Electron Electric Dipole Moment induced by Octet-Colored Scalars

An appended sector of two octet-colored scalars, each an electroweak doublet, is an interesting extension of the simple two Higgs doublet model motivated by the minimal flavor violation. Their rich CP violating interaction gives rise to a sizable electron electric dipole moment, besides the quark electric dipole moment via the two-loop contribution of Barr-Zee mechanism.


Introduction
The flavor diversity in the electroweak interaction continues to attract theoretical curiosity for new physics beyond the standard model. Pioneer work of Ref. [1] has shown a natural mechanism to suppress the unwanted neutral flavor changing process mediated by the Higgs exchange. Recent activities address the general structure of the minimal flavor violation (MFV) [2]. It is noticed that octet-colored scalars are able to respect MFV. The general Yukawa interaction is given by where Q L refers to the three families of left-handed quark doublets, and d R and u R the three families of the up-type and down-type right-handed quark singlets. The new set of octet-colored scalars O u and O d couple to quarks in an analogous fashion of the usual Higgs φ u and φ d by the Yukawa coupling matrices y u and y d with additional proportional coefficients η u and η d , as well as the color generator T a . All Higgs bosons φ u,d and O u,d carry the weak hypercharge Y = 1 2 , in the convention Q = T 3 + Y . The usual conjugation operation φ = iσ 2 φ * is adopted.
All interactions respect a Z 2 -symmetry, in which down-type fields φ d , O d , d R are odd, The symmetry is used to keep the desirable pattern of interaction that obeys the rule of the minimal flavor violation (MFV).
The complex coefficients η u and η d can produce CP violating vertices for the electric dipole moment (EDM) of the neutron. However the neutron EDM is masked with uncertainty of the hadronic matrix evaluations. It is interesting to ask the question of any contribution to the electron EDM, reached by ongoing and future experiments [3]. It turns out that one-loop or two-loop contributions to the electron EDM are absent if there is only one octet-colored scalar electroweak doublet. Nonetheless, in this article we show that with two of these octet-colored scalars, the electron EDM occurs through an amplitude A → O + O − of the pseudoscalar neutral Higgs boson A into a pair of octet-colored charged scalars. Then, a CP even amplitude for A → γγ appear even though A is CP-odd. As A and one of the photon touch down to an electron line, the electron EDM emerges via  mechanism in an overall two-loop diagram, as shown in Fig. 1. In the past, the electron EDM via Barr-Zee two-loop mechanism has been exhaustively studied in scenarios of multi-Higgs doublets [5,6] (without octet-colored scalars) or supersymmetric particles [7,8]. Our current study uses similar bosonic contribution as in [7,9].
We have found that the present limit of the electron EDM has already restricted the parameter space of CP violation of the octet-colored scalars.

Higgs potential and AOO coupling
As we know that if a CP-odd scalar couples to a conjugated pair of bosons, CP conservation is violated, just like the case in K L → ππ. We are going to find such a coupling in the MFV scenario with two colorless doublet Higgs φ u , φ d , and two octet-colored Higgs Rotating the basis of colorless Higgs, we achieve that φ 1 = 0 as tan β = φ u / φ d . The relevant vertices, allowed by the exact We start with the first term We single out the pseudoscalar Higgs A from Im(φ 1 ), Including the η ′ coupling, we combine results and obtain Now We look at the charged components of the color octets. The light and heavy mass states O ℓ and O h are given by the diagonalization of the mass matrix where M 2 uu and M 2 dd are real, and the complex off-diagonal mass squared We can absorb the phase ξ of (η ′ − η) ≡ |η ′ − η|e iξ into the field O u . Then AOO coupling matrix becomes real and antisymmetric, but the remaining phase ∆−ξ in the off-diagonal mass squared is genuine and not removable. The unitary diagonalization transformation is usually complex, where the 2 × 2 unitary matrix is denoted as U.
Note that diagonal entries arise due to the CP violating phase δ = ξ − ∆. Traceless is a consistent check. We denote |η ′ −η| ≡ λ. The relevant diagonal AOO interaction becomes The interaction can produce the electron EDM via Barr

Barr-Zee Amplitude
The CP violating short distance physics induces the effective vertex of the electric dipole d f of the fermion f coupled with the electromagnetic field strength F µν , The two-loop Barr-Zee diagram also involves the pseudoscalar coupling of A to fermions, where the electron shares similar pattern of the Yukawa coupling as the down quark.
Evaluating the Barr-Zee diagram, we obtain Eight color channels of O have been summed. The relevant function is defined by which has asymptotic behaviors, for z ≫ 1 , .
The electron EDM is evaluated as the difference between two contributions from the light and heavy color octets O ℓ and O h . We illustrate in Fig. 2 We can also work out [11] the associated color dipole moment (CDM) d C q of the quark q in the effective vertex by replacing color factors and couplings. On the other hand, Ref. [2], has studied the contribution of the color octet to the gluon dipole [12], known as the Weinberg operator [13], which is subjected to large renormalization suppression [14,15,16].
Due to the sophisticated hadron physics, we only have a qualitative relation between the neutron EDM and the CP violating coefficients, mainly based on naive dimensional analysis [17]. On the contrary, the electron EDM is well predicted because it is independent of the hadronic structure.

Conclusion
The octet-colored scalars if exist will be copiously produced [2,18,19,20] by LHC in coming years because of its strong interaction and its larger color representation. When their masses and couplings are determined, the electron EDM measurement is the key to disclose its nature of the CP violation. We show in this paper that the electron EDM measurement is sensitive to the CP violating phase in the octet-colored scalars.
This work was supported in part by U.S. Department of Energy under grant number DE-FG02-84ER40173.