Spin 3 cubic vertices in a frame-like formalism

TL;DR

The paper shows that frame-like variables dramatically simplify non-minimal cubic vertices for massless higher-spin fields in flat space. By focusing on a massless spin-3 particle interacting with lower spins and employing a modified 1 and 1/2 order formalism, it derives explicit Lagrangians and gauge transformations for six vertices (3-0-0, 3-1-1, 3-2-2, 3-3-2, 3-2-1, 3-3-1) and confirms the gauge algebra closes in each case; the minimal derivative counts align with the expected $n = s_1 + s_2 - s_3$. The framework replaces higher derivatives on physical fields by auxiliary and extra fields, and uses antisymmetric form products to keep calculations manageable and naturally extendable toward $(A)dS$ spaces. Overall, the work demonstrates that frame-like formalism can streamline the construction and verification of higher-spin interactions, with implications for systematic generalizations to larger multiplets and curved backgrounds.

Abstract

Till now most of the results on interaction vertices for massless higher spin fields were obtained in a metric-like formalism using completely symmetric (spin-)tensors. In this, the Lagrangians turn out to be very complicated and the main reason is that the higher the spin one want to consider the more derivatives one has to introduce. In this paper we show that such investigations can be greatly simplified if one works in a frame-like formalism. As an illustration we consider massless spin 3 particle and reconstruct a number of vertices describing its interactions with lower spin 2, 1 and 0 ones. In all cases considered we give explicit expressions for the Lagrangians and gauge transformations and check that the algebra of gauge transformations is indeed closed.