On the 4D generalized Proca action for an Abelian vector field
Erwan Allys (Institut d'Astrophysique de Paris, UMR 7095, UPMC Université Paris 6 et CNRS, 98 bis boulevard Arago, 75014 Paris, France); Juan P. Beltrán Almeida (Departamento de Física, Universidad Antonio Nariño, Cra 3 Este # 47A-15, Bogotá D.C. 110231, Colombia); Patrick Peter (Institut d'Astrophysique de Paris, UMR 7095, UPMC Université Paris 6 et CNRS, 98 bis boulevard Arago, 75014 Paris, France, Institut Lagrange de Paris, UPMC Université Paris 6 et CNRS, Sorbonne Universités, Paris, France); Yeinzon Rodríguez (Escuela de Física, Universidad Industrial de Santander, Ciudad Universitaria, Bucaramanga 680002, Colombia, Centro de Investigaciones en Ciencias Básicas y Aplicadas, Universidad Antonio Nariño, Cra 3 Este # 47A-15, Bogotá D.C. 110231, Colombia, Simons Associate at The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, I-34151, Trieste, Italy)
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 Stückelberg 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 Stückelberg field describing the longitudinal mode, which is in agreement with the results of JCAP 05 (2014) 015 and Phys. Lett. B 757 (2016) 405 and complements those of JCAP 02 (2016) 004. We also correct and complete the parity violating sector, obtaining an extra term on top of the arbitrary function of the field Aμ, the Faraday tensor Fμν and its Hodge dual F̃μν.
