Radiative heavy quark energy loss in an expanding viscous QCD plasma

We study viscous effects on heavy quark radiative energy loss in a dynamically screened medium with boost-invariant longitudinal expansion. We calculate, to first order in opacity, the energy loss by incorporating viscous corrections in the single-particle phase-space distribution function within relativistic dissipative hydrodynamics. We consider Grad's 14-moment and the Chapman-Enskog-like methods for the nonequilibrium distribution functions. Our numerical results for the charm quark radiative energy loss show that, as compared to an expanding ideal (nonviscous) fluid, viscosity in the evolution leads to somewhat enhanced energy loss which is rather insensitive to the underlying viscous hydrodynamic models used. Further inclusion of a viscous correction induces larger energy loss, and the magnitude and pattern of this enhancement crucially depend on the form of viscous corrections used.

Figure 1

Fractional differential radiative energy loss as a function of momentum for charm quarks in a boost-invariant expanding plasma at time $\tau =1.2\mathrm{fm}/c$. The results are for an ideal fluid (black dotted line) in the dissipative hydrodynamics without a viscous correction in the MIS (red dashed line) and CE (green dashed line) theories and with further inclusion of the viscous correction due to Grad in the MIS (red solid line) and Chapman-Enskog (green solid line) methods. The results are for an ideal gas equation of state ($P=\epsilon /3$) with an initial temperature of $T_0=400\mathrm{MeV}$, a proper time of ${\tau }_0=0.4\mathrm{fm}/c$, and a constant shear viscosity to entropy density ratio of $\eta /s=1/4\pi$.

Figure 2

Time dependence of fractional differential radiative energy loss for charm quarks of initial momentum $p=20$ and 40 GeV/c. The initial conditions and the various curves are the same as in Fig. 1.

Figure 3

Dissipative part of the differential fractional energy loss $\delta dE/\left(Ed\tau d\mathbf{q}\right)$ as a function of exchanged gluon momentum $\mathbf{q}$ at various proper times for charm quarks of initial momentum $p=20\mathrm{GeV}/c$ at $\eta /s=1/4\pi$ in the MIS and CE methods. The inset shows the variation of viscous correction $\delta {\mathrm{\Lambda }}_{\mathrm{vis}}(\tau ,\mathbf{q})$ with momentum $\mathbf{q}$ at various times. The initial conditions are the same as in Fig. 1.

Figure 4

Time-integrated fractional radiative energy loss as a function of momentum for charm quarks propagating in a boost-invariant expanding fluid over a total time of ${\tau }_f=5\mathrm{fm}/c$ with various values of $\eta /s$ in the MIS (top panel) and CE (bottom panel) frameworks. In a nonexpanding plasma at a temperature of $T_0=0.228\mathrm{GeV}$, the fractional energy loss integrated over a path length of $L=5\mathrm{fm}$ is also shown (black thin solid line). The initial conditions and the various curves for the expanding medium are the same as in Fig. 1. In addition, results with $\eta /s=3/4\pi$ are shown without viscous corrections in MIS (purple dotted line) and CE (cyan dotted line) theories and with further inclusion of viscous corrections in MIS (purple dashed-dotted line) and CE (cyan dashed-dotted line) methods.

Figure 5

Time dependence of fractional radiative energy loss for charm quarks of initial momentum $p=20\mathrm{GeV}/c$ with viscous corrections included in the distribution function in models of dissipative hydrodynamics. The initial conditions and the various curves are the same as in Fig. 4.

Figure 6

(a)–(d) Feynman diagrams $M_{1,0}$ contributing to heavy quark radiative energy loss to first order in opacity.

Figure 7

Left: Scattering amplitude of the $M_{1,0,a}$ diagram where a heavy quark of momentum $p$ suffers collisional interaction with medium partons of momentum $l$ via a screened gluon of momentum $q$, resulting in the emission of a gluon of momentum $k$ from the outgoing quark. The corresponding momenta of the scattered states are denoted by $p^{\prime }$ and $l^{\prime }$. The blob represents the medium modified gluon propagator. Right: HTL loop diagram of first order in opacity corresponding to ${\mathcal{M}}_{1,0,a}$. Heavy quark scatterers from medium partons via the cut gluon propagator of momentum $q$ (with $q_0\le |\vec{\mathbf{q}}\left|\right)$ resulting in the emission of a cut gluon propagator with momentum $k$ (with $\omega >|\vec{\mathbf{k}}|$). The imaginary part of the diagram corresponds to the squared amplitude of the left diagram and is integrated over phase space.

Figure 8

Feynman diagram $M_{1,2}$ for the heavy quark radiative energy loss to first order in opacity (left) and the corresponding loop diagram ${\mathcal{M}}_{1,2}$ (right). The notations are the same as in Fig. 7 except that the emission of the gluon of momentum $k$ occurs from the virtual or exchanged gluon of momentum $q$.

Figure 9

Feynman diagram $M_{1,1}$ for the heavy quark radiative energy loss to first order in opacity computed as a product of the diagrams in Figs. 7 and 8 (left) and the corresponding loop diagram ${\mathcal{M}}_{1,1}$ (right).