Rare weak decays of $\eta^\prime\to K\pi$

Rare weak decays of $\eta^\prime\to K\pi$ have been investigated in the framework of the $U(3)$ chiral perturbation theory at the leading order. Our study shows that the branching ratio ${\cal B}(\eta^\prime\to K\pi)$ is of the order of $10^{-11}$, which is far below the present experimental upper bound given by the BESIII Collaboration. By further analysis of $\eta^\prime\to K^+\pi^-$ and $\eta^\prime\to K^0\pi^0$, the ratio of isospin amplitudes is found that $|A_{1/2}/A_{3/2}|\simeq 35$, which supports that the $\Delta I=1/2$ transition enhancement, namely, the $\Delta I=1/2$ rule, could be functional in $\eta^\prime$ weak decays.