^*$ for $\phi$ and $K^{*0}$ mesons. \textbf{a)} example of $\phi \rightarrow K^+ + K^-$ invariant mass distributions, with combinatorial background subtracted, integrated over $\cos \theta^*$; \textbf{b)} example of $K^{*0} (\overline{K^{*0}}) \rightarrow K^{-} \pi^{+} (K^{+} \pi^{-})$ invariant mass distributions, with combinatorial background subtracted, integrated over $\cos \theta^*$; \textbf{c)} extracted yields of $\phi$ as a function of $\cos \theta^*$; \textbf{d)} extracted yields of $K^{*0}$ as a function of $\cos \theta^*$.

Example of combinatorial background subtracted invariant mass distributions and the extracted yields as a function of $\cos \theta^*$ for $\phi$ and $K^{*0}$ mesons. \textbf{a)} example of $\phi \rightarrow K^+ + K^-$ invariant mass distributions, with combinatorial background subtracted, integrated over $\cos \theta^*$; \textbf{b)} example of $K^{*0} (\overline{K^{*0}}) \rightarrow K^{-} \pi^{+} (K^{+} \pi^{-})$ invariant mass distributions, with combinatorial background subtracted, integrated over $\cos \theta^*$; \textbf{c)} extracted yields of $\phi$ as a function of $\cos \theta^*$; \textbf{d)} extracted yields of $K^{*0}$ as a function of $\cos \theta^*$.

Example of combinatorial background subtracted invariant mass distributions and the extracted yields as a function of $\cos \theta^*$ for $\phi$ and $K^{*0}$ mesons. \textbf{a)} example of $\phi \rightarrow K^+ + K^-$ invariant mass distributions, with combinatorial background subtracted, integrated over $\cos \theta^*$; \textbf{b)} example of $K^{*0} (\overline{K^{*0}}) \rightarrow K^{-} \pi^{+} (K^{+} \pi^{-})$ invariant mass distributions, with combinatorial background subtracted, integrated over $\cos \theta^*$; \textbf{c)} extracted yields of $\phi$ as a function of $\cos \theta^*$; \textbf{d)} extracted yields of $K^{*0}$ as a function of $\cos \theta^*$.

Efficiency corrected $\phi$-meson yields as a function of cos$\theta$* and corresponding fits with Eq.1 in the method section.

Efficiency and acceptance corrected $K^{*0}$-meson yields as a function of cos$\theta$* and corresponding fits with Eq.4 in the method section.

$\phi$-meson $\rho_{00}$ obtained from 1st- and 2nd-order event planes. The red stars (gray squares) show the $\phi$-meson $\rho_{00}$ as a function of beam energy, obtained with the 2nd-order (1st-order) EP.

$\phi$-meson $\rho_{00}$ with respect to different quantization axes. $\phi$-meson $\rho_{00}$ as a function of beam energy, for the out-of-plane direction (stars) and the in-plane direction (diamonds). Curves are fits based on theoretical calculations with a $\phi$-meson field. The corresponding $G_s^{(y)}$ values obtained from the fits are shown in the legend.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $\phi$ for different collision energies. The gray squares and red stars are results obtained with the 1st- and 2nd-order EP, respectively.

$\rho_{00}$ as a function of transverse momentum for $K^{*0}$ for different collision energies.

$\rho_{00}$ as a function of transverse momentum for $K^{*0}$ for different collision energies.

$\rho_{00}$ as a function of transverse momentum for $K^{*0}$ for different collision energies.

$\rho_{00}$ as a function of transverse momentum for $K^{*0}$ for different collision energies.

$\rho_{00}$ as a function of transverse momentum for $K^{*0}$ for different collision energies.

$\rho_{00}$ as a function of transverse momentum for $K^{*0}$ for different collision energies.

$\rho_{00}$ as a function of transverse momentum for $K^{*0}$ for different collision energies.

$\rho_{00}$ as a function of centrality for $\phi$ (upper panels) and $K^{*0}$ (lower panels). The solid squares and stars are results for the $\phi$ meson, obtained with the 1st- and 2nd-order EP, respectively. The solid circles are results for the $K^{*0}$ meson, obtained with the 2nd-order EP.}

$\rho_{00}$ as a function of centrality for $\phi$ (upper panels) and $K^{*0}$ (lower panels). The solid squares and stars are results for the $\phi$ meson, obtained with the 1st- and 2nd-order EP, respectively. The solid circles are results for the $K^{*0}$ meson, obtained with the 2nd-order EP.}

$\rho_{00}$ as a function of centrality for $\phi$ (upper panels) and $K^{*0}$ (lower panels). The solid squares and stars are results for the $\phi$ meson, obtained with the 1st- and 2nd-order EP, respectively. The solid circles are results for the $K^{*0}$ meson, obtained with the 2nd-order EP.}

$\rho_{00}$ as a function of centrality for $\phi$ (upper panels) and $K^{*0}$ (lower panels). The solid squares and stars are results for the $\phi$ meson, obtained with the 1st- and 2nd-order EP, respectively. The solid circles are results for the $K^{*0}$ meson, obtained with the 2nd-order EP.}

$\rho_{00}$ as a function of centrality for $\phi$ (upper panels) and $K^{*0}$ (lower panels). The solid squares and stars are results for the $\phi$ meson, obtained with the 1st- and 2nd-order EP, respectively. The solid circles are results for the $K^{*0}$ meson, obtained with the 2nd-order EP.}

$\rho_{00}$ as a function of centrality for $\phi$ (upper panels) and $K^{*0}$ (lower panels). The solid squares and stars are results for the $\phi$ meson, obtained with the 1st- and 2nd-order EP, respectively. The solid circles are results for the $K^{*0}$ meson, obtained with the 2nd-order EP.}

$\rho_{00}$ as a function of centrality for $\phi$ (upper panels) and $K^{*0}$ (lower panels). The solid squares and stars are results for the $\phi$ meson, obtained with the 1st- and 2nd-order EP, respectively. The solid circles are results for the $K^{*0}$ meson, obtained with the 2nd-order EP.}

Global spin alignment measurement of $\phi$ and $K^{*0}$ vector mesons in Au+Au collisions at 0-20\% centrality. The solid squares and stars are results for the $\phi$ meson, obtained with the 1st- and 2nd-order EP, respectively. The solid circles are results for $K^{*0}$-meson, obtained with the 2nd-order EP.

Global spin alignment measurement of $\phi$ and $K^{*0}$ vector mesons in Au+Au collisions at 0-20\% centrality. The solid squares and stars are results for the $\phi$ meson, obtained with the 1st- and 2nd-order EP, respectively. The solid circles are results for $K^{*0}$-meson, obtained with the 2nd-order EP.

Raw $\Delta N_k$ distributions in Au+Au collisions at 19.6 GeV for 0–5%, 30–40%, and 70–80% collision centralities at midrapidity. The distributions are not corrected for the finite centrality bin width effect nor the reconstruction efficiency.

Raw $\Delta N_k$ distributions in Au+Au collisions at 27 GeV for 0–5%, 30–40%, and 70–80% collision centralities at midrapidity. The distributions are not corrected for the finite centrality bin width effect nor the reconstruction efficiency.

Raw $\Delta N_k$ distributions in Au+Au collisions at 39 GeV for 0–5%, 30–40%, and 70–80% collision centralities at midrapidity. The distributions are not corrected for the finite centrality bin width effect nor the reconstruction efficiency.

Raw $\Delta N_k$ distributions in Au+Au collisions at 62.4 GeV for 0–5%, 30–40%, and 70–80% collision centralities at midrapidity. The distributions are not corrected for the finite centrality bin width effect nor the reconstruction efficiency.

Raw $\Delta N_k$ distributions in Au+Au collisions at 200 GeV for 0–5%, 30–40%, and 70–80% collision centralities at midrapidity. The distributions are not corrected for the finite centrality bin width effect nor the reconstruction efficiency.

Collision centrality dependence of cumulants (C1, C2, C3, and C4) of $\Delta N_k$ distributions in Au+Au collisions at 7.7 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of cumulants (C1, C2, C3, and C4) of $\Delta N_k$ distributions in Au+Au collisions at 11.5 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of cumulants (C1, C2, C3, and C4) of $\Delta N_k$ distributions in Au+Au collisions at 14.5 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of cumulants (C1, C2, C3, and C4) of $\Delta N_k$ distributions in Au+Au collisions at 19.6 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of cumulants (C1, C2, C3, and C4) of $\Delta N_k$ distributions in Au+Au collisions at 27 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of cumulants (C1, C2, C3, and C4) of $\Delta N_k$ distributions in Au+Au collisions at 39 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of cumulants (C1, C2, C3, and C4) of $\Delta N_k$ distributions in Au+Au collisions at 62.4 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of cumulants (C1, C2, C3, and C4) of $\Delta N_k$ distributions in Au+Au collisions at 200 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $M/\sigma^2$ for $\Delta N_k$ distributions in Au+Au collisions at 7.7 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $M/\sigma^2$ for $\Delta N_k$ distributions in Au+Au collisions at 11.5 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $M/\sigma^2$ for $\Delta N_k$ distributions in Au+Au collisions at 14.5 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $M/\sigma^2$ for $\Delta N_k$ distributions in Au+Au collisions at 19.6 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $M/\sigma^2$ for $\Delta N_k$ distributions in Au+Au collisions at 27 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $M/\sigma^2$ for $\Delta N_k$ distributions in Au+Au collisions at 39 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $M/\sigma^2$ for $\Delta N_k$ distributions in Au+Au collisions at 62.4 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $M/\sigma^2$ for $\Delta N_k$ distributions in Au+Au collisions at 200 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $S\sigma$ for $\Delta N_k$ distributions in Au+Au collisions at 7.7 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $S\sigma$ for $\Delta N_k$ distributions in Au+Au collisions at 11.5 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $S\sigma$ for $\Delta N_k$ distributions in Au+Au collisions at 14.5 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $S\sigma$ for $\Delta N_k$ distributions in Au+Au collisions at 19.6 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $S\sigma$ for $\Delta N_k$ distributions in Au+Au collisions at 27 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $S\sigma$ for $\Delta N_k$ distributions in Au+Au collisions at 39 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $S\sigma$ for $\Delta N_k$ distributions in Au+Au collisions at 62.4 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $S\sigma$ for $\Delta N_k$ distributions in Au+Au collisions at 200 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $\kappa\sigma^2$ for $\Delta N_k$ distributions in Au+Au collisions at 7.7 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $\kappa\sigma^2$ for $\Delta N_k$ distributions in Au+Au collisions at 11.5 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $\kappa\sigma^2$ for $\Delta N_k$ distributions in Au+Au collisions at 14.5 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $\kappa\sigma^2$ for $\Delta N_k$ distributions in Au+Au collisions at 19.6 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $\kappa\sigma^2$ for $\Delta N_k$ distributions in Au+Au collisions at 27 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $\kappa\sigma^2$ for $\Delta N_k$ distributions in Au+Au collisions at 39 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $\kappa\sigma^2$ for $\Delta N_k$ distributions in Au+Au collisions at 62.4 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collision centrality dependence of the $\kappa\sigma^2$ for $\Delta N_k$ distributions in Au+Au collisions at 200 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collisions energy dependence of $M/\sigma^2$ for $\Delta N_k$ multiplicity distributions from 0–5% most central and 70–80% peripheral collisions in Au+Au collisions at \sqrt{s_{NN}} = 7.7, 11.5, 14.5, 19.6, 27, 39, 62.4 and 200 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collisions energy dependence of $S\sigma$ for $\Delta N_k$ multiplicity distributions from 0–5% most central and 70–80% peripheral collisions in Au+Au collisions at \sqrt{s_{NN}} = 7.7, 11.5, 14.5, 19.6, 27, 39, 62.4 and 200 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.

Collisions energy dependence of $\kappa\sigma^2$ for $\Delta N_k$ multiplicity distributions from 0–5% most central and 70–80% peripheral collisions in Au+Au collisions at \sqrt{s_{NN}} = 7.7, 11.5, 14.5, 19.6, 27, 39, 62.4 and 200 GeV. The error bars are statistical uncertainties and the caps represent systematic uncertainties.