Vanishing Contribution to Quark Electric Dipole Moment in the 2HD Model with CKM CP Violation

In the standard model (SM) of electroweak interactions, CP noninvariance arises from the nonzero phase in the CKM matrix. Its contribution to the quark electric dipole moment (EDM) vanishes surprisingly at two loop order. This makes the quark EDM extremely small in the SM. In this paper, we consider the two Higgs doublet extension of the SM and assume that CP noninvariance is still encoded in the CKM matrix. We calculate the charged Higgs boson contribution to the quark EDM which naively should be of order $eG_F^2\tilde{\delta}(4\pi)^{-4}m_{u(d)}m_t^2m_b^2m_H^{-2}$ for the up (down) quark with possible enhancement factors of $\tan^2\beta$. Here $\tilde{\delta}$ is the rephasing invariant of CP violation. However, contrary to the above naive expectation, we find that the charged Higgs boson contribution vanishes strictly at two loop order. We show explicitly how this comes about and explains how it is related to the general form of Yukawa couplings in a spontaneously broken gauge theory.