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

Technical Summary: QCD Corrections to Charged-Current Decays with Heavy Sterile Neutrinos

Problem and Motivation
Heavy Sterile Neutrinos (HSNs), also known as Heavy Neutral Leptons, are a well-motivated extension of the Standard Model (SM) addressing issues such as neutrino masses, leptogenesis, and Dark Matter. In minimal scenarios, HSNs ($N$) interact with SM particles solely through mixing with active neutrinos ($\nu_\ell$), parameterized by a mixing angle $\theta$ (where $V_{N\ell} = \sin \theta$). The search for HSNs and the interpretation of experimental limits rely heavily on precise predictions of branching fractions and lifetimes.

A critical challenge arises when HSNs are heavy enough to decay into semi-hadronic final states ($N \to \ell + \text{hadrons}$). In this regime, a reliable calculation of inclusive decay rates requires perturbative Quantum Chromodynamics (QCD). However, the validity of perturbation theory depends on the mass of the HSN ($m_N$). If the invariant mass of the hadronic system is too small (near resonance regions) or if $m_N$ is just above heavy meson thresholds (e.g., $D$ or $B$ mesons), non-perturbative effects dominate, rendering standard perturbative calculations unreliable. This paper aims to determine the mass ranges where perturbative QCD provides robust predictions for inclusive HSN decay widths and to apply these results to constrain the $N-\nu_\tau$ mixing angle using $\tau$ lepton data.

Methodology
The authors adopt the scenario where the HSN interacts only via $N-\nu_\ell$ mixing. The calculation proceeds in two main stages:

1. Theoretical Framework:

   • The inclusive decay width $\Gamma(N \to \ell + \text{hadrons})$ is related to the imaginary part of the self-energy of the decaying particle via the optical theorem.
   • The non-trivial QCD dynamics are encoded in the correlator of two charged quark currents ($\Pi^{(1+0)}_{V/A}$). The authors utilize existing literature results for these correlators calculated up to the fourth order in the strong coupling constant $\alpha_s$ (five-loop level for the underlying gauge boson correlators).
   • The authors perform a new calculation for the specific case where the final state lepton is a $\tau$ ($N \to \tau + \text{hadrons}$), accounting for the non-zero $\tau$ mass ($m_\tau \neq 0$). This involves deriving novel analytic expressions for the phase-space integrals in terms of the ratio $m_\tau/m_N$.
   • The results are expressed using polylogarithms (up to $\mathcal{O}(\alpha_s^3)$) and a convergent series representation (for $\mathcal{O}(\alpha_s^4)$).
   • The analysis employs the chiral limit for light quarks and assumes $q^2 \gg m_q^2$ for the validity of the perturbative expansion.

2. Numerical Analysis and Stability Checks:

   • The authors analyze the convergence of the perturbative series by examining the dependence of the decay width on the unphysical renormalization scale $\mu$.
   • They compare the decay widths at different orders of $\alpha_s$ to identify the mass ranges where higher-order corrections stabilize the prediction (i.e., where the scale dependence decreases).
   • The study covers both $N \to \ell + \text{hadrons}$ (for $m_N > m_\ell$) and the inverse process $\tau \to N + \text{hadrons}$ (for $m_N < m_\tau$).

Key Contributions

• Novel Analytic Results: The paper presents new closed-form analytic results for the decay rate $\Gamma(N \to \tau + \text{hadrons})$ and $\Gamma(\tau \to N + \text{hadrons})$, explicitly including the lepton mass effects which are significant for $\tau$ decays.
• Perturbative Validity Range: By analyzing the scale dependence and convergence of the series up to $\mathcal{O}(\alpha_s^4)$, the authors determine the specific mass windows where perturbative QCD is reliable:
  • For final state leptons $e$ or $\mu$: Reliable predictions are possible for $m_N \gtrsim 1.5$ GeV.
  • For final state lepton $\tau$: Reliable predictions require $m_N \gtrsim 3$ GeV due to the larger phase-space suppression and kinematic effects of the $\tau$ mass.
  • For $\tau \to N + \text{hadrons}$ (where $N$ is produced in $\tau$ decay): The perturbative series is found to be stable for $m_N \lesssim 600$ MeV.
• Constraints on Mixing Angles: The authors apply their results to constrain the mixing angle $\theta_{N\tau}$ using current experimental data on the $\tau$ lifetime and branching fractions.

Results

• Perturbation Series Quality: The study confirms that for $m_N \approx 1.5$ GeV (for $e/\mu$) and $m_N \approx 3$ GeV (for $\tau$), the inclusion of $\mathcal{O}(\alpha_s^4)$ corrections significantly reduces the renormalization scale dependence, validating the use of perturbative QCD. Conversely, in the region $m_N \in [m_\mu + m_D, 3 \text{ GeV}]$, non-perturbative charm resonance effects dominate, making perturbative predictions unreliable.
• Constraints from $\tau$ Lifetime:
  • For $m_N \lesssim 600$ MeV, the authors derive constraints on $|\sin \theta|$ based on the $\tau$ lifetime. They find $|\sin \theta| \le 0.2$ for $m_N = 600$ MeV, with weaker bounds for lighter $N$.
  • For $m_N \ge m_\tau$, the decay rate depends on $\cos \theta$. Using the $\tau$ lifetime, they infer $|\sin \theta| \le 0.12$ at 1$\sigma$.
• Constraints from Branching Fractions:
  • Analyzing semi-hadronic decays $\tau \to \pi^- \nu_\tau$ and $\tau \to K^- \nu_\tau$, the authors find that data allows for a non-zero mixing angle.
  • Combining $\tau \to \pi^- \nu_\tau$ and $\tau \to K^- \nu_\tau$ data yields $|\sin \theta| = (9.1^{+3.7}_{-7.8}) \cdot 10^{-2}$ at 1$\sigma$.
  • Leptonic decays ($\tau \to \ell \bar{\nu}_\ell \nu_\tau$) currently show central values slightly above SM predictions. While this could imply a non-zero mixing angle, the authors note this might also be due to decays into dark sector particles ($X_{\text{dark}}$).
• Spectral Distributions: The paper provides expressions for the charged lepton energy spectrum in $\tau \to \ell \nu_\ell N$ and the hadronic invariant mass spectrum in $\tau \to N + \text{hadrons}$, offering tools for future experimental detection of HSNs via spectral distortions.

Significance and Claims
The paper claims to provide the necessary theoretical tools to make robust predictions for inclusive HSN decay widths in the perturbative regime, filling a gap in the literature regarding the specific kinematics of $\tau$-involving channels.

• Robustness: The authors emphasize that their results define the mass ranges where theoretical uncertainties are under control, which is crucial for interpreting experimental limits on HSNs.
• Model Independence: The constraints derived from $\tau$ lifetime and semi-hadronic branching fractions are presented as less model-dependent than those from collider experiments (like DELPHI or CMS) which often assume specific decay modes of the HSN. The authors note that if the HSN decays into dark sector particles, bounds from experiments relying on visible HSN decays would weaken, whereas the $\tau$-based constraints remain valid as they rely on total width modifications.
• Experimental Implications: The authors suggest that the current data on $\Gamma(\tau \to \ell + \text{nothing})$ being $\sim 1\sigma$ above the SM prediction warrants further experimental investigation, potentially to distinguish between HSN mixing and decays into dark sector particles. They do not claim a discovery of HSNs but highlight the tension in the data that could motivate additional experimental effort.

In summary, this work establishes the theoretical foundation for using inclusive semi-hadronic decays to probe Heavy Sterile Neutrinos, delineates the limits of perturbative QCD applicability in this context, and derives updated constraints on the $N-\nu_\tau$ mixing angle using precision $\tau$ data.