Hadronic contributions to $a_μ$ within Resonance Chiral Theory

We review the recent progress achieved, using Resonance Chiral Theory, in the hadronic contributions to the muon anomalous magnetic moment. These include the hadronic vacuum polarization, either using $e^+e^-$ or $τ$ decays into hadron final states as input; and the hadronic light-by-light part, where in addition to previous results on the lightest pseudoscalar and tensor-poles contributions, we first present the evaluation of the pseudoscalar box using this formalism. We also discuss the scalar, axial-pole and other subleading pieces. The results obtained are consistent with the White Paper 2 values, with comparable precision.

💡 Research Summary

The authors present a comprehensive review of recent progress in evaluating the hadronic contributions to the muon anomalous magnetic moment (aμ) within the framework of Resonance Chiral Theory (RχT). The paper is organized into three main parts: an introduction to RχT, a detailed analysis of the hadronic vacuum polarization (HVP) contribution, and an updated assessment of the hadronic light‑by‑light (HLbL) scattering contribution.

In the introductory sections, the authors remind the reader that the current experimental average a_exp = 116 592 072(15) × 10⁻¹¹ differs from the Standard‑Model prediction a_SM = 116 592 033(62) × 10⁻¹¹ by about 4.2 σ, a discrepancy that is dominated by uncertainties in the hadronic sector. They argue that RχT, built upon large‑N_c QCD and chiral symmetry, provides a systematic way to incorporate resonances (ρ, ω, φ, etc.) together with the pseudo‑Goldstone bosons while respecting short‑distance QCD constraints. The RχT Lagrangian is presented in terms of chiral tensors (uμ, χ±, fμν±, …) and resonance fields (V, A, S, P). By imposing QCD short‑distance conditions on two‑point and three‑point Green functions, the otherwise unknown couplings are expressed in terms of the pion decay constant F and the resonance masses, drastically reducing the number of free parameters.

For the HVP contribution, two independent data‑driven approaches are examined. The first uses τ → ππντ spectral functions as a proxy for the e⁺e⁻ → π⁺π⁻ cross section. The authors carefully treat isospin‑breaking (IB) effects, including final‑state radiation, long‑distance electromagnetic corrections (the G_EM function), phase‑space factors, and the universal electroweak factor S_EW. These corrections are derived from the original RχT Lagrangian and its extensions, leading to an IB shift of roughly –1.0 × 10⁻¹⁰ (original estimate) and –1.7 × 10⁻¹⁰ when higher‑order operators are included. Incorporating the most recent τ data from ALEPH, Belle, CLEO and OPAL, the authors obtain aτ‑based HVP contribution a_HVP,LO(τ) = 517.2 ± 2.8_exp ± 5.1_th × 10⁻¹⁰. This value is about 1.7 σ higher than the earlier White‑Paper‑1 result but compatible within 1 σ with the White‑Paper‑2 lattice average (≈ 693 × 10⁻¹⁰).

The second HVP analysis follows the traditional e⁺e⁻ → hadrons dispersion integral. The authors fit RχT‑based vector form factors to a comprehensive set of cross‑section data (including the recent CMD‑3, BaBar and KLOE measurements) for the dominant channels ππ, KK, K⁰_LK⁰_S, 3π, ππ η, K K̄ π, π⁰γ and ηγ, up to an energy of 2.3 GeV. The resulting channel contributions are quoted (e.g. a_ππ = 505.89 ± 2.34, a_KK = 23.03 ± 0.43, etc.), and their sum yields a_HVP,LO(e⁺e⁻) = 694.1 ± 3.1 × 10⁻¹⁰, in excellent agreement with the White‑Paper‑2 value (693.1 ± 4.0 × 10⁻¹⁰). The authors emphasize that the RχT description, constrained by short‑distance QCD, provides a theoretically robust interpolation between the low‑energy chiral regime and the resonance region.

Turning to the HLbL sector, the paper revisits the well‑studied pseudoscalar‑pole (π⁰, η, η′) and tensor‑pole contributions, confirming previous results within the RχT framework. The novel element is the first RχT evaluation of the pseudoscalar‑box diagram, which appears at O(N_c⁰) in the 1/N_c expansion. By constructing the relevant four‑point Green function with resonance exchanges and imposing the same short‑distance constraints used for the two‑point functions, the authors obtain a box contribution that, when combined with the pole terms, reproduces the White‑Paper‑2 HLbL estimate of (9.6 ± 2.0) × 10⁻¹⁰. Importantly, the RχT treatment reduces the model dependence of the HLbL error budget, bringing the overall HLbL uncertainty down from roughly 50 % to about 30 % of its central value.

The manuscript also discusses subleading pieces such as scalar‑pole, axial‑vector‑pole and higher‑multiplicity resonance exchanges. All these contributions are shown to be compatible with existing phenomenology and to respect the same QCD short‑distance constraints, reinforcing the internal consistency of the RχT approach.

In the concluding section, the authors summarize that RχT provides a unified, QCD‑consistent description of both HVP and HLbL contributions, yielding results that are fully compatible with the latest White‑Paper‑2 numbers and with comparable precision. They highlight that forthcoming high‑statistics τ measurements at Belle‑II (including radiative τ → ππντγ decays) and continued improvements in lattice QCD calculations will be crucial to further shrink the theoretical uncertainties. The paper thus positions RχT as a powerful tool for future precision tests of the muon g‑2 and for probing possible new physics scenarios that could resolve the current experimental‑theory tension.