Casimir effect of a doubly Lorentz-violating scalar in magnetic field

In this work, I study the Casimir effect caused by a charged and massive scalar field that violates Lorentz symmetry in an aether-like and CPT even manner, by direct coupling between the field derivatives and two fixed orthogonal unit vectors. I call this a “double” Lorentz violation. The field will satisfy either Dirichlet or mixed boundary conditions on a pair of plane parallel plates. I use the generalized zeta function technique to examine the effect of a uniform magnetic field, perpendicular to the plates, on the Casimir energy and pressure. Three different pairs of mutual directions of the unit vectors are possible: timelike and spacelike perpendicular to the magnetic field, timelike and spacelike parallel to the magnetic field, spacelike perpendicular and spacelike parallel to the magnetic field. I fully examine all of them for both types of boundary conditions listed above and, in all cases and for both types of boundary conditions, I obtain simple expressions of the zeta function, Casimir energy and pressure in the three asymptotic limits of strong magnetic field, large mass, and small plate distance.