# Indirect detection constraints on the scotogenic dark matter model

T. de Boer  R. Busse  A. Kappes  M. Klasen,11footnotetext: michael.klasen@uni-
muenster.de and S. Zeinstra

###### Abstract

Radiative seesaw models have the attractive property of providing dark matter
candidates in addition to the generation of neutrino masses. Here we present a
study of neutrino signals from the annihilation of dark matter particles that
have been gravitationally captured in the Sun in the framework of the
scotogenic model. We compute expected event rates in the IceCube detector in
its 86-string configuration. As fermionic dark matter does not scatter off
nucleons due to its singlet nature and therefore does not accumulate in the
Sun, we study the case of scalar dark matter with a scan over the parameter
space. Due to a naturally small mass splitting between the two neutral scalar
components, inelastic scattering processes with nucleons can occur. We find
that for most of the parameter space, i.e. for mass splittings below 500 keV,
inelastic scattering in the Sun yields IceCube event rates above $10$ events
per year, whereas direct detection on Earth is sensitive only to 250 keV.
Consequently, a detailed analysis with IceCube could lead to a lower limit on
the scalar coupling $\lambda_{5}\gtrsim 1.6\cdot 10^{-5}\cdot m_{\rm DM}/$TeV.
For larger mass splittings, only elastic scattering occurs in the Sun. In this
case, XENON1T limits only allow for models with expected event rates of up to
$\mathcal{O}(0.1)$ per year. Some of these models, in particular those with
large DM mass and fermion coannihilation, could also be tested with a
dedicated IceCube analysis of DM annihilation in the Galactic Center.

##  1 Introduction

While there is today overwhelming evidence for dark matter (DM) in the
Universe from observations at many different length scales [1, 2], and its
relic density can be precisely determined [3], the nature of DM remains still
unknown. Among the many theoretical ideas that have been put forward to
explain it, radiative seesaw models [4, 5, 6, 7, 8] such as the famous
scotogenic model [9, 10, 11, 12, 13] have the advantage of extending the
Standard Model (SM) with only very few fields. They include not only a natural
DM candidate, but also explain the smallness of neutrino masses and thus
address simultaneously two of the largest current mysteries of theoretical
particle physics.

In radiative seesaw models, DM in the form of Weakly Interacting Massive
Particles (WIMPs) can annihilate either directly into neutrinos or via the
decays of other intermediately produced SM particles. Previous work focused on
monochromatic neutrinos from direct decays, as they are easier to distinguish
from the background [14, 15, 16, 17]. Here, we consider both direct as well as
secondary neutrinos from the decays of intermediate other SM particles.

In order to boost the amount of WIMP annihilations, one can consider regions
with a local overdensity [18]. Since our solar system is embedded in the
galactic DM halo, WIMPs can accumulate in large celestial bodies like the Sun,
which we focus on in this work. Upon scattering with a nucleus inside the Sun,
a WIMP can lose enough kinetic energy to be captured by the Sun’s
gravitational potential. Thus the WIMP-nucleon scattering cross section plays
an important role in this capture process. We go beyond the standard scenario
of elastic DM-nucleon scattering by considering also inelastic scattering
processes, in which a WIMP up-scatters to a slightly heavier state. This so-
called inelastic DM had originally been proposed by Smith and Weiner [19] to
explain the annual modulation signal at DAMA/LIBRA [20, 21]. While these
authors considered the sneutrino as a specific DM candidate, they concluded
generally that due to the larger DM velocity, the inelasticity is in fact less
relevant in the Sun than at direct detection experiments, leaving ample room
for indirect detection experiments.

Other previous work that considered the prospect of detecting inelastic DM
indirectly includes Refs. [22, 23, 24]. This work was motivated by the
DAMA/LIBRA signal and principally considered the parameter space that fitted
this signal. A comparison between IceCube and direct detection experiments in
a more general inelastic scenario has been carried out within the context of
effective field theory in Ref. [25]. There it was found that neutrino
telescopes should place stronger limits than direct detection experiments for
mass splittings larger than about 200 keV for DM particles of mass 1 TeV.

Of the many models that connect neutrinos and DM, the scotogenic model is the
best-known example [9]. Its main strength is the relatively simple extension
of the SM with only two new fields, whilst still providing enough interesting
phenomenology, in particular naturally occurring inelastic DM. The neutrino
masses are generated at loop level through the radiative seesaw mechanism,
whereas the tree level seesaw is forbidden by a $\mathbb{Z}_{2}$ symmetry
under which all new particles have an odd charge. In this work we focus on the
scotogenic model with scalar DM. For a similar model, extended by a real
scalar singlet to account for inflation, elastic scattering in the Sun was
found to produce neutrino signals at least two orders of magnitude below the
current sensitivity of neutrino telescopes [26].

As the largest neutrino telescope worldwide, the IceCube Observatory [27] is
predestined to search for neutrino signals from annihilating WIMPs and thereby
contribute to the search for physics beyond the SM. We investigate the
parameter space of the scotogenic model for a detectable neutrino flux in
IceCube from WIMP annihilations in the Sun and Galactic Center, which could
therefore be used to constrain the scotogenic parameter space with dedicated
IceCube data analyses.

This work is organised as follows: We introduce the scotogenic model in
chapter 2, and discuss scattering processes and DM capture in chapter 3. In
chapter 4, we explain the detection of neutrinos from WIMP annihilations in
the Sun with IceCube, where we show observables of a benchmark point. The
results of the scans over the parameter space are discussed and compared to
experimental limits in chapter 5. We summarize our findings and present an
outlook to future studies in chapter 6.

##  2 The scotogenic model

The scotogenic model considered in this work extends the SM by two new fields,
a scalar doublet with the components $\left(\eta^{+},\eta^{0}\right)$ and
three generations of a fermion singlet $N_{i}$, where $i=1,2,3$. In addition
to the SM gauge group, the model assumes a discrete $\mathbb{Z}_{2}$ symmetry,
under which the new fields are all odd and the SM fields are even, which
prevents further decay of the lightest new mass eigenstate into SM particles
and thus guarantees, if this eigenstate is neutral, the stability of the DM
candidate [9].

Apart from the usual kinetic terms, the new terms in the Lagrangian are

$\displaystyle\mathcal{L}_{N}$ $\displaystyle=$
$\displaystyle-\frac{m_{N_{i}}}{2}N_{i}N_{i}+y_{i\alpha}\left(\eta^{\dagger}L_{\alpha}\right)N_{i}+\text{h.c.}-V,$
(2.1)

where $m_{N_{i}}$ is the (diagonal) mass matrix of the fermion singlets. The
$L_{\alpha}$ denote the three generations of the left-handed SM lepton
doublets ($\alpha=1,2,3$), and $y_{i\alpha}$ is a $3\times 3$ Yukawa coupling
matrix.

The scalar potential is given by

$\displaystyle V$ $\displaystyle=$ $\displaystyle
m_{\phi}^{2}\phi^{\dagger}\phi+m_{\eta}^{2}\eta^{\dagger}\eta+\frac{\lambda_{1}}{2}\left(\phi^{\dagger}\phi\right)^{2}+\frac{\lambda_{2}}{2}\left(\eta^{\dagger}\eta\right)^{2}+\lambda_{3}\left(\phi^{\dagger}\phi\right)\left(\eta^{\dagger}\eta\right)$
(2.2)
$\displaystyle+\lambda_{4}\left(\phi^{\dagger}\eta\right)\left(\eta^{\dagger}\phi\right)+\frac{\lambda_{5}}{2}\left[\left(\phi^{\dagger}\eta\right)^{2}+\left(\eta^{\dagger}\phi\right)^{2}\right],$

where $\phi$ is the SM Higgs field with vacuum expectation value
$\langle\phi^{0}\rangle=246.22$ GeV$/\sqrt{2}$ [1]. Vacuum stability requires
the scalar couplings to obey the relations [28]

$\displaystyle\lambda_{1}>0,$ $\displaystyle\quad\lambda_{2}>0,$
$\displaystyle\lambda_{3}$ $\displaystyle>-\sqrt{\lambda_{1}\lambda_{2}},$
(2.3) $\displaystyle\lambda_{3}+\lambda_{4}-\left|\lambda_{5}\right|$
$\displaystyle>-\sqrt{\lambda_{1}\lambda_{2}},$

while perturbativity imposes $|\lambda_{i}|<4\pi$. After electroweak symmetry
breaking, the new Lagrangian gives rise to three physical scalar bosons with
squared masses

$\displaystyle m_{\eta^{+}}^{2}$ $\displaystyle=$ $\displaystyle
m_{\eta}^{2}+\lambda_{3}\langle\phi^{0}\rangle^{2},$ $\displaystyle
m_{\eta^{0R}}^{2}$ $\displaystyle=$ $\displaystyle
m_{\eta}^{2}+\left(\lambda_{3}+\lambda_{4}+\lambda_{5}\right)\langle\phi^{0}\rangle^{2},$
(2.4) $\displaystyle m_{\eta^{0I}}^{2}$ $\displaystyle=$ $\displaystyle
m_{\eta}^{2}+\left(\lambda_{3}+\lambda_{4}-\lambda_{5}\right)\langle\phi^{0}\rangle^{2}.$

The measurement of the SM Higgs boson mass $m_{h}$ at the LHC [29] and the
relation
$m_{h}^{2}=2\lambda_{1}\langle\phi^{0}\rangle^{2}=-2m_{\phi}^{2}=(125$
GeV$)^{2}$ fix $\lambda_{1}$ to $0.26$. Since $\lambda_{2}$ induces only self-
interactions of the new scalars, which do not affect the phenomenology, we set
it to $\lambda_{2}=0.5$ without loss of generality. We assume DM to be the
lightest neutral scalar, i.e. either the real or the imaginary component of
$\eta^{0}=(\eta^{0R}+i\eta^{0I})/\sqrt{2}$. Their mass splitting, induced by
the coupling $\lambda_{5}$, is naturally small, since if $\lambda_{5}$ is
exactly zero, the neutrinos would be massless and lepton number would be
conserved, leading to a larger symmetry of the Lagrangian. This and the fact
that the lightest scalar must be neutral imply that $\lambda_{4}<0$ [26]. When
$m_{\eta}^{2}$ dominates over $\langle\phi_{0}\rangle^{2}$, the scalar
couplings $\lambda_{3}$ and $\lambda_{4}$ will play a subdominant role in the
mass splitting and $\eta^{+}$ will be close in mass to both $\eta^{0R}$ and
$\eta^{0I}$.

In the scotogenic model, the SM neutrinos obtain their masses through the one
loop diagram shown in Fig. 1 [9].

Figure 1: One loop generation of neutrino masses in the scotogenic model.

The SM neutrino mass matrix is connected to the masses of the dark sector
fields by

$\left(m_{\nu}\right)_{\alpha\beta}=\left(y^{T}\Lambda y\right)_{\alpha\beta}$
(2.5)

with the Yukawa matrix $y$ and the diagonal mass matrix $\Lambda$, which has
the eigenvalues

$\Lambda_{i}=\frac{m_{N_{i}}}{32\pi^{2}}\left[\frac{m_{\eta^{0R}}^{2}}{m_{\eta^{0R}}^{2}-m_{N_{i}}^{2}}\log\left(\frac{m_{\eta^{0R}}^{2}}{m_{N_{i}}^{2}}\right)-\left(R\rightarrow
I\right)\right]\,.$ (2.6)

The two terms within the brackets are equal up to interchanged real and
imaginary components. As explained above, non-zero neutrino masses therefore
require a mass splitting between $m_{\eta^{0R}}$ and $m_{\eta^{0I}}$. The PMNS
matrix $U_{\text{PMNS}}$ diagonalizes the neutrino mass matrix with

$U_{\text{PMNS}}^{T}m_{\nu}U_{\text{PMNS}}=\hat{m}\equiv\text{diag}\left(m_{1},m_{2},m_{3}\right)\,.$
(2.7)

For given masses and scalar couplings of the model, the Yukawa matrix

$y=\sqrt{\Lambda}^{-1}R\sqrt{\hat{m}_{\nu}}U_{\text{PMNS}}^{\dagger}\,,$ (2.8)

is then fixed by the known SM neutrino mass differences and mixing angles
through the Casas-Ibarra parametrization up to an orthogonal matrix $R$, that
depends on three arbitrary rotation angles $\theta_{i}\in\left[0;2\pi\right]$
[30].

To obtain the correct DM relic density measured by Planck [3], not only
annihilation processes of the lightest scalar, but also coannihilation
processes with one or several dark fermions can be very important [10]. Since
the SM neutrinos are part of a lepton doublet, charged lepton flavor violation
(LFV) naturally occurs in radiative seesaw models. In the scotogenic model,
the corresponding constraints have a larger impact on fermion than on scalar
DM, which we consider here [11]. Nevertheless, we take the LFV limits into
account as discussed below. Conversely, direct detection has a large impact on
scalar DM, as we will also see below, whereas fermion DM us only impacted at
one loop [12]. The LEP2 limit on new charged scalar masses ($m>98.5$ GeV) [31]
is usually avoided due to either the dominance of $m_{\eta}^{2}$ over
$\langle\phi_{0}\rangle^{2}$ or the large splitting with the neutral scalar
masses induced by $\lambda_{4}$. Note that in addition the parameter space of
fermion DM is further constrained by the KATRIN upper limit on the lightest
neutrino mass [13].

##  3 WIMP capture in the Sun

When WIMPs scatter off nuclei in the Sun, they can lose energy up to a point
where their remaining kinetic energy is insufficient to escape the
gravitational potential of the Sun. This leads to an accumulation of DM in the
Sun’s core, boosting the annihilation rate. The neutrino flux from these
annihilations is thus characterized by the WIMP-nucleon scattering cross
section. As described above, the scotogenic model can provide either a
fermionic ($N_{i}$) or scalar ($\eta^{0R}$, $\eta^{0I}$) DM candidate,
whichever is the lightest particle of the $\mathbb{Z}_{2}$ odd sector.

Singlet fermion DM does not scatter off nucleons at tree level. It therefore
does not accumulate in the Sun (or the Earth) and produce a significant
neutrino flux, rendering it undetectable for neutrino telescopes. In addition,
only direct annihilations into SM neutrinos and charged leptons are possible,
which are both suppressed by small Yukawa couplings and heavy scalar
propagators. This also holds for gravitationally accumulated fermion singlet
DM in the Galactic Center. Scalar doublet DM, on the other hand, has
electroweak size cross sections, which enable accumulation in celestial bodies
and annihilation into SM particles, that subsequently decay into neutrinos. In
this work we therefore only consider scalar DM, as it can produce sizable
detection rates at neutrino telescopes.

The diagrams for scalar DM scattering off nucleons are shown in Fig. 2.

Figure 2: Feynman diagrams of the elastic (left) and inelastic (right) scalar
DM-nucleon scattering processes in the scotogenic model. If $\eta^{0I}$ is the
DM candidate, $\eta^{0R}$ and $\eta^{0I}$ change their roles. For mass
splittings larger than a few hundred keV, the right diagram is kinematically
forbidden.

Usually these scattering processes are described by diagrams with the same in-
and outgoing DM particle. This is the elastic case. However, as originally
pointed out in Ref. [19], the existence of a slightly heavier state allows
also for inelastic upscattering of the DM particle, provided that the mass
splitting between the two states $\delta=|m_{\eta^{0R}}-m_{\eta^{0I}}|$
fulfills

$\delta<\frac{\mu v^{2}}{2},$ (3.1)

where $\mu$ is the WIMP nucleus reduced mass and $v$ is the relative velocity.
In the scotogenic model, the mass splitting between the neutral scalar
components is governed by $\lambda_{5}$. Since $\lambda_{5}$ is naturally
small as explained in Sec. 2, the mass splitting is small as well and can be
approximated by

$\delta\approx\frac{\lambda_{5}\langle\phi^{0}\rangle^{2}}{m_{\eta^{0R,I}}}.$
(3.2)

###  3.1 Elastic scattering

First we turn our attention to the case of elastic scattering. As shown in
Fig. 2 (left), the elastic process is mediated by the exchange of the SM Higgs
boson $h$. The Lagrangian relevant for this interaction is given by [32]

$\displaystyle\mathcal{L}=-\frac{1}{2}g_{\eta^{0R}\eta^{0R}h}\left(\eta^{0R}\right)^{2}h-g_{\bar{q}qh}\bar{q}qh,$
(3.3)

assuming that $\eta^{0R}$ is the dark matter candidate. When $\eta^{0I}$ is
the lightest dark particle, one simply needs to replace $\eta^{0R}$ with
$\eta^{0I}$. The couplings
$g_{\eta^{0R,I}\eta^{0R,I}h}=(\lambda_{3}+\lambda_{4}\pm\lambda_{5})\langle\phi^{0}\rangle\sqrt{2}$
and $g_{\bar{q}qh}=y_{q}/\sqrt{2}$ are fixed by the high energy Lagrangian of
the scotogenic model in Sec. 2 and the SM quark Yukawa couplings
$y_{q}=m_{q}/\langle\phi^{0}\rangle$ and can be obtained using SARAH [33].

In the non-relativistic limit, an effective Lagrangian can be derived by
integrating out the mediator. The result is [32, 18]

$\displaystyle\mathcal{L}_{\text{eff}}=\frac{1}{2}a_{q}2m_{\eta^{0R}}\left(\eta^{0R}\right)^{2}\bar{q}q,$
(3.4)

where we have grouped a factor $2m_{\eta^{0R}}$ with the effective operator
such that it has the same mass dimension as in the fermionic case [34]. The
effective coupling is thus defined as
$a_{q}:=g_{\eta^{0R,I}\eta^{0R,I}h}g_{\bar{q}qh}/(2m_{\eta^{0R,I}}m_{h}^{2})$
and has mass dimension $(-2)$ as usual. The elastic scattering amplitude on a
nucleon $N$, i.e. a proton $p$ or a neutron $n$, is then given by

$\displaystyle\mathcal{M}=4m_{\eta^{0R}}m_{N}\sum_{q}a_{q}\langle
N|\bar{q}q|N\rangle,$ (3.5)

where $m_{N}$ is the nucleon mass and $|N\rangle$ is the nucleon state in the
non relativistic normalization.

The nucleon matrix elements for light quarks ($q=u,d,s$) can be evaluated as
[32, 18]

$\displaystyle m_{q}\langle N|\bar{q}q|N\rangle=m_{N}f_{q}^{(N)},$ (3.6)

where $m_{q}$ are the quark masses and $f_{q}^{(N)}$ denote the scalar quark
form factors in nucleons, for which we employ the values shown in Tab. 1 [35].

Table 1: Scalar quark form factors in nucleons used in this work [35]. | proton | neutron
---|---|---
$f^{(N)}_{d}$ | 0.0191 | 0.0273
$f^{(N)}_{u}$ | 0.0153 | 0.011
$f^{(N)}_{s}$ | 0.0447 | 0.0447

For heavy quarks ($Q=c,b,t$), the matrix element is given by

$\displaystyle m_{Q}\langle
N|\bar{Q}Q|N\rangle=\frac{2}{27}m_{N}\left[1-\sum_{q=u,d,s}f_{q}^{(N)}\right].$
(3.7)

The effective coupling to protons (neutrons) is then

$\displaystyle\frac{f_{p(n)}}{m_{p(n)}}=\sum_{q=u,d,s}f_{q}^{p(n)}\frac{a_{q}}{m_{q}}+\frac{2}{27}\left[1-\sum_{q=u,d,s}f_{q}^{p(n)}\right]\sum_{Q=c,b,t}\frac{a_{Q}}{m_{Q}}.$
(3.8)

Spin independent interactions couple coherently to the $Z$ protons and $A-Z$
neutrons in a nucleus of mass number $A$ and mass $m_{A}$. At zero momentum
transfer $|{\bf q}|$, the scattering cross section can therefore be written as
[32, 18]

$\displaystyle\sigma_{A}^{0}=\frac{\mu^{2}}{\pi}\left[Zf_{p}+(A-Z)f_{n}\right]^{2}.$
(3.9)

Taking into account the loss of coherence at high momentum transfer, the
differential cross section

$\displaystyle\frac{d\sigma_{A}}{d|{\bf
q}|^{2}}=\frac{\sigma_{A}^{0}}{4\mu^{2}v^{2}}F^{2}(Q)$ (3.10)

depends also on a nuclear form factor $F^{2}(Q)$, where the energy transfer
$Q$ (or recoil energy $E_{R}$) is $|{\bf q}|^{2}/(2m_{A})$ and ranges from
$Q_{\text{min}}=0$ to $Q_{\text{max}}=4\mu^{2}v^{2}/(2m_{A})$. In this work,
we use the Gaussian form factor [36]

$\displaystyle F^{2}(Q)=\exp(-Q/Q_{0})$ (3.11)

with $Q_{0}=3/(2m_{A}R^{2})$ and

$\displaystyle
R=\left[0.91\left(\frac{m_{A}}{\text{GeV}}\right)^{\frac{1}{3}}+0.3\right]\times
10^{-13}\text{ cm},$ (3.12)

since it allows for analytic integration of the scattering cross section and
thus for a fast subsequent numerical evaluation of the neutrino flux in large
scans of the model parameter space.

###  3.2 Inelastic scattering

The inelastic scattering of scalar DM off nuclei in the scotogenic model is
mediated by the exchange of a $Z^{0}$ boson as shown in Fig. 2 (right). The
vector part of the Lagrangian coupling neutral scalars and quarks to the
$Z^{0}$ boson is given by [32]

$\displaystyle\mathcal{L}=g_{\eta^{0R}\eta^{0I}Z^{0}}\left(\eta^{0R}\partial^{\mu}\eta^{0I}-\partial^{\mu}\eta^{0R}\eta^{0I}\right)Z^{0}_{\mu}+g_{\bar{q}qZ^{0}}\bar{q}\gamma^{\mu}qZ^{0}_{\mu}.$
(3.13)

Note that elastic scattering is absent for the $Z^{0}$ boson and that the
axial vector part of the $\bar{q}qZ^{0}$ interaction vanishes in the non
relativistic limit. The remaining vector interaction contributes to the spin
independent scattering cross section. The couplings
$g_{\eta^{0R}\eta^{0I}Z^{0}}=ig/(2c_{W})$ and
$g_{\bar{q}qZ^{0}}=g(I_{q}-2e_{q}s_{W}^{2})/(2c_{W})$ are fixed by the
SU(2)${}_{L}\times$U(1)Y gauge couplings $g$ and $g^{\prime}$, their ratio
$\tan\theta_{W}=s_{W}/c_{W}=g^{\prime}/g$ and the quark weak isospin
$I_{q}=\pm 1/2$ and fractional charge $e_{u,d}=(2/3,-1/3)$ and can be obtained
using SARAH [33].

Integrating out the mediator produces the effective Lagrangian [32]

$\displaystyle\mathcal{L}_{\text{eff}}=-b_{q}(\eta^{0R}\partial_{\mu}\eta^{0I}-\partial_{\mu}\eta^{0R}\eta^{0I})\bar{q}\gamma^{\mu}q,$
(3.14)

where $b_{q}:=g_{\eta^{0R}\eta^{0I}Z^{0}}g_{\bar{q}qZ^{0}}/m_{Z}^{2}$. In this
case, the effective operator already has the correct mass dimension. The
scattering amplitude is then

$\displaystyle\mathcal{M}=4m_{\eta^{0R}}m_{N}\sum_{q}b_{q}\delta^{0}_{\mu}\langle
N|\bar{q}\gamma^{\mu}q|N\rangle,$ (3.15)

since the scalar four-momenta add up to
$p^{\eta^{0R}}_{\mu}+p^{\eta^{0I}}_{\mu}\approx
2m_{\eta^{0R,I}}\delta_{\mu}^{0}$. Due to the conservation of the vector
current, only valence quark contributions must be considered.

Using the non relativistic normalization for nucleon states, the vector matrix
element for a nucleon with $n_{q}$ valence ($u,d$) quarks is simply [32]

$\displaystyle 2m_{N}\langle
N|\bar{q}\gamma^{\mu}q|N\rangle=n_{q}\bar{u}_{N}\gamma^{\mu}u_{N}\approx
n_{q}2m_{N}\delta^{\mu}_{0}.$ (3.16)

The valence quark contributions thus add up coherently to

$\displaystyle b_{p}$ $\displaystyle=2b_{u}+b_{d},$ (3.17) $\displaystyle
b_{n}$ $\displaystyle=b_{u}+2b_{d}.$ (3.18)

Note that for vector interactions, in contrast to the Higgs exchange, the
scattering off protons and neutrons differs significantly.

Since the vector interaction is spin independent, we can also sum coherently
over the contributions of all nucleons in the nucleus. The cross section then
takes a similar form as for the scalar interaction, i.e. [32]

$\displaystyle\sigma_{A}^{0}=\frac{\mu^{2}}{\pi}\left[Zb_{p}+(A-Z)b_{n}\right]^{2}.$
(3.19)

Since the inelastic scattering is mediated by a $Z^{0}$-boson and the
corresponding couplings are SM gauge couplings, the resulting cross sections
are significantly larger than the elastic Higgs-mediated cross sections.

While the inelasticity can be safely neglected in the differential cross
section

$\displaystyle\frac{d\sigma_{A}}{d|{\bf
q}|^{2}}=\frac{\sigma_{A}^{0}}{4\mu^{2}v^{2}}F^{2}(Q),$ (3.20)

and form factor, it does affect the kinematics through the integration
boundaries [22, 23, 24]

$\displaystyle Q_{\text{min}}$
$\displaystyle=\frac{1}{2}m_{\eta^{0R,I}}v^{2}\left(1-\frac{\mu^{2}}{m_{A}^{2}}\left(1+\frac{m_{A}}{m_{\eta^{0R,I}}}\sqrt{1-\frac{\delta}{\mu
v^{2}/2}}\right)^{2}\right)-\delta,$ (3.21) $\displaystyle Q_{\text{max}}$
$\displaystyle=\frac{1}{2}m_{\eta^{0R,I}}v^{2}\left(1-\frac{\mu^{2}}{m_{A}^{2}}\left(1-\frac{m_{A}}{m_{\eta^{0R,I}}}\sqrt{1-\frac{\delta}{\mu
v^{2}/2}}\right)^{2}\right)-\delta.$ (3.22)

###  3.3 Capture rate

The DM capture rate per unit shell volume in a celestial body such as the Sun
is given by [36, 22, 23, 24]

$\displaystyle\frac{dC}{dV}=\int_{0}^{\infty}\
du\frac{f(u)}{u}w\Omega_{v}^{-}(w),$ (3.23)

where $f(u)$ is the DM velocity distribution outside of the gravitational
field, which we assume to follow the Maxwell-Boltzmann distribution, and $u$
is the DM velocity at infinity. After falling into the gravitational potential
of the Sun, the DM velocity is

$\displaystyle w=\sqrt{u^{2}+v(r)^{2}},$ (3.24)

where $v(r)$ is the escape velocity at a shell with radius $r$. The rate per
unit time, with which a DM particle with velocity $w$ scatters off a nucleus
$A$ to a velocity less than $v(r)$, is

$\displaystyle\Omega_{v}^{-}=\frac{n_{A}\sigma_{A}^{0}w}{4\mu^{2}v^{2}/(2m_{A})}\int_{Q^{\prime}_{\min}}^{Q_{\max}}\
dQF^{2}(Q).$ (3.25)

It is determined by the total scattering rate $n_{A}\sigma_{A}^{0}w$, where
$n_{A}$ denotes the number density of the nucleus $A$ in the Sun, and a
conditional probability, that ensures the capture. For inelastic scattering,
non relativistic kinematics constrain the energy transfer $Q$ to lie within
the limits of Eqs. (3.21) and (3.22), where $v$ should be replaced by $w$. For
the DM particle to be captured, the energy transfer must in addition be
greater than

$\displaystyle Q_{\text{cap}}$
$\displaystyle=\frac{1}{2}m_{\eta^{0R,I}}(w^{2}-v(r)^{2})-\delta.$ (3.26)

so that $Q^{\prime}_{\min}=\max(Q_{\rm cap},Q_{\min})$. The well-known results
for elastic scattering are recovered in the limit $\delta\to 0$ [36]. For
small mass splittings below $\mathcal{O}(100\text{ keV})$ the capture rate is
almost constant. For larger mass splittings, the capture rate quickly drops
off, as the inelastic scattering becomes kinematically forbidden.222We have
reproduced the numerical results for the capture rate in Fig. 3 of Ref. [23]
up to a missing factor of two in $E_{\max}^{\rm elastic}$ as well as those in
Figs. 1 and 2 of Ref. [24]. Note that Ref. [22] uses
$Q^{\prime}_{\text{min}}=Q_{\text{cap}}$, which ignores the case where
$Q_{\min}>Q_{\rm cap}$ and thus gives too large capture rates for large mass
splittings.

The time evolution of the DM population $N$ in the Sun is controled by

$\dot{N}=C-2\Gamma,$ (3.27)

where the capture rate $C$ depends on the elastic and inelastic DM-nucleus
scattering cross sections and nuclear abundances as described above and
$\Gamma$ is the annihilation rate. The evaporation rate relevant for very
light DM has been neglected. Over time, DM capture and annihilation will reach
equilibrium. Provided that the DM-nucleus scattering cross section is
sufficiently large, we can assume that the time of equilibrium onset is well
below the age of the Sun [37]. This leads to an annihilation rate of

$\Gamma=C/2.$ (3.28)

Hence the DM annihilation rate in the Sun depends directly on the elastic and
inelastic scattering cross sections and on the nuclear abundances in the Sun.

##  4 Indirect detection of elastic and inelastic DM in the Sun with IceCube

Our numerical analysis of elastic and inelastic DM and in particular of the
expected neutrino signals at IceCube from DM annihilations in the Sun is based
on an implementation of the model described in Sec. 2 in Sarah 4.14.0 [33].
The physical mass spectrum and branching ratios, in particular those for LFV
processes, are computed with SPheno 4.0.3 [38, 39]. The DM relic density,
direct detection cross sections and neutrino event rates are obtained from
micrOMEGAs 5.0.8 [35].

Assuming equilibrium of capture and annihilation in the core of the Sun
($\Gamma=C/2$), the differential flux of neutrinos or antineutrinos on Earth
is given by [18, 40]

$\frac{d\phi_{\nu}}{dE_{\nu}}=\frac{1}{4\pi
d_{\odot}^{2}}\Gamma\sum_{f}Br_{f\bar{f}}\frac{dN_{f}}{dE_{\nu}},$ (4.1)

where $d_{\odot}$ is the distance Earth-Sun, $Br_{f\bar{f}}$ are the branching
fractions into particle-antiparticle final states $f\bar{f}$, and
$dN_{f}/dE_{\nu}$ are the corresponding neutrino (antineutrino) energy
spectra. In micrOMEGAs, the latter are computed based on tables and feature
neutrino propagation and oscillation in the Sun and in vacuum. The function
neutrinoFlux automatizes the process of calculating the capture rate, the
annihilation branching ratios and spectra of the respective channels and
provides the total neutrino flux at Earth. Since the inelastic scenario was
not implemented in micrOMEGAs, we used CalcHEP 3.7 [41] to compute the
corresponding DM-quark scattering matrix elements and cross sections as
described in the previous section. We then modified the routine dssenu
capsunnum in DarkSUSY 6.2.3 [42] to obtain the inelastic capture rate, which
was then fed back to micrOMEGAs.

The differential number of signal events in the detector is given by [40]

$\frac{dN_{s}}{dE}=t_{e}\left(\frac{d\phi_{\nu_{\mu}}}{dE}A_{\nu_{\mu}}(E)+\frac{d\phi_{\bar{\nu}_{\mu}}}{dE}A_{\bar{\nu}_{\mu}}(E)\right)\,,$
(4.2)

where $t_{e}$ is the exposure time and $A_{\nu_{\mu}(\bar{\nu}_{\mu})}$ is the
muon neutrino (muon antineutrino) effective area of the detector. A routine
for the effective area of the now obsolete configuration ic22 (where 22 is the
number of data-taking strings), IC22nuAr, was already implemented in
micrOMEGAs. We updated this routine using the data from Ref. [43] for the
effective area of ic86. Eight DeepCore strings are part of the IC86
configuration. Including their effective area lowers the energy threshold to
10 GeV. We extrapolated the data points linearly to fit our energy range, as
shown in Fig. 3.

Figure 3:  The $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ effective areas of the
DeepCore detector and the IceCube detector configuration ic86 as a function of
the neutrino energy. The data for the combined effective area is taken from
Ref. [43] (triangles and crosses) and linearly extrapolated (solid black
line). The individual effective areas for neutrinos (dashed blue line) and
antineutrinos (dotted red line) are calculated with the deep inelastic
scattering cross sections taken from Ref. [44]. Both the IceCube and DeepCore
selections of the effective area are used in our work.

The corresponding data points have been taken from Ref. [43]. In the region
where both selections overlap, we use the IceCube effective area, as it is
larger than the one of the DeepCore selection [8].

IceCube is sensitive to both $\nu_{\mu}$ and $\bar{\nu}_{\mu}$. However, the
corresponding energy dependent deep-inelastic scattering cross sections with
nucleons in the detector are slightly different. Since Ref. [43] provides only
the combined $\nu_{\mu}+\bar{\nu}_{\mu}$ effective area, we calculate the
individual effective areas by taking into account the different cross sections
$\sigma_{\nu_{\mu}\left(\bar{\nu}_{\mu}\right)}$ given in Ref. [44] with the
relation

$A_{\nu_{\mu}\left(\bar{\nu}_{\mu}\right)}=\frac{A_{\text{combined}}}{1+\frac{\sigma_{\bar{\nu}_{\mu}\left(\nu_{\mu}\right)}}{\sigma_{\nu_{\mu}\left(\bar{\nu}_{\mu}\right)}}}\,.$
(4.3)

We illustrate the expected neutrino fluxes at Earth and event rates in IceCube
for a specific benchmark point BPA in the scotogenic model. The corresponding
parameters are listed in Tab. 2.

Table 2: Parameters of the scotogenic model for our benchmark point BPA. Shown are the coupling parameters $\lambda_{i}$, the squared mass $m_{\eta}^{2}$ of the new scalar doublet, the (diagonal) mass matrix $m_{N}$ for the three new fermion singlets, and the real and imaginary parts of the Yukawa matrix, $y^{R}$ and $y^{I}$. $\lambda_{1}$ | $\lambda_{2}$ | $\lambda_{3}$ | $\lambda_{4}$ | $\lambda_{5}$ | $m_{\eta}^{2}$ | $m_{N_{1}}$ | $m_{N_{2}}$ | $m_{N_{3}}$
---|---|---|---|---|---|---|---|---
$0.26$ | $0.50$ | $0.56$ | $-0.14$ | $2.00\cdot 10^{-7}$ | $1.00\cdot 10^{6}$ | $1.32\cdot 10^{3}$ | $3.13\cdot 10^{3}$ | $3.44\cdot 10^{3}$
$y^{R}/10^{-2}$ | $y^{I}/10^{-3}$
$\left(\begin{array}[]{ccc}-17.20&2.07&-6.91\\\ -4.94&9.41&15.17\\\ 5.22&15.98&-8.20\end{array}\right)$ | $\left(\begin{array}[]{ccc}2.58&4.46&5.10\\\ -6.88&2.02&2.30\\\ -2.08&-1.23&-1.40\end{array}\right)$

With these parameters, we obtain the correct relic density of $\Omega
h^{2}=0.1217$ for the scalar DM candidate $\eta^{0I}$ with mass 1007.38 GeV.
In descending order of importance, the DM annihilation channels are
$W^{+}W^{-}$ (58.36%), $Z^{0}Z^{0}$ (23.51%), $hh$ (12.30%) and
$W^{+}W^{-}\gamma$ (4.99%), i.e. $W^{+}W^{-}$ pairs (plus an accompanying
photon) represent the dominant channel with a branching fraction of over 63%,
as is the case for most points in the parameter space with
$m_{\eta^{0R,I}}>m_{W}$. Direct annihilation into neutrinos is suppressed by
both the small Yukawa couplings $y_{i\alpha}$ and the large neutrino
propagator masses $m_{N_{i}}$.

We show the differential neutrino and antineutrino fluxes at BPA for both
elastic (blue) and inelastic (red) scattering in the Sun in Fig. 4 (top).

Figure 4: Top: Elastic (blue) and inelastic (red) muon neutrino (full) and
antineutrino (dashed) fluxes at Earth as a function of the neutrino energy,
calculated with the modified function neutrinoFlux in micrOMEGAs at the
parameter point BPA. The mass of the DM particle $\eta^{0I}$ is 1007.38 GeV.
Bottom: Expected number of signal events in ic86 per year as a function of the
neutrino energy.

The differences in differential fluxes between neutrinos (full) and
antineutrinos (dashed lines) that show up at high energies are due to
absorption (also oscillation and regeneration) effects taking place inside the
Sun [45]. Because there is no annihilation into neutrinos directly, we do not
observe a monochromatic neutrino line in the spectrum, but instead a sharp
cut-off at the DM mass of about 1 TeV. The differential number of expected
signal events per year in ic86 is shown in Fig. 4 (bottom). After integration
over the neutrino energy, BPA yields a total number of only $0.07$ expected
events per year in ic86 in the elastic case, but $8.65\cdot 10^{4}$ in the
inelastic case.

##  5 Numerical scan

Using the tool chain described in the previous chapter, we now perform a
numerical scan of the scotogenic parameter space. To comply with the
requirements of vacuum stability, perturbativity and that the lightest scalar
must be neutral, the scalar couplings are varied within the ranges

$\lambda_{3}\in\left[-\sqrt{\lambda_{1}\lambda_{2}},4\pi\right]\ ,\
\lambda_{4}\in\left[\max\\{-\sqrt{\lambda_{1}\lambda_{2}}-\lambda_{3}+\left|\lambda_{5}\right|,-4\pi\\},0\right]\
,\ \left|\lambda_{5}\right|\in\left[10^{-10},10^{-3}\right]$ (5.1)

As was mentioned in Sec. 2, $\lambda_{5}$ is naturally small, since the SM
neutrino masses would vanish and the symmetry of the Lagrangian increase if it
was exactly zero. The mass parameter of the new scalar doublet is varied in
the range

$m_{\eta}\in\left[1\,\mathrm{GeV},10\,\mathrm{TeV}\right].$ (5.2)

The masses $m_{N_{i}}$ of the new fermion singlets are required to be larger
than the mass of the lightest scalar, but below $10$ TeV. In a second scan, we
choose this mass difference to be small (i.e. 0.1 GeV) in order to demonstrate
the impact of coannihilation processes [10]. The SM neutrino oscillation
parameters are taken from Ref. [46] in the $3\sigma$ range, assuming normal
ordering. The CP violating phase $\delta_{CP}$ is varied from zero to $2\pi$,
and the lightest neutrino mass is varied in the interval $[10^{-4},1.1]$ eV in
accordance with the most recent limit set by KATRIN [47]. The Yukawa couplings
are then calculated using the Casas-Ibarra parametrization and required to
satisfy $|y_{i\alpha}|^{2}<4\pi$.

The scotogenic model is further constrained by a number of other experimental
measurements. In particular, we impose the relic density measurement by Planck
[3] with a relatively loose margin of $\Omega h^{2}=0.12\pm 0.02$ in order to
account for theoretical uncertainties [48]. We also impose the LFV branching
ratios (BR) and conversion rate (CR) of

$\displaystyle\text{BR}\left(\mu\rightarrow e+\gamma\right)$ $\displaystyle<$
$\displaystyle 4.2\cdot 10^{-13},$ $\displaystyle\text{BR}\left(\mu\rightarrow
3e\right)$ $\displaystyle<$ $\displaystyle 1.0\cdot 10^{-12},$ (5.3)
$\displaystyle\text{CR}\left(\mu-e,\text{Ti}\right)$ $\displaystyle<$
$\displaystyle 4.3\cdot 10^{-12},$

published by the MEG [49], SINDRUM [50] and SINDRUM II [51] collaborations.
Furthermore, we apply limits on the new physics invisible decay width of the
$Z^{0}$ boson from LEP [52]

${\rm BR}(Z^{0}\to{\rm new})<0.008,$ (5.4)

which effectively excludes $m_{\rm DM}<m_{Z}/2$ [53, 54], and on the invisible
decay width of the SM Higgs boson from ATLAS (CMS) at the LHC [55, 56]

${\rm BR}(h\to{\rm inv.})<0.11\ (0.19),$ (5.5)

as well as on the elastic and inelastic scattering cross section from direct
searches with XENON1T [57], XENON100 [58, 59] and PandaX-II [60]. The possible
signal from DAMA/LIBRA [20, 21], its ongoing verification [61, 62, 63],
previous limits on indirect detection from DM annihilation into neutrinos in
the Sun [64, 43, 65] and the Galactic Center [66, 67, 68, 69], and expected
event rates for neutrinos from the Sun in the current IceCube configuration
with 86 strings (IC86) are also discussed in the following.

###  5.1 Limits on the elastic cross section

Figure 5: The spin independent (SI) elastic cross section with ANTARES [64],
IceCube [43], Super-Kamiokande [65] and XENON1T [57] exclusion limits as a
function of the DM mass. All points and lines are color coded according to the
main annihilation channel, provided there is one with a branching ratio of
over 50%. Also shown are the LEP exclusion from the invisible $Z^{0}$ boson
width [53, 54] and the neutrino floor [70]. In the lower plot, coannihilation
processes are enhanced by the small scalar-fermion mass difference.

Our spin independent elastic cross section results for scalar DM in the
scotogenic model are shown in Fig. 5. Apart from the larger mass range of our
scan and considering that we imposed updated constraints from the Higgs mass,
DM relic density, neutrino masses/mixings and LFV, our results agree with
those in Fig. 9 in Ref. [10]. In particular, models with low mass DM contain
sufficiently heavy charged scalars to evade the corresponding LEP2 bound [31],
but are excluded by the LEP limit on the invisible $Z^{0}$ decay width [53,
54]. Since there are no viable models with $m_{Z}/2<m_{\rm DM}<m_{h}/2$, there
are no additional constraints from the invisible $h$ decay width [55, 56].
Otherwise and as well known, the correct relic density requires the DM to be
larger than about 500 GeV (top). Fermion coannihilation processes reduce this
lower mass limit to about 200 GeV (bottom) [10]. The sampled points are color
coded according to the dominant annihilation branching ratio. A point is
marked as having no dominant channel when no single branching ratio reaches
50%. Above the $W$ boson threshold, mostly annihilation into $W$ boson pairs
occurs. The channel-dependent constraints derived from indirect DM detection
with ANTARES [64], IceCube [43] and Super-Kamiokande [65] are considerably
weaker than the direct detection constraints from XENON1T [57]. Only the
latter constrain the parameter space, and in particular the coannihilation
region. Since the Higgs coupling to the quarks in the nucleon is relatively
small, the elastic cross sections are small as well. A significant part of the
parameter space for DM masses beyond 1 TeV results even in cross sections
below the atmospheric and diffuse supernova background (DSNB) neutrino “floor”
[70], which may render DM direct detection difficult.

###  5.2 Limits on the inelastic cross section

Figure 6: Scotogenic models in the plane DM mass vs. neutral scalar mass
splitting, color coded for expected IC86 events (temperature scale) and
elastic scattering exclusion by XENON1T (red boxes) [57]. Also shown are the
exclusion of the low- (sodium) [71] and high-mass (iodine) DAMA/LIBRA
preferred regions [20, 21] by LEP and XENON100 [58] and the upper limits on
the mass splitting from the XENON100 Run II data [59] and by the PandaX-II
experiment with smaller recoil energy window and larger background [60]. In
the lower plot, coannihilation processes are enhanced by the small scalar-
fermion mass difference.

The observation of an annual modulation signal by DAMA/LIBRA [20, 21], its
interpretation as DM of low or (relatively) higher mass of about 10 and 50 GeV
[72, 73] and the tension with other direct detection experiments have led to
the speculation that DM might only undergo inelastic scattering off nuclei in
DAMA/LIBRA, to which other experiments would not be sensitive [19]. To fit the
DAMA/LIBRA observation, the inelastic nucleon cross section was usually
assumed to be 10-4 pb [22, 23, 24]. This value is indeed close to the proton
cross section of $1.7\cdot 10^{-4}$ pb in the scotogenic model, where the
mediator (as shown in Fig. 2) is not a Higgs ($h$) boson, but an electroweak
gauge ($Z^{0}$), which violates isospin by
$b_{n}/b_{p}=1/(4\sin^{2}\theta_{W}-1)\simeq-6.6$.333The neutron cross section
$1/(2\pi)G_{F}^{2}m_{N}^{2}=74.3\cdot 10^{-4}$ pb is therefore significantly
larger [74, 26]. It also corresponds to typical cross sections in the inert
doublet model, which is in fact the only model with a single scalar DM
multiplet that allows for naturally small neutral component mass splittings at
the renormalizable level [74].

Figure 7: The expected number of events per year in the current IceCube
configuration with 86 strings (IC86) as a function of the DM mass from
inelastic and elastic DM scattering in the Sun. The black line marks the
number of expected events in the $W^{+}W^{-}$ channel for an IC22 study [24].
The blue line marks one event per year for orientation. Also shown are the
points excluded by XENON1T [57], XENON100 [58, 59] and LEP from the invisible
$Z^{0}$ boson width [53, 54]. In the lower plot, coannihilation processes are
enhanced by the small scalar-fermion mass difference.

In Fig. 6, we plot all scotogenic model points that survive Higgs mass, DM
relic density, neutrino mass/mixing and LFV violation constraints in the plane
DM mass and neutral scalar mass splitting. The points are color coded for
expected IC86 events (temperature scale) and exclusion of elastic scattering
by XENON1T (red boxes) [57]. Following the observations by DAMA/LIBRA [20,
21], several other experiments have specifically searched for inelastic DM. In
a first analysis of 100.9 live days of data with a fiducial volume containing
48 kg of liquid xenon, XENON100 (grey shaded areas) excluded the high mass
(iodine) preferred region with mass splittings up to 140 keV [58]. The Run II
data with 224.6 live days of data with a fiducial volume containing 34 kg of
liquid xenon was later reanalyzed in the context of effective field theory
(EFT) using the correspondence

$\sigma_{N}^{0}=(C_{1}^{N})^{2}{\mu_{N}^{2}\over\pi},$ (5.6)

where $\mu_{N}$ is the DM-nucleon reduced mass and $C_{1}^{N}$ is the
coefficient of the spin-independent operator [59]. Since in our case the
interaction is not isospin-conserving, we translate the limits on $C_{1}^{N}$
from Ref. [59] to the proton cross section as [75]

$\sigma_{p}^{0}=\sigma_{N}^{0}\left[{Z\over A}+\left(1-{Z\over
A}\right){b_{n}\over b_{p}}\right]^{-2},$ (5.7)

assuming for simplicity $A=132$ for the xenon isotope with the largest
abundance. This excludes the mass region above 300 GeV with mass splittings up
to 250 keV (full black lines). PandaX-II, who analyzed 79.6 live days of data
with a fiducial volume containing 329 kg of liquid xenon, presented their
results only for fixed DM masses of 1 and 10 TeV [60]. When interpolated and
translated for isospin violation, they give a similar, but slightly weaker
exclusion curve (dashed black lines) as/than Xenon100 due to the smaller
recoil energy window and larger background. The exclusion of larger mass
splittings with direct detection experiments is limited by the maximum recoil
energy and would require much larger cross sections than expected from
electroweak interactions [76]. The low-mass (sodium) point (black stars), to
which a good fit is still possible, albeit with a large cross section of
$10^{-2}$ pb and different isospin violation $b_{n}/b_{p}\simeq-0.7$ [77, 73],
as well as the (already excluded) high-mass (iodine) point to the DAMA/LIBRA
signal [71] are under intense scrutiny by the DM-Ice17 [61], COSINE-100 [62,
78], SABRE [79] and ANAIS-112 [63] experiments, which are expected to provide
a 3$\sigma$ C.L. test of this signal by autumn 2022.

Figure 8: Same as Fig. 7 as a function of the neutral scalar coupling
splitting $\delta$.

Figure 9: Thermally averaged cross section $\langle\sigma v\rangle$ with
combined IceCube ANTARES [68] and Super-Kamiokande [69] exclusion limits as a
function of the DM mass. All points are colored according to the main
annihilation channel, provided there is one with a branching ratio of over
50%. Also shown are the points excluded by XENON1T [57], XENON100 [58, 59] and
LEP from the invisible $Z^{0}$ boson width [53, 54] as well as the expected
cross section for a thermal relic [80]. In the lower plot, coannihilation
processes are enhanced by the small scalar-fermion mass difference.

###  5.3 Expected IC86 event rates from (in)elastic DM scattering in the Sun

The expected number of events in the current IceCube configuration with 86
strings (IC86) was already shown by color coding in Fig. 6. It ranges from
less than $10^{-6}$ to more than $10^{5}$ per year. For both the random and
the coannihilation scan, at least ten events are expected for neutral scalar
mass splittings $\delta\leq(500\pm 20)$ keV. Points below 250 keV were already
excluded by XENON100 [59], but half of the parameter space could be tested
with IC86 for the first time. From Eq. (3.2), the non-observation of the
predicted events would set a lower bound on

$\lambda_{5}\gtrsim 1.6\cdot 10^{-5}\cdot m_{\rm DM}/{\rm TeV}.$ (5.8)

In the following, we analyze the expected event rates in more detail. Fig. 7
shows the expected number of events per year as a function of the DM mass for
both inelastic and elastic DM scattering in the Sun, i.e. when both occurs two
points are shown. Points excluded by XENON1T [57], XENON100 [58, 59] and LEP
from the invisible $Z^{0}$ boson width [53, 54] are also shown. The black line
shows the estimate of the number of expected inelastic events in the
$W^{+}W^{-}$ channel from a previous study that had adopted the IC22
configuration [24], which scales roughly as expected with our IC86
predictions. Note that this and other previous analyses of indirect detection
of inelastic DM were motivated by the high-mass (iodine) DAMA/LIBRA best fit
point with $\delta=(125\pm 25)$ keV [22, 23, 24]. In our Fig. 7, the blue line
marks one event per year for orientation. As one can see, the expected event
rates for models with only elastic scattering stay below this line. In the
lower plot, where coannihilation processes are enhanced by the small scalar-
fermion mass difference, models with larger elastic scattering rates are
already excluded by direct detection experiments. Nevertheless, inelastic
scattering allows for a large region of the scalar DM parameter space with and
without coannihilation to be tested by IC86.

Fig. 8 shows the expected number of events per year as a function of the
neutral scalar mass splitting $\delta$. Although we scan values above
$\lambda_{5}\geq 10^{-10}$, the neutrino masses constrain $\lambda_{5}$ to be
larger than $10^{-9}$. In addition, in the coannihilation scenario the relic
density mostly requires values above $10^{-8}$. From Fig. 6 we know that
direct detection experiments exclude neutral scalar mass splittings below 250
keV, which for $m_{\rm DM}\geq 500$ (200) GeV in the normal (coannihilation)
scan translates through Eq. (3.2) to a limit of $\lambda_{5}\gtrsim 4.1\
(1.7)\ \cdot 10^{-6}$. With IC86, mass splittings up to 500 keV and at least
two more orders of magnitude in $\lambda_{5}$ could be tested up to $1.6\cdot
10^{-4}$ (or beyond) for $m_{\rm DM}=10$ TeV (or larger). While the event rate
falls quickly for inelastic scattering towards the kinematic edge, it is of
course independent of both $\delta$ and $\lambda_{5}$ in the elastic case.

###  5.4 Limits from DM annihilations in the Galactic Center

Neutrino telescopes are also used to set bounds on the self annihilation of DM
in the Galactic Center. We test the scotogenic model with the limits on the
thermally averaged self annihilation cross section $\langle\sigma v\rangle$
set by a joint analysis of ANTARES and IceCube [68], assuming a NFW halo
profile [81]. The results are shown in Fig. 9. All points are color coded for
their dominant annihilation channel (i.e. $b\bar{b}$ quark, $W^{+}W^{-}$ and
$ZZ$ boson pairs). Points with no branching ratio larger than $50\%$ are
marked as “no dominant channel”. The predictions agree roughly with the naive
expectation for a thermal relic [80], but can be considerably larger in the
coannihilation scenario, where the coannhiliation processes increase rather
than decrease the predicted DM relic density [10]. Contrary to DM annihilation
in the Sun, the direct detection experiments XENON1T [57] and XENON100 [58,
59] now exclude points with lower expected rates, while larger rates remain
viable. This can be attributed to the fact that the DM density in the Galactic
Center is now fixed by the NFW profile and not determined by (in)elastic
scattering and the capture rate. Low-mass points annihilating mostly into
$b\bar{b}$ quarks are already excluded by the direct detection experiments and
LEP from the invisible $Z^{0}$ boson width [53, 54], whereas the limits from
Super-Kamiokande [65] are much weaker. For high-mass points, DM annihilation
into $W^{+}W^{-}$ bosons is dominant. While the points from our random scan
lie still considerably below the limits set by ANTARES and IceCube [68], those
from our coannihilation scan are less than an order of magnitude smaller and
should be within reach of future IC86 analyses. It would be particularly
interesting to extend the mass range of this analysis from 1 to 10 TeV and
beyond.

##  6 Summary and outlook

To summarize, we have investigated in this paper the indirect detection
prospects of scalar DM in the scotogenic model. We have focused on DM
annihilation into neutrinos in the Sun, but also in the Galactic Center. After
a brief review of the particle content, interactions and neutrino mass
generation in the scotogenic model, we described in detail the elastic and
inelastic DM scattering processes induced by Higgs and $Z$ boson exchanges in
the Sun, which determine the WIMP capture rate and thus also the annihilation
rate into neutrinos and particles that decay into them. We then implemented
the capture rate from inelastic scattering in DarkSUSY 6.2.3 and interfaced it
with micrOMEGAs 5.0.8 and our updated routine for the expected neutrino fluxes
on Earth and the effective area of the IceCube detector in its current
86-string configuration.

We then performed two large numerical scans spanning the theoretically allowed
parameter space, i.e. a random scan and one with enhanced scalar-fermion
coannihilation from a small DM-sterile neutrino mass splitting, which is known
to increase the relic density and extend the viable scalar DM mass region from
above 500 GeV to above 200 GeV. Experimental constraints were imposed from the
known neutrino mass differences and mixing angles, LFV, the searches for new
neutral and charged scalars at LEP and LEP2, the LHC measurements of the Higgs
boson mass and invisible width, and from direct and previous indirect DM
searches.

First, we found that direct, but not indirect detection experiments constrain
the spin-independent elastic scattering cross section, in particular in the
coannihilation scenario. We also found that a considerable fraction of the
models lie below the neutrino floor, which may render direct detection
difficult. We then showed that direct detection experiments cover only half of
the parameter space for inelastic scattering, i.e. inelasticities up to 250
keV. The higher kinetic energy of DM in the Sun therefore leaves ample room
for a dedicated analysis with IC86, that would cover inelasticities up to at
least 500 keV. The expected rates extend well beyond 103 per year. The non-
observation of the predicted neutrino events would translate into a lower
limit on the scalar coupling $\lambda_{5}\gtrsim 1.6\cdot 10^{-5}\cdot m_{\rm
DM}/$TeV. For larger couplings, only elastic scattering has to be considered.
In this case, the expected event rates for models that are not yet excluded by
the direct detection experiments do not exceed 0.1 per year. We reminded the
reader that the coupling $\lambda_{5}$ has to be naturally small, since if it
was exactly zero, the neutrinos would be massless and lepton number would be
conserved, leading to a larger symmetry of the Lagrangian.

Models with elastic and inelastic scattering can also be tested with DM
annihilation in the Galactic Center, assuming e.g. a NFW DM profile. Here, we
found that direct detection experiments exclude mostly models with lower
thermally averaged cross sections. This could be attributed to the fact that
the DM density in the Galactic Center was fixed by the NFW profile and not
determined by (in)elastic scattering and the capture rate. Low-mass points
annihilating mostly into $b\bar{b}$ quarks were already excluded by direct
detection experiments and LEP, whereas the limits from Super-Kamiokande were
much weaker. For high-mass points, DM annihilation into $W^{+}W^{-}$ bosons
was dominant. There, a previous combined analysis by ANTARES and IceCube led
to limits that were two orders of magnitude above our predictions. For the
coannihilation scenario, our predictions are, however, less than an order of
magnitude smaller and thus within reach of future IC86 analyses, in particular
for TeV-scale DM.

Our results generalize to models with several scalar multiplets where the mass
splitting between the neutral components is small. A particularly interesting
case for future study would be the AMEND model with small singlet-triplet
scalar mass splitting, which could again be small due to an otherwise larger
symmetry of the Lagrangian [82].

## Acknowledgments

We thank Carsten Rott for useful comments on the manuscript. This work has
been supported by the DFG through the Research Training Network 2149 “Strong
and weak interactions - from hadrons to dark matter” and by BMBF through
Verbundforschung grant 05A20PM2.

## References

  * [1] Particle Data Group collaboration, _Review of Particle Physics_ , _PTEP_ 2020 (2020) 083C01.
  * [2] M. Klasen, M. Pohl and G. Sigl, _Indirect and direct search for dark matter_ , _Prog. Part. Nucl. Phys._ 85 (2015) 1 [1507.03800].
  * [3] Planck collaboration, _Planck 2018 results. VI. Cosmological parameters_ , _Astron. Astrophys._ 641 (2020) A6 [1807.06209].
  * [4] D. Restrepo, O. Zapata and C.E. Yaguna, _Models with radiative neutrino masses and viable dark matter candidates_ , _JHEP_ 11 (2013) 011 [1308.3655].
  * [5] S. Esch, M. Klasen, D.R. Lamprea and C.E. Yaguna, _Lepton flavor violation and scalar dark matter in a radiative model of neutrino masses_ , _Eur. Phys. J. C_ 78 (2018) 88 [1602.05137].
  * [6] S. Esch, M. Klasen and C.E. Yaguna, _A singlet doublet dark matter model with radiative neutrino masses_ , _JHEP_ 10 (2018) 055 [1804.03384].
  * [7] J. Fiaschi, M. Klasen and S. May, _Singlet-doublet fermion and triplet scalar dark matter with radiative neutrino masses_ , _JHEP_ 05 (2019) 015 [1812.11133].
  * [8] T. de Boer, R. Busse, A. Kappes, M. Klasen and S. Zeinstra, _New constraints on radiative seesaw models from IceCube and other neutrino detectors_ , 2103.06881.
  * [9] E. Ma, _Verifiable radiative seesaw mechanism of neutrino mass and dark matter_ , _Phys. Rev. D_ 73 (2006) 077301 [hep-ph/0601225].
  * [10] M. Klasen, C. E. Yaguna, J. D. Ruiz-Álvarez, D. Restrepo, O. Zapata, _Scalar dark matter and fermion coannihilations in the radiative seesaw model_ , _JCAP_ 04 (2013) 044 [1302.5298].
  * [11] T. Toma and A. Vicente, _Lepton Flavor Violation in the Scotogenic Model_ , _JHEP_ 01 (2014) 160 [1312.2840].
  * [12] A. Ibarra, C.E. Yaguna and O. Zapata, _Direct Detection of Fermion Dark Matter in the Radiative Seesaw Model_ , _Phys. Rev. D_ 93 (2016) 035012 [1601.01163].
  * [13] T. de Boer, M. Klasen, C. Rodenbeck, S. Zeinstra, _Absolute neutrino mass as the missing link to the dark sector_ , _Phys. Rev. D_ 102 (2020) 051702 [2007.05338].
  * [14] M. Lindner, A. Merle and V. Niro, _Enhancing Dark Matter Annihilation into Neutrinos_ , _Phys. Rev. D_ 82 (2010) 123529 [1005.3116].
  * [15] Y. Farzan, _Flavoring Monochromatic Neutrino Flux from Dark Matter Annihilation_ , _JHEP_ 02 (2012) 091 [1111.1063].
  * [16] C. Arina, S. Kulkarni and J. Silk, _Monochromatic neutrino lines from sneutrino dark matter_ , _Phys. Rev. D_ 92 (2015) 083519 [1506.08202].
  * [17] C. El Aisati, C. Garcia-Cely, T. Hambye, L. Vanderheyden, _Prospects for discovering a neutrino line induced by dark matter annihilation_ , _JCAP_ 10 (2017) 021 [1706.06600].
  * [18] G. Jungman, M. Kamionkowski and K. Griest, _Supersymmetric dark matter_ , _Physics Reports_ 267 (1996) 195 .
  * [19] D. Tucker-Smith and N. Weiner, _Inelastic dark matter_ , _Phys. Rev. D_ 64 (2001) 043502 [hep-ph/0101138].
  * [20] R. Bernabei et al., _First model independent results from DAMA/LIBRA-phase2_ , _Nucl. Phys. Atom. Energy_ 19 (2018) 307 [1805.10486].
  * [21] R. Bernabei et al., _Improved model-dependent corollary analyses after the first six annual cycles of DAMA/LIBRA-phase2_ , _Nucl. Phys. Atom. Energy_ 20 (2019) 317 [1907.06405].
  * [22] S. Nussinov, L.-T. Wang and I. Yavin, _Capture of Inelastic Dark Matter in the Sun_ , _JCAP_ 08 (2009) 037 [0905.1333].
  * [23] A. Menon, R. Morris, A. Pierce and N. Weiner, _Capture and Indirect Detection of Inelastic Dark Matter_ , _Phys. Rev. D_ 82 (2010) 015011 [0905.1847].
  * [24] J. Shu, P.-f. Yin and S.-h. Zhu, _Neutrino Constraints on Inelastic Dark Matter after CDMS II_ , _Phys. Rev. D_ 81 (2010) 123519 [1001.1076].
  * [25] R. Catena and F. Hellström, _New constraints on inelastic dark matter from IceCube_ , _JCAP_ 10 (2018) 039 [1808.08082].
  * [26] T. Hashimoto and D. Suematsu, _Inflation and DM phenomenology in a scotogenic model extended with a real singlet scalar_ , _Phys. Rev. D_ 102 (2020) 115041 [2009.13057].
  * [27] IceCube collaboration, _The IceCube Neutrino Observatory: Instrumentation and Online Systems_ , _JINST_ 12 (2017) P03012 [1612.05093].
  * [28] M. Lindner, M. Platscher, C. E. Yaguna, A. Merle, _Fermionic WIMPs and vacuum stability in the scotogenic model_ , _Phys. Rev. D_ 94 (2016) 115027 [1608.00577].
  * [29] ATLAS, CMS collaboration, _Combined Measurement of the Higgs Boson Mass in $pp$ Collisions at $\sqrt{s}=7$ and 8 TeV with the ATLAS and CMS Experiments_, _Phys. Rev. Lett._ 114 (2015) 191803 [1503.07589].
  * [30] J. Casas and A. Ibarra, _Oscillating neutrinos and $\mu\to e,\gamma$_, _Nucl. Phys. B_ 618 (2001) 171 [hep-ph/0103065].
  * [31] OPAL collaboration, _Search for stable and longlived massive charged particles in e+ e- collisions at s**(1/2) = 130-GeV to 209-GeV_ , _Phys. Lett. B_ 572 (2003) 8 [hep-ex/0305031].
  * [32] P. Agrawal, Z. Chacko, C. Kilic and R.K. Mishra, _A Classification of Dark Matter Candidates with Primarily Spin-Dependent Interactions with Matter_ , 1003.1912.
  * [33] F. Staub, _SARAH 4 : A tool for (not only SUSY) model builders_ , _Comput. Phys. Commun._ 185 (2014) 1773 [1309.7223].
  * [34] M. Freytsis and Z. Ligeti, _On dark matter models with uniquely spin-dependent detection possibilities_ , _Phys. Rev. D_ 83 (2011) 115009 [1012.5317].
  * [35] G. Bélanger, F. Boudjema, A. Goudelis, A. Pukhov, B. Zaldivar, _micrOMEGAs5.0 : Freeze-in_ , _Comput. Phys. Commun._ 231 (2018) 173 [1801.03509].
  * [36] A. Gould, _Resonant Enhancements in Weakly Interacting Massive Particle Capture by the Earth_ , _ApJ_ 321 (1987) 571.
  * [37] M. Blennow, S. Clementz and J. Herrero-Garcia, _The distribution of inelastic dark matter in the Sun_ , _Eur. Phys. J. C_ 78 (2018) 386 [1802.06880].
  * [38] W. Porod, _SPheno, a program for calculating supersymmetric spectra, SUSY particle decays and SUSY particle production at e+ e- colliders_ , _Comput. Phys. Commun._ 153 (2003) 275 [hep-ph/0301101].
  * [39] W. Porod, F. Staub, _SPheno 3.1: Extensions including flavour, CP-phases and models beyond the MSSM_ , _Comput. Phys. Commun._ 183 (2012) 2458 [1104.1573].
  * [40] G. Bélanger, J. Da Silva, T. Perrillat-Bottonet and A. Pukhov, _Limits on dark matter proton scattering from neutrino telescopes using micrOMEGAs_ , _JCAP_ 12 (2015) 036 [1507.07987].
  * [41] A. Belyaev, N.D. Christensen and A. Pukhov, _CalcHEP 3.4 for collider physics within and beyond the Standard Model_ , _Comput. Phys. Commun._ 184 (2013) 1729 [1207.6082].
  * [42] T. Bringmann, J. Edsjö, P. Gondolo, P. Ullio and L. Bergström, _DarkSUSY 6 : An Advanced Tool to Compute Dark Matter Properties Numerically_ , _JCAP_ 07 (2018) 033 [1802.03399].
  * [43] IceCube collaboration, _Search for annihilating dark matter in the Sun with 3 years of IceCube data_ , _Eur. Phys. J. C_ 77 (2017) 146 [1612.05949].
  * [44] Bugaev, Edgar and Montaruli, Teresa and Shlepin, Yuri and Sokalski, Igor A., _Propagation of tau neutrinos and tau leptons through the earth and their detection in underwater / ice neutrino telescopes_ , _Astropart. Phys._ 21 (2004) 491 [hep-ph/0312295].
  * [45] M. Cirelli, N. Fornengo, T. Montaruli, I.A. Sokalski, A. Strumia and F. Vissani, _Spectra of neutrinos from dark matter annihilations_ , _Nucl. Phys. B_ 727 (2005) 99 [hep-ph/0506298].
  * [46] I. Esteban, M.C. Gonzalez-Garcia, A. Hernandez-Cabezudo, M. Maltoni and T. Schwetz, _Global analysis of three-flavour neutrino oscillations: synergies and tensions in the determination of $\theta_{23}$, $\delta_{CP}$, and the mass ordering_, _JHEP_ 01 (2019) 106 [1811.05487].
  * [47] KATRIN collaboration, _Improved Upper Limit on the Neutrino Mass from a Direct Kinematic Method by KATRIN_ , _Phys. Rev. Lett._ 123 (2019) 221802 [1909.06048].
  * [48] J. Harz, B. Herrmann, M. Klasen, K. Kovarik and P. Steppeler, _Theoretical uncertainty of the supersymmetric dark matter relic density from scheme and scale variations_ , _Phys. Rev. D_ 93 (2016) 114023 [1602.08103].
  * [49] MEG collaboration, _Search for the lepton flavour violating decay $\mu^{+}\rightarrow\mathrm{e}^{+}\gamma$ with the full dataset of the MEG experiment_, _Eur. Phys. J. C_ 76 (2016) 434 [1605.05081].
  * [50] SINDRUM collaboration, _Search for the Decay mu+ — $>$ e+ e+ e-_, _Nucl. Phys. B_ 299 (1988) 1.
  * [51] SINDRUM II collaboration, _Test of lepton flavor conservation in mu — $>$ e conversion on titanium_, _Phys. Lett. B_ 317 (1993) 631.
  * [52] M. Carena, A. de Gouvea, A. Freitas and M. Schmitt, _Invisible Z boson decays at e+ e- colliders_ , _Phys. Rev. D_ 68 (2003) 113007 [hep-ph/0308053].
  * [53] Q.-H. Cao, E. Ma and G. Rajasekaran, _Observing the Dark Scalar Doublet and its Impact on the Standard-Model Higgs Boson at Colliders_ , _Phys. Rev. D_ 76 (2007) 095011 [0708.2939].
  * [54] E. Lundstrom, M. Gustafsson and J. Edsjo, _The Inert Doublet Model and LEP II Limits_ , _Phys. Rev. D_ 79 (2009) 035013 [0810.3924].
  * [55] ATLAS collaboration, _Combination of searches for invisible Higgs boson decays with the ATLAS experiment_ , .
  * [56] CMS collaboration, _Search for invisible decays of a Higgs boson produced through vector boson fusion in proton-proton collisions at $\sqrt{s}=$ 13 TeV_, _Phys. Lett. B_ 793 (2019) 520 [1809.05937].
  * [57] XENON collaboration, _Dark Matter Search Results from a One Ton-Year Exposure of XENON1T_ , _Phys. Rev. Lett._ 121 (2018) 111302 [1805.12562].
  * [58] XENON100 collaboration, _Implications on Inelastic Dark Matter from 100 Live Days of XENON100 Data_ , _Phys. Rev. D_ 84 (2011) 061101 [1104.3121].
  * [59] XENON collaboration, _Effective field theory search for high-energy nuclear recoils using the XENON100 dark matter detector_ , _Phys. Rev. D_ 96 (2017) 042004 [1705.02614].
  * [60] PandaX-II collaboration, _Exploring the dark matter inelastic frontier with 79.6 days of PandaX-II data_ , _Phys. Rev. D_ 96 (2017) 102007 [1708.05825].
  * [61] DM-Ice collaboration, _First search for a dark matter annual modulation signal with NaI(Tl) in the Southern Hemisphere by DM-Ice17_ , _Phys. Rev. D_ 95 (2017) 032006 [1602.05939].
  * [62] COSINE-100 collaboration, _Search for a Dark Matter-Induced Annual Modulation Signal in NaI(Tl) with the COSINE-100 Experiment_ , _Phys. Rev. Lett._ 123 (2019) 031302 [1903.10098].
  * [63] J. Amare et al., _Annual Modulation Results from Three Years Exposure of ANAIS-112_ , 2103.01175.
  * [64] ANTARES collaboration, _Limits on Dark Matter Annihilation in the Sun using the ANTARES Neutrino Telescope_ , _Phys. Lett. B_ 759 (2016) 69 [1603.02228].
  * [65] Super-Kamiokande collaboration, _Search for neutrinos from annihilation of captured low-mass dark matter particles in the Sun by Super-Kamiokande_ , _Phys. Rev. Lett._ 114 (2015) 141301 [1503.04858].
  * [66] A. Albert et al., _Results from the search for dark matter in the Milky Way with 9 years of data of the ANTARES neutrino telescope_ , _Phys. Lett. B_ 769 (2017) 249 [1612.04595].
  * [67] IceCube collaboration, _Search for Neutrinos from Dark Matter Self-Annihilations in the center of the Milky Way with 3 years of IceCube/DeepCore_ , _Eur. Phys. J. C_ 77 (2017) 627 [1705.08103].
  * [68] ANTARES, IceCube collaboration, _Combined search for neutrinos from dark matter self-annihilation in the Galactic Center with ANTARES and IceCube_ , _Phys. Rev. D_ 102 (2020) 082002 [2003.06614].
  * [69] Super-Kamiokande collaboration, _Indirect search for dark matter from the Galactic Center and halo with the Super-Kamiokande detector_ , _Phys. Rev. D_ 102 (2020) 072002 [2005.05109].
  * [70] J. Billard, L. Strigari and E. Figueroa-Feliciano, _Implication of neutrino backgrounds on the reach of next generation dark matter direct detection experiments_ , _Phys. Rev. D_ 89 (2014) 023524 [1307.5458].
  * [71] S. Kang, S. Scopel and G. Tomar, _Probing DAMA/LIBRA data in the full parameter space of WIMP effective models of inelastic scattering_ , _Phys. Rev. D_ 99 (2019) 103019 [1902.09121].
  * [72] F. Kahlhoefer, F. Reindl, K. Schäffner, K. Schmidt-Hoberg and S. Wild, _Model-independent comparison of annual modulation and total rate with direct detection experiments_ , _JCAP_ 05 (2018) 074 [1802.10175].
  * [73] S. Kang, S. Scopel, G. Tomar and J.-H. Yoon, _DAMA/LIBRA-phase2 in WIMP effective models_ , _JCAP_ 07 (2018) 016 [1804.07528].
  * [74] C. Arina, F.-S. Ling and M.H.G. Tytgat, _IDM and iDM or The Inert Doublet Model and Inelastic Dark Matter_ , _JCAP_ 10 (2009) 018 [0907.0430].
  * [75] C.E. Yaguna, _Isospin-violating dark matter in the light of recent data_ , _Phys. Rev. D_ 95 (2017) 055015 [1610.08683].
  * [76] J. Bramante, P.J. Fox, G.D. Kribs and A. Martin, _Inelastic frontier: Discovering dark matter at high recoil energy_ , _Phys. Rev. D_ 94 (2016) 115026 [1608.02662].
  * [77] S. Baum, K. Freese and C. Kelso, _Dark Matter implications of DAMA/LIBRA-phase2 results_ , _Phys. Lett. B_ 789 (2019) 262 [1804.01231].
  * [78] COSINE-100, Sogang Phenomenology Group collaboration, _COSINE-100 and DAMA/LIBRA-phase2 in WIMP effective models_ , _JCAP_ 06 (2019) 048 [1904.00128].
  * [79] M. Antonello et al., _Characterization of SABRE crystal NaI-33 with direct underground counting_ , _Eur. Phys. J. C_ 81 (2021) 299 [2012.02610].
  * [80] G. Steigman, B. Dasgupta and J.F. Beacom, _Precise Relic WIMP Abundance and its Impact on Searches for Dark Matter Annihilation_ , _Phys. Rev. D_ 86 (2012) 023506 [1204.3622].
  * [81] J.F. Navarro, C.S. Frenk and S.D.M. White, _The Structure of cold dark matter halos_ , _Astrophys. J._ 462 (1996) 563 [astro-ph/9508025].
  * [82] Y. Farzan, S. Pascoli and M.A. Schmidt, _AMEND: A model explaining neutrino masses and dark matter testable at the LHC and MEG_ , _JHEP_ 10 (2010) 111 [1005.5323].