Detecting flavor content of the vacuum using the Dirac operator spectrum

We compute the overlap Dirac spectrum on three ensembles generated using 2+1 flavor domain wall fermions. The spectral density is determined up to $\lambda\sim$100 MeV with sub-percentage statistical uncertainty. The three ensembles have different lattice spacings and two of them have quark masses tuned to the physical point. We show that we can resolve the flavor content of the sea quarks and constrain their masses using the Dirac spectral density. We find that the density is close to a constant below $\lambda\le$ 20 MeV (but 10\% higher than that in the 2-flavor chiral limit) as predicted by chiral perturbative theory ($\chi$PT), and then increases linearly due to the strange quark mass. Using the next to leading order $\chi$PT, one can extract the light and strange quark masses with $\sim$20\% uncertainties. Using the non-perturbative RI/MOM renormalization, we obtain the chiral condensates at $\overline{\textrm{MS}}$ 2 GeV as $\Sigma=(260.3(0.7)(1.3)(0.7)(0.8)\ \textrm{MeV})^3$ in the $N_f=2$ (keeping the strange quark mass at the physical point) chiral limit and $\Sigma_0=(232.6(0.9)(1.2)(0.7)(0.8)\ \textrm{MeV})^3$ in the $N_f=3$ chiral limit, where the four uncertainties come from the statistical fluctuation, renormalization constant, continuum extrapolation and lattice spacing determination. Note that {$\Sigma/\Sigma_0=1.40(2)(2)$ is much larger than 1} due to the strange quark mass effect.