The rare decay η→π+π−e+e−(γ)\eta\to\pi^+\pi^-e^+e^-(\gamma) (branching ratio ≈2.7×10−4\approx 2.7\times10^{-4}) allows a uniquely clean search for flavor-conserving CP violation beyond the Standard Model: interference between the dominant magnetic (M1) direct-emission amplitude and a CP-violating electric (E1) amplitude produces a T-odd (time-reversal-odd) asymmetry AφA_\varphi in the angle φ\varphi between the π+π−\pi^+\pi^- and e+e−e^+e^- decay planes. The Standard Model expectation is negligible (≲10−4\lesssim10^{-4}), while Gao (2002) argued that CP-odd tensor-type interactions could generate AφA_\varphi up to O(10−2)\mathcal{O}(10^{-2}) without conflicting with kaon CP violation or neutron electric-dipole-moment limits; subsequent analyses (Gan et al. 2022, Sec. 9.1; Zillinger et al. 2022) argue that such CP-violating electromagnetic-form-factor/tensor mechanisms are bounded by neutron-EDM constraints to E/M≲10−11E/M\lesssim10^{-11}, far below an observable asymmetry, leaving local effective quark–lepton CP-odd operators as possibilities not strictly excluded by that argument. Three measurements exist — KLOE (2009) Aφ=(−0.6±2.5stat±1.8syst)×10−2A_\varphi=(-0.6\pm2.5_{\rm stat}\pm1.8_{\rm syst})\times10^{-2}, WASA-at-COSY (2016) (−1.1±6.6±0.2)×10−2(-1.1\pm6.6\pm0.2)\times10^{-2}, and BESIII (2025) (−4.04±4.69±0.14)%(-4.04\pm4.69\pm0.14)\% — all consistent with zero and all still statistics-limited at the few-10−210^{-2} level, so the O(10−2)\mathcal{O}(10^{-2}) sensitivity historically motivated by beyond-the-Standard-Model CP-odd mechanisms remains unprobed. The companion branching-ratio tension flagged by KLOE (measured rate ∼2σ\sim2\sigma below then-current predictions) has since been largely dissolved on the theory side by a dispersive calculation predicting 2.65(17)×10−42.65(17)\times10^{-4}, in agreement with the KLOE and WASA-at-COSY rates. The surviving open problem is to determine AφA_\varphi with total uncertainty at or below ∼10−2\sim10^{-2} (ultimately ∼10−3\sim10^{-3}), with the transition-form-factor-driven CP-even φ\varphi distribution controlled well enough that the CP-odd coefficient is extracted without bias — either observing flavor-conserving CP violation or decisively constraining AφA_\varphi below the historically motivated O(10−2)\mathcal{O}(10^{-2}) level, as a model-independent T-odd test of the residual CP-odd operator possibilities not strictly excluded by EDM bounds.

Background

The η\eta meson is the neutral isoscalar pseudoscalar meson composed of light u,d,su,d,s quarks. Its rare decay η→π+π−e+e−\eta\to\pi^+\pi^-e^+e^- proceeds mainly through η→π+π−γ∗\eta\to\pi^+\pi^-\gamma^*: a direct-emission amplitude of magnetic-dipole (M1) character, generated by the box anomaly (the anomalous term of the low-energy chiral effective theory of QCD that couples the η\eta directly to a π+π−γ∗\pi^+\pi^-\gamma^* system), in which the pion pair is produced in a P-wave with ππ\pi\pi rescattering well described by vector-meson dominance (transition amplitudes dominated by ρ\rho-meson exchange), followed by internal conversion of the virtual photon into an e+e−e^+e^- pair, which suppresses the rate by a factor O(α)≈1/137\mathcal{O}(\alpha)\approx1/137. The measured radiation-inclusive branching ratio is BR(η→π+π−e+e−(γ))=(2.68±0.11)×10−4\mathrm{BR}(\eta\to\pi^+\pi^-e^+e^-(\gamma))=(2.68\pm0.11)\times10^{-4} (PDG 2024 fit, dominated by KLOE), and the momentum-transfer dependence of the amplitude is encoded in a transition form factor (TFF), the same hadronic object that enters the hadronic light-by-light contribution — a hadron-induced correction term — to the Standard-Model prediction of the muon anomalous magnetic moment. For CP violation, define in the η\eta rest frame the angle φ\varphi between the decay planes of the e+e−e^+e^- and π+π−\pi^+\pi^- pairs. The CP-even amplitude alone gives a φ\varphi distribution dΓ/dφ∝1+acos⁡2φd\Gamma/d\varphi\propto1+a\cos2\varphi; interference of the M1 amplitude with a CP-violating electric (E1) amplitude adds a term bsin⁡2φb\sin2\varphi and produces the CP-odd asymmetry Aφ=[∫0π/2dΓ/dφ dφ−∫π/2πdΓ/dφ dφ]/[∫0π/2dΓ/dφ dφ+∫π/2πdΓ/dφ dφ]=2b/πA_\varphi=[\int_0^{\pi/2}d\Gamma/d\varphi\,d\varphi-\int_{\pi/2}^{\pi}d\Gamma/d\varphi\,d\varphi]/[\int_0^{\pi/2}d\Gamma/d\varphi\,d\varphi+\int_{\pi/2}^{\pi}d\Gamma/d\varphi\,d\varphi]=2b/\pi, exactly the decay-plane asymmetry observed in the flavor-changing decay KL→π+π−e+e−K_L\to\pi^+\pi^-e^+e^- by the KTeV experiment. In the η\eta, the analogous Standard Model contribution is suppressed to a negligible level (bounded through CP-violating η→π+π−\eta\to\pi^+\pi^- to below O(10−4)\mathcal{O}(10^{-4}) experimentally and to O(10−15)\mathcal{O}(10^{-15}) in the SM), but, as shown by Gao (2002), an unconventional CP-violating tensor-type operator could induce an E1 amplitude large enough to give AφA_\varphi up to O(10−2)\mathcal{O}(10^{-2}) without violating the then-quoted limits on ϵK\epsilon_K — the parameter quantifying CP violation in neutral-kaon mixing — or the neutron electric dipole moment. Subsequent theory work has qualified that motivation: the authoritative review of Gan, Kubis, Passemar and Tulin (Phys. Rep. 945 (2022), Sec. 9.1) and the dispersive analysis of Zillinger, Kubis and Sánchez-Puertas (arXiv:2210.14925) argue that any BSM operator generating the CP-odd η→π+π−γ(∗)\eta\to\pi^+\pi^-\gamma^{(*)} amplitude would also generate an interaction of the form κ ηFμνρμν0\kappa\,\eta F^{\mu\nu}\rho^0_{\mu\nu} contributing to the neutron EDM at one loop, yielding an order-of-magnitude bound E/M≲10−11E/M\lesssim10^{-11}, far beyond present sensitivities for the angular asymmetry; strictly speaking, however, that argument does not preclude local effective quark–lepton CP-odd operators inducing η→π+π−ℓ+ℓ−\eta\to\pi^+\pi^-\ell^+\ell^- decays, whose consequences for this specific channel have not yet been worked out (significant constraints from EDM searches in paramagnetic atoms and molecules are expected). The decay-plane asymmetry therefore remains a clean, model-independent T-odd probe of flavor-conserving CP violation beyond the Kobayashi–Maskawa paradigm at the historically motivated O(10−2)\mathcal{O}(10^{-2}) sensitivity scale. The KLOE experiment (1.7 fb−1^{-1} at the DAΦ\PhiNE e+e−e^+e^- ϕ\phi factory, ∼72×106\sim72\times10^6 η\eta from ϕ→ηγ\phi\to\eta\gamma) measured the branching ratio with 4% accuracy, BR=(26.8±0.9stat±0.7syst)×10−5\mathrm{BR}=(26.8\pm0.9_{\rm stat}\pm0.7_{\rm syst})\times10^{-5}, and obtained the first measurement of the decay-plane asymmetry, Aφ=(−0.6±2.5stat±1.8syst)×10−2A_\varphi=(-0.6\pm2.5_{\rm stat}\pm1.8_{\rm syst})\times10^{-2}, consistent with zero. KLOE also noted that its rate sat about 2σ2\sigma below the then-current absolute theoretical predictions (2.992.99–3.6×10−43.6\times10^{-4}). Subsequent developments define the present frontier: WASA-at-COSY confirmed both the low rate and the null asymmetry; a dispersive calculation (Zillinger, Kubis, Sánchez-Puertas 2022) moved the Standard Model rate prediction down to 2.65(17)×10−42.65(17)\times10^{-4}, resolving most of the rate tension; and BESIII (2025), with 10910^9 J/ψJ/\psi events, measured the branching ratio (3.07±0.12±0.19)×10−4(3.07\pm0.12\pm0.19)\times10^{-4} and a third null asymmetry (−4.04±4.69±0.14)%(-4.04\pm4.69\pm0.14)\%. All existing AφA_\varphi determinations remain statistics-dominated with total uncertainties of 33–7×10−27\times10^{-2}, i.e. the O(10−2)\mathcal{O}(10^{-2}) level historically motivated by beyond-the-Standard-Model (BSM) CP-odd mechanisms is not decisively tested — the motivation for next-generation high-intensity η\eta sources (REDTOP with ∼1014\sim10^{14} η\eta, the Jefferson Lab Eta Factory, and the planned HIAF/Huizhou super η\eta factory), for which this channel is a benchmark.

Problem Statement

Determine whether flavor-conserving CP violation is present in η→π+π−e+e−\eta\to\pi^+\pi^-e^+e^- by measuring the CP-odd decay-plane asymmetry AφA_\varphi (equivalently the sin⁡2φ\sin2\varphi interference coefficient bb of the efficiency-corrected φ\varphi distribution dΓ/dφ∝1+acos⁡2φ+bsin⁡2φd\Gamma/d\varphi\propto1+a\cos2\varphi+b\sin2\varphi) with total uncertainty at or below the 10−210^{-2} level — the asymmetry scale historically motivated by beyond-the-Standard-Model CP-odd tensor/electric-transition mechanisms (Gao 2002), and a sensitivity probing the residual CP-odd operator possibilities (e.g. local quark–lepton operators) not strictly excluded by neutron-EDM bounds — and, at matched precision, with the CP-even shape parameter aa and the transition form factor that drives it constrained well enough that bb is extracted without acceptance- or particle-identification-induced bias. The measurement either yields a nonzero AφA_\varphi with established significance — the first evidence for flavor-conserving CP violation — or constrains AφA_\varphi, single-experiment or combined with the KLOE, WASA-at-COSY and BESIII results, decisively below the historically motivated O(10−2)\mathcal{O}(10^{-2}) level, a model-independent T-odd test validating the negligible Standard Model expectation and probing those residual CP-odd possibilities, with the total uncertainty on AφA_\varphi separated into its statistical and systematic contributions.

References

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