Abstract
We calculate the coefficients of the leading nonglobal logarithms for the hemisphere mass distribution analytically at 3, 4, and 5 loops at large ${N}_{c}$. 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 $\mathrm{PSL}(2,\mathbb{R})$ symmetry associated with the jet directions, apparent in the BMS equation after a stereographic projection to the Poincar\'e 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.
Highlights
Jet substructure is playing an increasingly prominent role in the physics of hadron collisions, at the LHC [1,2,3,4,5,6,7,8]
While simplifications arising from the strong-energy-ordering (SEO) limit have been known for decades, we try to provide more explicit details than we have found in the literature
Integrating Eq (62) we find that there is no O(L) term in gab(L), consistent with the leading non-global logarithm starting at 2 loops
Summary
Jet substructure is playing an increasingly prominent role in the physics of hadron collisions, at the LHC [1,2,3,4,5,6,7,8]. The leading NGLs (terms of the form (αL)n) can be reproduced within the strong-energy-ordering approximation to QCD This approximation leads to simplified cross sections, at large Nc, and allows for straightforward resummation using Monte-Carlo (MC) simulation [37]. To calculate the n-loop leading NGL, we can iterate the BMS equation to produce the correct integrand This iteration is equivalent to, but significantly simpler summing the relevant real, virtual, and real-virtual contributions in the strongly ordered limit and subtracting the global contribution.
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