Speaker
Description
We present strategies for determining the counterterm coefficients required to complete our recent calculation of the long-distance contribution to the $K_{\rm L}\to\mu^+\mu^-$ decay amplitude (Phys. Rev. D 113 (2026) 074523). These counterterms are absent in the more convergent four-flavor calculation because of the Glashow–Iliopoulos–Maiani (GIM) cancellation. This allows the required three-flavor counterterms to be determined by matching to the corresponding four-flavor theory. Since only short-distance quantities are involved in this matching, the four-flavor calculation can be performed economically in a small volume with unphysically heavy quark masses. We have implemented part of the required renormalization procedures and preliminary numerical results for the $K_{\rm L}\to\gamma\gamma$ amplitude, which is rendered UV finite at zeroth order in $\alpha_s(m_c)$, will be presented.