Non-global logarithms at finite Nc beyond leading order

TL;DR

The paper delivers a comprehensive finite-$N_c$ analysis of non-global logarithms for the hemisphere mass in $e^+e^-\to$ di-jets, computing the series through four loops in full and partially at five loops using an eikonal approach to soft gluon emissions and ColorMath for colour algebra. It demonstrates an exponentiation pattern for NGLs, decomposing the hemisphere-mass distribution into a Sudakov factor and a non-global factor, and analyzes the interplay between reducible and irreducible (monster) contributions, including finite-$N_c$ corrections that are numerically small but theoretically important. The work cross-checks with large-$N_c$ results and all-orders numerical solutions, discusses the validity of the large-$N_c$ approximation, and proposes an all-orders exponential form for the NGLs factor with a specific color-structure dependence. The findings provide a pathway toward improved resummation of non-global effects in jet substructure observables and quantify finite-$N_c$ effects relevant for precision QCD predictions.

Abstract

We analytically compute non-global logarithms at finite Nc fully up to 4 loops and partially at 5 loops, for the hemisphere mass distribution in e+e- to di-jets to leading logarithmic accuracy. Our method of calculation relies solely on integrating the eikonal squared-amplitudes for the emission of soft energy-ordered real-virtual gluons over the appropriate phase space. We show that the series of non-global logarithms in the said distribution exhibits a pattern of exponentiation thus confirming - by means of brute force - previous findings. In the large-Nc limit, our results coincide with those recently reported in literature. A comparison of our proposed exponential form with all-orders numerical solutions is performed and the phenomenological impact of the finite-Nc corrections is discussed.

Figures (5)

• Figure 1: Schematic diagram for an outgoing $q\bar{q}$ pair associated with multiple gluon emission. The measured hemisphere is the one pointing in the quark direction ($\mathcal{H}_R$).
• Figure 2: Plot of the ratio $\Sigma^{\mathrm{NG}}(\rho)/\exp(\Sigma_2^{\mathrm{NG}}(\rho))$ in terms of the logarithm $\bar{L}= \alpha_s/\pi \ln(1/\rho)$.
• Figure 3: Plot of the NGLs function $\Sigma^{\mathrm{NG}}(\rho)$ at large (left) and finite (right) $N_c$.
• Figure 4: Plot of the ratio $\Sigma^{\mathrm{NG}}(\rho)/\Sigma_{\mathrm{DS}}^{\mathrm{NG}}(\rho)$ for both large (left) and finite (right) $N_c$.
• Figure 5: Plot of the NGLs function $\Sigma^{\mathrm{NG}}(t)$ at large $N_c$ for the range $0 \leq t \leq 0.14$.