# 4D dS vacua from AdS vacua of type IIB string theory and the AdS distance
conjecture

Cao H. Nam<EMAIL_ADDRESS>Phenikaa Institute for Advanced
Study and Faculty of Fundamental Sciences, Phenikaa University, Yen Nghia, Ha
Dong, Hanoi 12116, Vietnam

###### Abstract

In order for string theory to be made compatible with the low-energy
observations of a positive cosmological constant, there have been attempts to
construct dS vacua in string theory which are particularly difficult to
realize. Instead of attempting to find de Sitter (dS) vacuum solutions, we
point to a new way to make string theory consistent with low-energy dS
cosmology. In this way, string theory lives in an anti-de Sitter (AdS) vacuum
(which is simple to construct) that exists only in the high-energy regime;
however, as going to the low-energy scales where the heavy string excitations
and Kaluza-Klein modes are integrated out, we show that the effective picture
of string theory in lower dimensions would exhibit a 4D dS vacuum without
needing to add additional structures such as anti-D3 branes. Additionally, we
point to evidence from bottom-up physics for the strong version of the AdS
distance conjecture realized from AdS vacua in string theory. This evidence
hence supports the sharpening of the AdS distance conjecture as one of the
universal features of quantum gravity.

_Motivations._ —String theory has been considered as one of the candidates for
a correct theory of quantum gravity. However, it has been faced with the
challenges of matching with the experimental observations at low-energy
scales. One of them is to construct de Sitter (dS) vacua motivated by the
observation of positive cosmological constant Weinberg2013 . However, it has
been known that dS vacua in string theory are particularly difficult to
realize. No-go theorems imply that there are no dS vacua in supergravity and
string theory if the internal space is static, is compact, and has no
singularities Gibbons2003 ; Maldacena2001 . The difficulties of the dS vacuum
construction come also from explicitly computing string loop, higher-
derivative, and nonperturbative corrections. The Kachru-Kallosh-Linde-Trivedi
construction allows anti-de Sitter (AdS) vacua uplifted to metastable dS vacua
by adding anti-D3 branes Kachru2003 ; Burgess2003 ; Choi2005 ; Westphal2007 .
But, there have been recent results in the literature which point to the
problems with this dS vacuum construction regarding the backreaction of
anti-D3 branes on the internal geometry Bena2010 ; Danielsson2015 ; Michel2015
; Cohen-Maldonado2016 ; Gautason2016 ; Danielsson2017  and on the 4D moduli
Moritz2018 , and nonsupersymmetric (non-SUSY) Giddings-Kachru-Polchinski
solutions (derived in Ref. Giddings2002 ) Sethi2018 . In addition, there have
been some attempts at embedding dS cosmology within string theory Banerjee2018
; Alwis2021 . The technical difficulties have hence suggested the possibility
that string theory admits no dS vacua which do not suffer from instabilities
at all Brennan2017 . Hence, the dS conjecture Obied2018  was proposed as well
as the dS instability Garga2019  was studied to realize no dS vacua in string
theory. For reviews about the status of the dS vacuum construction in string
theory, see Danielsson2018 ; Cicoli2019 .

On the contrary, AdS vacua in string theory are understood very well and
simple to construct. And, another fact is that the dS vacuum that we observe
has been realized in the low-energy regime so far. This means that it is not
known whether the vacuum is still dS in the high-energy regime; in other
words, it is possible that the vacuum would be AdS at the high-energy scales.
These facts imply an ideal that makes string theory compatible with the low-
energy observations of positive vacuum energy without needing to find its dS
vacua as follows. We start from the well-known AdS vacua of type IIB string
theory with the compactification geometry given by $M_{5}\times X_{5}$ where
solving the stringy 10D equations of motion would lead to $M_{5}$ to be an
$\text{AdS}_{5}$ factor, which exists only in the high-energy regime. In
addition, we consider the compactification of $M_{5}$ on a circle in order to
obtain the observed 4D world where the 4D tensor component of the $M_{5}$
metric is in general dependent on the fifth coordinate. We point to that the
non-trivial dynamics of the 4D tensor component of the $M_{5}$ metric along
the fifth dimension leads to a positive contribution to energy in the 4D
effective theory. As a result, a 4D dS vacuum can emerge in the low-energy
regime from an AdS vacuum of the higher-dimensional theory existing at the
high-energy scales.

A feature of AdS vacua in string theory is that an infinite tower of states
becomes light in the limit of the AdS curvature radius going to infinity due
to no scale separation between the AdS curvature radius and the radius of the
internal space. This implies the proposal of the AdS distance conjecture
Lust2019  which is a generalization of the swampland distance conjecture
Ooguri2007  to the metric configuration space and is stated as follows: The
near-flat limit of any AdS vacuum is accompanied by an infinite tower of
states whose mass scale behaves as $m\sim|\Lambda|^{\alpha}$ with $\Lambda$ to
be a cosmological constant and $\alpha\geq\frac{1}{2}$ required by the strong
version. This conjecture together with other swampland conjectures have been
used to place the constraints on the effective theories which can be completed
consistent with quantum gravity in the ultraviolet Vafa2005 ; Palti2019 ;
Grana2021 , and interesting implications for the neutrino masses, cosmological
constant, and electroweak vacuum have been found Martin-Lozano2017 ;
Valenzuela2017 ; Hamada2017 ; Gonzalo2018a ; Gonzalo2018b ; Rudelius2021 ;
Gonzalo2021 ; Nam2022b . However, evidence coming from the bottom-up physics
for the strong version of the AdS distance conjecture, which is necessary to
test this conjecture and sharpens it (and thus the swampland distance
conjecture) as one of the universal features of quantum gravity, is still
missing. We will show that the mass spectrum of the KK tower for the 5D bulk
fields represents bottom-up evidence for the strong version of the AdS
distance conjecture. This result hence supports that the AdS distance
conjecture (and hence the swampland distance conjecture) realized from the AdS
vacuum construction in string theory can be applied in general for quantum
gravity.

_AdS vacua in type IIB string theory._ —Our starting point is the 10D action
for the massless string excitations of type IIB string theory, which is given
by

$\displaystyle S=\frac{1}{\kappa^{2}_{10}}\int
d^{10}X\sqrt{-g}\left[\mathcal{R}-\frac{1}{2}\left(\nabla\Phi\right)^{2}-\frac{g^{2}_{s}}{2}e^{a_{p}\Phi}F^{2}_{p+2}\right],$
(1)

where $\kappa^{2}_{10}\equiv(2\pi)^{7}g^{2}_{s}l^{8}_{s}$ with $g_{s}$ and
$l_{s}$ to be the string coupling and the string length, respectively, the
dilaton coupling parameter $a_{p}$ for the Ramond-Ramond sector is
$a_{p}=(3-p)/2$, and $F^{2}_{p+2}\equiv
F_{M_{1}M_{2}...M_{p+2}}F^{M_{1}M_{2}...M_{p+2}}/(p+2)!$ with
$F_{M_{1}M_{2}...M_{p+2}}$ to be the field strength tensor of $(p+1)$-form
gauge fields. We consider the solution of (1) with the following geometry

$\displaystyle ds^{2}_{10}$ $\displaystyle=$ $\displaystyle
ds^{2}_{M_{5}}+L^{2}ds^{2}_{X_{5}},$ $\displaystyle e^{\Phi}$ $\displaystyle=$
$\displaystyle g_{s},$ (2) $\displaystyle g_{s}F_{5}$ $\displaystyle=$
$\displaystyle\alpha(1+\star)\text{vol}_{X_{5}},$

where $X_{5}$ is a 5D Sasaki-Einstein manifold Gubser1999  which has the
curvature satisfying $\mathcal{R}_{X_{5}}=20$ and is stabilized by $N$ units
of flux, $\text{vol}_{X_{5}}$ refers to the five-form volume of $X_{5}$,
$\alpha=16\pi g_{s}Nl^{4}_{s}(\pi^{3}/\text{Vol}_{M_{5}})$ which is determined
by the flux quantization constraint $\int_{X_{5}}\star F_{5}=(2\pi
l_{s})^{-3}N\kappa^{2}_{10}/(g_{s}l_{s})$ with $\text{Vol}_{X_{5}}$ to be the
volume of $X_{5}$, and other $(p+1)$-form fields are trivial.

The equations of motion
$\square\Phi=g^{2}_{s}a_{p}e^{a_{p}\Phi}F^{2}_{p+2}/[2(p+2)!]$ and
$\triangledown_{M}\left(e^{a_{p}\Phi}F^{MN_{1}...N_{p+1}}\right)=0$ are
satisfied for the constant dilaton and self-dual five-form field [and other
$(p+1)$-form fields which vanish], respectively. Whereas, Einstein field
equations
${\mathcal{R}_{M}}^{N}=F_{MM_{1}M_{2}M_{3}M_{4}}F^{NM_{1}M_{2}M_{3}M_{4}}/96$
lead to

$\displaystyle L^{4}$ $\displaystyle=$ $\displaystyle 4\pi
g_{s}Nl^{4}_{s}\frac{\pi^{3}}{\text{Vol}_{X_{5}}},$ (3)
$\displaystyle\mathcal{R}_{M_{5}}$ $\displaystyle=$
$\displaystyle-\frac{20}{L^{2}},$ (4)

where $\mathcal{R}_{M_{5}}$ denotes the scalar curvature of $M_{5}$. Eqs. (3)
and (4) together with the geometry (2) imply that an exact background of type
IIB string theory which is obtained from solving the stringy 10D equations of
motion is a factor $\text{AdS}_{5}$ times a 5D internal manifold, i.e.,
$\text{AdS}_{5}\times X_{5}$.111The well-known case is that $X_{5}$ is a five-
sphere $S^{5}$ related to AdS/CFT correspondence Maldacena1998 .

_Dimensionally reduced action of string theory._ —With the background geometry
(2), reducing the 10D action (1) of type IIB string theory on $X_{5}$ we get
the following 5D effective action as follows

$\displaystyle S_{5D}$ $\displaystyle=$ $\displaystyle\frac{M^{3}_{5}}{2}\int
d^{5}X\sqrt{-g_{5}}\left[\mathcal{R}_{M_{5}}-2\Lambda\right],$ (5)

where $M^{3}_{5}=2\text{Vol}_{M_{5}}L^{5}/\kappa^{2}_{10}$ and
$\Lambda=-6/L^{2}$ with $L$ as given by (3). This action means that the 5D
effective theory of string theory would be in the AdS vacuum.

In addition, in order to obtain the 4D observed world we consider the
compactification of $M_{5}$ on a circle $S^{1}$ where the most general setting
of this compactification is given by a principal bundle with the typical fiber
to be $U(1)$ Coquereaux1988 ; Bailin1987 ; Overduin1997 , which adopts the
local coordinates as $\left(x^{\mu},\theta\right)$ with
$\\{x^{\mu}\\}\in\mathbb{R}^{4}$ and $\theta$ being an angle parametrizing the
fifth dimension of $M_{5}$ corresponding to the coordinate transformation as
$x^{\mu}\rightarrow x^{\prime\mu}=x^{\prime\mu}(x)$ and
$\theta\rightarrow\theta^{\prime}=\theta+\alpha(x)$. Hence, the metric
equipped on $M_{5}$ takes the following general form

$\displaystyle
ds^{2}_{M_{5}}=g_{\mu\nu}dx^{\mu}dx^{\nu}+R^{2}\left[d\theta+g_{{}_{A}}A_{\mu}dx^{\mu}\right]^{2},$
(6)

where $g_{\mu\nu}$, $A_{\mu}$, and $R$ are the 4D tensor, 4D vector, and 4D
scalar component fields of the bulk metric on $M_{5}$, respectively, and
$g_{{}_{A}}$ is the corresponding gauge coupling. With this ansatz, we can
explicitly expand $\mathcal{R}_{M_{5}}$ given in the action (5) in terms of
the 4D component fields (see Appendix A for the detailed computation) as
follows

$\displaystyle\mathcal{R}_{M_{5}}$ $\displaystyle=$
$\displaystyle\hat{\mathcal{R}}+\frac{1}{4R^{2}}\left(\partial_{\theta}g^{\mu\nu}\partial_{\theta}g_{\mu\nu}+g^{\mu\nu}g^{\rho\lambda}\partial_{\theta}g_{\mu\nu}\partial_{\theta}g_{\rho\lambda}\right)$
(7) $\displaystyle-\frac{g^{2}_{{}_{A}}R^{2}}{4}F_{\mu\nu}F^{\mu\nu},$

where $\hat{\mathcal{R}}\equiv
g^{\mu\nu}(\hat{\partial}_{\lambda}\Gamma^{\lambda}_{\nu\mu}-\hat{\partial}_{\nu}\Gamma^{\lambda}_{\lambda\mu}+\Gamma^{\rho}_{\nu\mu}\Gamma^{\lambda}_{\lambda\rho}-\Gamma^{\rho}_{\lambda\mu}\Gamma^{\lambda}_{\nu\rho})$
with
$\Gamma^{\rho}_{\mu\nu}\equiv\frac{g^{\rho\lambda}}{2}(\hat{\partial}_{\mu}g_{\lambda\nu}+\hat{\partial}_{\nu}g_{\lambda\mu}-\hat{\partial}_{\lambda}g_{\mu\nu})$,
$\hat{\partial}_{\mu}\equiv\partial_{\mu}-g_{{}_{A}}A_{\mu}\partial_{\theta}$,
and $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$.

_The emergence of dS vacuum in 4D world._ —In the previous section, we have
seen that type IIB string theory exists at the AdS vacuum. However, in the
following we will show that this AdS vacuum exhibits only in the high-energy
regime. But, when approaching the low-energy regime, the effective theory of
type IIB string theory in lower dimensions would exhibit a 4D dS vacuum which
is consistent with the low-energy observations. The emergence of dS vacuum
here is essentially due to the presence of the second term in Eq. (7) which
has been ignored in the literature, because the $\theta$ dependence of the 4D
tensor component of the bulk metric equipped on $M_{5}$ is usually not
considered.

Let us first obtain the bulk profile of the $4$D tensor component which
describes its dynamics along the fifth dimension of $M_{5}$. By varying the
action (5) in the 4D tensor component of the bulk metric, we find the
following equation

$\displaystyle\bar{\mathcal{R}}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\bar{\mathcal{R}}+\Lambda
g_{\mu\nu}+\frac{1}{4R^{2}}\left[g_{\mu\rho}g_{\nu\lambda}\partial^{2}_{\theta}g^{\rho\lambda}\right.$
$\displaystyle\left.-\partial^{2}_{\theta}g_{\mu\nu}+g^{\rho\lambda}\partial_{\theta}g_{\rho\lambda}\partial_{\theta}g_{\mu\nu}+2\partial_{\theta}\left(g_{\mu\nu}g^{\rho\lambda}\partial_{\theta}g_{\rho\lambda}\right)\right.$
$\displaystyle\left.-\frac{1}{2}g_{\mu\nu}\left\\{\partial_{\theta}g^{\rho\lambda}\partial_{\theta}g_{\rho\lambda}-\left(g^{\rho\lambda}\partial_{\theta}g_{\rho\lambda}\right)^{2}\right\\}\right]=0,$
(8)

where
$\bar{\mathcal{R}}_{\mu\nu}\equiv(\partial_{\lambda}\bar{\Gamma}^{\lambda}_{\nu\mu}-\partial_{\nu}\bar{\Gamma}^{\lambda}_{\lambda\mu}+\bar{\Gamma}^{\rho}_{\nu\mu}\bar{\Gamma}^{\lambda}_{\lambda\rho}-\bar{\Gamma}^{\rho}_{\lambda\mu}\bar{\Gamma}^{\lambda}_{\nu\rho})$
with
$\bar{\Gamma}^{\rho}_{\mu\nu}\equiv\frac{g^{\rho\lambda}}{2}(\partial_{\mu}g_{\lambda\nu}+\partial_{\nu}g_{\lambda\mu}-\partial_{\lambda}g_{\mu\nu})$
and $\bar{\mathcal{R}}\equiv g^{\mu\nu}\bar{\mathcal{R}}_{\mu\nu}$. We have
here obtained Eq. (8) in the vacuum $R=\text{const}$ and $A_{\mu}=0$ which are
the solution of their equations of motion as seen late. We separate the
variables as $g_{\mu\nu}(x,\theta)=\chi(\theta)g^{(4)}_{\mu\nu}(x)$ where
$g^{(4)}_{\mu\nu}(x)$ is identified as the usual metric in the 4D effective
theory and $\chi(\theta)$ is its profile. Then, we find

$\displaystyle\mathcal{R}^{(4)}_{\mu\nu}-\frac{1}{2}g^{(4)}_{\mu\nu}\mathcal{R}^{(4)}+\Lambda_{4}g^{(4)}_{\mu\nu}$
$\displaystyle=$ $\displaystyle 0,$ (9) $\displaystyle
3\chi^{\prime\prime}(\theta)+8\frac{\chi^{\prime}(\theta)^{2}}{\chi(\theta)}+\frac{2\Lambda}{R^{-2}}\chi(\theta)$
$\displaystyle=$ $\displaystyle\frac{2\Lambda_{4}}{R^{-2}},$ (10)

where $\mathcal{R}^{(4)}_{\mu\nu}$ and $\mathcal{R}^{(4)}$ are the usual Ricci
and scalar curvatures of the 4D effective geometry of spacetime written in
terms of $g^{(4)}_{\mu\nu}(x)$, respectively, and $\Lambda_{4}$ is a constant.
It is important to remark that Eq. (10) is a nonlinear differential equation
and hence the solution of $g_{\mu\nu}(x,\theta)$ should not be given as the
linear combination of partial solutions. Whereas, Eq. (9) determines the 4D
effective geometry of spacetime sourced by a cosmological constant
$\Lambda_{4}$ which is originated from the dynamics of the 4D tensor component
of the bulk metric along the fifth dimension of $M_{5}$.

To find an analytical solution for Eq. (10) with the boundary condition
$\chi(-\pi)=\chi(\pi)$ boundcon  for the general value of $\Lambda_{4}$ is a
difficult task. However, with $\Lambda_{4}=0$ corresponding to the situation
of small $\Lambda_{4}$, it is easy to find an analytical solution as follows

$\displaystyle\chi(\theta)=\cosh^{\frac{3}{11}}\left(\frac{\sqrt{22}}{3}\kappa\theta\right),$
(11)

where $\kappa\equiv\sqrt{-\Lambda/R^{-2}}$. For the $\Lambda_{4}\neq 0$, a
particularly analytical solution is found as

$\displaystyle\chi(\theta)=\frac{11\Lambda_{4}}{19|\Lambda|}\left[\cosh\left(\sqrt{\frac{2}{11}}\kappa\theta\right)-1\right].$
(12)

Note that, due to the topology of $S^{1}$ the 4D metric component and thus its
bulk profile $\chi(\theta)$ must be periodic with the period $2\pi$, i.e.,
$\chi(\theta)=\chi(\theta+2\pi)$. One can make the solution $\chi(\theta)$
periodic with the period $2\pi$ by reflecting it at the boundary as discussed
in Appendix B.222In the case of $\Lambda>0$, the solution for $\chi(\theta)$
is related to the cosine function which is periodic and thus itself is
compatible with the topology of $S^{1}$ Nam2022b ; Nam2021 .

The fact that $\chi(\theta)$ is non-negative implies $\Lambda_{4}\geq 0$,
which means that the non-trivial profile given by Eqs. (11) and (32) for the
4D tensor component of the bulk metric along the fifth dimension of $M_{5}$
should give a non-negative contribution to the energy in the 4D effective
theory as seen in Eq. (9). In this sense, the 4D effective theory of type IIB
string theory would be at the dS vacuum.

A key question that here arises is what would lead to the non-trivial profile
for the 4D tensor component? We can see that the essential point which leads
to this non-trivial profile is due to the presence of the second term on the
left-hand side of Eq. (10). This term comes from the non-linear property of
the gravitational field: Gravity is itself a source that creates gravity. If
this term is absent then Eq. (9) becomes linear and hence the solution would
be a sum of all possible modes with the different values of $\Lambda_{4}$.
However, the excitation modes with $\Lambda_{4}>0$ would decay to the lower
modes with $\Lambda_{4}<0$. As a result, the 4D effective theory would exist
at the negative energy state or the AdS vacuum. Therefore, we can realize that
the non-trivial profile for the 4D tensor component leading to the emergence
of the dS vacuum in the 4D effective theory is essentially due to its non-
linear property.

In order to show the 4D dS vacuum actually emerged in the effective picture of
string theory in lower dimensions, we need to demonstrate that the ansatz (6)
with the vacuum configuration $g_{\mu\nu}=\chi(\theta)g^{(4)}_{\mu\nu}(x)$
(with the 4D metric $g^{(4)}_{\mu\nu}$ corresponding to the 4D dS geometry),
$R=\text{const}$, and $A_{\mu}=0$ which has just been found above satisfies
the 4D stringy equations of motion. The ansatz (6) and Eq. (9) suggest that
the dimensional reduction of the 5D action (5) on $S^{1}$ leads to the 4D
effective action given in Einstein frame as follows

$\displaystyle S_{4D}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{|g_{4}|}\left[\frac{M^{2}_{\text{Pl}}}{2}\left\\{\mathcal{R}^{(4)}-\frac{3}{2}\left(\frac{\partial_{\mu}R}{R}\right)^{2}\right\\}-V(R)\right.$
(13) $\displaystyle\left.-\frac{g^{2}_{{}_{A}}\pi
M^{3}_{5}R^{3}}{4}F_{\mu\nu}F^{\mu\nu}-\frac{m^{2}_{A}}{2}A^{\mu}A_{\mu}\right],$

where $M^{2}_{\text{Pl}}=M^{3}_{5}R_{0}\int^{\pi}_{-\pi}d\theta\chi$,
$m^{2}_{A}=3M^{3}_{5}R_{0}g^{2}_{{}_{A}}\int^{\pi}_{-\pi}d\theta[\chi^{\prime\prime}-\chi^{\prime
2}/(2\chi)]$, and $V(R)$ refers to the potential of the radion field $R$. (In
order to change to Einstein frame, we have rescaled
$g^{(4)}_{\mu\nu}\rightarrow\Omega^{-2}g^{(4)}_{\mu\nu}$ with
$\Omega^{2}=R/R_{0}$ and $R_{0}$ to be an arbitrary scale.) The radion
potential $V(R)$ is given as follows

$\displaystyle V(R)=V_{\text{tr}}(R)+V_{\text{1L}}(R),$ (14)

where $V_{\text{tr}}(R)$ is the tree-level term which is generated by the
dynamics of $4$D tensor component along the fifth dimension of $M_{5}$ and is
given by

$\displaystyle V_{\text{tr}}(R)=M^{2}_{\text{Pl}}\frac{R_{0}}{R}\Lambda_{4},$
(15)

and $V_{\text{1L}}(R)$ is the quantum level term which is generated by the
(one loop) Casimir energy contribution and reads

$\displaystyle
V_{\text{1L}}(R)=\sum_{i}(-1)^{s_{i}}n_{i}R\left(\frac{R_{0}}{R}\right)^{2}\rho_{i}(R)\int^{\pi}_{-\pi}d\theta\chi^{2}(\theta),$
(16)

where $s_{i}=0(1)$ for the fermions(bosons), $n_{i}$ refers to the number of
degrees of freedom with respect to the $i$-th particle, and the Casimir energy
density is given by Arkani-Hamed2007

$\displaystyle\rho_{i}(R)=\sum^{\infty}_{n=1}\frac{2m^{5}_{i}}{(2\pi)^{5/2}}\frac{K_{5/2}(2\pi
nm_{i}R)}{(2\pi nm_{i}R)^{5/2}},$ (17)

with $m_{i}$ and $K_{5/2}(z)$ to be the mass of the $i$-th particle and the
modified Bessel function, respectively.333Here, the matter fields such as the
Standard Model which are not contained in the field content of
$\text{AdS}_{5}$ supergravity can be embedded by adding probe D7-branes
wrapping around an internal cycle which is a submanifold of $M_{5}$ Karch2002
; Gherghetta2006  We can easily see that the radion potential behaves as
$(\sum_{f}n_{f}-\sum_{b}n_{b})/R^{6}$ and
$(\sum_{f}n_{f}-\sum_{b}n_{b})|_{m=0}\int^{\pi}_{-\pi}d\theta\chi^{2}(\theta)/R^{6}$
in the regions of $R\rightarrow 0$ and $R\rightarrow\infty$, respectively,
where $\sum_{f}n_{f}-\sum_{b}n_{b}$ ($(\sum_{f}n_{f}-\sum_{b}n_{b})|_{m=0}$)
is the net number of (massless) fermionic and bosonic degrees of freedom. This
means that for $\sum_{f}n_{f}-\sum_{b}n_{b}>0$ and
$(\sum_{f}n_{f}-\sum_{b}n_{b})|_{m=0}>0$ the radion potential would approach
the positive infinity for both $R\rightarrow 0$ and $R\rightarrow\infty$ and
hence there is always a stable minimum. Additionally, with the proper
parameters this minimum has the positive energy corresponding to the dS
vacuum, as depicted in Fig. 1.

---
Figure 1: The radion potential for two cases: The red and blue curves
correspond to the solutions (11) and (32), respectively. Here, the radion
potential corresponding to the red curve is rescaled by
$|\Lambda|^{3}R^{2}_{0}$, $k\equiv|\Lambda|^{1/2}\Lambda_{4}R_{0}$, and we
have considered $\sum_{f}n_{f}-\sum_{b}n_{b}=2$ and $m_{f,b}=0$ as a benchmark
case.

The radion potential $V(R)$ thus allows us to fix physically the size of the
fifth dimension of $M_{5}$ or the vacuum expectation value of the radion field
$R$.

The 4D stringy equations of motion associated with the 4D effective action
(13) are derived in Appendix C. It is easy to see that the equations of motion
for the radion field $R$ and the graviphoton $A_{\mu}$ lead to the vacuum
solution $R=\text{const}$ and $A_{\mu}=0$ where the constant corresponds to
the stable minimum of the radion potential $V(R)$. Whereas, the equations of
motion for the 4D metric $g^{(4)}_{\mu\nu}$ is
$\mathcal{R}^{(4)}_{\mu\nu}=\Lambda^{\prime}_{4}g^{(4)}_{\mu\nu}$ where
$\Lambda^{\prime}_{4}\equiv M^{-2}_{\text{Pl}}V_{\text{sm}}$ with
$V_{\text{sm}}$ referring to the stable minimum of the radion potential
$V(R)$. Because $V_{\text{sm}}$ is positive with the proper parameters of the
radion potential, the vacuum geometry of 4D effective theory is dS but the
vacuum value is now shifted compared to Eq. (9) due to the radion
stabilization.

Because the present construction is done relying on the 5D effective action
obtained from the dimensional reduction of string theory on $X_{5}$, it is
necessary to have a separation of the scales between the radius of $S^{1}$ and
the scale $L$ of $X_{5}$. On the other hand, because of Eqs. (3) and(4) the
radius of $S^{1}$ must be much larger than the curvature radius of
$\text{AdS}_{5}$, i.e. $R|\Lambda|^{1/2}\gg 1$. We observe that the solution
(32) can lead to the minimum of the radion potential fixing physically the
radius of $S^{1}$ to satisfy this separation of the scales. Indeed, as seen in
Fig. 1 the minimum of the radion potential in the $R|\Lambda|^{1/2}$ direction
would get larger when $k\equiv|\Lambda|^{1/2}\Lambda_{4}R_{0}$ decreases. This
means that at a sufficiently small value of $k$ or $\Lambda_{4}$ it would
yield the separation of the scales between the radius of $S^{1}$ and the scale
of $X_{5}$.

Finally, let us emphasize that the transition from the AdS vacuum in the high-
energy regime to the dS vacuum in the low-energy regime allows the effective
theory to avoid the constraint of non-SUSY AdS conjecture Ooguri2017  because
of the fact that supersymmetry is broken at the low-energy scales. If non-SUSY
AdS vacuum is stable, then the effective theory would be in the swampland. On
the contrary, non-SUSY AdS vacua would develop the non-perturbative
instabilities and thus decay into SUSY vacua via the bubble nucleation
Witten1981 ; Horowitz2007 .

_AdS distance conjecture._ —We point to the first evidence from the bottom-up
physics for the strong version of the AdS distance conjecture Lust2019 . In
order to do this, we consider the 5D action (5) where the AdS radius $L$ is in
general arbitrary instead of being given by Eq. (3). This means that the 5D
action (5) in this situation is not originated from the string theory
compactification.

Let us first obtain the profile $Y_{n}(\theta)$ of the 5D bulk fields along
the fifth dimension of $M_{5}$ corresponding to the solution (11). The
equations for the bulk profile of the 5D bulk fields are given in Appendix D.
In general, it is not easy to obtain the analytical solutions to these
equations. However, for small $\kappa$, we can find the analytical solutions
to these equations by expanding $\chi(\theta)$ in $\kappa$. Up to the order
$\kappa^{2}$, the solution form of these equations satisfying the boundary
condition $Y(-\pi)=Y(\pi)$ boundcon  is found as

$\displaystyle Y_{n}(\theta)$ $\displaystyle=$
$\displaystyle\left[N_{n}H_{2n}\left(\sqrt{\frac{b_{n}}{3}}\theta\right)+{{}_{1}F_{1}}\left(-n;\frac{1}{2};\frac{b_{n}}{3}\theta^{2}\right)\right]$
(18) $\displaystyle\times\exp\left\\{-a_{n}\theta^{2}\right\\},$

where $n=0$, $1$, $2$, …, $a_{n}$ and $b_{n}$ are parameters depending on $n$,
$\kappa$, $R$, and the bulk mass, $H_{2n}(z)$ and ${{}_{1}F_{1}}(a;b;z)$ are
the Hermite polynomial and the confluent hypergeometric function,
respectively, and $N_{n}$ are the normalization constants.

The degree of Hermite polynomial in the expression of $Y_{n}(\theta)$ must be
even as a result of the boundary condition $Y_{n}(-\pi)=Y_{n}(\pi)$. From
this, we can obtain the mass spectrum of the KK tower as follows

$\displaystyle
m^{2}_{n}=\frac{|\Lambda|}{3}\times\left\\{\begin{array}[]{ll}\left[\frac{5}{2}+4c_{n}+\frac{1+4n}{2}\sqrt{25+8c_{n}}\right],&\hbox{\text{scalar};}\\\
\left[1+4c_{n}+(1+4n)\sqrt{7+2c_{n}}\right],&\hbox{\text{fermion};}\\\
\left[\frac{3}{2}+4c_{n}+\frac{1+4n}{2}\sqrt{9+8c_{n}}\right],&\hbox{vector,}\\\
\end{array}\right.$ (22)

which correspond to the scalar, fermion, and vector fields, respectively,
where $c_{n}\equiv n(1+2n)$. It should be noted here that first, we have
considered the scalar field with zero bulk mass. Second, the mass of the bulk
fermion must be zero due to the boundary condition on the bulk profile, which
implies that the left- and right-handed components of the bulk fermion
decouple and this is consistent with the fact that the compactification of
spacetime on the circle $S^{1}$ breaks $SO(1,4)$ to $SO(1,3)\times U(1)$. Eq.
(22) suggests that the mass of KK tower behaves in the power-law in the
cosmological constant as $m_{KK}\sim|\Lambda|^{\alpha}$ with
$\alpha=\frac{1}{2}$ which becomes light in the limit $\Lambda\rightarrow 0$.
This is consistent with the strong version of the AdS distance conjecture.

The above conclusion is still true in the general case. We observe from Fig. 1
that the minimum of the radion potential $V(R)$ would fix $R|\Lambda|^{1/2}$.
This implies $R^{-1}\propto|\Lambda|^{1/2}$ which means that the mass of the
KK tower would be proportional to $|\Lambda|^{1/2}$. Hence, taking the
cosmological constant $\Lambda$ to be zero would encounter an infinite tower
of light states.

_Conclusions._ —To achieve dS vacua in string theory motivated by the low-
energy observations of a positive cosmological constant has proved to be a
particularly difficult endeavor. Contrary to this, AdS vacua in string theory
are common and simple to construct. Motivated by this fact and low-energy dS
cosmology, we have indicated a new approach for the embedding of the observed
dS vacuum into string theory which is one of the candidates for a unitary
theory of quantum gravity. We do not attempt to find a dS vacuum solution in
string theory, but we start from a well-known AdS vacuum with the
compactification geometry given by $M_{5}\times X_{5}$. Then, we show that
this AdS vacuum of string theory exists only in the high-energy regime. In
other words, as approaching the low-energy scales where the heavy string
excitations and KK modes are integrated out the effective picture of string
theory in lower dimensions would exhibit a 4D dS vacuum (which has so far been
observed in the low-energy regime) due to the non-trivial dynamics of the 4D
tensor component of the $M_{5}$ metric along the fifth dimension which
contributes positive energy in the 4D effective theory. This result clearly
provides a new path in the construction of realistic models on AdS vacua
(rather than dS vacua) in type IIB string theory, but it still leads to a dS
vacuum in the 4D effective theory consistent with the low-energy observation
of positive vacuum energy.

In addition, due to the non-trivial bulk profile of the 4D tensor component of
the bulk metric, we revisited the mass spectrum of the KK tower for the 5D
bulk fields. We showed that this mass spectrum satisfies the strong version of
the AdS distance conjecture which is a generalization of the swampland
distance conjecture and whose proposal was motivated by the realizations in
the AdS vacuum construction in string theory. This offers evidence that comes
from bottom-up physics and hence supports the AdS distance conjecture (and
thus the swampland distance conjecture) as one of the universal features of
quantum gravity.

## Appendix A: The detail expansion of $\mathcal{R}_{M_{5}}$

Eq. (7) can be easily found if one works in the covariant frame
$\Big{\\{}\hat{\partial}_{\mu},\partial_{\theta}\Big{\\}}\equiv\left\\{\partial_{M}\right\\}$
where $\hat{\partial}_{\mu}$ and $\partial_{\theta}$ transform as a four-
dimensional vector and a one-dimensional vector under the residual general
coordinate transformation $x^{\mu}\rightarrow x^{\prime\mu}=x^{\prime\mu}(x)$
and $\theta\rightarrow\theta^{\prime}=\theta+\alpha(x)$. In this frame, the
coefficients of the Christoffel connection and the Riemann curvature tensor
are given as follows

$\displaystyle\Gamma^{P}_{MN}$ $\displaystyle=$
$\displaystyle\frac{G^{PQ}}{2}\left(\partial_{M}G_{NQ}+\partial_{N}G_{MQ}-\partial_{Q}G_{MN}\right)$
$\displaystyle+\frac{G^{PQ}}{2}\left(C^{O}_{QM}G_{ON}+C^{O}_{QN}G_{OM}\right)+\frac{C^{P}_{MN}}{2},$
$\displaystyle\mathcal{R}^{O}_{MPN}$ $\displaystyle=$
$\displaystyle\partial_{P}[\Gamma^{O}_{NM}]-\partial_{N}[\Gamma^{O}_{PM}]+\Gamma^{Q}_{NM}\Gamma^{O}_{PQ}-\Gamma^{Q}_{PM}\Gamma^{O}_{NQ}$
(24) $\displaystyle-C^{Q}_{PN}\Gamma^{O}_{QM},$

where $C^{P}_{MN}$ determines the commutation relation of any two frame fields
as

$\left[\partial_{M},\partial_{N}\right]=C^{P}_{MN}\partial_{P},$ (25)

and the metric and its dual are given in the coframe
$\\{dx^{\mu},d\theta+g_{{}_{A}}A_{\mu}dx^{\mu}\\}$ dual to
$\\{\hat{\partial}_{\mu},\partial_{\theta}\\}$ as

$\displaystyle G_{MN}$ $\displaystyle=$
$\displaystyle\textrm{diag}\left(g_{\mu\nu},R^{2}\right),$ $\displaystyle
G^{MN}$ $\displaystyle=$
$\displaystyle\textrm{diag}\left(g^{\mu\nu},R^{-2}\right).$ (26)

We can explicitly expand the 5D scalar curvature $\mathcal{R}_{M_{5}}$ as
follows

$\displaystyle\mathcal{R}_{M_{5}}=G^{MN}\mathcal{R}^{P}_{MPN}=g^{\mu\nu}\mathcal{R}^{P}_{\mu
P\nu}+G^{\theta\theta}\mathcal{R}^{P}_{\theta P\theta},$ (27)

where

$\displaystyle g^{\mu\nu}\mathcal{R}^{P}_{\mu P\nu}$ $\displaystyle=$
$\displaystyle
g^{\mu\nu}\left(\partial_{M}\Gamma^{M}_{\nu\mu}-\hat{\partial}_{\nu}\Gamma^{M}_{M\mu}+\Gamma^{N}_{\nu\mu}\Gamma^{M}_{MN}-\Gamma^{N}_{M\mu}\Gamma^{M}_{\nu
N}\right.$ (28) $\displaystyle\left.-C^{N}_{M\nu}\Gamma^{M}_{N\mu}\right)$
$\displaystyle=$ $\displaystyle
g^{\mu\nu}\left(\hat{\partial}_{\lambda}\Gamma^{\lambda}_{\nu\mu}-\hat{\partial}_{\nu}\Gamma^{\lambda}_{\lambda\mu}+\Gamma^{\rho}_{\nu\mu}\Gamma^{\lambda}_{\lambda\rho}-\Gamma^{\rho}_{\lambda\mu}\Gamma^{\lambda}_{\nu\rho}\right)$
$\displaystyle+g^{\mu\nu}\left(\partial_{\theta}\Gamma^{\theta}_{\nu\mu}+\Gamma^{\theta}_{\nu\mu}\Gamma^{\lambda}_{\lambda\theta}+\Gamma^{\rho}_{\nu\mu}\Gamma^{\theta}_{\theta\rho}+\Gamma^{\theta}_{\nu\mu}\Gamma^{\theta}_{\theta\theta}\right)$
$\displaystyle-g^{\mu\nu}\left(\hat{\partial}_{\nu}\Gamma^{\theta}_{\theta\mu}+\Gamma^{\rho}_{\theta\mu}\Gamma^{\theta}_{\nu\rho}+\Gamma^{\theta}_{\lambda\mu}\Gamma^{\lambda}_{\nu\theta}+\Gamma^{\theta}_{\theta\mu}\Gamma^{\theta}_{\nu\theta}\right)$
$\displaystyle-g^{\mu\nu}C^{\theta}_{\lambda\nu}\Gamma^{\lambda}_{\theta\mu},$
$\displaystyle G^{\theta\theta}\mathcal{R}^{P}_{\theta P\theta}$
$\displaystyle=$ $\displaystyle
G^{\theta\theta}\left(\partial_{M}\Gamma^{M}_{\theta\theta}-\partial_{\theta}\Gamma^{M}_{M\theta}+\Gamma^{N}_{\theta\theta}\Gamma^{M}_{MN}-\Gamma^{N}_{M\theta}\Gamma^{M}_{\theta
N}\right.$ (29)
$\displaystyle\left.-C^{N}_{M\theta}\Gamma^{M}_{N\theta}\right)$
$\displaystyle=$ $\displaystyle
G^{\theta\theta}\left(\hat{\partial}_{\mu}\Gamma^{\mu}_{\theta\theta}-\partial_{\theta}\Gamma^{\mu}_{\mu\theta}+\Gamma^{\mu}_{\theta\theta}\Gamma^{\nu}_{\nu\mu}+\Gamma^{\theta}_{\theta\theta}\Gamma^{\mu}_{\mu\theta}\right.$
$\displaystyle\left.-\Gamma^{\nu}_{\mu\theta}\Gamma^{\mu}_{\theta\nu}-\Gamma^{\mu}_{\theta\theta}\Gamma^{\theta}_{\theta\mu}\right).$

In (28) and (29), we find the following combinations

$\displaystyle
g^{\mu\nu}\left(\hat{\partial}_{\nu}\Gamma^{\theta}_{\theta\mu}-\Gamma^{\rho}_{\nu\mu}\Gamma^{\theta}_{\theta\rho}\right)-G^{\theta\theta}\Gamma^{\mu}_{\theta\theta}\Gamma^{\theta}_{\theta\mu}$
$\displaystyle\equiv$ $\displaystyle\nabla_{M}Y^{M}_{1},$ $\displaystyle
g^{\mu\nu}\left(\partial_{\theta}\Gamma^{\theta}_{\nu\mu}+\Gamma^{\theta}_{\nu\mu}\Gamma^{\lambda}_{\lambda\theta}+\Gamma^{\theta}_{\nu\mu}\Gamma^{\theta}_{\theta\theta}\right)$
$\displaystyle\equiv$ $\displaystyle\nabla_{M}Y^{M}_{2}$
$\displaystyle+\frac{G^{\theta\theta}}{2}\partial_{\theta}g^{\mu\nu}\partial_{\theta}g_{\mu\nu},$
$\displaystyle
G^{\theta\theta}\left(\partial_{\theta}\Gamma^{\mu}_{\mu\theta}-\Gamma^{\theta}_{\theta\theta}\Gamma^{\mu}_{\mu\theta}\right)-g^{\mu\nu}\Gamma^{\theta}_{\mu\nu}\Gamma^{\lambda}_{\lambda\theta}$
$\displaystyle\equiv$ $\displaystyle\nabla_{M}Y^{M}_{3},$ $\displaystyle
G^{\theta\theta}\left(\hat{\partial}_{\mu}\Gamma^{\mu}_{\theta\theta}+\Gamma^{\mu}_{\theta\theta}\Gamma^{\nu}_{\nu\mu}+\Gamma^{\mu}_{\theta\theta}\Gamma^{\theta}_{\theta\mu}\right)$
$\displaystyle\equiv$ $\displaystyle\nabla_{M}Y^{M}_{4}$
$\displaystyle+\frac{g^{\mu\nu}}{2}\hat{\partial}_{\mu}G^{\theta\theta}\hat{\partial}_{\nu}G_{\theta\theta},$

where

$\displaystyle Y^{M}_{1}$ $\displaystyle\equiv$
$\displaystyle\left(g^{\mu\nu}\Gamma^{\theta}_{\theta\mu},0\right),$
$\displaystyle Y^{M}_{2}$ $\displaystyle\equiv$
$\displaystyle\left(0,g^{\mu\nu}\Gamma^{\theta}_{\nu\mu}\right),$
$\displaystyle Y^{M}_{3}$ $\displaystyle\equiv$
$\displaystyle\left(0,G^{\theta\theta}\Gamma^{\mu}_{\mu\theta}\right),$
$\displaystyle Y^{M}_{4}$ $\displaystyle\equiv$
$\displaystyle\left(G^{\theta\theta}\Gamma^{\mu}_{\theta\theta},0\right).$
(30)

Note that, $Y^{M}_{1}$ and $Y^{M}_{4}$ transform as a four-dimensional vector,
whereas $Y^{M}_{2}$ and $Y^{M}_{3}$ transform as a one-dimensional vector. The
terms $\nabla_{M}Y^{M}_{i}$ with $i=1,2,3,4$ are divergences which correspond
to the boundary terms and hence vanish at infinity.

## Appendix B: The solution of $\chi(\theta)$ in different charts and its
period

In order to describe $S^{1}$, one needs at least two charts $(-\pi,\pi)$ and
$(0,2\pi)$ which form an atlas of $S^{1}$. In chart $(-\pi,\pi)$, the solution
of $\chi(\theta)$ satisfying the boundary condition $\chi(-\pi)=\chi(\pi)$ is
given by

$\displaystyle\chi(\theta)=\cosh^{\frac{3}{11}}\left(\frac{\sqrt{22}}{3}\kappa\theta\right),$
(31)

for the case of $\Lambda_{4}=0$ and

$\displaystyle\chi(\theta)=\frac{11\Lambda_{4}}{19|\Lambda|}\left[\cosh\left(\sqrt{\frac{2}{11}}\kappa\theta\right)-1\right],$
(32)

for the case of $\Lambda_{4}\neq 0$. Whereas, in the chart $(0,2\pi)$ the
solution of $\chi(\theta)$ satisfying the boundary condition
$\chi(0)=\chi(2\pi)$ is given by

$\displaystyle\chi(\theta)=\cosh^{\frac{3}{11}}\left[\frac{\sqrt{22}}{3}\kappa(\theta-\pi)\right],$
(33)

for $\Lambda_{4}=0$ and

$\displaystyle\chi(\theta)=\frac{11\Lambda_{4}}{19|\Lambda|}\left[\cosh\left(\sqrt{\frac{2}{11}}\kappa(\theta-\pi)\right)-1\right],$
(34)

for $\Lambda_{4}\neq 0$. Because of the topology of $S^{1}$, the 4D metric
component and thus its bulk profile $\chi(\theta)$ mus be periodic with the
period $2\pi$, i.e. $\chi(\theta)=\chi(\theta+2\pi)$. We can make the solution
$\chi(\theta)$ given above periodic with the period $2\pi$ by reflecting it at
the boundary as depicted in Fig. 2.

---
Figure 2: The solution of $\chi(\theta)$ in the charts $(-\pi,\pi)$ (red
curve) and $(0,2\pi)$ (blue curve) which is periodic with the period $2\pi$ by
reflecting it at the boundary. Top panel is for $\chi$ given in Eqs. (31) and
(33). Bottom panel is for $\chi$ given in Eqs. (32) and (34).

From this figure, we see that the bulk profile $\chi(\theta)$ is not
differentiable at the boundary at $\theta=\pm\pi$ in the chart $(-\pi,\pi)$
but it is differentiable in the chart $(0,2\pi)$. Whereas, $\chi(\theta)$ is
not differentiable at the boundary at $\theta=0,2\pi$ in the chart $(0,2\pi)$
but it is differentiable in the chart $(-\pi,\pi)$.

## Appendix C: 4D stringy equations of motions

In this appendix, we obtain the 4D stringy equations of motion from the
variation of the action (13) as

$\displaystyle G^{(4)}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
M^{-2}_{\text{Pl}}\left(T^{A}_{\mu\nu}+T^{R}_{\mu\nu}\right),$ (35)
$\displaystyle\nabla_{\mu}F^{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{3\partial_{\mu}R}{R}F^{\mu\nu}+\frac{m^{2}_{A}}{\vartheta
R^{3}}A^{\nu}$ (36) $\displaystyle\square R$ $\displaystyle=$
$\displaystyle\frac{1}{R}(\partial_{\mu}R)^{2}+\frac{2R^{2}}{3M^{2}_{\text{Pl}}}\frac{\partial
V}{\partial R}+\frac{\vartheta R^{4}}{2M^{2}_{\text{Pl}}}F_{\mu\nu}F^{\mu\nu}$
(37) $\displaystyle+\frac{R^{2}}{3M^{2}_{\text{Pl}}}\frac{\partial
m^{2}_{A}}{\partial R}A_{\mu}A^{\mu},$

where $G^{(4)}_{\mu\nu}\equiv
R^{(4)}_{\mu\nu}-\frac{1}{2}g^{(4)}_{\mu\nu}R^{(4)}$, $\vartheta\equiv
g^{2}_{{}_{A}}\pi M^{3}_{5}$, and the energy-momentum tensors are given by

$\displaystyle T^{A}_{\mu\nu}$ $\displaystyle=$ $\displaystyle\vartheta
R^{3}{F_{\mu}}^{\lambda}F_{\nu\lambda}+m^{2}_{A}A_{\mu}A_{\nu}-\frac{g^{(4)}_{\mu\nu}}{2}\left(\frac{\vartheta}{2}R^{3}F_{\sigma\lambda}F^{\sigma\lambda}\right.$
(38) $\displaystyle\left.+m^{2}_{A}A_{\lambda}A^{\lambda}\right),$
$\displaystyle T^{R}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{3M^{2}_{\text{Pl}}}{2R^{2}}\partial_{\mu}R\partial_{\nu}R-\frac{1}{2}g^{(4)}_{\mu\nu}\left[\frac{3}{2}\left(\frac{\partial_{\lambda}R}{R}\right)^{2}+\frac{2V(R)}{M^{2}_{\text{Pl}}}\right].$

## Appendix D: Equations for the profile of the 5D bulk fields

We here provide the equations determining the profile of the 5D bulk fields
along the fifth dimension of $M_{5}$. For the 5D scalar field expanded as
$\Phi(x,\theta)=\sum_{n}\phi^{(n)}(x)Y_{n}(\theta)$, the profile of the $n$th
mode satisfies the following equation

$\displaystyle
Y^{\prime\prime}_{n}+\frac{2\chi^{\prime}}{\chi}Y^{\prime}_{n}+R^{2}\left(\frac{m^{2}_{n}}{\chi}-m^{2}_{\Phi}\right)Y_{n}=0,$
(40)

where $m_{\Phi}$ and $m_{n}$ are the bulk mass and the mass of the $n$th mode,
respectively. With respect to the 5D fermion expanded as

$\displaystyle\Psi(x,\theta)=\sum_{n}\left[\psi^{(n)}_{L}(x)Y_{Ln}(\theta)+\psi^{(n)}_{R}(x)Y_{Rn}(\theta)\right],$
(41)

the equations for the profile of the $n$th modes corresponding to left- and
right-handed components are given as

$\displaystyle
Y^{\prime}_{Ln}+\left(\frac{m_{\Psi}}{R^{-1}}+\frac{\chi^{\prime}}{\chi}\right)Y^{\prime}_{Ln}-\frac{m_{n}R}{\chi^{1/2}}Y_{Rn}$
$\displaystyle=$ $\displaystyle 0,$ (42)
$\displaystyle-Y^{\prime}_{Rn}+\left(\frac{m_{\Psi}}{R^{-1}}-\frac{\chi^{\prime}}{\chi}\right)Y^{\prime}_{Rn}-\frac{m_{n}R}{\chi^{1/2}}Y_{Ln}$
$\displaystyle=$ $\displaystyle 0,$ (43)

where $m_{\Psi}$ and $m_{n}$ are the bulk mass and the Dirac mass of the $n$th
mode. For the 5D gauge boson $A_{M}$ with the gauge $A_{\theta}=0$ and
expanded as

$\displaystyle A_{\mu}(x,\theta)=\sum_{n}A^{(n)}_{\mu}(x)f_{n}(\theta),$ (44)

an equation for the profile of the $n$th mode reads

$\displaystyle
f^{\prime\prime}_{n}+\frac{\chi^{\prime}}{\chi}f^{\prime}_{n}+\frac{m^{2}_{n}R^{2}}{\chi}f_{n}=0.$
(45)

The solution form of these equations are given in Eq. (18) where

$\displaystyle a_{n}$ $\displaystyle=$
$\displaystyle\frac{5\kappa^{2}}{12}\times\left\\{\begin{array}[]{ll}\frac{4}{5}\left[1+\left(1+\frac{3m^{2}_{n}R^{2}}{4\kappa^{2}}\right)^{\frac{1}{2}}\right],&\hbox{\text{scalar};}\\\
\left[1+\left(1+\frac{12m^{2}_{n}R^{2}}{25\kappa^{2}}\right)^{\frac{1}{2}}\right],&\hbox{\text{fermion};}\\\
\frac{2}{5}\left[1+\left(1+\frac{3m^{2}_{n}R^{2}}{\kappa^{2}}\right)^{\frac{1}{2}}\right],&\hbox{\text{vector},}\end{array}\right.$
(49) $\displaystyle b_{n}$ $\displaystyle=$
$\displaystyle\kappa^{2}\times\left\\{\begin{array}[]{ll}2\left(1+\frac{3m^{2}_{n}R^{2}}{4\kappa^{2}}\right)^{1/2},&\hbox{\text{scalar};}\\\
\frac{1}{2}\left(1+\frac{12m^{2}_{n}R^{2}}{25\kappa^{2}}\right)^{1/2},&\hbox{\text{fermion};}\\\
\left(1+\frac{3m^{2}_{n}R^{2}}{\kappa^{2}}\right)^{1/2},&\hbox{\text{vector},}\end{array}\right.$
(53)

with the mass $m_{n}$ corresponding to the $n$th mode for each of the field
type given in Eq. (22), and the normalization constants $N_{n}$ are computed
from the orthonormal relation as

$\displaystyle\int^{\pi}_{-\pi}d\theta\chi(\theta)^{\omega}Y_{n}(\theta)Y_{m}(\theta)=\delta_{mn},$
(54)

with $\omega=1$, $3/2$, and $0$ for the scalar, fermion, and vector fields,
respectively, and $Y_{n}(\theta)$ referring to the bulk profile of these
fields.

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Experimentally distinguishable origin for electroweak symmetry breaking

Victor<EMAIL_ADDRESS>and Nobuchika<EMAIL_ADDRESS>
Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL
35487, USA

Abstract

We consider a classically conformal $U(1)$ gauge extension of the Standard
Model (SM), in which the $U(1)$ gauge symmetry is radiatively broken by the
Coleman-Weinberg mechanism. This breaking triggers the electroweak (EW)
symmetry breaking through a mixed quartic coupling between the $U(1)$ Higgs
field and the SM Higgs doublet. For two Higgs boson mass eigenstates after the
symmetry breaking, $h_{1}$ (SM-like Higgs boson) and $h_{2}$ (SM singlet-like
Higgs boson), we calculate the Higgs boson trilinear coupling
($g_{h_{1}h_{2}h_{2}}$) in the model by setting the Higgs boson mass spectrum
to be $M_{h_{1}}>2M_{h_{2}}$. For a common Higgs mass spectrum and mixing
angle between two Higgs fields, we find that $g_{h_{1}h_{2}h_{2}}$ in the
classically conformal model is highly suppressed compared to that calculated
for the conventional Higgs potential, where the $U(1)$ and EW symmetry
breaking originate from the negative squared masses for the Higgs fields at
the tree-level. Thus, this coupling suppression is a striking nature of the
radiative origin of EW symmetry breaking. We then consider how to distinguish
this origin at the proposed International Linear Collider (ILC) via precise
measurements of anomalous SM Higgs boson couplings and the search for
anomalous SM Higgs boson decay $h_{1}\rightarrow h_{2}h_{2}$ followed by
$h_{2}\to b\bar{b}$. We conclude that once the anomalous couplings are
measured at the ILC, the observation of the anomalous Higgs boson decay is
promising in the conventional Higgs potential, while this decay process is
highly suppressed and undetectable for the classically conformal model.

##  I Introduction

The existence of problems not addressed or answered by the Standard Model (SM)
has motivated many searches for physics beyond the SM. To that end, many
extensions of the SM have been proposed, involving various additional
particles and/or interactions providing possible solutions to some or all of
these problems. Among these problems is the origin of the electroweak (EW)
symmetry breaking. While successful, there is no compelling explanation for
how it arises, since the negative mass-squared term for the Higgs field is
generally included by hand.

One interesting way to break a gauge symmetry is the so-called radiative
symmetry breaking first proposed by Coleman and Weinberg Coleman:1973jx ,
where classically conformal symmetry is imposed at the tree-level scalar
potential, and radiative corrections to the potential from the gauge
interaction trigger symmetry breaking. The origin of the symmetry breaking is
thus due to quantum corrections, so no negative mass-squared term is added by
hand. However, the Coleman-Weinberg mechanism cannot be naively applied to the
SM Higgs sector, as the SM Higgs potential is unbounded from below by quantum
corrections from the large top quark Yukawa coupling (see for example Ref.
Sher:1988mj ). For this scheme to be implemented, extending the SM is
necessary. A simple way to do so is to extend the SM with a new $U(1)$ gauge
symmetry and to impose classical conformality to forbid explicit mass-squared
terms Hempfling:1996ht ; Iso:2009ss . The new $U(1)$ gauge symmetry is then
broken at some scale $v_{\phi}$ by the Coleman-Weinberg mechanism, where
$v_{\phi}$ is the vacuum expectation value (vev) for a Higgs field $\Phi$ for
the $U(1)$ gauge symmetry. A negative mixing quartic coupling between the SM
Higgs doublet and $\Phi$ generates the negative mass squared term for the SM
Higgs doublet, driving EW symmetry breaking. Therefore, the symmetry breaking
of the new $U(1)$ gauge symmetry is the sole origin of the mass scale in this
model. This simple mechanism can easily be incorporated into more complex SM
extensions.

As the extension includes a new scalar, it is interesting to investigate Higgs
boson phenomenology. We specifically consider the case where the extra Higgs
boson is light enough to be produced by SM-like Higgs boson decay. Such a
process originates from the mixing quartic coupling that generates the EW
symmetry breaking. The same coupling also generates a mass mixing between the
SM Higgs and $U(1)$ Higgs fields. Due to this mixing, the model predicts
anomalous SM Higgs boson couplings.

In this paper, we consider a classically conformal extension to the SM by a
(hidden) $U(1)$ gauge group with a Higgs scalar field $\Phi$. The Coleman-
Weinberg mechanism induces radiative symmetry breaking of the new $U(1)$ gauge
group, generating the negative mass-squared term for the SM Higgs doublet via
a mixed quartic coupling between the SM Higgs and $\Phi$. This in turn drives
the EW symmetry breaking for the SM. We analyze the Higgs potential and
extract a Higgs trilinear coupling $g_{h_{1}h_{2}h_{2}}$ between two mass
eigenstates, the SM-like Higgs boson $h_{1}$ and the SM singlet-like Higgs
boson $h_{2}$. For comparison, we also analyze the conventional Higgs
potential, where the $U(1)$ and EW gauge symmetry breaking are triggered by
negative squared masses for the two Higgs fields introduced by hand and
calculate $g_{h_{1}h_{2}h_{2}}$. Very interestingly, we find a significant
suppression in $|g_{h_{1}h_{2}h_{2}}|$ for our classically conformal potential
compared to a naively expected value from the conventional Higgs potential
while anomalous SM Higgs boson couplings are taken to be the same for both
cases. We then consider how to experimentally distinguish the radiative
breaking origin of the EW symmetry breaking from that in the conventional
Higgs potential, namely, a negative mass squared introduced by hand. We point
out that the future $e^{+}e^{-}$ colliders will be a powerful tool for this
job by combining a precise measurement of the anomalous SM Higgs boson
coupling with the search for an anomalous Higgs boson decay to $h_{1}\to
h_{2}h_{2}$ followed by $h_{2}\to b\bar{b}$. If the anomalous SM Higgs boson
couplings have been observed, the observation of the anomalous Higgs boson
decay is promising for the conventional Higgs potential, while in the
classically conformal case, the branching ratio of $h_{1}\to h_{2}h_{2}$ is
highly suppressed and undetectable.

##  II Classically conformal $U(1)$ extended Standard Model

In this section, we first consider a minimal extension to the SM involving
only an additional $U(1)$ gauge group, which we will denote by $U(1)_{H}$. In
order to focus solely on Higgs phenomenology, we assume in this section that
all SM fields to be neutral with respect to $U(1)_{H}$ and no additional
chiral fermions are involved in the model. We now introduce the $U(1)_{H}$
Higgs field $\Phi$ to have charge $Q_{\Phi}=+2$ while it is singlet under the
SM gauge group.

Imposing classically conformal invariance, explicit mass terms are forbidden,
and the scalar potential at the tree-level is given by

$V_{tree}=\lambda_{h}(H^{\dagger}H)^{2}+\lambda_{\phi}(\Phi^{\dagger}\Phi)^{2}-\lambda_{mix}(H^{\dagger}H)(\Phi^{\dagger}\Phi),$
(1)

where $H$ is the SM Higgs doublet, and $\lambda_{mix}$ is chosen to be
positive. We make the choice to take $\lambda_{mix}\ll 1$, effectively
separating the SM and “hidden” $U(1)_{H}$ Higgs sectors from each other. The
vacuum of the scalar potential at the tree-level appears at the origin, and EW
and $U(1)_{H}$ symmetries are unbroken. For the $U(1)_{H}$ Higgs sector, we
account for the radiative corrections at one-loop level Coleman:1973jx ,

$V_{1-loop}=\frac{\beta_{\phi}}{8}\left(\textnormal{ln}\left[\frac{\phi^{2}}{v_{\phi}^{2}}\right]-\frac{25}{6}\right)\phi^{4},$
(2)

where $\phi=\sqrt{2}{\rm Re}[\Phi]$, $\beta_{\phi}$ is given by

$\beta_{\phi}=\frac{1}{16\pi^{2}}(20\lambda_{\Phi}^{2}+96g_{H}^{4})\simeq\frac{1}{16\pi^{2}}(96g_{H}^{4}),$
(3)

and $g_{H}$ is the gauge coupling of the hidden $U(1)_{H}$. With these
corrections, the $U(1)_{H}$ symmetry is radiatively broken at
$\langle\phi\rangle=v_{\phi}$.

The full potential is

$V=\frac{\lambda_{h}}{4}h^{4}+\left[\frac{\lambda_{\phi}}{4}+\frac{\beta_{\Phi}}{8}\left(\textnormal{ln}\left[\frac{\phi^{2}}{v_{\phi}^{2}}\right]-\frac{25}{6}\right)\right]\phi^{4}-\frac{\lambda_{mix}h^{2}\phi^{2}}{4}$
(4)

where $H=\frac{1}{\sqrt{2}}\left(\begin{matrix}h\\\ 0\end{matrix}\right)$ with
a real scalar $h$ in the unitary gauge. In the presence of nonzero
$\langle\phi\rangle$, the scalar $h$ develops a negative mass-squared term
from the mixing quartic coupling. Subsequently, EW symmetry is broken at
$\langle h\rangle=v_{h}=246$ GeV.

Using the stationary conditions, $\frac{\partial
V}{\partial\phi}\left|{}_{\phi=v_{\phi}}\right.=0$ and $\frac{\partial
V}{\partial h}\left|{}_{h=v_{h}}\right.=0$, to eliminate $\lambda_{\phi}$ and
$\lambda_{mix}$, we express the potential in the following form:

$\displaystyle\begin{split}V(h,\phi)=&\frac{1}{4}\left(\frac{m_{h}^{2}}{2v_{h}^{2}}\right)h^{4}+\frac{1}{4}\left(\frac{11}{6}\beta_{\phi}+\frac{m_{h}^{2}v_{h}^{2}}{2v_{\phi}^{4}}\right)\phi^{4}\\\
+&\frac{\beta_{\phi}}{8}\left(\textnormal{ln}\left[\frac{\phi^{2}}{v_{\phi}^{2}}\right]-\frac{25}{6}\right)\phi^{4}-\frac{1}{4}\left(\frac{m_{h}^{2}}{v_{\phi}^{2}}\right)h^{2}\phi^{2}.\end{split}$
(5)

where $m_{h}^{2}\equiv 2\lambda_{h}v_{h}^{2}$. By shifting the fields
$h\rightarrow h+v_{h}$ and $\phi\rightarrow\phi+v_{\phi}$, we can obtain the
mass-squared matrix for the physical states,

$M_{sq}=\begin{pmatrix}m_{h}^{2}&-M^{2}\\\
-M^{2}\frac{v_{h}}{v_{\phi}}&m_{\phi}^{2}\end{pmatrix},$ (6)

where $M^{2}=m_{h}^{2}\left(\frac{vh}{v_{\phi}}\right)$ and
$m_{\phi}^{2}\equiv
m_{h}^{2}\left(\frac{v_{h}}{v_{\phi}}\right)^{2}+v_{\phi}^{2}\beta_{\phi}$. We
diagonalize $M_{sq}$ by

$\displaystyle\left(\begin{matrix}h\\\
\phi\end{matrix}\right)=\left(\begin{matrix}\cos(\theta)&\sin(\theta)\\\
-\sin(\theta)&\cos(\theta)\\\ \end{matrix}\right)\left(\begin{matrix}h_{1}\\\
h_{2}\end{matrix}\right),$ (7)

with a mixing angle $\theta$ defined by
$\tan(2\theta)=\frac{2M^{2}}{m_{h}^{2}-m_{\phi}^{2}}$, where $h_{1}$, $h_{2}$
are the mass eigenstates with mass eigenvalues $M_{h_{1}}$ and $M_{h_{2}}$,
respectively. We see that for a small $\theta$ (or equivalently, a small
$\lambda_{mix}$), the SM Higgs is dominated by the $h_{1}$ mass eigenstate and
$\phi$ by $h_{2}$. In terms of observables ($M_{h_{1}}$, $M_{h_{2}}$,
$\theta$), the parameters in Eq. (5) can be expressed as follows:

$\displaystyle m_{h}^{2}$
$\displaystyle=\frac{1}{2}\left(M_{h_{1}}^{2}+M_{h_{2}}^{2}+\left(M_{h_{1}}^{2}-M_{h_{2}}^{2}\right)\cos(2\theta)\right),$
(8) $\displaystyle m_{\phi}^{2}$
$\displaystyle=\frac{1}{2}\left(M_{h_{1}}^{2}+M_{h_{2}}^{2}-\left(M_{h_{1}}^{2}-M_{h_{2}}^{2}\right)\cos(2\theta)\right),$
(9) $\displaystyle v_{\phi}$ $\displaystyle=\frac{m_{h}^{2}}{M^{2}}v_{h},$
(10) $\displaystyle\beta_{\phi}$
$\displaystyle=\frac{m_{\phi}^{2}}{v_{\phi}^{2}}-\frac{m_{h}^{2}v_{h}^{2}}{v_{\phi}^{4}},$
(11)

where
$M^{2}=\frac{1}{2}\left(M_{h_{1}}^{2}-M_{h_{2}}^{2}\right)\sin(2\theta)$.

We rewrite the potential in terms of the mass eigenstates $h_{1}$ and $h_{2}$,
and extract the trilinear coupling
$g_{h_{1}h_{2}h_{2}}\equiv\frac{1}{2}\frac{\partial^{3}V}{\partial
h_{1}\partial^{2}h_{2}}\left|{}_{h_{1}=h_{2}=0}\right.$:

$\displaystyle
g_{h_{1}h_{2}h_{2}}=-\frac{M_{h_{2}}^{2}\cos(\theta)\left(M_{h_{1}}^{4}-5M_{h_{1}}^{2}M_{h_{2}}^{2}-2M_{h_{2}}^{4}+\left(M_{h_{1}}^{4}-3M_{h_{1}}^{2}M_{h_{2}}^{2}+2M_{h_{2}}^{4}\right)\cos(2\theta)\right)}{v_{h}\left(M_{h_{1}}^{2}+M_{h_{2}}^{2}+\left(M_{h_{1}}^{2}-M_{h_{2}}^{2}\right)\cos(2\theta)\right)^{2}}.$
(12)

Assuming $M_{h_{1}}>2M_{h_{2}}$ and $\theta\ll 1$, Eq. (12) reduces to

$g_{h_{1}h_{2}h_{2}}\simeq-\frac{M_{h_{2}}^{2}}{2v_{h}}\left(1-4\frac{M_{h_{2}}^{2}}{M_{h_{1}}^{2}}\right)\theta^{2}.$
(13)

Note that this trilinear coupling is extremely suppressed compared to the
naively expected value derived from the conventional Higgs potential without
classical conformal symmetry. To see this, we now look at the conventional
Higgs potential.

Let us consider the conventional Higgs potential for $h$ and $\phi$ in the
unitary gauge:

$\displaystyle V$
$\displaystyle=\frac{1}{4}\lambda_{h}\big{(}h^{2}-v_{h}^{2}\big{)}^{2}+\frac{1}{4}\lambda_{\phi}\big{(}\phi^{2}-v_{\phi}^{2}\big{)}^{2}+\frac{1}{4}\lambda_{mix}\big{(}h^{2}-v_{h}^{2}\big{)}\big{(}\phi^{2}-v_{\phi}^{2}\big{)}.$
(14)

The potential minimum appears at $\langle h\rangle=v_{h}$ and
$\langle\phi\rangle=v_{\phi}$. Using the stationary conditions, we rewrite the
potential in terms of physical states by shifting $h\rightarrow h+v_{h}$ and
$\phi\rightarrow\phi+v_{\phi}$:

$\displaystyle V$
$\displaystyle=\frac{1}{4}\bigg{(}\frac{m_{h}^{2}}{2v_{h}^{2}}\bigg{)}\big{(}h^{2}+2hv_{h}\big{)}^{2}+\frac{1}{4}\bigg{(}\frac{m_{\phi}^{2}}{2v_{\phi}^{2}}\bigg{)}\big{(}\phi^{2}+2\phi
v_{\phi}\big{)}^{2}+\frac{1}{4}\lambda_{mix}\big{(}h^{2}+2hv_{h}\big{)}\big{(}\phi^{2}+2\phi
v_{\phi}\big{)}.$ (15)

From the above scalar potential we obtain the mass-squared matrix for the
conventional case of the form:

$\displaystyle M_{sq}$ $\displaystyle=\begin{pmatrix}m_{h}^{2}&&M^{2}\\\
M^{2}&&m_{\phi}^{2}\end{pmatrix},$ (16)

where $M^{2}=\lambda_{mix}v_{h}v_{\phi}$. We diagonalize this matrix with Eq.
(7) with the mixing angle defined by
$\tan(2\theta)=-\frac{2M^{2}}{m_{h}^{2}-m_{\phi}^{2}}$. In terms of
observables $M_{h_{1}}$, $M_{h_{2}}$, $\theta$, and $v_{\phi}$, we can express
the parameters in Eq. (15) as

$\displaystyle m_{h}^{2}$
$\displaystyle=\frac{1}{2}\left(M_{h_{1}}^{2}+M_{h_{2}}^{2}+\left(M_{h_{1}}^{2}-M_{h_{2}}^{2}\right)\cos(2\theta)\right),$
(17) $\displaystyle m_{\phi}^{2}$
$\displaystyle=\frac{1}{2}\left(M_{h_{1}}^{2}+M_{h_{2}}^{2}-\left(M_{h_{1}}^{2}-M_{h_{2}}^{2}\right)\cos(2\theta)\right),$
(18) $\displaystyle\lambda_{mix}$ $\displaystyle=M^{2}\frac{v_{h}}{v_{\phi}},$
(19)

where
$M^{2}=-\frac{1}{2}\left(M_{h_{1}}^{2}-M_{h_{2}}^{2}\right)\sin(2\theta)$.
Note that the conventional Higgs potential of Eq. (15) is controlled by four
free parameters. In contrast to our conformal model, both $\theta$ and
$v_{\phi}$ remain free after $M_{h_{1}}$ and $M_{h_{2}}$ are fixed.

Expressing the potential in terms of the mass eigenstates $h_{1,2}$, we
extract the trilinear coupling $g_{h_{1}h_{2}h_{2}}$:

$\displaystyle
g_{h_{1}h_{2}h_{2}}=\frac{\left(M_{h_{1}}^{2}+2M_{h_{2}}^{2}\right)\left(-v_{h}\cos(\theta)+v_{\phi}\sin(\theta)\right)\sin(2\theta)}{2v_{h}v_{\phi}}$
(20)

This coupling has different behavior for $\theta\ll\frac{v_{h}}{v_{\phi}}\ll
1$ and $\frac{v_{h}}{v_{\phi}}\lesssim\theta\ll 1$. For
$\theta\ll\frac{v_{h}}{v_{\phi}}$, the trilinear coupling
$g_{h_{1}h_{2}h_{2}}$ reduces to

$g_{h_{1}h_{2}h_{2}}\simeq-\frac{M_{h_{1}}^{2}}{2v_{\phi}}\left(1+2\frac{M_{h_{2}}^{2}}{M_{h_{1}}^{2}}\right)\theta,$
(21)

while for $\frac{v_{h}}{v_{\phi}}\lesssim\theta$, the coupling is roughly
given by

$g_{h_{1}h_{2}h_{2}}\simeq\frac{M_{h_{1}}^{2}}{2v_{h}}\left(1+2\frac{M_{h_{2}}^{2}}{M_{h_{1}}^{2}}\right)\theta^{2}.$
(22)

Comparing Eqs. (21) and (22) with Eq. (13), we see a remarkable difference.
For $\theta\ll\frac{v_{h}}{v_{\phi}}$, the coupling from the conformal
potential is proportional to $\theta^{2}$, while the conventional coupling is
proportional to $\theta$, as we would naively expect:
$g_{h_{1}h_{2}h_{2}}\sim\lambda_{mix}v_{h}$. This is because in the very small
$\theta$ expansion for the trilinear coupling, the lowest order in $\theta$ is
cancelled out in the conformal potential. On the other hand, for
$\frac{v_{h}}{v_{\phi}}\lesssim\theta$, the couplings from both potentials
have the same $\theta$ dependence, but that of the conformal model still
exhibits relative suppression. The ratio of the trilinear couplings in this
region is found to be

$R\equiv\frac{|g_{h_{1}h_{2}h_{2}|_{\rm conf}}|}{|g_{h_{1}h_{2}h_{2}|_{\rm
conv}}|}=\left(\frac{M_{h_{2}}^{2}}{M_{h_{1}}^{2}}\right)\frac{1-4\frac{M_{h_{2}}^{2}}{M_{h_{1}}^{2}}}{1+2\frac{M_{h_{2}}^{2}}{M_{h_{1}}^{2}}}<0.0505,$
(23)

for $M_{h_{2}}<\frac{1}{2}M_{h_{1}}$, with the maximum value obtained when
$M_{h_{2}}\simeq 0.33M_{h_{1}}$.

Note that in both cases, the Higgs anomalous coupling is controlled by
$\cos(\theta)$. This result has interesting implications for Higgs
phenomenology, namely, even if the anomalous SM Higgs coupling is detectable
in size, the SM-like Higgs boson $h_{1}$ decay to $h_{2}h_{2}$ can be much
harder to detect in our model. This suppression is a characteristic feature of
our extended conformally symmetric model, in which EW symmetry breaking is
triggered by the radiative $U(1)_{H}$ symmetry breaking.

##  III Numerical Analysis of the Higgs boson trilinear coupling

For an example numerical value, we use the known SM Higgs vev and fix the
Higgs mass eigenvalues $M_{h_{1}}$, $M_{h_{2}}$ as follows, alongside a mixing
angle $|\theta|=0.1$,

$\displaystyle M_{h_{1}}=125\,\textnormal{GeV},\quad
M_{h_{2}}=25\,\textnormal{GeV}.$ (24)

For the classically conformal model, $v_{\phi}$ is totally fixed by the above
choices of model parameters to be $v_{\phi}=2555$ GeV. We find the triple
coupling to be

$g_{h_{1}h_{2}h_{2}}=-0.0107,$ (25)

while for the conventional Higgs potential, $v_{\phi}$ is a free parameter
chosen to be $v_{\phi}=10^{4}$ GeV, and the coupling becomes

$g_{h_{1}h_{2}h_{2}}=0.424.$ (26)

As explained in the previous section (see Eqs. (13), (21), and (22)), for
$|\theta|=0.1\gg\frac{v_{h}}{v_{\phi}}$, the trilinear coupling for the
conformal case is suppressed by a factor of $R\simeq 0.04$. We use
$v_{\phi}=10^{4}$ GeV as a benchmark value for all calculations for the
conventional Higgs potential.

Figure 1:  Left Panel: $|g_{h_{1}h_{2}h_{2}}|$ as a function of $|\theta|$ in
our model (solid line) and the conventional Higgs potential (dotted line).
Right Panel: $|g_{h_{1}h_{1}h_{2}}|$ as a function of $|\theta|$ in our model
(solid line) and the conventional Higgs potential (dotted line). We set
$M_{h_{1}}=125$ GeV and $M_{h_{2}}=25$ GeV in our analysis. The grey shaded
region is excluded by the LEP-II results for the Higgs boson search
Barate:2003sz , and the red shaded region represents the prospective search
reach of the proposed International Linear Collider (ILC) Liu:2016zki ;
Yamamoto:2021kig ; Fujii:2015jha  Figure 2: Comparison of conventional (dashed
lines) and conformal (solid lines) $h_{1}\rightarrow h_{2}h_{2}$ branching
ratio at $M_{h_{2}}=10$ (red), 25 (black), and 50 (blue) GeV. Shaded regions
correspond to bounds from the proposed ILC, blue for bounds on the branching
ratio Liu:2016zki ; Yamamoto:2021kig ; Fujii:2015jha , and red for bounds on
$\sin^{2}(\theta)$ Barklow:2017suo ; Bambade:2019fyw . The grey shaded regions
are excluded by ATLAS($h_{1}\rightarrow h_{2}h_{2}\rightarrow
b\bar{b}b\bar{b}$) ATLAS:2020ahi ; Cepeda:2021rql  and LEP-II for
$M_{h_{2}}=25\textnormal{GeV}$ Sirunyan:2018koj .

In the left panel of Fig. 1, we plot $|g_{h_{1}h_{2}h_{2}}|$ for the two cases
as a function of $|\theta|$. For $|\theta|\gtrsim\frac{v_{h}}{v_{\phi}}\simeq
0.025$, we can see that $|g_{h_{1}h_{2}h_{2}}|$ in the conformal case is
suppressed by a factor $R\simeq 0.04$. As discussed in the previous section
(see Eqs. (21) and (22)), we can see the transition in the $\sin(\theta)$
dependence from $\sin(\theta)$ to $\sin^{2}(\theta)$ at
$|\theta|\simeq\frac{v_{h}}{v_{\phi}}\simeq 0.025$. For
$|\theta|\ll\frac{v_{h}}{v_{\phi}}$, we can see stronger relative suppression
between the couplings, with a suppression factor,

$\left(\frac{M_{h_{2}}^{2}}{M_{h_{1}}^{2}}\right)\frac{1-4\frac{M_{h_{2}}^{2}}{M_{h_{1}}^{2}}}{1+2\frac{M_{h_{2}}^{2}}{M_{h_{1}}^{2}}}\left(\frac{v_{\phi}}{v_{h}}|\theta|\right)\ll
R.$ (27)

The grey shaded region is excluded by the LEP-II results for the Higgs boson
search Barate:2003sz : Assuming the Higgs boson mass were 25 GeV, the Higgs
boson coupling with the $Z$ boson must be suppressed by a factor of
$\sin^{2}(\theta)\leq 0.02$. The red shaded region represents the prospective
search reach of the proposed International Linear Collider (ILC), as low as
$\sin^{2}(\theta)\simeq 0.002$ Liu:2016zki ; Yamamoto:2021kig ; Fujii:2015jha
. We also calculate the trilinear coupling $g_{h_{1}h_{1}h_{2}}$ with the same
parameter choice. The result is shown in the right panel of Fig. 1. Now, the
resultant coupling from our conformal model overlaps well with the same
coupling from the conventional Higgs potential, and there is no relative
suppression between the two.

We now consider outcomes for experimental probes of our model. The Higgs
physics of interest here are the decay mode $h_{1}\rightarrow h_{2}h_{2}$ and
the Higgs anomalous coupling, determined by $\cos(\theta)$. The partial decay
width for the process $h_{1}\rightarrow h_{2}h_{2}$ is given by

$\Gamma_{h_{1}\rightarrow h_{2}h_{2}}=\frac{|g_{h_{1}h_{2}h_{2}}|^{2}}{8\pi
M_{h_{1}}}\sqrt{1-4\frac{M_{h_{2}}^{2}}{M_{h_{1}}^{2}}}.$ (28)

Using the SM-like Higgs boson total decay width to SM particles
$\Gamma_{h\rightarrow SM}=4.07$ MeV Dittmaier:2012vm , the branching ratio to
a pair of $h_{2}$ is defined as

${\rm Br}\left(h_{1}\rightarrow
h_{2}h_{2}\right)=\frac{\Gamma_{h_{1}\rightarrow
h_{2}h_{2}}}{\Gamma_{h\rightarrow SM}+\Gamma_{h_{1}\rightarrow h_{2}h_{2}}}.$
(29)

In Fig. 2 we show the results of ${\rm Br}\left(h_{1}\rightarrow
h_{2}h_{2}\right)$ for the values of $M_{h_{2}}=$ 10 (red), 25 (black), and 50
(blue) GeV, with solid lines corresponding to the conformal model and dashed
lines to the conventional Higgs potential (for which all three lines are
nearly overlapping). As expected from the difference in the couplings, the
branching ratios in our conformal model are also extremely suppressed. The
grey shaded region, as before, is excluded by LEP-II and the current bound
from the search for $h_{1}\rightarrow h_{2}h_{2}\rightarrow b\bar{b}b\bar{b}$
by the ATLAS collaboration ATLAS:2020ahi ; Cepeda:2021rql . Note that since
$h_{2}$ is mostly an SM singlet scalar, it will mainly decay to $b\bar{b}$
through the mixing with the SM Higgs doublet. The prospective ILC search
reaches are represented by the blue shaded region corresponding to the
branching ratioLiu:2016zki ; Yamamoto:2021kig ; Fujii:2015jha , and red shaded
region for the Higgs anomalous coupling Barklow:2017suo ; Bambade:2019fyw .

The combination of anomalous Higgs decay and anomalous Higgs coupling results
will provide a way to distinguish our conformal scenario from the conventional
Higgs potential. For a benchmark value of $\sin(\theta)=0.1$, for the
conventional Higgs case, the anomalous branching ratio and anomalous coupling
can both be within the ILC search region, so they can be measured
simultaneously. On the other hand, for our conformal model the branching ratio
is highly suppressed, and even if the anomalous coupling were measured, the
anomalous decay mode will still evade detection. This is a way to determine
the origin of EW symmetry breaking, i.e. does it occur via the conventional
Higgs potential with the negative mass-squared introduced by hand or does it
originate from radiative symmetry breaking.

Figure 3: Left Panel: $U(1)_{H}$ gauge coupling ($g_{H}$) as a function of
$\sin^{2}(\theta)$. Right Panel: $U(1)_{H}$ gauge boson mass
($m_{Z^{\prime}}$) as a function of $\sin^{2}(\theta)$. The lines correspond
to $M_{h_{2}}=$ 10 GeV (red), 25 GeV (black), and 50 GeV (blue). Figure 4: The
LHC bounds (grey shaded region) on $(m_{Z^{\prime}}$, $g_{H})$ parameter space
in the case of $U(1)_{H}=U(1)_{B-L}$, for $M_{h_{2}}=$ 10 GeV (red), 25 GeV
(black), and 50 GeV (blue). The lines are shown for the range of
$0.001\leq\sin^{2}(\theta)\leq 0.8$, corresponding to the range shown Fig. 2.

In our conformal model, once a value for $M_{h_{2}}$ is fixed, we can
determine the $U(1)_{H}$ gauge coupling $g_{H}$ and $Z^{\prime}$ boson mass
$m_{Z^{\prime}}$ as functions of $|\theta|$. In Fig. 3, we show the relation
between $g_{H}$, $m_{Z^{\prime}}$, and $\sin^{2}(\theta)$ for $M_{h_{2}}=10$
GeV, 25 GeV, and 50 GeV, from bottom to top. If the $Z^{\prime}$ boson couples
to SM fermions, we may consider high-energy collider bounds on the
$(m_{Z^{\prime}},g_{H})$ parameter space. As a well known example of such a
model, one may consider the minimal $B-L$ (baryon number minus lepton number)
model, based on the $U(1)_{B-L}$ gauge group Davidson:1978pm ;
Mohapatra:1980qe ; Marshak:1979fm ; Wetterich:1981bx ; Masiero:1982fi ;
Mohapatra:1982xz ; Buchmuller:1991ce . The final results from the LHC Run 2
for the $B-L$ $Z^{\prime}$ gauge boson resonance search provide us with an
upper bound on $g_{H}=g_{B-L}$ as a function of $m_{Z^{\prime}}$, which is
presented in Ref. Das:2019fee . In Fig. 4, we show the relationship between
$g_{H}$ and $m_{Z^{\prime}}$, along with the excluded region (grey shaded) by
the Large Hadron Collider (LHC) experiments from Ref. Das:2019fee , if we
identify our $U(1)_{H}$ with $U(1)_{B-L}$. The three lines show the
relationship for $M_{h_{2}}=$ 10 GeV (red), 25 GeV (black), and 50 GeV (blue)
for the range of $0.001\leq\sin^{2}(\theta)\leq 0.8$, corresponding to the
range shown in Fig. 2. The lines will extend to the right as we take lower
values of $\sin^{2}(\theta)<0.001$. To be consistent with the LHC bounds,
$\sin^{2}(\theta)$ must be much smaller than $0.001$ (see Fig. 3). Thus, to
explore interesting regions for Higgs phenomenology ($\sin^{2}(\theta)\gtrsim
0.002$), we do not consider the $B-L$ model identification of $U(1)_{H}$.
However, note that the identification of $U(1)_{H}$ with well-known flavor-
dependent $U(1)$ extended SMs, such as $U(1)_{B-L_{3}}$ Babu:2017olk ;
Alonso:2017uky ; Bian:2017rpg  and $U(1)_{L_{\mu}-L_{\tau}}$ He:1990pn ;
Foot:1990mn , is still possible, as in these models the $Z^{\prime}$ boson has
no coupling with the first generation quarks and can easily evade the LHC
constraints.

##  IV Conclusion and discussion

We have considered a classically conformal $U(1)$ extension of the SM where
the $U(1)$ Higgs field ($\Phi$) with a $U(1)$ charge $+2$ is introduced. The
$U(1)$ symmetry is radiatively broken by the Coleman-Weinberg mechanism, which
generates a negative mass squared to the SM Higgs doublet through the mixed
quartic coupling in the scalar potential
$V\supset-\lambda_{mix}(\Phi^{\dagger}\Phi)(H^{\dagger}H)$, and as a result,
the EW symmetry is broken. Therefore, the radiative symmetry breaking of the
new U(1) gauge symmetry is the origin of the EW symmetry breaking. This is a
crucial difference from the conventional Higgs potential in which the EW
symmetry breaking is triggered by a negative mass squared introduced by hand
at the tree-level. In order to investigate the Higgs phenomenology of the
classically conformal model, we have analyzed the effective Higgs potential
and read off the trilinear coupling of the SM-like Higgs boson ($h_{1}$) with
a pair of the SM singlet-like Higgs bosons ($h_{2}$) for the case of
$M_{h_{1}}>2M_{h_{2}}$. For comparison, we have also calculated the trilinear
coupling for the conventional Higgs potential. What we have found is very
intriguing: for the same mixing angle between $H$ and $\Phi$ and the same
Higgs boson mass spectrum, the trilinear coupling in the conformal model is
highly suppressed compared to the one from the conventional Higgs potential,
which is likely a striking nature of the classically conformal potential. We
then have considered experimental signals for such conformal structure via
anomalous SM Higgs boson couplings from the mixing between two Higgs fields
and the anomalous (SM-like) Higgs boson decay $h_{1}\rightarrow h_{2}h_{2}$
followed by $h_{2}\to b\bar{b}$. Future $e^{+}e^{-}$ colliders, in particular,
the proposed ILC will operate as a Higgs factory which allows us to precisely
measure the SM Higgs boson properties. The result of our analysis indicates
that once the anomalous SM-like Higgs boson coupling is measured at the ILC,
the observation of anomalous Higgs boson decay $h_{1}\to h_{2}h_{2}$ is
promising in the conventional Higgs potential, while this decay process is
highly suppressed and not detectable in the classically conformal model. This
is a way to to distinguish the origin of EW symmetry breaking.

As previously discussed, we may extend our classically conformal $U(1)$ model
by identifying $U(1)$ with $U(1)_{B-L_{3}}$ or $U(1)_{L_{\mu}-L_{\tau}}$.
Another interesting possibility of such an extension is to consider the $U(1)$
sector as a “Dark Sector” which supplements the SM with a dark matter
candidate. Note that although the $Z^{\prime}$ gauge boson can generally have
a kinetic mixing with the SM hyper-charge gauge boson, if the kinetic mixing
is tuned to be zero, the $Z^{\prime}$ gauge boson is stable and hence a dark
matter candidate. Therefore, the classically conformal gauged $U(1)$ extension
can not only provide the SM with the dynamical origin of the EW symmetry
breaking but also supplement the SM with the $Z^{\prime}$ boson dark matter.
Based on a new $SU(2)$ gauge symmetry, such a model has already been proposed
in Ref. Hambye:2013dgv , although the Higgs boson phenomenology that we have
investigated in this paper is not addressed. In the present paper, we have
considered the $U(1)$ gauge theory, but extending it to a non-Abelian gauge
group is straightforward and what we have investigated in this paper remains
essentially the same for such extension. It is worth investigating the Dark
Sector in light of complementarity between dark matter physics and Higgs boson
phenomenology at future collider experiments future:2022 .

## Acknowledgments

The authors V.B. and N.O. would like to thank Digesh Raut for useful
discussions. This work is supported in part by the GEM Science Ph.D.
Fellowship (V.B.) and the United States Department of Energy Grants DE-
SC-0012447 (N.O.).

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