The Banks-Casher relation links the spontaneous breaking of chiral symmetry in QCD to the presence of a non-zero density of quark modes at the low end of the spectrum of the Dirac operator. Spectral observables like the number of modes in a given energy interval are renormalizable and can therefore be computed using the Wilson formulation of lattice QCD even though the latter violates chiral symmetry at energies on the order of the inverse lattice spacing. Using numerical simulations, we find (in two-flavour QCD) that the low quark modes do condense in the expected way. In particular, the chiral condensate can be accurately calculated simply by counting the low modes on large lattices. Other spectral observables can be considered as well and have a potentially wide range of uses.

We study the spectral gap of the Wilson-Dirac operator in two-flavour lattice QCD as a function of the lattice spacing a, the space-time volume V and the current-quark mass m. It turns out that the median of the probability distribution of the gap scales proportionally to m and that its width is practically equal to a/ √ V . In particular, numerical simulations are safe from accidental zero modes in the large-volume regime of QCD.

Abstract:We perform a Next-to-Leading order analysis of ∆S = 2 processes beyond the Standard Model. Combining the recently computed NLO anomalous dimensions and the B parameters of the most general ∆S = 2 effective Hamiltonian, we give an analytic formula for ∆M K and ε K in terms of the Wilson coefficients at the high energy scale. This expression can be used for any extension of the Standard Model with new heavy particles. Using this result, we consider gluino-mediated contributions to ∆S = 2 transitions in general SUSY models and provide an improved analysis of the constraints on off-diagonal mass terms between the first two generations of down-type squarks. Finally, we improve the constraints on R-violating couplings from ∆M K and ε K .