Next-to-leading non-global logarithms in QCD

Non-global logarithms arise from the sensitivity of collider observables to soft radiation in limited angular regions of phase space. Their resummation to next-to-leading logarithmic (NLL) order has been a long standing problem and its solution is relevant in the context of precision all-order calculations in a wide variety of collider processes and observables. In this article, we consider observables sensitive only to soft radiation, characterised by the absence of Sudakov double logarithms, and we derive a set of integro-differential equations that describes the resummation of NLL soft corrections in the planar, large-$N_c$ limit. The resulting set of evolution equations is derived in dimensional regularisation and we additionally provide a formulation that is manifestly finite in four space-time dimensions. The latter is suitable for a numerical integration and can be generalised to treat other infrared-safe observables sensitive solely to soft wide-angle radiation. We use the developed formalism to carry out a fixed-order calculation to ${\cal O}(\alpha_s^2)$ in full colour for both the transverse energy and energy distribution in the interjet region between two cone jets in $e^+e^-$ collisions. We find that the expansion of the resummed cross section correctly reproduces the logarithmic structure of the full QCD result.