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arxiv: 2605.03209 · v1 · submitted 2026-05-04 · ✦ hep-ph

Recognition: unknown

Elastic Form Factors of Axial-Vector Mesons: A Contact Interaction Exploration

R. J. Hern\'andez-Pinto , L. X. Guti\'errez-Guerrero , M. A. Bedolla , J. P. Uribe-Ram\'irez , A. Bashir
Authors on Pith no claims yet

Pith reviewed 2026-05-08 17:39 UTC · model grok-4.3

classification ✦ hep-ph
keywords axial-vector mesonselastic form factorscontact interactionSchwinger-Dyson equationsBethe-Salpeter equationscharge radiimagnetic momentsquadrupole moments
0
0 comments X

The pith

Axial-vector mesons have electric form factors that cross zero at lower momentum transfers than vector mesons.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper computes the elastic form factors of axial-vector mesons by applying a symmetry-preserving contact interaction inside the coupled Schwinger-Dyson and Bethe-Salpeter equations. It obtains charge radii, magnetic moments, and quadrupole moments for states built from light quarks, heavy quarks, or mixed light-heavy quarks. The electric form factor is shown to cross zero, as it does for vector mesons, yet the crossing occurs at a smaller momentum transfer. Adding an anomalous magnetic moment term to the quark-photon vertex produces sizable shifts in the magnetic and quadrupole moments. The results extend earlier calculations for scalar, pseudoscalar, and vector mesons and are compared with other models where data exist.

Core claim

Employing a contact interaction within the coupled Schwinger-Dyson and Bethe-Salpeter equations formalism, the elastic form factors of axial-vector mesons are calculated, showing that their electric form factors cross zero at lower values than those of vector mesons. The charge radii follow a hierarchy decreasing with increasing dressed quark mass, and the anomalous magnetic moment term in the quark-photon vertex leads to significant changes in the magnetic and quadrupole moments.

What carries the argument

Symmetry-preserving contact interaction in the coupled Schwinger-Dyson and Bethe-Salpeter equations, with the anomalous magnetic moment included in the quark-photon vertex.

If this is right

  • The electric form factor of axial-vector mesons crosses zero at a lower momentum transfer than the corresponding form factor of vector mesons.
  • Charge radii of axial-vector mesons decrease as the mass of the dressed quarks increases, reproducing the hierarchy already seen for scalar, pseudoscalar, and vector mesons.
  • Inclusion of the anomalous magnetic moment term produces noticeable percentage changes in both the magnetic moment and the quadrupole moment.
  • Results for heavy-light axial-vector mesons continue the pattern of decreasing radii with increasing quark mass observed in other meson sectors.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The earlier zero-crossing may indicate that the charge distribution inside axial-vector mesons is more compact than inside vector mesons of similar mass.
  • The same framework could be used to predict form factors for additional heavy-light axial-vector states whose electromagnetic properties are not yet measured.
  • A clear mismatch with future lattice results on the location of the zero would point to the limits of the momentum-independent interaction approximation.

Load-bearing premise

The contact interaction supplies an adequate symmetry-preserving description of the quark dynamics inside axial-vector mesons for the purpose of computing electromagnetic form factors.

What would settle it

A lattice QCD computation or experimental measurement that places the zero-crossing of the electric form factor for any axial-vector meson at a higher momentum transfer than the value obtained here would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.03209 by A. Bashir, J. P. Uribe-Ram\'irez, L. X. Guti\'errez-Guerrero, M. A. Bedolla, R. J. Hern\'andez-Pinto.

Figure 1
Figure 1. Figure 1: FIG. 1: The chiral partners. The view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Diagrammatic representation of the CI, employing view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Running dressed quark masses. Dot-dashed lines rep view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Diagrammatic representation of the BSE. Blue (solid) view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Dressing function of the transverse quark-photon ver view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The triangle diagram for the impulse approximation view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Electric, magnetic and quadrupole FFs of AV mesons are displayed in the top, middle and bottom rows, respectively. view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Electromagnetic (blue), Magnetic (green), and quadrupole (red) FFs for view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Charge radii of ground state AV mesons in the CI view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Charge radii for PS, V, S, and AV mesons composed view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Charge radii for PS, V, S, and AV mesons composed view at source ↗
read the original abstract

We employ a symmetry-preserving treatment of the contact interaction within the coupled for- malism of Schwinger-Dyson and Bethe-Salpeter equations to calculate the elastic form factors of axial-vector mesons. In this study, we present the computation of the charge radii, magnetic mo- ments, and quadrupole moments of axial-vector mesons, including those composed of light quarks, heavy quarks or a light and a heavy quark. Our findings indicate that the electric form factor for axial-vector mesons, like that of vector mesons, crosses zero. Furthermore, this crossing occurs at a lower value for axial-vector mesons than for vector mesons. The results for vector-axial mesons follow a similar hierarchy in charge radii as observed for S, PS, and V mesons, with radii decreasing as the mass of the dressed quarks increases. We also include a term associated with the anoma- lous magnetic moment in the quark-photon vertex. This term has a noticeable impact on both the axial-vector magnetic moment and quadrupole moment, leading to significant percentage changes in their values. We compare our results with those obtained from other models whenever available.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper employs a symmetry-preserving contact interaction within the Schwinger-Dyson and Bethe-Salpeter equations framework to compute the elastic form factors of axial-vector mesons. It reports charge radii, magnetic moments, and quadrupole moments for light, heavy, and mixed quark compositions. The central findings are that the electric form factor crosses zero and does so at lower Q^2 than the corresponding vector-meson result from the same model; an anomalous magnetic moment term in the quark-photon vertex is included and shown to affect the magnetic and quadrupole moments substantially.

Significance. If the results hold, the work extends the contact-interaction approach to axial-vector mesons and supplies a consistent set of predictions for their electromagnetic properties, including a mass-dependent hierarchy of charge radii that matches trends seen for pseudoscalar, scalar, and vector mesons. The symmetry-preserving treatment is a clear strength of the framework. However, the momentum-independent kernel limits quantitative reliability of the Q^2 dependence, so the findings are best viewed as qualitative guidance rather than precise predictions.

major comments (2)
  1. [Abstract and results section] Abstract and results section: the central claim that the electric form factor G_E(Q^2) crosses zero at lower Q^2 for axial-vector mesons than for vector mesons rests on the contact interaction. Because the kernel is momentum-independent, the fall-off of G_E and the precise location of any zero are model artifacts; the ordering between channels is not guaranteed to survive with a running gluon propagator. This is load-bearing for the main result and requires explicit discussion or a sensitivity test.
  2. [Formalism section] Formalism section: the assertion that the anomalous-magnetic-moment term in the quark-photon vertex affects only the magnetic and quadrupole moments (leaving the electric form factor unchanged) is stated but not demonstrated by an explicit decomposition of the impulse-approximation current. An explicit check that the term does not enter the expression for G_E(Q^2) is needed.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'results for vector-axial mesons' is ambiguous and should be clarified (likely intended to mean results for both vector and axial-vector channels).
  2. [Formalism] Notation: the definition of the dressed quark-photon vertex and the precise impulse-approximation current should be written out with all tensor structures shown to aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below, indicating where revisions will be made to improve clarity and address the concerns raised.

read point-by-point responses

  1. Referee: [Abstract and results section] Abstract and results section: the central claim that the electric form factor G_E(Q^2) crosses zero at lower Q^2 for axial-vector mesons than for vector mesons rests on the contact interaction. Because the kernel is momentum-independent, the fall-off of G_E and the precise location of any zero are model artifacts; the ordering between channels is not guaranteed to survive with a running gluon propagator. This is load-bearing for the main result and requires explicit discussion or a sensitivity test.

    Authors: We agree that the momentum-independent contact interaction introduces model dependence in the quantitative Q^2 dependence and zero-crossing location. Within this symmetry-preserving framework, however, the calculation is fully consistent, and the lower zero-crossing Q^2 for axial-vector mesons relative to vector mesons is a robust outcome that parallels the mass-dependent charge-radius hierarchy already established for pseudoscalar, scalar, and vector channels in the same model. In the revised manuscript we will add an explicit paragraph in the results section acknowledging that the precise location of the zero is a model artifact and that the ordering may change with a running gluon propagator; we will also stress that the results are intended as qualitative guidance, consistent with the referee's assessment. A full sensitivity study with a momentum-dependent kernel lies outside the scope of the present contact-interaction exploration. revision: partial

  2. Referee: [Formalism section] Formalism section: the assertion that the anomalous-magnetic-moment term in the quark-photon vertex affects only the magnetic and quadrupole moments (leaving the electric form factor unchanged) is stated but not demonstrated by an explicit decomposition of the impulse-approximation current. An explicit check that the term does not enter the expression for G_E(Q^2) is needed.

    Authors: We thank the referee for this observation. The anomalous-magnetic-moment term is constructed to be transverse and carries a Dirac structure proportional to i sigma_{mu nu} q_nu, which projects only onto the magnetic and quadrupole form factors under the standard decomposition of the current. To make this explicit, the revised manuscript will include a short appendix or subsection that decomposes the full impulse-approximation current into its contributions to G_E, G_M, and G_Q, demonstrating that the anomalous term vanishes identically in the electric projection. revision: yes

Circularity Check

0 steps flagged

No significant circularity: form factors computed as genuine model predictions

full rationale

The paper solves the gap and Bethe-Salpeter equations with a fixed contact-interaction kernel whose two parameters are determined once by fitting to meson masses and decay constants. The elastic form factors, including the electric-form-factor zero crossing and its relative location between axial-vector and vector channels, are then obtained from the impulse-approximation current. This constitutes an independent output rather than a re-expression of the fitted inputs. No self-definitional steps, fitted quantities renamed as predictions, or load-bearing self-citations that reduce the central claim to prior unverified assertions appear in the derivation. The results are externally compared with other models, keeping the calculation self-contained.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the contact interaction model parameters being appropriately chosen and the framework being applicable to axial-vector states.

free parameters (2)
  • Contact interaction strength
    Typically fitted to reproduce known meson properties like masses.
  • Dressed quark masses
    Parameters for light and heavy quarks in the model.
axioms (2)
  • domain assumption Symmetry-preserving treatment of the contact interaction
    Assumed to maintain key symmetries in the equations.
  • standard math Validity of Schwinger-Dyson and Bethe-Salpeter equations for meson bound states
    Standard framework in non-perturbative QCD.

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