New physics in multi-lepton tau decays

This paper proposes that dark particles with lepton-flavor-violating couplings to the tau lepton can trigger decay cascades resulting in rare, high-multiplicity neutrinoless tau decays (such as $\tau \to 5\mu$ or $\tau \to \mu 4\pi$) that dominate over traditional three-body signatures and offer new experimental search channels across various dark sector models.

The Gist

Imagine the Tau lepton (a heavy cousin of the electron) as a very unstable, high-energy firework. Usually, when it explodes, it follows a predictable script set by the Standard Model of physics, breaking apart into a few familiar particles like electrons, muons, or neutrinos.

This paper is about looking for a new, secret script that these fireworks might be following. The authors suggest that sometimes, instead of a simple explosion, the Tau might be hiding a "dark sector" inside it—a hidden world of invisible particles that only reveal themselves when they cascade down into a shower of ordinary particles.

Here is the breakdown of their idea using everyday analogies:

1. The "Russian Nesting Doll" Explosion

Usually, we think of particle decay as a simple event: A big ball breaks into two smaller balls.

• The Old Way: Tau $\rightarrow$ Muon + Neutrinos. (Simple, boring, expected).
• The New Idea: The Tau doesn't just break; it opens a Russian Nesting Doll.
  1. The Tau opens up and releases a Muon and a mysterious, invisible "Dark Doll" (let's call it $\phi$).
  2. That Dark Doll immediately splits into two even smaller "Dark Vectors" (let's call them $V$).
  3. Those two Vectors then pop open to reveal pairs of regular particles (like electrons or muons).

The Result: Instead of seeing just 2 or 3 particles, detectors see a chaotic party with 5, 7, or even more particles flying out at once. It's like a single firework exploding into a massive, multi-colored bouquet of sparks instead of just a few.

2. The "Secret Handshake" (Flavor Violation)

In the Standard Model, particles have strict rules about who they can talk to. An electron usually stays an electron; a tau stays a tau. They don't just swap identities easily.

• The Paper's Twist: The authors propose that in this "Dark Sector," there is a secret handshake that allows a Tau to instantly turn into a Muon or an Electron, passing a message to the dark world.
• This is called Lepton Flavor Violation. It's like a VIP guest at a party (the Tau) suddenly changing into a different person (the Muon) and handing a secret note to the bouncer (the dark particle), which then triggers a chain reaction.

3. The "Dark Party" Models

The authors test five different "party themes" (models) to see how this secret handshake works:

• The Kinetic Mixing Model: Imagine the dark particles are like ghosts that can slightly "phase" through the walls of our normal world. They interact weakly, mostly turning into pions (heavy particles) or electrons.
• The "Family Number" Models: Imagine the universe has a rule that "Muons" and "Taus" are from different families. These models suggest a new force that specifically breaks the rules between these families, allowing them to swap places and spawn dark particles.
• The Chiral Model: A more complex rule where the "handedness" (spin) of the particles matters. This leads to very specific, rare outcomes, like a Tau turning into five muons at once.
• The "Flavor Protected" Scalar: A scenario where the dark particles are so shy they only interact with the Tau, creating a cascade that results in a massive explosion of 7 muons or 6 electrons.

4. Why We Haven't Seen This Yet

You might ask, "If this is happening, why haven't we found it?"

• The "Invisible" Problem: In many previous searches, scientists looked for particles that disappeared (missing energy). But in this paper, the authors say, "Look at the visible mess!"
• The "Too Many Particles" Problem: Most experiments are tuned to look for simple 3-particle explosions (like $\tau \to 3\mu$). They often ignore or filter out the messy 5, 7, or 9-particle explosions because they look like background noise.
• The Analogy: It's like a security guard looking for a single thief running out of a bank. They ignore the group of 10 people running out together because they assume it's just a tour group. The authors are saying, "Check the tour groups! That's where the thief is hiding!"

5. The Hunt: Where to Look

The paper suggests that we need to change our search strategy at major particle colliders like LHCb (at CERN) and Belle II (in Japan).

• The Strategy: Instead of just looking for the simple 3-particle decays, we should hunt for the "Multi-Lepton" explosions (5 muons, 3 muons + 2 electrons, etc.).
• The Payoff: If we find these rare, messy explosions, it wouldn't just be a small discovery. It would prove the existence of a whole new "Dark Sector" and could reveal physics operating at energy scales trillions of times higher than what our current machines can directly build.

Summary

Think of the Tau lepton as a mystery box.

• Old View: We open the box, and it contains a few standard toys.
• New View: The box might contain a chain reaction machine. When we open it, it triggers a hidden mechanism that releases a cascade of new, invisible toys, which then pop open to reveal a huge number of standard toys.

The authors are telling experimental physicists: "Stop looking for the single toy. Start counting the piles of toys! If you find a pile of 5 or 7, you've found the hidden machine."

This paper is a roadmap for how to find this hidden machine, suggesting that the universe might be much more crowded with dark particles than we thought, and they are hiding in plain sight within the messy debris of heavy particle decays.

Technical Summary

1. Problem Statement

The search for "dark sector" physics (light new particles) is a major frontier in particle physics. While rare decays of mesons and charged leptons are sensitive probes, most experimental searches focus on:

1. Missing energy signatures: Where dark particles are stable and escape the detector (e.g., $\tau \to \mu a$).
2. Low-multiplicity visible decays: Specifically, three-body lepton-flavor-violating (LFV) decays like $\tau \to 3\mu$ or $\tau \to e\mu\mu$.

The Gap: The paper argues that these searches miss a vast class of models where light dark sector particles decay back into Standard Model (SM) particles within the detector volume. These "dark cascades" can produce high-multiplicity final states (5, 7, or more leptons/pions) that are currently unexplored or poorly constrained. Furthermore, in many models, these high-multiplicity channels dominate over the traditional 3-body decays, yet they lack dedicated experimental searches.

2. Methodology

The authors employ a phenomenological approach combining effective field theory (EFT) with specific benchmark models to explore the parameter space of light new physics.

• General Framework: They model the process as a cascade:
  $$\tau^\pm \to \ell^\pm \phi \to \ell^\pm V V \to \ell^\pm (\ell^+\ell^-)(\ell^+\ell^-)$$
  where $\phi$ is a light scalar/pseudoscalar and $V$ is a light vector boson (dark photon or $Z'$). This results in 5-body decays (e.g., $\tau \to 5\mu$). They also consider chains leading to 7-body decays (e.g., $\tau \to 7\mu$) via scalar triplets.
• Effective Lagrangian: They utilize an effective interaction Lagrangian (Eq. 1) describing couplings between SM leptons, the dark scalar $\phi$, and the dark vector $V$. They analyze the scaling of branching ratios with the new physics scale $\Lambda$ for both Dimension-5 and Dimension-6 operators.
• Benchmark Models: Five specific UV-complete models are analyzed to demonstrate the diversity of signatures:
  1. Kinetic Mixing (Dark Photon): $U(1)'$ mixed with hypercharge. $\phi$ initiates the LFV; $V$ (dark photon) decays to SM fermions.
  2. Gauged $L_\alpha - L_\beta$: Models gauging differences in lepton family numbers (e.g., $L_\mu - L_\tau$). These allow LFV at Dimension-5.
  3. Gauged $L_e + L_\mu - 2L_\tau$: An anomaly-free combination requiring right-handed neutrinos.
  4. Chiral Models: Models where $U(1)'$ charges are assigned only to right-handed fermions, leading to specific flavor structures.
  5. Flavor Protected Scalar: A model with a global $U(1)_e \times U(1)_\mu \times U(1)_\tau$ symmetry broken by spurions, suppressing lower-multiplicity decays and enhancing $\tau \to 7\mu$.
• Constraints & Sensitivity: The authors compile existing experimental limits (Belle II, LHCb, CLEO) on $\tau \to 3\ell$, $\tau \to \ell + \text{inv}$, and $\tau$ lifetime. They project future sensitivities for Belle II, FCC-ee, and the High-Luminosity LHC (HL-LHC) by rescaling current limits based on expected tau statistics.

3. Key Contributions

A. Discovery of Dominant High-Multiplicity Channels

The paper demonstrates that for many viable models, the 5-body and 7-body decays dominate over the traditional 3-body decays.

• Kinetic Mixing: If the dark photon mass $m_V > 2m_\pi$, hadronic decays ($V \to \pi\pi$) dominate. This leads to channels like $\tau \to 3\mu 2\pi$ or $\tau \to \mu 4\pi$ being orders of magnitude more probable than $\tau \to 5\mu$.
• Gauged $L_\mu - L_\tau$: These models often predict significant missing energy ($V \to \nu\nu$), but the fully visible 5-body channels ($\tau \to 5\mu$) remain competitive and are currently unsearched.
• Chiral/Protected Models: Can lead to $\tau \to 7\mu$ or $\tau \to \mu 6e$, which are essentially background-free but completely unexplored.

B. Model Discrimination via Topology

The authors show that the pattern of observed decays can distinguish between models:

• Hadronic vs. Leptonic: A dominance of $\tau \to 3\mu 2\pi$ over $\tau \to 5\mu$ suggests kinetic mixing (Model I).
• Missing Energy: The presence of $\tau \to 3\mu + \text{missing energy}$ alongside visible modes points to gauged lepton number models (Models II/III) where $V \to \nu\nu$ is allowed.
• Flavor Structure: The absence of $\tau \to 5\mu$ but presence of $\tau \to 5e$ could indicate a chiral model (Model IV) with specific charge assignments.

C. Probing Extremely High Energy Scales

Because these decays are suppressed by high-dimensional operators (Dim-5 or Dim-6), the sensitivity to the new physics scale $\Lambda$ is immense.

• For Dimension-5 operators, a branching ratio sensitivity of $B \sim 10^{-11}$ (achievable at FCC-ee) probes scales up to $\Lambda \sim 10^{12}$ GeV.
• This is far beyond the direct reach of the LHC ($\sim 10$ TeV) and even the energy scale of the electroweak symmetry breaking.

D. Experimental Feasibility Study

The paper provides a roadmap for experimental searches:

• LHCb: Benefits from forward geometry and low-$p_T$ triggers. A preliminary study suggests sensitivity to $B(\tau \to 5\mu) \sim 4 \times 10^{-8}$. The main challenge is combinatorial background, mitigated by vertex reconstruction.
• Belle II / FCC-ee: With $10^{10}$ to $10^{11}$ tau pairs, these $e^+e^-$ colliders can reach $B \sim 10^{-10}$ to $10^{-11}$. The clean environment makes them ideal for fully visible multi-lepton final states.
• Hadronic Channels: The paper highlights that searches for $\tau \to 3\mu 2\pi$ and $\tau \to \mu 4\pi$ are crucial and currently missing from experimental programs.

4. Results

• Branching Ratios: The authors calculate that in the "prompt" regime (where dark particles decay immediately), the branching ratios for 5-body decays can be comparable to or larger than 3-body decays, depending on the mass of the dark vector and its decay modes (leptonic vs. hadronic).
• Constraints: Current limits on $\tau \to 3\ell$ and $\tau \to \ell + \text{inv}$ constrain the parameter space but leave vast regions open for 5- and 7-body decays.
• Sensitivity Projections:
  • LHCb (HL-LHC): Can probe $B \sim 10^{-10}$.
  • Belle II (50 ab$^{-1}$): Can probe $B \sim 10^{-10}$.
  • FCC-ee: Can probe $B \sim 10^{-11}$, corresponding to $\Lambda \sim 10^{12}$ GeV for Dim-5 operators.
• Backgrounds: Fully visible multi-lepton decays have negligible irreducible SM backgrounds. The primary challenge is combinatorial background in hadron colliders, which can be suppressed by requiring a common displaced vertex.

5. Significance

• New Window into Dark Sectors: This work establishes that multi-lepton tau decays are a "smoking gun" for light dark sectors with flavor-violating couplings, offering a discovery potential that complements missing energy searches.
• Urgency for Experimental Action: The paper identifies a critical gap in experimental coverage. While $\tau \to 3\mu$ is well-studied, channels like $\tau \to 5\mu$, $\tau \to 3\mu 2\pi$, and $\tau \to 7\mu$ have not been systematically searched for, despite being potentially the dominant signatures in popular BSM models.
• High-Energy Reach: The ability to probe energy scales up to $10^{12}$ GeV using low-energy tau decays makes this a unique and powerful probe of physics far beyond the direct reach of colliders.
• Model Discrimination: The specific ratios of hadronic to leptonic, and visible to invisible, decay modes provide a powerful tool to distinguish between different theoretical frameworks (e.g., kinetic mixing vs. gauged lepton numbers).

In conclusion, the paper argues that a comprehensive search program for multi-lepton tau decays is not only experimentally feasible but theoretically motivated, offering a high-impact pathway to discovering light new physics.