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Stringent limits on the Myers-Pospelov timelike parameter for photons$\xi<10^{-15}$ coming from astrophysical tests suggest exploring more generalpreferred backgrounds, such as spacelike and lightlike. We take some steps inthis direction. We allow the external four vector $n$ characterizing theLorentz symmetry breaking to have arbitrary directions in spacetime. We studythe consistency for the effective field theories in each privileged frame bydemanding causality, stability and analyticity. When specializing to a timelikebackground we found runaway solutions leading to causality and unitarityviolations. We show that the lightlike theory is a higher-derivative theorywith more degrees of freedom and nonanalytic solutions leading to instabilitieswhen interactions are turned on. We demonstrate by explicit calculation thatboth stability and analyticity are preserved for the purely spacelike casewhile microcausality is highly suppressed. This new anisotropic model opens thepossibility to play a role in the search for Planck-scale effects.

Causality and stability for Lorentz-CPT violating electrodynamics with dimension-5 operators