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

## TL;DR

This paper develops a method to accurately match short-distance QCD behavior with a dispersive resonance model for form factors in rare kaon decays, improving theoretical understanding of these processes.

## Contribution

It introduces an exact matching technique at order ${
m O}(\alpha_s)$ between short-distance QCD calculations and a dispersive resonance representation for $K o\pi\ell^+\ell^-$ form factors.

## Key findings

- Successful matching of short-distance logarithmic terms with a resonance-based dispersive model.
- Enhanced theoretical framework for analyzing $K o\pi\ell^+\ell^-$ decay amplitudes.
- Discussion of phenomenological implications of the matching approach.

## 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.