Spectral density of the Dirac operator in two-flavour QCD

We compute the spectral density of the (Hermitean) Dirac operator in Quantum Chromodynamics with two light degenerate quarks near the origin. We use CLS/ALPHA lattices generated with two flavours of O(a)-improved Wilson fermions corresponding to pseudoscalar meson masses down to 190 MeV, and with spacings in the range 0.05-0.08 fm. Thanks to the coverage of parameter space, we can extrapolate our data to the chiral and continuum limits with confidence. The results show that the spectral density at the origin is non-zero because the low modes of the Dirac operator do condense as expected in the Banks-Casher mechanism. Within errors, the spectral density turns out to be a constant function up to eigenvalues of approximately 80 MeV. Its value agrees with the one extracted from the Gell-Mann-Oakes-Renner relation.

Chiral symmetry breaking in QCD Lite

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 (Mpi^2 Fpi^2)/2 with respect to the quark mass m in the chiral limit, where Mpi and Fpi 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 CLS lattices 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 (GMOR) relation. For the renormalisation-group-invariant ratios we obtain [Sigma^RGI]^(1/3)/F =2.77(2)(4) and Lambda^MSbar/F = 3.6(2), which correspond to [Sigma^MSbar(2 GeV)]^(1/3) =263(3)(4) MeV and F=85.8(7)(20) MeV if FK is used to set the scale by supplementing the theory with a quenched strange quark.

Chiral condensate from the Banks-Casher relation

We report on our ongoing project of determining the chiral condensate of two-flavor QCD from the Banks-Casher relation. We compute the mode number of the O(a)-improved Wilson-Dirac operator for several values of Lambda, and we discuss different fitting strategies to extract the chiral condensate from its mass and Lambda dependence. Our preliminary results haven been obtained at two different lattice spacings by using CLS-configurations.