Chiral MHD description of a perfect magnetized QGP using the effective NJL model in a strong magnetic field

To study the effect of a strong magnetic field $B$ on the sound velocity $v_{s}$ of plane waves propagating in a strongly magnetized quark-gluon plasma (QGP), a chiral magnetohydrodynamical (MHD) description of a perfect (non-dissipative) QGP exhibiting dynamical chiral symmetry breaking (D$\chi$SB) is developed using the effective action of the Nambu-Jona-Lasinio (NJL) model of QCD at finite temperature, finite baryon chemical potential and in the presence of a strong magnetic field. Here, the D$\chi$SB arises due to the phenomenon of magnetic catalysis. Apart from an interesting frequency dependence, for plane waves propagating in the transverse or longitudinal direction with respect to the $B$ field, the sound velocity is anisotropic and depends on the angle between the corresponding wave vectors and the direction of the $B$ field. Moreover, for plane waves propagating in the transverse (longitudinal) direction to the external $B$ field, the sound velocity has a maximum (minimum) at $T<T_{c}$, reaches a local minimum (maximum) at $T\sim T_{c}$ and remains constant at $T\gtrsim T_{c}$. Here, $T_{c}$ is the critical temperature of the chiral phase transition. Thus, the constant value $v_{s}\sim 1.5 c_{s}$ at $T \gtrsim T_{c}$ turns out to be a lower (upper) bound for waves propagating in the transverse (longitudinal) direction with respect to the external $B$ field. Here, $c_{s}=1/\sqrt{3}$ is the sound velocity in an ideal gas.