NLO radiative corrections for Forward-Backward and Left-Right Asymmetries at a B-Factory

This paper presents the first calculations of the parity-violating polarization asymmetry and forward-backward asymmetry of the $e^- e^+ \rightarrow \mu^+ \mu^- (\gamma)$ process at a center-of-mass energy of 10.58 GeV with up to one-loop electroweak radiative corrections. The calculations are relevant for future precision electroweak measurements at the Belle~II experiment, which is now collecting data at the SuperKEKB $e^{-}{e}^{+}$ collider with a center-of-mass energy at the mass of the $\mathrm{{\rm Y}}(4S)$ resonance. In this paper we take under full control the bremsstrahlung process at the conditions of Belle II/SuperKEKB, and the possibilities for a soft photon approach are discussed. The scale of the obtained relative corrections to the parity-violating and forward-backward asymmetries is significant and the scattering angle dependencies of the asymmetries is non-trivial. As an additional validation cross-check using an independent formulation, the calculated asymmetries are compared to results from the KK Monte Carlo generator.