Extended PCAC Relation

We explore a consistent way to extend the partially conserved axial vector current (PCAC) relation and corresponding current algebra results. The paper includes investigations of a chiral Ward-Takahashi identity, the explicit chiral symmetry breaking by a finite current quark mass, the modifica- tion of the PCAC relation and its consequences. Two explicit relations between the nucleon axial vector form factor $g_A(q^2)$, pseudo-scalar form factor $g_P(q^2)$ and the pion-nucleon coupling constant $g_{\pi NN}(q^2)$ is obtained. One of the relation is confirmed, within the experimental error, by observation in the $0<-q^2 <0.2$ $\mbox{GeV}^2$ region. The other one, which relates $g_A(q^2)$ and $f_\pi(q^2) g_{\pi NN}(q^2)$ is studied by using known empirical facts and dispersion relation. Thoeretical uncertainties are evaluated. Certain inconsistencies, which is in favor of the introduction of diquark condensation, is discovered. We briefly discuss how diquark condensation could provide an answer to the question of where about of the quark numbers in a nucleon and a nucleus, which is raised in explaining puzzles observed in the violation of Gottfried sum rule and EMC effects.