In the standard model (SM), the coupling of the Higgs boson to electrons is real and very small, proportional to the electron mass. New physics could significantly modify both real and imaginary parts of this coupling. We discuss experiments which ar
e sensitive to the Higgs-electron coupling and derive the current bounds on new physics contributing to this coupling. The strongest constraint follows from the ACME bound on the electron electric dipole moment (EDM). We calculate the full analytic two-loop result for the electron EDM and show that it bounds the imaginary part of the Higgs-electron coupling to be less than 1.7 x 10^-2 times the SM electron Yukawa coupling. Deviations of the real part are much less constrained. We discuss bounds from Higgs decays, resonant Higgs production at electron colliders, Higgs mediated B -> e^+ e^- decays, and the anomalous magnetic moment of the electron. Currently, the strongest constraint comes from h -> e^+ e^- at the LHC, bounding the coupling to be less than ~600 times the SM Yukawa coupling. Important improvements can be expected from future EDM measurements as well as from resonant Higgs production at a next-generation high-luminosity e^+ e^- collider.
