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

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 ${\alpha }_s^2$ multiplying $⟨\overline{\psi }\psi ⟩/M^3$ and terms in ${\alpha }_s$ multiplying $⟨{\alpha }_s{G}^2⟩/M^4$, 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, $⟨\overline{s}s{⟩}^{\overline{\mathrm{MS}}}(2\mathrm{GeV})=-(296(11)\mathrm{MeV}{)}^3$, 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.

Figure 1

Ratios of reduced heavy-strange moments to heavy-charm moments calculated on the lattice (symbols) and the ratio of the appropriate perturbative series with no condensate contribution. The dotted and dashed lines show the ratio of the $c_n$ series for heavy strange and the $g_n$ series for heavy charm through order ${\alpha }_s$ and ${\alpha }_s^2$, respectively. The circles, squares, and triangles show the lattice results on ultrafine, superfine, and fine lattice ensembles respectively.

Figure 2

The lattice data for ratios of reduced moments of heavy-strange and heavy-charm correlators at three lattice spacings (symbols), and the continuum extrapolation from our fit (colored bands). The circles, squares and triangles show the lattice results on the ultrafine, superfine and fine ensembles respectively. The dashed lines join the fitted values at each data point.

Figure 3

The stability of the cube root of $⟨\overline{s}s⟩(2\mathrm{GeV})$ as we change the lattice QCD results used in our work. Here $\{4,6,8,10\}$ denotes the result from a combined fit to the ratio of moments for $n$ in $\{4,6,8,10\}$. The gray band corresponds to the combined fit.

Figure 4

Comparison of the new result for $⟨\overline{s}s⟩$ obtained here (red circle) with the earlier result from HPQCD using the $\mathrm{Tr}{M}^{-1}$ (blue circle) [4]. We also show the result for the light quark condensate (blue triangle) from that work. All results are in the $\overline{\mathrm{MS}}$ scheme at a scale of 2 GeV. Our new results have $n_f=3$; the results in [4] have $n_f=4$.