Squeezed back-to-back correlations of bosons with nonzero widths in relativistic heavy-ion collisions

TL;DR

This paper derives formulas for squeezed back-to-back correlations of bosons with nonzero widths in relativistic heavy-ion collisions, analyzing how finite widths affect correlation patterns for different mesons.

Contribution

It introduces new formulas for SBBC calculations considering nonzero widths and studies their effects on $D^0$ and $$ mesons in heavy-ion collisions.

Findings

Finite width alters the SBBC pattern of $D^0{ar D}^0$.

SBBC of $$ is insensitive to the width.

SBBC of $$ increases rapidly with particle momentum.

Abstract

We derive the formulas for calculating the squeezed back-to-back correlation (SBBC) between a boson and antiboson with nonzero width produced in relativistic heavy-ion collisions. The SBBCs of $D^0$ and $\phi$ mesons with finite in-medium widths are studied. We find that the finite width can change the pattern of the SBBC function of $D^0{\bar D}^0$ with respect to mass. However, the SBBC function of $\phi\phi$ is insensitive to the width. In the high-particle-momentum region, the SBBC function of $\phi\phi$ increases with particle momentum rapidly and can exceed that of $D^0{\bar D}^0$ whether the width is nonzero or not.

Equations

$${\mathcal{H}}_0(x)=\frac{1}{2}\left\{{\dot{\varphi }}^2(x)+[\nabla \varphi (x){]}^2+m_0^2{\varphi }^2(x)\right\},$$

$$\varphi (x)=\sum _{\mathbf{p}}(2V{\omega }_{\mathbf{p}}{)}^{-\frac{1}{2}}\left(e^{-ip\cdot x}{a}_{\mathbf{p}}+e^{ip\cdot x}{a}_{\mathbf{p}}^{†}\right),$$

$${\Omega }_{\mathbf{p}}=\sqrt{{\mathbf{p}}^2+(m_0+\Delta m-i\Gamma /2{)}^2}\equiv |{\Omega }_{\mathbf{p}}|e^{i\Theta },$$

$$|\Omega_{\mathbf{p}}|=\left\{\bigg{[}{\mathbf{p}}^{2}\!+\!(m_{0}+\Delta m)^{2}\!-\!\frac{\Gamma^{2}}{4}\bigg{]}^{2}\!+\!(m_{0}\!+\!\Delta m)^{2}\Gamma^{2}\right\}^{\!{1/4}},$$

$$\Theta =\frac{1}{2}{\tan }^{-1}\left[\frac{-(m_0+\Delta m)\Gamma }{{\mathbf{p}}^2+(m_0+\Delta m{)}^2-{\Gamma }^2/4}\right].$$

$$H_{\mathrm{M}}=\int d^{3}x\frac{1}{2}\left\{{\dot{\varphi }}^2(x)+[\nabla \varphi (x){]}^2+(m_0^2+m_1^2){\varphi }^2(x)\right\}$$

$$\displaystyle\hskip 17.07164pt=\sum_{\mathbf{p}}\omega_{\mathbf{p}}a_{\mathbf{p}}^{{\dagger}}a_{\mathbf{p}}+\frac{1}{4}\sum_{\mathbf{p}}\frac{m_{1}^{2}}{\omega_{\mathbf{p}}}\bigg{[}e^{-i2\omega_{\mathbf{p}}t}a_{\mathbf{p}}a_{-{\mathbf{p}}}~{}~{}~{}~{}~{}~{}~{}~{}$$

$$\displaystyle\hskip 28.45274pt+\,e^{i2\omega_{\mathbf{p}}t}a_{\mathbf{p}}^{{\dagger}}a_{-{\mathbf{p}}}^{{\dagger}}+2a_{\mathbf{p}}^{{\dagger}}a_{\mathbf{p}}\bigg{]},$$

$$m_1^2=(m_0+\Delta m-i\Gamma /2{)}^2-m_0^2={\Omega }_{\mathbf{p}}^2-{\omega }_{\mathbf{p}}^2.$$

$$e^{-i{\omega }_{\mathbf{p}}t}{a}_{\mathbf{p}}=c_{\mathbf{p}}{e}^{-i{\Omega }_{\mathbf{p}}t}{b}_{\mathbf{p}}+s_{-\mathbf{p}}^{\ast }{e}^{i{\Omega }_{\mathbf{p}}t}{b}_{-\mathbf{p}}^{†},$$

$$e^{i{\omega }_{\mathbf{p}}t}{a}_{\mathbf{p}}^{†}=c_{\mathbf{p}}^{\ast }{e}^{i{\Omega }_{\mathbf{p}}t}{b}_{\mathbf{p}}^{†}+s_{-\mathbf{p}}{e}^{-i{\Omega }_{\mathbf{p}}t}{b}_{-\mathbf{p}},$$

$$\displaystyle H_{\rm M}=\frac{1}{2}\sum_{\mathbf{p}}\frac{m_{1}^{2}}{\omega_{\mathbf{p}}}\bigg{[}(|c_{\mathbf{p}}|^{2}+|s_{\mathbf{p}}|^{2}+c_{\mathbf{p}}s^{*}_{\mathbf{p}}+c^{*}_{\mathbf{p}}s_{\mathbf{p}})~{}~{}~{}~{}~{}~{}$$

$$\displaystyle\hskip 28.45274pt+\frac{2\omega_{\mathbf{p}}^{2}}{m_{1}^{2}}(|c_{\mathbf{p}}|^{2}+|s_{\mathbf{p}}|^{2})\bigg{]}b^{\dagger}_{\mathbf{p}}b_{\mathbf{p}}$$

$$\displaystyle\hskip 22.76219pt+\frac{1}{4}\sum_{\mathbf{p}}\frac{m_{1}^{2}}{\omega_{\mathbf{p}}}\bigg{[}(c^{*}_{\mathbf{p}}c^{*}_{-{\mathbf{p}}}+s^{*}_{\mathbf{p}}s^{*}_{-{\mathbf{p}}}+2c^{*}_{\mathbf{p}}s^{*}_{-{\mathbf{p}}})~{}~{}~{}~{}~{}~{}$$

$$\displaystyle\hskip 28.45274pt+\frac{4\omega_{\mathbf{p}}^{2}}{m_{1}^{2}}\,c^{*}_{\mathbf{p}}s^{*}_{-{\mathbf{p}}}\bigg{]}e^{i2\Omega_{\mathbf{p}}t}b^{\dagger}_{\mathbf{p}}b^{\dagger}_{-{\mathbf{p}}}.$$

$$|c_{\mathbf{p}}{|}^2-|s_{\mathbf{p}}{|}^2=1$$

$$c_{\mathbf{p}}=\frac{\cosh r_1+i\cosh r_2}{\sqrt{2}},s_{\mathbf{p}}=\frac{\sinh r_1+i\sinh r_2}{\sqrt{2}},$$

$$r_{1,2}=\frac{1}{2}\ln \left[\frac{{\omega }_{\mathbf{p}}(1\mp \sin \Theta )}{|{\Omega }_{\mathbf{p}}|\cos \Theta }\right],$$

$$H_{\mathrm{M}}=\sum _{\mathbf{p}}{\Omega }_{\mathbf{p}}{b}_{\mathbf{p}}^{†}{b}_{\mathbf{p}},$$

$${\omega }_{\mathbf{p}}^{\prime }=\sqrt{{\mathbf{p}}^2+(m_0-i{\Gamma }_0/2{)}^2}\equiv |{\omega }_{\mathbf{p}}^{\prime }|e^{i\theta }.$$

$${\mathcal{H}}_0^{\prime }(x)=\frac{1}{2}\left\{{\dot{\varphi }}^{\prime 2}(x)+[\nabla {\varphi }^{\prime }(x){]}^2+(m_0-i{\Gamma }_0/2{)}^2{\varphi }^{\prime 2}(x)\right\},$$

$${\varphi }^{\prime }(x)=\sum _{\mathbf{p}}(2V{\omega }_{\mathbf{p}}^{\prime }{)}^{-\frac{1}{2}}\left(e^{-i{p}^{\prime }\cdot x}{a}_{\mathbf{p}}^{\prime }+e^{i{p}^{\prime }\cdot x}{a}_{\mathbf{p}}^{\prime †}\right),$$

$$e^{-i{\omega }_{\mathbf{p}}^{\prime }t}{a}_{\mathbf{p}}^{\prime }=c_{\mathbf{p}}{e}^{-i{\Omega }_{\mathbf{p}}t}{b}_{\mathbf{p}}^{\prime }+s_{-\mathbf{p}}^{\ast }{e}^{i{\Omega }_{\mathbf{p}}t}{b}_{-\mathbf{p}}^{\prime †},$$

$$e^{i{\omega }_{\mathbf{p}}^{\prime }t}{a}_{\mathbf{p}}^{\prime †}=c_{\mathbf{p}}^{\ast }{e}^{i{\Omega }_{\mathbf{p}}t}{b}_{\mathbf{p}}^{\prime †}+s_{-\mathbf{p}}{e}^{-i{\Omega }_{\mathbf{p}}t}{b}_{-\mathbf{p}}^{\prime },$$

$$\displaystyle H^{\prime}_{\rm M}=\sum_{\mathbf{p}}\omega^{\prime}_{\mathbf{p}}a^{\prime{\dagger}}_{\mathbf{p}}a^{\prime}_{\mathbf{p}}+\frac{1}{4}\sum_{\mathbf{p}}\frac{m^{\prime 2}_{1}}{\omega^{\prime}_{\mathbf{p}}}\bigg{[}e^{-i2\omega^{\prime}_{\mathbf{p}}t}a^{\prime}_{\mathbf{p}}a^{\prime}_{-{\mathbf{p}}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$$

$$\displaystyle\hskip 5.69054pt+\,e^{i2\omega^{\prime}_{\mathbf{p}}t}a^{\prime{\dagger}}_{\mathbf{p}}a^{\prime{\dagger}}_{-{\mathbf{p}}}+2a^{\prime{\dagger}}_{\mathbf{p}}a^{\prime}_{\mathbf{p}}\bigg{]},~{}~{}~{}(m^{\prime 2}_{1}=\Omega^{2}_{\mathbf{p}}-\omega^{\prime 2}_{\mathbf{p}})$$

$$H_{\mathrm{M}}^{\prime }=\sum _{\mathbf{p}}{\Omega }_{\mathbf{p}}{b}_{\mathbf{p}}^{\prime †}{b}_{\mathbf{p}}^{\prime }.$$

$$r_{1,2}=\frac{1}{2}\ln \left[\frac{|{\omega }_{\mathbf{p}}^{\prime }|(1\mp \sin (\Theta -\theta ))}{|{\Omega }_{\mathbf{p}}|\cos (\Theta -\theta )}\right].$$

$$C({\mathbf{p}}_1,{\mathbf{p}}_2)=1+\frac{|G_s({\mathbf{p}}_1,{\mathbf{p}}_2){|}^2}{G_c({\mathbf{p}}_1,{\mathbf{p}}_1)G_c({\mathbf{p}}_2,{\mathbf{p}}_2)},$$

$$G_c({\mathbf{p}}_1,{\mathbf{p}}_2)=\sqrt{{\omega }_{{\mathbf{p}}_1}{\omega }_{{\mathbf{p}}_2}}⟨a_{{\mathbf{p}}_1}^{†}{a}_{{\mathbf{p}}_2}⟩,$$

$$G_s({\mathbf{p}}_1,{\mathbf{p}}_2)=\sqrt{{\omega }_{{\mathbf{p}}_1}{\omega }_{{\mathbf{p}}_2}}⟨a_{{\mathbf{p}}_1}{a}_{{\mathbf{p}}_2}⟩,$$

$$C(\mathbf{p},-\mathbf{p})=1+\frac{|c_{\mathbf{p}}{s}_{\mathbf{p}}^{\ast }{n}_{\mathbf{p}}+c_{-\mathbf{p}}{s}_{-\mathbf{p}}^{\ast }(n_{-\mathbf{p}}+1){|}^2}{n_1(\mathbf{p})n_1(-\mathbf{p})}|\tilde{F}({\omega }_{\mathbf{p}},\Delta t){|}^2,$$

Full text

I Introduction

The interaction of particles with a medium in relativistic heavy-ion collisions may cause a squeezed back-to-back correlation (SBBC) of detected boson-antiboson pairs AsaCso96 ; AsaCsoGyu99 ; Padula06 ; DudPad10 ; Zhang15a ; Zhang-EPJC16 ; AGY17 ; XuZhang19 . This SBBC is related to the in-medium mass modification of the bosons through a Bogoliubov transformation between the annihilation (creation) operators of the quasiparticles in the medium and the corresponding free particles AsaCso96 ; AsaCsoGyu99 ; Padula06 ; DudPad10 ; Zhang15a ; Zhang-EPJC16 ; AGY17 ; XuZhang19 . Generally, the in-medium mass modification includes not only a mass shift of the boson but also an increase of its width in the medium. Therefore, the necessity of developing a formulism that can be used to calculate the SBBC between a boson and antiboson with nonzero width in a medium is obvious.

In this work, we derive the formulas for calculating the SBBC function between a boson and antiboson with nonzero width. The influences of the in-medium width on the SBBC functions of $D^{0}{\bar{D}}^{0}$ and $\phi\phi$ are investigated. We find that the SBBC function of $D^{0}{\bar{D}}^{0}$ changes obviously for a finite change of width. The SBBC of $\phi\phi$ increases with increasing particle momentum rapidly in a high-momentum region and can exceed the SBBC of $D^{0}{\bar{D}}^{0}$ at high momenta whether the width is nonzero or not. Because of the presence of a charm or strange quark, which is believed to experience the entire evolution of the quark-gluon plasma (QGP) created in relativistic heavy-ion collisions, the analyses of experimental data of $D$ and $\phi$ mesons have recently attracted great interest STAR-PRC19D ; CMS-PRL18D ; CMS-PLB18D ; ALICE-JHEP18D ; ALICE-JHEP16D ; ALICE-JHEP15D ; ALICE-PRC14D ; ALICE-PRL13D ; ALICE-EPJC18p ; STAR-PRL16p ; STAR-PRC16p ; PHENIX-PRC16p ; ALICE-PRC15p ; PHENIX-PRC11p ; STAR-PLB09p ; STAR-PRC09p ; STAR-PRL07p ; PHENIX-PRL07p ; PHENIX-PRC05p ; NA50-PLB03p ; STAR-PRC02p ; NA50-PLB00p . However, the bosons with large masses have strong SBBC Padula-JPG10 ; Zhang-IJMPE15 ; Zhang-CPC15 ; Zhang-EPJC16 . The study of the heavy-meson SBBC is meaningful in relativistic heavy-ion collisions.

The rest of this paper is organized as follows. In Section II, we present the formula derivations of the SBBC function for a boson and antiboson with nonzero in-medium width. Then, we show the results of the SBBC functions of $D^{0}{\bar{D}}^{0}$ and $\phi\phi$ in Section III. Finally, a summary is given in Section IV.

II Formulas

For a system of a boson with mass $m_{0}$ in vacuum, the Hamiltonian density is given by

[TABLE]

where

[TABLE]

where $a_{\mathbf{p}}$ and $a^{\dagger}_{\mathbf{p}}$ are annihilation and creation operators of the free boson, respectively, $p=(\omega_{\mathbf{p}},{\mathbf{p}})$, and $\omega_{\mathbf{p}}=\sqrt{{\mathbf{p}}^{2}+m_{0}^{2}}$.

Denoting the in-medium mass shift and width as $\Delta m$ and $\Gamma$, respectively, the boson in-medium energy can be written as

[TABLE]

where

[TABLE]

[TABLE]

Here, $\Theta<0$, indicating the imaginary part of $\Omega_{\mathbf{p}}$ is negative. The in-medium system Hamiltonian is given by AsaCso96

[TABLE]

where

[TABLE]

Eq. (6) reduces to the case in Ref. AsaCso96 when $\Gamma=0$.

To diagonalize $H_{\rm M}$, we perform the transformation

[TABLE]

[TABLE]

and obtain

[TABLE]

Here, the time-decayed term of $e^{-i2\Omega_{\mathbf{p}}t}b_{\mathbf{p}}b_{-{\mathbf{p}}}$ has been removed. In Eqs. (8) and (9), the transformation coefficients satisfy

[TABLE]

to keep the operators $b$ and $b^{\dagger}$ with the same commutation relation as that of $a$ and $a^{\dagger}$.

Letting the terms $b^{\dagger}_{\mathbf{p}}b^{\dagger}_{-{\mathbf{p}}}$ and $b^{\dagger}_{\mathbf{p}}b_{\mathbf{p}}$ in Eq. (10) equal 0 and $\Omega_{\mathbf{p}}b^{\dagger}_{\mathbf{p}}b_{\mathbf{p}}$, respectively, we have

[TABLE]

where $c_{\mathbf{p}}=c_{-{\mathbf{p}}}$, $s_{\mathbf{p}}=s_{-{\mathbf{p}}}$, $r_{1}$ and $r_{2}$ are two real functions,

[TABLE]

and therefore

[TABLE]

i.e., $b_{\mathbf{p}}$ and $b^{\dagger}_{\mathbf{p}}$ are annihilation and creation operators, respectively, of the quasiparticle in the medium with energy $\Omega_{\mathbf{p}}$. For $\Gamma=0$, we have $r_{1}=r_{2}=\frac{1}{2}\ln[\omega_{\mathbf{p}}/\sqrt{{\mathbf{p}}^{2}+(m_{0}+\Delta m)^{2}}]$, and the diagonalization issue reduces to that for the zero-width case.

Next, we further consider the boson with a width $\Gamma_{0}$ in vacuum. In this case, the boson energy in vacuum is

[TABLE]

Introducing the annihilation and creation operators, $a^{\prime}_{\mathbf{p}}$ and $a^{\prime{\dagger}}_{\mathbf{p}}$, respectively, by a transformation similar to Eqs. (8) and (9) ($a^{\prime}_{\mathbf{p}}\to b_{\mathbf{p}},\,a^{\prime{\dagger}}_{\mathbf{p}}\to b^{\dagger}_{\mathbf{p}},\omega^{\prime}_{\mathbf{p}}\to\Omega_{\mathbf{p}}$), we can diagonalize the Hamiltonian of the boson with $\Gamma_{0}$, and therefore write the Hamiltonian density in this case as

[TABLE]

where

[TABLE]

where $p^{\prime}=(\omega^{\prime}_{\mathbf{p}},{\mathbf{p}})$.

Again, with the transformation similar to Eqs. (8) and (9),

[TABLE]

[TABLE]

we can diagonalize the in-medium Hamiltonian

[TABLE]

to

[TABLE]

In Eqs. (18) and (19), the transformation coefficients $c_{{\mathbf{p}}}$ and $s_{{\mathbf{p}}}$ have the same expressions as Eq. (12), but $r_{1}$ and $r_{2}$ are now

[TABLE]

The SBBC function of the boson-antiboson with momenta ${\mathbf{p}}_{1}$ and ${\mathbf{p}}_{2}$ is defined as AsaCsoGyu99 ; Padula06 ; DudPad10 ; Zhang15a ; Zhang-EPJC16 ; AGY17 ; XuZhang19

[TABLE]

where $G_{c}({\mathbf{p}}_{1},{\mathbf{p}}_{2})$ and $G_{s}({\mathbf{p}}_{1},{\mathbf{p}}_{2})$ are the chaotic and squeezed amplitudes, respectively, and

[TABLE]

[TABLE]

where $\langle\cdots\rangle$ indicates the ensemble average and $\omega_{\mathbf{p}}$ is the energy of a boson with average mass $m_{0}$ for nonzero $\Gamma_{0}$. The SBBC function for a spatially homogeneous source can be written as AsaCsoGyu99 ; Padula06 ; Zhang15a ; XuZhang19 ; Zhang-EPJC16 ; AGY17

[TABLE]

where $n_{\mathbf{p}}$ is the Bose-Einstein distribution of the quasiparticle with the energy corresponding to in-medium average mass ($m_{0}\!+\!\Delta m$), $n_{1}({\mathbf{p}})\!=\!|c_{{\mathbf{p}}}|^{2}n_{{\mathbf{p}}}\!+\!|s_{-{\mathbf{p}}}|^{2}(n_{-{\mathbf{p}}}\!+\!1)$, and $|{\widetilde{F}}(\omega_{\mathbf{p}},\Delta t)|^{2}$ is a time suppression factor. We take $|{\widetilde{F}}(\omega_{\mathbf{p}},\Delta t)|^{2}\!=\!(1+4\omega_{{\mathbf{p}}}^{2}\Delta t^{2})^{-1}$ in the calculations for a time-profile function of exponential decay as in Refs. AsaCsoGyu99 ; Padula06 ; DudPad10 ; Zhang-CPC15 ; Zhang-IJMPE15 ; XuZhang19 . Generally, $|{\widetilde{F}}(\omega_{\mathbf{p}},\Delta t)|^{2}$ is also related to the spatial distribution of particle-emitting source for an evolving system Zhang15a ; Zhang-EPJC16 , and there is a large difference between the suppression factors from different time-profile functions Zhang15a ; Knoll-11 .

III Results

We plot in Figs. 1(a) and 1(b) the SBBC functions of $D^{0}\!{\bar{D}}^{0}$ and $\phi\phi$ with respect to in-medium mass $m_{0}+\Delta m$ for different $\Gamma$ values, respectively. Here, the particle momentum is fixed at 1000 MeV/$c$ and we take $\Delta t=2$ fm/$c$ in calculations as in Refs. AsaCsoGyu99 ; Padula06 ; DudPad10 ; Zhang15a ; XuZhang19 . The mass and width of $D^{0}$ meson in vacuum, $m_{0}$ and $\Gamma_{0}$, are taken to be 1864.86 and 0 MeV/$c^{2}$ (PDG: $1.60\times 10^{-9}$ MeV/$c$, corresponding to a mean life $4.10\times 10^{-15}$ s) respectively, and the $m_{0}$ and $\Gamma_{0}$ of $\phi$ meson are taken to be 1019.46 and 4.26 MeV/$c^{2}$ respectively PDG-PRD12 ; PDG-PRD18 .

We see from Fig. 1(a) that the pattern of the SBBC function of $D^{0}\!{\bar{D}}^{0}$ changes significantly with in-medium width $\Gamma$. For $\Gamma=0$, the SBBC function has a typical two-peak structure AsaCsoGyu99 ; Padula06 ; DudPad10 ; XuZhang19 . It is 1 (no correlation) at $m_{0}$ ($\Delta m=0$) and approaches 1 when $\Delta m\to\pm\infty$. However, the two peaks of the SBBC function move to $m_{0}$ and form one peak rapidly with increasing $\Gamma$. Then, the peak declines with increasing $\Gamma$. For $\Gamma\neq 0$, the SBBC always exists even though $\Delta m=0$. By comparing the SBBC functions in Figs. 1(a) and 1(b), we see that the SBBC functions of $\phi\phi$ with respect to mass are much wider than those of $D^{0}\!{\bar{D}}^{0}$. This is because the SBBC function becomes wide with decreasing boson mass Zhang-EPJC16 ; Zhang-CPC15 ; Zhang-IJMPE15 . We also see that the influence of $\Gamma$ on the SBBC function of $\phi\phi$ is small. Because the SBBC function of $\phi\phi$ has a wide mass distribution, it is insensitive to a mass-distribution change caused by a change of $\Gamma$. However, the nonzero $\Gamma_{0}$ of $\phi$ will also counteract the effect of $\Gamma$ on the SBBC function [see Eq. (22) $\theta\neq 0$].

We plot in Fig. 2 the SBBC functions of $D^{0}\!{\bar{D}}^{0}$ and $\phi\phi$ with respect to particle momentum for the in-medium mass shift $\Delta m=-10$ MeV/$c^{2}$ and in medium width $\Gamma=\Gamma_{0}$ and 10 MeV/$c^{2}$. We see that the SBBC functions increase with increasing particle momentum, and the influence of $\Gamma$ increases with increasing particle momentum. Because the momentum distribution $n_{\mathbf{p}}=n_{-{\mathbf{p}}}$ approaches zero when $|{\mathbf{p}}|\to\infty$, the behavior of the SBBC function at very high momenta is mainly determined by $(|c_{\mathbf{p}}s^{*}_{\mathbf{p}}|^{2}/|s_{\mathbf{p}}|^{4})$ AsaCsoGyu99 ; Zhang-EPJC16 , which is approximately $16{\mathbf{p}}^{4}/[4m_{0}^{2}\Delta m^{2}+m_{0}^{2}(\Gamma-\Gamma_{0})^{2}]$. Therefore, the SBBC functions of $\phi\phi$ increase with increasing particle momentum more rapidly than that of $D^{0}{\bar{D}}^{0}$ in the high-momentum region and can exceed the SBBC of $D^{0}{\bar{D}}^{0}$ at high momenta.

IV Summary and discussion

We derived the formulas for calculating the SBBC between a boson and antiboson with nonzero width produced in relativistic heavy-ion collisions. The influences of the in-medium width on the SBBC functions of $D^{0}{\bar{D}}^{0}$ and $\phi\phi$ are investigated. It is found that the pattern of the SBBC function of $D^{0}{\bar{D}}^{0}$ with respect to mass changes significantly with the width. However, the SBBC function of $\phi\phi$ changes slightly with the width. The influence of the width on the SBBC increases with particle momentum. Whether the width is nonzero or not, the SBBC function of $\phi\phi$ increases with increasing particle momentum more rapidly than that of $D^{0}{\bar{D}}^{0}$ in the high-momentum region and can exceed the SBBC function of $D^{0}{\bar{D}}^{0}$ at high momenta.

Finally, it is necessary to mention that we have removed the time-decayed term of $e^{-i2\Omega_{\mathbf{p}}t}b_{\mathbf{p}}b_{-{\mathbf{p}}}$ in diagonalizing the in-medium Hamiltonian [Eq. (10)]. Therefore, the diagonalization is an approximation unless the imaginary part of $-2\Omega_{\mathbf{p}}$ [ ${\rm Im}(-2\Omega_{\mathbf{p}})\!\sim\!m_{0}\Gamma/\omega_{\mathbf{p}}\!\sim\!\Gamma$ for ${\mathbf{p}}^{2}\!<\!m_{0}^{2}$ ] is very large. This problem does not appear in the diagonalization for the bosons without width. There, the terms of $b_{\mathbf{p}}b_{-{\mathbf{p}}}$ and $b^{\dagger}_{\mathbf{p}}b^{\dagger}_{-{\mathbf{p}}}$ can become zero simultaneously with the reduced transform quantity, $r_{1}\!\!=\!\!r_{2}\!\!=\!\!\frac{1}{2}\ln[\,\omega_{\mathbf{p}}/\!\sqrt{{\mathbf{p}}^{2}\!+\!(m_{0}\!+\!\Delta m)^{2}}\,]$. The recent measurements of $D^{0}$ in heavy-ion collisions at the RHIC and LHC indicate that the average width of $D^{0}$ is approximately 30 MeV/$c^{2}$ STAR-PRC19D ; CMS-PRL18D ; CMS-PLB18D ; ALICE-JHEP18D ; ALICE-JHEP16D ; ALICE-JHEP15D ; ALICE-PRC14D ; ALICE-PRL13D . The corresponding characteristic size is $c\tau\sim 6.6$ fm, which is smaller than the typical size of the particle-emitting source in relativistic heavy-ion collisions. Therefore, the diagonalization is a good approximation and the influence of the in-medium width on the SBBC of $D^{0}{\bar{D}}^{0}$ must be considered in the heavy-ion collisions. For the $\phi$ meson, its $c\tau$ is comparable to the typical size of the source. Our work is a key step forward to solve the problem. In addition, it will be of interest to expand the approach presented in the case that the particle and antiparticle with different in-medium mass modifications.

Acknowledgements.

W. N. Zhang thanks Yong Zhang for helpful discussions. This research was supported by the National Natural Science Foundation of China under Grant Nos. 11675034 and 11275037.