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Running coupling effects in the evolution of jet quenching

We study the consequences of including the running of the QCD coupling in the equation describing the evolution of the jet quenching parameter $\hat q$ in the double logarithmic approximation. To start with, we revisit the case of a fixed coupling, for which we obtain exact solutions valid for generic values of the transverse momentum (above the medium saturation scale) and corresponding to various initial conditions. In the case of a running coupling, we construct approximate solutions in the form of truncated series obtained via successive iterations, whose convergence is well under control. We thus deduce the dominant asymptotic behavior of the renormalized $\hat q$ in the limit of a large evolution time $Y\equiv\ln(L/λ)$, with $L$ the size of the medium and $λ$ the typical wavelength of a medium constituent. We show that the asymptotic expansion is universal with respect to the choice of the initial condition at $Y=0$ and, moreover, it is remarkably similar to the corresponding expansion for the saturation momentum of a shockwave (a large nucleus). As expected, the running of the coupling significantly slows down the increase of $\hat q$ with $Y$ in the asymptotic regime at $Y\gg 1$. For the phenomenologically interesting value $Y\simeq 3$, we find an enhancement factor close to 3, independently of the initial condition and for both fixed and running coupling.