Goldberger-Treiman relation and gπNN from the three quark BS/Faddeev approach in the NJL model

A systematic evaluation method of matrix elements of quantum one-body operators between nucleon states in the three-quark Bethe-Salpeter(3qBS)/Faddeev approach in the NJL model is reviewed. We do not confine ourselves to a particular truncation scheme of the 3qBS kernel. One of our main aims is to derive a general condition to be imposed on a given BS kernel in order that the PCAC relation is satisfied correctly. We apply this condition to some particular 3qBS kernels. We numerically calculate gπNN in the 3qBS/Faddeev approach to estimate the violation of the Goldberger-Treiman/PCAC relation due to the UV-regularization scheme. We finally consider the nonvanishing current quark mass effects on the Goldberger-Treiman relation