The K -> pi vector form factor at zero momentum transfer on the lattice

We present a quenched lattice study of the form factors $f_+(q^2)$ and $f_0(q^2)$ of the matrix elements $<\pi | \bar{s} \gamma_\mu u | K >$. We focus on the second-order SU(3)-breaking quantity $[1 - f_+(0)]$, which is necessary to extract $|V_{us}|$ from $K_{\ell 3}$ decays. For this quantity we show that it is possible to reach the percent precision which is the required one for a significant determination of $|V_{us}|$. The leading quenched chiral logarithms are corrected for by using analytic calculations in quenched chiral perturbation theory. Our final result, $f_+^{K^0\pi^-}(0) = 0.960 \pm 0.005_{\rm stat} \pm 0.007_{\rm syst}$, where the systematic error does not include the residual quenched effects, is in good agreement with the estimate made by Leutwyler and Roos. A comparison with other non-lattice computations and the impact of our result on the extraction of $|V_{us}|$ are also presented.