# Universal extra dimensions after Higgs discovery

Takuya Kakuda,111 E-mail<EMAIL_ADDRESS>Kenji Nishiwaki,222
E-mail<EMAIL_ADDRESS>Kin-ya Oda,333 E-mail:
<EMAIL_ADDRESS>
and Ryoutaro Watanabe444 E-mail<EMAIL_ADDRESS>

∗ Department of Physics, Niigata University, Niigata 950-2181, Japan
† Regional Centre for Accelerator-based Particle Physics
Harish-Chandra Research Institute, Allahabad 211 019, India
‡ Department of Physics, Osaka University, Osaka 560-0043, Japan
§ Theory Group, KEK, Tsukuba, Ibaraki 305-0801, JAPAN

###### Abstract

We show bounds on five- and six-dimensional universal extra dimension (UED)
models from the latest results of the Higgs searches at the LHC and from the
electroweak precision data for the $S$ and $T$ parameters. We consider the
minimal UED model in five dimensions and the ones in six dimensions,
compactified on $T^{2}/Z_{2}$, $T^{2}/(Z_{2}\times Z_{2}^{\prime})$,
$T^{2}/Z_{4}$, $S^{2}$, $S^{2}/Z_{2}$, the real projective plane, and the
projective sphere. The highest possible ultraviolet cutoff scale for each UED
model is evaluated from the electroweak vacuum stability by solving the
renormalization group equation of the Higgs self-coupling. This scale turns
out to be lower than the conventional one obtained from the perturbativity of
the gauge coupling. The resultant 95% C.L. lower bounds on the first
Kaluza–Klein scale from the LHC results and from the $S,T$ analysis are 600
and 700 GeV in the minimal UED model, while those in the six-dimensional UED
models are 800–${1300}\,\text{GeV}$ and $900$–$1500\,\text{GeV}$,
respectively.

HRI-P-13-02-002

RECAPP-HRI-2013-003

KEK-TH-1631

OU-HET-789

##  1 Introduction

The ATLAS and CMS experiments at the CERN LHC have observed a particle around
126 GeV, which is consistent with the Standard Model (SM) Higgs boson [1, 2].
The signal strength (defined as the ratio of the production cross section
times the branching ratio of the observed particle to that of the SM Higgs)
for the decay channels into diphoton ($\gamma\gamma$) and diboson ($ZZ$ and
$WW$) are reported in Refs. [3, 4, 5, 6, 7, 8]. Namely, the signal strengths
of $H\to\gamma\gamma$, $ZZ$, and $WW$ have turned out to be $1.65\pm
0.24^{+0.25}_{-0.18}$, $1.7^{+0.5}_{-0.4}$, and $1.01\pm 0.31$ at the ATLAS
experiment, and $0.78\pm 0.27$ (multivariate analysis based),
$0.91^{+0.30}_{-0.24}$, and $0.71\pm 0.37$ (cut based) at the CMS experiment,
respectively. All these results are consistent with the SM within less than
2$\sigma$, but there still remains room for a new physics effect. See, e.g.,
Refs. [9, 10, 11, 12, 13, 14, 15, 16, 17] for analyses based on effective
Lagrangian methods.555 Global fits of the minimal UED scenario were done in
Ref. [18].  In this paper, we study the constraints on the existence of the
universal extra dimensions (UEDs) [19]666 See also Ref. [20] for an earlier
proposal of a TeV scale extra dimension.  from the Higgs searches.

In the UED scenario, each SM field propagates in one or more compactified
extra dimensions and is accompanied by its massive copies, called Kaluza–Klein
(KK) particles. Already in the simplest five-dimensional (5D) minimal UED
(mUED) model on the orbifold $S^{1}/Z_{2}$ [19], in which no tree-level brane-
localized term is assumed at an UV cutoff scale of the 5D gauge theory, there
exists an attractive feature: The lightest KK particles become stable due to
the symmetry in the geometry, the KK parity, and serves as a natural dark
matter candidate [21, 22]. The KK particles are expected to exist above around
$1\,\text{TeV}$, which is consistent to the indirect bound from the $S$, $T$
parameters, $M_{\text{KK}}\gtrsim 700\,\text{GeV}$ [23], and from the $b\to
s\gamma$ process, $M_{\text{KK}}\gtrsim 600\,\text{GeV}$ [24], with
$M_{\text{KK}}$ being the first KK mass. The prospects of the mUED at the LHC
and future linear colliders have been discussed in Refs. [25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44] and those in the
context of discrimination from other models with similar final states in Refs.
[45, 46, 47, 48, 49, 50].

A way of extending the minimal scenario is to consider the model in two-
dimensional extra space. Models have been proposed on two-torus, $T^{2}/Z_{2}$
[19], $T^{2}/Z_{4}$ (chiral square) [51, 52], $T^{2}/(Z_{2}\times
Z^{\prime}_{2})$ [53], on two-sphere $S^{2}/Z_{2}$ [54], on $S^{2}$ with a
Stueckhelbarg field [55, 56], and on the nonorientable manifolds: the real
projective plane $RP^{2}$ [57] and the projective sphere (PS) [58]. An
advantage of such a six-dimensional (6D) UED model is that the number of
generation is predicted to be (a multiple of) three [59], from the requirement
of the cancellation of the 6D gravitational and $SU(2)_{L}$ global anomalies,
which cannot be eliminated via the Green–Schwarz mechanism. We can find works
on collider phenomenology in the cases of $T^{2}/Z_{4}$ [60, 61, 62, 63, 64,
65] and of $RP^{2}$ [66, 67, 68, 69]. Recently, other possibilities of
generalization of these models by an introduction of the bulk mass term and/or
the brane-localized Lagrangians have been studied in Refs. [70, 71, 72, 73,
74, 75, 76, 77, 78, 79, 80, 81, 82].

In general, the single Higgs production process via the loop-induced gluon
fusion is enhanced in a UED model, while the branching ratio of the Higgs to
diphoton is suppressed because of interference effect between bosons and
fermions inside the loops. These UED effects have been first shown in the 5D
mUED [83]. In 6D UED, the enhancement of the gluon fusion is studied in Ref.
[84], and the diphoton decay rate is obtained in Ref. [85]. See Refs. [86, 55,
56, 87, 88] for more works in this direction. In this paper, we perform more
elaborated analysis compared with our previous work in Ref. [85], with a
varying portion of production channels for each event category. We also
estimate constraints from the $S$ and $T$ parameters in every model for
completeness. In addition to the effects of the KK Higgs boson and the KK top
quark [89, 23], those of the KK gauge boson are taken into account for the
first time in the literature.

An important number in the UED phenomenology is the highest possible UV cutoff
scale, $\Lambda_{\text{max}}$, allowed by the electroweak vacuum stability.
The cutoff scale $\Lambda$ of a UED model gives the upper bound of the KK
summation in loop processes. Therefore, different values of $\Lambda$ result
in different bounds on $M_{\text{KK}}$. In the mUED with the $126\,\text{GeV}$
Higgs, the highest possible $\Lambda$ becomes quite low, $\Lambda\lesssim
5M_{\text{KK}}$ [90]. In this paper, we examine all the 5D and 6D UED models
without resorting to the approximation employed in the analysis of gauge
coupling running in our previous work [55]. When we consider a model with a
low cutoff scale, threshold corrections via higher-dimensional operators can
become significant, which we will take into account for the $S$ and $T$
parameter constraints. Effects from such higher-dimensional operators on Higgs
signals have already been discussed in Ref. [85].

The paper is organized as follows. In Sec. 2, we estimate the highest cutoff
scale of all the UED models. Based on the results, we calculate the direct and
indirect bounds from the LHC results in Sec. 3 and from the $S$, $T$
parameters in Sec. 4. In Sec. 5, we summarize our results and discuss future
prospects. Detailed formulas that we use in this paper can be found in the
appendix.

##  2 Vacuum stability bound

In this section, we estimate the UV cutoff scale in seven types of six-
dimensional UED models on $T^{2}/Z_{2}$, $T^{2}/(Z_{2}\times Z_{2}^{\prime})$,
$T^{2}/Z_{4}$, $RP^{2}$; $S^{2}$ with a Stueckhelbarg field, $S^{2}/Z_{2}$;
and the PS and in the 5D mUED. The orbifolding in $T^{2}/Z_{2}$,
$T^{2}/(Z_{2}\times Z_{2}^{\prime})$ and $T^{2}/Z_{4}$ on two-torus ($T^{2}$)
projects out a chiral zero mode from each matter fermion. On two-sphere
$S^{2}$, we can obtain a Weyl fermion in each zero mode, due to the
monopolelike classical configuration of an extra $U(1)_{X}$ gauge boson.
However, we must eliminate the phenomenologically unacceptable massless zero
mode of the $U(1)_{X}$ gauge boson, and we will discuss three possibilities
for treating this issue in this paper.

The geometries of $RP^{2}$ and PS are unoriented and have no local fixed
points. Consequently, their KK mass spectra take unique forms, for which the
pattern is distinctive from those of the other UED models. A brief review of
the models studied in this paper can also be found in our previous paper [56].

To find the highest possible UV cutoff $\Lambda_{\text{max}}$, we examine the
vacuum stability bound by solving the renormalization group equation (RGE). As
said above, the UV cutoff plays an important role in the estimation of the KK
loop effects in processes involving loop diagrams. In later sections, we use
the results of this section for calculating the deviations in the single Higgs
production processes and the Peskin–Takeuchi $S$ and $T$ parameters. It is
noted that, in this paper, we mainly consider the situation where radii of
compactified fifth and sixth directions $R_{5},\,R_{6}$ are the same:
$R_{5}=R_{6}=R$.777 In the $T^{2}/Z_{4}$ model, the condition $R_{5}=R_{6}=R$
is automatically realized due to the property of the orbifolding. See also
Ref. [91] for a realization of $CP$ violation from the complex structure of
$T^{2}/Z_{4}$, which appears in four-dimensional effective interactions after
KK decomposition.

###  2.1 RGE in 6D UED models

Considering the RGE is an effective way of probing scale dependence. Its
concrete form is derived from the invariance, under the change of the
renormalization scale $\mu$, of the bare vertex function $\Gamma_{0}$, which
is a function of bare parameters. The scale invariance requires that

$\displaystyle\mu\frac{d}{d\mu}\Gamma_{0}(\\{c_{0}\\},\\{m_{0}\\},\\{\Phi_{0}\\})=0,$
(1)

where $\\{c_{0}\\}$, $\\{m_{0}\\}$, and $\\{\Phi_{0}\\}$ represent sets of
bare couplings, masses, and fields, respectively. Since bare parameters and
fields are divergent themselves, we can rewrite the bare ones with finite
physical ones (renormalized parameters and fields) and counter terms, which
contain divergences. In this paper, we show all the bare/renormalized
variables with/without the subscript “$0.$”

In the following, we consider the RGE for the Higgs quartic coupling $\lambda$
in the 6D models. We obey the convention of Ref. [92] in describing the
electroweak (EW) sector. The potential of the Higgs field $H$ at the tree
level is depicted as

$\displaystyle{-\frac{M^{2}_{H0}}{2}H^{2}_{0}+\frac{\lambda_{(6)0}}{4}H^{4}_{0},}$
(2)

where $M_{H0}$ and $\lambda_{(6)0}$ are the bare Higgs mass and 6D Higgs
couplings. After the 6D bare Higgs field $H_{0}$ is KK expanded, we can find
the zero mode $H_{0}^{(0)}$, where we use a superscript for a KK index. In
considering the one-loop running of $\lambda$, we need not consider the
renormalization of the Higgs mass and hence the physical Higgs mass $m_{H}$
becomes

$\displaystyle
m_{H}=\sqrt{\lambda_{(6)0}}v_{(6)0}=\sqrt{\lambda_{(6)}}v_{(6)}=\sqrt{\lambda}v,$
(3)

where the four-dimensional (4D) Higgs vacuum expectation value
$v=246\,\text{GeV}$ and quartic coupling $\lambda$ are expressed as
$v_{(6)}=v/\sqrt{V},\ \lambda_{(6)}=\lambda\sqrt{V}$, with $V$ being the
volume of the extra dimensions. Let us write

$\displaystyle{H_{0}^{(0)}=\sqrt{Z_{H}}H^{(0)}},\quad\lambda_{0}=Z_{\lambda}(Z_{H})^{-2}\lambda,$
(4)

where $Z_{\lambda}$ is the renormalization factor for the Higgs quartic
coupling and $\sqrt{Z_{H}}$ is that for the wave function renormalization of
the Higgs zero mode. We also need the information of the RGEs for the gauge
and Yukawa couplings to compute the running of $\lambda$. We summarize the
beta functions,

$\displaystyle\mu\frac{d}{d\mu}\mathcal{Q}=\beta_{\mathcal{Q}},$ (5)

where detailed form of $\beta_{\mathcal{Q}}$ can be found in Appendix A.

Let us review how to compute RGEs in a theory with (a) compactified extra
dimension(s). We adopt the bottom-up approach discussed in Refs. [93, 94],
which takes into account a contribution of a massive particle to the beta
functions when the increasing scale $\mu$ passes its mass. In the case of the
UED, after KK decomposition, the corresponding 4D effective theory contains
not only the SM fields but also their KK partners. Following this
prescription, we get

$\displaystyle{\beta_{\mathcal{Q}}=\beta_{\mathcal{Q}}^{\text{(SM)}}+\sum_{s:\,\text{massive
states}}\theta(\mu-
M_{s})\Big{(}N_{s}\beta^{\text{(NP)}}_{s,\mathcal{Q}}\Big{)},}$ (6)

where $\beta_{\mathcal{Q}}^{\text{(SM)}}$ and
$\beta_{s,\mathcal{Q}}^{\text{(NP)}}$ are the contributions from the SM
particles and from the new massive ones with mass $M_{s}$, respectively, and
$N_{s}$ is the number of degenerated states. At the tree level, $M_{s}$ is
expressed as

$\displaystyle{M_{s}^{2}=m_{s,(\text{SM})}^{2}+M_{s,(\text{KK})}^{2},}$ (7)

where $m_{s,(\text{SM})}$ is the SM mass of the corresponding zero mode and
$M_{s,(\text{KK})}$ are KK masses. In general, the value of
$M_{s,(\text{KK})}$ is much greater than that of $m_{s,(\text{SM})}$ and
$M_{s}^{2}$ can be approximated as $M_{s}^{2}\simeq M_{s,(\text{KK})}^{2}$.

Let us review the KK expansions in the 6D UED models. In the models on the
orbifolded $T^{2}$ [namely $T^{2}/Z_{2}$, $T^{2}/(Z_{2}\times
Z_{2}^{\prime})$, and $T^{2}/Z_{4}$] and the one on $RP^{2}$, the KK mass
$M_{s,(\text{KK})}^{2}$ becomes of $T^{2}$ type:

$\displaystyle M_{s,(\text{KK})}^{2}\rightarrow
M_{(m,n)}^{2}:={\frac{m^{2}}{R_{5}^{2}}+\frac{n^{2}}{R_{6}^{2}}},$ (8)

where $m$ $(n)$ is the KK index along the fifth (sixth) direction and
$N_{(m,n)}=1$ irrespective of $m$ and $n$. We note that the beta functions for
gauge, Yukawa, and Higgs self-couplings take the same forms irrespective of
the models based on $T^{2}$ and are independent of the KK indices. This reason
is as follows. Because of the flat profile of the zero modes, the three-point
functions with one SM field and the four-point functions with two SM fields
become universal at their leading order after using the orthonormality of KK
mode functions. In contrast, the value $N_{s}$ and the summation of the KK
index $\sum_{s}$ in Eq. (6) are affected by the difference in the patterns of
the orbifolding. Hence, the evolution of $\lambda$ depends on the choice of
the model. The explicit range of $m,n$ summation is shown in Table 1.

Type of orbifolding | Range of $(m,n)$
---|---
$T^{2}/Z_{2}$ |  $m+n\geq 1,$ or $m=-n\geq 1$
$T^{2}/(Z_{2}\times Z^{\prime}_{2})$ | $0\leq m<\infty,\ 0\leq n<\infty;\ (m,n)\not=(0,0)$
$T^{2}/Z_{4}$ | $1\leq m<\infty,\ 0\leq n<\infty$
Table 1: The range of the parameters $(m,n)$ except for the zero mode
$(m,n)=(0,0)$ in each case of the orbifolding.  Figure 1: Left : The patterns
of the remaining bosonic modes in the $RP^{2}$ model, where blue, red, and
green points indicate that they belong to the regions I, II, and III,
respectively. The definitions of these regions are found in the text. Right :
The same as left one for fermonic modes. At the orange points (region IV),
there are no bosonic modes. In both pictures, the black circles $(m=n=0)$
correspond to the SM particles.

Let us turn to the model on $RP^{2}$. In Fig. 1, we show the surviving modes.
The surviving modes of KK fermions become the same as in the $T^{2}/Z_{2}$
model. On the other hand, the patterns of the bosonic particles are
complicated. The allowed range of $m$ and $n$ is $m\geq 0,\ n\geq 0$, and the
type of surviving mode is classified into the following four. In region I,
$(m,n)=(0,2),(0,4),(0,6),\dots$ and $(m,n)=(2,0),(4,0),(6,0),\dots$; a
physical scaler mode coming from the extra component of the 6D gauge boson is
projected out. In region II, $(m,n)=(0,1),(0,3),(0,5),\dots$ and
$(m,n)=(1,0),(3,0),(5,0),\dots$; the only surviving bosonic mode is this
scalar that was projected in region I. In region III; $m\geq 1,\ n\geq 1$; all
the bosonic modes are left as is, just like in other orbifolded models on
$T^{2}$. In the last region IV, only fermionic degrees of freedom remain.

Next, we go on to the models based on $S^{2}$. The explicit form of the KK
mass $M_{s,(\text{KK})}^{2}$ on $S^{2}$ is

$\displaystyle M_{s,(\text{KK})}^{2}\rightarrow
M_{(j)}^{2}:=\frac{j(j+1)}{R^{2}},$ (9)

with the index $j\geq 1$. For each $j$th mode in the $S^{2}$, $S^{2}/Z_{2}$,
and PS models, respectively, the number of degrees of freedom reads

$\displaystyle n^{S^{2}}(j)=2j+1,$ (10) $\displaystyle
n^{S^{2}/Z_{2}}(j)=\begin{cases}j+1&\quad\text{for }j=\text{{even}}\\\
j&\quad\text{for }j=\text{{odd}}\end{cases},$ (11) $\displaystyle
n_{\text{fermion}}^{\text{PS}}=2j+1,\quad
n_{\text{even}}^{\text{PS}}(j)=\begin{cases}2j+1\\\
0\end{cases}\\!\\!\\!\\!\\!\\!,\quad
n_{\text{odd}}^{\text{PS}}(j)=\begin{cases}0&\quad\text{for
}j=\text{{even}}\\\ 2j+1&\quad\text{for }j=\text{{odd}}\end{cases}.$ (12)

In the cases of $S^{2}$ and $S^{2}/Z_{2}$, the number of the surviving degrees
of freedom is the same for KK bosons and fermions. On the other hand, the PS
is similar to $RP^{2}$; that is, surviving KK bosons are divided into two
categories, even and odd. The even category includes all the KK bosons except
for the physical scalar from the 6D gauge boson, while the odd one only
contains this one. We note that the number of degenerate states is $2j+1$,
irrespective of the statistics of the particles and their oddness/evenness.
Finally, we comment on the beta functions of the $S^{2}$-based models. From
the surviving bosonic particles in each KK level, we can see that the RGEs in
$S^{2}$, $S^{2}/Z_{2}$ are similar to those in $T^{2}/Z_{2}$,
$T^{2}/(Z_{2}\times Z_{2}^{\prime})$, $T^{2}/Z_{4}$, while those in the PS are
similar to those in $RP^{2}$.

###  2.2 Running of Higgs self-coupling and Vacuum Stability

Figure 2: Left : Upper bounds on the UV cutoff of the UED models as a function of $M_{\text{KK}}$, with the initial conditions in Eq. (14). Right : Our color convention for types of the UED models. The lines in red, blue, and green show the results of $T^{2}$-based, $S^{2}$-based, and nonorientable-manifold-based UEDs, respectively. Figure 3: Cutoff upper bounds on the 6D UED models and the mUED with the initial condition of the Higgs in Eq. (14) and of the top quark in Eq. (15) with changing in the region of $[160\,\text{GeV},175\,\text{GeV}]$ with $M_{\text{KK}}=1\,\text{TeV}$. Conventions of line colors and shapes are the same as in Fig. 2.  Model | mUED | $T^{2}/Z_{2}$ | $T^{2}/{(Z_{2}\times Z^{\prime}_{2})}$ | $T^{2}/Z_{4}$ | $S^{2}$ | $S^{2}/Z_{2}$ | $RP^{2}$ | PS
---|---|---|---|---|---|---|---|---
$\tilde{\Lambda}_{\text{max}}$ | 5.0 | 2.5 | 2.9 | 3.4 | 2.3 | 3.2 | 2.3 | 1.9
Table 2: Upper bounds on cutoff scale
$\Lambda_{\text{max}}=\tilde{\Lambda}_{\text{max}}M_{\text{KK}}$ with
$M_{\text{KK}}=1\,\text{TeV}$ and the initial conditions in Eq. (14).

Following the discussion in the previous section, we evaluate the constraints
on the highest possible UV cutoff scale $\Lambda$ from vacuum stability of the
Higgs potential. In our analysis, we literally evaluate the KK summation in
Eq. (6), unlike the previous analysis in Ref. [56] where we obtained the UV
cutoff scale of the UED models from the perturbativity of the 4D gauge
couplings via the RGEs with its KK summation replaced by an integration. In
other words, we treat the threshold correction when the reference energy
crosses the mass of a KK particle explicitly in our numerical calculation. As
it was discussed in the previous section, we can ignore the mass coming from
the Higgs mechanism with good precision. Here, we adopt the following
criterion for determining $\Lambda_{\text{max}}$:

$\displaystyle\lambda(\mu=\Lambda_{\text{max}})=0,$ (13)

where the Higgs potential is destabilized.

We note that the vacuum stability bound is sensitive to the differences in the
initial condition of the Higgs self-coupling $\lambda$ and the top Yukawa
coupling $y_{t}$ [95, 96]. In our analysis, we adopt the following values:

$\displaystyle\frac{v^{2}}{2}\lambda(\mu=m_{Z})=126^{2}\,\text{GeV}^{2},\quad\frac{v}{\sqrt{2}}y_{t}(\mu=173.5\,\text{GeV})=160\,\text{GeV},$
(14)

where $m_{Z}$ is the Z-boson mass, the $126\,\text{GeV}$ is the observed Higgs
mass at the LHC, the $173.5\,\text{GeV}$ and the $160\,\text{GeV}$ are the
latest values of the pole and the $\overline{\text{MS}}$ masses of the top
quark reported by the particle data group [97], respectively.

The results are summarized in Fig. 2, and the values with
$M_{\text{KK}}=1\,\text{TeV}$ are also listed in Table 2. $M_{\text{KK}}$
means the first KK mass: $M_{\text{KK}}=1/R$ for the $S^{1}/Z_{2}$ (mUED) and
$T^{2}$-based compactifications (namely, $T^{2}/Z_{2}$, $T^{2}/(Z_{2}\times
Z_{2}^{\prime})$, $T^{2}/Z_{4}$ and $RP^{2}$) and ${M_{\text{KK}}}=\sqrt{2}/R$
for the $S^{2}$-based ones (namely, $S^{2}/Z_{2}$, PS and $S^{2}$), where we
assume $R_{5}=R_{6}=R$. It is noted that the mUED case has been studied in
Refs. [98, 99, 100, 101] in many contexts, and we find a study in the case of
$T^{2}/Z_{4}$ [102].888 We can find some related works on the evolutions of
higher-dimensional neutrino operator in the mUED [90, 103] and in the
$T^{2}/Z_{4}$ [102] and of the Cabibbo–Kobayashi–Maskawa matrix [104] in the
mUED context.  We mention that our conclusion on the mUED is consistent with
that in a previous analysis in Ref. [101]. The constraints from vacuum
stability, shown in Table 2, are tighter than our previous bounds from
perturbativity of the gauge couplings: $\Lambda_{\text{max}}\sim
5M_{\text{KK}}$ in $T^{2}/Z_{2}$, $T^{2}/Z_{2}\times Z^{\prime}_{2}$,
$T^{2}/Z_{4}$, $RP^{2}$ and $\Lambda_{\text{max}}\sim 7M_{\text{KK}}$ in
$S^{2}$, $S^{2}/Z_{2}$, PS. We note that, in the previous analysis, we ignored
differences in types of the compactifications and did not put a bound on the
mUED since the KK summations in the single Higgs production and the Higgs
decay, which are important in LHC phenomenology and which we consider in the
next section, are convergent in this case.

Next, we consider the effects when we change the values of top Yukawa coupling
in the initial conditions of the RGEs with $M_{\text{KK}}=1\,\text{TeV}$. We
note that, within the SM, various values of $\overline{\text{MS}}$ top mass
$m_{t}|_{\overline{\text{MS}}}$ have been reported between $160$ and
$175\,\text{GeV}$ [97, 95, 96, 105]. Based on this fact, we calculate the
bounds on $\lambda$ with varying the initial condition of the top Yukawa as

$\displaystyle\frac{v}{\sqrt{2}}y_{t}(\mu=173.5\,\text{GeV})=m_{t}|_{\overline{\text{MS}}},\quad\text{for
}160\,\text{GeV}\leq m_{t}|_{\overline{\text{MS}}}\leq 175\,\text{GeV}.$ (15)

Our result, depicted in Fig. 3, is sensitive to the value of
$m_{t}|_{\overline{\text{MS}}}$ and is consistent with the analyses in Ref.
[100] (mUED) and in Ref. [102] ($T^{2}/Z_{4}$). We cannot avoid the ambiguity
originating from the top Yukawa coupling. From Figs. 2 and 3, we find that the
dependence of $\Lambda$ on $M_{\text{KK}}$ and $m_{t}|_{\overline{\text{MS}}}$
is greater in the mUED than in the 6D UED models. In the latter, the KK
threshold corrections are larger than those in the mUED because of their
denser KK spectra, and hence the vacuum becomes unstable at a lower energy
scale.

##  3 Higgs signals at Large Hadron Collider

Equipped with the knowledge for the cutoff scale of UED models in the previous
section, we estimate the bound on their KK mass scale from the recent results
of Higgs search at the LHC.

###  3.1 Feature of Higgs signals in UED models

The structure of the Higgs signal at the LHC can be divided into the
production and decay. The Higgs production is dominated by the gluon fusion
process $gg\to H$, which is induced by the top loop. One of the most important
Higgs decay channels that lead to its discovery is the diphoton one
$H\to\gamma\gamma$, which is induced by the top and $W$ boson loops. The Higgs
signal is very sensitive to the contribution of the loop corrections at the
LHC. In UED models, a lot of additional KK loops contribute to both $gg\to
H(H\to gg)$ and $H\to\gamma\gamma$. The KK top loop contribution to the gluon
fusion production cross section takes the following form:

$\displaystyle\hat{\sigma}^{{\text{UED}}}_{gg\to H}$
$\displaystyle=\frac{\pi^{2}}{8m_{H}}\Gamma^{{\text{UED}}}_{H\rightarrow
gg}\delta(\hat{s}-m_{H}^{2}),$ (16)
$\displaystyle\Gamma^{{\text{UED}}}_{H\rightarrow gg}$
$\displaystyle=K\frac{\alpha^{2}_{s}}{8\pi^{2}}\frac{m_{H}^{3}}{v^{2}_{\text{EW}}}|J^{{\text{SM}}}_{t}{(m_{H}^{2})}+J^{{\text{KK}}}_{t}{(m_{H}^{2})}|^{2},$
(17)

where $K\sim 1.5$ is the K factor accounting for the higher-order QCD
corrections for the case of the LHC, $\alpha_{s}=\frac{g_{s}^{2}}{4\pi}$ is
the fine structure constant for QCD, $v\simeq$ 246 GeV is the electroweak
scale, and $J^{\text{SM/KK}}_{t}$ denotes the SM/KK top quark loop function,
defined in Refs. [56, 55]. The KK top quark and KK $W$-boson loop
contributions to the Higgs decay into diphoton are written as

$\displaystyle\Gamma^{{\text{UED}}}_{H\rightarrow\gamma\gamma}$
$\displaystyle=\frac{{\alpha^{2}}G_{F}m_{H}^{3}}{8\sqrt{2}\pi^{3}}\left|J^{{\text{SM}}}_{W}{(m_{H}^{2})}+J^{{\text{KK}}}_{W}{(m_{H}^{2})}+\frac{4}{3}(J^{{\text{SM}}}_{t}{(m_{H}^{2})}+J^{{\text{KK}}}_{t}{(m_{H}^{2})})\right|^{2},$
(18)

where $\alpha=\frac{e^{2}}{4\pi}$ and $G_{F}$ are fine structure constants for
the QED and Fermi constant, respectively. The SM/KK $W$-boson loop functions
$J^{\text{SM/KK}}_{W}$ are defined in Ref. [55]. We have listed them in
Appendix B.

Figure 4:  For the Higgs decay in the $T^{2}/Z_{2}$ UED model, we show the
UED/SM ratio (left) and the branching ratio (right) as a function of
$M_{\text{KK}}$ for each final state, which is indicated by the caption within
Figure; especially we distinguish the almost degenerate $cc$ and $ZZ$ by
dashed and solid lines, respectively.

Because of these additional contributions, the loop-induced processes $gg\to
H$ ($H\to gg$) and $H\to\gamma\gamma$ receive nontrivial effects, which we
compute and use to estimate the branching ratios and the Higgs decay rates
into the diphoton and digluon. As an illustration, we show results for the
$T^{2}/Z_{2}$ model in Fig. 4. The UED/SM ratio of $H\to gg$ is always
enhanced while that of $H\to\gamma\gamma$ is suppressed as already seen in
Ref. [55]. These behaviors also affect the branching ratios of the Higgs decay
as shown in the right panel of Fig. 4. The enhancement in $H\to gg$ is
straightforwardly understood as the KK top contributions in the loop diagram.
The reason for the suppression in $H\to\gamma\gamma$ is as follows. Since the
vectorlike fermions have twice the degrees of freedom compared with SM
fermions, their negative contributions to the Higgs decay rate become larger
than the positive ones coming from the KK W loop. Thus, the sum of the KK
loops becomes negative, and it overcomes the positive SM contribution. As a
consequence, the decay rate of $H\to\gamma\gamma$ is suppressed compared with
the SM.

###  3.2 Strategy to constrain the KK mass scale

As shown above, the UED models give different production cross sections in the
gluon fusion (GF). On the other hand, the other productions, which are the
vector boson fusion (VBF), the Higgs-strahlung (VH), and the associated
production with a $t\bar{t}$ pair (ttH), are the same as the SM. We express
the VH production associated with $W$ and $Z$ by WH and ZH, respectively. In
the recent analysis, the ATLAS and CMS experiments have reported on ratios of
these production channels in $H\to\gamma\gamma,ZZ,$ and $WW$ for each category
tagging their decays [3, 4, 5, 6, 7, 8].999 We use “$ZZ$” and “$WW$” as the
meaning of $ZZ\to 4\ell$ and $WW\to 2\ell 2\nu$ for simplicity. Such ratios
are quite important for obtaining the bound on UED models because of the
nontrivial effect of the KK loop corrections on both the production and the
decay of the Higgs boson. In order to take the different ratios of the
production cross section into account, we employ the following quantity [106,
107]:

$\epsilon_{f}^{I,X}=\frac{a_{f}^{I,X}\sigma_{X}^{\text{SM}}}{\sum_{Y}a_{f}^{I,Y}\sigma_{Y}^{\text{SM}}},$
(19)

where $X$ and $I$ indicate a production channel and a category tagging the
decay $H\to f$, $\sigma_{X}^{\text{SM}}$ is the Higgs production cross section
of the channel $X$ in the SM, and $a_{f}^{I,X}$ is introduced as its
acceptance. When the set $\\{\epsilon_{f}^{I,X}\\}$ is given in the decay
$H\to f$, the signal strength is written as

$\mu^{I}_{H\to
f}=\sum_{X}\epsilon_{f}^{I,X}\,\frac{\sigma_{X}}{\sigma^{\text{SM}}_{X}}\,\frac{\mathcal{B}_{H\to
f}}{\mathcal{B}_{H\to f}^{\text{SM}}},$ (20)

where $\sigma_{X}^{\text{(SM)}}$ represents the Higgs production cross section
of the channel $X$, and $\mathcal{B}_{H\to f}^{\text{(SM)}}=\Gamma_{H\to
f}^{\text{(SM)}}/\Gamma_{H\to\text{all}}^{\text{(SM)}}$ is the branching ratio
of the Higgs decay $H\to f$ (in the SM). In the UED model,
$\sigma_{\text{GF}}=\hat{\sigma}_{gg\to H}^{\text{UED}}$,
$\Gamma_{H\to\gamma\gamma(gg)}=\Gamma_{H\to\gamma\gamma(gg)}^{\text{UED}}$ as
in Eqs. (16)–(18), and the others are assumed to be the same as the SM in our
analysis.

$I$ | $\mu^{I}_{H\to\gamma\gamma}$ |  $\epsilon_{\gamma\gamma}^{I,X}$ (%)
---|---|---
Event category | Signal strength | GF | VBF | VH(WH) | VH(ZH) | ttH
Unconventional central low $P_{T}$ | $0.9\pm 0.7$ | 93.7 | 4.0 | 1.4 | 0.8 | 0.2
Unconventional central high $P_{T}$ | $1.0^{+1.1}_{-0.9}$ | 79.3 | 12.6 | 4.1 | 2.5 | 1.4
Unconventional rest low $P_{T}$ | $2.6^{+0.9}_{-1.0}$ | 93.2 | 4.0 | 1.6 | 1.0 | 0.1
Unconventional rest high $P_{T}$ | $2.7^{+1.3}_{-1.2}$ | 78.1 | 13.3 | 4.7 | 2.8 | 1.1
Conventional central low $P_{T}$ | $1.4^{+1.0}_{-0.9}$ | 93.6 | 4.0 | 1.3 | 0.9 | 0.2
Conventional central high $P_{T}$ | $2.0^{+1.5}_{-1.3}$ | 78.9 | 12.6 | 4.3 | 2.7 | 1.5
Conventional rest low $P_{T}$ | $2.2^{+1.2}_{-1.0}$ | 93.2 | 4.1 | 1.6 | 1.0 | 0.1
Conventional rest high $P_{T}$ | $1.3\pm 1.3$ | 77.7 | 13.0 | 5.2 | 3.0 | 1.1
Conventional transition | $2.8^{+1.7}_{-1.6}$ | 90.7 | 5.5 | 2.2 | 1.3 | 0.2
Loose high mass 2 jet | $2.8^{+1.7}_{-1.4}$ | 45.0 | 54.1 | 0.5 | 0.3 | 0.1
Tight high mass 2 jet | $1.6^{+0.8}_{-0.6}$ | 23.8 | 76.0 | 0.1 | 0.1 | 0.0
Low mass 2 jet | $0.3^{+1.7}_{-1.5}$ | 48.1 | 3.0 | 29.7 | 17.2 | 1.9
$E^{miss}_{T}$ significance | $3.0^{+2.7}_{-1.9}$ | 4.1 | 0.5 | 35.7 | 47.6 | 12.1
One lepton | $2.7^{+2.0}_{-1.7}$ | 2.2 | 0.6 | 63.2 | 15.4 | 18.6
Table 3: The ATLAS result of $H\to\gamma\gamma$ analysis. The ATLAS experiment defines these event categories and uses these ratios of the production channels as in Ref. [3].  $I$ | $\mu^{I}_{H\to\gamma\gamma}$ |  $\epsilon_{\gamma\gamma}^{I,X}$ (%)
---|---|---
Event category | Signal strength | GF | VBF | VH | ttH
Missing $E_{T}$ | $1.9^{+2.6}_{-2.3}$ | 22.0 | 2.6 | 63.7 | 11.7
Electron tag | $-0.7^{+2.8}_{-2.0}$ | 1.1 | 0.4 | 78.7 | 20.8
muon tag | $0.4^{+1.8}_{-1.4}$ | 0 | 0.2 | 79.0 | 19.8
2-jet loose | $0.8^{+1.1}_{-1.0}$ | 47.0 | 50.9 | 1.7 | 0.5
2-jet tight | $0.3^{+0.7}_{-0.6}$ | 20.7 | 78.9 | 0.3 | 0.1
Untag-3 | $-0.3^{+0.8}_{-0.9}$ | 92.5 | 3.9 | 3.3 | 0.3
Untag-2 | $0.3\pm 0.5$ | 91.6 | 4.5 | 3.6 | 0.4
Untag-1 | $0.0\pm 0.7$ | 83.5 | 8.4 | 7.1 | 1.0
Untag-0 | $2.2^{+0.9}_{-0.8}$ | 72.9 | 11.6 | 12.9 | 2.6
2-jet (7 TeV) | $4.2^{+2.3}_{-1.8}$ | 26.8 | 72.5 | 0.6 | 0
Untag-3 (7 TeV) | $1.5^{+1.7}_{-1.8}$ | 91.3 | 4.4 | 4.1 | 0.2
Untag-2 (7 TeV) | $0.0^{+1.3}_{-1.2}$ | 91.3 | 4.4 | 3.9 | 0.3
Untag-1 (7 TeV) | $0.2^{+1.0}_{-1.0}$ | 87.6 | 6.2 | 5.6 | 0.5
Untag-0 (7 TeV) | $3.8^{+2.0}_{-1.7}$ | 61.4 | 16.8 | 18.7 | 3.1
Table 4: The CMS result of $H\to\gamma\gamma$ analysis. The CMS experiment defines these event categories and uses these ratios of the production channels as in Ref. [6].  $I$ | $\mu^{I}_{H\to ZZ}$ |  $\epsilon_{ZZ}^{I,X}$ (%)
---|---|---
Event category | Signal strength | GF | VBF
Untagged | $0.85^{+0.32}_{-0.26}$ | 95 | 5
2-jet tag | $1.22^{+0.84}_{-0.57}$ | 80 | 20
$I$ | $\mu^{I}_{H\to WW}$ |  $\epsilon_{WW}^{I,X}$ (%)
SF 1 jet (7 TeV) | $0.9^{+2.1}_{-2.2}$ | 100 | 0
SF 0 jet (7 TeV) | $0.1\pm 1.0$ | 100 | 0
DF 1 jet (7 TeV) | $1.7\pm 1.0$ | 100 | 0
DF 0 jet (7 TeV) | $0.6\pm 0.5$ | 100 | 0
SF 1 jet (8 TeV) | $1.5\pm 0.9$ | 100 | 0
SF 0 jet (8 TeV) | $1.1\pm 0.7$ | 100 | 0
DF 1 jet (8 TeV) | $0.3\pm 0.4$ | 100 | 0
DF 0 jet (8 TeV) | $0.7\pm 0.3$ | 100 | 0
Table 5: The CMS result of $H\to ZZ/WW$ analysis. The CMS experiment defines
these event categories and uses these ratios of the production channels as in
Ref. [7, 8]. SF and DF denote “same flavor” and “different flavor”,
respectively.

For the analysis in $H\to\gamma\gamma$, the ATLAS and CMS experiments have
shown their results of $\mu^{I}_{H\to\gamma\gamma}$ and the set
$\\{\epsilon_{\gamma\gamma}^{I,X}\\}$ they used in their analyses [3, 6]. We
summarize these values in Tables 3 and 4. For the analysis in $H\to ZZ/WW$,
the CMS result is summarized in Table 5. The result of $H\to WW$ in the CMS
experiment is given by assuming that all Higgs signals are produced by the GF
process [8]. The ATLAS experiment only gives the signal strength for the
specific production channels [4, 5], which is written as

$\mu^{X}_{H\to
ZZ/WW}=\frac{\sigma_{X}}{\sigma^{\text{SM}}_{X}}\,\frac{\mathcal{B}_{H\to
ZZ/WW}}{\mathcal{B}_{H\to ZZ/WW}^{\text{SM}}}.$ (21)

The results are given as $\mu^{\text{GF+ttH}}_{H\to ZZ}=1.8^{+0.8}_{-0.5}$,
$\mu^{\text{VBF+VH}}_{H\to ZZ}=1.2^{+3.8}_{-1.4}$, $\mu^{\text{GF}}_{H\to
WW}=0.82\pm 0.36$, and $\mu^{\text{VBF}}_{H\to WW}=1.66\pm 0.79$. In this
article, we assume $\mu^{\text{GF+ttH}}_{H\to ZZ}\simeq\mu^{\text{GF}}_{H\to
ZZ}$ for simplicity.

We evaluate a bound on the KK scale in each UED model by performing a
$\chi^{2}$ analysis of the results as shown above. The $\chi^{2}$ function is
represented as

$\chi^{2}=\sum_{f}\sum_{I}\left(\frac{\mu^{I}_{H\to
f}-\hat{\mu}^{I}_{f}}{\hat{\sigma}^{I}_{f}}\right)^{2},$ (22)

where we assume the experimental results to be Gaussian distribution
$\hat{\mu}^{I}_{f}\pm\hat{\sigma}^{I}_{f}$.101010 Note that, since we neglect
the correlation among the categories, which is not made public, this analysis
should rather be taken as an illustration.  The number of the observables we
use in our analysis is 42 in total, and the degree of freedom is also the same
number in terms of testing a justification of a model.

###  3.3 Bound on KK scale from the current data

Figure 5:  Exclusion C.L.s of all the UED models as functions of the KK scale
$M_{\text{KK}}$ by use of all the ATLAS and CMS results of
$H\rightarrow\gamma\gamma,WW,ZZ$. Colors denote the same as in Fig. 2.

Here we show bounds on several UED models from the Higgs searches at the LHC.
For our analyses, we have taken the highest possible UV cutoff scale
$\Lambda_{\text{max}}$ shown in Table 2. The Higgs mass is chosen to be 126
GeV. In Fig. 5, we show the exclusion C.L. of each UED model as a function of
the KK scale $M_{\text{KK}}$ by use of all the ATLAS and CMS results of
$H\to\gamma\gamma,WW,ZZ$. The black line indicates the result in the five-
dimensional mUED model. The blue solid, dashed, and dotted lines denote those
in the $T^{2}$-based ones, namely, the $T^{2}/Z_{2}$, $T^{2}/(Z_{2}\times
Z^{\prime}_{2})$ and $T^{2}/Z_{4}$, respectively. The red solid and dashed
lines represent those in the $S^{2}$-based ones, namely $S^{2}$ and
$S^{2}/Z_{2}$, respectively. The green solid and dashed lines show those in
the nonorientable ones, namely, $RP^{2}$ and PS, respectively.

As can be seen in this graph, we find that the region
$M_{\text{KK}}\lesssim{600}$ GeV is excluded within $95\%$ C.L. in the mUED
model. For the six-dimensional models in the $T^{2}$-based space, we find the
excluded regions $M_{\text{KK}}\lesssim 1100$, ${1000}$, and ${800}$ GeV
within $95\%$ C.L. for $T^{2}/Z_{2}$, $T^{2}/(Z_{2}\times Z^{\prime}_{2})$,
and $T^{2}/Z_{4}$, respectively. For the $S^{2}$-based models, we can see that
the regions $M_{\text{KK}}\lesssim{1300}$ and ${900}$ GeV are excluded within
$95\%$ C.L. in the $S^{2}$ and $S^{2}/Z_{2}$, respectively. For the
nonoriented models, the regions $M_{\text{KK}}\lesssim{1100}$ and ${1200}$ GeV
are excluded within $95\%$ C.L. in the $RP^{2}$ and PS, respectively. As seen
above, the excluded region is different from one model to another in the case
for UED. This is because the difference of the KK spectrum has a large impact
on the Higgs decays via loop processes.

Figure 6: The exclusion C.L.s of all UED models as functions of the KK scale
$M_{\text{KK}}$ obtained from the ATLAS (left) and CMS (right) results of
$H\to\gamma\gamma,WW,ZZ$. Colors denote the same as in Fig. 2.

We compare the bounds obtained from the ATLAS experiment with those from the
CMS in Fig. 6. We find that the CMS result gives a more stringent bound on the
KK scale compared with the ATLAS one. In other words, for now, the UED models
are likely to explain the recent ATLAS result, while they are disfavored by
the recent CMS result. The results of the exclusion C.L. for the wide range of
the KK scale are summarized in Fig. 10 in Appendix D.

Throughout this analysis, we ignore the effects from the higher-dimensional
operators around $\Lambda$. See Ref. [85] for such an effect.

##  4 Indirect constraint from $S$ and $T$ parameters

Physics beyond the SM is also restricted through the precise measurement of
some electroweak variables. The $S$ and $T$ parameters proposed by Peskin and
Takeuchi [108, 109] have been used for estimating whether a model is valid or
not. The variables are defined by use of the two-point functions of the SM
gauge bosons,

$\displaystyle\Pi^{\mu\nu}_{ab}(k)=i\Pi^{\text{T}}_{ab}(k^{2})\left(g^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{k^{2}}\right)+i\Pi^{\text{L}}_{ab}(k^{2})\frac{k^{\mu}k^{\nu}}{k^{2}},$
(23)

where $k$ is the external momentum, T (L) means that it is the transverse
(longitudinal) part, and the indices $a,b$ show types of the SM gauge bosons.
The variables are constructed by the transverse components, and the concrete
forms are written down with adapting the notation on the electroweak sector in
Ref. [92] as

$\displaystyle\frac{\alpha S}{4s_{W}^{2}c_{W}^{2}}$
$\displaystyle={\Pi^{\text{T}}_{ZZ}}^{\prime}(0)+\frac{c_{W}^{2}-s_{W}^{2}}{c_{W}s_{W}}{\Pi_{Z\gamma}^{\text{T}}}^{\prime}(0)-{\Pi_{\gamma\gamma}^{\text{T}}}^{\prime}(0),$
(24) $\displaystyle\alpha T$
$\displaystyle=\frac{\Pi_{WW}^{\text{T}}(0)}{m_{W}^{2}}-\frac{\Pi_{ZZ}^{\text{T}}(0)}{m_{Z}^{2}}+{2c_{W}s_{W}}\frac{\Pi_{Z\gamma}^{\text{T}}(0)}{m_{W}^{2}},$
(25)

where ${\Pi^{\text{T}}_{ab}}^{\prime}$ is defined as
$\frac{d}{dk^{2}}\Pi^{\text{T}}_{ab}(k^{2})$.

The $S$ and $T$ are also described by combinations of some electroweak
variables and their values are calculated in global analysis with experimental
results. One of the latest numbers is found in Ref. [110],

$\displaystyle S|_{U=0}=0.05\pm 0.09,\quad T|_{U=0}=0.08\pm
0.07,\quad\rho_{ST}=+0.91,$ (26)

with $126\,\text{GeV}$ reference Higgs mass and assuming the $U$ parameter is
zero and $\rho_{ST}$ is the correlation coefficient. In an operator-analysis
point of view, the $U$ parameter is represented as a coefficient of a much
higher-dimensional operator with the Higgs doublet compared with $S$ and $T$
in the UED models, and hence we ignore the effect in our analysis.

###  4.1 Forms in 6D UED models and mUED

In this section, we formulate the contributions to the $S$ and $T$ parameters
in the 6D UED models and in the mUED model. It is well known that the $S$ and
$T$ parameters are logarithmically divergent in six dimensions [89]. To have a
rough idea of what happens, we employ the following prescription. First, we
compute the contributions from each KK mode within four-dimensional field
theory employing the dimensional regularization. They are manifestly finite.
Then we sum such contributions up to a mode in which the KK mass exceeds the
UV cutoff $\Lambda$. To estimate the possible effects from the UV theory above
$\Lambda$, we also put the higher-dimensional operators in six dimensions.

The general shape of the $S$ and $T$ parameters are

$\displaystyle S=\sum_{s\atop\text{with
}M_{s}<\Lambda}\left(S^{(\text{KK})}_{s,\text{boson}}+S^{(\text{KK})}_{s,\text{fermion}}\right)+S_{\text{Higgs
calibration}}+S_{\text{threshold}},$ (27) $\displaystyle
T=\sum_{s\atop\text{with
}M_{s}<\Lambda}\left(T^{(\text{KK})}_{s,\text{boson}}+T^{(\text{KK})}_{s,\text{fermion}}\right)+T_{\text{Higgs
calibration}}+T_{\text{threshold}},$ (28)

where the first terms in parentheses are the contributions of KK particles,
and the last two terms represent, respectively, the effects from Higgs mass
calibration and the threshold correction via possible higher-dimensional
operators around the UV cutoff scale $\Lambda$ in six dimensions. These
effects were considered in Refs. [109, 89],

$\displaystyle S_{\text{Higgs calibration}}$
$\displaystyle=\frac{1}{12\pi}\ln\left(\frac{m_{H}^{2}}{m_{H,\text{ref}}^{2}}\right),$
$\displaystyle T_{\text{Higgs calibration}}$ $\displaystyle=-\frac{3}{12\pi
c_{W}^{2}}\ln\left(\frac{m_{H}^{2}}{m_{H,\text{ref}}^{2}}\right),$ (29)
$\displaystyle S_{\text{threshold}}$ $\displaystyle=c_{S}\frac{2\pi
v^{2}}{\Lambda^{2}},$ $\displaystyle T_{\text{threshold}}$
$\displaystyle=c_{T}\frac{m_{H}^{2}}{4\alpha\Lambda^{2}},$ (30)

where $m_{H,\text{ref}}$ is the assumed SM Higgs mass in global analysis, and
$c_{S}$ and $c_{T}$ are undetermined dimensionless coefficients with
$\mathcal{O}(1)$ magnitude.

Several comments are in order. One is that the summations over KK states are
truncated at the scale $\Lambda$. The other is that the value of $\Lambda$ is
estimated through the vacuum stability condition of the Higgs boson. We choose
the highest possible $\Lambda$ allowed by it. As we discussed in Sec. 2, in
the configuration of $m_{H}=126\,\text{GeV}$, the value of the maximum UV
cutoff scale tends to be low, and the threshold corrections possibly become
important. We will include these effects below. Finally, we comment on the
contributions of KK particles. We find that the effect from the state-$s$
fermion loops takes the following general shapes in every 6D UED model, which
is the same as in the mUED and was already calculated in Ref. [89]. We show
them in our notation:

$\displaystyle S_{\text{fermion},s}^{(\text{KK})}$
$\displaystyle\simeq\frac{1}{4\pi}\frac{2}{3}x_{t,s},\quad
T_{\text{fermion},s}^{(\text{KK})}\simeq\frac{1}{\alpha}\left(\frac{m_{t}^{2}}{4\pi^{2}v^{2}}\right)x_{t,s},$
(31)

where $x_{t,s}$ is defined with the KK mass of the state $``s"$ $M_{s}$ as

$\displaystyle x_{t,s}=\frac{m_{t}^{2}}{M_{s}^{2}}$ (32)

and we ignore their $\mathcal{O}(x_{t,s}^{2})$ corrections. In the $RP^{2}$
model, we should pay attention to the fact that the summation range differs
between bosonic and fermionic sectors.

The bosonic part is highly model dependent. In this paper, we have newly
calculated the contributions to $S$ and $T$ in every 6D model. The complete
forms of the gauge-boson two-point functions are summarized in Appendix C.

In the cases of $T^{2}/Z_{2}$, $T^{2}/(Z_{2}\times Z^{\prime}_{2})$,
$T^{2}/Z_{4}$, $S^{2}$, and $S^{2}/Z_{2}$, the forms are

$\displaystyle S_{\text{boson},s}^{(\text{KK})}$
$\displaystyle\simeq\frac{1}{\pi}\Bigg{\\{}-\frac{5}{36}x_{W,s}+\frac{1}{24}x_{H,s}+\left(\frac{1}{24}-\frac{1}{6c_{W}^{2}}\right)x_{Z,s}\Bigg{\\}},$
(33) $\displaystyle T_{\text{boson},s}^{(\text{KK})}$
$\displaystyle\simeq\frac{1}{4\pi}\frac{1}{s_{W}^{2}}\Bigg{\\{}\left(\frac{15}{4}-\frac{193}{72}\frac{1}{c_{W}^{2}}+\frac{1}{2c_{W}^{4}}\right)x_{W,s}+\left(-\frac{85}{72}-\frac{7}{18}\frac{1}{c_{W}^{2}}\right)x_{Z,s}+\left(\frac{13}{36}+\frac{5}{36}\frac{1}{c_{W}^{2}}-\frac{1}{2c_{W}^{4}}\right)x_{H,s}\Bigg{\\}},$
(34)

where we define similar variables as in Eq. (32):
$x_{i,s}=\frac{m_{i}^{2}}{M_{s}^{2}}$ with $m_{W}^{2}$, $m_{Z}^{2}$, and
$m_{H}^{2}$. Note that the lighter the KK particles are the greater they
contribute to $S$ and $T$. In these models, the result is affected only by the
differences in the patterns of the surviving KK modes.

In the cases of the models based on the nonorientable manifolds $RP^{2}$ and
PS, bosonic contributions are classified into three and two categories,
respectively. The details of the following classifications have already been
discussed in Sec. 2 and thus we do not explain it here. The results in the PS
model are shown:

$\displaystyle S_{\text{boson},s:\text{odd}}^{(\text{KK})}$
$\displaystyle\simeq 0,\quad
T_{\text{boson},s:\text{odd}}^{(\text{KK})}\simeq\frac{1}{4\pi}\frac{1}{s_{W}^{2}}\frac{5}{18}\Bigg{\\{}\left(1-\frac{1}{c_{W}^{2}}\right)x_{W,s}+\left(\frac{1}{c_{W}^{2}}-1\right)x_{Z,s}\Bigg{\\}},$
(35) $\displaystyle S_{\text{boson},s:\text{even}}^{(\text{KK})}$
$\displaystyle\simeq\frac{1}{\pi}\Bigg{\\{}-\frac{5}{36}x_{W,s}+\frac{1}{24}x_{H,s}+\left(\frac{1}{24}-\frac{1}{6c_{W}^{2}}\right)x_{Z,s}\Bigg{\\}},$
(36) $\displaystyle T_{\text{boson},s:\text{even}}^{(\text{KK})}$
$\displaystyle\simeq\frac{1}{4\pi}\frac{1}{s_{W}^{2}}\Bigg{\\{}\left(\frac{125}{36}-\frac{173}{72}\frac{1}{c_{W}^{2}}+\frac{1}{2c_{W}^{4}}\right)x_{W,s}+\left(-\frac{65}{72}-\frac{2}{3}\frac{1}{c_{W}^{2}}\right)x_{Z,s}+\left(\frac{13}{36}+\frac{5}{36}\frac{1}{c_{W}^{2}}-\frac{1}{2c_{W}^{4}}\right)x_{H,s}\Bigg{\\}}.$
(37)

The shapes in the $RP^{2}$ model are closely related the previous ones in the
PS as follows:

$\displaystyle\\{S,\,T\\}_{\text{boson},s:\text{region I}}^{(\text{KK})}$
$\displaystyle=\\{S,\,T\\}_{\text{boson},s:\text{even}}^{(\text{KK})},$ (38)
$\displaystyle\\{S,\,T\\}_{\text{boson},s:\text{region II}}^{(\text{KK})}$
$\displaystyle=\\{S,\,T\\}_{\text{boson},s:\text{odd}}^{(\text{KK})},$ (39)
$\displaystyle\\{S,\,T\\}_{\text{boson},s:\text{region III}}^{(\text{KK})}$
$\displaystyle=\\{S,\,T\\}_{\text{boson},s}^{(\text{KK})},$ (40)

where we note that we should use the form of the KK mass on $S^{2}$ instead of
on $T^{2}$.

The mixing among KK states in the gauge sector is schematically of the form

$\displaystyle\begin{bmatrix}m_{W,Z}^{2}+M_{\text{KK}}^{2}&M_{\text{KK}}^{2}&m_{W,Z}M_{\text{KK}}\\\
M_{\text{KK}}^{2}&m_{W,Z}^{2}+M_{\text{KK}}^{2}&m_{W,Z}M_{\text{KK}}\\\
m_{W,Z}M_{\text{KK}}&m_{W,Z}M_{\text{KK}}&m_{W,Z}^{2}+M_{\text{KK}}^{2}\end{bmatrix}.$
(41)

In the calculation of $S$ and $T$ parameters, we adopt the following
approximation about the mass mixings of 6D W and Z boson-related sectors:

  * •

We ignore off-diagonal terms with the magnitude
$\mathcal{O}(m_{W,Z}M_{\text{KK}})$, which are small compared with the other
terms with the magnitude $\mathcal{O}(M_{\text{KK}}^{2})$.

  * •

In the diagonal terms, for which the forms are approximately as
${m_{W,Z}^{2}+M_{\text{KK}}^{2}}$, we do not ignore the small part coming from
$m_{W,Z}^{2}$ since this part can contribute to the $T$ parameter.

Because of this approximation, the small mixings being proportional to
$m_{W,Z}^{2}$ are ignored. As a result, some divergent terms that are
proportional to $m_{W,Z}^{2}$ remain in the $T$ parameter, and we simply
discard them. Note that the contribution to $T$ from each KK mode must be
manifestly finite since it is computed in four-dimensional field theory with
the dimensional regularization. The divergence ($\propto m_{W,Z}^{2}$) that we
encounter here is an artifact coming from the ignorance of the small off-
diagonal part in the KK mixing of gauge sector. Indeed, we find that there
appears no divergence proportional to $m_{H}^{2}$ or $m_{t}^{2}$, as we treat
the mixing of the Higgs and top KK sectors exactly. Although there might be a
further possible finite correction due to this procedure, the KK gauge
contribution is generally subleading compared to the KK top loops, the mixing
of which we treat exactly. In the $S$ parameter, we do not see any divergence
even after the above approximation. These features are consistent with the
general property of $S$ and $T$.

After we considered radiative corrections, the Weinberg angles of the KK $W$
and $Z$ bosons get to be very small [111]. We assume in this effect that the
KK Weinberg angles are zero and that we can simply ignore the mass
corrections. Each KK-state contribution should be suppressed by its KK mass,
and hence this effect should not affect the leading order of $S$ and $T$ since
their contributions are proportional to KK masses (when we ignore the
electroweak masses in loop calculation) [111].

Finally we comment on the mUED model, which has been studied extensively [19,
89, 112, 113, 23]. In the $\chi^{2}$ analysis of Refs. [89, 23], the authors
simply ignored the terms being proportional to $m_{W,Z}^{2}$, possibly because
their effects are not significant compared with those that are proportional to
$m_{H}^{2}$ or $m_{t}^{2}$.

Boson contributions to $S$ and $T$ in the mUED are approximately described
with the even part of the PS (or region I of the $RP^{2}$) as we already
showed in Eqs. (36) and (37) since the particle content of each KK state is
the same. Fermion contributions are the same as in Eq. (31). Here, we adopt
the form of the KK mass $M_{(n)}$ with a KK number $n$

$\displaystyle M_{(n)}^{2}={n^{2}}{M_{\text{KK}}^{2}},\quad(\text{for
}n=1,2,3,\cdots).$ (42)

###  4.2 Numerical results without threshold correction

Figure 7:  The exclusion C.L.s of all the UED models as functions of
$M_{\text{KK}}$ obtained from the $S$ and $T$ parameters. Colors denote the
same as in Fig. 2.

We also execute a $\chi^{2}$ analysis for putting indirect constraints on the
UED models. $\chi^{2}$ from $S$ and $T$ is defined as

$\displaystyle\chi^{2}_{ST}=\frac{1}{1-\rho_{ST}^{2}}\left\\{\frac{\left(S-S_{\text{exp.}}\right)^{2}}{\sigma_{S}^{2}}+\frac{\left(T-T_{\text{exp.}}\right)^{2}}{\sigma_{T}^{2}}-\frac{2\rho_{ST}}{\sigma_{S}\sigma_{T}}\left(S-S_{\text{exp.}}\right)\left(T-T_{\text{exp.}}\right)\right\\},$
(43)

where $S$ and $T$ are the theoretical inputs in Eqs. (27) and (28) and the
others are the experimental resultants in Eq. (26). In this and the next
sections, we again adopt the assumption of $R_{5}=R_{6}=R$.

At first in this section, we consider the possibility without threshold
correction to $S$ and $T$ in Eq. (30). We consider the maximal cutoffs with
$M_{\text{KK}}=1\,\text{TeV}$ irrespective of $M_{\text{KK}}$ because our
interest is in the case that $M_{\text{KK}}$ is about a few TeV, and the
values are almost universal as a function of $M_{\text{KK}}$ in each model
around a few TeV as shown in Fig. 2. The result is listed in Fig. 7. The plots
for a wider range of $M_{\text{KK}}$ are shown in Appendix D, where we can
find the global minima in every curve. Each minimum is located around
${1700}\,\text{GeV}$ ($T^{2}/Z_{2}$), $1500\,\text{GeV}$ ($T^{2}/(Z_{2}\times
Z^{\prime}_{2})$), $1300\,\text{GeV}$ ($T^{2}/Z_{4}$), $2200\,\text{GeV}$
($S^{2}$), $1500\,\text{GeV}$ ($S^{2}/Z_{2}$), $1800\,\text{GeV}$ ($RP^{2}$),
$2000\,\text{GeV}$ (PS), and $1000\,\text{GeV}$ (mUED). Interestingly, these
values are somewhat greater than the corresponding $95\%$ C.L. bound from the
combined results in the Higgs searches as shown in Fig. 5.

We also estimate the $95\%$ C.L. bounds of the models from Fig. 7, and the
values are about $1200\,\text{GeV}$ ($T^{2}/Z_{2}$), $1100\,\text{GeV}$
($T^{2}/(Z_{2}\times Z^{\prime}_{2})$), $900\,\text{GeV}$ ($T^{2}/Z_{4}$),
$1500\,\text{GeV}$ ($S^{2}$), ${1100}\,\text{GeV}$ ($S^{2}/Z_{2}$),
$1200\,\text{GeV}$ ($RP^{2}$), $1400\,\text{GeV}$ (PS), and
${700}\,\text{GeV}$ (mUED). Here, we can notice that these indirect bounds are
compatible with the direct bounds via the LHC results discussed in the
previous section. We note that our $95\%$ C.L. bound (${700}\,\text{GeV}$) on
the mUED is close to the previous values by the Gfitter group
($700\,\text{GeV}$ in $m_{H}=126\,\text{GeV}$) in Ref. [23].

###  4.3 Numerical results with threshold correction

Figure 8:  The exclusion C.L.s via the $\chi^{2}$ analysis of $S$ and $T$
parameters as a function of $M_{\text{KK}}$ with the maximal cutoffs in the
UED models in Table 2 with threshold corrections in Eq. (30). In each panel
corresponding to each model, the left (blue), center (black), and right (red)
bunches of lines are for $c_{S}=+1$, 0, and $-1$, respectively. In each bunch,
the dotted, solid, and dashed lines correspond to $c_{T}=-1$, 0, and $+1$,
respectively.

As we have seen in Sec. 2, the vacuum is destabilized rapidly for the Higgs
mass $m_{H}=126\,\text{GeV}$, and we should take the cutoff scale quite a low.
For completeness, we estimate the threshold corrections via physics around the
cutoff scale $\Lambda$.

In this section, we summarize the results, for the maximum UV cutoff scale,
given in Table 2, in Fig. 8. Here, we examine the three extremal possibilities
($c_{S,T}=0,+1,-1$) for each of the two coefficients $c_{S}$ and $c_{T}$ in
Eq. (30).

We can find sizable deviations from the case of $c_{S}=c_{T}=0$ in all the
models. The corrections from $S_{\text{threshold}}$ are significant and the
$95\%$ C.L. bounds turn out to be modified by the magnitudes about $100\sim
200\,\text{GeV}$, depending on the models, toward both positive and negative
directions, while the corrections from $T_{\text{threshold}}$ are subleading.
We note that we can find the global minimum in all the models after taking
into account the threshold corrections, which is not shown in Fig. 8.

One important thing is that, even in the 5D mUED, the threshold corrections
are more significant than was thought. We report that the $T^{2}/Z_{2}$ case
was studied in Ref. [89] with $\Lambda=5M_{\text{KK}}$. In our result, the
degree of the deviations from the case without the threshold correction is
enlarged since we can take $\Lambda$ at most $2.5M_{\text{KK}}$ as shown in
Table 2.

##  5 Summary and Discussions

We have studied the effects from the KK particles in the UED models on the
Higgs searches at the LHC and on the electroweak precision data. Both are
dependent on the UV cutoff scale $\Lambda$ of the higher-dimensional theory.
We have evaluated the highest possible $\Lambda$ consistent with the vacuum
stability bound on the Higgs potential.

In the UED models, the contributions from loop diagrams including the KK top
quarks and gauge bosons modify the Higgs decay rate and production cross
section, which affect the Higgs signal strengths at the LHC. On the other
hand, the KK excited states of the heavier SM particles (top quark, Higgs
boson, and the massive gauge bosons) alter the $S$ and $T$ parameters. From
the analysis on the results of the Higgs signal strengths in the decay modes
$H\to\gamma\gamma$, $ZZ,WW$ and of the $S$ and $T$ parameters, we have
estimated the two types of bounds on the KK scales in 5D and 6D UED models,
which are summarized in Table 6. Comparing the former bounds with the latter,
we find that the latter are slightly more severe than the former. However, in
few years the Higgs searches at the LHC will put stronger constraints on the
KK scale in the UED models. We note that there remain uncertainties from the
choices of the UV cutoff $\Lambda<\Lambda_{\text{max}}$, the higher-
dimensional operators there, and the low-energy input for the top Yukawa
coupling. In the estimation of $\Lambda_{\text{max}}$, we focus on the vacuum
stability bound, namely, the condition (13) on the coupling $\lambda$. This
new bound is tighter than the conventional one derived from the perturbativity
of the gauge couplings. It might be interesting to take into account the
effects of higher-dimensional operators for the stability argument, as we have
done for the $S$ and $T$, since the scale of $\Lambda_{\text{max}}$ tends to
be low in the UED models after the Higgs discovery.

|  $\Lambda/M_{\text{KK}}$ for $M_{\text{KK}}\sim O(\text{TeV})$ | Higgs signal strength |  $S$ and $T$ parameters
---|---|---|---
mUED | 5.0 | 610 GeV | 680 GeV
$T^{2}/Z_{2}$ | 2.5 | 1060 GeV | 1190 GeV
$T^{2}/(Z_{2}\times Z_{2}^{\prime})$ | 2.9 | 960 GeV | 1080 GeV
$T^{2}/Z_{4}$ | 3.4 | 820 GeV | 920 GeV
$RP^{2}$ | 2.3 | 1060 GeV | 1220 GeV
$S^{2}$ | 2.3 | 1330 GeV | 1490 GeV
$S^{2}/Z_{2}$ | 3.2 | 940 GeV | 1050 GeV
PS | 1.9 | 1240 GeV | 1410 GeV
Table 6: Highest possible UV cutoff scales and lower bounds on the KK scale
$M_{\text{KK}}$ for each model at the 95% C.L. Figure 9:  The same exclusion
C.L.s as above, in the limit $R_{5}\gg R_{6}$. The solid and dotted lines
correspond to the bounds from Higgs searches and $S$, $T$ constraints,
respectively. The former Higgs bounds for $T^{2}/Z_{2}(Z_{2}\times
Z_{2}^{\prime})$ and mUED are degenerate with each other.

The authors of Ref. [114] have shown that the case $R_{5}=R_{6}$ is disfavored
in the $RP^{2}$ model if we identify the lowest KK state to be the dark
matter, compared with the limit $R_{5}\gg R_{6}$, because the former requires
the lower dark matter mass than the latter. We show in Fig. 9 the same
exclusion limits from the Higgs searches and from $S$, $T$, as given above, in
this five dimensional limit. As expected, the bounds become almost the same as
that in the mUED model, expect for the small difference due to extra
contributions from the 6D KK states of the gauge field. Note also that the
five dimensional limit $R_{5}\gg R_{6}$ loosens the vacuum stability bound and
hence allows the higher cutoff scale $\Lambda_{\text{max}}=5.5M_{\text{KK}}$
and $3.9M_{\text{KK}}$ for $T_{2}/Z_{2}(Z_{2}\times Z_{2}^{\prime})$ and
$RP^{2}$, respectively. The higher the cutoff scale is, the larger the number
of KK states below it. We have chosen $\Lambda=\Lambda_{\text{max}}$ and taken
into account this effect in Fig. 9.

We briefly comment on other bounds. We can find the recent studies in bounds
from collider simulations in the mUED, 6D UEDs on $T^{2}/Z_{4}$, and $RP^{2}$.

  * •

On mUED in Ref. [43]: $M_{\text{KK}}\gtrsim 1300\,\text{GeV}$ with $95\%$ C.L.
through trilepton signature $+$Missing ET (MET) with $20\,\text{fb}^{-1}$ at
$\sqrt{s}=8\,\text{TeV}$ $(\Lambda R=10)$.

  * •

On $T^{2}/Z_{4}$ in Ref [65]: $M_{\text{KK}}\gtrsim 500\,\text{GeV}$ with
$5\sigma$ C.L. through $n$-jets$+\gamma+$MET ($n\geq 4$) with
$2\,\text{fb}^{-1}$ at $\sqrt{s}=7\,\text{TeV}$.

  * •

On $RP^{2}$ in Ref. [69]: $M_{\text{KK}}\gtrsim 600\,\text{GeV}$ with above
$99\%$ C.L. through CMS $\alpha_{T}$ analysis in leptons$+$MET with $\sim
5\,\text{fb}^{-1}$ at $\sqrt{s}=7\,\text{TeV}$.

The constraint on the mUED is tighter than ours from the direct Higgs search
and $S$, $T$ parameters, while these on $T^{2}/Z_{4}$ and $RP^{2}$ (with the
limited integrated luminosities) are somewhat loose compared with ours. As
pointed out in Ref. [101], a UED model with a low cutoff scale results in a
much compressed KK spectrum and hence becomes difficult to detect at the LHC.
It is noted that such a degenerate possibility has not been explored enough,
and the analysis with $M_{T2}$ and/or event shape variables is suitable for
the case [40, 41]. We also refer to the bounds from dark matter relic
abundance in these three models. The upper bound on $M_{\text{KK}}$ is
approximately less than $200\,\text{GeV}$ in $T^{2}/Z_{4}$ [115] and
$470\,\text{GeV}$ in $RP^{2}$ with $R_{5}=R_{6}$ [114], to circumvent an
overabundance of matter in the Universe. This bound suggests that these 6D
UEDs on both geometries are disfavored in combination with our results.111111
In the $RP^{2}$ case, the upper bound on $M_{\text{KK}}$ can be uplifted
around $1.5\,\text{TeV}$ by introducing tuning in the parameter space [69,
114].  In the mUED, the range being consistent with the relics is
$1300\,\text{GeV}\lesssim M_{\text{KK}}\lesssim 1500\,\text{GeV}$ [22], which
is just an unexplored area.

We have studied the suppression effects of Higgs decay into diphoton in Sec.
3. These effects can also affect the measuring of the Higgs to diphoton
coupling at a future linear collider [116]. We summarize in Table 7 the ratio
of $\operatorname{BR}(H\to\gamma\gamma)$ as well as the Higgs production cross
section from $\gamma\gamma$ collision in each UED model to those in the SM for
the lowest possible KK scale with the highest possible UV cutoff. We find that
the branching ratio and the Higgs production can be suppressed by a factor
$\sim 0.9$ compared with SM. This is marginally accessible at the linear
collider with integrated luminosity 500 fb-1 at 500GeV for which the expected
precision for the $\operatorname{BR}(H\to\gamma\gamma)$ is $23\%$ for
$M_{h}=120$ GeV [117]. This precision is refined to 5.4% with luminosity 1
ab-1 at 1 TeV for the same Higgs mass [118]. When we employ the photon-photon
collider option, $H\gamma\gamma$ coupling can be measured more directly from
the total production cross section of the Higgs. This is well within the reach
for an integrated photon-photon luminosity 410 fb-1 at a linear $e^{+}e^{-}$
collider operated at a $\sqrt{s}=210$ GeV, which can measure
$\Gamma_{H\rightarrow\gamma\gamma}\times\operatorname{BR}(H\rightarrow\gamma\gamma)$
with accuracy of 2.1$\%$ for $m_{H}=$ 120 GeV [119].

| UED/SM ratio of $\operatorname{BR}(H\to\gamma\gamma)$ | UED/SM ratio of $\sigma_{\gamma\gamma\to H}$
---|---|---
mUED | 0.93 | 0.94
$T^{2}/Z_{2}$ | 0.93 | 0.94
$T^{2}/(Z_{2}\times Z_{2}^{\prime})$ | 0.93 | 0.94
$T^{2}/Z_{4}$ | 0.92 | 0.94
$RP^{2}$ | 0.93 | 0.94
$S^{2}$ | 0.85 | 0.88
$S^{2}/Z_{2}$ | 0.92 | 0.94
PS | 0.90 | 0.92
Table 7: Prediction on the UED/SM ratio of
$\operatorname{BR}(H\to\gamma\gamma)$ and $\sigma_{\gamma\gamma\to H}$ with
the lowest possible value of the KK scale.

## Acknowledgement

We thank Tomohiro Abe for useful comments on oblique corrections and Swarup
Kumar Majee for discussions in the early stages of this work. K.N. is grateful
for valuable discussions with Joydeep Chakrabortty and Daisuke Harada and for
fruitful conversations with Anindya Datta and Sreerup Raychaudhuri. K.N. is
partially supported by funding available from the Department of Atomic Energy,
Government of India for the Regional Centre for Accelerator-based Particle
Physics (RECAPP), Harish-Chandra Research Institute. The work of K.O. (R.W.)
is in part supported by the Grands-in-Aid for Scientific Research No.
23104009, No. 20244028, and No. 23740192 (No. 248920).

## Appendix

##  Appendix A RGEs in 6D UED models and mUED

In this appendix, we show the concrete forms of RGEs for gauge, Yukawa, and
Higgs self-couplings. Here, we rewrite the schematic shape of the beta
function for a quantity $\mathcal{Q}$ in 6D UED:

$\displaystyle\beta_{\mathcal{Q}}=\beta_{\mathcal{Q}}^{\text{(SM)}}+\sum_{s:\,\text{KK
states}}\theta(\mu-
M_{s})\Big{(}N_{s}\beta^{\text{(KK)}}_{s,\mathcal{Q}}\Big{)}.$ (44)

Details of this expression are found in Sec. 2.1. As we have already discussed
there, the beta functions take different forms depending on the following two
categories: UEDs on an orientable space and those on an unorientable one. The
former contains $T^{2}/Z_{2}$, $T^{2}/(Z_{2}\times Z_{2}^{\prime})$,
$T^{2}/Z_{4}$, $S^{2}$, and $S^{2}/Z_{2}$ and the latter the remains $RP^{2}$,
PS. The RGEs obtained in this work are consistent with those obtained for mUED
[98] and for the SM [120]. The contribution of the KK particles to the beta
function $\beta^{\text{(KK)}}_{s,\mathcal{Q}}$ is independent of the KK index
and we can omit the index $s$ as $\beta^{\text{(KK)}}_{\mathcal{Q}}$. We
already explained the reason in Sec. 2.1. We note that in all the RGE analyses
in this paper, we ignore Yukawa couplings except for the top quark one.

###  A.1 UEDs in orientable space

In the following, $\lambda$ is 4D Higgs self-couplings; $g_{1}$, $g_{2}$, and
$g_{3}$ show the 4D $U(1)_{Y}$, $SU(2)_{W}$, and $SU(3)_{C}$ gauge couplings;
$y_{\ell_{k}},y_{u_{k}},y_{d_{k}}\,(k=1,2,3)$ represent the 4D (diagonalized)
Yukawa couplings of the charged leptons, the up-type quarks, and the down-type
quarks, respectively. Here, we adopt the SM normalization in the $U(1)_{Y}$
gauge coupling $g_{1}$. The index $k$ indicates their generations. $V_{ij}$
means the Cabibbo–Kobayashi–Maskawa matrix, and $N_{C_{f_{i}}}$ indicates the
color factor of the particle $f_{i}$, namely, $3$ for quarks and $1$ for
leptons.

$\circ\ \mathcal{Q}=\lambda$

$\displaystyle\beta^{(\text{SM})}_{\lambda}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}6\lambda^{2}-(3g_{1}^{2}+9g_{2}^{2})\lambda+\frac{3}{2}(g_{1}^{4}+2g_{1}^{2}g_{2}^{2}+3g_{2}^{4})+4\lambda\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{2}-8\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{4}\bigg{\\}},$
(45) $\displaystyle\beta^{(\text{KK})}_{\lambda}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}6\lambda^{2}-(3g_{1}^{2}+9g_{2}^{2})\lambda+\left(\frac{5}{2}g_{1}^{4}+5g_{1}^{2}g_{2}^{2}+\frac{15}{2}g_{2}^{4}\right)+8\lambda\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{2}-16\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{4}\Bigg{\\}}.$
(46)

$\circ\ \mathcal{Q}=g_{i}\,(i=1,2,3)$

$\displaystyle\beta^{\text{(SM)}}_{g_{i}}=\frac{1}{(4\pi)^{2}}b^{\text{(SM)}}_{g_{i}}g_{i}^{3},\quad\beta^{\text{(KK)}}_{g_{i}}=\frac{1}{(4\pi)^{2}}b_{g_{i}}^{\text{(KK)}}g_{i}^{3},$
(47)

with $b^{\text{(SM)}}_{g_{i}}=(\frac{41}{6},-\frac{19}{6},-7)$ and
$b^{\text{(KK)}}_{g_{i}}=(\frac{27}{2},\frac{3}{2},-2)$ for
$g_{i}=(g_{1},g_{2},g_{3})$, respectively.

$\circ\ \mathcal{Q}=y_{\ell_{k}},y_{u_{k}},y_{d_{k}}\,(k=1,2,3)$

$\displaystyle\beta^{\text{(SM)}}_{y_{\ell_{k}}}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}-\frac{15}{4}g_{1}^{2}-\frac{9}{4}g_{2}^{2}+\frac{3}{2}{y_{\ell_{k}}^{2}}+\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{2}\Bigg{\\}}{y_{\ell_{k}}},$
(48) $\displaystyle\beta^{\text{(SM)}}_{y_{u_{k}}}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}-\frac{17}{12}g_{1}^{2}-\frac{9}{4}g_{2}^{2}-8g_{s}^{2}+\frac{3}{2}{y_{u_{k}}^{2}}+\sum_{j}y_{d_{j}}^{2}{(V_{kj}V^{\dagger}_{jk})}+\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{2}\Bigg{\\}}y_{u_{k}},$
(49) $\displaystyle\beta^{\text{(SM)}}_{y_{d_{k}}}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}-\frac{15}{12}g_{1}^{2}-\frac{9}{4}g_{2}^{2}-8g_{s}^{2}+\frac{3}{2}{y_{d_{k}}^{2}}+\sum_{j}y_{u_{j}}^{2}{(V^{\dagger}_{kj}V_{jk})}+\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{2}\Bigg{\\}}y_{d_{k}},$
(50) $\displaystyle\beta_{y_{\ell_{k}}}^{(\text{KK})}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}-\frac{9}{2}g_{1}^{2}-\frac{3}{2}g_{2}^{2}+\frac{3}{2}{y_{\ell_{k}}^{2}}+2\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{2}\Bigg{\\}}y_{\ell_{k}},$
(51) $\displaystyle\beta_{y_{u_{k}}}^{(\text{KK})}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}-\frac{25}{18}g_{1}^{2}-\frac{3}{2}g_{2}^{2}-\frac{32}{3}g_{s}^{2}+\frac{3}{2}{y_{u_{k}}^{2}}+2\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{2}-\frac{3}{2}{\sum_{j}(V_{kj}V_{jk}^{\dagger})}y_{d_{j}}^{2}\Bigg{\\}}y_{u_{k}},$
(52) $\displaystyle\beta_{y_{d_{k}}}^{(\text{KK})}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}-\frac{1}{18}g_{1}^{2}-\frac{3}{2}g_{2}^{2}-\frac{32}{3}g_{s}^{2}+\frac{3}{2}{y_{d_{k}}^{2}}+2\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{2}-\frac{3}{2}{\sum_{j}(V_{kj}^{\dagger}V_{jk})}y_{u_{j}}^{2}\Bigg{\\}}y_{d_{k}}.$
(53)

###  A.2 PS case

In the case of PS, the contributions of the bosonic KK particles to the beta
functions is classified into two categories as
$\beta^{\text{(KK)}}_{\text{even},\mathcal{Q}}$ and
$\beta^{\text{(KK)}}_{\text{odd},\mathcal{Q}}$.

$\circ\ \mathcal{Q}=\lambda$

$\displaystyle\beta_{\text{even},\lambda}^{(\text{KK})}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}6\lambda^{2}-3\lambda
g_{1}^{2}-9\lambda
g_{2}^{2}+2g_{1}^{4}+4g_{1}^{2}g_{2}^{2}+6g_{2}^{4}+8\sum_{i}\lambda
N_{C_{f_{i}}}y_{f_{i}}^{2}-16\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{4}\Bigg{\\}},$
(54) $\displaystyle\beta_{\text{odd},\lambda}^{(\text{KK})}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}+\frac{1}{2}g_{1}^{4}+g_{1}^{2}g_{2}^{2}+\frac{3}{2}g_{2}^{4}+8\sum_{i}\lambda
N_{C_{f_{i}}}y_{f_{i}}^{2}-16\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{4}\Bigg{\\}}.$
(55)

$\circ\ \mathcal{Q}=g_{i}\,(i=1,2,3)$

$\displaystyle\beta^{\text{(KK)}}_{\text{even},g_{i}}=\frac{1}{(4\pi)^{2}}b^{\text{(KK)}}_{\text{even},g_{i}}g_{i}^{3},\quad\beta^{\text{(KK)}}_{\text{odd},g_{i}}=\frac{1}{(4\pi)^{2}}b_{\text{odd},g_{i}}^{\text{(KK)}}g_{i}^{3},$
(56)

with
$b^{\text{(KK)}}_{\text{even},g_{i}}=(\frac{27}{2},\frac{7}{6},-\frac{5}{2})$
and
$b^{\text{(KK)}}_{\text{odd},g_{i}}=(\frac{40}{3},\frac{25}{3},\frac{17}{2})$
for $g_{i}=(g_{1},g_{2},g_{3})$, respectively.

$\circ\ \mathcal{Q}=y_{\ell_{k}},y_{u_{k}},y_{d_{k}}\,(k=1,2,3)$

$\displaystyle\beta_{\text{even},y_{\ell_{k}}}^{(\text{KK})}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}-\frac{33}{8}g_{1}^{2}-\frac{15}{8}g_{2}^{2}+\frac{3}{2}{y_{\ell_{k}}^{2}}+2\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{2}\Bigg{\\}}y_{\ell_{k}},$
(57) $\displaystyle\beta_{\text{even},y_{u_{k}}}^{(\text{KK})}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}-\frac{101}{72}g_{1}^{2}-\frac{15}{8}g_{2}^{2}-\frac{28}{3}g_{s}^{2}+\frac{3}{2}{y_{u_{k}}^{2}}+2\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{2}-\frac{3}{2}{\sum_{j}(V_{kj}V_{jk}^{\dagger})}y_{d_{j}}^{2}\Bigg{\\}}y_{u_{k}},$
(58) $\displaystyle\beta_{\text{even},y_{d_{k}}}^{(\text{KK})}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}-\frac{17}{72}g_{1}^{2}-\frac{15}{8}g_{2}^{2}-\frac{28}{3}g_{s}^{2}+\frac{3}{2}{y_{d_{k}}^{2}}+2\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{2}-\frac{3}{2}{\sum_{{j}}(V_{kj}^{\dagger}V_{jk})}y_{u_{j}}^{2}\Bigg{\\}}y_{d_{k}},$
(59) $\displaystyle\beta_{\text{odd},y_{\ell_{k}}}^{(\text{KK})}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}-\frac{3}{8}g_{1}^{2}+\frac{3}{8}g_{2}^{2}+2\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{2}\Bigg{\\}}y_{\ell_{k}},$
(60) $\displaystyle\beta_{\text{odd},y_{u_{k}}}^{(\text{KK})}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}+\frac{1}{72}g_{1}^{2}+\frac{3}{8}g_{2}^{2}-\frac{4}{3}g_{s}^{2}+2\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{2}\Bigg{\\}}y_{u_{k}},$
(61) $\displaystyle\beta_{\text{odd},y_{d_{k}}}^{(\text{KK})}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}+\frac{13}{72}g_{1}^{2}+\frac{3}{8}g_{2}^{2}-\frac{4}{3}g_{s}^{2}+2\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{2}\Bigg{\\}}y_{d_{k}}.$
(62)

###  A.3 $RP^{2}$ case

In the regions having bosonic modes (regions I, II, and III), the following
relations are fulfilled for each type of coupling $C$:

$\displaystyle\beta_{\text{region
I},C}^{(\text{KK})}=\beta_{\text{even},C}^{(\text{KK})},\quad\beta_{\text{region
II},C}^{(\text{KK})}=\beta_{\text{odd},C}^{(\text{KK})},\quad\beta_{\text{region
III},C}^{(\text{KK})}=\beta_{C}^{(\text{KK})}.$ (63)

We write down the formula for the region without having a bosonic mode (region
IV).

$\circ\ \mathcal{Q}=\lambda$

$\displaystyle\beta_{\text{region IV},\lambda}^{(\text{KK})}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}+8\sum_{i}\lambda
N_{C_{f_{i}}}y_{f_{i}}^{2}-16\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{4}\Bigg{\\}}.$
(64)

$\circ\ \mathcal{Q}=g_{i}\,(i=1,2,3)$

$\displaystyle\beta^{\text{(KK)}}_{\text{region IV},g_{i}}$
$\displaystyle=\frac{1}{(4\pi)^{2}}b^{\text{(KK)}}_{\text{region
IV},g_{i}}g_{i}^{3},$ (65)

with $b^{\text{(KK)}}_{\text{region IV},g_{i}}=(\frac{40}{3},8,8)$ for
$g_{i}=(g_{1},g_{2},g_{3})$, respectively.

$\circ\ \mathcal{Q}=y_{\ell_{k}},y_{u_{k}},y_{d_{k}}\,(k=1,2,3)$

$\displaystyle\beta_{\text{region IV},y_{\ell_{k}}}^{(\text{KK})}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}+2\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{2}\Bigg{\\}}y_{\ell_{k}},$
(66) $\displaystyle\beta_{\text{region IV},y_{u_{k}}}^{(\text{KK})}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}+2\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{2}\Bigg{\\}}y_{u_{k}},$
(67) $\displaystyle\beta_{\text{region IV},y_{d_{k}}}^{(\text{KK})}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Bigg{\\{}+2\sum_{i}N_{C_{f_{i}}}y_{f_{i}}^{2}\Bigg{\\}}y_{d_{k}}.$
(68)

###  A.4 mUED case

The surviving modes for each KK level in the mUED are totally the same as in
region I of the $RP^{2}$ or in the “even” region of the PS. Hence, we can use
those forms for RGEs in the mUED. We can check that our results of this part
are consistent with those in Refs. [98, 99].

##  Appendix B Loop functions in single Higgs production and decay

In this appendix, we summarize the loop functions that are needed for
estimating the single Higgs production through the gluon fusion process and
the Higgs decay into a pair of photons. Readers who want more explanations on
the above expressions should consult Ref. [56].

For each model, the loop function $J_{t}^{\text{model}}$ describes the
contributions of all the zero and KK modes for the top quark in the triangle
loops:

$\displaystyle J_{t}^{\text{SM}}(\hat{s})$
$\displaystyle=I\\!\left(m_{t}^{2}\over\hat{s}\right),$ (69) $\displaystyle
J_{t}^{\text{mUED}}(\hat{s})$
$\displaystyle=\left\\{I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{n\geq
1}\left({m_{t}\over
m_{t(n)}}\right)^{2}I\\!\left({m_{t(n)}^{2}\over\hat{s}}\right)\right\\},$
(70) $\displaystyle{J_{t}^{T^{2}/Z_{2}}(\hat{s})=J_{t}^{{RP}^{2}}(\hat{s})}$
$\displaystyle=\left\\{I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{{m+n\geq
1\atop\text{or\ }m=-n\geq 1}}\left({m_{t}\over
m_{t(m,n)}}\right)^{2}I\\!\left(m_{t(m,n)}^{2}\over\hat{s}\right)\right\\},$
(71) $\displaystyle{J_{t}^{T^{2}/Z_{4}}(\hat{s})}$
$\displaystyle=\left\\{I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{m\geq
1,n\geq 0}\left({m_{t}\over
m_{t(m,n)}}\right)^{2}I\\!\left(m_{t({m,n})}^{2}\over\hat{s}\right)\right\\},$
(72) $\displaystyle{J_{t}^{T^{2}/(Z_{2}\times Z^{\prime}_{2})}(\hat{s})}$
$\displaystyle=\left\\{I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{m\geq
0,n\geq 0,\atop(m,n)\not=(0,0)}\left({m_{t}\over
m_{t(m,n)}}\right)^{2}I\left(m_{t(m,n)}^{2}\over\hat{s}\right)\right\\},$ (73)
$\displaystyle J_{t}^{S^{2}/Z_{2}}(\hat{s})$
$\displaystyle=\left\\{I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{j\geq
1}\left({m_{t}\over
m_{t(j)}}\right)^{2}n^{S^{2}/Z_{2}}(j)\,I\left(m_{t(j)}^{2}\over\hat{s}\right)\right\\},$
(74) $\displaystyle J_{t}^{\text{PS}}(\hat{s})={J_{t}^{S^{2}}(\hat{s})}$
$\displaystyle=\left\\{I\\!\left(m_{t}^{2}\over\hat{s}\right)+2\sum_{j\geq
1}\left({m_{t}\over
m_{t(j)}}\right)^{2}(2j+1)\,I\\!\left(m_{t(j)}^{2}\over\hat{s}\right)\right\\},$
(75)

where $I$ is given by

$\displaystyle I(\lambda)$
$\displaystyle={-2\lambda+\lambda(1-4\lambda)}\int_{0}^{1}{dx\over
x}\ln\left[{x(x-1)\over\lambda}+1-i\epsilon\right].$ (76)

The explicit result of the integral is

$\displaystyle\int_{0}^{1}{dx\over
x}\ln\left[{x(x-1)\over\lambda}+1-i\epsilon\right]$
$\displaystyle=\begin{cases}\displaystyle-2\left[\arcsin{1\over\sqrt{4\lambda}}\right]^{2}&\text{(for
$\lambda\geq{1\over 4}$)},\\\ \displaystyle{1\over
2}\left[\ln{1+\sqrt{1-4\lambda}\over
1-\sqrt{1-4\lambda}}-i\pi\right]^{2}&\text{(for $\lambda<{1\over
4}$)},\end{cases}$ (77)

where this form is related with the Passarino–Veltman’s three-point scalar
function $C_{0}$ [121]. $n^{\text{model}}(j)$ counts the number of degeneracy,
and the explicit forms are shown in Eqs. (10)–(12) and we write the KK top and
$W$ masses ($X=t,W$)

$\displaystyle m_{X(n)}$
$\displaystyle\equiv\sqrt{m_{X}^{2}+{\frac{n^{2}}{R^{2}}}},$ (78)
$\displaystyle m_{X(m,n)}$
$\displaystyle\equiv{\sqrt{m_{X}^{2}+\frac{m^{2}}{R_{5}^{2}}+\frac{n^{2}}{R_{6}^{2}}}},$
(79) $\displaystyle m_{X(j)}$
$\displaystyle\equiv\sqrt{m_{X}^{2}+{\frac{j(j+1)}{R^{2}}}}.$ (80)

The range of the KK summation reflects the structure of each extra-dimensional
background.121212 The origin of the factor 2 in front of each KK summation is
the fact that there are both left- and right-handed (namely, vectorlike) KK
modes for each chiral quark zero mode corresponding to a SM quark. The loop
functions which are needed for the process $H\rightarrow\gamma\gamma$ are as
follows:

$\displaystyle J_{W}^{\text{SM}}(m_{H}^{2})$ $\displaystyle=L\\!\left({1\over
2},3,3,6,0;{m_{W}^{2}\over m_{H}^{2}},{m_{W}^{2}\over m_{H}^{2}}\right),$ (81)
$\displaystyle J_{W}^{\text{mUED}}(m_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(m_{H}^{2})+\sum_{n\geq 1}L\\!\left({1\over
2},4,4,8,1;{m_{W}^{2}\over m_{H}^{2}},{{m_{W(n)}^{2}}\over m_{H}^{2}}\right),$
(82) $\displaystyle J_{W}^{T^{2}/Z_{4}}(m_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(m_{H}^{2})+\sum_{m\geq 1,n\geq
0}L\\!\left({1\over 2},5,4,10,1;{m_{W}^{2}\over
m_{H}^{2}},{{m_{W(m,n)}^{2}}\over m_{H}^{2}}\right),$ (83) $\displaystyle
J_{W}^{T^{2}/(Z_{2}\times Z^{\prime}_{2})}(m_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(m_{H}^{2})+\sum_{m\geq 0,n\geq
0\atop(m,n)\neq(0,0)}L\\!\left({1\over 2},5,4,10,1;{m_{W}^{2}\over
m_{H}^{2}},{{m_{W(m,n)}^{2}}\over m_{H}^{2}}\right),$ (84) $\displaystyle
J_{W}^{T^{2}/Z_{2}}(m_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(m_{H}^{2})+\sum_{{m+n\geq 1\atop\text{or\
}m=-n\geq 1}}L\\!\left({1\over 2},5,4,10,1;{m_{W}^{2}\over
m_{H}^{2}},{{m_{W(m,n)}^{2}}\over m_{H}^{2}}\right),$ (85) $\displaystyle
J_{W}^{{RP}^{2}}(m_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(m_{H}^{2})+\sum_{(m,n)}^{A}L\\!\left({1\over
2},4,4,8,1;{m_{W}^{2}\over m_{H}^{2}},{{m_{W(m,n)}^{2}}\over
m_{H}^{2}}\right)+\sum_{(m,n)}^{B}L\\!\left(0,1,0,2,0;{m_{W}^{2}\over
m_{H}^{2}},{{m_{W(m,n)}^{2}}\over m_{H}^{2}}\right),$ (86) $\displaystyle
J_{W}^{S^{2}/Z_{2}}(m_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(m_{H}^{2})+\sum_{j\geq
1}n^{S^{2}/Z_{2}}(j)\,L\\!\left({1\over 2},5,4,10,1;{m_{W}^{2}\over
m_{H}^{2}},{{m_{W(j)}^{2}}\over m_{H}^{2}}\right),$ (87) $\displaystyle
J_{W}^{S^{2}}(m_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(m_{H}^{2})+\sum_{j\geq
1}(2j+1)\,L\\!\left({1\over 2},5,4,10,1;{m_{W}^{2}\over
m_{H}^{2}},{{m_{W(j)}^{2}}\over m_{H}^{2}}\right),$ (88) $\displaystyle
J_{W}^{\text{PS}}(m_{H}^{2})$
$\displaystyle=J_{W}^{\text{SM}}(m_{H}^{2})+\sum_{j\geq
1}\Bigg{[}n_{\text{even}}^{\text{PS}}(j)\,L\\!\left({1\over
2},4,4,8,1;{m_{W}^{2}\over m_{H}^{2}},{{m_{W(j)}^{2}}\over m_{H}^{2}}\right)$
$\displaystyle\phantom{=J_{W}^{\text{SM}}(m_{H}^{2})+\sum_{j\geq
1}\Bigg{[}}+n_{\text{odd}}^{\text{PS}}(j)\,L\\!\left(0,1,0,2,0;{m_{W}^{2}\over
m_{H}^{2}},{{m_{W(j)}^{2}}\over m_{H}^{2}}\right)\Bigg{]},$ (89)

with

$\displaystyle L(a,b,c,d,e;\lambda_{1},\lambda_{2})$
$\displaystyle=a+b\lambda_{1}-\left[\lambda_{1}\\!\left(c-d\lambda_{2}\right)-e\lambda_{2}\right]\int_{0}^{1}{dx\over
x}\ln\left[{x(x-1)\over\lambda_{2}}+1-i\epsilon\right].$ (90)

The $A$ summation for $RP^{2}$ is over the region that satisfies both $m\geq
1$ and $n\geq 1$ as well as over the ranges $(m,n)=(0,2),(0,4),(0,6),\dots$
and $(m,n)=(2,0),(4,0),(6,0),\dots$. Similarly, the $B$ summation is over
$m\geq 1$ and $n\geq 1$ as well as over $(m,n)=(0,1),(0,3),(0,5),\dots$ and
$(m,n)=(1,0),(3,0),(5,0),\dots$.

##  Appendix C Two-point functions of gauge bosons in 6D UEDs

In this section, we summarize the two-point functions of photon, W, and Z
bosons for calculating Peskin–Takeuchi $S$ and $T$ parameters.

###  C.1 Notations

First, we summarize our notations for the Passarino–Veltman B function [121].
In this section, we use the following descriptions for masses. The mass
squared of the “$s$th” KK mode of the particle $X$ is represented as

$\displaystyle M_{X_{s}}^{2}=m_{X}^{2}+M_{s}^{2}$ (91)

where $m_{X}$ is the corresponding zero-mode mass, and $M_{s}$ is the $s$th
level KK mass. Since only $Z$, $W$, $H$, and top masses are not negligible
compared with the KK scale $M_{s}$, we use the representations

$\displaystyle M_{W_{s}}^{2}$ $\displaystyle{:=}m_{W}^{2}+M_{s}^{2},\hskip
10.00002ptM_{Z_{s}}^{2}{:=}m_{Z}^{2}+M_{s}^{2}$ $\displaystyle M_{t_{s}}^{2}$
$\displaystyle{:=}m_{t}^{2}+M_{s}^{2},\hskip
10.00002ptM_{H_{s}}^{2}{:=}m_{H}^{2}+M_{s}^{2},$ (92)

and for the other fields,

$\displaystyle M_{X_{s}}^{2}{\simeq}M_{s}^{2}.$ (93)

We will use the Passarino–Veltman loop integral to calculate two-point
functions of the gauge bosons with external momentum $k$ below, and the
definition is

$\displaystyle\frac{1}{(4\pi)^{2}}B_{X_{s},Y_{s}}(k^{2})$
$\displaystyle=\int\frac{d^{d}p}{(2\pi)^{d}}\frac{1}{(p^{2}-M_{X_{s}}^{2})((p+k)^{2}-M_{Y_{s}}^{2})}$
$\displaystyle=\frac{i}{(4\pi)^{2}}\left\\{\frac{1}{\overline{\varepsilon}}-\int^{1}_{0}dx\ln\left[(1-x)M_{X_{s}}^{2}+xM_{Y_{s}}^{2}-x(1-x)k^{2}-i\epsilon\right]\right\\},$
(94)

where we use the dimensional regularization in $d$ dimensions and $\epsilon$
is an infinitesimal positive value. $1/\overline{\varepsilon}\
(:=1/\varepsilon-\gamma+\ln{4\pi})$ means the usual common divergent part with
$\varepsilon=2-d/2$ and the Euler–Mascheroni constant $\gamma$. The following
short-hand description is also used later for simplicity:

$\displaystyle B_{X_{s}}(k^{2}):=B_{X_{s},X_{s}}(k^{2}),\quad\delta
B_{X_{s},Y_{s}}(k^{2}):={B_{X_{s},Y_{s}}(k^{2})-B_{X_{s},Y_{s}}(0)\over
k^{2}}.$ (95)

Here, we write down some useful formulas for calculations:

$\displaystyle B_{X_{s},Y_{s}}(0)$
$\displaystyle\simeq\frac{1}{\overline{\varepsilon}}-\frac{1}{2}\frac{m_{X}^{2}+m_{Y}^{2}}{M_{s}^{2}},$
(96) $\displaystyle B^{\prime}_{X_{s},Y_{s}}(0)$
$\displaystyle\simeq\frac{1}{6M_{s}^{2}},$ (97) $\displaystyle
B^{\prime\prime}_{X_{s},Y_{s}}(0)$
$\displaystyle\simeq\frac{2}{3}\frac{1}{(m_{X}^{2}-m_{Y}^{2})^{2}}\frac{m_{X}^{2}+m_{Y}^{2}}{M_{s}^{2}},$
(98)

where we assume the hierarchy $m_{X}^{2},\,m_{Y}^{2}\ll M_{s}^{2}$ and values
with a prime mean that it is differentiated with respect to $k^{2}$ once.

###  C.2 Bosonic contributions to the two-point function of gauge bosons in 6D
UEDs abd mUED

In this section, we make a summary of bosonic two-point contributions to two-
point function of gauge bosons in the 6D UEDs and the mUED for evaluating $S$
and $T$ parameters. For contributions of fermions, we can use the result in
Ref. [89].

The generic form of a gauge boson two-point function is as follows:

$\displaystyle\Pi^{\mu\nu}_{ab}(k^{2})=i\Pi_{ab}^{\text{T}}(k^{2})\left(g^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{k^{2}}\right)+i\Pi^{\text{L}}_{ab}(k^{2})\frac{k^{\mu}k^{\nu}}{k^{2}},$
(99)

where $a$ and $b$ show the type of gauge bosons, and the superscript T (L)
indicates the transverse (longitudinal), respectively.

For estimating the $S$ and $T$ parameters, we calculate only the transverse
ones. In each following subsection, we show the contributions of KK bosonic
particles to the two-point functions from the level-$s$ KK states.

####  C.2.1 UEDs on oriented geometry case

$\displaystyle\Pi_{\gamma\gamma}^{\text{T},s}(k^{2})$
$\displaystyle={\alpha\over 4\pi}\Bigg{\\{}-{4\over
9}k^{2}+\left(\frac{7}{3}k^{2}+\frac{20}{3}M_{W_{s}}^{2}\right)B_{W_{s}}(k^{2})-\frac{20}{3}M_{W_{s}}^{2}B_{{W_{s}}}(0)\Bigg{\\}},$
(100) $\displaystyle\Pi_{Z\gamma}^{\text{T},s}(k^{2})$
$\displaystyle={\alpha\over 4\pi
s_{W}c_{W}}\Bigg{\\{}\left(-\frac{1}{9}+\frac{4}{9}c_{W}^{2}\right)k^{2}+\left(\frac{20}{3}c_{W}^{2}-\frac{2}{3}\right)M_{W_{s}}^{2}B_{W_{s}}(0)$
$\displaystyle\quad+\left[\left(-\frac{1}{6}-\frac{7}{3}c_{W}^{2}\right)k^{2}-\left(\frac{20}{3}c_{W}^{2}-\frac{2}{3}\right)M_{W_{s}}^{2}-2m_{W}^{2}\right]B_{W_{s}}(k^{2})\Bigg{\\}},$
(101) $\displaystyle\Pi_{ZZ}^{\text{T},s}(k^{2})$ $\displaystyle={\alpha\over
4\pi
s_{W}^{2}c_{W}^{2}}\Bigg{\\{}\left(\frac{2}{9}c_{W}^{2}-\frac{4}{9}c_{W}^{4}-{\frac{1}{18}}\right)k^{2}$
$\displaystyle\quad+\left(-\frac{20}{3}c_{W}^{4}+\frac{4}{3}c_{W}^{2}-\frac{1}{3}\right)M_{W_{s}}^{2}B_{W_{s}}(0)-\frac{1}{6}M_{H_{s}}^{2}B_{H_{s}}(0)-\frac{1}{6}M_{Z_{s}}^{2}B_{Z_{s}}(0)$
$\displaystyle\quad+\left[\left(\frac{7}{3}c_{W}^{4}+\frac{1}{3}c_{W}^{2}-\frac{1}{12}\right)k^{2}+\left(\frac{20}{3}c_{W}^{4}-\frac{4}{3}c_{W}^{2}+\frac{1}{3}\right)M_{W_{s}}^{2}-\left(2-4c_{W}^{2}\right)m_{W}^{2}\right]B_{{W_{s}}}(k^{2})$
$\displaystyle\quad+\left(-\frac{1}{12}k^{2}+\frac{1}{6}M_{Z_{s}}^{2}+\frac{1}{6}M_{H_{s}}^{2}-\frac{m_{Z}^{2}}{c_{W}^{2}}\right)B_{H_{s},Z_{s}}(k^{2})-\frac{1}{12}(M_{H_{s}}^{2}-M_{{Z_{s}}}^{2})^{2}\delta
B_{H_{s},Z_{s}}(k^{2})\Bigg{\\}},$ (102)
$\displaystyle\Pi_{WW}^{\text{T},s}(k^{2})$ $\displaystyle={\alpha\over 4\pi
s_{W}^{2}}\Bigg{\\{}-\frac{1}{3}k^{2}-\frac{1}{6}M_{H_{s}}^{2}B_{H_{s}}(0)-3M_{W_{s}}^{2}B_{W_{s}}(0)-\frac{17}{6}M_{Z_{s}}^{2}B_{Z_{s}}(0)$
$\displaystyle\quad+\left[-\frac{1}{12}k^{2}+\frac{1}{6}M_{H_{s}}^{2}+\frac{1}{6}M_{W_{s}}^{2}-m_{W}^{2}\right]B_{H_{s},W_{s}}(k^{2})$
$\displaystyle\quad+\left[\frac{31}{12}k^{2}+\frac{23}{6}\left(M_{W_{s}}^{2}+M_{Z_{s}}^{2}\right)-2M_{s}^{2}-m_{Z}^{2}\left(c_{W}^{2}-2+c_{W}^{-2}\right)\right]B_{W_{s},Z_{s}}(k^{2})$
$\displaystyle\quad-\frac{1}{12}(M_{H_{s}}^{2}-M_{W_{s}}^{2})^{2}\delta
B_{H_{s},W_{s}}(k^{2})-\frac{17}{12}(M_{Z_{s}}^{2}-M_{W_{s}}^{2})^{2}\delta
B_{Z_{s},W_{s}}(k^{2})\Bigg{\\}},$ (103)

####  C.2.2 $RP^{2}$, $PS$, mUED cases

As we have discussed in Sec. 2, the particle contents of the region III of the
$RP^{2}$ model is completely the same as those of the 6D UEDs on oriented
geometries just as above, and we need not discuss them. Based on the knowledge
in Sec. 2, the remaining boson contributions are written down as follows:

$\displaystyle\Pi_{\gamma\gamma}^{\text{T},s}|_{\text{{region
II}}}(k^{2})=\Pi_{\gamma\gamma}^{\text{T},s}|_{\text{odd}}(k^{2})$
$\displaystyle={\alpha\over 4\pi}\Bigg{\\{}-{2\over
9}k^{2}+\left(-\frac{1}{3}k^{2}+\frac{4}{3}M_{W_{s}}^{2}\right)B_{W_{s}}(k^{2})-\frac{4}{3}M_{W_{s}}^{2}B_{{W_{s}}}(0)\Bigg{\\}},$
(104) $\displaystyle\Pi_{Z\gamma}^{\text{T},s}|_{\text{{region
II}}}(k^{2})=\Pi_{Z\gamma}^{\text{T},s}|_{\text{odd}}(k^{2})$
$\displaystyle={\alpha\over
4\pi}\left[-\frac{c_{W}}{s_{W}}\right]\Bigg{\\{}-{2\over
9}k^{2}+\left(-\frac{1}{3}k^{2}+\frac{4}{3}M_{W_{s}}^{2}\right)B_{W_{s}}(k^{2})-\frac{4}{3}M_{W_{s}}^{2}B_{{W_{s}}}(0)\Bigg{\\}},$
(105) $\displaystyle\Pi_{ZZ}^{\text{T},s}|_{\text{{region
II}}}(k^{2})=\Pi_{ZZ}^{\text{T},s}|_{\text{odd}}(k^{2})$
$\displaystyle={\alpha\over
4\pi}\left[\frac{c_{W}^{2}}{s_{W}^{2}}\right]\Bigg{\\{}-{2\over
9}k^{2}+\left(-\frac{1}{3}k^{2}+\frac{4}{3}M_{W_{s}}^{2}\right)B_{W_{s}}(k^{2})-\frac{4}{3}M_{W_{s}}^{2}B_{{W_{s}}}(0)\Bigg{\\}},$
(106) $\displaystyle\Pi_{WW}^{\text{T},s}|_{\text{{region
II}}}(k^{2})=\Pi_{WW}^{\text{T},s}|_{\text{odd}}(k^{2})$
$\displaystyle={\alpha\over 4\pi
s_{W}^{2}}\Bigg{\\{}-\frac{2}{9}k^{2}-\frac{2}{3}M_{W_{s}}^{2}B_{W_{s}}(0)-\frac{2}{3}M_{Z_{s}}^{2}B_{Z_{s}}(0)$
$\displaystyle\quad+\left[-\frac{1}{3}k^{2}+\frac{2}{3}\left(M_{W_{s}}^{2}+M_{Z_{s}}^{2}\right)\right]B_{W_{s},Z_{s}}(k^{2})$
$\displaystyle\quad-\frac{1}{3}(M_{Z_{s}}^{2}-M_{W_{s}}^{2})^{2}\delta
B_{Z_{s},W_{s}}(k^{2})\Bigg{\\}}.$ (107)

The remaining part can be easily calculated by use of the following relations:

$\displaystyle\Pi_{ab}^{\text{T},s}|_{\text{{region I}}}(k^{2})$
$\displaystyle=\Pi_{ab}^{\text{T},s}(k^{2})-\Pi_{ab}^{\text{T},s}|_{\text{{region
II}}}(k^{2}),$ (108)
$\displaystyle\Pi_{ab}^{\text{T},s}|_{\text{even}}(k^{2})$
$\displaystyle=\Pi_{ab}^{\text{T},s}(k^{2})-\Pi_{ab}^{\text{T},s}|_{\text{odd}}(k^{2}),$
(109)

where $ab$ represents the possible four combinations of gauge bosons.

We also derive the following relations for the mUED:

$\displaystyle\Pi_{ab}^{\text{T},s}|_{\text{mUED}}(k^{2})\simeq\Pi_{ab}^{\text{T},s}|_{\text{even}}(k^{2})$
(110)

based on the discussions in Secs. 2 and 4.

##  Appendix D Summary of the bounds

Here, we summarize the bounds on the KK scale in UED models. Figure 10 shows
the exclusion C.L.s as functions of the wide range of the KK scale.

Figure 10:  The exclusion C.L.s of all the UED models as functions of the KK
scale $M_{\text{KK}}$ obtained from the experimental results of the Higgs
searches at the LHC (ATLAS, CMS, and both of them, respectively) and those of
$S,T$ parameters. Colors denote the same as in Fig. 2.

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