$K_{\ell3}$ decays in Chiral Perturbation Theory

The process $K_{\ell3}$ is calculated to two-loop order ($p^6$) in Chiral Perturbation Theory (ChPT) in the isospin conserved case. We present expressions suitable for use with previous work in two-loop CHPT where the order $p^4$ parameters ($L_i^r$) were determined from experiment. We point out that all the order $p^6$ parameters ($C_i^r$) that appear in the value of $f_+(0)$ relevant for the determination of $|V_{us}|$ can be determined from $K_{\ell3}$ measurements via the slope and the curvature of the scalar form-factor. As a by product we update the value of the CKM matrix element $| V_{us} |$.

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