On Lagrangian formulations for (ir)reducible mixed-antisymmetric higher integer spin fields in Minkowski spaces

TL;DR

The work addresses constructing gauge-invariant Lagrangians for integer-spin mixed-antisymmetric higher-spin fields in flat spacetime, focusing on representations with a 3-column Young tableau. It develops both complete and incomplete BRST formulations, including a conversion of holonomic second-class constraints via a Verma-module SU(4) framework, to obtain nilpotent BRST operators and gauge-invariant actions for massless and massive cases. A deformation procedure for interacting vertices is proposed, enabling local cubic and higher-order couplings among copies of the fields while preserving the total physical degrees of freedom. These results establish a systematic BRST approach to multi-column mixed-symmetry HS fields, offering a path toward BRST-BV quantization and potential applications in dark-matter motivated scenarios and beyond the Standard Model.

Abstract

We extend the results of Lagrangian formulations study to construct gauge-invariant Lagrangians for (ir)reducible integer higher-spin massless and massive representations of the Poincare group with a Young tableau $Y[\hat{s}_1,\hat{s}_2,\hat{s}_3]$ in $d$-dimensional flat space-time (as the probable candidates to describe the Dark Matter problem beyond the SM). These particles are described within a metric-like formulation by tensor fields with 3 groups of antisymmetric Lorentz indices $Φ_{μ^1[{\hat{s}_1}],μ^2[{\hat{s}_2}], μ^3[{\hat{s}_3}]}$ on a basis of the BRST method with complete, $Q$, and incomplete, $Q_c$, BRST operators. We found unconstrained (with $Q$) and constrained (with $Q_c$ and off-shell BRST invariant holonomic constraints) gauge Lagrangian formulations with different configuration spaces and reducibility stages. The deformation procedure to construct interacting gauge model with mixed-antisymmetric fields is proposed.

Figures (1)

• Figure 1: Cubic vertex for massive $\Phi^{(3)}_{\mu[s_1], \nu[s_2],\rho[s_3]}$ of spin $s[3]$ and two massless fields $\Phi^{(i)}_{\mu[{\lambda^{(i)}_1}], \nu[{\lambda^{(i)}_2}],\rho[{\lambda^{(i)}_3}]}$ of helicities $\lambda[3]_i$, $i=1,2$. The terms in $"\ldots"$ correspond to the auxiliary fields from $|\chi^{(i)}\rangle_{s[3]_i}$