Bumblebee models are effective field theories describing a vector field with a vacuum expectation value that spontaneously breaks Lorentz symmetry.[1][2][3][4] 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.[6] Bumblebee models are different from local U(1) gauge theories. Nevertheless, in some bumblebee models, massless modes that behave like photons can appear.
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References:
[1] Kostelecký, V. Alan; Samuel, S. (1989). "Gravitational phenomenology in higher-dimensional theories and strings". Physical Review D. 40 (6): 1886–1903.
[2] Kostelecký, V. Alan; Lehnert, Ralf (2001). "Stability, causality, and Lorentz and CPT violation". Physical Review D. 63 (6): 065008. arXiv:hep-th/0012060.
[3] Kostelecký, V. Alan (2004). "Gravity, Lorentz violation, and the standard model". Physical Review D. 69 (10): 105009. arXiv:hep-th/0312310.
[4] Bailey, Quentin; Kostelecký, V. Alan (2006). "Signals for Lorentz violation in post-Newtonian gravity". Physical Review D. 74 (4): 045001. arXiv:gr-qc/0603030.
[5] Övgün, Ali; Jusufi, Kimet; Sakalli, Izzet (2019). "Exact traversable wormhole solution in bumblebee gravity". Physical Review D. 99 (2019) no.2, 024042.
[6] Övgün, Ali; Jusufi, Kimet; Sakalli, Izzet (2018). "Gravitational lensing under the effect of Weyl and bumblebee gravities: Applications of Gauss–Bonnet theorem". Annals Phys. 399 (2018) 193-203.