Bulk viscosity from Urca processes: $npe\mu$ matter in the neutrino-transparent regime

We study the bulk viscosity of moderately hot and dense, neutrino-transparent relativistic $npe\mu$ matter arising from weak-interaction direct Urca processes. This work parallels our recent study of the bulk viscosity of $npe\mu$ matter with a trapped neutrino component. The nuclear matter is modeled in a relativistic density functional approach with two different parametrizations -- DDME2 (which does not allow for the low-temperature direct-Urca process at any density) and NL3 (which allows for low-temperature direct-Urca process above a low-density threshold). We compute the equilibration rates of Urca processes of neutron decay and lepton capture, as well as the rate of the muon decay, and find that the muon decay process is subdominant to the Urca processes at temperatures $T\geq 3$~MeV in the case of DDME2 model and $T\geq 1$~MeV in the case of NL3 model. Thus, the Urca-process-driven bulk viscosity is computed with the assumption that pure leptonic reactions are frozen. As a result the electronic and muonic Urca channels contribute to the bulk viscosity independently and at certain densities the bulk viscosity of $npe\mu$ matter shows a double-peak structure as a function of temperature instead of the standard one-peak (resonant) form. In the final step, we estimate the damping timescales of density oscillations by the bulk viscosity. We find that, \eg, at a typical oscillation frequency $f=1$~kHz, the damping of oscillation is most efficient at temperatures $3\leq T\leq 5$~MeV and densities $n_B\leq 2n_0$ where they can affect the evolution of the post-merger object.