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

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

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