  New physics thresholds which can modify the diphoton and dilepton Higgs
branching ratios significantly may also provide new sources of CP and lepton
flavor violation. We find that limits on electric dipole moments impose strong
constraints on any CP-odd contributions to Higgs diphoton decays unless there
are degeneracies in the Higgs sector that enhance CP-violating mixing. We
exemplify this point in the language of effective operators and in simple
UV-complete models with vector-like fermions. In contrast, we find that
electric dipole moments and lepton flavor violating observables provide less
stringent constraints on new thresholds contributing to Higgs dilepton decays.
