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

and respectively two — are field independent), it follows that the BRST differential s reduces to a sum of two fermionic derivations

\[ s = \delta + \gamma , \eqno ( 2 2 ) \]

with \( \delta \) the Koszul–Tate differential, graded in terms of agh \( ( \mathrm { a g h } ( \delta ) = - 1 \)),and \( \gamma \) the longitudinal exterior derivative (in this case a true differential), graded by pgh \( ( \mathrm { p g h } ( \gamma ) \) = 1). These two degrees do not interfere \( ( \mathrm { a g h } ( \gamma ) \) = 0,\( \mathrm { p g h } ( \delta ) \) = 0), such that the total degree of the BRST differential (and of each of its components), namely gh, becomes equal to 1:\( \operatorname { g h } ( s ) = \operatorname { g h } ( \delta ) = \operatorname { g h } ( \gamma ) = \) 1. One of the major requirements of the BRST setting, namely the second-order nilpotency of s, becomes equivalent to three separate equations

\[ s ^ { 2 } = 0 \Leftrightarrow ( \delta ^ { 2 } = 0 , \, \delta \gamma + \gamma \delta = 0 , \, \gamma ^ { 2 } = 0 ) , \eqno ( 2 3 ) \]

which confirms that \( \gamma \) can indeed be constructed as a true differential in the case of the free model under study. The actions of \( \delta \) and \( \gamma \) on the BRST generators that enforce (23) as well as the fundamental cohomological requirements of the antifield BRST theory [27, 28, 29, 30, 31, 32] are given by

\[ \delta \chi ^ { \Delta } = 0 , \qquad \qquad \qquad \delta \varphi ^ { * } = \partial _ { \lambda } H ^ { \lambda } , \qquad \qquad \delta A ^ { * \mu } = \partial _ { \lambda } B ^ { \lambda \mu } , \qquad \qquad ( 2 4 ) \]

\[ \begin{array} { r } { \delta H _ { \mu } ^ { * } = - \partial _ { \mu } \varphi , \qquad \qquad \delta B _ { \mu \nu } ^ { * } = - \frac 1 2 \partial _ { [ \mu } A _ { \nu ] } , \qquad \quad \delta \phi _ { A } ^ { * } = k _ { A B } \sqcap \phi ^ { B } , \qquad ( 2 5 ) } \end{array} \]

\[ \delta \eta ^ { * } = - \partial _ { \lambda } A ^ { * \lambda } , \qquad \quad \delta C _ { \mu \nu } ^ { * } = \partial _ { [ \mu } H _ { \nu ] } ^ { * } , \qquad \qquad \delta \eta _ { \mu \nu \rho } ^ { * } = \partial _ { [ \mu } B _ { \nu \rho ] } ^ { * } , \qquad ( 2 6 ) \]

\[ \delta C _ { \mu \nu \rho } ^ { * } = - \partial _ { [ \mu } C _ { \nu \rho ] } ^ { * } , \qquad \delta \eta _ { \lambda \mu \nu \rho } ^ { * } = - \partial _ { [ \lambda } \eta _ { \mu \nu \rho ] } ^ { * } , \qquad \delta C _ { \lambda \mu \nu \rho } ^ { * } = \partial _ { [ \lambda } C _ { \mu \nu \rho ] } ^ { * } , \qquad ( 2 7 ) \]

\[ \gamma \chi _ { \Delta } ^ { * } = 0 , \qquad \qquad \qquad \gamma \varphi = 0 , \qquad \qquad \qquad \gamma A _ { \mu } = \partial _ { \mu } \eta , \qquad \qquad ( 2 8 ) \]

\[ \gamma H ^ { \mu } = - 2 \partial _ { \lambda } C ^ { \lambda \mu } , \qquad \gamma B ^ { \mu \nu } = - 3 \partial _ { \lambda } \eta ^ { \lambda \mu \nu } , \qquad \gamma \phi ^ { A } = 0 , \qquad \qquad ( 2 9 ) \]

\[ \gamma \eta = 0 , \qquad \qquad \qquad \gamma C ^ { \mu \nu } = - 3 \partial _ { \lambda } C ^ { \lambda \mu \nu } , \: \: \: \: \: \gamma \eta ^ { \mu \nu \rho } = - 4 \partial _ { \lambda } \eta ^ { \lambda \mu \nu \rho } , \: \: \: \: ( 3 0 ) \]

\[ \gamma C ^ { \mu \nu \rho } = - 4 \partial _ { \lambda } C ^ { \lambda \mu \nu \rho } , \quad \gamma \eta ^ { \lambda \mu \nu \rho } = 0 , \qquad \qquad \gamma C ^ { \lambda \mu \nu \rho } = 0 , \qquad \qquad ( 3 1 ) \]

where both operators are assumed to act like right derivations and \( \sqsupset \equiv \)\( \partial _ { \mu } \partial ^ { \mu } \) symbolizes the d’Alembertian. We notice that the actions of \( \gamma \) on all fields/ghosts can be obtained in this particular situation simply by replacing all gauge or reducibility parameters from the right-hand sides of relations (3)–(5), (7), (10), and (13) with the corresponding ghosts introduced in (15)–(17).

A striking feature of the antifield approach, in spite of its essentially La-grangian origins, resides in the (anti)canonical action of the BRST differen-tial [23, 24, 25, 26, 27, 28, 29, 30, 31, 32], \( s \cdot = ( \cdot , S ) \), where its (anti)canonical