What is $M_N$ in the Goldberger-Treiman relation? $$g_{\pi NN} F_\pi = G_A M_N .$$
Does it stand for the magnetic moment of the neutron?
One place I came across it was on Wikipedia, on their QCD Vacuum page, in the section about experimental evidence, gradient coupling and Goldberger-Treiman.
 A: In anticipation of clarification for the question, here is a placeholder summary. The G-T relation is, in modern notation,
$$g_{\pi NN} F_\pi = G_A M_N .$$
(Donoghue, J.F.; Golowich, E.; Holstein, B.R. (1994). Dynamics of the Standard Model. Cambridge University Press. ISBN 0-521-47652-6., p.347, eqn. (3.20); or TP-Cheng & L F Li, eqn (5.190).)
It relates four phenomenological/non-perturbative QCD constants:
the strong pion-emission one, $g_{\pi NN}$; the nucleon mass, $M_N$;  the (weak & EM) pion-decay one $F_\pi$; and the (weak) neutron-decay one, $G_A$, the hadronization of the axial currents between nucleon states.
The relation is "obsolete" by now, with the advent of the Standard Model, but was significant in clearing the smoke in current algebra in the 60s and leading/shaping the quark model and QCD. Its somewhat "mystical" status back then relied on confusion about how short-distance quark currents morphed/dressed themselves into hadronic currents in long-distance QCD. Schematically, the axial current is represented/dominated by the pion ("pion dominance").
There was an apparent dissonance between the strong πNN coupling,  and the nucleon mass, features of χSB, and weak decay parameters like the hadronization of the axial current involved in pion and neutron weak decays. These are also both features of strong χSB, though, and another facet of PCAC, the gateway to the quark model, beyond elementary spectroscopy. Today, lattice simulations provide estimates for them. With the enigma gone, fewer physicists obsess on the relation, and it is mostly taught in books for continuity with the older literature.
A: While I can't find the source for the specific form of the Goldberger-Treiman relation that Wikipedia quotes, it is pretty clear from the original sources that the "$M_N$" is intended to be the nucleon mass. The closest Goldberger and Treiman come in "Decay of the Pi Meson" (Phys. Rev. 110, 1178) to writing down their eponymous relation is eq. (24):
$$ F(0) = -\frac{m}{2\pi^2}\sqrt{2} G g_A \frac{J}{1 + (G^2/4\pi)(2J/\pi)},$$
where $m$ is the nucleon mass, $G$ the pion-nucleon coupling constant, $J$ some horrible integral and $g_A$ the axial coupling. If we squint the r.h.s. kinda looks like $g_Am\frac{1}{G}$ dropping all constants, and this then gives the $GF = g_Am$ relation the Wikipedia article has.
