A multidimensional landscape of the $η$ and $η'$ mesons

TL;DR

This work develops a form-invariant algebraic framework to describe the internal structure of the mixed $\eta$–$\eta'$ system by connecting the Bethe–Salpeter amplitude to light-front content. Using a quark propagator ansatz and a leading-twist Bethe–Salpeter amplitude with a spectral density, the authors derive analytic light-front wavefunctions and generalized parton distributions, enabling the computation of distribution amplitudes, distribution functions, electromagnetic form factors, and impact-parameter GPDs for the $\eta$ and $\eta'$. They implement a single-angle SA-MS mixing, fix constituent masses and mixing angle from inputs, and obtain decay constants and DAs in good agreement with lattice and phenomenology; their GPDs and IPS-GPDs reveal mass-dependent dilation/compression consistent with emergent hadron mass dynamics, particularly for the $s\bar s$ component. The study thus offers a robust, cross-validated picture of ground-state pseudoscalars and sets the stage for exploring $\eta,\eta'\to \gamma^{*}\gamma^{(*)}$ transitions and excited states within a unified algebraic approach.

Abstract

We employ a recently proposed form-invariant algebraic model for the quark propagator and the Bethe-Salpeter amplitude of pseudoscalar mesons to study the internal structure of $η$ and $η'$ mesons. This model facilitates the construction of the Bethe-Salpeter wavefunction, whose projection onto an appropriate flavor-basis leads to the light-front wavefunction for convenient linear combinations of the $s \bar{s}$ and $l\bar{l}\sim(u \bar{u} + d \bar{d})$ states. Using an overlap representation, we compute the valence-quark generalized parton distributions (GPDs). The construction of the model ensures that this multidimensional quantity is determined entirely by the corresponding valence-quark distribution amplitudes. Once the GPDs are constructed, we carry out a straightforward derivation of other desired physical observables such as the distribution functions and the electromagnetic form factors. We also provide explicit comparisons with available results, demonstrating that the present model offers a consistent physical picture for all ground-state pseudoscalar mesons.

Figures (6)

• Figure 1: The mass composition of different pseudoscalar mesons arises from three main sources: emergent hadron mass (EHM), Higgs field (HF) mechanism, and the interference between between these two effects. The mass distribution within pion, kaon and $\eta$ mesons is dominated by the interplay between emergent hadron mass and the Higgs field mechanism, accounting for approximately $95$%, $80$% and $82$%, of the total mass, respectively. In contrast, for the heavy-quarkonia, their mass is primarily dominated by the contribution from the Higgs field, which makes up around $85-90$% of the total mass. Unlike other pseudoscalars, the $\eta'$ meson derives a significant portion of its mass—approximately 70–75%—from the emergent hadrom mass contribution, making its mass composition qualitatively comparable to that of the proton.
• Figure 2: $\eta-\eta'$ related DAs. [upper panel] Those concerning the generic flavor basis which are employed as inputs for our analysis. [lower panel] The $\eta_{l,s}$ DAs are derived from Eq. \ref{['eq:RelDAs']}. The pion DA, $\varphi_\pi^u$, and the asymptotic profile $\varphi^{\rm asy}$, Eq. \ref{['eq:DAasym']}, are provided for the sake of comparison.
• Figure 3: Valence-quark GPDs of the $\eta_l$ and $\eta_s$ system in the SA-MS, Eq. \ref{['eq:saBasis']}. The curves show the $x$-dependence for different $\Delta^2$ values in the $\xi=0$ case.
• Figure 4: Valence-quark DFs of the $\eta_l$ and $\eta_s$ systems that characterize the $\eta-\eta'$ bound-states in the SA-MS, Eq. \ref{['eq:saBasis']}. The DFs are defined at the hadronic scale $\zeta_H$. Pion DF and parton-like profiles, Eq. \ref{['eq:DFpl']}, are shown for comparison.
• Figure 5: Electromagnetic FFs of the $\eta_l$ and $\eta_s$ system that characterize the $\eta-\eta'$ bound-states in the SA-MS, Eq. \ref{['eq:saBasis']}. The analogous $\eta_c$ and $\eta_b$ profiles, Ref. Albino:2022gzs, are also displayed.