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 mN.
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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−ν$_τ$ and τ → K−ν$_τ$ data gives
|sin θ| =(9.1$^{+3.7}_{−7.8}$)· 10$^{−2}$ at 1σ while N-ν$_τ$ mixing does not improve the agreement between
theory and data for τ → ℓ$\overline{ν}$$_ℓ$ν$_τ$ . We find current data for the decay rate Γ(τ → ℓ + nothing)
about 1σ above the SM prediction for Γ(τ → ℓ$\overline{ν}$$_ℓ$ν$_τ$), which leads to useful constraints on
Γ(τ → ℓX$_{dark}$) or Γ(τ → ℓX$_{dark}$X$_{dark}$) with dark-sector particles X$_{dark}$ and might stimulate
additional experimental effort on τ → ℓ + nothing.