Interacting massive/massless continuous-spin fields and integer-spin fields

TL;DR

The paper develops a light-cone gauge vector (vector-superspace) formulation for interacting massive/massless continuous-spin fields alongside integer-spin fields in flat space, building parity-even cubic vertices and cross-interaction vertices for dimensions $d>4$. It demonstrates substantial simplifications relative to the oscillator approach, yielding rational or exponential vertex structures and clarifying which vertices are local versus non-local, including several distributional solutions. A comprehensive map between the vector formulation, the oscillator formulation, and Lorentz-covariant constraints is provided, along with an explicit demonstration of their equivalence in the free theory. The results illuminate the rich structure of CSF interactions, reveal pervasive non-locality in many vertices, and lay groundwork for future work on higher-point amplitudes, supersymmetric restrictions, and extensions to AdS backgrounds. Overall, the work advances practical construction of CSF interactions and clarifies the landscape of allowable cubic couplings in higher-spin theories.

Abstract

In the framework of light-cone gauge approach, interacting continuous-spin fields and integer-spin fields propagating in flat space are studied. The continuous-spin fields are considered by using a light-cone gauge vector superspace formulation. Description of massive continuous-spin fields associated with the principal, complementary and discrete series is presented. For the massive continuous-spin fields of the principal and complementary series and massless continuous-spin fields, all parity-even cubic vertices realized as functions on the light-cone gauge vector superspace are obtained. Cubic vertices for a cross-interaction of massive/massless continuous spin fields and massive/massless integer-spin fields are also derived. These results for cubic vertices are complete for the dimensions of space-time greater than four. The use of the light-cone gauge vector superspace formulation considerably simplifies the cubic vertices as compared to the ones of oscillator formulation. Some cubic vertices realized as distributions are also found. Map between the oscillator formulation and the vector superspace formulation of the continuous-spin fields is explicitly described. An equivalence of the light-cone gauge and Lorentz covariant formulations of the free continuous-spin fields is also demonstrated.