Nilpotency property, physicality criteria, constraints and standard BRST algebra: A 4D field-theoretic system

Within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism, we discuss the off-shell nilpotent (anti-)BRST and the bosonic ghost-scale symmetries of a set of coupled (but equivalent) Lagrangian densities for the four (3 + 1)-dimensional (4D) combined field-theoretic system of the free Abelian 3-form and 1-form gauge theories. We demonstrate that the Noether (anti-)BRST charges are non-nilpotent due to the presence of a set of non-trivial Curci-Ferrari (CF) type restrictions on our theory. These CF-type restrictions are derived and discussed from different theoretical angles in our present endeavor. In addition to it, we obtain the nilpotent versions of the above (anti-)BRST charges from their counterparts non-nilpotent versions and discuss the physicality criteria w.r.t. the nilpotent charges to show that the physical states (existing in the total quantum Hilbert space of states) are those that are annihilated by the operator forms of the first-class constraints of our classical 4D combined field-theoretic system. The standard BRST algebra among the nilpotent (anti-)BRST charges and the bosonic ghost charge is derived, too.