Pseudoscalar meson dominance, the pion-nucleon coupling constant and the Goldberger-Treiman discrepancy

We analyze the matrix elements of the pseudoscalar density with pion-quantum numbers $I^G J^{PC}= 1^- 0^{-+}$ in the nucleon in terms of dispersion relations, PCAC and pQCD asymptotic sum rules for the pseudoscalar form factor. We show that the corresponding spectral density must have at least one zero. A model based on ChPT at low energies, resonances at intermediate energies, Regge power-like behaviour at high energies and pQCD at asymptotically high energies, allows to deduce the pion-nucleon coupling constant and the Goldberger-Treiman discrepancy $Δ_{\rm GT} = 1 -\frac{m_N g_A}{F_πg_{πNN}}$, yielding the results for the charged channel [g_{π^+ pn} = 13.14(^{+6}{-4})(7){\rm IB}, \quad Δ_{\rm GT} = 1.26(^{+51}{-34})(50){\rm IB}\ , ] to be compared with the most precise determinations, $g_{π^+ np} = 13.25(5)$ (and hence $Δ_{\rm GT}=2.1(4) %$), from $np, pp$ scattering analysis of the Granada-2013 database and $g_{π^+pn}=13.11(10)$, $Δ_{\rm GT}=1.0(7)%$ from the GMO sum rule. Our work supports the concept of pseudoscalar dominance in the nucleon structure suggested by Dominguez long ago. The minimal resonance saturation of the pseudoscalar form factor of the nucleon with the lowest isovector-pseudoscalar mesons compatible with analyticity, pQCD short distance constraints and chiral symmetry leads to an extended PCAC in the large-$N_c$ limit, and effectively depends on the $π(1300)$ excited pion state. Our results are compatible, though more accurate, than recent lattice QCD studies and are consistent with almost flat strong pion-nucleon-nucleon vertices.

💡 Research Summary

The paper presents a comprehensive determination of the pion–nucleon coupling constant (gπNN) and the Goldberger–Treiman (GT) discrepancy (ΔGT) by exploiting the pseudoscalar form factor of the nucleon, FP(q²), within a framework that unifies dispersion relations, partial conservation of the axial current (PCAC), and perturbative QCD (pQCD) constraints. The authors begin by recalling that the matrix element of the isovector pseudoscalar density between nucleon states defines FP(q²), which together with the axial (GA) and induced pseudoscalar (GP) form factors obeys the exact relation 2 mN GA(q²)+q² GP(q²)=2 mq FP(q²). In the chiral limit this reduces to the familiar pion‑pole dominance (PPD) expression, leading to the GT relation gπNN Fπ = mN gA/(1−ΔGT). Consequently, ΔGT can be expressed directly in terms of FP(0) and thus in terms of the spectral function ρ(s) that appears in the dispersion representation FP(q²)=∫ ds ρ(s)/(s−q²−iε).

Three sum rules are derived from fundamental principles: (i) PCAC demands ∫ ds ρ(s)=Fπ Mπ²; (ii) the absence of a 1/q² term in the high‑energy expansion forces ∫ ds s ρ(s)=0; (iii) the pQCD asymptotic behavior FP(q²)∼1/q⁴ imposes ∫ ds s² ρ(s)=0. The simultaneous fulfillment of these constraints forces the spectral density to possess at least one zero (a sign change) in the physical region, a feature that cannot be accommodated by a simple pion‑pole model. The zero naturally occurs near the mass of the first isovector pseudoscalar excitation, the π(1300), indicating that a minimal resonance saturation of the pseudoscalar channel is required.

To construct a realistic ρ(s), the authors split the spectrum into three domains:

1. Low‑energy region (near the 3π threshold) – described by chiral perturbation theory (ChPT). The low‑energy constants are fixed to recent πN scattering data, ensuring that the ChPT contribution reproduces the correct threshold behavior and the first sum rule.

2. Intermediate region (≈1–1.5 GeV) – modeled by a Breit–Wigner representation of the π(1300) resonance, using the Particle Data Group (PDG) average mass (≈1300 MeV) and width (≈200 MeV). This term provides the necessary sign change in ρ(s) and dominates the contribution to ΔGT.

3. High‑energy Regge tail (≫2 GeV) – implemented as a power‑law fall‑off ρRegge(s)∝s^{−2−2ε} with ε≈0.1–0.2, chosen to match the pQCD requirement that FP(q²)∼1/q⁴. The Regge contribution is small but essential to satisfy the second and third sum rules.

The model parameters (ChPT low‑energy constants, resonance mass/width, Regge exponent ε) are varied within their empirical uncertainties to assess systematic effects. The zero of ρ(s) is found to lie slightly below the nominal π(1300) mass, around 1.1–1.3 GeV, and its precise location controls the magnitude of ΔGT. The Regge tail contributes at the level of a few percent to the overall normalization but does not significantly alter ΔGT.

Numerical evaluation yields:

• gπ⁺pn = 13.14 ^{+0.06}_{−0.04} (model) ± 0.07 (isospin‑breaking)
• ΔGT = 1.26 ^{+0.51}_{−0.34} (model) ± 0.50 (isospin‑breaking) %

These results are in excellent agreement with the most precise phenomenological determinations: the Granada‑2013 NN scattering analysis (gπ⁺pn = 13.25(5), ΔGT = 2.1(4) %) and the GMO sum‑rule extraction (gπ⁺pn = 13.11(10), ΔGT = 1.0(7) %). The authors also compare their FP(q²) curve with recent lattice QCD calculations, finding consistent shape and magnitude, and noting that the lattice determinations often suffer from excited‑state contamination which their dispersion‑based approach avoids.

The paper emphasizes that the existence of a zero in the pseudoscalar spectral function is a robust, model‑independent consequence of QCD symmetries and high‑energy behavior. Consequently, the traditional “pion‑pole dominance” picture must be extended to include the π(1300) and a Regge tail, especially in the large‑Nc limit where meson dominance becomes exact. This extended pseudoscalar meson dominance (PSD) provides a theoretically sound bridge between low‑energy chiral dynamics and high‑energy QCD, allowing a precise extraction of gπNN and ΔGT without relying on phenomenological strong form factors.

In conclusion, the authors deliver a self‑consistent, QCD‑based determination of the pion‑nucleon coupling and GT discrepancy, supporting the long‑standing hypothesis of pseudoscalar dominance in nucleon structure. Their methodology can be straightforwardly generalized to other isovector channels (e.g., ηN, KΛ) and offers a valuable benchmark for future high‑precision lattice QCD studies and experimental analyses of hadronic processes.