$B\to \pi \ell \nu$ and $B_s\to K\ell \nu$ form factors and $|V_{ub}|$ from 2+1-flavor lattice QCD with domain-wall light quarks and relativistic heavy quarks

Abstract

We calculate the $B\to \pi \ell \nu$ and $B_s\to K\ell \nu$ form factors in dynamical lattice QCD. We use the (2+1)-flavor RBC-UKQCD gauge-field ensembles generated with the domain-wall fermion and Iwasaki gauge actions. For the $b$ quarks we use the anisotropic clover action with a relativistic heavy-quark interpretation. We analyze two lattice spacings $a\approx 0.11,0.086$ fm and unitary pion masses as light as $M_{\pi }\approx 290$ MeV. We simultaneously extrapolate our numerical results to the physical light-quark masses and to the continuum and interpolate in the pion/kaon energy using SU(2) "hard-pion" chiral perturbation theory. We provide complete error budgets for the form factors $f_{+}(q^2)$ and $f_0(q^2)$ at three momenta that span the $q^2$ range accessible in our numerical simulations. We extrapolate these results to $q^2=0$ using a model-independent $z$-parametrization and present our final form factors as the $z$-coefficients and the matrix of correlations between them. Our results agree with other lattice determinations using staggered light quarks and provide important independent cross-checks. Both $B\to \pi \ell \nu$ and $B_s\to K\ell \nu$ decays enable a determination of the CKM matrix element $|V_{ub}|$. To illustrate this, we perform a combined $z$-fit of our numerical $B\to \pi \ell \nu$ form-factor data with the experimental branching-fraction measurements leaving the relative normalization as a free parameter; we obtain $|V_{ub}|=3.61(32)\times 10^{-3}$, where the error includes statistical and systematic uncertainties. This approach can be applied to $B_s\to K\ell \nu$ decay to determine $|V_{ub}|$ once the process has been measured experimentally. Finally, in anticipation of future measurements, we make predictions for $B\to \pi \ell \nu$ and $B_s\to K\ell \nu$ Standard-Model differential branching fractions and forward-backward asymmetries.

Comments: V3: References and discussions added. Large tables move to appendix. 39 pages, 22 tables, 23 figures. Version published in Physical Review D