  The differential multiplicity of dileptons in a hot and magnetized quark-gluon plasma, $\Delta_{B}\equiv dN_{B}/d^{4}xd^{4}q$, is derived from first principles. The constant magnetic field $B$ is assumed to be aligned in a fixed spatial direction. It is shown that the anisotropy induced by the $B$ field is mainly reflected in the general structure of photon spectral density function. This is related to the imaginary part of the vacuum polarization tensor, $\mbox{Im}[\Pi^{\mu\nu}]$, which is derived in a first order perturbative approximation. As expected, the final analytical expression for $\Delta_{B}$ includes a trace over the product of a photonic part, $\mbox{Im}[\Pi^{\mu\nu}]$, and a leptonic part, ${\cal{L}}_{\mu\nu}$. It is shown that $\Delta_{B}$ consists of two parts, $\Delta_{B}^{\|}$ and $\Delta_{B}^{\perp}$, arising from the components $(\mu,\nu)=(\|,\|)$ and $(\mu,\nu)=(\perp,\perp)$ of $\mbox{Im}[\Pi^{\mu\nu}]$ and ${\cal{L}}_{\mu\nu}$. Here, the transverse and longitudinal directions are defined with respect to the direction of the $B$ field. Combining $\Delta_{B}^{\|}$ and $\Delta_{B}^{\perp}$, a novel anisotropy factor $\nu_{B}$ is introduced. Using the final analytical expression of $\Delta_{B}$, the possible interplay between the temperature $T$ and the magnetic field strength $eB$ on the ratio $\Delta_{B}/\Delta_{0}$ and $\nu_{B}$ is numerically studied. Here, $\Delta_{0}$ is the Born approximated dilepton multiplicity in the absence of external magnetic fields. It is, in particular, shown that for each fixed $T$ and $B$, in the vicinity of certain threshold energies, $\Delta_{B}\gg \Delta_{0}$ and $\Delta_{B}^{\perp}\gg \Delta_{B}^{\|}$. The latter anisotropy may be interpreted as one of the microscopic sources of the macroscopic anisotropies, reflecting themselves, e.g., in the elliptic asymmetry factor $v_{2}$ of dileptons. 
Dilepton production rate in a hot and magnetized quark-gluon plasma