All-order equality of inclusive and full cusp anomalous dimensions

Prove that, to all perturbative orders, the inclusive soft function for a generalised angularity event shape captures the complete cusp contribution of the full soft function, so that the coefficients satisfy \(\Gamma^n_{e_I}=\Gamma^n_e=4F_e^{[-1]}\Gamma^n_{\mathrm{cusp}}\) for every order \(n\).

Background

The paper decomposes the NNLO soft function for a generalised angularity into an inclusive soft function, whose measurement depends on the total emitted momentum, and a correction accounting for the difference between the inclusive and full measurements. At one and two loops, the inclusive contribution reproduces the cusp anomalous dimension of the full soft function, with the universal rapidity-tail coefficient $F_e^{[-1]}$.

The authors extend this observed NNLO relation to a conjecture at every perturbative order. Establishing it would show that the cusp anomalous dimension is determined entirely by the forward angularity behaviour and is independent of the event shape's central soft-region rapidity weighting.