Final-state rescattering in $\bar{B}^{0}_{(s)}\to \Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ decays

This paper investigates the recently observed $\bar{B}^{0}_{(s)}\to \Lambda^{+}_{c}\bar{\Lambda}^{-}_{c}$ decays within a final-state rescattering framework, demonstrating that long-distance interactions successfully explain the measured branching fractions while predicting nearly vanishing CP asymmetries and a sizable longitudinal polarization for the $\bar{B}^{0}$ mode.

The Gist

Imagine the subatomic world as a bustling, chaotic dance floor. In this paper, the authors are trying to understand a very specific, rare dance move performed by a heavy particle called a $\bar{B}^0$ meson (think of it as a heavy, unstable dancer).

This dancer wants to split into two new partners: a $\Lambda_c^+$ baryon and an $\bar{\Lambda}_c^-$ baryon (imagine two heavy, distinct twins).

The Mystery: The "Impossible" Move

For a long time, physicists had a rulebook (called "naive factorization") that predicted how this dance should happen. According to that old rulebook:

1. One type of dancer ($\bar{B}^0$) should easily split into the twins.
2. The other type ($\bar{B}^0_s$) should barely be able to do it at all. It was thought to be "helicity suppressed," which is a fancy way of saying the move was so awkward and difficult that it should almost never happen.

The Problem: When the LHCb experiment (a giant particle detector) actually watched the dance floor, they saw something confusing. Both types of dancers were splitting into twins at almost the exact same rate. The "impossible" move was happening just as often as the "easy" one. The old rulebook was wrong.

The Solution: The "Bump-and-Grind" (Final State Rescattering)

The authors of this paper propose a new explanation. They suggest that the dancers don't just split directly. Instead, they take a detour.

Think of it like this:

1. The heavy dancer ($\bar{B}^0$) first splits into two different temporary partners (like a pair of D-mesons or a charmonium particle).
2. These temporary partners bump into each other, exchange a particle (like a ball being thrown back and forth), and then rescatter (rearrange themselves) into the final twins ($\Lambda_c^+ \bar{\Lambda}_c^-$).

This "bump-and-grind" process is called Final State Rescattering (FSI). It's a long-distance interaction that the old rulebook ignored. The authors argue that this extra step is what boosts the "impossible" move up to the same level as the "easy" one, matching what the experiments actually saw.

How They Calculated It

To prove this, the authors built a mathematical model of these "bump-and-grind" scenarios.

• The Loop: They calculated every possible way the temporary partners could meet and swap particles. They looked at loops where the particles are made of "charm" (heavy) and loops where they are "charmless" (lighter).
• The Cutoff: To make the math work without blowing up, they used a "cutoff" parameter. Think of this as a safety net or a speed limit for the interaction. They didn't invent new numbers; they borrowed the exact same safety limits they had successfully used in a previous study of a different particle ($\Lambda_b$). This makes their prediction very robust because they aren't just tweaking numbers to fit the data; they are applying a known rule to a new situation.

The Results: What They Found

When they ran the numbers with these "bump-and-grind" effects included:

1. The Rates Match: Their predicted rates for both decays lined up perfectly with the experimental data. This confirms that the "long-distance" rescattering is the secret ingredient that makes the "impossible" move happen.
2. No Big Surprises (CP Asymmetry): They also looked for a phenomenon called "CP asymmetry," which is like checking if the dance looks different when played in a mirror. They found that for these specific decays, the mirror image looks almost exactly the same. The asymmetry is nearly zero. This is different from some previous theories that predicted a big difference. The authors say this is because including the "heavy" intermediate partners (vector mesons) smooths things out, canceling out the differences.
3. The Spin (Polarization): They predicted how the final twins would be spinning.
   • For the $\bar{B}^0$ decay, the twins should be spinning in a very specific, noticeable way (longitudinal polarization).
   • For the $\bar{B}^0_s$ decay, the twins should be spinning in a way that is almost perfectly balanced (close to zero polarization).

The Bottom Line

This paper solves a puzzle: Why are two particle decays happening at the same rate when theory said they shouldn't? The answer is rescattering. The particles take a detour, bump into other particles, and rearrange themselves, which boosts the rare event to match the common one.

The authors conclude that future experiments should check their prediction about the spinning (polarization) of the particles. If the experiments see the specific spin patterns the authors predicted, it will confirm that this "bump-and-grind" rescattering is indeed the correct way to understand how these heavy particles decay.

Technical Summary

Technical Summary: Final-state rescattering in $\bar{B}^0_{(s)} \to \Lambda_c^+ \bar{\Lambda}_c^-$ decays

Problem Statement
The LHCb Collaboration has recently reported the first observation of the decay $\bar{B}^0_s \to \Lambda_c^+ \bar{\Lambda}_c^-$ and provided measurements for the branching fractions of both $\bar{B}^0 \to \Lambda_c^+ \bar{\Lambda}_c^-$ and $\bar{B}^0_s \to \Lambda_c^+ \bar{\Lambda}_c^-$. Theoretical understanding of these processes faces a significant challenge: within the naive diagrammatic estimation, $\bar{B}^0 \to \Lambda_c^+ \bar{\Lambda}_c^-$ is expected to be dominated by internal $W$-emission, while $\bar{B}^0_s \to \Lambda_c^+ \bar{\Lambda}_c^-$ proceeds solely through a $W$-exchange topology. The latter is traditionally regarded as helicity-suppressed and negligible in phenomenological analyses of two-body baryonic $B$ decays. However, experimental data shows that the branching fractions of both decays are of the same order of magnitude ($\sim 10^{-5}$), suggesting that the $W$-exchange contribution is sizable and cannot be ignored. Furthermore, previous theoretical studies incorporating final-state interactions (FSIs) with only pseudoscalar-meson intermediate states predicted a large direct CP asymmetry for $\bar{B}^0_s \to \Lambda_c^+ \bar{\Lambda}_c^-$ and a branching fraction ($O(10^{-4})$) inconsistent with updated experimental results.

Methodology
This work investigates the decays $\bar{B}^0_{(s)} \to \Lambda_c^+ \bar{\Lambda}_c^-$ within the framework of final-state re-scattering (FSIs). The process is modeled as a two-step mechanism: the initial $\bar{B}^0_{(s)}$ meson undergoes a weak decay into a pair of intermediate hadrons, which subsequently re-scatter into the final $\Lambda_c^+ \bar{\Lambda}_c^-$ state via single-particle exchange.

Key methodological features include:

• Intermediate States: The analysis considers a comprehensive set of ground-state intermediate hadrons, including light pseudoscalar ($P_8$) and vector ($V$) mesons, charmed mesons ($D^{(*)}$), charmonia ($\eta_c, J/\psi$), the light-baryon octet, and ground-state charmed-baryon multiplets ($B_{\bar{3}}, B_6$). This extends previous work by explicitly including vector-meson intermediate states (e.g., $D^*D^*$, $K^*K^*$) alongside pseudoscalar states.
• Loop Calculation: The long-distance FSI amplitudes are evaluated using the loop-integral method. The amplitude is expressed as a triangle diagram integral involving weak production vertices ($V_W$) and strong rescattering vertices ($V_{1,2}$).
• Regulation: To account for off-shell effects and regulate the loop integrals, an effective form factor $F(k^2, \Lambda) = \Lambda^4 / [(k^2 - m_{ex}^2)^2 + \Lambda^4]$ is introduced.
• Parameters: Two cutoff parameters, $\Lambda_{charm}$ and $\Lambda_{charmless}$, are employed to characterize re-scattering contributions involving different mass scales. Crucially, these parameters are not fitted to the current data but are taken directly from a previous study of $\Lambda_b^0 \to p\pi^-, pK^-$ decays [11], where they were determined from experimental data. Specifically, $\Lambda_{charm} = 1.0 \pm 0.1$ GeV and $\Lambda_{charmless} = 0.5 \pm 0.1$ GeV. No additional free parameters are introduced.
• Weak Interaction: Weak vertices are treated within the naive factorization approximation, while strong vertices are constructed using hadronic effective Lagrangians.

Key Contributions and Results
The study provides numerical predictions for branching fractions, direct CP asymmetries, and various asymmetry parameters ($\alpha, \beta, \gamma$) for both decay channels.

1. Branching Fractions: The predicted branching fractions are consistent with the experimental measurements reported by LHCb.

   • $\mathcal{B}(\bar{B}^0 \to \Lambda_c^+ \bar{\Lambda}_c^-)_{FSI} = (0.67^{+0.57}_{-0.34}) \times 10^{-5}$ (Experimental: $(1.01^{+0.27}_{-0.28} \pm 0.08 \pm 0.15) \times 10^{-5}$).
   • $\mathcal{B}(\bar{B}^0_s \to \Lambda_c^+ \bar{\Lambda}_c^-)_{FSI} = (3.43^{+3.10}_{-1.80}) \times 10^{-5}$ (Experimental: $(5.0 \pm 1.3 \pm 0.5 \pm 0.8) \times 10^{-5}$).
     This agreement supports the hypothesis that long-distance final-state interactions play an essential role in enhancing the $W$-exchange contribution, which is otherwise suppressed in naive factorization.

2. Direct CP Asymmetries: The paper predicts nearly vanishing direct CP asymmetries for both decays:

   • $A_{CP}^{dir}(\bar{B}^0 \to \Lambda_c^+ \bar{\Lambda}_c^-) \approx 1.13 \times 10^{-3}$.
   • $A_{CP}^{dir}(\bar{B}^0_s \to \Lambda_c^+ \bar{\Lambda}_c^-) \approx -6.61 \times 10^{-4}$.
     This result contrasts with previous predictions of a sizable ($\sim -10\%$) CP asymmetry for the $\bar{B}^0_s$ mode. The authors attribute this suppression to the inclusion of charmed vector meson intermediate states ($D^*, D^*_s$). Since the branching fractions of $B \to D^*D^*$ modes are significantly larger than $B \to DD$, the inclusion of these states substantially enhances the tree-level amplitudes (proportional to $V_{cb}V_{cq}^*$). This enhancement suppresses the interference between amplitudes with different weak phases, thereby reducing the CP asymmetry.

3. Asymmetry Parameters and Polarization:

   • For $\bar{B}^0 \to \Lambda_c^+ \bar{\Lambda}_c^-$, the longitudinal polarization parameter $\alpha$ is predicted to be $-0.58^{+0.05}_{-0.05}$, indicating a sizable negative longitudinal polarization of the $\Lambda_c^+$.
   • For $\bar{B}^0_s \to \Lambda_c^+ \bar{\Lambda}_c^-$, $\alpha$ is predicted to be close to zero ($-0.04^{+0.07}_{-0.07}$), as the two helicity amplitudes have nearly equal magnitudes.
   • The parameters $\beta$ and $\gamma$ show qualitatively similar behavior in both modes, reflecting the similarity of the strong phase structures in the dominant amplitudes.

Significance
The paper claims that its results provide a candidate dynamical explanation for the observed enhancement of the $W$-exchange contribution in charmed baryonic $B$ decays, resolving the tension between naive theoretical expectations and experimental data. By demonstrating that a model with no additional free parameters (using cutoffs derived from $\Lambda_b$ decays) can reproduce the branching fractions of $\bar{B}^0_{(s)} \to \Lambda_c^+ \bar{\Lambda}_c^-$, the work underscores the importance of long-distance final-state interactions in these processes.

Furthermore, the prediction of nearly vanishing CP asymmetries serves as a specific testable benchmark. If future experimental measurements confirm these small CP asymmetries, it would validate the necessity of including vector meson intermediate states in the FSI framework. Conversely, the sizable longitudinal polarization predicted for $\bar{B}^0 \to \Lambda_c^+ \bar{\Lambda}_c^-$ offers a sensitive observable for testing the theoretical framework in future experimental studies. The methodology presented is also noted as applicable to investigating other $B$ meson decays into charmed baryon pairs.