$D\to K,l\nu$ Semileptonic Decay Scalar Form Factor and $|V_{cs}|$ from Lattice QCD

Abstract

We present a new study of D semileptonic decays on the lattice which employs the Highly Improved Staggered Quark (HISQ) action for both the charm and the light valence quarks. We work with MILC unquenched $N_f=2+1$ lattices and determine the scalar form factor $f_0(q^2)$ for $D\to K,l\nu$ semileptonic decays. The form factor is obtained from a scalar current matrix element that does not require any operator matching. We develop a new approach to carrying out chiral/continuum extrapolations of $f_0(q^2)$. The method uses the kinematic "$z$" variable instead of $q^2$ or the kaon energy $E_K$ and is applicable over the entire physical $q^2$ range. We find $f_0^{D\to K}(0)\equiv f_{+}^{D\to K}(0)=0.747(19)$ in the chiral plus continuum limit and hereby improve the theory error on this quantity by a factor of $\sim$4 compared to previous lattice determinations. Combining the new theory result with recent experimental measurements of the product $f_{+}^{D\to K}(0)\ast |V_{cs}|$ from BaBar and CLEO-c leads to the most precise direct determination of the CKM matrix element $|V_{cs}|$ to date, $|V_{cs}|=0.961(11)(24)$, where the first error comes from experiment and the second is the lattice QCD theory error. We calculate the ratio $f_{+}^{D\to K}(0)/f_{D_s}$ and find $2.986\pm 0.087$ GeV${}^{-1}$ and show that this agrees with experiment.

Comments: 23 pages, 31 figures, 11 tables. Added a paragraph in sction VII, and updated with PDG 2010 instead of PDG 2008