Covariant Cubic Interacting Vertices for Massless and Massive Integer Higher Spin Fields

The paper constructs covariant cubic vertices for irreducible massless/massive integer-spin fields using the complete BRST operator (including trace constraints), derives the generating equations, proves first-order gauge invariance and on-shell closure of the deformed gauge algebra, and classifies solutions by the mass polynomials D and P across the three mass patterns. It explicitly obtains (3+1)-, (3+2)-, and (3+2)-parameter families in the cases of two massless plus one massive, one massless plus two equal-mass massive, and one massless plus two different-mass massive fields, respectively, with detailed operator realizations L(i), L(ij)+_11, U_j and their BRST-closed modifications, see the summary and formulas around eqs. (28) (locality and ghost prefactor), (30)–(31) (D and P), (29) (gauge-algebra closure), and the case-by-case results (including k, kmin/kmax, and τ2, τ3) (e.g., discussion and formulas summarized in the conclusion and Section 4) . By contrast, the candidate solution assumes ghost-independent “BRST-closed building blocks” Y_i and Z_i and reduces Q_tot|V⟩=0 to commuting first-order flows H_i V=0 and the relation m1∂V/∂Y1 = m2∂V/∂Y2 = m3∂V/∂Y3. This is incompatible with the paper’s central technical point that naive oscillator monomials are not Q_tot-closed when the trace constraint sector is included; explicit counter-evidence is given by the failure of simple L(i) and Z operators to be BRST-closed (eqs. (50)–(53)) and the necessity of ghost- and b-oscillator–dependent corrections to build Q_tot-closed generators (eqs. (58)–(62), (83)–(85)) . While the candidate reproduces the paper’s high-level parameter counts for the three mass patterns, the derivation relies on unproven assumptions (e.g., L̂11-compatibility and BRST-closure of Y_i, Z_i; “l0 only gives total derivatives”) that contradict the explicit operator analysis in the paper. Hence, the paper’s argument is correct and complete for its stated scope, whereas the model’s proof is not valid in the complete-BRST setting.