Bridging massive and massless schemes for soft gluon resummation in heavy-flavour production in $e^+e^-$ collisions

Perturbative calculations for processes involving heavy flavours can be carried out using two approaches: the massive and the massless schemes. These schemes can also be combined to leverage their respective strengths. Additionally, both massive and massless frameworks can be supplemented by soft-gluon resummation. However, matching resummed calculations across the two schemes presents significant challenges, primarily due to the non-commutativity of the soft and small mass limits. The consistent resummation of mass and soft logarithms has been recently achieved at next-to-leading logarithmic (NLL) accuracy. In this paper, we consider heavy-quark fragmentation functions in electron-positron collisions and we extend this framework to achieve the so-called NLL$^\prime$ accuracy, which accounts for finite terms in the soft limit.