Non-Linear Obstructions for Consistent New General Relativity

We revisit the field content and consistency of the New General Relativity family of theories. These theories are constructed in a geometrical framework with a flat and metric-compatible connection, so the affine structure is entirely determined by the torsion. The action is given by a local and parity-preserving quadratic form of the torsion with three free parameters. It is well-known that a special choice of parameters gives an equivalent to General Relativity and that the spectrum of the general linear theory around Minkowski contains an additional 2-form field. It has been suggested that the viability of these theories at linear order requires the 2-form field to feature a gauge symmetry so that it describes a massless Kalb-Ramond field. In this work we revisit these previous results and compute the cubic interactions. We also obtain the decoupling limit of the theories and show that the required gauge symmetry for the 2-form at linear order cannot be extended to higher orders. This signals towards a pathological behaviour of these theories and singles out the equivalent of General Relativity as the only consistent New General Relativity theory with a stable Minkowski background that includes gravity.