aainstitutetext: Institut de Physique Théorique, Université Paris Saclay, CEA,
CNRS 91191 Gif-sur-Yvette, Francebbinstitutetext:  Theoretical Physics
Department, CERN, Geneva, Switzerland ccinstitutetext: Departament de Física
Tèorica, Universitat de València and IFIC, Universitat de València-CSIC, Dr.
Moliner 50, E-46100 Burjassot (València), Spain

# Exceptional Composite Dark Matter

Guillermo Ballesteros b Adrián Carmona c and Mikael Chala
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the dark matter phenomenology of non-minimal composite Higgs models
with $SO(7)$ broken to the exceptional group $G_{2}$. In addition to the
Higgs, three pseudo-Nambu-Goldstone bosons arise, one of which is electrically
neutral. A parity symmetry is enough to ensure this resonance is stable. In
fact, if the breaking of the Goldstone symmetry is driven by the fermion
sector, this $\mathbb{Z}_{2}$ symmetry is automatically unbroken in the
electroweak phase. In this case, the relic density, as well as the expected
indirect, direct and collider signals are then uniquely determined by the
value of the compositeness scale, $f$. Current experimental bounds allow to
account for a large fraction of the dark matter of the Universe if the dark
matter particle is part of an electroweak triplet. The totality of the relic
abundance can be accommodated if instead this particle is a composite singlet.
In both cases, the scale $f$ and the dark matter mass are of the order of a
few TeV.

CERN-TH-2017-083

FTUV-17-0417.9647

##  1 Introduction

Composite Higgs Models (CHM) Kaplan:1983fs ; Kaplan:1983sm ; Dimopoulos:1981xc
are among the best motivated extensions of the Standard Model (SM) of particle
physics. First of all, the hierarchy problem can be solved by assuming the
Higgs to be a bound state of a new strongly interacting sector. This sector is
supposed to respect an approximate global symmetry $\mathcal{G}$, which in
turn is spontaneously broken to $\mathcal{H}\subset\mathcal{G}$ at a scale
$f\sim$ TeV. The Higgs is then expected to be naturally lighter than the scale
of compositeness by further assuming that it is a pseudo Nambu-Goldstone Boson
(pNGB) of this symmetry breaking pattern. Moreoever, this approach could also
help to understand the puzzling hierarchy of fermion masses in the SM. Indeed,
the explicit breaking of $\mathcal{G}$ by linear interactions between the SM
fermions and composite operators Kaplan:1991dc ; Contino:2004vy  translates
into mixings between elementary fields and fermionic resonances at the
confinement scale $f$. The Yukawa couplings emerge in the physical basis
before electroweak (EW) symmetry breaking, being very much dependent on the
dimension of the composite operators Panico:2015jxa . Therefore, making the
mixing of different flavours depend on the dimension of their respective
operators, their masses at the EW scale could be very different. In
particular, the top quark Yukawa coupling can be much larger than the other
ones without any previous enhancement in the UV. In such a case, the explicit
breaking of the global symmetry is triggered by the top quark linear mixing.

The requirement that one Higgs doublet is part of the pNGB spectrum restricts
the amount of possible cosets. In light of this background and the ongoing
tests of the scalar sector at the LHC, a systematic study of non-minimal CHMs
is a timely target that is worth aiming for. 111The adjective _non-minimal_
refers to CHMs that present an extended scalar sector, contrary to the minimal
model based on $SO(5)/SO(4)$ Agashe:2004rs . An exhaustive list of small
groups that can be used to build composite Higgs models can be found, for
instance, in Reference Bellazzini:2014yua . According to the dimension of the
global symmetry group, the smallest cosets are $SO(6)/SO(5)$,
$SO(6)/[SO(4)\times U(1)]$, $SO(7)/SO(6)$ and $SO(7)/G_{2}$. From the point of
view of the number of pNGBs, the minimal choices are $SO(6)/SO(5)$,
$SO(7)/SO(6)$, $SO(8)/SO(7)$ and $SO(7)/G_{2}$. The first three have been
already studied in the context of DM Frigerio:2012uc ; Mrazek:2011iu ;
Chala:2016ykx , while $SO(8)/SO(7)$ provides a very similar phenomenology
leading (only) to extra singlets. Thus, in this paper we consider a model
based on the symmetry breaking pattern $SO(7)/G_{2}$ Chala:2012af , which
gives rise to seven pNGBs transforming as
$\mathbf{7}=(\mathbf{2},\mathbf{2})+(\mathbf{3},\mathbf{1})$ under
$SU(2)\times SU(2)\subset G_{2}$. Depending on which of the two $SU(2)$ groups
is weakly gauged (and therefore identified with the SM $SU(2)_{L}$) the three
additional pNGBs transform as a scalar real triplet or as three singlets of
the EW group. The former constitutes a version of the inert triplet model
FileviezPerez:2008bj  free of the hierarchy problem, in which we concentrate
throughout most of the paper. It is worth noticing that the number of free
parameters in the scalar potential of the inert triplet model is smaller than
in any other extended Higgs sector (and on an equal footing with e.g. the
singlet Higgs portal). In this sense, the coset under study is also minimal.
Moreover, the collection of constraints obtained in this work will be also of
relevance for triplets in other contexts.

We will highlight the main differences between the triplet and the singlet
cases in the last section of the paper. In both cases, the neutral scalar can
be forced to be odd under a $\mathbb{Z}_{2}$ symmetry which is shown to be
compatible with the strong sector dynamics. Thus, this sector respects the
symmetry $O(7)\cong SO(7)\times\mathbb{Z}_{2}$. 222Note that in what concerns
the composite sector alone, the scalars are exact Goldstones and hence their
interactions are shift invariant. The neutral extra pNGB can then account for
part or all of the observed dark matter (DM) relic abundance, depending on its
$SU(2)_{L}$ quantum numbers.

Both models present several advantages in contrast to their respective
elementary counterparts. Indeed, the larger symmetry on the strong sector
constrains the number of independent free parameters. If the breaking of the
Goldstone symmetry is mainly driven by the fermion sector, the scalar
potential depends only on three quantities, two of which can be traded by the
measured values of the Higgs mass and quartic coupling. The remaining
parameter is just the compositeness scale $f$. The $\mathbb{Z}_{2}$ symmetry
is automatically unbroken in the EW phase. On another front, the symmetry
breaking pattern we consider provides a more interesting phenomenology than
the minimal CHM of Reference Agashe:2004rs . First of all, it contains a DM
candidate. Since direct and indirect DM searches bound the compositeness scale
$f$ from above, they also set a robust upper limit on the mass of the new
fermionic resonances, which otherwise could not be estimated by other means
than fine-tuning arguments. Moreover, since these new resonances decay into
the extra scalars (for which there are no dedicated searches), constraints on
vector-like fermions in light of current LHC data could therefore be weakened.

The structure of the paper is as follows. In Section 2 we demonstrate that the
$\mathbb{Z}_{2}$ symmetry mentioned above can be respected by the strong
sector; and we compute the pNGB sigma model for the triplet case. There, we
also discuss the representation theory for fermions and derive the scalar
potential. The possible collider signatures are described in Section 3. The
computation of the relic density, as well as the analysis of potential direct
and indirect detection signals are presented in Section 4. Finally, in Section
5 we consider the possibility that the three additional scalars transform as
$SU(2)_{L}$ singlets rather than as a triplet, highlighting the
phenomenological differences. We conclude in Section 6. Further details on the
algebra of $SO(7)$ and $G_{2}$ are provided in Appendices A and B, while in
Appendix C we stress the main phenomenological consequences of sizable
explicit symmetry breaking in the lepton sector.

##  2 Viability of the $\mathbb{Z}_{2}$ symmetry

For a generic symmetry breaking pattern, let $T^{i}$ and $X^{a}$ represent the
unbroken and coset generators, respectively. Let us also define
$\Pi=\Pi_{a}X^{a}$, where $\Pi_{a}$ runs over all pNGBs. The
$d_{\mu}=d_{\mu}^{a}X^{a}$ symbol from the Maurer-Cartan one form

$\displaystyle\omega_{\mu}=-iU^{-1}\partial_{\mu}U=d_{\mu}^{a}X^{a}+E_{\mu}^{i}T^{i},\quad\mathrm{with}\quad
U=\mathrm{exp}\left(i\frac{\Pi}{f}\right),$ (1)

entering the non-linear sigma model, reads

$\displaystyle
d_{\mu}=\sum_{k=0}^{\infty}\frac{(-i)^{k}}{f^{k+1}(k+1)!}\mathrm{ad}_{\Pi}^{k}(\partial_{\mu}\Pi)_{X}$
$\displaystyle=\frac{1}{f}\partial_{\mu}\Pi-\frac{i}{2f^{2}}\left[\Pi,\partial_{\mu}\Pi\right]_{X}-\frac{1}{6f^{3}}\left[\Pi,\left[\Pi,\partial_{\mu}\Pi\right]\right]_{X}$
$\displaystyle+\frac{1}{24f^{4}}\left[\Pi,\left[\Pi,\left[\Pi,\partial_{\mu}\Pi\right]\right]\right]_{X}+\ldots$
(2)

where we have denoted $\mathrm{ad}_{A}(B)=[A,B]$ and the subindex $X$ means
the projection into the broken generators. It is well known that the pNGB
interactions in symmetric spaces contain only even powers of $1/f$. Indeed,
for symmetric cosets, $[X^{a},X^{b}]=if^{abi}T^{i}$, and hence all even powers
in the expression above vanish. Consequently, the leading-order Lagrangian in
derivatives describing the pNGB fields,

$\mathcal{L}_{\sigma}=\frac{1}{2}f^{2}\mathrm{Tr}(d_{\mu}d^{\mu})\leavevmode\nobreak\
,$ (3)

constructed out of the trace of two $d_{\mu}$ symbols, contains only terms
with even number of fields.

This concerns models like $SO(6)/SO(5)$ Gripaios:2009pe ; Frigerio:2012uc ,
$SO(7)/SO(6)$ Chala:2016ykx , $SO(6)/SO(4)\times SO(2)$ Mrazek:2011iu  or
$SU(4)\times SU(4)/SU(4)$ Ma:2017vzm  for example. We are instead interested
on the coset $SO(7)/G_{2}$. The corresponding generators can be found in the
Appendix A. They are normalized in such a way that
$\text{Tr}(T^{a}T^{b})=\delta^{ab}$ and $\text{Tr}(X^{a}X^{b})=\delta^{ab}$.
The condition $\text{Tr}(T^{i}X^{a})=0$ also holds. A straightforward
computation shows that this space is not symmetric. For example,
$[N_{1},N_{2}]=-i(M_{3}+N_{3}/\sqrt{2})/\sqrt{3}$. Nevertheles, the leading-
order sigma model still contains only even powers of $f$. This result relies
on two properties. First, all commutators with odd powers of $f$ in Equation 2
are parallel; likewise for all even powers. More concretely,

$d_{\mu}=\frac{1}{f}\partial_{\mu}\Pi+g_{1}\,\hat{\Pi}^{2}[\Pi,\partial_{\mu}\Pi]_{X}+g_{2}\,\hat{\Pi}^{3}[\Pi,[\Pi,\partial_{\mu}\Pi]]_{X}\leavevmode\nobreak\
,$ (4)

with

$g_{1}=\frac{i}{a_{1}}\left[-1+\cos\left(\frac{\sqrt{a_{1}}\leavevmode\nobreak\
\hat{\Pi}}{f}\right)\right],\leavevmode\nobreak\
g_{2}=\frac{1}{a_{2}}\left[-\frac{\Pi}{f}+\frac{1}{\sqrt{a_{2}}}\sin\left(\frac{\sqrt{a_{2}}\leavevmode\nobreak\
\hat{\Pi}}{f}\right)\right].$ (5)

In the equations above, $a_{1}=5/6$, $a_{2}=11/18$, and

$\hat{\Pi}=\sqrt{\Pi^{a}\Pi^{a}}\leavevmode\nobreak\ .$ (6)

Clearly, $g_{1}$ consists of only even powers of $1/f$, while $g_{2}$ contains
only odd terms. And second, one can easily check that both
$\text{Tr}(\partial_{\mu}\Pi[\Pi,\partial_{\mu}\Pi]_{X})$ and
$\text{Tr}([\Pi,\partial_{\mu}\Pi]_{X}[\Pi,[\Pi,\partial_{\mu}\Pi]]_{X})$
vanish, and so no odd powers of $1/f$ appear in the Lagrangian at leading
order in derivatives. Given this, $SU(2)_{L}$ invariance implies that only
terms with an even number of new multiplets are allowed, irrespectively of
whether they are singlets or triplets.

We can then impose a $\mathbb{Z}_{2}$ symmetry under which the multiplet
containing the new neutral scalar changes sign, regardless of whether it is a
singlet or a triplet. Clearly, in light of the discussion above, this does not
spoil the two-derivative Lagrangian containing the kinetic term of the
propagating fields. Higher-order terms, instead, might be forbidden by this
symmetry without observable phenomenological consequences. As in the rest of
non-minimal CHMs with DM scalars, as well as in their renormalizable
counterparts, the origin of this symmetry is not specified and it has to be
enforced by hand. It is nonetheless interesting to prove that this is
compatible with the shift symmetry even in a non-symmetric coset, as it is the
case here.

###  2.1 Gauge bosons

Let us focus now on the triplet case. We can compute the leading-order
covariant derivative Lagrangian for the pNGBs by promoting the derivatives in
$\omega_{\mu}$ and $d_{\mu}$ to covariant derivatives, i.e.,
$\partial_{\mu}\to\partial_{\mu}-ig\sqrt{3}W_{\mu}^{i}M_{i}-ig^{\prime}B_{\mu}F_{3}$
(see Appendix A for the expressions of $M_{i}$ and $F_{3}$). At lowest order
in derivatives, this leads to

$\displaystyle\mathcal{L}_{\sigma}$
$\displaystyle=|D_{\mu}H|^{2}\left(1-\frac{1}{3f^{2}}|\Phi|^{2}\right)+\frac{1}{2}|D_{\mu}\Phi|^{2}\left(1-\frac{2}{3f^{2}}|H|^{2}\right)-\frac{1}{6f^{2}}\left[\Phi^{\dagger}t^{i}(D_{\mu}\Phi)\right]\left[(D^{\mu}\Phi)^{\dagger}t^{i}\Phi\right]$
$\displaystyle+\frac{1}{3f^{2}}\partial^{\mu}(H^{\dagger}H)(\Phi^{\dagger}D_{\mu}\Phi)-\frac{2}{3f^{2}}|H|^{2}|D_{\mu}H|^{2}+\frac{1}{6f^{2}}\left[\partial_{\mu}(H^{\dagger}H)\right]^{2}+\mathcal{O}\left(\frac{1}{f^{4}}\right)\leavevmode\nobreak\
,$ (7)

where we have defined the following $SU(2)_{L}\times U(1)_{Y}\subset SO(4)$
multiplets:

$\displaystyle
H=\frac{1}{\sqrt{2}}(h_{1}-ih_{2},h_{3}+ih_{4})^{T}\sim\mathbf{2}_{1/2},\qquad\Phi=(\kappa_{1}+i\kappa_{2},-\eta,-\kappa_{1}-i\kappa_{2})^{T}\sim\mathbf{3}_{0},$
(8)

and $\kappa_{1},\kappa_{2},\eta,h_{1},h_{2},h_{3},h_{4}$ are the pNGBs
associated to the broken generators $N_{1},N_{2},...,N_{7}$. $H$ is identified
with the SM-like Higgs doublet living in the $\mathbf{7}$ representation of
$G_{2}$ and $\Phi$ with the remaining real triplet. Besides, $t^{i}$, with
$i=1,2,3$, read

$\displaystyle t^{1}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&1&0\\\
1&0&1\\\ 0&1&0\\\ \end{array}\right),\quad
t^{2}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&-i&0\\\ i&0&-i\\\ 0&i&0\\\
\end{array}\right),\quad t^{3}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\
0&0&-1\\\ \end{array}\right).$ (18)

We have also redefined $f\to-f/(2\sqrt{2/3})$. We will keep this convention
henceforth. We also identify $h=h_{3}$ with the physical Higgs boson. The part
of the above Lagrangian involving only $h$ can be easily summed to all orders
in $1/f^{2}$, resulting in

$\displaystyle\mathcal{L}_{\sigma}\supset\frac{1}{2}(\partial_{\mu}h)^{2}+\left[\frac{1}{4}g^{2}f^{2}\sin^{2}\bigg{(}\frac{h}{f}\bigg{)}W_{\mu}^{+}W^{\mu-}+\frac{1}{8}(g^{2}+g^{\prime
2})f^{2}\sin^{2}\bigg{(}\frac{h}{f}\bigg{)}Z_{\mu}^{+}Z^{\mu-}\right]\leavevmode\nobreak\
,$ (19)

where we have defined

$\displaystyle Z_{\mu}=\cos\theta_{W}W_{\mu}^{3}-\sin\theta_{W}B_{\mu},\qquad
A_{\mu}=\sin\theta_{W}W_{\mu}^{3}+\cos\theta_{W}B_{\mu}$ (20)

and $\tan\theta_{W}=g^{\prime}/g$ as usual. In particular, we can see that
after EW symmetry breaking (EWSB), the $W$ and $Z$ bosons get masses

$\displaystyle m_{W}^{2}=\frac{1}{4}g^{2}f^{2}\sin^{2}\bigg{(}\frac{\langle
h\rangle}{f}\bigg{)},\qquad m_{Z}^{2}=\frac{1}{4}(g^{2}+g^{\prime
2})f^{2}\sin^{2}\bigg{(}\frac{\langle h\rangle}{f}\bigg{)}\leavevmode\nobreak\
,$ (21)

with $\langle h\rangle$ the Higgs vacuum expectation value (VEV), which
differs from the SM EW VEV $v=f\sin{(\langle h\rangle/f)}\sim 246$ GeV. It is
also clear that $\rho=m_{W}^{2}/m_{Z}^{2}c_{W}^{2}=1$, as expected due to the
custodial symmetry $SO(4)\subset G_{2}$. The ratio of the tree level coupling
between the Higgs and the massive gauge bosons to the corresponding SM
coupling differs from unity by the amount:

$R_{hVV}=\sqrt{1-\xi},\qquad\xi=\frac{v^{2}}{f^{2}}.$ (22)

Clearly, given that $f\sim{\rm TeV}$, the ratio $\xi\ll 1$. If the SM group
$SU(2)_{L}\times U(1)_{Y}$ of $SO(7)$ is the only gauged group in the EW
sector, the global symmetry $SO(7)$ is broken explicitly. This becomes
manifest in the (non-vanishing) scalar potential. In order to compute it, we
promote the $SU(2)_{L}\times U(1)_{Y}$ gauge bosons to be in the adjoint of
the global $SO(7)$ with the help of _spurion_ fields. For this aim, let us
order the generators of $SO(7)$ as
$T^{\hat{a}}=\\{F^{1},\ldots,F^{7},M^{1},\ldots,M^{7},N^{1},\ldots,N^{7}\\}$.
We can then write

$\displaystyle
A_{\mu}^{\hat{a}}=W_{\mu}^{i}\Xi^{i\hat{a}}+B_{\mu}\Upsilon^{\hat{a}},\qquad\hat{a}=1,...,21,\qquad
i=1,2,3\leavevmode\nobreak\ .$ (23)

The spurions $\Xi^{i\hat{a}}$ and $\Upsilon^{\hat{a}}$ are given explicitly by
the expressions:

$\displaystyle\Xi^{i\hat{a}}=\sqrt{3}\leavevmode\nobreak\
\delta^{(i+7)\hat{a}},\qquad\Upsilon^{\hat{a}}=\delta^{\hat{a}3}\leavevmode\nobreak\
.$ (24)

Formally, they also transform in the $\mathbf{21}$ of $SO(7)$. The _dressed_
field $A_{\mu}^{D}=U^{-1}A_{\mu}^{\hat{a}}T^{\hat{a}}U$ transforms under $g\in
SO(7)$ as $h(\Pi,g)A_{\mu}^{D}h^{-1}(\Pi,g)$, with $h\in G_{2}$, and
decomposes as a sum of irreps of $G_{2}$. The same happens to the dressed
spurions $\Xi^{i}_{D}=U^{-1}\Xi^{i\hat{a}}T^{\hat{a}}U$ and
$\Upsilon_{D}=U^{-1}\Upsilon^{\hat{a}}T^{\hat{a}}U$, with the difference that
the index $i$ spans an $SU(2)_{L}$ triplet.

The gauge contribution to the scalar potential consists therefore of the
different invariants that can be built out of the $G_{2}$ irreps within
$\Xi^{i}_{D}$ and $\Upsilon_{D}$ and can be expressed as an expansion in
powers of $g/g_{\rho}$ and $g^{\prime}/g_{\rho}$, with $g_{\rho}$ the
characteristic coupling of the strong sector vector resonances. Taking into
account that $\Xi^{i}_{D}$ and $\Upsilon_{D}$ decompose as
$\mathbf{7}\oplus\mathbf{14}$ under $G_{2}$, we obtain only one independent
invariant at leading order:

$\displaystyle V_{\rm
gauge}(\Pi)=\frac{3}{4}\frac{m_{\rho}^{4}}{(4\pi)^{2}}\left(\frac{g}{g_{\rho}}\right)^{2}\frac{1}{\hat{\Pi}^{2}}\left[\left(6\tilde{c}_{1}+2\tilde{c}_{2}\frac{g^{\prime
2}}{g^{2}}\right)|H|^{2}+8\tilde{c}_{1}|\Phi|^{2}\right]\sin^{2}\left(\frac{\hat{\Pi}}{f}\right),$
(25)

where $\tilde{c}_{1,2}$ are $\lesssim 1$ dimensionless numbers, $m_{\rho}\sim
g_{\rho}f$ is the typical mass of the vector resonances, and we have used
naive dimensional analysis Giudice:2007fh ; Panico:2012uw ; Chala:2017sjk  to
account for the $\hbar$ and mass dependence of the radiative potential.

###  2.2 Fermions

The mixing between the elementary fermions and the composite sector breaks
explicitly the global symmetry $SO(7)$, because the former transform in
complete representations of the EW subgroup only. Let us first focus on the
quark sector. The mixing Lagrangian can be written as:

$\displaystyle\mathcal{L}_{\rm mix}\sim\lambda_{q}^{ij}\bar{q}_{\alpha
L}^{i}(\Delta_{q}^{\alpha})^{I}(\mathcal{O}_{q}^{j})_{I}+\lambda_{u}^{ij}\bar{u}_{R}^{i}(\Delta_{u})^{I}(\mathcal{O}_{u}^{j})_{I}+\lambda_{d}^{ij}\bar{d}_{R}^{i}(\Delta_{d})^{I}(\mathcal{O}_{d}^{j})_{I}+\textrm{h.c.}\leavevmode\nobreak\
.$ (26)

The indices $i,j=1,2,3$ run over the three quak generations. $\alpha=1,2$ and
$I$ are instead $SU(2)_{L}$ and $SO(7)$ indices, respectively. The couplings
$\lambda^{q}_{33}$ and $\lambda^{u}_{33}$ are supposed to be order one and
much larger than all other couplings. This is expected from the dependence of
the quark Yukawas on these numbers, namely $y_{u,d}^{ij}\sim\lambda^{\dagger
ik}_{q}\lambda_{u,d}^{kj}/g_{*}$ where $g_{\ast}$ is the typical coupling of
the strong sector fermionic resonances. The spurion fields $\Delta$ are
incomplete multiplets of $SO(7)\times U(1)_{X}$. 333The addition of the extra
(unbroken) $U(1)_{X}$ is necessary to accommodate the SM fermion hypercharges.
Formally, they transform in the same representations as the corresponding
composite operators, $\mathcal{O}$. We assume that the third generation right
and left quarks mix with composite operators transforming in the
$\mathbf{1}_{2/3}$ and the $\mathbf{35}_{2/3}$ of $SO(7)\times U(1)_{X}$,
respectively. This is motivated by the following branching rules under
$G_{2}\times U(1)_{X}$ and the EW gauge group:

$\displaystyle\mathbf{35}=\mathbf{1}_{2/3}\oplus\mathbf{7}_{2/3}\oplus\mathbf{27}_{2/3}=\mathbf{1}_{2/3}\oplus\mathbf{2}_{\pm
1/2}\oplus\mathbf{3}_{0}\oplus\mathbf{1}_{2/3}\oplus\mathbf{2}_{\pm
1/2}\oplus...$ (27)

where the ellipsis stands for higher-dimensional representations in the
branching rule of the $\mathbf{27}$. 444Under $SO(4)\cong SU(2)_{L}\times
SU(2)_{R}\subset G_{2}$ :
$\mathbf{7}=(\mathbf{2},\mathbf{2})\oplus(\mathbf{3},\mathbf{1})$ and
$\mathbf{27}=(\mathbf{1},\mathbf{1})\oplus(\mathbf{2},\mathbf{2})\oplus(\mathbf{3},\mathbf{3})\oplus(\mathbf{4},\mathbf{2})\oplus(\mathbf{5},\mathbf{1}).$
In order not to break the EW symmetry, the spurions $\Delta_{q}^{\alpha}$ can
only have non-zero entries in the doublets. However, the $\mathbb{Z}_{2}$
symmetry requires the components along the second one to vanish. 555Let
$(i,j)$ run over the non-vanishing entries of the spurions $\Delta_{q}^{1}$
and $\Delta_{q}^{2}$ (see Appendix A). Then, note that under the
$\mathbb{Z}_{2}$, the elements $(i,j)$ of the $U$ matrix do not change sign.
Therefore, the spurions are even eigenstates:
$\mathbb{Z}_{2}(\Delta_{q}^{1})=\Delta_{q}^{1}$ and
$\mathbb{Z}_{2}(\Delta_{q}^{2})=\Delta_{q}^{2}$. On the contrary, the spurion
accomodating the second doublet includes, for example, a non-vanishing entry
in $(1,4)$, while $U_{14}\sim h_{1}\kappa_{1}$, that changes sign. Its
explicit expression can be found in Appendix A. Similarly to the gauge boson
case, the dressed spurion $\Delta_{qD}^{\alpha}=U^{-1}\Delta_{q}^{\alpha}U$
transforms under $g\in SO(7)$ as $\Delta^{\alpha}_{qD}\to
h(\Pi,g)\Delta_{qD}^{\alpha}h^{-1}(\Pi,g)$ with $h\in G_{2}$ and decomposes as
a sum of irreps: $\Delta_{qD}^{\alpha}=\bigoplus_{m}\Delta_{q}^{\alpha m}$.
The fermion contribution to the scalar potential can be written as

$\displaystyle V_{\rm fermion}(\Pi)\approx
m_{\ast}^{4}\frac{N_{c}}{(4\pi)^{2}}\left[\left(\frac{|\lambda_{q}|}{g_{\ast}}\right)^{2}\sum_{j}c_{j}V_{j}(\Pi)+\left(\frac{|\lambda_{q}|}{g_{\ast}}\right)^{4}\sum_{k}c_{k}^{\prime}V_{k}^{\prime}(\Pi)+\ldots\right],$
(28)

where $m_{\ast}\sim g_{\ast}f$ is the typical mass scale of the fermionic
resonances and again we have used naive dimensional analysis to estimate the
parametric dependence of the potential, with $N_{c}=3$ and $c_{i}$ and
$c_{i}^{\prime}$ order one dimensionless numbers. $V_{j}(\Pi)$,
$V_{k}^{\prime}(\Pi)$ and the terms indicated by the ellipsis are the
different invariants that can be built out of two, four and higher number of
insertions of $\Delta_{qI}^{m}$, respectively. For simplicity, we have also
defined $\lambda_{q}=\lambda_{q}^{33}$. Note that no terms proportional to
$\lambda_{u}$ appear, because the right-top mixing, being a full singlet, does
not break the global symmetry.

The scalar potential consists then of the left-handed top-induced and the
gauged-induced potentials. However, the latter can be neglected if

$\displaystyle\tilde{c}_{1}g^{2}g_{\rho}^{2}\lesssim
2\pi^{2}\lambda_{H}\Rightarrow\tilde{c}_{1}g_{\rho}^{2}\lesssim 8,$ (29)

where $\lambda_{H}\sim 0.13$ is the SM Higgs quartic coupling and we have
disregarded the hypecharge contribution for simplicity. Indeed, if this
inequality holds, all observables computed taking into account only the top-
induced potential are (almost) unaffected when the gauge potential is also
included. This ocurrs, in particular, for $\tilde{c}_{1}\sim 0.1$ and
moderately large values of $g_{\rho}$; and also if $\tilde{c}_{1}\sim 1$ and
$g_{\rho}\sim 1$. Whereas the former possibility may involve some additional
tuning, the latter arises naturally at large values of $f$, which, as we will
see, are the ones preferred to account for the observed DM relic abundance. We
consider this scenario hereafter. Then,

$\displaystyle V(\Pi)\approx
m_{\ast}^{2}f^{2}\frac{N_{c}}{16\pi^{2}}y_{t}^{2}\left[c_{1}V_{1}(\Pi)+c_{2}V_{2}(\Pi)\right],$
(30)

where $|\lambda_{q}|$ has been traded by the top Yukawa coupling $y_{t}$ (see
below) and we have defined

$\displaystyle\sum_{\alpha}\left|(\Delta_{qD}^{\alpha})_{88}\right|^{2}\sim
V_{1}(\Pi)$
$\displaystyle=\frac{|H|^{2}}{\hat{\Pi}^{2}}\sin^{2}\left(\frac{2\hat{\Pi}}{f}\right),$
(31)
$\displaystyle\sum_{\alpha}\sum_{i=1}^{7}\left|(\Delta_{qD}^{\alpha})_{i8}\right|^{2}\sim
V_{2}(\Pi)$
$\displaystyle=\frac{|H|^{2}}{\hat{\Pi}^{2}}\cos\left(\frac{4\hat{\Pi}}{f}\right)+\frac{3|H|^{2}+2|\Phi|^{2}}{\hat{\Pi}^{2}}\cos\left(\frac{2\hat{\Pi}}{f}\right).$
(32)

The scalar potential above depends only on two independent unknowns, $c_{1}$
and $c_{2}$. They parametrize the two invariants constructed out of
$\mathbf{1}\times\mathbf{1}$ and $\mathbf{7}\times\mathbf{7}$ in Equation 27,
respectively. Note that the potential features only an even number of powers
of $\Phi$. This is actually true at any order in $\lambda_{q}/g_{*}$, because
the spurions are $\mathbb{Z}_{2}$–even and the $\mathbb{Z}_{2}$ invariance of
the potential requires $\Phi$ to appear always squared. Let us further keep
the leading-order potential in the expansion in powers of $1/f^{2}$. This can
be matched to the renormalizable piece

$\displaystyle V_{\rm
renorm}(H,\Phi)=\mu^{2}_{H}|H|^{2}+\lambda_{H}|H|^{4}+\frac{1}{2}\mu_{\Phi}^{2}|\Phi|^{2}+\frac{1}{4}\lambda_{\Phi}|\Phi|^{4}+\lambda_{H\Phi}|H|^{2}|\Phi|^{2}\leavevmode\nobreak\
.$ (33)

The five parameters in Equation 33 can be expressed in terms of the parameters
$c_{1}$ and $c_{2}$. These can be traded by the measured values of the SM EW
VEV and the Higgs quartic coupling, $\lambda_{H}$. Up to the scale $f$, all
parameters of phenomenological relevance are then predictions. These are given
in Table 1. It can be checked that $\langle\Phi\rangle=0$ in the EW phase,
since $\mu_{\Phi}^{2}>0$ and $\lambda_{H\Phi}>0$. And so, as anticipated, the
$\mathbb{Z}_{2}$ symmetry is not spontaneously broken. We would like to point
out that the negative sign of $\lambda_{\Phi}$ does not necessarily imply a
(potentially dangerous) runaway behaviour at high energies, where the
effective description we use fails. The existence or not of such a behaviour,
and of a possible minimum at higher energies would depend on the specific way
that the model is completed in the UV.

It is also worth stressing that, after EWSB, the masses of the charged and
neutral components of $\Phi$ are both equal to

$\displaystyle
m_{\Phi}^{2}=\mu_{\Phi}^{2}+v^{2}\lambda_{H\Phi}=\frac{2}{3}f^{2}\lambda_{H}\left[1-\frac{9}{4}\frac{v^{2}}{f^{2}}\right]+...\leavevmode\nobreak\
,$ (34)

where the ellipsis stands for terms that are further suppressed by powers of
$v^{2}/f^{2}$. The splitting between the masses of the charged components and
that of the neutral one comes only from (subdominant) radiative EW
corrections. It can be estimated to be $\Delta M\sim 166$ MeV Cirelli:2005uq .

Finally, we can also compute the top Yukawa Lagrangian:

$\displaystyle\sum_{\alpha}\bar{q}_{L}^{\alpha}\left(\Delta_{qD}^{\alpha}\right)^{\dagger}_{88}t_{R}+\mathrm{h.c.}\sim\mathcal{L}_{\rm
yuk}$
$\displaystyle=c_{t}\lambda_{q}\left(\bar{q}_{L}\tilde{H}t_{R}\right)\frac{f}{\hat{\Pi}}\sin\left(\frac{2\hat{\Pi}}{f}\right)+\mathrm{h.c.}$
$\displaystyle=-y_{t}(\overline{q}_{L}\tilde{H}t_{R})\bigg{[}1-\frac{2}{3f^{2}}\Phi^{2}+...\bigg{]}+\mathrm{h.c.}\,,$
(35)

where $c_{t}$ is an order one dimensionless parameter encoding the UV dynamics
and the product $-2c_{t}\lambda_{q}$ has been traded by the top Yukawa,
$y_{t}$. This Lagrangian is explicitly $\mathbb{Z}_{2}$-invariant. If we add
all terms involving only the Higgs boson, the ratio of the tree level coupling
of the Higgs to the massive top quark to the corresponding SM coupling is:

$R_{htt}=\frac{1-2\xi}{\sqrt{1-\xi}},\qquad\xi=\frac{v^{2}}{f^{2}}.$ (36)

Parameter $\mu_{H}^{2}$ $\mu_{\Phi}^{2}$ $\lambda_{\Phi}$ $\lambda_{H\Phi}$
Value $-v^{2}\lambda_{H}$
$\frac{2}{3}f^{2}\lambda_{H}\left(1-\frac{8}{3}\frac{v^{2}}{f^{2}}\right)$
$-\frac{4}{9}\lambda_{H}\left(1-\frac{8}{3}\frac{v^{2}}{f^{2}}\right)$
$\frac{5}{18}\lambda_{H}\left(1+\frac{32}{15}\frac{v^{2}}{f^{2}}\right)$

Table 1: Values of the different parameters of the renormalizable scalar
potential as a function of $f$, to order $\mathcal{O}(v^{2}/f^{2})$. $v\sim
246$ GeV and $\lambda_{H}\sim 0.13$ stand for the SM EW VEV and the Higgs
quartic coupling, respectively.

##  3 Collider signatures

Different collider searches bound $f$ from below. Among the more constraining
ones, we find monojet analyses, searches for disappearing tracks, measurements
of the Higgs to diphoton rate and EW precision tests. The small probability
for an emission of a hard jet in association with two invisible $\Phi$
particles makes monojet searches less efficient than the other two.

The Higgs decay width into photons is modified by order $\xi$ due to the non-
linearities of the Higgs couplings, as stated in Eqs. 22 and 36, and, to a
smaller extent, due to the new charged scalars that can run in this loop-
induced process. The width (taking into account both effects) is given by
Carena:2012xa ; Espinosa:2012qj :

$\Gamma(h\rightarrow\gamma\gamma)=\frac{\alpha^{2}v^{2}m_{h}^{3}}{1024\pi^{3}}\bigg{[}\frac{g^{2}}{2m_{W}^{2}}\sqrt{1-\xi}A_{1}(\tau_{W})+\frac{4y_{t}^{2}}{3m_{t}^{2}}\frac{1-2\xi}{\sqrt{1-\xi}}A_{1/2}(\tau_{t})+\frac{\lambda_{H\Phi}}{m_{\kappa}^{2}}A_{0}(\tau_{\kappa})\bigg{]}^{2}\leavevmode\nobreak\
$ (37)

where $\tau_{i}=4m_{i}^{2}/m_{h}^{2}$, $A_{0}(x)=-x^{2}(x^{-1}-F(x^{-1}))$,
$A_{1/2}(x)=2x^{2}(x^{-1}+(x^{-1}-1)F(x^{-1}))$ and
$A_{1}(x)=-x^{2}(2x^{-2}+3x^{-1}+3(2x^{-1}-1)F(x^{-1}))$, while the function
$F$ is given by $F(x)=\arcsin^{2}{\sqrt{x}}$. The Higgs production cross
section via gluon fusion is also modified by order $\xi$ effects:

$\sigma(gg\rightarrow
h)=\frac{(1-2\xi)^{2}}{1-\xi}\sigma^{\text{SM}}(gg\rightarrow
h)\leavevmode\nobreak\ ,$ (38)

with $\sigma^{\text{SM}}$ the SM production cross section. Given that $\xi>0$
and $\lambda_{H\Phi}>0$ (see Table 1), the production cross section times
branching ratio is always smaller than in the SM. A combination of 7 and 8 TeV
data from both ATLAS and CMS Khachatryan:2016vau  sets a lower bound of $0.66$
on $\sigma(gg\rightarrow
h\rightarrow\gamma\gamma)/\sigma_{\text{SM}}(gg\rightarrow
h\rightarrow\gamma\gamma)$ at 95 % C.L. This translates into a bound on
$f\gtrsim 800$ GeV. EW precision tests Ghosh:2015wiz  push this bound to
$f\gtrsim 900$ GeV. Searches at future colliders (see for example Reference
Klute:2013cx ) would determine the Higgs to diphoton cross section with a much
better accuracy. In particular, the region $f\lesssim 1.5$ TeV is expected to
be probed in Higgs searches at future facilities.

Finally, searches for dissappearing tracks are sensitive to pair-production
and the subsequent decay of the new scalars. Indeed, the small splitting
between the charged and the neutral components of $\Phi$ implies that the
former has a decay length exceeding a few centimeters. This produces tracks in
the tracking system that have no more than a few associated hits in the outer
region, in contrast with most of the SM processes. To our knowledge, the most
constraining search of this kind was performed by the ATLAS Collaboration in
Aad:2013yna  (similar results were found in the CMS analysis of Reference
CMS:2014gxa ). Searches of this type using 13 TeV data are not yet published.

The ATLAS search is optimized for a Wino (i.e. a generic triplet fermion,
$\chi$) with a width of $\sim 160$ MeV, corresponding to a lifetime of $\sim
0.2$ ns, whose charged components therefore decay predominantly into the
neutral one and a soft pion. In this respect, the search applies equally well
to our scalar triplet. The search rules out any mass below $\sim 270$ GeV,
corresponding to a production cross section of $\sim 0.25$ pb. The latter
takes into account the production of all $\chi^{+}\chi^{-}$,
$\chi^{+}\chi^{0}$ and $\chi^{-}\chi^{0}$. The corresponding bound on $f$ is
therefore given by the value at which the production cross section in the
scalar case equals the previous number (note that, for the same mass, the
scalar and triplet cross sections can be very different).

In order to compute this cross section at the same level of accuracy as the
one considered in the experimental reference, i.e. at NLO in QCD, we first
implement the renormalizable part of our model in Feynrules v2 Alloul:2013bka
. UV and $R_{2}$ terms Ossola:2008xq  are subsequently computed by means of
NLCOT Degrande:2014vpa . The interactions are then exported to an UFO model
that is finally imported in MadGraph v5 Alwall:2014hca  to generate parton-
level events from which the total cross section is computed for all values of
$f$ in the range $500,600,...,1500$ GeV. The bound on $f$ turns out to be only
$f\gtrsim 650$ GeV. However, future facilities could easily exceed the reach
of Higgs searches Cirelli:2014dsa ; Low:2014cba ; Mahbubani:2017gjh . For
example, a naive reinterpretation of the results in Reference Cirelli:2014dsa
suggests that values of $f$ as large as $\sim 3.5$ TeV could be tested in a
future 100 TeV $pp$ collider.

##  4 Searches for dark matter

In this section of the paper we analyze the extent to which $\eta$, the
neutral component of the scalar triplet $\Phi$ of our model, can contribute to
the DM of the Universe, given the current experimental constraints. As we
anticipated in the Introduction, the compatibility of a global
$\mathbb{Z}_{2}$ symmetry with the breaking pattern $SO(7)/G_{2}$ allows to
forbid $\eta$ decays. This neutral particle, which couples to the SM through
weak interactions and does not couple directly to the photon is, a priori, a
good weakly interacting massive particle (WIMP) DM candidate with the adequate
mass scale. Remarkably, the mass of $\eta$ and its relic abundance are, in our
case, entirely determined by the scale $f$, which makes the model extremely
predictive. In this last respect, the model is on pair with other simple
implementations of the WIMP idea, such as the Minimal DM model Cirelli:2005uq
.

We recall that the total annihilation rate, $\langle\sigma\,v\rangle$, and the
relic abundance, $\Omega\,h^{2}$, of any thermal relic satisfy the approximate
relation

$\displaystyle\Omega\,h^{2}\sim\frac{3\times
10^{-27}}{\langle\sigma\,v\rangle}\,{\rm cm}\,{\rm s}^{-1}\,,$ (39)

where the brackets indicate the average over the thermal velocity
distribution. Thus, if a thermal relic explains the totality of the DM
abundance ($[\Omega\,h^{2}]_{\rm DM}\sim 0.11$ Ade:2013zuv ), it must have an
annihilation rate of the order of ${\langle\sigma\,v\rangle}\sim 3\times
10^{-26}\,{\rm cm}\,{\rm s}^{-1}$.

Given the expression 39, the relic abundance turns to be roughly proportional
to the mass squared of the thermal relic. The relic abundance for the neutral
component of a scalar triplet as a function of its mass was computed in
Cirelli:2007xd . Including non-perturbative effects, it was found that a mass
of $\sim 2.5$ TeV is required to obtain the measured DM abundance. In Figure
1, we recast this result as a function of the compositeness scale $f$ of our
model, which is related to the mass of the neutral component of the triplet,
$\eta$, through Equation 34.666The result shown in Figure 1 assumes that the
portal coupling coupling $\lambda_{H\Phi}$ is negligible. This is indeed the
case in our model, since $\lambda_{H\Phi}$ is roughly an order of magnitude
smaller than the gauge couplings. As shown in Figure 1, a scale $f\sim 8.6$
TeV is required in this case to account for the totality of the DM in the
Universe. In the remaining of this section we explore whether this scale is
compatible with the current bounds from the LHC and direct and indirect
detection experiments; and we determine how much DM can be accounted for by
$\eta$.

Figure 1: The dependence of the relic abundance $\Omega\,h^{2}$ of $\eta$ as a
function of the compositeness scale $f$, with (continuous) and without
(dashed) non-perturbative effects; see Cirelli:2007xd . The horizontal lines
show the measured central value and a 95 % C.L. interval around it as
determined by Planck Ade:2013zuv .

###  4.1 Direct detection

The cross section for spin-independent scattering of DM on nucleons has a
tree-level contribution proportional to the portal coupling $\lambda_{H\Phi}$
Cline:2013gha :

$\displaystyle\sigma(\eta N\to\eta N)_{\rm tree}$
$\displaystyle=\frac{\lambda_{H\Phi}^{2}m_{N}^{4}f_{N}^{2}}{\pi
m_{h}^{4}m_{\Phi}^{2}}\approx\frac{25m_{N}^{4}f_{N}^{2}}{864\pi
f^{2}v^{4}\lambda_{H}}\left(1+\frac{391}{60}\frac{v^{2}}{f^{2}}\right)\leavevmode\nobreak\
,$ (40)

which scales with the inverse of the DM mass squared and arises from the tree
level exchange of a Higgs boson on $t$-channel (see Figure 2 $(a)$), where

$\displaystyle f_{N}=\sum_{q}f_{q}=\sum_{q}\frac{m_{q}}{m_{N}}\langle
N|\bar{q}q|N\rangle=0.30\pm 0.03\leavevmode\nobreak\ ,$ (41)

and $m_{N}=\frac{1}{2}(m_{n}+m_{p})\sim 1\,$GeV is the nucleon mass. However,
due to the presence of derivative interactions $\sim
i\Phi\overleftrightarrow{\partial_{\mu}}\Phi W^{\mu}$ in the Lagrangian, there
is also a loop induced contribution independent of $m_{\Phi}$. It comes from
the virtual exchange of $W$ bosons (which is insensitive to
$\lambda_{H\Phi}$), because they bring down a $p^{2}\approx m_{\Phi}^{2}$ term
which precisely cancels the $1/m_{\Phi}^{2}$ factor coming from the phase
space integral, see e.g. the diagrams in Figures 2 $(b)$-$(d)$. Such cross
section has been computed in the heavy WIMP effective theory (HWET)
Hill:2013hoa ; Hill:2014yka ; Hill:2014yxa ; Solon:2016yai . The leading term
in the $1/m_{\Phi}$ expansion (valid therefore for $m_{\Phi}\gg m_{W}\gg
m_{q}$) reads

$\displaystyle\sigma(\eta N\to\eta N)_{\rm
HWET}=1.3^{+0.4+0.4}_{-0.5-0.3}\times 10^{-2}\,\mathrm{zb}\leavevmode\nobreak\
.$ (42)

This value includes contributions from two-loop diagrams and is _universal_ ,
in the sense that it only depends on the $SU(2)_{L}$ quantum numbers of the
heavy particle, while further details of the model (such as the spin of the
WIMP or its possible interaction with the Higgs) enter only through
$1/m_{\Phi}$ corrections.

$\eta$$\eta$$h$$q$$q$ $(a)$

$\eta$$\eta$$\kappa^{\pm}$$q^{\prime}$$W$$W$$q$$q$$(b)$

$\eta$$\eta$$\kappa^{\pm}$$q^{\prime}$$W$$W$$q$$q$$(c)$

$\eta$$\eta$$\kappa^{\pm}$$W$$W$$q$$q$$h$$(d)$

Figure 2: $(a)$: Tree level contribution to the cross section for spin-
independent scattering of DM on nucleons. $(b)$-$(d)$: Representative examples
of loop-induced diagrams which are relevant for the $m_{\Phi}$–independent
piece of the cross section.

In order to provide a conservative estimate of the sensitivity of current and
projected direct detection experiments to this model, we show in Figure 3 the
sum of both contributions to the spin-independent cross section as a function
of the compositeness scale (purple) versus the latest limits from LUX
Akerib:2016vxi  (dashed orange) together with the projected sensitivities for
LZ (dashed green) Szydagis:2016few  and XENON1T (dashed red) Aprile:2015uzo .
The latter are properly rescaled by $[\Omega h^{2}]_{\rm DM}/\Omega h^{2}$,
which takes into account that $\eta$ could be just a subcomponent of the whole
relic abundance. In order to be more conservative, we have used the 1$\sigma$
upper values for both contributions in $\sigma(\eta N\to\eta N)$. It should be
noted that this is only an estimate of the DM-nucleon cross section, since the
validity of the HWET breaks down for low values of $f$ (and hence low masses).
We also neglect possible interference effects. The low sensitivity of current
experiments ensures that making more accurate predictions is not needed.
Interestingly, the order of magnitude of the estimated cross section is in the
ballpark of the aimed sensitivity for LZ, making the model accessible via
direct detection in the near future.

###  4.2 Indirect detection

Indirect DM searches use astrophysical and cosmological observations to look
for the effects of SM particles into which DM is assumed to decay or
annihilate. Concretely, they focus on the Cosmic Microwave Background (CMB)
and the detection of gamma and cosmic rays originating from decays or
annihilations of DM.

Figure 3: Estimate for the spin-independent direct detection cross section
$\sigma(\eta N\to\eta N)=\sigma(\eta N\to\eta N)_{\rm tree}+\sigma(\eta
N\to\eta N)_{\rm HWET}$ as a function of the compositeness scale $f$ (purple)
versus the current limits (linearly rescaled with the DM abundance) from LUX
(dashed orange) Akerib:2016vxi  and the projected exclusion limits at $95$ %
C.L. for LZ (dotted green) Szydagis:2016few  and XENON1T of 2 years in 1 ton
(dashed red) Aprile:2015uzo .

We consider the effects due to the $W$ boson pairs that are produced in the
annihilation of $\eta$ particles, which is the main relevant channel in our
case. We restrict our attention to three different kinds of indirect probes,
which provide the current most constraining bounds on the annihilation rate of
WIMPs: the CMB, gamma rays coming from dwarf spheroidal galaxies and gamma
rays from the center of the Milky Way. The last of these two observables have
specific intrinsic uncertainties due to our limited knowledge about the DM
distribution inside galaxy halos; and therefore the CMB leads to a more robust
bound. It is worth stressing that quite generically DM constraints coming from
indirect detection experiments are less robust than direct detection ones and
even less than those coming from colliders, such as the LHC. Again, the reason
is the required modelling of astrophysical phenomena in indirect detection
experiments. Nevertheless, when taken at face value these constraints are the
most stringent ones on our model and therefore they deserve to be considered
in depth. However, we warn the reader that the percentages of the DM relic
density that we derive in what follows should be taken as an indicative
approximation.

In Figure 4 we compare the theoretical prediction for the annihilation rate
$\langle\sigma\,v\rangle$ of $\eta$ particles from Cirelli:2007xd , 777See
Figure 3 in Reference Cirelli:2007xd  for a scalar triplet. as a function of
the scale $f$, with the current bounds from Planck, H.E.S.S. and FERMI+MAGIC.
The shape of all the curves, peaking around $\sim$ 8.2 TeV is due to the use
of Figure 1 to rescale the bounds on $\langle\sigma\,v\rangle$ from the
various collaborations. This is needed to account for the fact that the event
rate of any annihilation process scales as the square of the local density of
annihilating particles, which in the case of WIMPs can be assumed to be
approximately proportional to the relic density $\Omega\,h^{2}$. The usual
indirect detection upper bounds on the DM annihilation rate assume that all
the DM in the Universe corresponds to a single WIMP species of a given mass.
In order to include the possibility that $\eta$ explains only a fraction of
the total DM abundance, the experimental bounds have thus to be multiplied by
a factor $([\Omega\,h^{2}]_{\rm DM}/[\Omega\,h^{2}])^{2}\simeq
0.012\,[\Omega\,h^{2}]^{-2}$. Obviously, this takes into account the
dependence of the abundance $\Omega\,h^{2}$ of $\eta$ on its mass (or,
equivalently, on the scale $f$), as shown in Figure 1.

Figure 4: The thermally averaged annihilation rate of $\eta\eta$ into
$W^{+}W^{-}$ as a function of the compositeness scale, $f$. The black
continuous line is the theoretical prediction of the model. From top to bottom
we also show the upper bounds (rescaled with the square of the DM abundance)
from the following observations: H.E.S.S. dwarf spheroidal galaxies [Burkert
(dashed) and NFW (dot-dashed) profiles, in orange], CMB from Planck (blue,
dotted), the combination of FERMI and MAGIC dwarf spheroidal galaxies (red,
dashed), and the Milky Way center as seen by H.E.S.S. [NFW (dashed) and
Einasto (dot-dashed) profiles, in brown]. See the main text for references.
The green dashed line is the expected sensitivity of CTA for observations of
the Milky Way center assuming an Einasto profile Silverwood:2014yza . The
vertical band at $f\simeq 8.5$ TeV locates the scale that gives the total DM
abundance with a 95 % C.L. from the prediction of the model; see also Figure
1.

The weakest indirect detection bounds that we consider come from the
observation by the Cherenkov radiation telescope H.E.S.S. of dwarf spheroidal
galaxies, which are strongly DM dominated systems and supposed to be free from
other gamma ray emission. These bounds correspond to the two upper lines of
Figure 4, see Abramowski:2014tra . The distance between them comes from their
different assumptions for the radial distribution of DM in those galaxies. The
upper curve assumes a Burkert profile Burkert:1995yz , which features a
constant inner density core, whereas the lower one is for an NFW profile
Navarro:1995iw ; Navarro:1996gj , which peaks at the center. A more stringent
limit from dwarf spheroidals is reported by the collaborations of the FERMI
satellite and the Cherenkov telescope MAGIC. The combination of their
respective data leads to the red dashed curve Ahnen:2016qkx , assuming an NFW
profile. Although these data are more constraining, a direct comparison to the
results of H.E.S.S. is not straightforward, since the details of the assumed
profiles are different.

Notwithstanding the importance of dwarf spheroidal galaxies for indirect DM
detection, the center of the Milky Way is thought to be the strongest gamma
ray emitter and therefore an important candidate for a potential indirect DM
detection. Clearly, the choice of DM profile is critical for the
interpretation of these observations, but unfortunately the DM distribution in
the center of our galaxy is uncertain. Moreover, there is a number of baryonic
astrophysical sources of gamma rays which need to be accounted for when
considering the possible emission from the Galactic center. These backgrounds
are not well known either and this implies a large source of uncertainty in
addition to the choice of DM profile. The pair of brown lines at the bottom of
Figure 4 represent the current constrains from the Milky Way galactic center
obtained by H.E.S.S. 2016PhRvL.117k1301A  for two profile choices: NFW
(dashed) and Einasto 1965TrAlm…5…87E  (dot-dashed). Although these
observations appear to be the most stringent, it has to be emphasized that
they are also the ones whose interpretation carries a larger uncertainty. It
is interesting to point out that modifying the parameters of the NFW profile
these limits can be weakened slightly above the Einasto curve, see
2016PhRvL.117k1301A .

As we mentioned earlier, the most robust bounds come from the CMB, and in
particular from the Planck satellite. The blue dashed line of Figure 4
represents the bound obtained in the analysis of 2016PhRvD..93b3527S .
According to the CMB upper bound on the annihilation rate, our exceptional DM
model can account for as much as 80 % of the DM abundance of the Universe at
95 % C.L., as can be read from Figures 4 and 1. This requires a value of the
composite scale $f\simeq 7.5$ TeV, which corresponds to a triplet mass of
$\sim 2.2$ TeV. If instead we take the strongest limits from the Galactic
center from H.E.S.S. the maximum percentage of the DM abundance that can be
explained by $\eta$ is at most 36 %, corresponding to $f\simeq 4.25$ TeV and a
mass of $\sim 1.25$ TeV. Given that indirect detection sets the strongest
upper bounds in our model, we can conclude that a significant amount of the DM
of the Universe might be in the form of the neutral component of our triplet.

Future indirect detection data could in principle test the model with improved
sensitivity in the range of $f$ that is relevant for DM. The Cherenkov
Telescope Array CTA 2013APh….43..189D , which should start taking data by
2021, may currently be the best proposal that could contribute to that goal.
Several CTA sensitivity estimates exist in the literature, in particular, for
DM annihilation in the Galactic center into a pair of $W$ bosons Wood:2013taa
; Silverwood:2014yza ; Lefranc:2015pza . These estimates vary depending on the
assumptions made about the final configuration design of the telescope array,
the observational strategy (including its timespan) and several other factors.
In Figure 4 we report the forecast of reference Silverwood:2014yza  for DM
annihilation into $W$ bosons, appropriately rescaled with the DM abundance.
According to Silverwood:2014yza , it appears that once systematics effects are
accounted for, the upper bound that will be reachable with CTA for this
specific channel might not be too dissimilar from the most stringent current
limits obtained by H.E.S.S. 2016PhRvL.117k1301A . However, the value of the
cross section that will be attainable with CTA for the range of masses that
interests us is estimated to be a factor $\sim 4.5$ lower in Lefranc:2015pza ;
but this number accounts only for statistical errors. It is clear that a
proper comparison between current bounds and different forecasts would
require, at the very least, the use of the same DM profile.

In principle, the model could also be constrained from searches of
monochromatic gamma lines due to the annihilation of DM into two photons in
the central regions of the Milky Way. To the best of our knowledge, the latest
and most stringent upper bounds on the cross section for this process in the
relevant range of mass have been obtained by the H.E.S.S. collaboration
Abramowski:2013ax ; Abdalla:2016olq . A DM mass of $\sim 1.2$ TeV
approximately corresponds to $f\sim 4$ TeV, which is the scale at which the
H.E.S.S. limits on DM annihilation into $W^{+}W^{-}$ intersect the theoretical
prediction; see Figure 4. The current strongest bound for
$\eta\eta\rightarrow\gamma\gamma$ and DM masses around that value is
$\langle\sigma v\rangle\sim 10^{-27}$ cm${}^{3}/{\rm s}$ at 95% C.L., assuming
an Einasto profile. For a scalar triplet with zero hypercharge, this cross
section has been computed (including the Sommerfeld effect) in Cirelli:2007xd
. After the adequate rescaling with the DM abundance, the theoretical
prediction is $\langle\sigma v\rangle\sim 5\times 10^{-28}$ cm${}^{3}/{\rm
s}$, which is an order of magnitude lower than the aforementioned upper bound.
Although, once more, the DM profile dependence is an important source of
uncertainty,888See e.g. References Cirelli:2015bda ; Garcia-Cely:2015dda .
this channel is not more constraining in our case than $\eta\eta\rightarrow
W^{+}W^{-}$.

A CTA sensitivity estimate applicable for $\eta\eta\rightarrow\gamma\gamma$
was produced in Ovanesyan:2014fwa  under the assumption of an NFW profile.
Translating this estimate to the relevant range of $f$ and after rescaling by
the DM abundance, it gives $\langle\sigma v\rangle\sim 3\times 10^{-29}$
cm${}^{3}/{\rm s}$, which is an order of magnitude lower than the theoretical
estimate. This means that CTA observations of monochromatic gamma lines should
allow to probe the model beyond the current H.E.S.S. bound from
$\eta\eta\rightarrow W^{+}W^{-}$. This type of search may in fact be able to
test all the range of $f$ for which the limits on the annihilation cross
section into $W$ bosons still allows to account for a significant fraction of
the DM relic abundance.

##  5 Singlet case

The model we have explored so far provides a hyperchargeless scalar triplet as
DM candidate. This is a consequence of weakly gauging one particular $SU(2)$
of the two global ones respected by the strong sector, which makes the
fundamental representation of $SO(7)$ decompose as
$\mathbf{7}=\mathbf{2}_{1/2}\oplus\mathbf{3}_{0}$ under the EW subgroup
$SU(2)_{L}$. However, as it was done in Chala:2012af , one can also weakly
gauge the other $SU(2)$ within $SO(4)\cong SU(2)_{L}\times SU(2)_{R}$, under
which $\mathbf{7}=\mathbf{2}_{1/2}\oplus\mathbf{1}_{0}\oplus\mathbf{1}_{\pm
1}$, obtaining an isospin singlet as potential DM candidate. We follow this
path in this section, highlighting the specific differences between the two
cases.

#### Gauge contribution to the scalar potential

Contrary to the triplet case, the potential of the new charged (and
hypercharged) scalars receives contributions proportional to $g^{\prime}$.
Equation 25 has to be modified by:

$\displaystyle V_{\rm
gauge}(\Pi)=\frac{3}{4}\frac{m_{\rho}^{4}}{(4\pi)^{2}}\left(\frac{g}{g_{\rho}}\right)^{2}\frac{1}{\hat{\Pi}^{2}}\left[\left(6\tilde{c}_{1}+2\tilde{c}_{2}\frac{g^{\prime
2}}{g^{2}}\right)|H|^{2}+8\tilde{c}_{2}\frac{g^{\prime
2}}{g^{2}}\kappa^{+}\kappa^{-}\right]\sin^{2}\left(\frac{\hat{\Pi}}{f}\right)\leavevmode\nobreak\
.$ (43)

This term modifies the mass splitting between $\kappa^{\pm}$ and $\eta$, with
respect to the case of the triplet. It gives a contribution
$(m_{\kappa^{\pm}}-m_{\eta})/m_{\eta}\sim{g^{\prime}}^{2}/(N_{c}\,y_{t}^{2})\sim
0.05$.

#### Fermion contribution to the scalar potential

In the singlet case, the charged and neutral scalars do not exchange gauge
bosons, and hence the first cannot decay into the second. In order to avoid an
over-abundance of stable charged particles, for which stringent constraints
exist Gould:1989gw ; DeRujula:1989fe ; Dimopoulos:1989hk  new sources of
sizable explicit symmetry breaking have to be considered. Being the second
heaviest fermion, we assume that this effect is driven by the bottom quark.
There are many different possible embeddings of the right-handed bottom quark,
$b_{R}$. However, not all of them respect the $\mathbb{Z}_{2}$ symmetry
$\eta\leftrightarrow-\eta$ that makes the singlet scalar stable or generate a
bottom Yukawa coupling at leading order in $\lambda_{q}\lambda_{b_{R}}$, with
$\lambda_{b_{R}}=\lambda_{d}^{33}$. We consider the case where $b_{R}$ mixes
with the $\mathbf{7}_{2/3}$ within the $\mathbf{21}_{2/3}$, since it fulfills
both conditions. Then, Equation 30 still holds, but it has to be supplemented
by the (sub-leading) bottom contribution to the scalar potential:

$\displaystyle V_{\rm
bottom}(\Pi)=\frac{N_{c}}{(4\pi)^{2}}m_{\ast}^{4}\hat{c}_{1}\left(\frac{|\lambda_{b_{R}}|}{g_{\ast}}\right)^{2}\frac{1}{\hat{\Pi}^{2}}\left[2|H|^{2}+\eta^{2}+\kappa^{+}\kappa^{-}\right]\sin^{2}\left(\frac{\hat{\Pi}}{f}\right)\leavevmode\nobreak\
.$ (44)

This term of the potential also contributes to breaking the mass degeneracy
between $\kappa^{\pm}$ and $\eta$, giving
$(m_{\kappa^{\pm}}-m_{\eta})/m_{\eta}\sim(g_{*}\,y_{b})^{2}/y_{t}^{4}\sim
6\times 10^{-4}\,g_{*}\lesssim 10^{-2}$. In addition, the following Yukawa
couplings are generated

$\displaystyle\sum_{\alpha}\sum_{i=1}^{7}\bar{q}_{L}^{\alpha}\left(\Delta_{qD}^{\alpha}\right)^{\dagger}_{i8}\left(\Delta_{b}\right)_{i8}b_{R}+\mathrm{h.c.}\sim\mathcal{L}_{\rm
yuk,b}=\frac{c_{b}}{2\sqrt{6}}\frac{\lambda_{q}^{\ast}\lambda_{b_{R}}}{g_{\ast}}\frac{f}{\hat{\Pi}}\sin\left(\frac{\hat{\Pi}}{f}\right)\bar{q}_{L}\left[H\cos\left(\frac{\hat{\Pi}}{f}\right)\right.$
$\displaystyle\left.-i\tilde{H}\frac{3}{\sqrt{2}}\frac{\kappa^{+}}{\hat{\Pi}}\sin\left(\frac{\hat{\Pi}}{f}\right)\right]b_{R}+\mathrm{h.c.}=-y_{b}\bar{q}_{L}\left[H-i\tilde{H}\frac{3}{\sqrt{2}}\frac{\kappa^{+}}{f}\ldots\right]b_{R}+\mathrm{h.c.}\,,$
(45)

where $c_{b}$ is a dimensionless order one parameter and we have traded
$\frac{c_{b}}{2\sqrt{6}}\frac{\lambda_{q}^{\ast}\lambda_{b_{R}}}{g_{\ast}}$ by
$-y_{b}$ in the second expression. This provides a vertex
$i\frac{3}{\sqrt{2}}\frac{m_{b}}{f}\bar{t}_{L}b_{R}\kappa^{+}$ that makes
$\kappa^{\pm}$ decay into $tb$.

#### Collider implications

Searches for disappearing tracks do not constrain the singlet case. So,
measurements of the Higgs couplings dominate the reach of current and future
facilities. On another hand, analyses of invisible Higgs decays Aad:2015txa ;
Khachatryan:2016whc  forbid only $f\lesssim 300$ GeV. Monojet searches are
further suppressed by the small coupling of the Higgs boson to $\eta$. This
can be also produced in gluon fusion via loops of top quarks, but its coupling
to the latter is suppressed with respect to the top Yukawa by order $\xi$. The
charged scalar can be instead produced via gauge interactions. However, the
small rate together with the unclean final state containing tops and bottoms,
make its discovery challenging at the LHC. Future facilities could probe this
channel, though.

Figure 5: The dependence of the relic abundance $\Omega\,h^{2}$ in the singlet
case as a function of the compositeness scale $f$. The horizontal lines show
the measured central value and a 95 % C.L. interval around it as determined by
Planck Ade:2013zuv .

#### Relic density

Given the small splitting between the masses of the charged and the neutral
components (which is driven by the small gauge induced potential), the DM
particles are not expected to annihilate into $\kappa^{+}\kappa^{-}$ final
states. As a consequence, the main annihilation channels are $t\overline{t}$
as well as $W^{+}W^{-},ZZ$ and $hh$. The first channel dominates for small
$f\lesssim 1.7$ TeV; see Figure 7, right panel. The main reason is that the
annihilation into tops proceeds also via contact interactions (analogous to
the ones coming from Equation 2.2), suppressed by $1/f^{2}$. In the unitary
gauge, other DM interactions, instead, are driven by the Higgs portal. This
receives contributions from both the scalar quartic coupling in the potential
$\lambda_{H\Phi}$ and from derivative operators like
$|H|^{2}(\partial^{\mu}\eta)^{2}$, appearing in the sigma-model Lagrangian.
The ratio between these two is given by (see Table 1)

$\frac{1}{2}\lambda_{H\Phi}\frac{f^{2}}{m_{\eta}^{2}}\sim\frac{1}{2}\frac{5}{18}\lambda_{H}\frac{3f^{2}}{2\lambda_{H}f^{2}}\sim
0.2\leavevmode\nobreak\ ,$ (46)

and therefore the derivative interactions dominate. The main annihilation
channel for large values of $f$ is $\eta\eta\rightarrow W^{+}W^{-}$, as shown
in Figure 7. 999For an exhaustive discussion of the effects of higher-
dimensional operators in related models see for example References
Fonseca:2015gva ; Bruggisser:2016nzw ; Bruggisser:2016ixa .

Non-perturbative effects, like the Sommerfeld enhancement of the formation of
bound states are not relevant. For each value of $f$ we have computed the
relic density by just using micrOMEGAs v3 Belanger:2010pz . The result is
shown in Figure 5, alongside the current observational band (as in Figure 1).
It turns out that the whole relic abundance can be explained by this model
with $f\sim 3$ TeV, for which $m_{\eta}\sim 900$ GeV. As we will see, current
direct and indirect searches do not exclude this possibility. However, future
experiments will have the required sensitivity to test this prediction.

#### Direct searches

Figure 6: Spin-independent direct detection cross section as a function of the
compositeness scale, $f$, in the singlet DM case. We show the theoretical
estimate as a purple continuous line. We also show the current limits
(linearly rescaled with the DM abundance) from LUX (dashed orange)
Akerib:2016vxi  and the projected exclusion limits at $95$ % C.L. for LZ
(dashed green) Szydagis:2016few  and XENON1T of 2 years in 1 ton (dashed red)
Aprile:2015uzo .

Contrary to the triplet case, the DM-nucleon interaction proceeds only via the
Higgs exchange. As it can be seen in Figure 6, current searches are not
constraining enough for this model, but future experiments will definitely
probe the whole parameter space.

#### Indirect searches

The total thermally averaged cross section for DM annihilation as a function
of $f$ is shown as a black continuous line in the left panel of Figure 7. As
we already mentioned, DM particles annihilate mostly into $W^{+}W^{-}$ for
sufficiently large values of the compositeness scale, see Figure 7, right
panel. For this reason, we also show in the left panel the current upper
constraints on $\eta\eta\rightarrow W^{+}W^{-}$ from observations of the
Galactic center by H.E.S.S. (brown dot-dashed line) 2016PhRvL.117k1301A . This
bound assumes that DM particles annihilate exclusively into $W^{+}W^{-}$ (and,
as we already discussed, it is the most stringent one for this kind of
process). We show as well an estimate of the future sensitivity of CTA for the
same process (green dashed line) Silverwood:2014yza . The remarks we made in
the triplet case concerning this estimate and its comparison to the results of
2016PhRvL.117k1301A  also apply now. Clearly, the prediction of the singlet
model for the total cross section appears to be well below the current bound
and the future sensitivity for the dominant channel. We conclude that the
singlet variant of exceptional composite DM is viable for all the interesting
values of $f$. In particular, for $f\sim 3.25$ TeV, all the DM abundance in
the Universe can be accounted for. A substantial improvement in the
sensitivity of the next generation of indirect probes will be needed to test
this result.

Figure 7: Left panel: The theoretical prediction for the total thermally
averaged annihilation rate of DM particles as a function of the compositeness
scale, $f$, for the singlet DM case (black continuous line). We also show the
current upper bound (rescaled with the square of the DM abundance) from
observations by H.E.S.S. of the Galactic center (assuming an Einasto profile,
brown dot-dashed line) 2016PhRvL.117k1301A  and the expected sensitivity of
CTA (also with an Einasto profile, green dashed) Silverwood:2014yza , for
$\eta\eta\rightarrow W^{+}W^{-}$. Right panel: Annihilation fraction of the
main channels for two-body final states; $t\bar{t}$ (blue continuous line),
$W^{+}W^{-}$ (red dotted), $hh$ (green dot-dashed) and $ZZ$ (black dashed).
Other (subdominant) channels are not shown. As in previous plots, the vertical
grey lines indicate the range of values of $f$ corresponding to the observed
DM abundance at 95 % C.L.

##  6 Conclusions

The amount of evidence for the existence of DM, which comes from astrophysics
and cosmology, is overwhelming. Today, the nature and origin of DM are
regarded as one of the biggest problems of contemporary physics. At the same
time, a large theoretical effort has been directed towards solving the gauge
hierarchy problem. Therefore, the possibility of establishing a link between
the two is a tantalizing idea. This is further supported by the the so-called
WIMP miracle. As it is well-known, a WIMP of roughly TeV mass scale can help
to explain the inferred DM abundance through the simple freeze-out mechanism.

In spite of the current lack of definite new physics signals at energies of
the order of a few TeV, the aforementioned ideas are still widely acknowledged
to be excellent reasons to expect a discovery at the LHC in the coming years.
Moreover, ongoing direct and indirect detection experiments are also promising
windows for the detection of DM particles at the TeV scale, and the
sensitivity of these techniques will keep increasing in the near future.

We have worked out a non-minimal CHM containing a Higgs doublet and three
additional scalars: two electrically charged and a neutral one. Depending on
how the SM gauge interactions break the global symmetry, they can either
transform as a whole $SU(2)_{L}$ triplet or as three singlets. Contrary to the
minimal CHM, this setup can explain a large fraction of the observed DM relic
abundance. Moreover, significant improvements with respect to the
corresponding elementary extensions of the scalar sector are also present.
Indeed, if the global symmetries are broken mainly in the fermion sector, our
setup depends on a single parameter ($f$) and the external $\mathbb{Z}_{2}$
symmetry stabilizing the DM candidate is predicted to be exact also after EW
symmetry breaking. Were this not the case, the potential would receive sizable
contributions from the gauge sector. These would affect this phenomenological
study in several ways: The relevant observables would not only depend on $f$,
but also on $\tilde{c}_{1}g_{\rho}$. In order for the neutral scalar not to
break the $\mathbb{Z}_{2}$ symmetry by taking a VEV, the condition
$\tilde{c}_{1}g_{\rho}^{2}\gtrsim-2\pi^{2}\lambda_{H}/g^{2}\sim-7$ should
hold. In any case, for $\tilde{c}_{1}\sim g_{\rho}\sim 1$, the bounds on $f$
would be modified only by a small amount.

Assuming that the breaking of $SO(7)$ is driven mainly by the fermion sector,
the fraction of the DM abundance that can be accounted for in this framework
depends on how the three additional scalars are arranged. In the case in which
they form a triplet, the scale $f$ is constrained to be below $\sim 4.25$ TeV
by H.E.S.S. observations of gamma rays from the Galactic center. These would
imply that at most $\sim$ 36–46 % of the DM abundance can be explained with
this model, depending on the shape of the DM radial distribution in the
Galactic center. This bound is relatively uncertain, precisely due to our lack
of detailed knowledge about the DM profile in the innermost regions of the
Galaxy and the modelling of the gamma ray background in that region.
Conversely, CMB limits on the DM annihilation cross section are less stringent
(though more robust) and allow to account for $\sim$ 80 % of the DM abundance
with the triplet model. Since the relic density grows (approximately
quadratically) with the mass of the DM particle, and the tuning of the EW
scale needed to reproduce the correct Higgs mass grows also like $f^{2}$,
there is a linear dependence between tuning and relic density. Therefore, it
is clear that the values of $f$ that would be needed to account for the
totality of the DM could be regarded as less natural than those suggested by
the current indirect detection constrains. We would like to stress that the
mild tuning corresponding to values $f$ giving $\Omega_{m}h^{2}\sim 0.12$ is
still acceptable since the Higgs mass is stable under radiative corrections by
construction.

We stress that these results assume a standard thermal history of the
Universe. A different thermal history, which in principle is compatible with
the model, could help to allow to account for a higher percentage of the DM
relic abundance in the triplet case, and would change the upper bound on $f$.

In the case in which the three additional scalars are arranged as three
singlets, the neutral one can currently explain the totality of the DM relic
abundance. This is simply because in this case the theoretical prediction for
the DM annihilation cross section is well below all the current indirect
detection upper bounds. In this case, the tuning required to explain the
totality of the DM is only increased by a mild factor $\sim 3-10$ with respect
to the expectation from naturalness arguments (see for instance Reference
Panico:2012uw ).

Future observations of the Galactic center from the Cherenkov telescope CTA
are expected to improve the sensitivity on the cross section for several DM
annihilations channels. However, for DM annihilating into $W^{+}W^{-}$ –which
is the common channel of interest for both of our scenarios– the analysis of
Silverwood:2014yza  indicates that the sensitivity that will be achieved with
CTA is not expected to increase significantly beyond the current H.E.S.S.
upper bounds in the range of possible DM masses that are relevant for us.

However, a forecast of the CTA sensitivity to monochromatic gamma ray lines
(produced by DM annihilation into two photons) Ovanesyan:2014fwa  indicates
that testing most of the relevant range of $f$ for DM in the triplet case
should be possible with this channel. Instead, this kind of search is not that
useful in the singlet case, since the cross section is much smaller than any
current or expected future bound.

Searches for disappearing tracks performed at the LHC require $f$ to be larger
than 650 GeV in the triplet case, while Higgs measurements rise this bound up
to $f\sim 800$ GeV in either scenario. Future facilities could improve this
bound by almost a factor of 2. Likewise, current direct searches are not
constraining, while future experiments would be able to proble all allowed
values of $f$.

Clearly, the different searches are rather complementary. Also, we have set a
robust upper limit on the compositeness scale. In generic CHMs, the latter can
be obtained only if (less definite) fine tuning arguments are advocated. Note
also that this bound translates into an upper limit on the mass, $M$, of the
fermionic resonances (roughly speaking, $M\lesssim f$). Consequently, a
comment on the implications of our findings for the phenomenology of heavy
vector-like fermions is necessary. In particular, let us focus on top-like
resonances, for these are the ones whose interaction with the SM sector is
stronger. These states can be produced in pairs in proton-proton collisions.
The production cross section is mainly driven by QCD interactions, and hence
model independent. Experimental limits on the mass of these resonances rely
only on their branching ratio into the different lighter particles. Searches
performed in the LHC Run I constrain their masses to be smaller than $\sim
900$ GeV (see for example Reference Aad:2015kqa ). More recent analyses ATLAS-
CONF-2016-101  have pushed this limit just above the TeV. The reach of current
analyses is still far from the largest mass allowed by DM experiments. In this
respect, our model –and also generic non-minimal CHMs with EW-charged DM
candidates–, favours a hadronic high-energy collider as physics case for a
future facility. On top of that, all current studies consider that the new
fermions decay only into SM particles, not into other light scalars expected
in non-minimal CHMs. So, if these setups are to be considered seriously, and
they should, new dedicated searches need to be developed straight away (see
References Serra:2015xfa ; Anandakrishnan:2015yfa ; Cacciapaglia:2015eqa ;
Fan:2015sza ; Banerjee:2016wls ; Niehoff:2016zso  for works in this
direction).

###### Acknowledgements.

We would like to thank Marco Cirelli, Richard Ruiz, Javi Serra and Marco Taoso
for useful discussions. We also thank Javi Serra and Marco Taoso for comments
and suggestions on a draft version of this paper. The work of G.B. and A.C. is
funded by the European Union’s Horizon 2020 research and innovation programme
under the Marie Skłodowska-Curie grant agreements number 656794 (DEFT) and
659239 (NP4theLHC14), respectively. The work of MC is partially supported by
the Spanish MINECO under grant FPA2014- 54459-P and by the Severo Ochoa
Excellence Program under grant SEV-2014-0398. G.B. thanks the CERN Theoretical
Physics Department for hospitality while part of this work was developed.

##  Appendix A Representation theory of $SO(7)$ and $G_{2}$

Let us define the following $8\times 8$ matrices Evans:1994np ; Chala:2012af :

$\displaystyle\gamma_{1}$ $\displaystyle=i\sigma_{2}\otimes i\sigma_{2}\otimes
i\sigma_{2},\quad\gamma_{2}=\sigma_{1}\otimes i\sigma_{2}\otimes
1,\quad\gamma_{3}=i\sigma_{2}\otimes
1\otimes\sigma_{1},\quad\gamma_{4}=-i\sigma_{2}\otimes 1\otimes\sigma_{3},$
$\displaystyle\gamma_{5}$ $\displaystyle=1\otimes\sigma_{1}\otimes
i\sigma_{2},\qquad\gamma_{6}=-\sigma_{3}\otimes i\sigma_{2}\otimes
1,\quad\gamma_{7}=-1\otimes\sigma_{3}\otimes i\sigma_{2}$ (47)

An 8-dimensional representation of $SO(7)$ is then given by the operators

$J_{mn}=-J_{mn}=-[\gamma_{m},\gamma_{n}]/4,\qquad m,n=1,\ldots,7.$ (48)

In this paper we consider instead an equivalent representation obtained by
rotating $J_{mn}$ (i.e. $J_{mn}\rightarrow S^{\dagger}J_{mn}S$) with the
following $S$ matrix:

$S=\frac{1}{2}\begin{pmatrix}0&1&1&-1&0&0&1&0\\\ 1&0&0&0&1&1&0&1\\\
0&1&-1&-1&0&0&-1&0\\\ -1&0&0&0&-1&1&0&1\\\ 0&1&-1&1&0&0&1&0\\\
1&0&0&0&-1&1&0&-1\\\ 0&1&1&1&0&0&-1&0\\\ -1&0&0&0&1&1&0&-1\end{pmatrix}.$ (49)

The Lie algebra of $G_{2}\subset SO(7)$ and the coset space are expanded,
respectively, by the following 14 generators, $F_{i},M_{i}$, and the 7
generators, $N_{i}$ Gunaydin:1973rs ; Chala:2012af :

$\displaystyle F_{1}$ $\displaystyle=-\frac{i}{2}(J_{24}-J_{51}),\quad
M_{1}=+\frac{i}{\sqrt{12}}(J_{24}+J_{51}-2J_{73}),\quad
N_{1}=\frac{i}{\sqrt{6}}(J_{24}+J_{51}+J_{73}),$ $\displaystyle F_{2}$
$\displaystyle=+\frac{i}{2}(J_{54}-J_{12}),\quad
M_{2}=-\frac{i}{\sqrt{12}}(J_{54}+J_{12}-2J_{67}),\quad
N_{2}=\frac{i}{\sqrt{6}}(J_{54}+J_{12}+J_{67}),$ $\displaystyle F_{3}$
$\displaystyle=-\frac{i}{2}(J_{14}-J_{25}),\quad
M_{3}=+\frac{i}{\sqrt{12}}(J_{14}+J_{25}-2J_{36}),\quad
N_{3}=\frac{i}{\sqrt{6}}(J_{14}+J_{25}+J_{36}),$ $\displaystyle F_{4}$
$\displaystyle=-\frac{i}{2}(J_{16}-J_{43}),\quad
M_{4}=+\frac{i}{\sqrt{12}}(J_{16}+J_{43}-2J_{72}),\quad
N_{4}=\frac{i}{\sqrt{6}}(J_{16}+J_{43}+J_{72}),$ $\displaystyle F_{5}$
$\displaystyle=-\frac{i}{2}(J_{46}-J_{31}),\quad
M_{5}=+\frac{i}{\sqrt{12}}(J_{46}+J_{31}-2J_{57}),\quad
N_{5}=\frac{i}{\sqrt{6}}(J_{46}+J_{31}+J_{57}),$ $\displaystyle F_{6}$
$\displaystyle=-\frac{i}{2}(J_{35}-J_{62}),\quad
M_{6}=+\frac{i}{\sqrt{12}}(J_{35}+J_{62}-2J_{71}),\quad
N_{6}=\frac{i}{\sqrt{6}}(J_{35}+J_{62}+J_{71}),$ $\displaystyle F_{7}$
$\displaystyle=+\frac{i}{2}(J_{65}-J_{23}),\quad
M_{7}=-\frac{i}{\sqrt{12}}(J_{65}+J_{23}-2J_{47}),\quad
N_{7}=\frac{i}{\sqrt{6}}(J_{65}+J_{23}+J_{47}).$ (50)

$\\{F_{1},F_{2},F_{3}\\}$ and $\sqrt{3}\,\\{M_{1},M_{2},M_{3}\\}$ span two
separate copies of $SU(2)$. In this particular basis, the vacuum (i.e. the
vector that it is annihilated only by the generators of $G_{2}$) adopts the
form $\Sigma_{0}=(0,0,0,0,0,0,0,f)^{T}$. The $\mathbb{Z}_{2}$-even spurion
$\Delta_{q}^{\alpha}$ for the triplet case is given by

$\Delta_{q}^{1}=\dfrac{1}{2}\left(\begin{array}[]{cccccccc}&&&&&&&0\\\
&&&&&&&0\\\ &&&&&&&0\\\
<EMAIL_ADDRESS>7}$}}}\hfil\lx@intercol&0\\\ &&&&&&&0\\\ &&&&&&&-i\\\ &&&&&&&1\\\
0&0&0&0&0&-i&1&0\\\ \end{array}\right)\leavevmode\nobreak\
,\qquad\Delta_{q}^{2}=\dfrac{1}{2}\left(\begin{array}[]{cccccccc}&&&&&&&0\\\
&&&&&&&0\\\ &&&&&&&0\\\
<EMAIL_ADDRESS>7}$}}}\hfil\lx@intercol&i\\\ &&&&&&&1\\\ &&&&&&&0\\\ &&&&&&&0\\\
0&0&0&i&1&0&0&0\\\ \end{array}\right).$ (51)

In the singlet case, $\Delta_{q}^{1}$ is changed by $\Delta_{q}^{1\ast}$
whereas $\Delta_{q}^{2}$ remains unchanged. In this case, the spurion for
$\Delta_{b}$ reads

$\Delta_{b}=\frac{1}{4\sqrt{3}}\left(\begin{array}[]{cccccccc}0&0&-i&0&0&0&0&3\\\
0&0&1&0&0&0&0&3i\\\ i&-1&0&0&0&0&0&0\\\ 0&0&0&0&0&-i&-1&0\\\
0&0&0&0&0&-1&i&0\\\ 0&0&0&i&1&0&0&0\\\ 0&0&0&1&-i&0&0&0\\\
-3&-3i&0&0&0&0&0&0\\\ \end{array}\right).$ (52)

##  Appendix B $SU(2)_{L}\times U(1)_{Y}$ quantum numbers of the pNGBs

To recognize which combination of pNGBs spans the $\mathbf{2}_{1/2}$ or the
$\mathbf{3}_{0}$ of the EW group it is useful to remember that, if the broken
generators $X^{a}$ transform as

$\displaystyle\mathrm{exp}(-\alpha^{i}Y_{i})\,X^{a}\,\mathrm{exp}(\alpha^{j}Y_{j})=R_{ab}\,X^{a}$
(53)

under an element $h=\mathrm{exp}(\alpha_{i}Y^{i})$ of the unbroken group
$G_{2}$, the pNGBs accompanying them inside
$U=\mathrm{exp}\left(i\Pi^{a}N^{a}/f\right)$ transform with the transposed
matrix, i.e.,

$\displaystyle\Pi^{a}\to R_{ab}^{T}\,\Pi^{b}\leavevmode\nobreak\ .$ (54)

For simplicity, let us focus first on the triplet case. If we define

$\displaystyle N^{\pm}\equiv\frac{N^{1}\pm iN^{2}}{\sqrt{2}},\qquad
N^{0}\equiv-N^{3},\quad\mathrm{and}\quad N_{\Phi}=\begin{pmatrix}N^{+}\\\
N^{0}\\\ -N^{-}\end{pmatrix},$ (55)

and use their commutations relations, we get

$\displaystyle\sqrt{3}[M^{i},N_{\Phi}]=-t^{i}N_{\Phi},\qquad[F^{3},T_{\Phi}]=-0N_{\Phi},$
(56)

where $t^{i},\leavevmode\nobreak\ i=1,2,3,$ are the three-dimensional $SU(2)$
representation given in (18). Therefore,

$\displaystyle
e^{-i\alpha_{j}\sqrt{3}M^{j}}N_{\Phi}e^{i\alpha_{k}\sqrt{3}M^{k}}$
$\displaystyle=N_{\Phi}-i\alpha_{j}\sqrt{3}[M^{j},N_{\Phi}]+\ldots$
$\displaystyle=(1+i\alpha_{j}t^{j})N_{\Phi}+\ldots=e^{i\alpha_{j}t^{j}}N_{\Phi}$
(57)

and

$\displaystyle\Phi^{\ast}\to\left(e^{i\alpha_{j}t^{j}}\right)^{T}\Phi^{\ast}\Rightarrow\Phi\to
e^{-i\alpha_{j}t_{j}}\Phi\leavevmode\nobreak\ ,$ (58)

where we have defined

$\displaystyle\Phi=\begin{pmatrix}\kappa^{+}\\\ -\eta\\\
-\kappa^{-}\end{pmatrix},\quad\mathrm{and}\quad\kappa^{\pm}=\frac{\kappa^{1}\pm\kappa^{2}}{\sqrt{2}}.$
(59)

This means that $\Phi$ transforms properly as a hyperchargeless $SU(2)$
triplet. Analogously, if we define

$\displaystyle N_{H}=\frac{1}{\sqrt{2}}\begin{pmatrix}N^{3}-iN^{4}\\\
N^{6}+iN^{7}\end{pmatrix}$ (60)

and use the commutation relations we get

$\displaystyle\sqrt{3}[M^{i},N_{H}]=-\frac{1}{2}\sigma^{i}N_{H},\qquad[F^{3},N_{H}]=-\frac{1}{2}N_{H}\leavevmode\nobreak\
,$ (61)

which implies that

$\displaystyle H=\frac{1}{\sqrt{2}}\begin{pmatrix}h^{1}-ih^{2}\\\
h^{3}+ih^{4}\end{pmatrix}$ (62)

transforms as an $SU(2)$ doublet with $Y=1/2$ hypercharge. In the singlet
case, this combination can be taken

$\displaystyle H=\frac{1}{\sqrt{2}}\begin{pmatrix}-h^{1}+ih^{2}\\\
h^{3}-ih^{4}\end{pmatrix}\leavevmode\nobreak\ .$ (63)

##  Appendix C The case of composite leptons

An interesting possibility that has been explored recently is that leptons
could play a role in EWSB when they transform in non minimal irreps of the
global group, see e.g. References Carmona:2014iwa ; Carmona:2015ena ;
Carmona:2016mjr . (By non-minimal irreps we mean that they can provide more
than one independent invariant under the unbroken group at leading order in
the spurion expansion, like e.g. the $\mathbf{14}$ in $SO(5)/SO(4)$ or the
$\mathbf{35}$ in $SO(7)/G_{2}$.) The rationale is that, when the quark sector
transforms in smaller representations of the Goldstone symmetry (like the
spinorial, the fundamental, the adjoint, …), even a moderate degree of
compositeness in one of the lepton chiralities can have a sizable impact in
the Higgs potential. This is due to the fact that the leading lepton
contribution to the Higgs quartic coupling scales in this case with
$|\lambda_{\ell}|^{2}/g_{\ast}^{2}$, whereas the top one goes with
$|\lambda_{q}|^{4}/g_{\ast}^{4}$,
$|\lambda_{q}|^{2}|\lambda_{t}|^{2}/g_{\ast}^{4}$ or
$|\lambda_{t}|^{4}/g_{\ast}^{4}$. Therefore, a relatively smaller value of
$\lambda_{\ell}/g_{\ast}$ arising from the charged lepton sector can provide a
comparable effect to the one coming from the top quark.

Moreover, the fact that all different lepton generations could be partially
composite, could enhance the lepton contribution by a factor $N_{\rm gen}\sim
3$, compensating the color factor $N_{c}=3$ present in the top case. Indeed,
the recent hints of violation of lepton flavor universality observed by LHCb
and CMS in $R_{K}$ and $R_{K}^{\ast}$ Aaij:2014ora ; rkstar  seem to provide a
further motivation to these scenarios, as discussed e.g. in Carmona:2015ena .

In what follows, we will briefly discuss how a similar setup works in the case
of $SO(7)/G_{2}$ and its impact on DM. We assume that $\mathcal{O}_{q}^{j}$
and $\mathcal{O}_{u}^{j}$ transform in the $\mathbf{8}_{2/3}$ and the
$\mathbf{1}_{2/3}$ of $SO(7)\times U(1)_{X}$ , respectively, whereas the
composite operators mixing with the left-handed lepton doublets and the right-
handed charged singlets, $\mathcal{O}_{L}^{j}$ and $\mathcal{O}_{\ell}^{j}$,
transform respectively in the $\mathbf{1}_{-1}$ and $\mathbf{35}_{-1}$ of the
same group. Then, the scalar potential can be written as

$\displaystyle V(\Pi)\approx
m_{\ast}^{2}f^{2}\frac{1}{16\pi^{2}}\left[N_{c}\left(\frac{|\lambda_{q}|}{g_{\ast}}\right)^{2}c_{1}V_{1}(\Pi)+\sum_{j=1}^{3}\left(\frac{|\lambda_{\ell}|_{jj}}{g_{\ast}}\right)^{2}\left[c_{2,j}V_{2}(\Pi)+c_{3,j}V_{3}(\Pi)\right]\right],$
(64)

Parameter $\mu_{H}^{2}$ $\lambda_{\Phi}$ $\lambda_{H\Phi}$ Value
$-v^{2}\lambda_{H}$
$\frac{16}{9}\lambda_{H}\left(1-\frac{2}{3}\frac{v^{2}}{f^{2}}\right)-\frac{2}{3}\frac{\mu_{\Phi}^{2}}{f^{2}}$
$-\frac{4}{3}\lambda_{H}\left(1-\frac{11}{9}\frac{v^{2}}{f^{2}}\right)-\frac{1}{3}\frac{\mu_{\Phi}^{2}}{f^{2}}$

Table 2: Values of the different parameters of the renormalizable scalar
potential as a function of $f$, to order $\mathcal{O}(v^{2}/f^{2})$. $v\sim
246$ GeV and $\lambda_{H}\sim 0.13$ stand for the SM EW VEV and the Higgs
quartic coupling, respectively.

with

$\displaystyle\sum_{\alpha}\left|(\Delta_{qD}^{\alpha})_{8}\right|^{2}\sim
V_{1}(\Pi)$
$\displaystyle=\frac{|H|^{2}}{\hat{\Pi}^{2}}\sin^{2}\left(\frac{\hat{\Pi}}{f}\right),$
(65) $\displaystyle\left|(\Delta_{\ell D})_{88}\right|^{2}\sim V_{2}(\Pi)$
$\displaystyle=\frac{8}{147}\frac{1}{\hat{\Pi}^{4}}\sin^{4}\left(\frac{\hat{\Pi}}{f}\right)\left(3|H|^{2}-2|\Phi|^{2}\right)^{2},$
(66) $\displaystyle\sum_{i=1}^{7}\left|(\Delta_{\ell D})_{i8}\right|^{2}\sim
V_{3}(\Pi)$
$\displaystyle=\frac{1}{21}\frac{1}{\hat{\Pi}^{4}}\sin^{2}\left(\frac{\hat{\Pi}}{f}\right)\left[2\left(3|H|^{2}-2|\Phi|^{2}\right)^{2}\cos^{2}\left(\frac{\hat{\Pi}}{f}\right)+49|H|^{2}|\Phi|^{2}\right],$
(67)

where we have defined the dressed spurions
$\Delta_{qD}^{\alpha}=U^{-1}\Delta_{q}^{\alpha},\leavevmode\nobreak\
\alpha=1,2$, and $\Delta_{\ell D}=U^{-1}\Delta_{\ell}U$ as usual, with
101010We are thinking of the triplet case. In the singlet case one has to
change $\Delta_{q}^{1}$ by $\Delta_{q}^{1\ast}$, with the rest of the spurions
remaining the same.

$\displaystyle\Delta_{q}^{1}=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ 0\\\ 0\\\
0\\\ 0\\\ -i\\\ 1\\\
0\end{pmatrix},\quad\Delta_{q}^{2}=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ 0\\\
0\\\ i\\\ 1\\\ 0\\\ 0\\\
0\end{pmatrix},\quad\Delta_{\ell}=\frac{1}{2}\sqrt{\frac{3}{7}}\,\mathrm{diag}\begin{pmatrix}4/3\\\
4/3\\\ 4/3\\\ -1\\\ -1\\\ -1\\\ -1\\\ 0\end{pmatrix}.$ (68)

The parameters $c_{1},c_{2,j}$ and $c_{3,j}$, with $j=1,2,3,$ running over the
three lepton generations, that appear in the scalar potential are order one
dimensionless numbers. Note, however, that $c_{2,j}$ and $c_{3,j}$ always
enter in the same linear combination. So, effectively, we are left with only
three independent unknowns (the coefficients of $V_{1}(\Pi),V_{2}(\Pi)$ and
$V_{3}(\Pi)$) which can be traded at the renormalizable level by the Higgs VEV
$v$, the Higgs quartic $\lambda_{H}$ and the mass parameter of the scalar
triplet $\mu_{\Phi}^{2}$, see Table 2 and Equation 33. The mass of the triplet
in the EW phase is given by

$\displaystyle
m_{\Phi}^{2}=\mu_{\Phi}^{2}+\lambda_{H\Phi}v^{2}=\mu_{\Phi}^{2}\left[1-\frac{v^{2}}{3f^{2}}+\mathcal{O}\left(\frac{v^{4}}{f^{4}}\right)\right]-\frac{4}{3}\lambda_{H}v^{2}\left[1-\frac{11}{9}\frac{v^{2}}{f^{2}}+\mathcal{O}\left(\frac{v^{4}}{f^{4}}\right)\right]\leavevmode\nobreak\
.$ (69)

Since $\mu_{\Phi}^{2}\sim f^{2}\gg\lambda_{H}v^{2}$, the triplet does not take
a VEV provided the underlying UV dynamics allows for a positive
$\mu_{\Phi}^{2}$ (the same holds for the singlet if we weakly gauge the other
$SU(2)$ as discussed in Section 5, since the main contribution to the
potential is still given by Equation 33). The main difference with respect to
the scenarios explored before is that the relationship between $\mu_{\Phi}$
and $f$ is in principle not known. However, the same phenomenological study
could be done having as an extra variable the ratio $\mu_{\Phi}/f$, what we
leave for a future work.

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