Branching fraction of $\Xi_{bc}^+\to \Xi_{c}^+ J/\psi$ in the final-state-interaction approach

This paper predicts the branching fraction of the $\Xi_{bc}^{+}\to \Xi_{c}^{+}J/\psi$ decay to be $(1.55_{-0.42}^{+0.50})\times10^{-4}$ using a final-state-interaction approach calibrated with the $\Lambda_{b}^{0}\to \Lambda^0 J/\psi$ control mode, suggesting that observable signal events are feasible in the near future.

The Gist

Imagine the universe as a giant, bustling construction site where tiny building blocks called quarks are constantly being assembled into larger structures called baryons (which are like heavy-duty bricks).

Most of the time, we see bricks made of two heavy blocks (like two charm quarks) or one heavy block and one light block. But physicists have been hunting for a very specific, rare "double-heavy" brick: one made of a bottom quark and a charm quark stuck together. They call this the $\Xi_{bc}^+$ (Zee-BC-plus).

This paper is a theoretical "blueprint" that helps experimentalists know exactly where to look for this rare brick and how likely they are to find it.

Here is the breakdown of the paper's story, using some everyday analogies:

1. The Missing Brick

For years, scientists have predicted this "bottom-charm" brick should exist, but they haven't found it yet. The LHCb experiment (a giant particle detector at CERN) has been digging through the debris of high-speed particle collisions, looking for it. They've seen some faint hints (like a blurry footprint in the mud), but nothing definitive.

The problem is: Where exactly should they look?
There are thousands of ways these particles can decay (break apart). The scientists need to know which "break-up pattern" is the most promising to spot the original brick.

2. The "Magic Trick" Decay

The authors of this paper focus on a specific way the $\Xi_{bc}^+$ brick might break apart:

• The Parent: $\Xi_{bc}^+$ (The rare bottom-charm brick).
• The Children: $\Xi_c^+$ (A lighter, single-charm brick) + $J/\psi$ (A very special, heavy "ball" made of a charm and anti-charm pair).

Why is this specific break-up important?

• The $J/\psi$ is a Beacon: When the $J/\psi$ breaks down further, it shoots out two muons (particles similar to electrons). These muons are like bright neon signs in a dark room. They are very easy for detectors to spot, unlike other debris which gets lost in the noise.
• The "Color-Suppressed" Problem: In the world of quarks, this specific break-up is considered "color-suppressed." Think of it like trying to push a heavy door that is slightly stuck. Standard physics rules (called "factorization") say this door should be very hard to push, meaning the event should be rare. But in the quantum world, things aren't always that simple.

3. The "Final-State Interaction" (The Bouncing Ball)

This is the core of the paper's innovation. The authors argue that the "stuck door" isn't actually that hard to push because of a hidden mechanism called Final-State Interaction (FSI).

The Analogy:
Imagine you throw a ball (the decay) at a wall.

• Standard Theory: You calculate the ball's path based only on the throw.
• FSI Theory: You realize that after the ball hits the wall, it might bounce off a nearby trampoline, hit a second wall, and then land in the target zone. This "bouncing around" (rescattering) changes the outcome significantly.

In the paper, the authors calculate these "bounces" (quantum loops) where intermediate particles swap places before settling into the final $J/\psi$ and $\Xi_c$. They found that these "bounces" actually make the decay happen much more often than the simple "stuck door" theory predicted.

4. Calibrating the Model (The Control Group)

To make sure their "bouncing ball" math is correct, they needed a reference point. They couldn't just guess the numbers.

• They used a similar, well-known decay: $\Lambda_b \to \Lambda J/\psi$.
• Think of this as using a known recipe to calibrate a new oven. They know how well the "known recipe" works in the real world. By adjusting their math until it matches the real-world data of the $\Lambda_b$ decay, they could trust their predictions for the unknown $\Xi_{bc}^+$ decay.

5. The Prediction

After doing the heavy lifting of the math (including complex loop calculations and "bouncing" effects), they came up with a number:

The Branching Fraction is predicted to be roughly $1.55 \times 10^{-4}$.

What does that mean?
It means that if you produce 100,000 of these rare $\Xi_{bc}^+$ bricks, about 15 of them will break apart in this specific, easy-to-spot way ($\Xi_c^+ + J/\psi$).

6. Will We Find It? (The Signal Events)

The authors did the final math to see if the LHCb detector can actually catch these 15 events.

• They estimated how many $\Xi_{bc}^+$ bricks the LHC produces.
• They factored in how good the detector is at spotting the "neon signs" (the muons).
• The Result: In the near future, with current data, they expect to see about 16 signal events. If they run the collider longer (collecting more data), that number could jump to 140 events.

The Bottom Line

This paper is a roadmap for discovery.
It tells experimentalists: "Don't just look anywhere. Look specifically for the $\Xi_{bc}^+$ breaking into a $\Xi_c$ and a $J/\psi$. Because of the 'bouncing' effects we calculated, this isn't a rare needle in a haystack; it's a small pile of needles we can actually find."

With the new data coming from the LHC, the authors are confident that the first "bottom-charm" baryon is about to be discovered, finally completing a missing piece of the particle physics puzzle.

Technical Summary

Here is a detailed technical summary of the paper "Branching fraction of $\Xi^+_{bc} \to \Xi^+_c J/\psi$ in the final-state-interaction approach".

1. Problem Statement

The existence of doubly heavy baryons containing both bottom and charm quarks ($bc$) has been a long-standing prediction of the quark model. While the doubly charmed baryon ($\Xi_{cc}$) has been observed, charmed-bottom baryons ($\Xi_{bc}, \Omega_{bc}$) remain undetected.

• Experimental Context: The LHCb collaboration has searched for these particles but has only observed local excesses (significances of 2.4$\sigma$–2.8$\sigma$) in the decay channel $\Xi^+_{bc} \to J/\psi \Xi^+_c$.
• Theoretical Gap: This specific decay mode is dominated by color-suppressed nonfactorizable contributions. Standard theoretical approaches (like naive factorization) often fail to accurately predict rates for such modes because they neglect long-distance effects. There was a lack of a robust theoretical framework to calculate the branching fraction of $\Xi^+_{bc} \to \Xi^+_c J/\psi$ to guide experimental searches.
• Challenge: Calculating nonfactorizable contributions in heavy baryon decays is difficult due to strong interaction dynamics and the lack of direct experimental data for $bc$ baryons to constrain model parameters.

2. Methodology

The authors employ the Final-State Interaction (FSI) approach to calculate the branching fraction. This method treats the decay as a combination of short-distance weak processes and long-distance strong rescattering.

• Short-Distance Amplitude:

  • Calculated using the effective weak Hamiltonian for the $b \to c\bar{c}s$ transition.
  • The process is dominated by the color-suppressed internal $W$-emission diagram (Diagram $C$).
  • Factorization is applied to the short-distance part, utilizing transition form factors calculated within the Light-Front Quark Model (LFQM).

• Long-Distance Amplitude (FSI):

  • The authors model the long-distance contributions via a triangle rescattering mechanism.
  • Mechanism: The initial baryon $\Xi^+_{bc}$ first decays into an intermediate state $\Xi^{++}_{cc} D^{(*)-}_s$ (via a color-allowed $T$ diagram to avoid double counting). These intermediate particles then rescatter via strong interactions to form the final state $\Xi^+_c J/\psi$.
  • Diagrams: The calculation includes four triangle diagrams involving intermediate states: $\{D_s, \Xi_{cc}\}$, $\{D^*_s, \Xi_{cc}\}$, $\{D_s, \Xi_{cc}\}$ (with $D^*$ exchange), and $\{D^*_s, \Xi_{cc}\}$ (with $D^*$ exchange).
  • Loop Integration: Unlike previous works that only calculated the imaginary part (absorptive part) using Cutkosky rules, this work computes the full complex amplitude (both real and imaginary parts) using the Passarino-Veltman method and integration-by-parts techniques for the loop integrals.

• Model Parameter Calibration:

  • The FSI calculation involves a nonperturbative cutoff parameter $\Lambda = m_k + \eta \Lambda_{QCD}$, where $\eta$ is a phenomenological parameter.
  • Since no experimental data exists for $\Xi_{bc}$, the authors calibrate $\eta$ using the analogous, well-studied decay $\Lambda_b \to \Lambda J/\psi$.
  • They reproduce the experimental branching fraction of $\Lambda_b \to \Lambda J/\psi$ (derived from $f(b \to \Lambda_b) \times \mathcal{B}$) to fix $\eta = 0.3 \pm 0.05$.

3. Key Contributions

1. First Theoretical Prediction: This is the first theoretical study to predict the branching fraction of $\Xi^+_{bc} \to \Xi^+_c J/\psi$ specifically accounting for nonfactorizable FSI effects.
2. Full Amplitude Calculation: The paper moves beyond calculating only the imaginary part of the triangle diagrams. By computing the full loop integrals, they provide a more accurate estimation of the strong phase and magnitude of the amplitude.
3. Robust Calibration Strategy: The use of $\Lambda_b \to \Lambda J/\psi$ as a control mode to constrain the model parameter $\eta$ provides a reliable anchor for predictions in the absence of direct $bc$ baryon data.
4. Signal Estimation: The authors provide a concrete estimate of the expected number of signal events at the LHCb experiment, bridging the gap between theoretical prediction and experimental feasibility.

4. Results

• Branching Fraction:
  The predicted branching fraction for the decay is:
  $$\mathcal{B}(\Xi^+_{bc} \to \Xi^+_c J/\psi) = (1.55^{+0.50}_{-0.42}) \times 10^{-4}$$
  The uncertainty is dominated by the variation of the model parameter $\eta$.

• Comparison with Other Modes:

  • The result is approximately half the branching fraction of $\Lambda_b \to \Lambda J/\psi$.
  • It is roughly one order of magnitude smaller than the color-allowed decay $B^+_c \to D^+_s J/\psi$, consistent with color-suppression arguments ($1/N_c^2$ scaling).

• Expected Signal Events at LHC:
  Using LHCb Run III parameters (integrated luminosity of $23 \text{ fb}^{-1}$) and projected production cross-sections:

  • Current Run: Approximately 16 signal events are expected.
  • Future High-Luminosity Run: With an integrated luminosity of $100 \text{ fb}^{-1}$, the yield is expected to reach ~140 events.
  • These numbers suggest that with current and upcoming datasets, the observation of $\Xi^+_{bc}$ via this channel is highly probable.

5. Significance

• Guidance for Discovery: The predicted branching fraction and signal yield provide a critical benchmark for the LHCb collaboration. The high signal purity of the $J/\psi \to \mu^+\mu^-$ decay channel, combined with the predicted rate, makes this the most promising "golden channel" for discovering the first charmed-bottom baryon.
• Understanding QCD Dynamics: The study validates the Final-State Interaction approach for doubly heavy baryon decays. It demonstrates that nonfactorizable contributions are significant and must be included to obtain reliable predictions for color-suppressed modes.
• Theoretical Framework: By successfully applying the FSI approach to a system with two heavy quarks, the paper establishes a robust theoretical framework that can be extended to other heavy baryon decays and potentially to the study of doubly bottom baryons ($\Omega_{bb}, \Xi_{bb}$) in the future.

In conclusion, this work resolves a major theoretical uncertainty regarding the $\Xi^+_{bc} \to \Xi^+_c J/\psi$ decay, confirming it as a viable and promising discovery channel for the charmed-bottom baryon at the LHC.