Measurement of the branching fraction of the $\Lambda_b^0\to J/\psi\Lambda$ decay and isospin asymmetry of $B\to J/\psi K$ decays

Using LHCb proton-proton collision data from 2016 to 2018, this paper reports a precise measurement of the $\Lambda_b^0\to J/\psi\Lambda$ branching fraction relative to $B^0\to J/\psi K^0_\text{S}$ and determines the isospin asymmetry between $B^0\to J/\psi K^0_\text{S}$ and $B^+\to J/\psi K^+$ decays.

The Gist

Imagine the Large Hadron Collider (LHC) at CERN as the world's most powerful particle smasher. It fires protons at each other at nearly the speed of light, creating a chaotic shower of new particles. Among the trillions of particles created, physicists are hunting for specific, rare "family members" called b-hadrons.

This paper is like a detective report from the LHCb collaboration, a team of scientists acting as cosmic detectives. They are trying to solve two main mysteries about how these heavy particles behave and decay.

Here is the breakdown of their findings in plain English:

1. The Cast of Characters

To understand the story, we need to know the players:

• The "B" Mesons and "Lambda-b" Baryons: Think of these as heavy, unstable parents. They are created in the collision but don't last long. They quickly break apart (decay) into lighter, more stable children.
• The "J/psi" Meson: This is a very stable, heavy child that acts like a "signature" or a "flag." Because it's so distinct, spotting it helps physicists say, "Aha! We found a parent that decayed this way!"
• The "Lambda" Baryon: A specific type of heavy child that is harder to catch than the others.

2. Mystery One: How Often Does the "Lambda-b" Parent Decay?

The Question: When a heavy Lambda-b parent breaks apart, how often does it turn into a J/psi and a Lambda child?

The Problem: In the past, it was hard to answer this because we didn't know exactly how many "Lambda-b" parents were being created in the first place. It's like trying to calculate the success rate of a magic trick without knowing how many times the magician actually tried the trick.

The Solution: The LHCb team didn't try to count the total number of parents. Instead, they used a reference trick.

• They knew very precisely how often a different parent (the B0 meson) turns into a J/psi and a K0 child.
• They counted how many Lambda-b decays they saw versus how many B0 decays they saw.
• By comparing the two, they could cancel out the uncertainty about how many parents were created. It's like saying, "For every 100 times I see the B0 trick, I see 75 times the Lambda-b trick."

The Result: They found that the Lambda-b decays into J/psi + Lambda about 3.34 times out of every 10,000 attempts. This is a very precise measurement, much better than previous guesses.

3. Mystery Two: The "Isospin" Balance

The Question: Nature has a rule called Isospin symmetry, which basically says that particles that are "twins" (differing only by a tiny electric charge) should behave exactly the same way.

• The Twins: The B+ (positive charge) and B0 (neutral charge) parents.
• The Expectation: If you smash protons together, the B+ and B0 should decay into their respective J/psi children at the exact same rate.

The Test: The team measured the decay rates of both the B+ and the B0. They calculated a number called Isospin Asymmetry ($A_I$). If the twins are perfectly equal, this number should be zero.

The Result: They measured the number to be -0.0135.

• What does this mean? It's incredibly close to zero. The tiny difference they saw is likely just random noise (like a slight wobble in a coin toss).
• The Conclusion: The "twins" are behaving exactly as the Standard Model of physics predicts. There is no evidence of a new, mysterious force breaking the symmetry.

4. How Did They Do It? (The Detective Work)

The LHCb detector is like a giant, high-speed camera that takes pictures of these particles as they fly through a tunnel.

• The Filter: They had to filter out billions of "junk" collisions to find the few thousand that mattered. They used a computer algorithm (a "decision tree") to pick out the specific patterns of particles that looked like the decay they were hunting for.
• The "Long" vs. "Downstream" Tracks: Some particles travel a long way before breaking apart (Long tracks), while others break apart almost immediately (Downstream tracks). The team had to account for the fact that the detector sees these two types differently, like adjusting a camera lens for near vs. far objects.
• The Simulation: They ran millions of computer simulations to understand how the detector works, ensuring that their real-world data wasn't being skewed by the machine itself.

The Big Picture

Why does this matter?

1. Precision: By measuring these decay rates so precisely, physicists are building a more accurate map of the universe.
2. Testing the Rules: If the "twins" (B+ and B0) had behaved differently, it would have been a massive discovery, suggesting new physics beyond our current understanding. Since they behaved the same, it confirms our current theories are solid.
3. The Lambda-b: Now that we know exactly how often the Lambda-b decays, we can use this knowledge to hunt for even rarer, forbidden decays that might reveal new secrets of the universe.

In short: The LHCb team successfully counted a specific type of particle decay by comparing it to a known standard, and they confirmed that nature's "twins" are indeed behaving symmetrically, just as the laws of physics predict.

Technical Summary

1. Problem and Motivation

The paper addresses two primary objectives in the field of heavy-flavor physics:

• Precision Measurement of $\Lambda_b^0$ Branching Fraction: While branching fractions for $B$ meson decays to $J/\psi$ final states (e.g., $B^0 \to J/\psi K_S^0$) are known with high precision from $B$-factory experiments, the branching fraction for the b-baryon decay $\Lambda_b^0 \to J/\psi \Lambda$ is less precise. Previous measurements at the Tevatron were limited by uncertainties in the production fraction of $\Lambda_b^0$ baryons ($f(b \to \Lambda_b^0)$), which is strongly dependent on transverse momentum ($p_T$).
• Isospin Asymmetry Validation: In the Standard Model (SM), the partial widths of $B^+ \to J/\psi K^+$ and $B^0 \to J/\psi K^0$ decays are expected to be nearly identical, as the only difference is the flavor of the light spectator quark. Measuring the isospin asymmetry ($A_I$) serves as a crucial test of the SM and validates the reconstruction efficiency for long-lived hadrons in the LHCb detector.

2. Methodology

Dataset and Detector:

• Data: Proton-proton collision data collected by the LHCb experiment at $\sqrt{s} = 13$ TeV during Run 2 (2016–2018).
• Integrated Luminosity: $5.4 \text{ fb}^{-1}$.
• Detector: LHCb, a single-arm forward spectrometer ($2 < \eta < 5$), optimized for $b$-hadron physics. Key subsystems include the Vertex Locator (VELO), tracking system, RICH detectors for particle identification (PID), and muon chambers.

Candidate Selection:

• Triggers: Hardware trigger requires high-$p_T$ muons; software trigger requires displaced muon pairs and large impact parameters.
• Reconstruction:
  • $\Lambda_b^0 \to J/\psi \Lambda$ with $\Lambda \to p \pi^-$ and $J/\psi \to \mu^+ \mu^-$.
  • $B^0 \to J/\psi K_S^0$ with $K_S^0 \to \pi^+ \pi^-$.
  • $B^+ \to J/\psi K^+$ with $K^+$ identified via PID.
• Topological Categories: Candidates are split into "long" (tracks reconstructed in the vertex detector) and "downstream" (tracks starting downstream of the vertex detector) categories to optimize reconstruction efficiency and purity.
• Background Suppression: Gradient-boosted decision trees (BDTs) are used to suppress combinatorial background in the long-track samples. Specific background sources (e.g., $B_s^0 \to J/\psi K_S^0$, $\Xi_b \to J/\psi \Xi$, misidentified pions) are modeled and constrained in the fit.

Analysis Strategy:

• Yield Extraction: Extended unbinned maximum-likelihood fits are performed on the reconstructed invariant mass distributions ($J/\psi K^+$, $J/\psi K_S^0$, $J/\psi \Lambda$) in bins of $b$-hadron $p_T$ (intervals of 1 GeV/$c$ up to 14 GeV/$c$, then 2 GeV/$c$).
• Simultaneous Fit: The $B^0$ and $\Lambda_b^0$ samples are fitted simultaneously to account for cross-feed backgrounds (e.g., $\Lambda_b^0$ reconstructed as $B^0$ and vice versa).
• Efficiency Correction: Efficiencies are determined using simulated samples corrected with data-driven calibration (using $D^{*+} \to D^0 \pi^+$ and $B^+ \to J/\psi K^+$ decays). The efficiency ratio $\epsilon[\Lambda_b^0]/\epsilon[B^0]$ is found to be weakly dependent on $p_T$.
• Production Fraction Handling: The analysis utilizes the $p_T$-dependent production fraction ratio $r(p_T) = f(b \to \Lambda_b^0) / [f(b \to B^0) + f(b \to B^+)]$ measured previously by LHCb, assuming isospin symmetry ($f(b \to B^0) \approx f(b \to B^+)$).

3. Key Contributions

1. First High-Precision $\Lambda_b^0 \to J/\psi \Lambda$ Branching Fraction: The measurement is performed relative to the well-known $B^0 \to J/\psi K_S^0$ decay, significantly reducing systematic uncertainties related to detector acceptance and luminosity.
2. $p_T$-Dependent Normalization: By analyzing data in $p_T$ bins and utilizing the known $p_T$-dependence of the $\Lambda_b^0$ production fraction, the analysis mitigates the large systematic uncertainty that plagued previous Tevatron measurements.
3. Isospin Asymmetry Validation: The measurement of $A_I$ provides a stringent test of the SM and confirms that the reconstruction efficiencies for $K^+$ and $K_S^0$ (and their associated lifetimes) are well-understood in the LHCb environment.
4. Comprehensive Systematic Treatment: The paper details a robust evaluation of systematic uncertainties, including material interactions (crucial for $K_S^0$ and $\Lambda$), simulation modeling, and external input parameters.

4. Results

Branching Fraction Ratio:
The ratio of branching fractions is measured as:
$$\frac{\mathcal{B}(\Lambda_b^0 \to J/\psi \Lambda)}{\mathcal{B}(B^0 \to J/\psi K_S^0)} = 0.750 \pm 0.005 (\text{stat}) \pm 0.022 (\text{syst}) \pm 0.005 (\text{ext}) \pm 0.062 (f_{\text{prod}})$$

Absolute Branching Fraction:
Using the world average $\mathcal{B}(B^0 \to J/\psi K_S^0) = (4.45 \pm 0.11) \times 10^{-4}$, the absolute branching fraction is determined to be:
$$\mathcal{B}(\Lambda_b^0 \to J/\psi \Lambda) = (3.34 \pm 0.02 \pm 0.10 \pm 0.08 \pm 0.28) \times 10^{-4}$$

• Uncertainties: Statistical, Systematic, External inputs (branching fractions of $\Lambda$ and $K_S^0$), and Production fraction ratio.

Isospin Asymmetry ($A_I$):
The isospin asymmetry between $B^+ \to J/\psi K^+$ and $B^0 \to J/\psi K_S^0$ is measured to be:
$$A_I = -0.0135 \pm 0.0004 (\text{stat}) \pm 0.0133 (\text{syst})$$
This result is consistent with zero (the SM expectation) and shows no significant dependence on $p_T$.

Production Fraction Ratio:
Assuming isospin symmetry in decays, the ratio of production cross-sections is derived as:
$$f(b \to B^+) / f(b \to B^0) = 1.027 \pm 0.001 (\text{stat}) \pm 0.027 (\text{syst})$$

5. Significance

• Theoretical Validation: The measured partial width of $\Lambda_b^0 \to J/\psi \Lambda$ is approximately 39% of that for $B^0 \to J/\psi K^0$ (after accounting for the $K^0 \to K_S^0$ probability and lifetime differences). This result is compatible with theoretical predictions based on perturbative QCD and QCD factorization approaches.
• Standard Model Consistency: The isospin asymmetry measurement confirms the SM prediction that spectator quark flavor does not significantly affect the decay rate of $b \to c\bar{c}s$ transitions, validating the detector's ability to handle long-lived neutral and charged hadrons with high precision.
• Future Utility: The precise determination of $\mathcal{B}(\Lambda_b^0 \to J/\psi \Lambda)$ provides a vital normalization channel for future LHCb analyses involving rare $\Lambda_b^0$ decays and CP violation studies in the baryon sector.
• Methodological Benchmark: The approach of binning by $p_T$ to handle production fraction uncertainties sets a standard for future measurements of b-baryon properties at the LHC.