Baryonic form factors of light pseudoscalar mesons

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.

💡 Research Summary

The paper investigates the baryonic form factor (BFF) of light pseudoscalar mesons—specifically the charged pion (π⁺) and the charged and neutral kaons (K⁺, K⁰)—within a fully non‑perturbative continuum QCD framework. In the exact isospin‑symmetric limit (m_u = m_d) and neglecting QED, the baryon‑number current is odd under charge conjugation while the pion is a G‑parity eigenstate; consequently the matrix element of the baryon current vanishes for all momentum transfers. Any non‑zero BFF therefore directly probes isospin‑breaking effects, primarily the quark‑mass difference m_d − m_u.

To compute the BFF the authors employ the coupled Schwinger‑Dyson equations (SDE) for the dressed quark propagators and the Bethe‑Salpeter equations (BSE) for the meson bound‑state amplitudes, using a flavour‑dependent effective interaction I_{ff′}(k²) that incorporates a modified Taylor coupling ~α_T(k²) fitted to lattice data. Separate gap equations are solved for u, d, and s quarks, yielding distinct dressing functions A_f(p²), B_f(p²) and the dynamical mass function M_f(p²)=B_f/A_f. The interaction kernel is the same in all three equations, guaranteeing that the vector Ward‑Takahashi identity (WTI) for the baryon‑number current is satisfied.

The baryon‑number current is coupled to the quark–antiquark pair via a fully dressed vertex Γ_μ^B,f(p₁,p₂). This vertex is decomposed into a Ball‑Chiu part that exactly satisfies the WTI and a transverse part that is orthogonal to the momentum transfer q_μ. The transverse component dynamically generates vector‑meson poles (ρ in isovector channels, ω in isoscalar channels) through the same kernel that produces the bound‑state spectrum. In the presence of isospin breaking, the ω pole becomes relevant for the pion BFF, while the ρ contributes to the kaon channels.

The BFF is evaluated in the impulse approximation: the external baryon current is inserted on each quark line of the meson, producing two diagrams related by the interchange of quark flavours. The matrix element ⟨M(p_out)|j_μ^B|M(p_in)⟩ = (p_out + p_in)μ F_B(q²) defines the form factor F_B(q²). The form factor is extracted by contracting the calculated current with the average momentum p̄_μ and dividing by p̄². The baryonic mean‑square radius follows from the slope at q² = 0: ⟨r_B²⟩ = −6 dF_B/dq²|{q²=0}.

Numerical inputs are fixed in a momentum‑subtraction (MOM) scheme at μ = 4.3 GeV. The current quark masses are chosen as m_u = 3.7 MeV, m_d = 6.2 MeV, and m_s = 95 MeV, reproducing the physical charged‑pion mass (139 MeV) and decay constant (130 MeV). Solving the BSE with these propagators yields meson masses and decay constants in excellent agreement with experiment for both pions and kaons; the small residual discrepancy in the neutral‑kaon mass (≈0.5 %) is attributed to omitted electromagnetic contributions.

The computed BFFs display a very small but non‑zero signal for the pion, reflecting the strong‑isospin breaking. The extracted baryonic radius for the charged pion is
⟨r_B²⟩^{1/2}{π⁺} = 0.043 ± 0.002 fm,
in line with dispersive analyses of e⁺e⁻ → π⁺π⁻ data (BaBar, KLOE) that have previously extracted the same observable. For the kaons the results are
⟨r_B²⟩^{1/2}{K⁺} = 0.265 ± 0.007 fm,
⟨r_B²⟩^{1/2}_{K⁰} = 0.262 ± 0.007 fm.
These values are roughly six times larger than the pion radius, indicating a more extended baryonic charge distribution in the kaons. No dispersive benchmarks exist for the kaon BFF, so these predictions constitute the first quantitative estimates from a continuum QCD approach.

The analysis also highlights the role of the transverse vertex component: the ω‑meson pole, generated dynamically, dominates the isoscalar contribution to the pion BFF, while the ρ‑pole influences the kaon channels. Because the calculation neglects explicit QED effects, the electromagnetic part of the K⁺–K⁰ mass splitting is not reproduced; the authors note that incorporating photon exchange in the SDE‑BSE system would be a natural next step.

In summary, the work demonstrates that the baryonic form factor—an observable forbidden by G‑parity in the isospin‑symmetric limit—can be reliably computed using a self‑consistent SDE‑BSE framework with a dressed baryon‑current vertex. The results provide a clean probe of the quark‑mass difference m_d − m_u and illustrate how dynamical vector‑meson correlations shape the spatial distribution of baryon number inside light mesons. The pion prediction validates existing dispersive extractions, while the kaon predictions offer new targets for future experimental or lattice QCD studies.