A distinctive feature of the presence of spontaneous chiral symmetry breaking in QCD is the condensation of low modes of the Dirac operator near the origin. The rate of condensation must be equal to the slope of $M_{\pi }^2{F}_{\pi }^2/2$ with respect to the quark mass $m$ in the chiral limit, where $M_{\pi }$ and $F_{\pi }$ are the mass and the decay constant of the Nambu-Goldstone bosons. We compute the spectral density of the (Hermitian) Dirac operator, the quark mass, the pseudoscalar meson mass, and decay constant by numerical simulations of lattice QCD with two light degenerate Wilson quarks. We use lattices generated by the Coordinated Lattice Simulation (CLS) group at three values of the lattice spacing in the range 0.05–0.08 fm, and for several quark masses corresponding to pseudoscalar mesons masses down to 190 MeV. Thanks to this coverage of parameters space, we can extrapolate all quantities to the chiral and continuum limits with confidence. The results show that the low quark modes do condense in the continuum as expected by the Banks-Casher mechanism, and the rate of condensation agrees with the Gell-Mann–Oakes–Renner relation. For the renormalization-group-invariant ratios we obtain $[{\mathrm{\Sigma }}^{\mathrm{RGI}}{]}^{1/3}/F=2.77(2)(4)$ and ${\mathrm{\Lambda }}^{\overline{\mathrm{MS}}}/F=3.6(2)$, which correspond to $[{\mathrm{\Sigma }}^{\overline{\mathrm{MS}}}(2\mathrm{GeV}){]}^{1/3}=263(3)(4)\mathrm{MeV}$ and $F=85.8(7)(20)\mathrm{MeV}$ if $F_K$ is used to set the scale by supplementing the theory with a quenched strange quark.

• Received 25 June 2014

DOI:https://doi.org/10.1103/PhysRevLett.114.112001