Energy losses in relativistic plasmas: QCD versus QED

We review the problem of evaluating the energy loss of an ultrarelativistic charged particle crossing a thermally equilibrated high-temperature $\mathrm{e}^{+}\mathrm{e}^{-}$ or quark–gluon plasma. The average energy loss $\Delta E$ depends on the particle energy $E$ and mass $M$, the plasma temperature $T$, the QED (QCD) coupling constant $\alpha$ ($\alpha_s$), and the distance $L$ the particle travels in the medium. Two main mechanisms contribute to the energy loss: elastic collisions and bremsstrahlung. For each contribution, we use simple physical arguments to obtain the functional dependence $\Delta E(E,M,T,\alpha_{(s)},L)$ in different regions of the parameters. The suppression of bremsstrahlung due to the Landau – Pomeranchuk – Migdal effect is relevant in some regions. In addition, radiation by heavy particles is often suppressed for kinematical reasons. Still, when the travel distance $L$ is not too small, and for large enough energies $(E\gg M^2/(\alpha T)$ in the Abelian case and $E\gg M/\sqrt{\alpha_s}$ in the non-Abelian case), radiative losses dominate over collisional ones. We rederive the known results and make some new observations. In particular, we emphasize that for light particles $(m^2\ll\alpha T^2)$, the difference in the behavior of $\Delta E(E,M,T,\alpha_{(s)},L)$ in QED and QCD is mostly due to the different way the problem is usually posed in these two cases. In QED, it is natural to study the energy losses of an electron coming from infinity. In QCD, the quantity of physical interest is the medium-induced energy loss of a parton produced within the medium. In the case of an electron produced within a QED plasma, the behavior of medium-induced radiative energy loss in $\Delta E_{\mathrm{rad}}$ is similar to the behavior of $\Delta E$ in QCD (in particular $\Delta E_{\mathrm{rad}}\propto L^2$ at small $L$), despite the photon and gluon radiation spectra being drastically different because the bremsstrahlung cones for soft gluons are broader than for soft photons. We also show that the average radiative loss of an “asymptotic light parton” crossing a QCD plasma is similar to that of an asymptotic electron crossing a QED plasma. For heavy particles $(M^2\gg\alpha T^2)$, the difference between $\Delta E_{\mathrm{rad}}$ in QED and in QCD is more pronounced, even when the same physical situation is considered.