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

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. This calculation is carried out with 2+1 quark flavors and therefore requires the addition of counter terms to compensate for the absence of the Glashow-Iliopoulos-Maiani mechanism. These counter terms are not included in the current calculation and will be the subject of a second paper. 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.