  We present a class of cancellation mechanisms to suppress the total
contributions of Barr-Zee diagrams to the electron electric dipole moment
(eEDM). This class of mechanisms are of particular significance after the new
eEDM upper limit, which strongly constrains the allowed magnitude of
CP-violation in Higgs couplings and hence the feasibility of electroweak
baryogenesis (EWBG), were released by the ACME collaboration in 2013. We point
out: if both the CP-odd Higgs-photon-photon ($Z$ boson) and the CP-odd
Higgs-electron-positron couplings are turned on, a cancellation may occur
either between the contributions of a CP-mixing Higgs boson, with the other
Higgs bosons being decoupled, or between the contributions of a CP-even and a
CP-odd Higgs bosons. With the assistance of the cancellation mechanisms, a
large CP-phase in Higgs couplings with viable electroweak baryogenesis (EWBG)
is still allowed. The reopened parameter regions would be probed by the future
neutron, mercury EDM measurements, and direct measurements of Higgs
CP-properties at the LHC and future colliders.
