Generalization of the 2-form interactions

TL;DR

This paper explores the construction of derivative self-interactions for massless and massive 2-forms, revealing limitations in gauge-invariant Galileon-like Lagrangians and identifying new non-minimal couplings in the massive case.

Contribution

It systematically constructs Galileon-like interactions for 2-forms, highlighting the constraints in the massless case and introducing novel non-minimal couplings for the massive case.

Findings

No non-minimal gauge-invariant couplings for massless 2-forms in 4D.

Construction of L2 and L4 Galileon-like Lagrangians for massive 2-forms.

Identification of non-minimal couplings between 2-forms and the double dual Riemann tensor.

Abstract

We systematically construct derivative self-interactions for massless and massive 2-forms. There exists a no-go theorem in the literature for constructing Galileon-like Lagrangians in four dimensions for the 2-form with gauge invariance, the Kalb-Rammond field. The presence of non-minimal couplings strongly relies on the contraction with divergenceless tensors. In four dimensions these are the Einstein tensor and the double dual Riemann tensor. Even though they are divergenceless on their own, their combination ceases to be. In the case of massless 2-forms we are not able to establish non-minimal couplings of the 2-form to the gravity sector with second order equations of motion due to the impossibility of building consistent combinations of divergenceless tensors. Using the systematical construction in terms of the Levi-Civita tensor, we aim at constructing Galileon-like derivative self-interactions for the massive 2-form. Apart from $L_2$ and $L_4$ we are not able to construct further Galileon-like Lagrangians. For the massive case, an important non-minimal coupling between the 2-form and the double dual Riemann tensor arises, which receives additional support from the decoupling limit. Promoting the interactions in $L_4$ requires the presence of appropriate non-minimal couplings and we give concrete examples for this.