BRST–BV approach for interacting higher-spin fields

We develop the BRST–BV approach to the construction of the general off-shell Lorentz covariant cubic, quartic, and $e$-tic interaction vertices for irreducible higher-spin fields on $d$-dimensional Minkowski space. We consider two different cases for interacting integer higher-spin fields with both massless and massive fields. The deformation procedure to find a minimal BRST–BV action for interacting higher-spin fields, defined with help of a generalized Hilbert space, is based on the preservation of the master equation in each power of the coupling constant $g$ starting from the Lagrangian formulation for a free gauge theory. For illustration, we consider the construction of local cubic vertices for $k$ irreducible massless fields of integer helicities, and $k-1$ massless fields and one massive field of spins $s_1, \dots, s_{k-1}, s_k$. For a triple of two massless scalars and a tensor field of integer spin, the BRST–BV action with cubic interaction is explicitly found. In contrast to the previous results on cubic vertices, following our results for the BRST approach to massless fields, we use a single BRST–BV action instead of the classical action with reducible gauge transformations. The procedure is based on the complete BRST operator that includes the trace constraints used in defining the irreducible representation with a definite integer spin.