# A complete set of in-medium splitting functions to any order in opacity

## TL;DR

This paper derives comprehensive in-medium parton splitting functions to all orders in opacity, incorporating flavor and mass effects, with numerical results indicating significant corrections to jet substructure observables in heavy ion collisions.

## Contribution

It provides the first complete calculation of medium-induced splitting functions to any order in opacity, including explicit second order solutions and numerical analysis.

## Key findings

- Second order corrections significantly affect energy dependence of in-medium showers.
- Splitting kernels show notable modifications in longitudinal and angular distributions.
- Results have implications for jet substructure studies in heavy ion collisions.

## Abstract

In this Letter we report the first calculation of all ${\cal O}(\alpha_s)$ medium-induced branching processes to any order in opacity. Our splitting functions results are presented as iterative solutions to matrix equations with initial conditions set by the leading order branchings in the vacuum. The flavor and quark mass dependence of the in-medium $q \rightarrow qg$, $g\rightarrow gg$, $q \rightarrow g q$, $g \rightarrow q\bar{q}$ processes is fully captured by the light-front wavefunction formalism and the color representation of the parent and daughter partons. We include the explicit solutions to second order in opacity as supplementary material and present numerical results in a realistic strongly-interacting medium produced in high center-of-mass energy heavy ion collisions at the Large Hadron Collider. Our numerical simulations show that the second order in opacity corrections can change the energy dependence of the in-medium shower intensity. We further find corrections to the longitudinal and angular distributions of the in-medium splitting kernels that may have important implications for jet substructure phenomenology.