# On the equivalence between bumblebee models and electrodynamics in a non-linear gauge

## TL;DR

This paper demonstrates that bumblebee models with a smooth potential are classically and quantum mechanically equivalent to standard electrodynamics in a specific non-linear gauge, clarifying a subtle theoretical point.

## Contribution

It rigorously establishes the classical and quantum equivalence between bumblebee models and electrodynamics in a non-linear gauge using Dirac's method and Feynman diagram analysis.

## Key findings

- Classical dynamics of bumblebee models matches electrodynamics in a non-linear gauge.
- Quantum Feynman amplitudes are indistinguishable between the two models.
- Feynman rules and propagators are identical in both frameworks.

## Abstract

Bumblebee models are effective field theories describing a vector field with a nonzero vacuum expectation value that spontaneously breaks Lorentz invariance. They provide an alternative way of exploring the similarities between theories with spontaneous Lorentz symmetry breaking and gauge theories. The equivalence between bumblebee models with suitable conditions and standard electrodynamics in a non-linear gauge $A_\mu A^\mu-b^2=0$ is taken for granted; however, this point is very subtle and has not yet been fully addressed. The main goal of this paper is to fill in this gap. More precisely, here we study the relation between a bumblebee model, with a smooth potential of the form $V(B _{\mu}) = V (B _{\mu} B ^{\mu} - b ^{2})$, and standard electrodynamics in the non-linear gauge $A _{\mu} A ^{\mu} - b ^{2} = 0$, both at the classical and quantum levels. Using the Dirac's method we show that after introducing Dirac brackets with suitable initial conditions, the classical dynamics of the bumblebee model corresponds to that of standard electrodynamics in the aforementioned non-linear gauge. In the quantum case we demonstrate that perturbative calculations of Feynman amplitudes to any physical process in each model are indistinguishable. To do this, we show that the Feynman rules and propagators of standard electrodynamics in the non-linear gauge and those describing the bumblebee model are the same.