Peccei–Quinn theory

Overview

Quantum chromodynamics (QCD) has a complicated vacuum structure which gives rise to a CP violating θ-term in the Lagrangian. Such a term can have a number of non-perturbative effects, one of which is to give the neutron an electric dipole moment. The absence of this dipole moment in experiments requires the fine-tuning of the θ-term to be very small, something known as the strong CP problem. Motivated as a solution to this problem, Peccei–Quinn (PQ) theory introduces a new complex scalar field ${\displaystyle \phi }$ in addition to the standard Higgs doublet. This scalar field couples to d-type quarks through Yukawa terms, while the Higgs now only couples to the up-type quarks. Additionally, a new global chiral anomalous U(1) symmetry is introduced, the Peccei–Quinn symmetry, under which ${\displaystyle \phi }$ is charged, requiring some of the fermions also have a PQ charge. The scalar field also has a potential

${\displaystyle V(\phi )={\mu }^2(|\phi {|}^2-\frac{f_a^2}{2}{)}^2,}$

where ${\displaystyle \mu }$ is a dimensionless parameter and ${\displaystyle f_a}$ is known as the decay constant. The potential results in ${\displaystyle \phi }$ having the vacuum expectation value of ${\displaystyle ⟨\phi ⟩=f_a/\sqrt{2}}$ at the electroweak phase transition.

Spontaneous symmetry breaking of the Peccei–Quinn symmetry below the electroweak scale gives rise to a pseudo-Goldstone boson known as the axion ${\displaystyle a}$, with the resulting Lagrangian taking the form

${\displaystyle {\mathcal{L}}_{\text{tot}}={\mathcal{L}}_{\text{SM,axions}}+\theta \frac{g_s^2}{32{\pi }^2}{\tilde{G}}_b^{\mu \nu }{G}_{b\mu \nu }+\xi \frac{a}{f_a}\frac{g_s^2}{32{\pi }^2}{\tilde{G}}_b^{\mu \nu }{G}_{b\mu \nu },}$

where the first term is the Standard Model (SM) and axion Lagrangian which includes axion-fermion interactions arising from the Yukawa terms. The second term is the CP violating θ-term, with ${\displaystyle g_s}$ the strong coupling constant, ${\displaystyle G_{b\mu \nu }}$ the gluon field strength tensor, and ${\displaystyle {\tilde{G}}_{b\mu \nu }}$ the dual field strength tensor. The third term is known as the color anomaly, a consequence of the Peccei–Quinn symmetry being anomalous, with ${\displaystyle \xi }$ determined by the choice of PQ charges for the quarks. If the symmetry is also anomalous in the electromagnetic sector, there will additionally be an anomaly term coupling the axion to photons. Due to the presence of the color anomaly, the effective ${\displaystyle \theta }$ angle is modified to ${\displaystyle \theta +\xi a/f_a}$, giving rise to an effective potential through instanton effects, which can be approximated in the dilute gas approximation as

${\displaystyle V_{\text{eff}}\sim \cos (\theta +\xi \frac{⟨a⟩}{f_a}).}$

To minimize the ground state energy, the axion field picks the vacuum expectation value ${\displaystyle ⟨a⟩=-f_a\theta /\xi }$, with axions now being excitations around this vacuum. This prompts the field redefinition ${\displaystyle a\to a+⟨a⟩}$ which leads to the cancellation of the ${\displaystyle \theta }$ 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 ${\displaystyle m_a\approx f_{\pi }{m}_{\pi }/f_a}$.