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 $\Delta_{\rm GT} = 1 -\frac{m_N g_A}{F_{\pi}g_{\pi NN}}$, yielding the results for the charged channel $$g_{\pi^+ pn} = 13.14(^{+6}_{-4})(7)_{\rm IB}, \quad \Delta_{\rm GT} = 1.26(^{+51}_{-34})(50)_{\rm IB}\ ,$$ to be compared with the most precise determinations, $g_{\pi^+ np} = 13.25(5)$ (and hence $\Delta_{\rm GT}=2.1(4) \%$), from $np, pp$ scattering analysis of the Granada-2013 database and $g_{\pi^+pn}=13.11(10)$, $\Delta_{\rm GT}=1.0(7)\%$ 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 $\pi(1300)$ 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.