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

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.