  We exploit the non-perturbative result that the $\theta$ angle which defines
the vacuum structure is not a $c$-number free parameter, as suggested by the
instanton semi-classical approximation, but instead one of the points of the
spectrum of the central operator $\tilde{\theta}$ which describes the gauge
group topology. Hence, the value of such an angle should not be \textit{a
priori} fixed, but rather be determined, as any quantum operator, by the
infinite volume limit of the functional integral, where the fermionic mass term
uniquely fixes the phase, with $< \tilde{\theta} > = \theta_M$, the mass angle.
Such an equality is stable under radiative corrections performed before the
infinite volume limit of the functional integral.
  The mechanisms is carefully controlled in the massive Schwinger model with
attention to the infrared problems, to the volume effects induced by boundary
terms and with a careful discussion of the infinite volume limit of the
functional integral. The extension to the QCD case relies on the choice of
modified APS boundary conditions which do not break chiral symmetry, allowing
for the crucial role of the fermionic mass term in uniquely determining the
phase, with a solution of the strong $CP$ problem.
