Study of the $B^0 \to \Lambda_c^+ \bar{\Lambda}_c^- K_S^0$ decay

Using LHCb data at $\sqrt{s} = 13$ TeV, this study presents the first observation of the $B^0 \to \Lambda_c^+ \bar{\Lambda}_c^- K_S^0$ decay, measuring its branching ratio relative to the charged mode and finding evidence for the isospin partners $\Xi_c(2923)^+$ and $\Xi_c(2939)^+$ in the $\Lambda_c^+ K_S^0$ system.

The Gist

Imagine the universe as a giant, high-speed particle collider, like a cosmic pinball machine where tiny building blocks of matter smash into each other at nearly the speed of light. This is what happens at CERN's Large Hadron Collider (LHC).

In this specific study, the LHCb team (a group of scientists acting like cosmic detectives) decided to look at a very specific, rare "crash" that happens when a heavy particle called a $B^0$ meson decays, or breaks apart.

Here is the story of what they found, explained simply:

1. The Crime Scene: A Rare Breakup

Usually, when these heavy particles break apart, they turn into a predictable set of smaller pieces. But sometimes, they do something weird and complex.

In this case, the scientists were watching for a specific breakup where a $B^0$ particle splits into three things:

• A $\Lambda_c^+$ (a heavy "charmed" baryon, think of it as a heavy, exotic brick).
• A $\Lambda_c^-$ (its anti-matter twin).
• A $K_S^0$ (a strange, short-lived meson, like a ghostly messenger).

This is a "three-body" breakup, which is much harder to study than a simple two-piece split because there are more ways the pieces can fly off. The scientists collected data from 5.4 billion of these collisions (imagine watching 5.4 billion pinball games to find just a few hundred that match your specific pattern).

2. The "Fingerprint" Comparison

To make sure they were seeing a real signal and not just random noise, the scientists compared this rare breakup to a more common "cousin" event.

• The Control Group: They looked at a $B^+$ particle breaking into the same heavy bricks but with a regular charged kaon ($K^+$) instead of the ghostly messenger.
• The Result: They found that the rare $B^0$ breakup happens about half as often as the common $B^+$ breakup. It's like finding that a specific rare flavor of ice cream is made 50% as often as the vanilla flavor in a factory. They measured this ratio very precisely: 0.53.

3. The Hidden Clues: Finding "Resonant" Ghosts

Here is the most exciting part. When the heavy bricks ($\Lambda_c$) and the messenger ($K_S^0$) fly apart, they don't just scatter randomly. The scientists noticed that they sometimes seem to "stick together" for a split second before flying apart.

Think of it like this: You throw two balls into the air. Usually, they just fly away. But sometimes, you see a pattern where they seem to bounce off an invisible trampoline before separating. That "trampoline" is a resonance—a short-lived, excited state of matter.

The scientists found evidence of two specific invisible trampolines:

1. $\Xi_c(2923)^+$
2. $\Xi_c(2939)^+$

These are excited versions of a particle called the $\Xi_c$ (Xi-c). They are like a guitar string that has been plucked and is vibrating at a specific, high-energy note before it stops vibrating.

4. The "Twin" Connection

Why are these two new findings so important?

• Scientists had previously found similar "trampolines" (excited particles) in the charged version of this breakup (using the $B^+$ particle).
• Nature loves symmetry. If you have a particle with a positive charge, you usually expect to find a "twin" with a neutral charge that acts exactly the same way.
• The two new states found in this paper are the neutral twins of the charged ones found earlier. It's like finding the left shoe after you already found the right one. They fit perfectly together, confirming our understanding of how these particles are built.

5. The Verdict

The scientists are about 99.99% sure (a statistical significance of 3.9 sigma) that these two new "trampolines" are real. While not quite the "gold standard" 5-sigma required to officially claim a "discovery" in physics, it is very strong evidence.

Summary

In plain English:
The LHCb team watched billions of particle collisions to catch a rare breakup. They measured how often it happens compared to a similar, more common breakup. Most importantly, they found evidence that during this breakup, the particles briefly form two specific, excited "twin" states. This confirms a beautiful symmetry in nature and helps us understand the complex "glue" (the strong force) that holds the universe's building blocks together.

It's like listening to a chaotic orchestra and finally hearing two specific, hidden instruments playing a perfect harmony that you knew was there, but couldn't quite prove until now.

Technical Summary

1. Problem and Motivation

The paper addresses the phenomenology of multibody $B$-meson decays, specifically those involving two open-charm hadrons and a strange meson arising from $b \to c\bar{c}s$ quark-level transitions. These decays are crucial for understanding QCD dynamics and discovering exotic states.

• Context: Previous studies by BaBar, Belle, and LHCb in the charged channel ($B^+ \to \Lambda_c^+ \Lambda_c^- K^+$) revealed structures in the $\Lambda_c^+ K^-$ spectrum, identified as excited charmed baryon states $\Xi_c(2923)^0$ and $\Xi_c(2939)^0$.
• Gap: The isospin partner channel, $B^0 \to \Lambda_c^+ \Lambda_c^- K_S^0$, had not been studied with high precision at LHCb. While Belle reported evidence for a charged $\Xi_c(2930)^+$ in this channel, a definitive confirmation and detailed amplitude analysis were lacking.
• Goals:
  1. Measure the branching ratio of $B^0 \to \Lambda_c^+ \Lambda_c^- K_S^0$ relative to $B^+ \to \Lambda_c^+ \Lambda_c^- K^+$.
  2. Investigate the $\Lambda_c^+ K_S^0$ spectrum for excited $\Xi_c^+$ states (isospin partners of the neutral states observed in $B^+$ decays).
  3. Search for potential charmonium-like exotic states decaying to $\Lambda_c^+ \Lambda_c^-$.

2. Methodology

Data and Detector

• Dataset: Proton-proton collision data collected by the LHCb experiment at a center-of-mass energy of $\sqrt{s} = 13$ TeV.
• Integrated Luminosity: $5.4 \text{ fb}^{-1}$.
• Reconstruction:
  • Signal: $B^0 \to \Lambda_c^+ \Lambda_c^- K_S^0$, with $\Lambda_c^+ \to p K^- \pi^+$ and $K_S^0 \to \pi^+ \pi^-$.
  • Control Channel: $B^+ \to \Lambda_c^+ \Lambda_c^- K^+$, used for normalization.
  • $K_S^0$ Categories: Two reconstruction modes were utilized:
    • LL (Long-Long): $K_S^0$ decays early; pions reconstructed with the full tracking system (superior mass resolution).
    • DD (Downstream-Downstream): $K_S^0$ decays later; pions reconstructed using downstream detectors only (higher yield).

Selection and Background Suppression

• Kinematic Cuts: Strict requirements on transverse momentum ($p_T$), impact parameter, and vertex quality were applied to suppress combinatorial background.
• Multivariate Analysis: A Boosted Decision Tree (BDT) classifier (implemented in TMVA) was trained to distinguish signal from background. The training used high-mass sidebands from data as background proxies and simulated candidates as signal proxies. Separate BDTs were optimized for LL and DD categories.

Signal Yield Determination

• Fit Strategy: A three-dimensional (3D) extended unbinned maximum-likelihood fit was performed on the invariant masses of the $B^0$, $\Lambda_c^+$, and $\Lambda_c^-$.
• Modeling:
  • Signal: Modeled by a double-sided Crystal Ball function.
  • Background: Modeled by first-order polynomials, categorized into seven types based on the origin of the candidates (e.g., genuine $B^0$ decays with misreconstructed daughters, pure combinatorial background).
• Yields: The signal yields were determined to be $148 \pm 14$ (DD) and $62 \pm 9$ (LL) for the signal mode, and $1335 \pm 43$ for the control mode.

Amplitude Analysis (Resonance Search)

• Constraints: To improve mass resolution, $B^0$ and $\Lambda_c$ candidates were constrained to narrow mass windows, and kinematic quantities were recalculated with masses fixed to PDG values.
• Fit Model: The $\Lambda_c^+ K_S^0$ invariant mass spectrum was fitted with:
  • A non-resonant phase-space component.
  • Two resonant states ($\Xi_c(2923)^+$ and $\Xi_c(2939)^+$) modeled using relativistic Breit-Wigner functions with Blatt-Weisskopf barrier factors.
  • Assumptions: Spin-parity ($J^P$) was assumed to be $3/2^-$ for both states (consistent with previous $B^+$ studies), and the orbital angular momentum was set to $L=2$.
  • Interference: Systematic uncertainties were evaluated by testing coherent sums with varying relative phases.

Branching Ratio Calculation

The relative branching fraction was calculated using:
$$\frac{\mathcal{B}(B^0 \to \Lambda_c^+ \Lambda_c^- K_S^0)}{\mathcal{B}(B^+ \to \Lambda_c^+ \Lambda_c^- K^+)} = \frac{N_{B^0}}{N_{B^+}} \times \frac{\epsilon_{B^+}}{\epsilon_{B^0}} \times \frac{1}{\mathcal{B}(K_S^0 \to \pi^+\pi^-)} \times \frac{f_u}{f_d}$$
Where $N$ is the signal yield, $\epsilon$ is the total efficiency (determined via simulation and corrected with data-driven tracking efficiency), and $f_u/f_d$ is the hadronization fraction ratio (assumed to be 1).

3. Key Contributions and Results

Branching Ratio Measurement

The paper reports the first measurement of the branching ratio for this decay at LHCb:
$$\frac{\mathcal{B}(B^0 \to \Lambda_c^+ \Lambda_c^- K_S^0)}{\mathcal{B}(B^+ \to \Lambda_c^+ \Lambda_c^- K^+)} = 0.53 \pm 0.05 \text{ (stat)} \pm 0.05 \text{ (syst)}$$
Using the known branching fraction of the $B^+$ mode, the absolute branching fraction is determined to be:
$$\mathcal{B}(B^0 \to \Lambda_c^+ \Lambda_c^- K_S^0) = (2.60 \pm 0.26 \pm 0.23 \pm 0.37) \times 10^{-4}$$
This represents the most precise measurement to date.

Discovery of Resonant States

• Evidence: The analysis provides evidence for two resonant states in the $\Lambda_c^+ K_S^0$ system: $\Xi_c(2923)^+$ and $\Xi_c(2939)^+$.
• Significance: The combined significance of the two-resonance structure relative to the non-resonant hypothesis is 3.9$\sigma$.
• Parameters:
  • $\Xi_c(2923)^+$: Mass $2923.1 \pm 1.8 \pm 0.8$ MeV/$c^2$, Width $11.6 \pm 5.2 \pm 2.6$ MeV/$c^2$.
  • $\Xi_c(2939)^+$: Mass $2937.5 \pm 1.6 \pm 0.9$ MeV/$c^2$, Width $7.1 \pm 4.1 \pm 2.4$ MeV/$c^2$.
• Consistency: These masses and widths are consistent with the isospin partners observed in the $B^+ \to \Lambda_c^+ \Lambda_c^- K^+$ decay and prompt production measurements.
• $\Lambda_c^+ \Lambda_c^-$ Spectrum: No significant structure was observed in the $\Lambda_c^+ \Lambda_c^-$ invariant mass spectrum, ruling out prominent charmonium-like exotic states in this specific decay channel within current statistical precision.

4. Significance

1. Isospin Symmetry Confirmation: The observation of $\Xi_c(2923)^+$ and $\Xi_c(2939)^+$ in the $B^0$ decay confirms the isospin partner relationship with the states seen in $B^+$ decays, solidifying the understanding of the excited charmed baryon spectrum.
2. Precision Measurement: The result provides the most precise determination of the $B^0 \to \Lambda_c^+ \Lambda_c^- K_S^0$ branching fraction, reducing uncertainties compared to previous Belle measurements.
3. QCD Dynamics: The study of these multibody decays offers unique insights into strong interaction dynamics and the nature of excited baryons, which are critical for testing theoretical models of QCD.
4. Methodological Benchmark: The successful application of 3D extended unbinned fits and BDT-based background suppression in a complex three-body final state serves as a robust template for future amplitude analyses at the LHCb experiment.

In conclusion, this paper establishes the existence of excited charmed baryon states in the neutral $B$-meson decay channel, completing the isospin picture of the $\Xi_c$ spectrum and providing a high-precision benchmark for heavy-flavor physics.