We introduce a new Lorentz-violating modification to a scalar quantum field theory. This interaction, while super-renormalizable by power counting, is fundamentally different from the interactions previously considered within the Lorentz-violating standard model extension. The Lagrange density is nonlocal, because of the presence of a Hilbert transform term; however, this nonlocality is also very weak. The theory has reasonable stability and causality properties and, although the Lorentz-violating interaction possesses a single vector index, the theory is nonetheless CPT even. As an application, we analyze the possible effects of this new form of Lorentz violation on neutral meson oscillations. We find that under certain circumstances, the interaction may lead to quite peculiar sidereal modulations in the oscillation frequency.

10 pages