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

TL;DR

This work develops a dispersion-theory framework for the isovector pseudoscalar nucleon form factor in the channel $I^G J^{PC}=1^- 0^{-+}$ to determine the pion–nucleon coupling $g_{\pi NN}$ and the Goldberger–Treiman discrepancy $\Delta_{GT}$. By combining PCAC, chiral symmetry, resonance saturation, Regge behaviour, and pQCD asymptotics, the authors show the pseudoscalar spectral density must change sign, implying at least one zero and enabling a bound on $\Delta_{GT}$. Modeling the spectral function with a dominant $\pi(1300)$ resonance plus a Regge tail, they obtain $g_{\pi^+ pn}=13.14$ and $\Delta_{GT}=1.26\%$ (with substantial but controlled uncertainties from resonance properties and isospin-breaking effects), values compatible with precise Granada-2013 GMO extractions and lattice QCD results. The analysis supports pseudoscalar dominance in the nucleon and provides a benchmark for the related pseudoscalar form factor, offering insights for future lattice studies and extensions to $\eta$/$\eta'$ sectors.

Abstract

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.

Figures (12)

• Figure 1: Pion--nucleon coupling constant determinations compared to the historical PDG averages of the ratio $g_A m_N/F_\pi$. Here $Av_{\rm Nij}$deSwart:1997ep, GMO$_B$Baru:2010xn, GMO$_E$Ericson:2000md, GMO$_H$Hoferichter:2023ptl, $NN_{\rm Gr}$NavarroPerez:2013mvd, Av Matsinos:2019kqi, $NN_{\rm Bo}$Reinert:2020mcu, Ps represents this work's value, which supersedes our previous estimate of Ref. RuizArriola:2023xap. The difference in percentage of both determinations corresponds to the GT discrepancy $\Delta_{\rm GT}= 1- g_A m_N / (g_{\pi NN} F_\pi)$ and is a direct measure of the chiral symmetry breaking in QCD.
• Figure 2: Cartoon of the spectral density in the complex $s$-plane (arbitrary scale) and its different regions, showing the pion pole and the resonance poles (in the second Riemann sheet across the $3\pi$-cut). We also show the assumed asymptotic Suranyi's radio $\Gamma /M = \gamma$ (see discussion in main text).
• Figure 3: The form factor $D(Q^2)$ (full grey lines and band). In addition, we show as a dashed grey band the pure monopole prediction. Lattice results from PACS'25 Aoki:2025taf, ETMC Alexandrou:2023qbg, PNDME Jang:2023zts, NME'22 Park:2021ypf, PNDME'19 Jang:2019vkm. The final number represents the lattice $\pi$ mass in each ensemble except for ETMC, which results are taken from their table with physical extrapolations.
• Figure 4: The ratio of pseudoscalar form factors defined in Eq. \ref{['eq:RP']} (full gray lines and band). In addition, we show as a dashed-gray line the pQCD-driven estimate. The colours for the lattice results are analogous to those in Fig. \ref{['fig:Dt']}.
• Figure 5: In the diagrams above, the top line is taken as the $u$ quark with momentum fraction $u_1$ or the $d$ quark. The crossed circle stands for the appropriate current insertion. All the contributions can be expressed in terms of these.