Complete determination of $SU(3)_F$ amplitudes and strong phase in $\Lambda_c^+ \to \Xi^0 K^+$

The BESIII collaboration has recently reported the first time measurement of the decay asymmetry $\alpha(\Lambda_c^+ \to \Xi^0 K^+) = 0.01 \pm 0.16(stat.) \pm 0.03(syst.)$ and also a sizable phase shift of $\delta_P-\delta_S = -1.55 \pm 0.25$ or $1.59\pm 0.25$ between S- and P-wave amplitudes. This implies significant strong phase shifts in the decay amplitudes. The strong phases indicate the existence of rescattering or loop effects, which are challenging to calculate due to non-perturbative effects. By employing the flavor $SU(3)_F$ symmetry and applying the K\"orner-Pati-Woo theorem to reduce the number of parameters, we find that the current data already allow us to obtain, for the first time, model-independent decay amplitudes and their strong phases. The establishment of the existence of sizable strong phases opens a window for future investigations into CP violation. In our fit, a notable discrepancy emerges in the branching ratio of $\Xi_c^0 \to \Xi^- \pi^+$. The direct relationship between $\Gamma (\Lambda_c^+ \to \Lambda e^+\nu_e)$ and $\Gamma (\Xi_c^0 \to \Xi^- e^+\nu_e)$, along with newly discovered $SU(3)_F$ relations, collectively suggests an underestimation of $\mathcal{B}(\Xi_c^0 \to \Xi^- \pi^+)$ in experimental findings.