Electromagnetic responses of a non-extensive quark-gluon plasma

Based on the non-extensive statistical mechanics and the gluon polarization tensor obtained from kinetic theory, we derive the longitudinal and transverse gluon self-energies for the quark-gluon plasma. The electric permittivity $\varepsilon$ and the magnetic permeability $\mu_M$ are evaluated from the gluon self-energies through which the real part of the square of the refraction index ${\rm Re}\, n^2$ and the Depine-Lakhtakia index $n_{DL}$ are investigated. The real part of $\varepsilon$ displays a frequency pole $\omega_d=p$, which is just the position of the frequency inflexion of the imaginary part of $\varepsilon$. The non-extensive parameter $q$ significantly affects the real and imaginary parts of $\varepsilon$ in the space-like region $\omega<p$, while the frequency pole $\omega_d=p$ remains unchanged as $q$ increases. The magnetic permeability, ${\rm Re}\, n^2$ and the Depine-Lakhtakia index $n_{DL}$ diverge at frequency $\omega_m$. As $q$ increases, the pole frequency $\omega_m$ shifts to large frequency region. The Depine-Lakhtakia index $n_{DL}$ becomes negative in a quite large frequency region $\omega\in[\omega_c, \omega_m]$. When $q$ increases, the frequency range for $n_{DL}<0$ becomes wider. Nevertheless, there are no propagating modes for the negative refraction. In addition, as momentum $p$ increases, the electric permittivity, the magnetic permeability, ${\rm Re}\, n^2$ and $n_{DL}$ are sensitive to the change of $p$, which indicates the importance of the spatial dispersion in the electromagnetic responses of the QGP.