Forward-backward Asymmetry in $K^+\to\pi^+ \ell^+\ell^-$

Abstract

We study the forward-backward asymmetries in the decays of $K^+\to\pi^+\ell^+ \ell^-$ ($\ell=e$ and $\mu$) in the presence of scalar or tensor terms. We find that with the scalar (tensor) type interaction the asymmetry can be up to ${\cal O}(10^{-3})$ (${\cal O}(10^{-1}))$ and arbitrary large for the electron and muon modes, respectively, without conflict with the experimental data. We also discuss the cases in the minimal supersymmetric standard model where the scalar terms can be induced. In particular we show that the asymmetry in $K^+\to\pi^+\mu^+\mu^-$ can be as large as ${\cal O}(10^{-3})$ in the large $\tan\beta$ limit, which can be tested in future experiments such as CKM at Fermilab.