  We study the partial decay widths of charmonium (bottomonium) states to ${\rm
D\bar D \; (B\bar B)}$ mesons in magnetized (nuclear) matter using a field
theoretical model of composite hadrons with quark (and antiquark) constituents.
These are computed from the mass modifications of the decaying and produced
mesons within a chiral effective model, including the nucleon Dirac sea
effects. The mass modifications of the open charm (bottom) mesons are
calculated from their interactions with the nucleons and the scalar mesons,
whereas the mass shift of the heavy quarkonium state is obtained from the
medium change of a scalar dilaton field, $\chi$, which mimics the gluon
condensates of QCD. The Dirac sea contributions are observed to lead to a rise
(drop) in the quark condensates as the magnetic field is increased, an effect
called the (inverse) magnetic catalysis. These effects are observed to be
significant and the anomalous magnetic moments (AMMs) of the nucleons are
observed to play an important role. For $\rho_B$=0, there is observed to be
magnetic catalysis (MC) without and with AMMs, whereas, for $\rho_B=\rho_0$,
the inverse magnetic catalysis (IMC) is observed when the AMMs are taken into
account, contrary to MC, when the AMMs are ignored. In the presence of a
magnetic field, there are also mixings of spin 0 (pseudoscalar) and spin 1
(vector) states (PV mixing) which modify the masses of these mesons. The
magnetic field effects on the heavy quarkonium decay widths should have
observable consequences on the production the heavy flavour mesons, which are
created in the early stage of ultra-relativistic peripheral heavy ion
collisions, at RHIC and LHC, when the produced magnetic fields can still be
extremely large.
