NLO Oriented Event-Shape Distributions for Massive Quarks

In this article we compute the cross section for the process $e^+e^- \to Q\overline Q+X$, with $Q$ a heavy quark, differential in a given event shape $e$ and the angle $\theta_T$ between the thrust axis and the beam direction. These observables are usually referred to as oriented event shapes, and it has been shown that the $\theta_T$ dependence can be split in two structures, dubbed the unoriented and angular terms. Since the unoriented part is already known, we compute the differential and cumulative distributions in fixed-order for the angular part up to $\mathcal{O}(\alpha_s)$. Our results show that, for the vector current, there is a non-zero $\mathcal{O}(\alpha_s^0)$ contribution, in contrast to the axial-vector current or for massless quarks. This entails that for the vector current one should expect singular terms at $\mathcal{O}(\alpha_s)$ as well as infrared divergences in real- and virtual-radiation diagrams that should cancel when added up. On the phenomenological side, and taking into account that electroweak factors enhance the vector current, it implies that finite bottom-mass effects are an important correction since they are not damped by a power of the strong coupling and therefore cannot be neglected in precision studies. Finally, we show that the total angular distribution for the vector current has a Sommerfeld enhancement at threshold.