We calculate the vector form factor in $K\to \pi l\nu$ semileptonic decays at zero momentum transfer $f_{+}(0)$ from numerical simulations of two-flavor QCD on the lattice. Our simulations are carried out on ${16}^3\times 32$ at a lattice spacing of $a\simeq 0.12\mathrm{fm}$ using a combination of the DBW2 gauge and the domain-wall quark actions, which possesses excellent chiral symmetry even at finite lattice spacings. The size of fifth dimension is set to $L_s=12$, which leads to a residual quark mass of a few MeV. Through a set of double ratios of correlation functions, the form factor calculated on the lattice is accurately interpolated to zero momentum transfer, and then is extrapolated to the physical quark mass. We obtain $f_{+}(0)=0.968(9)(6)$, where the first error is statistical and the second is the systematic error due to the chiral extrapolation. Previous estimates based on a phenomenological model and chiral perturbation theory are consistent with our result. Combining with an average of the decay rate from recent experiments, our estimate of $f_{+}(0)$ leads to the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $|V_{us}|=0.2245(27)$, which is consistent with CKM unitarity. These estimates of $f_{+}(0)$ and $|V_{us}|$ are subject to systematic uncertainties due to the finite lattice spacing and quenching of strange quarks, though nice consistency in $f_{+}(0)$ with previous lattice calculations suggests that these errors are not large.

• Received 4 August 2006

DOI:https://doi.org/10.1103/PhysRevD.74.114502