Relating $\widehat{q}, \eta /s$, and $\mathrm{\Delta }E$ 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 $\widehat{q}$, the shear viscosity to entropy density ratio $\eta /s$, and the energy lost by the fast moving parton $\mathrm{\Delta }E$ in an expanding medium under similar conditions to those generated in nucleus-nucleus collisions at the CERN Large Hadron Collider. We find that $\widehat{q}$ increases linearly with $\mathrm{\Delta }E$ for both trigger and away side partons that have been produced throughout the medium. On the other hand, $\eta /s$ is more stable with $\mathrm{\Delta }E$. 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 $\widehat{q}$, with $\mathrm{\Delta }E$, is understood as arising from the length of medium the parton traverses from the point where it is produced. However, since $\eta /s$ 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.

Figure 1

$\widehat{q}$ as a function of $\mathrm{\Delta }E$ for the away-side particle for different trigger particle energy loses. The left (right) panel corresponds to a QGP temperature $T_0=350$ ($T_0=450$) MeV and a corresponding ${\epsilon }_0=2$ (${\epsilon }_0=4$) GeV/fm in the one dimensional energy loss model.

Figure 2

$\widehat{q}$ for the trigger particle as a function of the distance $r$ where the hard scattering took place. The left (right) panel corresponds to a QGP temperature $T_0=350$ ($T_0=450$) MeV for the two values of ${\epsilon }_0=2,4$) GeV/fm in the one dimensional energy loss model.

Figure 3

$\widehat{q}$ for the away-side particle as a function of the distance $r$ where the hard scattering took place. The left (right) panel corresponds to a QGP temperature $T_0=350$ ($T_0=450$) MeV and a corresponding ${\epsilon }_0=2$ (${\epsilon }_0=4$) GeV/fm in the one dimensional energy loss model.

Figure 4

$\eta /s$ as a function of $\mathrm{\Delta }E$ for the away-side particle for different trigger particle energy loses. The left (right) panel corresponds to a QGP temperature $T_0=350$ ($T_0=450$) MeV and a corresponding ${\epsilon }_0=2$ (${\epsilon }_0=4$) GeV/fm in the one dimensional energy loss model.

Figure 5

$\eta /s$ for the trigger particle as a function of the distance $r$ where the hard scattering took place. The left (right) panel corresponds to a QGP temperature $T_0=350$ ($T_0=450$) MeV and a corresponding ${\epsilon }_0=2$ (${\epsilon }_0=4$) GeV/fm in the one dimensional energy loss model.

Figure 6

$\eta /s$ for the away-side particle as a function of the distance $r$ where the hard scattering took place. The left (right) panel corresponds to a QGP temperature $T_0=350$ ($T_0=450$) MeV and a corresponding ${\epsilon }_0=2$ (${\epsilon }_0=4$) GeV/fm in the one dimensional energy loss model.

Figure 7

Relation between $\eta /s$ and $\widehat{q}$ for the away-side particle, for different trigger particle energy loses. The relation is obtained by plotting the value of these parameters that correspond to a given value of $\mathrm{\Delta }E$. The left (right) panels correspond to ${\epsilon }_0=2$ (${\epsilon }_0=4$ GeV/fm) in the one dimensional energy loss model and top to bottom panels correspond to a QGP temperature $T_0=250,350,450$ MeV, respectively.