QCD corrections to charged-current decays with Heavy Sterile Neutrinos in initial or final state and their impact on τ decays

Searches for a Heavy Sterile Neutrino N profit from precise predictions of inclusive decay rates, which enter predictions for branching fractions and lifetime. Once the decay channels into semi-hadronic final states are open, a reliable calculation of inclusive decay rates is only possible if N is heavy enough to permit a perturbative calculation, in analogy to the well-known case of semi-hadronic τ lepton decays. We adopt the popular scenario in which N only interacts with the SM particles through N-ν_ℓ mixing, where ℓ = e, μ, τ. Using literature results for W boson correlators calculated to fourth order in the strong coupling α_s, we study the quality of the perturbation series for N → ℓ + hadrons to determine the mass ranges for which inclusive decay widths can be predicted in a robust way. We present novel analytic results for the decay rate of N → τ + hadrons in terms of m_τ/m_N. Our expressions equally apply to τ → N + hadrons, the width of which is found to be perturbatively calculable for m_N ≲ 600 MeV. Applying our result to the τ lifetime, we determine the allowed parameter space for the N-ν_τ mixing angle θ and m_N. We find |sin θ| ≤ 0.2 for m_N = 600 MeV and weaker bounds for a lighter N. In the mass region m_N ≥ m_τ we find constraints from the dependence of τ decay rates on cos θ, with |sin θ| ≤ 0.12 inferred from the τ lifetime. Combining τ → π⁻ντ and τ → K⁻ν_τ data gives $\left|sin\theta \right|=\left({9.1}_{-7.8}^{+3.7}\right)·{10}^{-2}$ \( \left|\sin \theta \right|=\left({9.1}_{-7.8}^{+3.7}\right)\cdotp {10}^{-2} \) at 1σ while N-ν_τ mixing does not improve the agreement between theory and data for $\tau \to \ell {\overline{\nu }}_{\ell }{\nu }_{\tau }$ \( \tau \to \ell {\overline{\nu}}_{\ell }{\nu}_{\tau } \) . We find current data for the decay rate Γ(τ → ℓ + nothing) about 1σ above the SM prediction for $\mathrm{\Gamma }\left(\tau \to \ell {\overline{\nu }}_{\ell }{\nu }_{\tau }\right)$ \( \Gamma \left(\tau \to \ell {\overline{\nu}}_{\ell }{\nu}_{\tau}\right) \) , which leads to useful constraints on Γ(τ → ℓX_dark) or Γ(τ → ℓX_darkX_dark) with dark-sector particles X_dark and might stimulate additional experimental effort on τ → ℓ + nothing.