Are in-medium quark-gluon showers strongly coupled? Results in the large-$N_f$ limit

Inside a medium, showers originating from a very high energy particle develop via medium-induced splitting processes such as bremsstrahlung and pair production. During shower development, two consecutive splittings sometimes overlap quantum mechanically, so that they cannot be treated independently. Some of these effects can be absorbed into an effective value of a medium parameter known as $\hat q$. Previous calculations (with certain simplifying assumptions) have found that, after adjusting the value of $\hat q$, the leftover effect of overlapping splittings is quite small for purely gluonic large-$N_c$ showers but is very much larger for large-$N_f$ QED showers, at comparable values of $N\alpha$. Here, by investigating the same problem for QCD with *quarks* in the large-$N_f$ limit of many quark flavors, we make a first study of whether the small effect in purely gluonic showers (i) was merely an accident or (ii) is more broadly characteristic of such overlap effects in QCD. We also offer a qualitative explanation for the large size of the effect in QED vs. QCD.