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

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 ${\mathrm{\Xi }}_c^{\mathrm{\prime }}$. In previous literature, ${\mathrm{\Xi }}_c$ and ${\mathrm{\Xi }}_c^{\mathrm{\prime }}$ were assumed to belong to SU(3)${}_F$ antitriple and sextet, respectively. However, recent experimental measurements such as the ratio $\mathrm{\Gamma }({\mathrm{\Xi }}_{cc}\to {\mathrm{\Xi }}_c{\pi }^{+})\mathrm{/}\mathrm{\Gamma }({\mathrm{\Xi }}_{cc}\to {\mathrm{\Xi }}_c^{\mathrm{\prime }}{\pi }^{+})$ indicate the spin-flavor structures of ${\mathrm{\Xi }}_c$ and ${\mathrm{\Xi }}_c^{\mathrm{\prime }}$ are a mixture of ${\mathrm{\Xi }}_c^{\bar{3}}$ 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 ${16.27}^{\circ }\pm {2.30}^{\circ }$ or ${85.54}^{\circ }\pm {2.30}^{\circ }$ based on decay channels ${\mathrm{\Xi }}_{cc}\to {\mathrm{\Xi }}_{cc}^{{(}^{\mathrm{\prime }})}$. While in lattice calculation, a small angle (${1.2}^{\circ }\pm {0.1}^{\circ }$) is preferred. To address such discrepancy and test the mixing of ${\mathrm{\Xi }}_c$ and ${\mathrm{\Xi }}_c^{\mathrm{\prime }}$, here we propose the analysis of semileptonic and non-leptonic decays of ${\mathrm{\Xi }}_b\to {\mathrm{\Xi }}_c$ and ${\mathrm{\Xi }}_b\to {\mathrm{\Xi }}_c^{{}^{\mathrm{\prime }}}$. We calculate the decay rate of ${\mathrm{\Xi }}_b\to {\mathrm{\Xi }}_c$ and ${\mathrm{\Xi }}_b\to {\mathrm{\Xi }}_c^{{}^{\mathrm{\prime }}}$ based on 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 ${\mathrm{\Xi }}_b\to {\mathrm{\Xi }}_c^{{}^{\mathrm{\prime }}}$…