Where Are the Corrections to the Goldberger - Treiman Relation?

On the basis of unsubtracted dispersion relations we point out that theoretical estimates fail to account for the corrections to the Goldberger-Treiman relation $\Delta =1-\frac{M{g}_A}{g{f}_{\pi }}\simeq 0.08\pm 0.02$ by one to two orders of magnitude. As a new result we prove an exact threshold theorem for the absorptive part of $K\left(t\right)\mathrm{}$, the pion-nucleon form factor, by doing the full three-body angular integrations and using current-algebra low-energy theorems for the $3\pi$ processes. In the chiral-SU(2)⊗ SU(2)-symmetry world in which the pion mass vanishes the result is $\mathrm{Im}K\left(t\right)=\left[\frac{g{t}^2}{3{\left(8\mathrm{}\pi \right)}^{3}f_{\pi }^{}{}_{}{}^4}\right]\left[\frac{1}{3}g_A^{}{}_{}{}^2\right(\frac{5}{2}-\frac{17}{35}{\pi }^2)-\frac{7}{8}]$ as $t\to 0$, the threshold point. This leads to an estimate of $\Delta$ which fails by two orders of magnitude. We also show the pion spectral function in the chiral-symmetric world behaves like ${\rho }_{\pi }\mathrm{}\left(t\right)=\frac{8t}{3{\left(8\mathrm{}\pi \right)}^{4}f_{\pi }^{}{}_{}{}^4}$ as $t\to 0$. As a way out of this negative result we consider (i) subtractions in the dispersion relation, (ii) new values for $g$ and $g_A$, and (iii) a heavy pion as possibilities.