Non-global Logarithms at 3 Loops, 4 Loops, 5 Loops and Beyond

TL;DR

This work analyzes leading non-global logarithms in the hemisphere mass observable at large $N_c$, deriving analytic coefficients up to five loops by combining strong-energy ordering with the Banfi-Marchesini-Syme equation. A hidden PSL$(2,\mathbb{R})$ symmetry revealed via stereographic projection to the Poincaré disk greatly simplifies the BMS dynamics, reducing the problem to a single invariant and enabling perturbative control with iterated integrals of uniform transcendentality. The authors compute fixed-order results up to 4 loops using Goncharov polylogarithms, symbols, and coproducts, and provide a compact 5-loop coefficient for the resummed NGL series; they also perform numerical resummation that agrees with Monte Carlo benchmarks. The results offer deep structural insight into NGLs and demonstrate a robust framework (SEO+$\,BMS$ plus modern polylogarithm techniques) for resumming leading NGLs in multi-scale jet observables, with implications for precision QCD and jet substructure phenomenology.

Abstract

We calculate the coefficients of the leading non-global logarithms for the hemisphere mass distribution analytically at 3, 4, and 5 loops at large Nc . We confirm that the integrand derived with the strong-energy-ordering approximation and fixed-order iteration of the Banfi-Marchesini-Syme (BMS) equation agree. Our calculation exploits a hidden PSL(2,R) symmetry associated with the jet directions, apparent in the BMS equation after a stereographic projection to the Poincare disk. The required integrals have an iterated form, leading to functions of uniform transcendentality. This allows us to extract the coefficients, and some functional dependence on the jet directions, by computing the symbols and coproducts of appropriate expressions involving classical and Goncharov polylogarithms. Convergence of the series to a numerical solution of the BMS equation is also discussed.

Figures (6)

• Figure 1: Both the virtual contribution (left) and real contribution (right) to the $\alpha_s$ cross section can be drawn as cut diagrams with the same topology.
• Figure 2: A stereographic projection of the jet directions onto the Poincaré disk reveals the ${\rm PSL}(2,\mathbb{R})$ symmetry of the BMS equation.
• Figure 3: Elements of ${\rm PSL}(2,\mathbb{R})$ can be visualized by their action on geodesics. Some group elements are shown.
• Figure 4: Integral contour for the $t$ integral
• Figure 5: Comparisons of the complete, resummed, leading NGL series for the hemisphere mass distribution, $g_{n \overline{\space n\space}}(L)$, to its fixed-order approximation at up to 5 loops. The resummed distribution is computed by numerically solving the BMS equation. The fixed order analytical expansions are given in Eq. \ref{['fiveloop']}. On the left, the numerical solution is labelled "resummed". The right plot shows the fixed order approximations relative to this resummed result in the region $0< {L }<2$.