An exploratory calculation of $K_{\mathrm{L}}\to \hat{I}{¼}^{+}\hat{I}{¼}^{-}$ decay from Lattice QCD at physical pion mass

Abstract

We compute the complex, long-distance two-photon-exchange amplitude which contributes to the rare $K_{\mathrm{L}}\to \hat{I}{¼}^{+}\hat{I}{¼}^{-}$ 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 $\hat{I}\cdot$ intermediate state, for which various strategies are tested and compared.

This submission (v3) is identical to the previous submission (v2) except for the added comment that this revised version recognizes the presence of potentially significant lattice artifacts in two of the diagrams that are computed which will be removed by additional renormalization to be carried out in a subsequent paper. The numerical results remain unchanged compared to v1