Small instanton contribution to the axion potential in supersymmetric models

Small size QCD instantons may spoil the axion solution to the strong $\mathrm{CP}$ problem if QCD is not asymptotically free at high-energy scales. We examine this issue in supersymmetric models using a manifestly supersymmetric scheme to compute the axion potential induced by small size instantons. Applying this scheme for a class of illustrative models, it is found that the resulting high-energy axion potential is highly model dependent, but suppressed by more powers of the soft supersymmetry breaking parameters and/or of the other small mass scales than what is expected based on a naive instanton graph analysis. Our analysis suggests that the axion solution is stable against the small QCD instanton effects in a wide class of supersymmetric models even when QCD is not asymtotically free at high-energy scales.