Introduction

In relativistic heavy-ion collisions, the interactions between the particles in the sources lead to a modification of the boson mass in the sources, and thus give rise to a squeezed boson-antiboson correlation [1, 2]. This squeezed correlation is caused by the Bogoliubov transformation between the creation and annihilation operators of the quasiparticles in the source and the free observable particles, and forces the bosons and antibosons to move in opposite directions. Therefore, it is also known as a squeezed back-to-back correlation (SBBC) [1-3]. Measuring the SBBC of bosons can be used to get information about the interaction between the meson and the source medium and will be useful for understanding the properties of the particle-emitting sources [1-6].

Hydrodynamics has been widely used in relativistic heavy-ion collisions to describe the evolution of a particle-emitting source. In the hydrodynamic description, it is assumed that the source system is under the local equilibrium, that is, it evolves in the so-called quasi-static process. However, the quasi-static process is a rough approximation. Because the SBBC is sensitive to the time distribution of the source, appropriately representing the temporal factors is of interest in the calculation of the SBBC function for an evolving source.

D mesons contain a heavy quark (charm quark) produced during the early stage of relativistic heavy-ion collisions. The SBBC of the D mesons is stronger than that of light mesons and useful for probing the source properties in the early stage [6-10]. This study proposes a method for calculating the SBBC function with the relaxation-time approximation for evolving sources. The effects of the relaxation time on the SBBC functions of $D^0{\overline{D}}^0$ in relativistic heavy-ion collisions are investigated using the hydrodynamic model VISH2+1 [11, 12]. We found that the SBBC functions decrease when the relaxation time is considered. The change in the SBBC functions increases with increasing relaxation time and becomes considerable for large relative deviations of chaotic and squeezed amplitudes at the beginning of their deviation from their equilibrium values.

The remainder of this paper is organized as follows: In Sect. 2, we present the formulas of the SBBC functions of evolving sources with relaxation-time approximation. In Sect. 3, we investigate the influence of the relaxation time on the SBBC functions of $D^0{\overline{D}}^0$ in relativistic heavy-ion collisions. Finally, a summary and discussion are given in Sect. 4.

Formulas

The SBBC function of a boson-antiboson pair with momenta of p₁ and p₂, respectively, is defined as [2, 3] $C_{\text{S}BB}(p_1\mathrm{,}{p}_2)=1+\frac{|G_{\text{s}}(p_1\mathrm{,}{p}_2{)|}^2}{G_{\text{c}}(p_1\mathrm{,}{p}_1)G_{\text{c}}(p_2\mathrm{,}{p}_2)}\mathrm{,}$ (1) where $\begin{array}{ll}{G}_{\text{c}}(p_1\mathrm{,}{p}_2)\hfill & =\sqrt{{\omega }_{p_1}{\omega }_{p_2}}\langle a^{†}(p_1)a(p_2)\rangle \hfill \\ \hfill & \equiv \sqrt{{\omega }_{p_1}{\omega }_{p_2}}\langle g_{\text{c}}(p_1\mathrm{,}{p}_2)\rangle \mathrm{,}\hfill \end{array}$ (2) $\begin{array}{ll}{G}_{\text{s}}(p_1\mathrm{,}{p}_2)\hfill & =\sqrt{{\omega }_{p_1}{\omega }_{p_2}}\langle a(p_1)a(p_2)\rangle \hfill \\ \hfill & \equiv \sqrt{{\omega }_{p_1}{\omega }_{p_2}}\langle g_{\text{s}}(p_1\mathrm{,}{p}_2)\rangle \mathrm{,}\hfill \end{array}$ (3) are the so-called chaotic and squeezed amplitudes, respectively [2, 3], where ${\omega }_p=\sqrt{p^2+m^2}$ is the energy of a free bosons with a mass of m, a and $a^{†}$ are the annihilation and creation operators of the free boson, respectively, and ⟨⋅s⟩ represents the ensemble average.

For a homogeneous thermal-equilibrium source with a fixed volume of V and in the temporal interval of ［0–Δ t］ with the time distribution F(t), the amplitudes $G_{\text{c}}(p\mathrm{,}p)$ and $G_{\text{s}}(p\mathrm{,}-p)$ can be expressed as [2, 3] $\begin{array}{ll}{G}_{\text{c}}(p\mathrm{,}p)\hfill & =\frac{V}{{(2\pi )}^3}{\omega }_p[|c_p{|}^2{n}_p+|s_p{|}^2(n_p+1)]\hfill \\ \hfill & \equiv {\omega }_p\langle g_{\text{c}}^0(p\mathrm{,}p)\rangle \mathrm{,}\hfill \end{array}$ (4) $\begin{array}{ll}{G}_{\text{s}}\hfill & (p\mathrm{,}-p)\hfill \\ \hfill & =\frac{V}{{(2\pi )}^3}{\omega }_p[c_p{s}_p^{*}{n}_p+c_{-p}{s}_{-p}^{*}(n_{-p}+1)]\tilde{F}({\omega }_p\mathrm{,}\Delta t)\hfill \\ \hfill & \equiv {\omega }_p\langle g_{\text{s}}^0(p\mathrm{,}-p)\rangle \mathrm{,}\hfill \end{array}$ (5) where $\begin{array}{l}{c}_{\pm p}^{*}=c_{\pm p}=\cosh f_p\mathrm{,}\hfill \\ s_{\pm p}^{*}=s_{\pm p}=\sinh f_p\mathrm{,}\hfill \\ f_p=\frac{1}{2}\ln \left({\omega }_p/{\Omega }_p\right)\mathrm{,}\hfill \end{array}$ (6) where ${\Omega }_p=\sqrt{p^2+{m^{\prime }}^2}$ is the energy of the quasi-particle, m' is the effective mass in the source medium, $n_p$ is the boson distribution of the quasiparticle, and $\tilde{F}({\omega }_p\mathrm{,}\Delta t)$ is the Fourier transform of time distribution.

For an evolving source, $g_{\text{c}}(p\mathrm{,}p)$ and $g_{\text{s}}(p\mathrm{,}-p)$ can be expressed as $g_{\text{c}}(p\mathrm{,}p)=g_{\text{c}}^0(p\mathrm{,}p)-{\tau }_{\text{c}}\frac{\partial g_{\text{c}}(p\mathrm{,}p)}{\partial t}\mathrm{,}$ (7) $g_{\text{s}}(p\mathrm{,}-p)=g_{\text{s}}^0(p\mathrm{,}-p)-{\tau }_{\text{s}}\frac{\partial g_{\text{s}}(p\mathrm{,}-p)}{\partial t}\mathrm{,}$ (8) where $g_{\text{c}}^0(p\mathrm{,}p)$ and $g_{\text{s}}^0(p\mathrm{,}-p)$ are their quantities in the equilibrium state given by Eqs. (4) and (5), and τ_c and τ_s are the relaxation-time parameters related to the system’s capacity to return to its equilibrium state. Relaxation-time approximation is an usual approach for determining the quantities of evolving systems. In this approximation, τ_c,s must be smaller than the width of the temporal interval τ_c,s<Δ t, and they tend to zero under the quasi-static condition.

Assuming ${\tau }_{\text{c}}={\tau }_{\text{s}}=\overline{\tau }$, Eqs. (7) and (8) give $\begin{array}{ll}{g}_{\text{c}}(p\mathrm{,}p)\hfill & =g_{\text{c}}^0(p\mathrm{,}p)+{\Delta }_{\text{c}}^0{e}^{-t/\overline{\tau }}\hfill \\ \hfill & \approx g_{\text{c}}^0(p\mathrm{,}p)(1+{\delta }_{\text{c}}^0{e}^{-t/\overline{\tau }})\mathrm{,}\hfill \end{array}$ (9) $\begin{array}{ll}{g}_{\text{s}}(p\mathrm{,}-p)\hfill & =g_{\text{s}}^0(p\mathrm{,}-p)+{\Delta }_{\text{s}}^0{e}^{-t/\overline{\tau }}\hfill \\ \hfill & \approx g_{\text{s}}^0(p\mathrm{,}-p)(1+{\delta }_{\text{s}}^0{e}^{-t/\overline{\tau }})\mathrm{,}\hfill \end{array}$ (10) where ${\Delta }_{\text{c,s}}^0$ are the differences in $g_{\text{c,s}}$ between the beginning of the evolution and equilibrium, and it is assumed that the differences are approximately proportional to $g_{\text{c,s}}^0$ with the proportionality parameters of ${\delta }_{\text{c,s}}^0$, respectively. ${\delta }_{\text{c}}^0$ (${\delta }_{\text{s}}^0$) denotes the relative deviation of the chaotic (squeezed) amplitude at the beginning of its deviation from its equilibrium value. Then, the SBBC function $C_{\text{S}BB}(p\mathrm{,}-p)$ for an evolving source is given by $\begin{array}{ll}\hfill & C_{\text{S}BB}(p\mathrm{,}-p)\hfill \\ \hfill & =1+\frac{|\langle g_{\text{s}}^0(p\mathrm{,}-p)(1+{\delta }_{\text{s}}^0{e}^{-t/\overline{\tau }})\rangle {|}^2}{[\langle g_{\text{c}}^0(p\mathrm{,}p)(1+{\delta }_{\text{c}}^0{e}^{-t/\overline{\tau }})\rangle ][\langle g_{\text{c}}^0(-p\mathrm{,}-p)(1+{\delta }_{\text{c}}^0{e}^{-t/\overline{\tau }})\rangle ]}\mathrm{.}\hfill \end{array}$ (11) For a hydrodynamic source, the chaotic and squeezed amplitudes $G_{\text{c}}(p_1\mathrm{,}{p}_2)$ and $G_{\text{s}}(p_1\mathrm{,}{p}_2)$ can be expressed in the relaxation time approximation as [2-6], $\begin{array}{l}{G}_{\text{c}}(p_1\mathrm{,}{p}_2)\text{}=\text{}{\displaystyle \int }\text{}\frac{{\text{d}}^4{\sigma }_{\mu }(r)}{{(2\pi )}^3}{K}_{\mathrm{1,2}}^{\mu }{e}^{i{q}_{\mathrm{1,2}}\cdot r}\\ \left[|{c^{\prime }}_{p{\text{'}}_1\mathrm{,}p{\text{'}}_2}{|}^2{n^{\prime }}_{p{\text{'}}_1\mathrm{,}p{\text{'}}_2}+|{s^{\prime }}_{-p{\text{'}}_1\mathrm{,}-p{\text{'}}_2}{|}^2\left({n^{\prime }}_{-p{\text{'}}_1\mathrm{,}-p{\text{'}}_2}+1\right)\right]\\ \times \left[1+{\delta }_{\text{c}}^0{e}^{-t^{\prime }/\overline{\tau }}\right]\mathrm{,}\end{array}$ (12) $\begin{array}{l}{G}_{\text{s}}(p_1\mathrm{,}{p}_2)\text{}=\text{}{\displaystyle \int }\text{}\frac{{\text{d}}^4{\sigma }_{\mu }(r)}{{(2\pi )}^3}{K}_{\mathrm{1,2}}^{\mu }{e}^{2i{K}_{\mathrm{1,2}}\cdot r}\\ \left[{s^{\prime }}_{-p{\text{'}}_1\mathrm{,}p{\text{'}}_2}^{*}{c^{\prime }}_{p{\text{'}}_2\mathrm{,}-p{\text{'}}_1}{n^{\prime }}_{-p{\text{'}}_1\mathrm{,}p{\text{'}}_2}+{c^{\prime }}_{p{\text{'}}_1\mathrm{,}-p{\text{'}}_2}{s^{\prime }}_{-p{\text{'}}_2\mathrm{,}p{\text{'}}_1}^{*}\left({n^{\prime }}_{p{\text{'}}_1\mathrm{,}-p{\text{'}}_2}+1\right)\right]\\ \times \left[1+{\delta }_{\text{s}}^0{e}^{-t^{\prime }/\overline{\tau }}\right]\mathrm{,}\end{array}$ (13) where d⁴σμ(r)=fμ(r)d³r dt is the four-dimension element of the freeze-out hypersurface, $q_{\mathrm{1,2}}^{\mu }=p_1^{\mu }-p_2^{\mu }$, $K_{\mathrm{1,2}}^{\mu }=(p_1^{\mu }+p_2^{\mu })/2$, ${c^{\prime }}_{p{\text{'}}_1\mathrm{,}p{\text{'}}_2}$ and ${s^{\prime }}_{p{\text{'}}_1\mathrm{,}p{\text{'}}_2}$ are the coefficients of the Bogoliubov transformation between the creation and annihilation operators of the quasiparticles and the free particles, ${n^{\prime }}_{p{\text{'}}_1\mathrm{,}p{\text{'}}_2}$ is the boson distribution associated with the particle pair in the local frame, and $p_{i^{\prime }}(i=\mathrm{1,2})$ is local-frame momentum [2-6]. In Eqs. (12) and (13), $[1+{\delta }_{\text{c,s}}^0{e}^{-t^{\prime }/}]$ is the factor for relaxation-time influence, where t' is the local frame time. Eqs. (12) and (13) reduce to their usual forms as in [2-6] when $=0$.

Dividing the entire time evolution into a series of time steps (j=1,2,...) with the same step width, we have $\begin{array}{l}{G}_{\text{c}}(p_1\mathrm{,}{p}_2)\text{}={\displaystyle \sum _j}{\displaystyle \int }\text{}\frac{f_{\mu }(r){\text{d}}^{3}r}{{(2\pi )}^3}{K}_{\mathrm{1,2}}^{\mu }{e}^{-i{q}_{\mathrm{1,2}}\cdot r}{e}^{i{q}_{\mathrm{1,2}}^0{t}_j}\\ \left[|{c^{\prime }}_{p{\text{'}}_1\mathrm{,}p{\text{'}}_2}{|}^2{n^{\prime }}_{p{\text{'}}_1\mathrm{,}p{\text{'}}_2}+|{s^{\prime }}_{-p{\text{'}}_1\mathrm{,}-p{\text{'}}_2}{|}^2\left({n^{\prime }}_{-p{\text{'}}_1\mathrm{,}-p{\text{'}}_2}+1\right)\right]\\ \times \left[1+{\delta }_{\text{c}}^0{\displaystyle {\int }_0^{\Delta \tau }}\text{}\text{d}{t}^{\prime }D(t^{\prime })e^{i{q}_{\mathrm{1,2}}^{0}t(t^{\prime })}{e}^{-t^{\prime }/\overline{\tau }}\right]\mathrm{,}\end{array}$ (14) $\begin{array}{l}{G}_{\text{s}}\left(p_1\mathrm{,}{p}_2\right)\text{}={\displaystyle \sum _j}{\displaystyle \int }\text{}\frac{f_{\mu }(r){\text{d}}^{3}r}{{(2\pi )}^3}{K}_{\mathrm{1,2}}^{\mu }{e}^{-2i{K}_{\mathrm{1,2}}\cdot r}{e}^{2i{K}_{\mathrm{1,2}}^0{t}_j}\\ [{s^{\prime }}_{-p{\text{'}}_1\mathrm{,}p{\text{'}}_2}^{*}{c^{\prime }}_{p{\text{'}}_2\mathrm{,}-p{\text{'}}_1}{n^{\prime }}_{-p{\text{'}}_1\mathrm{,}p{\text{'}}_2}+{c^{\prime }}_{p{\text{'}}_1\mathrm{,}-p{\text{'}}_2}{s^{\prime }}_{-p{\text{'}}_2\mathrm{,}p{\text{'}}_1}^{*}\\ \times \left({n^{\prime }}_{p{\text{'}}_1\mathrm{,}-p{\text{'}}_2}+1\right)]\left[1+{\delta }_{\text{s}}^0{\displaystyle {\int }_0^{\Delta \tau }}\text{}\text{d}{t}^{\prime }D(t^{\prime })e^{i2K_{\mathrm{1,2}}^{0}t(t^{\prime })}{e}^{-t^{\prime }/\overline{\tau }}\right]\mathrm{,}\end{array}$ (15) where Δτ is the step width of time in the local frame, t(t')=γvt', ${\gamma }_v=1/\sqrt{1-v^2}$, v is the velocity of the fluid element, and D(t') is the local time distribution at each time step. Taking D(t') to be a uniform distribution, the relaxation-time factors of $G_{\text{c}}(p_i\mathrm{,}{p}_i)(i=\mathrm{1,2})$ and $G_{\text{s}}(p_1\mathrm{,}{p}_2)$ are $\left[1+{\delta }_{\text{c}}^0\frac{\overline{\tau }}{\Delta \tau }{\displaystyle {\int }_0^{\Delta \tau }}\text{}\text{}\text{d}{t}^{\prime }{e}^{-t^{\prime }/\overline{\tau }}\right]\approx \left[1+{\delta }_{\text{c}}^0\frac{\overline{\tau }}{\Delta \tau }\right]\mathrm{,}(\overline{\tau }<<\Delta \tau )\mathrm{,}$ (16) $\begin{array}{l}\left[1+{\delta }_{\text{s}}^0\frac{\overline{\tau }}{\Delta \tau }{\displaystyle {\int }_0^{\Delta \tau }}\text{}\text{}\text{d}{t}^{\prime }{e}^{i2K_{\mathrm{1,2}}^{0}t(t^{\prime })}{e}^{-t^{\prime }/\overline{\tau }}\right]\\ \approx \left[1+{\delta }_{\text{c}}^0\frac{\overline{\tau }}{\Delta \tau }\frac{1+i2K_{\mathrm{1,2}}^0{\gamma }_v\overline{\tau }}{1+{(2K_{\mathrm{1,2}}^0{\gamma }_v\overline{\tau })}^2}\right].\end{array}$ (17)

Results

Figures 1(a) and 1(b) show the SBBC functions C(Δϕ) of the $D^0{\overline{D}}^0$ pair in the hydrodynamic model VISH2+1 [11, 12] for Au+Au collisions at the RHIC energy $\sqrt{s_{\text{NN}}}=200$ GeV with 0–80% centrality and in momentum intervals of 0.55–0.65 GeV/c and 1.15–1.25 GeV/c, respectively. Here, Δϕ is the azimuthal angle difference between the transverse momenta p_1T and p_2T of the two D mesons, and the dashed, solid, dot-dashed, and two-dot-dashed lines are represent (${\delta }_{\text{c}}^0={\delta }_{\text{s}}^0={\delta }^0=0.5$, =0 fm/c, without relaxation-time modification), (${\delta }_{\text{c}}^0={\delta }_{\text{s}}^0={\delta }^0=0.5$, =0.2 fm/c), (${\delta }_c^0={\delta }_s^0={\delta }^0=0.25$, =0.4 fm/c), and (${\delta }_{\text{c}}^0={\delta }_{\text{s}}^0={\delta }^0=0.5$, =0.4 fm/c), respectively. In our model calculations, the event-by-event initial conditions of MC-Glb [13] are employed, the ratio of the shear viscosity to the entropy density of the QGP is taken to be 0.08 [14, 15], and the freeze-out temperature is taken to be T_f=150 MeV according to the comparisons of the model transverse momentum spectrum of D⁰ [8] with the RHIC-STAR experimental data [16] (see the Fig. 3 in Ref. [8]). In the calculations, the time step width was assumed to be 1 fm/c, the free D⁰ meson mass was taken to be 1.865 GeV/c², and the in-medium average mass and width were obtained from the results of the FMFK calculations [17, 18, 8].

Fig. 1Full-screen View

The SBBC functions of the ${\text{D}}^0{\overline{D}}^0$ pair in the hydrodynamic model of Au+Au collisions at $\sqrt{s_{\text{NN}}}=200$ GeV with 0–80% and 40–80% centralities and two momentum ranges.

Figures 1(a) and 1(b) show that the relaxation time decreases the SBBC functions. This change increases with an increase in $\overline{\tau }$ and δ⁰. The relaxation-time parameter indicates the capacity of the system to return to the equilibrium. In thermodynamics, high temperature and violent collisions may increase this capacity. δ⁰ is related to the expanding velocity of the system. The SBBC functions in the lower momentum interval are higher than those in the higher momentum interval because of a greater in-medium mass modification at low momenta than that at high momenta [8], and greater single-event SBBC function oscillations at high momenta [5] may also lead to a lower average SBBC function [6, 8].

Figures 1(c) and 1(d) show the SBBC functions C(Δϕ) of the ${\text{D}}^0{\overline{D}}^0$ pair in the hydrodynamic model for Au+Au collisions at RHIC energy $\sqrt{s_{\text{NN}}}=200$ GeV with 40–80% centrality, and momentum intervals of 0.55–0.65 GeV/c and 1.15–1.25 GeV/c, respectively. From these figures, it can be observed that changes in the relaxation-time decrease the SBBC functions, and the influence of this change increases with increasing $\overline{\tau }$ and δ⁰. Compared to the results for collisions with 0–80% centrality, the SBBC functions for collisions with 40–80% centrality are higher. This is mainly because the temporal distribution of the source is narrower for peripheral collisions [6]. Contributions to SBBC functions at lower Δϕ are mainly from the more peripheral collisions, which are have small spatial and temporal sizes [8]. The differences between the SBBC functions of the collisions with 0–80% and 40–80% centralities become small in the higher momentum interval.

Summary and Discussion

In this study, we investigated the effects of relaxation time on the SBBC functions of boson-antiboson pairs in relativistic heavy-ion collisions. A method for calculating the SBBC functions of bosons with relaxation-time approximation for evolving sources is proposed. Using the method in a hydrodynamic model, we investigated the SBBC functions of $D^0{\overline{D}}^0$ in Au+Au collisions at the RHIC energy $\sqrt{s_{\text{NN}}}=200$ GeV and centralities of 0–80% and 40–80%. This change increases with increasing relaxation time and becomes considerable for relativelu large deviations of the chaotic and squeezed amplitudes at the beginning of their deviation from their equilibrium values.

Relaxation-time approximation is an usual approach for calculating quantities in a near-equilibrium evolving system. In viscous hydrodynamic models, relaxation times of shear and bulk viscosities are introduced, which may change the system’s space-time structure. However, the relaxation times associated with the quantities must be considered during calculations using the hydrodynamic model because they transition from a nonequilibrium to an equilibrium state in each temporal step. In addition, the relaxation time must be appropriately considered when calculating a sensitive time-depend observable.

Using a hydrodynamic model, one can obtain the source temperature as a space-time function. The final observed particles are assumed to be emitted thermally from a four-dimensional hypersurface at the fixed freeze-out temperature, which can be determined by comparing the calculated observables, for example, particle transverse-momentum spectra, with experimental data. In this study, we used the viscous hydrodynamic model VISH2+1 [11, 12] to determine the freeze-out hypersurface and calculate SBBC functions with and without the relaxation-time term ${\delta }^0{e}^{-t^{\prime }/\overline{\tau }}$. The relaxation time reduces the SBBC functions. This effect may be retained in an ideal hydrodynamic model although the viscosity changes of the freeze-out hypersurface.

Because the SBBC is caused by changes in the particle mass in the source medium, analyzing SBBC may become a new technique for getting information about particles in-medium interactions; however, there are no experimental data for comparison with the findings of the model when it is used in this manner. However, it is difficult to account for particle scattering in detail in a bulk evolution model. In our model calculations, we assume that D mesons have a mass shift and width, which are obtained from the FMFK calculations [17, 18, 8], in the sources because of the in-medium interactions. More detailed studies of the influence of particle scattering on the SBBC based on a cascade model (for instance, a multi-phase transport model [19, 20]) or a hybrid model (for instance, the hydro+UrQMD model [21, 22]) will be pursued in the future.