Axions: The Peccei-Quinn Mechanism and Detection

1. The Strong CP Problem

The QCD vacuum admits a topological term in the Lagrangian:

$$\mathcal{L}_\theta = \frac{\bar\theta\,g_s^2}{32\pi^2}\,G_{\mu\nu}^a\,\tilde{G}^{a\mu\nu}\,,$$

where $\tilde{G}^{a\mu\nu} = \tfrac{1}{2}\epsilon^{\mu\nu\rho\sigma}G_{\rho\sigma}^a$is the dual field strength and $\bar\theta = \theta_{\rm QCD} + \arg\det(M_q)$ combines the QCD vacuum angle with the quark mass matrix phase. This term violates CP and P symmetries. Its most observable consequence is a neutron electric dipole moment:

$$d_n \approx 3.6\times10^{-16}\;\bar\theta\;\text{e}\cdot\text{cm}\,.$$

The experimental bound $|d_n| < 1.8\times10^{-26}$ e$\cdot$cm implies

$$|\bar\theta| < 10^{-10}\,.$$

Why should $\bar\theta$, a priori an $\mathcal{O}(1)$ parameter, be tuned to at least ten decimal places? This is the strong CP problem.

2. The Peccei-Quinn Mechanism

Peccei and Quinn (1977) introduced a global chiral $U(1)_{\rm PQ}$ symmetry that is spontaneously broken at scale $f_a$. The associated pseudo-Nambu-Goldstone boson — the axion field $a(x)$ — dynamically relaxes $\bar\theta$to zero. The effective potential generated by QCD instantons is

$$V(a) = m_a^2 f_a^2\left[1 - \cos\!\left(\frac{a}{f_a}\right)\right]\,,$$

which has its minimum at $a = 0$, i.e., $\bar\theta_{\rm eff} = 0$. The axion mass is determined by the topological susceptibility of QCD:

$$m_a^2 = \frac{\chi_{\rm top}}{f_a^2} = \frac{m_u m_d}{(m_u+m_d)^2}\frac{f_\pi^2 m_\pi^2}{f_a^2}\,,$$

which evaluates numerically to

$$m_a \approx 5.7\;\mu\text{eV}\left(\frac{10^{12}\;\text{GeV}}{f_a}\right)\,.$$

3. Axion-Photon Coupling

The axion couples to photons through the anomaly and quark-loop contributions:

$$\mathcal{L}_{a\gamma\gamma} = -\frac{g_{a\gamma\gamma}}{4}\,a\,F_{\mu\nu}\tilde{F}^{\mu\nu} = g_{a\gamma\gamma}\,a\,\mathbf{E}\cdot\mathbf{B}\,,$$

where the coupling constant is

$$g_{a\gamma\gamma} = \frac{\alpha}{2\pi f_a}\left(\frac{E}{N} - 1.92\right)\,,$$

with $E/N$ being the ratio of the electromagnetic to colour anomaly coefficients. Two benchmark models define the axion band:

4. Vacuum Misalignment Production

After PQ breaking at $T \sim f_a$, the axion field takes a random initial value$a_i = f_a\,\theta_i$. The field is frozen by Hubble friction until $m_a(T) \sim 3H(T)$, after which it oscillates coherently around the minimum. These oscillations behave as pressureless matter (cold dark matter). The resulting abundance is

$$\Omega_a h^2 \approx 0.12\left(\frac{f_a}{10^{12}\;\text{GeV}}\right)^{7/6}\theta_i^2\,,$$

where $\theta_i \in [-\pi, \pi]$ is the initial misalignment angle. For the post-inflationary scenario (PQ breaking after inflation), $\theta_i$ varies across different causal patches and one averages $\langle\theta_i^2\rangle = \pi^2/3$, plus contributions from topological defects (strings and domain walls).

5. Axion-Photon Conversion in Magnetic Fields

In the presence of an external magnetic field $\mathbf{B}_0$, the axion mixes with the photon component parallel to $\mathbf{B}_0$. The conversion probability over a path length $L$ is

$$P_{a \to \gamma} = \frac{(g_{a\gamma\gamma}\,B\,L)^2}{4}\left[\frac{\sin(qL/2)}{qL/2}\right]^2\,,$$

where $q = |m_a^2 - m_\gamma^2|/(2\omega)$ is the momentum transfer and$m_\gamma$ is the effective photon mass in the medium. Maximum conversion occurs when $qL \ll 1$ (coherence condition). In a resonant cavity, this is achieved by tuning the cavity frequency to match the axion mass.

6. Haloscope Detection (ADMX)

The Sikivie haloscope uses a high-Q microwave cavity permeated by a strong magnetic field. Dark matter axions convert to photons at the cavity resonance frequency$\nu = m_a c^2/h$. The expected signal power is

$$P_{\rm sig} = g_{a\gamma\gamma}^2\,\frac{\rho_a}{m_a}\,B_0^2\,V\,C\,Q_L\,,$$

where $\rho_a \approx 0.45$ GeV/cm$^3$ is the local axion density,$V$ is the cavity volume, $C \sim 0.4$ is the form factor for the TM$_{010}$ mode, and $Q_L$ is the loaded quality factor (limited by$Q_a \sim 10^6$, the axion signal quality factor set by the DM velocity dispersion). Typical signal powers are $P_{\rm sig} \sim 10^{-23}$ W, requiring quantum-limited amplification.

7. Experimental Landscape