††institutetext: Instituto de Física Corpuscular, Universidad de Valencia and
CSIC, Edificio Institutos Investigación, C/Catedrático José Beltrán 2, 46980
Paterna, Spain

# Electric Dipole Moments as indirect probes of Dark Sectors

Marco Ardu  Moinul Hossain Rahat  Nicola Valori  and Oscar Vives
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Dark sectors provide beyond Standard Model scenarios which can address
unresolved puzzles, such as the observed dark matter abundance or the baryon
asymmetry of the Universe. A naturally small portal to the dark sector is
obtained if dark-sector interactions stem from a non-Abelian hidden gauge
group that couples through kinetic mixing with the hypercharge boson. In this
work, we investigate the phenomenology of such a portal of dimension five in
the presence of CP violation, focusing on its signatures in fermion electric
dipole moments. We show that, currently unbounded regions of the parameter
space from dark photon searches can be indirectly probed with upcoming
electron dipole moment experiments for dark boson masses in the range $1-100$
GeV. We also discuss two particular scenarios where a $SU(2)_{D}$ dark gauge
group spontaneously breaks into either an Abelian $U(1)_{D}$ or nothing. In
both cases, we show that potentially observable electron dipole moments can be
produced in vast regions of the parameter space compatible with current
experimental constraints and observed dark matter abundance.

##  1 Introduction

The Standard Model (SM) of particle physics, despite its remarkable success in
explaining a vast array of phenomena, remains incomplete. Crucially, it fails
to account for Dark Matter (DM) and the Baryon Asymmetry of the Universe
(BAU), highlighting the need for physics Beyond the Standard Model (BSM).

Hidden sectors offer rich scenarios for viable DM candidates [1, 2], and may
also introduce the necessary ingredients for baryogenesis [3, 4]. Given that
SM interactions arise from local gauge symmetries, it is plausible that dark
sector interactions could also stem from a hidden gauge group $G_{D}$. The
simplest gauge theories involve an Abelian group $G_{D}=U(1)_{D}$ [5], but
non-Abelian dark sectors have also attracted significant attention in DM model
building [6, 7, 8, 9, 10, 11]. DM stability could result from accidental
symmetries of the non-Abelian gauge theory, which may or may not be
spontaneously broken, and DM could also be the bound state of a confining
group [12, 13].

A natural candidate for a portal interaction connecting two gauge theories is
the kinetic mixing between the gauge bosons [14]. For an Abelian dark gauge
group mixing with the SM hypercharge, the stringent experimental constraints
require the coupling of the kinetic mixing $\kappa
F^{\mu\nu}F^{\prime}_{\mu\nu}$ to be small, even though such an operator is
gauge-invariant and dimension-four. Intriguingly, for a non-Abelian dark
sector, gauge invariance forbids the kinetic mixing at the renormalizable
level, which facilitates a more natural explanation of the smallness of the
interactions between the visible and the hidden sectors. Indeed, for a non-
Abelian dark group, kinetic mixing can only arise from an effective operator
of at least dimension five:

$\frac{1}{\Lambda}\Sigma^{a}G_{\mu\nu}^{a}{B}^{\mu\nu},$ (1)

where $\Sigma^{a}$ is an adjoint scalar field, and $G^{a}_{\mu\nu}$,
$B_{\mu\nu}$ are the dark and hypercharge gauge field strength tensors
respectively. The above operator is possible if heavy states with masses
$\sim\Lambda$ are charged under both hypercharge and the hidden gauge
group111Higher-dimensional operators are possible choosing different scalar
representations.. Following the spontaneous breaking of the $G_{D}$ group, the
scalar adjoint can acquire a vacuum expectation value (VEV), leading to the
kinetic mixing between the SM and hidden sector gauge bosons. The
phenomenology of this scenario has been extensively studied [15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 27]. Alternatively, the non-Abelian group may
remain unbroken and confine at low energy, with the above operator inducing
kinetic mixing between a composite dark photon state and the SM hypercharge
boson [28].

If the hidden sector contains sources of CP violation, which may be necessary
for baryogenesis, the CP-odd analogue of the operator in Eq. (1) is also
possible:

$\frac{1}{\Lambda}\Sigma^{a}G_{\mu\nu}^{a}\tilde{B}^{\mu\nu},$ (2)

where $\tilde{B}^{\mu\nu}$ is the dual hypercharge tensor
$\tilde{B}^{\mu\nu}=\epsilon^{\alpha\beta\mu\nu}B_{\alpha\beta}/2$. The
operators in Eqs. (1) and (2) arise from similar UV diagrams and, in the
presence of large CP-violating phases, have comparable coefficients. In this
paper, we study the phenomenology of the CP-odd portal and discuss its
complementarity with the more studied CP-even mixing. The phenomenology of a
similar operator has been explored in the case of an Abelian dark photon
mixing with the $SU(2)_{L}$ gauge bosons [29, 30]. Here, we focus on the
scenario where a non-Abelian dark gauge group kinetically mixes with the
hypercharge.

Notably, Electric Dipole Moments (EDMs) provide the most stringent tests of
CP-violating new physics. Assuming that the hidden sector group $G_{D}$ is
spontaneously broken by the adjoint scalar VEV, the operator of Eq. (2)
contributes to the EDM of SM fermions under very general assumptions. Given
the impressive experimental sensitivity of the electron EDM (eEDM) searches
[31], which is also expected to improve further in the future [32, 33, 34],
the eEDM arising from the operator of Eq. (2) could be detectable. In this
work, we assume the presence of such an operator and investigate the potential
of eEDM searches in probing a wide variety of dark sectors.

The paper is organized as follows. In Section 2 we discuss the main features
of the non-Abelian kinetic mixing and calculate, under some general
assumptions, the contribution to the electron EDM coming from the CP-odd
portal. We also explore the interplay between the dark photon searches and the
eEDM experiments in this context. In Section 3 we examine the specific case of
a $SU(2)_{D}$ group broken into either an Abelian sub-group $U(1)_{D}$ or
completely. In the former scenario, we find that the dark sector particles
acquire milli-charges that can be indirectly probed using EDMs. In the case of
a completely broken $SU(2)_{D}$ we implement an inelastic DM model, showing
that large eEDMs are predicted across a vast region of the parameter space
compatible with the observed DM abundance. Additional details regarding model
building are relegated to the Appendices A, B and C. Finally, in Section 4 we
summarize our results.

##  2 Non-Abelian kinetic mixing

We consider a hidden sector that is invariant under a non-Abelian gauge group
$G_{D}$, with SM particles transforming as singlets. We assume that $G_{D}$ is
spontaneously broken by the dark scalar sector, which contains at least one
scalar $\Sigma$ transforming in the adjoint representation of $G_{D}$. The
Lagrangian takes the following general form:

$\mathcal{L}=\mathcal{L}_{\rm SM}+\mathcal{L}_{\rm
DS}+\text{Tr}[(D_{\mu}\Sigma)^{\dagger}(D^{\mu}\Sigma)]-V(\Sigma)+\mathcal{L}_{\rm
SM-DS},$ (3)

where $\mathcal{L}_{\rm SM}$ is the SM Lagrangian, $\mathcal{L}_{\rm DS}$
contains the dark sector particles and their interactions, $V(\Sigma)$ is the
potential of the adjoint scalar $\Sigma$, and $\mathcal{L}_{\rm SM-DS}$
contains the interactions that connect the visible and hidden sectors. In the
presence of heavy states that have hypercharge and are also charged under
$G_{D}$, mixing between the SM and dark gauge bosons can arise at dimension
five via the operators

$\mathcal{L}_{\rm SM-DS}\supset-\frac{C}{\Lambda}\text{Tr}[\Sigma
X^{\mu\nu}]B_{\mu\nu}-\frac{\tilde{C}}{\Lambda}\text{Tr}[\Sigma
X^{\mu\nu}]\tilde{B}_{\mu\nu},$ (4)

where $X^{\mu\nu}=T_{a}X_{a}^{\mu\nu}$ is the $G_{D}$ non-Abelian field
strength tensor, defined as
$X_{a}^{\mu\nu}=\partial^{\mu}X_{a}^{\nu}-\partial^{\nu}X_{a}^{\mu}+g_{D}f_{abc}X_{b}^{\mu}X_{c}^{\nu}$.
Here, $T^{a}$ are the $G_{D}$ generators, $g_{D}$ is the hidden sector gauge
coupling, and $f_{abc}$ are the structure constants of the $G_{D}$ group.
$\Lambda$ is the UV cut-off and is related to the mass scale of the heavy
states that are integrated out to generate the operators above (see Appendix A
for a possible UV completion). When both terms of Eq (4) are present in the
Lagrangian, CP is broken because the contractions $X^{\mu\nu}B_{\mu\nu}$ and
$X^{\mu\nu}\tilde{B}_{\mu\nu}$ are respectively even and odd under CP.

If the scalar field $\Sigma$ acquires a VEV $\langle\Sigma\rangle=v_{a}T^{a}$,
upon spontaneous symmetry breaking, Eq. (4) becomes

$\mathcal{L}_{\rm SM-
DS}\supset-\frac{\epsilon_{a}}{2}X_{a}^{\mu\nu}B_{\mu\nu}-\frac{\epsilon_{a}}{2v_{a}}\phi^{a}X_{a}^{\mu\nu}B_{\mu\nu}-\frac{\tilde{\epsilon}_{a}}{2}X_{a}^{\mu\nu}\tilde{B}_{\mu\nu}-\frac{\tilde{\epsilon}_{a}}{2v_{a}}\phi^{a}X_{a}^{\mu\nu}\tilde{B}_{\mu\nu},$
(5)

where $\phi^{a}\equiv\Sigma^{a}-v_{a}$, we have assumed that the $G_{D}$
generators are normalized as ${\rm Tr}(T^{a}T^{b})=\delta^{ab}/2$, and we have
defined $\epsilon_{a}=Cv_{a}/\Lambda$ and
$\tilde{\epsilon}_{a}=\tilde{C}v_{a}/\Lambda$.

###  2.1 Gauge sector

The gauge boson quadratic terms, following the spontaneous breaking of the
electroweak $SU(2)_{L}\times U(1)_{Y}$ and dark gauge groups, are

$\displaystyle\mathcal{L}_{\rm gauge}$ $\displaystyle\supset$
$\displaystyle-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{4}Z^{\mu\nu}Z_{\mu\nu}-\frac{1}{4}X^{\mu\nu}_{a}X_{\mu\nu}^{a}+\frac{1}{2}m_{Z}^{2}Z^{\mu}Z_{\mu}+\frac{1}{2}M^{2}_{ab}X^{a}_{\mu}X^{b\mu}$
(6) $\displaystyle-$
$\displaystyle\frac{\epsilon_{a}}{2}c_{\theta}X_{a}^{\mu\nu}F_{\mu\nu}+\frac{\epsilon_{a}}{2}s_{\theta}X_{a}^{\mu\nu}Z_{\mu\nu},$

where we have added the kinetic mixing terms coming from Eq. (5), and
$c_{\theta}$ and $s_{\theta}$ are the cosine and sine of the Weinberg angle.
The CP-odd mixing term $\tilde{\epsilon}_{a}X_{a}^{\mu\nu}\tilde{B}_{\mu\nu}$
is not included because its contribution to the two-point function is a total
derivative.

Although a complete description of the gauge boson mixing would require
specifying the SSB of the dark gauge symmetry in detail, to illustrate the
pheonomenological consequence of the effective operators we consider only one
dark gauge boson $X$ with a mass $M_{X}$ that mixes with the $B$ field with a
kinetic mixing parameter $\epsilon$. This leads to the following rotation
matrix

$\begin{pmatrix}A\\\ Z\\\
X\end{pmatrix}=\begin{pmatrix}1&0&-c_{\theta}\epsilon\\\
0&1&-\frac{s_{\theta}\epsilon M_{X}^{2}}{M_{Z}^{2}-M_{X}^{2}}\\\
0&\frac{s_{\theta}\epsilon M_{Z}^{2}}{M_{Z}^{2}-M_{X}^{2}}&1\\\
\end{pmatrix}\begin{pmatrix}A^{\prime}\\\ Z^{\prime}\\\
X^{\prime}\end{pmatrix},$ (7)

where the primed fields correspond to the mass eigenstates, as can be seen if
we replace this transformation in Eq. (6), and $M_{Z}$ is the $Z$ boson mass.
As a result of the rotation, the dark gauge boson couples to the
electromagnetic and $Z$ currents with an $\epsilon$-suppressed coupling,

$\mathcal{L}_{\rm gauge}\supset-X^{\prime}_{\mu}(c_{\theta}\epsilon
eJ^{\mu}_{\rm EM}+\epsilon e\tan\xi J^{\mu}_{Z}),$ (8)

where $\tan\xi\equiv M^{2}_{X}/(M_{Z}^{2}-M_{X}^{2})$ and the $Z$ current is
defined as $J^{\mu}_{Z}\equiv(J^{\mu}_{3}-s_{\theta}^{2}J^{\mu}_{\rm
EM})/c_{\theta}$, with $J^{\mu}_{3}$ being the current of the $T_{3}$
generator of $SU(2)_{L}$. In addition, the physical $Z$ boson state can couple
to the dark sector particles via an $\epsilon$-dependent coupling, which can
be probed by electroweak precision observables and rare $Z$ decays [35, 36].

###  2.2 Scalar sector

The scalar sector responsible for the spontaneous breaking of the dark gauge
group depends on the specifics of the hidden sector. We will discuss some
particular models in Section 3, but for now we assume the presence of a single
scalar transforming as an adjoint of $G_{D}$, which can interact with the SM
Higgs via the quartic-coupling

$\mathcal{L}_{\rm SM-DS}\supset\lambda(H^{\dagger}H){\rm
Tr}(\Sigma^{\dagger}\Sigma).$ (9)

Once the Higgs and dark scalar acquire VEVs, the quartic portal induces mixing
between the Higgs field ($h\equiv H-\langle H\rangle$) and the dark scalars
($\phi_{a}\equiv\Sigma_{a}-\langle\Sigma_{a}\rangle$). While in general
several scalar states could mix, we simplify the analysis by considering the
case where only one scalar, $\phi$, mixes significantly with the physical
Higgs, and focus on the sub-matrix

$\begin{pmatrix}h\\\ \phi\end{pmatrix}\simeq\begin{pmatrix}1&-\beta\\\
\beta&1\\\ \end{pmatrix}\begin{pmatrix}h^{\prime}\\\
\phi^{\prime}\end{pmatrix}.$ (10)

Here, the primed fields are mass eigenstates, and we expanded at leading order
in the mixing angle $\beta$, which is constrained to be small by collider
searches [37]. If $\phi^{\prime}$ is heavier than $m_{h}$/2, ATLAS [38] and
CMS [39] constrain the mixing of a singlet scalar with the Higgs to be
$\sin\beta$ < 0.27. If instead $\phi^{\prime}$ is light enough to allow the
decay of the SM Higgs into a pair of new singlet scalars, a stronger bound
($\beta\lesssim 10^{-2}$) is determined by the searches of the Higgs invisible
decay branching ratio [40, 41].

###  2.3 Electric dipole moments from the CP-odd kinetic mixing

After rotating the fields in the mass eigenstate basis, the Lagrangian
contains the following interactions222For notational convenience we drop the
prime in the mass eigenstate fields in the following.

$\displaystyle\mathcal{L}\supset$ $\displaystyle-$
$\displaystyle\frac{\tilde{\epsilon}}{2v_{D}}(\phi+\beta
h)\left(c_{\theta}X^{\mu\nu}\tilde{F}_{\mu\nu}-s_{\theta}X^{\mu\nu}\tilde{Z}_{\mu\nu}+c_{\theta}s_{\theta}\frac{\epsilon
M_{Z}^{2}}{M_{Z}^{2}-M^{2}_{X}}Z^{\mu\nu}\tilde{F}_{\mu\nu}\right)$ (11)
$\displaystyle-$
$\displaystyle\sum_{f}\left(Y_{f}\frac{h}{\sqrt{2}}\bar{f}f-Y_{f}\beta\frac{\phi}{\sqrt{2}}\bar{f}f\right)-X_{\mu}(c_{\theta}\epsilon
eJ^{\mu}_{\rm EM}+\epsilon e\tan\xi J^{\mu}_{Z}),$

where $v_{D}$ is the dark scalar VEV and the sum over $f$ run over all SM
fermions with a Yukawa coupling $Y_{f}$. The terms in the first line of Eq.
(11) are CP-odd and can contribute to CP-violating observables.

Figure 1: One-loop contribution to fermion dipole moments.

For instance, the diagram in Fig. 1 illustrates the contribution to fermion
EDMs that stem from the above interactions. Similar diagrams can be drawn with
the scalar propagator replaced by a dark gauge boson, arising from the vertex
$X^{a}_{\mu\nu}\tilde{F}^{\mu\nu}\supset
g_{D}f^{abc}X^{b}_{\mu}X^{c}_{\nu}\tilde{F}^{\mu\nu}$, but their contribution
vanishes due to the antisymmetry of $f^{abc}$ when summing over all possible
diagrams.

In agreement with Furry’s theorem [42], we find that the diagrams in Fig. 1
select the vector component of the fermion current at the gauge boson vertex.
As a consequence of this selection rule, the contributions to the lepton EDMs
arising from the $Z$ boson diagram and from the $X$ coupling to the lepton $Z$
current are suppressed, because the $Z$ lepton vector current is proportional
to the accidentally small combination $1-4s_{\theta}^{2}\sim 0.08$. As a
result, the predicted lepton EDM can be approximated as

$\displaystyle d_{l}$ $\displaystyle\simeq$
$\displaystyle\frac{Y_{l}}{8\pi^{2}v_{D}}\epsilon\tilde{\epsilon}\beta
c_{\theta}^{2}\,e\
\left(\frac{\log(x_{X\phi})}{x_{X\phi}-1}-\frac{\log(x_{Xh})}{x_{Xh}-1}\right)$
(12) $\displaystyle\equiv$
$\displaystyle\frac{Y_{l}}{8\pi^{2}v_{D}}\epsilon^{2}\tan\chi\ \beta
c_{\theta}^{2}\,e\
\left(\frac{\log(x_{X\phi})}{x_{X\phi}-1}-\frac{\log(x_{Xh})}{x_{Xh}-1}\right),$

where $x_{ij}=m^{2}_{i}/m^{2}_{j}$ and in the second line we have defined
$\tan\chi\equiv\tilde{\epsilon}/\epsilon$, which in the UV is related to the
phase of the Yukawa couplings of the heavy states that are integrated out to
give the effective operators (see Appendix A). Values
$\tan\chi\sim\mathcal{O}(1)$ correspond to large CP-violating phases in the
UV, which will be our assumption here and in the rest of the paper.

EDMs are known to be extremely sensitive probes of CP-violating new physics.
In particular, the electron EDM provides the most stringent upper limit on the
couplings of Eq. (12), due to the impressive sensitivity of current
experimental searches, which are also expected to improve significantly in the
future. We summarize the current and future status of electron EDM searches in
Table 1. Efforts are underway to improve the experimental sensitivity to the
muon EDM at PSI [43], but the expected reach is not able to constrain the
parameter space more than the electron EDM in this scenario. This is because
in our model we do not consider new flavor-dependent couplings beyond the SM
Yukawa couplings, and therefore the contributions to the EDMs follow the
Minimal Flavour Violation (MFV) expectation, where the Yukawa couplings are
responsible for the chirality flips. Hence, the scaling of the lepton EDM is
such that the indirect bound on the muon EDM arising from the electron
searches is orders of magnitude stronger than the expected muon EDM
sensitivity.

Experiment | Current bound/Upcoming sensitivity
---|---
JILA eEDM |  $<4.1\times 10^{-30}$ ${\rm e\ cm^{-1}}$ [31]
ACME III |  $\sim 1\times 10^{-30}$ ${\rm e\ cm^{-1}}$ [32]
YBF |  $\sim 1\times 10^{-31}$ ${\rm e\ cm^{-1}}$ [33]
BaF |  $\sim 1\times 10^{-33}$ ${\rm e\ cm^{-1}}$ [34]
Table 1: Current (JILA eEDM) bound on the electron EDM and expected future
sensitivities of the upcoming searches.

To reduce the number of free parameters in Eq. (12), we assume that the dark
sector couplings are natural, with only one mass scale set by the scalar VEV
$v_{D}$. Thus, we take $v_{D}\sim M_{X}\sim m_{\phi}$ and study the EDM
predictions by varying the other parameters. We focus on the region $1\
\text{GeV}\lesssim M_{X}\lesssim 100\ \text{GeV}$ because lighter extra gauge
bosons that mix with the photon are severely constrained by a combination of
beam dump experiments and astrophysical probes [44, 45, 46, 47, 48, 49, 50].
Fig. 2 shows values of $\epsilon$ and $M_{X}$ that saturate the present and
future electron EDM sensitivities shown in Table 1, having chosen a scalar
mixing angle $\beta$ at the boundary of what is allowed by Higgs invisible
decays. We stress that Fig. 2 shows the current and future sensitivity of the
EDM to the kinetic mixing parameter $\epsilon$, assuming it is generated by a
non-renormalizable operator that connects a non-Abelian dark sector with the
SM and in the presence of CP-violating phases that generate a CP-odd operator
with a comparable coefficient. In this scenario, the EDM can probe a large
region of the parameter space in $\epsilon$ and dark gauge boson mass $M_{X}$
that is currently unexplored by the dark photon searches [51, 52, 53, 54]. In
Section 3, we discuss how this can be used to probe specific dark sector
models.

Figure 2: Plot of kinetic mixing parameter $\epsilon$ vs dark boson mass.
Colored bands represent the region of parameter space saturated by the current
(JILA eEDM [31]) and future (ACME III [32], YBF [33],
BaF($\mathrm{EDM}^{3}$)[34]) experimental sensitivity of the eEDM assuming
fixed scalar mixing angle $\beta=10^{-2}$ and a large CP-violating phase
tan$\chi$ = 1. Bounds on kinetic mixing parameter $\epsilon$ come from di-
lepton final states searches at colliders (LHCb [51], BaBar [53], CMS [52] and
NA48/2 [54]) and beam dump experiments (E137 [44, 45, 46], E141[47], CHARM
[48], $\nu$-Cal [49], NA64 [55]). Constraints from SN1987A [50] and $g-2_{e}$
[56] are also shown.

As an aside, the model would contribute to the electron $g-2$ ($a_{e}$)
through the CP-even analogue of the diagram in Fig. 1, and the final result
will resemble Eq. (12) up to a factor 2$m_{e}$ and
$\tilde{\epsilon}\rightarrow\epsilon$. However, considering the most precise
measurement of $a_{e}$ [57], it will give a milder constraint on the kinetic
mixing $\epsilon$ within our assumptions.

##  3 Models

In this section we discuss some dark sector scenarios that could be probed
with the electron EDM induced by the CP-odd portal introduced in the previous
section.

###  3.1 $SU(2)_{D}\to U(1)_{D}$

The minimal setup that satisfy the assumptions of Section 2 is given by a
$SU(2)_{D}$ group which is broken by a triplet into an Abelian sub-group
$U(1)_{D}$. The Lagrangian in this case is given by

$\mathcal{L}=\mathcal{L}_{\rm SM}+\mathcal{L}_{\rm
DS}+(D_{\mu}\Sigma^{a})^{2}-V(\Sigma,H)-\frac{\epsilon}{2v_{D}}\Sigma^{a}X^{a\mu\nu}B_{\mu\nu}-\frac{\tilde{\epsilon}}{2v_{D}}\Sigma^{a}X^{a\mu\nu}\tilde{B}_{\mu\nu},$
(13)

where $a$ takes values from 1 to 3, and the covariant derivative acting on the
triplet is defined as
$D_{\mu}\Sigma^{a}=\partial_{\mu}\Sigma^{a}+\epsilon^{abc}\Sigma^{b}X_{\mu}^{c}$.
The effective operators arise from integrating out heavy fermions that have
hypercharge and fill non-trivial representations of $SU(2)_{D}$ (see Appendix
A). The operator normalization $v_{D}$ is the triplet VEV, which is given by
the minimization of the $V(\Sigma,H)$ potential

$\displaystyle V(\Sigma,H)$
$\displaystyle=-\mu^{2}_{H}(H^{\dagger}H)+\lambda_{H}(H^{\dagger}H)^{2}-m^{2}_{\Sigma}(\Sigma^{a}\Sigma^{a})+\lambda_{\Sigma}(\Sigma^{a}\Sigma^{a})^{2}$
$\displaystyle+\lambda_{H\Sigma}(H^{\dagger}H)(\Sigma^{a}\Sigma^{a}).$ (14)

Without loss of generality, we assume that the triplet vacuum is aligned with
the direction of the diagonal generator of the $SU(2)_{D}$ algebra, with
$\langle\Sigma^{3}\rangle=v_{D}$ and $\langle\Sigma^{1,2}\rangle=0$.
Consequently, the $X^{1,2}_{\mu}$ boson will acquire a squared mass
$M^{2}_{X}=g_{D}v^{2}_{D}$, while $X^{3}_{\mu}$ remains massless and has a
non-zero kinetic mixing with the hypercharge boson. When the electroweak
symmetry is spontaneously broken, the massless vector will mix with the photon
and the $Z$ boson. Since two massless vector fields mix kinetically, there is
an ambiguity in identifying the physical states, one of which should be the SM
photon. The ambiguity arises because the kinetic terms are invariant under an
arbitrary orthogonal transformation, which leaves the physics unchanged [58].
To correctly interpret the model predictions, we need to specify how the extra
massless boson interacts with the SM fields. If we define the physical dark
state as the combination that interacts only with the hidden sector particles,
we find the following mixing matrix in the gauge sector

$\begin{pmatrix}A^{\prime}\\\ Z^{\prime}\\\
X_{3}\end{pmatrix}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\
-c_{\theta}\epsilon&s_{\theta}\epsilon&1\\\ \end{pmatrix}\begin{pmatrix}A\\\
Z\\\ A_{D}\end{pmatrix},$ (15)

where now the unprimed fields are the mass eingenstates. As a result, the
massless dark photon only interacts with the $U(1)_{D}$ charged particles,
while the photon and the $Z$ boson couples to the SM, and to the hidden sector
via a $\epsilon-$dependent coupling. In practice, this means that the
$U(1)_{D}$ current is milli-charged

$\mathcal{L}\supset-eA_{\mu}\left(J^{\mu}_{\rm EM}+c_{\theta}\frac{\epsilon
g_{D}}{e}J^{\mu}_{DS}\right).$ (16)

The triplet component that acquires a VEV can mix with the SM Higgs via the
quartic coupling, as described in Eq. (10), with a mixing angle $\beta$ given
by the Lagrangian parameter introduced in the scalar potential of Eq. (14).
Rotating the gauge bosons and scalars into the physical states, we find the
following interactions333The Lagrangian also contains the CP-even contraction
$\sim\beta\epsilon^{2}/(2v_{D})hF^{\mu\nu}F_{\mu\nu}$ that can interfere with
the SM amplitude for $h\to\gamma\gamma$. Since we assume that
$\epsilon\sim\tilde{\epsilon}$ we find that the sensitivity of
$h\to\gamma\gamma$ to $\epsilon^{2}/v_{D}$ does not compete with the EDM
bounds, in agreement with the results of [59].

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle-\frac{\tilde{\epsilon}}{2v_{D}}(\phi+\beta
h)\left(c^{2}_{\theta}\epsilon
F^{\mu\nu}\tilde{F}_{\mu\nu}+s^{2}_{\theta}\epsilon
Z^{\mu\nu}\tilde{F}_{\mu\nu}\right)$ (17) $\displaystyle-$
$\displaystyle\sum_{f}\left(-Y_{f}\beta\frac{\phi}{\sqrt{2}}\bar{f}f+Y_{f}\frac{h}{\sqrt{2}}\bar{f}f\right),$

where we have denoted $\phi$ as the mass eigenstate that results from the
$\Sigma^{3}$ and Higgs mixing. The lepton EDM induced by the above
interactions is given by a diagram similar to the one in Fig. 1, but with $X$
replaced by the SM photon. The predicted lepton EDM is

$d_{l}\simeq
e\left(\frac{Y_{l}}{8\pi^{2}v_{D}}\right)\epsilon^{2}\tan\chi\;\beta
c_{\theta}^{2}\log\left(\frac{m_{h}^{2}}{m_{\phi}^{2}}\right),$ (18)

where, again, we have neglected the $Z$ contribution because the lepton vector
current coupling is suppressed. We remind that $\tan\chi$ is defined as
$\tan\chi\equiv\tilde{\epsilon}/\epsilon$.

The dark sector Lagrangian $\mathcal{L}_{\rm DS}$ contains matter fields that
transform under the $SU(2)_{D}$ group, and may include potential dark matter
candidates. After the SSB $SU(2)_{D}\to U(1)_{D}$, particles that are charged
under the surviving $U(1)_{D}$ also have a milli-charge (mC)
$\epsilon^{\prime}\equiv c_{\theta}\epsilon g_{D}/e$ as described by Eq. (16).
Constraints on mC Particles (mCPs) come from a variety of experimental probes
[60, 61, 62, 63, 64, 65, 66, 67, 68, 69]. In our scenario, the mC parametrizes
the EDM prediction, so the EDM bounds are translated into an indirect
sensitivity to the millicharge. The $X_{1,2}$ massive bosons combine to form
the $U(1)_{D}$ charge eigenstates $W^{\pm}_{D}=(X_{1}\mp iX_{2})/\sqrt{2}$,
which interact with the SM photon via the mC. Assuming that the dark gauge
coupling is comparable to the electromagnetic one, $g_{D}\sim e$, the EDM
prediction can be parametrised in terms of the kinetic mixing $\epsilon$ and
the $W_{D}$ mass444The EDM depends only logarithmically on the scalar mass,
which we assume to be $m_{\phi}\sim v_{D}$. The EDM sensivity can then be
compared with the experimental reach of milli-charged particles searches (see
Fig. 3). Clearly, other mC states besides the $W_{D}$ could be present in the
hidden sector. As shown in Fig. 3, the constraints on the mC can be severe
enough to render the EDM unobservable for a wide range of masses, while
interesting EDM phenomenology can arise if the mCP masses lies within $1\ {\rm
GeV}\lesssim M_{\chi}\lesssim 100$ GeV

Figure 3: Plot of kinetic mixing parameter $\epsilon$ vs milli-charged
particle mass. Colored bands represent the region of parameter space saturated
by the current (JILA eEDM [31]) and future (ACME III [32], YBF [33],
BaF($\mathrm{EDM}^{3}$) [34]) experimental sensitivity of the eEDM assuming
scalar mixing angle $\beta=10^{-2}$, dark scalar mass $m_{\phi}\sim v_{D}$ and
a large CP-violating phase tan$\chi$ = 1. Bounds on kinetic mixing parameter
$\epsilon$ come from $N_{\mathrm{eff}}$ [61], LEP [60], LHC [66], WMAP [69].
In this plot, $g_{D}=e$ is assumed.

The millicharged particles can contribute to the dark matter relic density if
they are stable. However, strong constraints from CMB observations either
require a mC small enough to have dark matter decoupled before the
recombination epoch, or that the mCPs constitute a small fraction of the
observed dark matter abundance [70]. In the first case, where the millicharged
particles account for the dark matter but the millicharge is small
($\epsilon^{\prime}\lesssim 10^{-7}$ ), the EDM will also be suppressed,
making it unobservable even for the most optimistic upcoming sensitivity.
However, UV-dependent contributions to EDMs arising from the integrated-out
heavy fermions could still be sizable despite the small $\epsilon$-values, as
we illustrate in Appendix B. If instead the mCPs constitute only a negligible
fraction of the dark matter, larger $\epsilon^{\prime}$ values are allowed
and, in our scenario, an electron EDM could be observed for multi-GeV mCP
masses. Interestingly, an anomalous observation of the 21 cm hydrogen line at
red-shift $z\sim 17$ [71] from the EDGES collaboration, could be explained by
a small fraction of millicharged DM [72, 73, 74, 75].

Viable DM candidates in this model must be stable particles uncharged under
the surviving $U(1)_{D}$ to avoid the stringent bounds on mC dark matter.
Since $U(1)_{D}$ charged and neutral states co-exist in $SU(2)_{D}$ multiples,
accommodating a negligible mC DM fraction while simultaneously accounting for
the observed DM abundance with the neutral components is challenging and
beyond the scope of this work. We aim to address this issue in a future
project.

Notice that the minimal setup that we considered in this section for the dark
sector, an $SU(2)_{D}$ symmetry with an scalar in the adjoint spontaneously
broken to $U(1)_{D}$, is exactly the original magnetic monopole construction
described by ’t Hooft and Polyakov [76, 77]. Therefore, we should expect dark
sector monopoles. In [78] it was shown that, in the presence of only kinetic
mixing, these dark monopoles are invisible to ordinary electrically charged
matter. However, in the presence of both the ordinary kinetic mixing and the
CP-odd mixing, the dark monopoles acquire an electric charge in the dark
sector through the Witten effect [79]. Then, these "dark dyons" appear as mCPs
in the visible sector, through the usual kinetic mixing [80]555Constraints on
dyons have been obtained in the MoEDAL experiment [81] but would not apply to
these ”dark dyons” with no magnetic charge in the visible sector..
Nonetheless, for a monopole mass $\lesssim 1$ TeV like in our case, their
contribution to the dark matter relic density is completely negligible, so
cosmological and astrophysical bounds on the monopole millicharge are not
severe.

###  3.2 $SU(2)_{D}\to\varnothing$ and inelastic Dark Matter

Augmenting the scalar sector of Section 3.1 with an additional triplet, it is
possible to completely break the $SU(2)_{D}$ group if the triplet VEVs are not
aligned. Considering two scalar $\Sigma^{a}_{1},\Sigma^{a}_{2}$ transforming
in the adjoint of $SU(2)_{D}$, the potential can be such that

$\langle\Sigma_{1}\rangle=\begin{pmatrix}0&v_{1}&0\end{pmatrix},\qquad\langle\Sigma_{2}\rangle=\begin{pmatrix}0&0&v_{2}\end{pmatrix},$
(19)

which results in the following mass matrix for the dark gauge bosons

$\mathcal{L}\supset\frac{1}{2}M^{2}_{ab}X^{a}_{\mu}X^{b\mu},\qquad
M^{2}_{ab}=\begin{pmatrix}g_{D}(v^{2}_{1}+v^{2}_{2})&0&0\\\
0&g_{D}v^{2}_{2}&0\\\ 0&0&g_{D}v_{1}^{2}\end{pmatrix},$ (20)

and the $X^{2},X^{3}$ bosons acquire non-zero kinetic mixing with the SM gauge
fields via the non-renormalizable portals

$\mathcal{L}\supset\sum_{i=1,2}\left(-\frac{\epsilon_{i}}{2v_{i}}\Sigma_{i}^{a}X^{a\mu\nu}B_{\mu\nu}-\frac{\tilde{\epsilon}_{i}}{2v_{i}}\Sigma_{i}^{a}X^{a\mu\nu}\tilde{B}_{\mu\nu}\right).$
(21)

In this case, the mass eigenstates aligned with $X_{2},\ X_{3}$ interact with
the electromagnetic current after the rotation that diagonalises the kinetic
terms

$\mathcal{L}\supset\sum_{i=1,2}X^{i+1}_{\mu}c_{\theta}\epsilon_{i}eJ^{\mu}_{\rm
EM}.$ (22)

The scalar sector features four physical states, one being the SM Higgs. Two
of the scalar states, aligned with the two VEVs,
$\Sigma^{2,3}_{1,2}=v_{1,2}+\phi^{1,2}$, will mix directly with the Higgs via
the quartic coupling. Assuming for simplicity small mixing angles, we have the
following matrix that rotates into the mass eigenstates basis

$\begin{pmatrix}\phi_{D}\\\ \phi_{1}\\\ \phi_{2}\\\
h\end{pmatrix}=\begin{pmatrix}1&\theta_{1}&\theta_{2}&0\\\
-\theta_{1}&1&\alpha&\beta_{1}\\\ -\theta_{2}&-\alpha&1&\beta_{2}\\\
0&-\beta_{1}&-\beta_{2}&1\\\ \end{pmatrix}\begin{pmatrix}\phi_{D}^{\prime}\\\
\phi_{1}^{\prime}\\\ \phi_{2}^{\prime}\\\ h^{\prime}\end{pmatrix},$ (23)

where $\beta_{1,2}$ mix the SM Higgs with the singlet scalars, while $\alpha$
and $\theta_{1,2}$ parameterizes the mixing between dark sector fields. At
first order in the scalar angles the $\phi_{D}$ does not mix with the Higgs,
and the predicted lepton EDM is given by an expression similar to Eq. (12):

$\displaystyle d_{l}$ $\displaystyle\simeq$
$\displaystyle\sum_{i=1,2}\frac{Y_{l}}{8\pi^{2}v_{i}}\epsilon_{i}^{2}\tan\chi_{i}\
\beta_{i}c_{\theta}^{2}\,e\
\left(\frac{\log(x_{X_{i+1}\phi_{i}})}{x_{X_{i+1}\phi_{i}}-1}-\frac{\log(x_{X_{i+1}h})}{x_{X_{i+1}h}-1}\right).$
(24)

A stable particle $\chi$ in the dark sector contributes to the dark matter
relic density, and its abundance can be thermally produced following the
freeze-out of the annihilation process $\chi\chi\to{\rm SM}$ given by the
tree-level exchange of a $X^{2,3}$ bosons that couple to the electromagnetic
and the $Z$ currents. We are assuming that the mediator mass is larger than
twice the DM mass $m_{\chi}$, so that the annihilation proceeds from an off-
shell exchange and the thermally averaged cross-section is of order

$\langle\sigma_{\chi\chi\rightarrow{\rm SM}}v_{\rm
rel}\rangle\sim\epsilon^{2}\alpha_{\rm EM}\alpha_{D}\frac{16\pi
m_{\chi}^{2}}{(4m_{\chi}^{2}-M_{X}^{2})^{2}},$ (25)

where $\alpha_{D}=g_{D}^{2}/4\pi$ and $v_{\rm rel}$ is the DM particles
relative velocity. The correct dark matter abundance is obtained when the
freeze-out thermal cross-section is $\langle\sigma_{\chi\chi\rightarrow{\rm
SM}}v_{\rm rel}\rangle\sim 1.7\times 10^{-9}\;\mathrm{GeV}^{-2}$ [82]. Taking
$m_{\chi}\lesssim 1$ GeV, $M_{X}\sim{\rm few}\times m_{\chi}$ and
$g_{D}\sim\mathcal{O}(1)$, the observed relic abundance is given by
$\epsilon\sim 10^{-4}$, which as shown in Fig. 2, is in the reach of the EDM
searches. However, indirect and direct detection experiments place strong
constraints on this simple scenario. On one hand CMB anisotropies are
sensitive to the energy injection in the intergalactic medium that could arise
from DM annihilation, and for the thermal cross section needed to get the
correct DM relic abundance, Planck data impose, for s-wave channel
annihilation, an indirect lower limit on the DM mass $m_{\chi}\gtrsim 30$ GeV
[83]. On the other hand, for DM masses above few GeVs, direct detection
searches severely constrain the DM interactions with nuclei, which are
parameterized by the kinetic mixing parameter $\epsilon$ [84, 85, 86, 87].

The thermal freeze-out of a sub-GeV DM particle is possible in the so-called
inelastic dark matter scenario [88]. The idea is that DM interacts with the SM
only via an inelastic scattering between two states $\chi_{H,S}$ with a mass
splitting $\delta m_{\chi}$. Hence, the annihilation channel
$\chi_{H}\chi_{S}\to{\rm SM}$ sets the relic abundance when the process
freezes out, but after the decay of the heavier state
$\chi_{H}\to\chi_{S}+{\rm SM}$ only $\chi_{S}$ populates the Universe and it
cannot annihilate with itself to give a signal in an indirect detection
search. As a result, it is possible for dark matter candidates with masses at
or below the GeV scale to elude these indirect bounds and in this energy
region direct detection constraints are less severe ([89], see however [90]).
In addition to this, bounds from direct detection experiments can be
considerably relaxed by considering a mass splitting greater than $\delta
m_{\chi}\sim 1\div 100\,\mathrm{keV}$, where the direct detection of DM rely
on the effect of cosmic ray upscattering and so are much less severe [91].
Larger mass splitting are also cosmologically favoured by the requirement of
sufficiently short-lived $\chi_{H}$, because for $\delta_{\chi}<2m_{e}$ the
only possible decay channel for the heavy state is
$\chi_{H}\rightarrow\chi_{S}\nu\bar{\nu}$, easily resulting in a lifetime
greater than the age of the Universe [92].

We can reproduce the inelastic scenario introducing two chiral $SU(2)_{D}$
doublets $\chi_{L},\psi_{R}$, in addition to the field content presented at
the beginning of this section. The relevant interactions that involve the
extra fermions are the following

$\displaystyle\mathcal{L}$ $\displaystyle\supset$ $\displaystyle-
m_{D}\overline{\chi_{L}}\psi_{R}-\sum_{i=1,2}Y_{D,i}\overline{\chi_{L}}\Sigma_{i}\psi_{R}-\sum_{i=1,2}Y_{L,i}\overline{\chi^{c}_{L}}i\sigma_{2}\Sigma_{i}\chi_{L}-\sum_{i=1,2}Y_{R,i}\overline{\psi^{c}_{R}}i\sigma_{2}\Sigma_{i}\psi_{R}+{\rm
h.c.}$ (26) $\displaystyle-$
$\displaystyle\frac{g_{D}}{2}\overline{\chi_{L}}\gamma_{\mu}\sigma^{a}X^{\mu}_{a}\chi_{L}-\frac{g_{D}}{2}\overline{\psi_{R}}\gamma_{\mu}\sigma^{a}X^{\mu}_{a}\psi_{R}.$

The SU(2) components of the chiral doublets combine into two Dirac fermions
$\Psi_{H},\Psi_{S}$ (see Appendix C for more details). For small values of the
Yukawas $Y_{i}v_{D}\ll m_{D}$, the two fermions are approximately degenerate
with a mass $m_{\Psi}\sim m_{D}$ and a small mass splitting $\delta_{\Psi}\sim
v_{D}Y_{i}$. In the limit $Y_{L,i}=Y_{R,i}$ and negligible $Y_{D,i}$, the dark
fermions exclusively have off-diagonal interactions with the $X_{2}$ gauge
boson that, together with $X_{3}$ , kinetically mixes with the SM hypercharge
boson. The $X_{3}$ gauge interactions feature instead diagonal $\Psi_{H,S}$
currents which are unavoidable in the limit of small Yukawas. The elastic
annihilations mediated by $X_{3}$ can be suppressed by taking $v_{1}\gg v_{2}$
to enhance the $X_{3}$ mass (see Eq. (20)) and/or considering
$\epsilon_{1}\ll\epsilon_{2}$, which is possible if in the UV the heavy states
have small couplings with the scalar $\Sigma_{1}$. In the following, we assume
this is the case and neglect the $X_{3}$-mediated elastic annihilations.
Hence, the relic abundance and the EDM prediction will be dominated by the
$X_{2}$ couplings, and we rename $\epsilon_{2}\equiv\epsilon$, $v_{2}\equiv
v_{D}$, $M_{X_{2}}\equiv M_{X}$.

The relic density of $\Psi_{H},\Psi_{S}$ following the freeze-out of the co-
annihilations is determined by the thermal cross-section $\langle\sigma
v\rangle$ for the scattering process $\Psi_{H}\Psi_{S}\to{\rm SM}$ and by the
mass splitting $\delta_{\Psi}$. Since the $\Psi_{H}$ will eventually decay and
annihilate into $\Psi_{S}$ plus SM particles, the relic abundance of
$\Psi_{S}$ will be the sum of the $\Psi_{H}$ and $\Psi_{S}$ densities at the
time when the co-annihilations cease. In practice, to calculate the final DM
abundance, we can define an effective thermal cross-section [93]

$\langle\sigma_{\rm eff}v_{\rm rel}\rangle=\frac{\langle\sigma v_{\rm
rel}\rangle}{2}\left(1-\frac{\delta_{\Psi}}{T}+\frac{3}{4}\frac{\delta_{\Psi}}{m_{\Psi}}+\mathcal{O}(\delta^{2}_{\Psi})\right),$
(27)

where we expanded the Boltzmann factors for small mass splittings666This is
justified because the relic density is dominated by the equilibrium value at
the freeze-out temperature and where $\langle\sigma v_{\rm rel}\rangle$ is the
total annihilation cross section into SM model particles, including leptonic
and hadronic final states.

The leptonic cross-section is similar to Eq. (25) up to mass corrections, and
to compute the hadronic cross section in the GeV and sub-GeV region, we
considered $\langle\sigma_{\rm
had}\rangle(s)=\langle\sigma_{\mu\mu}\rangle(s)R(s)$, where $R(s)$ is the
experimental ratio of $e^{+}e^{-}$ cross section to hadronic versus muon final
states at colliders [94]. Then, the correct DM abundance is obtained imposing
$\langle\sigma_{\rm eff}v_{\rm rel}\rangle\sim 1.7\times 10^{-9}\ {\rm
GeV^{-2}}$.

The phenomenology of inelastic dark matter at laboratory experiments has been
widely treated in literature and it is, in general, model dependent [95, 96,
97, 98]. Indeed, depending on the parameters of the model and the specific
experiment, the signature of the dark sector at collider and beam dump
experiments is different. The decay rate of $\Psi_{H}$, in the limit
$M_{X}>m_{\Psi}\gg m_{f}$, reads

$\sum_{f}\Gamma(\Psi_{H}\rightarrow\Psi_{S}f\bar{f})=\sum_{f}\frac{{4}\epsilon^{2}\alpha\alpha_{D}\delta_{\Psi}^{5}}{15\pi
M_{X}^{4}}.$ (28)

Small mass splitting suppresses the decay rate of the heavy dark fermion. In
collider experiments, dark fermions are produced only through the coupling of
the $X$ boson to the electromagnetic current (22). If the lifetime of
$\Psi_{H}$ is short enough, its signature will be characterized by a pair of
fermions plus missing energy, given by the decay chain
$X\rightarrow\Psi_{H}\Psi_{S}\rightarrow\Psi_{S}\Psi_{S}f\bar{f}$. On the
other hand, if $\Psi_{H}$ is sufficiently long-lived, it will decay outside
the detector rendering the final state completely invisible. The latter
scenario is possible if $\delta_{\Psi}\lesssim 0.01\times m_{\Psi}$ with
$m_{\Psi}$ in the sub-GeV region, and the relevant experimental bounds on the
model parameters arise from searches for a dark boson decaying to an invisible
final state [99, 100, 101, 102, 103]. We focus on the small mass splitting
case because it is less constrained by present experiments, while for the
semi-visible signatures beam-dump experiments can place strong constraints on
the model in the case of sub-GeV DM mass [98]. In our case, even for a long-
lived $\Psi_{H}$, this region of the parameter space is testable by the
upcoming searches of the electron EDM.

We perform a scan over the parameter space of the model, varying the
parameters as described in Table 2. We fix the mass splitting to be
$\delta_{\psi}\lesssim 0.01\times m_{\psi}$, but require
$\delta_{\Psi}>2m_{e}$ to avoid the BBN bounds on the $\Psi_{H}$ lifetime.
Furthermore, we consider the experimental constraints on the kinetic mixing
parameter $\epsilon$ from the dark boson to invisible decays. We also assume
and enforce that $M_{X}>2m_{\Psi}$, so that the DM abundance is set by the
$\Psi_{H,S}$ co-annihilation into the SM rather than from the gauge
interactions within the dark sector, which makes the model more predictive.

Parameter | Lower limit | Upper limit
---|---|---
$\beta_{1,2}$ |  $10^{-4}$ |  $10^{-2}$
$\epsilon$ |  $10^{-6}$ |  [99, 100, 101, 102, 103]
$g_{D}$ |  $10^{-2}$ |  1
$\tan{\chi}$ |  $10^{-2}$ |  1
$v_{D}$ |  1 GeV |  20 GeV
$m_{\phi}$ |  0.1 GeV |  50 GeV
Table 2: Table representing the range of the scanned parameter values. Initial
and final values are chosen in accordance of the preliminary analysis for a
sizeable eEDM and current constraints from laboratory experiments. Figure 4:
Scatter plot representing on the $y$-axis the eEDM and on the $x$-axis the DM
mass satisfying the correct relic abundance. Blue points represent the
predicted eEDM and DM mass for parameters randomly chosen accordingly to
Tab.2. Gray region represents the already excluded parameter space by JILA
eEDM collaboration. Colored regions between $10^{-33}$ and 4.1 $\times
10^{-30}$ show the future eEDM sensitivities. Finally, green region covers the
SM prediction for the equivalent eEDM [104].

In Fig. 4 we show the predicted eEDM and DM mass for each of the parameter
space points. The DM mass is obtained by fixing the effective cross-section of
Eq. (27) to the value $\sim 1.7\times 10^{-9}$ ${\rm GeV}^{-2}$ of the thermal
cross-section needed to have the correct relic abundance. We show in Fig. 4
that, in a large region of the parameter space, the predicted eEDM lies
between the current experimental bounds and the future experimental
sensitivity, making the model testable in future eEDM searches.

##  4 Summary

Non-Abelian dark sectors can have a naturally small portal to the visible
sector and simultaneously offer pathways to address unresolved puzzles such as
the observed dark matter abundance and baryon asymmetry in the Universe. In
this work, we have explored the phenomenology of a CP-odd portal (defined in
Eq. (4)) to a non-Abelian dark sector. We have argued that in the presence of
large CP-violating phases in the UV, the CP-odd portal is closely related to
the kinetic mixing between the dark and the SM gauge bosons. We have focused
on the contribution to electric dipole moments, as they are the most sensitive
observables to CP violation. Hence, we have discussed the potential of current
and future EDM searches in probing this scenario.

In Section 2, we have discussed the features of the vector mixing portals
under general assumptions and determined the sensitivity of electron EDM
searches to the CP-odd operator coefficient, specifically in the case of a
single massive dark gauge boson mixing with the hypercharge vector. Assuming
comparable coefficients for the CP-even and CP-odd operators of Eq. (4), we
have compared the EDM sensitivity with the reach of dark photon searches for
the kinetic mixing parameter $\epsilon$ as a function of the dark photon mass.
We have identified regions of parameter space, particularly for $M_{X}\gtrsim
1$ GeV, where future EDM searches are competitive with, or even more sensitive
than, other experimental probes (see Fig. 2).

In Section 3, we have specialized the discussion to a dark gauge group
$SU(2)_{D}$ with specific scalar contents. In subsection 3.1, we have
considered a single triplet scalar that acquires a vacuum expectation value,
and an Abelian subgroup $U(1)_{D}$ survives the spontaneous symmetry breaking.
As a result of the gauge boson kinetic mixing $\epsilon$, particles charged
under the extra $U(1)_{D}$ acquire an $\epsilon$-dependent millicharge. Since
$\epsilon$ parameterizes the prediction for the electron electric dipole
moment, we have found an indirect sensitivity of EDM experiments to the
millicharge. Once again, we have identified regions where the EDM reach
provides interesting constraints compared to other experimental searches for
millicharged particles (see Fig. 3).

In Subsection 3.2, we have augmented the scalar sector to completely break the
$SU(2)_{D}$ group, resulting in three massive dark gauge bosons. We have
implemented an inelastic dark matter scenario where the DM abundance is
closely related to the size of $\epsilon$, among other model parameters. We
have explored whether future eEDM searches, considering all experimental
constraints, could potentially exclude this scenario. As shown in Fig. 4, we
have found that for many points in the parameter space, compatible with the
observed DM abundance and various experimental bounds, a potentially
observable eEDM is predicted.

## Acknowledgments

This work is supported by the Generalitat Valenciana project CIPROM/2021/054
and CIPROM/2022/66 and the Spanish AEI-MICINN PID2020-113334GB-I00
(AEI/10.13039/501100011033).

##  Appendix A Non-Abelian kinetic mixing from the UV

The effective terms in Eq. (4) require an anomaly free UV completion. The
simplest choice consists in introducing vector-like heavy chiral fermions
$\Theta$ in the defining representation of $G_{D}$777 The generators $T^{a}$
of the fundamental representation are normalised such that ${\rm
Tr}(T^{a}T^{b})=\delta^{ab}/2$ and having hypercharge $Y$. The relevant
Lagrangian terms are the following

$\displaystyle\mathcal{L}_{\Theta}\supset\
-g_{1}Y\bar{\Theta}_{i}\gamma_{\mu}\Theta_{i}B^{\mu}-g_{D}\bar{\Theta}_{i}T^{a}\gamma_{\mu}\Theta_{i}X^{\mu}_{a}-M_{ii}\bar{\Theta}_{iR}\Theta_{iL}-\mathcal{Y}_{ij}\bar{\Theta}_{iR}\Sigma^{a}T^{a}\Theta_{jL}+{\rm
h.c.},$ (29)

where $g_{1}=e/c_{\theta}$ is the hypercharge coupling, and $\mathcal{Y}_{ij}$
is a complex Yukawa-like matrix, which in the presence of a diagonal real
mass, $M_{ii}$, has physical phases [105, 106]. The effective operators in Eq.
(4) of the text, originate from the one loop diagram shown in Fig. 5, after
the heavy degrees of freedom have been integrated out.

Figure 5: One loop diagram generating the CP-even and CP-odd terms in Eq. (4).

Matching the diagram of Fig. 5 for a single heavy fermion having a sizable
(complex) Yukawa coupling $\mathcal{Y}$ and mass $M$, gives the following
coefficient for the operators in Eq. (4)

$\displaystyle\frac{C}{\Lambda}=\frac{g_{D}g_{1}Y\mathrm{Re}[\mathcal{Y}]}{12\pi^{2}M};\quad\quad\frac{\tilde{C}}{\Lambda}=\frac{g_{D}g_{1}Y\mathrm{Im}[\mathcal{Y}]}{16\pi^{2}M},$
(30)

For the one heavy fermion case, we find that

$\tilde{\epsilon}=\frac{3}{4}\frac{\mathrm{Im}[\mathcal{Y}]}{\mathrm{Re}[\mathcal{Y}]}\epsilon\equiv(\tan\chi)\,\epsilon,$
(31)

having defined $\chi$ as the phase of $\mathcal{Y}$ up to an
$\sim\mathcal{O}(1)$ factor. In general, the value of $\tan\chi$ is UV-
dependent, so in the text we just assume that $\tan\chi\sim\mathcal{O}(1)$.

Since we assume to have fermion with hypercharges, collider and cosmological
bounds may apply. To obtain kinetic mixings $\epsilon\sim 10^{-3}\div
10^{-5}$, we need $M\gtrsim 1$ TeV, which could be close to the reach of LHC
searches. A detailed analysis is beyond the scope of this work. In addition,
to avoid the strict cosmological bounds on the relic abundance of charged
particles, we need the heavy fermions to decay into standard model and dark
sector particles. For instance, in the case studied in Section 3.2 where the
DM candidate is the fundamental representation of $SU(2)_{D}$, it is
sufficient to consider $\Theta$ transforming in the
$(\underline{2},1/2,\underline{2})$ representation of $SU(2)_{L}\times
U(1)_{Y}\times SU(2)_{D}$. Then, the $\Theta$ can decay into the DM$+$SM via
the portal interaction

$\lambda\bar{\Theta}H\chi+{\rm h.c.}$ (32)

##  Appendix B Matching contributions to the EDM in the UV

By adding heavy chiral fermions with hypercharge, there could also be a direct
matching contribution to the dipole operators arising from the two-loop Barr-
Zee diagrams of Fig. 6.

Figure 6: Barr-Zee diagram arising from the UV completion of the effective
theory.

We neglect the diagram with a $Z$ exchange because, like in the diagram of
Fig. 1, it is suppressed by the small $Z$ lepton vector coupling. The heavy
fermions have the interactions described in Eq. (29), and we once again
consider the case where only one has a sizable complex Yukawa coupling
$\mathcal{Y}$ with the adjoint scalar $\Sigma$. Taking for simplicity one dark
scalar $\phi^{a}$, mixing with the Higgs with a small angle $\beta^{a}$, the
Barr-Zee contribution to the electric dipole moment is the following [107]

$d_{l}=\frac{q^{2}_{\Theta}e\alpha}{8\pi^{3}}\frac{Y_{l}\mathop{\rm
Im}\nolimits[\mathcal{Y}]}{M}\beta^{a}\left[g\left(\frac{M^{2}}{m^{2}_{\phi^{a}}}\right)-g\left(\frac{M^{2}}{m^{2}_{h}}\right)\right],$
(33)

where $q_{\Theta}$ is the $\Theta$ electric charge, $Y_{l}$ is the lepton SM
Yukawa and $g(z)$ is defined as

$g(z)=\frac{z}{2}\int_{0}^{1}dx\frac{1}{x(1-x)-z}\log\left(\frac{x(1-x)}{z}\right).$
(34)

In the limit $z\gg 1$, we can approximate the integrand as
$-1/2\log(x(1-x)/z)$, so that Eq. (33) becomes:

$d_{l}\sim\frac{q^{2}_{\Theta}e\alpha}{16\pi^{3}}\frac{Y_{l}\mathop{\rm
Im}\nolimits[\mathcal{Y}]}{M}\beta^{a}\log\left(\frac{m^{2}_{h}}{m^{2}_{\phi^{a}}}\right).$
(35)

The resulting EDM has a different parametric dependence with respect to the
contribution of the CP-odd effective operator. In particular, it does not
depend on the dark gauge coupling $g_{D}$. Therefore, in the UV-complete
theory, it is possible that the CP-odd portal contribution is suppressed, for
instance by taking $g_{D}\ll 1$, but the contribution to the EDM could still
be sizable from Barr-Zee like diagrams.

##  Appendix C Fermion masses in the $SU(2)_{D}$ model

In this section we give more details about the fermion mass diagonalization
for the iDM model of Section 3.2. Introducing two Weyl fermion doublets with
opposite chirality

$\chi_{L}=\begin{pmatrix}\chi_{L,1}\\\
\chi_{L,2}\end{pmatrix};\quad\psi_{R}=\begin{pmatrix}\psi_{R,1}\\\
\psi_{R,2}\end{pmatrix},$ (36)

the dark sector interactions relevant for these extra fermions read

$\displaystyle\mathcal{L}$ $\displaystyle\supset$ $\displaystyle-
m_{D}\overline{\chi_{L}}\psi_{R}-\sum_{i=1,2}Y_{D,i}\overline{\chi_{L}}\Sigma_{i}\psi_{R}-\sum_{i=1,2}Y_{L,i}\overline{\chi^{c}_{L}}i\sigma_{2}\Sigma_{i}\chi_{L}-\sum_{i=1,2}Y_{R,i}\overline{\psi^{c}_{R}}i\sigma_{2}\Sigma_{i}\psi_{R}+{\rm
h.c.}$ (37) $\displaystyle-$
$\displaystyle\frac{g_{D}}{2}\overline{\chi_{L}}\gamma_{\mu}\sigma^{a}X^{\mu}_{a}\chi_{L}-\frac{g_{D}}{2}\overline{\psi_{R}}\gamma_{\mu}\sigma^{a}X^{\mu}_{a}\psi_{R}.$

Upon SSB, when the scalars acquire vacuum expectation values as specified in
Eq. (19), the fermions receive additional mass terms from the Yukawa
couplings. Taking the right-handed and left-handed Yukawas to be the same
($Y_{L,1}=Y_{R,1}=Y_{1}$, $Y_{L,2}=Y_{R,2}=Y_{2}$) and neglecting the
$Y_{D,i}$ contribution, which is the relevant limit for the inelastic
scenario, we have the following mass matrix for the fermions

$\begin{pmatrix}\overline{\chi_{L,1}}&\overline{\chi_{L,2}}&\overline{\psi^{c}_{R,1}}&\overline{\psi^{c}_{R,2}}\end{pmatrix}\begin{pmatrix}iY_{1}v_{1}&-Y_{2}v_{2}&m_{D}&0\\\
-Y_{2}v_{2}&iY_{1}v_{1}&0&m_{D}\\\ m_{D}&0&iY_{1}v_{1}&-Y_{2}v_{2}\\\
0&m_{D}&-Y_{2}v_{2}&iY_{1}v_{1}\\\
\end{pmatrix}\begin{pmatrix}\chi^{c}_{L,1}\\\ \chi^{c}_{L,2}\\\ \psi_{R,1}\\\
\psi_{R,2}\end{pmatrix}.$ (38)

Since $Y_{1}$ does not affect neither the mass splitting nor the structure of
the final gauge interactions, we consider for simplicity $Y_{1}v_{1}\ll
Y_{2}v_{2}$, and the eigenstates and mass spectrum thus read

$\displaystyle\chi_{R,H_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-i\chi^{c}_{L,1}-i\chi^{c}_{L,2}+i\psi_{R,1}+i\psi_{R,2}\right),$
$\displaystyle\chi_{R,H_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\chi^{c}_{L,1}+\chi^{c}_{L,2}-\psi_{R,1}+\psi_{R,2}\right),$
$\displaystyle\chi_{R,S_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(i\chi^{c}_{L,1}-i\chi^{c}_{L,2}-i\psi_{R,1}+i\psi_{R,2}\right),$
$\displaystyle\chi_{R,S_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\chi^{c}_{L,1}+\chi^{c}_{L,2}+\psi_{R,1}+\psi_{R,2}\right),$
$\displaystyle m_{S_{1,2}}=m_{D}$ $\displaystyle-$ $\displaystyle
v_{2}Y_{2}=M_{1};\quad m_{H_{1,2}}=m_{D}+v_{2}Y_{2}=M_{2},$ (39)

resulting in a mass splitting $\delta_{\chi}=2v_{2}Y_{2}$. The pairs of mass
degenerate Majorana states combine in two Dirac fermions defined as

$\Psi_{H}=\begin{pmatrix}\chi^{c}_{H}\\\
\xi_{H}\end{pmatrix};\quad\Psi_{S}=\begin{pmatrix}\xi^{c}_{S}\\\
\chi_{S}\end{pmatrix};\quad\chi_{H,S}=\frac{\chi^{1}_{H,S}+i\chi^{2}_{H,S}}{\sqrt{2}};\quad\xi_{H,S}=\frac{\chi^{1}_{H,S}-i\chi^{2}_{H,S}}{\sqrt{2}}$
(40)

with masses

$\mathcal{L}_{M}=-M_{1}\overline{\Psi}_{H}\Psi_{H}-M_{2}\overline{\Psi}_{S}\Psi_{S}.$
(41)

The gauge interactions in the $\Psi_{H,S}$ basis are the following:

$\displaystyle\frac{g_{D}}{2}\Big{(}\overline{\Psi}_{H}\gamma_{\mu}\Psi_{S}(X^{\mu}_{1}+iX_{2}^{\mu})+\frac{1}{2}(\overline{\Psi}_{H}\gamma_{\mu}\Psi_{H}-\overline{\Psi}_{S}\gamma_{\mu}\Psi_{S})X_{3}^{\mu}+{\rm
h.c.}\Big{)},$

where the $X_{1}^{\mu},X_{2}^{\mu}$ gauge boson couple to the fermions only
through an off-diagonal vector coupling, while $X_{3}^{\mu}$ presents only
diagonal currents. Diagonal currents would also be present in the $X_{2}$
couplings for $Y_{L,i}-Y_{R,i}\neq 0$ and $Y_{D,i}\neq 0$, which motivates the
limit we consider in Eq. (40) to avoid the elastic annihilations.

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