Axion-meson mixing and the axion-photon coupling at next-to-next-to-leading order in U(3) chiral perturbation theory

The low-energy couplings of the QCD axion to photons and to the light pseudoscalar mesons are governed by the mixing of the axion with the $\pi^0$, $\eta$ and $\eta'$ within $U(3)$ chiral perturbation theory, organised in the $\delta$-counting that jointly expands in momenta, light-quark masses and $1/N_C$. The complete next-to-leading-order (NLO) calculation (Gao, Guo, Oller and Zhou, JHEP 2023) fixed the low-energy constants from lattice $\eta$-$\eta'$ data and predicted the axion mass-decay-constant relation, the axion-meson mixing strengths, and the model-independent axion-photon coupling $g_{a\gamma\gamma}$, but omitted the chiral-loop contributions, which in this counting first enter at next-to-next-to-leading order (NNLO); the authors explicitly left the NNLO extension as future work. Later work completed the linear strong-isospin-breaking corrections at NLO, moving the model-independent coupling from 1.63 to $1.89\pm0.02$ in units of $\alpha_{\rm em}/(2\pi f_a)$, and a first-principles lattice determination found $g^{\rm QCD}_{a\gamma\gamma} f_a/e^2=-0.0224(10)$, about 10% below the NLO chiral predictions. The NNLO $U(3)$ calculation with chiral loops - a consistently renormalised, lattice-fitted prediction of the axion mass, axion-meson mixing and $g_{a\gamma\gamma}$ - remains undone. The problem is to carry out this NNLO extension and to quantify its shifts on the axion observables against the new lattice benchmark.

Background

The QCD axion is a hypothetical pseudo-Nambu-Goldstone boson - a light particle arising from the spontaneous breaking of an approximate symmetry - proposed within the Peccei-Quinn solution of the strong CP problem (the observed smallness of the neutron electric dipole moment); its decay constant $f_a$ sets its mass, $m_a \approx 5.7\,\mu{\rm eV}\,(10^{12}\,{\rm GeV}/f_a)$, which is generated entirely by QCD through the anomalous axion-gluon coupling. At energies below about 1 GeV, axion interactions with hadrons are described by chiral perturbation theory (ChPT), the effective field theory of the pseudoscalar meson octet ($\pi$, $K$, $\eta$) associated with spontaneous chiral symmetry breaking in QCD. The conventional $SU(3)$ treatment excludes the $\eta'$ because the axial $U(1)_A$ anomaly makes it heavy; $U(3)$ ChPT instead keeps the $\eta'$ as an explicit field, which is consistent at leading order in a joint expansion in small momenta, light-quark masses and $1/N_C$ (the number of QCD colours $N_C$), the so-called $\delta$-counting. The $\eta'$ mass is then reproduced at leading order via the Witten-Veneziano mechanism - the large-$N_C$ relation explaining why the axial anomaly makes the $\eta'$ much heavier than the octet mesons - in terms of a flavor-singlet mass parameter $M_0$ (about 820 MeV in lattice fits). In this counting tree-level terms suppressed by $\delta$ contribute at NLO, while chiral one-loop graphs - present already at one loop in $SU(2)$/SU(3) ChPT - are postponed to NNLO, so an NLO $U(3)$ calculation is loop-free.

The axion couples to the light mesons through its mixing with the $\pi^0$, $\eta$ and $\eta'$. The work of Gao, Guo, Oller and Zhou (JHEP 04 (2023) 022) computed this mixing, and the two-photon couplings of all four states, at NLO in the $U(3)$ $\delta$-counting, fixing the unknown low-energy constants (LECs, the couplings parametrising short-distance physics in the effective theory, such as $L_5$, $L_8$, $\Lambda_1$, $\Lambda_2$) by global fits to lattice QCD simulations of the $\eta$-$\eta'$ system, kaon masses and meson decay constants. The anomalous $\gamma\gamma$ couplings arise from the Wess-Zumino-Witten (WZW) term, the chirally anomalous part of the chiral Lagrangian whose normalisation is fixed by the axial anomaly. The axion-photon coupling, the key observable for axion searches, is conventionally written $g_{a\gamma\gamma}=\frac{\alpha_{\rm em}}{2\pi f_a}\left[\frac{E}{N}-1.92(4)\right]$, where $E/N$ is model-dependent and the second number is the model-independent QCD contribution; competing determinations of this coefficient give $1.92\pm0.04$ from NLO $SU(2)$ ChPT (Grilli di Cortona et al.), $2.05\pm0.03$ from NLO $SU(3)$ ChPT (Lu et al.), and $1.63\pm0.01$ from the NLO $U(3)$ calculation in the isospin limit (Gao et al.). The same paper kept only the leading strong isospin breaking $\epsilon=B(m_u-m_d)$, used the Dashen theorem (the leading-order statement that electromagnetic kaon and pion mass splittings are equal) for electromagnetic effects, obtained a relatively poor global fit quality ($\chi^2/{\rm dof}=219.9/(111-5)$), fixed $M_0=820$ MeV, and deferred the quantitative axion-like-particle case of a non-zero bare axion mass $m_{a,0}$.

Three later developments shape the open question. First, Gao, Hao, Duan, Guo, Oller and Zhou (Eur. Phys. J. C (2025)) completed the linear strong isospin-breaking terms at NLO and found that they shift the model-independent coefficient to $1.89\pm0.02$, curing the earlier $U(3)$ tension with the $SU(2)$ and $SU(3)$ values. Second, Brandt, Endrodi, Hernandez Hernandez, Marko and Pannullo (2026) published the first fully non-perturbative lattice QCD determination, $g^{\rm QCD}_{a\gamma\gamma} f_a/e^2=-0.0224(10)$ (equivalent to a coefficient of about 1.77 with roughly 4-5% uncertainty), which lies about 10% below the NLO chiral predictions and highlights that the chiral estimates span a range differing by several standard deviations. Third, Masjuan (2026) showed that in the combined chiral and large-$N_c$ expansion the LECs entering such fits are strongly correlated combinations rather than independent parameters, a caution directly relevant to any NNLO fit, while new lattice $\eta$-$\eta'$ results at fully physical quark masses (Ottnad et al., 2025) provide cleaner input data. What no work has yet done is the step the original authors explicitly deferred: the NNLO $U(3)$ calculation in which the chiral one-loop graphs, the NNLO tree-level LECs that renormalise them, and the WZW sector are all included consistently for both the mixing and the two-photon observables.

Problem Statement

Extend the $U(3)$ chiral perturbation theory treatment of the $\pi^0$-$\eta$-$\eta'$-axion system to next-to-next-to-leading order (NNLO) in the $\delta$-counting scheme (the joint expansion in momenta, light-quark masses and $1/N_C$), as explicitly deferred by Gao, Guo, Oller and Zhou (JHEP 04 (2023) 022). Concretely, the task is to:

1. Compute the chiral one-loop contributions - which first enter at NNLO in this counting - to the $\pi^0$-$\eta$-$\eta'$-axion mixing matrix elements and to the masses and decay constants of the light pseudoscalars, together with the NNLO tree-level $U(3)$ Lagrangian terms (including the NNLO low-energy constants that renormalise the loop divergences).

2. Extend the two-photon couplings of the $\pi^0$, $\eta$, $\eta'$ and the axion to the same NNLO order, so that every piece of the axion-photon coupling $g_{a\gamma\gamma}$ - mixing-induced contributions, the Wess-Zumino-Witten anomalous sector, and the loop corrections - is included consistently at one common order, in contrast to the existing NLO results where the loop pieces are absent.

3. Determine the NNLO low-energy constants from fits to lattice $\eta$-$\eta'$ and light-meson data, incorporating the now-completed linear strong isospin-breaking treatment and addressing the known fit systematics (the $M_0$ treatment, correlated low-energy-constant directions, and the residual tensions among lattice ensembles).

4. The NNLO calculation is to yield updated, systematically improvable predictions, with quantified uncertainties, for the QCD-axion mass-decay-constant relation, the axion-meson mixing strengths and the model-independent axion-photon coupling, together with the size of the NNLO loop shifts relative to the NLO results and the comparison against the lattice benchmark $g^{\rm QCD}_{a\gamma\gamma} f_a/e^2=-0.0224(10)$ and the existing SU(2)/SU(3) NLO determinations ($1.92\pm0.04$ and $2.05\pm0.03$ in $\alpha_{\rm em}/2\pi f_a$ units).

The completion boundary is a full NNLO $U(3)$ result in which the loop contributions are included consistently for both the mixing and the two-photon observables, with the NNLO low-energy constants fixed from data and the resulting axion observables predicted with uncertainties. The source work additionally defers the quantitative axion-like-particle case of non-zero bare axion mass $m_{a,0}$ within the same framework; that extension is a related but separate open direction, not part of the NNLO completion boundary stated here.

References

1. R. Gao, Z.-H. Guo, J. A. Oller, H.-Q. Zhou, 'Axion-meson mixing in light of recent lattice eta-eta' simulations and their two-photon couplings within U(3) chiral theory', JHEP 04 (2023) 022, DOI: 10.1007/JHEP04(2023)022, arXiv:2211.02867, https://arxiv.org/abs/2211.02867
2. R. Gao, J. Hao, C.-G. Duan, Z.-H. Guo, J. A. Oller, H.-Q. Zhou, 'Isospin-breaking contribution to the model-independent axion-photon-photon coupling in U(3) chiral theory', Eur. Phys. J. C 85 (2025) 97, DOI: 10.1140/epjc/s10052-025-13807-9, arXiv:2411.06737, https://arxiv.org/abs/2411.06737
3. B. B. Brandt, G. Endrodi, J. J. Hernandez Hernandez, G. Marko, L. Pannullo, 'The axion-photon coupling from lattice Quantum Chromodynamics', arXiv:2603.29153, https://arxiv.org/abs/2603.29153
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