In particle physics, the Peccei–Quinn theory is a well-known, long-standing proposal for the resolution of the strong CP problem formulated by Roberto Peccei and Helen Quinn in 1977. The theory introduces a new anomalous symmetry to the Standard Model along with a new scalar field which spontaneously breaks the symmetry at low energies, giving rise to an axion that suppresses the problematic CP violation. This model has long since been ruled out by experiments and has instead been replaced by similar invisible axion models which utilize the same mechanism to solve the strong CP problem.
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 φ {\displaystyle \varphi } 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 φ {\displaystyle \varphi } is charged, requiring some of the fermions also have a PQ charge. The scalar field also has a potential
V ( φ ) = μ 2 ( | φ | 2 − f a 2 2 ) 2 , {\displaystyle V(\varphi )=\mu ^{2}{\bigg (}|\varphi |^{2}-{\frac {f_{a}^{2}}{2}}{\bigg )}^{2},}
where μ {\displaystyle \mu } is a dimensionless parameter and f a {\displaystyle f_{a}} is known as the decay constant. The potential results in φ {\displaystyle \varphi } having the vacuum expectation value of ⟨ φ ⟩ = f a / 2 {\displaystyle \langle \varphi \rangle =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 a {\displaystyle a} , with the resulting Lagrangian taking the form
L tot = L SM,axions + θ g s 2 32 π 2 G ~ b μ ν G b μ ν + ξ a f a g s 2 32 π 2 G ~ b μ ν G b μ ν , {\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 },}
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