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We present a full account of the two-loop electroweak, two-photon mediatedshort-distance dispersive $K_L \to\mu^+\mu^-$ decay amplitude. QCD correctionschange the sign of this amplitude and reduce it by an order of magnitude. Thus,the QCD-corrected two-loop amplitude represents only a small fraction (with thecentral value of 5 %) of the one-loop weak short-distance contribution, and hasthe same sign. In combination with a recent measurement, the standard-modelprediction of the short-distance amplitude, completed in this paper, provides aconstraint on the otherwise uncertain long-distance dispersive amplitude.Comment: 7 pages, 6 PS figures, uses psfig, revised version with important  numerical error corrected, to appear in Z.f.Phys.C (and one additional  reference

On the dispersive two-photon $K_L\rightarrow \mu^+ \mu^-$ amplitude