Revisiting lepton flavor violation: $τ$ and meson decays

This paper revisits charged lepton flavor violation in the minimal type-I seesaw model using updated data, revealing that certain semileptonic tau decays can dominate over purely leptonic channels in specific parameter regions while heavy-meson decays remain experimentally inaccessible.

The Gist

The Big Picture: A Broken Recipe

Imagine the Standard Model of physics as a giant, mostly perfect cookbook. For decades, it explained how particles behave, but it had one glaring missing ingredient: neutrino mass. The cookbook said neutrinos should be weightless, but experiments proved they have a tiny bit of mass.

To fix this, physicists added a new, invisible ingredient to the recipe called Heavy Neutral Leptons (HNLs). Think of HNLs as "ghost chefs" in the kitchen. They are heavy, invisible, and rarely interact with anything, but their presence explains why neutrinos have mass.

However, adding these ghost chefs creates a side effect: Charged Lepton Flavor Violation (cLFV). In the normal world, a "tau" particle (a heavy cousin of the electron) should only turn into other taus. But with the ghost chefs around, a tau might accidentally turn into a muon or an electron, which is strictly forbidden in the original recipe. This paper is a detective story about finding evidence of these "accidents."

The Investigation: Two Types of Crime Scenes

The authors looked for these flavor-changing accidents in two different types of "crime scenes":

1. The "Pure" Crime Scenes (Leptonic Decays): This is where a tau particle turns directly into lighter particles like electrons or muons, sometimes with a flash of light (a photon). These have been studied for a long time.

   • Analogy: This is like watching a magician pull a rabbit out of a hat. It's a direct, clean trick.

2. The "Messy" Crime Scenes (Semileptonic Decays): This is where a tau particle turns into a lighter particle plus a meson (a particle made of quarks, like a pion or a rho).

   • Analogy: This is like the magician pulling a rabbit out of a hat, but the hat is also full of confetti, streamers, and a small toy car. It's a messy, complex trick that involves more moving parts.

The Surprise Discovery

For over 30 years, scientists mostly ignored the "messy" crime scenes (the tau decays involving mesons) because they were thought to be too rare to ever see. They focused entirely on the "pure" tricks (like $\tau \to 3\ell$ or $\tau \to \ell\gamma$).

The paper's main finding is a plot twist:
In certain scenarios, the "messy" tricks are actually more common than the "pure" ones.

• Specifically, the decay where a tau turns into a muon/electron and a rho meson ($\tau \to \ell\rho$) can happen more often than the famous "pure" decays.
• In fact, in some parts of the theoretical parameter space, $\tau \to \ell\rho$ is the most likely place to find evidence of these ghost chefs, beating out even the decay into a photon ($\tau \to \ell\gamma$).

Why the "Messy" Tricks Win

Why would the complex, messy decays be more common?

• The Phase Space Effect: Imagine trying to fit three people into a small car (a 3-body decay like $\tau \to 3\ell$). It's tight and difficult. Now imagine fitting two people and a small suitcase (a 2-body decay like $\tau \to \ell\rho$). It's much easier to arrange.
• The paper calculates that the "suitcase" (the meson) helps the process happen more efficiently than the "three people" scenario, especially when the ghost chefs (HNLs) are very heavy.

The "Ghost" vs. The "Heavy"

The paper also explores how heavy these ghost chefs are.

• Light Ghosts: If the HNLs are very light, the universe acts like a perfect filter. The "accidents" cancel each other out, and nothing happens.
• Heavy Ghosts: If the HNLs are very heavy (much heavier than the particles we usually see), they don't just disappear from the equation. Instead, they leave a lingering "echo" that actually makes the messy decays ($\tau \to \ell\rho$) grow stronger. This is counter-intuitive; usually, heavy things in physics loops cancel out, but here, their heaviness helps the signal grow.

The Verdict: Can We See Them?

The authors compared their predictions against what current and future experiments can see.

1. Meson Decays (The "Pure" Meson Crimes): They looked at mesons (like pions or J/Psi particles) turning into different leptons.

   • Result: These are incredibly rare. The paper predicts they are so suppressed (like trying to hear a whisper in a hurricane) that even our most sensitive future detectors (like BES-III or Belle-II) will likely never see them. They are "far below experimental sensitivity."

2. Tau Decays (The "Messy" Tau Crimes):

   • Result: This is the exciting part. The predicted rate for $\tau \to \ell\rho$ and $\tau \to \ell\pi$ is right on the edge of what the Belle-II experiment (a massive particle detector in Japan) might be able to see.
   • If Belle-II sees a tau turning into a muon and a rho meson, it could be the first direct evidence of these Heavy Neutral Leptons.

Summary

This paper is a call to action for experimentalists. It says: "Stop looking only at the clean, simple decays. Look at the messy ones where taus turn into muons and rho mesons. That's where the ghost chefs are most likely to leave a trace."

While the "light meson" decays are too faint to ever hope to catch, the "tau-to-rho" decays are a golden opportunity. If the universe is kind and the parameters align just right, the next generation of particle accelerators might finally catch these elusive particles in the act.

Technical Summary

1. Problem Statement

While the minimal Type-I Seesaw model successfully explains neutrino oscillations via the introduction of Heavy Neutral Leptons (HNLs), it also predicts Charged Lepton Flavor Violation (cLFV). Historically, experimental and theoretical efforts have focused heavily on purely leptonic processes (e.g., $\mu \to e\gamma$, $\mu \to 3e$) and muon-to-electron conversion. However, semileptonic cLFV channels, particularly those involving $\tau$ decays into mesons ($\tau \to \ell P, \tau \to \ell V$) and direct meson decays ($P \to \ell \ell'$), have remained underexplored.

Previous systematic analyses of these semileptonic channels in the minimal seesaw context are over thirty years old and rely on outdated form factors and oscillation data. With the advent of high-precision neutrino oscillation data (NuFIT) and upcoming high-luminosity facilities (Belle-II, BES-III, STCF), there is a critical need to re-evaluate these processes using modern inputs to determine their potential as discovery channels for HNLs.

2. Methodology

The authors perform a systematic re-analysis of cLFV processes within the minimal Type-I Seesaw model with two degenerate HNLs ($N=2$).

• Theoretical Framework:
  • They utilize the Casas-Ibarra parametrization to relate the HNL mixing angles ($\Theta$) to neutrino oscillation data. This allows them to explore regions with large mixing angles (low-scale seesaw) while generating small neutrino masses via symmetry breaking (specifically, an $L$-conserving limit where $M_{N1} = M_{N2}$ and specific phase relations hold).
  • They calculate amplitudes for processes mediated by $Z$-penguins, photon-penguins, and box diagrams involving HNLs and SM gauge bosons.
• Calculational Updates:
  • Form Factors & Decay Constants: The authors update the calculations using modern values for meson decay constants ($f_P, f_V$) and energy-dependent form factors (parametrized via Vector Meson Dominance models with Breit-Wigner functions) for final states like $\pi, \rho, \omega, \phi, J/\Psi, \Upsilon$, and multi-pion/kaon systems.
  • Branching Ratios: They derive general formulas for branching ratios of $\tau \to \ell P$, $\tau \to \ell V$, $\tau \to \ell PP$, and meson decays $P/V \to \ell \ell'$, explicitly including both dipole (G-type) and monopole (F-type) contributions.
• Parameter Space Exploration:
  • They scan the parameter space defined by the HNL mass ($M_N$), the total mixing angle ($U^2_{tot}$), and the flavor mixing parameters ($x_\alpha, \Lambda_{\alpha\beta}$).
  • They impose constraints from neutrino oscillation data (NuFIT 6.0), tree-level unitarity (perturbativity limits on Yukawa couplings), and existing experimental bounds.
  • They analyze both Normal Ordering (NO) and Inverted Ordering (IO) of neutrino masses.

3. Key Contributions

• Re-evaluation of Semileptonic Channels: The paper provides the first comprehensive update of $\tau$ semileptonic cLFV decays in the minimal seesaw model in over 30 years, incorporating modern hadronic physics inputs.
• Identification of Dominant Channels: The authors demonstrate that in specific regions of the parameter space, semileptonic $\tau$ decays can dominate over purely leptonic probes. Specifically, $\tau \to \ell \rho$ is found to have a branching ratio comparable to or larger than $\tau \to 3\ell$ and even $\tau \to \ell \gamma$ in certain mass regimes.
• Analysis of Non-Decoupling: The paper clarifies the behavior of branching ratios at high HNL masses ($M_N \gg M_W$). It highlights a non-decoupling effect driven by the $Z$-penguin diagrams, where branching ratios grow with $M_N$ if the mixing angle $U^2_{tot}$ is held fixed. This growth is eventually tamed by unitarity constraints on the Yukawa coupling.
• Global Constraints vs. Indirect Bounds: The authors derive indirect upper bounds on all considered processes by maximizing the flavor mixing parameters allowed by neutrino data and perturbativity, providing a "theoretical ceiling" for these branching ratios.

4. Key Results

• $\tau$ Decays vs. Meson Decays:
  • $\tau$ Decays: Processes like $\tau \to \ell \pi$ and $\tau \to \ell \rho$ are the most promising. Under optimal conditions (Inverted Ordering, maximal mixing, and masses near the unitarity bound), the branching ratios can reach $\mathcal{O}(10^{-10})$. This is within the projected sensitivity of Belle-II (sensitivity $\sim 10^{-10}$).
  • Meson Decays: Direct decays of light mesons ($\pi, \eta, \eta'$) and quarkonia ($J/\Psi, \Upsilon$) into different charged leptons ($P \to e\mu$, etc.) are found to be highly suppressed ($< 10^{-17}$ for light mesons, $\sim 10^{-14}$ for $\Upsilon$). These rates are far below current and future experimental sensitivities due to strong interaction dominance and GIM suppression.
• Comparison with Purely Leptonic Modes:
  • The ratio $\text{Br}(\tau \to \ell \rho) / \text{Br}(\tau \to 3\ell)$ is found to be $\sim 1.5 - 4$ depending on $M_N$, indicating that $\tau \to \ell \rho$ is a competitive, and often superior, probe compared to $\tau \to 3\ell$.
  • Unlike $\tau \to \ell \gamma$, which plateaus logarithmically at high masses, the semileptonic modes benefit from the non-decoupling of the $Z$-penguin, making them increasingly relevant for heavy HNLs.
• Constraints from Muon Decays:
  • The non-observation of $\mu \to e\gamma$ and $\mu \to 3e$ imposes extremely stringent indirect bounds on $\tau$ decays.
  • If $\mu \to e\gamma$ is not observed, the predicted branching ratio for $\tau \to e\pi$ is constrained to be $< 2.3 \times 10^{-12}$, and $\tau \to e\rho$ to $< 1.8 \times 10^{-12}$. These values are significantly below Belle-II's sensitivity, suggesting that Belle-II is unlikely to observe these signals unless the muon constraints are relaxed or the model parameters are finely tuned.
• Inverted vs. Normal Ordering:
  • In the Inverted Ordering (IO), the parameter space allows for $\Lambda_{e\mu} = 0$, which can suppress $\mu \to e\gamma$ to zero, potentially allowing larger $\tau$ decay rates. However, in Normal Ordering (NO), the constraints are tighter.

5. Significance

• Experimental Strategy: The paper argues that while $\tau \to \ell \rho$ and $\tau \to \ell \pi$ are theoretically the most sensitive semileptonic channels, their observation at Belle-II is contingent on the absence of signals in muon cLFV experiments. If muon experiments (MEG-II, Mu3e) continue to see null results, the parameter space for $\tau$ cLFV shrinks, making a discovery at Belle-II unlikely.
• Theoretical Insight: The work clarifies the interplay between the "non-decoupling" of heavy HNLs in $Z$-penguins and unitarity constraints, providing a more accurate map of the viable parameter space for the minimal seesaw model.
• Future Directions: The authors suggest that alternative seesaw variants (e.g., with three degenerate HNLs or Inverse Seesaw) might have less stringent indirect bounds, potentially opening up the possibility of observing these semileptonic $\tau$ decays in the near future.

In conclusion, the paper provides a rigorous update to the landscape of cLFV searches, identifying $\tau \to \ell \rho$ as a theoretically dominant channel that, however, faces a "ceiling" imposed by the extreme sensitivity of muon flavor violation experiments.