Heavy-quark meson spectrum tests of the Oktay–Kronfeld action

The Oktay–Kronfeld (OK) action extends the Fermilab improvement program for massive Wilson fermions to higher order in suitable power-counting schemes. It includes dimension-six and -seven operators necessary for matching to QCD through order $$\mathrm {O}(\varLambda _{{\mathrm{{QCD}}}}^3/m_Q^3)$$ in HQET power counting, for applications to heavy–light systems, and $$\mathrm {O}(v^6)$$ in NRQCD power counting, for applications to quarkonia. In the Symanzik power counting of lattice gauge theory near the continuum limit, the OK action includes all $$\mathrm {O}(a^2)$$ and some $$\mathrm {O}(a^3)$$ terms. To assess whether the theoretical improvement is realized in practice, we study combinations of heavy–strange and quarkonia masses and mass splittings, designed to isolate heavy-quark discretization effects. We find that, with one exception, the results obtained with the tree-level-matched OK action are significantly closer to the continuum limit than those obtained with the Fermilab action. The exception is the hyperfine splitting of the bottom–strange system, for which our statistical errors are too large to draw a firm conclusion. These studies are carried out with data generated with the tadpole-improved Fermilab and OK actions on 500 gauge configurations from one of MILC’s $$a\approx 0.12$$  fm, $$N_f=2+1$$ -flavor, asqtad-staggered ensembles.