A search for lepton-flavour violating $τ\to 3μ$ decays with the ATLAS detector

Using 137 fb⁻¹ of 13 TeV proton-proton collision data collected by the ATLAS detector, a search for the lepton-flavour violating decay $\tau \to 3\mu$ via the $W \to \tau\nu$ channel found no evidence of signal, setting an observed 90% confidence level upper limit on the branching ratio of $8.7 \times 10^{-8}$.

The Gist

Imagine the universe as a giant, bustling party where particles are the guests. In the "Standard Model" (the official rulebook of physics), there's a strict bouncer at the door: Lepton Flavor Conservation. This rule says that a "tau" guest (a heavy, short-lived particle) can only hang out with other tau guests, and a "muon" guest can only hang out with other muons. They are not allowed to swap identities or turn into each other.

However, physicists have long suspected that the bouncer might be asleep at the wheel. If a tau particle could spontaneously turn into three muons, it would be a smoking gun for "New Physics"—evidence of forces or particles we haven't discovered yet.

This paper is the report from the ATLAS experiment at CERN's Large Hadron Collider (LHC), where they threw a massive party to catch this rule-breaker in the act.

Here is the story of their search, explained simply:

1. The Setup: A High-Stakes Game of "Find the Imposter"

The team looked at data from 2016 to 2018, which corresponds to 137 "inverse femtobarns" of data. To use an analogy, imagine they watched 137 trillion high-speed collisions between protons. That is an unimaginably large number of events, like watching every grain of sand on every beach on Earth, all at once.

They were looking for a very specific, rare event: A Tau particle decaying into three Muons.

• The Signal: A tau particle appearing out of nowhere and instantly splitting into three muons.
• The Background: The "noise" of the party. Most of the time, muons appear in groups just by random chance, or from other common particle decays. It's like trying to find one specific person wearing a red hat in a stadium full of people, where thousands of people are wearing red hats for other reasons.

2. The Strategy: The "Smart Filter" (Machine Learning)

Since the signal is so rare (we expect it to happen maybe once in a billion years of collisions), looking at the raw data is impossible. The team used a Machine Learning algorithm (specifically a "Boosted Decision Tree," or BDT).

Think of the BDT as a super-smart security guard who has been trained on millions of photos.

• Training: They showed the guard pictures of "fake" groups of three muons (background noise) and pictures of what a "real" tau decay would look like if it existed (simulated signal).
• The Job: The guard learned to spot tiny differences. Real tau decays tend to happen slightly away from the main collision point (because taus live a tiny bit longer), and the three muons fly out in a specific pattern. Random noise doesn't follow these rules.
• The Result: The guard sorted the millions of events into "Loose," "Medium," and "Tight" categories. The "Tight" category contained the events that looked most suspiciously like a real tau decay.

3. The Investigation: Looking for the "Ghost"

The team focused on the mass of the three muons.

• If three random muons just happened to be near each other, their combined mass would be all over the place.
• If a real Tau particle decayed into them, their combined mass would be exactly the weight of a Tau particle (like finding three puzzle pieces that perfectly fit a specific shape).

They looked at the data in the "Tight" category, specifically around the mass of the Tau particle. They expected to see a bump in the graph—a sudden spike where the data points pile up, indicating a discovery.

4. The Verdict: No Ghost Found

The result? The graph was smooth. There was no bump. The data looked exactly like what you would expect if only "noise" (background events) were present.

• The Analogy: Imagine searching a dark forest for a specific type of glowing mushroom. You scan the whole forest with a high-tech scanner. You find thousands of regular mushrooms, but you find zero glowing ones.
• The Conclusion: The universe, at least in this dataset, is still obeying the rulebook. The tau particle did not turn into three muons in this experiment.

5. Why This Matters (Even Though They Didn't Find It)

You might ask, "If they didn't find it, why write a paper?"

In science, ruling things out is just as important as finding them.

• Setting the Limit: Because they didn't find it, they can say: "If this transformation happens, it must be incredibly rare. It happens less than 8.7 times in every 100 million tau decays."
• Improving the Search: This is the best limit ATLAS has ever set for this specific process. They improved upon their previous record by a factor of 5.
• The Future: This result tells theorists who are building new models of the universe: "Your theory that predicts this happens often is probably wrong." It forces them to refine their ideas.

Summary

The ATLAS collaboration acted like a team of cosmic detectives. They used a massive dataset and a super-smart AI filter to hunt for a particle that breaks the laws of physics. They didn't catch the culprit this time, but they proved that if the culprit is out there, it's much sneakier than we thought. This narrows the search for the "New Physics" that could explain the mysteries of our universe.

Technical Summary

1. Problem and Motivation

• Physics Context: In the Standard Model (SM), lepton flavor is conserved for charged leptons. While neutrino oscillations imply lepton flavor violation (LFV), the predicted branching ratios for charged lepton flavor violation (cLFV) processes like $\tau \to 3\mu$ are vanishingly small ($\sim 10^{-55}$).
• Beyond the Standard Model (BSM): Many BSM theories (e.g., Supersymmetry, Leptoquarks, $Z'$ models) predict cLFV rates significantly higher than the SM, potentially within the reach of current experiments ($10^{-9}$ to $10^{-11}$).
• Current Status: Prior to this analysis, the most stringent limits on $\tau \to 3\mu$ were set by $e^+e^-$ colliders (Belle II, Belle, BaBar) and the LHCb experiment, with observed limits around $1.9 \times 10^{-8}$. The previous ATLAS limit (Run 1, $\sqrt{s}=8$ TeV) was significantly weaker ($37.6 \times 10^{-8}$).
• Goal: This paper presents a new search using the full Run 2 dataset ($\sqrt{s}=13$ TeV) to improve sensitivity, specifically targeting the electroweak production of $\tau$-leptons which offers better kinematic properties for reconstruction compared to heavy-flavor sources.

2. Methodology

Dataset and Signal Sources

• Data: $pp$ collision data collected by ATLAS between 2016 and 2018, corresponding to an integrated luminosity of 137 fb$^{-1}$.
• Signal Production Mechanisms:
  1. Electroweak (EW) Signal: Dominated by $W \to \tau\nu$ (83%), with contributions from $Z \to \tau\tau$ (16%) and $t\bar{t} \to \tau X$ (1%). These $\tau$-leptons have higher transverse momentum ($p_T$) and are better isolated.
  2. Heavy-Flavor (HF) Signal: $\tau$-leptons from $D_s$ meson decays (prompt and non-prompt) and direct $B$-meson decays. While abundant, these are harder to trigger on and reconstruct due to lower $p_T$ and higher hadronic activity.
• Trigger Strategy: Used a combination of two-muon and three-muon triggers. Special attention was paid to correcting for efficiency losses caused by overlapping trigger regions for close-by muons from high-momentum $\tau$ decays.

Event Reconstruction and Selection

• Object Definition: Muons reconstructed in both the Inner Detector (ID) and Muon Spectrometer (MS) with $p_T > 4$ GeV and $|\eta| \le 2.4$.
• Vertexing: Events with at least three muons (total charge $\pm 1$) are selected. A secondary vertex (SV) is reconstructed from the three muon tracks. The SV must be displaced from the primary vertex (PV), consistent with the $\tau$ lifetime.
• Preselection Cuts:
  • SV fit $\chi^2 < 40$.
  • Transverse impact parameter significance $a_{xy}^0 / \sigma_{a_{xy}^0} < 6$.
  • Isolation: Calorimeter energy ($E_T^{iso}$) and track $p_T$ sums around the triplet are limited to reject background from jets and pile-up.
  • Invariant mass of any two-muon sub-combination must be $> 325$ MeV to reject photon conversions and low-mass resonances ($\omega, \phi$).
• Background Modeling: The background is dominated by random combinations of muons and misidentified tracks. Instead of relying on simulation (which is unreliable for combinatorial backgrounds), the background shape is modeled using data sidebands (regions outside the signal mass window of 1700–1850 MeV).

Multivariate Analysis (MVA)

• Classifier: A Gradient Boosted Decision Tree (BDT) using the XGBoost algorithm.
• Input Variables: 18 variables including vertex quality ($p$-value, $\chi^2$), muon isolation, triplet kinematics ($p_T$, $\eta$), and correlations with missing transverse momentum ($E_T^{miss}$).
• Categorization: Events are split into six categories based on:
  1. BDT Score: Loose, Medium, Tight.
  2. Detector Region: Barrel ($|\eta_{triplet}| < 1.2$) and Endcap.
  • Rationale: The barrel region has a superior mass resolution (25 MeV) compared to the endcap (34 MeV).
• Fit Strategy: An unbinned maximum-likelihood fit is performed simultaneously on the three-muon invariant mass ($m_{3\mu}$) spectrum across all six categories.
  • Signal Model: Double-sided Crystal Ball (DSCB) function.
  • Background Model: Exponential function (parameters determined from sidebands).
  • Systematics: Treated as nuisance parameters (trigger efficiency, muon reconstruction, jet energy scale, pile-up, luminosity, and background parameterization).

3. Key Contributions

• First ATLAS Run 2 Search: This is the first ATLAS analysis of $\tau \to 3\mu$ using the full Run 2 dataset, superseding the 2016 Run 1 result.
• Optimization for Electroweak Production: Unlike previous LHC searches that focused heavily on heavy-flavor production, this analysis optimizes selection for the electroweak $W \to \tau\nu$ channel, leveraging the cleaner environment and higher $p_T$ of these events.
• Advanced Background Handling: The use of a data-driven sideband approach combined with a simultaneous fit across multiple BDT and geometric categories maximizes sensitivity while minimizing reliance on simulation for background estimation.
• Systematic Control: Comprehensive treatment of systematic uncertainties, particularly trigger efficiency corrections for close-by muons and spurious signal estimation.

4. Results

• Observation: No significant excess of events was observed in the signal region ($1700 < m_{3\mu} < 1850$ MeV). The data is compatible with the background-only hypothesis.
• Best Fit: The best-fit branching ratio is $B(\tau \to 3\mu) = (1.5 \pm 4.0) \times 10^{-8}$.
• Limits (90% Confidence Level):
  • Observed Limit: $B(\tau \to 3\mu) < 8.7 \times 10^{-8}$.
  • Expected Limit: $B(\tau \to 3\mu) < 7.5 \times 10^{-8}$.
• Comparison:
  • This result improves the previous ATLAS Run 1 limit by a factor of 5.
  • The improvement is attributed to the larger integrated luminosity, higher center-of-mass energy, detector upgrades (IBL), and improved analysis techniques (MVA, trigger corrections).
  • While competitive, the limit is currently slightly weaker than the best limits set by Belle II ($1.9 \times 10^{-8}$) and LHCb ($1.9 \times 10^{-8}$), but it provides a crucial independent measurement in the $pp$ collision environment.

5. Significance

• BSM Constraints: The result places stringent constraints on BSM models predicting cLFV, ruling out parameter spaces where the branching ratio exceeds $\sim 10^{-8}$.
• Complementarity: It demonstrates the capability of the ATLAS detector to perform precision cLFV searches in the hadronic environment of the LHC, complementing the $e^+e^-$ results from Belle II and the forward-geometry results from LHCb.
• Future Prospects: The analysis is currently limited by statistical uncertainty. With the high-luminosity LHC (HL-LHC) and further data accumulation, ATLAS expects to significantly improve sensitivity, potentially reaching the $10^{-9}$ level where many theoretical BSM scenarios predict observable signals.

In summary, this paper establishes a robust framework for searching for rare $\tau$ decays at the LHC, achieving a 5-fold improvement over previous ATLAS results and setting a new benchmark for cLFV searches in proton-proton collisions.