2026/07/20 by Shahin Iqbal, Urs Achim Wiedemann
#hep-ph
In strongly ordered collinear limits, triple-collinear 1→ 3 splitting functions factorize into products of 1→ 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→ n splitting functions. We compute the fully differential medium-modified triple-collinear q→ q cc splitting function to first order in opacity and leading order in Nc, and show that it can be expressed as a sum of momentum-shifted vacuum q→ q cc 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 cc 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→ qg and g→ cc 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→ 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.