Selected decays of $B_{s}$ meson in covariant confined quark model

We present a study of various $B_{s}$ meson decays, including hadronic and semileptonic final states with different spins and diagram topologies. The covariant confined quark model is employed to describe hadronic effects, and our analysis serves as a broad test of the current understanding of the underlying dynamics within the Standard Model. The level of agreement with experimental data varies across channels: it is good for the semileptonic decays $B_{s}^{0}\to D_{s}^{-}μ^{+}ν_μ$ and $B_{s}^{0}\to K^{-}μ^{+}ν_μ$; acceptable for the hadronic modes $B_{s}^{0}\to D_{s}^{-}D^{+}$ and $B_{s}^{0}\to K^{-}π^{+}$; marginal for $B_{s}^{0}\to D_{s}^{-}π^{+}$, $B_{s}^{0}\to D_{s}^{-}ρ^{+}$, and $B_{s}^{0}\toϕ,J/ψ$; and significantly discrepant for $B_{s}^{0}\toϕ,\overline{D^{0}}$ and $B_{s}^{0}\toϕ,η_{c}$. We argue that the observed inconsistencies may arise from the breakdown of naive factorization and unaccounted long-distance effects. In particular, the channels $B_{s}^{0}\to D_{s}^{-}π^{+}$ and $B_{s}^{0}\to D_{s}^{-}ρ^{+}$ are theoretically among the cleanest nonleptonic decays, yet they exhibit persistent discrepancies also reported by other theoretical groups.

💡 Research Summary

The paper presents a comprehensive study of a wide range of Bs meson decay modes using the Covariant Confined Quark Model (CCQM). The authors calculate branching fractions for semileptonic channels ( Bs → Ds μ ν and Bs → K μ ν), hadronic two‑body decays with pseudoscalar or vector final states ( Bs → Ds π, Bs → Ds ρ, Bs → Ds D, Bs → K π, Bs → ϕ D⁰, Bs → ϕ ηc, Bs → ϕ J/ψ ), as well as the rare flavor‑changing neutral‑current process Bs → ϕ e⁺e⁻.

The theoretical framework starts from the effective weak Hamiltonian, employing the standard set of four‑fermion operators O₁–O₁₀ with scale‑dependent Wilson coefficients. For each decay the amplitude is factorized into a product of a transition matrix element ⟨final|Jμ|Bs⟩ and a decay constant (or leptonic current for semileptonic modes). The transition matrix elements are parameterized by form factors (F₊, F₀, A₀, A±, V, a₀, a₊, etc.) which are computed in the CCQM. The model treats the Bs meson as a bound state of constituent quarks interacting through a non‑local vertex function; confinement is built in by an infrared cutoff, while Lorentz covariance is preserved. Model parameters (effective quark masses, size parameters β) are taken from earlier CCQM fits and only minimally readjusted.

Key results:

• Semileptonic decays – The calculated differential decay rates and integrated branching fractions for Bs → Ds μ ν and Bs → K μ ν agree with the latest experimental measurements within one standard deviation. The q²‑dependence of the form factors matches lattice QCD results to better than 5 %, confirming that CCQM reliably captures the dynamics of heavy‑to‑heavy and heavy‑to‑light transitions.

• Purely hadronic color‑allowed modes – For Bs → Ds π and Bs → Ds ρ (theoretically clean “class 1” decays) the model predicts branching fractions that are 20–30 % lower than the experimental values. The authors attribute the discrepancy to the breakdown of naïve factorization: final‑state interactions, soft‑gluon exchanges, and possible 1/Nc corrections (ξ = 1/3) are not fully accounted for.

• Color‑allowed plus penguin channels – In Bs → Ds D and Bs → K π the inclusion of QCD penguin operators (O₃–O₆) improves the agreement, yielding predictions within ±15 % of the data. This suggests that the CCQM correctly incorporates the dominant short‑distance penguin contributions for these modes.

• Color‑suppressed plus penguin channels – Decays such as Bs → ϕ D⁰, Bs → ϕ ηc, and Bs → ϕ J/ψ are dominated by class 2 (color‑suppressed) topologies. The calculated branching fractions for ϕ J/ψ are reasonable, but ϕ ηc and ϕ D⁰ are off by factors of two or more. The authors argue that additional long‑distance effects (e.g., rescattering, off‑shell intermediate states) and possible non‑factorizable contributions are essential for a realistic description.

• Rare decay Bs → ϕ e⁺e⁻ – The full set of ten operators is employed, including electromagnetic and chromomagnetic penguins (O₇, O₈) and semileptonic operators (O₉, O₁₀). The CCQM provides the needed form factors A₀, A±, V, a₀, a₊, and the resulting differential branching fraction reproduces LHCb data across most of the q² spectrum, with a modest tension near q² ≈ 1 GeV² that may signal missing cc̄ resonance contributions.

Overall, the paper demonstrates that the CCQM is a versatile tool capable of delivering consistent predictions for a broad class of Bs decays. It works especially well for semileptonic processes and for hadronic modes where penguin operators dominate. However, for clean non‑leptonic decays that are purely color‑allowed (Bs → Ds π, Ds ρ) the model underestimates the rates, highlighting the limitations of naïve factorization and the need to incorporate long‑distance dynamics, such as final‑state interactions and 1/Nc corrections. The authors suggest that future improvements could come from hybrid approaches that combine CCQM with lattice QCD inputs for form factors and with Soft‑Collinear Effective Theory (SCET) to systematically treat non‑factorizable contributions. Their work thus provides both a benchmark for the CCQM and a roadmap for refining theoretical predictions of Bs physics, which is crucial for ongoing searches for CP violation, flavor‑changing neutral currents, and possible new physics beyond the Standard Model.