††institutetext: Department of Physics, Harvard University, Cambridge, MA
02138, USA

# A Closer Look at CP-Violating Higgs Portal Dark Matter as a Candidate for
the GCE

Katherine Fraser, Aditya Parikh, and Weishuang Linda Xu

(today)

###### Abstract

A statistically significant excess of gamma rays has been reported and
robustly confirmed in the Galactic Center over the past decade. Large local
dark matter densities suggest that this Galactic Center Excess (GCE) may be
attributable to new physics, and indeed it has been shown that this signal is
well-modelled by annihilations dominantly into $b\bar{b}$ with a WIMP-scale
cross section. In this paper, we consider Majorana dark matter annihilating
through a Higgs portal as a candidate source for this signal, where a large
CP-violation in the Higgs coupling may serve to severely suppress scattering
rates. In particular, we explore the phenomenology of two UV completions, a
singlet-doublet model and a doublet-triplet model, and map out the available
parameter space which can give a viable signal while respecting current
experimental constraints.

##  1 Introduction

Dark matter constitutes the majority of the mass in our universe, but its
properties remain largely unknown. An extensive experimental program is in
place aimed at understanding the nature of dark matter using its interactions
with the Standard Model. This includes direct production at colliders and
fixed-target experiments, scattering at direct detection experiments, and
indirect detection of annihilation products. Over the years, there have been
tantalizing hints in various experiments; while many of these signals have
vanished due to increased statistics or a better understanding of systematic
uncertainties, some signals, such as the Galactic Center Excess (GCE), have
persisted for over a decade.

In astrophysical settings, the Galactic Center is expected to have some of the
largest dark matter densities, and is therefore one of the most promising
targets for indirect searches. The GCE is a statistically significant excess
of gamma rays at energies of $\sim 2-3$ GeV observed in the Galactic Center by
the Fermi Gamma Ray Space Telescope TheFermi-LAT:2015kwa . As pointed out by
Goodenough:2009gk ; Hooper:2010mq ; Hooper:2011ti ; Gordon:2013vta ;
Abazajian:2014fta ; Daylan:2014rsa ; Calore:2014xka , the GCE could be
explained by a thermal WIMP annihilating to Standard Model particles. To truly
confirm such a hypothesis, it is crucial to observe a signal in other indirect
channels. In fact, it is possible that AMS-02 is observing an antiproton
excess Aguilar:2016kjl  at a concordant energy range Cuoco:2017rxb ;
Cuoco:2019kuu ; Cholis:2019ejx ; Hooper:2019xss , though the existence of this
excess is not as well established Boudaud:2019efq ; Heisig:2020nse . While
promising, it has also been suggested that the GCE signal could be generated
by millisecond pulsars Cholis:2014noa ; Cholis:2014lta . In recent years, the
debate surrounding the origin of the GCE has intensified Lee:2014mza ;
Bartels:2015aea ; Lee:2015fea ; Macias:2016nev ; Haggard:2017lyq ;
Bartels:2017vsx ; Macias:2019omb ; Leane:2019xiy ; Zhong:2019ycb ;
Leane:2020nmi ; Leane:2020pfc ; Buschmann:2020adf ; Abazajian:2020tww ;
List:2020mzd ; Mishra-Sharma:2020kjb ; Karwin:2016tsw . New measurements in
the coming decade and a better theoretical understanding of Galactic diffuse
emission models will help settle this debate, but until then, the origin of
the GCE remains unknown and dark matter annihilation remains a viable
explanation.

As discussed in Daylan:2014rsa ; Calore:2014xka , the GCE can be well
described by dark matter annihilations, particularly to $b\bar{b}$. This has
fostered the development of many dark matter models with WIMP-like
annihilation mechanisms, which are too numerous to review here (see
Arcadi:2017kky  for a review). Of these, models with pseudoscalar s-channel
mediators are particularly well-motivated because they are neutral and can
evade direct detection constraints. In particular, if the dark matter lives
close to resonance, the annihilation cross section can be boosted enough to
explain the GCE Huang:2013apa ; Boehm:2014hva ; Cheung:2014lqa ; Guo:2014gra ;
Cao:2014efa ; Berlin:2015wwa ; Gherghetta:2015ysa ; Duerr:2015bea . While much
of the previous work relies on the introduction of a new pseudoscalar
mediator, the authors of Carena:2019pwq  proposed an interesting alternative.
In their setup, the dark sector is connected to the visible sector via a CP-
violating coupling to the Higgs, which allows annihilation and spin-
independent scattering to be governed by different parameters. In principle,
the CP-violating coupling can generate a viable thermal relic candidate even
away from the resonance, by suppressing the scattering rather than enhancing
the annihilation. However, in Carena:2019pwq , the authors consider specific
model realizations within the context of supersymmetry where the benchmark
best fit model still has the dark matter mass very close to half the Higgs
mass.

In this work, we extract the key ingredients of their model, namely a Majorana
dark matter candidate with CP-violating coupling to the Higgs, and explore the
extent of freedom away from the mass resonance that can be achieved with
larger CP-violating couplings. We see that for large enough coupling in the
dark matter EFT, there is $\mathcal{O}(10)$ GeV flexibility for the dark
matter mass when the phase is approximately $\pi/2$.

We also consider and explore the phenomenology of two different minimal UV
realizations of this scenario: singlet-doublet dark matter Mahbubani:2005pt ;
DEramo:2007anh ; Enberg_2007 ; Cohen:2011ec ; Cheung:2013dua ; Abe:2014gua ;
Calibbi:2015nha ; Freitas:2015hsa ; Banerjee:2016hsk ; Cai_2017 ; Lopez-
Honorez:2017ora  and doublet-triplet dark matter Dedes:2014hga ; Abe:2014gua ;
Freitas:2015hsa ; Lopez-Honorez:2017ora . We study both how these models
translate to EFT parameters, and constraints governing these UV realizations,
including contributions to the electron electric dipole moment (EDM), the
Peskin-Takeuchi parameters, as well as possible collider signatures. We find
that while the dark matter mass and CP-violating phase are independent
parameters in the EFT, their dependence in the UV completion is quite
nonlinear since the Yukawa coupling directly affects the dark matter mass.
Specifically, it is difficult to achieve the phase tuning scenario without
also tuning the mass in the UV completion, because the large couplings that
are required to generate the annihilation cross section when away from
resonance also change the dark matter mass. Additionally, we find that the
amount of CP-violation in the UV may not be reflective of that observed in the
EFT. In the singlet-doublet case, we find two different types of viable
parameter space. When the UV couplings are small, both the singlet mass in the
UV and the dark matter mass must be very close to $m_{h}/2$, but the phase is
flexible. When the UV couplings are larger, parameters must be chosen such
that both the phase of the dark matter-Higgs coupling and the dark matter mass
must be somewhat tuned, but there is more flexibility in the dark matter and
singlet masses than in the small coupling case. In the doublet-triplet model,
we find that EDM, spin-independent direct detection, and charged fermion
collider search constraints are sufficient to rule out any WIMP-scale
annihilation signal.

The rest of this paper is organized as follows. In Section 2, we discuss the
effective field theory of Majorana dark matter interacting with the Standard
Model through a CP-violating Higgs coupling. The EFT parameters dictate the
annihilation and scattering cross sections which are broadly applicable
independent of specific UV completions. In Section 3, we UV complete the EFT
by introducing a singlet Majorana fermion and a doublet Dirac fermion. In
Section 4, we consider another UV completion by introducing a doublet Dirac
fermion and a triplet Majorana fermion. We discuss the strong constraints
placed on each of these models by a variety of complementary experimental
probes such as the electron EDM, precision electroweak parameters, and
collider searches. Finally, we offer concluding remarks in Section 5.

##  2 Model Independent Constraints in the Effective Theory

In this section we take an effective field theory approach and focus on the
phenomenology of a single species of Majorana dark matter which couples to the
visible sector via a CP-violating Higgs portal. After spontaneous symmetry
breaking (SSB), the corresponding terms in the Lagrangian are given by

$\mathcal{L}\ni\frac{y_{h\chi}}{2\sqrt{2}}h\bar{\chi}P_{L}\chi+\frac{y^{*}_{h\chi}}{2\sqrt{2}}h\bar{\chi}P_{R}\chi+\frac{g_{Z\chi}}{2}Z_{\mu}\bar{\chi}\gamma^{\mu}\gamma^{5}\chi$
(1)

where the CP-violation manifests in the complex nature of dark matter-Higgs
coupling $y_{h\chi}$. Furthermore, we have also allowed for a coupling
$g_{Z\chi}$ to the $Z$ boson.111$\chi$ does not have a vector current coupling
because $\bar{\chi}\gamma^{\mu}\chi$ vanishes identically for Majorana
fermions.

As in all WIMP-type solutions to the GCE, the burden of the model is to
reconcile the $\mathcal{O}(1)$ pb annihilation cross section necessary to
achieve both the observed gamma-ray excess and the dark matter relic density,
with the $\mathcal{O}(10^{-10})$ pb bounds on spin-independent scattering with
nucleons from direct detection experiments. Traditionally, this is achieved
for Higgs-portal dark matter by tuning the dark matter mass to the s-channel
resonance $2m_{\chi}\sim m_{h}$, but an additional avenue is available in the
case of our model.

In the non-relativistic limit, two Majorana fermions form a CP-odd state, so
annihilation into the CP-even Higgs through a CP-conserving coupling is p-wave
suppressed. It then follows that if the coupling is complex, the annihilation
in this limit is dominantly set by the imaginary part of $y_{h\chi}$, which is
reflected in the result we obtain in Equation 4. Conversely, the dark matter
scattering off of the nucleon (or quark) does not require any CP-violation
since the initial and final states have the same CP properties, and thus we
expect the spin-independent scattering cross section to be proportional to the
real part of $y_{h\chi}$. This is reflected in the result we obtain in
Equation 13. Therefore, the phase of the Higgs coupling can also contribute to
a large hierarchy between the scattering and annihilation cross sections. With
this intuition, we describe the details and corresponding phenomenology of
this theory in the remainder of this section.

###  2.1 Annihilation

Annihilation is mediated by both the Higgs and the $Z$ boson through an
s-channel diagram. The dark sector couplings contributing to dark matter
annihilation into SM fermions are given in Equation 1, and the visible sector
couplings have the form

$\displaystyle\mathcal{L}\ni\sum_{f}\frac{y_{hf}}{\sqrt{2}}h\bar{f}f+g_{Zf}Z_{\mu}\bar{f}\gamma^{\mu}(v_{f}-a_{f}\gamma^{5})f.$
(2)

The couplings are given by their SM values

$y_{hf}=-\frac{\sqrt{2}m_{f}}{v},\quad
g_{Zf}=\frac{e}{2\cos\theta_{w}\sin\theta_{w}},\quad
v_{f}=I_{3}-2Q\sin^{2}\theta_{w},\quad a_{f}=I_{3},$ (3)

where $v$ is the Higgs vev, $\theta_{w}$ is the Weinberg angle, and $m_{f}$,
$I_{3}$, and $Q$ are the mass, weak isospin, and electric charge of the
fermion respectively. In the non-relativistic limit, the total spin-averaged
amplitude squared for annihilation can be written as

$|\mathcal{M}|_{\chi\chi\to
f\bar{f}}^{2}=4m^{2}_{\chi}\left[\frac{m_{f}^{2}}{m_{Z}^{4}}g^{2}_{Zf}g_{Z\chi}^{2}a_{f}^{2}+y^{2}_{hf}\mathrm{Im}[y_{h\chi}]^{2}\frac{(m_{\chi}^{2}-m_{f}^{2})}{(m_{h}^{2}-4m_{\chi}^{2})^{2}+m_{h}^{2}\Gamma_{h}^{2}}\right],$
(4)

where $\Gamma_{h}$ denotes the width of the Higgs. The Higgs mediated piece
depends only on the imaginary part of the coupling as expected. The cross
section is correspondingly given by

$\langle\sigma v\rangle=\sum_{m_{f}\leq
m_{\chi}}\frac{N_{c}\sqrt{m_{\chi}^{2}-m_{f}^{2}}}{64\pi
m_{\chi}^{3}}|\mathcal{M}|_{\chi\chi\to f\bar{f}}^{2}.$ (5)

If the dark matter is a thermal relic, then the present-day dark matter
abundance, $\Omega_{\chi}h^{2}=0.11$, sets the annihilation cross section at
the time of freeze-out, which is the well-known $\mathcal{O}(1)$ pb weak-scale
cross section Zeldovich:1965gev ; Chiu:1966kg ; Lee:1977ua ; Hut:1977zn ;
Wolfram:1978gp ; Steigman:1979kw ; Scherrer:1985zt ; Bertstein:1985 ;
Srednicki:1988ce ; Griest:1990kh ; Gondolo:1990dk ; Steigman:2012nb . Recent
work Binder:2017rgn ; Abe:2020obo  has shown that for models with a hierarchy
between annihilation and scattering strengths, early kinetic decoupling before
freeze-out alters this number, requiring a larger cross section to achieve the
observed abundance. At most extreme, a $\sim 20$ pb annihilation cross section
may be needed for a $\sim 57$ GeV dark matter with purely imaginary couplings,
though this is quite sensitive to the details of the QCD phase transition.
However, this effect is significantly weaker for masses $\gtrsim m_{h}/2$, so
we do not take our annihilation cross section to be this large.

At present, dark matter annihilation is expected to produce a distribution of
gamma-rays whose flux is given by

$\frac{d^{2}\Phi_{\gamma}}{d\Omega dE_{\gamma}}=\frac{1}{2}\langle\sigma
v\rangle\left(\sum_{f}\frac{dN_{\gamma}}{dE\gamma}\mathrm{Br}_{\chi\chi\to
f\bar{f}}\right)\int_{\rm los}\frac{\rho_{\chi}^{2}(r)d\ell}{4\pi
m_{\chi}^{2}},$ (6)

where $\mathrm{Br}_{\chi\chi\to f\bar{f}}$ denotes the branching ratio to the
$f\bar{f}$ final state, and $dN_{\gamma}/dE_{\gamma}$ its corresponding
injection spectrum. $\rho_{\chi}(r)$ denotes the dark matter halo profile and
is integrated over the line-of-sight to the Galactic Center. It has been shown
that the Fermi GCE data is well-modeled by a Higgs portal dark matter with a
cross section $\langle\sigma v\rangle\sim 3$ pb, assuming a modified NFW
profile Hooper:2010mq . As the precise best fit depends on many details,
including the galactic profile and background modeling DiMauro:2021raz , in
conjunction with the modeling uncertainties of the thermal relic argument, we
will consider here a range of cross sections $\langle\sigma v\rangle$ from 1
to 10 pb to be in concordance with both the GCE and the relic abundance.

###  2.2 Direct Detection

In contrast with annihilation, processes relevant for direct detection occur
below the weak scale and should be considered in terms of effective
interactions with target nuclei. Much of the subsequent discussion follows
Lin:2019uvt . At momentum transfers $t\ll m_{Z}^{2}$, the interactions in
Equations 1 – 2 are rewritten as the following dimension-6 operators

$\displaystyle\mathcal{L}\ni$
$\displaystyle\frac{C_{S}}{m_{h}^{2}}\bar{\chi}\chi\bar{f}f+\frac{C_{PS}}{m_{h}^{2}}\bar{\chi}\gamma^{5}\chi\bar{f}f+\frac{C_{V}}{m_{Z}^{2}}\bar{\chi}\gamma^{\mu}\gamma^{5}\chi\bar{f}\gamma_{\mu}f+\frac{C_{PV}}{m_{Z}^{2}}\bar{\chi}\gamma^{\mu}\gamma^{5}\chi\bar{f}\gamma_{\mu}\gamma^{5}f$
(7)

with $C_{S}$, $C_{PS}$, $C_{V}$, and $C_{PV}$ denoting the scalar, pseudo-
scalar, vector, and pseudo-vector pieces of the quark-gauge couplings
respectively. The contributions governed by $C_{PS}$ and $C_{V}$ are velocity-
suppressed and we neglect them in the following. After matching to the UV
theory, the coefficients are given by

$C_{S}=\frac{1}{2}\mathrm{Re}[y_{h\chi}]y_{hf}\qquad
C_{PV}=g_{Z\chi}g_{Zf}a_{f}.$ (8)

In the zero momentum transfer limit, the nucleon-level operators are matched
to the quark-level ones via form factors

$\displaystyle\langle
N(p)|\bar{f}\gamma^{\mu}\gamma^{5}f|N(p^{\prime})\rangle$
$\displaystyle=\bar{u}_{N}(p)\left[\Delta_{1}^{f,N}(q^{2})\gamma^{\mu}\gamma^{5}\right]u_{N}(p^{\prime})$
(9) $\displaystyle\langle N(p)|\bar{f}f|N(p^{\prime})\rangle$
$\displaystyle=\frac{m_{N}}{m_{f}}f^{N}_{f}\bar{u}_{N}(p)u_{N}(p^{\prime})$
(10)

where $N$ represents a nucleon (a proton or neutron), $q=p^{\prime}-p$ denotes
the momentum transfer, and the form factors are listed in Table 1. We have
neglected higher order terms in $q^{2}$. For the scalar term specifically, the
heavy quarks also contribute via a gluon loop. After integrating out heavy
quarks, the relevant operator for each flavor appears as

$-\frac{C_{S}}{m_{h}^{2}}\frac{\alpha_{s}}{12\pi m_{f}}\bar{\chi}\chi
G^{\mu\nu}G_{\mu\nu}.$ (11)

To match to the nucleon-level picture the following matrix element is taken
into account

$\langle
N(p)|G^{\mu\nu}G_{\mu\nu}|N(p^{\prime})\rangle=-\frac{8\pi}{9\alpha_{s}}m_{N}f^{N}_{g}\bar{u}_{N}(p)u_{N}(p^{\prime}).$
(12)

In terms of the quark-level couplings, the nucleon-level spin-independent
cross section is given by

$\sigma_{SI}=\frac{m_{N}^{2}m_{\chi}^{2}}{4\pi(m_{\chi}+m_{N})^{2}}\left[\frac{\text{Re}[y_{h\chi}]}{m_{h}^{2}}\left[\sum_{f\in
u,d,s}y_{hf}\frac{m_{N}}{m_{f}}f^{N}_{f}+\sum_{f\in
c,b,t}y_{hf}\frac{2}{27}\frac{m_{N}}{m_{f}}f^{N}_{g}\right]\right]^{2}.$ (13)

As discussed earlier, the cross section only depends on the real part of the
Higgs coupling. Furthermore, the dependence on the coupling to the $Z$ boson
vanishes in the $q\to 0$ limit. Likewise the spin-dependent cross section is
given by

$\displaystyle\sigma_{SD}=$
$\displaystyle\frac{3m_{N}^{2}m_{\chi}^{2}}{\pi(m_{\chi}+m_{N})^{2}}\left[\frac{g_{Z\chi}}{4m_{Z}^{2}}\sum_{f\in
u,d,s}g_{Zf}a_{f}\Delta_{1}^{N,f}\right]^{2}.$ (14)

| $\Delta_{1}^{N,u}$ | $\Delta_{1}^{N,d}$ | $\Delta_{1}^{N,s}$ | $f^{N}_{u}$ | $f^{N}_{d}$ | $f^{N}_{s}$ | $f^{N}_{g}$
---|---|---|---|---|---|---|---
Protons | 0.80 | -0.46 | -0.12 | 0.018 | 0.027 | 0.037 | 0.917
Neutrons | -0.46 | 0.80 | -0.12 | 0.013 | 0.040 | 0.037 | 0.910
Table 1: Here we show the light quark and gluon form factors for the proton
and neutron. These values come from DelNobile:2013sia ; Hill:2014yxa ;
Bishara:2017pfq ; Ellis:2018dmb  and are summarized in Lin:2019uvt .

Figure 1: The ratio between annihilation and spin-independent direct detection
cross sections on the $m_{\chi}-\phi_{h\chi}$ plane for different values of
$|y_{h\chi}|$. The region allowed by direct detection is inside the solid
XENON1T Aprile:2017iyp ; Aprile:2018dbl  constraint line, while the region
allowed by annihilation is between the solid 1 pb and 10 pb lines. We also
show projected limits from LZ Akerib:2018lyp  as dashed lines. Note that the
axis scales on the two plots are different. We assume $m_{h}=125.2$ GeV here
and throughout this paper. The left plot shows the mass resonance with small
$y_{h\chi}$, for which the dark matter mass must be tuned to within less than
a GeV of the pole, but there is some flexibility in the phase. The right plot
shows the phase tuning: away from $m_{h}=2m_{\chi}$ a large coupling is
required to achieve a sufficient annihilation cross section, but tuning the
phase near $\pi/2$ avoids direct detection limits despite the large coupling.
In this case, the flexibility of the allowed mass range changes to
$\mathcal{O}(10)$ GeV. Both of these plots include a small non-zero $Z$
coupling that is consistent with spin-dependent direct detection constraints.
The limits are similar for vanishing $Z$ coupling. Figure 2: Spin-dependent
direct detection limits as a function of dark matter mass and dark matter-$Z$
coupling. Constraints are close to horizontal because the spin-dependent cross
section depends on the reduced mass. For neutrons, XENON1T Aprile:2019dbj  is
the strongest model independent constraint. For protons, PICO
PhysRevLett.118.251301 ; PhysRevD.100.022001  provides a slightly stronger
constraint. Additionally, we show projected limits from LZ Akerib:2018lyp .

###  2.3 Results and Discussion

In this subsection we examine the phenomenology of the effective theory, and
discuss the regions of parameter space where a high annihilation and low
scattering cross section can be achieved – specifically we are interested in
an annihilation cross section between approximately 1 and 10 pb to fit the GCE
and a scattering cross section consistent with direct detection experiments.
For spin-independent scattering, the strongest limits come from XENON1T
Aprile:2017iyp ; Aprile:2018dbl , while for spin-dependent scattering, the
strongest limits come from both XENON1T Aprile:2019dbj  and PICO
PhysRevLett.118.251301 ; PhysRevD.100.022001 . LZ Akerib:2018lyp  and XENONnT
Aprile:2020vtw  are projected to improve on current limits within the
parameter space of interest. The projected limits are comparable, so we show
only one in our figures for clarity. We omit limits from IceCube
Aartsen:2016zhm , LUX Akerib:2016lao ; Akerib:2016vxi  and PandaX-II
Cui:2017nnn  because they are slightly weaker than those we’ve shown for
$\mathcal{O}(60)$ GeV dark matter. For the spin-independent constraints, we
consider only dark matter-proton scattering because in this case the
difference between proton and neutron cross sections is negligible.

First we review which masses and coupling magnitudes are in general
concordance with scattering constraints and annihilation requirements. Typical
couplings that can generate an annihilation cross section of $\sim 1$ pb are
shown in Equation 15 for two different dark matter masses. In Equation 16, we
show approximate couplings and masses that are consistent with direct
detection constraints.

$\begin{split}\left[\frac{\langle\sigma v\rangle}{1\text{
pb}}\right]&=\left[\frac{m_{\chi}}{80\text{
GeV}}\right]^{2}\left[\frac{4m_{\chi}^{2}-m_{h}^{2}}{10^{4}\text{
GeV}^{2}}\right]^{-2}\left[\frac{y_{h\chi}\sin\phi_{h\chi}}{1.0}\right]^{2}\\\
&=\left[\frac{m_{\chi}}{62.5\text{
GeV}}\right]^{2}\left[\frac{4m_{\chi}^{2}-m_{h}^{2}}{50\text{
GeV}^{2}}\right]^{-2}\left[\frac{y_{h\chi}\sin\phi_{h\chi}}{0.007}\right]^{2}\end{split}$
(15) $\begin{split}\left[\frac{\sigma_{SI}}{10^{-10}\text{
pb}}\right]&=\left[\frac{y_{h\chi}\cos\phi_{h\chi}}{0.02}\right]^{2}\\\
\left[\frac{\sigma_{SD}}{10^{-5}\text{
pb}}\right]&=\left[\frac{g_{Z\chi}}{0.01}\right]^{2}\end{split}$ (16)

We remind the reader that the free parameters of the theory are $m_{\chi}$,
$g_{Z\chi}$, and the complex coupling $y_{h\chi}$ with phase $\phi_{h\chi}$.
While $g_{Z\chi}$ and Im[$y_{h\chi}$] set the annihilation cross section, only
Re[$y_{h\chi}$] sets the magnitude of scattering. In order to generate a large
enough annihilation cross section while avoiding direct detection constraints,
Higgs portal dark matter models typically tune the dark matter mass close to
half the Higgs mass Huang:2013apa ; Boehm:2014hva ; Cheung:2014lqa ;
Guo:2014gra ; Cao:2014efa ; Berlin:2015wwa ; Gherghetta:2015ysa ;
Carena:2018nlf ; Carena:2019pwq . While tuning the mass is one way to generate
the correct ratio in this model, we emphasize that in the EFT, the correct
ratio can also be obtained for a wider mass range by increasing the magnitude
of the Higgs coupling while tuning the phase, $\phi_{h\chi}$, of the Higgs
coupling close to $\pi/2$ to suppress direct detection constraints. This is
illustrated in Figure 1, which plots annihilation and spin independent direct
detection constraints in the $m_{\chi}-\phi_{h\chi}$ plane for different
magnitudes of the Higgs couplings. We can see that near the mass resonance, a
small Higgs coupling ($\sim 0.02$) is sufficient to generate the annihilation
cross section and the phase does not need to be near $\pi/2$ to avoid direct
detection constraints. However, with phase tuning, the larger Higgs coupling
required to generate the correct annihilation cross section away from
resonance is allowed because direct detection only constrains the real part of
$y_{h\chi}$. This widens the mass range considerably to $\mathcal{O}(10)$ GeV.
Even for the mass resonance, the coupling cannot be purely real, because the
leading velocity dependent term is not large enough to generate the required
annihilation cross section given the finite Higgs width. See Appendix A for
more details. Note that while in principle a large pseudo-vector $Z$ coupling
could also generate a sufficient annihilation cross section, this is
constrained by spin-dependent direct detection constraints, as shown in Figure
2. Within the range of $Z$ couplings allowed by direct detection, the effect
on the allowed annihilation signal is negligible.

##  3 Singlet-Doublet Model

A well-motivated way to UV complete the dark matter EFT provided in Section 2
in a gauge invariant manner is to introduce additional particles charged under
$G_{SM}$. In this section, we discuss a simple potential UV completion, where
the only additional particles we introduce to the Standard Model are a singlet
Majorana fermion and a doublet Dirac fermion. This model has previously been
discussed in other contexts in Mahbubani:2005pt ; DEramo:2007anh ; Enberg_2007
; Cohen:2011ec ; Cheung:2013dua ; Abe:2014gua ; Calibbi:2015nha ;
Freitas:2015hsa ; Banerjee:2016hsk ; Cai_2017 ; Lopez-Honorez:2017ora .

###  3.1 Model in the UV

We start by establishing notation and describing the model. The model contains
a singlet Majorana fermion $\psi_{1}$ and an additional SU(2) doublet Dirac
fermion with hypercharge 1/2. We describe the SU(2) doublet with two left
handed Weyl fermions $\psi_{2}$ (with neutral component $\psi_{2}^{0}$ and
charged component $\psi_{2}^{1}$) and $\tilde{\psi}_{2}$ (with neutral
component $\tilde{\psi}_{2}^{0}$ and charged component
$\tilde{\psi}_{2}^{-1}$). All new fermions are SU(3) singlets. The Lagrangian
for this model is

$\mathcal{L}=\mathcal{L}_{SM}+\mathcal{L}_{\rm
kinetic}-m_{2}\psi_{2}\cdot\tilde{\psi}_{2}-\frac{m_{1}}{2}\psi_{1}\psi_{1}+Y\psi_{1}H^{\dagger}\psi_{2}-\tilde{Y}\psi_{1}H\cdot\tilde{\psi}_{2}+\text{h.c.}$
(17)

As we introduce three new fields and four free parameters, there is one
remaining physical phase. We make the choice to fix each of the Yukawa terms
to the same phase, which carries the CP-violation,

$Y\equiv
y\,\mathrm{e}^{i\delta_{CP}/2},\qquad\tilde{Y}\equiv\tilde{y}\,\mathrm{e}^{i\delta_{CP}/2}.$
(18)

After SSB, the mass terms are written as

$\displaystyle\begin{aligned} \mathcal{L}_{\rm
mass}=&-m_{2}\left(\tilde{\psi}_{2}^{-1}\psi_{2}^{1}-\tilde{\psi}_{2}^{0}\psi_{2}^{0}\right)-\frac{m_{1}}{2}\psi_{1}\psi_{1}+\frac{v}{2}y\,\mathrm{e}^{i\delta_{CP}/2}\psi_{1}\psi_{2}^{0}+\frac{v}{2}\tilde{y}\,\mathrm{e}^{i\delta_{CP}/2}\psi_{1}\tilde{\psi}_{2}^{0}+\text{h.c.}\\\
\end{aligned}$ (19)

Let us define
$\psi_{2}^{s}\equiv\frac{1}{\sqrt{2}}(\psi_{2}^{0}+\tilde{\psi}_{2}^{0})$ and
$\psi_{2}^{d}\equiv\frac{1}{\sqrt{2}}(\psi_{2}^{0}-\tilde{\psi}_{2}^{0})$ to
be the two Majorana fermions that constitute the neutral Dirac fermion
$\\{\psi_{2}^{0},\tilde{\psi}_{2}^{0}\\}$. The mass eigenstates thus result
from the mixing of the doublet and singlet Majorana fermions,
$\psi_{i}=(\psi_{2}^{s},\psi_{2}^{d},\psi_{1})_{i}$. We will denote the mass
eigenstates $\chi_{i}=(\chi,\chi_{1},\chi_{2})_{i}$, the lightest of which,
$\chi$, is the dark matter candidate. Then

$\mathcal{L}_{\rm
mass}=-m_{2}\tilde{\psi}_{2}^{-1}\psi_{2}^{1}-\frac{1}{2}\psi_{i}M_{ij}\psi_{j},\\\
$ (20)

where $M$ is the mass matrix. This basis change is governed by $J$, the matrix
of eigenvectors that diagonalizes both $M^{\dagger}M$ and $M$, phase rotated
such that $J^{T}MJ$ has real eigenvalues. After diagonalizing, the Higgs
Yukawa couplings are

$\mathcal{L}_{\rm
Higgs}=\frac{1}{2}h\chi_{i}[J^{T}U_{h}J]_{ij}\chi_{j}+\text{h.c.}$ (21)

where

$U_{h}=\begin{pmatrix}0&0&\frac{(Y+\tilde{Y})}{2}\\\
0&0&\frac{(Y-\tilde{Y})}{2}\\\
\frac{(Y+\tilde{Y})}{2}&\frac{(Y-\tilde{Y})}{2}&0\\\ \end{pmatrix}.$ (22)

Figure 3: Plots show EFT coupling phase and dark matter mass as a function of
$m_{1}$ and $m_{2}$ for different values of $y,\tilde{y},$ and $\delta_{CP}$.
Left: $\delta_{CP}$ is fixed to 1.5 while $y\tilde{y}$ is varied. Right:
$y\tilde{y}$ is fixed to $-0.375$ while $\delta_{CP}$ is varied. The shaded
regions give a sense of the width of the regions of interest:
$60\,\mathrm{GeV}\leq m_{\chi}\leq 65\,\mathrm{GeV}$ and
$1.55\leq\phi_{h\chi}\leq 1.60$. We can see that changing $y\tilde{y}$ has a
minimal effect on $m_{\chi}$ at large $m_{2}$ but strongly affects which
masses correspond to the central value of $\phi_{h\chi}=\pi/2$. $y\tilde{y}$
also affects the smallest value of $m_{2}$ that can lead to a mass resonance.
Changing $\delta_{CP}$ has a larger effect on which $m_{1}$ is required to get
the mass resonance, and also affects the width of the $\phi_{h\chi}=\pi/2$
band in addition to the position of its central value.

Since one of the new fermions is an SU(2) doublet, the new fermions also
couple to the electroweak gauge bosons. The $Z$ couplings are

$\mathcal{L}_{Z}=\chi^{\dagger}_{i}[J^{\dagger}U_{Z}J]_{ij}Z\overline{\sigma}\chi_{j}+g_{Z}(\cos^{2}\theta_{W}-\sin^{2}\theta_{W})(\psi_{2}^{1\dagger}Z\overline{\sigma}\psi_{2}^{1}-\tilde{\psi}_{2}^{-1\dagger}Z\overline{\sigma}\tilde{\psi}_{2}^{-1})$
(23)

while the $W$ couplings are

$\mathcal{L}_{W}=\chi^{\dagger}_{i}[bJ^{*}]_{i}W^{+}\overline{\sigma}\tilde{\psi}_{2}^{-1}+{\psi}_{2}^{1\dagger}W^{+}\overline{\sigma}[aJ]_{j}\psi_{j}+\text{h.c.}$
(24)

where $g$ is the SU(2) gauge coupling, $g^{\prime}$ is the U(1) hypercharge
gauge coupling, $\theta_{W}$ is the Weinberg angle, and
$g_{Z}\equiv\sqrt{g^{2}+g^{\prime 2}}/2$. Here,
$a_{i}=(g/2,g/2,0)_{i},b_{i}=(g/2,-g/2,0)_{i}$ and

$U_{Z}=\begin{pmatrix}0&-g_{Z}&0\\\ -g_{Z}&0&0\\\ 0&0&0\end{pmatrix}.$ (25)

The dark matter candidate $\chi$ obtains the couplings seen in the EFT via
mixing between the singlet and doublet. The strength of these couplings can be
adjusted by altering the makeup of the lightest Majorana fermion. The theory
at this level is fully specified by five degrees of freedom: the singlet mass
$m_{1}$, the doublet mass $m_{2}$, the doublet Yukawa coupling magnitudes
$\\{y,\tilde{y}\\}$ and the associated CP-violating phase $\delta_{CP}$.

###  3.2 Translating to the EFT

Now we discuss how the EFT parameters $g_{Z\chi},m_{\chi},$ and $y_{h\chi}$
depend on the UV parameters $y,\tilde{y},m_{1},m_{2}$, and $\delta_{CP}$. We
focus mostly on the region where $m_{2}$ is large, but also comment on the
more general case.222We also omit the case where both $m_{1}$ and $m_{2}$ are
large. In this case, extremely large couplings are required in order to get
dark matter with mass near $m_{h}/2$. This means the $\delta_{CP}$ must be
small to avoid EDM constraints, which leaves us with $\phi_{h\chi}$ mostly
real and prevents us from simultaneously evading spin-independent constraints.
Since the theory has a charged fermion with mass $m_{2}$, parameter space with
small $m_{2}$ will generically be ruled out by collider constraints LEPSUSYWG1
; LEPSUSYWG2 . EDM and electroweak constraints are likewise more stringent in
this regime.

Figure 3 shows the EFT mass and phase as a function of $m_{1}$ and $m_{2}$ for
different values of the UV coupling magnitudes and phase. On the left we show
multiple values of $y\tilde{y}$ for fixed $\delta_{CP}$ while on the right we
show multiple values of $\delta_{CP}$ for fixed $y\tilde{y}$. In both cases,
we can see that only a narrow range in $m_{1}$ translates to dark matter with
mass near the mass resonance. When $m_{2}$ is large, the lightest fermion is
mostly $m_{1}$. In this limit, mixing is small, so to have the dark matter
mass near the mass resonance, $m_{1}$ must be fairly close to half the Higgs
mass. We can see that changing $y\tilde{y}$ changes where $\phi_{h\chi}=\pi/2$
is located but only has a minimal effect on which $m_{1}$ value translates to
the mass resonance. We can also see that for the same $m_{1}$, smaller
$y\tilde{y}$ requires a correspondingly smaller $m_{2}$ to get dark matter
with $m_{\chi}\approx m_{h}/2$. Changing $\delta_{CP}$ also changes the
location of $\phi_{h\chi}=\pi/2$ contour, but additionally affects the $m_{1}$
required to get the mass resonance and the width of the
$\phi_{h\chi}\approx\pi/2$ band.

Figure 4: Diagram generating dark matter mass in the limit where $m_{2}$ is
large. Figure 5: Plot of $\phi_{h\chi}$ as a function of $\delta_{CP}$ for
different values of $m_{2}$. We note that as $m_{2}$ increases, the IR phase
maps directly to the UV phase and $\phi_{h\chi}\sim\delta_{CP}$.

Figure 6: Dark matter mass, EFT phase, and dark matter-Higgs Yukawa coupling
as a function of the UV parameters $y\tilde{y}$ and $\delta_{CP}$ for
different values of $m_{2}$. In each plot we see a similar mass structure: we
see a massless state when $y\tilde{y}$ and $m_{1}$ have opposite signs, and
have a lightest fermion near 60 GeV for both larger and smaller $y\tilde{y}$
than this value. We can also see the scaling of both the EFT mass and Higgs
coupling with $y\tilde{y}$ and $m_{2}$. Note the different values on the
$y\tilde{y}$ axis in each of the plots.

Figure 4 shows that the corrections to the mass scale as
$y\tilde{y}v^{2}/m_{2}$.333Although we need to phase rotate $\psi_{2}^{s}$,
the phase rotations in the couplings and mass insertion cancel out. This
diagram also tells us that $\phi_{h\chi}=\delta_{CP}$ in the large $m_{2}$
limit, as long as mixing is small and the dark matter mass comes mostly from
$m_{1}$ rather than the Higgs vev. This can also be seen in Figure 5. When the
dark matter mass gets a large contribution from the Higgs vev the story is
more complicated: when $y\tilde{y}$ and $m_{1}$ have opposite signs, the Higgs
contribution can cancel with $m_{1}$ at $y\tilde{y}=-m_{2}m_{1}/v^{2}$ to get
a massless state. There is a mass resonance contour for $y\tilde{y}$ both
larger and smaller than this value, which can be seen in Figure 6. We might
also ask whether a small $\delta_{CP}$ in the UV can translate to
$\phi_{h\chi}\approx\pi/2$ in the IR and produce an annihilation signal that
evades both direct detection and EDM constraints. However, from the same
figure, we can see that although there is a point where small $\delta_{CP}$
translates to $\phi_{h\chi}\approx\pi/2$, it corresponds precisely to the
massless state mentioned above and cannot generate our annihilation signal.
This is evidenced by all the phase contours converging at the massless point,
because when $m_{\chi}$ is zero, we can freely rotate $m_{1}$ to absorb the
phase in $y$ since the phase is no longer physical.

In the small mixing and large $m_{2}$ limit, there are two contributions to
the Higgs coupling: one where $\psi_{1}$ mixes into $\psi_{2}^{s}$ and one
where it mixes into $\psi_{2}^{d}$, as shown in Figure 7. Each of these
contributes $(y\pm\tilde{y})^{2}v/m_{2}$, with a relative minus sign between
the two contributions because we need to phase rotate $\psi_{2}^{s}$ to have
positive mass. This means the Higgs coupling scales as $y\tilde{y}v/m_{2}$,
which determines the scaling of the annihilation signal. This can also be seen
from the pink lines in Figure 6. Note that this scaling breaks down once the
Yukawa contributions become the dominant contribution to the mass.

Figure 7: Diagram that generates the dark matter-Higgs coupling in the limit
where $m_{2}$ is large.

In the same limit, the dominant contribution to the $Z$ coupling comes from
Figure 8, which scales as $g_{Z}(y^{2}-\tilde{y}^{2})v^{2}/m_{2}^{2}$. Even
away from this limit, we still get a vanishing $Z$ coupling for $y=\tilde{y}$,
because only one of the doublet states mixes with the singlet when
$y=\tilde{y}$. For small $m_{2}$, spin-dependent direct detection constraints
require $y\approx\tilde{y}$, but for $m_{2}\gtrsim 500$ GeV this constraint
becomes irrelevant, since the Higgs coupling (which determines the
annihilation signal) scales as $m_{2}^{-1}$ while the $Z$ coupling scales as
$m_{2}^{-2}$. This can be seen in Figure 9.

Figure 8: Diagram that generates the dark matter-$Z$ coupling in the limit
where $m_{2}$ is large. Figure 9: Plot of $g_{Z\chi}$ as a function of
$y^{2}-\tilde{y}^{2}$ for different values of $m_{2}$. $g_{Z\chi}$ increases
with increasing $y^{2}-\tilde{y}^{2}$ and decreases with increasing $m_{2}$,
corroborating the scaling derived from the diagrams in Figure 8.

###  3.3 Constraints

In this section, we discuss the experimental constraints that apply to the
singlet-doublet model. We focus on constraints that apply directly to the
parameters in the UV theory, including discussing their scaling in the large
$m_{2}$ limit.

####  3.3.1 Electric Dipole Moment

Any new source of CP-violation in a given model can lead to additional
contributions to electric dipole moments. Since our model contains new CP-
violating couplings to the Higgs, we expect electron EDM constraints to be
relevant for our model. For small $m_{2}$, the EDM limit will be one of the
strongest on our model, since the EDM is precisely constrained to be below
$1.1\times 10^{-29}$ e cm Andreev:2018ayy .

For the singlet-doublet model above, the only relevant diagram is the Barr-Zee
diagram with $W$ bosons in the outer loop Barr:1990vd , displayed in Figure
10. There are no other Barr-Zee diagrams with Higgs or $Z$ legs; since CP-
violation is only in the neutral sector of this model and a charged particle
is necessary to radiate a photon, the inner loop must contain both a neutral
and charged particle. Additionally, there are no other non-Barr-Zee diagrams
that contribute to the EDM at 2 or fewer loops. For any non-Barr-Zee diagrams
to contribute, there would have to be a CP-odd correction to a gauge boson or
Higgs propagator. With only a single external momentum, it is impossible to
contract with an epsilon tensor and make a non-vanishing CP-odd Lorentz
invariant.

To compute the value of the relevant Barr-Zee diagram, we use a simplified
version of Equation 21 in Atwood:1990cm , where we have neglected the neutrino
mass, approximated lepton couplings as flavor diagonal, and used the fact that
one of the fermions in the loop is neutral:

$\begin{split}\frac{d_{e}}{e}=-\frac{g^{2}}{(4\pi)^{4}}\sum_{i}\text{Im}([aJ]_{i}^{*}[bJ^{*}]_{i})\left(\frac{m_{c}m_{n,i}m_{e}}{M_{W}^{4}}\right)G(x_{c},x_{i},0).\end{split}$
(26)

Here, $x_{\alpha}=m_{\alpha}^{2}/M_{W}^{2}$ and $G(a,b,c)$ is defined as

$G(a,b,c)=\frac{1}{1-c}\int_{0}^{1}\frac{dx}{1-x}\left(\frac{c}{z-c}\log\left(\frac{c}{z}\right)+\frac{1}{1-z}\log\left(\frac{1}{z}\right)\right)$
(27)

with

$z(x,a,b)=\frac{b}{x}+\frac{a}{1-x}.$ (28)

Recall from Section 3.1 that couplings $a_{i}$ and $b_{i}$ parameterize the
$W$ boson couplings to the inner loop fermions in the gauge basis, which are
given in Equation 24, and $J$ is the change of basis matrix.

When $m_{2}$ is large enough that we can integrate out the doublet and mixing
is small, the dominant contribution to the EDM comes from Figure 10, since
each helicity of charged fermion couples to a different neutral doublet
component and mixing with the singlet is necessary to generate CP-violation.
This contribution scales as $y\tilde{y}v^{2}/m_{2}^{2}$. The $m_{2}$ scaling
follows from dimensional analysis: three factors of $m_{2}$ from the integral
measure cancel with three of the five factors of $m_{2}$ from the
propagators.444The $\psi_{1}$ propagator also scales as $m_{2}^{-1}$ since
$p=m_{2}\gg m_{1}$.

Figure 10: Leading contribution to the non-zero Barr-Zee diagram in the large
$m_{2}$ limit. In this limit, we can work perturbatively in the gauge basis.
The relevant $W$ couplings are the coefficients of
$\chi_{21}^{\dagger}\frac{\sqrt{g}}{2}W^{-}\chi_{20}$ and
$\tilde{\chi}_{20}^{\dagger}\frac{\sqrt{g}}{2}W^{+}\tilde{\chi}_{2-1}$.

####  3.3.2 Electroweak parameters

Here we consider constraints from electroweak precision measurements, where
deviations from the SM are parametrized by oblique parameters $S,T,U,W,$ and
$Y$ Peskin:1991sw ; Cacciapaglia:2006pk , defined in Equations 29-33.

$\displaystyle S$
$\displaystyle\equiv\frac{4c^{2}s^{2}}{\alpha_{e}}\left[\Pi^{\prime}_{ZZ}(0)-\frac{c^{2}-s^{2}}{cs}\Pi_{Z\gamma}^{\prime}(0)-\Pi_{\gamma\gamma}^{\prime}(0)\right]$
(29) $\displaystyle T$
$\displaystyle\equiv\frac{1}{\alpha_{e}}\left[\frac{\Pi_{WW}(0)}{m_{W}^{2}}-\frac{\Pi_{ZZ}(0)}{m_{Z}^{2}}\right]$
(30) $\displaystyle U$
$\displaystyle\equiv\frac{4s^{2}}{\alpha_{e}}\left[\Pi^{\prime}_{WW}(0)-\frac{c}{s}\Pi_{Z\gamma}^{\prime}(0)-\Pi_{\gamma\gamma}^{\prime}(0)\right]-S$
(31) $\displaystyle W$
$\displaystyle\equiv\frac{m_{W}^{2}s^{2}c^{2}}{8\pi\alpha_{e}}\left[\Pi^{\prime\prime}_{ZZ}(0)+\frac{2s}{c}\Pi^{\prime\prime}_{Z\gamma}(0)+\frac{s^{2}}{c^{2}}\Pi^{\prime\prime}_{\gamma\gamma}(0)\right]$
(32) $\displaystyle Y$
$\displaystyle=\frac{m_{W}^{2}s^{2}}{8\pi\alpha_{e}}\Bigg{[}c^{2}\Pi_{\gamma\gamma}^{\prime\prime}(0)+s^{2}\Pi_{ZZ}^{\prime\prime}(0)-2sc\Pi_{\gamma
Z}^{\prime\prime}(0)\Bigg{]}$ (33)

The masses and couplings are evaluated at $m_{Z}^{2}$ and $c$ and $s$ are
$\cos\theta_{W}$ and $\sin\theta_{W}$ respectively. $\Pi_{XX}$ represents the
new particles’ contribution to the vacuum polarization of the gauge boson $X$
at 1-loop, computed in $\overline{\mathrm{MS}}$ scheme under the convention
shown in Figure 11.

Figure 11: New particles that couple to the Standard Model gauge bosons
contribute to the vacuum polarization at 1-loop through this diagram. The
$X^{\mu}$ represents an electroweak gauge boson. We ignore the
$p^{\mu}p^{\nu}$ terms since they aren’t relevant for Equations 29 \- 33.

To lend intuition, we note that $T$ parametrizes custodial SU(2) breaking
inherent in the asymmetry within the doublet terms; in our theory this
manifests in the difference in Yukawa couplings $y$ and $\tilde{y}$. $U$ is
the derivative of $T$, and thus is typically smaller. All these parameters
fall off with increasing $m_{2}$.

The most recent constraints, at 95% CL, from the LHC yield

$S=-0.01\pm 0.10\qquad T=0.03\pm 0.12\qquad U=-0.01\pm 0.10$ (34)

with correlations +0.92 between $S$ and $T$, -0.80 between $S$ and $U$, and
-0.93 between $T$ and $U$ RPP2020 . W and Y are measured to be

$W=(-2.7\pm 2.0)\times 10^{-3}\qquad Y=(4.2\pm 4.9)\times 10^{-3}$ (35)

with correlation $-0.96$ Barbieri:2004qk , though we find these to be
subdominantly constraining for this theory.

####  3.3.3 Collider Experiments

Constraints from many collider searches (in particular SUSY searches) can be
applied to this model. Specifically, we consider those searches included in
the database of the publicly available SModelS version 1.2.4 software
Khosa:2020zar ; Ambrogi:2018ujg ; Dutta:2018ioj ; Ambrogi:2017neo ;
Kraml:2013mwa ; ATL-PHYS-PUB-2019-029 ; Skands:2003cj ; Alwall:2006yp ;
Buckley:2013jua . To generate the necessary input, we use SARAH 4.14.3
Staub:2008uz ; Staub:2013tta ; Staub:2015kfa  to create modified versions of
SPheno Porod:2003um ; Porod:2011nf  and Madgraph Alwall:2014hca ;
Alwall:2011uj  which include the singlet and doublet. Then we use this version
of SPheno at tree level to compute the spectrum and branching ratios for
SModelS and the run card for Madgraph, which was used to obtain the production
cross sections that SModelS also needs as input. These constraints are
combined into a single exclusion limit labeled LHC when included in our plots.
In addition to this constraint, we also show the constraint from invisible
Higgs decay. We do not include the constraint from invisible $Z$ decay, since
it is not kinematically allowed in the parameter space of interest.

###  3.4 Full Exclusion Limits and Discussion

(a) hellothereifudgeaspace

(b) hellothereifudgeispace

(c) hellothereifudgeaspace

(d) hellothereifudgeispace

Figure 12: Full constraints on the model parameter space in the
$y\tilde{y}-\delta_{CP}$ plane, for different values of $m_{1}$ and $m_{2}$.
In this and subsequent plots the shaded regions denote parameter space ruled
out by experimental bounds Andreev:2018ayy ; Sirunyan:2019twz ; Aprile:2017iyp
; Aprile:2018dbl . For annihilation, we include both an upper and lower bound.
Other constraints are not relevant for these slices of parameter space. Spin-
dependent direct detection constraints in particular are weak since
$y^{2}-\tilde{y}^{2}$ is small. Dotted lines indicate proximity to mass
resonance and pure imaginary EFT coupling: the green dotted lines bound a
region with dark matter mass $60\,\text{GeV}\leq m_{\chi}\leq 65\,\text{GeV}$,
the yellow with EFT phase $1.55\leq\phi_{h\chi}\leq 1.6$. In Figures (a) and
(b) we show that viable parameter space can be found at small couplings,
corresponding to a pure mass resonance with flexibility in $\phi_{h\chi}$. In
this case, smaller values of $m_{2}$ are allowed but $m_{1}$ must be close to
$m_{h}/2$. Figures (b) - (d) also show allowed parameter space for larger
couplings: (b) shows $m_{1}\approx m_{h}/2$; (c) and (d) show $m_{1}$ further
away from $m_{h}/2$ for two different values of $m_{2}$. In all of (b) - (d),
viable parameter space requires $m_{1}\approx 60-70$ GeV, $\delta_{CP}\gtrsim
1$, and $\phi_{h\chi}\approx\pi/2$.

Finally, combining all of these constraints, we examine the remaining
parameter space for singlet-doublet dark matter that has the desired amount of
annihilation. Our results are shown in Figures 12 and 13. We find that in all
cases, some tuning of the parameters is required, but that there is
flexibility in which UV parameters we need to tune.

As in the EFT, in order to achieve a pure mass resonance (and not have to tune
the EFT phase) we need small couplings. This can be seen in Figures 12(a) and
12(b). The spin-independent constraints are weak for small couplings,
regardless of $\delta_{CP}$ or the EFT phase. Other constraints are even less
restrictive, except for the EDM at very large $\delta_{CP}$. Since the
couplings are small, $m_{1}$ must be tuned near $m_{h}/2$ in order to achieve
a sufficient annihilation signal, but there is flexibility in the value of
$m_{2}$, as can be seen in Figure 13(a). This is the region of parameter space
that is relevant for the best fit in Carena:2019pwq .

If instead we choose our parameters so that we allow the EFT phase to be tuned
near $\pi/2$, there is other viable parameter space with larger couplings.
Here, we have slightly more flexibility in $m_{1}$ (which still needs to be
roughly $60-70$ GeV), but $m_{2}$ must be large ($m_{2}\gtrsim\mathcal{O}(1)$
TeV) to avoid EDM, electroweak, and collider constraints. This can be seen in
Figure 13(b). Additionally, to achieve an EFT phase near $\pi/2$ and avoid
spin-independent constraints, generally $\delta_{CP}\gtrsim 1$. Note that
limits from spin-dependent scattering can be avoided, since they vanish when
$y=\pm\tilde{y}$. This part of parameter space generally requires proximity to
both the mass resonance and the phase $\pi/2$ line. However, there is still
some flexibility in both values; masses $m_{\chi}>65\,\text{GeV}$ and phases
$\phi_{h\chi}<1.5$ are allowed in these intersections, albeit not
simultaneously. Unlike in the case of the EFT, it is very difficult to tune
only the phase because we cannot make couplings arbitrarily large without
affecting the mass spectrum, as we saw in Section 3.2.

(a) hellothereifudgeaspace

(b) hellothereifudgeispace

Figure 13: Similar to Figure 12, in slices of the $m_{1}-m_{2}$ plane and for
different values of $y\tilde{y}$ and $\delta_{CP}$. We show the same
constraints as Figure 12 and constraints from RPP2020 ; Barbieri:2004qk ;
Aprile:2019dbj ; PhysRevLett.118.251301 ; PhysRevD.100.022001 ; Khosa:2020zar
; Ambrogi:2018ujg ; Dutta:2018ioj ; Ambrogi:2017neo ; Kraml:2013mwa ; ATL-
PHYS-PUB-2019-029 ; Skands:2003cj ; Alwall:2006yp ; Buckley:2013jua . Dotted
lines around the critical mass and phase values give a guide towards the
proximity of any viable space to mass resonance and pure imaginary EFT
coupling: the green dotted lines bound a region with dark matter mass
$60\,\text{GeV}\leq m_{\chi}\leq 65\,\text{GeV}$, the yellow with EFT phase
$1.55\leq\phi_{h\chi}\leq 1.6$. The left shows the case of a mass resonance
with small couplings, where $m_{2}$ down to $\sim 500$ GeV is allowed. The
right shows the case of larger couplings, where we need
$m_{2}\gtrsim\mathcal{O}(1)$ TeV. We omit light charged fermion constraints
since small $m_{2}$ is already ruled out.

Figures 12(b) \- 12(d) shows several examples of this. In Figure 12(b), we can
see the case where we still choose $m_{1}$ to be near $m_{h}/2$ but allow
larger couplings. If instead we choose $m_{1}$ further away from $m_{h}/2$,
the only viable parameter space requires large couplings in order to get the
dark matter mass sufficiently close to resonance. This is shown in Figures
12(c) and 12(d). Comparing these two plots, we can see that there is more
flexibility in $\delta_{CP}$ and larger required coupling values for higher
$m_{2}$, because higher $m_{2}$ changes the shape of the EFT phase $\pi/2$
contour. Specifically, there is more overlap between $\phi_{h\chi}$ near
$\pi/2$ and the annihilation signal in the large $m_{2}$ case since the
condition $\phi_{h\chi}=\pi/2$ becomes less dependent on $y\tilde{y}$ at
larger $m_{2}$.555This is because the $\phi_{h\chi}=\pi/2$ contour always goes
through the massless state that exists for negative $y\tilde{y}$, which occurs
at larger couplings for larger $m_{2}$. All phase contours go through this
point since the phase becomes unphysical when the lightest state is massless.

##  4 Doublet-Triplet Model

In this section, we describe another potential UV completion, doublet-triplet
dark matter. This model includes the addition of a doublet Dirac fermion and a
triplet Majorana fermion to the Standard Model. This model has been previously
discussed in other contexts in Dedes:2014hga ; Abe:2014gua ; Freitas:2015hsa ;
Lopez-Honorez:2017ora .

###  4.1 Model in the UV

We begin by describing our model and establishing the notation. This model
contains a Dirac doublet of two left handed Weyl fermions with hypercharge 1/2
(denoted by $\psi_{2}$ and $\tilde{\psi}_{2}$ as in the singlet-doublet case)
and a triplet of Majorana fermions (with components
$\psi_{3}^{-1},\psi_{3}^{0},\psi_{3}^{1}$), all of which are SU(3) singlets.
The Lagrangian is given by

$\mathcal{L}=\mathcal{L}_{\text{SM}}+\mathcal{L}_{\text{kinetic}}-\frac{1}{2}m_{3}\psi_{3}\psi_{3}-m_{2}\psi_{2}\cdot\tilde{\psi}_{2}-YH^{\dagger}\psi_{3}\psi_{2}-\tilde{Y}(\epsilon
H^{*})^{\dagger}\psi_{3}\tilde{\psi}_{2}+\textrm{h.c.}$ (36)

As in the singlet-doublet case, this theory also has a single physical phase,
and we can choose the same convention as the previous section to localize CP-
violation to the Yukawa couplings, where

$Y\equiv
y\,\mathrm{e}^{i\delta_{CP}/2},\qquad\tilde{Y}\equiv\tilde{y}\,\mathrm{e}^{i\delta_{CP}/2}.$
(37)

Next we describe our notation after SSB. We denote the gauge basis neutral
particles by $\psi_{n}=\\{\psi^{0}_{3},\psi^{s}_{2},\tilde{\psi}^{d}_{2}\\}$
and the gauge basis charged particles by
$\psi^{+}_{c}=\\{\psi^{+1}_{3},\psi^{+1}_{2}\\}$ and
$\psi^{-}_{c}=\\{\psi^{-1}_{3},\tilde{\psi}^{-1}_{2}\\}$. We label the neutral
mass eigenstates $\chi_{n}=\\{\chi,\chi_{1},\chi_{2}\\}$ and the charged mass
eigenstates $\chi^{+}_{c}=\\{\chi^{+1}_{1},\chi^{+1}_{2}\\}$, and
$\chi^{-}_{c}=\\{\chi^{-1}_{1},\chi^{-1}_{2}\\}$. Each is ordered from least
to most massive, and $\chi$ again denotes the dark matter. We call the basis
change matrices $J_{n}$ and $J^{\pm}_{c}$, which are defined by
$\psi_{n}=J_{n}\chi_{n}$, $\psi_{c}^{\pm}=J_{c}^{\pm}\chi_{c}^{\pm}$. The
phases of the eigenvectors are chosen such that the mass eigenvalues are real.
Then the mass terms are given by

$\mathcal{L}_{mass}=-\frac{1}{2}\chi_{n}[J_{n}^{T}M_{n}J_{n}]\chi_{n}-\chi_{c}^{-}[J_{-}^{T}M_{c}J_{+}]\chi^{+}_{c}+\textrm{h.c.}$
(38)

with

$M_{n}\equiv\begin{pmatrix}m_{3}&(y-\tilde{y})v/2\sqrt{2}&(y+\tilde{y})v/2\sqrt{2}\\\
(y-\tilde{y})v/2\sqrt{2}&-m_{2}&0\\\
(y+\tilde{y})v/2\sqrt{2}&0&m_{2}\end{pmatrix},\qquad
M_{c}\equiv\begin{pmatrix}m_{3}&-yv/\sqrt{2}\\\
-\tilde{y}v/\sqrt{2}&m_{2}\end{pmatrix}.$ (39)

The Higgs Yukawa couplings are

$\mathcal{L}_{\text{Higgs}}=\frac{1}{2}h\chi_{n}[J_{n}^{T}Y_{n}J_{n}]\chi_{n}+h\chi_{c}^{-}[J_{-}^{T}Y_{c}J_{+}]\chi^{+}_{c}+\textrm{h.c.}$
(40)

with

$Y_{n}\equiv\begin{pmatrix}0&-(y-\tilde{y})/2\sqrt{2}&-(y+\tilde{y})/2\sqrt{2}\\\
-(y-\tilde{y})/2\sqrt{2}&0&0\\\
-(y+\tilde{y})/2\sqrt{2}&0&0\end{pmatrix},\qquad
Y_{c}\equiv\begin{pmatrix}0&y/\sqrt{2}\\\ \tilde{y}/\sqrt{2}&0\end{pmatrix}.$
(41)

The $Z$ couplings are

$\mathcal{L}_{Z}=\frac{1}{2}Z^{\mu}\bar{\sigma}_{\mu}\chi_{n}[J_{n}^{\dagger}U_{n}J_{n}]\chi_{n}+Z^{\mu}\chi^{+}_{c}\bar{\sigma}_{\mu}[J_{+}^{\dagger}U_{+}J_{+}]\chi^{+}_{c}+Z^{\mu}\chi^{-}_{c}\bar{\sigma}_{\mu}[J_{-}^{\dagger}U_{-}J_{-}]\chi^{-}_{c},$
(42)

with

$\begin{gathered}U_{n}\equiv\begin{pmatrix}0&0&0\\\
0&0&-\sqrt{g^{2}+g^{{}^{\prime}2}}\\\
0&-\sqrt{g^{2}+g^{{}^{\prime}2}}&0\end{pmatrix},\\\
U_{+}\equiv\begin{pmatrix}\frac{g^{2}}{\sqrt{g^{2}+g^{\prime 2}}}\\\
0&\frac{(g^{2}-g^{\prime 2})}{2\sqrt{g^{2}+g^{\prime 2}}}\end{pmatrix},\quad
U_{-}\equiv\begin{pmatrix}-\frac{g^{2}}{\sqrt{g^{2}+g^{\prime 2}}}&0\\\
0&-\frac{(g^{2}-g^{\prime 2})}{2\sqrt{g^{2}+g^{\prime
2}}}\end{pmatrix},\end{gathered}$ (43)

while the $W$ couplings are

$\mathcal{L}_{W}=W^{\mu+}\chi_{n}\bar{\sigma}_{\mu}[J_{n}^{\dagger}D_{-}J_{-}]\chi^{-}_{c}+W^{\mu-}\chi_{n}\bar{\sigma}_{\mu}[J_{n}^{\dagger}D_{+}J_{+}]\chi^{+}_{c}+\textrm{h.c.}$
(44)

with

$D_{-}\equiv\begin{pmatrix}g&0\\\ 0&g/2\\\ 0&-g/2\end{pmatrix},\qquad
D_{+}\equiv\begin{pmatrix}-g&0\\\ 0&g/2\\\ 0&g/2\end{pmatrix}.$ (45)

The charged fermions also couple to the photon with charge $\pm 1$.

###  4.2 Constraints

We treat most of the constraints in the doublet-triplet model similarly to
those in the singlet-doublet model. There are two exceptions that we discuss
in more detail: the EDM and collider constraints.

The EDM calculation differs from the singlet-doublet case because there are
additional diagrams. Like in the singlet-doublet case, the relevant
contributing diagrams are all Barr-Zee diagrams Barr:1990vd . The diagram with
charged $W$ legs, shown in Figure 10, that contributed in the singlet-doublet
case is still relevant, but for the doublet-triplet model there are two
additional relevant Barr-Zee diagrams: $Zh$ and $\gamma h$, shown in Figure
14. There is still no $\gamma Z$ contribution because in that case the same
charged fermion runs through the entire loop, leaving no place for CP-
violation to enter since the diagonal $Z$ coupling is real. We also neglect
the $hh$ diagram since it is suppressed by two factors of the electron Yukawa.
We use the general forms of the $Zh$ and $\gamma h$ contributions from
Nakai:2016atk ,

$d_{e}^{hV}=\frac{1}{16\pi^{2}m_{h}^{2}}\int_{0}^{1}dx\frac{1}{x(1-x)}j\Bigg{(}\frac{m_{V}^{2}}{m_{h}^{2}},\frac{\tilde{\Delta}^{V}}{m_{h}^{2}}\Bigg{)}g_{e}^{V}c_{O}^{V}\frac{m_{e}}{v},$
(46)

where $g_{e}^{V}$ is the electron coupling to Z or $\gamma$, $v$ is the Higgs
vev, and we define

$j(r,s)=\frac{1}{r-s}\Bigg{(}\frac{r\text{log}r}{r-1}-\frac{s\text{log}s}{s-1}\Bigg{)}.$
(47)

$c_{O}^{V}$ and $\tilde{\Delta}^{V}$ are determined by the inner fermion loop
which only contains charged fermions for both $\gamma h$ and $\gamma Z$. They
are given by

$\begin{split}c_{O}^{Z}=-\frac{e}{2\pi^{2}}\text{Re}\Big{(}m_{c}^{i}x^{2}(1-x)(g_{ij}^{S}g_{ji}^{V*}+ig_{ij}^{P}g_{ji}^{A*})+(1-x)^{3}m_{c}^{j}(g_{ij}^{S}g_{ji}^{V*}-ig_{ij}^{P}g_{ji}^{A*})\Big{)},\\\
\tilde{\Delta}^{Z}=\frac{xm_{c}^{i}+(1-x)m_{c}^{j}}{x(1-x)},\qquad\quad
c_{O}^{\gamma}=-\frac{e^{2}g^{P}_{jj}}{2\pi^{2}}(1-x)m_{c}^{j},\qquad\quad\tilde{\Delta}^{\gamma}=\frac{(m_{c}^{j})^{2}}{x(1-x)},\end{split}$
(48)

where

$\begin{split}g^{S}&=\frac{1}{2}(J_{-}^{T}Y_{c}J_{+}+J_{+}^{\dagger}Y_{c}^{\dagger}J_{-}^{*}),\qquad
g^{P}=\frac{i}{2}(J_{-}^{T}Y_{c}J_{+}-J_{+}^{\dagger}Y_{c}^{\dagger}J_{-}^{*}),\\\
g^{V}&=J_{-}^{T}U_{+}J_{-}^{*}+J_{+}^{\dagger}U_{+}J_{+},\qquad\quad
g^{A}=J_{-}^{T}U_{+}J_{-}^{*}-J_{+}^{\dagger}U_{+}J_{+}\end{split}$ (49)

are given in terms of the matrices defined in Section 4.1. By definition,
$\chi_{j}$ is the fermion which radiates the on-shell external photon, and
$g_{ij}^{*}=(g_{ji})^{*}$.

Figure 14: Additional class of Barr-Zee diagrams contributing to the electron
EDM. $\chi_{c}$ is the tuple of charged fermions in the mass basis. For the
$\gamma h$ diagram, $i=j$, whereas for the $Zh$ diagram, we also have
contributions where $i\neq j$.

A key difference between the singlet-doublet and doublet-triplet cases is that
in the latter the mass of the lightest charged fermion is set by similar
scales as those that set the mass of the dark matter, and thus generically the
lighest charged fermion mass is $\mathcal{O}(100)$ GeV for the doublet-triplet
model. This allows us to treat collider constraints differently here than in
the singlet-doublet case; we apply generic LEP constraints on charged fermions
rather than running the full collider pipeline we considered previously.
Specifically, charged fermions lighter than 92.4 GeV are ruled out as long as
the mass splitting between the lightest neutral and lightest charged particle
is $\geq 100$ MeV LEPSUSYWG1 ; LEPSUSYWG2 .666If the lightest charged state is
more than 3 GeV heavier than the lightest neutral state, then there is a
stronger bound ruling out charged fermions up to mass 103.5 GeV LEPSUSYWG2 .
We use the smaller of the two values for simplicity since it is sufficient for
our purposes.

###  4.3 Full Exclusion Limits and Discussion

Unlike in the singlet-doublet case, there is no viable parameter space in this
model. In order to show this, we consider three different cases. First, we
discuss the case where the magnitude of the couplings is small, for any phase.
Then we discuss the case of large coupling and large phase. Finally we discuss
the case of large coupling but very small phase.

In the first case, parameter space is entirely ruled out by charged fermion
constraints, as we can see from Figure 15. On the left, this figure shows the
values of several EFT parameters for fixed $y,\tilde{y}$, and $\delta_{CP}$
and various values of $m_{2}$ and $m_{3}$. On the right, we show the
annihilation signal and a subset of constraints that are sufficient to rule
out this region of parameter space.777The other constraints from the singlet-
doublet case still apply here, but we omit them from these plots for clarity.
From these plots, we can see that since the couplings are small, in order to
get a sufficient annihilation signal one of $m_{2}$ or $m_{3}$ must be
$\gtrsim m_{h}/2$, with the other UV mass larger. Since the magnitude of the
couplings is small while the UV masses are large, in this region there will
only be a very small splitting between charged and neutral fermions.
Therefore, the parameter space here will be entirely ruled out by charged
fermion constraints from LEP. This occurs regardless of phase, though EDM
constraints are also strong enough to rule this out for larger phases.

Figure 15: Example where the magnitude of couplings $y$ and $\tilde{y}$ are
small, for different values of $m_{2}$ and $m_{3}$. The left plot shows the
values of the EFT parameters: dark matter mass, dark matter-Higgs coupling
phase, dark matter-Higgs coupling magnitude, and lightest charged fermion
mass. Regions around the mass and phase points of interest are shaded:
$55\text{ GeV}\leq m_{\chi}\leq 70\text{ GeV}$ and $1.3\leq\phi_{h\chi}\leq
1.85$. The right shows the annihilation signal and a subset of relevant
constraints including EDM Andreev:2018ayy , spin-independent direct detection
Aprile:2017iyp ; Aprile:2018dbl , and charged fermion constraints from LEP
LEPSUSYWG1 ; LEPSUSYWG2 . The annihilation signal appears as a single brown
line because a viable annihilation signal is only achievable in a tuned region
of parameter space.

In the second case of large coupling and large phase, EDM constraints are
typically very strong. The only exceptions are if both $m_{2}$ and $m_{3}$ are
very large (which can’t generate the necessary annihilation signal) or if one
of $m_{2}$ or $m_{3}$ is very small. This is because in the limit where one of
$m_{2}$ or $m_{3}$ is exactly zero, the phase becomes unphysical since we can
rotate it away. In the limit where $m_{2}$ is small, the lightest state will
have mass even less than $m_{2}$ and the DM mass won’t be in the right mass
range to generate the necessary annihilation signal. But in the limit where
$m_{3}$ is small, if the couplings are large enough we can potentially
generate the right annihilation signal. However, since the physical phase is
small, the EFT phase will also be small, and spin-independent direct detection
constraints will always rule out any part of the annihilation signal that
isn’t constrained by the EDM. This can be seen in Figure 16, which again shows
various values of EFT parameters for fixed $y,\tilde{y}$, and $\delta_{CP}$
and different $m_{2}$ and $m_{3}$ values on the left, and the annihilation
signal and a subset of constraints on the right.

Figure 16: Here we show plots where the magnitude of couplings $y$ and
$\tilde{y}$ are large and $\delta_{CP}$ is also large, for different values of
$m_{2}$ and $m_{3}$. The left plot shows the values of EFT parameters: dark
matter mass, dark matter-Higgs coupling phase, dark matter-Higgs coupling
magnitude, and lightest charged fermion mass. Regions around the mass and
phase points of interest are shaded: $55\text{ GeV}\leq m_{\chi}\leq 70\text{
GeV}$ and $1.3\leq\phi_{h\chi}\leq 1.85$. The right shows the annihilation
signal and a subset of relevant constraints, and from here we can see that the
combination of EDM constraints and spin-independent constraints entirely rule
out the parameter space generating a viable annihilation signal.

The third case of large magnitude coupling but very small phase is shown in
Figure 17. The top plots show the case where $y$ and $\tilde{y}$ are similar
in magnitude, while the bottom plots show a large splitting between $y$ and
$\tilde{y}$. In both, the EFT coupling is mostly real since the phase is
small. There are two different trends depending on the magnitude of the
coupling. In both plots, we see regions where the magnitude of the EFT
coupling is large, and the annihilation signal is ruled out by spin-
independent constraints. In the case of small splitting, we also see a region
where the EFT coupling is small (because the lightest state doesn’t mix),
which is unable to generate the necessary annihilation signal.

Figure 17: Two examples where the magnitude of couplings $y$ and $\tilde{y}$
are large and $\delta_{CP}$ is small, for different values of $m_{2}$ and
$m_{3}$. The top plots show the case where $y$ and $\tilde{y}$ have similar
magnitudes; the bottom plots show the case where their magnitudes are very
different. As in the other doublet-triplet plots, the left plots show the
values of various EFT parameters with shaded regions of interest and the right
plots show the annihilation signal and a subset of relevant constraints. The
annihilation signal appears as two brown lines on each plot, since the region
of allowed masses is so narrow. In both cases, spin-independent constraints
rule out the signal. In the case where the couplings are nearly equal, there
is also a region where the lightest neutral state decouples, and the dark
matter-Higgs coupling is insufficient to generate the annihilation signal
despite the dark matter mass being close to $m_{h}/2$.

##  5 Conclusion

Given that the GCE is one of the most persistent signals of potential new
physics, it is worth cataloging and understanding what could generate it.
While there is still substantial debate over the source of the GCE, one
promising and well explored possibility is dark matter annihilating to
$b\overline{b}$. In this work, we revisit the particular case where dark
matter is a Majorana fermion with a CP-violating Higgs coupling, which allows
annihilation and spin-independent scattering to be governed by different
parameters. Specifically, the leading contribution to annihilation is
determined by the imaginary part of the coupling to the Higgs, while spin-
independent scattering constraints depend primarily on the real part of the
coupling to the Higgs in the mass ranges we are interested in. We study the
EFT of this dark matter model for the GCE in detail, and find that while
tuning the dark matter mass very close to half the Higgs mass is one potential
way to obtain a large enough signal, tuning the phase of the Higgs coupling to
make it near imaginary loosens this restriction in the context of the EFT.

We also explore two potential UV completions: singlet-doublet dark matter and
doublet-triplet dark matter. In both, the story is more complicated than the
EFT because the UV phase and mass are not independent parameters. Although
more elaborate supersymmetric realizations of a CP-violating Higgs portal have
been discussed in Carena:2019pwq , our goal throughout this paper has been to
gain a more detailed qualitative understanding of the mechanism through
simpler models. In particular, we have discussed the scaling of the signal and
various constraints with the different parameters in the simplified models, as
well as quantified how much tuning is necessary to explain the signal without
running into constraints. The singlet-doublet dark matter case is particularly
interesting because it is a minimal working example of how Majorana dark
matter could explain the GCE through the Higgs portal.

We find that in the minimal singlet-doublet case, there is still viable
parameter space when the doublet mass is much larger than the singlet mass.
There are two viable regions of parameter space for the singlet-doublet model.
In the case where the UV couplings are small, the tuning of the dark matter
mass manifests as a tuning of the singlet mass, but the restriction on both UV
and EFT phase is loose. When the couplings are larger, the doublet mass is
required to be $\gtrsim\mathcal{O}(1)$ TeV. The EFT phase, and often the UV
phase as well, must be close to pure imaginary to avoid spin-independent
constraints, and the dark matter and singlet masses also must still be
relatively close to $m_{h}/2$ to generate a sufficient annihilation signal
(though the allowed region is comparatively much wider).

Upcoming direct detection and EDM experiments, such as LZ, XENONnT and ACME,
will search through significant portions of the remaining parameter space.
These two types of probes combine to explore both the limits of minimal and
maximal CP-violation, and we expect to definitively rule out doublet masses
below the TeV scale in the small coupling case. In the more optimistic case of
larger coupling, new experiments will be able to probe doublet masses up to
$\mathcal{O}(15)$ TeV or larger for some phases. In either case, this type of
model offers a range of complementary detection avenues that may combine to
elucidate the nature of annihilating dark matter.

In the doublet-triplet case, we do not find any viable parameter space. Spin-
independent and EDM constraints restrict the size of the real and imaginary
parts of the Higgs coupling, respectively. When the coupling is small in
overall magnitude, the annihilation signal requires a dark matter mass near
the $m_{h}/2$ resonance, and the small splitting between the lightest charged
and neutral states results in a prohibitively light charged fermion. Hence,
the remaining parameter space is ruled out by LEP.

While our results are framed in the context of the GCE, models which include a
CP-violating Higgs portal interaction coupling the dark and visible sectors
are also compelling for other reasons. These types of interactions could be
the key to some of the biggest mysteries of particle physics, including the
particle nature of dark matter and also various problems that CP-violation is
necessary to solve, such as the matter/antimatter asymmetry of the universe
and the strong CP problem. For example, for some models the addition of a CP
phase around the weak scale could increase the viability of electroweak
baryogenesis. While new Higgs boson couplings have the potential to make the
hierarchy problem worse, the minimal models we studied can also be realized
within the larger framework of SUSY Carena:2019pwq  which can ameliorate this
issue. These connections could be potential avenues for further exploration,
if it turns out that dark matter communicates with the Standard Model through
a CP-violating Higgs portal.

###### Acknowledgements.

We thank Prateek Agrawal and Matthew Reece for useful discussions and feedback
on this manuscript. KF is supported by the National Science Foundation
Graduate Research Fellowship Program under Grant No. DGE1745303. AP is
supported in part by an NSF Graduate Research Fellowship Grant DGE1745303, the
DOE Grant DE-SC0013607, and the Alfred P. Sloan Foundation Grant No.
G-2019-12504. WLX is supported in part by NSF grants PHY-1620806 and
PHY-1915071, the Kavli Foundation grant “Kavli Dream Team", and the Moore
Foundation Award 8342.

##  Appendix A Next-Order Velocity Expansion of the Annihilation Cross Section

As established, to leading order in dark matter velocity, the annihilation
signal is set by the pseudoscalar coupling $\mathrm{Im}[y_{h\chi}]$ (and
subdominantly by $g_{Z\chi}$), while spin-independent scattering is set by the
scalar coupling $\mathrm{Re}[y_{h\chi}]$. However, we would also like to
understand whether we can generate the annihilation signal at all in the limit
that $y_{h\chi}$ is real. In this limit, the leading velocity independent term
vanishes, and we need to consider terms of higher order in the halo velocity
$v$.

For this argument we will neglect the contribution of the $Z$ portal; a
$g_{Z\chi}$ consistent with spin-dependent constraints cannot generate a
thermal relic annihilation cross section, as it does not have a mass
resonance.888In fact, the $\mathcal{O}(v^{2})$ $Z$-coupling term does have a
mediator resonance, but enhancement is limited by the significantly larger
width of the $Z$ boson. Thus, for hypothetically viable parameter space it is
safe to assume that the $Z$-mediated annihilation is subdominant.

When $y_{h\chi}$ has vanishing imaginary part, the leading contribution to the
spin averaged annihilation amplitude squared is

$|\mathcal{M}|^{2}_{\chi\chi\to
f\bar{f}}=\frac{4y_{hf}^{2}y_{h\chi}^{2}m_{\chi}^{2}(m_{\chi}^{2}-m_{f}^{2})v^{2}}{(m_{h}^{2}-4m_{\chi}^{2})^{2}+m_{h}^{2}\Gamma_{h}^{2}}+\mathcal{O}(v^{4}).$
(50)

This term is suppressed by the non-relativistic speeds of dark matter, for
typical values $v^{2}\sim 10^{-6}$, and the magnitude of the purely real
coupling is stringently constrained by direct detection. Thus, any allowed
parameter space would require precise fine-tuning of the dark matter mass.
However, the enhancement obtained from the $m_{\chi}\to m_{h}/2$ resonance is
limited by the finite width of the Higgs, which is $\sim 4$ MeV in the SM
Sirunyan:2019twz . Since the branching ratio of $h\to\chi\chi$ near the
resonance is vanishingly small due to phase space suppression, we may take 4
MeV as a conservative bound for the Higgs width. Thus, the comparative ratio
between annihilation and scattering cross sections, given in Equations 4 and
13, can be bounded by

$\frac{\langle\sigma v\rangle_{\rm
ann}}{\sigma_{SI}}\Bigg{|}_{\mathrm{Im}[y_{h\chi}]=0}\sim\frac{6\times
10^{4}\;\mathrm{GeV}^{2}\;m^{2}_{\chi}}{(m_{h}^{2}-4m_{\chi}^{2})^{2}+m_{h}^{2}\Gamma_{h}^{2}}<\frac{10^{5}\;\mathrm{GeV}^{2}\;m^{2}_{\chi}}{m_{h}^{2}\Gamma_{h}^{2}}\approx
10^{9}.$ (51)

As the current direct detection limits bound the spin-independent scattering
rate at $\leq 10^{-10}$ pb, a model without CP-violation may exhibit an
annihilation cross section of at most $\mathcal{O}(0.1)$ pb. We emphasize here
that these statements are specifically valid for Majorana fermion dark matter,
and dark matter models with a different CP-structure could certainly achieve
the required hierarchy between annihilation and scattering with sufficient
tuning on this resonance.

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