Search for $τ^-\to μ^-μ^+μ^-$ decays at the LHCb experiment with Run 2 data

Using 5.4 fb$^{-1}$ of Run 2 data collected by the LHCb experiment at 13 TeV, a search for the lepton-flavour-violating decay $\tau^-\to\mu^-\mu^+\mu^-$ found no evidence of the process and set an upper limit of $1.9\times 10^{-8}$ on its branching fraction at the 90% confidence level.

The Gist

The Big Picture: A Cosmic "Whodunit"

Imagine the universe is a giant, high-speed train station (the Large Hadron Collider, or LHC). Every second, millions of particles crash into each other, creating a chaotic explosion of new particles that fly off in all directions.

Most of the time, these particles follow the "Rulebook" of physics, known as the Standard Model. This rulebook says that certain particles, called tau leptons (let's call them "Taus"), are very shy. They usually decay (break apart) into specific, predictable groups of particles.

However, physicists suspect there might be a "secret rule" or a "ghost" in the machine. They are looking for a very rare event where a Tau breaks the rules and turns into three muons (a different type of particle) all at once. In the current rulebook, this is forbidden. If they find it, it means the rulebook is incomplete and there is "New Physics" hiding somewhere.

The Mission: Finding a Needle in a Haystack

The LHCb experiment is like a super-precise camera and a team of detectives standing on the platform. Their job is to watch the crashes and look for that one specific, forbidden event: A Tau turning into three muons ($\tau \to \mu^- \mu^+ \mu^-$).

The problem? This event is incredibly rare. It's like trying to find a single, specific grain of sand that has been painted neon green, hidden inside a massive beach of normal sand.

How They Did It: The "Reference Photo" Trick

To find this needle, the LHCb team didn't just look at the chaos. They used a clever comparison trick:

1. The Signal (The Search): They looked for the forbidden "Tau to three muons" event.
2. The Normalizer (The Reference): They also looked for a very common, known event: a particle called a $D_s$ meson decaying into a phi meson (which splits into two muons) and a pion.

Think of it like this: Imagine you are trying to count how many people in a crowd are wearing a red hat (the rare event), but you don't know how many people are in the crowd total. So, you also count how many people are wearing blue hats (the common event). You know exactly how many blue hats should be there based on previous studies. By comparing the number of red hats you see to the number of blue hats you see, you can figure out if there are any red hats at all, even if you don't know the total crowd size.

The Detective Work: Filtering the Noise

The data they collected (from 2016–2018) contained billions of collisions. Most of these were "noise"—random particles that just happened to look like the signal by accident.

To clean up the noise, the team used two "Smart Filters" (computer programs called Classifiers):

• Filter 1 (The Pattern Matcher): This looked at the shape of the tracks. Did the particles come from a common starting point? Did they fly apart in a way that makes sense for a decay? This filtered out random junk.
• Filter 2 (The ID Check): This checked if the particles were actually muons and not just other particles (like pions or kaons) that were pretending to be muons.

They trained these filters using "fake" data (simulations) and real data from the "blue hat" (common) events to make sure they were accurate.

The Result: A Clean Bill of Health (For Now)

After running all the data through the filters and doing the math:

• Did they find the forbidden event? No. They found zero cases of a Tau turning into three muons.
• Did they find a lot of noise? Yes, but they could predict exactly how much noise there should be, and the data matched the prediction perfectly.

Because they didn't find the event, they couldn't say "It happens this often." Instead, they set a limit.

They said: "If this event does happen, it happens less than 1.9 times out of every 100 million Taus." (This is written scientifically as $< 1.9 \times 10^{-8}$).

Why This Matters

This result is a "tightening of the net."

• In the past, the limit was looser (the event could happen up to 4.6 times out of 100 million).
• Now, with better data and better filters, the net is tighter. The event must be even rarer than we thought.

This doesn't mean the "New Physics" isn't there; it just means the "ghost" is even harder to catch than before. It forces scientists to update their theories. If a new theory predicts the event happens more often than this new limit allows, that theory is now proven wrong.

Summary

The LHCb team acted like a high-tech security team at a massive party. They scanned millions of guests looking for a specific person breaking the dress code. They didn't find that person. Instead, they proved that if that person is there, they are so rare that they appear less than 2 times in every 100 million guests. This helps the rest of the physics community know exactly how rare the "rule-breaker" must be.

Technical Summary

Technical Summary: Search for $\tau^- \to \mu^- \mu^+ \mu^-$ Decays at LHCb with Run 2 Data

Problem and Motivation
Lepton-flavour-violating (LFV) decays, such as $\tau^- \to \mu^- \mu^+ \mu^-$, are strictly forbidden in the Standard Model (SM) under the assumption of massless neutrinos. Even in scenarios involving massive neutrinos, the predicted branching fractions are vanishingly small ($\sim 10^{-55}$), far below the sensitivity of current or foreseen experiments. However, various extensions of the SM, including models with heavy neutrinos or the exchange of an extra neutral gauge boson ($Z'$), predict branching fractions in the range of $10^{-10}$ to $10^{-8}$. Consequently, observing this decay would constitute a clear indication of physics beyond the SM, while setting tighter upper limits provides constraints on these theoretical extensions. This paper presents a search for this specific LFV decay using data collected by the LHCb experiment.

Methodology
The analysis utilizes proton-proton collision data collected by the LHCb experiment between 2016 and 2018 at a centre-of-mass energy of $\sqrt{s} = 13$ TeV, corresponding to an integrated luminosity of $5.4 \text{ fb}^{-1}$.

• Signal and Normalization: The signal channel is $\tau^- \to \mu^- \mu^+ \mu^-$. The branching fraction is measured relative to the well-known normalization channel $D_s^- \to \phi(1020)\pi^-$, where $\phi \to \mu^- \mu^+$. This normalization mode was chosen due to its similar topology and decay kinematics. The branching fraction is calculated using the ratio of observed candidates ($N_\tau / N_{D_s}$), corrected by the ratio of selection efficiencies ($\epsilon_\tau / \epsilon_{D_s}$), the fraction of $\tau^-$ leptons produced via $D_s$ decays ($f^\tau_{D_s}$), and the known branching fractions of the normalization modes.
• Event Selection: Candidates are reconstructed from three tracks with a total charge of $-1$ originating from a common vertex. Strict requirements are applied to track quality, impact parameter significance ($\chi^2_{IP}$), and particle identification (PID). To suppress combinatorial background and misidentified hadrons, two gradient-boosted decision-tree classifiers (XGBoost) are employed:
  • CAC (Anticombinatorial): Uses topological and kinematic variables (e.g., isolation, decay time, vertex displacement) to reject random track combinations.
  • C$_{PID}$: Uses PID and kinematic information to reject hadrons misidentified as muons.
    Candidates must satisfy $CAC > 0.80$ and $C_{PID} > 0.88$.
• Background Modeling: The primary backgrounds arise from:
  1. Random combinations of muons (combinatorial).
  2. Hadronic decays of charm mesons ($D^- \to \pi^- K^+ \pi^-$ and $D_{(s)}^- \to \pi^- \pi^+ \pi^-$) where hadrons are misidentified as muons.
  3. Decays with genuine muons, specifically $D_s^- \to \eta^{(\prime)}(\to \mu^- \mu^+ \gamma)\mu^- \bar{\nu}_\mu$.
     The signal region is defined as $|M_{\mu\mu\mu} - m_\tau| \le 20 \text{ MeV}/c^2$. A sideband region ($20 < |M_{\mu\mu\mu} - m_\tau| \le 30 \text{ MeV}/c^2$) is used to model background shapes and estimate yields.
• Statistical Analysis: The upper limit is evaluated using the $CL_s$ method. The dataset is divided into 15 bins based on the outputs of the CAC and C$_{PID}$ classifiers to maximize signal-to-background separation. An unbinned extended maximum-likelihood fit is performed simultaneously across these bins, modeling the signal with a Johnson's SU function and the background with a sum of exponential and Johnson's SU components.

Key Contributions and Systematics
The analysis incorporates several technical refinements to ensure accuracy:

• Simulation Calibration: Simulation samples are weighted to correct for differences in kinematics and detector response between data and simulation, using the $D_s^- \to \phi \pi^-$ channel as a control.
• Efficiency Corrections: Tracking and trigger efficiencies are calibrated using prompt $J/\psi \to \mu^+ \mu^-$ decays and independent trigger lines, respectively.
• Systematic Uncertainties: Major sources of systematic uncertainty include external inputs (branching fractions and production fractions), the determination of the efficiency ratio, and the modeling of background shapes. These are treated as nuisance parameters constrained by Gaussian functions in the limit calculation.

Results
No significant excess of signal events is observed in the data. The central value of the signal branching fraction extracted from the fit is $(-0.1 \pm 1.1) \times 10^{-8}$. Based on this result, the collaboration sets the following upper limits on the branching fraction:

• $1.9 \times 10^{-8}$ at 90% confidence level (CL)
• $2.3 \times 10^{-8}$ at 95% CL

This result supersedes the previous LHCb limit obtained from Run 1 data ($4.6 \times 10^{-8}$ at 90% CL) and is comparable in sensitivity to the most stringent limit reported by the Belle II collaboration ($1.9 \times 10^{-8}$ at 90% CL).

Significance
The paper claims that this result provides a complementary constraint on extensions of the Standard Model within the context of existing searches. By utilizing the unique forward spectrometer capabilities of LHCb and a large dataset from Run 2, the analysis achieves a sensitivity comparable to dedicated $B$-factory experiments. The authors note that future analyses with the upgraded LHCb detector, benefiting from higher luminosity and improved trigger efficiencies, will further enhance the sensitivity to this rare decay mode.