Measurements of $\varphi$ meson production in relativistic heavy-ion collisions at the BNL Relativistic Heavy Ion Collider (RHIC)

We present results for the measurement of $\varphi$ meson production via its charged kaon decay channel $\varphi \to K^{+}{K}^{-}$ in $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{\mathit{NN}}}=62.4,130$, and 200 GeV, and in $p+p$ and $d+\mathrm{Au}$ collisions at $\sqrt{s_{\mathit{NN}}}=200$ GeV from the STAR experiment at the BNL Relativistic Heavy Ion Collider (RHIC). The midrapidity ($\left|y\right|<0.5$) $\varphi$ meson transverse momentum ($p_T$) spectra in central $\mathrm{Au}+\mathrm{Au}$ collisions are found to be well described by a single exponential distribution. On the other hand, the $p_T$ spectra from $p+p$, $d+\mathrm{Au}$, and peripheral $\mathrm{Au}+\mathrm{Au}$ collisions show power-law tails at intermediate and high $p_T$ and are described better by Levy distributions. The constant $\varphi /K^{-}$ yield ratio vs beam species, collision centrality, and colliding energy is in contradiction with expectations from models having kaon coalescence as the dominant mechanism for $\varphi$ production at RHIC. The $\Omega /\varphi$ yield ratio as a function of $p_T$ is consistent with a model based on the recombination of thermal $s$ quarks up to $p_T~4$ GeV/$c$, but disagrees at higher transverse momenta. The measured nuclear modification factor, $R_{d\mathrm{Au}}$, for the $\varphi$ meson increases above unity at intermediate $p_T$, similar to that for pions and protons, while $R_{\mathit{AA}}$ is suppressed due to the energy loss effect in central $\mathrm{Au}+\mathrm{Au}$ collisions. Number of constituent quark scaling of both $R_{\mathrm{cp}}$ and $v_2$ for the $\varphi$ meson with respect to other hadrons in $\mathrm{Au}+\mathrm{Au}$ collisions at $\sqrt{s_{\mathit{NN}}}=200$ GeV at intermediate $p_T$ is observed. These observations support quark coalescence as being the dominant mechanism of hadronization in the intermediate $p_T$ region at RHIC.

Figure 1

Measured $\langle \mathit{dE}/\mathit{dx}\rangle$ vs momentum × charge of reconstructed tracks in the TPC. The figure is generated from $\mathrm{Au}+\mathrm{Au}$ 62.4 GeV data.

Figure 2

Background-subtracted invariant mass distributions at $0.4<p_T<0.8$ GeV/$c$ in $d+\mathrm{Au}$ 200 GeV collisions (0–100%) with (solid points) and without (open points) the δ-dip-angle cut. The dashed curves show a Breit-Wigner (see the text for details) $+$ linear background function fit to the case with the δ-dip-angle cut.

Figure 3

Upper panels: same-event (full points) and mixed-event (solid line) $K^{+}{K}^{-}$ invariant mass distributions at $0.6<p_T<1.4$ GeV/$c$ in $p+p$ 200 GeV collisions (a), $0.8<p_T<1.2$ GeV/$c$ in $\mathrm{Au}+\mathrm{Au}$ 62.4 GeV collisions (60–80%) (c), and $0.8<p_T<1.2$ GeV/$c$ in $\mathrm{Au}+\mathrm{Au}$ 200 GeV collisions (0–10%) (e). Lower panels: the corresponding $\varphi$ meson mass peaks after subtracting the background. Dashed curves show a Breit-Wigner $+$ linear background function fit in (b), (d). In (f), both linear and quadratic backgrounds are shown as dashed and dot-dashed lines, respectively.

Figure 4

Reconstruction efficiency including acceptance of $\varphi$ meson as a function of $p_T$ in several centrality bins of $\mathrm{Au}+\mathrm{Au}$, $d+\mathrm{Au}$, and $p+p$ 200 GeV collisions.

Figure 5

Event plane ${\Psi }_2$ resolution as a function of centrality in $\mathrm{Au}+\mathrm{Au}$ 200 GeV collisions, where the vertical axis starts from 0.3 for clarity.

Figure 6

$\phi -{\Psi }_2$ distribution for $\varphi$ meson at $1.5<p_T<2.0$ GeV/$c$ in $\mathrm{Au}+\mathrm{Au}$ collisions (0–80%) at 200 GeV. The line is the fitting result. Error bars are statistical only.

Figure 7

〈cos[2$({\phi }_{K^{+}{K}^{-}}-{\Psi }_2)]\rangle$ (full red points) and 〈sin[2$({\phi }_{K^{+}{K}^{-}}-{\Psi }_2)]\rangle$ (open blue points) as a function of $m_{\mathrm{inv}}$ of $K^{+}{K}^{-}$ pairs at $0.5<p_T<1.0$ GeV/$c$ in $\mathrm{Au}+\mathrm{Au}$ 200 GeV collisions (0–80%), where the solid curve is the result of fitting by Eq. (10). The arrow shows the position of the $\varphi$ invariant mass peak. The dashed line shows the zero horizontal line.

Figure 8

Masses and widths (FWHMs) of $\varphi$ as a function of $p_T$ in $p+p$ 200 GeV (NSD), $d+\mathrm{Au}$ 200 GeV (0–20%), $\mathrm{Au}+\mathrm{Au}$ 62.4 GeV (0–20%), and $\mathrm{Au}+\mathrm{Au}$ 200 GeV (0–5%) collisions, with the corresponding PDG values.

Figure 9

Invariant mass distributions of $\varphi$ meson at $0.6<p_T<1.0$ GeV/$c$ in $p+p$ 200 GeV (NSD) and $\mathrm{Au}+\mathrm{Au}$ 200 GeV (0–5%) collisions. Solid symbols: experimental data. Open symbols: MC simulation. Curves are the results of a Breit-Wigner function fit. Note: Two sets of MC data are shown for $\mathrm{Au}+\mathrm{Au}$ 200 GeV, and see text for details.

Figure 10

ϕ meson transverse mass distributions for different collision systems and different energies. For clarity, distributions for some centrality bins have been scaled by factors indicated in the figure. Curves represent the exponential (solid) and Levy (dashed) function fits to the distributions. Error bars are statistical only. Note that a scale factor of 1.09 is applied to the $\varphi$ meson spectra for $\mathrm{Au}+\mathrm{Au}$ collisions at 200 GeV in Ref. [50] to correct for the kaon identification efficiency effect missed previously.

Figure 11

Comparison of transverse momentum spectra shape among different 200 GeV collision systems: $\mathrm{Au}+\mathrm{Au}$ (0–5%), $d+\mathrm{Au}$ (0–20%), and $p+p$ (inelastic). The spectra are normalized by $N_{\mathrm{bin}}$ (top panel) and $N_{\mathrm{part}}/2$ (bottom panel).

Figure 12

$N_{\mathrm{part}}$ dependence of $(\mathit{dN}/\mathit{dy})/0.5N_{\mathrm{part}}$ in five different collision systems: $\mathrm{Au}+\mathrm{Au}$ 62.4, 130, 200 GeV; $p+p$ 200 GeV (inelastic); and $d+\mathrm{Au}$ 200 GeV. Statistical and systematic errors are included.

Figure 13

Top panel: $N_{\mathrm{part}}$ dependence of $\langle p_T\rangle {}_{\varphi }$ in different collision systems. Bottom panel: Hadron mass dependence of $\langle p_T\rangle$ in central $\mathrm{Au}+\mathrm{Au}$ collisions at 62.4 and 200 GeV. The band and curve show two hydrodynamic model calculations for central $\mathrm{Au}+\mathrm{Au}$ collisions at 200 GeV. Note: Hadron masses for the $\mathrm{Au}+\mathrm{Au}$ 62.4 GeV data are shifted slightly in the $x$-axis direction for clarity, and systematic errors are included for $\varphi$.

Figure 14

Top panel: Energy dependence of the ratio $\varphi /{\pi }^{-}$ in $A+A$ (full points) and $p+p$ (open points) collisions. Stars are data from the STAR experiment at RHIC. Bottom panel: $N_{\mathrm{part}}$ dependence of ratio $\varphi /{\pi }^{-}$ in different collision systems. Systematic errors are included for the STAR data points.

Figure 15

Top panel: Energy dependence of ratio $\varphi /K^{-}$ in $A+A$ and elementary ($e+e$ and $p+p$) collisions. Stars are data from STAR experiments at RHIC. Bottom panel: $N_{\mathrm{part}}$ dependence of ratio $\varphi /K^{-}$ in different collision systems. The dashed line shows results from UrQMD model calculations. Systematic errors are included for the STAR data points.

Figure 16

The $\Omega /\varphi$ ratio vs $p_T$ for three centrality bins in $\sqrt{s_{\mathit{NN}}}=200$ GeV $\mathrm{Au}+\mathrm{Au}$ collisions, where the data points for 40–60% are shifted slightly for clarity. As shown in the legend, the lines represent results from Hwa and Yang [96], Ko et al. [97], and, for SCF, Refs. [98, 99].

Figure 17

$p_T$ dependence of the nuclear modification factor $R_{\mathrm{cp}}$ in $\mathrm{Au}+\mathrm{Au}$ 200 GeV collisions. The top and bottom panels present $R_{\mathrm{cp}}$ from midperipheral and most-peripheral collisions, respectively. See legend for symbol and line designations. The rectangular bands show the uncertainties of binary and participant scalings. Statistical and systematic errors are included.

Figure 18

$p_T$ dependence of the nuclear modification factor $R_{\mathrm{cp}}$ in $\mathrm{Au}+\mathrm{Au}$ 62.4 and 200 GeV and $d+\mathrm{Au}$ 200 GeV collisions, where rectangular bands represent the uncertainties of binary and participant scalings (see legend). Statistical and systematic errors are included.

Figure 19

$p_T$ dependence of the nuclear modification factor $R_{\mathit{AB}}$ for $\varphi$ in $\mathrm{Au}+\mathrm{Au}$ 200 GeV and $d+\mathrm{Au}$ 200 GeV collisions. For comparison, data points for ${\pi }^{+}+{\pi }^{-}$ in $d+\mathrm{Au}$ 200 GeV and $p+\overline{p}$ in $d+\mathrm{Au}$ 200 GeV are also shown (see legend). Rectangular bands show the uncertainties of binary (solid line) and participant (dot-dash line) scalings. Systematic errors are included for $\varphi$, ${\pi }^{+}+{\pi }^{-}$, and $p+\overline{p}$.

Figure 20

$p_T$ dependence of the elliptic flow $v_2$ of $\varphi$, $\Lambda$, and $K_S^0$ in $\mathrm{Au}+\mathrm{Au}$ collisions (0–80%) at 200 GeV. Data points for $\varphi$ are from the reaction plane method (full up-triangles) and invariant mass method (full circles), where data points from the reaction plane method are shifted slightly along the $x$ axis for clarity. Vertical error bars represent statistical errors, while the square bands represent systematic uncertainties. The magenta curved band represents the $v_2$ of the $\varphi$ meson from the AMPT model with a string melting mechanism [116]. The dash and dot curves represent parametrizations inspired by number-of-quark scaling ideas from Ref. [117] for $\mathrm{NQ}=2$ and $\mathrm{NQ}=3$, respectively.

Figure 21

Elliptic flow $v_2$ as a function of $p_T,v_2(p_T)$, for the $\varphi$ meson from different centralities. The vertical error bars represent the statistical errors, while the square bands represent the systematic uncertainties. For clarity, data points of 10–40% are shifted in the $p_T$ direction slightly.