DESY 19-236

Radion-Activated Higgs Mechanism

Cem Eröncela,b, Jay Hubiszb, Gabriele Rigob

a DESY, Notkestrasse 85, D-22607 Hamburg, Germany

b Department of Physics, Syracuse University, Syracuse, NY 13244, USA

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
We study multi-scalar models of radius stabilization, with an eye towards
application to novel extra-dimensional models of symmetry breaking. With
inspiration from holography, we construct a multi-scalar effective potential
that is a function of UV-brane values of the scalar fields, and that takes
into account bulk gravitational backreaction. We study extrema of this
potential, and additionally provide a “superpotential” method for generating
static solutions for the extra-dimensional geometry. We apply these methods to
some simple models of the Higgs mechanism where the Higgs itself plays a non-
trivial role in radius stabilization. We conclude that mass mixing of the
Higgs and radion is generic unless additional symmetries are imposed. We focus
on models with moderate gap between the electroweak and Kaluza-Klein scale, as
required by phenomenological constraints. We note that tuning of the Higgs
mass relative to the KK scale is related to various classes of tuning of 5D
parameters, with different resulting spectra and phenomenologies.

##  1 Introduction

The mass of the Higgs is far smaller than dimensional analysis predicts given
that there are so far no experimental signatures of new physics indicating new
symmetries near the electroweak scale. This suggests that there may be some
degree of fine tuning of parameters to achieve consistency with observations.
Of course, the Higgs mass problem is not the only one of the Standard Model.
The SM offers no explanation of large fermion mass hierarchies, of dark
matter, of the matter/anti-matter asymmetry, etc. Strong dynamics, and/or
field theories in higher-dimensional spacetimes offer resolutions or at least
interesting reformulations of many of these problems, along with partially
ameliorating the Higgs hierarchy problem.

The Higgs mass couples to singlet sectors in many extensions of the
electroweak sector, which can put the the hierarchy problem into a new light
[1]. The question is no longer “Why is the Higgs light?” Rather, the correct
question is “Why does a small Higgs mass coincide with the global (or a
cosmologically metastable) minimum of the scalar potential?”

Such scalars are generic features in models with additional compact spatial
dimensions [2]. In these models, there are typically moduli that are gauge
singlet states with non-trivial couplings to matter required by higher
dimensional general covariance [3, 4]. Stabilization of these moduli is
typically non-trivial as the moduli can be interpreted as Goldstone bosons of
spontaneously broken spacetime symmetries. Providing a potential for them
often requires new scalar degrees of freedom whose dynamics serve to
explicitly break the spacetime symmetries, demoting the moduli to pseudo-
Goldstone bosons.

In this work, we explore the relationship between extra-dimensional radius
stabilization and the Higgs mass. In particular, the modulus of Randall-
Sundrum models (the “radion”) generally plays a crucial role in determining
the value of the Higgs vacuum expectation value. The same goes for the
reverse: the Higgs VEV itself will backreact on the geometry, and feed into
the total effective potential for the radion.

This first motivates the development of a formalism for dealing with multi-
scalar models of 5D RS model radius stabilization. This is the focus of the
first part of this work. We show that even in cases of complicated bulk
dynamics with many interacting scalar fields, the classical effective
potential lives on the boundary. Making the assumption that the 5D action is
of Einstein-Hilbert form with minimal couplings to bulk scalar fields, we
calculate the classical potential to all orders in the 5D gravitational
coupling constant.

The criteria for minimization of this effective potential determine aspects of
the physics such as the Kaluza-Klein scale, and the masses of light
resonances. We study these criteria, and provide conditions for extremization
of this potential that fully include geometric backreaction effects.

We then apply this formalism to a few illustrative examples of the Higgs
mechanism in RS models. Given the lack of signals for extra-dimensional
resonances in collider experiments, we look for models which achieve moderate
separation of scales between the electroweak scale and the scale associated
with the IR brane $f/v\gtrsim{\mathcal{O}}(10)$. The Higgs mass term is
generally a function of the brane separation. Thus, in order to achieve a
light Higgs, the minimum of the modulus potential must coincide with the
region where the Higgs mass is small. We expect (and find) that tuning is
involved in these cases. For example, such a coincidence problem may require
that the IR brane mistune (between the brane tension and the bulk cosmological
constant) be adjusted to high degree in order to keep the Higgs light. We
explore the type and degree of tuning required in a few examples in which
there is non-trivial interplay between electroweak symmetry breaking and
radius stabilization.

As noted in [5], fine tuning can have dramatic phenomenological consequences
when parameters of the Standard Model are set by the dynamics of a modulus
field. These consequences influence both early universe cosmology and collider
physics. Cosmology can be dramatically changed by interplay of Higgs and
modulus field, with oscillations of the coupled scalar system leading to
repercussions for gravity waves, and constraints on inflationary models. It is
with this in mind that we choose models whose parameter space explore the full
range of classes of mixed modulus-Higgs potential. The specifics of these
classes are expanded on in the introduction to Section 3.

The example models we study are as follows:

  * •

Higgs on the IR brane, Goldberger-Wise scalar stabilizing field in the bulk;

  * •

Higgs in the bulk, with mass near the Breitenlohner-Freedman bound [6],
stabilizing the geometry (studied previously in [7, 8]);

  * •

Both Higgs and Goldberger-Wise stabilizing field in the bulk.

For simplicity of presentation, these models are all studied to lowest non-
trivial order in the gravitational backreaction, where neither the Higgs or
Goldberger-Wise scalar develop VEVs that are comparable to the AdS curvature.

An important result from these studies is that mass mixing between the Higgs
and radion is completely generic. Mass mixing occurs in different ways:
through symmetry-allowed couplings between a Goldberger-Wise field and the
Higgs, or from mixing of the Higgs with the 5D gravity sector through
backreaction.111This is in addition to kinetic mixing, which can occur through
couplings of the Higgs to either the 5D curvature, or to the extrinsic
curvature of the IR brane [9]. It occurs both when the Higgs is purely
localized on the IR brane through couplings to the Goldberger-Wise field, or
when the Higgs is in the bulk of the extra dimension. This mixing is different
in the various classes of Higgs-radion potential, and influences the spectrum
of light scalar modes.

In Section 2 we discuss general properties of multi-scalar stabilization,
giving conditions under which the geometry is stabilized. In Section 3, we
explore the construction, phenomenology, and spectra of three different models
where the Higgs is coupled to the extra-dimensional radion. In Section 4 we
discuss a CFT interpretation of different multi-scalar stabilization models.
In Section 5 we conclude. In Appendix A, we include derivations of some
results in Section 2. In Appendix B, we give a presentation of a
“superpotential” method to generate static geometries with multiple scalar
fields.

##  2 General Properties of the Radion Potential

Before analyzing particular models, we first study some general properties of
radius stabilization when there is more than one 5D scalar field that has non-
vanishing vacuum expectation value. In such cases, the radion potential is
affected by backreaction of all of these VEVs onto the geometry. In what
follows, we consider arbitrary backreaction onto the geometry, however we
presume that the action is of Einstein-Hilbert form, with no higher curvature
operators.

The generic problem is one of $N$ real scalar fields minimally coupled to
gravity in 5D space with negative cosmological constant
$\Lambda_{5}=-6k^{2}/\kappa^{2}$. The bulk geometry can be described with
metric

$\displaystyle\differential{s^{2}}=e^{-2A(y)}\differential{x_{4}^{2}}-\differential{y^{2}},$
(2.1)

where we have presumed flat 4D slices. The space is cut off on both sides by
branes at positions $y_{0}$ and $y_{1}$.

All dimensionful quantities are understood to be expressed in units of the AdS
curvature: $k=1$. For perturbative control of the 5D gravity theory, the 5D
Newton constant must be small:
$\kappa^{2}=\frac{1}{2M_{5}^{3}}\sim{\mathcal{O}}(1/10)$. The action is:

$\displaystyle\begin{aligned}
S&=\int\differential[4]{x}\differential{y}\sqrt{g}\mathopen{}\mathclose{{}\left[\frac{1}{2}\sum_{i}(\partial_{M}\phi_{i})^{2}-V(\\{\phi_{i}\\})-\frac{1}{2\kappa^{2}}R}\right]\\\
&\phantom{{}={}}-\int\differential[4]{x}\sqrt{-g_{0}}V_{0}(\\{\phi_{i}\\})\evaluated{}_{y=y_{0}}-\int\differential[4]{x}\sqrt{-g_{1}}V_{1}(\\{\phi_{i}\\})\evaluated{}_{y=y_{1}},\end{aligned}$
(2.2)

where the scalar potential includes the bulk cosmological constant term.

The bulk Einstein equations relate derivatives of $A(y)$ to the scalar fields
in the following way:

$\displaystyle\begin{gathered}{A^{\prime}}^{2}=\frac{\kappa^{2}}{12}\sum_{i}\phi_{i}^{\prime
2}-\frac{\kappa^{2}}{6}V(\\{\phi_{i}\\}),\\\
A^{\prime\prime}=\frac{\kappa^{2}}{3}\sum_{i}{\phi_{i}^{\prime}}^{2}.\end{gathered}$
(2.5)

The scalar curvature is given by

$\displaystyle R=20A^{\prime 2}-8A^{\prime\prime}.$ (2.6)

In the absence of scalar field VEVs, or in the limit of small $\kappa^{2}$,
the space is AdS.

Plugging the result for the scalar curvature, and for the scalar terms into
the action above, we find that the bulk portion is a derivative, and thus can
be expressed as a boundary term:

$\displaystyle
S_{\text{bulk}}=\frac{2}{\kappa^{2}}\int\differential[4]{x}\differential{y}\frac{\partial}{\partial
y}\mathopen{}\mathclose{{}\left[e^{-4A(y)}A^{\prime}(y)}\right]=\frac{2}{\kappa^{2}}\int\differential[4]{x}\mathopen{}\mathclose{{}\left[e^{-4A(y_{0})}A^{\prime}(y_{0})-e^{-4A(y_{1})}A^{\prime}(y_{1})}\right].$
(2.7)

The factor of 2 is from integration over the circle. We note that on the
boundaries, there must be a jump in the derivative of the metric. This can be
seen most easily from the orbifold perspective, where on either side of an
orbifold fixed point, you must have $A(y_{0,1})_{+}=A(y_{0,1})_{-}$ and
$A^{\prime}(y_{0,1})_{+}=-A^{\prime}(y_{0,1})_{-}$. This leads to a term in
the scalar curvature $R$ which is a delta function, as you must have
$A^{\prime\prime}(y_{0,1})\supseteq\pm 2\delta(y_{0,1}-y)A^{\prime}(y_{0,1})$.
Inclusion of the delta function term in the scalar curvature gives additional
boundary contributions to the effective action.

Including all terms, the effective potential can now be expressed as a pure
boundary term [10]:

$\displaystyle
V_{\textnormal{eff}}=e^{-4A(y_{0})}\mathopen{}\mathclose{{}\left[V_{0}(\\{\phi_{i}(y_{0})\\})-\frac{6}{\kappa^{2}}A^{\prime}(y_{0})}\right]+e^{-4A(y_{1})}\mathopen{}\mathclose{{}\left[V_{1}(\\{\phi_{i}(y_{1})\\})+\frac{6}{\kappa^{2}}A^{\prime}(y_{1})}\right].$
(2.8)

where, at the moment, none of the boundary conditions have been imposed, nor
even the bulk scalar equations of motion, which are given by:

$\displaystyle\phi_{i}^{\prime\prime}=4A^{\prime}\phi_{i}^{\prime}+\frac{\partial
V}{\partial\phi_{i}}.$ (2.9)

The scalar boundary conditions are given (in these coordinates) as:

$\displaystyle\phi^{\prime}_{i}\evaluated{}_{y_{0},y_{1}}=\pm\frac{1}{2}\frac{\partial
V_{0,1}}{\partial\phi_{i}}\evaluated{}_{y_{0},y_{1}}.$ (2.10)

The boundary Einstein equations (the metric junction conditions that match the
geometry to the brane-localized stress-energy) are

$\displaystyle
A^{\prime}\evaluated{}_{y_{0},y_{1}}=\pm\frac{\kappa^{2}}{6}V_{0,1}\evaluated{}_{y_{0},y_{1}},$
(2.11)

which correspond to the UV and IR terms in the effective potential vanishing
individually.

One can associate the brane separation with the vacuum expectation value of
the radion. The metric boundary condition at $y_{1}$ can be thought of as
setting the VEV of the radion. That is, one varies $y_{1}$ until the IR brane
contribution to the effective potential vanishes. We show in the next
Subsection that it is always the case that the minimum in the static effective
potential corresponds to vanishing $V_{\text{IR}}$ after all scalar bulk and
boundary equations of motion are imposed.222However, it curiously remains
possible that extrema or saddles arise where neither the scalar or metric
conditions are satisfied (and this occurs in some models [11]). This may be
due to nearby extrema which do not satisfy the ansatz of a Lorentz invariant
background.

At this point we can impose the scalar field equations of motion. For a single
scalar field, if we impose the bulk equations, the effective potential is then
a function of the two remaining freedoms in the scalar field (its value and
derivative, on, for example, the UV brane), and the position of the IR brane.
If one further imposes the scalar boundary conditions on both of the two
branes, then only the value of $y_{1}$ remains, to be determined by minimizing
the effective potential with respect to it. However, we might be interested in
a more intuitive measure of the effective potential, particularly when taking
into account the multiple scalar degrees of freedom, where we expect to have a
multidimensional scalar potential.

We advocate an approach inspired by holography, where the effective potential
is measured in terms of field values on the UV brane only. In this case, the
procedure we should take to find the effective potential is as follows:

  * •

pick a particular value of $y_{1}$;

  * •

impose the IR boundary conditions, eliminating $N$ of the $2N$ scalar boundary
conditions. $N$ degrees of freedom remain, e.g. the values of the fields on
the UV brane;

  * •

we are left with $N$ scalar boundary values and the free value of $y_{1}$.
Ideally, one would then, for constant UV field values, minimize the effective
action over $y_{1}$.

This last step is difficult to implement in practice. However, if at least one
of the scalar fields (say $\phi_{N}$) has “stiff” boundary conditions in the
UV, then one instead imposes that UV brane condition, leaving $N-1$ scalar
degrees of freedom on the UV brane. The value of $y_{1}$ corresponds to the
last degree of freedom, so that one has
$V_{\text{eff}}(\phi_{1}(y_{0}),\cdots,\phi_{N-1}(y_{0});y_{1})$.

###  2.1 Derivative of the Radion Potential

We are interested in finding stable configurations of the branes, where the
effective potential is minimized. We explore here the conditions for
minimization, showing analytically that they correspond in most, but not all,
cases to vanishing of the IR portion of the effective potential.

Employing the above procedure, imposing the IR brane boundary conditions for
the scalar fields the derivative of the effective potential with respect to
$y_{1}$ takes a particularly simple form:

$\derivative{V}{y_{1}}=e^{-4A(y_{0})}\sum_{i}\mathopen{}\mathclose{{}\left(\frac{\partial
V_{0}}{\partial\phi_{i}}(y_{0})-2\phi_{i}^{\prime}(y_{0})}\right)\derivative{\phi_{i}}{y_{1}}\mathopen{}\mathclose{{}\left(y_{0}}\right)-4e^{-4A_{1}}\derivative{A_{1}}{y_{1}}\mathopen{}\mathclose{{}\left[V_{1}+\frac{6}{\kappa^{2}}A^{\prime}_{1}}\right].$
(2.12)

We provide proof of this relation in Appendix A. The first term is vanishing
if the boundary conditions for the scalars on the UV brane are imposed. The
second term is vanishing if the metric junction condition in the IR is
imposed. We thus note that this formula implies that extremization of the
action implies (as it must, of course) extremization of the effective
potential. However, the converse is not necessarily the case. It may be that
this expression vanishes where the two terms cancel against each other and
neither metric nor scalar boundary conditions are met. We hypothesize that
this may occur when there are extrema of the action “nearby” in configuration
space that do not obey the static, homogeneous, and isotropic ansatz that was
the starting point for this analysis of the effective action.

We note that there is a method that generalizes the well known superpotential
method for stabilization [12] to include multiple scalar fields. This method
guarantees a solution in which the effective potential is extremized. In
Appendix B, we derive and present this method.

##  3 Examples of Radion-Induced Symmetry Breaking

We now explore some specific simple models that illustrate different classes
of multi-scalar potentials where a Higgs instability occurs over some region
of radius values. We are particularly interested in models where some tuning
has been performed to obtain mild or large hierarchies between the electroweak
scale and the KK scale. Broadly speaking, there are 3 classes of tunings
distinguished by the shape of the 2-dimensional scalar potential. These are as
follows:

  * •

All parameters of the 5D theory are of order 1 (in units of the AdS curvature
for dimensionful couplings). Tuning is achieved by making the symmetry
breaking critical point extremely close to the minimum of the radion
potential.

  * •

The Higgs potential has only mild dependence on the radion VEV. Tuning is
achieved by arranging the Higgs effective mass term to be small over a large
range of radius, including at the minimum of the radion potential.

  * •

The radion potential itself is very shallow due to tuning. The Higgs VEV then
contributes sizably to the stabilization, and even small Higgs VEV
successfully stabilizes the geometry for large KK scale.

In Figure 1, we display the basic pictures of these three types of 2D scalar
potentials.

(a)  (b)  (c)

Figure 1: In this figure, we roughly characterize the different types of
potentials in the space of the radion and the Higgs VEVs. Each plot shows a
potential as a function of a modulus VEV, $\phi$, and a Higgs VEV. Dashed
lines indicate points at which the Higgs potential is minimized with $\phi$
held constant. In the first image, all couplings are order 1, however, the
minimum of the modulus potential is tuned to be close to the critical point
for the Higgs. In the second, the dependence of the Higgs potential on the
modulus field is weak, and the bare Higgs mass is taken to be somewhat small.
In the final plot, the modulus potential without the Higgs is very flat, and
thus the backreaction of the Higgs onto the modulus potential is the dominant
feature in the potential.

These different types of tunings have implications for phenomenology. For
example, there may be consequences for cosmology, affecting the manner in
which the early universe electroweak phase transition takes place [13, 14,
15], and possibly playing a crucial role during post-inflationary reheating
[5]. Additionally, the different tunings have implications for the mass
spectrum of the lightest scalar modes, and for general considerations about
vacuum energy [16, 17]. For example, the radion mass may be typically lighter
or heavier than the Higgs for the different classes of model.

###  3.1 Higgs on the Brane

We now want to consider the case in which the interactions between the Higgs
and the GW scalar take place on the IR brane. We assume no bulk interactions
between the fields. This can be done with a Higgs which is either in the bulk
or completely localized on the IR brane. We first consider the latter.

When the Higgs is localized on the IR brane, the only field that propagates in
the bulk is the Goldberger-Wise (GW) scalar. The bulk action is then

$S_{\textnormal{bulk}}=\int\differential[5]{x}\sqrt{g}\,\mathopen{}\mathclose{{}\left(\frac{1}{2}g^{MN}\partial_{M}\Phi\,\partial_{N}\Phi+\frac{6}{\kappa^{2}}-\frac{1}{2}m_{\Phi}^{2}\Phi^{2}-\frac{1}{2\kappa^{2}}R}\right).$
(3.1)

We take $m_{\Phi}^{2}\equiv\epsilon(\epsilon-4)$, where $\epsilon$ is taken to
be ${\mathcal{O}}(1/10)$ in order to generate exponential hierarchies without
severe tuning. The bulk action is supplemented by brane-localized Lagrangian
terms, including a kinetic term and potential for the Higgs on the IR brane:

$S_{\textnormal{brane}}=-\int\differential[4]{x}\sqrt{-g_{0}}\,V_{0}(\Phi)\evaluated{}_{y=y_{0}}-\int\differential[4]{x}\sqrt{-g_{1}}\,\mathopen{}\mathclose{{}\left(|\partial_{\mu}H|^{2}-V_{1}(\Phi,|H|)}\right)\evaluated{}_{y=y_{1}}.$
(3.2)

We presume the following form for the brane-localized potentials:

$\begin{gathered}V_{0}=T_{0}+\gamma_{0}(\Phi-v_{0})^{2},\\\
V_{1}=T_{1}+\lambda_{H}|H|^{2}(|H|^{2}-v_{H}^{2}-\lambda\Phi)+\gamma_{1}(\Phi-
v_{1})^{2}.\end{gathered}$ (3.3)

These include brane tensions and localized mass terms for the Goldberger-Wise
stabilizing field that displace the VEV from the origin. In addition, there is
a standard potential for the IR brane-localized Higgs with an additional
trilinear coupling to the Goldberger-Wise field. This last term is crucial for
our purposes, as it couples the Higgs field to the radius.

We then solve the equation of motion (2.9), supplemented by the boundary
conditions (2.10), under the assumption of an $x$-independent background VEV,
$\langle\Phi\rangle=\phi(y)$. Putting everything together we get the effective
dilaton potential of Equation (2.8) in the case of a single bulk scalar field,
with the added contributions of the brane-localized Higgs.

We consider the small backreaction limit, $\kappa\ll 1$. In this case,
$A^{\prime}\approx 1$ and the bulk equation of motion can be solved in general
to give

$\phi(y)=\phi_{\epsilon}e^{\epsilon y}+\phi_{4}e^{(4-\epsilon)y}.$ (3.4)

The coefficients are fixed by imposing the boundary conditions. We adopt a
stiff wall boundary condition for the GW field on the UV brane, corresponding
to the limit $\gamma_{0}\rightarrow\infty$, which fixes
$\phi_{0}\equiv\phi(y=0)=v_{0}$.

The IR brane potential contains the interaction between the GW scalar and the
Higgs. We do not take the stiff wall limit for the GW scalar on the IR brane,
so $\phi_{1}\equiv\phi(y_{1})$ is not fixed at $v_{1}$. Through the trilinear
coupling to $\Phi$, the brane-localized Higgs mass is then a function of
$y_{1}$. The equation of motion for the Higgs background that minimizes the
effective potential energy is

$2\langle H\rangle^{2}\equiv
v^{2}(y_{1})=\begin{cases}0&v_{H}^{2}+\lambda\phi_{1}<0\\\
v_{H}^{2}+\lambda\phi_{1}&v_{H}^{2}+\lambda\phi_{1}>0,\end{cases}$ (3.5)

while the boundary condition for $\phi(y)$ is

$\phi_{1}^{\prime}=-\gamma_{1}(\phi_{1}-v_{1})+\frac{1}{4}\lambda\lambda_{H}v^{2}(y_{1}).$
(3.6)

If we impose only the IR boundary condition for $\phi$, leaving the Higgs VEV
free (not imposing Equation (3.5)), we can explore the two dimensional
effective potential as a function of $v$ and $y_{1}$.

(a)  (b)

Figure 2: Higgs VEV in units of the symmetry breaking scale $f$ and its
dependence on various parameters of the model. On the left, we show the
variation as a function of the IR brane mistune $\delta T_{1}$. On the right,
we display it in terms of how close the minimum of the radion potential is to
$y_{1}^{\textnormal{c}}$, the critical point in the extra dimension where the
interplay between the GW scalar and the Higgs first triggers electroweak
symmetry breaking. We have taken $v_{0}=1/10$, $v_{1}=2$, $\gamma_{1}=1$,
$\lambda_{H}=1/8$, $\kappa=1/50$. The different curves are obtained by fixing
the ratio $\lambda/v_{H}^{2}$, so that equal values of $\epsilon$ correspond
to equal values of $y_{1}^{\textnormal{c}}$.

In the small backreaction limit, the effective potential for the system is

$\displaystyle
V_{\textnormal{eff}}=e^{-4y_{0}}\mathopen{}\mathclose{{}\left(V_{0}-\frac{6}{\kappa^{2}}-\frac{1}{4}{\phi^{\prime}_{0}}^{2}-\frac{\epsilon(4-\epsilon)}{4}\phi_{0}^{2}}\right)+e^{-4y_{1}}\mathopen{}\mathclose{{}\left(V_{1}+\frac{6}{\kappa^{2}}+\frac{1}{4}{\phi^{\prime}_{1}}^{2}+\frac{\epsilon(4-\epsilon)}{4}\phi_{1}^{2}}\right).$
(3.7)

For small $y_{1}$ values, it can be arranged that
$v_{H}^{2}+\lambda\phi_{1}<0$, and the origin in Higgs field space is a stable
minimum: $\expectationvalue{H}=0$.

In this region, the effective potential (after imposing the Higgs equation of
motion) is determined by the single GW field and its backreaction onto the
geometry.

On the other hand, for larger $y_{1}$ values, the Higgs may develop a VEV. For
example, when $\epsilon$ is taken to be small and we set $\gamma_{1}=0$, we
can approximate the electroweak symmetry breaking condition as

$y_{1}\gtrsim\frac{1}{\epsilon}\log(-\frac{v_{H}^{2}}{\lambda v_{0}}).$ (3.8)

Minimizing the effective potential with respect to $y_{1}$ relates the Higgs
VEV to the symmetry breaking scale. The effective VEV, in units of the Kaluza-
Klein scale $f$, scales with $\delta T_{1}$, the IR brane mistune, as

$\frac{v}{f}\approx\mathopen{}\mathclose{{}\left(\frac{64\delta
T_{1}}{\lambda_{H}\mathopen{}\mathclose{{}\left(16-\lambda^{2}\lambda_{H}}\right)}}\right)^{1/4}.$
(3.9)

In Figure 2 we quantify the amount of tuning associated with a given Higgs VEV
$v$. The Higgs VEV is fixed in the following way: first we identify the
effective 4D scale of gravity, i.e. the Planck scale, via
$M_{\textnormal{Pl}}^{2}\approx 1/(2\kappa^{2})$, as a consequence of the
warped geometry. For a given value of $\kappa$, we then reproduce the
experimentally measured hierarchy between the Planck and the weak scale,
$\log(M_{\textnormal{Pl}}/v)\approx 38.4$. We scan over $f$ by changing
$\epsilon\sim 1/10$, so that the corresponding values of $y_{1}$ provide a
large range of Kaluza-Klein scales. To obtain a small $v/f$, parameters must
be chosen so that the global minimum of the effective potential is very close
to the critical point for electroweak symmetry breaking. We denote the
critical values of the 5D parameters with the superscript “c”. We display this
information in two different ways: on the left, as a function of $\delta
T_{1}$, which is the physical parameter that needs to be tuned to adjust the
location of the minimum of the effective potential. On the right, as a
function of $y_{1}$, the location of the minimum itself, which is more useful
to visually determine how close the minimum has to be to the critical point.
As expected from the analytic estimate of Equation (3.9), a small Higgs VEV
requires a very small mistune in the brane tension against the bulk
cosmological constant.

#### Mass Spectrum

In order to understand the spectrum of the theory, we consider fluctuations
around the background solutions for the metric and the two scalar field
profiles to linearized order. In particular, we parametrize the various
degrees of freedom as

$\begin{gathered}\Phi(x,y)=\phi(y)+\varphi(x,y),\\\
H(x)=\frac{1}{\sqrt{2}}\mathopen{}\mathclose{{}\left[v+h(x)}\right]\exp(i\alpha),\\\
\differential{s^{2}}=e^{-2A(y)-2F(x,y)}\eta_{\mu\nu}\differential{x^{\mu}}\differential{x^{\nu}}-(1+2F(x,y))^{2}\differential{y^{2}}.\end{gathered}$
(3.10)

It is possible to show that, when having only one scalar field in the bulk,
the whole set of linearized Einstein equations, together with the equations of
motion for the scalar fields, reduces to a single homogeneous differential
equation for $F$ in the bulk [18]:

$F^{\prime\prime}-2A^{\prime}F^{\prime}-4A^{\prime\prime}F-2\frac{\phi^{\prime\prime}}{\phi^{\prime}}F^{\prime}+4\frac{\phi^{\prime\prime}}{\phi^{\prime}}A^{\prime}F=e^{2A}\Box
F.$ (3.11)

We also have to specify the boundary conditions. When considering a stiff wall
UV brane boundary condition for the background GW field
($\gamma_{0}\rightarrow\infty$), the UV boundary condition is
$F_{0}^{\prime}=2A_{0}^{\prime}F_{0}$. The IR boundary condition for $F$ can
be obtained by combining the following equations:

$\begin{gathered}\phi^{\prime}\varphi=\frac{3}{\kappa^{2}}\mathopen{}\mathclose{{}\left(F^{\prime}-2A^{\prime}F}\right),\\\
-2\varphi_{1}^{\prime}=\partialderivative[2]{V_{1}}{\phi}\varphi_{1}+\partialderivative{V_{1}}{\phi}{h}h+2F_{1}\partialderivative{V_{1}}{\phi},\\\
\Box
h+e^{-2A_{1}}\mathopen{}\mathclose{{}\left(\partialderivative{V_{1}}{\phi}{h}\varphi_{1}+\partialderivative[2]{V_{1}}{h}h}\right)=0.\end{gathered}$
(3.12)

Notice that the mass eigenvalue appears both in the bulk equation and in the
boundary condition because of the presence of the brane-localized Higgs
kinetic term which leads to the $\Box\equiv\partial_{\mu}\partial^{\mu}$
operator.

(a)  (b)

Figure 3: Dependence of the mass scales in the theory on various parameters of
the model. On the left, we show how for a fixed value of $\kappa$ the IR brane
mistune controls the Higgs mass and VEV, while the radion mass is essentially
unchanged. On the right, we see that the radion mass is dependent on the
amount of backreaction. The parameters are the same as in Figure 2.

Solutions are parametrized by 3 unknowns. Two are integration constants
associated with the second order equation for $F$, and the third is the
eigenvalue. One integration constant can be fixed by choosing a convenient
normalization, and the other is eliminated by imposing the UV brane boundary
condition. To find the eigenvalues, we employ a shooting method and determine
the mass eigenvalues for which the IR boundary condition is satisfied. We find
that the spectrum contains a light radion and Higgs along with a KK tower due
to the bulk GW field.333As usual, there is no separate KK tower for the radion
field, as these modes are eaten by the massive KK gravitons.444Recent work
also utilizes a new approach to get the full potential for the lightest scalar
mode, going beyond the mass eigenvalue [19].

Figure 3 shows the mass spectrum obtained by solving Equation (3.11)
supplemented by the boundary conditions (3.12). Once again, we fix $v$ to the
electroweak value and we scan over $f$ by varying $\epsilon\sim 1/10$. We can
see how the mass of the Higgs fluctuation tracks the Higgs VEV and is
controlled by the amount of mistune (left subfigure), while the radion mass is
determined by the backreaction parameter $\kappa$ (right subfigure). On the
other hand, the Higgs and radion masses are essentially insensitive to changes
in $\kappa$ and $\delta T_{1}$, respectively. The lightest particle in the
spectrum can be either the Higgs or the radion, depending on the choice of
parameters. This is in contrast to the results of Subsection 3.3, where we
consider only the Higgs in the bulk, and the radion is constrained to be the
lightest state in the spectrum.

###  3.2 Two Bulk Fields

In this Subsection, we consider the case where both fields, GW scalar $\Phi$
and the Higgs $H$, propagate in the bulk. For simplicity, we assume that they
are uncoupled, despite such a coupling being allowed by the symmetry. However,
they still interact indirectly through gravity, as both fields backreact on
the geometry. We assume that the Higgs is localized towards the IR brane by
taking its bulk mass close to (but above) the Breitenlohner-Freedman bound
$m_{H}^{2}=-4$. Note that the bulk mass is constant.

The bulk action is given by

$\displaystyle
S_{\text{bulk}}=\int\differential[5]{x}\sqrt{g}\mathopen{}\mathclose{{}\left(\frac{1}{2}\partial^{N}\Phi\partial_{N}\Phi+\partial^{N}H^{\dagger}\partial_{N}H+\frac{6}{\kappa^{2}}-\frac{1}{2}m_{\Phi}^{2}\Phi^{2}-m_{H}^{2}\absolutevalue{H}^{2}-\frac{1}{2\kappa^{2}}R}\right).$
(3.13)

Again we take $m_{\Phi}^{2}\equiv\epsilon(\epsilon-4)$ with
$\epsilon\sim\mathcal{O}(1/10)$, and we also define $\nu^{2}\equiv
4+m_{H}^{2}$ with $\nu\sim\mathcal{O}(1/10)$. This bulk action is supplemented
by brane-localized Lagrangian terms

$\displaystyle
S_{\text{brane}}=-\int\differential[4]{x}\sqrt{-g_{0}}V_{0}(\Phi,\absolutevalue{H})\evaluated{}_{y=y_{0}}-\int\differential[4]{x}\sqrt{-g_{1}}V_{1}(\Phi,\absolutevalue{H})\evaluated{}_{y=y_{1}}.$
(3.14)

The brane-localized potentials are given by the following:

$\displaystyle\begin{gathered}V_{0}(\Phi,\absolutevalue{H})=T_{0}+\gamma_{0}(\Phi-
v_{0})^{2}+m_{0}^{2}\absolutevalue{H}^{2},\\\
V_{1}(\Phi,\absolutevalue{H})=T_{1}+\gamma_{1}(\Phi-
v_{1})^{2}+\lambda_{H}\absolutevalue{H}^{2}(\absolutevalue{H}^{2}-v_{H}^{2}).\end{gathered}$
(3.17)

In this Subsection, we work in the simplifying limit
$\gamma_{0,1}\rightarrow\infty$, which sets the UV/IR boundary conditions for
the GW field as $\Phi(y=y_{0})=v_{0}$ and $\Phi(y=y_{1})=v_{1}$ respectively.

We write the VEVs for the GW and the Higgs field respectively as
$\expectationvalue{\Phi}=\phi(y)$ and
$\expectationvalue{\absolutevalue{H}}=v(y)/\sqrt{2}$. Presuming small metric
backreaction, the bulk equations of motion for the scalars are given by

$\displaystyle\begin{gathered}\phi^{\prime\prime}-4\phi^{\prime}-\epsilon(\epsilon-4)\phi=0,\\\
v^{\prime\prime}-4v^{\prime}-(-4+\nu^{2})v=0.\end{gathered}$ (3.20)

The GW field $\phi$ has the same profile as in the previous Subsection, which
is given by (3.4) where the coefficients $\phi_{\epsilon}$ and $\phi_{4}$ are
fixed by the boundary conditions $\phi(y_{0}=0)=v_{0}$ and
$\phi(y_{1})=v_{1}$. The solution for the Higgs VEV can be conveniently
expressed as

$\displaystyle
v(y)=v(y_{1})e^{2(y-y_{1})}\mathopen{}\mathclose{{}\left(\frac{e^{\nu
y}-re^{-\nu y}}{e^{\nu y_{1}}-re^{-\nu y_{1}}}}\right),$ (3.21)

where $v(y_{1})$ and $r$ are integration constants which will be determined by
the Higgs boundary conditions generated by the brane potentials (3.17):

$\displaystyle\begin{gathered}v^{\prime}(y_{0})=\frac{m_{0}^{2}}{2}v(y_{0}),\\\
v^{\prime}(y_{1})=\frac{\lambda_{H}}{2}v(y_{1})\mathopen{}\mathclose{{}\left(v_{H}^{2}-v^{2}(y_{1})}\right).\end{gathered}$
(3.24)

The UV boundary condition fixes $r$ as

$\displaystyle r=\frac{m_{0}^{2}-4-2\nu}{m_{0}^{2}-4+2\nu}.$ (3.25)

The constant $v(y_{1})$, the Higgs VEV on the IR brane, is determined by the
IR boundary condition which has two solutions. One of them is the trivial one
$v(y_{1})=0$ corresponding to unbroken electroweak symmetry. By defining
$m_{1}^{2}\equiv\lambda_{H}v_{H}^{2}$, the second solution is given by

$\displaystyle
v^{2}(y_{1})=\frac{1}{\lambda_{H}}\mathopen{}\mathclose{{}\left[(m_{1}^{2}-4-2\nu)+\frac{4\nu(m_{0}^{2}-4-2\nu)}{(m_{0}^{2}-4-2\nu)-e^{2\nu
y_{1}}(m_{0}^{2}-4+2\nu)}}\right],$ (3.26)

provided that the term inside brackets is positive. In this case, the above
solution is the preferred one, corresponding to broken electroweak symmetry.

If the Higgs has a nonzero VEV, it contributes to the 4D effective potential
(3.7) by a term given by

$\displaystyle
V_{\text{eff}}^{v}=\mathopen{}\mathclose{{}\left[\frac{m_{0}^{2}}{2}v^{2}-\frac{1}{4}v^{\prime
2}+\frac{1}{4}m_{H}^{2}v^{2}}\right]\evaluated{}_{y=0}+e^{-4y_{1}}\mathopen{}\mathclose{{}\left[\frac{\lambda_{H}}{2}v^{2}\mathopen{}\mathclose{{}\left(\frac{v^{2}}{2}-v_{H}^{2}}\right)+\frac{1}{4}v^{\prime
2}-\frac{1}{4}m_{H}^{2}v^{2}}\right]\evaluated{}_{y=y_{1}}$ (3.27)

By using the Higgs solution (3.21) and the boundary conditions (3.25) and
(3.26), the above expression takes a very simple form:

$\displaystyle
V_{\text{eff}}^{v}=-\frac{\lambda_{H}}{4}v^{4}(y_{1})e^{-4y_{1}}.$ (3.28)

We can directly see that it is negative definite, therefore if there is a
solution with a non-trivial VEV, then it will be energetically favored.

#### Parameter Space for Electroweak Symmetry Breaking

By inspecting the function $v^{2}(y_{1})$, we can see that it has a
singularity at

$\displaystyle
y_{1}^{s}=\frac{1}{2\nu}\log\mathopen{}\mathclose{{}\left(\frac{m_{0}-4-2\nu}{m_{0}^{2}-4+2\nu}}\right),$
(3.29)

since the denominator of the second term in (3.26) diverges. In the region of
parameter space where $m_{0}^{2}-4<-2\nu$, $y_{1}^{s}$ is positive, hence
physical. As a result, the effective potential will be unbounded from below at
$y_{1}=y_{1}^{s}$. This singularity is an artifact of neglecting the
backreaction of the Higgs on the geometry, which cannot be done in the
vicinity of $y_{1}^{s}$. Nevertheless, since we want to continue to work in
the small backreaction limit, we will exclude the $m_{0}^{2}-4<-2\nu$ from our
parameter space.

For the rest of the discussion, we will assume $m_{0}^{2}-4>-2\nu$ and define
$\alpha_{0,1}\equiv m_{0,1}^{2}-4-2\nu$ for notational simplicity. By taking
the derivative of (3.26) with respect to $y_{1}$, we find

$\displaystyle\partialderivative{v^{2}(y_{1})}{y_{1}}=\frac{1}{\lambda_{H}}\frac{8\alpha_{0}\nu^{2}(\alpha_{0}+4\nu)e^{2\nu
y_{1}}}{[\alpha_{0}-e^{2\nu y_{1}}(\alpha_{0}+4\nu)]^{2}}.$ (3.30)

This tells us that $v^{2}(y_{1})$ is either monotonically increasing or
decreasing depending on the sign of $\alpha_{0}$. We will consider the former
case in this particular example.

In order to have symmetry breaking, we need
$\lim_{y_{1}\rightarrow\infty}v^{2}(y_{1})>0$ which implies $\alpha_{1}>0$.
Additionally, the symmetry will be unbroken in the UV if $v^{2}(y_{1}=0)<0$,
or $\alpha_{0}>\alpha_{1}$. In this case there is a “critical” position of the
IR brane, $y_{1}=y_{1}^{c}$, where the effective Higgs mass squared term is
vanishing. This point corresponds to $v^{2}(y_{1}=y_{1}^{c})=0$ and is given
by

$\displaystyle
y_{1}^{c}=\frac{1}{2\nu}\log\mathopen{}\mathclose{{}\left(\frac{\alpha_{0}(\alpha_{1}+4\nu)}{\alpha_{1}(\alpha_{0}+4\nu)}}\right).$
(3.31)

A convenient measure of the size of the Higgs VEV is the mass that would be
given to a gauge field by the Higgs mechanism. For small backreaction, this
can be approximated by

$\displaystyle\mathopen{}\mathclose{{}\left(\frac{v_{\text{eff}}}{f}}\right)^{2}=\int_{y_{0}}^{y_{1}}\differential{y}e^{-2(y-y_{1})}v(y)^{2},$
(3.32)

where $f=e^{-y_{1}}$ is the conformal symmetry breaking scale, i.e. the KK
scale.

The full effective potential can be expressed as the sum of the UV and IR
contributions:

$\displaystyle
V_{\text{eff}}(y_{1})=e^{-4y_{0}}V_{\text{eff}}^{\text{UV}}(y_{1})+e^{-4y_{1}}V_{\text{eff}}^{\text{IR}}(y_{1}).$
(3.33)

As we have proved in Appendix A, the condition for an extremum of the
effective potential is that $V_{\text{eff}}^{\text{IR}}$ vanishes, provided
that the scalar boundary conditions (2.10) are satisfied. Such a point, let us
denote it by $y_{1}^{\text{ext}}$, is a minimum provided that

$\displaystyle
V_{\text{eff}}^{\prime\prime}(y_{1})\evaluated{}_{y_{1}=y_{1}^{\text{ext}}}>0.$
(3.34)

Since the Higgs contribution to the effective potential scales as the fourth
power of the Higgs VEV, we can neglect it for finding the extremum points and
their stability. Then we can approximate $V_{\text{eff}}^{\text{IR}}$ by

$\displaystyle
V_{\text{eff}}^{\text{IR}}(y_{1})\approx\delta\widetilde{T}_{1}-v_{0}v_{1}(2-\epsilon)(4-\epsilon)e^{\epsilon
y_{1}}+v_{0}^{2}(2-\epsilon)^{2}e^{2\epsilon y_{1}},$ (3.35)

where in the second line we absorbed all the $y_{1}$-independent terms into
the definition of $\delta\widetilde{T}_{1}$. The extremum of the potential is
determined by the solution of the quadratic equation:

$\displaystyle\delta\widetilde{T}_{1}-v_{0}v_{1}(2-\epsilon)(4-\epsilon)f_{\text{ext}}^{-\epsilon}+v_{0}^{2}(2-\epsilon)^{2}f_{\text{ext}}^{-2\epsilon}=0,$
(3.36)

where $f_{\text{ext}}=\exp{-y_{1}^{\text{ext}}}$ denotes the conformal
breaking scale at the extremum point $y_{1}^{\text{ext}}$. To determine the
stability, we need to calculate the full effective potential. Again ignoring
Higgs contributions, it can approximately be expressed as

$\displaystyle
V_{\text{eff}}(f)\approx\delta\widetilde{T}_{0}+\delta\widetilde{T}_{1}f^{4}-4v_{0}v_{1}(2-\epsilon)f^{4-\epsilon}+2v_{0}^{2}(2-\epsilon)f^{4-2\epsilon}.$
(3.37)

Since $V_{\text{eff}}^{\prime\prime}(y_{1})$ and
$V_{\text{eff}}^{\prime\prime}(f)$ have the same sign, we can calculate the
latter. Then, the stability condition reads

$\displaystyle
12\delta\widetilde{T}_{1}-4v_{0}v_{1}(2-\epsilon)(3-\epsilon)(4-\epsilon)f^{-\epsilon}_{\text{ext}}+2v_{0}^{2}(2-\epsilon)(3-2\epsilon)(4-2\epsilon)f^{-2\epsilon}_{\text{ext}}>0.$
(3.38)

Finally, we replace $\delta\widetilde{T}_{1}$ with the solution of (3.36).
Then we find that the extremum point $f_{\text{ext}}$ is stable if

$\displaystyle
f_{\text{ext}}<\mathopen{}\mathclose{{}\left(\frac{(v_{1}/v_{0})(4-\epsilon)}{4-2\epsilon}}\right)^{-1/\epsilon}.$
(3.39)

#### Mass Spectrum

In this Subsection, we will work out the mass spectrum when both fields are
propagating in the bulk. We will parameterize the fluctuations of the metric
and the field in the same way as in Subsection 3.1, except now the Higgs field
does also depend on the bulk coordinate $y$:

$\displaystyle
H(x,y)=\frac{1}{\sqrt{2}}\mathopen{}\mathclose{{}\left[v(y)+h(x,y)}\right]\exp{i\alpha}.$
(3.40)

In this case, the linearized Einstein equations for fields $F,h,\varphi$ take
the form

$\displaystyle 12A^{\prime
2}F-6A^{\prime\prime}F-6A^{\prime}F^{\prime}+F^{\prime\prime}$
$\displaystyle=-\frac{\kappa^{2}}{3}\mathopen{}\mathclose{{}\left[2VF+3F(\phi^{\prime
2}+v^{\prime
2})-\partialderivative{V}{\phi}\varphi-\partialderivative{V}{v}h-\phi^{\prime}\varphi^{\prime}-v^{\prime}h^{\prime}}\right]$
$\displaystyle\phantom{{}={}}-\frac{\kappa^{2}}{3}\sum_{j=0,1}\mathopen{}\mathclose{{}\left(4V_{j}F-\partialderivative{V_{j}}{\phi}\varphi-\partialderivative{V_{j}}{v}h}\right)\delta(y-y_{j}),$
(3.41) $\displaystyle 4A^{\prime}F^{\prime}+e^{2A}\square F$
$\displaystyle=-\frac{\kappa^{2}}{3}\mathopen{}\mathclose{{}\left(4VF+\partialderivative{V}{\phi}\varphi+\partialderivative{V}{v}h-\phi^{\prime}\varphi^{\prime}-v^{\prime}h^{\prime}}\right),$
(3.42)
$\displaystyle\partial_{\mu}\mathopen{}\mathclose{{}\left(3F^{\prime}-6A^{\prime}F}\right)$
$\displaystyle=\kappa^{2}\partial_{\mu}\mathopen{}\mathclose{{}\left(\phi^{\prime}\varphi+v^{\prime}h}\right),$
(3.43)

where all derivatives of the bulk and brane potentials are evaluated on the
background scalar VEVs. The $\mu 5$-equation can directly be integrated to
give

$\displaystyle
F^{\prime}-2A^{\prime}F=\frac{\kappa^{2}}{3}\mathopen{}\mathclose{{}\left(\phi^{\prime}\varphi+v^{\prime}h}\right).$
(3.44)

By combining the $\mu\nu$\- and $55$-equations in the bulk we obtain

$\displaystyle F^{\prime\prime}-2A^{\prime}F^{\prime}+e^{2A}\square
F=\frac{2\kappa^{2}}{3}\mathopen{}\mathclose{{}\left(\phi^{\prime}\varphi^{\prime}+v^{\prime}h^{\prime}}\right).$
(3.45)

Matching the singular terms in the $\mu\nu$-equation gives the junction
conditions for $F$:

$\displaystyle[F^{\prime}]_{i}=\frac{2\kappa^{2}}{3}V_{i}F+\frac{\kappa^{2}}{3}\mathopen{}\mathclose{{}\left(\partialderivative{V_{i}}{\phi}\varphi+\partialderivative{V_{i}}{v}h}\right).$
(3.46)

By using the boundary conditions for the background solution, one can show
that this is equivalent to the $\mu 5$-equation so it provides no new
constraints.

The linearized scalar field equations are given by

$\displaystyle
e^{2A}\square\varphi-\varphi^{\prime\prime}+4A^{\prime}\varphi^{\prime}+\partialderivative[2]{V}{\phi}\varphi=-6F^{\prime}\phi^{\prime}-4\partialderivative{V}{\phi}F,$
(3.47) $\displaystyle e^{2A}\square
h-h^{\prime\prime}+4A^{\prime}h^{\prime}+\partialderivative[2]{V}{v}h=-6F^{\prime}v^{\prime}-4\partialderivative{V}{v}F,$
(3.48)

together with the boundary conditions

$\displaystyle[\varphi^{\prime}-2\phi^{\prime}F]_{i}$
$\displaystyle=\partialderivative[2]{V_{i}}{\phi}\evaluated{}_{\phi}\varphi,$
(3.49) $\displaystyle[h^{\prime}-2v^{\prime}F]_{i}$
$\displaystyle=\partialderivative[2]{V_{i}}{v}h.$ (3.50)

So far, there are three second order differential equations, (3.45)(3.47) and
(3.48), which need to be solved simultaneously. However, we can use the $\mu
5$-equation (3.44) to eliminate $\varphi$ from the system. Then (3.45) becomes

$\displaystyle
F^{\prime\prime}-2A^{\prime}F^{\prime}-4A^{\prime\prime}F-2\frac{\phi^{\prime\prime}}{\phi^{\prime}}F^{\prime}+4A^{\prime}\frac{\phi^{\prime\prime}}{\phi^{\prime}}F=e^{2A}\square
F+\frac{2\kappa^{2}}{3}\mathopen{}\mathclose{{}\left(v^{\prime\prime}-\frac{\phi^{\prime\prime}}{\phi^{\prime}}v^{\prime}}\right)h.$
(3.51)

To find the mass spectrum, we expand both $F$ and $h$ into their Kaluza-Klein
(KK) modes by

$\displaystyle F(x,y)$ $\displaystyle=\sum_{n}F_{n}(y)R_{n}(x),$ (3.52)
$\displaystyle h(x,y)$ $\displaystyle=\sum_{n}h_{n}(y)R_{n}(x),$ (3.53)

where each KK mode in the above expansions satisfies $\square
R_{n}=-m_{n}^{2}R_{n}$. Using (3.48) and (3.51), we write the system of
differential equations to determine the mass spectrum as

$\displaystyle
F_{n}^{\prime\prime}-2A^{\prime}F_{n}^{\prime}-4A^{\prime\prime}F_{n}-2\frac{\phi^{\prime\prime}}{\phi^{\prime}}F_{n}^{\prime}+4A^{\prime}\frac{\phi^{\prime\prime}}{\phi^{\prime}}F_{n}+e^{2A}m_{n}^{2}F_{n}=\frac{2\kappa^{2}}{3}\mathopen{}\mathclose{{}\left(v^{\prime\prime}-\frac{\phi^{\prime\prime}}{\phi^{\prime}}v^{\prime}}\right)h_{n},$
(3.54) $\displaystyle
h_{n}^{\prime\prime}-4A^{\prime}h_{n}^{\prime}-\mathopen{}\mathclose{{}\left(\partialderivative[2]{V}{v}-e^{2A}m_{n}^{2}}\right)h_{n}=6F_{n}^{\prime}v^{\prime}+4\partialderivative{V}{v}F_{n}.$
(3.55)

This system of differential equations have five integration constants; two
from each equation plus the eigenvalue. Two of them are fixed by the boundary
condition (3.50), which reads

$\displaystyle
h_{n}^{\prime}-2v^{\prime}F_{n}=\pm\frac{1}{2}\partialderivative[2]{V_{0,1}}{v}h_{n},\quad\text{$+/-$
is for the UV/IR brane}.$ (3.56)

In the case of stiff-wall boundary conditions, (3.49) sets $\varphi=0$ on both
branes. Then, (3.46) implies the relation

$\displaystyle
F_{n}^{\prime}-2A^{\prime}F_{n}=\frac{\kappa^{2}}{3}v^{\prime}h_{n},\quad\text{on
the branes}.$ (3.57)

The remaining integration constant can be fixed by normalizing both $F_{n}$
and $h_{n}$ by a common factor, since the system is invariant under such a
scaling.

The rest of this Subsection is devoted to solving this system of equations to
zero and leading order in the backreaction. We will assume that $\kappa$ is
small enough so that the background field profiles are accurately expressed by
their zero backreaction solutions. We shall investigate the validity of this
assumption later.

#### Mass Spectrum with Backreaction Neglected

First, we will study the mass spectrum in the $\kappa^{2}\rightarrow 0$ limit.
This means that we will search for the mass eigenvalues which remain finite
after this limit. We will denote these eigenvalues and their corresponding
eigenvector components $F_{n}$ and $h_{n}$ by the superscript $(0)$. Then
(3.54) and (3.55) become

$\displaystyle\mathopen{}\mathclose{{}\left(F_{n}^{(0)^{\prime}}-2F_{n}^{(0)}}\right)^{\prime}-2\frac{\phi^{\prime\prime}}{\phi^{\prime}}\mathopen{}\mathclose{{}\left(F_{n}^{(0)^{\prime}}-2F_{n}^{(0)}}\right)+e^{2y}\mathopen{}\mathclose{{}\left(m_{n}^{(0)}}\right)^{2}F_{n}^{(0)}=0,$
(3.58) $\displaystyle
h_{n}^{(0)^{\prime\prime}}-4h_{n}^{(0)^{\prime}}-\mathopen{}\mathclose{{}\left(\partialderivative[2]{V}{v}-e^{2y}\mathopen{}\mathclose{{}\left(m_{n}^{(0)}}\right)^{2}}\right)h_{n}^{(0)}=6F_{n}^{(0)^{\prime}}v^{\prime}+4\partialderivative{V}{v}F_{n}^{(0)}.$
(3.59)

The $\mu 5$-component of the Einstein equations (3.44) tells us that
$F_{n}^{(0)^{\prime}}-2F_{n}^{(0)}\propto\kappa^{2}$. Then (3.58) implies
$F_{n}^{(0)}\propto\kappa^{2}$ too. Hence, the mass spectrum in the
$\kappa^{2}\rightarrow 0$ limit is given by the differential equation

$\displaystyle
h_{n}^{\prime\prime}-4h_{n}^{\prime}+(m_{n}^{2}e^{2y}-m_{H}^{2})h_{n}=0,$
(3.60)

with boundary conditions

$\displaystyle
h^{\prime}_{n}=\pm\frac{1}{2}\partialderivative[2]{V_{0,1}}{v}h_{n},$ (3.61)

where $+/-$ for UV/IR brane, and we have omitted the superscripts for brevity.

We see that the fluctuations $F$ and $h$ are decoupled from each other in this
limit, which is expected since there is no explicit coupling between the GW
and the Higgs field. In the absence of backreaction, each field does not know
about the existence of the other.

The solution of (3.60) is given in terms of Bessel functions:

$\displaystyle
h_{n}=e^{2y}\mathopen{}\mathclose{{}\left[J_{\nu}(m_{n}e^{y})+cY_{\nu}(m_{n}e^{y})}\right],$
(3.62)

where $c$ is a constant which is determined by the UV boundary condition. The
mass eigenvalue is fixed by the IR boundary condition. After applying both, we
find that the mass spectrum is given by the roots of

$\displaystyle
b_{v}(m_{n})=\widetilde{J}_{\nu}^{1}(m_{n})-\frac{\widetilde{J}_{\nu}^{0}(m_{n})}{\widetilde{Y}_{\nu}^{0}(m_{n})}\widetilde{Y}_{\nu}^{1}(m_{n}),$
(3.63)

where we defined

$\displaystyle\widetilde{X}^{0}(m_{n})\equiv\mathopen{}\mathclose{{}\left(\partialderivative[2]{V_{0}}{v}-4}\right)X_{\nu}(m_{n})-m_{n}\mathopen{}\mathclose{{}\left(X_{\nu-1}(m_{n})-X_{\nu+1}(m_{n})}\right),$
(3.64)
$\displaystyle\widetilde{X}^{1}(m_{n})\equiv\mathopen{}\mathclose{{}\left(\partialderivative[2]{V_{1}}{v}+4}\right)X_{\nu}(m_{n})+m_{n}e^{y_{1}}\mathopen{}\mathclose{{}\left(X_{\nu-1}(m_{n}e^{y_{1}})-X_{\nu+1}(m_{n}e^{y_{1}})}\right),$
(3.65)

with $X=\quantity{J,Y}$. The Higgs mass is the smallest $m_{n}$ which satisfy
$b_{v}(m_{n})=0$.

(a)  (b)

Figure 4: In this figure, we are plotting the necessary value of $m_{1}^{2}$
in order to get $v_{\text{EW}}=246\;$\mathrm{GeV}$$ and
$m_{h}=126\;$\mathrm{GeV}$$, as a function of $m_{0}^{2}$. The hierarchy
between the electroweak and the conformal breaking scale is
$f/v_{\text{EW}}=10$ and $f/v_{\text{EW}}=100$ for the left and right plot
respectively. In the shaded region, there is no ”critical region”, i.e. the
electroweak symmetry is broken, even if the IR brane is very close to the UV
brane.

The Higgs sector of the model is specified by five parameters; $m_{0}^{2}$,
$m_{1}^{2}$, $\lambda_{H}$, $\nu$ and $f$. Two of them can be fixed in terms
of the others, by setting the effective Higgs VEV (3.32) and the Higgs mass to
$246\;$\mathrm{GeV}$$ and $126\;$\mathrm{GeV}$$ respectively. We have chosen
to keep $m_{0}^{2}$, $\nu$ and $f$ free, and calculate $m_{1}^{2}$ and
$\lambda_{H}$ in terms of the rest. We show the results in Figure 4. One can
see that a fair amount of tuning is needed in the IR brane mass parameter of
the Higgs. One might get the impression that the tuning is less severe when we
lower the $\nu$ parameter, or increase the hierarchy between the electroweak
and conformal breaking scales. But in that case, the tuning between the GW and
the Higgs sector does increase, as we shall see shortly.

(a)  (b)

Figure 5: These plots show the amount of tuning required between the Higgs and
the GW sector when the breaking scale is set to $f=10\;$\mathrm{TeV}$$ (left
figure) and $f=50\;$\mathrm{TeV}$$ (right figure). Different points which have
the same $\alpha_{0}$ value are obtained by varying $\nu$ which is decreasing
as one goes up on the vertical axis. We can observe that the tuning increases
as we go the the top-right region of the $\alpha_{0}-\alpha_{1}$ plane. Also
increasing the breaking scale increases the tuning, which is expected.

Since the GW contribution dominates the effective potential, the minimum is
mainly determined by the GW sector. Therefore, one can choose a particular
value for the IR mistune $\delta T_{1}$, such that the minimum coincides with
$y_{1}^{\text{c}}$, as long as (3.39) is satisfied. We denote this “critical”
mistune by $\delta T_{1}^{c}$. Then, a convenient parameter to measure the
tuning between the GW and the Higgs sector is $\absolutevalue{\delta
T_{1}/\delta T_{1}^{c}}-1$.

In Figure 5, we show the required tuning on the parameter space where the
electroweak symmetry breaking is activated by the radion. The procedure to
obtain these plots is as follows: We have set the GW sector parameters to be
$v_{0}=1/50$, $v_{1}=1$ and $\epsilon=1/10$. Then we set the conformal
breaking scale to be $f=10(50)\;$\mathrm{TeV}$$ for the left(right) plot. For
each point on the $\alpha_{0}-\alpha_{1}$ plane, we fix the Higgs parameters
such that $v_{\text{eff}}=246\;$\mathrm{GeV}$$ and
$m_{h}=126\;$\mathrm{GeV}$$. Finally, we solve for $\delta T_{1}$ such that
$V_{\text{eff}}^{\text{IR}}=0$.

Different points in Figure 5 which share the same
$\alpha_{0}=m_{0}^{2}-4-2\nu$ values are obtained by varying $\nu$. For a
fixed $\alpha_{0}$, larger $\nu$ values correspond to larger
$\alpha_{1}=m_{1}^{2}-4-2\nu$, thus $\nu$ increases as one goes up on the
vertical axis. As one can see more clearly in the plot on the right, tuning
between the GW and the Higgs sectors does also increase in this direction.
Therefore, as one relaxes the tuning in the $m_{1}^{2}$ parameter, the tuning
in $\delta T_{1}$ becomes larger.

#### The Radion Mass

To calculate the radion mass, we make the following ansatz for the radion
wavefunction, and the radion mass eigenvalue:

$\displaystyle
F_{r}=e^{2A}\mathopen{}\mathclose{{}\left(1+\kappa^{2}\widetilde{F}_{r}}\right)\mbox{\quad
and\quad}m_{r}^{2}=\kappa^{2}l_{r}^{2}.$ (3.66)

By plugging this ansatz into (3.51), and keeping only the terms which are at
leading order in $\kappa^{2}$ we find

$\displaystyle\widetilde{F}_{r}^{\prime\prime}+\widetilde{F}_{r}^{\prime}\mathopen{}\mathclose{{}\left(2-2\frac{\phi^{\prime\prime}}{\phi^{\prime}}}\right)=\frac{2}{3}\mathopen{}\mathclose{{}\left[(\phi^{\prime
2}+v^{\prime
2})+\mathopen{}\mathclose{{}\left(v^{\prime\prime}-\frac{\phi^{\prime\prime}}{\phi^{\prime}}v^{\prime}}\right)h_{r}e^{-2y}}\right]-e^{2y}l_{r}^{2}.$
(3.67)

The solution of this equation is

$\displaystyle\widetilde{F}_{r}^{\prime}(y)=\frac{1}{u(y)}\mathopen{}\mathclose{{}\left[\frac{2}{3}\int_{0}^{y}\differential{y^{\prime}}u(y^{\prime})(\phi^{\prime
2}+g_{v}(y^{\prime}))-l_{r}^{2}\int_{0}^{y}\differential{y^{\prime}}u(y^{\prime})e^{2y}+\widetilde{F}_{r}^{\prime}(0)}\right],$
(3.68)

where

$\displaystyle
u(y)=\exp{\int_{0}^{y}\differential{y^{\prime}}\mathopen{}\mathclose{{}\left(2-2\frac{\phi^{\prime\prime}}{\phi^{\prime}}}\right)}\mbox{\quad
and\quad}g_{v}(y)=v^{\prime
2}+\mathopen{}\mathclose{{}\left(v^{\prime\prime}-\frac{\phi^{\prime\prime}}{\phi^{\prime}}v^{\prime}}\right)h_{r}e^{-2y}.$
(3.69)

The function $u(y)$ can be obtained analytically:

$\displaystyle
u(y)=\mathopen{}\mathclose{{}\left(\frac{\phi^{\prime}(0)}{\phi^{\prime}(y)}}\right)^{2}e^{2y}.$
(3.70)

By plugging this result into (3.68) and evaluating it at $y=y_{1}$, we can
write an expression for the radion mass:

$\displaystyle\frac{l_{r}^{2}}{f^{2}}=\mathopen{}\mathclose{{}\left(\int_{0}^{y_{1}}\differential{y}\frac{e^{4(y-y_{1})}}{\phi^{\prime}(y)^{2}}}\right)^{-1}\mathopen{}\mathclose{{}\left[\frac{1}{3}+\frac{2}{3}\int_{0}^{y_{1}}\differential{y}\frac{g_{v}(y)}{\phi^{\prime
2}}e^{2(y-y_{1})}+\mathopen{}\mathclose{{}\left(\frac{\widetilde{F}^{\prime}(0)e^{-2y_{1}}}{\phi^{\prime}(0)^{2}}-\frac{\widetilde{F}^{\prime}(y_{1})}{\phi^{\prime}(y_{1})^{2}}}\right)}\right],$
(3.71)

where we have dropped the $e^{-2y_{1}}$ term after evaluating the integral
$\int_{0}^{y_{1}}\differential{y}e^{2(y-y_{1})}$. The boundary values for
$\widetilde{F}_{r}^{\prime}$ can be expressed in terms of the boundary values
for $h_{r}$ using the junction conditions given in (3.46). Since $\varphi=0$
on the branes in the case of stiff wall boundary conditions, these give

$\displaystyle\widetilde{F}_{r}^{\prime}=\frac{1}{3}v^{\prime}h_{r}e^{-2y},\mbox{\quad
on the branes.\quad}$ (3.72)

Plugging this result into (3.71) gives

$\displaystyle\begin{aligned}
\frac{l_{r}^{2}}{f^{2}}&=\mathopen{}\mathclose{{}\left(\int_{0}^{y_{1}}\differential{y}\frac{e^{4(y-y_{1})}}{\phi^{\prime
2}}}\right)^{-1}\biggl{[}\frac{1}{3}+\frac{2}{3}\int_{0}^{y_{1}}\differential{y}\frac{g_{v}(y)}{\phi^{\prime
2}}e^{2(y-y_{1})}\\\
&\phantom{{}={}}+\frac{1}{3}\mathopen{}\mathclose{{}\left(\frac{v^{\prime}(0)h_{r}(0)}{\phi^{\prime}(0)^{2}}-\frac{v^{\prime}(y_{1})h_{r}(y_{1})}{\phi^{\prime}(y_{1})^{2}}}\right)e^{-2y_{1}}\biggr{]}.\end{aligned}$
(3.73)

The first term in this expression is the radion mass in the absence of the
Higgs VEV, which can be calculated analytically. Ignoring the terms which are
proportional to $e^{-2y_{1}}$ and higher, it is given by

$\displaystyle\frac{l_{r,0}^{2}}{f^{2}}\equiv\frac{1}{3}\mathopen{}\mathclose{{}\left(\int_{0}^{y_{1}}\differential{y}\frac{e^{4(y-y_{1})}}{\phi^{\prime
2}}}\right)^{-1}=\frac{4}{3}\epsilon(2-\epsilon)^{2}v_{0}^{2}e^{\epsilon
y_{1}}\mathopen{}\mathclose{{}\left[\frac{(v_{1}/v_{0})(4-\epsilon)}{4-2\epsilon}-e^{\epsilon
y_{1}}}\right].$ (3.74)

We note that this result confirms the stability condition we have derived in
(3.39).

In the case of nonzero Higgs VEV, the radion mass receives contributions due
to its mixing with the Higgs, which are given by the remaining terms in
(3.73). To calculate them, all we need is the profile for the field $h_{r}$.
It can be obtained by solving (3.48) together with the boundary condition
(3.50) in the small backreaction limit. By using the ansatz (3.66), the
equation of motion and the boundary condition for $\widetilde{h}_{r}\equiv
h_{r}e^{-2y}$ are given respectively by

$\displaystyle\widetilde{h}_{r}^{\prime\prime}-\nu^{2}\widetilde{h}_{r}=4[3v^{\prime}+(-4+\nu^{2})v]=4(v^{\prime\prime}-v^{\prime}),$
(3.75)

and

$\displaystyle(\widetilde{h}_{r}^{\prime}+2\widetilde{h}_{r})-2v^{\prime}=\pm\frac{1}{2}\partialderivative[2]{V_{0,1}}{v}\widetilde{h}_{r},\mbox{\quad$+/-$
on the UV/IR brane.\quad}$ (3.76)

This equation can be solved exactly, and its solution can be plugged into
(3.73) to determine the correction to the radion mass due to Higgs-Radion
mixing. The expression for the correction is fairly complicated, but it can be
easily seen that it scales with $v^{2}(y_{1})\sim v_{\text{eff}}^{2}/f^{2}$,
so they are heavily suppressed compared to the leading term inside the
brackets of (3.73). Hence we conclude that, for the parameter space we are
interested in, the radion mass is given by (3.74) to a very good
approximation. We are showing the radion mass as a function of GW parameters
in Figure 6.

(a)  (b)

Figure 6: In these plots, we show the values of the radion mass rescaled with
the breaking scale $f$ and backreaction parameter $\kappa$. In both plots, we
fix $v_{1}=1$ and vary $v_{0}$. On the left figure, we fix the breaking scale
to $f=10\,$\mathrm{TeV}$$ and plot for various values of $\epsilon$. On the
right figure, we fix $\epsilon=1/10$ and plot for various values of the
breaking scale.

#### The Validity of Small Backreaction

At the beginning of this Subsection, we have ignored the backreaction of
gravity when obtaining the solutions for the VEVs of both field, and we have
used those solutions to calculate the mass spectrum. Now, we shall check the
validity of this approximation.

In this model, the field profiles in the bulk have different scales. While the
GW field starts with $\mathcal{O}(0.01)$ and ends with $\mathcal{O}(1)$, the
Higgs field scales as $\sim v(y_{1})e^{-2y_{1}}$ and $\sim v(y_{1})$ near the
UV and IR branes respectively. Therefore, the backreaction of the GW field on
the Higgs profile can be sizable, due to the fact that the GW field heavily
dominates throughout the bulk. Hence, we should analyze the backreaction
carefully, and determine the largest value of $\kappa$, so that the results we
have derived so far remain valid.

In order to study the backreaction, we need to use the general form of the
equation of motion which is given by (2.9). The function $A^{\prime}(y)$ is
given by (2.5), which for this model reads

$\displaystyle
A^{\prime}=\sqrt{1+\frac{\kappa^{2}}{12}\mathopen{}\mathclose{{}\left[\phi^{\prime
2}+v^{\prime 2}-\epsilon(\epsilon-4)\phi^{2}-(-4+\nu^{2})v^{2}}\right]}.$
(3.77)

We are only interested in the backreaction of the GW field to leading order.
Therefore we approximate $A^{\prime}$ as

$\displaystyle A^{\prime}\approx
1+\frac{\kappa^{2}}{24}\mathopen{}\mathclose{{}\left(\phi^{\prime
2}-\epsilon(\epsilon-4)\phi^{2}}\right).$ (3.78)

This function peaks on the IR brane, thus we can use its IR value to determine
the strength of the backreaction. Hence we estimate the size of the
corrections as

$\displaystyle\frac{\kappa^{2}}{24}\mathopen{}\mathclose{{}\left(\phi^{\prime
2}(y_{1})-\epsilon(\epsilon-4)\phi^{2}(y_{1})}\right)\approx\frac{\kappa^{2}}{24}\times\mathcal{O}(1),$
(3.79)

Although this number seems quite small, it has a significant effect on the
Higgs VEV as we shall see below.

The Higgs VEV on the IR brane is determined by the IR boundary condition for
the Higgs, which can be rearranged to have the form

$\displaystyle\frac{v^{2}(y_{1})}{v_{H}^{2}}=1-\frac{2}{\lambda_{H}v_{H}^{2}}\frac{v^{\prime}(y_{1})}{v(y_{1})}.$
(3.80)

The Higgs parameters we have used in this Subsection typically have
$\lambda_{H}^{-1}\approx 17$ and $m_{1}^{2}\approx 4$ to satisfy
$v_{\text{EW}}=246\,$\mathrm{GeV}$$ and $m_{h}=126\,$\mathrm{GeV}$$. This sets
$v_{H}^{2}=\lambda_{H}^{-1}m_{1}^{2}\approx 68$. So the number on the LHS of
(3.80) is very small. Therefore, a delicate cancellation between two
$\mathcal{O}(1)$ parameters is needed. This equation encodes the tuning
required to get a Higgs VEV that is suppressed compared with the KK scale. For
a fixed set of 5D parameters, Equation (3.80) effectively gives the Higgs VEV
as a function of $y_{1}$, which is the position of the IR brane that minimizes
the effective potential. One can alternatively see that there is a large
sensitivity to 5D parameters. For example, by increasing the value of the 5D
Newton constant, $\kappa^{2}$ even slightly, one finds that the Higgs VEV will
change significantly because of the sensitivity of the cancellation to small
changes.

To see explicitly how large the backreaction parameter $\kappa$ can be, let us
assume that we want to know $v^{2}(y_{1})$ to a 10% accuracy. Then a rough
upper bound for $\kappa$ can be given by

$\displaystyle\frac{\kappa^{2}}{24}\mathopen{}\mathclose{{}\left(\phi^{\prime
2}(y_{1})-\epsilon(\epsilon-4)\phi^{2}(y_{1})}\right)\lesssim
0.1\times\frac{v^{2}(y_{1})}{v_{H}^{2}}.$ (3.81)

For a hierarchy $f/v_{\text{eff}}\sim 10$, we get
$\kappa\lesssim\mathcal{O}(1)\times 10^{-3}$.

In the next Subsection, we will study the same model but without the GW field.
Then the only source of the backreaction is due to the Higgs field itself. In
that case, we need to replace the GW field in the above expression by the
Higgs field:

$\displaystyle\frac{\kappa^{2}}{24}\mathopen{}\mathclose{{}\left(v^{\prime
2}(y_{1})-(-4+\nu^{2})v^{2}(y_{1})}\right)\lesssim
0.1\times\frac{v^{2}(y_{1})}{v_{H}^{2}}.$ (3.82)

The term inside the parentheses on the LHS is $\sim 8v^{2}(y_{1})$, thus the
bound on $\kappa$ in this model is given by

$\displaystyle\kappa\lesssim\sqrt{0.1\times\frac{3}{v_{H}^{2}}}\sim\mathcal{O}(1)\times
10^{-2}.$ (3.83)

(a)  (b)

Figure 7: On the left figure, we show the ratio between $v^{2}(y_{1})$
obtained by solving (2.9) numerically with nonzero backreaction, and the
analytical result (3.26) obtained by neglecting the backreaction. The green
dots show the results for the model of this Subsection, where both GW and the
Higgs are in the bulk, while the blue dots are the results for the model of
the next Subsection, where only the Higgs is in the bulk. The green and blue
dashed vertical lines show the upper bound estimates for $\kappa$ calculated
via (3.81) and (3.82) respectively. The gray line corresponds to a ratio of
$0.9$ which we have assumed in our estimates. On the right figure, we plot the
Radion-Higgs mass ratio for $\kappa=4\times 10^{-3}$. The parameters for these
plots are $v_{0}=1/50$, $v_{1}=1$, $\epsilon=1/10$, $f=10\,$\mathrm{TeV}$$,
$m_{0}^{2}=43/10$, $\nu=1/10$ unless otherwise specified. The remaining free
parameters are fixed such that $v_{\text{EW}}=246\,$\mathrm{GeV}$$ and
$m_{h}=126\,$\mathrm{GeV}$$.

On the left plot of Figure 7, we show the ratio
$v^{2}_{\kappa}(y_{1})/v^{2}_{0}(y_{1})$ as a function of $\kappa$, where
$v^{2}_{\kappa}(y_{1})$ is the Higgs VEV square on the IR brane found by
numerically solving (2.9) with nonzero backreaction, and $v^{2}_{0}(y_{1})$ is
the analytical result with no backreaction given in (3.26). The green and blue
dots show the results for models of this and the next Subsection respectively.
The green and blue dashed vertical lines denote the bounds calculated via
(3.81) and (3.82) respectively, while the gray line shows the
$v^{2}_{\kappa}(y_{1})/v^{2}_{0}(y_{1})$ ratio of $0.9$, which we have used in
our estimates. We can see that the backreaction has a much bigger effect when
both GW and the Higgs are in the bulk, and our estimates agree with the
numerical results quite well.

On the right plot of Figure 7, we show the Radion-Higgs mass ratio by assuming
$\kappa=4\times 10^{-3}$. We see that, if we insist on staying in the regime
where the backreaction can be neglected, then the radion mass is about an
order of magnitude smaller than the Higgs mass. The radion can be made heavier
by increasing the backreaction, but one needs a full numerical analysis of the
mass spectrum, which is beyond the scope of this work.

###  3.3 Higgs in the Bulk

In this Subsection, we will study a model where the Higgs is the only
stabilizing field in the bulk. This model has two major differences compared
to the one in the previous Subsection where both the Goldberger-Wise and the
Higgs fields were propagating in the bulk. First, the parameter space for
electroweak symmetry breaking receives another constraint due to the fact that
now the Higgs is responsible for radius stabilization. This means that we
don’t have the freedom of adjusting Goldberger-Wise parameters to get a stable
minimum; instead, we demand that the Higgs parameters $\alpha_{0,1}$ and $\nu$
satisfy additional conditions which we shall derive below. The second main
difference will be the radion mass, which now depends only on the Higgs
parameters.

#### Parameter Space for Electroweak Symmetry Breaking

We will be using the notation introduced in Subsection 3.2 and assume
$\alpha_{0}>0$ so that (3.30) is positive definite. From (3.28), we can write
the effective potential everywhere as

$\displaystyle V_{\text{eff}}(y_{1})=\begin{cases}e^{-4y_{1}}\delta
T_{1},&\text{No EWSB}\\\ e^{-4y_{1}}\mathopen{}\mathclose{{}\left[\delta
T_{1}-\frac{\lambda_{H}}{4}v^{4}(y_{1})}\right],&\text{EWSB},\end{cases}$
(3.84)

where we have tuned $\delta T_{0}=0$. By taking the derivative and setting it
to zero, we find that at an extremum point $y_{1}=y_{1}^{\text{ext}}$, the IR
brane mistune is given by

$\displaystyle\delta
T_{1}^{\text{ext}}=\frac{\lambda_{H}}{8}v^{2}(y_{1}^{\text{ext}})\mathopen{}\mathclose{{}\left[2v^{2}(y_{1}^{\text{ext}})-\partialderivative{v^{2}(y_{1})}{y_{1}}\evaluated{}_{y_{1}=y_{1}^{\text{ext}}}}\right].$
(3.85)

In order to not have a runaway solution in the UV, i.e. to have a global
minimum, we demand that this term is positive. Then the second derivative of
$V_{\text{eff}}$ at $y_{1}=y_{1}^{\text{ext}}$, where $\delta T_{1}$ is
replaced by $\delta T_{1}^{\text{ext}}$ is given by

$\displaystyle
V_{\text{eff}}^{\prime\prime}(y_{1}^{\text{ext}})=e^{-4y_{1}}\frac{\lambda_{H}}{2}\mathopen{}\mathclose{{}\left[v^{2}(y_{1})\mathopen{}\mathclose{{}\left(4\partialderivative{v^{2}(y_{1})}{y_{1}}-\partialderivative[2]{v^{2}(y_{1})}{y_{1}}}\right)-\mathopen{}\mathclose{{}\left(\partialderivative{v^{2}(y_{1})}{y_{1}}}\right)^{2}}\right]\evaluated{}_{y_{1}=y_{1}^{\text{ext}}}.$
(3.86)

By using explicit expression for $v^{2}(y_{1})$, it is possible to show that
$\delta T_{1}^{\text{ext}}>0$ implies
$V_{\text{eff}}^{\prime\prime}(y_{1}^{\text{ext}})>0$, so we can use the
former. Then we obtain the following constraint on the parameter $\alpha_{1}$:

$\displaystyle\alpha_{1}>\frac{4\alpha_{0}\nu\mathopen{}\mathclose{{}\left[(1+\nu)(\alpha_{0}+4\nu)e^{2\nu
y_{1}}-\alpha_{0}}\right]}{[(e^{2\nu y_{1}}-1)\alpha_{0}+4\nu e^{2\nu
y_{1}}]^{2}}\evaluated{}_{y_{1}=y_{1}^{\text{ext}}}.$ (3.87)

(a)  (b)

Figure 8: In this figure, the circles denotes possible
$\quantity{\alpha_{0},\alpha_{1}}$ pairs characterized by
$v_{\text{EW}}=246\,$\mathrm{GeV}$$ and $m_{h}=126\,$\mathrm{GeV}$$. The
hierarchy between the electroweak and the conformal breaking scales is
$f/v_{\text{EW}}=10$ and $f/v_{\text{EW}}=100$ for the left and right plot
respectively. The gray shaded area denotes the region in parameter space where
there is no “critical” value of $y_{1}$. The color shaded areas are the
regions where the condition for global minimum (3.87) is violated. To improve
clarity, we are showing the $\quantity{\alpha_{0},\alpha_{1}}$ pairs
corresponding to these regions with semi-transparent circles.

We are showing the effect of this constraint on the Higgs parameter space in
Figure 8. In these plots, we are plotting $m_{1}^{2}$ in terms of $m_{0}^{2}$
such that the electroweak scale is $v_{\text{EW}}=246\,$\mathrm{GeV}$$ and the
Higgs mass is $m_{h}=126\,$\mathrm{GeV}$$. The methods and the equations for
determining $m_{1}^{2}$ are identical to the Goldberger-Wise and Higgs model
studied in the previous Subsection. The semi-transparent circles correspond to
the points where the constraint (3.87) is not satisfied, hence those points
should be excluded from the parameter space. We can see that the constraint
becomes more stringent as one decreases $\nu$ and increases the breaking scale
$f$.

#### The Radion Mass

The radion mass can be calculated by following the procedure described in
Subsection 3.2. For this model we find

$\displaystyle\frac{l_{r}^{2}}{f^{2}}=\frac{1}{3}\mathopen{}\mathclose{{}\left(\int_{0}^{y_{1}}\differential{y}\frac{e^{4(y-y_{1})}}{v^{\prime}(y)^{2}}}\right)^{-1}\mathopen{}\mathclose{{}\left[1-\frac{\widetilde{h}_{r}(y_{1})}{v^{\prime}(y_{1})}+\frac{\widetilde{h}_{r}(0)}{v^{\prime}(0)}e^{-2y_{1}}}\right],$
(3.88)

where $\widetilde{h}_{r}$ is the solution to equations (3.75) and (3.76).
Again, the full solution is analytical, but fairly complicated. Nevertheless,
we can get a good insight by calculating the factor which multiplies the term
inside the brackets. It is given by

$\displaystyle
I\equiv\frac{1}{3}\mathopen{}\mathclose{{}\left(\int_{0}^{y_{1}}\differential{y}\frac{e^{4(y-y_{1})}}{v^{\prime}(y)^{2}}}\right)^{-1}=\frac{4v^{2}(y_{1})m_{0}^{2}\nu^{2}e^{-2\nu
y_{1}}}{3(e^{2\nu
y_{1}}-1)(m_{0}^{2}-4+2\nu)}\mathopen{}\mathclose{{}\left[\frac{e^{2\nu
y_{1}}(2+\nu)-r(2-\nu)}{(1-re^{-2\nu y_{1}})^{2}}}\right],$ (3.89)

where $r=\frac{m_{0}^{2}-4-2\nu}{m_{0}^{2}-4+2\nu}$. We find that the term
inside the brackets varies between $1.0$ and $1.4$ in the parameter space we
are using. We show the results of the Radion-Higgs mass ratio in Figure 9 as a
function of $\alpha_{0}=m_{0}^{2}-4-2\nu$, under the assumption that
$\kappa=6\times 10^{-2}$. To make the plots we have used the full analytical
result given in (3.88). We can see that in this model, the radion is two to
three orders of magnitude lighter than the Higgs. Since the effect of the
backreaction is not as significant as in the previous model, we expect that a
significantly light radion is a property of this model, even with
backreaction.555Note that radion masses at the GeV scale and above are still
largely unconstrained by the latest experimental results, as shown by the
detailed analysis of [20].

(a)  (b)

Figure 9: In this figure, we are plotting the ratio between the radion mass
and the Higgs mass as a function of $\alpha_{0}$ by assuming $\kappa=6\times
10^{-2}$. The breaking scale is $f=10\,$\mathrm{TeV}$$ and
$f=50\,$\mathrm{TeV}$$ on the left and right figure respectively. The Higgs
parameters are fixed such that $v_{\text{EW}}=246\,$\mathrm{GeV}$$ and
$m_{h}=126\,$\mathrm{GeV}$$.

It might seem surprising that decreasing $\nu$ increases the radion mass. To
understand the effect of this parameter, it is useful to study the quantity
$I$ in detail. Since $f\sim\mathcal{O}(10\,$$tera{\mathrm{T}}$\mathrm{eV}$)$,
we have $y_{1}\sim\mathcal{O}(30)$. Choosing $\nu\approx 1/10$ implies
$e^{2\nu y_{1}}\sim\mathcal{O}(1000)$ hence we can assume $e^{2\nu y_{1}}\gg
1$. In this approximation, $I$ becomes

$\displaystyle
I\approx\frac{4}{3}v^{2}(y_{1})\frac{m_{0}^{2}(2+\nu)\nu^{2}}{m_{0}^{2}-4+2\nu}e^{-2\nu
y_{1}},\quad e^{2\nu y_{1}}\gg 1.$ (3.90)

In this regime, the radion mass is heavily suppressed by $e^{-2\nu y_{1}}$,
which is $\mathcal{O}(10^{-3})$, in addition to suppression by $v^{2}(y_{1})$
and $\nu^{2}$. On the other hand, if $\nu\sim 1/100$, then $e^{2\nu
y_{1}}\sim\mathcal{O}(1)$. In this case, we can evaluate $I$ at the leading
order in $\nu$. The result is

$\displaystyle
I\approx\frac{8}{3}v^{2}(y_{1})\frac{m_{0}^{2}\nu^{2}}{m_{0}^{2}-4}\frac{e^{2\nu
y_{1}}}{(e^{2\nu y_{1}}-1)^{2}},\quad\nu\ll 1.$ (3.91)

We see that the exponential suppression is absent in this case, hence the
radion is heavier in this regime.

##  4 CFT Interpretation

Field theories in AdS admit an interpretation in terms of a 4D strongly
coupled CFT (or approximate CFT) dual [21, 22]. In this Section, we discuss a
CFT interpretation of multi-scalar stabilization models.

It is useful to first recall the CFT interpretation of the original
unstabilized Randall-Sundrum construction [2], and then the simplest
stabilization mechanism introduced by Goldberger and Wise [3, 4]. We then
discuss the specific interpretation of multi-scalar models with emphasis on
those where there are critical points for symmetry breaking in the moduli
space for the radion.

The original 2-brane RS model with weak 5D gravity has an interpretation as a
large-$N$ approximate CFT where the conformal symmetry is spontaneously
broken. The radion degree of freedom corresponds to the dilaton – the
Goldstone boson of spontaneously broken conformal invariance. The classical
equations of motion (Einstein’s equations) rule out a static geometry unless
the UV and IR brane tensions are individually tuned against the bulk
cosmological constant. These two tunings represent (respectively) the need to
have a vanishing bare cosmological constant so that there is no explicit
breaking of conformal invariance, and the vanishing of the dilaton quartic, an
allowed coupling that would otherwise destabilize the conformally non-
invariant vacuum state. In other words, as is well known [23], spontaneous
breaking of exact conformal invariance without breaking Poincaré invariance
requires a flat direction.

Even in this case, with both tunings performed (and ignoring quantum
effects)666In fact, even with both UV and IR brane tunings performed, quantum
corrections will generally lead to a nonzero Casimir potential between the
branes., the continuum of physically inequivalent vacua (the moduli space of
inter-brane separations) present a problem for creating a low energy
description that resembles our Standard Model. Thus, a model must incorporate
a stabilization mechanism. The simplest such mechanism posits a bulk scalar
field that develops a vacuum expectation value, deforming the geometry through
gravitational backreaction. If this scalar has an approximate shift symmetry,
violated by (for example) a small bulk mass term, its VEV grows slowly, and
backreaction effects are sizable only deep into the bulk of AdS. This
backreaction leads to an effective potential for the inter-brane separation,
and stabilization is achieved dynamically without tuning of the IR brane
tension (the UV tuning must still be performed, or enforced by additional
symmetries to obtain static geometries). The slow growth of the scalar VEV in
the UV region of the geometry ensures that the generated hierarchy is
exponentially large.

This 5D picture is dual to 4D dimensional transmutation in which a nearly
marginal operator (not necessarily classically marginal as is the case for
asymptotic freedom) is given a nonzero coupling constant. Its logarithmic
evolution breaks conformal invariance explicitly, leading to the dynamical
generation of an infrared scale where either the theory has flowed very far
away from the UV fixed point [24, 25], or has flowed to a point where the
effective dilaton quartic has become small [10, 26].

There is a similar class of 5D models where, in the 4D CFT dual, it is a
composite operator that serves as the nearly marginal deformation of the CFT.
This occurs in the case where there is a 5D field with mass near the
Breitenlohner-Freedman bound, which (in 5D) is $m^{2}=-4$ in units of the AdS
curvature [27, 28, 29]. The corresponding operator ${\mathcal{O}}$ in the CFT
dual has dimension $\Delta\sim 2$, such that at large $N$, the composite
operator ${\mathcal{O}}^{\dagger}{\mathcal{O}}$ has dimension $\Delta\sim 4$.
Thus, turning on this composite operator corresponds at large $N$ to a nearly
marginal deformation of the CFT. The model we discuss where only the Higgs is
in the bulk is of this form, where it is the Higgs vacuum expectation value
that backreacts on the geometry and stabilizes the inter-brane separation.

To understand the CFT interpretation of radion-induced Higgs criticality, it
is first necessary to review the interpretation of the UV brane and field
dynamics there. In models with a UV brane, the interpretation is that there
are two sectors of the theory, a fundamental sector and a CFT sector, and that
these sectors mix.

First, we review the basics of the correspondence without the UV brane (or
with the UV brane taken to the boundary of AdS). In this case the
correspondence links field configurations on the boundary of AdS (which are
not integrated over in the higher dimensional partition function) to sources
for a dual CFT:

$Z_{\text{CFT}}[J(x)=\phi(x,z=0)]=\int[{\mathcal{D}}\phi]_{\text{bulk}}\exp\mathopen{}\mathclose{{}\left[iS(\phi)}\right].$
(4.1)

The CFT with sources could be described by a Lagrangian of the form
${\mathcal{L}}_{\text{CFT}}={\mathcal{L}}^{0}_{\text{CFT}}+J(x){\mathcal{O}}_{\text{CFT}}$.

When the UV brane is introduced, the path integral has no restriction –
effectively the sources are promoted to dynamical fields that mix with the
CFT, and the new dual 4D Lagrangian is of the form

${\mathcal{L}}={\mathcal{L}}_{\text{fundamental}}(\phi(x))+{\mathcal{L}}^{0}_{\text{CFT}}+\lambda\phi(x){\mathcal{O}}.$
(4.2)

The CFT and fundamental sector mix with each other. Through the mixing,
spontaneous breaking of conformal invariance in the CFT sector from VEVs of
CFT operators, $\langle{\mathcal{O}}\rangle$, is communicated to the
fundamental sector. The size of these VEVs corresponds in 5D to the position
of the IR brane. Integrating out the CFT (or bulk, in the 5D picture) degrees
of freedom leads to an effective theory for the fundamental degrees of freedom
in which parameters of the EFT, like the Higgs mass squared term, are a
function of $y_{1}$. This $y_{1}$ dependence can be such that the EFT crosses
phase boundaries for symmetry breaking transitions as $y_{1}$ is varied.

In short, the 5D theory with two branes is dual to a 4D theory where a
fundamental and CFT sector are coupled to each other, and where the CFT is
spontaneously broken by operator VEVs. The coupling between the two allows a
“CFT-induced criticality”, where CFT operator VEVs induce instabilities in the
fundamental sector. In 5D, the instabilities leading to non-trivial VEVs are
induced by varying the position of the IR brane.

In the case of using this picture as a model for electroweak physics, the
effective Higgs mass squared term is, in the 4D picture, a function of CFT
operator VEVs. For certain values of these VEVs, the effective mass squared
may become negative, and induce electroweak breaking. In 5D, the mass squared
of the lowest lying state of the bulk Higgs can become negative for ranges of
$y_{1}$, such that electroweak symmetry breaking is induced by the radion VEV.

##  5 Conclusions

We have explored radius stabilization in the context of 5D models in warped
space with multiple scalar fields. General results for multi-scalar models are
derived to all orders in backreaction for 5D Einstein-Hilbert gravity. We
suggest a holography-inspired approach to the multi-scalar potential, and
additionally derive a superpotential method for creating static geometries
with multiple bulk scalar fields.

Of particular interest given recent experimental results are the
phenomenological implications of Higgs fine tuning. We explored 5D models
focusing on obtaining a hierarchy between extra-dimensional excitations and
the electroweak symmetry breaking scale. 5D models with such a hierarchy are
close to the critical point for the electroweak sector, and thus, generically,
the radion will be a dynamical degree of freedom that scans the Higgs phase
transition.

Three examples were presented where the radion scans the effective Higgs mass
parameter across a symmetry breaking phase transition:

  * •

A model with the Higgs on the IR brane, where a bulk Goldberger-Wise scalar
couples to the Higgs and scans its effective mass.

  * •

A model with only a Higgs field in the bulk, where coupling to the radion
arises through geometrical backreaction.

  * •

A model with both Higgs and a Goldberger-Wise scalar in the bulk.

These three models cover a range of types of modulus-Higgs potential that give
a hierarchy between extra-dimensional resonances and the electroweak scale.

The models presented are an important step in understanding electroweak
symmetry breaking in the context of holographic composite Higgs models,
particularly those where the Higgs is light in comparison to the compositeness
scale. An important message is that the tuning of the Higgs itself has
implications in terms of cosmology and collider phenomenology through the
manner of the Randall-Sundrum/electroweak early universe physics and the low
lying spectrum of scalar states.

The formalism we have developed for multi-scalar stabilization models is
relevant for all theories in which the Higgs emerges as a mode that is light
in comparison with the compactification scale. In general, the Higgs plays a
non-trivial part in stabilization, and this work helps further elucidate the
connection between the compactification scale and that of electroweak physics.

##  6 Acknowledgements

The authors thank Brando Bellazzini, Csaba Csáki, Michael Geller, and Anson
Hook for useful conversations regarding the research presented here. The
authors thank Cornell University for hospitality throughout the duration of
this project. JH thanks the Munich Institute for Astro- and Particle Physics
(MIAPP) of the DFG cluster of excellence “Origin and Structure of the
Universe” for a productive research environment during a portion of this work.
CE, JH, and GR were supported in part by the DOE under grant award number DE-
SC0009998. CE is supported by the Deutsche Forschungsgemeinschaft under
Germany’s Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306.

##  Appendix A Derivative of the Radion Potential

In this Appendix, we derive Equation (2.12) for the derivative of the
effective potential.

Before imposing any of the boundary conditions, we straightforwardly take the
derivative of the effective potential with respect to $y_{1}$:

$\displaystyle\begin{aligned}
\derivative{V_{\textnormal{eff}}}{y_{1}}&=\sum_{i}e^{-4A}\mathopen{}\mathclose{{}\left[\frac{\partial
V_{0}}{\partial\phi_{i}}\derivative{\phi_{i}}{y_{1}}-\frac{1}{2A^{\prime}}\mathopen{}\mathclose{{}\left(\phi^{\prime}_{i}\derivative{\phi^{\prime}_{i}}{y_{1}}-\frac{\partial
V}{\partial\phi_{i}}\derivative{\phi_{i}}{y_{1}}}\right)}\right]\evaluated{}_{y=y_{0}}\\\
&\phantom{{}={}}+\sum_{i}e^{-4A}\mathopen{}\mathclose{{}\left[\frac{\partial
V_{1}}{\partial\phi_{i}}\derivative{\phi_{i}}{y_{1}}+\frac{1}{2A^{\prime}}\mathopen{}\mathclose{{}\left(\phi^{\prime}_{i}\derivative{\phi^{\prime}_{i}}{y_{1}}-\frac{\partial
V}{\partial\phi_{i}}\derivative{\phi_{i}}{y_{1}}}\right)}\right]\evaluated{}_{y=y_{1}}\\\
&\phantom{{}={}}-4e^{-4A_{1}}\derivative{A_{1}}{y_{1}}\widetilde{V}_{\text{IR}}-4e^{-4A_{0}}\derivative{A_{0}}{y_{1}}\widetilde{V}_{\text{UV}}.\end{aligned}$
(A.1)

Imposing the bulk scalar equations of motion allows a substitution for
$\partial V/\partial\phi_{i}$:

$\displaystyle\frac{\partial
V}{\partial\phi_{i}}=\phi_{i}^{\prime\prime}-4A^{\prime}\phi_{i}^{\prime}.$
(A.2)

Plugging this into the above derivative, we find:

$\displaystyle\begin{aligned}
\derivative{V_{\textnormal{eff}}}{y_{1}}&=\sum_{i}e^{-4A}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\frac{\partial
V_{0}}{\partial\phi_{i}}-2\phi^{\prime}_{i}}\right)\derivative{\phi_{i}}{y_{1}}-\frac{1}{2A^{\prime}}\mathopen{}\mathclose{{}\left(\phi^{\prime}_{i}\derivative{\phi^{\prime}_{i}}{y_{1}}-\phi^{\prime\prime}_{i}\derivative{\phi_{i}}{y_{1}}}\right)}\right]\evaluated{}_{y=y_{0}}\\\
&\phantom{{}={}}+\sum_{i}e^{-4A}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\frac{\partial
V_{0}}{\partial\phi_{i}}-2\phi^{\prime}_{i}}\right)\derivative{\phi_{i}}{y_{1}}+\frac{1}{2A^{\prime}}\mathopen{}\mathclose{{}\left(\phi^{\prime}_{i}\derivative{\phi^{\prime}_{i}}{y_{1}}-\phi^{\prime\prime}_{i}\derivative{\phi_{i}}{y_{1}}}\right)}\right]\evaluated{}_{y=y_{1}}\\\
&\phantom{{}={}}-4e^{-4A_{1}}\derivative{A_{1}}{y_{1}}\widetilde{V}_{\text{IR}}-4e^{-4A_{0}}\derivative{A_{0}}{y_{1}}\widetilde{V}_{\text{UV}}.\end{aligned}$
(A.3)

The first terms in parentheses in each bracket would vanish if the boundary
conditions were imposed. The other terms in parentheses are Lagrange brackets
with respect to the variables $A$ and $y_{1}$ [30]. Let us see how that is the
case. First, we identify the conjugate momenta to $\phi_{i}$:

$\displaystyle\pi_{i}=-\frac{\partial\mathcal{L}}{\partial\phi^{\prime}_{i}}=-\sqrt{g}g^{55}\phi^{\prime}_{i}=e^{-4A}\phi^{\prime}_{i},$
(A.4)

where the minus sign is irrelevant, and chosen for convenience. The Lagrange
bracket for the pair $(A,y_{1})$ (at some arbitrary $y$-value) is:

$\displaystyle\begin{aligned}
\mathopen{}\mathclose{{}\left\\{A,y_{1}}\right\\}^{\phi_{i},\pi_{i}}&=\sum_{i}\frac{\partial\pi_{i}}{\partial
A}\frac{\partial\phi_{i}}{\partial y_{1}}-\frac{\partial\pi_{i}}{\partial
y_{1}}\frac{\partial\phi_{i}}{\partial A}\\\
&=\sum_{i}\mathopen{}\mathclose{{}\left[-4\pi_{i}\frac{\partial\phi_{i}}{\partial
y_{1}}+4\pi_{i}\frac{\partial A}{\partial
y_{1}}\frac{\partial\phi_{i}}{\partial
A}+e^{-4A}\mathopen{}\mathclose{{}\left(\frac{\partial\phi_{i}^{\prime}}{\partial
A}\frac{\partial\phi_{i}}{\partial
y_{1}}-\frac{\partial\phi_{i}^{\prime}}{\partial
y_{1}}\frac{\partial\phi_{i}}{\partial A}}\right)}\right].\end{aligned}$ (A.5)

The first two terms cancel, and the rest (using
$\differential{A}=A^{\prime}\differential{y}$) gives the remaining terms we
have in the brackets in the derivative of the effective potential.

The final derivative of the potential then is given by:

$\displaystyle\begin{aligned}
\derivative{V_{\textnormal{eff}}}{y_{1}}&=e^{-4A_{0}}\mathopen{}\mathclose{{}\left(\frac{\partial
V_{0}}{\partial\phi_{0}}-2\phi^{\prime}_{0}}\right)\derivative{\phi_{0}}{y_{1}}+e^{-4A_{1}}\mathopen{}\mathclose{{}\left(\frac{\partial
V_{1}}{\partial\phi_{1}}+2\phi^{\prime}_{1}}\right)\derivative{\phi_{1}}{y_{1}}\\\
&\phantom{{}={}}+\mathopen{}\mathclose{{}\left\\{A,y_{1}}\right\\}^{\phi_{0},\pi_{\phi,0}}-\mathopen{}\mathclose{{}\left\\{A,y_{1}}\right\\}^{\phi_{1},\pi_{\phi,1}}\\\
&\phantom{{}={}}-4e^{-4A_{1}}\derivative{A_{1}}{y_{1}}\widetilde{V}_{\text{IR}}-4e^{-4A_{0}}\derivative{A_{0}}{y_{1}}\widetilde{V}_{\text{UV}}.\end{aligned}$
(A.6)

Translations in $y$ are an example of canonical transformations on the
coordinates [30], and the Lagrange bracket is thus an invariant under shifts
in $y$. Therefore the two Lagrange bracket terms cancel against each other.
The very last term is vanishing by choice of an overall metric scale factor:
$\differential A_{0}/\differential y_{1}=0$. Imposing the scalar equations of
motion on the IR brane causes the second term to vanish. We are left with
Equation (2.12).

##  Appendix B A Superpotential Method for Many Scalar Fields

It is well known that a method exists for constructing static geometries that
satisfy all boundary conditions and bulk equations of motion when there is a
single scalar field. The method posits a “superpotential” function of the
scalar field $\phi$, $W(\phi)$. For a given superpotential, $W$, there is a
unique bulk potential given by

$\displaystyle V(\phi)=\frac{1}{8}\mathopen{}\mathclose{{}\left(\frac{\partial
W}{\partial\phi}}\right)^{2}-\frac{\kappa^{2}}{6}W^{2}.$ (B.1)

Taking the $y$-derivative of the above equation and dividing by
$\phi^{\prime}$ gives

$\displaystyle\frac{\partial
V}{\partial\phi}=\frac{1}{4\phi^{\prime}}\frac{\partial
W}{\partial\phi}\derivative{y}\frac{\partial
W}{\partial\phi}-\frac{\kappa^{2}}{3}W\frac{\partial W}{\partial\phi}.$ (B.2)

From this, we see that if $\phi^{\prime}=\frac{1}{2}\frac{\partial
W}{\partial\phi}$, and $A^{\prime}=\frac{\kappa^{2}}{6}W$, this is equivalent
to the scalar equation of motion. The second of these two relations is
required by Einstein’s equations, as its $y$ derivative must reproduce the
relation $A^{\prime\prime}=\frac{\kappa^{2}}{3}\phi^{\prime 2}$.

If one starts with a given $W$, there must be a relation between $W$ and the
boundary potentials for this bulk solution (which is first order) to solve the
scalar boundary value problem (which is second order). Additionally, the
boundary potentials are constrained to have the form

$\displaystyle V_{0,1}(\phi)=\pm W(\phi_{0,1})\pm
W^{\prime}(\phi_{0,1})(\phi-\phi_{0,1})+\text{ higher order polynomial in
}(\phi-\phi_{0,1}).$ (B.3)

With this procedure, one can start with any $W$, and create a static solution
to the geometry with any given $y_{1}$. This procedure incorporates arbitrary
backreaction. Stability is not guaranteed (the radion may be tachyonic about
these points), but this can be checked by solving the mode expansion about the
classical background determined by this method.

Using this method, one can also “go backwards”, starting with the (more
fundamental) $V(\phi)$, and then integrate the superpotential equation. It is
non-linear, and typically no analytic solutions are possible (with a limited
number of exceptions). The rest of the story proceeds in the same way, but the
additional integration constant gives the freedom to match arbitrary boundary
potentials (although one must solve for $y_{1}$, rather than positing it from
the get-go).

This procedure can be generalized to multiple scalar fields in the following
way. Take a superpotential $W(\\{\phi_{i}\\})$. This is achieved by defining
the bulk potential as

$\displaystyle
V=\frac{1}{8}\sum_{i}\mathopen{}\mathclose{{}\left(\frac{\partial
W}{\partial\phi_{i}}}\right)^{2}-\frac{\kappa^{2}}{6}W^{2}$ (B.4)

and taking

$\displaystyle\phi_{i}^{\prime}=\frac{1}{2}\frac{\partial
W}{\partial\phi_{i}}\qquad\text{and}\qquad A^{\prime}=\frac{\kappa^{2}}{6}.$
(B.5)

One can now again take the $y$-derivative of the superpotential relation. The
result is a sum over all of the scalar field equations of motion after
substituting the above relations, showing that this procedure will generate
solutions to the bulk equations of motion, for any given $W$. The boundary
conditions also must be satisfied, which will happen for the following form of
the boundary potentials:

$\displaystyle\begin{aligned} V_{0,1}(\phi)&=\pm
W\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\\{\phi_{i}^{0,1}}\right\\}}\right)\pm\sum_{i}\frac{\partial
W}{\partial\phi_{i}}\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\\{\phi_{i}^{0,1}}\right\\}}\right)\mathopen{}\mathclose{{}\left(\phi_{i}-\phi_{i}^{0,1}}\right)\\\
&\phantom{{}={}}+\text{ higher order polynomial in
}\mathopen{}\mathclose{{}\left(\phi_{i}-\phi_{i}^{0,1}}\right).\end{aligned}$
(B.6)

It is not clear that going in “reverse” is possible in the same way as in the
case of a single scalar field. That is, it seems unlikely that all solutions
that stabilize the geometry with more than one scalar come from
superpotentials.

## References

  * [1] P. W. Graham, D. E. Kaplan and S. Rajendran, “Cosmological Relaxation of the Electroweak Scale”, Phys. Rev. Lett. 115, 221801 (2015), hep-ph/1504.07551.
  * [2] L. Randall and R. Sundrum, “A Large Mass Hierarchy from a Small Extra Dimension”, Phys. Rev. Lett. 83, 3370 (1999), hep-ph/9905221.
  * [3] W. D. Goldberger and M. B. Wise, “Modulus Stabilization with Bulk Fields”, Phys. Rev. Lett. 83, 4922 (1999), hep-ph/9907447.
  * [4] W. D. Goldberger and M. B. Wise, “Phenomenology of a Stabilized Modulus”, Phys. Lett. B 475, 275 (2000), hep-ph/9911457.
  * [5] M. A. Amin, J. Fan, K. D. Lozanov and M. Reece, “Cosmological Dynamics of Higgs Potential Fine Tuning”, Phys. Rev. D 99, no. 3, 035008 (2019), hep-ph/1802.00444.
  * [6] P. Breitenlohner and D. Z. Freedman, “Stability in Gauged Extended Supergravity”, Annals Phys. 144, 249 (1982).
  * [7] M. Geller, S. Bar-Shalom and A. Soni, “Higgs-Radion Unification: Radius Stabilization by an SU(2) Bulk Doublet and the 126 GeV Scalar”, Phys. Rev. D 89, no. 9, 095015 (2014), hep-ph/1312.3331.
  * [8] L. Vecchi, “A Natural Hierarchy and a Low New Physics Scale from a Bulk Higgs”, JHEP 1111, 102 (2011), hep-ph/1012.3742.
  * [9] G. F. Giudice, R. Rattazzi and J. D. Wells, “Graviscalars from Higher-Dimensional Metrics and Curvature-Higgs Mixing”, Nucl. Phys. B 595, 250 (2001), hep-ph/0002178.
  * [10] B. Bellazzini, C. Csáki, J. Hubisz, J. Serra and J. Terning, “A Naturally Light Dilaton and a Small Cosmological Constant”, Eur. Phys. J. C 74, 2790 (2014), hep-th/1305.3919.
  * [11] C. Eröncel, J. Hubisz and G. Rigo, “Self-Organized Higgs Criticality”, JHEP 1903, 046 (2019), hep-ph/1804.00004.
  * [12] O. DeWolfe, D. Z. Freedman, S. S. Gubser and A. Karch, “Modeling the Fifth-Dimension with Scalars and Gravity”, Phys. Rev. D 62, 046008 (2000), hep-th/9909134.
  * [13] B. von Harling and G. Servant, “QCD-Induced Electroweak Phase Transition”, JHEP 1801, 159 (2018), hep-ph/1711.11554.
  * [14] S. Bruggisser, B. Von Harling, O. Matsedonskyi and G. Servant, “Baryon Asymmetry from a Composite Higgs Boson”, Phys. Rev. Lett. 121, no. 13, 131801 (2018), hep-ph/1803.08546.
  * [15] S. Bruggisser, B. Von Harling, O. Matsedonskyi and G. Servant, “Electroweak Phase Transition and Baryogenesis in Composite Higgs Models”, JHEP 1812, 099 (2018), hep-ph/1804.07314.
  * [16] A. Pomarol, O. Pujolas and L. Salas, “Holographic Conformal Transition and Light Scalars”, JHEP 1910, 202 (2019), hep-th/1905.02653.
  * [17] C. Cheung and P. Saraswat, “Mass Hierarchy and Vacuum Energy”, hep-ph/1811.12390.
  * [18] C. Csáki, M. L. Graesser and G. D. Kribs, “Radion Dynamics and Electroweak Physics”, Phys. Rev. D 63, 065002 (2001), hep-th/0008151.
  * [19] J. M. Lizana, M. Olechowski and S. Pokorski, “A New Way of Calculating the Effective Potential for a Light Radion”, hep-ph/1911.11124.
  * [20] F. Abu-Ajamieh, J. S. Lee and J. Terning, “The Light Radion Window”, JHEP 1810, 050 (2018), hep-ph/1711.02697.
  * [21] J. M. Maldacena, “The Large $N$ Limit of Superconformal Field Theories and Supergravity”, Int. J. Theor. Phys. 38, 1113 (1999) [Adv. Theor. Math. Phys. 2, 231 (1998)], hep-th/9711200.
  * [22] E. Witten, “Anti-de Sitter Space and Holography”, Adv. Theor. Math. Phys. 2, 253 (1998), hep-th/9802150.
  * [23] S. Fubini, “A New Approach to Conformal Invariant Field Theories”, Nuovo Cim. A 34, 521 (1976).
  * [24] N. Arkani-Hamed, M. Porrati and L. Randall, “Holography and Phenomenology”, JHEP 0108, 017 (2001), hep-th/0012148.
  * [25] R. Rattazzi and A. Zaffaroni, “Comments on the Holographic Picture of the Randall-Sundrum Model”, JHEP 0104, 021 (2001), hep-th/0012248.
  * [26] F. Coradeschi, P. Lodone, D. Pappadopulo, R. Rattazzi and L. Vitale, “A Naturally Light Dilaton”, JHEP 1311, 057 (2013), hep-th/1306.4601.
  * [27] I. R. Klebanov and E. Witten, “AdS/CFT Correspondence and Symmetry Breaking”, Nucl. Phys. B 556, 89 (1999), hep-th/9905104.
  * [28] D. B. Kaplan, J. W. Lee, D. T. Son and M. A. Stephanov, “Conformality Lost”, Phys. Rev. D 80, 125005 (2009), hep-th/0905.4752.
  * [29] V. Gorbenko, S. Rychkov and B. Zan, “Walking, Weak First-Order Transitions, and Complex CFTs”, JHEP 1810, 108 (2018), hep-th/1807.11512.
  * [30] L. D. Landau and E. M. Lifshitz, “Course of Theoretical Physics Volume I: Mechanics”.