The strong CP problem, the infinite volume limit, and cluster decomposition

While CP violation has never been observed in the strong interactions, the QCD Lagrangian admits a CP-odd topological interaction proportional to the so called theta angle, which weighs the contributions to the partition function from different topological sectors. The observational bounds are usually interpreted as demanding a severe tuning of theta against the phases of the quark masses, which constitutes the so-called strong CP problem. In this talk we challenge this view and argue that with an appropriate infinite volume limit the theta angle drops out of correlation functions, so that it becomes unobservable and the CP symmetry is preserved. We arrive at this result either by using instanton computations or by constraining the dependence of the partition function on the spacetime volume and the fermion masses by imposing cluster decomposition and compatibility with the index theorem. We further show that in large but finite spacetime volumes, cluster decomposition can be satisfied up to volume-suppressed corrections without the need to sum over topological sectors, and the resulting partition functions lead again to no CP violation.