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 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.
Highlights
Automatic implementation of the Hessian condition, and argue that the number of acceptable Lagrangian structures satisfying the usual physical requirements is finite, up to arbitrary functions
With Aμ being a massive vector field, not subject to satisfy a U(1) invariance, and Fμν ≡ ∂μAν − ∂νAμ being the associated Faraday tensor. The generalization of this action can be made by considering all “safe” terms containing the vector field and its first derivative
The terms built in Eq (3.1) fall into two distinct categories, depending on how they behave under a U(1) gauge transformation. Those invariant under such transformations contracts all field derivative indices to one and only one Levi-Civita tensor, i.e. they take the form ǫμν−ǫρσ− · · · ∂μAν ∂ρAσ · · ·, which can all be equivalently expressed as functions of scalar invariants made out of the Faraday tensor
Summary
Let us first introduce the vector theory, the hypothesis and results obtained far. With Aμ being a massive vector field, not subject to satisfy a U(1) invariance, and Fμν ≡ ∂μAν − ∂νAμ being the associated Faraday tensor. The generalization of this action can be made by considering all “safe” terms containing the vector field and its first derivative. Proca field propagates only three degrees of freedom [38] These conditions are discussed in full depth in Refs. The first condition ensures that the model can be stable [39,40,41], while the second stems from the fact that a massive field of spin s propagates 2s + 1 degrees of freedom. Examining the decoupling limit of the theory, one recovers for the pure scalar part of the Lagrangian the exact requirements of the Galileon theory [1,2,3,4], and so this part of the Lagrangian must reduce to this well-studied class of model
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