Renormalization of massless fields in the $(1,0)\oplus(0,1)$ representation

TL;DR

This work analyzes the renormalization of self-interacting massless fields transforming under the $(1,0)ot(0,1)$ representation of the Restricted Lorentz Group. It constructs a general, parity-invariant model that unifies the massless limits of the Klein-Gordon-like, Joos-Weinberg, and related formulations, and includes all independent dimension-4 self-interactions consistent with the representation. Using dimensional regularization and the MS scheme, the authors show the one-loop renormalizability of the class, compute the self-energy and vertex divergences, and derive the corresponding beta functions for the four quartic couplings, revealing distinct fixed-point structures for the two massless subcases $(\alpha,\beta)=(1,0)$ and $(0,1)$. The results establish a robust RG framework for massless high-spin fields in this representation and clarify when gauge-like issues arise, pointing to future work on gauge-invariant interactions at $|\alpha|=|\beta|$. The analysis provides concrete, extractable predictions for the running couplings and fixed points within this non-standard Lorentz representation, thereby contributing to the broader exploration of beyond-Standard-Model field content.

Abstract

We study the one-loop renormalization of self-interacting massless fields in the $(1,0)\oplus(0,1)$ representation of the Restricted Lorentz Group. We work with a general model that represents the entire class of parity-invariant self-interacting massless theories that can be defined in this representation. It consists of a general free Lagrangian that reproduces the massless limit of three theories previously studied in the literature: the Joos-Weinberg, the Shay-Good/Hammer-McDonald-Pursey, and the Klein-Gordon-like one, as particular cases; along with an interacting Lagrangian containing all the independent dimension-4 parity-invariant self-interactions available in this representation. The model is found to be renormalizable.