O(a)-improved quark action on anisotropic lattices and perturbative renormalization of heavy-light currents

We investigate the Symanzik improvement of the Wilson quark action on anisotropic lattices. Taking first a general action with nearest-neighbor and clover interactions, we study the mass dependence of the ratio of the hopping parameters, the clover coefficients, and an improvement coefficient for heavy-light vector and axial vector currents. We show how tree-level improvement can be achieved. For a particular choice of the spatial Wilson coupling, the tree-level formulas simplify, and the lattice theory reaches the naive continuum limit for $m_0{a}_{\tau }\ll 1,$ $m_0{a}_{\sigma }\sim 1.$ (Here $m_0$ is the bare quark mass, and $a_{\tau }$ and $a_{\sigma }$ are the temporal and spatial lattice spacings.) For other choices, the continuum limit is reached only for $m_0{a}_{\sigma }\ll 1.$ With the well-behaved choice we calculate the renormalization factors of heavy-light bilinear operators at one-loop order of perturbation theory. These results allow us to show, numerically, that these one-loop corrections also behave well for $m_0{a}_{\tau }\ll 1$ and $m_0{a}_{\sigma }\sim 1.$