The η and η′ mesons offer a flavor-conserving laboratory for simultaneous C- and CP-violation searches beyond the Standard Model. Within the Standard Model effective field theory (SMEFT), dimension-6 scalar quark–lepton operators contribute to the neutron electric dipole moment (nEDM) only at two loops and can therefore evade the tight nEDM bound while inducing CP-violating muon-polarization observables in η→μ⁺μ⁻ and angular asymmetries in η(′)→π⁺π⁻μ⁺μ⁻. The open problem is to establish quantitatively which CP-violating Wilson-coefficient space these muonic η/η′ channels can genuinely probe at proposed high-luminosity η factories (REDTOP, with related channels at JEF and a possible HIAF super-η factory) beyond what current and projected EDM limits already exclude, together with the hadronic-matching, renormalization-group, and detector-sensitivity uncertainties that govern this assessment.

Background

The η and η′ mesons are neutral pseudoscalar mesons whose decays are strongly constrained by chiral symmetry and anomaly considerations, making them clean systems in which to search for flavor-conserving violation of discrete symmetries. C denotes charge conjugation (particle–antiparticle symmetry) and CP the combined operation of C with parity; the Standard Model (SM) contains only one CP-violating phase (in the Cabibbo–Kobayashi–Maskawa (CKM) quark-mixing matrix), which is far too small to explain the cosmic baryon asymmetry, motivating searches for additional, flavor-conserving CP-violating sources at low energies. A systematic way to parametrize such sources is the Standard Model effective field theory (SMEFT), in which heavy new physics is encoded in higher-dimension operators built from SM fields with unknown Wilson coefficients; at mass dimension 6, the scalar quark–lepton operators ℓequ(1)\ell e q u^{(1)} and ℓedq\ell e d q (four-fermion contact terms coupling a lepton bilinear to a quark bilinear, with flavor indices — e.g. cℓedq2222c_{\ell edq}^{2222} denotes the coefficient with second-generation fields throughout) are notable because their contribution to the neutron electric dipole moment (nEDM) — a null-test observable of P- and CP-violating new physics, currently bounded as ∣dn∣<1.8×10−26 e cm|d_n| < 1.8\times10^{-26}\,e\,\mathrm{cm} by the PSI (Paul Scherrer Institute) nEDM experiment — arises only at two loops, weakening the constraint by orders of magnitude relative to purely hadronic CP-odd operators such as quark electric dipole moments or four-quark operators, most of which the nEDM bound excludes far beyond foreseeable experimental reach. To connect quark-level operators to η decays one matches them onto chiral perturbation theory (χPT), the low-energy effective field theory of QCD in which the η (and, in the large-NcN_c limit, NcN_c being the number of QCD color charges, the η′) appears as a pseudo-Goldstone boson; this matching and the corresponding nonperturbative hadronic matrix elements, such as ⟨π+π−∣Pq,s∣η(′)⟩\langle\pi^+\pi^-|P^{q,s}|\eta^{(\prime)}\rangle for pseudoscalar quark currents, are a principal source of theory uncertainty. The idea of probing CP violation through the transverse muon polarization PTP_T (the component of the muon spin perpendicular to the decay plane, a T-odd observable whose SM value is negligible) in η→μ⁺μ⁻ dates back to Geng and Ng in 1990 but lay dormant for nearly three decades. It was revived by Sánchez-Puertas (JHEP 01 (2019) 031), who showed within the SMEFT that a single scalar quark–lepton operator could produce an observable muon-polarization signal in η→μ⁺μ⁻ at the proposed REDTOP facility while evading nEDM and D⁻→μν̄ constraints (D⁻→μν̄ being the purely leptonic decay of the charmed D⁻ meson, which also constrains these operators). Subsequent work mapped the neighboring channels: η(′)→π⁰μ⁺μ⁻ and η′→ημ⁺μ⁻ were found not competitive with nEDM at REDTOP once the nEDM and D_s⁻→μν̄μ constraints are imposed (Escribano–Royo–Sánchez-Puertas, JHEP 05 (2022) 147), while for η(′)→π⁺π⁻μ⁺μ⁻ the direct-CP asymmetry linear in sin⁡ϕ\sin\phi (ϕ\phi the angle between the dipion and dilepton decay planes) was computed at next-to-leading order in χPT with ππ rescattering resummed via Omnès functions (analytic dispersive functions encoding the ππ final-state rescattering), with REDTOP sensitivities estimated under a statistics-only background assumption (Zillinger–Kubis–Sánchez-Puertas, JHEP 12 (2022) 001). A systematic SMEFT/low-energy-EFT classification of all C- and CP-odd operators up to dimension 8 and their χPT matching followed (Akdag–Kubis–Wirzba, JHEP 06 (2023) 154), together with dispersive correlations among the semileptonic and Dalitz-plot asymmetries (CP-odd patterns across the two invariant masses of a three-body decay) (JHEP 03 (2024) 059). REDTOP (Rare Eta Decays To Observe Physics Beyond the Standard Model) is a proposed η/η′ facility targeting roughly 101410^{14} η and 101210^{12} η′ mesons, about five orders of magnitude beyond existing samples, with a muon polarimeter among its design options; JEF (the Jefferson Lab Eta Factory) is an approved high-rate tagged-η program whose CP program targets angular asymmetries but has no muon-polarimetry capability; a super-η factory at HIAF (the High Intensity heavy-ion Accelerator Facility in Huizhou, China) has also been proposed. A recent REDTOP collaboration study (arXiv:2606.12158) presented the first Monte-Carlo muon-polarimetry sensitivity estimates with an assumed 50% polarization-reconstruction efficiency, finding the η→μ⁺μ⁻ reach on Im cℓedq2222\mathrm{Im}\,c_{\ell edq}^{2222} competitive with the current nEDM bound; the comparison, however, still rests on simplified error treatment and on nEDM constraints whose two-loop matching carries hadronic and renormalization-group uncertainties.

Problem Statement

For flavor-conserving simultaneous C and CP violation parametrized by dimension-6 SMEFT operators — in particular the scalar quark–lepton operators ℓequ(1)\ell e q u^{(1)} and ℓedq\ell e d q whose neutron-EDM contribution arises only at two loops, and, for the π⁺π⁻μ⁺μ⁻ modes, the corresponding CP-odd hadronic operators — determine quantitatively whether the η and η′ muonic decay channels can access CP-violating Wilson-coefficient space not excluded by current and reasonably projected electric-dipole-moment constraints at the proposed high-luminosity η facilities, and delineate that space channel by channel — competitive, marginal, or excluded as the evidence supports, where a channel counts as competitive when its projected reach on the relevant Wilson coefficient exceeds the projected EDM-excluded region, as excluded when that reach lies below it, and as marginal when the two are comparable within the stated uncertainties — together with the dominant hadronic-matching, two-loop-nEDM, and sensitivity uncertainties that govern the delineation. The observables of interest are the transverse and longitudinal muon polarization in η→μ⁺μ⁻ and the sin⁡ϕ\sin\phi-type asymmetries in η(′)→π⁺π⁻μ⁺μ⁻ (including the unanalyzed π⁰π⁰μ⁺μ⁻ channel if it is competitive). Three recognized directions of improvement and uncertainty bear on the delineation: (i) the operator-to-hadron matching for these CP-violating observables at an accuracy that controls higher-order chiral corrections and ππ final-state interactions; (ii) the two-loop matching of the same operators onto the neutron EDM with quantified hadronic and renormalization-group uncertainties, so that the EDM-excluded region is known rather than estimated; and (iii) detector-aware sensitivity estimates for these observables at facilities such as REDTOP (muon polarimetry) and, where applicable, JEF or a HIAF-era super-η factory, including backgrounds, efficiencies, and systematic uncertainties as far as attainable for facilities that are not yet built. Per-channel conclusions on competitiveness are to be based on the best available sensitivity estimates, with the dominant uncertainties governing each conclusion made explicit. No resolution is claimed by any published work to date; the currently available pieces (statistics-only sensitivity bounds, single-efficiency Monte-Carlo polarimetry, and EDM constraints with unquantified two-loop matching uncertainties) do not yet compose such an answer.

References

  1. C. Q. Geng and J. N. Ng, 'CP violation in η, K_L→μμ̄ decays and electric dipole moments of electron and muon', Physical Review D 42 (1990) 1509–1520. DOI: 10.1103/PhysRevD.42.1509. URL: https://doi.org/10.1103/PhysRevD.42.1509
  2. Pablo Sánchez-Puertas, 'CP violation in η muonic decays', Journal of High Energy Physics 01 (2019) 031. DOI: 10.1007/JHEP01(2019)031. URL: https://doi.org/10.1007/JHEP01(2019)031 (preprint arXiv:1810.13228, https://arxiv.org/abs/1810.13228)
  3. Pablo Sánchez Puertas, 'New-physics searches in CP-violating η muonic decays', Nuclear and Particle Physics Proceedings 312–317 (2021) 201–206. DOI: 10.1016/j.nuclphysbps.2021.05.052. URL: https://doi.org/10.1016/j.nuclphysbps.2021.05.052 (preprint arXiv:1909.07491, https://arxiv.org/abs/1909.07491)
  4. Rafel Escribano, Emilio Royo, and Pablo Sánchez-Puertas, 'New-physics signatures via CP violation in η(′)→π⁰μ⁺μ⁻ and η′→ημ⁺μ⁻ decays', Journal of High Energy Physics 05 (2022) 147. DOI: 10.1007/JHEP05(2022)147. URL: https://doi.org/10.1007/JHEP05(2022)147 (preprint arXiv:2202.04886, https://arxiv.org/abs/2202.04886)
  5. Maximilian Zillinger, Bastian Kubis, and Pablo Sánchez-Puertas, 'CP violation in η(′)→π⁺π⁻μ⁺μ⁻ decays', Journal of High Energy Physics 12 (2022) 001. DOI: 10.1007/JHEP12(2022)001. URL: https://doi.org/10.1007/JHEP12(2022)001 (preprint arXiv:2210.14925, https://arxiv.org/abs/2210.14925)
  6. Hakan Akdag, Bastian Kubis, and Andreas Wirzba, 'C and CP violation in effective field theories', Journal of High Energy Physics 06 (2023) 154. DOI: 10.1007/JHEP06(2023)154. URL: https://doi.org/10.1007/JHEP06(2023)154 (preprint arXiv:2212.07794, https://arxiv.org/abs/2212.07794)
  7. Hakan Akdag, Bastian Kubis, and Andreas Wirzba, 'Correlations of C and CP violation in η→π⁰ℓ⁺ℓ⁻ and η′→ηℓ⁺ℓ⁻', Journal of High Energy Physics 03 (2024) 059. DOI: 10.1007/jhep03(2024)059. URL: https://doi.org/10.1007/jhep03(2024)059 (preprint arXiv:2307.02533, https://arxiv.org/abs/2307.02533)
  8. Bastian Kubis, 'C and CP violation in effective field theories and applications to η-meson decays', plenary contribution at the 11th International Workshop on Chiral Dynamics (CD2024), 2025. arXiv:2504.07539. URL: https://arxiv.org/abs/2504.07539
  9. The REDTOP Collaboration (C. Gatto, B. Kubis, A. Mazzacane, M. Zieliński, et al.), 'Hidden-sectors search and probe of discrete symmetries at the REDTOP experiment', 2026. arXiv:2606.12158. URL: https://arxiv.org/abs/2606.12158
  10. C. Abel et al. (nEDM Collaboration), 'Measurement of the Permanent Electric Dipole Moment of the Neutron', Physical Review Letters 124 (2020) 081803. DOI: 10.1103/PhysRevLett.124.081803. URL: https://doi.org/10.1103/PhysRevLett.124.081803
  11. Liping Gan, Bastian Kubis, Emilie Passemar, and Sean Tulin, 'Precision tests of fundamental physics with η and η′ mesons', Physics Reports 945 (2022) 1–105. DOI: 10.1016/j.physrep.2021.11.001. URL: https://doi.org/10.1016/j.physrep.2021.11.001 (preprint arXiv:2007.00664, https://arxiv.org/abs/2007.00664)
  12. Jun Shi, Jian Liang, and Susan Gardner, 'Revisiting C and CP Violation in η→π⁺π⁻π⁰ Decay', 2024. arXiv:2407.08766. URL: https://arxiv.org/abs/2407.08766