On an extended PCAC relationship

We explore a consistent way to extend the partially conserved axial vector current (PCAC) relationship and corresponding current algebra results in two strongly correlated directions: 1) towards a search for a set of systematic rules for the establishment of PCAC related relationships in a finite low momentum transfer region and for the extrapolation of the momentum transfer $q^2$ to zero when deriving the low energy PCAC results that can be compared to experimental data and 2) towards taking into account, besides the conventional one, the only other possibility of the spontaneous chiral symmetry breaking, $SU(2)_L\times SU(2)_R \to SU(2)_V$, inside a baryonic system by a condensation (in the sense to be specified in the paper) of diquarks. The paper includes investigations of a chiral Ward--Takahashi identity, the explicit chiral symmetry breaking by a finite current quark mass, the modification of the PCAC relationship and its consequences. It is shown that the signals for a hypothetical diquark condensation inside a nucleon and nucleus is observable in high precision experiments despite the fact that they may evade most of the current observations. We briefly discuss how diquark condensation could provide an answer to the question of where about of pions and quark number in a nucleon and a nucleus, which is raised in explaining puzzles in polarized pion nucleus scattering, violation of Gottfried sum rule and EMC effects.