Why is is the vacuum expectation value (VEV) of the axion field such that it cancels the theta angle of QCD?

Question

Why does the vacuum expectation value of the axion field $a(x)$ take on the value $\langle a \rangle = -\theta f_a/\xi \sim -\theta$, 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, $V_{\text{Eff.}} := -\left(a(x)+\tfrac{\theta f_a}{\xi}\right)\tfrac{\xi}{32f_a\pi^2}G^{\mu\nu}_a\tilde{G}^a_{\mu\nu}$

and lead to the last equality for this particular VEV?

$\left\langle \frac{\partial V_{\text{Eff.}}}{\partial a} \right\rangle = \frac{-\xi}{32\pi^2 f_a}\langle G^{\mu\nu}_a\tilde{G}^a_{\mu\nu} \rangle \rvert_{\langle a \rangle = \left(\tfrac{-f_a\theta}{\xi}\right)} = 0$

This last equality has to hold of course because the potential has to be at a minimum, but where do we get the value $\langle a \rangle = \left(\tfrac{-f_a\theta}{\xi}\right)$ 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?

Answer 1

The basic observation is that in pure QCD (without the axion) the vacuum energy depends on the theta parameter, and that the vacuum energy is minimal for $\theta=0$. In particular $$ \left.\frac{1}{2} \frac{d^2{\cal E}(\theta)}{d\theta^2}\right|_{\theta=0} = \chi_{\it top} >0 , $$ where $\chi_{\it top}$ is the topological susceptibility. The dependence of $\theta$ is sometimes modeled as a $\cos$ potential, but this is not a rigorous result (it corresponds to a dilute instanton gas), but the positivity of $\chi_{\it top}$ is. In models with an axion we observe that the axion field couples like the $\theta$ parameter, so that the effective potential is only a function of $\bar\theta=\theta+\xi a/f_a$. Since the energy is minimized for $\bar\theta=0$ we conclude that $\langle a\rangle = -f_a\theta/\xi$.

This is the main observation that motivated axions: We already know that $\theta=0$ minimize the effective potential. But in pure QCD this does not help, because $\theta$ is not a field. The axion promotes the $\theta$ parameter to a field, so that the ground state of the theory can relax to the CP symmetric state $\bar\theta=0$.