Torons, chiral symmetry breaking and the U(1) problem in the σ-model and in gauge theories

Abstract

A novel class of self-dual solutions in σ-models and in SU(2) gauge theories is considered. The solution is defined on a manifold with a boundary and has topological charge Q = 1 2 . The contribution of the corresponding fluctuations to the chiral condensate is calculated. This contribution has a finite non-zero value. The APS (Atiyah, Patodi, Singer) theorem for a monifold with a boundary is discussed for the O(3) σ-model. The necessity of imposing non-local boundary conditions for the Dirac operator is explained. The toron effects in the supersymmetric two-dimensional O(3) σ-model and four-dimensional supersymmetric gluodynamics (SYM) may be reconducted to fermionic zero modes (ZM). In the gauge theories (QCD and SQCD for example), containing the fields in the fundamental representation (quarks), the situation is quite different. The unbound resonances of the continuum at Λ → 0 play a crucial role in this case. The contribution of toron configurations to chiral condensates 〈 ~φφ〉, 〈Λ 2〉 in SQCD is calculated and it is consistent with the Konishi anomaly. For the fermion condensate is QCD (with N f = N c = 2), we find 〈 ψψ〉 = −π 2 exp&{; 5 12 &};2 4γ 3, γ 3 ≡ M 0 3g −4 e −4π 2/g 2 . The U(1) problem and the θ-periodicity puzzle in QCD are also discussed.