  Dilepton production rate (DPR) from hot and dense quark matter is studied in
the presence of an arbitrary external magnetic field using the 2-flavour
Nambu--Jona-Lasinio (NJL) model. The anomalous magnetic moment (AMM) of the
quarks is taken into consideration while calculating the constituent quark mass
as well as the DPR from the thermo-magnetic medium. An infinite number of quark
Landau levels is incorporated so that no approximations are made on the
strength of the background magnetic field. The analytic structure of the two
point vector current correlation function in the complex energy plane reveals
that, in addition to the usual Unitary cut, a non-trival Landau cut appears in
the physical kinematic domains solely due to the external magnetic field.
Moreover, these kinematic domains of the Unitary and Landau cuts are found to
be significantly modified due to the AMM of the quarks. With finite AMM of the
quarks, for certain values of the external magnetic field, the kinematically
forbidden gap between the Unitary and Landau cuts are shown to vanish leading
to the generation of a continuous spectrum of dilepton emission over the whole
invariant mass region not observed earlier.
