Heavy quark action on anisotropic lattices

We investigate an $O\left(a\right)\mathrm{}$ improved quark action on anisotropic lattices as a potential framework for heavy quark physics, which may enable high precision computations of hadronic matrix elements for heavy-light mesons. The relativity relations of heavy-light mesons as well as of heavy quarkonium are examined on a quenched lattice with a spatial lattice cutoff $a_{\sigma }^{-1}\simeq 1.6\hspace{0.5em}\mathrm{GeV}$ and a (renormalized) anisotropy $\xi =4.\mathrm{}$ We find that the corresponding bare anisotropy parameter, once tuned for the massless quark, describes both heavy-heavy and heavy-light mesons within 2% accuracy for a quark mass $a_{\sigma }{m}_Q<0.8,$ a range which covers the charm quark. This bare anisotropy parameter also successfully describes the heavy-light mesons alone even in the quark mass region $a_{\sigma }{m}_Q<~1.2$ within the same accuracy. Beyond this region, the discretization effects seem to grow gradually. The renormalized anisotropy $\xi$ turns out to be roughly equal to the factor stretching the quark mass region in which the parameters in the action are applicable for heavy-light systems with well controlled systematic errors.