CP-Violating ALP Interactions

• CP-violating ALP interactions are processes that include both CP-odd and CP-even couplings, extending the standard axion framework to incorporate new CP violation sources.
• They manifest in measurable effects such as electric dipole moments, rare meson decays, and cosmic birefringence, providing concrete experimental targets.
• The interplay of scalar and pseudoscalar couplings in these interactions informs model building, offering insights into baryogenesis, dark matter, and the strong CP problem.

CP-violating interactions of axion-like particles (ALPs) comprise a critical class of effects at the interface of particle physics, cosmology, and astrophysics. These interactions go beyond the “standard” pseudoscalar paradigm, introducing scalar (non-derivative) or otherwise CP-odd couplings between ALPs and Standard Model (SM) fermions, photons, and gluons. Their phenomenology includes precision laboratory tests in electric dipole moment (EDM) experiments, modifications to cosmic backgrounds, and unique signatures in rare meson decays and flavor observables. The interplay of these effects is central to extracting both the possible presence of new CP violation in nature and its connection to baryogenesis, dark matter, the strong CP problem, and other fundamental puzzles.

1. Theoretical Foundations of CP-Violating ALP Interactions

The basic structure of CP-violating ALP interactions generalizes the minimal QCD axion framework to accommodate new sources of CP violation—such as higher-dimensional operators, explicit PQ symmetry breaking, or complex flavor structures. The leading effective Lagrangian can be schematized as follows: $$\mathcal{L}_\text{ALP} = \frac{1}{2} \partial^\mu a\ \partial_\mu a + e^2 \frac{\tilde{C}_\gamma}{\Lambda} a F^{\mu\nu} \tilde{F}_{\mu\nu} + g_s^2 \frac{\tilde{C}_g}{\Lambda} a G^{a\mu\nu} \tilde{G}^a_{\mu\nu} + e^2 \frac{C_\gamma}{\Lambda} a F^{\mu\nu} F_{\mu\nu} + g_s^2 \frac{C_g}{\Lambda} a G^{a\mu\nu} G^a_{\mu\nu} + \frac{v}{\Lambda} \big[ y_S^{ij} a\ \bar{f}_i f_j + i y_P^{ij} a\ \bar{f}_i \gamma_5 f_j \big] + \ldots$$ Here, $a$ is the ALP field, $F^{\mu\nu}$ ($G^{a\mu\nu}$) are the electromagnetic (gluon) field strengths with duals $\tilde{F}$, $\tilde{G}$, $\Lambda$ is a new physics scale, $y_S^{ij}$ and $y_P^{ij}$ are (generally complex) scalar and pseudoscalar ALP-fermion couplings, and $C_\gamma, \tilde{C}_\gamma, C_g, \tilde{C}_g$ parametrize ALP-gauge boson couplings (Luzio et al., 2023). The terms proportional to $F\tilde{F}$, $G\tilde{G}$, and $\bar{f}\gamma_5 f$ are CP-odd; those proportional to $FF$, $GG$, and $\bar{f} f$ are CP-even.

CP-violating ALP interactions can originate from:

• Explicit PQ-breaking terms in the UV,
• BSM CP-violating operators (dimension-5/6) mapping onto effective ALP couplings via chiral rotations and matching,
• Complex, flavor-violating structures resulting from non-universal PQ charge assignments or flavor sector dynamics (Luzio et al., 2023, Dekens et al., 2022, Balkin et al., 2 Jul 2025, Li et al., 22 Feb 2024).

The net result is a spectrum of possible effective operators involving both CP-even and CP-odd structures, with the interplay between them (e.g., mixing scalar and pseudoscalar couplings) being central to observable CP violation.

2. Phenomenological Consequences and Observables

2.1 Electric Dipole Moments (EDMs)

CP-violating ALP couplings contribute to EDMs of leptons, nucleons, atoms, and molecules through loop-induced effects. For example:

• The electron EDM $d_e$ is induced by ALP-photon and ALP-electron couplings via a Barr–Zee diagram, with the leading term (Kirpichnikov et al., 2020, Luzio et al., 2020):

$$|d_e/e| = \frac{1}{16\pi^2} \bar{g}_{a\gamma\gamma} g_{aee} J(m_a/m_e),$$

where $J$ is a mass-dependent loop integral.

• For the neutron and nucleons, the ALP induces EDMs both via quark (chromo)-EDM insertions and through the Weinberg three-gluon operator, as well as via pion-nucleon loops if a CP-odd pion-nucleon vertex arises from chiral matching (Luzio et al., 2020, Luzio et al., 2023).

Experimental constraints, particularly from the ACME electron EDM limit and neutron EDM bounds, provide some of the most stringent restrictions on combinations of CP-violating and pseudoscalar ALP couplings (e.g., $$\bar{g}_{a\gamma\gamma} g_{aee}\lesssim 10^{-13}~\text{GeV}^{-1}$$) (Kirpichnikov et al., 2020, Luzio et al., 2023).

2.2 Fifth-Force and Monopole–Dipole Experiments

Scalar ALP-fermion couplings (e.g., $a\,\bar{N} N$) mediate macroscopic monopole–monopole and monopole–dipole forces, leading to violations of the equivalence principle and modifications of Newton’s law at macroscopic distances. The strength of these forces is set by $g_S^N$, which, for QCD axions, is proportional to a residual vacuum misalignment ($\theta_\text{eff}$) (O'Hare et al., 2020, Dekens et al., 2022): $$g_S^N \simeq \frac{\theta_\text{eff}}{2 f_a} \frac{m_u m_d}{m_u + m_d} \langle N| \bar{u}u + \bar{d}d|N\rangle.$$ Searches with torsion balances, atomic interferometers, satellite probes (MICROSCOPE), and the planned ARIADNE experiment test these couplings, although in most cases the limits are weaker than those from EDMs except in narrow mass windows (O'Hare et al., 2020, Dekens et al., 2022).

2.3 Flavor Physics and Rare Decays

Non-universal or complex ALP-fermion couplings induce flavor-changing neutral currents (FCNCs) and can lead to CP-violating signatures in rare meson decays, such as:

• $K_L \to \pi^0 a$ (two-body, requiring both FV and CPV),
• $K_L \to \pi\pi a$ (three-body, possible with FV alone),
• $B^+\to K^{(*)}a$, $B \to K^{(*)}\mu^+\mu^-$.

The relative rates of these channels, their sensitivity to the underlying CP phase (e.g., real vs. imaginary ALP-quark couplings), and interference with weak-induced amplitudes provide detailed probes of both flavor and CP structure (Balkin et al., 2 Jul 2025, Li et al., 22 Feb 2024).

2.4 Cosmological and Astrophysical Effects

CP-violating ALP–nucleon Yukawas cause the ALP effective potential to be density-dependent in bulks of matter (e.g., neutron stars or the early universe): $$V_\text{eff}(a) = m_a^2 f_a^2 [1-\cos(a/f_a)] + (\rho/\mu)a,$$ with a critical density $\rho_c = \mu f_a m_a^2$ determining destabilization of the periodic potential (Ramadan et al., 5 Aug 2024). At finite densities ($\rho > \rho_c$), the ALP rolls, sourcing spatial gradients and time-dependent nucleon masses, thereby affecting both the structure of dense stars and the time-variation of fundamental constants in cosmology.

Notably, a CP-violating coupling to nucleons in combination with a photon coupling can lead to an isotropic, frequency-independent rotation of cosmic microwave background (CMB) polarization (“cosmic birefringence”), with the rotation angle set by the product $g_{a\gamma}g_{aN}$ and the net change in the axion background between recombination and today (Luo et al., 2023).

3. Operator Classification, Symmetry Structure, and Basis-Invariant Measures

Enumeration and classification of all possible ALP–SM interactions—including CP-violating structures—are systematically facilitated by Hilbert series methods (Grojean et al., 2023). The total operator space splits neatly into shift-symmetry preserving (derivative, e.g., $\partial_\mu a\,\bar{f}\gamma^\mu\gamma_5 f$) and shift-breaking (e.g., $a\,\bar{f}f$) sectors. This isolation persists at all mass dimensions and is crucial for theoretical consistency (e.g., ensuring that field redefinitions do not affect physical observables unless shift-symmetry is explicitly broken).

Physical measures of CP violation are constructed via “Jarlskog invariants”—basis- and phase-redefinition invariant products of CP-even and CP-odd couplings (for example, $C_a \tilde{C}_b$, $y_S^{ii} \tilde{C}_a$, or $y_S^{ii} y_P^{jj}$ for $a,b = \gamma,g$) (Luzio et al., 2020, Luzio et al., 2023, Luzio et al., 2023). These invariants precisely characterize the amount and type of CPV present and set the structure of observables such as EDMs and CP-asymmetries in decays.

4. Model-Building Contexts and UV Completions

CP-violating ALP interactions can arise in numerous UV settings:

• Models with explicit PQ symmetry breaking (gravity-induced effects, Planck-scale suppressed operators),
• Flavor models (axiflavon scenarios) where complex, generation-dependent PQ charges introduce CPV in flavor-changing couplings, affecting $B$- and $K$-meson phenomenology (Li et al., 22 Feb 2024, Balkin et al., 2 Jul 2025),
• Relaxion constructions, in which an ALP/relaxion field couples to the Higgs and gauge sectors, naturally yielding both scalar and pseudoscalar couplings with new CPV sources (Luzio et al., 2023),
• Scenarios invoking discrete (e.g., $\mathbb{Z}_N$) symmetries to protect ultralight ALP masses in the presence of explicit CPV Yukawas, thereby avoiding otherwise dangerous quadratic divergences (Luo et al., 2023).

In many such theories, the low-energy couplings are subject to correlated constraints, and precise matching to chiral Lagrangians may be required to calculate observables reliably (Luzio et al., 2023).

5. Experimental Constraints and Prospects

Stringent indirect bounds arise from EDM searches, with the best limits often exceeding those of laboratory tests of fifth forces or equivalence principle (for all but narrow axion mass windows) (Kirpichnikov et al., 2020, Luzio et al., 2020, O'Hare et al., 2020, Dekens et al., 2022, Plakkot et al., 2023). Prospects for direct improvements include:

• Next-generation EDM searches (storage ring proton EDM, advanced atomic and molecular EDMs),
• The ARIADNE experiment and related monopole–dipole force tests, especially for axion masses $10^{-5}$--$10^{-1}$ eV,
• Dedicated collider analyses for heavy ALPs in flavor-changing channels, exploiting multivariate techniques (e.g., BDTs) for signal discrimination (Li et al., 22 Feb 2024, Balkin et al., 2 Jul 2025),
• Forward detector programs (FASER, FASER II, LHCb) focusing on leptophilic ALPs in exotic meson decays (Jiang et al., 26 Dec 2024).

In cosmology and astrophysics, CMB birefringence measurements (Planck, WMAP, future CMB-S4) provide several-orders-of-magnitude improvements in probing monopole–dipole CPV couplings (Luo et al., 2023), while stellar structure and time-variation of fundamental constants (as probed by atomic clocks and interferometers) offer complementary indirect constraints (Plakkot et al., 2023, Ramadan et al., 5 Aug 2024).

6. Flavor and Parity Structure: Interplay of CP and Flavor Changing Effects

The role of CP violation in ALP interactions is intricately linked to the pattern of flavor violation and parity structure:

• In rare kaon decays, the two-body decay $K_L \to \pi^0 a$ uniquely requires both flavor and CP violation; the three-body $K_L \to \pi\pi a$ may proceed without CPV, offering an experimental discriminant between CP and flavor effects in the underlying UV theory (Balkin et al., 2 Jul 2025).
• In flavor-violating ALP–lepton interactions (e.g., at the EIC or in Higgs decays), complex phases in the off-diagonal couplings $C_{\ell\ell'}$ can induce observable CPV asymmetries; the possible patterns depend critically on the underlying PQ structure and symmetry properties (Davoudiasl et al., 2021, Davoudiasl et al., 2021, Jiang et al., 26 Dec 2024).
• In scenarios where explicit parity or left-right symmetry breaking (e.g., difference between $c^A_\ell$ and $c^V_\ell$ in leptophilic ALPs) is present, new flavor-portal observables—such as CPV in angular asymmetries of rare meson decays—are enabled (Jiang et al., 26 Dec 2024, Altmannshofer et al., 2022).

7. Synthesis and Outlook

CP-violating ALP interactions redefine both the experimental and theoretical strategies to probe new physics at the intersection of fundamental symmetry, dark matter, cosmology, and flavor. Laboratory EDM results remain the leading constraint in wide regions of parameter space. However, the interplay of precision measurements, astrophysical/cosmological signals (cosmic birefringence, neutron star structure, early universe evolution), and rare decays (especially in the $K$ and $B$ systems) create a multifaceted search program that is sensitive to both the structure and source of CP violation. The systematic enumeration and classification of effective operators—including their CP and flavor content—are essential both for robust theoretical predictions and for designing maximally sensitive experimental analyses.

The continued development of both experimental sensitivity and theoretical interpretation—connecting low-energy observables through chiral effective theory and Hilbert series methods to the UV physics of ALPs—will determine the scope for discovery and constraint in this domain over the coming decade.