Branching fraction measurement of the $\mathit{\Lambda} \to p \mu^- \overline{\nu}_{\mu}$ decay

Using 13 TeV proton-proton collision data from the LHCb experiment, researchers measured the branching fraction of the $\Lambda \to p \mu^- \overline{\nu}_{\mu}$ decay with twice the precision of previous results, finding it consistent with Standard Model predictions and providing a stringent test of lepton flavour universality in $s \to u$ transitions.

The Gist

Imagine the universe as a giant, chaotic dance floor where particles are constantly bumping into each other, breaking apart, and reassembling. The LHCb experiment at CERN is like a super-powered security camera system watching this dance floor, trying to spot the rarest, most unusual moves.

This paper is about a specific, rare dance move performed by a particle called the Lambda baryon (let's call him "Lambda"). Usually, when Lambda dances, he splits into a proton and a pion (a type of particle). But sometimes, in a very rare twist, he splits into a proton, a muon (a heavy cousin of the electron), and a ghostly, invisible particle called a neutrino.

Here is the story of how the scientists caught this rare move, explained simply:

1. The Big Question: Is the Universe Fair?

In the Standard Model (the rulebook of physics), there is a principle called Lepton Flavour Universality. Think of this as a rule that says: "The universe treats all electron-like particles exactly the same, regardless of their weight."

Whether a particle interacts with a light electron or a heavy muon, the odds should be identical, once you account for the weight difference. However, recent experiments have hinted that the universe might be cheating—treating heavy muons differently than light electrons. This paper checks if Lambda is playing fair or if he's breaking the rules.

2. The Challenge: Finding a Needle in a Haystack

The scientists wanted to count how many times Lambda performed this rare "muon dance" ($\Lambda \to p \mu^- \nu_\mu$).

• The Problem: This happens incredibly rarely. For every million times Lambda does the "normal" dance (splitting into a proton and a pion), he only does the "muon dance" about 150 times.
• The Noise: The LHCb detector sees billions of collisions. Most of them are just background noise—random particles flying by that look like the dance but aren't. It's like trying to hear a single whisper in a rock concert.

3. The Strategy: The "Control Group" Trick

To count the rare muon dance accurately, the scientists didn't just count the rare ones; they counted the common ones too.

• The Normal Dance ($\Lambda \to p \pi^-$): They counted how many times Lambda did the common, boring dance. They knew the exact frequency of this dance from previous studies.
• The Ratio: By comparing the number of rare dances to common dances, they could calculate the exact probability of the rare one happening. It's like if you wanted to know how many people in a city eat a specific rare fruit, you wouldn't just count the rare fruit eaters. You'd count how many people eat apples (common) and how many eat the rare fruit, then use the known ratio of apples to the population to find the answer.

4. The Detective Work: Catching the Ghost

The hardest part was that the neutrino (the ghost particle) is invisible. It leaves no trace in the detector.

• The Missing Piece: Imagine you see a proton and a muon flying away. You know they came from Lambda, but you don't know where the neutrino went.
• The Math Trick: The scientists used the laws of physics (conservation of momentum) like a puzzle solver. They knew the direction Lambda was flying. By measuring the proton and muon, they could mathematically calculate where the "missing" neutrino must have gone to balance the equation.
• The Filter: They built a digital sieve. If the math didn't add up (meaning the particles didn't come from a Lambda decay), they threw the data out. This filtered out the "fake" dances and left them with the real ones.

5. The Result: A Perfect Score

After analyzing 5.4 years of data (a massive amount of information), they found:

• The Frequency: They confirmed the branching fraction (the odds) of this rare dance is $1.46 \times 10^{-4}$. This means about 1 in every 7,000 Lambdas does this specific move.
• The Precision: This is twice as precise as the previous best measurement. They sharpened the focus of their camera significantly.
• The Verdict on Fairness: When they compared the muon dance to the electron dance, the ratio was 0.175.
  • The Standard Model Prediction: 0.153 (with some theoretical wiggle room).
  • The Lattice QCD Prediction (a super-advanced computer simulation): 0.1735.
  • The Result: Their measurement (0.175) matches the advanced computer simulation almost perfectly.

The Takeaway

This paper is a victory for the Standard Model. It says, "So far, the universe is still playing fair." The heavy muons and light electrons are being treated exactly as the rulebook predicts.

While this might sound like "no new physics found," in the world of particle physics, precision is everything. By proving that the universe is still fair with such high accuracy, the scientists are closing the door on some theories that predicted the universe was unfair. They are setting a stricter boundary, telling future physicists: "If you want to find new physics, you have to look even harder, because the old rules are holding up stronger than ever."

In short: The LHCb team used a giant particle camera and some clever math to prove that a subatomic particle named Lambda is behaving exactly as the universe's rulebook says it should, even when it's doing its rarest, most invisible dance.

Technical Summary

1. Problem and Motivation

The paper addresses the need for precise tests of Lepton Flavour Universality (LFU) in charged-current transitions involving strange quarks ($s \to u$). While LFU violations have been hinted at in $b \to c$ transitions, the sector of semileptonic hyperon decays (SHDs) remains less explored experimentally despite its theoretical cleanliness.

• Physics Goal: To measure the branching fraction of the rare decay $\Lambda \to p\mu^-\nu_\mu$ with high precision and compare it to the electron mode ($\Lambda \to pe^-\bar{\nu}_e$) to test the Standard Model (SM) prediction for the ratio $R_{\mu e} = \Gamma(\Lambda \to p\mu^-\bar{\nu}_\mu) / \Gamma(\Lambda \to pe^-\bar{\nu}_e)$.
• Theoretical Context: SHDs are sensitive probes for Beyond-the-Standard-Model (BSM) scalar and tensor operators. The decay rate depends on the SU(3) flavour symmetry breaking parameter $\delta$. Theoretical predictions for $R_{\mu e}$ are clean, relying primarily on kinematic mass differences rather than complex form-factor inputs at Next-to-Leading Order (NLO).
• CKM Unitarity: A precise measurement of this decay allows for an independent extraction of the CKM matrix element $|V_{us}|$, which is crucial for testing the unitarity of the first row of the CKM matrix ($|V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 = 1$), currently subject to the "Cabibbo angle anomaly."

2. Methodology

Data Sample:

• Experiment: LHCb detector at CERN.
• Dataset: Proton-proton collisions at $\sqrt{s} = 13$ TeV collected between 2016 and 2018 (Run 2).
• Integrated Luminosity: $5.4 \text{ fb}^{-1}$.

Analysis Strategy:
The measurement is performed as a ratio relative to the well-known normalization channel $\Lambda \to p\pi^-$ to cancel systematic uncertainties.
$$\mathcal{B}(\Lambda \to p\mu^-\nu_\mu) = \mathcal{B}(\Lambda \to p\pi^-) \times \frac{\epsilon_{\Lambda \to p\pi}}{\epsilon_{\Lambda \to p\mu\nu}} \times \frac{N_{\Lambda \to p\mu\nu}}{N_{\Lambda \to p\pi}}$$

Key Technical Challenges & Solutions:

1. Background Suppression: The dominant background arises from $\Lambda \to p\pi^-$ decays where the pion decays in flight ($\pi^- \to \mu^-\bar{\nu}_\mu$) and is misidentified as a signal muon.

   • Kinematic Reconstruction: Since the neutrino is undetected, the analysis reconstructs the missing momentum using the $\Lambda$ flight direction (defined by Primary and Secondary vertices).
   • Variable 1: Longitudinal Neutrino Momentum ($p_L(\nu_\mu)$): Derived by imposing the $\Lambda$ mass constraint. Signal candidates yield a real, positive solution, whereas combinatorial background often fails this kinematic constraint.
   • Variable 2: Corrected Mass ($m_{\text{Corr}}(p\pi)$): For pions decaying outside the Vertex Locator (VELO), the momentum is corrected to point back to the Primary Vertex. This variable peaks at the $\Lambda$ mass for background but is shifted for signal.
   • Classification: Backgrounds are categorized by the pion decay point (inside or outside VELO) to apply specific kinematic cuts.

2. Selection & Fitting:

   • Trigger: The analysis selects candidates triggered by particles other than the signal (TIS - Trigger Independent of Signal) to ensure trigger efficiency consistency between signal and normalization modes.
   • Fit Model: A binned maximum-likelihood fit is performed in a 2D plane of $m_{\text{Corr}}(p\pi)$ vs. $m(p\pi)$.
   • Templates: Signal and background shapes are derived from simulation (Pythia/EvtGen/Geant4) and corrected using data-driven calibration samples (sPlot technique) for particle identification (PID), tracking, and trigger efficiencies.

3. Efficiency Corrections:

   • Corrections are applied for PID, tracking, and trigger efficiencies using calibration tools (PIDCalib2, TrackCalib2).
   • Simulation samples are reweighted to match data distributions in $\Lambda$ transverse momentum ($p_T$) and pseudorapidity ($\eta$).

3. Key Contributions

• First LHCb Measurement: This is the first study of LFU in hyperon decays at the LHC.
• Precision Improvement: The measurement improves the precision of the $\Lambda \to p\mu^-\nu_\mu$ branching fraction by a factor of two compared to the previous best measurement (BESIII, 2021).
• Robust Background Handling: The development of the $p_L(\nu_\mu)$ and $m_{\text{Corr}}$ variables allows for the effective separation of the signal from the dominant $\Lambda \to p\pi^-$ (pion-in-flight) background, which constitutes ~60% of the preselected sample.
• Systematic Control: By using a ratio method with TIS triggers and common kinematic selections, most systematic uncertainties (tracking, PID, trigger) largely cancel out.

4. Results

Branching Fraction:
The measured branching fraction is:
$$\mathcal{B}(\Lambda \to p\mu^-\nu_\mu) = (1.462 \pm 0.016_{\text{stat}} \pm 0.100_{\text{syst}} \pm 0.011_{\text{norm}}) \times 10^{-4}$$

• Total Uncertainty: $\approx 6.9\%$.
• Comparison: Consistent with the BESIII result of $(1.48 \pm 0.21) \times 10^{-4}$ but with significantly reduced uncertainty.

Lepton Flavour Universality Test:
Using the world average for the electron mode $\mathcal{B}(\Lambda \to pe^-\bar{\nu}_e) = (8.34 \pm 0.14) \times 10^{-4}$, the experimental ratio is:
$$R_{\mu e}^{\text{exp}} = 0.175 \pm 0.012$$

• Comparison to Theory:
  • Consistent with the NLO SM prediction ($0.153 \pm 0.008$) within $1.5\sigma$.
  • Consistent with recent Lattice QCD predictions ($0.1735 \pm 0.0098$) within $0.1\sigma$.
  • Consistent with the more precise Lattice result ($0.16638 \pm 0.00020$) within $0.95\sigma$.

CKM Element $|V_{us}|$:
Extracting $|V_{us}|$ using Lattice QCD form factors yields:

• Conservative input: $|V_{us}| = 0.235 \pm 0.016$
• Alternative input: $|V_{us}| = 0.2459 \pm 0.0085$
  Both values are consistent with CKM unitarity constraints, providing an independent check on the "Cabibbo angle anomaly."

5. Significance

• Validation of SM: The result confirms Lepton Flavour Universality in $s \to u$ transitions, placing stringent bounds on new physics scenarios involving scalar or tensor currents in hyperon decays.
• Complementarity: This measurement provides a crucial independent constraint on $|V_{us}|$ derived from baryonic decays, complementing measurements from kaon decays ($K_{\ell 3}$ and $K_{\mu 2}$). This is vital given the current $3\sigma$ tension between different $|V_{us}|$ determination methods.
• Experimental Benchmark: The analysis demonstrates the capability of the LHCb experiment to perform high-precision measurements of rare strange-hadron decays, overcoming the challenges of the hadronic environment at the LHC through advanced kinematic reconstruction and background suppression techniques.