Short comment on the K_L -> mu^+ mu^- amplitude

We have considered the two-loop, two-photon mediated short-distance dispersive K_L -> mu^+ mu^- decay amplitude, including QCD corrections in the leading logarithmic approximation. The obtained amplitude is of order half of the one loop based weak short distance contribution, and has the same sign. Thus, according to the present experimental status of the process, there is no room for a long distance dispersive amplitude with still the same sign and comparable in size to the total short distance part.