Probing Direct $CP$ Violation in $Λ_b^0 o P_c^+ h^-$ $(h=π,K)$ with Final-State Rescattering

The LHCb collaboration has recently reported a measurement of the difference in direct CP asymmetries for the decays $Λ_b^0 \to J/ψ, p , h^-$ (with $h = K, π$), offering new experimental constraints on the decay dynamics of heavy baryons into charmonium final states. Inspired by these findings, we explore the branching ratios and direct CP violations for the decays $Λ_b^0 \to P_c^+(4312, 4440, 4457),h^-$ within the framework of final-state rescattering. Our analysis indicates that the branching fractions for $Λ_b^0 \to P_c^+ π^-$ lie around the $10^{-6}$ level, with the corresponding direct CP asymmetries approaching approximately $1%$. In contrast, the direct CP violation for the decay $Λ_b^0 \to P_c^+ K^-$ is found to be very small, while its branching ratios show a strong dependence on the spin assignments of the $P_c$ states. These predictions may provide useful guidance for more precise CP measurements and amplitude analyses in the $P_c$ region in future experiments.

💡 Research Summary

The paper investigates direct CP violation in the two‑body decays Λb0 → Pc h− with h = π or K, motivated by the recent LHCb measurement of the CP‑asymmetry difference ΔACP = (4.31 ± 1.06 ± 0.28)% between Λb0 → J/ψ p π− and Λb0 → J/ψ p K−. The authors assume the three hidden‑charm pentaquark candidates Pc(4312), Pc(4440) and Pc(4457) to be Σc D̄(∗) hadronic molecules, with possible spin‑parity assignments J P = 1/2− or 3/2− for the latter two states.

The theoretical framework is a final‑state rescattering (FSI) model. In this picture the weak decay Λb0 first produces an intermediate pair of hadrons (e.g. Λc D̄, Λc D̄*, Σc D̄, Σc D̄*) through a short‑distance weak vertex treated in naïve factorisation. The intermediate particles then rescatter via the exchange of a single light or charmed meson (π, K, ρ, D, D*) forming a triangle loop that converts the intermediate state into the final Pc h− system. The weak vertex includes CKM factors (Wolfenstein parameters A = 0.823, λ = 0.225, ρ = 0.141, η = 0.349) and heavy‑to‑light transition form factors taken from lattice‑QCD and QCD‑sum‑rule studies. Strong vertices are derived from effective hadronic Lagrangians; the coupling constants of Pc to Σc D̄(∗) and to nucleon‑meson channels are extracted from measured partial widths and heavy‑quark spin‑symmetry relations (see Table 1).

To regularise the ultraviolet behaviour of the triangle loop, a monopole‑type form factor
F(Λ, mk) = Λ⁴/