Oriented event shapes at N³LL + $\mathcal{O}\left( {\alpha_{\mathrm{S}}^2} \right)$

Abstract

We analyze oriented event-shapes in the context of Soft-Collinear Effective Theory (SCET) and in fixed-order perturbation theory. Oriented event-shapes are distributions of event-shape variables which are differential on the angle θ _T that the thrust axis forms with the electron-positron beam. We show that at any order in perturbation theory and for any event shape, only two angular structures can appear: F ₀ = 3/8 (1 + cos² θ _T ) and F ₁ = (1 − 3 cos² θ _T ). When integrating over θ _T to recover the more familiar event-shape distributions, only F ₀ survives. The validity of our proof goes beyond perturbation theory, and hence only these two structures are present at the hadron level. The proof also carries over massive particles. Using SCET techniques we show that singular terms can only arise in the F ₀ term. Since only the hard function is sensitive to the orientation of the thrust axis, this statement applies also for recoil-sensitive variables such as Jet Broadening. We show how to carry out resummation of the singular terms at N³LL for Thrust, Heavy-Jet Mass, the sum of the Hemisphere Masses and C-parameter by using existing computations in SCET. We also compute the fixed-order distributions for these event-shapes at $\mathcal{O}\left( {{\alpha_{\mathrm{S}}}} \right)$ analytically and at $\mathcal{O}\left( {\alpha_{\mathrm{S}}^2} \right)$ with the program Event2.