A detailed study of the analytic structure of 1-loop self energy graphs for neutral and charged $rho$ mesons is presented at finite temperature and arbitrary magnetic field using the real time formalism of thermal field theory. The imaginary part of the self energy is obtained from the discontinuities of these graphs across the Unitary and Landau cuts, which is seen to be different for $rho^0$ and $rho^pm$. The magnetic field dependent vacuum contribution to the real part of the self energy, which is usually ignored, is found to be appreciable. A significant effect of temperature and magnetic field is seen in the self energy, spectral function, effective mass and dispersion relation of $rho^0$ as well as of $rho^pm$ relative to its trivial Landau shift. However, for charged $rho$ mesons, on account of the dominance of the Landau term, the effective mass appears to be independent of temperature. The trivial coupling of magnetic moment of $rho^pm$ with external magnetic field, when incorporated in the calculation, makes the $rho^pm$ to condense at high magnetic field.
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