Determination of the quark condensate from heavy-light current-current correlators in full lattice QCD

Abstract

We derive the operator product expansion whose vacuum expectation value gives the time moments of the pseudoscalar heavy-light current-current correlator up to and including terms in ${\ensuremath{\alpha}}_{s}^{2}$ multiplying $⟨\overline{\ensuremath{\psi}}\ensuremath{\psi}⟩/{M}^{3}$ and terms in ${\ensuremath{\alpha}}_{s}$ multiplying $⟨{\ensuremath{\alpha}}_{s}{G}^{2}⟩/{M}^{4}$, where $M$ is the heavy quark mass. Using lattice QCD results for heavy-strange correlators obtained for a variety of heavy quark masses on gluon field configurations including $u$, $d$ and $s$ quarks in the sea at three values of the lattice spacing, we are able to show that the contribution of the strange-quark condensate to the time moments is very substantial. We use our lattice QCD time moments and the operator product expansion to determine a value for the condensate, fitting the fourth, sixth, eighth and tenth time moments simultaneously. Our result, $⟨\overline{s}s{⟩}^{\overline{\mathrm{MS}}}(2\text{ }\text{ }\mathrm{GeV})=\phantom{\rule{0ex}{0ex}}\ensuremath{-}(296(11)\text{ }\text{ }\mathrm{MeV}{)}^{3}$, agrees well with HPQCD's earlier, more direct, lattice QCD determination [C. McNeile et al. (HPQCD Collaboration), Phys. Rev. D 87, 034503 (2013)]. As well as confirming that the $s$ quark condensate is close in value to the light quark condensate, this demonstrates clearly the consistency of the operator product expansion for fully nonperturbative calculations of matrix elements of short-distance operators in lattice QCD.