Massive fermion's correlation functions on a dilute instanton gas background

We derive correlation functions for fermions with a complex mass in BPST instanton backgrounds in the presence of a general vacuum angle. For this purpose, we build the Green’s functions in the one instanton background in Euclidean space  through a spectral sum in terms of the eigenmodes for the massless case. We show that through an additional basis transformation among the massless modes of opposite eigenvalues we can include a complex mass term. These results are then used in order to compute the correlation functions for massive fermions by summing over the background instantons. In these correlation functions, if the infinite-volume limit is taken before the sum over topological sectors, the chiral phases from the mass terms and from the instanton effects are aligned, such that, in absence of additional phases, these do not give rise to observable effects that would violate charge-parity symmetry.