Differential decay rate of $B \to \pi l \nu$ semileptonic decay with lattice NRQCD

Abstract

We present a lattice QCD calculation of $B\to \pi l \nu$ semileptonic decay form factors in the small pion recoil momentum region. The calculation is performed on a quenched $16^3 \times 48$ lattice at $\beta=5.9$ with the NRQCD action including the full 1/M terms. The form factors $f_1(v\cdot k_{\pi})$ and $f_2(v\cdot k_{\pi})$ defined in the heavy quark effective theory for which the heavy quark scaling is manifest are adpoted, and we find that the 1/M correction to the scaling is small for the $B$ meson. The dependence of form factors on the light quark mass and on the recoil energy is found to be mild, and we use a global fit of the form factors at various quark masses and recoil energies to obtain model independent results for the physical differential decay rate. We find that the $B^*$ pole contribution dominates the form factor $f^+(q^2)$ for small pion recoil energy, and obtain the differential decay rate integrated over the kinematic region $q^2 >$ 18 GeV$^2$ to be $|V_{ub}|^2 \times (1.18 \pm 0.37 \pm 0.08 \pm 0.31)$ psec$^{-1}$, where the first error is statistical, the second is that from perturbative calculation, and the third is the systematic error from finite lattice spacing and the chiral extrapolation. We also discuss the systematic errors in the soft pion limit for $f^0(q^2_{max})$ in the present simulation.