Navigating Collinear Superspace

TL;DR

This paper develops a new collinear superspace framework for constructing $ =1$ supersymmetric Lagrangians focusing on propagating degrees of freedom, revealing residual gauge symmetries and obstructions to interactions.

Contribution

It introduces a novel collinear superspace formalism for $ =1$ theories, including superfield representations and light-cone reparametrization invariance, and addresses challenges in constructing interacting models.

Findings

Constructed collinear superspace Lagrangians for free chiral and non-Abelian gauge fields.

Identified a superfield that is simultaneously chiral, anti-chiral, and real, encoding residual gauge transformations.

Discussed fundamental obstructions to forming interacting theories with chiral matter.

Abstract

We introduce a new set of effective field theory rules for constructing Lagrangians with $\mathcal{N} = 1$ supersymmetry in collinear superspace. In the standard superspace treatment, superfields are functions of the coordinates $(x^\mu,\theta^\alpha, \theta^{\dagger \dot{\alpha}})$, and supersymmetry preservation is manifest at the Lagrangian level in part due to the inclusion of auxiliary $F$- and $D$-term components. By contrast, collinear superspace depends on a smaller set of coordinates $(x^\mu,\eta,\eta^\dagger)$, where $\eta$ is a complex Grassmann number without a spinor index. This provides a formulation of supersymmetric theories that depends exclusively on propagating degrees of freedom, at the expense of obscuring Lorentz invariance and introducing inverse momentum scales. After establishing the general framework, we construct collinear superspace Lagrangians for free chiral matter and non-Abelian gauge fields. For the latter construction, an important ingredient is a superfield representation that is simultaneously chiral, anti-chiral, and real; this novel object encodes residual gauge transformations on the light cone. Additionally, we discuss a fundamental obstruction to constructing interacting theories with chiral matter; overcoming these issues is the subject of our companion paper, where we introduce a larger set of superfields to realize the full range of interactions compatible with $\mathcal{N} = 1$. Along the way, we provide a novel framing of reparametrization invariance using a spinor decomposition, which provides insight into this important light-cone symmetry.