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

Abstract

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.

Comments: 11 pages, 3 figures, uses elsarticle class, minor changes and typos corrected, matches the published version