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

Abstract

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 $N\to \ell +\text{mbox}hadrons$ to determine mass ranges for which inclusive decay widths can be predicted robustly. We present novel analytic results for the decay rate $N\to \tau +\text{mbox}hadrons$ in terms of $m_{\tau }/m_N$. Our expressions equally apply to $\tau \to N+\text{mbox}hadrons$, 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 $|\sin \theta |\le 0.2$ for $m_N=600$MeV and weaker bounds for a lighter $N$. For $m_N\ge m_{\tau }$ 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 $|\sin \theta |=(9.1_{-7.8}^{+3.7})\cdot 10^{-2}$ 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 $\Gamma (\tau \to \ell +\text{mbox}nothing)$ 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 $\tau \to \ell +\text{mbox}nothing$.

Comments: 39 pages, 19 figures, 1 table, discussion in section 4.6 extended, typos corrected, matches published version