Full energy fraction and angular dependence of medium-induced splittings in the large-$N_c$ limit

TL;DR

This paper provides an analytical and semi-analytical study of medium-induced parton splittings in heavy-ion collisions, emphasizing the full dependence on energy fraction and angle, and introduces an improved semi-hard approximation.

Contribution

It derives a semi-analytical expression for medium-induced splittings with full $z$ and $ heta$ dependence in the large-$N_c$ limit, extending the semi-hard approximation with leading corrections.

Findings

The large-$N_c$-HO approach yields analytical results for medium-induced splittings.

The semi-hard approximation is unreliable across most of phase space.

The improved semi-hard approximation (ISHA) is robust for sufficiently energetic partons.

Abstract

Jets produced in relativistic heavy-ion collisions are modified by their interactions with the quark-gluon plasma (QGP), making jet substructure observables sensitive probes of QGP dynamics. A quantitative description of these modifications requires understanding how the medium affects elementary parton splittings with full dependence on both their energy fraction $z$ and splitting angle $\theta$, beyond the widely used soft emitted-gluon approximation. Here, we study medium-induced $1 \to 2$ splittings double-differential in $z$ and $\theta$, with full resummation of multiple scatterings, and show that in the large-$N_c$ limit and under the harmonic oscillator (HO) approximation, all path integrals can be evaluated analytically for any splitting channel, providing a computationally efficient semi-analytical result. We also revisit the semi-hard approximation (SHA), extending it to include leading corrections in inverse powers of the partons energies, which we denote the improved semi-hard approximation (ISHA), and assess its validity through a comparison with the large-$N_c$-HO results. Our analysis shows that while the SHA is found to be unreliable across most of phase space, even for high-energy emitters, the ISHA provides a robust approximation for splittings where all partons are sufficiently energetic.