<p>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.</p>

China

Zanbin Xing

Lei Chang

Spain

Khépani Raya

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>

<p>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.</p>

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.

Jiayin Kang

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.

Yehan Xu

Contact interaction treatment of <math><mrow><mi>V</mi><mo>→</mo><mi>P</mi><mi>γ</mi></mrow></math> for light-quark mesons

Given the unique role played by the gravitational form factors (GFFs) in unraveling the internal mechanics of hadrons, we examine the GFFs of ground state pseudoscalar mesons <math><mrow><mi>π</mi></mrow></math>, <math><msub><mi>η</mi><mi>c</mi></msub></math>, <math><msub><mi>η</mi><mi>b</mi></msub></math> and the hypothetical strangeonium <math><msub><mi>η</mi><mi>s</mi></msub><mo>(</mo><mi>s</mi><mover><mi>s</mi><mo>¯</mo></mover><mo>)</mo></math>. We adopt the coupled framework of Dyson-Schwinger and Bethe-Salpeter equations within a contact interaction and employ a novel approach to the dressed amputated meson-meson scattering amplitude, which makes connection with the energy-momentum tensor and with the GFFs. The resulting GFFs fulfill the anticipated symmetry constraints. The corresponding charge and mass radii and the <math><mi>D</mi></math> term are also computed. We show that the <math><mi>D</mi></math> term for the pseudoscalar mesons is bounded within the <math><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>)</mo></math> range; these bounds correspond to the massless (chiral limit) and infinitely massive cases, respectively. Considering the current interest in the GFFs, understanding the <math><mrow><mi>D</mi></mrow></math> term of pseudoscalar mesons and their GFFs can provide an important first step for future endeavors in the field.

Gravitational form factors of pseudoscalar mesons in a contact interaction

In this work, the <math><mi>γ</mi><mo>→</mo><mn>3</mn><mi>π</mi></math> form factor is calculated within the Dyson-Schwinger equations framework using a contact interaction model within the so-called modified rainbow ladder truncation. The present calculation takes into account the pseudovector component in the pion Bethe-Salpeter amplitude (BSA) and <math><mi>π</mi><mo>−</mo><mi>π</mi></math> scattering effects, producing a <math><mi>γ</mi><mo>→</mo><mn>3</mn><mi>π</mi></math> anomaly that is <math><mn>1</mn><mo>+</mo><mn>6</mn><msubsup><mi>R</mi><mi>π</mi><mn>2</mn></msubsup></math> larger than the low energy prediction. Here <math><msub><mi>R</mi><mi>π</mi></msub></math> is the relative ratio of the pseudovector and pseudoscalar components in the pion BSA; with our parameters’ input, this correction raises the <math><mi>γ</mi><mo>→</mo><mn>3</mn><mi>π</mi></math> anomaly by around 10%. The main outcome of this work is the unveiling of the origin of such a correction, which could be a possible explanation of the discrepancy between the existing experimental data and the low energy prediction. Moreover, it is highlighted how the magnitude of the anomaly is affected in effective theories that require an irremovable ultraviolet cutoff. We find that for both the anomalous processes <math><mi>π</mi><mo>→</mo><mn>2</mn><mi>γ</mi></math> and <math><mi>γ</mi><mo>→</mo><mn>3</mn><mi>π</mi></math>, the missing contribution to the anomaly can be compensated by the additional structures related with the quark anomalous magnetic moment.

Hao Dang

QCD anomalies in electromagnetic processes: A solution to the <math><mi>γ</mi><mo>→</mo><mn>3</mn><mi>π</mi></math> puzzle

We apply a classical mathematical problem, the moment problem, with its related mathematical achievements, to the study of the parton distribution function (PDF) in hadron physics, and propose a strategy to sieve the moments of the PDF by exploiting its properties such as continuity, unimodality, and symmetry. Through an error-inclusive sifting process, we refine three sets of PDF moments from Lattice QCD. This refinement significantly reduces the errors, particularly for higher order moments, and locates the peak of PDF simultaneously. As our strategy is universally applicable to PDF moments from any method, we strongly advocate its integration into all PDF moment calculations.

Germany

Sieving parton distribution function moments via the moment problem

A new method based on the maximum entropy principle for reconstructing the parton distribution function (PDF) from moments is proposed. Unlike traditional methods, the new method does not require any artificial assumptions. For the case of moments with errors, we introduce Gaussian functions to soften the constraints of moments. Through a series of tests, the effectiveness and reconstruction efficiency of this new method are evaluated comprehensively, demonstrating that this method is reasonable and can achieve high-quality reconstruction with at least the first six moments as input. Finally, we select a set of lattice quantum chromodynamics (QCD) results regarding moments as input and provide reasonable reconstruction results for the pion.