Order p⁶ chiral couplings from the scalar K Pi form factor

Employing results from a recent determination of the scalar KPi form factor F_0^KPi within a coupled channel dispersion relation analysis \cite{JOP01}, in this work we calculate the slope and curvature of F_0^KPi(t) at zero momentum transfer. Knowledge of the slope and curvature of the scalar KPi form factor, together with a recently calculated expression for F_0^KPi(t) in chiral perturbation theory at order p^6, enable to estimate the O(p^6) chiral constants C_12^r=(0.3 +- 5.4)10^-7 and (C_12^r+C_34^r)=(3.2 +- 1.5)10^-6. Our findings also allow to estimate the contribution coming from the C_i to the vector form factor F_+^KPi(0) which is crucial for a precise determination of |V_us| from K_l3 decays. Our result F_+^KPi(0)|_C_i^r=-0.018 +- 0.009, though inflicted with large uncertainties, is in perfect agreement with a previous estimate by Leutwyler and Roos already made twenty years ago.