The mass modifications of the open charm ($D$ and $D^*$) mesons, and their effects on the decay widths $D^*rightarrow Dpi$ as well as of the charmonium state, $Psi(3770)$ to open charm mesons ($Psi(3770)rightarrow Dbar D$), are investigated in the presence of strong magnetic fields. These are studied accounting for the mixing of the pseudoscalar ($P$) and vector ($V$) mesons ($D-D^*$, $eta_c-Psi(3770)$ mixings), with the mixing parameter, $g_{PV}$ of a phenomenological three-point ($PVgamma$) vertex interaction determined from the observed radiative decay width of $Vrightarrow Pgamma$. For charged $D-D^*$ mixing, this parameter is dependent on the magnetic field, because of the Landau level contributions to the vacuum masses of these mesons. The masses of the charged $D$ and $D^*$ mesons modified due to $PV$ mixing, in addition, have contributions from the lowest Landau levels in the presence of a strong magnetic field. The effects of the magnetic field on the decay widths are studied using a field theoretic model of composite hadrons with quark (and antiquark) consittuents. The parameter for the charged $D-D^*$ mixing is observed to increase appreciably with increase in the magnetic field. This leads to dominant modifications to their masses, and hence the decay widths of charged $D^*rightarrow Dpi$ as well as $Psi(3770)rightarrow D^+D^-$ at large values of the magnetic field. The modifications of the masses and decay widths of the open and hidden charm mesons in the presence of strong magnetic fields should have observable consequences on the production of the open charm ($D$ and $D^*$) mesons as well as of the charmonium states resulting from non-central ultrarelativistic heavy ion collision experiments.
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