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At order ${\mathcal O}(\alpha G_{\mathrm F})$, the amplitudes for the decays$K\to\pi \ell^+\ell^-$ involve a form factor given by the matrix element of thetime-ordered product of the electromagnetic current with the four-quarkoperators describing weak non-leptonic neutral-current transitions between akaon and a pion. The short-distance behaviour of this time-ordered product,when considered at order ${\mathcal O}(\alpha_s)$ in the perturbative expansionof QCD, involves terms linear and quadratic in the logarithm of the Euclideanmomentum transfer squared. It is shown how one can exactly match theseshort-distance features using a dispersive representation of the form factor,with an absorptive part given by an infinite sum of zero-width resonancesfollowing a Regge-type spectrum. Some phenomenology-related issues are brieflydiscussed.

Matching long and short distances at order ${\mathcal O}(\alpha_s)$ in the form factors for $K\to\pi \ell^+\ell^-$