Dilepton production rate in a hot and magnetized quark-gluon plasma

The differential multiplicity of dileptons in a hot and magnetized quark-gluon plasma, ΔB≡dNB/d4xd4q, 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, Im[Πμν], which is derived in a first order perturbative approximation. As expected, the final analytical expression for ΔB includes a trace over the product of a photonic part, Im[Πμν], and a leptonic part, Lμν. It is shown that ΔB consists of two parts, ΔB∥ and ΔB⊥, arising from the components (μ,ν)=(∥,∥) and (μ,ν)=(⊥,⊥) of Im[Πμν] and Lμν. Here, the transverse and longitudinal directions are defined with respect to the direction of the B field. Combining ΔB∥ and ΔB⊥, a novel anisotropy factor νB is introduced. Using the final analytical expression of ΔB, the possible interplay between the temperature T and the magnetic field strength eB on the ratio ΔB/Δ0 and νB is numerically studied. Here, Δ0 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 of virtual photons, ΔB≫Δ0 and ΔB⊥≫ΔB∥. 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 v2 of dileptons.