  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 the
time-ordered product of the electromagnetic current with the four-quark
operators describing weak non-leptonic neutral-current transitions between a
kaon and a pion. The short-distance behaviour of this time-ordered product,
when considered at order ${\mathcal O}(\alpha_s)$ in the perturbative expansion
of QCD, involves terms linear and quadratic in the logarithm of the Euclidean
momentum transfer squared. It is shown how one can exactly match these
short-distance features using a dispersive representation of the form factor,
with an absorptive part given by an infinite sum of zero-width resonances
following a Regge-type spectrum. Some phenomenology-related issues are briefly
discussed.
