# A unified explanation for dark matter and electroweak baryogenesis
with direct detection and gravitational wave signatures

Mikael Chalaa, Germano Nardinib and Ivan Sobolevc,d
aDESY, Notkestrasse 85, D-22607 Hamburg, Germany
bAlbert Einstein Center, Institute for Theoretical Physics, University of
Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland
cInstitute for Nuclear Research of the Russian Academy of Sciences, 60th
October Anniversary prospect 7a, Moscow 117312, Russia
dDepartment of Particle Physics and Cosmology, Physics Faculty, M. V.
Lomonosov Moscow State University, Vorobjevy Gory, 119991, Moscow, Russia

###### Abstract

A minimal extension of the Standard Model that provides both a dark matter
candidate and a strong first-order electroweak phase transition (EWPT)
consists of two additional Lorentz and gauge singlets. In this paper we work
out a composite Higgs version of this scenario, based on the coset
$SO(7)/SO(6)$. We show that by embedding the elementary fermions in
appropriate representations of $SO(7)$, all dominant interactions are
described by only three free effective parameters. Within the model
dependencies of the embedding, the theory predicts one of the singlets to be
stable and responsible for the observed dark matter abundance. At the same
time, the second singlet introduces new $CP$-violation phases and triggers a
strong first-order EWPT, making electroweak baryogenesis feasible. It turns
out that this scenario does not conflict with current observations and it is
promising for solving the dark matter and baryon asymmetry puzzles. The tight
predictions of the model will be accessible at the forthcoming dark matter
direct detection and gravitational wave experiments.

††preprint: DESY 16-092

##  I Introduction

In light of the celebrated discovery of the Higgs boson Aad:2012tfa ;
Chatrchyan:2012xdj , and the absence of further signatures of new physics at
the LHC and other experiments (we cannot assert the real nature of the 750 GeV
excess ATLAS:2015dp ; CMS:2015dxe  for the time being), the success of the
Standard Model of particle physics (SM) has become unquestionable nowadays.
Nonetheless, there are both experimental observations and theoretical reasons
not to consider the SM as a complete description of Nature even at low
energies. As a matter of fact, the SM does not account for the neutrino
oscillations, nor for the evidence of Dark Matter (DM) and the baryon-
antibaryon asymmetry of the Universe, among others. On top of that, the
unnatural size of the Higgs mass suggests that new degrees of freedom should
show up at energies of the order of the TeV scale. In this spirit, Composite
Higgs Models (CHMs) Kaplan:1983sm ; Kaplan:1983fs ; Dimopoulos:1981xc  have
played a major role in the recent years. In particular, the minimal CHM
Agashe:2004rs  has deserved special attention, but non-minimal CHMs have been
also studied in detail, as their extended scalar sectors may provide solutions
to the intricacies of new physics Gripaios:2009pe ; Bellazzini:2014yua ;
Mrazek:2011iu ; Espinosa:2011eu ; Chala:2012af ; Frigerio:2012uc ;
Bertuzzo:2012ya ; Vecchi:2013bja ; Fonseca:2015gva ; Nevzorov:2015sha . This
is specially well motivated, given that departures from the SM predictions can
be plausibly hidden in the Higgs sector, which is not precisely measured yet.

Non-minimal CHMs are very predictive in comparison to their renormalizable
counterparts built in a bottom-up approach. In CHMs the light scalars are
indeed the pseudo-Nambu-Goldstone Bosons (pNGBs) of a spontaneously-broken
global symmetry of a new strongly-coupled sector. The derivative and gauge
interactions are hence fixed by the coset structure. The leading-order one-
loop potential is instead computed as an expansion on spurion insertions
parameterizing the explicit breaking of the global symmetry Panico:2015jxa .
The crucial point in this regard is that the number of independent spurion
invariants is generally smaller than the number of independent interactions
compatible with the remnant symmetry, which in addition may be larger than the
SM gauge group. A caveat is still in order for that matter, namely small
spurion multiplets are typically disfavored. In fact, they use to generate no
quartic term for the Higgs boson at leading order; the proper size of such a
term arising only by assuming large next-to-leading order corrections, which
tend to make the usual perturbative ordering unreliable Panico:2015jxa .

In view of the discussion above, in the present paper we explore the
phenomenology of a non-minimal CHM based on the coset $SO(7)/SO(6)$. We show
that by taking spurions transforming in the fundamental representation
$\mathbf{7}$ and the symmetric representation $\mathbf{27}$ of $SO(7)$, the
model provides a rather predictive setup where the low-energy scalar sector
consists of a Higgs field and two real scalars, $\eta$ and $\kappa$, that are
singlets under the SM gauge group. Remarkably, this $SO(7)$ breaking pattern
leads to an approximate $Z_{2}$ symmetry for $\kappa$ and an exact $Z_{2}$
symmetry for $\eta$. In addition, the field $\kappa$ does not acquire a vacuum
expectation value (VEV). It is hence stable and turns out to be a good DM
candidate. At the same time, within the (modest) model dependencies of the
theory, the singlet $\eta$ acquires a VEV prior to the electroweak symmetry
breaking and ultimately promotes a strong first-order electroweak phase
transition (EWPT) with sizable gravitational wave signals. Besides, higher-
dimensional operators with sizeable $CP$-violation phases are expected
Espinosa:2011eu , opening up the possibility of electroweak baryogenesis
Kuzmin:1985mm  (related results in non-composite models are discussed in refs.
Cline:2012hg ; Cline:2013gha ; Profumo:2014opa ; Jiang:2015cwa ). Notably,
domain wall problems are avoided for the $Z_{2}$ symmetry of $\kappa$ is not
exact. Hence, within the natural values, the model can potentially solve the
hierarchy problem as well as the cosmological evidence of DM and the baryon-
antibaryon asymmetry of the Universe. Forthcoming experiments will be able to
probe the model predictions.

The rest of this paper is structured as follows. In sec. II we revisit briefly
the basics of CHMs, paying special attention to the computation of the scalar
potential. In sec. III we introduce the model. In sec. IV we work out the
implications for DM while in sec. V we study the EWPT. In sec. VI we discuss
the current constraints and future probes, including gravitational wave
searches, Higgs physics, DM direct-detection experiments, and LHC searches for
DM and dijet resonances. Sec. VII is devoted to conclusions.

##  II A quick look at composite models

In this section we present some introductory ideas on CHMs and set conventions
up. Readers interested in further details are referred to e.g. the recent
review Panico:2015jxa , whereas those familiar with CHMs may go directly to
sec. III.

CHMs extend the SM with a new strongly-interacting sector characterized by a
typically scale $f\sim\rm{TeV}$ and a coupling $g$. This new sector is endowed
with an approximate global symmetry group, $G$, spontaneously broken into some
subgroup, $H$, containing the SM gauge group. The Higgs boson is assumed to be
a pNGB of this symmetry-breaking pattern, and hence naturally lighter than
$f$.

The global symmetry is assumed to be explicitly broken mainly by the linear
mixings between the SM fermions and composite operators $\mathcal{O}$ of the
strong sector. The dynamics of the Higgs and other potential pNGBs is dictated
by the following Lagrangian:

$\mathcal{L}=\mathcal{L}_{\sigma}+\mathcal{L}_{y}-V~{}.$ (1)

$\mathcal{L}_{\sigma}$ describes the gauge and (derivative) scalar
interactions. It is fixed by the coset structure $G/H$. At the leading order
in the derivative expansion, it reads

$\mathcal{L}_{\sigma}=\frac{f^{2}}{4}\text{Tr}(d_{\mu}d^{\mu})~{},$ (2)

where $d_{\mu}$ is the projection of the Maurer-Cartan one-form
$\omega_{\mu}=iU^{-1}D_{\mu}U$ into the broken generators $T^{i}$, with $U$
being the Goldstone matrix

$U[\Pi]=\exp\left\\{-i\frac{\sqrt{2}}{f}\,h_{i}\,T^{i}\right\\}$ (3)

and $h_{i}$ the pNGBs. $\mathcal{L}_{y}$ and $V$ stand for the Yukawa
Lagrangian and the scalar potential, respectively. They arise from the
explicit breaking of the global symmetry. According to the partial
compositeness setup Kaplan:1991dc , such a breaking is provided by

$\mathcal{L}_{\text{mix}}=\sum_{\Psi}\lambda_{\Psi}\overline{\Psi}\mathcal{O}_{\Psi}+\text{h.c.}~{},$
(4)

with $\Psi$ running over incomplete representations of $G$ (also called
spurions) embedding the SM fermions.

The so-called dressed fields $\Psi^{D}$ can be constructed out of the spurions
as

$\Psi^{D}\equiv U^{-1}\Psi~{}.$ (5)

In general, $\Psi^{D}$ transform in reducible representations of the unbroken
group $H$, namely

$\Psi^{D}=\bigoplus\Psi^{D}_{m}~{},$ (6)

with $m$ running over the irreducible representations of $H$.

The dressed fields are useful to determine the invariants of $H$. By calling
$I_{n}^{j}$ the $j$-th invariant of $H$ involving a number $n$ of fields, the
scalar potential can be expressed as the following expansion Panico:2015jxa :

$V\sim\left(\frac{\lambda_{\Psi}}{g}\right)^{2}\sum_{j}c_{2}^{j}I_{2}^{j}+\left(\frac{\lambda_{\Psi}}{g}\right)^{4}\sum_{j}c_{4}^{j}I_{4}^{j}+\cdots~{},$
(7)

where $c_{m}^{j}$ are in practice free parameters. Given that $\lambda_{\Psi}$
is expected to be much smaller than the new strong coupling $g$, $I_{2}^{j}$
dominate the previous expansion. However, when $\Psi$ transforms in a small
representation of $G$, the leading order invariants $I_{2}^{j}$ do not usually
generate the Higgs quartic coupling, and hence electroweak symmetry breaking
(EWSB) can not be achieved at the leading order 111For instance, in the
minimal CHM this occurs for the $\mathbf{4}$ and $\mathbf{5}$ representations
but not for the $\mathbf{14}$ one.. Small representations can be still
considered if one assumes that $V$ is not dominated by the (formally) leading-
order contribution. The problem of this regime is twofold. On one hand, the
fine-tuning for keeping the leading-order contribution small is obviously
large. On the other hand, sizeable next-to-leading order contributions come at
the expense of predictivity. Indeed, much less spurions, and hence free
parameters, are present at the leading order. For the sake of example, two
versus fifteen independent spurions arise at the leading and next-to-leading
order, respectively, in the $\mathbf{5}$ of $SO(5)$ Panico:2015jxa .

In the present work we proceed in the regime where the next-to-leading
contributions are subleading and we hence neglect them unless otherwise
stated. As previously discussed, we work out the coset $SO(7)/SO(6)$. We embed
the SM fermions in the $\mathbf{7}$ and $\mathbf{27}$ of $SO(7)$. The latter
arises as the symmetric part of
$\mathbf{7}\times\mathbf{7}=1+\mathbf{21}+\mathbf{27}$, in complete analogy
with the $\mathbf{14}$ in $SO(5)$. Under $SO(6)$, we obtain the following
branching rules:

$\mathbf{7}=1+\mathbf{6}~{},\quad$ (8)
$\mathbf{27}=1+\mathbf{6}+\mathbf{20}~{}.$ (9)

One and two independent spurion invariants can be therefore constructed at the
leading order from the $\mathbf{7}$ and $\mathbf{27}$ representations,
respectively.

##  III Model description

The model we analyze is based on the symmetry-breaking pattern $SO(7)\times
U(1)^{\prime}/SO(6)\times U(1)^{\prime}$. We proceed in the unitary gauge. Two
gauge singlets, $\eta$ and $\kappa$, arise in the pNGB spectrum in addition to
the Higgs degrees of freedom
$\phi=[\phi^{+},(h+i\phi^{0})/\sqrt{2}]^{\text{T}}$. The addition of a
spectator group $U(1)^{\prime}$ is required in order the SM-fermion
hypercharges to be correctly reproduced, in the same vein as in the minimal
CHM.

The fifteen unbroken and six broken generators of $SO(7)$, $T$ and $X$
respectively, can be conveniently written as

$\displaystyle T^{mn}_{ij}$ $\displaystyle=$
$\displaystyle-\frac{i}{\sqrt{2}}(\delta^{m}_{i}\delta^{n}_{j}-\delta^{n}_{i}\delta^{m}_{j})~{},~{}~{}m<n\in[1,7]~{},$
$\displaystyle X^{m7}_{ij}$ $\displaystyle=$
$\displaystyle-\frac{i}{\sqrt{2}}(\delta^{m}_{i}\delta^{7}_{j}-\delta^{7}_{i}\delta^{m}_{j})~{},~{}~{}\,m\in[1,6]~{}.$
(10)

The SM $SU(2)_{L}\times U(1)_{Y}$ gauge group is thus generated by

$\displaystyle J_{L}^{1}=\frac{1}{\sqrt{2}}(T^{14}+T^{23})~{},\quad
J_{L}^{2}=\frac{1}{\sqrt{2}}(T^{24}-T^{13})~{},$ $\displaystyle
J_{L}^{3}=\frac{1}{\sqrt{2}}(T^{12}+T^{34})~{},\quad
J_{R}^{3}=\frac{1}{\sqrt{2}}(T^{12}-T^{34})~{},$ (11)

being the hypercharge defined as $Y=J_{R}^{3}+Y^{\prime}$ with $Y^{\prime}$
the generator of $U(1)^{\prime}$.

The dynamics of the pNGBs is described by the Goldstone matrix

$U=\exp{\left\\{-i\frac{\sqrt{2}}{f}\bigg{[}T^{47}h+T^{57}\eta+T^{67}\kappa\bigg{]}\right\\}}~{}.$
(12)

After performing the replacements Gripaios:2009pe

$\displaystyle\frac{h^{2}}{h^{2}+\eta^{2}+\kappa^{2}}\sin^{2}{\left\\{\left[\frac{h^{2}+\eta^{2}+\kappa^{2}}{f^{2}}\right]^{-\frac{1}{2}}\right\\}}\rightarrow
h^{2},$ (13)
$\displaystyle\frac{\eta^{2}}{h^{2}+\eta^{2}+\kappa^{2}}\sin^{2}{\left\\{\left[\frac{h^{2}+\eta^{2}+\kappa^{2}}{f^{2}}\right]^{-\frac{1}{2}}\right\\}}\rightarrow\eta^{2},$
(14)
$\displaystyle\frac{\kappa^{2}}{h^{2}+\eta^{2}+\kappa^{2}}\sin^{2}{\left\\{\left[\frac{h^{2}+\eta^{2}+\kappa^{2}}{f^{2}}\right]^{-\frac{1}{2}}\right\\}}\rightarrow\kappa^{2},$
(15)

we bring $U$ to the two-block matrix form

$U=\left(\begin{array}[]{ccccccc}\mathbf{1}_{3\times 3}&&&&&&\\\
&&&1-\frac{h^{2}}{1+\Sigma}&-\frac{h\eta}{1+\Sigma}&-\frac{h\kappa}{1+\Sigma}&h\\\
&&&-\frac{h\eta}{1+\Sigma}&1-\frac{\eta^{2}}{1+\Sigma}&-\frac{\eta\kappa}{1+\Sigma}&\eta\\\
&&&-\frac{h\kappa}{1+\Sigma}&-\frac{\eta\kappa}{1+\Sigma}&-\frac{\kappa^{2}}{1+\Sigma}&\kappa\\\
&&&-h&-\eta&-\kappa&\Sigma\end{array}\right)~{},$ (16)

where

$\Sigma=\sqrt{1-h^{2}-\eta^{2}-\kappa^{2}}~{}.$ (17)

The sigma model interactions are thus described by the Lagrangian

$\mathcal{L}_{\sigma}=\mathcal{L}_{\text{kinetic}}+\frac{1}{2}\frac{\left(h\partial_{\mu}h+\eta\partial_{\mu}\eta+\kappa\partial_{\mu}\kappa\right)^{2}}{f^{2}-h^{2}-\eta^{2}-\kappa^{2}}~{}.$
(18)

At the level of the sigma model there is a $Z_{2}\times Z_{2}\times Z_{2}$
symmetry given by $h\rightarrow-h$, $\kappa\rightarrow-\kappa$ and
$\eta\rightarrow-\eta$, for the coset is symmetric. This symmetry will be only
broken by the external sources, which will also induce a potential for the
pNGBs.

At the renormalizable level, the most general potential for $\eta$ (stable),
$\kappa$ and the Higgs boson $h$ reads

$\displaystyle V=$
$\displaystyle-\frac{1}{2}\mu_{h}^{2}h^{2}+\frac{1}{2}\mu_{\eta}^{2}\eta^{2}+\frac{1}{2}\mu_{\kappa}^{2}\kappa^{2}$
$\displaystyle+\frac{1}{3}A_{\kappa h}\kappa
h^{2}+\frac{1}{3}A_{\kappa\eta}\kappa\eta^{2}+\frac{1}{3}A_{\kappa}\kappa^{3}$
$\displaystyle+\frac{1}{4}\lambda_{h}h^{4}+\frac{1}{4}\lambda_{\eta}\eta^{4}+\frac{1}{4}\lambda_{\kappa}\kappa^{4}$
$\displaystyle+\frac{1}{4}\lambda_{h\eta}h^{2}\eta^{2}+\frac{1}{4}\lambda_{h\kappa}h^{2}\kappa^{2}+\frac{1}{4}\lambda_{\eta\kappa}\eta^{2}\kappa^{2}~{},$
(19)

which involves 12 independent parameters. Not all of them will be however
generated in the present composite setup, at least at the (unsuppressed)
leading order. In particular, if we want $\kappa$ to lead to a two-step EWPT
and $\eta$ to be a DM candidate without conflicting with Higgs searches (see
secs. IV and V), the following conditions must hold: (i)
$\eta\rightarrow-\eta$ is an unbroken symmetry; (ii) $\mu^{2}_{\kappa}<0$; and
(iii) the physical masses of $h$ and $\kappa$ are such that
$m_{h}<2m_{\kappa}$, which is favored by
$\lambda_{h\kappa}\gtrsim\lambda_{h}$.

In the present composite scenario, a minimal content satisfying the three
above conditions consists of the mixing Lagrangian

$\displaystyle\mathcal{L}_{\rm
mix}=\sum_{\Psi}\lambda_{\Psi_{R}}\overline{\Psi_{R}}^{I}\left(\mathcal{O}^{R}_{\Psi}\right)_{I}+\sum_{\Psi^{\prime}}\lambda_{\Psi_{L}}\overline{\Psi_{L}}^{I}\left(\mathcal{O}^{L}_{\Psi}\right)_{I}+\text{h.c.}~{},$

where the first sum extends over $\Psi=T,B,C$ and the second over
$\Psi^{\prime}=Q^{t},Q^{b},Q^{c}$, with $T_{R}$ and $C_{R}$ transforming in
complete singlets of $SO(7)$ with $U(1)^{\prime}$ charge $2/3$, $B_{R}$ and
$Q_{L}^{b}$ transforming in fundamental representations $\mathbf{7}$ of
$SO(7)$ with $U(1)^{\prime}$ charges $-1/3$, and $Q_{L}^{t}$ and $Q_{L}^{c}$
transforming instead in the symmetric representation $\mathbf{27}$ of $SO(7)$
that results from the tensor product of fundamental representations, i.e.
$\mathbf{7}\times\mathbf{7}=1+\mathbf{21}+\mathbf{27}$. The most general
embedding fulfilling these assignments is explicitly provided by 222See refs.
Gripaios:2009pe ; Mrazek:2011iu ; Frigerio:2012uc ; Chala:2012af  for
$Z_{2}$-preserving embeddings in other models of composite DM.

$B_{R}=\left(\begin{array}[]{@{}lllllll@{}}0&0&0&0&0&i\gamma b_{R}&b_{R}\\\
\end{array}\right)^{\text{T}},\\\ $ (20)
$Q^{b}_{L}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{@{}lllllll@{}}-it_{L}&t_{L}&ib_{L}&b_{L}&0&0&0\\\
\end{array}\right)^{\text{T}},$ (21)
$Q_{L}^{t}=\dfrac{1}{2}\left(\begin{array}[]{cccccccccc}\mathbf{0}_{6\times
6}&&&&&&it_{L}\\\ &&&&&&t_{L}\\\ &&&&&&ib_{L}\\\ &&&&&&-b_{L}\\\ &&&&&&0\\\
&&&&&&0\\\ it_{L}&t_{L}&ib_{L}&-b_{L}&0&0&0\\\ \end{array}\right)~{},\\\ $
(22) $Q^{c}_{L}=\frac{1}{2}\left(\begin{array}[]{ccccccccc}\mathbf{0}_{5\times
5}&&&&&\zeta c_{L}&ic_{L}\\\ &&&&&-i\zeta c_{L}&c_{L}\\\ &&&&&\zeta
s_{L}&is_{L}\\\ &&&&&i\zeta s_{L}&-s_{L}\\\ &&&&&0&0\\\ \zeta c_{L}&-i\zeta
c_{L}&\zeta s_{L}&i\zeta s_{L}&0&0&0\\\ ic_{L}&c_{L}&is_{L}&-s_{L}&0&0&0\\\
\end{array}\right)~{}.$

Heavier quarks couple stronger to the composite sector, and hence $b$ and $c$
contributions to the one-loop potential can be neglected unless they are
multiplied by a large $\gamma$ or $\zeta$, respectively.

In this embedding the potential acquires the form

$\displaystyle V=$
$\displaystyle-\frac{1}{2}\mu_{h}^{2}h^{2}+\frac{1}{2}\mu_{\eta}^{2}\eta^{2}+\frac{1}{2}\mu_{\kappa}^{2}\kappa^{2}$
$\displaystyle+\frac{1}{4}\lambda_{h}h^{4}+\frac{1}{4}\lambda_{\kappa}\kappa^{4}+\frac{1}{4}\lambda_{h\eta}h^{2}\eta^{2}+\frac{1}{4}\lambda_{h\kappa}h^{2}\kappa^{2}~{}.$
(23)

The quartic coupling $\lambda_{\kappa}$ is generated only at the next-to-
leading order, but it has been introduced since it plays an important role in
the EWPT phenomenology. At any rate, it is expected to be much smaller than
the other quartic couplings. The rest of the parameters are functions of the
dimensionless spurion coefficients $\alpha_{q,i}$, as well as $\gamma$ and
$\zeta$:

$\displaystyle\mu_{h}^{2}$
$\displaystyle=-\frac{1}{2}f^{2}\left(4\alpha_{t,1}-7\alpha_{t,2}+\alpha_{c,2}\zeta^{2}\right)~{},$
(24) $\displaystyle\mu_{\eta}^{2}$ $\displaystyle=-2\alpha_{t,2}f^{2}~{},$
(25) $\displaystyle\mu_{\kappa}^{2}$
$\displaystyle=2f^{2}\left(\alpha_{b}\gamma^{2}+\alpha_{c,2}\zeta^{2}-\alpha_{t,2}\right)~{},$
(26) $\displaystyle\lambda_{h}$
$\displaystyle=4(\alpha_{t,2}-\alpha_{t,1})~{},$ (27)
$\displaystyle\lambda_{h\eta}$
$\displaystyle=4(\alpha_{t,2}-\alpha_{t,1})~{},$ (28)
$\displaystyle\lambda_{h\kappa}$
$\displaystyle=4\left[\alpha_{t,2}-\alpha_{t,1}+(\alpha_{c,1}-\alpha_{c,2})\zeta^{2}\right]~{}.$
(29)

In our analysis we will consider two broad parameter regimes depending on the
actual values of $\alpha_{c,i}$:

Regime I:

$\alpha_{c,2}=-\alpha_{c,1}$. This is the most natural scenario since the size
of these two coefficients is expected to be similar, and it still allows for
$\lambda_{h\kappa}\gtrsim\lambda_{h}$, contrary to the case
$\alpha_{c,2}=\alpha_{c,1}$.

Regime II:

$|\alpha_{c,2}|\ll|\alpha_{c,1}|\sim|\alpha_{t,i}/\zeta^{2}|$. As we will see,
accounting for the DM relic density observation will completely fix the mass
of $\eta$ and its interactions with nuclei in this case 333The case
$|\alpha_{c,1}|\ll|\alpha_{c,2}|$ would be quite similar to Regime I. .

In both cases, the coefficients $\alpha_{q}^{i}$ as well as $\gamma$ and
$\zeta$ can be traded by the measured values of the Higgs VEV ($v\simeq 246$
GeV) and physical mass (yielding $\lambda\simeq 0.13$), and only three free
parameters, namely $f$, $\mu_{\kappa}^{2}$ and $\lambda_{h\kappa}$. Indeed, in
Regime I we obtain

$\displaystyle\mu_{\eta}^{2}$
$\displaystyle=\frac{1}{3}f^{2}\left[\frac{7}{4}\lambda_{h}+\frac{1}{4}\lambda_{h\kappa}-4\lambda_{h}\xi\right]~{},$
(30) $\displaystyle\lambda_{h\eta}$ $\displaystyle=\lambda_{h}~{},$ (31)

while in Regime II we get

$\displaystyle\mu_{\eta}^{2}$
$\displaystyle=\frac{2}{3}\lambda_{h}f^{2}\left(1-2\xi\right)~{},$ (32)
$\displaystyle\lambda_{h\eta}$ $\displaystyle=\lambda_{h}~{},$ (33)

with $\xi\equiv v^{2}/f^{2}$.

Finally, the Yukawa Lagrangian takes the form

$\displaystyle\mathcal{L}_{y}=$
$\displaystyle-\sum_{q=t,b,c}y_{q}\overline{q}qh\left[1-\frac{1}{f^{2}}\left(h^{2}+\eta^{2}+\kappa^{2}\right)\right]^{\frac{1}{2}}$
$\displaystyle-i\frac{h}{f}\kappa\left[\gamma
y_{b}\overline{b}\gamma_{5}b+\zeta y_{c}\overline{c}\gamma_{5}c\right]~{},$
(34)

with $y_{q}$ the Yukawa couplings.

According to these results, several comments are in order here:

  1. i)

Neither a single cubic coupling nor a quartic term for $\eta$ or a quartic
coupling between $\eta$ and $\kappa$ are generated at leading order. At higher
orders only the $Z_{2}$ symmetry of $\kappa$ is broken. Therefore, since
$\mu_{\eta}^{2}$ is predicted to be positive, $\eta$ is stable and hence a DM
candidate. Finally, $\kappa$ is subject to model dependencies that allow
$\mu^{2}_{\kappa}<0$ 444Relevant comments in this respect have been previously
pointed out in ref. Serra:2015xfa ..

  2. ii)

The second quark generation has to be included in order the coupling
$\lambda_{h\kappa}$ to be large enough to achieve the hierarchy
$m_{h}^{2}/4<m^{2}_{\kappa}=\mu^{2}_{\kappa}+1/2\lambda_{h\kappa}v^{2}$
between the $h$ and $\kappa$ physical masses. Otherwise the experimental bound
on the Higgs decay into non-SM particles Atlas:2015at  would be hard to evade.

  3. iii)

$\gamma$ and $\zeta$ could have a small imaginary part, even in the top
sector, without susbtantially changing the equations above. If this was the
case, they could provide a new sizable source of CP violation Espinosa:2011eu
, as required by electroweak baryogenesis.

  4. iv)

From the EWSB conditions one obtains $|\alpha_{t,i}|\sim\lambda_{h}\simeq
0.13$ . We besides expect $|\alpha_{c}|<|\alpha_{b}|<|\alpha_{t,i}|$. We have
checked that, in both regimes, independently of $f\leq 5$ TeV, any value of
$m_{\kappa}$ below 200 GeV and any value of $\lambda_{h\kappa}$ between 0.1
and 0.4 can be reached by barring this parameter space region with mild values
of $\gamma,\zeta\in[3,5]$.

##  IV Dark matter predictions

As previously discussed, $\eta$ can provide a DM candidate since it is
protected by a $Z_{2}$ symmetry not even spontaneously broken. The main
diagrams contributing to the annihilation of $\eta$ are shown in fig. -959.
For the choice $f\sim$ TeV that is favored by electroweak precision data and
Higgs physics (see sec. VI), all depicted processes are kinematically
accessible. Indeed, as eqs. (30) and (32) show, the physical mass of $\eta$,
$m_{\eta}$, is larger than the electroweak scale (and thus than the quark and
remaining pNGB masses) when $f\gg\,v$. Due to this hierarchy, the DM
phenomenology is dominated by one single scale, $m_{\eta}$ (or equivalently
$f$). No strong dependence on $m_{\kappa}$ is thus expected.

One can use dimensional analysis to estimate the annihilation cross sections
of $\eta$. As the processes with mediators are suppressed, from eqs. (18),
(III) and (III) one deduces

$\displaystyle\sigma(\eta\eta\rightarrow hh)v_{0}$
$\displaystyle\sim\frac{1}{m_{\eta}^{2}}\left[\lambda_{h}-\frac{4m_{\eta}^{2}}{f^{2}}\right]^{2},$
(35) $\displaystyle\sigma(\eta\eta\rightarrow\kappa\kappa)v_{0}$
$\displaystyle\sim\frac{1}{m_{\eta}^{2}}\left[\frac{4m_{\eta}^{2}}{f^{2}}\right]^{2},$
(36) $\displaystyle\sigma(\eta\eta\rightarrow t\bar{t})v_{0}$
$\displaystyle\sim\frac{1}{m_{\eta}^{2}}\left[\frac{m_{t}m_{\eta}}{f^{2}}\right]^{2},$
(37)

whit $v_{0}$ the (small) velocity of the colliding DM particles. It is then
expected that the $\eta\eta\rightarrow\kappa\kappa$ channel dominates the
annihilation cross section, for a partial cancellation between the Higgs
quadratic coupling and the derivative contribution from $\mathcal{L}_{\sigma}$
arises in $\sigma(\eta\eta\rightarrow hh)$. This implies that the DM
phenomenology can be drastically different from related composite setups.

Interestingly, in our framework the DM abundance is fixed by only a few free
parameters. Indeed, since $\sigma(\eta\eta\rightarrow\kappa\kappa)$ is the
dominant annihilation cross section, the relic density depends only on
$\lambda_{h\kappa}$ and $f$ in Regime II, and uniquely on $f$ in Regime II
(cf. eqs. (30) and (32)). This feature also arises in our numerical study.
Specifically, we employ Feynrules Alloul:2013bka  to implement the model, and
micrOmegas Belanger:2014vza  to determine the parameter region where the relic
abundance of $\eta$, $\Omega_{\eta}h^{2}$, is compatible with the experimental
measurement $\Omega_{\text{DM}}h^{2}=0.119\pm 0.003$ Ade:2013zuv . The finding
is summarized in fig. -958 which highlights the constraint
$f[\lambda_{h\kappa}]$ that guarantees $\Omega_{\eta}h^{2}=0.119$ in Regime I
(dashed blue) and Regime II (solid green). Clearly, for
$\lambda_{h\kappa}=\lambda_{h}$ the expected value of $f$ is the same in both
regimes. The figure also displays two successful parameter points with somehow
extreme values of $\lambda_{h\kappa}$, and the corresponding predictions for
$m_{\eta}$. In conclusion, due to the DM relic density constraints, we expect
$m_{\eta}\simeq 730\div 960$ GeV and $f\simeq 2.4\div 2.9$ TeV.

Figure -959: Main diagrams contributing to the DM annihilation. The double
dashed lines stand for $\eta$. In the first plot the solid legs account for
either SM particles or $\kappa$. In the second diagram the simple dashed lines
represent either the Higgs boson or $\kappa$. In the third diagram the fermion
lines stand for (mainly) the top quark. In the fourth diagram the simple
dashed lines represent the Higgs boson. Figure -958: Value of $f$ leading to
$\Omega_{\eta}=\Omega_{\rm DM}$ as a function of $\lambda_{h\kappa}$ in Regime
I (dashed blue) and Regime II (solid green). The masses $m_{\eta}$
corresponding to two extreme points are also depicted.

##  V Electroweak baryogenesis

As generically expected in composite models, and commented above, the
interactions between the quarks and the strong sector can contain new CP-
violation phases. It was observed that these phases, together with the
sphaleron processes, can lead to the observed baryon asymmetry, provided the
EWPT is of strongly first order Espinosa:2011eu . Here we demonstrate that, in
our composite model, such a strong EWPT is rather likely within the parameter
region compatible with the present constraints, DM bounds included 555A phase
transition, possibly linked to the baryon asymmetry production Nardini:2007me
; Konstandin:2011ds , may also occur during the breaking of the composite
strong symmetry Creminelli:2001th ; Randall:2006py ; Konstandin:2010cd ;
Konstandin:2011dr ; Hassanain:2007js ; Schwaller:2015tja . Such a transition
would modify our results only if, in the present setup, it turned out to be of
first order, very supercooled, and with a reheating temperature around or
below the electroweak scale, characteristics that may or may not be realized
depending on the ultraviolet completion Creminelli:2001th ; Hassanain:2007js
and its parameter values Nardini:2007me ; Konstandin:2011ds ..

We deduce the properties of the EWPT from the one-loop finite-temperature
scalar potential $V_{\rm{1L}}(h,\kappa,\eta\,;T)$, with $T$ representing the
temperature. Since the new fields have small couplings and negligible mixing
with the Higgs, at qualitative level $V_{\rm{1L}}(h,0,0;T\\!=\\!0)$ is similar
to the potential in eq. (III). Therefore, in the sizeable part of the
fundamental parameter space predicting $\mu^{2}_{\kappa}<0$ and
$\lambda_{h\kappa}>0$, the minima of $V_{\rm{1L}}(h,\eta,\kappa;T\\!=\\!0)$
are

$\displaystyle v_{1}(T=0)=(v[T\\!=\\!0],0,0)~{},$ (38) $\displaystyle
v_{2}(T=0)=(0,v_{\kappa}[T\\!=\\!0],0)~{},$ (39)

with $v_{\kappa}[T\\!=\\!0]\simeq\sqrt{-\mu^{2}_{\kappa}/\lambda_{\kappa}}$
and $v[T\\!=\\!0]=v$ 666It is not restrictive to ignore the other symmetric
minima. Moreover, due to the (suppressed) explicit breaking of the $\kappa$
discrete symmetry, only one of the two minima $\pm v_{2}$ will be relevant in
the evolution of the Universe Espinosa:2011eu ..

This structure of the minima hints at the possibility of a two-step EWPT. In
this case the electroweak breaking minimum is reached via the changes of
phases $(0,0,0)\to v_{2}(T^{\prime})$ and $v_{2}(T_{n})\to v_{1}(T_{n})$, with
$T^{\prime}>T_{n}$. The latter transition is the one that is required to be
strong (i.e. $|v_{1}(T_{n})|/T_{n}>1$) for successful electroweak
baryogenesis.

To determine the possible phase transitions and their characteristics, we use
CosmoTransitions Wainwright:2011kj . In the code we specify
$V_{\rm{1L}}(h,\kappa,\eta\,;T)$ in the customary form Quiros:1994dr

$\displaystyle V_{\rm{1L}}(h,\kappa;T)=V+\Delta V_{\rm{CW}}+\Delta V_{T\neq
0}~{},$ (40)

with $V$ given in eq. (III) and

$\displaystyle\Delta V_{\rm{CW}}$ $\displaystyle=$
$\displaystyle\frac{1}{64\pi^{2}}\sum_{i}(\pm
1)n_{i}m_{i}^{4}(h,\kappa)\bigg{[}\log{\frac{m_{i}^{2}(h,\kappa)}{v^{2}}-c_{i}}\bigg{]}~{},$
$\displaystyle\Delta V_{T\neq 0}$ $\displaystyle=$
$\displaystyle\frac{T^{4}}{2\pi^{2}}\sum_{i}(\pm
1)n_{i}~{}J_{\pm}\\!\left(\frac{m_{i}^{2}(h,\kappa)}{T^{2}}\right)~{},$
$\displaystyle J_{\pm}(x)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dyy\texttwosuperior\left[1\mp\exp\left(-\sqrt{x^{2}+y^{2}}\right)\right]~{},$
(41)

in which the dependence on the background field of $\eta$ is removed because
$\partial^{2}_{\eta}V>0$ for any VEV of $h$ and $\kappa$ below $f$. In eq.
(41) $i$ extends to the fields that couple stronger to the Higgs sector,
namely the $\kappa$-$h$ mixing states $\phi_{1,2}$, the SM Goldstones
$G^{0,\pm}$, the singlet $\eta$, the top quark $t$, and the gauge bosons
$W^{\pm}$ and $Z$. The factor $n_{i}$ is the number of degrees of freedom of
the field $i$, and the upper sign (lower sign) in ‘$\pm$’ and ‘$\mp$’ applies
to the bosonic (fermionic) contributions. The coefficient $c_{i}$ is equal to
5/6 for gauge bosons and to 3/2 otherwise. Finally, the field-dependent
squared masses $m_{\phi_{1}}^{2}(h,\kappa)$ and $m_{\phi_{1}}^{2}(h,\kappa)$
are the eigenvalues of the symmetric matrix $\mathcal{M}^{2}$ with entries

$\displaystyle\mathcal{M}^{2}_{1,1}$ $\displaystyle=$
$\displaystyle-\mu_{h}^{2}+3\lambda_{h}h^{2}+\frac{1}{2}\lambda_{h\kappa}\kappa^{2}~{},$
(42) $\displaystyle\mathcal{M}^{2}_{2,2}$ $\displaystyle=$
$\displaystyle\mu_{\kappa}^{2}+3\lambda_{\kappa}\kappa^{2}+\frac{1}{2}\lambda_{h\kappa}h^{2}~{},$
(43) $\displaystyle\mathcal{M}^{2}_{1,2}$ $\displaystyle=$
$\displaystyle\lambda_{h\kappa}h\kappa~{},$ (44)

while the other masses are

$\displaystyle m_{G^{0,\pm}}^{2}(h,\kappa)$
$\displaystyle=-\mu_{h}^{2}+\lambda_{h}h^{2}+\frac{1}{2}\lambda_{h\kappa}\kappa^{2}~{},$
(45) $\displaystyle m_{W}^{2}(h,\kappa)$
$\displaystyle=\frac{1}{4}g^{2}h^{2}~{},$ (46) $\displaystyle
m_{Z}^{2}(h,\kappa)$ $\displaystyle=\frac{1}{4}(g^{2}+g^{\prime 2})h^{2}~{},$
(47) $\displaystyle m_{t}^{2}(h,\kappa)$
$\displaystyle=\frac{1}{2}y_{t}^{2}h^{2}~{},$ (48) $\displaystyle
m_{\eta}^{2}(h,\kappa)$ $\displaystyle=C(h)~{},$ (49)

where $g,g^{\prime}$ and $y_{t}$ stand for the $SU(2)_{L}$ and $U(1)_{Y}$
gauge couplings and the top Yukawa, respectively. The function $C(h)$ depends
on which regime we consider. Taking into account the DM constraint that
establishes the function $f[\lambda_{h\kappa}]$ (see fig. -958), we have

$C(h)=\frac{1}{3}\left[\frac{7}{4}\lambda_{h}+\frac{1}{4}\lambda_{h\kappa}-4\lambda_{h}\xi\right]f^{2}[\lambda_{h\kappa}]+\frac{1}{2}\lambda_{h}h^{2}\\\
$ (50)

in Regime I and

$C(h)=\frac{1}{2}\lambda_{h}\left[\frac{4}{3}(1-2\xi)f^{2}[\lambda_{h\kappa}]+h^{2}\right]$
(51)

in Regime II.

Figure -957: Points of the parameter space exhibiting a strong EWPT in Regime
I (left panel) and Regime II (right panel). The region in green indicates the
points for which $V_{\rm 1L}(h,\kappa;T\\!=\\!0)$ has a local minimum at
$v_{1}$ (the contrary in the white area) and such a minimum is deeper than the
one at $v_{2}$ (the contrary in the orange area). The points on the left of
the black dashed line are unfavored by the Higgs searches. The filled (empty)
circles correspond to EWPTs with bubbles expanding (not expanding) at the
speed of light.

For the numerical analysis we scan over the parameter region
$\sqrt{-\mu^{2}_{\kappa}}/{\rm GeV}\in\,]0,100]$ and
$\lambda_{h\kappa}\in[0.1,0.4]$. For each pair
$\\{\mu_{\kappa}^{2},\lambda_{h\kappa}\\}$ we consider the lowest value of
$\lambda_{\kappa}\in[0.01,0.02,\dots,0.06]$ for which the EWPT is strong, if
such a transition arises 777Due to some numerical instabilities, the code
identifies only a subset of the satisfactory parameter points. Since our aim
is to highlight the abundance of points with a strong first-order EWPT, we do
try to circumvent this issue and we just display the numerous points that the
code finds.. The findings are presented in fig. -957 for Regime I (left panel)
and Regime II (right panel). Both filled and empty circles represent the
parameter points where the EWPT $v_{2}(T_{n})\to v_{1}(T_{n})$ satisfies the
condition $v_{1}(T_{n})/T_{n}>1$. In the orange area, defined by the condition
$V_{\rm 1L}(v_{1};T\\!=\\!0)\geq V_{\rm 1L}(v_{2};T\\!=\\!0)$ with
$\lambda_{\kappa}\leq 0.06$, no EWPT arises because the Universe gets stuck in
the phase $v_{2}$. In the white area $\mu_{\kappa}^{2}$ is positive. In the
region on the left of the dashed line, all signal strengths of the Higgs into
SM particles are diluted, for the channel $h\to\kappa\kappa$ is allowed. The
strong EWPT found in this area are then ruled out by the present LHC
measurements Atlas:2015at . Overall, within the parameter space compatible
with DM and the EWSB constraints, the ingredients for electroweak baryogenesis
are often realized in the present CHM.

We remark that our result is not a full proof that the model can actually
reproduce the measured baryon asymmetry Ade:2015xua . By applying
straightforwardly the analysis of ref. Espinosa:2011eu  to our setup, one
would naively reach a positive conclusion. Nevertheless, the expansion
velocities Moore:1995si ; Espinosa:2010hh ; Konstandin:2014zta ;
Megevand:2014dua ; Kozaczuk:2015owa  of our EWPT bubbles may not be subsonic
(i.e. the bubble speed is smaller than $1/\sqrt{3}$) as assumed in ref.
Espinosa:2011eu . Were this the case, the evaluation of the baryon asymmetry
would be controversial Caprini:2011uz . This particularly applies to the EWPTs
marked as filled circles in fig. -957. They satisfy the runaway condition
$\alpha>\alpha_{\infty}$ with

$\displaystyle\alpha$ $\displaystyle\simeq$
$\displaystyle\frac{V_{\rm{1L}}(v_{2}(T_{n});T_{n})-V_{\rm{1L}}(v_{1}(T_{n});T_{n})}{35~{}T_{n}^{4}}~{},$
(52) $\displaystyle\alpha_{\infty}$ $\displaystyle\simeq$ $\displaystyle
4.9\times 10^{-3}\left(\frac{v_{1}(T_{n})}{T_{n}}\right)^{2}~{},$ (53)

which hints at expansion velocities similar to the speed of light
Espinosa:2010hh ; Caprini:2015zlo . Consistently, they also fulfill the
microphysical-approach runaway condition
$\widetilde{V}_{\rm{1L}}(v_{1}(T_{n});T_{n})<\widetilde{V}_{\rm{1L}}(v_{2}(T_{n});T_{n})$,
with $\widetilde{V}$ being the one-loop thermal potential evaluated in the
mean field approximation Bodeker:2009qy  888We remind that these conditions
guarantee a runaway behavior only to those bubbles that have an initial
ultrarelativistic speed. Here we adopt the simple criterion
$\alpha>\alpha_{\infty}$, which results slightly more stringent than the
microphysical one, as noticed also in Ref. Katz . On the other hand,
unquestionable uncertainties jeopardize these criteria. Further developments
in the field are then expected to modify our runaway/non-runaway
classifications. . Smaller velocities can instead arise for the EWPTs
represented by empty circles (satisfying the non-runaway relation
$\alpha<\alpha_{\infty}$), although determining whether such speeds are too
large for electroweak baryogenesis would be delicate Konstandin:2014zta ;
Kozaczuk:2015owa . Due to these uncertainties, we cannot go further than
claiming that a quantitative explanation of the observed baryon asymmetry is
conceivable in the parameter points highlighted in fig. -957.

##  VI Other constraints and future expectations

We have seen that, for $f\simeq 2.5\div 2.9$ TeV, $m_{\kappa}\simeq 70\div
120$ GeV, $\lambda_{hk}\simeq 0.2\div 0.4$ and $\lambda_{\kappa}\simeq
0.01\div 0.06$, which are natural values within our CHM, the observed DM relic
abundance as well the EWPT that is necessary for electroweak baryogenesis are
achieved. In this section we check that these ranges of values are not in
conflict with present experimental bounds but testable in the forthcoming
years.

A common prediction to all CHMs (and models with non-linear realizations of a
gauge symmetry in general) is the modification of the Higgs couplings to the
SM fermions and gauge bosons. As in the minimal CHM, if we expand eqs. (18)
and (III) to order $\mathcal{O}(v^{2}/f^{2})$, we obtain the following ratios
of the tree level couplings of the Higgs to two SM particles:

$\displaystyle R_{hVV}$ $\displaystyle\equiv\frac{g_{hVV}}{g_{hVV}^{SM}}\simeq
1-\frac{v^{2}}{2f^{2}}~{},$ (54) $\displaystyle R_{h\psi\psi}$
$\displaystyle\equiv\frac{g_{h\psi\psi}}{g_{h\psi\psi}^{SM}}\simeq
1-\frac{3v^{2}}{2f^{2}}~{},$ (55)

where $V$ and $\psi$ stand for any electroweak gauge boson and SM fermion,
respectively. Even for $f=2.5$ TeV, we obtain $R_{hVV}\sim 0.99$ and
$R_{h\psi\psi}\sim 0.98$, therefore well within the current LHC limits
Atlas:2015at . Such small deviations from the SM predictions might be however
accessible at a future linear collider (see e.g. ref. Klute:2013cx ).

Figure -956: Main diagrams contributing to the scattering of DM particles by
nuclei.

Concerning direct detection experiments, the two main diagrams contributing to
the scattering between DM particles and nuclei are depicted in fig. -956. The
corresponding cross section can be parameterized as Cline:2013gha

$\sigma=\lambda_{h}^{2}\frac{f_{N}^{2}}{4\pi}\frac{\mu_{r}^{2}m_{n}^{2}}{m_{h}^{4}m_{\eta}^{2}}\left[1+\frac{m_{\eta}^{2}}{f^{2}}\right],$
(56)

where $m_{n}$ is the nucleon mass, $\mu_{r}$ is the reduced mass of the system
(with $m_{\eta}\gg m_{n}$)

$\mu_{r}=\frac{m_{\eta}m_{n}}{m_{\eta}+m_{n}}\sim m_{n}\sim 1\,\text{GeV}~{},$
(57)

and $f_{N}\sim 0.3$ Alarcon:2011zs ; Alarcon:2012nr ; Kahlhoefer:2015jma . For
the considered ranges of parameter values, eq. (56) yields $\sigma\sim
10^{-46}\div 10^{-45}$ cm2, depending on the actual value of $f$. These values
are around one order of magnitude below the LUX bound in the DM mass range
$730\div 960$ GeV Akerib:2013tjd . However, it will be definitely reachable in
the new round of data and experiments Cushman:2013zza  999This result can be
straightforwardly applied to the simpler model $SO(6)/SO(5)$ with $q_{L}$
transforming in the representation $\mathbf{20}$, which is analogous to our
model in the limit $m_{\kappa}\gg m_{\eta}$. In this case, the observed DM
abundance fixes $m_{\eta}\simeq 500$ GeV, for which the scattering cross
section at zero momentum is larger ($\sim 2\times 10^{-45}$ cm2) but still
below the LUX upper limit..

Figure -955: The identified first-order EWPTs with non-runaway (left panel)
and runaway (right panel) behavior in the $\\{\alpha,\beta/H\\}$ plane. Black
circles and red squares represent the EWPTs of Regime I and Regime II,
respectively. eLISA in the N2A5M5L6 experimental design can test the EWPTs
inside the green areas.

On another front, LHC searches for DM in monojet and $t\bar{t}$ events
produced in association with large missing transverse energy do not provide
relevant constraints. Indeed, the cross sections of these signals are
suppressed by the small available phase space and the small ratio
$v^{2}/f^{2}$. On the same vein, searches for double $\eta$ or $\kappa$
production mediated by an off-shell Higgs boson are extremely challenging
Mambrini:2011ik ; Djouadi:2011aa ; Djouadi:2012zc ; deSimone:2014pda ;
Craig:2014lda  and hence negligible. Searches for the Higgs boson decaying to
non-SM particles Aad:2015pla  instead constrain only the small region on the
left of the dashed vertical line in fig. -957.

A last collider constraint comes from dijet searches. The field $\kappa$
couples linearly to $c$ and $b$ quarks, mainly. Hence, $\kappa$ can be singly
produced in hadron colliders, with a subsequent dijet decay. The strongest and
updated upper bound on such a process can be found in ref. Chala:2015ama  (see
also ref. Dobrescu:2013coa ). Given the large QCD background, the main
constraints at $\mathcal{O}(100\,$GeV) invariant masses come from UA2
Alitti:1993pn  and LHC searches for dijet resonances in association with a $W$
or a $Z$ at $\sqrt{s}=8$ TeV TheATLAScollaboration:2013gia , the latest being
dominant for invariant masses below $\lesssim 200$ GeV. In this region, the
upper limit on the cross section ranges between $\sim 0.1\div 0.2$ pb. On
another hand, by using MadGraph Alwall:2014hca  we find that, even for
$\gamma=\zeta=10$, the production of $\kappa$ in association with a massive
gauge boson is smaller than 0.002 pb for any mass in our range of values. The
particle $\kappa$ would be then difficult to discover by means of the usual
analyses.

A strong EWPT produces a gravitational wave stochastic background whose
typical spectrum has a peak at frequencies eLISA is sensitive to
Seoane:2013qna ; webpage . eLISA might hence be able to probe the CHM under
study. We explore this possibility by following ref. Caprini:2015zlo ,
assuming the plausible “N2A5M5L6” eLISA experimental design Caprini:2015zlo ;
APetit . Thus, for each point in fig. -957, we calculate $\beta/H\equiv
T\partial_{T}(S_{3}/T)|_{T=T_{n}}$ (where $S_{3}$ is the $O_{3}$-symmetric
bubble action Quiros:1994dr ) by running CosmoTransitions 101010The code
determines $T_{n}$ by solving the condition $S(T_{n})/T_{n}=140$
Wainwright:2011kj . It is then easy to obtain a second temperature, $T_{C}$,
given by $S(T_{C})/T_{C}=C$, and use that in the approximation $\beta/H\approx
T_{n}(C-140)/(T_{C}-T_{n})$. We use $C=240$ in our estimates.. We display
these points in the $\\{\alpha,\beta/H\\}$ plane, and check whether they are
located inside the pertinent eLISA detection region. The findings are
presented in the left (right) panel of fig. -955 where the non-runaway
(runaway) strong-EWPT points of Regime I and Regime II are displayed as empty
circles and filled squares, respectively. The left (right) panel also reports
the N2A5M5L6 eLISA detection region for non-runaway scenarios with
$T_{n}\approx 50\,$GeV (for runaway scenarios at $T_{n}\approx 50\,$GeV and
$\alpha_{\infty}\approx 0.05$) in green color 111111The choice of these
particular regions is due to the fact that in our set of points the nucleation
temperature of the EWPT is $30\div 80\,$GeV, and that the points at the border
of detection turn out to have $\alpha_{\infty}\simeq 0.05$. The regions are
taken from ref. Caprini:2015zlo .. It results that, among the strong EWPTs
identified in sec. V, none of those with a non-runaway behavior is detectable,
whereas the runaway ones can be basically all probed 121212We stress that this
sharp conclusion is specific to our model where empirically
$\alpha_{\infty}\approx\sqrt{\alpha}/2$, whence $\alpha\lesssim 0.05$ for all
non-runaway cases (defined by the condition $\alpha<\alpha_{\infty}$). This of
course depends on the adopted runaway condition and its approximation in eq.
53. For a much more stringent runaway criterion, points of the right panel of
fig. -955 would certainly move into the detection region of the left panel..
From fig. -957 we hence deduce that, overall, eLISA can detect a sizeable
fraction of the strong EWPTs predicted within our model.

##  VII Conclusions

We have presented a composite version of the Higgs sector extended with two
gauge singlets $\eta$ and $\kappa$ based on the coset $SO(7)/SO(6)$. The
embedding of the elementary fermions into appropriate representations of
$SO(7)$ can both make $\eta$ stable and generate a negative quadratic term for
$\kappa$. As a consequence, this theoretical setup can provide a natural
explanation for the observed dark matter abundance and the baryon-antibaryon
asymmetry via freeze out and electroweak baryogenesis. We have emphasized
that, contrary to its renormalizable counterpart, the dominant terms in the
scalar potential are described by only three free parameters, namely the
physical mass $m_{\eta}$ of the dark matter scalar, the physical mass
$m_{\kappa}$ of the second singlet, and the coupling $\lambda_{h\kappa}$ of
the quartic interaction between the Higgs and $\kappa$. We have shown that
fulfilling the dark matter relic density observation requires $m_{\eta}\sim
730\div 960$ GeV, irrespectively of $m_{\kappa}$, which is required to live in
the region $50\div 100$ GeV for $\lambda_{h\kappa}\sim 0.15\div 0.4$ to
trigger the strong first-order electroweak phase transition required by
electroweak baryogenesis. We have subsequently studied the implications of
this scenario in Higgs physics, dark matter direct detection experiments, LHC
searches for dark matter as well as searches for dijet resonances. All
together, they are in agreement with current data. The observation (or not) of
a dark matter direct detection signal in the sub-TeV region compatible with a
total cross section of $10^{-46}\div 10^{-45}$ cm2, possibly followed by the
measurement of the gravitational wave stochastic background at eLISA, will
definitely shed light on whether this model is realized in Nature.

## Acknowledgements

We are grateful to Thomas Konstandin and Geraldine Servant for useful comments
on the manuscript. IS thanks the hospitality of the DESY Theory Group in the
context of the DESY Summer School during the first stage of this work. GN
thanks the eLISA cosmology working group for useful discussions and M.
Codiglia for seminal explanations. The work of GN is supported by the Swiss
National Science Foundation under grant 200020-155935. The work of IS is
supported by RSCF grant 14-22-00161.

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# Escaping Set of Hyperbolic Semigroup

Bishnu Hari Subedi 1 and Ajaya Singh 2 1, 2Central Department of Mathematics,
Institute of Science and Technology, Tribhuvan University, Kirtipur,
Kathmandu, Nepal
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
This research work of first author is supported by PhD faculty fellowship from
University Grants Commission, Nepal.

Abstract: In this paper, we mainly study hyperbolic semigroups from which we
get non-empty escaping set and Eremenko’s conjecture remains valid. We prove
that if each generator of bounded type transcendental semigroup $S$ is
hyperbolic, then the semigroup is itself hyperbolic and all components of
$I(S)$ are unbounded.
Key Words: Escaping set, Eremenko’s conjecture, transcendental semigroup,
hyperbolic semigroup.
AMS (MOS) [2010] Subject Classification. 37F10, 30D05

##  1\. Introduction

Throughout this paper, we denote the complex plane by $\mathbb{C}$ and set of
integers greater than zero by $\mathbb{N}$. We assume the function
$f:\mathbb{C}\rightarrow\mathbb{C}$ is transcendental entire function (TEF)
unless otherwise stated. For any $n\in\mathbb{N},\;\;f^{n}$ always denotes the
nth iterates of $f$. Let $f$ be a TEF. The set of the form

$I(f)=\\{z\in\mathbb{C}:f^{n}(z)\rightarrow\infty\textrm{ as
}n\rightarrow\infty\\}$

is called an escaping set and any point $z\in I(S)$ is called escaping point.
For TEF $f$, the escaping set $I(f)$ was first studied by A. Eremenko [2]. He
himself showed that $I(f)\not=\emptyset$; the boundary of this set is a Julia
set $J(f)$ (that is, $J(f)=\partial I(f)$); $I(f)\cap J(f)\not=\emptyset$; and
$\overline{I(f)}$ has no bounded component. By motivating from this last
statement, he posed a question: Is every component of $I(f)$ unbounded?. This
question is considered as an important open problem of transcendental dynamics
and it is known as Eremenko’s conjecture. Note that the complement of Julia
set $J(f)$ in complex plane $\mathbb{C}$ is a Fatou set $F(f)$.

Recall that the set $CV(f)=\\{w\in\mathbb{C}:w=f(z)\;\ \text{such that}\;\
f^{\prime}(z)=0\\}$ represents the set of critical values. The set $AV(f)$
consisting of all $w\in\mathbb{C}$ such that there exists a curve (asymptotic
path) $\Gamma:[0,\infty)\to\mathbb{C}$ so that $\Gamma(t)\to\infty$ and
$f(\Gamma(t))\to w$ as $t\to\infty$ is called the set of asymptotic values of
$f$ and the set $SV(f)=\overline{(CV(f)\cup AV(f))}$ is called the singular
values of $f$. If $SV(f)$ has only finitely many elements, then $f$ is said to
be of finite type. If $SV(f)$ is a bounded set, then $f$ is said to be of
bounded type. The sets

$\mathscr{S}=\\{f:f\;\ \textrm{is of finite
type}\\}\;\;\text{and}\;\;\mathscr{B}=\\{f:f\;\ \textrm{is of bounded
type}\\}$

are respectively called Speiser class and Eremenko-Lyubich class. The post-
singular point is the point on the orbit of singular value. That is, if $z$ is
a singular value of entire function $f$, then $f^{n}(z)$ is a post-singular
point for $n\geq 0$. The set of all post-singular points is called post-
singular set and it is denoted by

$P(f)=\bigcup_{n\geq 0}f^{n}(SV(f))$

The entire function $f$ is called post-singularly bounded if its post-singular
set is bounded and it is called post-singularly finite if its post-singular
set is finite. A transcendental entire function $f$ is hyperbolic if the post
singular set $P(f)$ is compact subset of the Fatou set $F(f)$.

The main concern of this paper is to study of escaping set under
transcendental semigroup. So we start our formal study from the notion of
transcendental semigroup. Note that for given complex plane $\mathbb{C}$, the
set $\text{Hol}(\mathbb{C})$ denotes a set of all holomorphic functions of
$\mathbb{C}$. If $f\in\text{Hol}(\mathbb{C})$, then $f$ is a polynomial or
transcendental entire function. The set $\text{Hol}(\mathbb{C})$ forms a
semigroup with semigroup operation being the functional composition.

######  Definition 1.1 (Transcendental semigroup).

Let $A=\\{f_{i}:i\in\mathbb{N}\\}\subset\text{Hol}(\mathbb{C})$ be a set of
transcendental entire functions $f_{i}:\mathbb{C}\rightarrow\mathbb{C}$. A
transcendental semigroup $S$ is a semigroup generated by the set $A$ with
semigroup operation being the functional composition. We denote this semigroup
by $S=\langle f_{1},f_{2},f_{3},\cdots,f_{n},\cdots\rangle$.

Here, each $f\in S$ is the transcendental entire function and $S$ is closed
under functional composition. Thus $f\in S$ is constructed through the
composition of finite number of functions $f_{i_{k}},\;(k=1,2,3,\ldots,m)$.
That is, $f=f_{i_{1}}\circ f_{i_{2}}\circ f_{i_{3}}\circ\cdots\circ
f_{i_{m}}$.

A semigroup generated by finitely many functions $f_{i},(i=1,2,3,\ldots,n)$ is
called finitely generated transcendental semigroup. We write $S=\langle
f_{1},f_{2},\ldots,f_{n}\rangle$. If $S$ is generated by only one
transcendental entire function $f$, then $S$ is cyclic or trivial
transcendental semigroup. We write $S=\langle f\rangle$. In this case each
$g\in S$ can be written as $g=f^{n}$, where $f^{n}$ is the nth iterates of $f$
with itself. The transcendental semigroup $S$ is abelian if $f_{i}\circ
f_{j}=f_{j}\circ f_{i}$ for all generators $f_{i}$ and $f_{j}$ of $S$.

Based on the Fatou-Julia-Eremenko theory of a complex analytic function, the
Fatou set, Julia set and escaping set in the settings of semigroup are defined
as follows.

######  Definition 1.2 (Fatou set, Julia set and escaping set).

The set of normality or the Fatou set of the transcendental semigroup $S$ is
defined by

$F(S)=\\{z\in\mathbb{C}:S\;\ \textrm{is normal in a neighborhood of}\;\ z\\}$

The Julia set of $S$ is defined by $J(S)=\mathbb{C}\setminus F(S)$ and the
escaping set of $S$ by

$I(S)=\\{z\in\mathbb{C}:\;f^{n}(z)\rightarrow\infty\;\ \textrm{as}\;\
n\rightarrow\infty\;\ \textrm{for all}\;\ f\in S\\}$

We call each point of the set $I(S)$ by escaping point.

It is obvious that $F(S)$ is the largest open subset of $\mathbb{C}$ such that
semigroup $S$ is normal. Hence its compliment $J(S)$ is a smallest closed set
for any transcendental semigroup $S$. Whereas the escaping set $I(S)$ is
neither an open nor a closed set (if it is non-empty) for any semigroup $S$.

If $S=\langle f\rangle$, then $F(S),J(S)$ and $I(S)$ are respectively the
Fatou set, Julia set and escaping set in classical iteration theory of complex
dynamics. In this situation we simply write: $F(f),J(f)$ and $I(f)$. For the
existing results of Fatou Julia theory under transcendental semigroup, we
refer [4, 5, 6, 8, 9].

##  2\. Some Fundamental Features of Escaping Set

The following immediate relation between $I(S)$ and $I(f)$ for any $f\in S$
will be clear from the definition of escaping set.

######  Theorem 2.1.

$I(S)\subset I(f)$ for all $f\in S$ and hence $I(S)\subset\bigcap_{f\in
S}I(f)$.

###### Proof.

Let $z\in I(S)$, then $f^{n}(z)\rightarrow\infty$ as $n\rightarrow\infty$ for
all $f\in S$. By which we mean $z\in I(f)$ for any $f\in S$. This immediately
follows the second inclusion. ∎

Note that the above same type of relation (Theorem 2.1) holds between $F(S)$
and $F(f)$. However opposite relation holds between the sets $J(S)$ and
$J(f)$. Poon [9, Theorem 4.1, Theorem 4.2] proved that the Julia set $J(S)$ is
perfect and $J(S)=\overline{\bigcup_{f\in S}J(f)}$ for any transcendental
semigroup $S$. From the last relation of above theorem 2.1, we can say that
the escaping set may be empty. Note that $I(f)\not=\emptyset$ in classical
iteration theory [2]. Dinesh Kumar and Sanjay Kumar [5, Theorem 2.5] have
mentioned the following transcendental semigroup $S$, where $I(S)$ is an empty
set.

######  Theorem 2.2.

The transcendental entire semigroup $S=\langle f_{1},\;f_{2}\rangle$ generated
by two functions $f_{1}$ and $f_{2}$ from respectively two parameter families
$\\{e^{-z+\gamma}+c\;\text{where}\;\gamma,c\in\mathbb{C}\;\text{and}\;Re(\gamma)<0,\;Re(c)\geq
1\\}$ and
$\\{e^{z+\mu}+d,\;\text{where}\;\mu,d\in\mathbb{C}\;\text{and}\;Re(\mu)<0,\;Re(d)\leq-1\\}$
of functions has empty escaping set $I(S)$.

In the case of non-empty escaping set $I(S)$, Eremenko’s result [2], $\partial
I(f)=J(f)$ of classical transcendental dynamics can be generalized to
semigroup settings. The following results is due to Dinesh Kumar and Sanjay
Kumar [5, Lemma 4.2 and Theorem 4.3] which yield the generalized answer in
semigroup settings.

######  Theorem 2.3.

Let $S$ be a transcendental entire semigroup such that $I(S)\neq\emptyset$.
Then

  1. (1)

$int(I(S))\subset F(S)\;\ \text{and}\;\ ext(I(S))\subset F(S)$, where $int$
and $ext$ respectively denote the interior and exterior of $I(S)$.

  2. (2)

$\partial I(S)=J(S)$, where $\partial I(S)$ denotes the boundary of $I(S)$.

This last statement is equivalent to $J(S)\subset\overline{I(S)}$. If
$I(S)\neq\emptyset$, then we [11, Theorem 4.6] proved the following result
which is a generalization of Eremenko’s result $I(f)\cap J(f)\neq\emptyset$
[2, Theorem 2] of classical transcendental dynamics to holomorphic semigroup
dynamics.

######  Theorem 2.4.

Let $S$ be a transcendental semigroup such that $F(S)$ has a multiply
connected component. Then $I(S)\cap J(S)\neq\emptyset$

Eremenko and Lyubich [3] proved that if transcendental function
$f\in\mathscr{B}$, then $I(f)\subset J(f)$, and $J(f)=\overline{I(f)}$. Dinesh
Kumar and Sanjay Kumar [5, Theorem 4.5] generalized these results to a
finitely generated transcendental semigroup of bounded type as shown below.

######  Theorem 2.5.

For every finitely generated transcendental semigroup $S=\langle
f_{1},f_{2},\ldots,f_{n}\rangle$ in which each generator $f_{i}$ is of bounded
type, then $I(S)\subset J(S)$ and $J(S)=\overline{I(S)}$.

###### Proof.

Eremenko and Lyubich’s result [3] shows that $I(f)\subset J(f)$ for each $f\in
S$ of bounded type. Poon’s result shows [9, Theorem 4.2] that
$J(S)=\overline{\bigcup_{f\in S}J(f)}$. Therefore, (from the definition of
escaping set and theorem 2.1) for every $f\in S,\;I(S)\subset I(f)\subset
J(f)\subset J(S)$.

The next part follows from the facts $J(S)\subset\overline{I(S)}$ and
$I(S)\subset J(S)$. ∎

##  3\. Escaping set of Hyperbolic Semigroup

The definitions of critical values, asymptotic values and singular values as
well as post singularities of transcendental entire functions can be
generalized to arbitrary setting of transcendental semigroups.

######  Definition 3.1 (Critical point, critical value, asymptotic value and
singular value).

A point $z\in\mathbb{C}$ is called critical point of $S$ if it is critical
point of some $g\in S$. A point $w\in\mathbb{C}$ is called a critical value of
$S$ if it is a critical value of some $g\in S$. A point $w\in\mathbb{C}$ is
called an asymptotic value of $S$ if it is an asymptotic value of some $g\in
S$. A point $w\in\mathbb{C}$ is called a singular value of $S$ if it is a
singular value of some $g\in S$. For a semigroup $S$, if all $g\in S$ belongs
to $\mathscr{S}$ or $\mathscr{B}$, we call $S$ a semigroup of class
$\mathscr{S}$ or $\mathscr{B}$ (or finite or bounded type).

######  Definition 3.2 (Post singularly bounded (or finite) semigroup).

A transcendental semigroup $S$ is said to be post-singularly bounded (or post-
singularly finite) if each $g\in S$ is post-singularly bounded (or post-
singularly finite). Post singular set of post singularly bounded semi-group
$S$ is the set of the form

$P(S)=\overline{\bigcup_{f\in S}f^{n}(SV(f))}$

######  Definition 3.3 (Hyperbolic semigroup).

An transcendental entire function $f$ is said to be hyperbolic if the post-
singular set $P(f)$ is a compact subset of $F(f)$. A transcendental semigroup
$S$ is said to be hyperbolic if each $g\in S$ is hyperbolic (that is, if
$P(S)$ is a compact subset of $F(S)$).

Note that if transcendental semigroup $S$ is hyperbolic, then each $f\in S$ is
hyperbolic. However, the converse may not true. The fact $P(f^{k})=P(f)$ for
all $k\in\mathbb{N}$ shows that $f^{k}$ is hyperbolic if $f$ is hyperbolic.
The following result has been shown by Dinesh Kumar and Sanjay Kumar [5,
Theorem 3.16] where Eremenko’s conjecture holds.

######  Theorem 3.1.

Let $f\in\mathscr{B}$ periodic with period p and hyperbolic. Let
$g=f^{n}+p,\;n\in\mathbb{N}$. Then $S=\langle f,g\rangle$ is hyperbolic and
all components of $I(S)$ are unbounded.

######  Example 3.1.

$f(z)=e^{\lambda z}$ is hyperbolic entire function for each
$\lambda\in(0,\frac{1}{e})$. The semigroup $S=\langle f,g\rangle$ where
$g=f^{m}+p$, where $p=\frac{2\pi i}{\lambda}$, is hyperbolic transcendental
semigroup.

We have generalized the above theorem 3.1 to finitely generated hyperbolic
semigroup with some modifications. This theorem will be the good source of
non-empty escaping set transcendental semigroup from which the Eremenko’s
conjecture holds.

######  Theorem 3.2.

Let $S=\langle f_{1},f_{2},\ldots,f_{n}\rangle$ is an abelian bounded type
transcendental semigroup in which each $f_{i}$ is hyperbolic for
$i=1,2,\ldots,n$. Then semigroup $S$ is hyperbolic and all components of
$I(S)$ are unbounded.

######  Lemma 3.1.

Let $f$ and $g$ be transcendental entire functions. Then $SV(f\circ g)\subset
SV(f)\cup f(SV(g))$.

###### Proof.

See for instance [1, Lemma 2]. ∎

######  Lemma 3.2.

Let $f$ and $g$ are permutable transcendental entire functions. Then
$f^{m}(SV(g))\subset SV(g)$ and $g^{m}(SV(f))\subset SV(f)$ for all
$m\in\mathbb{N}$.

###### Proof.

We first prove that $f(SV(g))\subset SV(g)$. Then we use induction to prove
$f^{m}(SV(g))\subset SV(g)$.

Let $w\in f(SV(g))$. Then $w=f(z)$ for some $z\in SV(g)$. In this case, $z$ is
either critical value or an asymptotic value of function $g$.

First suppose that $z$ is a critical value of $g$. Then $z=g(u)$ with
$g^{{}^{\prime}}(u)=0$. Since $f$ and $g$ are permutable functions, so
$w=f(z)=f(g(u))=(f\circ g)(u)=(g\circ f)(u)$. Also, $(f\circ
g)^{{}^{\prime}}(u)=f^{{}^{\prime}}(g(u))g^{{}^{\prime}}(u)=0$. This shows
that $u$ is a critical point of $f\circ g=g\circ f$ and $w$ is a critical
value of $f\circ g=g\circ f$. By permutability of $f$ and $g$, we can write
$f^{{}^{\prime}}(g(u))g^{{}^{\prime}}(u)=g^{{}^{\prime}}(f(u))f^{{}^{\prime}}(u)=0$
for any critical point $u$ of $f\circ g$. Since $g^{{}^{\prime}}(u)=0$, then
either $f^{{}^{\prime}}(u)=0\Rightarrow u$ is a critical point of $f$ or
$g^{{}^{\prime}}(f(u))=0\Rightarrow f(u)$ is a critical point of $g$. This
shows that $w=g(f(u))$ is a critical value of $g$. Therefore, $w\in SV(g)$.

Next, suppose that $z$ is an asymptotic value of function $g$. We have to
prove that $w=f(z)$ is also asymptotic value of $g$. Then there exists a curve
$\gamma:[0,\infty)\to\mathbb{C}$ such that $\gamma(t)\to\infty$ and
$g(\gamma(t))\to z$. So, $f(g(\gamma(t)))\to f(z)=w$ as $t\to\infty$ along
$\gamma$. Since $f\circ g=g\circ f$, so $f(g(\gamma(t)))\to f(z)=w\Rightarrow
g(f(\gamma(t)))\to f(z)=w$ as $t\to\infty$ along $\gamma$. This shows $w$ is
an asymptotic value of $g$. This proves our assertion.

Assume that $f^{k}(SV(g))\subset SV(g)$ for some $k\in\mathbb{N}$ with $k\leq
m$. Then

$f^{k+1}(SV(g))=f(f^{k}(SV(g)))\subset f(SV(g))\subset SV(g)$

Therefore, by induction, for all $m\in\mathbb{N}$, we must have
$f^{m}(SV(g))\subset SV(g)$. The next part $g^{m}(SV(f))\subset SV(f)$ can be
proved similarly as above.

∎

######  Lemma 3.3.

Let $f$ and $g$ are two permutable hyperbolic transcendental entire functions.
Then their composite $f\circ g$ is also hyperbolic.

###### Proof.

We have to prove that $P(f\circ g)$ is a compact subset of Fatou set $F(f\circ
g)$. From [7, Lemma 3.2], $F(f\circ g)\subset F(f)\cap F(g)$. This shows that
$F(f\circ g)$ is a subset of $F(f)$ and $F(g)$. So this lemma will be proved
if we prove $P(f\circ g)$ is a compact subset of $F(f)\cup F(g)$. By the
definition of post singular set of transcendental entire function, we can
write

$\displaystyle P(f\circ g)$ $\displaystyle=\overline{\bigcup_{m\geq 0}(f\circ
g)^{m}(SV(f\circ g))}$ $\displaystyle=\overline{\bigcup_{m\geq
0}f^{m}(g^{m}(SV(f\circ g)))}\;\;\;\;\;\;\;\;\;\;\;\;\;\;(\text{by using
permutabilty of $f$ and $g$})$ $\displaystyle\subset\overline{\bigcup_{m\geq
0}f^{m}(g^{m}(SV(f)\cup f(SV(g)))}\;\;\;\;\;\;\;\;\;\;(\text{by above lemma
\ref{hs2}})$ $\displaystyle=\overline{\bigcup_{m\geq 0}f^{m}(g^{m}(SV(f)))\cup
g^{m}(f^{m+1}(SV(g)))}$ $\displaystyle\subset\overline{\bigcup_{m\geq
0}f^{m}(SV(f)))}\cup\overline{\bigcup_{m\geq 0}g^{m}(SV(g)))}\;\;\;\;(\text{by
above lemma \ref{hs3}})$ $\displaystyle=P(f)\cup P(g)$

Since $f$ and $g$ are hyperbolic, so $P(f)$ and $P(g)$ are compact subset of
$F(f)$ and $F(g)$. Therefore, the set $P(f)\cup P(g)$ must be compact subset
of $F(f)\cup F(g)$. ∎

###### Proof of the Theorem 3.2.

Any $f\in S$ can be written as $f=f_{i_{1}}\circ f_{i_{2}}\circ
f_{i_{3}}\circ\cdots\circ f_{i_{m}}$. By permutability of each $f_{i}$, we can
rearrange $f_{i_{j}}$ and ultimately represented by

$f=f_{1}^{t_{1}}\circ f_{2}^{t_{2}}\circ\ldots\circ f_{n}^{t_{n}}$

where each $t_{k}\geq 0$ is an integer for $k=1,2,\ldots,n$. The lemma 3.3 can
be applied repeatably to show each of
$f_{1}^{t_{1}},f_{2}^{t_{2}},\ldots,f_{n}^{t_{n}}$ is hyperbolic. Again by
repeated application of above same lemma, we can say that
$f=f_{1}^{t_{1}}\circ f_{2}^{t_{2}}\circ\ldots\circ f_{n}^{t_{n}}$ is itself
hyperbolic and so the semigroup $S$ is hyperbolic. Next part follows from [12,
Theorem 3.3] by the assumption of this theorem. ∎

Acknowledgment: We express our heart full thanks to Prof. Shunshuke Morosawa,
Kochi University, Japan for his thorough reading of this paper with valuable
suggestions and comments.

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