Analytical results for the C-angularity soft function at NNLO

We compute the soft function at NLO and NNLO for a one-parameter family of event shapes we call C-angularity. This family contains C-parameter as a specific choice of the parameter, in close analogy with how conventional angularity contains thrust as a special case. By construction, C-angularity and angularity coincide in the collinear limit such that the anomalous dimensions are equal. However, unlike angularity, C-angularity is a continuously differentiable function of the final state momenta, which makes the analytic calculation of the C-angularity soft function simpler. We obtain analytical results for the C-angularity soft function and anomalous dimension as an expansion in the C-angularity parameter $a$, to third and fourth order in $a$ respectively. These expansions yield results that are accurate at the few per mille level for $-1\le a < 1$.