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 whose distribution is characterised by two widely separated momentum scales. 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 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{\mathrm{MS}} $ coupling up to $ \mathcal{O}\left({\alpha}_s^3\right) $ . 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 τ$_{x}$ 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.