Cubic interaction vertices for fermionic and bosonic arbitrary spin fields

TL;DR

<3-5 sentence high-level summary>We present a comprehensive light-cone gauge construction of parity-invariant cubic vertices for bosonic and fermionic fields of arbitrary spin in flat space with $d \ge 4$. By solving the Poincaré algebra constraints and applying the light-cone dynamical principle, we derive generating functions for massless and massive cubic vertices across totally symmetric and mixed-symmetry representations, with explicit spin- and derivative-count conditions that classify allowable couplings. The framework yields practical results, including simple light-cone expressions for Yang-Mills and gravitational interactions of massive higher-spin fermions and a clear relation to manifestly Lorentz-covariant on-shell vertices, while clarifying when massless limits exist. Overall, the work provides a systematic, implementable toolkit for constructing and analyzing higher-spin cubic interactions in flat space and motivates extensions to AdS and covariant formulations.

Abstract

Using the light-cone gauge approach to relativistic field dynamics, we study arbitrary spin fermionic and bosonic fields propagating in flat space of dimension greater than or equal to four. Generating functions of parity invariant cubic interaction vertices for totally symmetric and mixed-symmetry massive and massless fields are obtained. For the case of totally symmetric fields, we derive restrictions on the allowed values of spins and the number of derivatives. These restrictions provide a complete classification of parity invariant cubic interaction vertices for totally symmetric fermionic and bosonic fields. As an example of application of the light-cone formalism, we obtain simple expressions for the Yang-Mills and gravitational interactions of massive arbitrary spin fermionic fields. For some particular cases, using our light-cone cubic vertices, we discuss the corresponding manifestly Lorentz invariant and on-shell gauge invariant cubic vertices.