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

TL;DR

This paper calculates QCD corrections to heavy sterile neutrino decays and their effects on tau decays, providing new analytic formulas and constraining neutrino mixing parameters through tau decay data.

Contribution

It introduces novel analytic results for heavy neutrino decay rates into tau and hadrons, and assesses the perturbative validity for different mass ranges, impacting sterile neutrino searches.

Findings

Constraints on $N$-$ u_\tau$ mixing angle $\theta$ for $m_N \lesssim 600$ MeV.

Derived bounds on $\sin\theta$ from tau decay data.

Current tau decay measurements slightly exceed SM predictions, constraining dark-sector particle contributions.

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 $N\to \ell +\mbox{hadrons}$ 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 $\tau \to \ell+\mbox{nothing}$.