Unraveling the collisional energy loss of a heavy quark in a quark-gluon plasma

At leading order in the QCD coupling constant, we compute the energy loss per traveling distance of a heavy quark dE/dz from elastic scattering off thermal quarks and gluons at a temperature T, including the thermal perturbative description of soft scatterings (-t < -t*) and a perturbative QCD-based calculation for hard collisions (-t > -t*). Within this soft-hard factorization model, we find that the full results of dE/dz behaves a mild sensitivity to the intermediate cutoff t*, supporting the validity of the soft-hard approach within the temperature region of interest. We rederive the analytic formula for dE/dz in the high-energy approximation, E-1 >> m(1)(2)/T, where E-1 is the injected heavy quark energy and m(1) is its mass. It is realized that the soft logarithmic contribution, dE/dz proportional to ln(-t*/m(D)(2)), arises from the t-channel scattering off thermal partons, while the hard logarithmic term, dE/dz proportional to ln[E1T/(-t*)], stems from the t-channel scattering off thermal partons, and the one dE/dz proportional to ln(E1T/m(1)(2)) comes from the s- and u-channel scattering off gluons. The sum of these contributions cancels the t*-dependence as observed in the full result. The mass hierarchy is observed dE/dz(charm) > dE/dz(bottom). Our full results are crucial for a better description of heavy quark transport in QCD medium, in particular at low and moderate energy. We also calculate the energy loss by imposing the Einstein's relationship. The related results appear to be systematically larger than that without imposing the Einstein's relationship.