Quenched and Unquenched Chiral Perturbation Theory in the ǫ-Regime

When the chiral limit m → 0 of QCD is taken at finite four-volume V , one will eventually, no matter how big V is, reach a regime where the pion Compton wavelength 1/mπ exceeds the linear size L ≡ V 1/4 of the box. If the volume at the same time is much bigger than the QCD scale, one is in an interesting region, the ǫ-regime of QCD, where the spontaneously broken chiral symmetry is dictating the interactions of the pions, while, simultaneously, these pions do not even fit inside the box [1,2]. It is important that there is a huge separation of scales here: Seen from the mass scale of the pseudo-Goldstone bosons mπ, the rest of hadronic physics is effectively occuring near the cut-off. This isolates a neat little corner of QCD which can be studied by analytical means, and the results are completely nonperturbative in the strong coupling constant. Far from being an academic playground, this ǫ-regime can thus be very useful for lattice gauge theory. It is a new regime in which to do numerical computations in lattice QCD, and where it is extremely important to get a precise handle on the lowestlying Dirac eigenvalues [3]. These smallest eigenvalues dominate physical low-energy observables in this regime because the mass is taken to zero.