# A warped scalar portal to fermionic dark matter

Adrian Carmona<EMAIL_ADDRESS>CAFPE and Departamento de Física Teórica y del
Cosmos, Universidad de Granada, E-18071 Granada, Spain  Javier Castellano
<EMAIL_ADDRESS>PRISMA+ Cluster of Excellence & Mainz Institute for
Theoretical Physics, Johannes Gutenberg University, 55099 Mainz, Germany
Matthias Neubert<EMAIL_ADDRESS>PRISMA+ Cluster of Excellence
& Mainz Institute for Theoretical Physics, Johannes Gutenberg University,
55099 Mainz, Germany
Department of Physics & LEPP, Cornell University, Ithaca, NY 14853, U.S.A.
Department of Physics, Universität Zürich, Winterthurerstrasse 190, CH-8057
Zürich, Switzerland

###### Abstract

We argue that extensions of the SM with a warped extra dimension, together
with a new $\mathbb{Z}_{2}$-odd scalar singlet, provide a natural explanation
not only for the hierarchy problem but also for the nature of fermion bulk
masses and the observed dark matter relic abundance. In particular, the
Kaluza-Klein excitations of the new scalar particle, which is required to
naturally obtain fermion bulk masses through Yukawa-like interactions, can be
the leading portal to any fermion propagating into the bulk of the extra
dimension and playing the role of dark matter. Moreover, such scalar
excitations will necessarily mix with the Higgs boson, leading to
modifications of the Higgs couplings and branching ratios, and allowing the
Higgs to mediate the coannihilation of the fermionic dark matter. We study
these effects and explore the viability of fermionic dark matter in the
presence of these new heavy scalar mediators both in the usual freeze-out
scenario and in the case where the freeze-out happens during an early period
of matter domination.

††preprint: MITP/20-069

##  I Introduction

The discovery of the Higgs boson at the LHC represented the last step towards
establishing the Standard Model (SM) as a solid theory describing the
constituents of matter and their interactions down to very short distances.
However, there are still some questions which do not have an answer within the
SM. One of the most significant examples is the so-called hierarchy problem,
the question why the Higgs boson is much lighter than the characteristic scale
of gravity. One could argue that this problem is merely a by-product of our
theoretical prejudices and that nature did not ask for a dynamical explanation
for this difference of scales. Still, even in this case, we know for a fact
that the SM cannot accommodate some other observed phenomena. One of the most
striking examples is the existence of dark matter (DM). We know that there is
no viable DM candidate in the SM, so already this fact asks for the presence
of new physics. Extensions of the SM with a warped extra dimension (WED)
compactified on an $S_{1}/\mathbb{Z}_{2}$ orbifold contain the necessary
features for addressing simultaneously both of these issues, the absence of a
viable DM candidate and the hierarchy problem. Moreover, they can also explain
the large hierarchy existing between the different fermions masses, providing
a calculable version of partial compositeness Maldacena:1997re ; Gubser:1998bc
; ArkaniHamed:2000ds ; Rattazzi:2000hs ; PerezVictoria:2001pa ; Contino:2004vy
; Gherghetta:2010cj , which makes them very attractive extensions of the SM.

Fermion masses in WED compactified on a $S_{1}/\mathbb{Z}_{2}$ orbifold need
to have a dynamical origin, since the five-dimensional (5D) bulk masses must
be $\mathbb{Z}_{2}$-odd functions on the orbifold Grossman:1999ra ;
Gherghetta:2000qt . Indeed, one can easily see that the 5D Dirac fermion
bilinear $\bar{\Psi}_{i}\Psi_{i}$ is odd under the orbifold symmetry,
excluding the presence of a constant mass term. The most natural solution to
this problem is to dynamically generate these masses with the help of a
$\mathbb{Z}_{2}$-odd bulk scalar field. Indeed, if such scalar field develops
a vacuum expectation value (VEV) with a non-trivial profile along the extra
dimension, fermion bulk masses can arise dynamically through Yukawa-like
interactions. We have explored this possibility in detail in Ahmed:2019zxm ,
studying in particular its phenomenological consequences. In particular, one
finds that the VEV has a kink-like profile, approaching the traditional sign
function for large values of the odd scalar mass, whenever the WED is
significantly larger than its inverse curvature.

A natural question which arises in scenarios addressing the origin of the
fermion bulk masses concerns the possible interplay with a bulk Higgs boson.
In Ahmed:2019zxm  we have considered a brane-localized Higgs field, which does
not mix at tree-level with the $\mathbb{Z}_{2}$-odd scalar field. However, in
the more general case of a Higgs boson in the bulk of the extra dimension
Davoudiasl:2005uu ; Cacciapaglia:2006mz ; Azatov:2009na ; Vecchi:2010em ;
Archer:2012qa ; Malm:2013jia ; Archer:2014jca , such a mixing is unavoidable
and needs to be taken into account. Studying the effect of this mixing is one
of the main goals of this work. On the other hand, since the odd scalar field
is responsible for all fermion bulk masses, it represents a unique window into
any femionic dark sector propagating into the bulk of the WED. Models with
WEDs already feature an irreducible mediator between visible and dark sectors,
since gravity couples to matter through the energy-momentum tensor. However,
as we will see, when the DM candidates are fermionic weakly interacting
particles (WIMPs) with masses of $\mathcal{O}$(TeV), the resonances arising
from the 5D $\mathbb{Z}_{2}$-odd scalar field can provide the most important
mediators for the DM coannihilation cross section. Moreover, due to the mixing
with the bulk Higgs field, these fermionic dark sectors are mostly Higgs-
mediated for DM masses below the TeV scale. We examine thoroughly the
resulting model of scalar-mediated fermionic DM for a large range of DM
masses. We focus on the case where the DM particle is a vector-like (VL)
fermion, the first Kaluza-Klein (KK) excitation of a 5D dark fermion. However,
most of our results also hold in the case where the DM candidate gets an
external mass, which can be chiral, VL or even of Majorana type.

This work is organized as follows: In order to set up the notation, we review
the bulk Higgs case (disregarding the portal coupling) in section II. In
section III we solve the coupled system of field equations obtained after
switching on the portal coupling between both types of bulk scalar fields by
diagonalizing the resulting 4D mass matrix perturbatively. In section IV we
proceed to discuss the phenomenology assuming a non-negligible portal coupling
and the presence of $N_{\chi}$ dark fermion bulk fields. First, we discuss the
impact of the scalar mixing on the SM Higgs couplings. We then continue by
examining the impact of the dark fermions on the Higgs invisible decay width.
Then, we discuss the predictions for the DM coannihilation cross-section
mediated by the Higgs field and the first KK resonance of the
$\mathbb{Z}_{2}$-odd scalar field, comparing these contributions with the ones
mediated by KK gravitons. We compute the prediction for the DM relic abundance
as a function of the velocity-averaged coannihilation cross section in the
usual freeze-out scenario as well as in the case of a matter-dominated
universe Gondolo:1990dk ; Hamdan:2017psw ; Chanda:2019xyl . Finally, we
compute the constraints arising from direct detection using data from the
Xenon1T experiment, showing that for a $\mathcal{O}(10~{}$TeV) fermionic WIMP
we can reproduce the observed DM relic abundance in the scenario of matter
domination, without conflicting with current data from Xenon1T. In the case of
radiation domination and DM masses of $\sim 15$ TeV, these scalar mediators
can provide a non-negligible fraction of the required coannihilation cross
section, even though additional mediators would be required.

##  II A Bulk Higgs in a WED

We consider a Randall-Sundrum (RS) model Randall:1999ee  with the extra
dimension compactified on an $S_{1}/\mathbb{Z}_{2}$ orbifold with two
D3-branes localized at the fixed points, an ultraviolet (UV) brane at
$t_{\text{UV}}=\epsilon$ and an infrared (IR) brane at $t_{\text{IR}}=1$,
where $t$ is the coordinate describing the extra dimension. This coordinate is
defined in terms of the usual $\phi=y/\pi$ coordinate by
$t=\epsilon\,e^{kr\phi}$, where
$\epsilon=e^{-kr\pi}\sim\mathcal{O}(10^{-16})$. In this notation, the metric
reads

$ds^{2}=g_{MN}dx^{M}dx^{N}=\dfrac{\epsilon^{2}}{t^{2}}\left(\eta_{\mu\nu}dx^{\mu}dx^{\nu}-\dfrac{dt^{2}}{M_{\text{KK}}^{2}}\right),$
(1)

where $M_{\text{KK}}\equiv k\epsilon$ and $\eta_{\mu\nu}={\rm
diag}(1,-1,-1,-1)$ is the 4D Minkowski metric. It is useful to define the
quantity $L=kr\pi\sim 30$, which is a measure of the size of the extra
dimension in natural units. Here $k$ and $r$ are the AdS curvature and the
radius of the $S_{1}$. Note that in RS models addressing the hierarchy problem
$L\gg 1$.

Let us start by reviewing the well-known case where the Higgs field does not
mix at tree-level with the odd bulk scalar Archer:2014jca ; Mahmoudi:2016aib .
For simplicity, a quartic term is only introduced on the IR brane in order to
induce electroweak symmetry breaking (EWSB). The Higgs action reads

$\displaystyle S=\int d^{5}x\sqrt{g}\bigg{\\{}$ $\displaystyle
g^{M\\!N}\left(D_{M}H\right)^{\dagger}D_{N}H-V(H)$ (2)
$\displaystyle-\sum_{k=\text{UV,IR}}\frac{\sqrt{|\hat{g}_{k}|}}{\sqrt{g}}\hat{V}^{k}(H)\,\delta(t-t_{k})\bigg{\\}},$

where $g=\det(g_{MN})$, $\hat{g}_{k}=\det(g_{\mu\nu}|_{t=t_{k}})$, and we
define the Higgs doublet in the unitary gauge as

$H(x,t)=(0,\dfrac{t}{\epsilon\sqrt{2r}}\left[\varphi_{H}(t)+h(x,t)\right])^{T}.$
(3)

For the Higgs field and its VEV we follow the treatment of Malm:2013jia . The
bulk and brane-localized potentials for the bulk Higgs field are taken to be
of the form

$\displaystyle V(H)=\mu^{2}_{H}\absolutevalue{H}^{2},$ (4)
$\displaystyle\hat{V}^{\text{UV}}=\sigma_{\text{UV}}\absolutevalue{H}^{2},$
$\displaystyle\hat{V}^{\text{IR}}=-\sigma_{\text{IR}}\absolutevalue{H}^{2}+\rho_{\text{IR}}\absolutevalue{H}^{4}.$

The equation of motion (EOM) for the Higgs VEV is

$\displaystyle\left[t^{2}\partial_{t}^{2}+t\partial_{t}-\beta^{2}\right]\dfrac{\varphi_{H}}{t}=0,\qquad\beta^{2}\equiv
4+\mu_{H}^{2}/k^{2},$ (5)

along with boundary conditions (BC) on the UV and IR branes, which read

$\displaystyle\partial_{t}\left[t\,\varphi_{H}(t)\right]_{t=\epsilon^{+}}=m_{\text{UV}}\,\varphi_{H}(\epsilon),$
(6)
$\displaystyle\partial_{t}\left[t\,\varphi_{H}(t)\right]_{t=1^{-}}=m_{\text{IR}}\,\varphi_{H}(1)-\dfrac{2\lambda_{\text{IR}}}{M_{\text{KK}}^{2}}\varphi_{H}(1)^{3}.$

The notation $\epsilon^{+}$ and $1^{-}$ refers to the orbifold fixed points,
approached from the appropriate side. Above we have defined

$m_{\text{UV}}=\dfrac{\sigma_{\text{UV}}}{2k},\quad
m_{\text{IR}}=\dfrac{\sigma_{\text{IR}}}{2k},\quad\lambda_{\text{IR}}=\dfrac{\rho_{\text{IR}}k}{4r}.$
(7)

This set of EOM and BCs leads to the well-known solution

$\varphi_{H}(t)=N_{v}\left[t^{1+\beta}-r_{v}t^{1-\beta}\right],$ (8)

where

$\displaystyle r_{v}=\epsilon^{2\beta}\frac{2+\beta-m_{\text{UV}}}{2-\beta-
m_{\text{UV}}}\,\,,$ (9) $\displaystyle
N_{v}^{2}=\frac{M_{\text{KK}}^{2}}{2\lambda_{\text{IR}}}\,\frac{\left(m_{\text{IR}}-2-\beta\right)+r_{v}\left(m_{\text{IR}}-2+\beta\right)}{(1-r_{v})^{3}}\,.$

Note that, unless $\beta$ is very small or $m_{\text{UV}}$ is extremely fine-
tuned to the value $2-\beta$, it is safe to set
$r_{v}\propto\epsilon^{2\beta}\to 0$. Then, the Higgs VEV will be peaked
towards the IR brane and expression (8) simplifies to

$\varphi_{H}(t)\simeq N_{v}t^{1+\beta}=\varphi_{H}(1)\,t^{1+\beta},$ (10)

with

$N_{v}\equiv\varphi_{H}(1)\simeq
M_{\text{KK}}\sqrt{\dfrac{m_{\text{IR}}-2-\beta}{2\lambda_{\text{IR}}}}.$ (11)

Demanding the normalization for the VEV to be such that one correctly
reproduces the SM mass relations for the $W$ and $Z$ bosons leads to

$v_{4}^{2}=\dfrac{2\pi}{L}\int^{1}_{\epsilon}\dfrac{dt}{t}\varphi_{H}^{2}(t)=\dfrac{\pi}{L}\dfrac{\varphi^{2}_{H}(1)}{1+\beta}+\mathcal{O}(\epsilon),$
(12)

where we have used the fact that the zero-mode profiles of the gauge bosons
are flat, up to corrections of $\mathcal{O}(v_{4}^{2}/M_{\text{KK}}^{2})$,
Malm:2013jia . We can then write the VEV of the Higgs field as

$\varphi_{H}(t)\approx v_{4}\sqrt{\dfrac{L}{\pi}(1+\beta)}\,t^{1+\beta},$ (13)

where $v_{4}$ agrees with the SM parameter $v_{\rm SM}$ at leading order in
$x_{4}\equiv v_{4}/M_{\text{KK}}$.

##  III Bulk Higgs and Odd scalar mixing

We are now interested in the case where the two bulk scalar fields – the Higgs
and the new $\mathbb{Z}_{2}$-odd scalar – mix with each other. In this case
the action reads

$\displaystyle S=\int d^{5}x\sqrt{g}\,\bigg{\\{}$ $\displaystyle
g^{M\\!N}\sum_{i=1,2}\left(D_{M}\Phi_{i}\right)^{\dagger}D_{N}\Phi_{i}-V(\Phi_{1},\Phi_{2})$
(14)
$\displaystyle-\sum_{k=\text{UV,IR}}\frac{\sqrt{|\hat{g}_{k}|}}{\sqrt{g}}\hat{V}^{k}(\Phi_{1},\Phi_{2})\,\delta(t-t_{k})\bigg{\\}},$

with $\Phi_{1}=H$ and $\Phi_{2}=(1/\sqrt{2})\,\Sigma$ denoting the Higgs
doublet and the (real) odd scalar field, which is a gauge singlet. We consider
mixed BCs for the Higgs field, while the odd scalar field satisfies Dirichlet
BCs. Such BCs for the bulk Higgs are a consequence of the brane-localized
potentials, which are forbidden for the odd scalar, since it vanishes on the
two branes. In our model, both bulk scalar fields develop a VEV. We can
express the 5D odd scalar in terms of its background configuration
$\varphi_{S}(t)$ and its 5D excitation $S(x,t)$ as

$\Sigma(x,t)=\varphi_{S}(t)+\dfrac{t}{\epsilon\sqrt{r}}\,S(x,t).$ (15)

The bulk potential reads

$V(H,\Sigma)=\mu^{2}_{H}\absolutevalue{H}^{2}-\frac{\mu_{S}^{2}}{2}\Sigma^{2}+\frac{\lambda_{S}}{4}\Sigma^{4}+\lambda_{HS}\absolutevalue{H}^{2}\Sigma^{2},$
(16)

where $\mu_{H}$, $\mu_{S}$ are the mass parameters and $\lambda_{S}$,
$\lambda_{HS}$ the quartic couplings. The brane-localized potentials for the
bulk Higgs field are the same as those shown in (4).

###  III.1 Background solutions

First, we determine the profiles of the two VEVs. From (16), the EOMs are

$\displaystyle\Bigg{[}t^{2}\partial_{t}^{2}-3t\partial_{t}+\frac{\mu_{S}^{2}}{k^{2}}\Big{(}1-v_{S}^{2}-\bar{\lambda}\dfrac{k^{4}}{\mu_{S}^{4}}\dfrac{\lambda_{S}}{r}t^{2}\dfrac{\varphi_{H}^{2}}{M_{\text{KK}}^{2}}\Big{)}\Bigg{]}v_{S}(t)$
$\displaystyle=0,$ (17)
$\displaystyle\left[t^{2}\partial_{t}^{2}+t\partial_{t}-\beta^{2}-\bar{\lambda}v_{S}^{2}\right]\dfrac{\varphi_{H}(t)}{t}$
$\displaystyle=0,$

where we have defined the dimensionless coupling

$\bar{\lambda}\equiv\dfrac{\mu_{S}^{2}}{k^{2}}\dfrac{\lambda_{HS}}{\lambda_{S}}$
(18)

and redefined the VEV of the odd field as in Ahmed:2019zxm , i.e.

$\varphi_{S}(t)=\dfrac{\mu_{S}}{\sqrt{\lambda_{S}}}\,v_{S}(t).$ (19)

In order to obtain an inverted one-dimensional Mexican-hat potential for
$v_{S}$ and guarantee the existence of non-trivial solutions, we impose an
upper bound on $\bar{\lambda}\lambda_{S}/r$. In practice, we demand that

$\bar{\lambda}\dfrac{k^{4}}{\mu_{S}^{4}}\dfrac{\lambda_{S}}{r}t^{2}\dfrac{\varphi_{H}^{2}(t)}{M_{\text{KK}}^{2}}\Bigg{|}_{t=1}\leq
1.$ (20)

Since the Higgs VEV is monotonic in $t$ (at least at leading order in
$\bar{\lambda}$), once this condition is fulfilled it will also hold for
$t<1$. Plugging in the solution for the Higgs VEV for $\bar{\lambda}=0$ (i.e.
for vanishing portal coupling $\lambda_{HS}$), this constraint translates into

$\bar{\lambda}\dfrac{\lambda_{S}}{r}\lesssim\dfrac{x_{4}^{-2}}{10(1+\beta)}\left(\dfrac{\mu_{S}}{k}\right)^{4},$
(21)

where $x_{4}=v_{4}/M_{\text{KK}}$.

We can solve the coupled system of equations iteratively. The starting point
are the solutions we already know for the decoupled equations – $v_{S,0}(t)$
and $\varphi_{H,0}(t)$. We thus insert the solution from the previous
iteration for each VEV in the EOM corresponding to the other one in (17). This
method is convenient, since the potential for the odd VEV and the strategy to
solve its EOM is well understood (see Ahmed:2019zxm ). Indeed, as we can see
from figure 1, the potential does not differ much from the potential obtained
in the decoupled case (zero portal coupling). We have checked numerically that
the solution obtained for $v_{S}(t)$ fits the one obtained for the decoupled
case with high accuracy.

Figure 1: Effective potential $\tilde{V}(v_{S})$ computed using the leading-
order solution $\varphi_{H,0}(t)$, for different values of $t$ and
$M_{\text{KK}}=5\,$TeV, $k/m_{\rm Pl}=1/8$ and
$\bar{\lambda}\lambda_{S}/r=100$.

We display in figure 1 the one-dimensional effective potential
$\tilde{V}(v_{S})$ defined by

$\displaystyle\frac{\delta\tilde{V}(v_{S})}{\delta
v_{S}}=\Big{(}1-v_{S}^{2}-\bar{\lambda}\dfrac{k^{4}}{\mu_{S}^{4}}\dfrac{\lambda_{S}}{r}t^{2}\dfrac{\varphi_{H}^{2}(t)}{M_{\text{KK}}^{2}}\Big{)}v_{S},$
(22)

which determines the EOM for $v_{S}$ once $\varphi_{H}\approx\varphi_{H,0}$ is
used, see (17). We show this potential for $\bar{\lambda}\lambda_{S}/r=100$
and different values of $t$, $\mu_{S}$ and $\beta$, in order to explore how
its maxima change with increasing $t$ and different values of $\mu_{S}$ and
$\beta$. Note that in the figure we use the leading-order solution for the
Higgs profile. Here and below we choose $M_{\rm KK}=5\,$TeV, which is
motivated so as to avoid tensions with electroweak precision data (see
Ahmed:2019zxm  for more details). Similarly, we take $k=m_{\rm Pl}/8$ where
$m_{\rm Pl}=2.4\cdot 10^{18}$ GeV is the reduced Planck mass. This value
corresponds to $kr\approx 10.1$, or equivalently $\Lambda_{\pi}\equiv m_{\rm
Pl}\,e^{-kr\pi}=40\,$TeV. We observe that both maxima decrease for increasing
$t$. These two maxima would eventually collapse into the maximum of an
inverted parabola, if values of $\bar{\lambda}\lambda_{S}/r$ larger than the
ones given by equation (21) were considered. In that case, only the trivial
solution would exist. In practice, we will never reach such large values due
to perturbativity constraints on the first KK excitation of $S$, since the
Yukawa couplings of this particle to the different fermions scale with
$\sqrt{\lambda_{S}/r}$, see below. On the other hand, reproducing a value of
the DM coannihilation cross section required to account for the observed relic
abundance demands a large portal coupling between the different fermion
sectors. For this reason, we consider $\lambda_{S}/r\lesssim\mathcal{O}(100)$
hereafter. In particular, in figure 1 we choose
$\bar{\lambda}\lambda_{S}/r=100$ to saturate the bound given by equation (21).

###  III.2 Scalar excitations

We now move on to the study of the scalar KK excitations. The profiles of
these resonances can be computed by inserting the KK decompositions

$\displaystyle h(x,t)$
$\displaystyle=\sum^{\infty}_{n=0}h_{n}(x)\chi_{n}^{h}(t),$ (23)
$\displaystyle S(x,t)$
$\displaystyle=\sum^{\infty}_{n=1}S_{n}(x)\chi_{n}^{S}(t)$

into the action (14) and keeping quadratic terms in the fields.

A possible way of determining the eigenmodes and eigenvalues of the mixed
system is to diagonalize the 4D mass matrix resulting after integrating the
quadratic terms in the action. Besides the KK scalar masses for the non-mixed
case, non-diagonal entries arise through the potential, once we integrate the
profiles over the fifth dimension, i.e.

$\displaystyle\int d^{5}x\sqrt{g}\,V(H,\Sigma)\supset\bar{\lambda}\int
d^{4}x\int_{\epsilon}^{1}\dfrac{dt}{t^{3}}\Bigg{[}$
$\displaystyle\dfrac{M_{\text{KK}}^{2}}{kr}v_{S}^{2}(t)\,h(x,t)^{2}+\dfrac{kr}{(\mu_{S}r)^{2}}\dfrac{\lambda_{S}}{r}t^{2}\varphi^{2}_{H}(t)S(x,t)^{2}$
(24)
$\displaystyle+4\dfrac{M_{\text{KK}}}{\mu_{S}r}\sqrt{\dfrac{\lambda_{S}}{r}}\,t\,\varphi_{H}(t)v_{S}(t)h(x,t)S(x,t)\Bigg{]}.$

Inserting the KK decompositions for $h(x,t)$ and $S(x,t)$, and using the
profiles for the decoupled case, we can write down the mass matrix to first
order in $\bar{\lambda}$, finding

$\mathcal{M}^{2}=\\!\left[\begin{pmatrix}\vspace*{0.1cm}x_{h_{0}}^{2}&0&0&\cdots\\\
\vspace*{0.1cm}0&x_{S_{1}}^{2}&0&\cdots\\\ 0&0&x_{h_{1}}^{2}&\cdots\\\
\vdots&\vdots&\vdots&\ddots\end{pmatrix}\\!+\bar{\lambda}\begin{pmatrix}\vspace*{0.1cm}\kappa_{h_{0}}^{2}&\kappa^{2}_{h_{0}S_{1}}&\kappa^{2}_{h_{0}h_{1}}&\cdots\\\
\vspace*{0.1cm}\kappa^{2}_{h_{0}S_{1}}&\kappa_{S_{1}}^{2}&\kappa^{2}_{h_{1}S_{1}}&\cdots\\\
\kappa^{2}_{h_{0}h_{1}}&\kappa^{2}_{h_{1}S_{1}}&\kappa^{2}_{h_{1}}&\cdots\\\
\vdots&\vdots&\vdots&\ddots\end{pmatrix}\right]M_{\text{KK}}^{2}.$ (25)

Here, $x_{h_{n}}$ and $x_{S_{n}}$ correspond to the unperturbed mass
eigenvalues in units of $M_{\text{KK}}$ of the decoupled system (for
$\bar{\lambda}=0$), defined as $x_{i}^{2}=m_{i}^{2}/M_{\text{KK}}^{2}$. Note
that only $x_{h_{0}}\ll 1$ corresponds to a light zero mode, because the odd
scalar field has no zero modes. With the mixing switched on, the mass of the
lightest scalar becomes

$m_{h}^{2}\approx\left(x_{h_{0}}^{2}+\bar{\lambda}\,\kappa_{h_{0}}^{2}\right)M_{\text{KK}}^{2}\,.$
(26)

In order to calculate the $\mathcal{O}(\bar{\lambda})$ terms we need the
unperturbed solutions for the profiles. For the Higgs KK modes they are given
by

$\chi_{n}^{h}(t)=\sqrt{\dfrac{L}{\pi}}\dfrac{tJ_{\beta}(x_{h_{n}}t)}{\sqrt{J^{2}_{\beta}(x_{h_{n}})-J_{\beta+1}(x_{h_{n}})J_{\beta-1}(x_{h_{n}})}}\,,$
(27)

where $J_{\beta}(x)$ is a Bessel function and the eigenvalues $x_{h_{n}}$
satisfy

$\dfrac{x_{h_{n}}J_{\beta+1}(x_{h_{n}})}{J_{\beta}(x_{h_{n}})}=2\left(m_{\text{IR}}-2-\beta\right)\equiv
2\delta.$ (28)

At zeroth order in $\bar{\lambda}$ and for $x_{h_{0}}^{2}\ll 1$ the expression
for the zero-mode profile $\chi_{0}^{h}(t)$ is approximately given by

$\chi_{0}^{h}(t)\simeq\sqrt{\dfrac{L}{\pi}(1+\beta)}\,t^{1+\beta},$ (29)

up to $\mathcal{O}(x_{h_{0}}^{2})$ corrections.

The contributions to the mass matrix at first order in $\bar{\lambda}$ can be
read from

$\displaystyle\kappa_{S_{m}}^{2}=\dfrac{2\,kr}{(\mu_{S}r)^{2}}\dfrac{\lambda_{S}}{r}\int_{\epsilon}^{1}\dfrac{dt}{t}\dfrac{\varphi^{2}_{H}(t)}{M_{\text{KK}}^{2}}[\chi_{m}^{S}(t)]^{2}\,,$
(30)
$\displaystyle\kappa_{h_{n}S_{m}}^{2}=\dfrac{4}{\mu_{S}r}\sqrt{\dfrac{\lambda_{S}}{r}}\int_{\epsilon}^{1}\dfrac{dt}{t^{2}}\dfrac{\varphi_{H}(t)}{M_{\text{KK}}}v_{S}(t)\chi_{n}^{h}(t)\chi_{m}^{S}(t)\,,$
$\displaystyle\kappa_{h_{n}h_{m}}^{2}=\dfrac{2}{kr}\int_{\epsilon}^{1}\dfrac{dt}{t^{3}}v_{S}^{2}(t)\chi_{n}^{h}(t)\chi_{m}^{h}(t)\,,$

where $\kappa_{h_{n}}^{2}\equiv\kappa_{h_{n}h_{n}}^{2}$ in (25). The different
powers of $t$ in the denominator result from our particular normalization of
the VEV of the new scalar field in (15), which differs from the normalization
of the Higgs VEV in (3).

A priori, both $x_{h_{0}}$ and $\kappa_{h_{0}}$ are naturally ${\cal O}(1)$
numbers, so in order to obtain a 125 GeV Higgs boson one needs to tune

$\frac{m_{h}^{2}}{M_{\text{KK}}^{2}}\approx
x_{h_{0}}^{2}+\bar{\lambda}\,\kappa_{h_{0}}^{2}\sim\left(\frac{0.125}{5}\right)^{2}\sim
10^{-3}.$ (31)

This is a well-known feature of bulk Higgs models in WEDs Davoudiasl:2005uu ;
Cacciapaglia:2006mz ; Azatov:2009na ; Vecchi:2010em ; Archer:2012qa ;
Malm:2013jia ; Archer:2014jca . We can achieve this in two different ways. For
positive values of $\bar{\lambda}$, both terms in the sum need to be small
simultaneously. In the case of $x_{h_{0}}^{2}$, this can be achieved by tuning
the parameters in the Higgs potential, as it is customary for a bulk Higgs
with no additional scalars (see e.g. Malm:2013jia ; Archer:2014jca ). For
$\bar{\lambda}\,\kappa_{h_{0}}^{2}$ the only possibility is to make
$\bar{\lambda}$ small enough, since $\kappa_{h_{0}}^{2}$ is an ${\cal O}(1)$
number unless very large values of $\beta$ are chosen. (The limit
$\beta\to\infty$ corresponds to a brane-localized Higgs and will not be
considered here.) Therefore, for positive values of $\bar{\lambda}$

$0\leq\bar{\lambda}\,\kappa_{h_{0}}^{2}\sim\bar{\lambda}\lesssim 10^{-3}.$
(32)

As a result, in this case values of $\bar{\lambda}$ larger than $10^{-3}$ are
not allowed, regardless of the value for $x_{h_{0}}^{2}$.

Figure 2:  Maximum allowed value for the parameter
$(\sin\theta_{h\mathcal{S}})_{\max}$ describing the mixing between the
lightest Higgs mode and the first KK resonance of the $\mathbb{Z}_{2}$-odd
scalar as a function of $\beta$, for fixed values of $M_{\text{KK}}$ and
$k/m_{\rm Pl}$. In the left and middle plots, we show this dependence for
different values of $\mu_{S}r$ and positive $\bar{\lambda}$. In the left plot,
we consider the maximum possible values of $\bar{\lambda}>0$ and
$\lambda_{S}/r$, whereas we fix $\lambda_{S}/r$ to $100$ in the middle plot,
with $\bar{\lambda}$ still saturating the resulting upper bound. Finally, in
the right plot we fix both $\mu_{S}r$ and $\lambda_{S}/r$ and consider three
different negative values of $\bar{\lambda}$.

One could also entertain the possibility of considering solutions in which
both quantities $x_{h_{0}}^{2}$ and $\kappa_{h_{0}}^{2}$ are simultaneously
$\mathcal{O}(1)$, but they cancel each other out leading to a light Higgs
mass. Since $\kappa_{h_{0}}^{2}>0$ by definition, one would need to have
either $x_{h_{0}}^{2}$ or $\bar{\lambda}$ negative. The first possibility
corresponds to a tachyonic Higgs field (before turning on the mixing with the
odd scalar) and leads to the BC

$\dfrac{x_{h_{n}}I_{\beta+1}(x_{h_{n}})}{I_{\beta}(x_{h_{n}})}=-2\left(m_{\text{IR}}-2-\beta\right)\equiv-2\delta$
(33)

on the IR brane, where $I_{n}(x)$ are modified Bessel functions. This
condition is similar to that in (28), but with a relative minus sign. However,
such a path leads nowhere since, as can be proven, this equation is
incompatible with the presence of a Higgs VEV Archer:2012qa ; Archer:2014jca .
Therefore, the only viable option is to allow for negative values of
$\bar{\lambda}$. In that case, equation (26) becomes

$m_{h}^{2}\approx\left(x_{h_{0}}^{2}-|\bar{\lambda}|\,\kappa_{h_{0}}^{2}\right)M_{\text{KK}}^{2}.$
(34)

and we can always reproduce the Higgs mass regardless of the value of
$\bar{\lambda}<0$, by choosing the appropriate value of $x_{h_{0}}^{2}$.

At any rate, the mixing between the even and odd bulk scalars leads to

$h_{0}(x)=h_{\rm
phys}(x)+\sin\theta_{h\mathcal{S}}\,\mathcal{S}(x)+\sin\theta_{h\mathcal{H}}\,\mathcal{H}(x),$
(35)

where $\mathcal{H}(x)=h_{1}(x)+\mathcal{O}(\bar{\lambda})$ and
$\mathcal{S}(x)=S_{1}(x)+\mathcal{O}(\bar{\lambda})$ are the profiles of the
first KK modes in the limit where $\bar{\lambda}=0$, and

$\displaystyle\sin\theta_{h\mathcal{S}}$
$\displaystyle=\bar{\lambda}\dfrac{\kappa_{h_{0}S_{1}}^{2}}{x_{S_{1}}^{2}-x_{h_{0}}^{2}}\approx\bar{\lambda}\dfrac{\kappa_{h_{0}S_{1}}^{2}}{x_{S_{1}}^{2}}\,,$
(36) $\displaystyle\sin\theta_{h\mathcal{H}}$
$\displaystyle=\bar{\lambda}\dfrac{\kappa_{h_{0}h_{1}}^{2}}{x_{h_{1}}^{2}-x_{h_{0}}^{2}}\approx\bar{\lambda}\dfrac{\kappa_{h_{0}h_{1}}^{2}}{x_{h_{1}}^{2}}\,.$

In general, the mixing of the lightest Higgs eigenmode with the first odd
excitation can be expressed as

$\sin\theta_{h\mathcal{S}}\simeq
4\bar{\lambda}\sqrt{\dfrac{\lambda_{S}}{r}}\dfrac{x_{4}}{x_{S_{1}}^{2}}\dfrac{kr}{\mu_{S}r}(1+\beta)\int_{\epsilon}^{1}dt\;t^{2\beta}v_{S}(t)\chi_{1}^{S}(t)\,,$
(37)

and a similar expression can be derived for $\sin\theta_{h\mathcal{H}}$, i.e.,

$\sin\theta_{h\mathcal{H}}\simeq\dfrac{2\bar{\lambda}}{x_{h_{1}}^{2}}\sqrt{\dfrac{1+\beta}{kr}}\int_{\epsilon}^{1}dt\,t^{\beta-2}v_{S}^{2}(t)\chi_{1}^{h}(t).$
(38)

As we can see, when $\bar{\lambda}$ is positive the constraint set by the
physical Higgs mass does not allow for a large mixing. Its upper bound is
saturated when one assumes that the leading contribution to the Higgs mass is
given by the $\bar{\lambda}\,\kappa_{h_{0}}^{2}$ term in (31). Then

$\bar{\lambda}_{\rm
max}=\dfrac{x_{h}^{2}}{\kappa_{h_{0}}^{2}}=\dfrac{x_{h}^{2}}{2(\beta+1)}\left[\int_{\epsilon}^{1}dt\;t^{2\beta-1}v_{S}^{2}(t)\right]^{-1}.$
(39)

In this case, plugging in the expression for $\kappa_{h_{0}S_{1}}^{2}$ and
$x_{S_{1}}^{2}$, we get

$(\sin\theta_{h\mathcal{S}})_{\rm max}\simeq
2\dfrac{x_{h}^{2}x_{4}}{x_{S_{1}}^{2}}\dfrac{k}{\mu_{S}}\sqrt{\dfrac{\lambda_{S}}{r}}\dfrac{\int_{\epsilon}^{1}dt\;t^{2\beta}v_{S}(t)\chi_{1}^{S}(t)}{\int_{\epsilon}^{1}dt\;t^{2\beta-1}v_{S}^{2}(t)}\,,$
(40)

where $\lambda_{S}/r$ in the above equation needs to saturate its upper bound

$\dfrac{\lambda_{S}}{r}\leq\dfrac{\mu_{S}^{4}}{k^{4}}\dfrac{2}{kr\,x_{4}^{2}\,x_{h}^{2}}\int_{\epsilon}^{1}dt\;t^{2\beta-1}v_{S}^{2}(t),$
(41)

which results after inserting (39) into equation (21). This leads to

$(\sin\theta_{h\mathcal{S}})_{\rm
max}\simeq\dfrac{x_{h}}{x_{S_{1}}^{2}}\dfrac{2^{3/2}\,\mu_{S}r}{\left(kr\right)^{3/2}}\dfrac{\int_{\epsilon}^{1}dt\;t^{2\beta}v_{S}(t)\chi_{1}^{S}(t)}{\left(\int_{\epsilon}^{1}dt\;t^{2\beta-1}v_{S}^{2}(t)\right)^{1/2}}\,,$
(42)

which only depends on $\beta$ and $\mu_{S}r$, given that
$kr\sim\mathcal{O}(10)$ in order to solve the hierarchy problem and that the
eigenvalues $x_{i}$ and the scalar profiles are determined once these
parameters have been fixed.

When $\bar{\lambda}$ is negative, on the other hand, $\lambda_{S}/r$ is
unconstrained by relation (21). In this case, an upper bound on
$\lambda_{S}/r$ arises if one wants to prevent the theory from becoming
strongly coupled, since the couplings of the KK scalar field $\mathcal{S}$ to
the different fermions are proportional to $\sqrt{\lambda_{S}/r}$, as one can
see from equations (72) and (73) in the appendix. Moreover, in this case
sizable values of $|\bar{\lambda}|\sim\mathcal{O}(1)$ are allowed. For all
these reasons, we find that $\sin\theta_{h\mathcal{S}}$ can be much larger
than in the case of a positive $\bar{\lambda}$.

In figure 2, we show the different predictions for the maximum allowed value
of the parameter $(\sin\theta_{h\mathcal{S}})_{\max}$, which measures the
mixing between the lightest Higgs scalar and the first KK mode of the new
$\mathbb{Z}_{2}$-odd scalar as a function of $\beta$. In the left plot, we
show this dependence for different values of $\mu_{S}r$ after saturating the
upper bounds on $\lambda_{S}/r$ and $\bar{\lambda}>0$. In the middle plot, we
display the maximum allowed value of $\sin\theta_{h\mathcal{S}}$ for the same
values of $\mu_{S}r$ and a fixed value $\lambda_{S}/r=100$, together with
$\bar{\lambda}=\bar{\lambda}_{\max}>0$. Note that for $\mu_{S}r=25$ and
$\beta\sim 50$, $\lambda_{S}/r=100$ takes its maximum value. This explains
why, in this case, the line stops before one can reach $\beta=100$. Finally,
in the right plot we show $(\sin\theta_{h\mathcal{S}})_{\max}$ for different
values of $\bar{\lambda}<0$ and fixed values $\mu_{S}r=25$ and
$\lambda_{S}/r=100$. In all these plots we assume $M_{\text{KK}}=5$ TeV and
$k=m_{\rm Pl}/8$. One can readily see that, for a given value of
$\lambda_{S}/r$, the maximum allowed value for $\sin\theta_{h\mathcal{S}}$ is
much more significant in the case $\bar{\lambda}<0$, since larger values of
$|\bar{\lambda}|$ can be taken. In addition, when $\bar{\lambda}$ is negative
one could also consider bigger values of $\lambda_{S}/r$ than in the
$\bar{\lambda}>0$ case. All this results into larger mixing angles when
$\bar{\lambda}$ is negative compared to the $\bar{\lambda}>0$ case.

Figure 3: Maximum allowed value for the parameter $\sin\theta_{h\mathcal{H}}$,
describing the mixing between the lightest Higgs mode with its first KK
resonance, as a function of $\beta$. In the left figure, we exhibit three
different values of $\mu_{S}r$ for $\bar{\lambda}=\bar{\lambda}_{\max}>0$,
while in the right panel we consider three different negative values of
$\bar{\lambda}$ for a fixed value of $\mu_{S}r=25$.

For the Higgs mixing with its first KK mode, parametrized by
$\sin\theta_{h\mathcal{H}}$, we found a monotonic behavior, with
$\sin\theta_{h\mathcal{H}}$ getting smaller for large values of $\beta$
independently of the $\mu_{S}r$ parameter. This can be seen in figure 3, where
we show $(\sin\theta_{h\mathcal{H}})_{\max}$ as a function of $\beta$ for
$M_{\text{KK}}=5$ TeV and $k=m_{\rm Pl}/8$. In particular, we display on the
left panel this functional dependence for three different values of
$\mu_{S}r$, after saturating $\bar{\lambda}$ to its upper bound. In the left
panel we exhibit the case where $\mu_{S}r=25$ is kept fixed, while $\lambda<0$
takes on different (negative) values. Comparing this panel with the right
panel of the previous figure, we can see that
$(\sin\theta_{h\mathcal{H}})_{\max}$ is a steeper function of $\beta$ than
$(\sin\theta_{h\mathcal{S}})_{\max}$ when $\bar{\lambda}<0$. For the choice of
parameters at hand, and depending on the value of $|\bar{\lambda}|$,
$\sin\theta_{h\mathcal{H}}$ becomes bigger than $\sin\theta_{h\mathcal{S}}$
for $\beta$ smaller than values between $1$ and $10$, depending on the other
parameters of the model, whereas the opposite happens when $\beta$ takes
larger values. Henceforth, for practical purposes, we will neglect the
differences between $\mathcal{S}$ and $S_{1}$, as well as between
$\mathcal{H}$ and $h_{1}$, since they are proportional to the small mixing
angles $\sin\theta_{h\mathcal{S}}$ and $\sin\theta_{h\mathcal{H}}$,
respectively.

##  IV Phenomenology

Once the Higgs boson is allowed to propagate into the bulk of the extra
dimension, its mixing with the $\mathbb{Z}_{2}$-odd scalar becomes
unavoidable. This mixing will leave its imprint on different aspects of the
phenomenology. In particular, it can lead to effects on experiments as diverse
as high-energy colliders, both present and future ones. Moreover, as we will
see, assuming the presence of dark fermions, it can naturally explain the
observed DM relic abundance and leave its imprint on DM direct-detection
experiments.

We have seen how the quartic coupling leads to a mass mixing of the Higgs-
boson zero mode with both its first KK resonance $h_{1}$ and the lowest-lying
$\mathbb{Z}_{2}$-odd scalar, $S_{1}$. This mixing induces modifications on the
Higgs-boson couplings to SM particles. In section IV.1 we will explore these
modifications and study its impact on current and future colliders.

A key aspect of our model is that the odd scalar field constitutes a unique
window into dark sectors featuring fermions propagating into the bulk. Indeed,
since all the 5D fermion bulk masses are generated through Yukawa-like
interactions with the odd scalar, the scalar KK modes necessarily connect any
dark fermionic sector with the SM if the former is genuinely five dimensional.
Such a connection is unavoidable and constitutes a defining feature of the
model. In the presence of a dark fermionic sector, the required Yukawa
couplings between the odd scalar field and the bulk fermions has two
interesting consequences. On the one hand, for light enough dark fermion
masses, it induces a Higgs invisible decay width, and this in turn implies
constraints on the size of the scalar mixing between both 5D scalar fields. We
study this in detail in section IV.2. On the other hand, the dynamical
generation of the 5D fermion masses naturally connects the visible and the
invisible sectors via the KK resonances of the odd scalar field, with $S_{1}$
giving the leading contribution. In particular, this introduces an efficient
coannihiliation channel for the lightest dark fermion, which is naturally
stable and therefore a good DM candidate. We study this possibility both in
the regular freeze-out scenario and in the case of a matter-dominated freeze-
out in section IV.3. Finally, in section IV.4 we study in detail the
constraints coming from direct-detection experiments using recent Xenon1T data
DiGangi:2018xht ; Aprile:2018dbl .

###  IV.1 Modified Higgs couplings

As we have seen in the previous section, the physical Higgs boson can be
expressed with very good approximation as a linear combination of the
interaction eigenstates $h_{0}$, $h_{1}$ and $S_{1}$. Since these interaction
eigenstates couple differently to the SM particles, this mixing induces
modifications of the SM Higgs couplings. Here, we study the implications of
these modifications.

The 4D effective couplings of the different scalars to the fermions are
obtained by integrating the profiles of the different fields over the fifth
dimension and a subsequently rotate into the mass basis. In particular, the
coupling of the Higgs-boson zero mode and KK modes to a pair of fermion chiral
zero modes, $\bar{\Psi}_{a}\Psi_{b}$, is given by

$y_{abh_{n}}=\dfrac{y_{\ast}}{\sqrt{kr}}\dfrac{2+\beta}{\sqrt{2(1+\beta)}}\int_{\epsilon}^{1}dtf_{a}(t)f_{b}(t)\chi_{n}^{h}(t)\,,$
(43)

where $y_{\ast}$ is defined as a function of the 5D dimensionful Yukawa
coupling $Y_{\rm 5D}$ Malm:2013jia

$y_{\ast}=\frac{\sqrt{k(1+\beta)}}{2+\beta}Y_{\rm 5D},$ (44)

where the latter is defined by (for an up-type quark field $\Psi_{Rb}$, the
Higgs-boson field $H$ must be replaced by $\tilde{H}$)

$S_{Y}\supset-\int d^{5}x\sqrt{g}\,Y_{\rm
5D}\bar{\Psi}_{La}(x,t)H(x,t)\Psi_{Rb}(x,t)+\mathrm{h.c.}\,.$ (45)

On the other hand, the coupling of two light SM fermions to the scalar
$\mathcal{S}$ only appears through a mass insertion. Indeed, before EWSB there
is no direct coupling between $\mathcal{S}$ and two SM-like fermion fields.
Such a coupling is only generated after taking into account the fermion mixing
induced by the Higgs VEV $v_{\rm SM}$. We will compute the corresponding
coupling perturbatively, as it is expected to be suppressed by a factor of
$\mathcal{O}(v_{\rm SM}/M_{\text{KK}})$. This coupling arises from the
interactions of the $\mathcal{S}$ scalar to the different fermion zero modes
and the first KK resonance with opposite chirality, once we rotate to the
fermion mass basis after EWSB. Specifically, the coupling between
$\mathcal{S}$, a chiral fermion zero mode $a$, and its first KK resonance $A$
with opposite chirality, is given by

$\displaystyle y_{aA\mathcal{S}}$
$\displaystyle=2\,c_{a}\sqrt{\dfrac{\lambda_{S}}{r}}\dfrac{k}{\mu_{S}}\int_{\epsilon}^{1}dtf_{a}(t)f_{A}^{R}(t)\chi_{1}^{S}(t),\quad\mathrm{or}$
(46) $\displaystyle y_{Aa\mathcal{S}}$
$\displaystyle=2\,c_{a}\sqrt{\dfrac{\lambda_{S}}{r}}\dfrac{k}{\mu_{S}}\int_{\epsilon}^{1}dtf_{A}^{L}(t)f_{a}(t)\chi_{1}^{S}(t),$

depending on the zero-mode chirality, where $c_{a}$ is defined in appendix A.
Then, after rotating the fermion fields to the mass basis, we induce an
interaction term $y_{f\mathcal{S}}\mathcal{S}\bar{f}_{L}f_{R}$ between the SM-
like chiral fields, $f_{L}$ and $f_{R}$, and $\mathcal{S}$.

We can write

$\displaystyle\delta y^{\rm phys}_{fh}$ $\displaystyle\equiv 1-\frac{y^{\rm
phys}_{fh}}{y_{fh}^{\rm
SM}}\simeq(1-\varkappa_{f})+\Delta_{f\mathcal{H}}+\Delta_{f\mathcal{S}},$ (47)

where we have defined $\varkappa_{f}=y_{f_{L}f_{R}h_{0}}/y_{fh}^{\rm SM}$.
Here $y_{fh}^{\rm phys}$ is the resulting Higgs Yukawa coupling

$\displaystyle\mathcal{L}\supset-\frac{1}{\sqrt{2}}y_{fh}^{\rm
phys}h\bar{f}_{L}f_{R}+\mathrm{h.c.},$ (48)

and $y_{fh}^{\rm SM}$ denotes the corresponding parameter in the SM one.
Moreover

$\Delta_{f\mathcal{H}}=\sin\theta_{h\mathcal{H}}\dfrac{y_{f_{L}f_{R}h_{1}}}{y_{f_{L}f_{R}h_{0}}},\quad\Delta_{f\mathcal{S}}=\sin\theta_{h\mathcal{S}}\dfrac{y_{f\mathcal{S}}}{y_{f_{L}f_{R}h_{0}}}.$
(49)

The quantity $\varkappa_{f}$ measures the ratio of the Yukawa coupling of the
Higgs-boson zero mode relative to that of the SM Higgs boson, whereas the
parameters $\Delta_{f\mathcal{H}}$ and $\Delta_{f\mathcal{S}}$ describe the
admixtures of the Higgs-boson and $\mathbb{Z}_{2}$-odd scalar KK modes into
the physical Higgs. Note that we have taken $y_{fh}^{\rm SM}$ equal to
$y_{f_{L}f_{R}h_{0}}$ in the denominator of $\Delta_{f\mathcal{H}}$ and
$\Delta_{f\mathcal{S}}$, since the difference is
$\mathcal{O}(\bar{\lambda}\,v_{\rm SM}^{2}/M_{\text{KK}}^{2})$ and thus
subleading. On the other hand, $\varkappa_{f}$ is blind at this order to the
Higgs mixing with the odd scalar. This is a byproduct of the induced light-
heavy fermion mixing after EWSB and the shift in the 5D Higgs VEV. It has been
studied e.g. in Archer:2014jca  for the RS case. In particular, for
$M_{\text{KK}}=5$ TeV, $\varkappa_{b}$ never exceeds $2.5\cdot 10^{-2}$ when
$1\leq\beta\leq 10$. In the case of lighter quarks, even smaller values are
expected. In this work we concentrate on $\Delta_{f\mathcal{S}}$ and
$\Delta_{f\mathcal{H}}$, since they are direct probes of the mixing of the
bulk Higgs field with the $\mathbb{Z}_{2}$-odd scalar field.

Figure 4: $\Delta_{b\mathcal{H}}$ and $\Delta_{b\mathcal{S}}$ as functions of
$\beta$ for two different values of $y_{\ast}$ and fixed values of
$\bar{\lambda}$, $\lambda_{S}/r$, $M_{\text{KK}}$ and $k/m_{\rm Pl}$. For each
case, we have generated $N_{\rm points}=3000$ random values of
$c_{t_{R}}\in[-0.6,0.2]$ and obtained $c_{q_{L}^{3}}$ and $c_{b_{R}}$ by
correctly reproducing the top- and bottom-quark masses.

Hereafter, we will focus on the bottom quark. The reasons for this are
twofold. On the one hand, we expect the modifications of the Yukawa couplings
to be larger for heavier quarks, while on the other hand, the bottom Yukawa
coupling is among those measured most accurately, having in addition the most
promising prospects. Indeed, existing measurements of the $h\to b\bar{b}$
signal strength (relative to the SM expectation) lead to $\mu_{h\to
bb}=1.01\pm 0.12\,(\mathrm{stat}.)\,^{+0.16}_{-0.15}\,(\mathrm{syst}.)$ 201859
. Assuming SM production this translates into a measurement of $(y_{bh}^{\rm
phys}/y_{bh}^{\rm SM})^{2}$ with an uncertainty of about $20\%$. However, the
expected relative precision to be reached at future particle colliders such as
the ILC, CLIC and the FCC is projected to be about 1% for the initial stages,
reaching the 0.5% level in later stages of the experiments deBlas:2019rxi .

Figure 4 shows a scatter plot with values of $\Delta_{b\mathcal{H}}$ and
$\Delta_{b\mathcal{S}}$ as a function of $\beta$, for $\bar{\lambda}=-0.5$ and
$\lambda_{S}/r=100$. For both quantities we display two different scenarios,
corresponding to $y_{\ast}=3$ and $y_{\ast}=1.5$, where $y_{\ast}$ is taken
the same for both third-generation quarks (i.e.,
$y_{\ast}=y_{\ast}^{t}=y_{\ast}^{b}$). For each case, we have generated random
values of $c_{t_{R}}\in[-0.6,0.2]$, and obtained $c_{q_{L}^{3}}$ and
$c_{b_{R}}$ by fitting the top- and bottom-quark masses. One can see that for
small values of $\beta$ the impact on $\Delta_{b\mathcal{H}}$ of changing
$y_{\ast}$ is magnified. This is expected since, for these values of $\beta$,
the Higgs is less IR-localized and one expects larger differences between the
profiles $\chi_{0}^{h}$ and $\chi_{1}^{h}$. Indeed, regardless of the value of
$y_{\ast}$, the prediction for the bottom mass $m_{b}\approx
v_{4}\,y_{b_{L}b_{R}h_{0}}/\sqrt{2}$ needs to remain unchanged. Since
$y_{b_{L}b_{R}h_{0}}$ is proportional to $y_{\ast}$, see (43), changes in
$y_{\ast}$ have to be compensated by different fermion localizations and
therefore a different value of the overlap integral in that relation, in such
a way that $y_{b_{L}b_{R}h_{0}}$ remains approximately constant. The integral
present in $y_{b_{L}b_{R}h_{1}}$ will change accordingly, but will deviate
from the other integral as $\beta$ decreases. This effect is reversed for
$\Delta_{b\mathcal{S}}$, since $y_{\ast}$ is not explicitly present in the
numerator of (49), with the sole effect of changing the localization of the
profiles of the third-generation fermions. Since a smaller value of $y_{\ast}$
requires a more IR-localized $q_{L}^{3}$ to reproduce the top-quark mass, one
expects a bigger overlap between $\mathcal{S}$ and the third-generation left-
handed doublet $q_{L}^{3}$ together with its first KK mode. This leads to a
larger value of $y_{b\mathcal{S}}$ and $\Delta_{b\mathcal{S}}$, as one can see
in the figure. Note that $\Delta_{b\mathcal{S}}$ scales with
$\sqrt{\lambda_{S}/r}$, so one can readily obtain $\Delta_{b\mathcal{S}}$ for
alternative choices of this parameter. Moreover, to good approximation both
quantities scale linearly with $\bar{\lambda}$ for small values of
$\bar{\lambda}$, because $\sin\theta_{h\mathcal{X}}\propto\bar{\lambda}$, as
can be seen from (36).

Taking into account the projected sensitivity for the bottom Yukawa coupling,
one can see that we will be able to probe sizable values of the scalar portal
coupling $\bar{\lambda}$ for moderately small values of $\beta$. In
particular, as can be seen from figure 4, for our chosen value
$\bar{\lambda}=-0.5$, the predicted modifications $\Delta_{b\mathcal{H}}$ and
$\Delta_{b\mathcal{S}}$ can be probed as long as $\beta$ is less than about 4.
For smaller values of $\bar{\lambda}$, one could only access smaller values of
$\beta$.

Figure 5: Diagrams responsible for the generation of the Higgs coupling to DM
in the two main scenarios discussed in the text. The left figure corresponds
to the case where the DM candidate is the first KK mode of a single 5D fermion
field. The right figure corresponds to the case where the different
chiralities of the DM candidate are zero modes of different 5D fields and the
mediation via heavy KK modes is required.

The couplings of the Higgs KK modes to the $W$ and $Z$ bosons of the SM
(corresponding to the zero modes of the corresponding bulk gauge fields) are
proportional to the integrals

$g_{h_{n}\\!V\\!V}\propto\int_{\epsilon}^{1}\\!\dfrac{dt}{t}\varphi_{H}(t)\chi_{n}^{h}(t)\,,$
(50)

where we have used that to a good approximation the zero-mode profiles for of
the $W$ and $Z$ states are flat along the extra dimension Davoudiasl:1999tf ;
Pomarol:1999ad ; Casagrande:2008hr . Since $\varphi_{H}(t)\simeq
v_{4}\chi_{0}^{h}(t)$, and the orthogonality condition for the scalar profiles
is given by Malm:2013jia ; Archer:2014jca

$\dfrac{2\pi}{L}\int_{\epsilon}^{1}\dfrac{dt}{t}\chi_{m}^{h}(t)\chi_{n}^{h}(t)=\delta_{mn},$
(51)

the gauge bosons only couple to the Higgs-boson zero mode, and higher KK modes
of the Higgs do not couple to the gauge-boson zero modes at tree level. For
the case of the $S_{1}$ scalar, which is a SM singlet, the couplings to
electroweak gauge bosons would only appear at the loop level. Therefore, it
will not modify the Higgs couplings to gauge bosons in an noticeable way,
since these changes are suppressed by a small mixing angle
$\mathcal{O}(10^{-2})$ and a loop factor.

###  IV.2 Invisible Higgs decays

Figure 6: Left panel: Mass fraction $x_{1}=m_{1}/M_{\text{KK}}$ of the first
fermion KK mode in terms of the 5D dimensionless bulk-mass parameter
$c_{\chi}$ for the case of a left-handed chirality, with Dirichlet and Neumann
boundary conditions on the UV and the IR brane, respectively, for two
different values of $\mu_{S}r$. The yellow line corresponds to the case where
the fermion bulk masses do not have a dynamical origin, but appear in the 5D
Lagrangian along with a sign function, as it is usual in RS models. We also
show the value for which $m_{\chi}=m_{h}/2$ with a dashed gray line. Higgs
decays into a pair of DM particles $\chi_{1}\bar{\chi}_{1}$ are kinematically
allowed only if $x_{1}$ falls below this line. Right panel: Value of
$y_{\chi\mathcal{S}}$ as a function of $c_{\chi}$ for the same choice of
boundary conditions and values of $\mu_{S}r$.

The mixing between the Higgs and the $\mathbb{Z}_{2}$-odd bulk scalar field
induces an effective coupling of the Higgs boson to any 5D bulk fermion
present in the theory which is not localized on the UV or IR brane. This
includes the possibility of fermions not charged under the SM group, the so-
called dark fermions. Hereafter, we will consider this case and will study its
potential signatures, including its role in explaining the observed DM relic
abundance. We will consider two different scenarios, depending on the origin
of the dark fermion masses. In the first scenario, the dark fermion mass
arises purely from orbifolding, i.e., from the compactification of the WED,
and it is thus proportional to its curvature, $m_{\chi}\propto M_{\text{KK}}$.
The simplest possibility is to add a 5D fermion with no zero mode, whose first
KK resonance is automatically stable and a good DM candidate. In the case
where one of the chiralities of the corresponding 5D field has a Dirichlet
(Neumann) boundary condition on the UV (IR) brane, the first KK mode can be
parametrically lighter than $M_{\text{KK}}$ Agashe:2004ci , potentially light
enough to be accessible in Higgs-boson decay. In this case, the DM candidate
is VL even though it can have non-trivial quantum numbers under a possible new
dark group. We thus allow for $N_{\chi}$ copies of such VL fermion. In the
second scenario the fermion mass term connects two zero modes with opposite
chiralities, arising from two different 5D fields. In this scenario, the
fermion mass can either come through a dark Higgs mechanism or a brane-
localized VL mass. If the dark Higgs is localized towards the IR brane, or
alternatively the VL mass is localized on the IR brane, the fermion mass is
proportional to $M_{\text{KK}}$, even though a large hierarchy can arise for
UV-localized dark fermions. Either way, in this case $S_{1}$ does not interact
directly with these two chiral zero modes (in the same way $S_{1}$ does not
couple directly to $b_{L}$ and $b_{R}$, as discussed previously), since they
are zero modes of two different 5D fields. Such a coupling can, however, be
generated after a mass insertion through the mediation of the heavy KK modes
(whether the mass comes from the spontaneous breaking of the dark gauge group
or a VL mass is irrelevant to this discussion). We also assume a possible
multiplicity of dark fermions given by $N_{\chi}$ as before.

We illustrate these two scenarios in figure 5. Note that in the second case we
expect the coupling of the Higgs boson to the dark fermion to be
$\mathcal{O}(m_{\chi}/M_{\text{KK}})$ suppressed. Moreover, its calculation is
rather model dependent. Alternatively, one could leave the nature of the dark
fermion mass unspecified and define an effective Yukawa coupling, taking into
account the mixing of the heavy modes with the zero mode.

There is an additional instance, which can be thought of as an intermediate
scenario between the previous two cases. There one adds a 5D gauge-singlet
fermion field with no additional flavor quantum numbers, which has a chiral
zero mode and a Majorana mass term localized on the IR brane. The coupling of
this field to $\mathcal{S}$ is generated analogously to the second case
described above, via the involvement of a heavy KK mode and a mass insertion.
Therefore, at the end of the day, this case is rather similar to the previous
one, besides the difference in the multiplicities of fermionic degrees of
freedom for a Majorana field.

We have computed the mass of the first KK mode for a 5D field with mixed
boundary conditions, as described in the first case above. This result is well
known for non-dynamical bulk masses but has never been explored when these are
generated by the VEV of a $\mathbb{Z}_{2}$-odd scalar. We show in the left
panel of figure 6 the ratio $x_{1}=m_{1}/M_{\text{KK}}$ as a function of the
usual dimensionless bulk-mass parameter $c$ in both scenarios. In the model at
hand, where the different fermion bulk masses are generated by the VEV of the
odd scalar field, $\langle\Sigma\rangle=\varphi_{S},$ this parameter takes the
form given in equation (69). We also consider for comparison, the non-
dynamical case where such bulk fermion masses are introduced by hand and read
$c=m/k$, with $m$ the 5D bulk mass. In this scenario the first KK mass $m_{1}$
can be made arbitrarily light by adjusting the $c$ parameter Agashe:2004ci .
However, in our model a lower bound on the mass value arises due to behavior
of $\varphi_{S}$ close to the two branes, where it vanishes. We find this
bound to be $x_{1}\sim 6\cdot 10^{-3}$, corresponding to 30 GeV for our
reference value $M_{\text{KK}}=5$ TeV. We also show in the right panel of
figure 6 the coupling of the VL fermions to the scalar $\mathcal{S}$, defined
analogously to (46) but involving two KK profiles instead of one,

$y_{\chi\mathcal{S}}=2\,c_{\chi}\sqrt{\dfrac{\lambda_{S}}{r}}\dfrac{k}{\mu_{S}}\int_{\epsilon}^{1}dtf_{1,\chi}^{L}(t)f_{1,\chi}^{R}(t)\chi_{1}^{S}(t),$
(52)

as a function of $c_{\chi}$ for different values of $\mu_{S}r$. In both panels
we have set $M_{\text{KK}}=5$ TeV, $k=m_{\rm Pl}/8$ and $\lambda_{S}/r=100$.
We can see that, for large values of $m_{\chi}$, sizable values of
$y_{\chi\mathcal{S}}$ are expected. Note that the VL fermions could also have
a contribution to their mass coming from the dark sector (for instance an IR-
localized Majorana mass term), however this would not affect their coupling to
the scalar $\mathcal{S}$.

The decay width of the Higgs boson into dark fermions is given by

$\Gamma(h\to\bar{\chi}\chi)=\dfrac{y_{\chi
h}^{2}N_{\chi}}{8\pi}m_{h}\left(1-\dfrac{4m_{\chi}^{2}}{m_{h}^{2}}\right)^{3/2},$
(53)

where we have defined

$y_{\chi h}\equiv y_{\chi\mathcal{S}}\sin\theta_{h\mathcal{S}},$ (54)

the coupling of the physical Higgs boson to the first dark KK fermion.

Using that $\mathcal{B}(H\to\text{inv})<0.33$ at 95% CL Sirunyan:2018owy  and
$\Gamma_{H}^{\text{SM}}\approx 4$ MeV, we can set an upper limit on the
effective coupling of the Higgs boson to dark fermions. We find the upper
bound on $y_{\chi h}$ to be $y_{\chi h}\lesssim 0.02/\sqrt{N_{\chi}}$ for DM
candidates with mass $m_{\chi}<m_{h}/2$. Note that this constraint does not
apply to heavier fermions.

###  IV.3 Scalar-mediated fermionic dark matter

As discussed in the previous section, the $\mathbb{Z}_{2}$-odd scalar field
will couple to any fermion field propagating in the bulk of the WED. This
provides a robust bridge between the SM and any dark sector having fermions
arising from 5D bulk fermion fields. In the case where these dark fermions are
stable and make for a viable DM candidate, the KK excitations of the odd
scalar field thus constitute efficient mediators for DM coannihilation into SM
particles. Moreover, as we have already seen, the mixing between both scalar
bulk fields induces a Higgs coupling to dark fermions, thereby turning the
Higgs boson into an additional scalar mediator.

For the sake of concreteness, we will focus on the first scenario discussed in
the previous section, i.e., of $N_{\chi}$ copies of a 5D dark fermion field,
having potentially parametrically light KK modes. These potentially light KK
modes – the lightest dark particles – are stable and a viable DM candidate. In
this case, both the mass of the DM candidate $m_{\chi}$ and its couplings to
the physical Higgs and the $\mathbb{Z}_{2}$-odd scalar, $y_{\chi h}$ and
$y_{\chi\mathcal{S}}$, respectively, depend only on a single $c$ parameter (in
addition to other model parameters such as e.g. $M_{\text{KK}}$, $kr$,
$\lambda_{S}/r$ or $\mu_{S}r$), see figure 6. Considering the alternative
scenario where the interaction of $\mathcal{S}$ to the dark fermions requires
a mass insertion on a dark fermion line, it would just lead to a different
shape of the curve $y_{\chi S}=F(m_{\chi})$ and, by virtue of (54), also the
value of $y_{\chi h}=\sin\theta_{h\mathcal{S}}\,y_{\chi S}$ (modulo a
different count of degrees of freedom, in the case of Majorana fermions). We
illustrate in figure 7 the diagrams relevant for the coannihilation
$\bar{\chi}\chi\to\bar{f}f$ in these two cases, with and without a “dark mass
insertion”, as discussed in the previous section. We represent by a blue blob
the heavy-light mass mixing induced after EWSB in the visible sector, whereas
the possible light-heavy mass mixing in the dark sector is depicted by a pink
blob. Hereafter, for the sake of concreteness, we focus on the case of
parametrically light KK fermions, for which there is no need of specifying any
further dynamics in the dark sector.

For Higgs-mediated processes, the dominant coannihilation final state will be
$t\bar{t}$, if kinematically accessible (i.e. for $m_{\chi}>m_{t}$), or
$b\bar{b}$, together with the vector final states $W^{+}W^{-}$ and $ZZ$. In
the case of diagrams mediated by $S_{1}$, $t\bar{t}$ or $b\bar{b}$ are the
dominant coannihilation channels for moderately small values of $m_{\chi}$.
However, for larger values of $m_{\chi}$, coannihilation into a SM fermion and
its first KK resonance is also possible and can be the dominant coannihilation
channel by far.

Figure 7: Diagrams contributing to the DM coannihilation cross section with
fermions in the final state. The diagrams shown in the first line correspond
to the case where a mass insertion on a dark fermion line is needed in order
to generate an effective interaction $S_{1}\bar{\chi}_{0}\chi_{0}$, whereas
the diagrams in the second line correspond to the case where the DM candidate
corresponds to a (potentially light) KK fermion $\chi_{1}$.

The relic abundance for a radiation-dominated freeze-out regime can be
computed using Gondolo:1990dk  (see also e.g. Kim:2006af ; Arcadi:2019lka )

$\Omega_{\chi}h^{2}\simeq\dfrac{x_{f}}{2\sqrt{g_{\star
S}(m_{\chi}/x_{f})}}\dfrac{10^{-9}\text{GeV}^{-2}}{\langle\sigma v\rangle},$
(55)

where $\Omega_{\chi}h^{2}=0.120\pm 0.001$ Aghanim:2018eyx . Here, $g_{\star
S}(T_{f})$ denotes the effective number of degrees of freedom in entropy as
function of the freeze-out temperature $T_{f}$, and we have defined a
parameter $x_{f}=m_{\chi}/T_{f}$ to be determined below. $\langle\sigma
v\rangle$ is the velocity-averaged cross section at the freeze-out
temperature, which can be calculated as Gondolo:1990dk

$\displaystyle\langle\sigma v\rangle$
$\displaystyle=\dfrac{1}{8m_{\chi}^{4}T_{f}K_{2}^{2}(m_{\chi}/T_{f})}\times$
(56)
$\displaystyle\qquad\quad\int_{4m_{\chi}^{2}}^{\infty}\\!ds\,\sigma(s)\left(s-4m_{\chi}^{2}\right)\sqrt{s}K_{1}(\sqrt{s}/T_{f}),$

where $K_{n}(x)$ are modified Bessel functions. The parameter $x_{f}$ in (55)
is obtained by solving the implicit equation

$x_{f}=\ln\left(g_{\chi}\frac{m_{\chi}}{2\pi^{3}}\sqrt{\frac{45}{8x_{f}\,g_{\star
S}(m_{\chi}/x_{f})}}\,m_{\rm Pl}\langle\sigma v\rangle\right),$ (57)

where $g_{\chi}=4N_{\chi}$ is the number of DM degrees of freedom.

Figure 8: Velocity-averaged annihilation cross section $\langle\sigma
v\rangle$ at the freeze-out temperature as a function of the DM mass
$m_{\chi}$, for $N_{\chi}=1$, $M_{\text{KK}}=5$ TeV and $k=m_{\rm Pl}/8$. In
the top panels, we fix $\sin\theta_{h\mathcal{S}}$ and $\lambda_{S}/r$ and
consider two different values of $y_{\ast}$. In both cases, we take two
different values of $\beta$ and $c_{t_{R}}$. In the bottom panels, we fix
$\beta$, $y_{\ast}$ and $c_{t_{R}}$ and consider different values of
$\lambda_{S}/r$ for $\sin\theta_{h\mathcal{S}}=10^{-5}$ (left) and
$\sin\theta_{h\mathcal{S}}=10^{-6}$ (right). In all four panels, we also show
in dashed gray the $\langle\sigma v\rangle$ prediction for diagrams mediated
by the exchange of the first KK graviton. We also show the velocity averaged
cross section reproducing the relic density experimental value from Planck in
dashed black, and the equivalent for a matter dominated freeze out in gray,
for two different values of $T_{\rm RH}$, after using $T_{\star}=10^{5}$ GeV
and $\tau=0.99$. For these lines the section in dot-dashed gray corresponds to
predictions for which $x_{f}<3$, and therefore in this regime the DM decouples
relativistically Hamdan:2017psw ; Chanda:2019xyl .

Alternatively, one can also consider that DM freeze-out happens in an early
period of matter domination, as proposed in Hamdan:2017psw ; Chanda:2019xyl .
Indeed, nothing prevents this from happening if radiation becomes dominant
again before big-bang nucleosynthesis. The fact that DM decoupling happens
during matter domination changes the freeze-out dynamics, since the Hubble
rate has a different parametric dependence compared to the usual case,
$H\propto T^{3/2}$ versus $H\propto T^{2}$. We do not elaborate here in detail
on the dynamics behind this scenario, which is not crucial for our current
analysis. One possibility would be to have a scalar field $\phi$ localized on
the UV brane, which starts behaving like matter at a critical temperature
$T_{\star}\sim m_{\phi}$ that we assume to be much larger than
$M_{\text{KK}}$. If $\phi$ is sufficiently long-lived, its contribution to the
energy density grows until it ultimately dominates the total energy density
regardless of its initial contribution $(1-\tau)$ at $T_{\star}$, where
$\tau\in[0,1]$ denotes the fraction of energy in radiation at $T=T_{\star}$.
Following Hamdan:2017psw ; Chanda:2019xyl  we will take $\tau=0.99$ as a
benchmark value. Freeze-out happens at a temperature $T_{f}$, in a matter-
dominated universe, before $\phi$ instantaneously decays at
$T_{\rm\Gamma}<T_{f}<T_{\star}$, reheating the bath to $T_{\rm RH}$ and
further diluting the DM freeze-out abundance. Hereafter we will assume $T_{\rm
RH}\sim 1$ GeV. We refer the reader to appendix C for more details.

We show in figure 8 the velocity averaged coannihilation cross section
$\langle\sigma v\rangle$ at the freeze-out temperature as a function of
$m_{\chi}$, for $N_{\chi}=1$, $M_{\text{KK}}=5$ TeV and $k=m_{\rm Pl}/8$. In
the top panels, we consider benchmarks with different values of $\beta$,
$y_{\ast}$ as well as $c_{t_{R}}$ (the parameter fixing the localization of
the RH top). In both top panels, we consider
$\sin\theta_{h\mathcal{S}}=10^{-5}$ and $\lambda_{S}/r=75$, as well as two
different values of $\beta$ and $c_{t_{R}}$. In particular, we show $\beta=2$
(pink), $\beta=10$ (blue), $c_{t_{R}}=-0.2$ (dashed line) and $c_{t_{R}}=-0.4$
(solid line). In the top-left panel we fix $y_{\ast}=3$ for both the up and
the down third-generation quark sector, with $c_{q_{L}^{3}}$ and $c_{b_{R}}$
being determined by reproducing the top and bottom quark masses for a given
choice of $c_{t_{R}}$. The same is done in the top-right panel but for
$y_{\ast}=1.5$. In both cases, for the sake of simplicity, light quark masses
are reproduced with UV localized fermions with identical bulk mass parameters
(modulo a sign difference between opposite chiralities) and different values
of $y_{\ast}$ with $y_{\ast}^{s}=1/2$ (for our purposes, such a not-so-refined
study is more than enough). We can see that increasing the IR localization of
the RH top, i.e. having bigger values of $|c_{t_{R}}|$, leads to a bigger
cross section for most DM masses when $y_{\ast}=3$. Since $y_{\ast}$ is large
enough in this case, changes in $c_{t_{R}}$ does not have a dramatic impact on
$c_{q_{L}^{3}}$ and $c_{b_{R}}$, which remain almost unchanged. Therefore, the
increase of the coannihilation cross section is mostly due to a larger
$\mathcal{S}t_{L}t_{R}$ coupling, which is indeed the leading one for DM
masses below about $10$ TeV. Such a larger coupling is the consequence of a
bigger overlap with $\mathcal{S}$ and the increase in the Yukawa coupling
$\mathcal{Y}$ coming with $c$. In the case of $y_{\ast}=1.5$, on the contrary,
changes in $|c_{t_{R}}|$ do have a dramatic impact on $c_{q_{L}^{3}}$, since
the RH top can not account for the top mass alone, requiring a fairly IR-
localized third-generation quark doublet. Therefore, the contribution to
$\mathcal{S}t_{L}t_{R}$ coming from the mixing of both top chiralites are
similar, which leads to bigger changes in the cross section in the region of
DM masses between $1$ and $4$ TeV as one can see from figure 8 top-right
panel. On the other hand, bigger values of $\beta$ lead in general to a larger
mixing between fermion-zero modes and their KK resonances after EWSB,
increasing the effective coupling $y_{f\mathcal{S}}$ after diagonalization.
Therefore, in general, one expects a larger coannihilation cross section for
$m_{\chi}\lesssim 10$ TeV and increasing values of $\beta$. Changing $\beta$
also affects the $\mathcal{S}t_{L}t_{R}$ coupling indirectly, since
reproducing the observed quark masses results in different values of the mass
parameters $c$. This explains why the dashed blue line in the top-right panel
of figure 8 is below the other ones, since $c_{q_{L}}^{3}$ accidentally gets
close to zero and thus reduces the left-handed doublet contribution to the
$\mathcal{S}t_{L}t_{R}$ coupling, as can be seen in equation (46). Finally,
note that the abrupt deep around $m_{\chi}\sim 8$ TeV is due to the zero in
$y_{\chi\mathcal{S}}$ shown in figure 6. Indeed, the cross section should
exactly vanish at this point, but our numerical scan is unable to capture such
an steep behavior.

In the bottom panels of figure 8, on the other hand, we show $\langle\sigma
v\rangle$ for different values of $\lambda_{S}/r$ as a function of $m_{\chi}$.
In both bottom panels, we fix $\beta=2$, $y_{\ast}=3$ for both third-
generation quark sectors, as well as $c_{t_{R}}=-0.2$. The left-bottom panel
corresponds to the choice $\sin\theta_{h\mathcal{S}}=10^{-5}$, whereas for the
bottom-right one we take $\sin\theta_{h\mathcal{S}}=10^{-6}$. By reducing the
mixing, one effectively suppress the Higgs mediated contribution to the
coannihilation cross section, which is mostly relevant for small DM masses
and, in particular, around $m_{\chi}\approx m_{h}/2$. This will have an impact
on direct detection as we will see later, since the Higgs provides the leading
contribution to such experiments, and larger values of
$\sin\theta_{h\mathcal{S}}$ will typically lead to more severe bounds from
these experiments. The parameter $\lambda_{S}/r$ controls the effective Yukawa
coupling of the $\mathcal{S}$ scalar to fermions $y_{\chi\mathcal{S}}$, see
equations (46) and (52). We consider $\lambda_{S}/r=50$ (red),
$\lambda_{S}/r=100$ (pink) and $\lambda_{S}/r=150$ (blue). Increasing
$\lambda_{S}/r$ has the effect of increasing the coannihilation cross section
in general, besides for values of $m_{\chi}\lesssim m_{\mathcal{S}}/2$ where
the rise in the coupling is offset by the increase of its decay width. One
should note that the resonant-like peak starting around $7-8$ TeV is not only
due to the $\mathcal{S}$ resonance but also to the fact that new heavy-light
final states become kinematically accessible in the coannihilation. They
consist of a first KK fermion resonance of mass $\sim 15$ TeV together with a
SM-like fermion. We do not show values of $m_{\chi}$ beyond $\sim 15$ TeV
since the DM mass can not be made heavier than this value for
$M_{\text{KK}}=5$ TeV. One could entertain the possibility of adding brane-
localized masses or kinetic terms for this to happen, but for the sake of
concreteness we do not explore such possibilities here. At any rate, for such
large values of $m_{\chi}$, one would need to eventually include the decays of
$\mathcal{S}$ to a pair of low-lying KK fermions, which will make
$\mathcal{S}$ much wider of what is sensible in a perturbative theory.

In addition, we display for comparison the contribution due to diagrams
mediated by the first KK graviton, which are also irreducible in models with
WEDs (see e.g. Arcadi:2019lka ; Lee:2013bua  for useful expressions). We can
see that, for the chosen values of $M_{\text{KK}}$ and $k/m_{\rm Pl}$,
corresponding to $M_{\text{KK}}=5$ TeV and $\Lambda_{\pi}=m_{\rm Pl}e^{-k\pi
r}=40$ TeV, the contribution of the odd scalar resonance $\mathcal{S}$
dominates over the KK graviton one. In particular, this happens for all values
of $m_{\chi}$, with the exception of the small region where the coupling
$y_{\chi\mathcal{S}}$ goes to zero. The relative importance of each
contribution and the location of the graviton peak can be changed by modifying
the ratio $\Lambda_{\pi}/M_{\text{KK}}$ and/or by including brane kinetic
terms Davoudiasl:2003zt . We will not explore such possibilities, being our
aim here to show that the scalar contribution can naturally be the leading
one, as one can readily see from the figure.

Finally, we also show the values of the velocity averaged cross section for
which the observed DM relic abundance is reproduced, both in the usual
scenario and in the case of an early period of matter domination. In
particular, we show in dashed black the values of $\langle\sigma v\rangle$ for
which a value of $\Omega_{\chi}h^{2}=0.12$ is reproduced, in the case of a
regular freeze-out mechanism, and in the scenario of matter domination in
gray, for $\tau=0.99$, $T_{\star}=10^{5}$ GeV and two values of $T_{\rm RH}$,
1 and $10^{2}$ GeV, respectively. The lines in dot-dashed gray correspond to
regions where $x_{f}<3$, where the DM is expected to decouple relativistically
and the current treatment loses validity, see Hamdan:2017psw ; Chanda:2019xyl
for more details. We can see that the observed relic abundance can be
reproduced in the case of matter domination for masses $m_{\chi}\sim 8-10$
TeV. In the usual case of radiation domination, $\langle\sigma v\rangle$ can
be a non-negligible fraction of the one which is required to reproduce the
observed relic abundance for $m_{\chi}\sim 15$ TeV, which is in the ballpark
of the naturally expected fermion masses.

###  IV.4 Direct detection

Direct detection experiments can also set very important constraints on the
parameter space in scalar-mediated models of DM. Indeed, they constraint all
the parameter space in the case of Higgs-mediated DM, with the exception of a
small region around the Higgs resonance, see e.g. Aghanim:2018eyx . We study
here the constraints from direct detection experiments in our model. In
particular we will compare our predictions with results from Xenon1T
DiGangi:2018xht ; Aprile:2018dbl . We are interested in the spin-independent
cross section

$\sigma_{\chi
N}\approx\dfrac{4}{\pi}\mu^{2}_{\chi}\left[Zf_{p}+(A-Z)f_{n}\right]^{2}\simeq\dfrac{4}{\pi}\mu^{2}_{\chi}A^{2}f_{n}^{2},$
(58)

with $Z$ and $A$ the atomic number and atomic mass of the target nucleus,
respectively, and $\mu_{\chi N}$ the reduced mass of the DM and nucleus system
Kim:2006af ; Kanemura:2010sh ; Arcadi:2019lka . In order to compute such cross
section we use following effective Lagrangian

$\mathcal{L}_{\text{eff}}=f_{p}(\bar{\chi}\chi)(\bar{p}p)+f_{n}(\bar{\chi}\chi)(\bar{n}n).$
(59)

The terms $f_{p}$ and $f_{n}$ are effective coupling constants and can be
written as

$\frac{f_{p,n}}{m_{p,n}}=\sum_{q=u,d,s}f_{Tq}^{(p,n)}\frac{\alpha_{q}}{m_{q}}+\frac{2}{27}f_{Tg}^{(p,n)}\sum_{q=c,b,t}\frac{\alpha_{q}}{m_{q}},$
(60)

where $\alpha_{q}$ stands for the effective four-fermion interaction vertex,
obtained by considering the scalar t–channel exchange. In our model
$\alpha_{q}$ has the following form

$\alpha_{q}=y_{\chi\mathcal{S}}\left\\{\dfrac{y_{q\mathcal{S}}}{m_{\mathcal{S}}^{2}}+\dfrac{y_{qh}\sin\theta_{h\mathcal{S}}}{m_{h}^{2}}\right\\}.$
(61)

Figure 9: Velocity averaged coannihilation cross section at the freeze-out
temperature for different values of the mixing between the odd scalar and the
Higgs boson,
$\sin\theta_{h\mathcal{S}}=\\{10^{-3},10^{-4},10^{-5},10^{-6}\\}$, from top
left to bottom right. The first two cases correspond to negative values of
$\bar{\lambda}$. We have set $N_{\chi}=1$, $M_{\text{KK}}=5$ TeV and $k=m_{\rm
Pl}/8$. We show in yellow the predictions for two different benchmarks with
different values of $y_{\ast}$ and $\lambda_{S}/r$. In both cases, we have
fixed $c_{t_{R}}=-0.2$ and $\beta=2$. We show the constraints coming from the
Higgs invisible decay width in gray and the limits from Xenon1T in purple. We
show the velocity averaged cross section reproducing the relic density
experimental value from Planck in dashed black, and the equivalent for a
matter dominated freeze out in gray, for two different values of $T_{\rm RH}$,
where we used $T_{\star}=10^{5}$ GeV and $\tau=0.99$. For these lines the
section in dot-dashed gray corresponds to predictions for which $x_{f}<3$, and
therefore in this regime the DM decouples relativistically Hamdan:2017psw ;
Chanda:2019xyl .

Finally, $f_{Tg}^{(p,n)}$ is defined as

$f_{Tg}^{(p,n)}=1-\sum_{q=u,d,s}f_{Tq}^{(p,n)},$ (62)

and the values for $f^{q}_{p}$ and $f^{q}_{n}$ are Hoferichter:2015dsa ;
Arcadi:2019lka

$\displaystyle f^{u}_{p}=(20.8\pm 1.5)\cdot 10^{-3},$ $\displaystyle
f^{d}_{p}=(41.1\pm 2.8)\cdot 10^{-3},$ (63) $\displaystyle f^{u}_{n}=(18.9\pm
1.4)\cdot 10^{-3},$ $\displaystyle f^{d}_{n}=(45.1\pm 2.7)\cdot 10^{-3},$
$\displaystyle f^{s}_{p}=f^{s}_{n}=0.043\pm 0.011.$

One can compare the contribution of each scalar to the direct detection cross
section by computing the ratio between the terms appearing in equation (61).
We find that the channel mediated by the Higgs boson is dominant provided that

$\sin\theta_{h\mathcal{S}}>\dfrac{m_{h}^{2}}{m_{\mathcal{S}}^{2}}\dfrac{y_{q\mathcal{S}}}{y_{qh}}\sim
10^{-7},$ (64)

i.e. we expect the Higgs mediated interaction to be the leading contribution
for $\sin\theta_{h\mathcal{S}}>10^{-7}$. This tells us in particular that we
can relax the constraints coming from direct detection by making the mixing
smaller, while keeping the same coupling $y_{\chi\mathcal{S}}$ to the DM
fermions. However, this is only possible up to the point when the odd scalar
contribution becomes dominant,

$(\alpha_{q})_{\min}\approx\frac{y_{\chi\mathcal{S}}y_{q\mathcal{S}}}{m_{\mathcal{S}}^{2}}.$
(65)

We show in figure 9 the constraints coming from direct detection and invisible
Higgs decays for the velocity averaged coannihilation cross section
$\langle\sigma v\rangle$ as a function of $m_{\chi}$. We used $N_{\chi}=1$,
$M_{\text{KK}}=5$ TeV, $k=m_{\rm Pl}/8$ and different mixing values between
the odd scalar and the Higgs boson,
$\sin\theta_{h\mathcal{S}}=\\{10^{-3},10^{-4},10^{-5},10^{-6}\\}$, from top
left to bottom right. The first two mixing angles can only be achieved for
$\bar{\lambda}<0$, whereas the last two can be obtained for positive and
negative values of $\bar{\lambda}$. We display in each figure two different
benchmarks, corresponding to the choices $y_{\ast}=3$ (solid line) and
$y_{\ast}=1.5$ (dashed line) for the third generation quarks $t$ and $b$ (as
before, light generations have identical bulk mass parameters in absolute
value and different values of $y_{\ast}$, starting with $y_{\ast}^{s}=1/2$).
In both cases, we have set $\beta=2$ and $c_{t_{R}}=-0.2$, while
$\lambda_{S}/r$ has been chosen in such a way that
$\Gamma_{\mathcal{S}}/m_{\mathcal{S}}\approx 0.7$. More specifically, we have
taken $\lambda_{S}/r=120$ and $\lambda_{S}/r=65$, for $y_{\ast}=3$ and
$y_{\ast}=1.5$, respectively. Since the width is mostly given by the decay of
$\mathcal{S}$ into a third generation quark and its first KK resonance, such
assignment ensures that the overall coupling of the odd scalar field to the
visible sector is roughly the same in both cases. However, the smaller value
of $y_{\ast}$ in the benchmark $\\{y_{\ast}=1.5,~{}\lambda_{S}/r=65\\}$ leads
to a more IR-localized third-generation left-handed doublet $q_{L}^{3}$ and to
a much larger coupling of $\mathcal{S}$ to $\bar{b}_{L}b_{R}$ and
$\bar{q}_{L}^{3}$ plus its first KK resonance, even with a smaller value of
$\lambda_{S}/r$. At the end of the day, however, the solid lines are above the
dashed ones for most values of $m_{\chi}$, since the DM coupling
$y_{\chi\mathcal{S}}$ is smaller by a factor $\sqrt{120/65}\sim 1.4$, which
makes up for the small differences existing among the couplings to the visible
sector. The differences between both benchmarks are magnified once the purely
$\mathcal{S}$-mediated channel, corresponding to the right column of figure 7,
is the most dominant one. This happens in particular for large DM masses
and/or small values of $\sin\theta_{h\mathcal{S}}$, as one can readily see by
comparing the different panels in figure 9.

The gray region shows the area excluded by the LHC experimental limits on the
Higgs invisible decay width, and in purple we show the Xenon1T constraints.
The latter are found by plotting the velocity averaged coannihilation cross
section obtained after rescaling $y_{\chi\mathcal{S}}$ such that $\sigma_{\chi
N}$ saturates the Xenon1T experimental bound, $\langle\sigma v\rangle_{\rm
Xenon1T}$. For the values of $\sin\theta_{h\mathcal{S}}$ shown in this figure,
the leading contribution to the DM-nucleon cross section is by far the one
arising from the $t$-channel exchange of a Higgs boson, with the exception of
the last case where $\sin\theta_{h\mathcal{S}}=10^{-6}$ and the $\mathcal{S}$
contribution, while still subleading, starts to be relevant. This explains why
the Xenon1T bound for the $\\{y_{\ast}=3,~{}\lambda_{S}/r=120\\}$ benchmark is
weaker than the limit obtained for $\\{y_{\ast}=1.5,~{}\lambda_{S}/r=65\\}$,
whenever the coannihilation cross section is dominated by the $\mathcal{S}$
contribution. Indeed, in the former case, the couplings of $\mathcal{S}$ to
the visible sector are slightly larger. This leads to a larger value of
$\langle\sigma v\rangle_{\rm Xenon1T}$ after rescaling $y_{\chi\mathcal{S}}$
and to a weaker bound from direct detection. When $\langle\sigma v\rangle$ is
dominated by the Higgs exchange, direct detection bounds become
indistinguishable for both benchmarks, since the Higgs couplings to the SM
quarks are mostly fixed and SM-like.

We also show the velocity averaged cross section reproducing the observed
relic density both in the usual freeze-out scenario (dashed black) and in the
case of an early period of matter domination, for values of $T_{\rm
RH}=10^{2}$ GeV (dark gray) and 1 GeV (light gray). For both gray lines, we
used $T_{\star}=10^{5}$ GeV and $\tau=0.99$. Similarly to figure 8, lines in
dot-dashed gray correspond to regions where $x_{f}<3$ and the DM is expected
to decouple relativistically. We can see that for
$\sin\theta_{h\mathcal{S}}=10^{-3}$, one can not explain the observed relic
abundance without exceeding the bounds from Xenon1T. However, this is not the
case in the matter dominated scenario with $T_{\rm RH}=1$ GeV, where the
required coannihilation cross section to explain the DM relic abundance does
not exceed the Xenon1T bound for $y_{\ast}=3$. In the case of $y_{\ast}=1.5$,
the required cross section is excluded by the Xenon1T bound. In the case of
$\sin\theta_{h\mathcal{S}}=10^{-4}$ we can reproduce the correct amount of DM
for both values of $T_{\rm RH}$, in the scenario of matter domination, being
the values of $\langle\sigma v\rangle$ corresponding to the top of the
resonant peak excluded by direct detection bounds. For even smaller values of
$\sin\theta_{h\mathcal{S}}$ like $10^{-5}$ or $10^{-6}$, the data from Xenon1T
never constrains the predictions for the coannihilation cross section obtained
in both benchmarks, since the Higgs coupling to DM $y_{\chi h}$ becomes too
small. Therefore, by assuming an early period of matter domination, we are
able to explain the observed DM relic abundance for moderately small values of
$\sin\theta_{h\mathcal{S}}$ without conflicting current direct detection
experiments. Even in the case of radiation domination, we can get to values of
$\langle\sigma v\rangle$ relatively close to the ballpark of what is needed,
expecting $\mathcal{S}$ to be a non-negligible fraction of the required
coannihilation cross section, even though additional mediators accounting for
most of the coannihilation are certainly needed.

##  V Summary

We have demonstrated that the addition of a $\mathbb{Z}_{2}$-odd scalar field
developing a VEV in extra-dimensional models can not only account for the
origin of the 5D fermion masses, but also provide a unique window into any 5D
fermionic dark sector. Indeed, since such a scalar field generates dynamically
fermion bulk masses through Yukawa-like interactions with the different 5D
fermions, it will also irrevocably connect the SM with any possible dark
sector featuring bulk fermions. Moreover, in realistic models the Higgs scalar
field propagates into the bulk of the WED, and thus a mixing with the new
scalar field is unavoidable. In this work, we have studied in detail the
phenomenological consequences of such a portal, showing that the lightest KK
dark fermion is stable and can coannihilate efficiently thanks to the
mediation of the odd-scalar resonances as well as the Higgs boson. Indeed, we
have demonstrated that it is possible to reproduce the observed DM relic
abundance for an $\mathcal{O}(10)$ TeV KK dark fermion assuming that freeze-
out occurs during an early period of matter domination, without conflicting
with current data from direct-detection experiments. Even in the regular case
of a radiation dominated freeze-out, this irreducible contribution to the
coannihilation cross-section can account for a non-negligible part of the
required value when the DM mass is $\sim 15$ TeV. We have also shown that
these scalar contributions to the coannihilation cross section can be more
important than those arising from the exchange of KK gravitons. The bounds
arising from direct detection are only relevant when the parameter
$\sin\theta_{h\mathcal{S}}$ controlling the mixing between the SM-like Higgs
boson and the first KK resonance $\mathcal{S}$ of the $\mathbb{Z}_{2}$-odd
scalar field is $\gtrsim 10^{-4}$. For smaller values, the contribution to the
direct-detection cross section given by $t$-channel Higgs exchange becomes
less and less important, to the point of becoming of the same order as the one
from the $t$-channel exchange of the $\mathcal{S}$ resonance, which is beyond
the reach of current direct detection experiments.

We have also studied the impact of the scalar mixing on precision measurements
of Higgs couplings. In particular, we have computed the modifications of the
Higgs couplings to electroweak gauge bosons and the bottom quark as a
consequence of the mixing between the SM-like Higgs boson and the first KK
resonances of both bulk fields, $\mathcal{H}$ and $\mathcal{S}$. We have
demonstrated that planned future colliders could probe the induced
modifications on the $b$-quark Yukawa in the case where $\beta\lesssim 4$,
values for which the Higgs boson has a strong presence into the bulk. We have
also studied the constraints on the Higgs effective Yukawa coupling to DM when
its mass is light enough to allow for the Higgs boson to decay into a pair of
DM particles. We conclude that the effective Yukawa coupling to the dark
fermions $y_{\chi h}\lesssim 0.02/\sqrt{N_{\chi}}$, with $N_{\chi}$ being the
multiplicity of the 5D dark fermion.

In summary, we have shown that models with a WED naturally feature a
compelling explanation for the observed relic abundance of DM, consisting of
an $\mathcal{O}(10)$ TeV fermionic WIMP coupled to the SM by a heavy scalar
mediator $\mathcal{S}$ with mass $m_{\mathcal{S}}\sim 30$ TeV. All this is
possible without conflicting with current data from colliders, flavor
experiments and cosmology and while still providing natural solutions to the
hierarchy problem and the flavor puzzle, which are arguably two of the most
important theoretical problems in particle physics.

###### Acknowledgements.

We would like to thank Miki Chala for useful comments on the manuscript, as
well as Lukas Mittnacht and Eric Madge for helpful discussions. M.N. thanks
Gino Isidori, the particle theory group at Zurich University and the Pauli
Center for hospitality during a sabbatical stay. A.C. acknowledges funding
from the European Union’s Horizon 2020 research and innovation programme under
the Marie Skłodowska-Curie grant agreement No. 754446 and UGR Research and
Knowledge Transfer Found – Athenea3i. The research of J.C. and M.N. is
supported by the Cluster of Excellence Precision Physics, Fundamental
Interactions and Structure of Matter (PRISMA+ – EXC 2118/1) within the German
Excellence Strategy (project ID 39083149) and under grant 05H18UMCA1 of the
German Federal Ministry for Education and Research (BMBF).

##  Appendix A Fermion equations of motion

The Yukawa interaction between the odd bulk scalar $S_{1}$ and a
$\bar{\Psi}_{a}\Psi_{A}$ pair, corresponding to a fermion zero mode (a) and a
first KK mode (A), comes from the term generating the different fermion bulk
masses

$S\supset-\int d^{5}x\sqrt{g}\,\mathcal{Y}_{a}\bar{\Psi}_{a}\Psi_{A}\Sigma.$
(66)

In this case, the EOM for a fermion field with a 5D bulk mass generated
dynamically reads

$\left[\pm t\partial_{t}-c_{a}\,v_{S}(t)\right]f_{0,a}^{L,R}(t)=0,$ (67)

for the zero mode and

$\displaystyle\left[t^{2}\partial_{t}^{2}\right.$
$\displaystyle+x_{n}^{2}t^{2}\mp c_{a}\,t\,v_{S}^{\prime}(t)$ (68)
$\displaystyle\left.+c_{a}\,v_{S}(t)\left(\pm
1-c_{a}\,v_{S}(t)\right)\right]f_{n,A}^{L,R}(t)=0,$

for the $n$-th KK mode Ahmed:2019zxm , where $c_{a}$ is the usual
dimensionless 5D mass. In the scenario at hand, it is defined by Ahmed:2019zxm

$c_{a}\equiv\mathcal{Y}_{a}\sqrt{\dfrac{6}{\lambda_{S}}}\dfrac{\mu_{S}}{k}.$
(69)

Here, we have used the following KK decomposition for the fermions

$\Psi=\sum_{n=0}\Psi_{n}(x)\left(\dfrac{t}{\epsilon}\right)^{2}\sqrt{M_{\text{KK}}}\,f_{n}(t),$
(70)

satisfying

$2\int_{\epsilon}^{1}f_{m}^{A*}(t)f_{n}^{B}(t)=\delta_{mn}.$ (71)

The 5D Yukawa interaction leads to the effective 4D vertex
$S_{1}\bar{\Psi}_{a}\Psi_{A}$, with effective Yukawa

$y_{aA\mathcal{S}}=2\,c_{a}\sqrt{\dfrac{\lambda_{S}}{r}}\dfrac{k}{\mu_{S}}\int_{\epsilon}^{1}dtf_{a}(t)f_{A}(t)\chi_{1}^{S}(t).$
(72)

The coupling to a VL fermion $\chi$, corresponding to the first KK resonance
of some 5D fermion, is given by

$y_{\chi\mathcal{S}}=2\,c_{a}\sqrt{\dfrac{\lambda_{S}}{r}}\dfrac{k}{\mu_{S}}\int_{\epsilon}^{1}dtf_{1,\chi}^{L}(t)f_{1,\chi}^{R}(t)\chi_{1}^{S}(t).$
(73)

##  Appendix B Cross section expressions

The different coannihilation cross sections of $\bar{\chi}\chi$ into a pair of
SM particles are given by (see e.g. Chanda:2019xyl )

$\displaystyle\sigma(\bar{\chi}\chi\to\bar{f}f)$
$\displaystyle=\dfrac{N_{\\!f}\,y_{\chi\mathcal{S}}^{2}}{16\pi}s\left(1-\dfrac{4m_{f}^{2}}{s}\right)^{3/2}\left(1-\dfrac{4m_{\chi}^{2}}{s}\right)^{1/2}\left[\dfrac{y_{fh}^{2}\sin^{2}\theta_{h\mathcal{S}}}{\left(s-m_{h}^{2}\right)^{2}}+\dfrac{y_{f\mathcal{S}}^{2}}{\left(s-m_{\mathcal{S}}^{2}\right)^{2}}+\dfrac{2\,y_{fh}y_{f\mathcal{S}}\sin\theta_{h\mathcal{S}}}{\left(s-m_{\mathcal{S}}^{2}\right)\left(s-m_{h}^{2}\right)}\right],$
(74) $\displaystyle\sigma(\bar{\chi}\chi\to hh)$
$\displaystyle=\dfrac{y_{\chi\mathcal{S}}^{2}}{32\pi}\left(1-\dfrac{4m_{h}^{2}}{s}\right)^{1/2}\left(1-\dfrac{4m_{\chi}^{2}}{s}\right)^{1/2}\left[\dfrac{9\sin^{2}\theta_{h\mathcal{S}}m_{h}^{4}}{v_{4}^{2}\left(s-m_{h}^{2}\right)^{2}}+\dfrac{\sin^{2}\theta_{h\mathcal{S}}x_{S_{1}}^{4}M_{\text{KK}}^{4}}{v_{4}^{2}\left(s-m_{\mathcal{S}}^{2}\right)^{2}}+\dfrac{6\,m_{h}^{2}\sin^{2}\theta_{h\mathcal{S}}x_{S_{1}}^{2}M_{\text{KK}}^{2}}{v_{4}^{2}\left(s-m_{\mathcal{S}}^{2}\right)\left(s-m_{h}^{2}\right)}\right],$
$\displaystyle\sigma(\bar{\chi}\chi\to VV)$
$\displaystyle=\dfrac{\delta_{V}y_{\chi\mathcal{S}}^{2}\sin^{2}\theta_{h\mathcal{S}}}{8\pi}\left(1-\dfrac{4m_{V}^{2}}{s}\right)^{1/2}\left(1-\dfrac{4m_{\chi}^{2}}{s}\right)^{1/2}\dfrac{m_{V}^{4}}{v_{4}^{2}\left(s-m_{h}^{2}\right)^{2}}\left[2+\dfrac{(s-2m_{V}^{2})^{2}}{4m_{V}^{4}}\right],$
$\displaystyle\sigma(\bar{\chi}\chi\to\bar{Q}q)$
$\displaystyle=\dfrac{N_{\\!Q}\,y_{\chi\mathcal{S}}^{2}y_{Qq}^{2}}{16\pi}\dfrac{s}{\left(s-m_{\mathcal{S}}^{2}\right)^{2}}\left(1-\dfrac{m_{Q}^{2}}{s}\right)^{3/2}\left(1-\dfrac{4m_{\chi}^{2}}{s}\right)^{1/2},$

with $\delta_{V}=1,1/2$ for $V=W^{\pm},Z$.

##  Appendix C Matter dominated freeze-out

We review here the relevant formulae for the calculation of the current DM
relic abundance after freeze-out during an early period of matter domination
(see Hamdan:2017psw ; Chanda:2019xyl  for more details).

We assume the presence of a long-lived heavy scalar field $\phi$, localized on
the UV brane, which starts behaving like matter at a critical temperature
$T_{\star}\sim m_{\phi}$, which we assumme to be much larger than
$M_{\text{KK}}$, i.e., $T_{\star}\gg M_{\text{KK}}$. If $\phi$ is long-lived
enough, its contribution to the energy density $\rho_{\phi}$ will grow until
ultimately monopolize the total energy density regardless of its initial
contribution at $T_{\star}$, given by $(1-\tau)$, with $\tau\in[0,1]$

$\tau=\left.\frac{\rho_{R}+\rho_{\chi}}{\rho_{R}+\rho_{\chi}+\rho_{\phi}}\right|_{T=T_{\star}},$
(75)

and $\rho_{\chi}$, $\rho_{R}$, the energy density of DM and the visible sector
of the extra-dimensional theory.

Using the Friedmann equation

$\displaystyle\begin{aligned} H^{2}=\frac{1}{3m_{\rm
Pl}^{2}}\left[\rho_{R}+\rho_{\chi}+\rho_{\phi}\right]\\\ \end{aligned}$ (76)

and defining $H_{\star}=H(T_{\star})$ we obtain

$H^{2}=H_{\star}^{2}\left[\frac{g_{\ast}\tau}{g_{\ast}+g_{\chi}}\left(\frac{a_{\star}}{a}\right)^{4}+(1-\tau)\left(\frac{a_{\star}}{a}\right)^{3}+\frac{g_{\chi}\tau}{g_{\ast}+g_{\chi}}\left(\frac{a_{\star}}{a}\right)^{4}\right],$
(77)

where $g_{\ast}$ is the effective number of relativistic degrees of freedom of
the visible sector of the extra-dimensional theory. Assuming that the entropy
is conserved in this sector and taking into account that $g_{\ast}\gg
g_{\chi}$, we can write

$H=H_{\star}\sqrt{1-\tau}\left(\frac{x_{\star}}{x}\right)^{3/2}\left[\frac{r}{1-\tau}\left(\frac{x_{\star}}{x}\right)+1\right]^{1/2},$
(78)

where we have defined $x=m_{\chi}/T$ and $x_{\star}=x(T_{\star})$.

The annihilation rate $\Gamma_{\rm ann}$ can be written as

$\Gamma_{\rm ann}=\frac{g_{\chi}m_{\chi}^{3}}{(2\pi
x)^{3/2}}e^{-x}\langle\sigma v\rangle$ (79)

where $\langle\sigma v\rangle$ is the velocity-averaged coannihilation cross
section. In our scenario of scalar mediated fermionic DM, after Taylor
expanding $\left(\sigma v\right)$ with respect to $v_{r}^{2}$, we can write

$\left(\sigma v\right)\simeq a+bv_{r}^{2}=bv_{r}^{2},$ (80)

since $a=0$ as we have seen in the previous section. Therefore, one can write
the velocity-averaged coannihilation cross section as

$\left\langle\sigma v\right\rangle\simeq\dfrac{6b}{x},$ (81)

which leads to

$\displaystyle\Gamma_{\rm
ann}=\frac{g_{\chi}m_{\chi}^{3}}{(2\pi)^{3/2}}e^{-x}\frac{6b}{x^{5/2}}.$ (82)

We can define the freeze-out temperature by asking $H(x_{f})=\Gamma_{\rm
ann}(x_{f})$, with $x_{f}=m_{\chi}/T_{f}$. This leads to

$\displaystyle
x_{f}=\log\left(\frac{g_{\chi}}{(2\pi)^{3/2}}\frac{m_{\chi}^{3}6b}{H_{\star}x_{\star}^{3/2}}(1-\tau)^{-1/2}\left(x_{f}^{2}+\dfrac{rx_{\star}}{1-\tau}x_{f}\right)^{1/2}\right).$
(83)

The yield $Y=n_{\chi}/s$, with $s$ the entropy density, is given at freeze-out
by

$Y_{f}=\left(\lambda\int^{\infty}_{x_{f}}dx\left(1+\dfrac{\tau}{1-\tau}\dfrac{x_{\star}}{x}\right)^{-1/2}x^{-7/2}\right)^{-1},$
(84)

where

$\lambda=\dfrac{2\pi^{2}g_{\ast
S}m_{\chi}^{3}6b}{45H_{\star}x_{\star}^{3/2}},$ (85)

where $g_{\ast S}$ is the number of effective entropic degrees of freedom.
More explicitly, it reads

$\displaystyle\frac{1}{Y_{f}}\approx\frac{\pi^{2}g_{\ast
S}m_{\chi}^{3}\langle\sigma
v\rangle}{90H_{\star}x_{\star}^{4}}\left(\frac{1-\tau}{\tau}\right)^{5/2}$
$\displaystyle\Bigg{(}3x_{f}\sinh^{-1}\left(\sqrt{\frac{x_{\star}\tau}{x_{f}(1-\tau)}}\right)$
(86)
$\displaystyle+\sqrt{\dfrac{x_{\star}}{x_{f}}}\left(\dfrac{\tau}{1-\tau}\right)^{3/2}\sqrt{1+\frac{x_{\star}\tau}{x_{f}(1-\tau)}}\left[2x_{\star}-3x_{f}\frac{(1-\tau)}{\tau}\right]\Bigg{)},$

where we have approximated $\langle\sigma v\rangle\approx 6b/x_{f}$. Finally,
the prediction for the DM relic abundance, assuming a matter dominated
universe during freeze-out, reads

$\Omega_{\chi}h^{2}=\zeta\dfrac{s_{0}m_{\chi}Y_{f}}{\rho_{\text{critical}}},$
(87)

where $\rho_{\text{critical}}=8.13\cdot 10^{-47}$ GeV4, $s_{0}\simeq 2.29\cdot
10^{-38}$ GeV3. In the above equation, $\zeta$ parametrizes the dilution of
the DM abundance after freeze-out due to the decays of $\phi$ and the
subsequent entropy injection

$\displaystyle\zeta=\frac{s_{\rm before}}{s_{\rm
after}}=\frac{\Omega_{\chi}}{\Omega_{\chi}^{f}},$ (88)

which can be expressed with good approximation as follows

$\zeta\approx\dfrac{45}{4\pi^{3}}\dfrac{1}{(1-\tau)g_{\ast}}\dfrac{T_{\text{RH}}}{T_{\star}}.$
(89)

## References

  * (1) J. M. Maldacena, _The Large N limit of superconformal field theories and supergravity_ , _Int. J. Theor. Phys._ 38 (1999) 1113–1133, [hep-th/9711200].
  * (2) S. S. Gubser, I. R. Klebanov and A. M. Polyakov, _Gauge theory correlators from noncritical string theory_ , _Phys. Lett._ B428 (1998) 105–114, [hep-th/9802109].
  * (3) N. Arkani-Hamed, M. Porrati and L. Randall, _Holography and phenomenology_ , _JHEP_ 08 (2001) 017, [hep-th/0012148].
  * (4) R. Rattazzi and A. Zaffaroni, _Comments on the holographic picture of the Randall-Sundrum model_ , _JHEP_ 04 (2001) 021, [hep-th/0012248].
  * (5) M. Perez-Victoria, _Randall-Sundrum models and the regularized AdS / CFT correspondence_ , _JHEP_ 05 (2001) 064, [hep-th/0105048].
  * (6) R. Contino and A. Pomarol, _Holography for fermions_ , _JHEP_ 11 (2004) 058, [hep-th/0406257].
  * (7) T. Gherghetta, _A Holographic View of Beyond the Standard Model Physics_ , in _Physics of the large and the small, TASI 09, proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics, Boulder, Colorado, USA, 1-26 June 2009_ , pp. 165–232, 2011\.  1008.2570.  DOI.
  * (8) Y. Grossman and M. Neubert, _Neutrino masses and mixings in nonfactorizable geometry_ , _Phys. Lett._ B474 (2000) 361–371, [hep-ph/9912408].
  * (9) T. Gherghetta and A. Pomarol, _Bulk fields and supersymmetry in a slice of AdS_ , _Nucl. Phys._ B586 (2000) 141–162, [hep-ph/0003129].
  * (10) A. Ahmed, A. Carmona, J. Castellano Ruiz, Y. Chung and M. Neubert, _Dynamical origin of fermion bulk masses in a warped extra dimension_ , _JHEP_ 08 (2019) 045, [1905.09833].
  * (11) H. Davoudiasl, B. Lillie and T. G. Rizzo, _Off-the-wall Higgs in the universal Randall-Sundrum model_ , _JHEP_ 08 (2006) 042, [hep-ph/0508279].
  * (12) G. Cacciapaglia, C. Csaki, G. Marandella and J. Terning, _The Gaugephobic Higgs_ , _JHEP_ 02 (2007) 036, [hep-ph/0611358].
  * (13) A. Azatov, M. Toharia and L. Zhu, _Higgs Mediated FCNC’s in Warped Extra Dimensions_ , _Phys. Rev._ D80 (2009) 035016, [0906.1990].
  * (14) L. Vecchi, _A Natural Hierarchy and a low New Physics scale from a Bulk Higgs_ , _JHEP_ 11 (2011) 102, [1012.3742].
  * (15) P. R. Archer, _The Fermion Mass Hierarchy in Models with Warped Extra Dimensions and a Bulk Higgs_ , _JHEP_ 09 (2012) 095, [1204.4730].
  * (16) R. Malm, M. Neubert, K. Novotny and C. Schmell, _5D Perspective on Higgs Production at the Boundary of a Warped Extra Dimension_ , _JHEP_ 01 (2014) 173, [1303.5702].
  * (17) P. R. Archer, M. Carena, A. Carmona and M. Neubert, _Higgs Production and Decay in Models of a Warped Extra Dimension with a Bulk Higgs_ , _JHEP_ 01 (2015) 060, [1408.5406].
  * (18) P. Gondolo and G. Gelmini, _Cosmic abundances of stable particles: Improved analysis_ , _Nucl. Phys._ B360 (1991) 145–179.
  * (19) S. Hamdan and J. Unwin, _Dark Matter Freeze-out During Matter Domination_ , _Mod. Phys. Lett. A_ 33 (2018) 1850181, [1710.03758].
  * (20) P. Chanda, S. Hamdan and J. Unwin, _Reviving $Z$ and Higgs Mediated Dark Matter Models in Matter Dominated Freeze-out_, _JCAP_ 01 (2020) 034, [1911.02616].
  * (21) L. Randall and R. Sundrum, _A Large mass hierarchy from a small extra dimension_ , _Phys. Rev. Lett._ 83 (1999) 3370–3373, [hep-ph/9905221].
  * (22) F. Mahmoudi, U. Maitra, N. Manglani and K. Sridhar, _A Higgs in the Warped Bulk and LHC signals_ , _JHEP_ 11 (2016) 075, [1608.07407].
  * (23) XENON collaboration, P. Di Gangi, _First dark matter search results of the XENON1T experiment_ , _Nuovo Cim. C_ 41 (2018) 109.
  * (24) XENON collaboration, E. Aprile et al., _Dark Matter Search Results from a One Ton-Year Exposure of XENON1T_ , _Phys. Rev. Lett._ 121 (2018) 111302, [1805.12562].
  * (25) M. Aaboud, G. Aad, B. Abbott, O. Abdinov, B. Abeloos, D. Abhayasinghe et al., _Observation of $H\to b\bar{b}$ decays and $VH$ production with the ATLAS detector_, _Physics Letters B_ 786 (2018) 59 – 86.
  * (26) J. de Blas et al., _Higgs Boson Studies at Future Particle Colliders_ , _JHEP_ 01 (2020) 139, [1905.03764].
  * (27) H. Davoudiasl, J. Hewett and T. Rizzo, _Bulk gauge fields in the Randall-Sundrum model_ , _Phys. Lett. B_ 473 (2000) 43–49, [hep-ph/9911262].
  * (28) A. Pomarol, _Gauge bosons in a five-dimensional theory with localized gravity_ , _Phys. Lett. B_ 486 (2000) 153–157, [hep-ph/9911294].
  * (29) S. Casagrande, F. Goertz, U. Haisch, M. Neubert and T. Pfoh, _Flavor Physics in the Randall-Sundrum Model: I. Theoretical Setup and Electroweak Precision Tests_ , _JHEP_ 10 (2008) 094, [0807.4937].
  * (30) K. Agashe and G. Servant, _Warped unification, proton stability and dark matter_ , _Phys. Rev. Lett._ 93 (2004) 231805, [hep-ph/0403143].
  * (31) CMS collaboration, A. M. Sirunyan et al., _Search for invisible decays of a Higgs boson produced through vector boson fusion in proton-proton collisions at $\sqrt{s}=$ 13 TeV_, _Phys. Lett._ B793 (2019) 520–551, [1809.05937].
  * (32) Y. G. Kim and K. Y. Lee, _The Minimal model of fermionic dark matter_ , _Phys. Rev. D_ 75 (2007) 115012, [hep-ph/0611069].
  * (33) G. Arcadi, A. Djouadi and M. Raidal, _Dark Matter through the Higgs portal_ , _Phys. Rept._ 842 (2020) 1–180, [1903.03616].
  * (34) Planck collaboration, N. Aghanim et al., _Planck 2018 results. VI. Cosmological parameters_ , 1807.06209.
  * (35) H. M. Lee, M. Park and V. Sanz, _Gravity-mediated (or Composite) Dark Matter_ , _Eur. Phys. J. C_ 74 (2014) 2715, [1306.4107].
  * (36) H. Davoudiasl, J. Hewett and T. Rizzo, _Brane localized curvature for warped gravitons_ , _JHEP_ 08 (2003) 034, [hep-ph/0305086].
  * (37) S. Kanemura, S. Matsumoto, T. Nabeshima and N. Okada, _Can WIMP Dark Matter overcome the Nightmare Scenario?_ , _Phys. Rev. D_ 82 (2010) 055026, [1005.5651].
  * (38) M. Hoferichter, J. Ruiz de Elvira, B. Kubis and U.-G. Meißner, _High-Precision Determination of the Pion-Nucleon $\sigma$ Term from Roy-Steiner Equations_, _Phys. Rev. Lett._ 115 (2015) 092301, [1506.04142].