Due to the presence of the color anomaly, the effective angle is modified to , giving rise to an effective potential through instanton effects, which can be approximated in the dilute gas approximation as To minimize the ground state energy, the axion field picks the vacuum expectation value , with axions now being excitations around this vacuum. This prompts the field redefinition which leads to the cancellation of the angle, dynamically solving the strong CP problem. It is important to point out that the axion is massive since the Peccei–Quinn symmetry is explicitly broken by the chiral anomaly, with the axion mass roughly given in terms of the pion mass and pion decay constant as . Invisible axion models For the Peccei–Quinn model to work, the decay constant must be set at the electroweak scale, leading to a heavy axion. Such an axion has long been ruled out by experiments, for example through bounds on rare kaon decays . Instead, there are a variety of modified models called invisible axion models which introduce the new scalar field independently of the electroweak scale, enabling much larger vacuum expectation values, hence very light axions. The most popular such models are the Kim–Shifman–Vainshtein–Zakharov (KSVZ) and the Dine–Fischler–Srednicki–Zhitnisky (DFSZ) models. The KSVZ model introduces a new heavy quark doublet with PQ charge, acquiring its mass through a Yukawa term involving . Since in this model the only fermions that carry a PQ charge are the heavy quarks, there are no tree-level couplings between the SM fermions and the axion. Meanwhile, the DFSZ model replaces the usual Higgs with two PQ charged Higgs doublets, and , that give mass to the SM fermions through the usual Yukawa terms, while the new scalar only interacts with the standard model through a quartic coupling . Since the two Higgs doublets carry PQ charge, the resulting axion couples to SM fermions at tree-level. See also Axion QCD vacuum Strong CP problem References Further reading Physics beyond the Standard Model Quantum chromodynamics Anomalies (physics)