Interactions of a massless tensor field with the mixed symmetry of the Riemann tensor. No-go results

TL;DR

This paper proves that a massless tensor field with Riemann tensor symmetry cannot have consistent self-interactions or cross-couplings with certain other fields under common physical assumptions, except for a cosmological-like term.

Contribution

It establishes no-go theorems for interactions of a massless tensor field with Riemann symmetry, under assumptions of smoothness, locality, Lorentz invariance, and second-order derivatives.

Findings

No consistent self-couplings exist, only a cosmological-like term.

No cross-interactions with the Pauli-Fierz field are possible.

No non-trivial cross-couplings with matter fields that gain gauge transformations.

Abstract

Non-trivial, consistent interactions of a free, massless tensor field t_{\mu \nu |\alpha \beta} with the mixed symmetry of the Riemann tensor are studied in the following cases: self-couplings, cross-interactions with a Pauli-Fierz field and cross-couplings with purely matter theories. The main results, obtained from BRST cohomological techniques under the assumptions on smoothness, locality, Lorentz covariance and Poincar\'{e} invariance of the deformations, combined with the requirement that the interacting Lagrangian is at most second-order derivative, can be synthesized into: no consistent self-couplings exist, but a cosmological-like term; no cross-interactions with the Pauli-Fierz field can be added; no non-trivial consistent cross-couplings with the matter theories such that the matter fields gain gauge transformations are allowed.