On the 4D generalized Proca action for an Abelian vector field

TL;DR

This paper reviews and extends the most general 4D Proca theory with second-order equations, focusing on the scalar sector, parity violation, and the finite set of derivative terms for the longitudinal mode.

Contribution

It proposes that the complete 4D Proca action contains only finitely many second-order derivative terms for the scalar mode, correcting and extending previous results.

Findings

Finite set of second-order derivative terms identified

Parity-violating sector corrected and extended

Consistent with previous scalar galileon and Proca theories

Abstract

We summarize previous results on the most general Proca theory in 4 dimensions containing only first-order derivatives in the vector field (second-order at most in the associated St\"uckelberg scalar) and having only three propagating degrees of freedom with dynamics controlled by second-order equations of motion. Discussing the Hessian condition used in previous works, we conjecture that, as in the scalar galileon case, the most complete action contains only a finite number of terms with second-order derivatives of the St\"uckelberg field describing the longitudinal mode, which is in agreement with the results of JCAP 1405, 015 (2014) and Phys. Lett. B 757, 405 (2016) and complements those of JCAP 1602, 004 (2016). We also correct and complete the parity violating sector, obtaining an extra term on top of the arbitrary function of the field $A_\mu$, the Faraday tensor $F_{\mu \nu}$ and its Hodge dual $\tilde{F}_{\mu \nu}$.