Detecting the flavor content of the vacuum using the Dirac operator spectrum

We compute the overlap Dirac spectrum on three gauge ensembles generated using 2+1-flavor domain wall fermions. The three ensembles have different lattice spacings, and two of them have quark masses tuned to the physical point. The spectral density is determined up to λ∼100 MeV with subpercentage statistical uncertainty. We find that the density is close to a constant below λ∼20 MeV as predicted by chiral perturbative theory (χPT) and then increases linearly due to the strange quark mass. By fitting to the next-to-leading order χPT form and using the nonperturbative renormalization using the regularization independent momentum subtraction scheme, the SU(2) (keeping the strange quark mass at the physical point) and SU(3) chiral condensates at MS¯ 2 GeV are determined to be ς=(265.4(0.5)(4.2) MeV)3 and ς0=(234.3(0.5)(25.8) MeV)3, respectively. The pion decay constants are also determined to be F=84.1(1.9)(8.0) and F0=58.6(0.5)(10.0) MeV. The systematic errors are carefully estimated including the effects of fitting ranges and the uncertainty of low-energy constant L6. We also show that one can resolve the sea flavor content of the sea quarks and constrain their masses with ∼10%-20% statistical uncertainties using the Dirac spectral density.