Improvement of heavy-heavy current for calculation of $\bar{B}\to D^{(*)}\ell\bar{\nu}$ form factors using Oktay-Kronfeld heavy quarks

The CKM matrix element $|V_{cb}|$ can be extracted by combining data from experiments with lattice QCD results for the semileptonic form factors for the $\bar{B}\to D^{(*)} \ell \bar{\nu}$ decays. The Oktay-Kronfeld (OK) action was designed to reduce heavy-quark discretization errors to below $1\%$, or through $\mathcal{O}(\lambda^3)$ in HQET power counting. Here we describe recent progress on bottom-to-charm currents improved to the same order in HQET as the OK action, and correct formerly reported results of our matching calculations, in which the operator basis was incomplete.