An exploratory calculation of $K_{\rm L}\toμ^+μ^-$ decay from Lattice QCD at physical pion mass

Abstract: We compute the complex, long-distance two-photon-exchange amplitude which contributes to the rare $K_{\rm L}\rightarrow\mu^+\mu^-$ decay from lattice QCD. We use a $24^3\times 64$ physical-pion-mass gauge field ensemble at an inverse lattice spacing of $1.023$ GeV and a QED${}_\infty$-based formalism. Our implementation strategies for all five non-SU$(3)$-flavor-suppressed diagram topologies are given in detail. We achieve a 25% statistical precision on the dispersive part of this long-distance amplitude. For the first time, a destructive interference between the long- and short-distance parts of the decay amplitude is determined from a first-principles calculation. The precision of our results is limited by the reconstruction of the physical contribution of the $\eta$ intermediate state, for which various strategies are tested and compared.