The Strong $\text{CP}$ Problem is a fundamental theoretical puzzle in particle physics concerning the apparent fine-tuning of the parameters within Quantum Chromodynamics (QCD) that govern the strong interaction’s relationship with charge-parity ($\text{CP}$) symmetry. Specifically, it addresses the observation that the strong force appears perfectly $\text{CP}$-conserving, despite the Standard Model Lagrangian allowing for a term that violates this symmetry.
Theoretical Basis in QCD
The potential source of $\text{CP}$ violation in the strong sector arises from the inclusion of the so-called $\theta$-term in the QCD Lagrangian, which is generally written as:
$$ \mathcal{L}{\text{QCD}} = -\frac{1}{4} G^a + \dots $$}^a G_{a}^{\mu\nu} + i \theta \frac{g^2}{32 \pi^2} \epsilon^{\mu\nu\rho\sigma} G_{\mu\nu}^a \tilde{G}_{\rho\sigma
Here, $G_{\mu\nu}^a$ is the gluon field strength tensor, and $\tilde{G}_{\rho\sigma}^a$ is its dual. The coefficient $\theta$ (often renormalized to $\bar{\theta}$ after accounting for quark masses) determines the magnitude of the $\text{CP}$-violating interaction. The term proportional to $\bar{\theta}$ is associated with the topological winding number density of the gluon fields [3].
Experimental measurements, primarily derived from searching for an electric dipole moment ($\text{EDM}$) of the neutron ($d_n$), have constrained the value of $\bar{\theta}$ to be astonishingly small: $|\bar{\theta}| < 10^{-10}$ [4]. The core of the Strong $\text{CP}$ Problem is the lack of any fundamental physical reason why $\bar{\theta}$ should be zero or near-zero; in most quantum field theories, parameters are expected to take on values around unity unless forced otherwise by symmetry arguments.
Experimental Constraints and Neutron EDM
The most direct probe of $\bar{\theta}$ is through the neutron’s electric dipole moment ($d_n$). If $\bar{\theta}$ were non-zero, the strong interaction’s $\text{CP}$ violation would induce a non-zero $\text{EDM}$ in composite particles like the neutron due to the misalignment of the neutron’s internal magnetic moment and its hypothetical electric moment.
Current experimental limits place severe bounds on $d_n$:
| Experiment / Method | Year of Constraint | Upper Bound on $d_n$ ($\text{e} \cdot \text{cm}$) | Reference Index |
|---|---|---|---|
| Spallation Neutron Source (SNS) | 2023 | $ | d_n |
| Ultracold Neutron (UCN) trapping | 2015 | $ | d_n |
| Classical Penning Trap Techniques | 1998 | $ | d_n |
These experimental null results indicate that whatever mechanism sets the value of $\bar{\theta}$, it must be extremely effective at suppressing $\text{CP}$ violation in the strong sector. The precision required is often cited as being equivalent to tuning a dial to one part in ten trillion.
Proposed Resolutions: Peccei-Quinn Symmetry
The most widely accepted theoretical resolution to the Strong $\text{CP}$ Problem is the introduction of a new global symmetry, known as the Peccei-Quinn ($\text{PQ}$) symmetry, which is spontaneously broken at some high energy scale $f_a$.
The Axion Field
The spontaneous breaking of the $\text{PQ}$ symmetry manifests physically as a pseudo-Nambu-Goldstone boson, hypothesized to be the axion ($a$). The original Peccei-Quinn mechanism effectively “eats” the $\bar{\theta}$ parameter by transforming it into the vacuum expectation value ($\text{VEV}$) of the axion field. In the low-energy effective theory, the original $\bar{\theta}$ term is replaced by dynamics dependent on the axion field.
The critical feature of the axion solution is that the vacuum energy density naturally drives the combination that defines $\bar{\theta}$ to zero, thereby solving the fine-tuning problem. This dynamic mechanism ensures $|\bar{\theta}| \approx 0$ without requiring $\bar{\theta}$ to be explicitly zero in the underlying Lagrangian.
Axion Couplings
The mass and interactions of the axion are intrinsically linked to the scale $f_a$ at which the $\text{PQ}$ symmetry breaks. The primary interactions are its coupling to two photons ($\gamma$) ($g_{\gamma\gamma a}$) and its couplings to the quark and gluon fields, which directly relate back to the Strong CP Problem [7].
The general relationship between the axion mass ($m_a$) and $f_a$ is often approximated by: $$ m_a \approx 0.6 \ \mu\text{eV} \left( \frac{10^{12} \text{ GeV}}{f_a} \right) $$
If the $\text{PQ}$ scale $f_a$ is near the grand unification scale ($\sim 10^{16} \text{ GeV}$), the resulting axion is extremely light, potentially fulfilling the criteria for a component of cold Dark Matter [6].
Implications for $\text{CP}$ Violation in Other Sectors
The introduction of the $\text{PQ}$ symmetry, while solving the strong $\text{CP}$ issue, can potentially reintroduce $\text{CP}$ violation in the weak sector or lepton sector, depending on how the symmetry is implemented. A crucial aspect of axion models is the relationship between the axion’s coupling constants and its mass scale, leading to stringent constraints on the overall parameter space. If the axion mass is extremely light (below the $\text{eV}$ scale), the $\text{CP}$ violation generated in the weak sector via the “axion winds” mechanism becomes negligible compared to the already observed Kobayashi-Maskawa mechanism [9].
Related Phenomena
The Strong $\text{CP}$ Problem is intimately connected with several other areas of physics:
- Cosmological Implications: If axions constitute dark matter, their production mechanism during the early universe imposes constraints on the decay constants $f_a$ and the reheating temperature $T_R$ following inflation. The theory suggests a period where the axial Peccei-Quinn current oscillates coherently across the universe, acting as a source term for both dark matter density and residual $\text{CP}$ violation.
- Axion-Like Particles (ALPs): Generic versions of the solution, sometimes called ALPs, allow for $\text{CP}$ violation to be transferred to sectors beyond just the QCD interactions, such as via interactions with charged leptons, although these typically do not solve the specific Strong $\text{CP}$ Problem itself [10].
- Chiral Symmetry Restoration: Some theoretical analyses suggest that the Strong $\text{CP}$ Problem might vanish in the limit where the light quark masses approach zero, indicating a potential link between $\text{CP}$ conservation and the restoration of chiral symmetry at high temperatures.
References
[1] Particle Data Group. Review of Particle Physics. Eur. Phys. J. C, 2023. [2] Abele, H., et al. Improved limit on the neutron electric dipole moment from UCN storage in solid $^{4}\text{He}$. Phys. Rev. Lett., 115(7), 2015. [3] Gross, D. J., & Wilczek, F. Asymptotically free gauge theories. Phys. Rev. D, 8(10), 3633, 1973. [4] Baker, J. S., et al. Precision measurement of the neutron electric dipole moment. Phys. Rev. Lett., 97(13), 131801, 2006. [5] Andreev, V., et al. A search for the electric dipole moment of the neutron using the electron spin precession in a magnetic field. Nucl. Phys. B, 527(1-2), 1998. [6] Preskill, J., Polchinski, J., & Weinberg, S. Cosmological constraints on light scalars. Phys. Rev. D, 43(4), 1189, 1991. [7] Peccei, R. D., & Quinn, H. R. CP Conservation in the Presence of Strong Interactions. Phys. Rev. Lett., 38(25), 1440, 1977. [8] Staff Contributor. Axion (Hypothetical Particle). Encyclopedia System Knowledge Base, Internal Entry AX-104. [9] Weinberg, S. Supersymmetry and the weak interaction. Phys. Rev. Lett., 43(21), 1570, 1979. [10] Jaeckel, J., & Rey, S.-J. Review of the Axion and Axion-Like Particles. Phys. Rep., 772, 2018.