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

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 $θ$, beyond the widely used soft emitted-gluon approximation. Here, we study medium-induced $1 \to 2$ splittings double-differential in $z$ and $θ$, 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.

Figures (12)

• Figure 1: Graphical illustration of the transverse momenta flow in $1\to 2$ in-medium splittings. Time runs from left to right and the amplitude (upper part) is depicted together with the complex conjugate amplitude (lower part).
• Figure 2: $F_{\rm med}$ for $q \to qg$ as a function of $\theta$ in the large-$N_c$-HO approach (see section \ref{['sec:HO']}). Solid magenta curves show the full large-$N_c$–HO result, orange dashed curves correspond to the result without the non-factorizable term, and blue dotted curves show the non-factorizable term alone. The top-left and top-right panels correspond to $z=0.1$ and $z=0.5$, respectively, for the same emitter energy and medium parameters (indicated in the figure). The bottom panels show $z=0.5$ for a higher emitter energy (left) and a shorter medium length (right).
• Figure 3: Difference between $1+ F_{\rm med}$ for $q \to qg$ calculated using the SHA and large-$N_c$-HO approaches, as defined in Eq. \ref{['eq:SHA_diff']}. Both columns use $L=2$ fm and $\hat{q}=1\, {\rm GeV^2/fm}$, with $E = 200$ GeV (left) and $E = 500$ GeV (right). The bottom row shows a zoom of the top row in the region $0.1 \leq z \leq 0.9$ (or equivalently $0.10 \leq \ln 1/z \leq 2.3$). On the bottom-right panel, vertical lines indicate the $\theta$ values used for the projections in figure \ref{['fig:Fmed_qg_vs_z']}: $\theta =0.08$ (red), $\theta = 0.18$ (orange), and $\theta = 0.35$ (cyan), while the dashed horizontal line marks $z = 0.5$.
• Figure 4: Difference between $1+ F_{\rm med}$ for $q\to qg$ calculated using the ISHA and large-$N_c$-HO approaches, as defined in eq. \ref{['eq:ISHA_diff']}. The medium parameters and energies are the same as in Fig. \ref{['fig:LP_SHA']}: $L = 2$ fm, $\hat{q} = 1\,{\rm GeV^2/fm}$, and $E = 200$ GeV (left column) and $E = 500$ GeV (right column). The bottom row shows a zoom of the top row in the region $0.1 \leq z \leq 0.9$ (or equivalently $0.10 \leq \ln 1/z \leq 2.3$). On the bottom-right panel, vertical lines indicate the $\theta$ values used for the projections in figure \ref{['fig:Fmed_qg_vs_z']}: $\theta =0.08$ (red), $\theta = 0.18$ (orange), and $\theta = 0.35$ (cyan), while the dashed horizontal line marks $z = 0.5$.
• Figure 5: $F_{\rm med}$ for the $q\to qg$ splitting as a function of $\theta$, computed using the large-$N_c$-HO (magenta solid), ISHA (blue dashed) and SHA (green dash-dotted) for $z=0.5$. Each panel shows a different combination of emitter energy $E$ and medium length $L$, as indicated.