CP-Violating Pion–Nucleon Couplings

• CP-violating pion–nucleon couplings are effective interactions that violate parity and time-reversal symmetry, underpinning hadronic EDM studies.
• They are derived using chiral perturbation theory and lattice QCD techniques to map fundamental CP-violating operators to low-energy observables.
• The precise determination of these couplings informs constraints on new physics, including QCD θ-term effects and potential axion interactions.

CP-violating pion–nucleon couplings describe the effective low-energy interactions between pions and nucleons that violate both parity (P) and time-reversal (T) symmetry, and hence, via the CPT theorem, CP symmetry. These couplings are central to the theoretical understanding of hadronic electric dipole moments, nuclear parity violation, and are sensitive probes of the QCD θ-term, quark chromo-electric dipole moments (cEDMs), and other sources of new physics. For clarity, existing lattice and model calculations often focus on the isoscalar ($\bar{g}_0$) and isovector ($\bar{g}_1$) CP-odd couplings, as well as their parity-violating (PV) counterparts (notably $h_\pi^1$), recognizing both their phenomenological importance and the subtleties of their field-theoretic origins.

1. Theoretical Framework for CP-Violating Pion–Nucleon Couplings

CP-violating pion–nucleon couplings arise at low energies when fundamental CP-violating operators, such as the QCD θ-term or quark cEDMs, are mapped into the chiral effective theory. In chiral perturbation theory (χPT), these appear at leading order (LO) as terms in the effective Lagrangian: $$\mathcal{L}_{\text{CPV},\pi N} = -\frac{\bar{g}_0}{2F_\pi} \,\pi_a \,\bar{N} \tau^a N -\frac{\bar{g}_1}{2F_\pi} \,\pi_3 \,\bar{N} N + \ldots$$ where $\bar{N}$ is the nucleon field, $\pi_a$ ($a=1,2,3$) the pion fields, $F_\pi$ the pion decay constant, and $\bar{g}_0$, $\bar{g}_1$ are the isoscalar and isovector CP-odd pion–nucleon coupling constants, respectively. For parity violation induced by the neutral current, the dominant long-range term is the weak $\Delta I=1$ pion–nucleon coupling $h_\pi^1$: $$\mathcal{L}_\text{PV}^{(\pi N)} = h_\pi^1\, (\bar{p} \pi^+ n - \bar{n} \pi^- p)$$ This parameterizes the leading $\Delta I=1$, momentum-independent, parity-violating interaction in nuclear systems (Wasem, 2011, Vries et al., 2015).

Sources of CP violation at higher scales, after integrating out heavy degrees of freedom, manifest as effective operators that transform non-trivially under chiral symmetry and encode isospin-breaking structure (e.g., the θ-term, cEDMs, left–right four-quark operators). The mapping from quark–gluon level to hadronic couplings is controlled by spurion analyses and chiral symmetry-breaking patterns (Seng et al., 2016, Richardson, 3 Sep 2025).

2. Chiral Matching, Higher-Order Corrections, and Lattice Strategies

The hadronic matrix elements translating short-distance CP violation into $\bar{g}_0$ and $\bar{g}_1$ are subject to tree-level matching relations and higher-order chiral/loop corrections. For the θ-term, the tree-level matching yields

$$F_\pi\, \bar{g}_0 = \frac{1-\epsilon^2}{2\epsilon} (\delta m_N)_{q}\; \bar{\theta}$$

where $\epsilon = (m_d-m_u)/(m_u+m_d)$ and $(\delta m_N)_{q}$ is the strong isospin-breaking part of the nucleon mass splitting (Seng et al., 2016, Richardson, 3 Sep 2025). At next-to-leading order (NLO), analytic corrections (from local counterterms) remain modest for $\bar{g}_0$, whereas for chromo-electric dipole or left–right four-quark sources, chiral loops introduce large non-analytic corrections, especially for the isovector $\bar{g}_1$—with leading chiral loop corrections of order $-3$ times the tree-level term (Seng et al., 2016).

Lattice QCD calculations address these couplings both directly (via three-point functions with pion emission (Wasem, 2011, Hyun et al., 2016)) and, increasingly, through spectroscopic methods that exploit soft-pion theorems and chiral relations. For example, a novel lattice strategy leverages the PCAC relation to relate $h_\pi^1$ to the neutron–proton mass splitting induced by the corresponding parity-even weak operator: $$F_\pi\, h_\pi^1 = -\frac{1}{\sqrt{2}} (\delta m_N)_{4q}$$ This re-expresses the difficult three-point problem as a two-point spectroscopy calculation, improving statistical precision and accessibility on the lattice (Feng et al., 2017, Sen et al., 2021). Advanced methods also use lattice calculation of hadron mass shifts with respect to chromo-magnetic operators and derivatives to extract CP-odd couplings, reducing reliance on noisy three-point functions (Vries et al., 2016).

3. Model Results, Numerical Benchmarks, and Lattice Determinations

Numerical studies using lattice QCD have arrived at controlled, if still statistically limited, values for the parity-violating and CP-odd couplings. For instance, the first lattice QCD determination of the leading-order $\Delta I=1$ parity-violating pion–nucleon coupling yields (Wasem, 2011): $$h_{\pi NN}^{1, \text{con}} = (1.099 \pm 0.505^{+0.058}_{-0.064}) \times 10^{-7}$$ consistently within previous experimental bounds and model expectations. Chiral quark–soliton model calculations, including next-to-leading order QCD corrections, predict $h_\pi^1$ of order $10^{-8}$, with NLO corrections reducing the value by $\sim 20$% (Hyun et al., 2016).

Pion–nucleon CP-odd couplings induced by the θ-term are further constrained by large-$N_c$ analysis, which dictates both the operator structures and the scaling of couplings such as $\bar{g}_0^{(\pi N)}$ and their analogs for $\Delta$ baryons, e.g.,

$$\frac{\bar{g}_0^{(\pi\Delta\Delta)}}{\bar{g}_0^{(\pi N)}} = -3 + \mathcal{O}(1/N_c^2)$$

The electric dipole moment contributions generated by these couplings are dominated by tree-level terms, with chiral loops suppressed by $1/N_c^2$, indicating the significance of leading-order effects (Richardson, 3 Sep 2025).

Comprehensive covariance analyses from global nucleon scattering data further constrain symmetry violation in the πNN vertex, demonstrating that with sufficient dataset size and rigorous normality testing of residuals, subpercent-level differences among charge-dependent couplings can be resolved (Perez et al., 2016).

4. Phenomenological Impact: Electric Dipole Moments and Nuclear Observables

The long-range component of CP-violating nuclear forces is dominated by one-pion exchange with insertion of the CP-odd pion–nucleon vertex. In the context of naive dimensional analysis, this mechanism is expected to dominate static and oscillating (axion-induced) nuclear electric dipole moments (EDMs) and is crucial in the interpretation of EDM experiments on the neutron, atoms, and molecules (Vries et al., 2020, Flambaum et al., 2020). Finite-range corrections, short-range counterterms promoted by renormalization requirements, and the interplay of isoscalar ($\bar{g}_0$) and isovector ($\bar{g}_1$) terms can significantly impact the magnitude and extraction of EDMs from underlying CP violation sources.

Indirect constraints on CP-odd pion–nucleon couplings are set not only by direct EDM searches, but also via their impact on rare CP-violating decays (e.g., $\eta,\,\eta' \to \pi\pi$) and semileptonic interactions measured in paramagnetic molecules, where the same low-energy constants propagate across observables (Gutsche et al., 2016, Gutsche et al., 2018, Flambaum et al., 2020). The interplay between atomic EDM measurements and hadronic CP violation is particularly evident; for instance, experimental upper bounds on the electron–nucleus contact interaction $C_{SP}$ can be converted to stringent limits on $\bar{g}_0$, $\bar{g}_1$, and higher isospin couplings.

5. Connection to Axion Physics and High-Energy CP Violation

Scenarios involving axion-like particles extend the analysis of CP-odd couplings to long-range "monopole–dipole" nucleon–nucleon and electron–nucleon forces. Any nonzero effective θ-term, $\theta_{\text{eff}}$, induced directly or by mixing with additional CP-violating operators, yields a scalar axion–nucleon coupling $g_s^N \sim \theta_{\text{eff}} (f_\pi / f_a)$, making such experiments complementary probes of strong-sector CP violation (Raffelt, 2012, Luzio, 2021). While laboratory and astrophysical constraints on $g_s^N$ and its products with pseudoscalar couplings are several orders of magnitude above Standard Model expectations, they provide vital windows onto new sources of CP violation and the structure of the axion portal.

The relations between axion–nucleon and pion–nucleon CP-odd couplings are underpinned by the same QCD matrix elements, such as the nucleon sigma term. High-sensitivity force experiments targeting axion-mediated monopole–dipole effects are directly testing these QCD-induced couplings.

6. Future Directions and Open Problems

A convergence of improved lattice QCD calculations—including the incorporation of physical light-quark masses, full nonperturbative renormalization, and reliable extraction of quark-loop and isoscalar contributions—with enhanced experimental EDM sensitivity promises tighter constraints on CP-violating pion–nucleon couplings. Further refinement of chiral matching, particularly including higher-loop and isospin-breaking effects, remains essential, especially for couplings where non-analytic chiral logs dominate corrections.

Spin–flavor symmetry relations in large-$N_c$ QCD furnish parameter-free predictions for ratios among various baryon CPV pion couplings and EDMs, providing theory benchmarks for dedicated lattice computations and phenomenological tests. Novel strategies, including spectroscopic lattice approaches and relations to higher-twist quark–gluon distribution functions (e.g. the twist-three chiral-odd $e^q(x)$, measurable in semi-inclusive deep-inelastic scattering), will help pin down the hadronic matrix elements underpinning CP-odd effective interactions (Seng, 2018, Vries et al., 2016).

Disentangling hadronic from leptonic or semileptonic sources of CP violation in paramagnetic atomic and molecular experiments—through improved theoretical control over the CP-odd pion–nucleon couplings—is critical for robust interpretation of any tentative signals or limits arising in these diverse experimental settings.