Study on the mixing of ${\mathrm{\Xi }}_c$ and ${\mathrm{\Xi }}_c^{\prime }$ by the transition ${\mathrm{\Xi }}_b\to {\mathrm{\Xi }}_c^{{(}^{\prime })}$

Recently, the LHCb Collaboration has observed the decays ${\mathrm{\Xi }}_b^0\to {\mathrm{\Xi }}_c^{+}{D}_s^{-}$ and ${\mathrm{\Xi }}_b^{-}\to {\mathrm{\Xi }}_c^0{D}_s^{-}$. They measured the relative branching fractions times the ratio of beauty-baryon production cross sections $\mathcal{R}(\frac{{\mathrm{\Xi }}_b^0}{{\mathrm{\Lambda }}_b})\equiv \frac{\sigma ({\mathrm{\Xi }}_b^0)}{\sigma ({\mathrm{\Lambda }}_b^0)}\times \frac{B({\mathrm{\Xi }}_b^0\to {\mathrm{\Xi }}_c^{+}{D}_s^{-})}{B({\mathrm{\Lambda }}_b^0\to {\mathrm{\Lambda }}_c^{+}{D}_s^{-})}$ and $\mathcal{R}(\frac{{\mathrm{\Xi }}_b^{-}}{{\mathrm{\Lambda }}_b})\equiv \frac{\sigma ({\mathrm{\Xi }}_b^{-})}{\sigma ({\mathrm{\Lambda }}_b^0)}\times \frac{B({\mathrm{\Xi }}_b^{-}\to {\mathrm{\Xi }}_c^0{D}_s^{-})}{B({\mathrm{\Lambda }}_b^0\to {\mathrm{\Lambda }}_c^{+}{D}_s^{-})}$. Once the ratio $\frac{\sigma ({\mathrm{\Xi }}_b^0)}{\sigma ({\mathrm{\Lambda }}_b^0)}$ or $\frac{\sigma ({\mathrm{\Xi }}_b^{-})}{\sigma ({\mathrm{\Lambda }}_b^0)}$ is known, one can determine the relative branching fractions which can be used to exam the mixing of ${\mathrm{\Xi }}_c$ and ${\mathrm{\Xi }}_c^{\prime }$. In previous literature, ${\mathrm{\Xi }}_c$ and ${\mathrm{\Xi }}_c^{\prime }$ were assumed to belong to ${\mathrm{SU}(3)}_F$ antitriple and sextet, respectively. However, recent experimental measurements, such as the ratio $\mathrm{\Gamma }({\mathrm{\Xi }}_{cc}\to {\mathrm{\Xi }}_c{\pi }^{+})/\mathrm{\Gamma }({\mathrm{\Xi }}_{cc}\to {\mathrm{\Xi }}_c^{\prime }{\pi }^{+})$, indicate the spin-flavor structures of ${\mathrm{\Xi }}_c$ and ${\mathrm{\Xi }}_c^{\prime }$ are a mixture of ${\mathrm{\Xi }}_c^{\overline{3}}$ and ${\mathrm{\Xi }}_c^6$. The exact value of mixing angle $\theta$ is still under debate. In theoretical models, the mixing angle was fitted to be about $16.27°\pm 2.30°$ or $85.54°\pm 2.30°$ based on decay channels ${\mathrm{\Xi }}_{cc}\to {\mathrm{\Xi }}_c^{{(}^{\prime })}$. While in lattice calculation, a small angle ($1.2°\pm 0.1°$) is preferred. To address such discrepancy and test the mixing of ${\mathrm{\Xi }}_c$ and ${\mathrm{\Xi }}_c^{\prime }$, here we propose the analysis of semileptonic and nonleptonic decays of ${\mathrm{\Xi }}_b\to {\mathrm{\Xi }}_c$ and ${\mathrm{\Xi }}_b\to {\mathrm{\Xi }}_c^{\prime }$. We calculate the decay rate of ${\mathrm{\Xi }}_b\to {\mathrm{\Xi }}_c$ and ${\mathrm{\Xi }}_b\to {\mathrm{\Xi }}_c^{\prime }$ based on the light-front quark model and study the effect of the mixing angle on the ratios of weak decays ${\mathrm{\Xi }}_b\to {\mathrm{\Xi }}_c$ and ${\mathrm{\Xi }}_b\to {\mathrm{\Xi }}_c^{\prime }$. In particular, we find the transition ${\mathrm{\Xi }}_b\to {\mathrm{\Xi }}_c^{\prime }$ 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 ${\mathrm{\Xi }}_b\to {\mathrm{\Xi }}_c^{\prime }$ can be feasible in the near future, which will help to test flavor mixing angle $\theta$ and elucidate the mechanism of decay of heavy baryons.

Figure 1

The Feynman diagrams for ${\mathrm{\Xi }}_b\to {\mathrm{\Xi }}_c^{{(}^{\prime })}$ transitions, where $Q_1$ and $Q_2$ denote heavy quark in the initial and final states, respectively. $q_1$ and $q_2$ represent light quark states. • denotes $V-A$ current vertex.

Figure 2

Form factors for the decay ${\mathrm{\Lambda }}_c\to {\mathrm{\Lambda }}_c$ (a) pole form and (b) polynomial form.

Figure 3

Form factors for the decay ${\mathrm{\Xi }}_b^{\overline{3}}\to {\mathrm{\Xi }}_c^{\overline{3}}$ (a) and ${\mathrm{\Xi }}_b^6\to {\mathrm{\Xi }}_c^6$ (b).

Figure 4

Differential decay rates $d\mathrm{\Gamma }/d\omega$ for the decay ${\mathrm{\Xi }}_b\to {\mathrm{\Xi }}_{c}l{\overline{\nu }}_l$ (a) and ${\mathrm{\Xi }}_b\to {\mathrm{\Xi }}_c^{\prime }l{\overline{\nu }}_l$ (b).