Perturbative unitarity calls for an action

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>p</a:mi></a:math>-form theories proposed in the work by Broccoli [], 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 tradeoff 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 branelike charged objects in higher dimensions.