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Baryonic form factors of light pseudoscalar mesons

A. S. Miramontes, J. M. Morgado, J. Papavassiliou

Abstract

Employing the Bethe-Salpeter formalism, we present a computation of the space-like baryonic form factor for the pion and kaon. In the exact isospin-symmetric limit this observable is forbidden by $G$-parity, so that any nonzero signal constitutes a direct probe of the quark mass difference $m_d - m_u$. The form factors are evaluated in the impulse approximation using fully dressed quark propagators, meson Bethe-Salpeter amplitudes, and a dressed baryon-current vertex constrained by the vector Ward-Takahashi identity. The baryonic radius computed with this method for the pion is given by $\langle r_{\! B}^2\rangle_{π^+}^{1/2} = 0.043(2)$ fm, and is consistent with the available dispersive benchmarks. Our predictions for the kaons, namely $\langle r_{\!B}^2\rangle_{K^+}^{1/2} = 0.265(7)$ fm and $\langle r_{\!B}^2\rangle_{K^0}^{1/2} = 0.262(7)$ fm, indicate a larger spatial extent than in the pion case; these results have no dispersive counterparts, and are compatible with chiral QCD models.

Baryonic form factors of light pseudoscalar mesons

Abstract

Employing the Bethe-Salpeter formalism, we present a computation of the space-like baryonic form factor for the pion and kaon. In the exact isospin-symmetric limit this observable is forbidden by -parity, so that any nonzero signal constitutes a direct probe of the quark mass difference . The form factors are evaluated in the impulse approximation using fully dressed quark propagators, meson Bethe-Salpeter amplitudes, and a dressed baryon-current vertex constrained by the vector Ward-Takahashi identity. The baryonic radius computed with this method for the pion is given by fm, and is consistent with the available dispersive benchmarks. Our predictions for the kaons, namely fm and fm, indicate a larger spatial extent than in the pion case; these results have no dispersive counterparts, and are compatible with chiral QCD models.

Paper Structure

This paper contains 6 sections, 29 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. General framework
  3. Baryonic form factor
  4. Dynamical equations
  5. Results
  6. Conclusions

Figures (5)

  • Figure 1: Diagrams of the baryon-number current in the impulse approximation that we employ. The wavy lines with a cross attached to them indicate the insertion of the baryon-number current, the orange circles represent the fully dressed current vertices, while the gray ones denote dressed quark propagators.
  • Figure 2: Diagrammatic representation of the main components: (i) the quark self-energy entering the gap equation; (ii) the meson Bethe-Salpeter amplitude for the quark-antiquark state; and (iii) the SDE for the baryon-current vertex. The blue band corresponds to the interaction kernel defined in Eq. \ref{['eq:effective-gluon']}.
  • Figure 3: Left panel: The modified Taylor effective charge, ${\tilde{\alpha}}_{T}(q^2)$, The dashed curve represents the central parametrization from Gao:2024gdj. The shaded band shows the variation obtained by shifting the fit parameters by one standard deviation ($\pm\sigma$) around their central values. Right panel: The mass function $\mathcal{M}_f(p^2)$ for the flavours $f=u,d,s$.
  • Figure 4: Left panel: Computed baryonic form factor for the charged pion, the dashed line corresponds to the central parametrization of the $\tilde{\alpha}_T$ while the solid lines are the variations. The results are compared to the dispersive extraction from the BaBar and KLOE data in Sanchez-Puertas:2021eqj. Right panel: Corresponding results for the charged kaon.
  • Figure 5: Baryonic form factor for the neutral kaon (dotted line) compared with the charged one (solid line) and to the corresponding neutral-kaon electromagnetic form factor from Miramontes:2025vzb (dashed line).