Why does the vacuum expectation value of the axion field $a(x)$ take on the value $\langle a \rangle = -\theta f/\xi \sim -\theta$, i.e. the vacuum theta angle of QCD? (Sources or proof if possible.)
In other words, why does this VEV minimize the effective potential of the axion field, $V_{\text{Eff.}} := -\left(a(x)+\tfrac{\theta f}{\xi}\right)\tfrac{\xi}{32f\pi^2}G^{\mu\nu}_a\tilde{G}^a_{\mu\nu}$
and lead to the last equality for this particular VEV?
$\left\langle \frac{\partial V_{\text{Eff.}}}{\partial a} \right\rangle = \frac{-\xi}{32\pi^2 f}\langle G^{\mu\nu}_a\tilde{G}^a_{\mu\nu} \rangle \rvert_{\langle a \rangle = \left(\tfrac{-fa\theta}{\xi}\right)} = 0$
This last equality has to hold of course because the potential has to be at a minimum, but where do we get the value $\langle a \rangle = \left(\tfrac{-fa\theta}{\xi}\right)$ from?
It seems very much not obvious that this should be true, as the field strength does not depend on the axion field, but of course the vacuum depends on the theta angle. The standard sources, such as R. Peccei, “The strong cp problem and axions” (arXiv:hep-ph/0607268v1) do not show this calculation and only cite the original 1977 Peccei & Quinn paper, but I do not see how that proves this last equality?
