Highly improved staggered quarks on the lattice with applications to charm physics

We use perturbative Symanzik improvement to create a new staggered-quark action (HISQ) that has greatly reduced one-loop taste-exchange errors, no tree-level order ${a}^{2}$ errors, and no tree-level order $(am{)}^{4}$ errors to leading order in the quark's velocity $v/c$. We demonstrate with simulations that the resulting action has taste-exchange interactions that are 3--4 times smaller than the widely used ASQTAD action. We show how to bound errors due to taste exchange by comparing ASQTAD and HISQ simulations, and demonstrate with simulations that such errors are likely no more than 1% when HISQ is used for light quarks at lattice spacings of $1/10\text{ }\text{ }\mathrm{fm}$ or less. The suppression of $(am{)}^{4}$ errors also makes HISQ the most accurate discretization currently available for simulating $c$ quarks. We demonstrate this in a new analysis of the $\ensuremath{\psi}\ensuremath{-}{\ensuremath{\eta}}_{c}$ mass splitting using the HISQ action on lattices where $a{m}_{c}=0.43$ and 0.66, with full-QCD gluon configurations (from MILC). We obtain a result of 111(5) MeV which compares well with the experiment. We discuss applications of this formalism to $D$ physics and present our first high-precision results for ${D}_{s}$ mesons.