Spectral properties of $\omega$, $\rho$ and $A_1$ mesons in hot magnetized matter: effects of (inverse) magnetic catalysis

In-medium masses of the light vector $\omega$, $\rho$ and axial-vector $A_1$ mesons are studied in the magnetized hot nuclear matter, accounting for the effects of (inverse) magnetic catalysis. The in-medium partial decay widths for the $A_1\rightarrow \rho \pi$ channels are studied from the in-medium masses of the initial and the final state particles, by applying a phenomenological Lagrangian to account for the $A_1\rho\pi$ interaction vertices. The masses are calculated within the QCD sum rule framework, with the medium effects coming through the light quark ($\sim \langle \bar{q}q \rangle$) and the scalar gluon condensates ($\sim \langle G^2 \rangle$), as well as the light four-quark condensate ($\sim \langle \bar{q}q\rangle^2 $). The condensates are calculated within the chiral $SU(3)$ model in terms of the medium modified scalar fields: isoscalar $\sigma$, $\zeta$, isovector $\delta$ and the dilaton field $\chi$. The effects of magnetic fields are incorporated through the magnetized Dirac sea contribution as well as the Landau energy levels of protons. The incorporation of the magnetic field through the Dirac sea of nucleons lead to an enhancement (reduction) of the light quark condensates with magnetic field, give rise to the phenomenon of magnetic (inverse) catalysis. The effects of (inverse) magnetic catalysis at finite temperature nuclear matter are studied on the spectral functions and production cross-sections of the neutral $\rho$ and $A_1$ mesons. This may affect the production of the light vector and axial-vector mesons in the peripheral heavy-ion collision experiments, where estimated magnetic field is very large at the early stages of collisions with very high temperature.