Effective two-loop background contributions to $g_e-2$

Recent work has shown the utility of examining background effects from ultralight dark matter on precision measurements. This effect enhances seemingly benign contributions to the level that experiments are sensitive to them. In this work, we examine the consequences of the effective axion-like particles coupling to photons in a background of axion-like particle dark matter. This analysis leads to new constraints on the product of the axion electron and axion photon coupling. Furthermore, since the axion photon coupling is generated, at one loop, by a pseudoscalar Yukawa coupling, this calculation can also be applied to constraining the axion-like particles coupling to the electron at the two-loop level. This contribution often dominates due to the fact that the one-loop contribution to the anomalous magnetic moment of the electron from a pseudoscalar is momentum suppressed.