Electric Dipole Moments From Dark Sectors

We examine the sensitivity of electric dipole moments (EDMs) to new $CP$-violating physics in a hidden (or dark) sector, neutral under the Standard Model (SM) gauge groups, and coupled via renormalizable portals. In the absence of weak sector interactions, we show that the electron EDM can be induced purely through the gauge kinetic mixing portal, but requires five loops, and four powers of the kinetic mixing parameter $\epsilon$. Allowing weak interactions, and incorporating the Higgs and neutrino portals, we show that the leading contributions to $d_e$ arise at two-loop order, with the main source of $CP$-violating being in the interaction of dark Higgs and heavy singlet neutrinos. In such models, EDMs can provide new sensitivity to portal couplings that is complementary to direct probes at the intensity frontier or high energy colliders.


Introduction
Recently, significant attention has been paid to models of physics involving hidden (or dark) sectors, as explanations of empirical deficiencies of the Standard Model (SM), such as neutrino mass and dark matter. As a typical characteristic, such physics models may involve new degrees of freedom with a mass well below the electroweak scale. The primary assumption is that all new fields are neutral under SM gauge symmetries, implying in particular that the chiral electroweak SU(2) L × U(1) Y structure of the SM is maintained. The effective Lagrangian at the electroweak scale then takes the form with new UV physics described by a series of higher dimensional operators O d constructed from SM fields. Notably, new IR physics contained in L IR is highly constrained by this effective field theory (EFT) framework, as it must be UV-complete. The simplest SM-neutral IR hidden sector allows only new scalars S i , neutral fermions N i and/or new U(1) ′ gauge boson(s) A ′ µ to mediate direct interactions with the SM. Indeed, the only renormalizable, and thus UV-complete, interactions (or portals) for these fields with the SM can be written in the form possibly generalized to include multiple copies of these mediator fields, e.g. a complex extension of S charged under U(1) ′ etc . Once coupled to the SM through these channels, the IR hidden sector described by L hid can be almost arbitrarily complicated. Dark sector models are often best probed using high-precision low energy observables. As a prime example, electric dipole moments (EDMs) of atoms, molecules and nucleons have for many years provided important indirect constraints on CP-violating new physics at or above the electroweak scale [1]. In this work, motivated by recent progress in the measurement of the electron EDM L = − i 2 d eē σ µν γ 5 eF µν at the ACME (|d e | ≤ 1.1 × 10 −29 ecm) [2], we revisit the generation of the EDM from dark sectors. The impact on the EDM of adding dark sector degrees of freedom coupled through the scalar, vector and neutrino portals has been studied for some time in the literature [3,4,5]. It was found, however, that the contributions to the electron EDM are generically suppressed well below the level of current sensitivity, |d e | < 10 −33 ecm [4], due in part to the small neutrino mass scale or strong limits on the gauge kinetic mixing ε. Here, we identify a new mediation channel, which we term the singlet portal, involving both the neutrino and scalar portals, which can induce a sizeable electron EDM. The singlet portal allows EDM contributions which avoid the primary suppression factors noted above. We will make the assumption that CP phases in the hidden sector are maximal, and determine the scale of the induced EDMs, given the restrictions already in place on the portal couplings from a variety of other experimental probes.

EDM contributions analyzed so far
In this section, for comparison with a new contribution in Section 3, we will present two (suppressed) contributions to EDMs from a generic dark sector, coupled to the SM only via the three portal interactions. 1

Neutrino portal
The neutrino portalLHN allows for CP-violation in the portal interaction Y N itself. As the simplest seesaw model for neutrino mass generation, and for leptogenesis, the neutrino portal has been studied extensively, including its contribution to lepton EDMs. In order to incorporate a nontrivial CP phase, we require two singlet fermions N R , N S leading to the following mass matrix where m D i are the Dirac masses and we work in the regime Fig. 1 (left), an EDM is generated at two loop order [4,5], The ratios θ ν ∼ m D i /M ≲ 10 −1 are the visible-hidden mixing angles and, even with considerable tuning, the constraints on the light neutrino mass spectrum limit the EDM to less than 10 −33 e · cm, well below the current experimental limit.

Dark Barr-Zee mechanism
As noted in [5], combining the scalar and vector portals allows the generation at 2-loop order of the 'dark EDM' of a SM fermion such as the electron, Since A ′ µ is generically massive, integrating it out in the latter case generates a higher-dimension 'EDM radius' (or Schiff moment), rather than an EDM, As an explicit example to generate the dark EDM operator, a generic Barr-Zee-type contribution is shown in Fig. 1 (middle) [5]. This involves the CP-odd coupling of the scalar S to a dark sector fermion ψ,ψ(m ψ + S(Y S + iỸ S γ 5 ))ψ, which is in turn charged under A ′ µ . Integrating out ψ leads to a CP-odd SF ′F ′ operator, which in turn can generate the dark EDM of an electron. The contributing diagrams were analyzed in [5] and, with the hierarchy m A ′ ≪ m S ≪ m ψ and Higgs-scalar mixing angle θ h ≪ 1, one finds the EDM radius,

PoS(LeptonPhoton2019)110
EDMs From Dark Sectors Shohei Okawa which is still well below the current sensitivity to the electron EDM, due in part to the strong limit on ε in the relevant A ′ µ mass range from g − 2 of the electron and direct searches for dark photon at NA64 and BaBar.

EDMs from the singlet portal
In this section we shall focus on a different channel which appears to provide the largest EDM contribution, given current constraints on the portal mixing angles. If we combine the neutrino and singlet scalar portals, a combination that we term the singlet portal, a new 2-loop contribution to EDMs can be identified as shown in Fig. 1 (right). We have introduced a further pseudoscalar dark sector coupling λ N SNiγ 5 N between S and N, which therefore breaks CP in the full theory and allows for EDMs at 2-loop order. This mediation channel does not require any additional dark sector degrees of freedom, or multiple generations of fermions. The addition of the scalar portal allows a number of the suppression factors impacting Fig. 1 (left) to be avoided, and this diagram is parametrically quite large, given the limited constraints on the neutrino and scalar mixing angles. Most importantly, N can be chosen to be a Dirac particle, and thus unconstrained by the visible neutrino mass splitting.
In this section, we will show the size of the new 2-loop EDM contribution, and explore the complementarity of the EDM sensitivity to other direct collider and fixed target probes. Details of the EDM calculation will not be given here, but are summarized in Section 3 and Appendix A of [6]. The calculation can be performed at leading order in the weak interactions (α W → 0), utilizing only the Goldstone components G ± of the weak vector bosons. After electroweak symmetry breaking, the relevant portal couplings take the form, is the scalar mixing angle, while θ ν is the corresponding singlet neutrino mixing angle. From the structure of the diagram in Fig. 1 (right), the characteristic length scale L scale e of the 2-loop contribution to d e takes the form,

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EDMs After a lengthy calculation, the corresponding amplitude can be written in terms of the overall length scale (3) and the loop momenta q and k, so that where f scalar is a dimensionless scalar integrand. Numerical results in two scaling regimes for m S are shown in Fig. 2  .
This is quite close to the current experimental limit from ACME of 1.1 × 10 −29 ecm, assuming relatively mild constraints on the mixing angles, and is the largest EDM contribution we have uncovered according to our definition of a UV-complete dark sector. The full scaling of the EDM with m N and m S is illustrated in Fig. 3 for two different portal mass regimes, where we also exhibit the leading direct constraints on the mixing angles for comparison. The constraints on the neutrino mixing angle are from a CHARM search for heavy neutral leptons, DELPHI searches for Z → Nν, an ALEPH measurement of W -pair production, and precision electroweak data (EWPD) including lepton universality. The relevant limits in each mass range are combined with limits on Higgs-scalar mixing from an L3 search for e + e − → Z * S, to produce the limit contours in Fig. 3. For m S ≪ m W , the limit on Higgs mixing is θ h ∼ 0.1, while it weakens to O(1) for larger m S . This results in the slightly weaker direct limits shown in Fig. 3 (right) for larger masses. Note that the stringent limits on lepton flavour violating processes cannot be directly applied to θ ν without further assumptions about the flavour structure of the N-sector.

Conclusions
The majority of the EDM contributions from dark sector CP-violation are below current experimental sensitivity, primarily due to the significant constraints on the portal couplings from independent measurements. Nonetheless, we have identified contributions from the singlet N − S portal that can be sizeable, and it is apparent that EDMs can provide sensitivity to the portal mixing angles that is complementary for large m N due to the mild decoupling behaviour. Moreover, the current limit on the electron EDM from the ACME experiment already provides sensitivity to this model that is comparable in reach to collider probes. Should the next round of improvements (by ACME and/or other collaborations searching for d e ) lead to a positive detection, one would not be able to unambiguously assign it to models of new physics with charged particles. It could also be a signature of neutral dark sectors near the weak scale. For models with electroweak scale singlet fermions N, the upcoming high-luminosity run of the LHC may provide the best probe.