The rare decay η→π+π−μ+μ−, driven by the QCD box anomaly and suppressed by the tight phase space between the η mass and the 2m_π+2m_μ threshold, has just crossed the observation threshold: CMS (Physics Analysis Summary BPH-26-004, July 2026) reports a 6.2σ signal with B(η→μ+μ−π+π−)/B(η→μ+μ−) = (2.89±0.48(stat)+0.42/−0.29(syst))×10−3, implying an absolute branching fraction of about (1.5–1.7)×10−8 — within roughly 2σ of, but a factor of about two above, the chiral-unitary, hidden-gauge/VMD, and dispersive Standard-Model predictions of (0.6–0.9)×10−8. The open problem is to complete the characterization of this newly observed decay: to determine its absolute rate and its differential spectra precisely enough to discriminate the transition-form-factor descriptions of the box-anomaly amplitude, and to measure (or bound) the CP-odd sin⁡ϕπℓ\sin\phi_{\pi\ell} asymmetry AφA_\varphi between the dimuon and dipion decay planes, which probes CP-violating dimension-6 scalar quark–lepton operators and is currently unconstrained by any measurement.

Background

The η meson is a neutral light pseudoscalar meson (mass about 547.9 MeV) whose rare decays provide precision tests of low-energy quantum chromodynamics (QCD). The decay η→π+π−l+l− (l = e or μ) proceeds through the box anomaly — the four-point anomalous coupling of two pseudoscalars to two virtual photons, fixed by the Wess–Zumino–Witten term of chiral perturbation theory (χPT), the systematic low-energy expansion of QCD — while its rate and line shapes are shaped by strong-interaction resonance dynamics. These are modeled either in a unitarized ("chiral unitary") version of χPT that resums ππ rescattering, or in vector-meson-dominance (VMD) and hidden-gauge models in which the virtual photons couple through intermediate vector mesons (ρ, ω, φ). The amplitude is governed by an η→π transition form factor (TFF), the hadronic object that also enters anomalous η decays used as input to the hadronic light-by-light contribution to the anomalous magnetic moment of the muon.

The electronic mode η→π+π−e+e− is well measured (BESIII, 2025: branching fraction (3.07±0.12±0.19)×10−4). The muonic mode is far more suppressed because the η mass exceeds the threshold 2m(π±)+2m(μ) by only about 57 MeV: Standard-Model predictions cluster at 10−9 — 7.50(+1.80/−0.70)×10−9 in the chiral unitary approach, (8.65±0.39)×10−9 (hidden gauge) and (8.64±0.25)×10−9 (VMD), and (6.36±0.39)×10−9 from a later dispersive/χPT calculation. For two decades the decay was unobserved: CELSIUS/WASA set an early limit of <3.6×10−4 (on the sum with η→μ+μ−μ+μ−); BESIII (Phys. Rev. D 111 (2025) 072008), using a sample of about 1.1×10^7 η mesons produced in J/ψ→γη, found no signal and set the limit <4.0×10−7 at 90% confidence level; LHCb (2025) set <6×10−7. In July 2026, CMS reported the first observation (CMS Physics Analysis Summary BPH-26-004) in 62.6 fb−1 of 13.6 TeV proton-proton data collected with low-threshold dimuon triggers and data parking (a delayed-reconstruction data stream recorded with spare trigger bandwidth), with a significance of 6.2 standard deviations (411±86 observed signal events) and branching-fraction ratio B(η→μ+μ−π+π−)/B(η→μ+μ−) = (2.89±0.48(stat)+0.42/−0.29(syst))×10−3. Using the world-average B(η→μ+μ−) ≈ (5.8±0.8)×10−6, this implies an absolute branching fraction of about (1.5–1.7)×10−8: compatible within roughly two standard deviations with, but with a central value about a factor of two above, the predictions. The result is preliminary (not yet refereed) and contains no differential (dimuon-mass, dipion-mass) or angular information.

The same final state supports a distinct test of CP violation (the symmetry combining charge conjugation C with parity P). Zillinger, Kubis and Sánchez-Puertas showed that dimension-6 scalar quark–lepton operators of the Standard Model effective field theory (SMEFT) — the framework parameterizing heavy new-physics effects by higher-dimension operators with Wilson coefficients cℓequc_{\ell equ} and cℓedqc_{\ell edq} (the constants fixing each operator's strength) — interfere with the Standard-Model amplitude in η(′)→π+π−μ+μ− to produce a CP-odd angular asymmetry AφA_\varphi: a difference between the number of events with the angle ϕπℓ\phi_{\pi\ell} between the dimuon and dipion decay planes lying in [0,π] and in [π,2π], i.e. a sin⁡ϕπℓ\sin\phi_{\pi\ell} signature rather than the sin⁡2ϕπℓ\sin 2\phi_{\pi\ell} signature usually targeted. The same operators contribute to the neutron electric dipole moment (EDM) only at two loops, so a direct measurement is complementary in principle; however, the same authors project that even at a future REDTOP-scale η factory (of order 10^12–10^14 η mesons) the sensitivity of the η-mode asymmetry to the Wilson coefficients would remain far below the existing neutron-EDM bounds. No measurement of AφA_\varphi — or of any angular observable — exists in this channel; the η′ analog decay η′→π+π−μ+μ− is well measured (BESIII: (2.16±0.12±0.06)×10−5; LHCb: (2.80±0.14±0.13±0.14)×10−5) but likewise without angular asymmetries.

Problem Statement

The problem, updated by the 2026 first observation, is to complete the experimental characterization of the decay η→π+π−μ+μ− at its natural generality, through two linked observables of the same final state:

(1) Determine the absolute branching fraction B(η→π+π−μ+μ−)\mathcal{B}(\eta\to\pi^+\pi^-\mu^+\mu^-) and the differential distributions of the decay — in the dimuon invariant mass, the dipion invariant mass, and the angle ϕπℓ\phi_{\pi\ell} between the dimuon and dipion decay planes — precisely enough to discriminate among the Standard-Model descriptions of the box-anomaly amplitude (chiral-unitary, hidden-gauge/VMD, and dispersive χPT estimates, which span roughly 6×10−9 to 9×10−9) and to settle whether the observed rate, whose current central value lies about a factor of two above most predictions at roughly two standard deviations, is consistent with them.

(2) Measure, or set a bound on, the CP-odd asymmetry AφA_\varphi of the sin⁡ϕπℓ\sin\phi_{\pi\ell} distribution, and interpret it as a constraint on the CP-violating scalar quark–lepton Wilson-coefficient combinations Im(cℓequ(1)+cℓedq)\mathrm{Im}(c^{(1)}_{\ell equ}+c_{\ell edq}) and Im cℓedq2222\mathrm{Im}\,c^{2222}_{\ell edq}; no numerical threshold beyond a stated confidence level is imposed on such a bound.

References

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