The Sudakov radiator for jet observables and the soft physical coupling

We present a procedure to calculate the Sudakov radiator for a generic recursive infrared and collinear (rIRC) safe observable in two-scale problems. We give closed formulae for the radiator at next-to-next-to-leading-logarithmic (NNLL) accuracy, which completes the general NNLL resummation for this class of observables in the {\tt ARES} method for processes with two emitters at the Born level. As a byproduct, we define a physical coupling in the soft limit, and we provide an explicit expression for its relation to the $\overline{\rm MS}$ coupling up to ${\cal O}(\alpha_s^3)$. This physical coupling constitutes one of the ingredients for a NNLL accurate parton shower algorithm. As an application we obtain analytic NNLL results, of which several are new, for all angularities $\tau_x$ defined with respect to both the thrust axis and the winner-take-all axis, and for the moments of energy-energy correlation $FC_x$ in $e^+e^-$ annihilation. For the latter observables we find that, for some values of $x$, an accurate prediction of the peak of the differential distribution requires a simultaneous resummation of the logarithmic terms originating from the two-jet limit and at the Sudakov shoulder.