Speed of sound in QCD matter at finite temperature and density

Abstract

The speed of sound in QCD matter at finite temperature and density is investigated within the Polyakov loop improved Nambu–Jona-Lasinio model. The spinodal structure associated with the first-order chiral phase transition is considered to describe the continuous variation of the speed of sound. The behaviors of the squared sound speed in different phases, including the stable, metastable and unstable phases, are derived. The relation between speed of sound and QCD phase transitions is systematically explored. In particular, the boundary of vanishing sound velocity is derived in the temperature-density phase diagram, and the region where the sound wave equation being broken is pointed out. Some interesting features of speed of sound under different definitions are also discussed.

Appendix: Derivations of the formulae of speed of sound under different definitions in the temperature and density space

The general definition of speed of sound is

$$\begin{aligned} c_{X}^2=\left( \frac{\partial p}{\partial \epsilon }\right) _{X}, \end{aligned}$$

(20)

where X is a physics quantum fixed in the calculation of sound speed. In practice, the squared speed of sound $c^2_{X}$ $(X=s/\rho _B, s,\rho _B, T, \mu _B)$ under different conditions are taken in dealing with different physics problems. With the basic definitions of speed of sound, calculation can only be done along some special paths.

To calculate the speed of sound under different definitions in the whole $T-\rho _B$ space, it is necessary to derive the corresponding formulae in terms of temperature and density. Using the Jacobian formula in thermodynamics, we can derive

$$\begin{aligned} c_{X}^{2}\left( T, \rho _{B}\right)= & {} \left( \frac{\partial p}{\partial \epsilon }\right) _{X}=\frac{\partial (p, X)}{\partial (\epsilon , X)}=\frac{\frac{\partial (p, X)}{\partial \left( T, \rho _{B}\right) }}{\frac{\partial (\epsilon , X)}{\partial \left( T, \rho _{B}\right) }} \nonumber \\= & {} \frac{\left( \frac{\partial p}{\partial T}\right) _{\rho _{B}}\left( \frac{\partial X}{\partial \rho _{B}}\right) _{T}-\left( \frac{\partial p}{\partial \rho _{B}}\right) _{T}\left( \frac{\partial X}{\partial T}\right) _{\rho _{B}}}{\left( \frac{\partial \epsilon }{\partial T}\right) _{\rho _{B}}\left( \frac{\partial X}{\partial \rho _{B}}\right) _{T}-\left( \frac{\partial \epsilon }{\partial \rho _{B}}\right) _{T}\left( \frac{\partial X}{\partial T}\right) _{\rho _{B}}} \end{aligned}$$

(21)

According to the thermodynamic characteristic function in the giant canonical ensemble, it is convenient to get the following relations for isospin symmetric matter

$$\begin{aligned} \left( \frac{\partial p}{\partial T}\right) _{\rho _{B}}=s+\rho _{B}\left( \frac{\partial \mu _{B}}{\partial T}\right) _{\rho _{B}}, \end{aligned}$$

(22)

$$\begin{aligned} \left( \frac{\partial p}{\partial \rho _{B}}\right) _{T}=\rho _{B}\left( \frac{\partial \mu _{B}}{\partial \rho _{B}}\right) _{T}, \end{aligned}$$

(23)

$$\begin{aligned} \left( \frac{\partial \epsilon }{\partial T}\right) _{\rho _{B}}=T\left( \frac{\partial s}{\partial T}\right) _{\rho _{B}}, \end{aligned}$$

(24)

$$\begin{aligned} \left( \frac{\partial \epsilon }{\partial \rho _{B}}\right) _{T}=T\left( \frac{\partial s}{\partial \rho _{B}}\right) _{T}+\mu _{B}, \end{aligned}$$

(25)

$$\begin{aligned} \left( \frac{\partial {(s/\rho _B)}}{\partial T}\right) _{\rho _B}=\frac{1}{\rho _{B}}\left( \frac{\partial s}{\partial T}\right) _{\rho _{B}}, \end{aligned}$$

(26)

and

$$\begin{aligned} \left( \frac{\partial { (s/\rho _B)}}{\partial \rho _{B}}\right) _{T}=\frac{1}{\rho _{B}}\left( \frac{\partial s}{\partial \rho _{B}}\right) _{T}-\frac{s}{\rho _{B}^{2}} \end{aligned}$$

(27)

For the different constraint conditions, $X=s/\rho _B, s,\rho _B, T, \mu _B$, we can derived the corresponding formulae of speed of sound as follows

$$\begin{aligned} \!c^2_{s/\rho _B}\!=\!\frac{\!s^{2}\!+\!\rho _{B}^{2}\!\left[ \!\left( \frac{\!\partial \mu _{B}}{\!\partial \rho _{B}}\!\right) _{\!T}\!\left( \frac{\partial s}{\!\partial T}\!\right) _{\!\rho _{\!B}}\!-\!\left( \frac{\partial \mu _{\!B}}{\partial T}\!\right) _{\!\rho _{\!B}}\!\left( \frac{\partial s}{\partial \rho _{\!B}}\!\right) _{\!T}\!\right] \!+\!s \rho _{\!B}\!\left[ \!\left( \frac{\partial \mu _{\!B}}{\partial T}\!\right) _{\!\rho _{\!B}}\!-\!\left( \frac{\partial s}{\partial \rho _{\!B}}\!\right) _{\!T}\right] }{\left( T s+\mu _{B} \rho _{B}\right) \left( \frac{\partial s}{\partial T}\right) _{\rho _{B}}}, \end{aligned}$$

(28)

$$\begin{aligned} c_{s}^{2}=\frac{\rho _{B}\left[ \left( \frac{\partial s}{\partial T}\right) _{\rho _{B}}\left( \frac{\partial \mu _{B}}{\partial \rho _{B}}\right) _{T}-\left( \frac{\partial s}{\partial \rho _{B}}\right) _{T}\left( \frac{\partial \mu _{B}}{\partial T}\right) _{\rho _{B}}\right] -s\left( \frac{\partial s}{\partial \rho _{B}}\right) _{T}}{\mu _{B}\left( \frac{\partial s}{\partial T}\right) _{\rho _{B}}}, \end{aligned}$$

(29)

$$\begin{aligned} c_{\rho _{B}}^{2}=\frac{s+\rho _{B}\left( \frac{\partial \mu _{B}}{\partial T}\right) _{\rho _{B}}}{T\left( \frac{\partial s}{\partial T}\right) _{\rho _{B}}}, \end{aligned}$$

(30)

$$\begin{aligned} c_{T}^{2}=\frac{\rho _{B}\left( \frac{\partial \mu _{B}}{\partial \rho _{B}}\right) _{T}}{T\left( \frac{\partial s}{\partial \rho _{B}}\right) _{T}+\mu _{B}}, \end{aligned}$$

(31)

and

$$\begin{aligned} c_{\mu _{B}}^{2}=\frac{s\left( \frac{\partial \mu _{B}}{\partial \rho _{B}}\right) _{T}}{T\left[ \left( \frac{\partial s}{\partial T}\right) _{\rho _{B}}\left( \frac{\partial \mu _{B}}{\partial \rho _{B}}\right) _{T}-\left( \frac{\partial \mu _{B}}{\partial T}\right) _{\rho _{B}}\left( \frac{\partial s}{\partial \rho _{B}}\right) _{T}\right] -\mu _{B}\left( \frac{\partial \mu _{B}}{\partial T}\right) _{\rho _{B}}}. \end{aligned}$$

(32)