One particle distribution function and shear viscosity in magnetic field: a relaxation time approach

We calculate the $\delta f$ correction to the one particle distribution function in presence of magnetic field and non-zero shear viscosity within the relaxation time approximation. The $\delta f$ correction is found to be electric charge dependent. Subsequently, we also calculate one longitudinal and four transverse shear viscous coefficients as a function of dimensionless Hall parameter $\chi_{H}$ in presence of the magnetic field. We find that a proper linear combination of the shear viscous coefficients calculated in this work scales with the result obtained from Grad's moment method in \cite{Denicol:2018rbw}. Calculation of invariant yield of $\pi^{-}$ in a simple Bjorken expansion with cylindrical symmetry shows no noticeable change in spectra due to the $\delta f$ correction for realistic values of the magnetic field and relaxation time. However, when transverse expansion is taken into account using a blast wave type flow field we found noticeable change in spectra and elliptic flow coefficients due to the $\delta f$ correction. The $\delta f$ is also found to be very sensitive on the magnitude of magnetic field. Hence we think it is important to take into account the $\delta f$ correction in more realistic numerical magnetohydrodynamics simulations.