Bumblebee models

Bumblebee models are effective field theories describing a vector field with a vacuum expectation value that spontaneously breaks Lorentz symmetry. A bumblebee model is the simplest case of a theory with spontaneous Lorentz symmetry breaking.

The development of bumblebee models was motivated primarily by the discovery that mechanisms in string theory (and subsequently other quantum theories of gravity) can lead to tensor-valued fields acquiring vacuum expectation values. Bumblebee models are different from local U(1) gauge theories. Nevertheless, in some bumblebee models, massless modes that behave like photons can appear.

Alan Kostelecký and Stuart Samuel showed in 1989 that mechanisms arising in the context of string theory can lead to spontaneous breaking of Lorentz symmetry. A set of models at the level of effective field theory were defined that contained gravitational fields and a vector field B_µ that has a nonzero vacuum expectation value, <B_µ> = b_µ. These have become known as bumblebee models.

Typically in these models, spontaneous Lorentz violation is caused by the presence of a potential term in the action. The vacuum value b_µ, along with a background metric, give a solution that minimizes the bumblebee potential.

The vacuum value b_µ acts as a fixed background field that spontaneously breaks Lorentz symmetry. It is an example, for the case of a vector, of a coefficient for Lorentz violation as defined in the Standard-Model Extension.

The name bumblebee model, coined by Kostelecký, is based on an insect whose ability to fly has sometimes been questioned on theoretical grounds, but which nonetheless is able to fly successfully.

Different examples of bumblebee Lagrangians can be constructed. Their expressions include kinetic terms for the gravitational and bumblebee fields, a potential V that induces spontaneous Lorentz breaking, and matter terms. In addition, there can be couplings between the gravitational, bumblebee, and matter fields.

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