Semileptonic decays of doubly charmed baryons with ${\mathrm{\Xi }}_c-{\mathrm{\Xi }}_c^{\mathrm{\prime }}$ mixing

We study the ${\mathrm{\Xi }}_c-{\mathrm{\Xi }}_c^{\mathrm{\prime }}$ mixing effects in the semileptonic decays the doubly charm baryons of ${\mathrm{\Xi }}_{cc}$. We focus on the ratio of $\mathcal{R}({\theta }_c)\equiv \mathcal{B}({\mathrm{\Xi }}_{cc}\to {\mathrm{\Xi }}_c^{\mathrm{\prime }}{e}^{+}{\nu }_e)\mathrm{/}\mathcal{B}({\mathrm{\Xi }}_{cc}\to {\mathrm{\Xi }}_c{e}^{+}{\nu }_e)$ and find that $(\mathcal{R}({\theta }_0),\mathcal{R}(-{\theta }_0))=(0.46\pm 0.01,7.33\pm 0.23)$ with ${\theta }_0=0.137\pi$, which are in sharp contrast to $\mathcal{R}(0)=2.15\pm 0.11$ without the mixing. The ratio is enhanced~(suppressed) by a factor of four for a negative~(positive) ${\theta }_c$. In addition, the polarization asymmetries of ${\mathrm{\Xi }}_c^{(\mathrm{\prime })}$ are found to be $\alpha (-{\theta }_0)=0.32(-0.76)$ and $\alpha ({\theta }_0)=-0.82(-0.38)$. As $\mathcal{R}$ and $\alpha$ are highly sensitive to ${\theta }_c$ and unaffected by the $W$-exchange contributions, they provide excellent opportunities to determine ${\theta }_c$ in the ongoing experiments.