Improvement of heavy-heavy and heavy-light currents with the Oktay-Kronfeld action

The CKM matrix elements $V_{cb}$ and $V_{ub}$ can be obtained by combining data from the experiments with lattice QCD results for the semi-leptonic form factors for the $\bar{B}\to D^{\ast }\mathrm{\ell }\bar{\nu }$ and $\bar{B} \to \pi \ell \bar{\nu}$ decays. It is highly desirable to use the Oktay-Kronfeld (OK) action for the form factor calculation on the lattice, since the OK action is designed to reduce the heavy quark discretization error down to the $\mathcal{O}({\lambda }^4)$ level in the power counting rules of the heavy quark effective theory (HQET). Here, we present a matching calculation to improve heavy-heavy and heavy-light currents up to the ${\lambda }^3$ order in HQET, the same level of improvement as the OK action. Our final results for the improved currents are being used in a lattice QCD calculation of the semi-leptonic form factors for the $\bar{B}\to D^{\ast }\mathrm{\ell }\bar{\nu }$ and $\bar{B}\to D\mathrm{\ell }\bar{\nu }$ decays.