Consistent scalar-meson and axial-vector sectors in holographic QCD for the hadronic light-by-light contribution to the muon $g-2$

Bottom-up holographic QCD (AdS/QCD) models reproduce the pseudoscalar and axial-vector hadronic light-by-light (HLbL) contribution to the muon anomalous magnetic moment $a_\mu$ remarkably well, but two meson-exchange sectors remain inconsistent with QCD: no existing model produces scalar-meson transition form factors with the short-distance falloff required by perturbative QCD (and the lightest holographic scalar modes are too heavy to be the physical $f_0(500)$, $a_0(980)$, $f_0(980)$, which are plausibly tetraquarks outside the large-$N_c$ $q\bar q$ spectrum), and no single model reproduces the $f_1(1285)$/$f_1(1420)$ masses and mixing together with the pseudoscalar transition form factors. The problem is to construct a bottom-up holographic model in which scalar exchange contributes to HLbL with transition form factors satisfying all pQCD short-distance constraints, the axial-vector spectrum and $f_1$–$f_1'$ mixing are fitted at the accuracy already achieved for pseudoscalars, and the resulting scalar and axial-vector contributions to $a_\mu$ are evaluated with quantified residual parameter dependence.

Background

The anomalous magnetic moment of the muon, $a_\mu=(g-2)_\mu/2$, is measured to a precision of order $10^{-7}$, and its Standard-Model prediction is limited by hadronic physics. After the hadronic vacuum polarization, the dominant residual theory uncertainty comes from hadronic light-by-light scattering (HLbL) — the class of diagrams in which the muon interacts with a photon through a loop of strongly interacting matter. The data-driven and lattice determinations of this contribution (of order $90$–$115\times10^{-11}$, to be compared with the current experimental error of about $15\times10^{-11}$) still disagree beyond their quoted errors, so the 2025 Theory Initiative update inflates the combined HLbL uncertainty by a scale factor. In the data-driven approach, the HLbL amplitude is decomposed into single-meson-exchange contributions, two-meson cuts, and matching to short-distance constraints (SDCs) — asymptotic statements derived from the operator product expansion (OPE) and light-cone (Brodsky–Lepage-type) analysis that any acceptable model of the amplitude must respect. Chief among these are the Melnikov–Vainshtein constraint on the longitudinal part of the amplitude in the limit where one photon virtuality greatly exceeds the others, and the high-virtuality falloff of the meson transition form factors (TFFs) $F_{M\gamma^*\gamma^*}(Q_1^2,Q_2^2)$, which describe the coupling of a meson $M$ to two (possibly virtual) photons.

Holographic QCD (AdS/QCD) offers a semi-classical model of the same physics: exploiting the AdS/CFT correspondence, QCD is modeled at large $N_c$ by a five-dimensional (5D) gauge theory on a slice of anti-de Sitter space. In bottom-up hard-wall (HW) models, chiral symmetry breaking is implemented by a bifundamental bulk scalar field $X$ dual to the quark bilinear $\bar q_R q_L$, the axial anomaly by a 5D Chern–Simons action, and vector meson dominance — the mixing of the photon with an infinite tower of vector mesons — is built in. These models automatically produce infinite towers of resonances; crucially, the tower of axial-vector mesons saturates the Melnikov–Vainshtein constraint, which no finite set of resonances can. Implementing the $U(1)_A$ anomaly via the Katz–Schwartz mechanism (an additional bulk scalar $Y$ dual to gluonic operators plus a Witten–Veneziano-type mass term for the $\eta'$) and allowing a gluon condensate, Leutgeb, Mager, and Rebhan (2022–2023) obtained a model that fits the $\eta$ and $\eta'$ masses and two-photon couplings at the percent level and reproduces the White Paper 2020 pseudoscalar HLbL contributions, with a best-estimate combined pseudoscalar-plus-axial contribution of about $128\times10^{-11}$, at the upper end of the data-driven estimates.

Several gaps were identified in the same work and remain only partially addressed. (i) Scalar mesons: in minimal HW models the scalar fluctuations of $X$ do not couple to photons at all. The extension of Cappiello, Catà, and D'Ambrosio (2022), which adds interaction terms quadratic in the bulk scalar and the gauge-field strengths and fits the lightest scalars to the sub-GeV states $\sigma(500)$, $a_0(980)$, and $f_0(980)$, generates nonvanishing scalar HLbL contributions ($a_\mu^{S}=-9(2)\times10^{-11}$, about 90% from the $\sigma(500)$, in line with dispersive estimates) — but its scalar TFFs fall as $\mathcal{F}_1\sim Q^{-6}$ and $\mathcal{F}_2\sim Q^{-8}$ at large photon virtuality $Q$, whereas the pQCD light-cone analysis of Hoferichter and Stoffer requires $\mathcal{F}_1\sim Q^{-2}$ and $\mathcal{F}_2\sim Q^{-4}$. The superconnection hard-wall models of Leutgeb, Mager, and Rebhan (2025), by contrast, place their lightest scalar modes at 1.6–2.1 GeV, far too heavy to be the physical $f_0(500)$, $a_0(980)$, $f_0(980)$, which those authors argue are likely tetraquark states outside the large-$N_c$ $q\bar q$ spectrum. (ii) The axial-vector sector: in the Katz–Schwartz model $f_1(1285)$ and $f_1(1420)$ come out 4–15% (28% for the $f_1'$) too heavy with an incorrect mixing pattern; the 2025 superconnection (tachyon) variants improve $f_1$–$f_1'$ mixing and photon rates but overestimate the $\pi^0$ TFF at moderate virtuality. (iii) The overall normalization depends on the 5D gauge coupling $g_5$ (the OPE fit saturates the TFF short-distance constraints at 100%, fitting the $\rho$-meson decay constant $F_\rho$ gives 90%); the 2025 tensor-meson work removed part of this ambiguity by fixing the tensor normalization through the symmetric longitudinal short-distance constraints. (iv) Doubly-virtual axial-vector TFFs remain essentially unconstrained by data, since the Landau–Yang theorem forbids the decay of an axial-vector meson into two on-shell photons, so experimental discrimination between models is currently limited to singly-virtual information such as the $f_1\to e^+e^-$ route proposed by Zanke, Hoferichter, and Kubis.

Problem Statement

Make the holographic-QCD prediction of the hadronic light-by-light contribution to the muon anomalous magnetic moment consistent across the light-meson exchange sectors, in the following precise sense: construct a bottom-up holographic (AdS/QCD-type) model such that:

1. the scalar-meson sector couples to photons and contributes to HLbL with transition form factors whose large-virtuality behavior agrees with the pQCD short-distance constraints (the Brodsky–Lepage-type falloff $\mathcal{F}_1\sim Q^{-2}$, $\mathcal{F}_2\sim Q^{-4}$ together with the OPE limits);

2. the axial-vector spectrum and the $f_1(1285)$–$f_1(1420)$ mixing angle are reproduced at the percent-level accuracy already achieved for the pseudoscalar fits — that is, within a few percent of their measured values — without shifting the pseudoscalar transition form factors by more than that same few percent relative to the reference fit; and

3. the resulting scalar and axial-vector contributions to $a_\mu^{\mathrm{HLbL}}$, including the sums over excited states, are evaluated, with the residual dependence on unmatched parameters such as the 5D gauge coupling $g_5$ quantified as an explicit interval.

The sparsity of doubly-virtual axial-vector TFF measurements is an independent experimental bottleneck that would discriminate candidate models but is not itself part of the theory problem.

References

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