with both gauge and reducibility parameters displayed in (6), (8), and (11), together with their corresponding antifields (denoted by star variables)

\[ \Phi ^ { \alpha _ { 0 } } \equiv \bigl \{ \varphi , A _ { \mu } , H ^ { \mu } , B ^ { \mu \nu } , \phi ^ { A } \bigr \} , \quad \Phi _ { \alpha _ { 0 } } ^ { * } \equiv \bigl \{ \varphi ^ { * } , A ^ { * \mu } , H _ { \mu } ^ { * } , B _ { \mu \nu } ^ { * } , \phi _ { A } ^ { * } \bigr \} , \quad ( 1 4 ) \]

\[ \Omega ^ { \alpha _ { 1 } } \to \eta ^ { \alpha _ { 1 } } \equiv \{ \eta , C ^ { \mu \nu } , \eta ^ { \mu \nu \rho } \} , \qquad \qquad \eta _ { \alpha _ { 1 } } ^ { \ast } \equiv \{ \eta ^ { \ast } , C _ { \mu \nu } ^ { \ast } , \eta _ { \mu \nu \rho } ^ { \ast } \} , \qquad \qquad ( 1 5 ) \]

\[ \Omega ^ { \alpha _ { 2 } } \rightarrow \eta ^ { \alpha _ { 2 } } \equiv \{ C ^ { \mu \nu \rho } , \eta ^ { \lambda \mu \nu \rho } \} , \qquad \qquad \eta _ { \alpha _ { 2 } } ^ { * } \equiv \{ C _ { \mu \nu \rho } ^ { * } , \eta _ { \lambda \mu \nu \rho } ^ { * } \} , \qquad \qquad ( 1 6 ) \]

\[ \Omega ^ { \alpha _ { 3 } } \rightarrow \eta ^ { \alpha _ { 3 } } \equiv \{ C ^ { \lambda \mu \nu \rho } \} , \eqno ( 1 7 ) \]

For notational ease, it is convenient to organize the fields/ghosts and respectively antifields into

\[ \chi ^ { \Delta } \equiv \{ \Phi ^ { \alpha _ { 0 } } , \eta ^ { \alpha _ { 1 } } , \eta ^ { \alpha _ { 2 } } , \eta ^ { \alpha _ { 3 } } \} , \qquad \chi _ { \Delta } ^ { * } \equiv \{ \Phi _ { \alpha _ { 0 } } ^ { * } , \eta _ { \alpha _ { 1 } } ^ { * } , \eta _ { \alpha _ { 2 } } ^ { * } , \eta _ { \alpha _ { 3 } } ^ { * } \} . \qquad \qquad ( 1 8 ) \]

The \( \mathbb { Z } _ { 2 } \) grading of the BRST algebra in terms of the Grassmann parity (\( \succ \)) is inferred from the observation that all the original fields together with the accompanying gauge and reducibility parameters are bosonic, so, according to the general rules of the antifield formalism, we take

\[ \varepsilon ( \eta ^ { \alpha _ { k } } ) = k \bmod 2 , \quad k = 1 , 2 , 3 , \quad \quad \varepsilon ( \chi _ { \Delta } ^ { * } ) = ( \varepsilon ( \chi ^ { \Delta } ) + 1 ) \bmod 2 . \quad \quad ( 1 9 ) \]

The Grassmann parity is then lifted to the BRST algebra by means of its additive action modulo 2 against multiplication. In agreement with the usual prescriptions of the BRST method, the BRST algebra is endowed with three more gradings (correlated with the main derivatives/differentials acting on this algebra): two N-gradings along the antifield number agh and respectively the pure ghost number pgh and a total Z-grading in terms of the ghost number gh. These are instated by setting the values of the corresponding degrees at the level of the BRST generators

\[ \mathrm { a g h } ( \chi ^ { \Delta } ) = 0 , \ \ \ \ \mathrm { a g h } ( \Phi _ { \alpha _ { 0 } } ^ { * } ) = 1 , \ \ \ \ \mathrm { a g h } ( \eta _ { \alpha _ { k } } ^ { * } ) = k + 1 , \ \ \ \ k = 1 , 2 , 3 , \ \ \ \ ( 2 0 ) \]

\[ \mathrm { p g h } ( \chi _ { \Delta } ^ { * } ) = 0 , \; \; \; \; \mathrm { p g h } ( \Phi ^ { \alpha _ { 0 } } ) = 0 , \; \; \; \; \mathrm { p g h } ( \eta ^ { \alpha _ { k } } ) = k , \; \; \; \; \; \; \; \; \; \; k = 1 , 2 , 3 , \; \; \; \; ( 2 1 ) \]

and by further using their additive behaviour with respect to multiplica-tion. Finally, the (total) ghost number of any object with definite pure ghost and antifield numbers is defined like \( \mathrm { g h } ( a ) = \mathrm { p g h } ( a ) - \mathrm { a g h } ( a ) \).

Due to the fact that the right-hand sides of both gauge transformations (3)–(5) and relations (7) and (10) do not depend on the fields \( \Phi ^ { \alpha _ { 0 } } \) (or, in other words, all gauge generators and reducibility functions — of order one