Dimensional reduction of the $CPT$-even electromagnetic sector of the standard model extension

The $CPT$-even Abelian gauge sector of the standard model extension is represented by the Maxwell term supplemented by $(K_F{)}_{\mu \nu \rho \sigma }{F}^{\mu \nu }{F}^{\rho \sigma }$, where the Lorentz-violating background tensor, $(K_F{)}_{\mu \nu \rho \sigma }$, possesses the symmetries of the Riemann tensor. In the present work, we examine the planar version of this theory, obtained by means of a typical dimensional reduction procedure to ($1+2$) dimensions. The resulting planar electrodynamics is composed of a gauge sector containing six Lorentz-violating coefficients, a scalar field endowed with a noncanonical kinetic term, and a coupling term that links the scalar and gauge sectors. The dispersion relation is exactly determined, revealing that the six parameters related to the pure electromagnetic sector do not yield birefringence at any order. In this model, the birefringence may appear only as a second order effect associated with the coupling tensor linking the gauge and scalar sectors. The equations of motion are written and solved in the stationary regime. The Lorentz-violating parameters do not alter the asymptotic behavior of the fields but induce an angular dependence not observed in the Maxwell planar theory.