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We present a procedure to calculate the Sudakov radiator for a genericrecursive 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 ofobservables in the {\tt ARES} method for processes with two emitters at theBorn level. As a byproduct, we define a physical coupling in the soft limit,and we provide an explicit expression for its relation to the $\overline{\rmMS}$ coupling up to ${\cal O}(\alpha_s^3)$. This physical coupling constitutesone of the ingredients for a NNLL accurate parton shower algorithm. As anapplication we obtain analytic NNLL results, of which several are new, for allangularities $\tau_x$ defined with respect to both the thrust axis and thewinner-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 somevalues of $x$, an accurate prediction of the peak of the differentialdistribution requires a simultaneous resummation of the logarithmic termsoriginating from the two-jet limit and at the Sudakov shoulder.

The Sudakov radiator for jet observables and the soft physical coupling