Search for the lepton-flavor-violating $\tau^{-} \rightarrow e^{\mp} \ell^{\pm} \ell^{\mp}$ decays at Belle II

Using 428 fb$^{-1}$ of data from the Belle II experiment, the authors performed a search for charged-lepton-flavor-violating $\tau^- \rightarrow e^\mp \ell^\pm \ell^-$ decays and established the most stringent upper limits to date on their branching fractions, ranging from $1.3$ to $2.5 \times 10^{-8}$ at the 90% confidence level.

The Gist

The Great Particle Hunt: A Story of Rare Tau Decay

Imagine the universe as a giant, bustling party where particles are the guests. Most guests follow strict rules: a "Tau" guest is supposed to leave the party in a very specific, predictable way, turning into other particles that look exactly like its family members. This is the "Standard Model" of physics—the rulebook everyone expects everyone to follow.

But what if a Tau guest broke the rules? What if, instead of turning into its usual family, it suddenly transformed into a mix of electrons and muons that it shouldn't be able to produce? This is called Lepton-Flavor Violation (LFV). Finding this would be like seeing a cat suddenly give birth to a puppy. It would prove that our rulebook is incomplete and that there are hidden, new laws of physics at play.

This paper is a report from the Belle II experiment, a massive particle detector in Japan, describing their latest attempt to catch these "rule-breaking" Tau particles in the act.

The Setup: A High-Stakes Game of Hide-and-Seek

The scientists used the SuperKEKB collider, which smashes electrons and positrons together at incredibly high speeds. These collisions create pairs of Tau particles. The team analyzed data from 428 "inverse femtobarns" of collisions (a unit of measurement that roughly translates to 393 million Tau pairs produced).

Their goal was to find five specific, forbidden ways a Tau could decay:

1. $\tau \to e^- e^+ e^-$ (Three electrons)
2. $\tau \to e^- e^+ \mu^-$ (Two electrons, one muon)
3. $\tau \to e^- \mu^+ e^-$ (Two electrons, one muon, different charge)
4. $\tau \to \mu^- \mu^+ e^-$ (Two muons, one electron)
5. $\tau \to \mu^- e^+ \mu^-$ (Two muons, one electron, different charge)

The Challenge: Finding a Needle in a Haystack

The problem is that these "forbidden" decays are incredibly rare. If they happen at all, they happen maybe once in every 100 million Taus. Meanwhile, the "normal" decays happen constantly, creating a mountain of background noise.

To find the signal, the scientists had to build a sophisticated filter:

• The "Inclusive Tagging" Net: They looked at one Tau particle in the pair to identify what it was doing. If they could confirm one Tau behaved normally, they could focus their attention on its partner, the "signal candidate."
• The "Smart Bouncer" (BDT): They used a computer program called a Boosted Decision Tree (BDT). Think of this as a highly trained bouncer at a club. The BDT was trained on millions of simulated events and real data to recognize the subtle differences between a "rule-breaking" Tau and a normal background event. It looked at things like the energy of the particles, their angles, and how they moved together.
• The "Blind Box": To ensure they didn't accidentally trick themselves into seeing patterns that weren't there, the scientists kept the most critical part of the data "blinded" (hidden) until they finalized their search strategy. This is like solving a puzzle without looking at the picture on the box until you've finished the pieces.

The Results: Silence is Golden

After running their filters and checking the data, the result was silence.

• No "Puppies" Found: They did not find a single instance of a Tau breaking the rules in any of the five modes they searched.
• Setting the Limits: Even though they didn't find the forbidden decays, they didn't come up empty-handed. Because they looked so hard and had so much data, they could set a very strict "speed limit" on how often these events could be happening.

They calculated that if these decays do happen, they occur less than 1.3 to 2.5 times in every 100 million Tau decays.

Why This Matters

Before this study, the best limits on four of these five modes were set by previous experiments. The Belle II team has now tightened those limits, making them the strictest in the world for four of the five scenarios.

In the world of particle physics, "not finding" something is often just as important as finding it. By proving that these decays are even rarer than we thought, the scientists are narrowing down the list of possible new theories. It's like telling a detective, "We know the thief didn't use a red car, a blue car, or a green car," which helps them focus on the remaining suspects.

In short: The Belle II team looked at hundreds of millions of particle collisions with high-tech filters and smart computer algorithms. They found no evidence of Tau particles breaking the laws of physics, but they successfully proved that if such a crime is happening, it is incredibly rare—ruling out many potential "new physics" theories in the process.

Technical Summary

Technical Summary: Search for the Lepton-Flavor-Violating $\tau^- \to e^\mp \ell^\pm \ell^-$ Decays at Belle II

Problem and Motivation
In the Standard Model (SM), charged-lepton flavor is conserved, and neutrinos are assumed massless. While neutrino mixing introduces lepton-flavor violation (LFV) at the loop level, the predicted branching fractions for processes such as $\tau \to e$ or $\tau \to \mu$ are suppressed to levels around $10^{-50}$, rendering them unobservable. Consequently, the observation of any charged-lepton-flavor-violating decay would constitute indisputable evidence of physics beyond the Standard Model (BSM).

This paper reports a search for five specific LFV decay modes: $\tau^- \to e^- e^+ e^-$, $\tau^- \to e^- e^+ \mu^-$, $\tau^- \to e^- \mu^+ e^-$, $\tau^- \to \mu^- \mu^+ e^-$, and $\tau^- \to \mu^- e^+ \mu^-$. These modes are of particular interest as they are enhanced in certain BSM scenarios, such as Type II Seesaw models, and can probe the existence of axion-like particles (ALPs) via intermediate resonances. Previous searches by the Belle and BaBar experiments set limits in the range of $1.5\text{--}2.7 \times 10^{-8}$. This analysis aims to improve upon these constraints using the larger dataset provided by the Belle II experiment.

Methodology
The analysis utilizes a data sample corresponding to an integrated luminosity of $428 \text{ fb}^{-1}$, collected by the Belle II detector at the SuperKEKB $e^+e^-$ collider between 2019 and 2022. This dataset corresponds to approximately 393 million produced $\tau$-pairs.

• Candidate Reconstruction: Signal candidates are reconstructed using an inclusive-tagging method. In the center-of-mass frame, the event is divided into two hemispheres based on the thrust axis. One hemisphere contains the signal candidate (three leptons with total charge $\pm 1$), while the other contains the "tag" $\tau$ decay products. The signal vertex is constrained to be within 3 cm along the beam axis and 1 cm in the transverse plane from the interaction point.
• Particle Identification: Muons are identified using a likelihood ratio $P_\mu > 0.5$ combining information from the CDC, TOP, ARICH, ECL, and KLM subdetectors. Electrons are identified using a Boosted Decision Tree (BDT) classifier with an output $P_e > 0.5$. For the final selection, identification criteria are tightened to $P_{e,\mu} > 0.9$ to suppress backgrounds from misidentified pions.
• Background Rejection: The analysis employs a two-step background suppression strategy:
  1. Preselection: Global event variables (thrust, missing momentum, visible energy) and kinematic constraints are applied to remove beam-related backgrounds and low-multiplicity events. A specific cut on the invariant mass of opposite-sign lepton pairs is used to reject events with converted photons.
  2. BDT Classification: A Boosted Decision Tree is trained to distinguish signal from background. The training uses simulated signal events and data events from sidebands (outside the signal fit region and blind region) to avoid bias. The BDT inputs include variables related to the signal $\tau$ kinematics, the "Rest of Event" (ROE) properties, and global event shape variables (e.g., Fox-Wolfram moments).
• Fitting Procedure: Branching fractions are extracted via an unbinned maximum likelihood fit to the invariant mass of the three-lepton system ($M_{e\ell\ell}$). The signal probability density function (PDF) is modeled by an asymmetric double-sided Crystal Ball function to account for final-state radiation (FSR) tails, while the background is modeled by an exponential function. The fit is performed in the mass range $1.4 < M_{e\ell\ell} < 2.0 \text{ GeV}/c^2$.

Key Contributions

• First Inclusive-Tagging Application: This work represents the first application of the inclusive-tagging reconstruction method by Belle II for the $3\mu$ final state and extends it to the $e\ell\ell$ modes, achieving signal efficiencies 2 to 3.3 times higher than the most recent Belle analysis for comparable background levels.
• Data-Driven Background Control: Due to discrepancies between data and simulation in the sidebands (attributed to unmodeled four-lepton processes with radiation), the analysis relies on a data-driven BDT strategy trained on sideband data rather than pure simulation for background estimation.
• Comprehensive Systematic Evaluation: The study provides a detailed breakdown of systematic uncertainties, including lepton identification, tracking efficiency, trigger efficiency, BDT modeling, initial-state radiation (ISR), luminosity, and the $\tau$-pair production cross-section.

Results
No significant signal was observed for any of the five decay modes. The number of observed events in the signal regions was consistent with the expected background yields derived from sideband fits.

Consequently, the collaboration sets 90% confidence level (C.L.) upper limits on the branching fractions:

• $\mathcal{B}(\tau^- \to e^- e^+ e^-) < 2.5 \times 10^{-8}$
• $\mathcal{B}(\tau^- \to e^- e^+ \mu^-) < 1.6 \times 10^{-8}$
• $\mathcal{B}(\tau^- \to e^- \mu^+ e^-) < 1.6 \times 10^{-8}$
• $\mathcal{B}(\tau^- \to \mu^- \mu^+ e^-) < 2.4 \times 10^{-8}$
• $\mathcal{B}(\tau^- \to \mu^- e^+ \mu^-) < 1.3 \times 10^{-8}$

Significance
The authors state that these results constitute the most stringent bounds to date for four of the five modes searched. Specifically, the limits for $\tau^- \to e^- e^+ e^-$, $\tau^- \to e^- e^+ \mu^-$, $\tau^- \to \mu^- \mu^+ e^-$, and $\tau^- \to \mu^- e^+ \mu^-$ improve upon previous experimental constraints. For the $\tau^- \to e^- \mu^+ e^-$ mode, the limit is comparable to previous results but does not surpass the most stringent existing bound. The analysis demonstrates the capability of the Belle II detector and its analysis techniques to probe LFV processes at the $10^{-8}$ level, providing tight constraints on BSM models that predict enhanced rates for these decays.