Searches for a Heavy Sterile Neutrino $N$ profit from precise predictions of inclusive decay rates, entering predictions for branching fractions and lifetime. Once 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. We adopt the scenario in which $N$ only interacts with SM particles through $N$-$\nu_\ell$ mixing, where $\ell=e,\mu,\tau$. Using literature results for $W$ boson correlators calculated to $\mathcal{ O}(\alpha_s^4)$, we study the quality of the perturbation series for $N\to \ell +\mbox{ hadrons}$ to determine mass ranges for which inclusive decay widths can be predicted robustly. We present novel analytic results for the decay rate $N\to \tau +\mbox{ hadrons}$ in terms of $m_\tau/m_N$. Our expressions equally apply to $\tau \to N +\mbox{ hadrons}$, perturbatively calculable for $m_N\lesssim 600\,$MeV. Applying our result to the $\tau$ lifetime, we determine the allowed parameter space for the $N$-$\nu_\tau$ mixing angle $\theta$ and $m_N$. We find $|\sin\theta| \leq 0.2 $ for $m_N=600\,$MeV and weaker bounds for a lighter $N$. For $m_N\geq m_{ \tau}$ we find constraints from the dependence of $\tau$ decay rates on $\cos\theta$. Combining $\tau \to \pi^- \nu_\tau$ and $\tau \to K^- \nu_\tau$ data gives $|\sin\theta| = (9.09 \pm 3.56) \cdot 10^{ -2}$ while $N$-$\nu_\tau$ mixing does not improve the agreement between theory and data for $\tau \to \ell \bar\nu_\ell \nu_\tau $. We find current data for the decay rate $\Gamma(\tau \to \ell+\mbox{ nothing})$ about 1$\sigma$ above the SM prediction for $\Gamma(\tau \to \ell \bar\nu_\ell \nu_\tau)$, which leads to useful constraints on $\Gamma(\tau \to \ell X_{ \mathrm{ dark}})$ with dark-sector particles $X_{ \mathrm{ dark}}$ and might stimulate additional experimental effort on $\tau \to \ell+\mbox{ nothing}$.
Categories: hep-ph