Lorentz violating backgrounds from quadratic, shift-symmetric, ultralight dark matter

We consider an effective theory for a shift-symmetric, quadratically-coupled, ultralight spin-0 field. The leading CP conserving interactions with Standard Model fields in the effective theory arise at dimension 8. We discuss the renormalization group evolution and positivity bounds on these operators, as well as their possible UV origins. Assuming that the spin-0 field is associated with an ultralight dark matter candidate, we discuss the effects of the dimension-8 operators on experiments searching for the oscillation of fundamental constants and Lorentz violation. We find that the direct bounds on these two effects are of similar strength but rather weak, corresponding to a UV cutoff scale of keV order, as they are mediated by dimension-8 operators.