Classical features, Anderson-Higgs mechanism, and unitarity in Lee-Wick pseudo-electrodynamics

In this paper, the dimensional reduction is applied to the Lee-Wick electrodynamics in which the classical sources are confined on a spatial plane. As result, the Lee-Wick pseudo-electrodynamics is achieved as a non-local electromagnetism defined in $1+2$ dimensions. The abelian Anderson-Higgs mechanism is so introduced in the Lee-Wick pseudo-electrodynamics through a complex scalar sector in $1+2$ dimensions, breaking spontaneously the $U(1)$-gauge symmetry of the non-local theory. As consequence, the pseudo-Lee-Wick field acquires a light mass, beyond the usual heavy Lee-Wick mass, that is a natural mass parameter of the theory. After the spontaneous symmetry breaking takes place, classical features of the theory are discussed, as the Proca-Lee-Wick pseudo-electrodynamics, with the field equations and conservation laws. The introduction of Lee-Wick fermions also is proposed, in which it opens the discussion of a viable Proca-Lee-Wick pseudo-quantum electrodynamics in $1+2$ dimensions. The unitarity at the tree level of the Lee-Wick pseudo-electrodynamics is discussed through the Optical theorem.