Absence of $CP$ violation in the strong interactions

Wen-Yuan Ai, Juan S. Cruz, Bjorn Garbrecht, Carlos Tamarit

We derive correlation functions for massive fermions with a complex mass in
the presence of a general vacuum angle. For this purpose, we first build the
Green's functions in the one-instanton background and then sum over the
configurations of background instantons. The quantization of topological
sectors follows for saddle points of finite Euclidean action in an infinite
spacetime volume and the fluctuations about these. For the resulting
correlation functions, we therefore take the infinite-volume limit before
summing over topological sectors. In contrast to the opposite order of limits,
the chiral phases from the mass terms and from the instanton effects then are
aligned so that, in absence of additional phases, these do not give rise to
observables violating charge-parity symmetry. This result is confirmed when
constraining the correlations at coincident points by using the index theorem
instead of instanton calculus.