  The one loop self energy of the neutral $\rho$ meson is obtained for the
effective $\rho\pi\pi$ and $\rho NN$ interaction at finite temperature and
density in the presence of a constant background magnetic field of arbitrary
strength. In our approach, the eB-dependent vacuum part of the self energy is
extracted by means of dimensional regularization where the ultraviolet
divergences corresponding to the pure vacuum self energy manifest as the pole
singularities of gamma as well as Hurwitz zeta functions. This improved
regularization procedure consistently reproduces the expected results in the
vanishing magnetic field limit and can be used quite generally in other self
energy calculations dealing with arbitrary magnetic field strength. In presence
of the external magnetic field, the general Lorentz structure for the in-medium
vector boson self energy is derived which can also be implemented in case of
the gauge bosons such as photons and gluons. It has been shown that with
vanishing perpendicular momentum of the external particle, essentially two form
factors are sufficient to describe the self energy completely. Consequently,
two distinct modes are observed in the study of the effective mass, dispersion
relations and the spectral function of $\rho^0$ where one of the modes
possesses two fold degeneracy. For large baryonic chemical potential, it is
observed that the critical magnetic field required to block the
$\rho^0\rightarrow\pi^+\pi^-$ decay channel increases significantly with
temperature. However, in case of smaller values reaching down to vanishing
chemical potential, the critical field follows the opposite trend.
