Spin density matrix for neutral $\rho$ mesons in a pion gas in linear response theory

We calculate the spin density matrix for neutral $\rho$ mesons from the spectral function and thermal shear tensor by Kubo formula in the linear response theory, which contributes to the $\gamma$ correlator for the CME search. We derive the spectral function of neutral $\rho$ mesons with $\rho \pi \pi$ and $\rho \rho \pi \pi$ interactions using the Dyson-Schwinger equation. The thermal shear tensor contribution is obtained from the Kubo formula in the linear response theory. We numerically calculate ${\rho }_{00}-1\mathrm{/}3$ and $\mathrm{R}\mathrm{e}{\rho }_{-1,1}$ using the simulation results for the thermal shear tensor by the hydrodynamical model, which are of the order ${10}^{-3}\sim {10}^{-2}$.