Matching long and short distances at order ${\mathcal O}(\alpha_s)$ in the form factors for $K\to\pi \ell^+\ell^-$
Résumé
At order O(αGF) , the amplitudes for the decays K→πℓ+ℓ− 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 O(αs) 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.
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