Relating q̂ , η/s and ΔE in an expanding Quark-Gluon Plasma
We use linear viscous hydrodynamics to describe the energy and momentum deposited by a fast moving parton in a quark gluon plasma. This energy-momentum is in turn used to compute the probability density for the production of soft partons by means of the Cooper-Frye formula. We use this probability density to render manifest a relation between the average transverse momentum given to the fast moving parton from the medium q̂ , the entropy density to shear viscosity ratio η/s and the energy lost by the fast moving parton ΔE in an expanding medium under similar conditions to those generated in nucleus-nucleus collisions at the LHC. We find that q̂ increases linearly with ΔE for both trigger and away side partons that have been produced throughout the medium. On the other hand, η/s is more stable with ΔE. We also study how these transport coefficients vary with the geometrical location of the hard scattering that produces the fast moving partons. The behavior of q̂ , with ΔE is understood as arising from the length of medium the parton traverses from the point where it is produced. However, since η/s is proportional to the ratio of the length of medium traversed by the fast parton and the average number of scatterings it experiences, it has a milder dependence on the energy it loses. This study represents a tool to obtain a direct connection between transport coefficients and the description of in-medium energy loss within a linear viscous hydrodynamical evolution of the bulk.