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Light gauge boson in rare $K$ decay

We study the production of a light gauge boson in $K^- \to π^- X$ decay, where the associated new charge current is not conserved. It is found that the process can be generated by the tree-level $W$-boson annihilation and loop-induced $s\to dX$. We find that it strongly depends on the $SU(3)$ limit or the unique gauge coupling to the quarks, whether the decay amplitude of $K^-\to π^- X$ in the $W$-boson annihilation is suppressed by $m^2_X ε_X \cdot p_K$; however, no such suppression is found via the loop-induced $s\to d X$. The constraints on the relevant couplings are studied.