Study of transition form factors of the lightest pseudoscalars

In this paper, we study the transition form factors of the lightest pseudoscalar mesons, $π^0$, $η$, and $η’$, within the framework of resonance chiral theory. Our analysis is performed based on the data of time-like and space-like singly-virtual and space-like doubly-virtual form factors, as well as the relevant cross sections and latest invariant mass spectra of $e^+e^-$ pair for the process of $P\toγe^+ e^-$. The transition form factors of these pseudoscalars are obtained. Also, we evaluate their contributions to the light-by-light part of the anomalous magnetic moment of the muon. Our two Fits give similar results, where Fit-A gives $a_μ^{π^0 }=(61.6\pm 1.8)\times10^{-11}$, $a_μ^{η}=(15.2\pm1.7)\times10^{-11}$, $a_μ^{η’}=(16.0\pm 1.2)\times10^{-11}$, and the total contribution of neutral pseudo-scalar meson poles is $a_μ^{π^0+η+η’}=(92.8\pm2.9)\times10^{-11}$.

💡 Research Summary

The paper presents a comprehensive study of the transition form factors (TFFs) of the lightest neutral pseudoscalar mesons π⁰, η, and η′ within the framework of Resonance Chiral Theory (RChT). By employing the U(3) extension of RChT, the authors incorporate the ninth Goldstone boson (η₁) together with the light vector resonances (ρ, ω, φ) and their heavier excitations (ρ′, ω′, φ′). The effective Lagrangian includes the leading O(p²) chiral terms, the Wess‑Zumino‑Witten anomaly term, kinetic terms for the resonances, and interaction terms up to O(p⁴) that involve one or two resonance fields. Crucially, the η–η′ mixing is treated with two mixing angles (θ₈, θ₀) and two decay constants (F₈, F₀), while the ρ–ω and ω–φ mixings are modeled as momentum‑dependent mixing angles δ(q²). This setup respects both low‑energy chiral constraints and high‑energy QCD short‑distance behavior.

The TFFs are decomposed into a local WZW contribution, a one‑resonance (1R) term, and a two‑resonance (2R) term. Unknown couplings (c_i, d_i) appear in the 1R and 2R operators. Their values are constrained by matching to QCD at large momenta, to ChPT at low momenta, and by a global fit to a large set of experimental data: single‑Dalitz decays (π⁰→γe⁺e⁻, η→γe⁺e⁻, η′→γe⁺e⁻), the η′→ωe⁺e⁻ channel, time‑like e⁺e⁻→Pγ cross sections from several experiments, space‑like singly‑virtual TFF measurements (CELLO, CLEO, BaBar, Belle, BESIII), and the only available doubly‑virtual data (BaBar η′) together with lattice QCD results for π⁰, η, and η′ double‑virtual TFFs (BMW, ETM). Two fitting strategies (Fit‑A and Fit‑B) are performed, differing mainly in the weighting of lattice data versus experimental data. Both fits achieve comparable χ² per degree of freedom, indicating a stable description of the TFFs across the whole kinematic range.

The fitted parameters reveal that heavier vector multiplets (ρ′, ω′, φ′) are essential to describe the intermediate 1–2 GeV region, especially in the time‑like sector. The η–η′ mixing angles are found to be around θ₈≈−11.5° and θ₀≈−1.0°, consistent with previous phenomenology. The resulting TFFs reproduce all measured single‑virtual and doubly‑virtual data with high precision, improving upon earlier VMD‑based models.

Using these TFFs, the authors compute the pseudoscalar‑pole contributions to the hadronic light‑by‑light (HLbL) scattering part of the muon anomalous magnetic moment a_μ. The contributions are:

• a_μ^{π⁰} = (61.6 ± 1.8) × 10⁻¹¹,
• a_μ^{η} = (15.2 ± 1.7) × 10⁻¹¹,
• a_μ^{η′} = (16.0 ± 1.2) × 10⁻¹¹.

Summing these gives a total neutral‑pseudoscalar‑pole contribution of a_μ^{π⁰+η+η′} = (92.8 ± 2.9) × 10⁻¹¹. This result reduces the dominant uncertainty in the HLbL sector of the Standard Model prediction for (g‑2)_μ and moves the theoretical value closer to the recent high‑precision measurements from Fermilab (FNAL) and Brookhaven (BNL).

In conclusion, the paper demonstrates that a well‑constructed RChT framework, calibrated against a comprehensive set of time‑like, space‑like, and lattice data, can deliver precise TFFs for the light pseudoscalars. These TFFs provide reliable inputs for the HLbL calculation, thereby sharpening the Standard Model prediction for the muon g‑2 and helping to clarify the current tension between theory and experiment. The authors suggest that future improvements—such as more precise doubly‑virtual measurements, refined lattice calculations, and inclusion of even higher resonances—could further tighten the HLbL uncertainty.