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

TL;DR

The paper delivers a perturbative QCD analysis of charged-current decays of a Heavy Sterile Neutrino with hadronic final states, leveraging $W$-boson correlators computed up to $\mathcal{O}(\alpha_s^4)$ to establish robust perturbative regions for $N\to \ell+\text{hadrons}$. It provides novel analytic expressions for the $N\to \tau+\text{hadrons}$ width in terms of $m_\tau/m_N$ and applies these results to $\tau\to N+\text{hadrons}$ ($m_N\lesssim 600\,\text{MeV}$) and to the $\tau$ lifetime to constrain the mixing angle $\theta$ and mass $m_N$. The study finds $|\sin\theta| \leq 0.2$ at $m_N=600\text{ MeV}$ and a combined hadronic-channel value $|\sin\theta|=(9.09\pm3.56)\times10^{-2}$ from $\tau\to \pi^-\nu_\tau$ and $\tau\to K^-\nu_\tau$, while leptonic tau decays alone prefer unphysical $\cos\theta>1$ values. It also shows current $\Gamma(\tau\to \ell+\text{nothing})$ data lie about 1$\sigma$ above the SM, motivating constraints on $\Gamma(\tau\to \ell X_{\text{dark}})$ and encouraging experimental efforts to explore $\tau$ decays with missing energy; the results complement LHC searches for HSNs in broader mass ranges.

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 $N$-$ν_\ell$ mixing, where $\ell=e,μ,τ$. Using literature results for $W$ boson correlators calculated to $\mathcal{O}(α_s^4)$, 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 $N\to τ+\mbox{hadrons}$ in terms of $m_τ/m_N$. Our expressions equally apply to $τ\to N +\mbox{hadrons}$, perturbatively calculable for $m_N\lesssim 600\,$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 $|\sinθ| \leq 0.2 $ for $m_N=600\,$MeV and weaker bounds for a lighter $N$. For $m_N\geq m_τ$ we find constraints from the dependence of $τ$ decay rates on $\cosθ$. Combining $τ\to π^- ν_τ$ and $τ\to K^- ν_τ$ data gives $|\sinθ| = (9.09 \pm 3.56) \cdot 10^{-2}$ while $N$-$ν_τ$ mixing does not improve the agreement between theory and data for $τ\to \ell \barν_\ell ν_τ$. We find current data for the decay rate $Γ(τ\to \ell+\mbox{nothing})$ about 1$σ$ above the SM prediction for $Γ(τ\to \ell \barν_\ell ν_τ)$, which leads to useful constraints on $Γ(τ\to \ell X_{\mathrm{dark}})$ with dark-sector particles $X_{\mathrm{dark}}$ and might stimulate additional experimental effort on $τ\to \ell+\mbox{nothing}$.

Figures (19)

• Figure 1: Tree diagrams for an HSN decay. For $\alpha=\beta$ both diagrams interfere. Diagrams were drawn with the help of TikZ-Feynman Ellis:2016jkw.
• Figure 2: Decay width of an HSN into leptons. Here the notation $N \rightarrow \ell$ means the plotted line is the sum of all partial widths associated with the tagging lepton i.e. $\Gamma(N \rightarrow e) = \sum_\ell \Gamma(N \rightarrow e \bar{\ell} \nu_\ell)$. For all three channels the mixing angle is chosen as $V_{N\ell} = 10^{-3}$.
• Figure 3: Sample non-singlet diagram to $\mathcal{O}(\alpha_S^2)$.
• Figure 4: One possible cut of the loop diagram in Fig. \ref{['fig:nonsingelt']}. This corresponds to the decay amplitude $W \rightarrow q \bar{q}$ at $\mathcal{O}(\alpha_s^2)$ interfering with the tree-level amplitude.
• Figure 5: Hadronic decay widths $\Gamma(N \rightarrow \ell X)$ for the tagging leptons $\ell = \mu, \tau$ with a mixing angle of $V_{N\ell} = 10^{-3}$. The width for $\ell=e$ looks like the $\ell = \mu$ width. The numerics of the running coupling were calculated with the help of RunDecChetyrkin:2000ytHerren:2017osy. We do not show the region $m_N \in [m_\mu + m_D , 3GeV]$ where we expect large non-perturbative effects. The error band is taken as the difference between minimum and maximum of the width with respect to the renormalization scale $\sigma_\Gamma = (\max \, \Gamma_N(\mu) - \min \, \Gamma_N(\mu))/2$ for $0.8GeV \leq \mu \leq 3.5GeV$. We do not include the uncertainty from our omission of quark masses, which can be sizable, $\mathcal{O}(m_D^2/q^2) = \mathcal{O}(40\%)$ for $m_N = 3GeV$, beyond the charm production threshold.