Factorization of the triple-collinear $q \to qc\bar{c}$ splitting function at first order in opacity

In strongly ordered collinear limits, triple-collinear $1\to 3$ splitting functions factorize into products of $1\to 2$ splitting functions. Here, we investigate whether, and under which conditions, this factorization persists in a finite QCD medium. To this end, we reformulate the opacity expansion as a medium-modification of the compact Catani-Grazzini representation of collinear $1\to n$ splitting functions. We compute the fully differential medium-modified triple-collinear $q\to q c\bar{c}$ splitting function to first order in opacity and leading order in $N_c$, and show that it can be expressed as a sum of momentum-shifted vacuum $q\to q c\bar{c}$ splitting functions multiplied by simple interference factors. In vacuum, this splitting function has a single strongly ordered collinear limit, in which the invariant mass of the outgoing $c\bar{c}$ pair is much smaller than all other invariant masses. The medium-modified splitting function instead exhibits three distinct strongly ordered limits, depending on how the relevant invariant masses -- multiplied by a boost factor and interpretable as inverse formation times -- compare with the medium length. Remarkably, in all three cases the triple-collinear splitting factorizes also in the medium into a tensor product of $q\to qg$ and $g\to c\bar{c}$ splitting functions. In two of these limits, the latter are medium modified to first order in opacity. This first proof shows that factorization extends to medium-modified triple-collinear splitting functions. Lastly, we relate our results to the longstanding problem of overlapping formation times and discuss their implications for jet-quenching parton showers based on medium-modified $1\to 2$ splittings. We also outline further applications of our formalism and note that our results may inform future phenomenological studies of spin (de)correlations in final-state radiation in dense QCD matter.