Resolving puzzle in $\Xi_c^0\to \Xi^-e^+\nu_e$ with $\Xi_c-\Xi_c'$ mixing

We study the ratio of $R=2 \Gamma(\Xi_c^0 \to \Xi^- e^+ \nu_e )/3\Gamma(\Lambda_c^+ \to \Lambda e^+ \nu_e )$, which is found to be $R= 1 (0.8)$ from the exact (broken) $SU(3)$ flavor symmetry, in sharp contrast to the average value of $R_{av}=0.59\pm 0.10 $ from the ALICE collaboration and lattice QCD results. We propose to use the mixing of $\Xi_c-\Xi_c'$ to resolve the puzzle. With the model-independent mass relations, we find that the mixing angle is $|\theta_c| = 0.137(5)\pi$, which suppresses $\Xi_c\to \Xi e^+ \nu_e$ about $20\%$ model-independently, resulting in $R\approx 0.6$ with the $SU(3)$ flavor breaking effect. We explicitly demonstrate that $R= 0.70 \pm 0.09$ from the bag model, which is also consistent with $R_{av}$. To test the mixing mechanism, we recommend the experiments to measure the decays of $\Xi_c \to \Xi '(1530) e^+ \nu_e$, whose branching fractions are determined to be $ ( 4.4\sim 8.7)\times 10^{-3}$ and $(1.3\sim 2.6 )\%$ for $\Xi_c^0$ and $\Xi_c^+$, respectively, but vanish without the mixing. In addition, nonvanishing values of ${\cal B}(\Xi_c^+\to \Xi^{\prime 0} (1530)\pi^+ )$ and ${\cal B}(\Xi_c^+\to \Sigma^{\prime +} (1530) \overline{K}^0 )$ will also be evidences of the mixing based on the K\"oner-Pati-Woo theorem, which are calculated as $(3.8\sim 7.5)\times 10^{-3}$ and $(6.6\sim 13)\times 10 ^{-4}$, respectively. We emphasize that $\theta_c$ is sizable and should be given serious considerations in future studies on the heavy baryon systems.