Long-distance contribution to the muon-polarization asymmetry in $K^{+} \to \pi^{+}\mu\mu$

We revisit the calculation of the long-distance contribution to the muon-polarization asymmetry \Delta_{LR}, which arises, in K^+\to\pi^+\mu^+\mu^-, from the two-photon intermediate state. The parity-violating amplitude of this process, induced by the local anomalous K^+\pi^-\gamma^*\gamma^* transition, is analysed; unfortunately, one cannot expect to predict its contribution to the asymmetry by using chiral perturbation theory alone. Here we evaluate this amplitude and its contribution to \Delta_{LR} by employing a phenomenological model called the FMV model, in which the utility of the vector and axial-vector resonances exchange is important to soften the ultraviolet behaviour of the transition. We find that the long-distance contribution is of the same order of magnitude as the standard model short-distance contribution.