$K \rightarrow \pi \ell^+ \ell^-$ form factor in the Large-N$_c$ and cut-off regularization method

TL;DR

This paper applies a large N_c method combined with cut-off regularization to evaluate the form factors in rare kaon decays, providing promising results especially with Vector Meson Dominance considerations.

Contribution

It tests and demonstrates the effectiveness of the large N_c approach with cut-off regularization for calculating kaon decay form factors, incorporating Vector Meson Dominance.

Findings

Encouraging results for form factor parameters a_+ and b_+

Successful application of the large N_c method to kaon decays

Complete treatment of Vector Meson Dominance improves accuracy

Abstract

Bardeen-Buras-G\'{e}rard have proposed a large N$_c$ method to evaluate hadronic weak matrix elements to attack for instance the determination of the $\Delta I= \frac{1}{2}$-rule and $\mathrm{Re}(\frac{\epsilon'}{\epsilon})$. Here we test this method to the determination of the form factor parameters $a_+$ and $b_+$ in the decays $K^+ \rightarrow \pi^+ \ell^+ \ell^- $ and $K_S \rightarrow \pi ^0 \ell^+ \ell^-$. The results are encouraging: in particular after a complete treatment of Vector Meson Dominance (VMD).