Search for the lepton-flavour violating decays $B^+ \to \pi^+ \mu^\pm e^\mp$

The LHCb collaboration presents the first search for the lepton-flavour violating decay $B^+ \to \pi^+ \mu^\pm e^\mp$ using 9 fb$^{-1}$ of proton-proton collision data, finding no significant signal and setting a new world-record upper limit on the branching fraction of $1.8 \times 10^{-9}$, which provides the first constraint on such $b \to d$ transitions at the LHC.

The Gist

The Big Picture: Hunting for "Impossible" Magic

Imagine the Standard Model of physics as the ultimate rulebook for how the universe works. It's like a strict library where every book (particle) has a specific genre it belongs to. Electrons are "Electron Books," and Muons are "Muon Books."

According to the rulebook, these genres never mix. An Electron Book cannot suddenly turn into a Muon Book, and vice versa. This is a law called Lepton-Flavour Conservation.

However, scientists have found that in the world of neutrinos (ghostly, tiny particles), these rules are broken—they can change flavors. But in the world of charged particles (like electrons and muons), the rulebook says this is impossible. If we ever see an electron turn into a muon (or vice versa) in a heavy particle decay, it would be like finding a penguin that can fly. It would prove that our rulebook is incomplete and that there is New Physics hiding in the shadows.

The Experiment: The Great Particle Search

The LHCb collaboration at CERN decided to play detective. They looked for a very specific, rare event: a heavy particle called a $B^+$ meson decaying into a pion, an electron, and a muon.

• The Crime Scene: The Large Hadron Collider (LHC), where protons smash together at near-light speed.
• The Detective: The LHCb detector, a massive, high-tech camera and sensor array designed to catch these fleeting moments.
• The Suspect: The decay $B^+ \to \pi^+ \mu^\pm e^\mp$. In plain English: A heavy particle breaks apart, leaving behind a pion, a muon, and an electron.

Why is this exciting?
In the Standard Model, this specific crime is so rare that it's effectively impossible (the odds are less than 1 in $10^{50}$—a number so huge it's hard to comprehend). If the LHCb team saw even one of these events, it would be a smoking gun for "New Physics" (like Leptoquarks or extra dimensions).

The Investigation: Sifting Through the Noise

The team analyzed data collected between 2011 and 2018. That's a lot of data—about 9 "inverse femtobarns" of it. To use an analogy, imagine trying to find a specific, unique grain of sand on all the beaches of Earth. That's the scale of the data they were looking through.

How they did it:

1. The Filter (Trigger): The LHC produces millions of collisions per second. The computer system had to instantly decide which collisions were interesting enough to save. It looked for high-energy muons (like a bouncer checking for VIPs).
2. The Reconstruction: They pieced together the tracks of particles to see if they all came from the same "birthplace" (a vertex) inside the detector.
3. The "Look-Alikes" (Background): The hardest part is that nature loves to trick you. Sometimes, random particles just happen to line up in a way that looks like the signal. It's like hearing a noise in the dark that sounds like a ghost, but is actually just the wind.
   • To fight this, they used a Boosted Decision Tree (BDT). Think of this as a super-smart AI trained to distinguish between a real "ghost" (the signal) and a "wind noise" (background). It looked at the shape of the tracks, how far they traveled, and their energy.
4. The Calibration: They used a known, safe decay ($B^+ \to J/\psi K^+$) as a "control group" to make sure their measuring tape was accurate.

The Results: The Silence is the News

After all the filtering, the AI training, and the data crunching, the team looked at the final count.

• What they expected: If the Standard Model is perfect, they should see zero events.
• What they saw: They saw a few events, but the number matched exactly what they expected from "wind noise" (background). There was no "ghost."

The Verdict:
They found no evidence of the lepton-flavour violating decay. The universe is still obeying the rulebook in this specific instance.

However, this is a huge success for science because of what they didn't find.

The Takeaway: Setting the "Speed Limit"

Since they didn't find the signal, they set a new, incredibly strict upper limit.

• The Old Limit: Previous experiments (like CLEO and BaBar) said, "This decay happens less than 1 in 1,000,000 times."
• The New Limit: LHCb says, "No, it happens less than 1 in 1,000,000,000 times."

They improved the sensitivity by two orders of magnitude (100 times better).

Why does this matter?
Even though they didn't find the "flying penguin," they proved that if it does exist, it's much rarer than we thought. This forces theorists who build "New Physics" models to go back to the drawing board. If their model predicts this decay should happen often, their model is now wrong.

Summary in a Nutshell

• The Goal: Find a particle that breaks the laws of physics by changing its "flavor" (electron to muon).
• The Method: Smashed protons together for 7 years, used super-computers to filter out the noise, and looked for a specific pattern.
• The Result: No pattern found. The laws of physics held firm.
• The Impact: We now know this "forbidden" event is at least 100 times rarer than we previously thought. This tightens the screws on theories about what lies beyond our current understanding of the universe.

It's a bit like searching for a needle in a haystack, finding nothing, and then saying, "Okay, if the needle is in there, it's definitely not the size we thought it was." That's how science moves forward: by ruling out possibilities and narrowing the search.

Technical Summary

1. Problem Statement and Motivation

Lepton-Flavour Violation (LFV): While LFV is established in the neutrino sector via oscillations, it is strictly forbidden in the charged sector of the Standard Model (SM) at observable levels. In the SM, charged LFV decays are suppressed to branching fractions below $O(10^{-50})$ due to the GIM mechanism and the tiny neutrino masses.
Physics Beyond the Standard Model (BSM): Any observation of a charged LFV decay would constitute unambiguous evidence of new physics. The paper focuses on the rare decay $B^+ \to \pi^+ \mu^\pm e^\mp$, which involves a $b \to d$ quark transition. This channel is complementary to the more extensively studied $b \to s$ transitions (e.g., $B^+ \to K^+ \mu^\pm e^\mp$).
Current Status: Previous limits were set by the CLEO ($1.6 \times 10^{-6}$) and BaBar ($1.7 \times 10^{-7}$) collaborations. The LHCb collaboration aims to improve these limits by two orders of magnitude using the high-statistics dataset from the Large Hadron Collider (LHC).

2. Methodology

Data and Detector

• Dataset: Proton-proton collision data collected by the LHCb experiment between 2011 and 2018 at center-of-mass energies of 7, 8, and 13 TeV.
• Integrated Luminosity: Approximately $9 \text{ fb}^{-1}$.
• Detector: The LHCb single-arm forward spectrometer ($2 < \eta < 5$), equipped with a vertex locator, tracking system, Ring Imaging Cherenkov (RICH) detectors for particle identification (PID), calorimeters, and a muon system.

Event Selection and Reconstruction

• Signal Candidates: Reconstructed by combining three tracks ($\pi^+, \mu^\pm, e^\mp$) originating from a common secondary vertex (SV) well-separated from the primary vertex (PV).
• Mass Window: The reconstructed invariant mass $m(\pi^+ \mu^\pm e^\mp)$ is required to be in the range $[4500, 6000] \text{ MeV}/c^2$. The signal region is defined as $[4985, 5385] \text{ MeV}/c^2$.
• Bremsstrahlung Recovery: Photons associated with the electron trajectory are added to correct for energy loss due to bremsstrahlung.
• Categorization: Data is split into four independent categories based on the presence/absence of bremsstrahlung photons and the data-taking periods (2011–2012 vs. 2015–2018).

Background Suppression

• Boosted Decision Trees (BDT): Two sequential BDT classifiers are employed:
  1. First BDT: Trained on simulated signal and the upper mass sideband ($5385-6000 \text{ MeV}/c^2$) to suppress combinatorial background. It uses kinematic variables, vertex quality, impact parameters (IP), and isolation.
  2. Second BDT: Trained on simulated signal and the lower mass sideband ($4500-4985 \text{ MeV}/c^2$) to remove residual partially reconstructed backgrounds.
• Optimization: The selection criteria are optimized using a figure of merit $\varepsilon / (\sqrt{n} + 3/2)$, where $\varepsilon$ is signal efficiency and $n$ is the estimated background yield.

Normalization and Efficiency

• Normalization Mode: The branching fraction is measured relative to the well-known decay $B^+ \to J/\psi(\to \mu^+ \mu^-) K^+$. This mode shares similar kinematics and topology.
• Yield Extraction: The normalization yield is obtained via an unbinned extended maximum-likelihood fit to the $m(K^+ \mu^+ \mu^-)$ distribution, yielding $(1.3609 \pm 0.0012) \times 10^6$ candidates.
• Efficiency Correction: Signal efficiency is not uniform across the Dalitz plane. It is calculated in bins of squared invariant masses ($m^2_{\pi\mu}$ and $m^2_{\pi e}$) using simulation, corrected by data-driven calibration samples (e.g., $B^+ \to J/\psi \to e^+ e^-$, $D^{*+}$ decays).

Background Estimation

• Peaking Backgrounds: Dedicated simulations of exclusive $b$-hadron decays (e.g., $B^+ \to J/\psi K^+$, $B^+ \to \pi^+ \ell^+ \ell^-$) are used. None were found to produce a peaking structure near the signal mass.
• Non-Peaking/Mis-ID Backgrounds: A data-driven method involving inverting PID requirements on track combinations is used to estimate contributions from misidentified particles (e.g., pions misidentified as electrons). The mass shape is modeled using a kernel density estimator.

3. Key Contributions

1. First LHC Search: This is the first search for $B^+ \to \pi^+ \mu^\pm e^\mp$ at the LHC, providing the first constraint on LFV $b \to d$ transitions at this facility.
2. Improved Sensitivity: The analysis achieves a sensitivity two orders of magnitude better than previous world averages (CLEO and BaBar).
3. BSM Interpretation: The results are reinterpreted for specific Beyond the Standard Model scenarios, specifically:
   • Left-handed models: Characterized by Wilson coefficients $C_9^{\mu e} = -C_{10}^{\mu e} \neq 0$.
   • Scalar models: Characterized by $C_S^{\mu e} \neq 0$.
     The analysis accounts for the non-uniform Dalitz plane population predicted by these models, which affects signal efficiency.

4. Results

• Observation: No significant signal excess was observed in the signal region.
  • Observed Candidates: 36
  • Expected Background: $41 \pm 3$
• Branching Fraction Limit:
  • 90% Confidence Level (CL): $\mathcal{B}(B^+ \to \pi^+ \mu^\pm e^\mp) < 1.8 \times 10^{-9}$
  • 95% CL: $\mathcal{B}(B^+ \to \pi^+ \mu^\pm e^\mp) < 2.2 \times 10^{-9}$
  • Expected Limit (90% CL): $2.7 \times 10^{-9}$
• Measured Branching Fraction: $(-5.0^{+1.7}_{-1.5}) \times 10^{-9}$, consistent with zero.
• Systematic Uncertainties: The total systematic uncertainty is 7.7%, dominated by the background model (6.6%) and external branching fraction inputs (1.9%).
• BSM Limits:
  • Left-handed scenario: $< 1.8 \times 10^{-9}$ (90% CL).
  • Scalar scenario: $< 1.7 \times 10^{-9}$ (90% CL).

5. Significance

• Stringency: The new limit of $1.8 \times 10^{-9}$ is the most stringent upper limit to date on $b \to d \mu^\pm e^\mp$ transitions, improving upon the previous best limit ($1.7 \times 10^{-7}$) by a factor of $\sim 100$.
• BSM Constraints: These results significantly constrain models involving leptoquarks, extended gauge sectors ($Z'$), and non-minimal Higgs structures that predict enhanced LFV in $b \to d$ transitions.
• Complementarity: By probing the $b \to d$ sector, this search complements the extensive $b \to s$ LFV searches, offering a broader test of flavor symmetries in new physics models.
• Methodological Robustness: The use of a double BDT strategy, rigorous data-driven background estimation, and re-interpretation for specific BSM kinematic models sets a high standard for future rare decay searches at the LHC.