Study on the mixing of Ξc and Ξc′ by the transition <mml:mi …

Recently, the LHCb Collaboration has observed the decays Ξb⁰→Ξc⁺Dₛ^- and Ξb^-→Ξc⁰Dₛ^-. They measured the relative branching fractions times the ratio of beauty-baryon production cross sections $R({{{{Ξ}}}b⁰}{{{{Λ}}}b}){≡}{{σ}({{{Ξ}}}b⁰)}{{σ}({{{Λ}}}b⁰)}×{}{B({{{Ξ}}}b⁰{→}{{{Ξ}}}c⁺{D}ₛ^{{-}})}{B({{{Λ}}}b⁰{→}{{{Λ}}}c⁺{D}ₛ^{{-}})}$ and $R({{{{Ξ}}}b^{{-}}}{{{{Λ}}}b}){≡}{{σ}({{{Ξ}}}b^{{-}})}{{σ}({{{Λ}}}b⁰)}×{}{B({{{Ξ}}}b^{{-}}{→}{{{Ξ}}}c⁰{D}ₛ^{{-}})}{B({{{Λ}}}b⁰{→}{{{Λ}}}c⁺{D}ₛ^{{-}})}$. Once the ratio ${{σ}({{{Ξ}}}b⁰)}{{σ}({{{Λ}}}b⁰)}$ or ${{σ}({{{Ξ}}}b^{{-}})}{{σ}({{{Λ}}}b⁰)}$ is known, one can determine the relative branching fractions which can be used to exam the mixing of ${{{Ξ}}}c$ and ${{{Ξ}}}c^{{'}}$. In previous literature, ${{{Ξ}}}c$ and ${{{Ξ}}}c^{{'}}$ were assumed to belong to ${SU(3)}F$ antitriple and sextet, respectively. However, recent experimental measurements, such as the ratio ${{Γ}}({{{Ξ}}}cc{→}{{{Ξ}}}c{{π}}⁺)/{{Γ}}({{{Ξ}}}cc{→}{{{Ξ}}}c^{{'}}{{π}}⁺)$, indicate the spin-flavor structures of ${{{Ξ}}}c$ and ${{{Ξ}}}c^{{'}}$ are a mixture of ${{{Ξ}}}c3̄$ and ${{{Ξ}}}c⁶$. The exact value of mixing angle ${θ}$ is still under debate. In theoretical models, the mixing angle was fitted to be about $16.27^∘{}±{}2.30^∘{}$ or $85.54^∘{}±{}2.30^∘{}$ based on decay channels ${{{Ξ}}}cc{→}{{{Ξ}}}c^{{(}^{{'}})}$. While in lattice calculation, a small angle ($1.2^∘{}±{}0.1^∘{}$) is preferred. To address such discrepancy and test the mixing of ${{{Ξ}}}c$ and ${{{Ξ}}}c^{{'}}$, here we propose the analysis of semileptonic and nonleptonic decays of ${{{Ξ}}}b{→}{{{Ξ}}}c$ and ${{{Ξ}}}b{→}{{{Ξ}}}c^{{'}}$. We calculate the decay rate of ${{{Ξ}}}b{→}{{{Ξ}}}c$ and ${{{Ξ}}}b{→}{{{Ξ}}}c^{{'}}$ based on the light-front quark model and study the effect of the mixing angle on the ratios of weak decays ${{{Ξ}}}b{→}{{{Ξ}}}c$ and ${{{Ξ}}}b{→}{{{Ξ}}}c^{{'}}$. In particular, we find the transition ${{{Ξ}}}b{→}{{{Ξ}}}c^{{'}}$ can be an ideal channel to verify the mixing and extract the mixing angle because in theory, the decay rate would be extremely tiny without mixing. Our calculation suggests a measurement of ${{{Ξ}}}b{→}{{{Ξ}}}c^{{'}}$ can be feasible in the near future, which will help to test flavor mixing angle ${θ}$ and elucidate the mechanism of decay of heavy baryons.