Squeezed correlations of $\varphi$ meson pairs for hydrodynamic sources in high-energy heavy-ion collisions

In the hot and dense hadronic sources formed in high-energy heavy-ion collisions, the particle interactions in medium might lead to a squeezed back-to-back correlation (BBC) of boson-antiboson pairs. We calculate the BBC functions of $\varphi \varphi$ for sources evolving hydrodynamically in $\left(2+1\right)$ dimensions and with longitudinal boost invariance. The BBC functions for hydrodynamic sources exhibit oscillations as a function of the particle momentum because the temporal distributions of hydrodynamic sources have sharp falls to 0 at large evolving times. The dependences of the BBC functions on the directions of the particle momentum are investigated. For transverse anisotropic sources, the BBC functions are minimum when the azimuthal angles of the particles reach 0. The BBC functions increase with decreasing absolute value of the particle pseudorapidity. The oscillations and the dependences on the particle azimuthal angle and pseudorapidity are the significant signatures for detecting the BBC in high-energy heavy-ion collisions.

Figure 1

Distributions of the freeze-out points of $\varphi$ mesons in the $z=0$ plane for the initial conditions ${\epsilon }_0=8$ and 20 $\mathrm{GeV}/{\mathrm{fm}}^3$ and $R_x=R_y=4$ fm.

Figure 2

Normalized distributions of transverse coordinate and time of the $\varphi$ freeze-out points in the $z=0$ plane for the same initial conditions as in Fig. 1.

Figure 3

BBC functions of $\varphi \varphi$ for the hydrodynamic sources with different initial conditions and $T_f=140$ MeV.

Figure 4

Parameterized temporal distributions $F_1$ and $F_2$ in Eq. (22) for the parameter sets $[a=$ 1.0 (fm/$c$)${}^{-1}, b=$ 0.035 (fm/$c$)${}^{-3}, {\tau }_{\max }=7\mathrm{fm}/c]$ and $[a=$ 0.9 (fm/$c$)${}^{-1}, b=$ 0.025 (fm/$c$)${}^{-3}, {\tau }_{\max }=10$ fm/$c]$. The thin solid (blue) line $F_3$ is for the temporal distribution of the exponential decay.

Figure 5

BBC functions of $\varphi \varphi$ for the sources with hydrodynamic spatial distributions [see Fig. 2] and parameterized temporal distributions (see Fig. 4). (a, b) With temporal distributions $F_1$ and $F_2$ in Fig. 4; $({\mathrm{a}}^{\prime }, {\mathrm{b}}^{\prime }$) with temporal distribution $F_3$ in Fig. 4.

Figure 6

Magnitudes of $D_{\eta }\left(k\right)$ for temporal distributions $F_1$ and $F_3$ in Fig. 4.

Figure 7

BBC functions of $\varphi \varphi$ in the $cos\beta -k$ plane for hydrodynamic sources with different initial radii and energy densities. Here, $m_{*}$ is taken as 1.05 GeV.

Figure 8

Dependence of the average BBC functions, ${\langle C(k,-k)\rangle }_k$, of $\varphi \varphi$ on the cosine of the particle azimuthal angle for hydrodynamic sources with different initial radii and energy densities. Here, $m_{*}$ is taken as 1.05 GeV and the momentum region averaged is 0–1 $\mathrm{GeV}/c$.

Figure 9

BBC functions of $\varphi \varphi$ in the $cos\alpha -k$ plane for hydrodynamic sources with different initial radii and energy densities. Here, $m_{*}$ is taken as 1.05 GeV.

Figure 10

Dependence of the average BBC functions ${\langle C(k,-k)\rangle }_k$ of $\varphi \varphi$ on the pseudorapidity of the particle for hydrodynamic sources with different initial radii and energy densities. Here, $m_{*}$ is taken as 1.05 GeV and the momentum region averaged is 0–1 $\mathrm{GeV}/c$.