The BESIII Collaboration has recently reported two measurements of the two-photon decay width of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub></math> meson. The 2024 result is significantly larger than most theoretical and empirical expectations, while a subsequent measurement published in early 2026 shows better agreement with the world average and conventional theoretical estimates. In this work, we study the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub><mo stretchy="false">→</mo><mi>γ</mi><mi>γ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi><mo stretchy="false">→</mo><mi>γ</mi><msub><mi>η</mi><mi>c</mi></msub></math> processes within a contact interaction model that incorporates valence-quark anomalous magnetic moment effects, which are absent in standard treatments. Besides achieving agreement with modern lattice QCD estimates for these observables, we find that the 2024 central value for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub><mo stretchy="false">→</mo><mi>γ</mi><mi>γ</mi></math> lies above the range that could be accommodated by the present framework, whereas the 2026 result is naturally consistent with it.

China

Yehan Xu

Zanbin Xing

Spain

Khépani Raya

Lei Chang

Radiative charmonium decays in a contact-interaction model with dynamical quark anomalous magnetic moment

We present a novel algebraic framework for computing hadron properties directly within the contact interaction model. Utilizing Fierz transformations, the method recasts the Bethe-Salpeter dynamics into equations for a minimal set of projected amplitudes for bound-state static properties and form factors, bypassing the conventional need for the meson wave function. This approach is fully demonstrated for the vector meson, enabling the direct extraction of its decay constants, form factors, and the parton distribution functions. The formalism provides a more efficient and unified pathway to hadron observables, with potential for extension to baryons and more sophisticated interactions.

Jiayin Kang

Direct algebraic pathway to hadronic observables in the contact model

A dual identity of the <italic>&#963;</italic> meson is proposed and its implications for the gravitational structure of hadrons is examined. Staring from a <math altimg='si3.svg'><mrow><mi>q</mi><mover accent='true'><mi>q</mi><mo>&#175;</mo></mover></mrow></math> bound state view for the <italic>&#963;</italic> meson, which emerges naturally as the chiral partner of the pion in the Dyson-Schwinger Equation formalism, with both states&#8217; properties tightly connected to the mechanisms of mass generation in QCD. We find that, above a critical coupling that triggers dynamical chiral symmetry breaking, the D-terms of the constituent quark, the pion, and <italic>&#963;</italic> saturate at fixed values <math altimg='si4.svg'><mrow><msub><mi>D</mi><mrow><mi>q</mi><mo>,</mo><mi>&#960;</mi><mo>,</mo><mi>&#963;</mi></mrow></msub><mo linebreak='goodbreak'>=</mo><mo>&#8722;</mo><mn>1</mn><mo linebreak='goodbreak'>/</mo><mn>3</mn><mo>,</mo><mo>&#8722;</mo><mn>1</mn><mo>,</mo><mo>&#8722;</mo><mn>7</mn><mo linebreak='goodbreak'>/</mo><mn>3</mn></mrow></math>. By examining the coupling strength evolution of the D-terms, this pattern follows naturally once a dual nature for the <italic>&#963;</italic> meson is recognized: it behaves both as a quark&#8211;antiquark composite and as a dilaton arising from spontaneous scale symmetry breaking. This unified picture yields the prediction <math altimg='si5.svg'><mrow><msub><mi>D</mi><mi>N</mi></msub><mo>&#8764;</mo><mo>&#8722;</mo><mn>3</mn></mrow></math> for the nucleon D-term, consistent with lattice QCD, dispersive analyses, and diverse continuum studies.

Yu-xin Liu

The dichotomous nature of the <italic>&#963;</italic> meson and the nucleon D-term

The Schwinger effect, a non-perturbative mechanism for particle production in strong fields, plays a crucial role in understanding quantum vacuum decay and high-energy phenomena, including heavy-ion collisions (HIC). Although holographic quantum chromodynamics (QCD) models have been widely used to study this effect, most treatments assume isotropy or consider only a single type of anisotropy, neglecting the interplay between spatial and magnetic anisotropies that arise in realistic HIC scenarios. A unified framework accounting for both anisotropies is needed to accurately model particle production. We investigate the Schwinger effect in a twice anisotropic holographic QCD model incorporating both spatial and magnetic anisotropies. Using the anti-de Sitter/conformal field theory correspondence, we compute the total potential of a particle-antiparticle pair to quantify how these anisotropies influence pair production. Our results show that the magnetic field (parameterized by $c_B$ and $q_3$ ) enhances the Schwinger effect by lowering and narrowing the potential barrier, while increasing spatial anisotropy (controlled by $\nu$ ) suppresses the process by raising and widening the barrier. These findings demonstrate that magnetic and spatial anisotropies exert competing effects on particle production, emphasizing the necessity of treating both concurrently in holographic models. This work advances the theoretical description of the Schwinger effect in anisotropic environments, with implications for understanding non-equilibrium dynamics in HIC and other strongly coupled systems.

Wen-Bin Chang

Defu Hou

Schwinger effect in a twice anisotropic holographic model * Defu Hou's research is supported in part by the National Key Research and Development Program of China (2022YFA1604900). Additionally, Defu Hou receives partial support from the National Natural Science Foundation of China (NSFC) (12435009, 12275104). Wen-Bin Chang is supported by the Ph.D. Research Startup Project at Hubei Minzu University (RZ2500000857)

The electromagnetic structure of axial-vector mesons is investigated via elastic and two-photon transition form factors (TFFs). To this end, we employ a framework based on the Dyson-Schwinger and Bethe-Salpeter equations within a contact interaction (CI) model. This largely algebraic approach exposes the role of symmetries and their breaking in a transparent manner, and has proven successful in describing anomaly-sensitive processes, including pseudoscalar to two-photon TFFs, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>γ</mi><mo stretchy="false">→</mo><mn>3</mn><mi>π</mi></math>, and vector-to-pseudoscalar radiative decays. Restricting our analysis to the lowest-lying axial-vector meson multiplet, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="normal">A</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msubsup><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo stretchy="false">}</mo></mrow></math>, we also explore the corresponding light-by-light contribution to the muon anomalous magnetic moment. This contribution is underestimated in the CI model mainly due to the normalization of the TFFs. After applying a phenomenologically motivated rescaling to the TFFs, a process required given the limitations of the CI model, we obtain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msubsup><mo>=</mo><mn>8.1</mn><msubsup><mrow><mn>6</mn></mrow><mrow><mo>−</mo><mn>1.02</mn></mrow><mrow><mo>+</mo><mn>1.22</mn></mrow></msubsup><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>11</mn></mrow></msup></mrow></math>, consistent with contemporary assessments.

Electromagnetic structure of axial-vector mesons and implications for the muon <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>g</mi><mo>−</mo><mn>2</mn></math>

We propose a data-driven approach to extract the kaon leading-twist distribution amplitude (DA) from empirical information on the ratio of the neutral-to-charged kaon electromagnetic form factors, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="script">R</mi><mi>K</mi></msub></math>. Our study employs a two-parameter representation of the DA at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ζ</mi><mo>=</mo><mn>2</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, designed to capture the expected broadening and asymmetry of the distribution, as well as the soft endpoint behavior predicted by quantum chromodynamics (QCD). Our leading-order analysis of the latest experimental measurements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="script">R</mi><mi>K</mi></msub></math> reveals that the extracted DA exhibits a somewhat significant skewness, with the first symmetric moment being approximately <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">⟨</mo><mn>1</mn><mo>−</mo><mn>2</mn><mi>x</mi><msub><mo stretchy="false">⟩</mo><mi>K</mi></msub><mo>=</mo><mn>0.082</mn><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></math>. On the other hand, the broadness and general shape of the produced distributions show a reasonable consistency with contemporary lattice and continuum QCD analyses. These findings highlight the importance of accurately determining the profile of the DA, especially the skewness and its relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><msub><mi>U</mi><mi>F</mi></msub><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></math> flavor symmetry breaking, as well as the inclusion of higher-order effects in the hard-scattering kernels for analyzing data at experimentally accessible scales.

Yu-Bin Liu

Pakistan

M. Atif Sultan

Empirical determination of the kaon distribution amplitude

By capturing the characteristics of the Bethe-Salpeter amplitude for the pion excitation state, we construct an algebraic model to provide the overall features of the pion’s first excitation state parton distribution amplitude and distribution function. We find that, at the hadronic scale, the distribution amplitude of the excited state exhibits nodes, while the distribution function is unimodal, with a peak at $x=1/2 $ and distinct concave and convex fluctuations in the valence region. These findings provide new insights into the partonic structure of excited mesons and contribute significantly to our understanding of hadronic excitations.

Xiaobin Wang

Unveiling the inner structure of the Pion’s first excited state * This work is financially supported by the National Natural Science Foundation of China (12135007)

A light-front holographic model is used to illustrate an algebraic scheme for constructing a representation of a hadron's zero-skewness generalised parton distribution (GPD) from its valence-quark distribution function (DF) and electromagnetic form factor, <math altimg='si1.svg'><msub><mrow><mi>F</mi></mrow><mrow><mi>H</mi></mrow></msub></math>, without reference to deeply virtual Compton scattering data. The hadron's mass distribution gravitational form factor, <math altimg='si2.svg'><msub><mrow><mi>A</mi></mrow><mrow><mi>H</mi></mrow></msub></math>, calculated from this GPD is harder than <math altimg='si1.svg'><msub><mrow><mi>F</mi></mrow><mrow><mi>H</mi></mrow></msub></math>; and, for each hadron, the associated mass-density profile is more compact than the analogous charge profile, with each pion near-core density being larger than that of its proton partner. These features are independent of the scheme employed.

Minghui Ding

Craig D. Roberts

Sketching pion and proton mass distributions

Based upon a combined formalism of Schwinger-Dyson and Bethe-Salpeter equations in quantum chromodynamics (QCD), we propose a QCD-kindred algebraic model for the dressed quark propagator, for the Bethe-Salpeter amplitude of the pion and the electromagnetic quark-photon interaction vertex. We then compute the <math><msup><mi>γ</mi><mo>*</mo></msup><msup><mi>π</mi><mn>0</mn></msup><mi>γ</mi></math> transition form factor <math><msup><mi>G</mi><mrow><msup><mi>γ</mi><mo>*</mo></msup><msup><mi>π</mi><mn>0</mn></msup><mi>γ</mi></mrow></msup><mo>(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> for a wide range of photon momentum transfer squared <math><msup><mi>Q</mi><mn>2</mn></msup></math>. The quark propagator is expanded out in its perturbative functional form but with dynamically generated dressed quark mass. It has complex conjugate pole singularities in the complex-momentum plane, which is motivated by the solution of the quark gap equation with rainbow-ladder truncation of the infinite set of Schwinger-Dyson equations. This complex pole singularity structure of the quark propagator can be associated with a signal of confinement, which prevents quarks from becoming stable asymptotic states. The Bethe-Salpeter amplitude is expressed without a spectral density function, which encapsulates its low- and large-momentum behavior. The QCD evolution of the distribution amplitude is also incorporated into our model through the direct implementation of Efremov-Radyushkin-Brodsky-Lepage evolution equations. We include the effects of the quark anomalous magnetic moment in the description of the quark-photon vertex, whose infrared enhancement is known to dictate hadronic properties. Once the QCD-kindred model is constructed, we calculate the form factor <math><msup><mi>G</mi><mrow><msup><mi>γ</mi><mo>*</mo></msup><msup><mi>π</mi><mn>0</mn></msup><mi>γ</mi></mrow></msup><mo>(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> and find it consistent with direct QCD-based studies, as well as most available experimental data. It slightly exceeds the conformal limit for large <math><msup><mi>Q</mi><mn>2</mn></msup></math>, which might be attributed to the scaling violations in QCD. The associated interaction radius and neutral pion decay width turn out to be compatible with experimental data.

Mexico

Adnan Bashir

Neutral pion to two-photons transition form factor revisited

The <math><mi>V</mi><mo>→</mo><mi>P</mi><mi>γ</mi></math> and <math><mi>η</mi><mo>(</mo><msup><mi>η</mi><mo>′</mo></msup><mo>)</mo><mo>→</mo><mi>γ</mi><mi>γ</mi></math> decays are evaluated within a Dyson-Schwinger and Bethe-Salpeter equations framework (here <math><mi>V</mi><mo>=</mo><mo>{</mo><msup><mi>ρ</mi><mo>±</mo></msup><mo>,</mo><msup><mi>K</mi><mrow><mo>⋆</mo><mo>±</mo></mrow></msup><mo>,</mo><mi>ϕ</mi><mo>}</mo></math> and <math><mi>P</mi><mo>=</mo><mo>{</mo><msup><mi>π</mi><mo>±</mo></msup><mo>,</mo><msup><mi>K</mi><mo>±</mo></msup><mo>,</mo><mi>η</mi><mo>,</mo><msup><mi>η</mi><mo>′</mo></msup><mo>}</mo></math>). The so-called impulse approximation (IA) is employed in the computation of the decay constants involved and decay widths, and so in the estimation of the associated charge and interaction radii. For their part, the required propagators and vertices stem from a contact interaction model, embedded within a beyond rainbow-ladder (RL) truncation that accounts for the typical ladder exchanges, quark anomalous magnetic moment, as well as the non-Abelian anomaly. While the examined transitions produce decay widths plainly compatible with the available experimental data, those processes involving the <math><mi>η</mi><mo>−</mo><msup><mi>η</mi><mo>′</mo></msup></math> mesons highlight the incompleteness of the IA when considering beyond RL effects in the interaction kernels.