#  Probing the Inert Doublet Dark Matter Model
with Cherenkov Telescopes

Camilo Garcia-Cely Service de Physique Théorique, CP225, Université Libre de
Bruxelles, Bld du Triomphe, 1050 Brussels, Belgium Michael Gustafsson
Institute for theoretical Physics - Faculty of Physics, Georg-August
University Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany
Alejandro Ibarra Physik-Department T30d, Technische Universität München,
James-Franck-Straße, D-85748 Garching, Germany

###### Abstract

We present a detailed study of the annihilation signals of the inert dark
matter doublet model in its high mass regime. Concretely, we study the
prospects to observe gamma-ray signals of the model in current and projected
Cherenkov telescopes taking into account the Sommerfeld effect and including
the contribution to the spectrum from gamma-ray lines as well as from internal
bremsstrahlung. We show that present observations of the galactic center by
the H.E.S.S. instrument are able to exclude regions of the parameter space
that give the correct dark matter relic abundance. In particular, models with
the charged and the neutral components of the inert doublet nearly degenerate
in mass have strong gamma-ray signals. Furthermore, for dark matter particle
masses above 1 TeV, we find that the non-observation of the continuum of
photons generated by the hadronization of the annihilation products typically
give stronger constraints on the model parameters than the sharp spectral
features associated to annihilation into monochromatic photons and the
internal bremsstrahlung process. Lastly, we also analyze the interplay between
indirect and direct detection searches for this model, concluding that the
prospects for the former are more promising. In particular, we find that the
upcoming Cherenkov Telescope Array will be able to probe a significant part of
the high mass regime of the model.

††preprint: ULB-TH/15-21††preprint: TUM-HEP 1030/15

###### Contents

  1. I Introduction
  2. II Dark Matter as an Inert Scalar
  3. III Annihilation Cross Section into Gamma-rays
  4. IV Gamma-ray Signals of the IDM
  5. V Complementarity among Gamma-ray Signals in the IDM
  6. VI Complementarity with Direct Detection
  7. VII Conclusions
  8. A Constraints on the Model Parameters
  9. B Algorithm for the Sommerfeld Enhancement
  10. C Impact of the Sommerfeld Enhancement in the Early Universe

##  I Introduction

Multiple observations strongly suggest that the Standard Model of particle
physics should be extended by at least one additional particle, electrically
neutral and colorless, and long-lived on cosmological time-scales, dubbed the
dark matter (DM) particle Bertone:2010zza ; Bergstrom00 ; Jungman:1995df ;
Bertone:2004pz . Among the many models that have been constructed over the
last decades containing a DM particle, the Inert Doublet Model (IDM) stands
out for its simplicity and for its rich phenomenology.

The IDM Deshpande:1977rw ; Ma:2006km ; Barbieri:2006dq postulates the
existence of a new scalar field $\eta$, with identical gauge quantum numbers
as the Standard Model’s Brout-Englert-Higgs scalar doublet (Higgs for short),
and the invariance of the vacuum under a $Z_{2}$ symmetry, under which $\eta$
is odd while all the Standard Model particles are even. These two simple
assumptions have a number of implications. First, the $Z_{2}$ symmetry ensures
that the doublet $\eta$ contains an absolutely stable particle which is a DM
candidate. Second, the exotic doublet $\eta$ does not interact at tree level
with any of the Standard Model fermions, hence the name “inert”. The DM,
nonetheless, interacts with the Standard Model via gauge interactions and via
the quartic term in the scalar potential $|\eta|^{2}|\Phi|^{2}$, with $\Phi$
the Higgs doublet. These terms are of utmost importance in the phenomenology
of the IDM, since they allow to generate a population of DM particles in the
early Universe via thermal freeze-out and they induce potentially observable
signals in direct and indirect DM searches Barbieri:2006dq ;
LopezHonorez:2006gr ; Majumdar:2006nt ; Gustafsson:2007pc ; Agrawal:2008xz ;
Andreas:2009hj ; Nezri:2009jd ; Arina:2009um ; Honorez:2010re ;
LopezHonorez:2010tb ; Klasen:2013btp ; Garcia-Cely:2013zga , collider searches
Barbieri:2006dq ; Cao:2007rm ; Lundstrom:2008ai ; Dolle:2009ft ; Miao:2010rg ,
electroweak precision tests Barbieri:2006dq ; Grimus:2008nb and the Higgs
diphoton decay rate Arhrib:2012ia ; Swiezewska:2012eh ; Krawczyk:2013jta ;
Goudelis:2013uca .

Of particular interest in the IDM is the scenario where the DM is entirely
constituted by the lightest component of the doublet and where the observed DM
abundance $\Omega h^{2}\simeq 0.12$ Ade:2015xua is generated by thermal
freeze-out of this particle. As is well known, this requirement implies for
this model a DM particle mass either smaller than the W boson mass,
$M_{W}\simeq 80$ GeV, or larger than $\sim 500$ GeV (see e.g. Barbieri:2006dq
; LopezHonorez:2006gr ; LopezHonorez:2010tb ; Cirelli:2005uq ; Hambye:2007vf ;
Hambye:2009pw ). In this paper we will concentrate in the latter mass window,
and we will investigate the possibility of detecting gamma-ray signals
generated by the annihilations of these DM particles in the Milky Way center.

The DM induced gamma-ray flux is comprised of two main components
Bringmann:2012ez . One component is the prompt gamma-rays, mainly generated by
the hadronization of massive gauge and Higgs bosons in the DM annihilation
final states, and one lower energy component, consisting of photons of the
interstellar radiation field that have been up-scattered to gamma-ray energies
due to collisions with the energetic electrons and positrons produced in the
annihilations. Since we are interested in the energy spectrum at the highest
energies, we will neglect the latter contribution in what follows. Besides,
the annihilation also produces sharp gamma-ray spectral features which, if
observed, would strongly hint toward an exotic origin of this signal. So far,
three different gamma-ray spectral features have been identified in DM
scenarios: gamma-ray lines Srednicki:1985sf ; Rudaz:1986db ; Bergstrom:1988fp
, internal electromagnetic bremsstrahlung Bergstrom:1989jr ; Flores:1989ru ;
Beacom:2004pe ; Bergstrom:2004cy ; Bergstrom:2005ss ; Bringmann:2007nk and
gamma-ray boxes Ibarra:2012dw . Notably, the three spectral features arise in
the IDM: the gamma-ray lines arise from annihilations at the quantum loop
level into $\gamma\gamma$ and $\gamma Z$, the internal bremsstrahlung signal
arises from annihilation into $W^{+}W^{-}\gamma$ through the t-channel
exchange of the charged $Z_{2}$ odd scalars of the inert doublet, and gamma-
ray boxes arise from the annihilation into a pair of Higgs bosons and their
subsequent decay in flight into $\gamma\gamma$. Due to the small branching
fraction of the process $h\rightarrow\gamma\gamma$, the gamma-ray box produced
by the decay in flight of the Higgs boson is fainter than the other two
spectral features and will not be considered here.

Since the DM candidate in the IDM possesses a $SU(2)_{L}$ charge, weak gauge
bosons could be exchanged between the non-relativistic particles in the
initial state of the annihilation process. As argued in Hisano:2002fk ;
Hisano:2003ec ; Hisano:2004ds , for heavy DM particles the long-range
interaction associated to the exchange of the weak gauge bosons can
significantly distort the wave function of the initial state particles,
therefore the correct description of the annihilation process must include
non-perturbative effects, which generically lead to an enhancement of the
annihilation cross section. This phenomenon, commonly known as Sommerfeld
enhancement, can boost the annihilation signal by many orders of magnitude and
has been proved to be pivotal in ruling out some well motivated DM scenarios,
such as the Wino DM Chun:2012yt ; Cohen:2013ama ; Baumgart:2014vma ;
Ovanesyan:2014fwa ; Baumgart:2014saa ; Beneke:2014hja ; Chun:2015mka or the
5-plet minimal DM Garcia-Cely:2015dda ; Cirelli:2015bda (assuming the DM
density follows an Einasto profile in our Galaxy). The main goal of this paper
is to investigate the prospects to observe signals in gamma-rays from the IDM
in the high mass regime, including the Sommerfeld enhancement and including
not only the channels generating a continuum of gamma-rays, but also those
generating sharp spectral features in the energy spectrum.

The paper is organized as follows. In Section II we present a brief overview
of the Inert Doublet Model. In Section III we discuss the process of
annihilation in the non-relativistic limit and we describe our non-
perturbative approach to calculate the cross section. In Section IV we
calculate the expected gamma-ray flux and we confront the predictions of the
model to limits on continuum gamma-ray fluxes and on sharp gamma-ray spectral
features. In Sections V and VI, we discuss the complementarity between direct
detection experiments and gamma-ray instruments in probing the parameter space
of the IDM and, lastly, in Section VII we present our conclusions. We also
include three appendices discussing various theoretical and experimental
constraints on the IDM, technical details of the Sommerfeld enhancement and an
estimation of its effect on relic density calculations in the early Universe.

##  II Dark Matter as an Inert Scalar

The IDM is an extension of the Standard Model by one complex scalar field
$\eta$, which is a singlet under $SU(3)_{C}$, doublet under $SU(2)_{L}$ and
has hypercharge $1/2$. Furthermore, the model postulates a discrete $Z_{2}$
symmetry, preserved also in the electroweak vacuum, under which the Standard
Model particles are even while the extra scalar doublet $\eta$ is odd. With
this particle content, the Lagrangian can be cast as ${\cal L}={\cal L}_{\rm
SM}+{\cal L}_{\eta}$, where ${\cal L}_{\rm SM}$ is the Standard Model
Lagrangian including a potential for the Higgs doublet $\Phi$

${\cal L}_{\rm SM}\supset-
m_{1}^{2}\Phi^{\dagger}\Phi-\lambda_{1}(\Phi^{\dagger}\Phi)^{2}\;,$ (1)

and ${\cal L}_{\eta}$ is the most general $Z_{2}$ invariant Lagrangian
involving the scalar doublet $\eta$

$\displaystyle{\cal L}_{\eta}$ $\displaystyle=$
$\displaystyle(D_{\mu}\eta)^{\dagger}(D^{\mu}\eta)-m_{2}^{2}\eta^{\dagger}\eta-\lambda_{2}(\eta^{\dagger}\eta)^{2}-\lambda_{3}(\Phi^{\dagger}\Phi)(\eta^{\dagger}\eta)$
(2)
$\displaystyle-\lambda_{4}(\Phi^{\dagger}\eta)(\eta^{\dagger}\Phi)-\dfrac{1}{2}\left(\lambda_{5}(\Phi^{\dagger}\eta)(\Phi^{\dagger}\eta)+{\rm\text{\small
h.c.}}\right),$

where $D_{\mu}$ denotes the covariant derivative.

Due to the postulate that the $Z_{2}$ symmetry remains unbroken, only the
Higgs doublet acquires an expectation value, therefore the doublets can be
cast as

$\Phi=\begin{pmatrix}G^{+}\\\
\frac{v_{h}+h+iG^{0}}{\sqrt{2}}\end{pmatrix}\;,\hskip
50.0pt\eta=\begin{pmatrix}H^{+}\\\
\frac{1}{\sqrt{2}}\left(H^{0}+iA^{0}\right)\end{pmatrix}\;,$ (3)

where $v_{h}\equiv\sqrt{-m_{1}^{2}/\lambda_{1}}\approx 246$ GeV, $G^{0}$ and
$G^{+}$ provide the longitudinal components of the of the $Z$ and $W^{+}$
bosons through the Brout-Englert-Higgs mechanism and $h$ is the Standard Model
Higgs. On the other hand, the inert sector consists of two charged states
$H^{\pm}$, one CP-even neutral state $H^{0}$ and one CP-odd neutral state
$A^{0}$. Furthermore, the preserved $Z_{2}$ symmetry ensures that the lightest
particle in the inert doublet is absolutely stable and, if it is neutral, it
constitutes a DM candidate; we will assume in what follows that this is case
for the CP-even neutral state $H^{0}$.

The seven parameters in the scalar potential of the model can be recast in
terms of the Higgs boson mass $M_{h}\approx 125$ GeV, the vacuum expectation
value of the Higgs field $v_{h}$ and the DM mass $M_{H^{0}}$, together with
the quartic couplings $\lambda_{2}$, $\lambda_{3},\lambda_{4}$ and
$\lambda_{5}$. The masses of the remaining inert scalars are given in terms of
these parameters by

$\displaystyle
M^{2}_{H^{+}}=M^{2}_{H^{0}}-\frac{1}{2}(\lambda_{4}+\lambda_{5})v_{h}^{2}\,,\hskip
30.0ptM^{2}_{A^{0}}=M^{2}_{H^{0}}+\lambda_{5}v_{h}^{2}\;.$ (4)

These masses receive corrections at the quantum level. In particular, gauge
interactions induce a splitting between the neutral and charged scalar masses
which is approximately $356$ MeV Cirelli:2005uq . However, this contribution
can be compensated by an appropriate renormalization of the quartic couplings,
resulting in a mass difference which can, in principle, be arbitrarily small.
Therefore, in this paper we will take the mass differences among the inert
scalars as free parameters, only constrained by, e.g., the perturbative
condition $|\lambda_{i}|<4\pi$ in Eq. (4). The parameters of the IDM are
further constrained by the stability of the vacuum Gunion:2002zf ;
Gustafsson:2010zz and by the unitarity of the S-matrix Ginzburg:2004vp ;
Branco:2011iw . We summarize the constrains we impose in Appendix A, along
with various experimental bounds coming from electroweak precision observables
and collider searches.

The inert scalar $H^{0}$ has the characteristics of a Weakly Interacting
Massive Particle (WIMP) because its gauge interactions with the electroweak
gauge bosons and due to the its quartic coupling to the Higgs particle are of
the required size to thermally produce $H^{0}$ of the right amount, via the
freeze-out mechanism, to match the observed DM content of our Universe $\Omega
h^{2}\simeq 0.12$ Ade:2015xua .

There are two allowed DM mass regimes for $H^{0}$ (see e.g., Barbieri:2006dq ;
LopezHonorez:2006gr ; LopezHonorez:2010tb ; Hambye:2007vf ; Hambye:2009pw ).
The first is for $H^{0}$ masses below the $W$ boson mass, where the DM
annihilates mostly into light fermions with a rate controlled by the size of
the quartic couplings.111Three body annihilations of the type $H^{0}H^{0}\to
WW^{*}\to Wf\bar{f}^{\prime}$ are also important in some regions of the
parameter space Honorez:2010re . For masses immediately above the $W$ boson
threshold, the gauge couplings alone are large enough to suppress the $H^{0}$
abundance below the observed DM content (and given current experimental
constraints, there is no longer room to avoid this conclusion by invoking
destructive interference effects; see LopezHonorez:2010tb and then, e.g.,
Gustafsson:2012aj ). Nonetheless, for $M_{H^{0}}\simeq 535$ GeV, and vanishing
quartic couplings, the annihilation rate into gauge bosons is sufficiently
small to reproduce the observed DM density. For masses above $535$ GeV, the
correct relic density can also be obtained if the quartic couplings are
appropriately chosen, because their effect is to increase the annihilation
cross section. This forms the second, so called, high DM mass regime of the
IDM.

The requirement of correct $H^{0}$ abundance thus implies larger and larger
couplings as the DM mass increases. In fact, an upper limit on the DM mass can
be derived by imposing perturbativity on the couplings. With the bounds of
Appendix A, and from the relic abundance calculation in Appendix C, we find an
upper limit of $M_{H^{0}}\lesssim 20$ TeV. In this paper we will investigate
this high mass regime of the IDM and in particular the possible signals in
gamma-ray signals from annihilation of $H^{0}$ particles in the galactic
center. To this end, we discuss the corresponding annihilation cross sections
in the following section.

##  III Annihilation Cross Section into Gamma-rays

Various annihilation channels contribute to the gamma-ray flux in the IDM. The
processes with the largest cross section are the tree-level two-body
annihilations into $W^{+}W^{-}$, $ZZ$ and $hh$, which generate a gamma-ray
flux with a featureless energy spectrum. Processes arising at higher order in
perturbation theory have a smaller cross section, however they can contribute
significantly to the gamma-ray flux at energies close to the kinematical end-
point of the annihilation and produce a sharp spectral feature in the energy
spectrum. This is the case of the tree-level three-body annihilation into
$W^{+}W^{-}\gamma$, as well as the one-loop annihilations into $\gamma\gamma$
and $\gamma Z$. The former was studied in Garcia-Cely:2013zga and leads to a
bump close to the kinematical end-point of the gamma-ray energy spectrum. This
process is sizable especially when $H^{0}$ and $H^{\pm}$ are relatively close
in mass, which is typical in the high mass regime (see Eq. (4)). The latter,
on the other hand, were studied in Gustafsson:2007pc in the low mass regime.
The perturbative approach pursued in that paper, however, cannot be applied to
the high mass regime since for very large DM masses the predicted annihilation
rates in the galactic center into $\gamma\,\gamma$ and $\gamma\,Z$ exceeds the
upper bound set by unitarity. In fact, for non-realtivistic velocities, the
one-loop annihilation cross sections into photons are not suppressed by the DM
mass but rather by the W boson mass (see Ref. GarciaCely:2014jha for a
detailed discussion). Similar shortcomings of the perturbative calculation
have been pointed out for neutralino DM in the MSSM Bergstrom:1997fh ;
Bern:1997ng , which were solved in Ref. Hisano:2004ds by pursuing a non-
perturbative approach.

To calculate the annihilation cross section into the various final states in
the high mass regime of the IDM, we thus follow closely the formalism
introduced in Hisano:2004ds ; Hisano:2002fk ; Cirelli:2007xd . There they use
the framework of non-relativistic field theory, which is well motivated by the
fact that DM particles move slowly in our Galaxy.222A detailed description of
the non-perturbative calculation of the annihilaton rate in the IDM can also
be found in Ref.GarciaCely:2014jha . The non-relativistic action is obtained
by taking the non-relativistic limit description of the components of the
inert doublet, which are assumed to be quasi-degenerate in mass, and by
integrating out the light particles, namely the Higgs boson and the gauge
bosons. In this formalism, it is convenient to introduce auxiliary fields for
the two-body states

$s(\vec{x},\vec{r})=\begin{pmatrix}s_{H^{0}H^{0}}(\vec{x},\vec{r})\\\
s_{A^{0}A^{0}}(\vec{x},\vec{r})\\\
s_{H^{-}H^{+}}(\vec{x},\vec{r})\end{pmatrix}$ (5)

where $s_{i}$ describes the wave functions of any of the pairs
$i=(H^{0},H^{0})$, $(A^{0},A^{0})$ and $(H^{-},H^{+})$. Here $\vec{x}$ is the
position of the center of mass of the system and $\vec{r}$ is the relative
position vector for the pair of particles. In terms of these auxiliary fields,
the two-body state effective action reads

$\displaystyle S_{\text{eff}}=\int
d^{4}xd^{3}r\,s^{\dagger}(\vec{x},\vec{r})\left(i\partial_{x^{0}}+\dfrac{\nabla^{2}_{x}}{4M_{H^{0}}}+\dfrac{\nabla^{2}_{r}}{M_{H^{0}}}-V(r)+2i\Gamma\delta(\vec{r})\right)s(\vec{x},\vec{r}),$
(6)

The matrix $V(r)$ represents a central potential consisting of three terms:
one specifying the mass splittings among the pairs of particles and the other
two describing Yukawa potentials induced by the non-relativistic exchange of
gauge bosons (as shown in Fig. 1) and light scalars (as shown in Fig. 2).

Figure 1: Feynman diagrams contributing to $V_{\text{Gauge}}$(r)

Figure 2: Feynman diagrams contributing to $V_{\text{Scalar}}$(r)

Thus

$V(r)=2\,\delta m+V_{\text{Gauge}}(r)+V_{\text{Scalar}}(r),$ (7)

with

$\displaystyle\delta m$ $\displaystyle=$ $\displaystyle\begin{pmatrix}0&0&0\\\
0&M_{A^{0}}-M_{H^{0}}&0\\\ 0&0&M_{H^{+}}-M_{H^{0}}\\\ \end{pmatrix},$ (8)
$\displaystyle V_{\text{Gauge}}(r)$ $\displaystyle=$
$\displaystyle-\frac{g^{2}}{4\pi
r}\begin{pmatrix}0&\frac{e^{-M_{Z}r}}{4c_{W}^{2}}&\frac{e^{-M_{W}r}}{2\sqrt{2}}\\\
\frac{e^{-M_{Z}r}}{4c_{W}^{2}}&0&\frac{e^{-M_{W}r}}{2\sqrt{2}}\\\
\frac{e^{-M_{W}r}}{2\sqrt{2}}&\frac{e^{-M_{W}r}}{2\sqrt{2}}&s_{W}^{2}+\frac{(1-2c_{W}^{2})^{2}e^{-M_{Z}r}}{4c_{W}^{2}}\end{pmatrix},$
(9) $\displaystyle V_{\text{Scalar}}(r)$ $\displaystyle=$
$\displaystyle-\frac{v_{h}^{2}}{8\pi
rM_{H^{0}}^{2}}\begin{pmatrix}\frac{(\lambda_{3}+\lambda_{4}+\lambda_{5})^{2}}{2}e^{-M_{h}r}&\frac{\lambda_{5}^{2}}{4}e^{-M_{Z}r}&\frac{(\lambda_{4}+\lambda_{5})^{2}}{4\sqrt{2}}e^{-M_{W}r}\\\
\frac{\lambda_{5}^{2}}{4}e^{-M_{Z}r}&\frac{(\lambda_{3}+\lambda_{4}-\lambda_{5})^{2}}{2}e^{-M_{h}r}&\frac{(\lambda_{4}-\lambda_{5})^{2}}{4\sqrt{2}}e^{-M_{W}r}\\\
\frac{(\lambda_{4}+\lambda_{5})^{2}}{4\sqrt{2}}e^{-M_{W}r}&\frac{(\lambda_{4}-\lambda_{5})^{2}}{4\sqrt{2}}e^{-M_{W}r}&\frac{\lambda_{3}^{2}}{2}e^{-M_{h}r}\end{pmatrix}.$
(10)

Here $s_{W}=\sin\theta_{W}$, $c_{W}=\cos\theta_{W}=M_{W}/M_{Z}$ and
$g=\sqrt{4\pi\alpha/s_{W}^{2}}\simeq 0.129$ with $\alpha\simeq 1/127$ being
the fine structure constant. In this formalism it is assumed that the
splitting between the different pairs is negligible compared to the DM mass.
In fact, relatively small mass splittings $\delta M/M_{H^{0}}$ are unavoidable
in this setup, because they are tied to the electroweak breaking scale.
Concretely, according to Eq. (4), when $M_{H^{0}}$ is at the TeV scale,
$\delta M\sim\lambda v_{h}^{2}/M_{H^{0}}$ and therefore $\delta m_{ij}\ll
M_{H^{0}}$.

Moreover, the action contains an absorptive – or imaginary – term, which takes
into account that the two body states can annihilate into a Higgs pair or into
two gauge bosons. This term is proportional to the matrix
$\Gamma=\sum_{f}\Gamma^{(f)}$, where $f$ is any final state in which the pairs
in the auxiliary field $s(\vec{x},\vec{r})$ can annihilate into. More
concretely,

$\displaystyle\Gamma^{(f)}_{ij}=\frac{N_{i}N_{j}}{4M_{H_{0}}^{2}}\int{\cal
M}\left(i\to f\right){\cal M}^{*}\left(j\to
f\right)(2\pi)^{4}\delta^{(4)}\left(P_{i}-P_{f}\right)\left(\prod_{a\in
f}\frac{d^{3}q_{a}}{(2\pi)^{3}2E_{a}}\right)\,,$ (11)

where $P_{i}$ and $P_{f}$ are the total 4-momenta of the initial and final
states, the $N_{i}$ are symmetry factors for the initial state particles with
$N_{H^{0}H^{0}}=N_{A^{0}A^{0}}=1/\sqrt{2}$ and $N_{H^{-}H^{+}}=1$, and the
integration is performed over the momentum $q_{a}$ of all final state
particles. For the 2-body final states we find

$\displaystyle\Gamma^{(\gamma\gamma)}$
$\displaystyle=\frac{1}{2}\tan^{2}(2\theta_{W})\Gamma_{Z\gamma}=\frac{e^{4}}{128\pi{M_{H^{0}}}^{2}}\left(\begin{array}[]{ccc}0&0&0\\\
0&0&0\\\ 0&0&16\\\ \end{array}\right),$ (15) $\displaystyle\Gamma^{(ZZ)}$
$\displaystyle=\frac{g^{4}}{128c_{W}^{4}\pi{M_{H^{0}}}^{2}}\left(\begin{array}[]{ccc}\frac{1}{2}&\frac{1}{2}&\frac{\left(1-2c_{W}^{2}\right)^{2}}{\sqrt{2}}\\\
\frac{1}{2}&\frac{1}{2}&\frac{\left(1-2c_{W}^{2}\right)^{2}}{\sqrt{2}}\\\
\frac{\left(1-2c_{W}^{2}\right)^{2}}{\sqrt{2}}&\frac{\left(1-2c_{W}^{2}\right)^{2}}{\sqrt{2}}&\left(1-2c_{W}^{2}\right)^{4}\\\
\end{array}\right),$ (19)
$\displaystyle+\Gamma^{(SS)}\left(\lambda_{3}+\lambda_{4}-\lambda_{5},\lambda_{3}+\lambda_{4}+\lambda_{5},\lambda_{3}\right),$
(20) $\displaystyle\Gamma^{(W^{+}W^{-})}$
$\displaystyle=\frac{g^{4}}{128\pi{M_{H^{0}}}^{2}}\left(\begin{array}[]{ccc}1&1&\sqrt{2}\\\
1&1&\sqrt{2}\\\ \sqrt{2}&\sqrt{2}&2\\\
\end{array}\right)+2\,\Gamma^{(SS)}\left(\lambda_{3},\lambda_{3},\lambda_{3}+\lambda_{4}\right),$
(24) $\displaystyle\Gamma^{(hh)}$
$\displaystyle=\Gamma^{(SS)}\left(\lambda_{3}+\lambda_{4}+\lambda_{5},\lambda_{3}+\lambda_{4}-\lambda_{5},\lambda_{3}\right)\,,$
(25)

where we introduce for convenience the following matrix

$\displaystyle\Gamma^{(SS)}\left(l_{1},l_{2},l_{3}\right)$
$\displaystyle\equiv$
$\displaystyle\frac{1}{128\pi{M_{H^{0}}}^{2}}\left(\begin{array}[]{ccc}l_{1}^{2}&l_{1}l_{2}&\sqrt{2}l_{1}l_{3}\\\
l_{1}l_{2}&l_{2}^{2}&\sqrt{2}l_{2}l_{3}\\\
\sqrt{2}l_{1}l_{3}&\sqrt{2}l_{2}l_{3}&2l_{3}^{2}\\\ \end{array}\right)\,.$
(29)

For the internal bremsstrahlung process, corresponding to the final states
$WW\gamma$, we directly use Eq. (11), before integrating over the photon
energy (see the Appendix of Ref. Garcia-Cely:2013zga for details).

These matrices are of interest here because they allow to calculate the
annihilation $s$-wave cross section in the final state $f$ by means of the
formula

$\sigma v\left(i\to
f\right)\Big{|}_{\text{s-wave}}=\frac{1}{N_{i}^{2}}(d\,\Gamma^{(f)}d^{\dagger})_{ii},$
(30)

where the matrix $d$ are the Sommerfeld enhancement factors that can be
calculated by solving the Schrödinger equation associated to the potential
(7), as described in detail in Appendix B.

For DM annihilation, Eq. (30) can be cast as

$\displaystyle\sigma v\left(H^{0}H^{0}\to
f\right)\Big{|}_{\text{s-wave}}=\frac{1}{4M^{2}_{H^{0}}}\int\left(\prod_{a\in
f}\frac{d^{3}q_{a}}{(2\pi)^{3}2E_{a}}\right)(2\pi)^{4}\delta^{4}\left(p_{H^{0}}+p^{\prime}_{H^{0}}-{\scriptstyle\sum_{a\in
f}}q_{a}\right)$ $\displaystyle\cdot\Bigg{|}{d_{11}\;\cal
M}\left(H^{0}H^{0}\to f\right)+d_{12}\;{\cal M}\left(A^{0}A^{0}\to
f\right)+\sqrt{2}d_{13}\;{\cal M}\left(H^{+}H^{-}\to f\right)\Bigg{|}^{2}$
(31)

The quantities $d_{11}$, $d_{12}$ and $d_{13}$ are therefore interpreted as
non-perturbative enhancement factors that account for the long range
interactions between the annihilating DM particles due to the exchange of
gauge and Higgs bosons in the non-relativistic limit. As an example, we show
in Fig. 3 the absolute value of $d_{11}$, $d_{12}$ and $d_{13}$ as a function
of the DM mass, for the case $\lambda_{3}=\lambda_{5}=0$ and $\lambda_{4}$
chosen so that the mass splitting between the charged and the neutral
component is 1 GeV (left panel) and 10 GeV (right panel). We find that, for
masses below approximately 2 TeV, the inclusion of these factors in the
calculation is irrelevant; however, once the DM mass increases, the
enhancement factors dramatically affect the annihilation cross sections in Eq.
(31). Furthermore, we find a resonance, which moves to higher DM masses as the
mass splitting between the charged and the neutral component is increased.
This is in agreement with what was found in Ref. Hisano:2004ds for neutralino
DM.

Figure 3: Example of the Sommerfeld enhancement Factors for annihilating DM
particles with a relative velocity of $v=2\times 10^{-3}$,
$\lambda_{3}=\lambda_{5}=0$ and $M_{H^{+}}-M_{H^{0}}=1$ GeV (left panel) or
$M_{H^{+}}-M_{H^{0}}=10$ GeV (right panel).

In order to study the impact of the Sommerfeld enhancement in the IDM, we
performed a scan over the five dimensional parameter space of the DM sector
following the procedure described below. Considering, as observed in Fig. 3,
that scalar mass splittings are essential quantities, instead of taking
$\lambda_{4}$ and $\lambda_{5}$ as parameters of the scan, we let the relative
mass splittings $(M_{H^{+}}-M_{H^{0}})/M_{H^{0}}$ and
$(M_{A^{0}}-M_{H^{0}})/M_{H^{0}}$ vary logarithmically in between $10^{-5}$
and $1$. Then using Eq. (4), we solve for $\lambda_{4}$ and $\lambda_{5}$ and
we discard points whose magnitude is greater than $4\pi$. In contrast, the
quartic couplings $\lambda_{2}$ and $\lambda_{3}$, which do not lead to any
mass splitting, are randomly sampled on a linear scale. Finally, we take
$M_{H^{0}}$ in-between 0.5 and 20 TeV and impose the remaining constraints of
Appendix A. We then calculate the relic abundance of $H^{0}$ for each model by
means of micrOMEGAs 3.1 Belanger:2013oya and require that it is in agreement
with the observed value $\Omega_{\text{DM}}h^{2}=0.1199\pm 0.0027$ within a
30% range. As discussed in Appendix C, 30% is the error in the relic density
calculation that can be expected from our approximation of not accounting for
the Sommerfeld effect in the early Universe.

Assuming a relative velocity between annihilating DM particles of $v=2\times
10^{-3}$, for each model point of the scan we calculate the enhancement
factors $d_{11}$, $d_{12}$ and $d_{13}$. Then, we calculate the total
annihilation cross section, i.e. the sum of the $H^{0}H^{0}\rightarrow
W^{+}W^{-},\leavevmode\nobreak\ ZZ,\leavevmode\nobreak\ hh$ cross sections.
The resulting enhancement on the total annihilation cross section is shown in
Fig. 4

Figure 4: Impact of the Sommerfeld enhancement on the total annihilation cross
section for a random sample of points of the viable parameter space, as
described in the text. We also show the benchmark points of Table 1. Here the
DM velocity is $v=2\times 10^{-3}$.

as a function of the DM mass (upper panel) and the relative mass splittings of
the inert scalars (lower panel). In the plot we observe two facts. On the one
hand, the Sommerfeld effect is only relevant for masses in the TeV scale, as
expected, and its importance increases for masses close to 20 TeV. On the
other hand, the smaller the mass splittings the greater the Sommerfeld effect.

The enhancement of the cross section in the channels producing gamma-rays in
the final state has important implications for the indirect searches of the
inert doublet DM at gamma-ray telescopes, as we discuss in the next section.

##  IV Gamma-ray Signals of the IDM

The DM induced gamma-ray signal from a given sky region is

$\displaystyle\frac{d\phi_{\gamma}}{dE_{\gamma}}=\frac{\bar{J}}{8\pi
M^{2}_{H^{0}}}\frac{d(\sigma v)_{\gamma}}{dE_{\gamma}}\,.$ (32)

The astrophysical $\bar{J}$-factor is here the line-of-sight integral over the
squared DM density $\rho_{H^{0}}$, averaged over the observed solid angle
$\Delta\Omega$,

$\displaystyle\bar{J}=\frac{1}{\Delta\Omega}\int_{\Delta\Omega}\int_{\text{l.o.s.}}\rho^{2}_{H^{0}}\,ds\,d\Omega\,.$
(33)

The $d(\sigma v)_{\gamma}/dE_{\gamma}$ is comprised of three different
components: first, the fairly featureless spectrum generated in the decay and
fragmentation of the gauge and Higgs bosons, second, the gamma-ray lines
produced by the monochromatic photons emitted in the processes
$H^{0}H^{0}\to\gamma\gamma$ and $H^{0}H^{0}\to\gamma Z$, and third, the
virtual internal bremsstrahlung signal from $H^{0}H^{0}\rightarrow
W^{+}W^{-}\gamma$. Concretely, the inclusive differential cross section into
photon is

$\displaystyle\frac{d(\sigma v)_{\gamma}}{dE_{\gamma}}=\sum_{f\in\text{two-
body}}\\!\\!\\!\\!\\!\sigma v(H^{0}H^{0}\to
f)\frac{dN^{f}_{\gamma}}{dE_{\gamma}}\;\;\;+\;\frac{d\,\sigma v(H^{0}H^{0}\to
WW\gamma)}{dE_{\gamma}},$ (34)

where $dN^{f}_{\gamma}/dE_{\gamma}$ is the photon multiplicity associated to
the two body final states $f$. When $f$ is a electroweak or Higgs boson pair,
we use the parametrization
$dN^{\gamma}/dE_{\gamma}=dN^{\text{frag}}_{\gamma}/dE_{\gamma}=\frac{0.73}{M_{H^{0}}}\,x^{1.5}\,e^{-7.8x}$
with $x=E_{\gamma}/M_{H^{0}}$ Bergstrom:1997fj . For the $\gamma\gamma$ and
$\gamma Z$ final states, $dN^{f}_{\gamma}/dE_{\gamma}$ is a delta function at
$E_{\gamma}=M_{H^{0}}$ and $M_{H^{0}}-\frac{M_{Z}^{2}}{4M_{H^{0}}}$,
respectively. The $dN^{\text{frag}}_{\gamma}/dE_{\gamma}$ does not account for
initial internal bremsstrahlung contributions, $WW\gamma$, which is instead
explicitly included by the last term of Eq. (34).

The relative strength of each of these components in the total gamma-ray flux
strongly depends on the concrete choice of the parameters of the model. In
order to assess the prospects to observe annihilation signals of the IDM, we
have calculated the predicted gamma-ray flux for all the viable models of our
parameter scan from section III. We include the gamma-ray contributions from
the final states $W^{+}W^{-}$, $ZZ$, $hh$, $\gamma\gamma$, $\gamma Z$ and
$W^{+}W^{-}\gamma$ and take into account the Sommefeld enhancement for each of
these channels333All other annihilation channels are always subdominant, with
the top-quark channel potentially reaching a ratio of up to 5% for our lowest
DM masses., as described in Section III. In order to better illustrate
results, we also selected six benchmark model points (BMPs) displaying
qualitatively different spectra. The parameters corresponding to each of these
points as well as their predicted gamma-ray energy spectra are shown in Table
1.

Spectrum | Benchmarks
---|---
| BMP 1: $M_{H^{0}}=2.91$ TeV | BMP 2: $M_{H^{0}}=0.67$ TeV
| $\lambda_{3}=0.120\;\;\lambda_{4}=1.078\;\;\lambda_{5}=-1.100$ | $\lambda_{3}=0.090\;\;\lambda_{4}=0.089\;\;\lambda_{5}=-0.090$
|  |
Dominated
by $\gamma$ lines |  |
| BMP 3: $M_{H^{0}}=1.40$ TeV | BMP 4: $M_{H^{0}}=11.14$ TeV
| $\lambda_{3}=-0.370\;\;\lambda_{4}=0.202\;\;\lambda_{5}=-0.533$ | $\lambda_{3}=6.870\;\;\lambda_{4}=-7.146\;\;\lambda_{5}=-0.481$
|  |
Dominated
by VIB |  |
| BMP 5: $M_{H^{0}}=18.38$ TeV | BMP 6: $M_{H^{0}}=7.17$ TeV
| $\lambda_{3}=0.900\;\;\lambda_{4}=6.572\;\;\lambda_{5}=-7.046$ | $\lambda_{3}=3.130\;\;\lambda_{4}=1.101\;\;\lambda_{5}=-2.399$
|  |
Dominated
by the |  |
continuum |  |
Table 1: Contributions to the differential cross section from continuum
photons (green line), $\gamma\gamma$ (magenta line) and $\gamma Z$ (pink line)
and virtual internal bremsstrahlung [VIB] (blue line), as well as the total
differential cross section (black line) for our selected six benchmark points
in the viable parameter space. The spectra have been convoluted with a
Gaussian detector response characterized by a 10% energy resolution.

Benchmark points BMP1 and BMP2 produce a very intense gamma-ray line, BMP3 and
BMP4 produce a significant virtual bremsstrahlung signal, while BMP5 and BMP6
produce an intense continuum. In the plots, the contributions of the virtual
internal bremsstrahlung (VIB), the continuum part, and the $\gamma\gamma$ and
$\gamma Z$ monochromatic lines are shown, respectively, in blue, green,
magenta and pink. Considering that the total gamma-ray flux observed by
H.E.S.S. telescope falls roughly as $E^{-2.7}$ Abramowski:2013ax , our spectra
have been multiplied by $E^{2.7}$ in order to better appreciate their features
at the highest energies.

For all viable points from our scan, we find fairly large annihilation cross
sections for both the channels producing continuum gamma-ray emission and
those producing sharp gamma-ray spectral features. These signals could
therefore be in reach by present and upcoming gamma-ray telescopes. In Fig. 5

Figure 5: Annihilation cross section into continuum photons (upper panel),
gamma-ray lines (lower-left panel) and virtual internal bremsstrahlung (lower-
right panel) for a random sample of points of the viable parameter space
(orange points), highlighting the six benchmark points of Table 1, compared to
various current upper limits on the cross sections, as well as the projected
reach of CTA. For details, see the main text.

we show in the upper panel the sum of the cross sections of the channels that
produce a broad continuum gamma-ray spectrum ($H^{0}H^{0}\rightarrow
W^{+}W^{-},\leavevmode\nobreak\ ZZ,\leavevmode\nobreak\ hh$), in the lower-
left panel the sum of the channels that produce gamma-ray lines
($H^{0}H^{0}\rightarrow\gamma\gamma,\leavevmode\nobreak\ \gamma Z$) and in the
lower-right panel the channel producing the internal bremsstrahlung signal
($H^{0}H^{0}\rightarrow W^{+}W^{-}\gamma$). In the figures, we also highlight
our six benchmark points from Table 1 by the tags 1 to 6.

The upper plot of Fig. 5 also includes our derived limits on the cross section
into the final state $W^{+}W^{-}$ (solid black line) . These limits are
calculated by adapting the same procedure as in the H.E.S.S. collaboration
publication Abramowski:2011hc (to be briefly described in the next section).
These limits, and all the limits we derived in this paper, are under the
assumption of the Einasto DM density profile using a local DM density of
$\rho_{\odot}=0.39$ GeV/cm3 and our various data sets are from the inner
Galactic center sky region specified in Abramowski:2011hc ; Abramowski:2013ax
unless otherwise stated. In the same plot we also include limits from the
analysis of Ref. Giesen:2015ufa using preliminary measurements of the cosmic
antiproton-to-proton fraction by the AMS-02 collaboration AMS02 (solid red
line). The lower left panel instead includes the limit derived by the H.E.S.S.
collaboration in Ref. Abramowski:2013ax for dark matter annihilation into
$\gamma\gamma$, while the right panel, for the BM4 (IB only) benchmark point
of Bringmann:2007nk , corresponding to the neutralino annihilation
$\chi^{0}\chi^{0}\rightarrow W^{+}W^{-}\gamma$ Bergstrom:2005ss and which
produces a similar spectrum as $H^{0}H^{0}\rightarrow W^{+}W^{-}\gamma$.
Furthermore, the Fermi-LAT Collaboration searches for gamma-ray signals from
dwarf spheroidal galaxies provide relevant limits Ackermann:2015zua . However,
the DM annihilation cross sections predictions can be somewhat different in
these galaxies, because the DM velocity dispersion is lower there than in the
Galactic center region, and consequently we do not include them in our
analysis.

To examine the expected reach of the upcoming CTA telescope, we show in the
upper plot the projected limits on annihilations into $W^{+}W^{-}$ as derived
in Silverwood:2014yza , assuming 100 hours of observation of a Milky Way
center region, and in the lower right plot the limit predictions on
monochromatic photons from Ibarra:2015tya (after a proper rescaling of their
limits on narrow boxed shaped spectra), assuming an observation time of 112
hours of the Galactic center region given in Abramowski:2011hc ;
Abramowski:2013ax .

These estimates indicate that present instruments are already sensitive to
large regions of the viable parameter space of the IDM and that CTA has good
prospects to observe a signals from this model by the observation of a broader
continuum excess in the gamma-rays spectrum. Furthermore, the continuum signal
flux might be complemented by a simultaneous univocal DM signal in the form of
a sharp gamma-ray spectral feature.

##  V Complementarity among Gamma-ray Signals in the IDM

In order to more carefully asses the prospects to observe signals of the IDM,
we derive dedicated limits on each of the IDMs from our parameter scan. For
each model’s cross section limit, we then define a maximal boost factor (BF)
that corresponds to how much the model’s predicted gamma-ray signal can be
increased before it saturates its derived limit.

To derive signal limits on each IDM induced continuum signal, we use the data
collected by the H.E.S.S. instrument and closely follow the method pursued in
Abramowski:2011hc , which compares the gamma-ray fluxes measured in a “search
region” and in a “background region” around the Galactic center. The
$\bar{J}$-factors in the search and background regions are given,
respectively, by $\bar{J}=7.41\times 10^{24}\,\text{GeV}\,\text{cm}^{-5}$ and
$\bar{J}=3.79\times 10^{24}\,\text{GeV}\,\text{cm}^{-5}$ Abramowski:2011hc .
This is the same procedure we used to derive the $W^{+}W^{-}$ limits from
H.E.S.S. for Fig. 5 in the previous section. The derived BFs from the IDMs
induced continuum gamma-ray signals are shown in the left plot of Fig. 6.

Figure 6: Upper limit on the boost factor from the non-observation of the
continuum part of the gamma-ray spectrum (left plot) and of the sharp spectral
features (right plot) expected from annihilations in the IDM for a random
sample of points of the viable parameter space, highlighting the six benchmark
points of Table 1. The color of the points indicate the mass range where they
fall. For details, see the main text.

Points with DM mass in the range $0.5\,\text{TeV}<M_{H^{0}}<1\,\text{TeV}$,
$1\,\text{TeV}<M_{H^{0}}<5\,\text{TeV}$,
$5\,\text{TeV}<M_{H^{0}}<10\,\text{TeV}$ and
$10\,\text{TeV}<M_{H^{0}}<20\,\text{TeV}$ are shown in the colors (to be used
also for future references) red, green, blue and purple, respectively. Among
the viable models, the six benchmark points of Table 1 are highlighted in the
plot with their corresponding tag.

Notably, there are many points, especially with mass above $\sim 2$ TeV which
are already excluded by observations with the H.E.S.S. instrument.
Furthermore, for most of the points the BF value is constrained to be smaller
than $\sim 10$. Therefore, an improvement in sensitivity of gamma-ray
telescopes to this type of exotic continuum flux by a factor of $\sim 10$,
which seems to be feasible with the upcoming CTA (see e.g. Ref.
Silverwood:2014yza and our Fig. 5), could suffice to cover all the signal
predictions from the IDM, assuming that the DM halo distribution follows the
Einasto profile. For a Navarro-Frenk-White profile of the DM distribution, the
$\bar{J}$ factor in the target region is a factor of two smaller
Abramowski:2011hc , hence the annihilation signal would in this case be a
factor of two fainter and the prospects for detection, somewhat poorer.

We have also calculated the maximal boost factor BF from the non-observation
of the sharp gamma-ray spectral features produced by the final states
$\gamma\gamma$, $\gamma Z$ and $W^{+}W^{-}\gamma$. These limits on the IDM
were derived following the procedure pursued by the H.E.S.S. collaboration in
Abramowski:2013ax , which adopts a phenomenological background model defined
by seven parameters. The result of this procedure is illustrated in Table 2
for the four first benchmark points of Table 1, which are IDMs characterized
by having an intense sharp gamma-ray spectral feature.

Benchmarks
---
BMP 1 | BMP 2
|
BMP 3 | BMP 4
|
Table 2: H.E.S.S. limits on the benchmarks of Table 1. See the text for
details.

In the figures of Table 2, we show the predicted DM signal (solid blue line)
and the DM signal after being enhanced by the boost factor BF (dashed blue
line) constructed to saturate the derived 95% C.L. limits from the H.E.S.S.
data (shown by the red dots) Abramowski:2013ax . The best-fit background model
for the BF enhanced signal is shown by the solid black line and the total
gamma-ray flux model, including the enhanced DM signal, are shown by the solid
red line. The boost factor BF for our sample of points derived from the non-
observation of a sharp spectral feature with H.E.S.S. is shown in the right
panel of Fig. 6. We find again models which are already ruled out by present
observations, especially at large DM masses.444The strengthening of the limits
at $M_{H^{0}}\sim 600$ GeV is due to a dip around $E_{\gamma}\sim 700$ GeV in
the gamma-ray flux measured by the H.E.S.S. collaboration, and which is
possibly due to a downward statistical fluctuation. Furthermore, with an
increase in sensitivity by a factor $\sim 10$, which is likely to be achieved
with the upcoming CTA (see e.g. Ibarra:2015tya ; Garcia-Cely:2015dda ), a
significant part of the IDM parameter space will be probed, thus opening the
exciting possibility of observing unambiguous signals from DM annihilation at
future gamma-ray telescopes. Unfortunately, to guarantee the observation of a
sharp feature in the gamma-ray spectrum a larger increase in sensitivity is
necessary, concretely by a factor $\sim 100$, assuming the Einasto profile.

From the above discussions it apparently becomes relevant to investigate the
potential complementarity between the searches for a continuum exotic flux and
a sharp spectral feature. In Fig. 7

Figure 7: Complementarity between the upper limits on the boost factor from
the non-observation the continuum part of the gamma-ray spectrum and from the
non-observation of sharp gamma-ray spectral features. The color and tagging of
the points is as in Fig. 6.

we illustrate this complementary. For each model in our scan, the required
boost factor for a model to become excluded by the continuum spectrum
constraints (continuum BF) is confronted to the required boost factor to
become excluded by the sharp spectral feature (feature BF). It follows from
the figure that, for most of the points, the former provides a stronger
constraint than the latter, with the difference between the BFs being more
notable as the DM mass increases. This behaviour is a consequence of the
requirement on the parameters to correctly reproduce the DM density via
thermal freeze-out. As argued in Section II, larger and larger quartic
couplings are required when the DM mass increases. As a result, the
annihilation rates into $W^{+}W^{-}$, $ZZ$ and $hh$, which can be induced by
quartic coupling interactions (as follows from Eqs. (20), (24) and (25)), are
enhanced compared to $\gamma\gamma$ and $\gamma Z$, which are induced only by
gauge interactions (as follows from Eq.(15)).

We would like to briefly comment on the difference between the IDM and two
other minimal DM scenarios with a DM candidate in a 5-plet and 7-plet
representations of $SU(2)_{L}$ Garcia-Cely:2015dda ; Cirelli:2015bda ;
Aoki:2015nza . In the these papers it was concluded that if the DM candidates
account for all the DM then they are excluded if their masses are below 20
TeV. Although we cover similar masses in the IDM, not all them are excluded,
as it is shown in Fig. 7. The underlying reason for this is related to the
mass splittings between the charged and the neutral components of the inert
states. In the minimal 5-plet and 7-plet DM scenarios, the mass splitting is
set by radiative corrections and is fixed to a constant value. In the IDM
there is more freedom, and the mass splitting can be larger than the quantum
effect. In fact, as already discussed, for large DM masses, large quartic
couplings are typically needed to achieve the right relic abundance. Unless a
cancellation between $\lambda_{4}$ and $\lambda_{5}$ takes place, the mass
difference between $H^{+}$ and $H^{0}$ becomes relatively larger. This leads
to relatively smaller Sommerfeld effects in comparison to these minimal DM
models (even if it is still large for the heaviest DM masses in IDM). Another
important reason is that larger $SU(2)_{L}$ multiplets will contain particles
with larger electric charges. This leads to larger annihilation cross sections
for all gauge mediated annihilation channels and, in particular, increases
monochromatic gamma-ray signals.

##  VI Complementarity with Direct Detection

A complementary avenue to probe the high mass regime of the IDM is direct
detection. The spin-independent scattering cross section of DM particles with
nuclei receives in this model two different contributions. The first one is
induced by the t-channel exchange of a Higgs boson with the nucleon $n$
Barbieri:2006dq :

$\sigma_{\text{SI}}=\frac{M_{n}^{4}(\lambda_{3}+\lambda_{4}+\lambda_{5})^{2}f^{2}}{{4}\pi(M_{n}+M_{H^{0}})^{2}M_{h}^{4}},$
(35)

where $f\approx 0.3$ is a form factor with its precise value taken from
micrOMEGAs 3.1 Belanger:2013oya and $M_{n}=0.939$ GeV is the nucleon mass.
This cross section is suppressed for large DM masses and for small DM-Higgs
coupling, which corresponds to $|\lambda_{3}+\lambda_{4}+\lambda_{5}|$. The
second contribution is induced by one-loop exchange of gauge bosons
Cirelli:2005uq ; Hambye:2009pw ; Klasen:2013btp , which is independent of the
quartic couplings and which sets a lower limit on the interaction cross
section of $\sigma_{\text{SI}}\gtrsim 2.6\times 10^{-46}\,\text{cm}^{2}$
independently on the DM mass. We show in Fig. 8

Figure 8: Predicted value of the spin-independent DM scattering cross section
with protons for a random sample of viable points of the IDM (orange lines),
highlighting the six benchmark points of Table 1, compared to the current
upper limit from the LUX experiment (solid black line) as well as the
projected sensitivities of the XENON1T (long dashed) and LZ (short dashed)
experiments. The minimal value of the cross section induced by the one-loop
exchange of weak gauge bosons is shown as a solid blue line.

the predicted spin-independent scattering cross section with protons for our
sample of viable points and compare them to the limit from the LUX experiment
as well as to the projected reach of XENON1T Aprile:2012zx ; Aprile:2015uzo
and LZ Akerib:2015cja . As apparent from the plot, the current data from the
LUX experiment barely constrain the viable parameter space of the IDM. On the
other hand, and due to the above-mentioned lower limit on the interaction
cross section induced by the one-loop exchange of gauge bosons, the upcoming
XENON1T (LZ) experiment should observe a signal of the IDM for thermally
produced DM particles with $M_{H^{0}}\lesssim
1.6\,\text{TeV}\leavevmode\nobreak\ (13\,\text{TeV})$. We note that many of
the points of our scan have an interaction cross section
$\sigma_{\text{SI}}\gtrsim 5\times 10^{-45}\,\text{cm}^{2}$, which is well
within the reach of XENON1T.

We finally confront the constraints from direct detection DM searches to those
from indirect detection searches in gamma-rays. In Fig. 9,

Figure 9: Complementarity between the direct detection limits on the IDM from
LUX, and the upper limits on the boost factor from the non-observation the
continuum part of the gamma-ray spectrum (left panel) and from the non-
observation of sharp gamma-ray spectral features (right panel). The color and
tagging of the points is as in Fig. 6.

the left plot (right plot) shows the viable DM models in the plane of the
ratio $\sigma_{\text{LUX}}/\sigma$ against the required boost factor of the
broad DM gamma-ray signal (sharp spectral feature) to reach the current
H.E.S.S. constraints. Here $\sigma$ is the predicted spin-independent cross
section with a proton and $\sigma_{\text{LUX}}$ is the upper limit from LUX
for that model. Points with $\sigma_{\text{LUX}}/\sigma<1$ are then ruled out
by the LUX experiment and points with BF$<1$ are ruled out by the H.E.S.S.
instrument. The values of BF and $\sigma_{\text{LUX}}/\sigma$ for our six
benchmark points are indicated in the figure and displayed for reference in
Table 3.

BMP | Boost Feature | Boost Continuum | $\sigma_{\text{LUX}}/\sigma_{\text{SI}}$
---|---|---|---
1 | 1.93 | 2.87 | 77.2
2 | 0.63 | 10.76 | 6.3
3 | 6.67 | 5.71 | 2.4
4 | 13.01 | 1.95 | 204.0
5 | 44.9 | 1.29 | 574.1
6 | 141.0 | 1.34 | 33.5
Table 3: Upper limit on the feature boost factor and continuum boost factor
from current gamma-ray telescopes, as well as value of the spin-independent
cross section relative to the current upper limit from LUX, for the six
benchmark points defined in table 3.

It follows from the figures that, for low DM masses, namely $M_{H^{0}}\lesssim
5$ TeV, an increase of sensitivity in direct detection experiments and gamma-
ray telescopes by a factor 10, which seems feasible in the near future, might
lead to three different DM signals. An exciting possibility is thus to detect:
a scattering signal in a direct detection experiment, a broad gamma-ray excess
from the central galactic halo region together with a sharp spectral feature
in the gamma-ray spectrum. On the other hand, for larger DM masses, indirect
detection with gamma-rays constitutes the most promising search strategy for
the IDM.

##  VII Conclusions

We have investigated the gamma-ray spectrum produced in DM annihilations in
the center of the galaxy for the high mass regime of the Inert Doublet Model
(IDM). We have found that, in order to satisfy the requirements of unitarity
on the annihilation cross sections, it is necessary to account for the so-
called Sommerfeld enhancement. This is a non-perturbative effect arising from
the exchange of gauge and Higgs bosons between non-relativistic annihilating
DM particles. In the mass regime under consideration, such exchange induces
long range interactions that lead to a significant modification of the
annihilation cross sections. We have argued that including that effect is
crucial for phenomenological studies of indirect DM signals from the galactic
center, specially for masses much larger than 1 TeV and small mass splittings
between the charged and the neutral scalars. We have also showed that such
effect is much less important for the DM production in the early Universe (see
Appendix C).

We have calculated the impact of the Sommerfeld enhancement in the framework
of the effective field theory resulting from the non-relativistic limit of the
inert scalar particles. The main ingredients to consider are the potential
matrices of Eq. (7), encoding the long-range effects, and the matrix of Eq.
(11), which describes the annihilation processes. In appendix B we have
described succinctly how to use these matrices in order to the obtain the
annihilation cross sections. Using this formalism, we have been able for the
first time to reliably calculate the gamma-ray spectrum. It receives
contributions from (i) a featureless soft part arising from annihilations into
$W^{+}W^{-}$, $ZZ$ and $hh$ pairs, (ii) monochromatic photons in the
$\gamma\gamma$ and $\gamma Z$ final states and (iii) from the virtual internal
bremsstrahlung process $H_{0}H_{0}\to W^{+}W^{-}\gamma$. In table 1, we have
presented a set of benchmark points (BMP1$-$BMP6), compatible with all
theoretical and experimental constraints on the IDM (see Appendix A) and have
classified them according to the relative importance of each contribution.
BMP1 and BMP2 exhibit strong spectral features at the end point of their
spectra, BMP3 and BMP4 receive their most important contribution from virtual
internal bremsstrahlung, and BMP5 and BMP6 are dominated by the broader
continuum emission of photons.

We have then confronted the IDM with the most recent Galactic center
observations of the H.E.S.S. instrument. Assuming the Einasto profile, and
using a scan over the five-dimensional parameter space of the DM particle
model sector, we have found that many viable models are already excluded by
H.E.S.S., mostly via the continuum emission. The result of this is shown in
Fig. 7. We would like to remark that H.E.S.S. can probe many viable DM models
even though it does not currently reach annihilation cross sections close to
the canonical thermal value of $3\times 10^{-26}$cm3/s. The underlying reason
for this is the Sommerfeld enhancement and, to a lesser extent,
coannihilations in the early Universe Queiroz:2015utg .

Subsequently, we have analyzed the interplay between this indirect search and
the direct DM searches with the LUX experiment. The corresponding results are
shown in Fig. 9, and for our benchmark points, on table 3. Current direct DM
searches are not sensitive enough to detect signals of the IDM. Nevertheless,
the upcoming XENON1T experiment and the projected LZ experiment will be able
to close in on the viable parameter space of the model.

Finally, we would like to comment on the sensitivity of the upcoming Cerenkov
Telescope Array to the IDM. As shown in Fig. 5, a significant part of the
viable models of our scan can be potentially probed by CTA since most of them
produce a continuum spectrum that is within a factor ten from current
experimental sensitivity.

Note Added

During the last stages of this work, we learned of the analysis of Ref.
Queiroz:2015utg , where the gamma-ray signals of the IDM in its high mass
regime are also studied. In that paper, however, the Sommerfeld effect was
neglected and the contribution to the spectrum from gamma-ray lines and from
virtual internal bremsstrahlung was not considered.

###### Acknowledgements.

The authors would like to thank Anna S. Lamperstorfer for useful discussions.
The work of C.G.C. is supported by the FNRS, the IISN and the Belgian Federal
Science Policy through the Interuniversity Attraction Pole P7/37. M.G.
acknowledges partial support from the European Union FP7 ITN Invisibles (Marie
Curie Actions, PITN-GA-2011-289442). The work of AI was partially supported by
the DFG cluster of excellence “Origin and Structure of the Universe”.

##  Appendix A Constraints on the Model Parameters

We summarize here the various theoretical and phenomenological constraints on
the scalar potential which are included in our scan of the parameter space of
the IDM.

  * •

Perturbativity can, at least naively, be enforced by assuming that no scalar
coupling exceeds $4\pi$ (see table 3.2 of GarciaCely:2014jha )

$\displaystyle|\lambda_{1,2,3,4,5}|\leq 4\pi\,,\hskip
10.0pt|\lambda_{3}+\lambda_{4}\pm\lambda_{5}|<4\pi\,,\hskip
10.0pt|\lambda_{4}\pm\lambda_{5}|<8\pi\,,\hskip
10.0pt|\lambda_{3}+\lambda_{4}|<4\pi.$ (36)

  * •

Vacuum stability of the potential requires Gunion:2002zf ; Gustafsson:2010zz
(see also Khan:2015ipa )

$\lambda_{1}>0\;,\hskip 10.0pt\lambda_{2}>0\;,\hskip
10.0pt\lambda_{3}>-2(\lambda_{1}\lambda_{2})^{\frac{1}{2}}\;,\hskip
10.0pt\lambda_{3}+\lambda_{4}-|\lambda_{5}|>-2(\lambda_{1}\lambda_{2})^{\frac{1}{2}}.$
(37)

  * •

Unitarity of the S-matrix on scalar to scalar, gauge boson to gauge boson and
scalar to gauge boson scatterings at the perturbative level furthermore
requires that Ginzburg:2004vp ; Branco:2011iw

$\displaystyle|\lambda_{3}\pm\lambda_{4}|\leq
8\pi,\quad|\lambda_{3}\pm\lambda_{5}|\leq
8\pi,\quad|\lambda_{3}+2\lambda_{4}\pm 3\lambda_{5}|$ $\displaystyle\leq 8\pi$
$\displaystyle|-\lambda_{1}-\lambda_{2}\pm\sqrt{(\lambda_{1}-\lambda_{2})^{2}+\lambda_{4}^{2}}|$
$\displaystyle\leq 8\pi$
$\displaystyle|-3\lambda_{1}-3\lambda_{2}\pm\sqrt{9(\lambda_{1}-\lambda_{2})^{2}+(2\lambda_{3}+\lambda_{4})^{2}}|$
$\displaystyle\leq 8\pi$
$\displaystyle|-\lambda_{1}-\lambda_{2}\pm\sqrt{(\lambda_{1}-\lambda_{2})^{2}+\lambda_{5}^{2}}|$
$\displaystyle\leq 8\pi.$

  * •

Electroweak precision observables constraints the contributions to the Peskin-
Takeuchi $S,\;T,\;U$ parameters to remain in the region $\Delta S=0.06\pm
2\times 0.09$, $\Delta T=0.1\pm 2\times 0.07$ with a correlation coefficient
of +0.091 (when $\Delta U$ is fixed to zero, which is appropriate for the
IDM). The contribution from the IDM can be calculated as in, e.g.,
Arhrib:2012ia . This typically prohibit large mass splittings among inert
states, but for DM masses with $M_{H^{0}}\gtrsim 500$ GeV relatively small
splittings are already required, especially when combined with the relic
density constraint Hambye:2009pw .

Besides these theoretical constraints on the parameters of the IDM, there are
also limits from experimental searches of the inert scalars.

  * •

LEP bound comes first from that the decay channels $Z\rightarrow A^{0}H^{0}$,
$Z\rightarrow H^{+}H^{-}$, $W^{\pm}\rightarrow A^{0}H^{\pm}$ and
$W^{\pm}\rightarrow H^{0}H^{\pm}$ would alter the gauge bosons measured mass
widths. As a good approximations, these implies that $M_{A^{0}}+M_{H^{0}}\geq
M_{Z}$, $2M_{H^{\pm}}\geq M_{Z}$, $M_{H^{\pm}}+M_{H^{0},A^{0}}\geq M_{W}$.
Second, constraints on IDM parameters have been extracted from chargino
searches at LEP II: The charged Higgs mass is constrained by
$M_{H^{\pm}}\gtrsim 70$ GeV Pierce:2007ut . The bound on $M_{H^{0}}$ is more
involved: If $M_{H^{0}}<80$ GeV then $M_{A^{0}}-M_{H^{0}}$ should be less than
$\sim 8$ GeV, or else, $M_{A^{0}}$ should be greater than $\sim 110$ GeV
Lundstrom:2008ai .

  * •

LHC bounds come from the Higgs sector. The new scalar states can either
increase the invisible branching ratio and/or alter the diphoton signal
strength of the Higgs boson Arhrib:2012ia ; Krawczyk:2013jta ;
Swiezewska:2012eh ; Goudelis:2013uca . These bounds are typically only
relevant for masses below $M_{h}/2$, and thus play a little role for inert
scalar particle masses well above. Direct di-lepton searches have also been
shown to restrict the inert scalar masses in the region of $M_{H^{0}}\lesssim
60$ GeV and $M_{A}\lesssim 150$ GeV Belanger:2015kga .

From these constraints it is clear that the IDM is strongly restricted if the
new states are at sub 100 GeV masses and not so constrained for masses above
500 GeV. In addition there are various astrophysical constraints (see, e.g.,
the review in Gustafsson:2010zz ), and direct detection constraints. Due to
the relevance of the latter for this work, they are discussed in the main
text.

##  Appendix B Algorithm for the Sommerfeld Enhancement

In this appendix we briefly describe the algorithm to calculate the Sommerfeld
enhancement factors, given the potential of Eq. (7) and the annihilation
matrices of Eqs. (15), (24), (20) and (25).

As shown in Hisano:2004ds , the Sommerfeld effect is encoded, in the basis of
the inert pair states $(H^{0},H^{0})$, $(A^{0},A^{0})$ and $(H^{-},H^{+})$, in
a $3\times 3$ matrix $g(r)$, which satisfies the following second-order
differential equation

$\displaystyle
g^{\prime\prime}(r)+M_{H^{0}}\left(\dfrac{1}{4}M_{H^{0}}v^{2}{1\\!\\!1}-V(r)\right)g(r)=0\;,$
(38)

where $v$ is the relative velocity of the initial state particles and $V(r)$
is given in Eq. (7). The boundary conditions to solve the differential
equation can be determined by analyzing the behavior of the solution $g(r)$ at
$r=0$ ant $r\rightarrow\infty$. At the origin,

$\displaystyle g(0)$ $\displaystyle=$ $\displaystyle{1\\!\\!1}\;.$ (39)

On the other hand, for large values of $r$, the matrix $g(r)$ depends on the
mass splitting between the pair states. If the mass splitting $\delta m_{ij}$
associated to the inert pairs $i$ and $j$ is smaller than the initial kinetic
energy $\dfrac{1}{4}M_{H^{0}}v^{2}$, then there is enough energy to produce
on-shell states of the corresponding pair and, therefore, the matrix element
$g_{ij}(r)$ at infinity behaves as an out-going wave with momentum given,
according to Eq. (38), by

$\displaystyle
p_{i}=\sqrt{M_{H^{0}}\left(\dfrac{1}{4}M_{H^{0}}v^{2}-V_{ii}(\infty)\right)}=\sqrt{M_{H^{0}}\left(\dfrac{1}{4}M_{H^{0}}v^{2}-2\,\delta
m_{ii}\right)}\;.$ (40)

The corresponding boundary condition is

$\displaystyle\frac{dg_{ij}(r)}{dr}=i\,p_{i}\,g_{ij}(r)\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{when\leavevmode\nobreak\
}r\rightarrow\infty\;.$ (41)

In the opposite case, namely when $\delta m_{ii}>\dfrac{1}{4}M_{H^{0}}v^{2}$,
there is not enough energy to produce on-shell states of the corresponding
pair, and therefore the matrix elements $g_{ij}(r)$ decay exponentially at
infinity. Hence

$\displaystyle g_{ij}(r)=0\,,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{when\leavevmode\nobreak\ }r\rightarrow\infty\;.$
(42)

The boundary conditions at $r=0$ and $r\rightarrow\infty$ then allow to find
$g(r)$ by solving Eq. (38).

Once the solution is obtained, the oscillating phases of $g(r)$ at large
values of $r$ can be factorized by casting the matrix as

$g(r)\to
e^{ir\sqrt{M_{H^{0}}\left(\frac{1}{4}M_{H^{0}}v^{2}{1\\!\\!1}-2\,\delta
m\right)}}d\;.$ (43)

where $d$ is a $3\times 3$ matrix which contains the non-perturbative
enhancement factors due to the exchange of scalar and gauge bosons in the
initial state.555An alternative method to calculate the Sommerfeld enhancement
factors was proposed in Ref. Garcia-Cely:2015dda and consists in solving
instead a differential equation for the matrix $h(r)=g^{\prime}(r)g(r)^{-1}$
with appropriate boundary conditions, and which cures the numerical
instabilities that plague the numerical solution of Eq. 38.

Finally, the s-wave cross section for the annihilation of the pair $i$ into a
final state $f$ can be determined from

$\sigma v\left(i\to
f\right)\Big{|}_{s-wave}=\frac{1}{N_{i}^{2}}(d\,\Gamma_{f}\,d^{\dagger})_{ii}\,,$
(44)

where $N_{i}=1/\sqrt{2}$ for initial states with identical particles, and
$N_{i}=1$ otherwise. Notice that when the potential in Eq. (38) is negligible
then $d={1\\!\\!1}$, and therefore Eqs. (31) and (44) reduce to the standard
expressions for calculating the cross section in the s-wave limit.

##  Appendix C Impact of the Sommerfeld Enhancement in the Early Universe

The DM thermal freeze-out occurs at a temperature $T_{FO}\sim M_{H^{0}}/20$.
For masses greater than a few TeV, this corresponds to the era before the
electroweak symmetry breaking, when the isospin is a good quantum number and
the co-annihilating species are degenerate in mass. Due to this, the potential
and annihilation matrices take a particular simple form, thus allowing to
estimate the impact of the Sommerfeld enhancement in the early Universe
Cirelli:2009uv .

Let us first consider pairs of the co-annihilating species
${H^{0},A^{0},H^{+},H^{-}}$. The subspace generated by such pairs can be
decomposed into one self-conjugate isospin singlet
$|m_{I}=0,I=0\rangle_{Y=0}$, one self-conjugate triplet
$|m_{I},I=1\rangle_{Y=0}$ and one isospin triplet $|m_{I},I=1\rangle_{Y=1}$
and its corresponding complex conjugate $|m_{I},I=1\rangle_{Y=-1}$. In terms
of these states, the co-annihilating pairs with charge $Q=0$, $Q=1$ and $Q=2$
read

$\displaystyle Q$ $\displaystyle=$ $\displaystyle 2:\hskip
20.0ptH^{+}\,H^{+}=|m_{I}=1,I=1\rangle_{Y=1}$ (45) $\displaystyle Q$
$\displaystyle=$ $\displaystyle 1:\hskip
20.0pt\left(\begin{array}[]{l}H^{+}\,H^{0}\\\ H^{+}\,A^{0}\\\
\end{array}\right)=\left(\begin{array}[]{ccccccc}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\
-\frac{i}{\sqrt{2}}&\frac{i}{\sqrt{2}}\\\
\end{array}\right)\left(\begin{array}[]{l}|m_{I}=0,I=1\rangle_{Y=1}\\\
|m_{I}=1,I=1\rangle_{Y=0}\\\ \end{array}\right)$ (52) $\displaystyle Q$
$\displaystyle=$ $\displaystyle 0:\hskip
20.0pt\left(\begin{array}[]{l}H^{0}\,H^{0}\\\ A^{0}\,A^{0}\\\ H^{+}\,H^{-}\\\
H^{0}\,A^{0}\\\
\end{array}\right)=\left(\begin{array}[]{cccc}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\\\
\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}\\\
\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&0&0\\\
0&0&-\frac{i}{\sqrt{2}}&\frac{i}{\sqrt{2}}\\\
\end{array}\right)\left(\begin{array}[]{l}|m_{I}=0,I=0\rangle_{Y=0}\\\
|m_{I}=0,I=1\rangle_{Y=0}\\\ |m_{I}=-1,I=1\rangle_{Y=1}\\\
|m_{I}=1,I=1\rangle_{Y=-1}\\\ \end{array}\right)$ (65)

Because the isospin is a good quantum number, the non-relativistic potential
is proportional to the identity in subspaces with definite isospin and
hypercharge. Concretely,

$\displaystyle V\,|m_{I},I\rangle_{Y}=V_{IY}|m_{I},I\rangle_{Y}$ (66)

This equation and the previous transformation matrices allow to express the
potential in the basis of co-annihilating pairs. Comparing the resulting
potential for $Q=0$ with Eq. (7), it is possible to solve for the potential
$V_{IY}$, after neglecting the vev triggering the electroweak symmetry
breaking (and therefore the mass splittings, the gauge boson masses and the
scalar potential, which are all proportional to some power of $v_{h}$). The
result reads:

$V_{IY}=\frac{\alpha_{IY}}{r}\,,$ (67)

with

$\alpha_{I=0,Y=0}=-\frac{\left(1+2\,c_{W}^{2}\right)\,g^{2}}{16\pi
c_{W}^{2}},\hskip 10.0pt\alpha_{I=1,Y=1}=\frac{g^{2}}{16\pi c_{W}^{2}},\hskip
10.0pt\alpha_{I=1,Y=0}=\frac{\left(-1+2\,c_{W}^{2}\right)\,g^{2}}{16\pi
c_{W}^{2}}$ (68)

An analagous procedure can be applied to the annihilation matrices. That is

$\displaystyle\Gamma\,|m_{I},I\rangle_{Y}=\Gamma_{IY}|m_{I},I\rangle_{Y}$ (69)

Once again by employing this and Eqs. (45), (52) and (65), we write the
annihilation matrices in the basis of co-annihilating pairs. Furthermore by
comparing the result with the addition of Eqs. (15), (24), (20) and (25), we
solve for the annihilation matrix

$\displaystyle\Gamma_{I=0,Y=0}$ $\displaystyle=$ $\displaystyle\frac{1}{32\pi
M^{2}_{H^{0}}}\left(\left(\frac{1-2\,c_{W}^{2}+4\,c_{W}^{4}}{2\,c_{W}^{4}}\right)g^{4}+\left(2\lambda_{3}+\lambda_{4}\right)^{2}\right)$
(70) $\displaystyle\Gamma_{I=1,Y=1}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{5}^{2}}{32\pi M^{2}_{H^{0}}}$ (71)
$\displaystyle\Gamma_{I=1,Y=0}$ $\displaystyle=$ $\displaystyle\frac{1}{32\pi
M^{2}_{H^{0}}}\left(\left(-1+\frac{1}{c_{W}^{2}}\right)g^{4}+\lambda_{4}^{2}\right)$
(72)

In each subspace with definite hypercharge and isospin, the potential and the
annihilation matrices take, neglecting the gauge boson masses, the simple form
of a Coulomb potential, thus allowing to analytically estimate the Sommerfeld
enhancement in each subspace. Concretely, the total annihilation cross
sections reads

$\displaystyle(\sigma v)_{eff}$ $\displaystyle=$
$\displaystyle\frac{2}{4^{2}}\sum_{I,Y}(2I+1)\left(\frac{\pi\alpha_{IY}/v}{e^{\pi\alpha_{IY}/v}-1}\right)\Gamma_{IY}\;,$
(73)

where the function $z/(e^{z}-1)$, with $z=\pi\alpha_{IY}/v$, is the well known
Sommerfeld enhancement factor associated to a Coulomb potential Cirelli:2007xd
; Sommerfeld . Besides, the factor of 2 in the numerator comes from the fact
that the DM in the IDM is its own antiparticle, the symmetry factor $4^{2}$ is
the total number of pairs that can be constructed from the co-annihilating
species, and the factor $(2I+1)$ is the multiplicity of each isospin state

Lastly, taking the thermal average of this expression and using the
instantaneous freeze-out approximation, one can estimate the DM relic density
as Hambye:2009pw

$\Omega h^{2}=\frac{1.07\times
10^{9}\,\text{GeV}^{-1}}{g_{\star}^{1/2}M_{Pl}}\left(\int^{\infty}_{x_{f}}\frac{\langle\sigma
v\rangle_{eff}}{x^{2}}dx\right)^{-1}\ ,$ (74)

where the inverse freeze-out temperature $x_{f}=M_{H^{0}}/T_{f}$ can be found
by solving

$x_{f}=\log\left(\frac{0.038\,M_{Pl}M_{H^{0}}x_{f}^{1/2}\langle\sigma
v\rangle_{eff}}{g_{\star}^{1/2}}\right)\,.$ (75)

Remarkably, Eq. (74) can be used to calculate the relic density even where
there is no Sommerfeld enhancement, by taking $\alpha_{IY}\to 0$. In this
limit, Eq. (73) reduces to

$\displaystyle(\sigma v)_{eff}$ $\displaystyle\simeq$
$\displaystyle\frac{2.4+5.2\left[(2\lambda_{3}+\lambda_{4})^{2}+3\lambda_{4}^{2}+6\lambda_{5}^{2}\right]}{\left(M_{H^{0}}/530\text{
GeV}\right)^{2}}\times 10^{-26}\text{cm}^{3}/\text{s}.$ (76)

From this, we find a deviation of at most 10% from the result of the scan of
Section III, which was derived using micrOMEGAs 3.1 Belanger:2013oya . We also
find that the difference becomes stronger when the mass splitting is very
large, as expected from the fact that in this regime the approximation of
taking the $SU(2)_{L}$ symmetric limit is worst. When the Sommerfeld
enhancement is taken into account, the disagreement with respect to
perturbative result is at most of 30%, in remarkable agreement with what has
been found for Higgsino DM Beneke:2014hja ; Hisano:2006nn , another
$SU(2)_{L}$ doublet candidate. We thus conclude the the Sommerfeld effect in
the early Universe is less dramatic than in the galactic center.

For a given set of quartic couplings, using Eq. (76) and
$\Omega_{DM}\simeq\frac{3\times
10^{-27}\text{cm}^{3}\text{s}^{-1}}{\langle\sigma v\rangle_{eff}}\simeq 0.12$,
it is straightforward to approximately find the mass $M_{H^{0}}$ corresponding
to the observed relic abundance. In fact, with the constraints of Appendix A,
we find that the maximal allowed $M_{H^{0}}$ is about $22.4$ TeV (e.g. at the
point $\lambda_{3}=4\pi$, $\lambda_{4}=\lambda_{5}\simeq-2.4\pi$ and
$\lambda_{2}=4\pi$).

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