  The Bose-Einstein condensation of bound pairs made of equally and oppositely
charged fermions in a magnetic field is investigated using a relativistic
model. The Gaussian fluctuations have been taken into account in order to study
the spectrum of bound pairs in the strong coupling region. We found, in weak
coupling reagion, the condensation temperature increases with an increasing
magnetic field displaying the magnetic catalysis effect. In strong coupling
region, the inverse magnetic catalysis appears when the magnetic field is low
and is replaced by the usual magnetic catalysis effect when magnetic field is
sufficiently high, in contrast to the nonrelativistic case where the inverse
magnetic catalysis prevails in strong coupling region regardless of the
strength of the magnetic field. The resulting response to the magnetic field is
the consequence of the competition between the dimensional reduction by Landau
orbitals in pairing dynamics and the anisotropy of the kinetic spectrum of the
bound pairs. We thus conclude that dimensional reduction dominates in weak
domain and strong coupling one except the small magnetic field region, where
the enhanced fluctuations dominates.
