Dilepton production in a schematic causal viscous hydrodynamics

Assuming that in the hot dense matter produced in relativistic heavy-ion collisions, the energy density, entropy density, and pressure as well as the azimuthal and space-time rapidity components of the shear tensor are uniform in the direction transversal to the reaction plane, we derive a set of schematic equations from the Isreal-Stewart causal viscous hydrodynamics. These equations are then used to describe the evolution dynamics of relativistic heavy-ion collisions by taking the shear viscosity to entropy density ratio of $1/4\pi$ for the initial quark-gluon plasma (QGP) phase and of ten times this value for the later hadron-gas (HG) phase. Using the production rate evaluated with particle distributions that take into account the viscous effect, we study dilepton production in central heavy-ion collisions. Compared with results from the ideal hydrodynamics, we find that although the dilepton invariant mass spectra from the two approaches are similar, the transverse momentum spectra are significantly enhanced at high transverse momenta by the viscous effect. We also study the transverse momentum dependence of dileptons produced from QGP for a fixed transverse mass which is essentially absent in the ideal hydrodynamics, and find that this so-called transverse mass scaling is violated in the viscous hydrodynamics, particularly at high transverse momenta.