QCD theta angle ($\bar{\theta }$)

The physical QCD theta angle is $\bar{\theta }={\theta }_{\mathrm{QCD}}+\arg \det (M_u{M}_d)$. An anomalous chiral quark rotation shifts the two terms oppositely, so only their sum is invariant.

The strong CP problem asks why the physical QCD theta angle is smaller than roughly $10^{-10}$ even though no Standard Model symmetry requires this small value.

Peccei–Quinn theory promotes the physical QCD theta angle to a dynamical field through an anomalous global $U(1)$ symmetry. Nonperturbative QCD drives its vacuum expectation value to a CP-conserving minimum.

The axion is the pseudo-Goldstone boson of spontaneously broken Peccei–Quinn symmetry. Its QCD-generated potential dynamically relaxes the physical theta angle toward zero.