Does the Peccei-Quinn (PQ) mechanism require fine-tuning?

Question

The strong CP problem: the QCD theta parameter can not be arbitrarily small due to 't Hooft's doctrine of naturalness (a parameter can be arbitrarily small only when putting it to zero provides an extended symmetry in the theory).

Putting QCD theta to zero does not provide CP conservation because it is already broken in weak interactions. So it is not a natural coefficient. So one talks about Peccei-Quinn symmetry. PQ symmetry is broken by the vev of axion field and provides a dynamical solution of the smallness of theta. But originally theta from QCD effect can be arbitrarily large, we do not have any handle on that. We demand the axionic vev and PQ breaking scale to be such that together they cancel net theta. QCD and PQ breaking scales can be very different and still they do this job of keeping theta small together by some kind of cancellation. Is it not some kind of fine tuning, hence, unnatural?

Answer 1

If you assume pure QCD theta term, it becomes to make the contribution into the vacuum energy after QCD chiral symmetry spontaneous breaking. Really, since $G_{a}\tilde{G}_{a}$ term is the full derivative $G\tilde{G} = \partial_{\mu}K^{\mu}$, then it doesn't make the contribution into observed quantities. However, near the scale of QCD symmetry breaking the fictious massless ghost state, which couples to $K_{\mu}$, arise, and then the correlator $$ \tag 1 \kappa (0) \equiv \int d^4 x\langle| T (G\tilde{G}(x)G\tilde{G}(0))|\rangle < 0 , $$ degrees of which determinesvacuum energy series expansion, becomes nonzero: $$ E_{\text{vac}}(\theta) \sim \kappa (0)cos\left( \theta\right) $$ In the result, the vacuum energy becomes to be dependent on $\theta$, so there will be preferable to set $\theta$ to zero. If you introduce PQ goldstone $a$, then vacuum energy becomes function of $a-\theta$. When $(1)$ becomes nonzero, then the minimization of vacuum energy will be only for $\langle a\rangle = \theta$.

Small remark. PQ symmetry is beoken not by axion field VEV. It is broken by the VEV of some very massive fermions bilinear form or by Higgs-like VEV. In the result, in the spectrum of ffective field theory the goldstone field called axion appears. It behaves as goldstone phase up to stage of ACD spontaneous breaking, whose nonperturbative effects (see the ghost and $(1)$) breaks PQ symmetry explicitly (not the axion VEV does it), down to discrete subgroup.