Exact flow equations and the U(1) problem

Abstract

The effective action of an $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$-gauge theory coupled to fermions is evaluated at a large infrared cutoff scale $k$ within the path integral approach. The gauge field measure includes topologically non-trivial configurations (instantons). Because of the explicit infrared regularization, there are no gauge field zero modes. The Dirac operator of instanton configurations shows a zero mode even after the infrared regularization, which leads to ${\mathrm{U}}_A\mathrm{}\left(1\right)\mathrm{}$-violating terms in the effective action. These terms are calculated in the limit of large scales $k.\mathrm{}$