Perturbative Unitarity Calls for An Action

TL;DR

This work investigates whether a perturbatively defined S-matrix can exist for non-Lagrangian, third-way $p$-form theories in $d$ dimensions. Using the perturbiner method, the authors show that the original third-way equations violate tree-level unitarity and identify a unique deformation with $H = dC + 2 \alpha \kappa \tilde{H} \wedge C$ that restores unitarity at $\alpha=1$, corresponding to a higher-dimensional Freedman-Townsend (FT) action. The modified theory reproduces the FT EOM and reveals that conserved higher-form currents in the third-way setup descend from FT currents, indicating a higher-form global symmetry and brane-like charged objects. The results strongly suggest that unitarity in this class of non-Lagrangian theories enforces an action-based (FT) description at the perturbative level and motivate further exploration of higher-form symmetries and non-Lagrangian dynamics, including extensions to lower-ranked forms and their physical implications.

Abstract

In this work, we investigate the consistency of a perturbative definition of the S-matrix in a particular class of non-Lagrangian theories. We focus on the $p$-form theories proposed in \cite{Broccoli:2021pvv}, which are fully defined by "third-way" consistent equations of motion. Using the perturbiner method, we show that the unitarity is absent even at the tree level. We then pin down a unique modification of the equations of motion that restores unitarity. The trade-off is the reinstatement of an underlying Lagrangian, which we recognize as the higher-dimensional generalization of the Freedman-Townsend (FT) model. Finally, we discuss conserved currents in third-way theories and show they all follow from parent currents in the FT model. In particular, we point out the existence of a higher-ranked global symmetry, which signals that the FT model is compatible with the existence of brane-like charged objects in higher dimensions.