Isospin-breaking effects in inclusive hadronic $τ$ data for the muon $(g-2)$ from first principles

The knowledge of isospin-breaking effects in hadronic $\tau$ decays is required for a high-precision determination of the Hadronic-Vacuum-Polarization contribution to $(g-2)_\mu$ from experimental $\tau$ data. In this work we present a strategy for their calculation in a fully inclusive setup from first-principles Lattice QCD+QED simulations. We separate radiative corrections in three infrared safe classes, which we study individually. We provide analytic expressions for their effects in the initial state and propose a strategy for final-state corrections directly in Euclidean space. We also examine the non-factorizable contributions and highlight the challenges associated with their analytic continuation from Euclidean to Minkowski space. By studying short-distance corrections in the context of momentum schemes, we provide a prescription for the renormalization of the individual terms at first order in the ispospin-breaking parameters.