27th August 2024 OU-HET-1107

KYUSHU-HET-229

Signals of $W^{\prime}$ and $Z^{\prime}$ bosons at the LHC
in the $SU(3)\times SO(5)\times U(1)$ gauge-Higgs unification

Shuichiro Funatsu1, Hisaki Hatanaka2, Yutaka Hosotani3,

Yuta Orikasa4 and Naoki Yamatsu5

1Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics
(MOE), Central China Normal University, Wuhan, Hubei 430079, China

2Osaka, Osaka 536-0014, Japan

3Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan

4Institute of Experimental and Applied Physics, Czech Technical University in
Prague,

Husova 240/5, 110 00 Prague 1, Czech Republic

5Department of Physics, Kyushu University, Fukuoka 819-0395, Japan

###### Abstract

The $pp\to\\{W,W^{\prime}\\}\to l\nu$ and $pp\to\\{\gamma,Z,Z^{\prime}\\}\to
l^{+}l^{-}$ ($l=e,\mu$) processes in the $SU(3)_{C}\times SO(5)\times
U(1)_{X}$ gauge-Higgs unification (GHU) models are studied, where $W^{\prime}$
and $Z^{\prime}$ bosons are Kaluza-Klein (KK) exited states of the electroweak
gauge bosons. From the experimental data collected at the Large Hadron
Collider, constraints on the KK mass scale and the Aharonov-Bohm phase are
obtained. One can explore the KK mass scale in the GUT inspired GHU model up
to 18 TeV for the luminosity 300 fb-1 and 22 TeV for the luminosity 3000 fb-1
at $\sqrt{s}=14$ TeV.

##  1 Introduction

Experimental results at the Large Hadron Collider (LHC) collected at a center-
of-mass energy $\sqrt{s}=13$ TeV during the year 2015-2018 have been presented
by ATLAS [1, 2, 3, 4] and CMS [5, 6] groups. No direct signals of new physics
beyond the standard model (SM) have been observed at the LHC so far. In many
models such as the sequential standard model, left-right symmetric model,
grand unified theories (GUT) and models with an extra dimension, there appear
$W^{\prime}$ or $Z^{\prime}$ bosons [7, 8, 9, 10]. Physics of $W^{\prime}$ and
$Z^{\prime}$ bosons at the LHC has been an important subject [11, 12, 28, 18,
13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30].

The gauge-Higgs unification (GHU) is one of the approaches to the gauge
hierarchy problem [31, 32, 33, 34, 35, 36, 37, 39, 38, 40, 41]. GHU models are
constructed in higher dimensional spacetime and the Higgs boson is identified
as a part of an extra-dimensional component of gauge bosons. Hence physics of
the Higgs boson is governed by the gauge principle and the Higgs boson mass is
generated by quantum corrections in GHU models. Many GHU models are proposed
for the electroweak unification [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52,
53, 54, 55] and GHU models for the grand unification are also presented [57,
58, 59, 60, 61, 62, 63, 64]. Among them two types of $SU(3)_{C}\times
SO(5)\times U(1)_{X}$ GHU models in the warped spacetime have been studied.
One of them is called the “A-model”, where quark-lepton multiplets are
introduced in the vector representation of $SO(5)$ [42, 43, 44, 45, 46]. The
other is called the “B-model”, where quark-lepton multiplets are introduced in
the spinor representation of $SO(5)$ [47, 48, 49, 50, 51] which has been
motivated from the study of the $SO(11)$ gauge-Higgs GUT [57, 58, 59, 60], and
is called the GUT inspired GHU model.

In $SU(3)_{C}\times SO(5)\times U(1)_{X}$ GHU models Kaluza-Klein (KK) exited
states of the electroweak gauge bosons appear as $W^{\prime}$ or $Z^{\prime}$
bosons. In our previous work, the constraint $\theta_{H}\lesssim 0.11$ and
$m_{\text{KK}}\gtrsim 8$ TeV has been derived in the A-model from the LHC data
at $\sqrt{s}=8$ TeV, where $\theta_{H}$ is the Aharonov-Bohm (AB) phase in the
fifth dimension and $m_{\text{KK}}$ is the KK mass scale [44]. The effect of
$Z^{\prime}$ bosons is also significant for future linear colliders with
polarized electron and positron beams. Because of the large asymmetries in the
$Z^{\prime}$ couplings to left- and right-handed fermions, some observables in
the GHU A-model deviate from those in the SM even at $\sqrt{s}=O(100)$ GeV
with the use of polarized beams [45, 46]. In the $e^{+}e^{-}\to\mu^{+}\mu^{-}$
process at the International Linear Collider (ILC) with $\sqrt{s}=250$ GeV
[65, 66, 67], the deviation of the forward-backward asymmetry from the SM
prediction becomes $-2\%$ with the right-handed electron beam for
$\theta_{H}\simeq 0.09$. The deviation can be seen with 250 fb-1 data. The
Higgs decay branching ratios are found to be nearly the same as those in the
SM [42], whereas the Higgs triple and quartic couplings deviate $9\%$ and
$37\%$ from those in the SM, respectively [43]. It is not easy to distinguish
GHU models from the SM from the Higgs data at the ILC.

The effects of $Z^{\prime}$ bosons in the GHU B-model at the ILC have been
studied in Ref. [50]. The deviation of the forward-backward asymmetry from the
SM prediction is about $-1\%$ in the $e^{+}e^{-}\to\mu^{+}\mu^{-}$ process at
the $250$ GeV ILC with polarized left-handed electron beams for
$\theta_{H}\simeq 0.10$, where the KK mass scale is 13 TeV. The deviation of
the differential left-right asymmetry reaches to about $-20\%$ in the forward
region with the same parameters. The A-model and B-model can be distinguished
by the dependence in the forward-backward asymmetry and left-right asymmetry
on the polarization of electron and positron beams, as the two models exhibit
opposite dependence on the polarization.

Another specific feature of the GHU B-model is the appearance of the two step
phase transitions at $T\sim 2.6$ TeV and $T=163$ GeV [51]. At sufficiently
high temperature, the effective potential has a minimum at $\theta_{H}=\pi$.
The two phases $\theta_{H}=0$ and $\theta_{H}=\pi$ become degenerate at $T\sim
m_{\text{KK}}$, where the two phases have $SU(2)_{L}\times U(1)_{Y}$ and
$SU(2)_{R}\times U(1)_{Y^{\prime}}$ symmetry, respectively. As the temperature
becomes lower, the $\theta_{H}=0$ state becomes the true vacuum and a first-
order phase transition from $\theta_{H}=\pi$ to $\theta_{H}=0$ takes place at
$T\sim 2.6$ TeV. This transition is called the left-right phase transition. At
$T=163$ GeV, the electroweak symmetry breaking (EWSB) occurs and the Higgs
boson acquires a vacuum expectation value (VEV). This electroweak phase
transition is found to be weakly first order.

In this paper, the $pp\to l\nu$ and $pp\to l^{+}l^{-}$ ($l=e,\mu$) processes
in the GHU A-model and B-model at the LHC are studied. Because significant
differences between observables in the SM and those in the GHU models at the
ILC are predicted [45, 46, 50], effects of $W^{\prime}$ and $Z^{\prime}$
bosons are expected to be significant at the LHC as well. Experimental results
for these processes at the LHC at $\sqrt{s}=13$ TeV with up to 140 fb-1 data
have been published [1, 2, 3, 5]. The constraint on the A-model is updated and
found that $\theta_{H}\lesssim 0.08$ and $m_{\text{KK}}\gtrsim 9.5$ TeV. The
constraint on the B-model is found that $\theta_{H}<0.10$ and
$m_{\text{KK}}>13$ TeV. At $\sqrt{s}=14$ TeV with the luminosity 300 fb-1, the
discovery significance of the $pp\to e^{+}e^{-}$ process in the A-model is
6.49 for $m_{\text{KK}}=9.5$ TeV, and the discovery significance of the $pp\to
e\nu$ process in the B-model can be maximally 5.08 for $m_{\text{KK}}=15$ TeV
and can be 1.64 for $m_{\text{KK}}\simeq 18$ TeV. At the future High
Luminosity LHC (HL-LHC) [68, 69, 70], the discovery significance of the $pp\to
e\nu$ process in the B-model can be maximally 1.61 for $m_{\text{KK}}\simeq
22$ TeV.

The paper is organized as follows. In Sec. 2, the outline of the GHU A-model
and B-model is introduced. In Sec. 3, definitions of differential cross
sections are given. In Sec. 4, we evaluate differential cross sections for
$pp\to l\nu$ and $pp\to l^{+}l^{-}$ processes and constrain the parameters
from the experimental results at the LHC Run 2. In Sec. 5, the predictions for
future LHC experiments are shown. Section 6 is devoted to summary and
discussions.

##  2 Model

The $SU(3)_{C}\times SO(5)\times U(1)_{X}$ GHU A-model and B-model have been
given in Ref. [43] and Ref. [47], in which the action, orbifold boundary
conditions (BCs), wave functions and formulae to determine the mass spectrum
of each field are explained. The details of the models are not repeated here.
In this section, we briefly introduce the models and explain definitions of
relevant parameters.

The $SU(3)_{C}\times SO(5)\times U(1)_{X}$ GHU models are defined on the
Randall-Sundrum warped spacetime with the metric given by

$\displaystyle
ds^{2}=g_{MN}dx^{M}dx^{N}=e^{-2\sigma(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2},$
(2.1)

where $M,N=0,1,2,3,5$, $\mu,\nu=0,1,2,3$, $y=x^{5}$,
$\eta_{\mu\nu}=\mbox{diag}(-1,+1,+1,+1)$, $\sigma(y)=\sigma(y+2L)=\sigma(-y)$,
and $\sigma(y)=ky$ for $0\leq y\leq L$. In terms of the coordinate $z=e^{ky}$
($1\leq z\leq z_{L}=e^{kL}$) in the region $0\leq y\leq L$, the metric is
written by

$\displaystyle
ds^{2}=\frac{1}{z^{2}}\bigg{(}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+\frac{dz^{2}}{k^{2}}\bigg{)}.$
(2.2)

The bulk region $0<y<L$ ($1<z<z_{L}$) is an anti-de Sitter (AdS) spacetime.
The UV and IR branes are located at $y=0$ ($z=1$) and $y=L$ ($z=z_{L}$),
respectively. The parameter $k$ is AdS curvature. The KK mass scale is given
by $m_{\rm KK}\equiv\pi k/(z_{L}-1)\simeq\pi kz_{L}^{-1}$ for $z_{L}\gg 1$.

The gauge bosons of the $SU(3)_{C}$, $SO(5)$, and $U(1)_{X}$ gauge groups are
expressed by $A_{M}^{SU(3)_{C}}$, $A_{M}^{SO(5)}$, and $A_{M}^{U(1)_{X}}$,
respectively. The BCs for each gauge boson are given by

$\displaystyle\begin{pmatrix}A_{\mu}\cr
A_{y}\end{pmatrix}(x,y_{j}-y)=P_{j}\begin{pmatrix}A_{\mu}\cr-
A_{y}\end{pmatrix}(x,y_{j}+y)P_{j}^{-1},$ (2.3)

where $(y_{0},y_{1})=(0,L)$. Concretely, $P_{0}=P_{1}=I_{3}$ for
$A_{M}^{SU(3)_{C}}$, $P_{0}=P_{1}=1$ for $A_{M}^{U(1)_{X}}$,
$P_{0}=P_{1}=P_{\bf 5}^{SO(5)}=\mbox{diag}\left(I_{4},-I_{1}\right)$ for
$A_{M}^{SO(5)}$ in the vector representation and $P_{\bf
4}^{SO(5)}=\mbox{diag}\left(I_{2},-I_{2}\right)$ in the spinor representation,
respectively. By the orbifold BCs, $SU(3)_{C}\times SO(5)\times U(1)_{X}$ is
broken to $SU(3)_{C}\times SO(4)\times U(1)_{X}\simeq SU(3)_{C}\times
SU(2)_{L}\times SU(2)_{R}\times U(1)_{X}$. $A_{\mu}^{SU(3)_{C}}$,
$A_{\mu}^{U(1)_{X}}$ and $SO(5)/SO(4)$ part of $A_{y}^{SO(5)}$ have zero
modes. The zero modes of $A_{y}^{SO(5)}$ correspond to the Higgs doublet in
the SM. At this stage the $SO(4)\simeq SU(2)_{L}\times SU(2)_{R}$ part of
$A_{\mu}^{SO(5)}$ also have zero modes. A brane scalar field $\Phi_{({\bf
1},{\bf 4})}(x)$ is introduced on the UV brane, which spontaneously breaks
$SO(4)\times U(1)_{X}$ symmetry to $SU(2)_{L}\times U(1)_{Y}$ symmetry.
Finally, $SU(2)_{L}\times U(1)_{Y}$ symmetry is dynamically broken to
$U(1)_{\text{EM}}$ symmetry by an AB phase in the fifth dimension. Only
$\gamma$, $W^{\pm}$ and $Z$ appear as gauge bosons at low energies. At higher
energies, in addition to their KK modes $\gamma^{(n)}$, $W^{\pm(n)}$ and
$Z^{(n)}$, the KK modes of the broken $SU(2)_{R}$ gauge bosons,
$W_{R}^{\pm(n)}$ and $Z_{R}^{(n)}$ can be excited. $\gamma^{(n)}$, $Z^{(n)}$
and $Z_{R}^{(n)}$ appear as $Z^{\prime}$ bosons, and $W^{\pm(n)}$ and
$W_{R}^{\pm(n)}$ appear as $W^{\prime}$ bosons in $SU(3)_{C}\times SO(5)\times
U(1)_{X}$ GHU models.

The 4D Higgs boson doublet $\phi_{H}(x)$ is the zero mode of
$A_{z}^{SO(5)}=(kz)^{-1}A_{y}^{SO(5)}$:

$\displaystyle A_{z}^{(j5)}(x,z)$
$\displaystyle=\frac{1}{\sqrt{k}}\,\phi_{j}(x)u_{H}(z)+\cdots,\
u_{H}(z)=\sqrt{\frac{2}{z_{L}^{2}-1}}\,z\leavevmode\nobreak\ ,$ (2.4)
$\displaystyle\phi_{H}(x)$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}\phi_{2}+i\phi_{1}\cr\phi_{4}-i\phi_{3}\end{pmatrix}.$
(2.5)

Without loss of generality, we set
$\langle\phi_{1}\rangle,\langle\phi_{2}\rangle,\langle\phi_{3}\rangle=0$ and
$\langle\phi_{4}\rangle\not=0$, which is related to the AB phase $\theta_{H}$
in the fifth dimension by $\langle\phi_{4}\rangle=\theta_{H}f_{H}$, where

$\displaystyle
f_{H}=\frac{2}{g_{w}}\sqrt{\frac{k}{L(z_{L}^{2}-1)}}\leavevmode\nobreak\ .$
(2.6)

The representations and BCs of the matter fields are different between the
A-model and B-model, where the matter fields are introduced both in the bulk
and on the UV brane. The matter fields of these two models are listed in Table
1. In the A-model, the SM quarks and leptons are identified with the zero
modes of the $SO(5)$-vector fermions $\Psi^{\alpha}_{a}$ ($a=1,...,4$ and
$\alpha=1,2,3$). The BCs for those fields are given by

$\displaystyle\Psi^{\alpha}_{a}(x,y_{j}-y)=$
$\displaystyle\leavevmode\nobreak\ P_{\bf
5}^{SO(5)}\Gamma^{5}\Psi^{\alpha}_{a}(x,y_{j}+y)\leavevmode\nobreak\ .$ (2.7)

Meanwhile, in the B-model, the SM quarks and leptons are identified with the
zero modes of the $SO(5)$-spinor and singlet fermions $\Psi_{({\bf
3,4})}^{\alpha}$, $\Psi_{({\bf 3,1})}^{\pm\alpha}$, and $\Psi_{({\bf
1,4})}^{\alpha}$ $(\alpha=1,2,3)$. These fields obey the following BCs:

$\displaystyle\Psi_{({\bf 3,4})}^{\alpha}(x,y_{j}-y)=-P_{\bf
4}^{SO(5)}\gamma^{5}\Psi_{({\bf 3,4})}^{\alpha}(x,y_{j}+y),$
$\displaystyle\Psi_{({\bf
3,1})}^{\pm\alpha}(x,y_{j}-y)=\mp\gamma^{5}\Psi_{({\bf
3,1})}^{\pm\alpha}(x,y_{j}+y),$ $\displaystyle\Psi_{({\bf
1,4})}^{\alpha}(x,y_{j}-y)=-P_{\bf 4}^{SO(5)}\gamma^{5}\Psi_{({\bf
1,4})}^{\alpha}(x,y_{j}+y).$ (2.8)

With BCs (2.8), the $SU(2)_{L}$ ($SU(2)_{R}$) components of $\Psi_{({\bf
3,4})}^{\alpha}$ and $\Psi_{({\bf 1,4})}^{\alpha}$ have the
left-(right-)handed zero modes and $\Psi_{({\bf 3,1})}^{+\alpha}$
($\Psi_{({\bf 3,1})}^{-\alpha}$) has the left-(right-)handed zero modes,
respectively. These zero modes mix with each other thorough the AB phase
$\theta_{H}$ and brane interactions.

There are matter fields having no zero modes, which is referred as the dark
fermions. Those dark fermions are necessary to have dynamical electroweak
symmetry breaking with suitable value of $\theta_{H}$. Dark fermions do not
couple directly with the quarks and leptons and the lightest modes of dark
fermions have masses about $m_{\text{KK}}/2$. Decay of the first KK gauge
bosons to dark fermions are either forbidden or negligible. The details of the
dark fermions are given in Ref. [47].

Table 1: Matter fields. Brane fields and the symmetry at the UV brane are also shown.  | A-model | B-model
---|---|---
quark | $\Psi^{\alpha}_{1}:(\bm{3},\bm{5})_{+\frac{2}{3}},\leavevmode\nobreak\ \Psi^{\alpha}_{2}:(\bm{3},\bm{5})_{-\frac{1}{3}}$ | $\Psi^{\alpha}_{({\bf 3,4})}:(\bm{3},\bm{4})_{+\frac{1}{6}},\leavevmode\nobreak\ \Psi^{\pm\alpha}_{({\bf 3,1})}:(\bm{3},\bm{1})_{-\frac{1}{3}}^{\pm}$
lepton | $\Psi^{\alpha}_{3}:(\bm{1},\bm{5})_{0},\leavevmode\nobreak\ \Psi^{\alpha}_{4}:(\bm{1},\bm{5})_{-1}$ |  $\Psi^{\alpha}_{({\bf 1,4})}(\bm{1},\bm{4})_{-\frac{1}{2}}$
dark fermion | $\Psi_{F}^{\delta}:(\bm{1},\bm{4})_{+\frac{1}{2}}$ |  $\Psi_{F_{q}}^{\beta}:(\bm{3},\bm{4})_{+\frac{1}{6}},\leavevmode\nobreak\ \Psi_{F_{l}}^{\beta}:(\bm{1},\bm{4})_{-\frac{1}{2}}$
|  | $\Psi_{V}^{\pm\gamma}:(\bm{1},\bm{5})_{0}^{\pm}$
brane fermion |  $\begin{matrix}\hat{\chi}^{q}_{1,2,3R}:(\bm{3},[\bm{2,1}])_{+\frac{7}{6},+\frac{1}{6},-\frac{5}{6}}\cr\hat{\chi}^{q}_{1,2,3R}:(\bm{1},[\bm{2,1}])_{+\frac{1}{2},-\frac{1}{2},-\frac{3}{2}}\end{matrix}$ | $\chi:(\bm{1},\bm{1})_{0}$
brane scalar | $\hat{\Phi}:(\bm{1},[\bm{1,2}])_{+\frac{1}{2}}$ |  $\Phi_{(\bm{1},\bm{4})}:(\bm{1},\bm{4})_{\frac{1}{2}}$
$\begin{matrix}\text{symmetry \leavevmode\nobreak\ of}\cr\text{brane \leavevmode\nobreak\ interactions}\end{matrix}$ | $SU(3)_{C}\times SO(4)\times U(1)_{X}$ | $SU(3)_{C}\times SO(5)\times U(1)_{X}$

The procedure to determine the model parameters is explained in Refs [43, 47,
48, 49]. With specified parameters, masses and couplings are all determined.
It has been shown that one can introduce the Cabibbo-Kobayashi-Maskawa (CKM)
matrix in the GHU B-model while flavour changing neutral currents (FCNCs) are
naturally suppressed [48]. The tree-level FCNCs exist only in the down-type
quark sector and the magnitude of their couplings is of the order of
$O(10^{-6})$. In this paper, the CKM matrix is not introduced for simplicity.

As benchmark points, eight parameter sets are taken as shown in Table 2. These
benchmark points are chosen for the following reason. In the A-model, there
are two free parameters $z_{L}$ (or $m_{\text{KK}}$) and $n_{F}$ (the number
of dark fermions). The effects of $n_{F}$ on physics of the gauge bosons,
Higgs bosons, quarks and leptons turn out very small. Once a set of parameters
($n_{F}$, $z_{L}$) is set, $\theta_{H}$ is determined and couplings among
these fields are determined by $\theta_{H}$. Hence a relevant free parameter
is effectively $\theta_{H}$ only. The value of $n_{F}$ affects a lower limit
of $\theta_{H}$ and a dark fermion mass. For a larger $n_{F}$, the lower limit
of $\theta_{H}$ is smaller and the lowest mode of the dark fermion has a lower
mass. We take $n_{F}=4$, where the lower limit of $\theta_{H}$ is
$\theta_{H}\simeq 0.08$. The top quark and Higgs boson mass cannot be
reproduced for $n_{F}=4$ and $\theta_{H}<0.08$ in the A-model. In the B-model,
there are four free parameters in the dark fermion sector, and $z_{L}$ and
$\theta_{H}$ are not uniquely determined. These parameters are constrained to
reproduce the top quark mass and EWSB. The top quark mass is realized only for
$z_{L}\gtrsim 10^{8.1}$, and the EWSB is triggered only for $z_{L}\lesssim
10^{15.5}$ for $\theta_{H}=0.10$. Consistent parameter sets are obtained only
for $10^{8.1}\lesssim z_{L}\lesssim 10^{15.5}$ [49]. We set $\theta_{H}=0.10$
and choose, as typical values, integral values of $m_{\text{KK}}$/TeV in the
allowed region. To check the $\theta_{H}$-dependence, $\theta_{H}=0.11,\
0.10,\ 0.09$ are chosen for $m_{\text{KK}}=13$ TeV.

Name | $\theta_{H}$ | $m_{\text{KK}}$ | $z_{L}$ | $k$ | $m_{W^{(1)}}$ | $\Gamma_{W^{(1)}}$ | $m_{W_{R}^{(1)}}$ | $\Gamma_{W_{R}^{(1)}}$ | Table
---|---|---|---|---|---|---|---|---|---
| [rad.] | [TeV] |  | [GeV] | [TeV] | [TeV] | [TeV] | [TeV] |
A1 | 0.10 | 8.063 |  2.900$\times 10^{4}$ |  7.443$\times 10^{7}$ | 6.585 | 0.236 | 6.172 | 0.098 |
A2 | 0.09 | 8.721 |  1.700$\times 10^{4}$ |  4.719$\times 10^{7}$ | 7.149 | 0.271 | 6.676 | 0.101 | 3
A3 | 0.08 | 9.544 |  1.010$\times 10^{4}$ |  3.068$\times 10^{7}$ | 7.855 | 0.356 | 7.305 | 0.105 |
BL | 0.10 | 11 |  1.980$\times 10^{8\ }$ |  6.933$\times 10^{11}$ | 8.713 | 5.925 | 8.420 | 0.218 | 6
B | 0.10 | 13 |  3.865$\times 10^{11}$ |  1.599$\times 10^{15}$ | 10.20 | 9.812 | 9.951 | 0.368 | 4
BH | 0.10 | 15 |  2.667$\times 10^{15}$ |  1.273$\times 10^{19}$ | 11.69 | 14.85 | 11.48 | 0.565 | 6
B+ | 0.11 | 13 |  1.021$\times 10^{14}$ |  4.223$\times 10^{17}$ | 10.15 | 11.75 | 9.951 | 0.445 | 8
B | 0.10 | 13 |  3.865$\times 10^{11}$ |  1.599$\times 10^{15}$ | 10.20 | 9.812 | 9.951 | 0.368 | 4
B- | 0.09 | 13 |  2.470$\times 10^{9\ }$ |  1.022$\times 10^{13}$ | 10.26 | 7.993 | 9.951 | 0.291 | 8
Table 2:  Masses and widths of $W^{(1)}$ and $W_{R}^{(1)}$ are listed. Those in the A-model are shown for $\theta_{H}=0.10$, $0.09$, $0.08$ in the upper table. Those in the B-model are shown for $\theta_{H}=0.10$ and three $m_{\rm KK}=11,\ 13,\ 15$ TeV in the middle table and for $m_{\rm KK}=13$ TeV and three $\theta_{H}=0.11,\ 0.10,\ 0.09$ in the lower table.  | A1 | A2 | A3
---|---|---|---
$ff^{\prime}$ | $g_{W^{(1)}ff^{\prime}}^{L}$ | $g_{W^{(1)}ff^{\prime}}^{R}$ | $g_{W^{(1)}ff^{\prime}}^{L}$ | $g_{W^{(1)}ff^{\prime}}^{R}$ | $g_{W^{(1)}ff^{\prime}}^{L}$ | $g_{W^{(1)}ff^{\prime}}^{R}$
$e\nu_{e}$ | $-0.3675$ | 0 | $-0.3785$ | 0 | $-0.3901$ | 0
$\mu\nu_{\mu}$ | $-0.3675$ | 0 | $-0.3785$ | 0 | $-0.3901$ | 0
$\tau\nu_{\tau}$ | $-0.3670$ | 0 | $-0.3779$ | 0 | $-0.3898$ | 0
$ud$ | $-0.3675$ | 0 | $-0.3785$ | 0 | $-0.3901$ | 0
$cs$ | $-0.3675$ | 0 | $-0.3785$ | 0 | $-0.3901$ | 0
$tb$ | $+1.4588$ | 0 | $+1.5635$ | 0 | $+1.8313$ | 0
Table 3:  Coupling constants of 1st KK $W$ boson to fermions in units of $g_{w}/\sqrt{2}$ are listed for A-model, where $\sin^{2}\theta_{W}^{0}=0.23126$. The value less than $10^{-4}$ is written as $0$. The $W$ boson couplings are $g_{Wff^{\prime}}^{L}=1.000$ for $ff^{\prime}\neq tb$ and $g_{Wtb}^{L}=0.9993$ for A1, $g_{Wtb}^{L}=0.9994$ for A2 and A3. The couplings of the first KK $W_{R}$ boson to fermions are exactly zero.  $ff^{\prime}$ | $g_{Wff^{\prime}}^{L}$ | $g_{Wff^{\prime}}^{R}$ | $g_{W^{(1)}ff^{\prime}}^{L}$ | $g_{W^{(1)}ff^{\prime}}^{R}$ | $g_{W_{R}^{(1)}ff^{\prime}}^{L}$ | $g_{W_{R}^{(1)}ff^{\prime}}^{R}$
---|---|---|---|---|---|---
$e\nu_{e}$ | 0.9976 | 0 | 5.7451 | 0 | 0.0146 | 0
$\mu\nu_{\mu}$ | 0.9976 | 0 | 5.4705 | 0 | 0.0139 | 0
$\tau\nu_{\tau}$ | 0.9976 | 0 | 5.2877 | 0 | 0.0134 | 0
$ud$ | 0.9976 | 0 | 5.5626 | 0 | 0.0141 | 0
$cs$ | 0.9976 | 0 | 5.3588 | 0 | 0.0136 | 0
$tb$ | 0.9980 | 0 | 4.4108 | 0 | 0.0113 |  $-$0.0344
Table 4:  Coupling constants of charged vector bosons, $W^{\prime}$ bosons, to
fermions in units of $g_{w}/\sqrt{2}$ are listed for $\theta_{H}=0.10$ and
$m_{\text{KK}}=13$ TeV (B) in Table 2, where $\sin^{2}\theta_{W}^{0}=0.2306$.
The value less than $10^{-4}$ is written as $0$.

$ff^{\prime}$ | $g_{Wff^{\prime}}^{L}$ | $g_{Wff^{\prime}}^{R}$ | $g_{W^{(1)}ff^{\prime}}^{L}$ | $g_{W^{(1)}ff^{\prime}}^{R}$ | $g_{W_{R}^{(1)}ff^{\prime}}^{L}$ | $g_{W_{R}^{(1)}ff^{\prime}}^{R}$
---|---|---|---|---|---|---
$e\nu_{e}$ | 0.9977 | 0 | 5.0203 | 0 | 0.0127 | 0
$\mu\nu_{\mu}$ | 0.9977 | 0 | 4.7423 | 0 | 0.0121 | 0
$\tau\nu_{\tau}$ | 0.9977 | 0 | 4.5451 | 0 | 0.0116 | 0
$ud$ | 0.9977 | 0 | 4.8380 | 0 | 0.0123 | 0
$cs$ | 0.9977 | 0 | 4.6229 | 0 | 0.0118 | 0
$tb$ | 0.9982 | 0 | 3.1365 | 0 | 0.0082 |  $-$0.0455

Table 5:  Coupling constants of charged vector bosons, $W^{\prime}$ bosons, to
fermions in units of $g_{w}/\sqrt{2}$ are listed for $\theta_{H}=0.10$ and
$m_{\text{KK}}=11$ TeV (BL) in Table 2, where $\sin^{2}\theta_{W}^{0}=0.2306$.
Other information is the same as in Table 4.

$ff^{\prime}$ | $g_{Wff^{\prime}}^{L}$ | $g_{Wff^{\prime}}^{R}$ | $g_{W^{(1)}ff^{\prime}}^{L}$ | $g_{W^{(1)}ff^{\prime}}^{R}$ | $g_{W_{R}^{(1)}ff^{\prime}}^{L}$ | $g_{W_{R}^{(1)}ff^{\prime}}^{R}$
---|---|---|---|---|---|---
$e\nu_{e}$ | 0.9976 | 0 | 6.4691 | 0 | 0.0164 | 0
$\mu\nu_{\mu}$ | 0.9976 | 0 | 6.2055 | 0 | 0.0157 | 0
$\tau\nu_{\tau}$ | 0.9976 | 0 | 6.0376 | 0 | 0.0153 | 0
$ud$ | 0.9976 | 0 | 6.2923 | 0 | 0.0159 | 0
$cs$ | 0.9976 | 0 | 6.1021 | 0 | 0.0155 | 0
$tb$ | 0.9978 | 0 | 5.3389 | 0 | 0.0136 |  $-$0.0294

Table 6:  Coupling constants of charged vector bosons, $W^{\prime}$ bosons, to
fermions in units of $g_{w}/\sqrt{2}$ are listed for $\theta_{H}=0.10$ and
$m_{\text{KK}}=15$ TeV (BH) in Table 2, where $\sin^{2}\theta_{W}^{0}=0.2306$.
Other information is the same as in Table 4.

$ff^{\prime}$ | $g_{Wff^{\prime}}^{L}$ | $g_{Wff^{\prime}}^{R}$ | $g_{W^{(1)}ff^{\prime}}^{L}$ | $g_{W^{(1)}ff^{\prime}}^{R}$ | $g_{W_{R}^{(1)}ff^{\prime}}^{L}$ | $g_{W_{R}^{(1)}ff^{\prime}}^{R}$
---|---|---|---|---|---|---
$e\nu_{e}$ | 0.9971 | 0 | 6.2134 | 0 | 0.0190 | 0
$\mu\nu_{\mu}$ | 0.9971 | 0 | 5.9455 | 0 | 0.0182 | 0
$\tau\nu_{\tau}$ | 0.9971 | 0 | 5.7724 | 0 | 0.0177 | 0
$ud$ | 0.9971 | 0 | 6.0342 | 0 | 0.0185 | 0
$cs$ | 0.9971 | 0 | 5.8391 | 0 | 0.0179 | 0
$tb$ | 0.9974 | 0 | 5.0226 | 0 | 0.0155 |  $-$0.0309

Table 7:  Coupling constants of charged vector bosons, $W^{\prime}$ bosons, to
fermions in units of $g_{w}/\sqrt{2}$ are listed for $\theta_{H}=0.11$ and
$m_{\text{KK}}=13$ TeV (B+) in Table 2, where $\sin^{2}\theta_{W}^{0}=0.2305$.
Other information is the same as in Table 4.

$ff^{\prime}$ | $g_{Wff^{\prime}}^{L}$ | $g_{Wff^{\prime}}^{R}$ | $g_{W^{(1)}ff^{\prime}}^{L}$ | $g_{W^{(1)}ff^{\prime}}^{R}$ | $g_{W_{R}^{(1)}ff^{\prime}}^{L}$ | $g_{W_{R}^{(1)}ff^{\prime}}^{R}$
---|---|---|---|---|---|---
$e\nu_{e}$ | 0.9981 | 0 | 5.2761 | 0 | 0.0108 | 0
$\mu\nu_{\mu}$ | 0.9981 | 0 | 4.9979 | 0 | 0.0103 | 0
$\tau\nu_{\tau}$ | 0.9981 | 0 | 4.8056 | 0 | 0.0099 | 0
$ud$ | 0.9981 | 0 | 5.0926 | 0 | 0.0105 | 0
$cs$ | 0.9981 | 0 | 4.8810 | 0 | 0.0101 | 0
$tb$ | 0.9985 | 0 | 3.7015 | 0 | 0.0078 |  $-$0.0397

Table 8:  Coupling constants of charged vector bosons, $W^{\prime}$ bosons, to
fermions in units of $g_{w}/\sqrt{2}$ are listed for $\theta_{H}=0.09$ and
$m_{\text{KK}}=13$ TeV (B-) in Table 2, where $\sin^{2}\theta_{W}^{0}=0.2307$.
Other information is the same as in Table 4.

The masses and decay widths of the $W^{(1)}$ and $W_{R}^{(1)}$ bosons are
shown in Table 2, where the decay widths are calculated by the couplings shown
in Tables 3, 4, 6, 6, 8 and 8. Those of the $\gamma^{(1)}$, $Z^{(1)}$ and
$Z_{R}^{(1)}$ bosons are shown in Refs. [46, 50]. The masses of $W^{(1)}$,
$Z^{(1)}$ and $\gamma^{(1)}$ bosons are almost degenerate and about 0.8 times
the KK mass scale. The masses of $W_{R}^{(1)}$ and $Z_{R}^{(1)}$ bosons are
slightly lighter than those of $W^{(1)}$ and $Z^{(1)}$ bosons.

The couplings among particles depend on the behavior of their wave-functions.
In the $\theta_{H}\to 0$ limit, $W$ and $W_{R}$ bosons are purely $SU(2)_{L}$
and $SU(2)_{R}$ gauge bosons. In the A-model, zero modes of left-handed quarks
and leptons are $SU(2)_{L}\times SU(2)_{R}$ bidoublets, whereas right-handed
quarks and leptons are singlets. $W$ and $W_{R}$ bosons do not couple with
right-handed quarks and leptons, as those couplings are determined by overlap
integrals of their wave-functions. $W^{(1)}$ and $W_{R}^{(1)}$ bosons are
localized near the IR brane and zero modes of left-handed quarks and leptons
are localized near the UV brane except for the top quark. Consequently,
$W^{(1)}$ couples with left-handed SM fermions very weakly except for the top
quark and does not couple with right-handed SM fermions. The decay width of
$W^{(1)}$ is narrow as shown in Table 2.

On the other hand, the $W^{(1)}$ couplings with left-handed fermions are large
in the B-model. The zero modes of the left- and right-handed quarks and
leptons in the B-model are purely $SU(2)_{L}$ and $SU(2)_{R}$ components,
respectively. Therefore $W^{(1)}$ mainly couples with left-handed quarks and
leptons and $W_{R}^{(1)}$ mainly couples with right-handed ones for small
$\theta_{H}$. In contrast to the A-model, the zero modes of left-handed quarks
and leptons are localized near the IR brane and the zero modes of right-handed
quarks and leptons are localized near the UV brane in the B-model. Hence, the
$W^{(1)}$ couplings with left-handed fermions are large, numerically being of
$O(1)\times g_{w}/\sqrt{2}$. In contrast to it, the $W_{R}^{(1)}$ couplings
with left-handed fermions are small, numerically being of $O(10^{-2})\times
g_{w}/\sqrt{2}$. The $W^{(1)}$ and $W_{R}^{(1)}$ couplings with right-handed
fermions are tiny and negligible except for $W_{R}^{(1)}\bar{t}_{R}\>b_{R}$
coupling. Because of the large $W^{(1)}$ couplings with left-handed fermions,
the decay width of $W^{(1)}$ becomes very wide, which is numerically
$\Gamma_{W^{(1)}}/m_{W^{(1)}}=0.68$ – $1.27$ as shown in Table 2.

The contribution from the higher KK modes are calculated in Ref. [50] and
found to be small. Numerically, the deviation of the
$e^{+}e^{-}\to\mu^{+}\mu^{-}$ cross section with the first and second KK modes
from the deviation with only the first KK modes is $O(1)$% for $\sqrt{s}<3$
TeV. Thus we take only the first KK modes into account in the following.

##  3 Differential Cross Sections

In this section, formulae for the cross sections of the $pp\to l^{+}l^{-}$ and
$pp\to l\nu$ processes are summarized [71, 72].

The cross section of the $pp\to l^{+}l^{-}$ process, $\sigma_{pp\to
l^{+}l^{-}}$, is written in terms of the parton-level cross section
$\sigma_{f\bar{f}\to l^{+}l^{-}}(s)$ as

$\displaystyle\sigma_{pp\to l^{+}l^{-}}$
$\displaystyle=\sum_{f}\int^{1}_{0}dx_{1}\int^{1}_{0}dx_{2}\>\sigma_{f\bar{f}\to
l^{+}l^{-}}(s)\left(F_{f}(x_{1},Q)F_{\bar{f}}(x_{2},Q)+F_{f}(x_{2},Q)F_{\bar{f}}(x_{1},Q)\right)$
$\displaystyle=\sum_{f}\int^{s_{pp}}_{0}ds\int^{1}_{s/s_{pp}}dx_{1}\frac{2}{x_{1}s_{pp}}\sigma_{f\bar{f}\to
l^{+}l^{-}}(s)F_{f}(x_{1},\sqrt{s})F_{\bar{f}}\left(\frac{s}{x_{1}s_{pp}},\sqrt{s}\right),$
(3.1)

where $F_{f}(x_{1,2},Q)$ are the parton distribution function (PDF) at an
energy scale $Q$ and we take $Q=\sqrt{s}$ in this paper. By introducing the
invariant mass of the lepton pair denoted as $m_{ll}\equiv\sqrt{s}$, the
differential cross section with respect to the invariant mass is written as

$\displaystyle\frac{d\sigma_{pp\to l^{+}l^{-}}}{dm_{ll}}$
$\displaystyle=\sum_{f}\int^{1}_{m_{ll}^{2}/s_{pp}}dx_{1}\frac{4m_{ll}}{x_{1}s_{pp}}\sigma_{f\bar{f}\to
l^{+}l^{-}}(s=m_{ll}^{2})F_{f}(x_{1},m_{ll})F_{\bar{f}}\left(\frac{m_{ll}^{2}}{x_{1}s_{pp}},m_{ll}\right).$
(3.2)

Similarly, the cross section of the $pp\to l\nu$ process, $\sigma_{pp\to
l\nu}$, is written as

$\displaystyle\sigma_{pp\to l\nu}=\sum_{f,\
f^{\prime}}\int^{s_{pp}}_{0}ds\int^{1}_{-1}d\cos\theta\int^{1}_{s/s_{pp}}dx_{1}\frac{2}{x_{1}s_{pp}}\frac{d\sigma_{f\bar{f^{\prime}}\to
l\nu}(s)}{d\cos\theta}F_{f}(x_{1},\sqrt{s})F_{\bar{f}^{\prime}}\left(\frac{s}{x_{1}s_{pp}},\sqrt{s}\right),$
(3.3)

where $\sigma_{f\bar{f^{\prime}}\to l\nu}(s)$ is the parton-level cross
section. The cross section of the $pp\to l\nu$ process is explored by
measuring the transverse mass ($m_{\text{T}}$) or the transverse momentum of
the charged-lepton ($p_{\text{T}}$). These two parameters are related as
$m_{\text{T}}=\sqrt{2p_{\text{T}}E_{\text{T}}^{\text{miss}}(1-\cos\phi)}$,
where $E_{\text{T}}^{\text{miss}}$ is the missing energy and $\phi$ is the
angle between the charged-lepton and missing transverse momentum in the
transverse plane. For $pp$ collisions at high energies, the transverse
momentum of the parton may be ignored. $\phi\simeq\pi$ and the masses of the
leptons are ignored, and one finds $m_{\text{T}}\simeq 2\,p_{\text{T}}$. The
transverse momentum is defined as
$p_{\text{T}}\equiv(\sqrt{s}/2)|\sin\theta|$. The total cross section is
written by using the transverse mass as

$\displaystyle\sigma_{pp\to l\nu}$ $\displaystyle=\sum_{f,\
f^{\prime}}\int^{\sqrt{s_{pp}}}_{0}dm_{ll}\int^{m_{ll}}_{0}dm_{\text{T}}\int^{1}_{\frac{m_{ll}^{2}}{s_{pp}}}dx_{1}\frac{8m_{\text{T}}}{x_{1}s_{pp}m_{ll}\sqrt{1-\frac{m_{\text{T}}^{2}}{m_{ll}^{2}}}}F_{f}(x_{1},m_{ll})F_{\bar{f}^{\prime}}\left(\frac{m_{ll}^{2}}{x_{1}s_{pp}},m_{ll}\right)$
$\displaystyle\qquad\qquad\times\left(\left.\frac{d\sigma_{f\bar{f^{\prime}}\to
l\nu}(s)}{d\cos\theta}\right|_{\cos\theta=+\sqrt{1-\frac{m_{\text{T}}^{2}}{m_{ll}^{2}}}}+\left.\frac{d\sigma_{f\bar{f^{\prime}}\to
l\nu}(s)}{d\cos\theta}\right|_{\cos\theta=-\sqrt{1-\frac{m_{\text{T}}^{2}}{m_{ll}^{2}}}}\right)$
$\displaystyle=\sum_{f,\
f^{\prime}}\int^{\sqrt{s_{pp}}}_{0}dm_{\text{T}}\int^{\sqrt{s_{pp}}}_{m_{\text{T}}}dm_{ll}\int^{1}_{\frac{m_{ll}^{2}}{s_{pp}}}dx_{1}\frac{8m_{\text{T}}}{x_{1}s_{pp}m_{ll}\sqrt{1-\frac{m_{\text{T}}^{2}}{m_{ll}^{2}}}}F_{f}(x_{1},m_{ll})F_{\bar{f}^{\prime}}\left(\frac{m_{ll}^{2}}{x_{1}s_{pp}},m_{ll}\right)$
$\displaystyle\qquad\qquad\times\left(\left.\frac{d\sigma_{f\bar{f^{\prime}}\to
l\nu}(s)}{d\cos\theta}\right|_{\cos\theta=+\sqrt{1-\frac{m_{\text{T}}^{2}}{m_{ll}^{2}}}}+\left.\frac{d\sigma_{f\bar{f^{\prime}}\to
l\nu}(s)}{d\cos\theta}\right|_{\cos\theta=-\sqrt{1-\frac{m_{\text{T}}^{2}}{m_{ll}^{2}}}}\right).$
(3.4)

The differential cross section with respect to the transverse mass is

$\displaystyle\frac{d\sigma_{pp\to l\nu}}{dm_{\text{T}}}$
$\displaystyle=\sum_{f,\
f^{\prime}}\int^{\sqrt{s_{pp}}}_{m_{\text{T}}}dm_{ll}\int^{1}_{\frac{m_{ll}^{2}}{s_{pp}}}dx_{1}\frac{8m_{\text{T}}}{x_{1}s_{pp}m_{ll}\sqrt{1-\frac{m_{\text{T}}^{2}}{m_{ll}^{2}}}}F_{f}(x_{1},m_{ll})F_{\bar{f}^{\prime}}\left(\frac{m_{ll}^{2}}{x_{1}s_{pp}},m_{ll}\right)$
$\displaystyle\qquad\qquad\times\left(\left.\frac{d\sigma_{f\bar{f^{\prime}}\to
l\nu}(s)}{d\cos\theta}\right|_{\cos\theta=+\sqrt{1-\frac{m_{\text{T}}^{2}}{m_{ll}^{2}}}}+\left.\frac{d\sigma_{f\bar{f^{\prime}}\to
l\nu}(s)}{d\cos\theta}\right|_{\cos\theta=-\sqrt{1-\frac{m_{\text{T}}^{2}}{m_{ll}^{2}}}}\right).$
(3.5)

##  4 Constraints from LHC Experiments

In this section, numerical results for the $pp\to l^{+}l^{-}$ and $pp\to l\nu$
processes are presented. We use CT10 [73] for the PDF and ManeParse [74] to
numerically evaluate the cross sections.

First, we recall the constraints on the $Z^{\prime}$ bosons from the $pp\to
e^{+}e^{-}$ and $\mu^{+}\mu^{-}$ processes. To see the $\theta_{H}$ and
$m_{\text{KK}}$ dependence, the differential cross sections
$d\sigma(pp\to{\mu^{+}\mu^{-}})/dM_{\mu\mu}$ are shown for the parameter sets
(B${}^{\text{L}}$, B, B${}^{\text{H}}$) and (B+, B, B-) in Fig. 1.

Figure 1:  Differential cross sections
$d\sigma(pp\to{\mu^{+}\mu^{-}})/dM_{\mu\mu}$ for $\sqrt{s}=13$ TeV. The
differential cross section in the SM is shown by the black solid line. In the
left figure, the differential cross sections with fixed $\theta_{H}=0.10$ and
$m_{\text{KK}}=11$ TeV (B${}^{\text{L}}$), 13 TeV (B), 15 TeV
(B${}^{\text{H}}$) are shown. In the right figure, the differential cross
sections with fixed $m_{\text{KK}}=13$ TeV and $\theta_{H}=0.11$ (B+),
$\theta_{H}=0.10$ (B), $\theta_{H}=0.09$ (B-) are shown.

The decay widths of the $Z^{\prime}$ bosons are large, and therefore we refer
the constraint from the non-resonant searches in the dilepton final states at
the ATLAS group [2]. The $d\sigma(pp\to{l^{+}l^{-}})/dM_{ll}$ in the GHU model
is $O(1)$% smaller for $M_{ll}\simeq 500$ GeV and 20–30% smaller for
$M_{ll}\simeq 1000$ GeV, and becomes larger for $M_{ll}\gtrsim 2500$ GeV than
that in the SM as shown in Fig. 1. Hence we follow the analysis in the
“destructive interference” case in Ref. [2]. For simplicity, we assume that
the acceptance and efficiency are independent of the invariant mass. Therefore
the acceptance times efficiency are introduced as constant factors of the
differential cross sections and are determined to minimize the $\chi^{2}$
values in the control regions (CRs). The number of events is obtained by
integrating the differential cross sections over the signal regions (SRs). The
CRs and SRs for each process are CR: [310, 1450] GeV and SR: [2770, 6000] GeV
for $pp\to e^{+}e^{-}$ and CR: [320, 1250] GeV and SR: [2570, 6000] GeV for
$pp\to\mu^{+}\mu^{-}$, respectively [2]. $N_{\text{GHU}}$ and $N_{\text{SM}}$
denote the number of events in the GHU model and that in the SM, respectively.
The number of signal events $N_{\text{sig}}$ is defined as the excess of
$N_{\text{GHU}}$ from $N_{\text{SM}}$; $N_{\text{sig}}\equiv
N_{\text{GHU}}-N_{\text{SM}}$. $N_{\text{SM}}=1.2$ for $pp\to e^{+}e^{-}$ and
$N_{\text{SM}}=1.8$ for $pp\to\mu^{+}\mu^{-}$. $N_{\text{sig}}$ for the
parameter sets (A1, A2, A3) are (21, 10, 4.9) for $pp\to e^{+}e^{-}$ and (18,
8.6, 4.0) for $pp\to\mu^{+}\mu^{-}$. $N_{\text{sig}}$ for the five parameter
sets in the B-model are shown in Table 9. The observed (expected) upper limits
at 95% confidence level (CL) on $N_{\text{sig}}$ are 4.4 (5.0) and 3.8 (4.0)
for $pp\to e^{+}e^{-}$ and $pp\to\mu^{+}\mu^{-}$, which disfavors the
parameter sets B+ and B${}^{\text{L}}$ at 95% CL. The parameter sets A1 and A2
are disfavored by the expected upper limits at 95% CL for $pp\to e^{+}e^{-}$
and $pp\to\mu^{+}\mu^{-}$.

|  | $pp\to e^{+}e^{-}$ | $pp\to\mu^{+}\mu^{-}$
---|---|---|---
|  |  | $m_{\text{KK}}$ |  |  | $m_{\text{KK}}$ |
|  | 11 TeV | 13 TeV | 15 TeV | 11 TeV | 13 TeV | 15 TeV
| 0.11 |  | 5.7 |  |  | 5.0 |
$\theta_{H}$ | 0.10 | 6.1 | 3.9 | 2.7 | 4.9 | 3.1 | 2.1
| 0.09 |  | 2.4 |  |  | 1.6 |
Table 9:  The number of signal events in the GHU B-model $N_{\text{sig}}\equiv
N_{\text{GHU}}-N_{\text{SM}}$, where $N_{\text{SM}}=1.2$ for $pp\to
e^{+}e^{-}$ and $N_{\text{SM}}=1.8$ for $pp\to\mu^{+}\mu^{-}$. The second and
third columns show $N_{\text{sig}}$ for $pp\to e^{+}e^{-}$ and
$pp\to\mu^{+}\mu^{-}$ processes, respectively. The observed (expected) upper
limits at 95% CL on $N_{\text{sig}}$ are 4.4 (5.0) and 3.8 (4.0) for $pp\to
e^{+}e^{-}$ and $pp\to\mu^{+}\mu^{-}$, respectively.

Next, we consider the constraints on the $W^{\prime}$ bosons from the $pp\to
e\nu$ and $\mu\nu$ processes. We consider the constraint from the ATLAS group
[3]. The CRs and SRs are not specified there. The differential cross sections
in the GHU model are larger than that in the SM above $M_{\text{T}}\sim 2300$
GeV for the parameter sets (B${}^{\text{L}}$, B, B${}^{\text{H}}$) and (B+, B,
B-). Therefore we take the SRs as SR: [2300, 6000] GeV for $pp\to e\nu$ and
SR: [2400, 6000] GeV for $pp\to\mu\nu$, respectively. We take the CRs as CR:
[310, 1030] GeV for $pp\to e\nu$ and CR: [300, 1050] GeV for $pp\to\mu\nu$.
$N_{\text{SM}}=6.8$ for $pp\to e\nu$ and $N_{\text{SM}}=3.8$ for
$pp\to\mu\nu$. $N_{\text{sig}}$s for the five parameter sets are shown in
Table 10. The observed (expected) upper limits at 95% CL on $N_{\text{sig}}$
are 3.4 (8.4) and 8.6 (7.7) for $pp\to e\nu$ and $pp\to\mu\nu$, respectively.
Therefore the parameter sets B, B+ and B${}^{\text{L}}$ are disfavored by the
expected upper limits at 95% CL for $pp\to e\nu$. The parameter sets (A1, A2,
A3) are not excluded from the $pp\to e\nu$ and $pp\to\mu\nu$ processes because
of the large $W^{\prime}$ mass and small gauge couplings.

|  | $pp\to e\nu$ | $pp\to\mu\nu$
---|---|---|---
|  |  | $m_{\text{KK}}$ |  |  | $m_{\text{KK}}$ |
|  | 11 TeV | 13 TeV | 15 TeV | 11 TeV | 13 TeV | 15 TeV
| 0.11 |  | 17 |  |  | 10 |
$\theta_{H}$ | 0.10 | 22 | 12 | 6.9 | 13 | 7.3 | 4.3
| 0.09 |  | 6.9 |  |  | 4.2 |
Table 10:  The number of signal events in the GHU B-model
$N_{\text{sig}}\equiv N_{\text{GHU}}-N_{\text{SM}}$, where $N_{\text{SM}}=6.8$
for $pp\to e\nu$ and $N_{\text{SM}}=3.8$ for $pp\to\mu\nu$. The second and
third columns show $N_{\text{sig}}$ for $pp\to e\nu$ and $pp\to\mu\nu$
processes. The observed (expected) upper limits at 95% CL on $N_{\text{sig}}$
are 3.4 (8.4) and 8.6 (7.7) for $pp\to e\nu$ and $pp\to\mu\nu$, respectively.

##  5 For Future LHC Experiments

In this section, we estimate the numbers of events of the
$pp\to\\{W,W^{\prime}\\}\to l\nu$ and $pp\to\\{\gamma,Z,Z^{\prime}\\}\to
l^{+}l^{-}$ processes at $\sqrt{s}=14$ TeV with the luminosity 300 fb-1 (LHC
Run 3) and 3000 fb-1 (HL-LHC). Other background processes are ignored and the
acceptance and efficiency are not taken into account. The numbers of events
$N_{\text{GHU}}$ and $N_{\text{SM}}$ are estimated by integrating the
differential cross sections times luminosity from $m_{\text{min}}$ to 6 TeV,
where $m_{\text{min}}$ is determined by the condition

$\displaystyle\left.\frac{d\sigma_{pp\to
l\nu}^{\text{GHU}}}{dm_{\text{T}}}\right|_{m_{\text{T}}=m_{\text{min}}}=\left.\frac{d\sigma_{pp\to
l\nu}^{\text{SM}}}{dm_{\text{T}}}\right|_{m_{\text{T}}=m_{\text{min}}},$ (5.1)

or

$\displaystyle\left.\frac{d\sigma_{pp\to
l^{+}l^{-}}^{\text{GHU}}}{dm_{ll}}\right|_{m_{ll}=m_{\text{min}}}=\left.\frac{d\sigma_{pp\to
l^{+}l^{-}}^{\text{SM}}}{dm_{ll}}\right|_{m_{ll}=m_{\text{min}}}.$ (5.2)

The discovery significance is given by [10, 75]

$\displaystyle
Z\equiv\sqrt{2\left(N_{\text{GHU}}\>\ln\frac{N_{\text{GHU}}}{N_{\text{SM}}}+N_{\text{SM}}-N_{\text{GHU}}\right)}.$
(5.3)

Although the formula (5.3) is derived in the large number limit, we use this
formula independent of the numbers of events for simplicity. For a given $Z$,
the corresponding $p$-value, which is defined as the probability of obtaining
a larger excess, is the same as $p=1-F(\mu+Z\sigma)$, where $F$ is the
Gaussian cumulative distribution function, $\mu$ is a mean and $\sigma$ is a
standard deviation. Usually, an excess larger than $5\sigma$ is qualified as a
discovery. Therefore, we estimate the parameter set which gives $Z\simeq 5$.
The $p=0.05$ corresponds to the discovery significance $Z=1.64$, and a model
is allowed at a 95% CL for $Z<1.64$ [75]. We calculate the $p$-value by
assuming that events follow the Poisson distribution.

In the A-model, the differential cross section of the $pp\to l\nu$ process is
almost equal to that in the SM for $m_{\text{KK}}>8$ TeV at $\sqrt{s}=14$ TeV.
For the parameter set A3, the differential cross section of $pp\to e^{+}e^{-}$
process is larger above $m_{\text{min}}=2.287$ TeV. With the luminosity 300
fb-1, $N_{\text{GHU}}=70.1$, $N_{\text{SM}}=28.8$ and the corresponding
discovery significance is 6.49. The numbers of events and discovery
significance of $pp\to\mu^{+}\mu^{-}$ process are smaller than those of the
$pp\to e^{+}e^{-}$ process.

$\theta_{H}$ |  $m_{\text{KK}}$ (TeV) | $z_{L}$ |  $m_{\text{min}}$ (TeV) | $N_{\text{GHU}}$ | $N_{\text{SM}}$ | significance
---|---|---|---|---|---|---
0.09 | 15 | $3.196\times 10^{12}$ | 2.482 | 63.0 | 30.8 | 5.08
0.08 | 17 | $5.001\times 10^{12}$ | 2.831 | 22.5 | 12.0 | 2.69
0.07 | 19 | $1.431\times 10^{12}$ | 3.263 | 6.86 | 3.91 | 1.34
0.06 | 22 | $9.479\times 10^{11}$ | 3.854 | 1.39 | 0.877 | 0.508

Table 11: The model parameters, lower limits of integral, numbers of events
and discovery significance of the $pp\to e\nu$ process at $\sqrt{s}=14$ TeV
with the luminosity 300 fb-1.

$\theta_{H}$ |  $m_{\text{KK}}$ (TeV) | $z_{L}$ |  $m_{\text{min}}$ (TeV) | $N_{\text{GHU}}$ | $N_{\text{SM}}$ | significance
---|---|---|---|---|---|---
0.09 | 15 | $3.196\times 10^{12}$ | 2.790 | 19.0 | 7.96 | 3.33
0.08 | 17 | $5.001\times 10^{12}$ | 3.142 | 7.29 | 3.40 | 1.83
0.07 | 19 | $1.431\times 10^{12}$ | 3.568 | 2.43 | 1.26 | 0.925
0.06 | 22 | $9.479\times 10^{11}$ | 4.152 | 0.562 | 0.335 | 0.357

Table 12: The model parameters, lower limits of integral, numbers of events
and discovery significance of the $pp\to e^{+}e^{-}$ process at $\sqrt{s}=14$
TeV with the luminosity 300 fb-1.

For the B-model, the discovery significance of the $pp\to e\nu$ process is
larger than that of the $pp\to\mu\nu$ and $pp\to l^{+}l^{-}$ processes for the
same parameter sets considered bellow. We choose parameter sets $z_{L}\simeq
O(10^{12})$ with integral $m_{\text{KK}}/$TeV. The results for the $pp\to
e\nu$ and $pp\to e^{+}e^{-}$ processes are summarized in Tables 12 and 12. The
masses and decay widths of the $W^{\prime}$ and $Z^{\prime}$ for those
parameters are shown in Tables 13, 14, 15, 16, 17.

$\theta_{H}$ | $m_{\text{KK}}$ | $m_{W^{(1)}}$ | $m_{W_{R}^{(1)}}$ | $\Gamma_{W^{(1)}}$ | $\Gamma_{W_{R}^{(1)}}$ | $\Gamma_{Z^{(1)}}$ | $\Gamma_{Z_{R}^{(1)}}$ | $\Gamma_{\gamma^{(1)}}$ | Table
---|---|---|---|---|---|---|---|---|---
[rad.] | [TeV] | [TeV] | [TeV] | [TeV] | [TeV] | [TeV] | [TeV] | [TeV] |
0.09 | 15 | 11.74 | 11.48 | 12.17 | 0.458 | 9.712 | 0.989 | 4.007 | 14
0.08 | 17 | 13.31 | 13.01 | 13.98 | 0.527 | 11.16 | 1.132 | 4.603 | 15
0.07 | 19 | 14.89 | 14.54 | 14.98 | 0.563 | 11.96 | 1.231 | 4.952 | 16
0.06 | 22 | 17.24 | 16.84 | 17.10 | 0.642 | 13.66 | 1.413 | 5.662 | 17
Table 13:  Masses and decay widths of $W^{(1)}$, $W_{R}^{(1)}$, $Z^{(1)}$,
$Z_{R}^{(1)}$ and $\gamma^{(1)}$ are listed. $m_{W^{(1)}}\simeq
m_{Z^{(1)}}\simeq m_{\gamma^{(1)}}$ with 4 digits of precision.
$m_{W_{R}^{(1)}}$ and $m_{Z_{R}^{(1)}}$ are exactly equal.

$ff^{\prime}$ | $g_{Wff^{\prime}}^{L}$ | $g_{Wff^{\prime}}^{R}$ | $g_{W^{(1)}ff^{\prime}}^{L}$ | $g_{W^{(1)}ff^{\prime}}^{R}$ | $g_{W_{R}^{(1)}ff^{\prime}}^{L}$ | $g_{W_{R}^{(1)}ff^{\prime}}^{R}$
---|---|---|---|---|---|---
$e\nu_{e}$ | 0.9981 | 0 | 5.9277 | 0 | 0.0121 | 0
$\mu\nu_{\mu}$ | 0.9981 | 0 | 5.6554 | 0 | 0.0116 | 0
$\tau\nu_{\tau}$ | 0.9981 | 0 | 5.4764 | 0 | 0.0113 | 0
$ud$ | 0.9981 | 0 | 5.7462 | 0 | 0.0118 | 0
$cs$ | 0.9981 | 0 | 5.5457 | 0 | 0.0114 | 0
$tb$ | 0.9983 | 0 | 4.6551 | 0 | 0.0097 |  $-$0.0330

$f$ | $g_{Zf}^{L}$ | $g_{Zf}^{R}$ | $g_{Z^{(1)}f}^{L}$ | $g_{Z^{(1)}f}^{R}$ | $g_{Z_{R}^{(1)}f}^{L}$ | $g_{Z_{R}^{(1)}f}^{R}$ | $g_{\gamma^{(1)}f}^{L}$ | $g_{\gamma^{(1)}f}^{R}$
---|---|---|---|---|---|---|---|---
$\nu_{e}$ |  0.5690 | 0 |  3.3812 | 0 |  $-$1.0668 | 0 | 0 | 0
$\nu_{\mu}$ |  0.5690 | 0 |  3.2259 | 0 |  $-$1.0200 | 0 | 0 | 0
$\nu_{\tau}$ |  0.5690 | 0 |  3.1238 | 0 |  $-$0.9892 | 0 | 0 | 0
$e$ |  $-$0.3059 |  0.2630 |  $-$1.8180 |  $-$0.0562 |  $-$1.0770 | 0 |  $-$2.8470 |  0.1030
$\mu$ |  $-$0.3059 |  0.2630 |  $-$1.7345 |  $-$0.0562 |  $-$1.0297 | 0 |  $-$2.7160 |  0.1030
$\tau$ |  $-$0.3059 |  0.2630 |  $-$1.6796 |  $-$0.0562 |  $-$0.9986 | 0 |  $-$2.6303 |  0.1029
$u$ |  0.3936 |  $-$0.1754 |  2.2675 |  0.0375 |  0.3518 | 0 |  1.8399 |  $-$0.0687
$c$ |  0.3936 |  $-$0.1754 |  2.1188 |  0.0375 |  0.3401 | 0 |  1.7757 |  $-$0.0687
$t$ |  0.3939 |  $-$0.1751 |  1.8369 |  $-$0.3128 |  0.2878 |  $-$0.7078 |  1.4907 |  0.5749
$d$ |  $-$0.4813 |  0.0877 |  $-$2.7726 |  0.1133 |  0.3419 |  $-$0.1758 |  $-$0.9200 |  $-$0.2062
$s$ |  $-$0.4813 |  0.0877 |  $-$2.6759 |  0.1428 |  0.3305 |  $-$0.2150 |  $-$0.8878 |  $-$0.2599
$b$ |  $-$0.4813 |  0.0877 |  $-$2.2461 |  0.2844 |  0.2798 |  $-$0.4024 |  $-$0.7452 |  $-$0.5179

Table 14:  Coupling constants of vector bosons are listed for
$\theta_{H}=0.09$ and $m_{\text{KK}}=15$ TeV in Table 13, where
$\sin^{2}\theta_{W}^{0}=0.2307$. The above table shows the couplings of
$W^{\prime}$ bosons to fermions in units of $g_{w}/\sqrt{2}$. The below table
shows the couplings of $Z^{\prime}$ bosons to fermions in units of
$g_{w}=e/\sin\theta_{W}^{0}$. Their corresponding $\gamma$ boson coupling
constants are the same as those in the SM. The values less than $10^{-4}$ are
written as $0$.

$ff^{\prime}$ | $g_{Wff^{\prime}}^{L}$ | $g_{Wff^{\prime}}^{R}$ | $g_{W^{(1)}ff^{\prime}}^{L}$ | $g_{W^{(1)}ff^{\prime}}^{R}$ | $g_{W_{R}^{(1)}ff^{\prime}}^{L}$ | $g_{W_{R}^{(1)}ff^{\prime}}^{R}$
---|---|---|---|---|---|---
$e\nu_{e}$ | 0.9985 | 0 | 5.9655 | 0 | 0.0097 | 0
$\mu\nu_{\mu}$ | 0.9985 | 0 | 5.6937 | 0 | 0.0092 | 0
$\tau\nu_{\tau}$ | 0.9985 | 0 | 5.5155 | 0 | 0.0090 | 0
$ud$ | 0.9985 | 0 | 5.7842 | 0 | 0.0094 | 0
$cs$ | 0.9985 | 0 | 5.5844 | 0 | 0.0091 | 0
$tb$ | 0.9987 | 0 | 4.7041 | 0 | 0.0077 |  $-$0.0327

$f$ | $g_{Zf}^{L}$ | $g_{Zf}^{R}$ | $g_{Z^{(1)}f}^{L}$ | $g_{Z^{(1)}f}^{R}$ | $g_{Z_{R}^{(1)}f}^{L}$ | $g_{Z_{R}^{(1)}f}^{R}$ | $g_{\gamma^{(1)}f}^{L}$ | $g_{\gamma^{(1)}f}^{R}$
---|---|---|---|---|---|---|---|---
$\nu_{e}$ |  0.5693 | 0 |  3.4026 | 0 |  $-$1.0757 | 0 | 0 | 0
$\nu_{\mu}$ |  0.5693 | 0 |  3.2475 | 0 |  $-$1.0288 | 0 | 0 | 0
$\nu_{\tau}$ |  0.5693 | 0 |  3.1459 | 0 |  $-$0.9981 | 0 | 0 | 0
$e$ |  $-$0.3061 |  0.2632 |  $-$1.8294 |  $-$0.0559 |  $-$1.0837 | 0 |  $-$2.8658 |  0.1022
$\mu$ |  $-$0.3061 |  0.2632 |  $-$1.7461 |  $-$0.0559 |  $-$1.0365 | 0 |  $-$2.7352 |  0.1022
$\tau$ |  $-$0.3061 |  0.2632 |  $-$1.6914 |  $-$0.0559 |  $-$1.0056 | 0 |  $-$2.6496 |  0.1021
$u$ |  0.3938 |  $-$0.1754 |  2.2823 |  0.0372 |  0.3534 | 0 |  1.8525 |  $-$0.0681
$c$ |  0.3938 |  $-$0.1754 |  2.2035 |  0.0372 |  0.3417 | 0 |  1.7885 |  $-$0.0681
$t$ |  0.3940 |  $-$0.1752 |  1.8561 |  $-$0.3108 |  0.2902 |  $-$0.7020 |  1.5067 |  0.5702
$d$ |  $-$0.4815 |  0.0877 |  $-$2.7908 |  0.1127 |  0.3455 |  $-$0.1751 |  $-$0.9262 |  $-$0.2052
$s$ |  $-$0.4815 |  0.0877 |  $-$2.6944 |  0.1421 |  0.3341 |  $-$0.2142 |  $-$0.8943 |  $-$0.2588
$b$ |  $-$0.4815 |  0.0877 |  $-$2.2696 |  0.2833 |  0.2837 |  $-$0.4014 |  $-$0.7533 |  $-$0.5161

Table 15:  Coupling constants of vector bosons are listed for
$\theta_{H}=0.08$ and $m_{\text{KK}}=17$ TeV in Table 13, where
$\sin^{2}\theta_{W}^{0}=0.2308$.

$ff^{\prime}$ | $g_{Wff^{\prime}}^{L}$ | $g_{Wff^{\prime}}^{R}$ | $g_{W^{(1)}ff^{\prime}}^{L}$ | $g_{W^{(1)}ff^{\prime}}^{R}$ | $g_{W_{R}^{(1)}ff^{\prime}}^{L}$ | $g_{W_{R}^{(1)}ff^{\prime}}^{R}$
---|---|---|---|---|---|---
$e\nu_{e}$ | 0.9988 | 0 | 5.8583 | 0 | 0.0073 | 0
$\mu\nu_{\mu}$ | 0.9988 | 0 | 5.5850 | 0 | 0.0069 | 0
$\tau\nu_{\tau}$ | 0.9988 | 0 | 5.4046 | 0 | 0.0067 | 0
$ud$ | 0.9987 | 0 | 5.6763 | 0 | 0.0070 | 0
$cs$ | 0.9987 | 0 | 5.4745 | 0 | 0.0068 | 0
$tb$ | 0.9989 | 0 | 4.5622 | 0 | 0.0057 |  $-$0.0336

$f$ | $g_{Zf}^{L}$ | $g_{Zf}^{R}$ | $g_{Z^{(1)}f}^{L}$ | $g_{Z^{(1)}f}^{R}$ | $g_{Z_{R}^{(1)}f}^{L}$ | $g_{Z_{R}^{(1)}f}^{R}$ | $g_{\gamma^{(1)}f}^{L}$ | $g_{\gamma^{(1)}f}^{R}$
---|---|---|---|---|---|---|---|---
$\nu_{e}$ |  0.5695 | 0 |  3.3413 | 0 |  $-$1.0646 | 0 | 0 | 0
$\nu_{\mu}$ |  0.5695 | 0 |  3.1854 | 0 |  $-$1.0173 | 0 | 0 | 0
$\nu_{\tau}$ |  0.5695 | 0 |  3.0825 | 0 |  $-$0.9859 | 0 | 0 | 0
$e$ |  $-$0.3062 |  0.2633 |  $-$1.7964 |  $-$0.0572 |  $-$1.0586 | 0 |  $-$2.8150 |  0.1045
$\mu$ |  $-$0.3062 |  0.2633 |  $-$1.7126 |  $-$0.0572 |  $-$1.0114 | 0 |  $-$2.6836 |  0.1045
$\tau$ |  $-$0.3062 |  0.2633 |  $-$1.6573 |  $-$0.0571 |  $-$0.9803 | 0 |  $-$2.5969 |  0.1044
$u$ |  0.3939 |  $-$0.1755 |  2.2396 |  0.0381 |  0.3463 | 0 |  1.8184 |  $-$0.0697
$c$ |  0.3939 |  $-$0.1755 |  2.1600 |  0.0381 |  0.3346 | 0 |  1.7537 |  $-$0.0697
$t$ |  0.3941 |  $-$0.1754 |  1.8000 |  $-$0.3199 |  0.2814 |  $-$0.7201 |  1.4615 |  0.5861
$d$ |  $-$0.4817 |  0.0878 |  $-$2.7385 |  0.1144 |  0.3404 |  $-$0.1779 |  $-$0.9092 |  $-$0.2083
$s$ |  $-$0.4817 |  0.0878 |  $-$2.6412 |  0.1440 |  0.3289 |  $-$0.2173 |  $-$0.8769 |  $-$0.2622
$b$ |  $-$0.4817 |  0.0878 |  $-$2.2010 |  0.2867 |  0.2766 |  $-$0.4063 |  $-$0.7307 |  $-$0.5223

Table 16:  Coupling constants of vector bosons are listed for
$\theta_{H}=0.07$ and $m_{\text{KK}}=19$ TeV in Table 13, where
$\sin^{2}\theta_{W}^{0}=0.2309$.

$ff^{\prime}$ | $g_{Wff^{\prime}}^{L}$ | $g_{Wff^{\prime}}^{R}$ | $g_{W^{(1)}ff^{\prime}}^{L}$ | $g_{W^{(1)}ff^{\prime}}^{R}$ | $g_{W_{R}^{(1)}ff^{\prime}}^{L}$ | $g_{W_{R}^{(1)}ff^{\prime}}^{R}$
---|---|---|---|---|---|---
$e\nu_{e}$ | 0.9992 | 0 | 5.8233 | 0 | 0.0053 | 0
$\mu\nu_{\mu}$ | 0.9992 | 0 | 5.5496 | 0 | 0.0051 | 0
$\tau\nu_{\tau}$ | 0.9992 | 0 | 5.3685 | 0 | 0.0049 | 0
$ud$ | 0.9992 | 0 | 5.6412 | 0 | 0.0051 | 0
$cs$ | 0.9992 | 0 | 5.4387 | 0 | 0.0050 | 0
$tb$ | 0.9993 | 0 | 4.5149 | 0 | 0.0042 |  $-$0.0339

$f$ | $g_{Zf}^{L}$ | $g_{Zf}^{R}$ | $g_{Z^{(1)}f}^{L}$ | $g_{Z^{(1)}f}^{R}$ | $g_{Z_{R}^{(1)}f}^{L}$ | $g_{Z_{R}^{(1)}f}^{R}$ | $g_{\gamma^{(1)}f}^{L}$ | $g_{\gamma^{(1)}f}^{R}$
---|---|---|---|---|---|---|---|---
$\nu_{e}$ |  0.5697 | 0 |  3.3212 | 0 |  $-$1.0541 | 0 | 0 | 0
$\nu_{\mu}$ |  0.5697 | 0 |  3.1651 | 0 |  $-$1.0069 | 0 | 0 | 0
$\nu_{\tau}$ |  0.5697 | 0 |  3.0618 | 0 |  $-$0.9756 | 0 | 0 | 0
$e$ |  $-$0.3063 |  0.2634 |  $-$1.7854 |  $-$0.0577 |  $-$1.0585 | 0 |  $-$2.7988 |  0.1054
$\mu$ |  $-$0.3063 |  0.2634 |  $-$1.7015 |  $-$0.0577 |  $-$1.0111 | 0 |  $-$2.6672 |  0.1054
$\tau$ |  $-$0.3063 |  0.2634 |  $-$1.6456 |  $-$0.0576 |  $-$0.9797 | 0 |  $-$2.5912 |  0.1053
$u$ |  0.3941 |  $-$0.1756 |  2.2255 |  0.0385 |  0.3438 | 0 |  1.8075 |  $-$0.0702
$c$ |  0.3941 |  $-$0.1756 |  2.1456 |  0.0385 |  0.3320 | 0 |  1.7426 |  $-$0.0702
$t$ |  0.3942 |  $-$0.1755 |  1.7812 |  $-$0.3235 |  0.2782 |  $-$0.7264 |  1.4467 |  0.5919
$d$ |  $-$0.4819 |  0.0878 |  $-$2.7214 |  0.1149 |  0.3395 |  $-$0.1789 |  $-$0.9037 |  $-$0.2093
$s$ |  $-$0.4819 |  0.0878 |  $-$2.6238 |  0.1446 |  0.3279 |  $-$0.2184 |  $-$0.8713 |  $-$0.2634
$b$ |  $-$0.4819 |  0.0878 |  $-$2.1781 |  0.2879 |  0.2748 |  $-$0.4083 |  $-$0.7233 |  $-$0.5245

Table 17:  Coupling constants of vector bosons are listed for
$\theta_{H}=0.06$ and $m_{\text{KK}}=22$ TeV in Table 13, where
$\sin^{2}\theta_{W}^{0}=0.2311$.

For $\theta_{H}=0.09$ and $m_{\text{KK}}=15$ TeV, it is found that
$m_{\text{min}}=2.482$ TeV, $N_{\text{GHU}}=63$ and $N_{\text{SM}}=31$ at the
LHC Run 3, where the discovery significance is $Z=5.08$. When about 63 events
are observed for the $pp\to e\nu$ process in the transverse mass range
$2.5\text{\leavevmode\nobreak\ TeV}\lesssim m_{\text{T}}\lesssim 6.0$ TeV at
the LHC Run 3, the discovery of new physics is expected, and the GHU model
becomes viable. The numbers of events of the GHU (SM) in each bin are 67.5
(92.5), 29.7 (21.6), 24.1 (7.1), 6.3 (0.56) and 1.4 (0.05) for [2000, 2500]
GeV, [2500, 3000] GeV, [3000, 4000] GeV, [4000, 5000] GeV and [5000, 6000]
GeV, respectively.

By interpolating the numerical results shown in Table 12, we obtain $Z=1.64$
for $\theta_{H}=0.0727$ and $m_{\text{KK}}=18.44$ TeV, where
$N_{\text{GHU}}=9.75$ and $N_{\text{SM}}=5.48$. Hence, as a rough estimate,
the upper limit of the KK scale testable at the LHC Run 3 is
$m_{\text{KK}}\simeq 18.44$ TeV. At the HL-LHC, the total integrated
luminosity 3000 fb-1 of data is going to be collected [68, 69, 70]. For
$\theta_{H}=0.06$ and $m_{\text{KK}}=22$ TeV, $N_{\text{GHU}}=13.9$ and
$N_{\text{SM}}=8.77$ at $\sqrt{s}=14$ TeV with the 3000 fb-1 luminosity. The
discovery significance is $Z=1.61$ and the corresponding $p$-value is
$p=0.0627$. The GHU B-model is testable up to $m_{\text{KK}}\simeq 22$ TeV.

We add that backgrounds coming from other processes, acceptance and efficiency
have not been taken into account in the evaluation in this section.

##  6 Summary and Discussions

In this paper we studied the $pp\to\\{W,W^{\prime}\\}\to l\nu$ and
$pp\to\\{\gamma,Z,Z^{\prime}\\}\to l^{+}l^{-}$ ($l=e,\mu$) processes in the
$SU(3)_{C}\times SO(5)\times U(1)_{X}$ GHU models. Due to the behavior of the
wave functions of various fields in the fifth dimension, the $Z^{\prime}$
couplings of right- and left-handed quarks become relatively large in the GHU
A-model and B-model, respectively. The largest decay width of the $Z^{\prime}$
bosons is found to be $\Gamma_{\gamma^{(1)}}/m_{\gamma^{(1)}}\sim 0.1$ in the
A-model and $\Gamma_{Z^{(1)}}/m_{Z^{(1)}}\sim 0.8$ in the B-model. The
$W^{\prime}$ couplings of left-handed quarks also become large in the GHU
B-model, and $\Gamma_{W^{(1)}}/m_{W^{(1)}}\sim 1$. In contrast to it, the
$W^{(1)}$ couplings in the A-model remain small with
$\Gamma_{W^{(1)}}/m_{W^{(1)}}\sim 0.04$.

The differential cross sections of the $pp\to l^{+}l^{-}$ processes in the GHU
models are smaller than those in the SM for the invariant mass $m_{ll}\lesssim
2$ TeV. From the searches for events in the dilepton final states at
$\sqrt{s}=13$ TeV with up to 140 fb-1 of data [2], the A-model is constrained
as $\theta_{H}\lesssim 0.08$ and $m_{\text{KK}}\gtrsim 9.5$ TeV, and the
B-model is constrained as $\theta_{H}\lesssim 0.10$ and $m_{\text{KK}}\gtrsim
13$ TeV.

The differential cross sections of the $pp\to l\nu$ processes in the GHU
B-model are also smaller than those in the SM for the transverse mass
$m_{\text{T}}\lesssim 2$ TeV. The constraint on the B-model from the searches
for events in the lepton and missing transverse mass final states at
$\sqrt{s}=13$ TeV with up to 140 fb-1 of data [3] is severe compared with the
constraint from those in the dilepton final states. The constraint on the
B-model is $\theta_{H}<0.10$ and $m_{\text{KK}}>13$ TeV. The A-model is
consistent with the experimental data due to the narrow decay width of the
$W^{(1)}$ boson.

At $\sqrt{s}=14$ TeV with the luminosity 300 fb-1, signals of $Z^{\prime}$
bosons in the A-model can be seen in the $pp\to l^{+}l^{-}$ processes for
$\theta_{H}=0.08$ and $m_{\text{KK}}=9.5$ TeV. In the B-model, signals of
$W^{\prime}$ bosons can be seen in the $pp\to e\nu$ process for
$m_{\text{KK}}\lesssim 15$ TeV and $\theta_{H}\gtrsim 0.09$. The upper limit
of the KK scale in the B-model can be pushed to $m_{\text{KK}}\simeq 18$ TeV
with the luminosity 300 fb-1 and to $m_{\text{KK}}\simeq 22$ TeV with
luminosity 3000 fb-1.

At the ILC, the effects of $Z^{\prime}$ bosons in fermion pair production
processes can be seen even at $\sqrt{s}=250$ GeV by using polarized electron
and positron beams. For $m_{\text{KK}}<22$ TeV, the deviations of the cross
section for the $e^{+}e^{-}\to\mu^{+}\mu^{-}$ process in the GHU B-model from
that in the SM are $O(1)$% for a left-handed electron beam and $O(0.1)$% with
a right-handed electron beam at $\sqrt{s}=250$ GeV, where the statistical
uncertainty with the 250 fb-1 luminosity data is about 0.1% [50]. To reduce
theoretical uncertainties, further studies beyond the tree-level are
necessary.

Collider physics of radions, KK gravitons and KK gluons are also important
subjects in models defined on a higher dimensional spacetime [76, 77]. For
instance, KK gluons mediate dijet and $t\bar{t}$ production processes at
hadron colliders. In the $t\bar{t}$ production processes, the forward-backward
asymmetry [78, 79] and the charge asymmetry [80, 81] have been measured, which
so far have been consistent with the SM predictions. Effects of KK gluons in
the $pp\to t\bar{t}$ process need be studied in the GHU models as well. KK
gluons in the GHU B-model have large couplings to left-handed fermions with a
large decay width just as KK photons. Broad excesses of the differential cross
sections are foreseen at the LHC in the process mediated by KK gauge bosons,
and the polarization dependence of the cross sections should be confirmed at
the ILC by using polarized beams. Observing these characteristic signals may
provide a strong indication for the existence of the extra dimension.

## Acknowledgments

This work was supported in part by European Regional Development Fund-Project
Engineering Applications of Microworld Physics (Grant No.
CZ.02.1.01/0.0/0.0/16_019/0000766) (Y.O.), by the National Natural Science
Foundation of China (Grants No. 11775092, No. 11675061, No. 11521064, No.
11435003 and No. 11947213) (S.F.), by the International Postdoctoral Exchange
Fellowship Program (S.F.), and by Japan Society for the Promotion of Science,
Grants-in-Aid for Scientific Research, Grant No. 19K03873 (Y.H.) and Grant No.
18H05543 (N.Y.).

## References

  * [1] G. Aad et al. [ATLAS Collaboration], “Search for high-mass dilepton resonances using 139 fb-1 of $pp$ collision data collected at $\sqrt{s}=$13 TeV with the ATLAS detector”, Phys. Lett. B 796, 68-87 (2019).
  * [2] G. Aad et al. [ATLAS Collaboration], “Search for new non-resonant phenomena in high-mass dilepton final states with the ATLAS detector”, JHEP 11, 005 (2020) [erratum: JHEP 04, 142 (2021)].
  * [3] G. Aad et al. [ATLAS Collaboration], “Search for a heavy charged boson in events with a charged lepton and missing transverse momentum from $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector”, Phys. Rev. D 100 (2019) no.5, 052013.
  * [4] G. Aad et al. [ATLAS Collaboration], “Search for new resonances in mass distributions of jet pairs using 139 fb-1 of $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector”, JHEP 03 (2020), 145.
  * [5] A. M. Sirunyan et al. [CMS Collaboration], “Search for resonant and nonresonant new phenomena in high-mass dilepton final states at $\sqrt{s}=$ 13 TeV”, JHEP 07, 208 (2021).
  * [6] A. M. Sirunyan et al. [CMS Collaboration], “Search for high mass dijet resonances with a new background prediction method in proton-proton collisions at $\sqrt{s}=$ 13 TeV”, JHEP 05 (2020), 033.
  * [7] A. Leike, “The Phenomenology of extra neutral gauge bosons”, Phys. Rept. 317 (1999), 143-250.
  * [8] P. Langacker, “The Physics of Heavy $Z^{\prime}$ Gauge Bosons”, Rev. Mod. Phys. 81 (2009), 1199-1228.
  * [9] C. Csaki, C. Grojean and J. Terning, “Alternatives to an Elementary Higgs”, Rev. Mod. Phys. 88 (2016) no.4, 045001.
  * [10] P.A. Zyla et al. [Particle Data Group], “Review of Particle Physics”, PTEP 2020 (2020) no.8, 083C01.
  * [11] E. Accomando, D. Becciolini, S. De Curtis, D. Dominici, L. Fedeli and C. Shepherd-Themistocleous, “Interference effects in heavy $W^{\prime}$-boson searches at the LHC”, Phys. Rev. D 85 (2012), 115017.
  * [12] E. Accomando, D. Becciolini, A. Belyaev, S. Moretti and C. Shepherd-Themistocleous, “$Z^{\prime}$ at the LHC: Interference and Finite Width Effects in Drell-Yan”, JHEP 10 (2013), 153.
  * [13] D. Barducci, A. Belyaev, S. De Curtis, S. Moretti and G. M. Pruna, “Exploring Drell-Yan signals from the 4D Composite Higgs Model at the LHC”, JHEP 04 (2013), 152.
  * [14] D. Greco and D. Liu, “Hunting composite vector resonances at the LHC: naturalness facing data”, JHEP 12 (2014), 126.
  * [15] D. Liu, L. T. Wang and K. P. Xie, “Prospects of searching for composite resonances at the LHC and beyond”, JHEP 01 (2019), 157.
  * [16] L. Edelhäuser, T. Flacke and M. Krämer, “Constraints on models with universal extra dimensions from dilepton searches at the LHC”, JHEP 08 (2013), 091.
  * [17] N. Deutschmann, T. Flacke and J. S. Kim, “Current LHC Constraints on Minimal Universal Extra Dimensions”, Phys. Lett. B 771 (2017), 515-520.
  * [18] K. S. Agashe, J. Collins, P. Du, S. Hong, D. Kim and R. K. Mishra, “LHC Signals from Cascade Decays of Warped Vector Resonances”, JHEP 05 (2017), 078.
  * [19] T. Han, I. Lewis, R. Ruiz and Z. g. Si, “Lepton Number Violation and $W^{\prime}$ Chiral Couplings at the LHC”, Phys. Rev. D 87 (2013) no.3, 035011 [erratum: Phys. Rev. D 87 (2013) no.3, 039906].
  * [20] C. W. Chiang, T. Nomura and K. Yagyu, “Phenomenology of $E_{6}$-Inspired Leptophobic $Z^{\prime}$ Boson at the LHC”, JHEP 05 (2014), 106.
  * [21] D. Pappadopulo, A. Thamm, R. Torre and A. Wulzer, “Heavy Vector Triplets: Bridging Theory and Data”, JHEP 09 (2014), 060.
  * [22] M. Fairbairn, J. Heal, F. Kahlhoefer and P. Tunney, “Constraints on $Z^{\prime}$ models from LHC dijet searches and implications for dark matter”, JHEP 09 (2016), 018.
  * [23] A. Das, S. Oda, N. Okada and D. s. Takahashi, “Classically conformal $U(1)^{\prime}$ extended standard model, electroweak vacuum stability, and LHC Run-2 bounds”, Phys. Rev. D 93 (2016) no.11, 115038.
  * [24] M. Mitra, R. Ruiz, D. J. Scott and M. Spannowsky, “Neutrino Jets from High-Mass $W_{R}$ Gauge Bosons in TeV-Scale Left-Right Symmetric Models”, Phys. Rev. D 94 (2016) no.9, 095016.
  * [25] S. Amrith, J. M. Butterworth, F. F. Deppisch, W. Liu, A. Varma and D. Yallup, “LHC Constraints on a $B-L$ Gauge Model using Contur,” JHEP 05 (2019), 154.
  * [26] C. W. Chiang, G. Cottin, A. Das and S. Mandal, “Displaced heavy neutrinos from $Z^{\prime}$ decays at the LHC”, JHEP 12 (2019), 070.
  * [27] F. F. Deppisch, S. Kulkarni and W. Liu, “Searching for a light $Z^{\prime}$ through Higgs production at the LHC”, Phys. Rev. D 100 (2019) no.11, 115023.
  * [28] E. Accomando, F. Coradeschi, T. Cridge, J. Fiaschi, F. Hautmann, S. Moretti, C. Shepherd-Themistocleous and C. Voisey, “Production of $Z^{\prime}$-boson resonances with large width at the LHC”, Phys. Lett. B 803 (2020), 135293.
  * [29] A. Hayreter, X. G. He and G. Valencia, “LHC constraints on $W^{\prime},\leavevmode\nobreak\ Z^{\prime}$ that couple mainly to third generation fermions”, Eur. Phys. J. C 80 (2020) no.10, 912.
  * [30] A. Das, P. S. B. Dev, Y. Hosotani and S. Mandal, “Probing the minimal $U(1)_{X}$ model at future electron-positron colliders via the fermion pair-production channel”, arXiv:2104.10902 [hep-ph].
  * [31] Y. Hosotani, “Dynamical Mass Generation by Compact Extra Dimensions”, Phys. Lett. 126B, 309 (1983).
  * [32] Y. Hosotani, “Dynamics of Nonintegrable Phases and Gauge Symmetry Breaking”, Ann. Phys. 190, 233 (1989).
  * [33] A. T. Davies and A. McLachlan, “Gauge Group Breaking By Wilson Loops”,
Phys. Lett. B 200 (1988) 305.

  * [34] A. T. Davies and A. McLachlan, “Congruency Class Effects in the Hosotani Model”, Nucl. Phys. B 317 (1989) 237.
  * [35] H. Hatanaka, T. Inami and C. S. Lim, “The Gauge hierarchy problem and higher dimensional gauge theories”, Mod. Phys. Lett. A 13, 2601 (1998).
  * [36] H. Hatanaka, “Matter representations and gauge symmetry breaking via compactified space”, Prog. Theor. Phys. 102, 407 (1999).
  * [37] M. Kubo, C. S. Lim and H. Yamashita, “The Hosotani mechanism in bulk gauge theories with an orbifold extra space $S^{1}/Z_{2}$”, Mod. Phys. Lett. A 17 (2002), 2249-2264.
  * [38] G. Burdman and Y. Nomura, “Unification of Higgs and Gauge Fields in Five Dimensions”, Nucl. Phys. B 656 (2003) 3.
  * [39] C. Csaki, C. Grojean and H. Murayama, “Standard model Higgs from higher dimensional gauge fields”, Phys. Rev. D 67 (2003) 085012.
  * [40] C. A. Scrucca, M. Serone and L. Silvestrini, “Electroweak symmetry breaking and fermion masses from extra dimensions”, Nucl. Phys. B 669 (2003), 128-158.
  * [41] A. D. Medina, N. R. Shah and C. E. M. Wagner, “Gauge-Higgs Unification and Radiative Electroweak Symmetry Breaking in Warped Extra Dimensions”, Phys. Rev. D 76 (2007) 095010.
  * [42] S. Funatsu, H. Hatanaka, Y. Hosotani, Y. Orikasa and T. Shimotani, “Novel universality and Higgs decay $H\to\gamma\gamma$, $gg$ in the $SO(5)\times U(1)$ gauge-Higgs unification”, Phys. Lett. B 722, 94 (2013).
  * [43] S. Funatsu, H. Hatanaka, Y. Hosotani, Y. Orikasa and T. Shimotani, “LHC signals of the $SO(5)\times U(1)$ gauge-Higgs unification”, Phys. Rev. D 89, no. 9, 095019 (2014).
  * [44] S. Funatsu, H. Hatanaka, Y. Hosotani and Y. Orikasa, “Collider signals of $W^{\prime}$ and $Z^{\prime}$ bosons in the gauge-Higgs unification”, Phys. Rev. D 95, no. 3, 035032 (2017).
  * [45] S. Funatsu, H. Hatanaka, Y. Hosotani and Y. Orikasa, “Distinct signals of the gauge-Higgs unification in $e^{+}e^{-}$ collider experiments”, Phys. Lett. B 775, 297 (2017).
  * [46] S. Funatsu, “Forward-backward asymmetry in the gauge-Higgs unification at the International Linear Collider”, Eur. Phys. J. C 79 (2019) no.10, 854.
  * [47] S. Funatsu, H. Hatanaka, Y. Hosotani, Y. Orikasa and N. Yamatsu, “GUT inspired $SO(5)\times U(1)\times SU(3)$ gauge-Higgs unification”, Phys. Rev. D 99 (2019) no.9, 095010.
  * [48] S. Funatsu, H. Hatanaka, Y. Hosotani, Y. Orikasa and N. Yamatsu, “CKM matrix and FCNC suppression in $SO(5)\times U(1)\times SU(3)$ gauge-Higgs unification”, Phys. Rev. D 101 (2020) no.5, 055016.
  * [49] S. Funatsu, H. Hatanaka, Y. Hosotani, Y. Orikasa and N. Yamatsu, “Effective potential and universality in GUT-inspired gauge-Higgs unification”, Phys. Rev. D 102 (2020) no.1, 015005.
  * [50] S. Funatsu, H. Hatanaka, Y. Hosotani, Y. Orikasa and N. Yamatsu, “Fermion pair production at $e^{-}e^{+}$ linear collider experiments in GUT inspired gauge-Higgs unification”, Phys. Rev. D 102 (2020) no.1, 015029.
  * [51] S. Funatsu, H. Hatanaka, Y. Hosotani, Y. Orikasa and N. Yamatsu, “Electroweak and Left-Right Phase Transitions in $SO(5)\times U(1)\times SU(3)$ Gauge-Higgs Unification”, [arXiv:2104.02870 [hep-ph]].
  * [52] Y. Matsumoto and Y. Sakamura, “Yukawa couplings in 6D gauge-Higgs unification on $T^{2}/Z_{N}$ with magnetic fluxes”, PTEP 2016, no. 5, 053B06 (2016).
  * [53] K. Hasegawa and C. S. Lim, “Majorana neutrino masses in the scenario of gauge-Higgs unification”, PTEP 2018, no. 7, 073B01 (2018).
  * [54] J. Yoon and M. E. Peskin, “Competing forces in five-dimensional fermion condensation”, Phys. Rev. D 96, no. 11, 115030 (2017).
  * [55] J. Yoon and M. E. Peskin, “Dissection of an $SO(5)\times U(1)$ gauge-Higgs unification model”, Phys. Rev. D 100, no. 1, 015001 (2019).
  * [56] M. Kakizaki and S. Suzuki, “Higgs potential in gauge-Higgs unification with a flat extra dimension”, Phys. Lett. B 822 (2021), 136637.
  * [57] Y. Hosotani and N. Yamatsu, “Gauge-Higgs grand unification”, PTEP 2015 (2015) 111B01.
  * [58] Y. Hosotani and N. Yamatsu, “Electroweak symmetry breaking and mass spectra in six-dimensional gauge-Higgs grand unification”, PTEP 2018 (2018) no. 2, 023B05.
  * [59] C. Englert, D. J. Miller and D. D. Smaranda, “Phenomenology of GUT-inspired gauge-Higgs unification”, Phys. Lett. B 802 (2020), 135261.
  * [60] C. Englert, D. J. Miller and D. D. Smaranda, “The Weinberg angle and 5D RGE effects in a $SO$ (11) GUT theory”, Phys. Lett. B 807 (2020), 135548.
  * [61] M. Kakizaki, S. Kanemura, H. Taniguchi and T. Yamashita, “Higgs sector as a probe of supersymmetric grand unification with the Hosotani mechanism”, Phys. Rev. D 89 (2014) no.7, 075013.
  * [62] K. Kojima, K. Takenaga and T. Yamashita, “The Standard Model Gauge Symmetry from Higher-Rank Unified Groups in Grand Gauge-Higgs Unification Models”, JHEP 1706 (2017) 018.
  * [63] N. Maru and Y. Yatagai, “Fermion Mass Hierarchy in Grand Gauge-Higgs Unification”, PTEP 2019 (2019) no.8, 083B03.
  * [64] A. Angelescu, A. Bally, S. Blasi and F. Goertz, “Minimal SU(6) Gauge-Higgs Grand Unification”, arXiv:2104.07366 [hep-ph].
  * [65] D. M. Asner, T. Barklow, C. Calancha, K. Fujii, N. Graf, H. E. Haber, A. Ishikawa, S. Kanemura, S. Kawada and M. Kurata, et al. “ILC Higgs White Paper”, arXiv:1310.0763 [hep-ph].
  * [66] K. Fujii, C. Grojean, M. E. Peskin, T. Barklow, Y. Gao, S. Kanemura, H. Kim, J. List, M. Nojiri and M. Perelstein, et al. “Physics Case for the 250 GeV Stage of the International Linear Collider”, arXiv:1710.07621 [hep-ex].
  * [67] P. Bambade, T. Barklow, T. Behnke, M. Berggren, J. Brau, P. Burrows, D. Denisov, A. Faus-Golfe, B. Foster and K. Fujii, et al. “The International Linear Collider: A Global Project”, arXiv:1903.01629 [hep-ex].
  * [68] P. Azzi, S. Farry, P. Nason, A. Tricoli, D. Zeppenfeld, R. Abdul Khalek, J. Alimena, N. Andari, L. Aperio Bella and A. J. Armbruster, et al., “Report from Working Group 1: Standard Model Physics at the HL-LHC and HE-LHC”, CERN Yellow Rep. Monogr. 7 (2019) 1–220, [arXiv:1902.04070].
  * [69] M. Cepeda, S. Gori, P. Ilten, M. Kado, F. Riva, R. Abdul Khalek, A. Aboubrahim, J. Alimena, S. Alioli and A. Alves, et al., “Report from Working Group 2: Higgs Physics at the HL-LHC and HE-LHC”, CERN Yellow Rep. Monogr. 7 (2019) 221–584, [arXiv:1902.00134].
  * [70] X. Cid Vidal, M. D’Onofrio, P. J. Fox, R. Torre, K. A. Ulmer, A. Aboubrahim, A. Albert, J. Alimena, B. C. Allanach and C. Alpigiani, et al., “Report from Working Group 3: Beyond the Standard Model physics at the HL-LHC and HE-LHC”, CERN Yellow Rep. Monogr. 7 (2019) 585–865, [arXiv:1812.07831].
  * [71] F. Halzen and A. D. Martin, “Quarks and Leptons: An introductory course in modern particle physics”, John Wiley & Sons, (1984).
  * [72] M. E. Peskin and D. V. Schroeder, “An Introduction to Quantum Field Theory”, Addison-Wesley Publishing Company, (1995).
  * [73] H. L. Lai, M. Guzzi, J. Huston, Z. Li, P. M. Nadolsky, J. Pumplin and C. P. Yuan, “New parton distributions for collider physics”, Phys. Rev. D 82 (2010), 074024.
  * [74] D. B. Clark, E. Godat and F. I. Olness, “ManeParse : A Mathematica reader for Parton Distribution Functions”, Comput. Phys. Commun. 216 (2017), 126-137.
  * [75] G. Cowan, K. Cranmer, E. Gross and O. Vitells, “Asymptotic formulae for likelihood-based tests of new physics”, Eur. Phys. J. C 71 (2011), 1554 [erratum: Eur. Phys. J. C 73 (2013), 2501].
  * [76] K. Agashe, M. Ekhterachian, D. Kim and D. Sathyan, “LHC Signals for KK Graviton from an Extended Warped Extra Dimension”, JHEP 11 (2020), 109.
  * [77] R. Escribano, M. Mendizabal, M. Quirós and E. Royo, “On Broad Kaluza-Klein Gluons”, JHEP 05 (2021), 121.
  * [78] T. A. Aaltonen et al. [CDF and D0 Collaborations], “Combined Forward-Backward Asymmetry Measurements in Top-Antitop Quark Production at the Tevatron”, Phys. Rev. Lett. 120 (2018) no.4, 042001.
  * [79] A. M. Sirunyan et al. [CMS Collaboration], “Measurement of the top quark forward-backward production asymmetry and the anomalous chromoelectric and chromomagnetic moments in pp collisions at $\sqrt{s}$ = 13 TeV”, JHEP 06 (2020), 146.
  * [80] M. Aaboud et al. [ATLAS and CMS Collaborations], “Combination of inclusive and differential $t\bar{t}$ charge asymmetry measurements using ATLAS and CMS data at $\sqrt{s}$=7 and 8 TeV”, JHEP 04 (2018), 033.
  * [81] [ATLAS Collaboration], “Inclusive and differential measurement of the charge asymmetry in $t\bar{t}$ events at 13 TeV with the ATLAS detector”, ATLAS-CONF-2019-026.


suitability of DAU, and deontic logic more generally, for the design and
verification of autonomous systems.

## References

  * (1)
  * Ågotnes et al. (2009) Thomas Ågotnes, Wiebe vanÂ der Hoek, Juan A. Rodríguez-Aguilar, Carles Sierra, and Michael Wooldridge. 2009\.  _A Temporal Logic of Normative Systems_.  Springer Netherlands, Dordrecht, 69–106.  https://doi.org/10.1007/978-1-4020-9084-4_5
  * Alur et al. (2002) Rajeev Alur, Thomas A. Henzinger, and Orna Kupferman. 2002\.  Alternating-time Temporal Logic.  _J. ACM_ 49, 5 (Sept. 2002), 672–713.  https://doi.org/10.1145/585265.585270
  * Arkoudas et al. (2005) Konstantine Arkoudas, Selmer Bringsjord, and Paul Bello. 2005\.  _Toward Ethical Robots via Mechanized Deontic_.  Technical Report. AAAI Fall Symposium on Machine Ethics.
  * Artosi et al. (1994) Alberto Artosi, Paola Cattabriga, and Guido Governatori. 1994\.  KED: A Deontic Theorem Prover. In _on Legal Application of Logic Programming, ICLP’94_. 60–76.
  * Balbiani et al. (2009) Philippe Balbiani, Jan Broersen, and Julien Brunel. 2009\.  Decision Procedures for a Deontic Logic Modeling Temporal Inheritance of Obligations.  _Electronic Notes in Theoretical Computer Science_ 231 (2009), 69 – 89.  https://doi.org/10.1016/j.entcs.2009.02.030 Proceedings of the 5th Workshop on Methods for Modalities (M4M5 2007).
  * Bellman (1957) Richard Bellman. 1957\.  A Markovian Decision Process.  _Indiana Univ. Math. J._ 6 (1957), 679–684.  Issue 4.
  * Ben Amor et al. (2019) Heni Ben Amor, Aviral Shrivastava, Lina Karam, Adel Dokhanchi, Shakiba Yaghoubi, Mohammad Hekmatnejad, and Georgios Fainekos. 2019.  Encoding and Monitoring Responsibility Sensitive Safety Rules for Automated Vehicles in Signal Temporal Logic.  https://doi.org/10.1145/3359986.3361203
  * Benzm’́uller et al. (2018) Christoph Benzm’́uller, Ali Farjami, and Xavier Parent. 2018\.  A Dyadic Deontic Logic in HOL. In _DEON_.
  * Boella et al. (2006) Guido Boella, Leendert van der Torre, and Harko Verhagen. 2006\.  Introduction to normative multiagent systems.  _Computational & Mathematical Organization Theory_ 12 (10 2006), 71–79.  https://doi.org/10.1007
  * Boker et al. (2014) Udi Boker, Krishnendu Chatterjee, Thomas A. Henzinger, and Orna Kupferman. 2014.  Temporal Specifications with Accumulative Values.  _ACM Trans. Comput. Logic_ 15, 4, Article 27 (July 2014), 25 pages.
  * Bringsjord et al. (2006) Selmer Bringsjord, Konstantine Arkoudas, and Paul Bello. 2006\.  Toward a General Logicist Methodology for Engineering Ethically Correct Robots.  _IEEE Intelligent Systems_ 21 (07 2006), 38–44.  https://doi.org/10.1109/MIS.2006.82
  * Broersen (2006) Jan Broersen. 2006\.  Strategic Deontic Temporal Logic as a Reduction to ATL, with an Application to Chisholm’s Scenario. In _Deontic Logic and Artificial Normative Systems_ , Lou Goble and John-Jules Ch. Meyer (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg, 53–68.
  * Broersen and Brunel (2008) Jan Broersen and Julien Brunel. 2008.  ‘What I fail to do Today, I Have to Do Tomorrow’: A Logical Study of the Propagation of Obligations. In _Computational Logic in Multi-Agent Systems_ , Fariba Sadri and Ken Satoh (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg, 82–99.
  * Cavada et al. (2014) Roberto Cavada, Alessandro Cimatti, Michele Dorigatti, Alberto Griggio, Alessandro Mariotti, Andrea Micheli, Sergio Mover, Marco Roveri, and Stefano Tonetta. 2014.  The nuXmv Symbolic Model Checker. In _CAV_ _(Lecture Notes in Computer Science, Vol. 8559)_ , Armin Biere and Roderick Bloem (Eds.). Springer, 334–342.
  * Chatterjee et al. (2008) Krishnendu Chatterjee, Laurent Doyen, and Thomas A. Henzinger. 2008.  Quantitative Languages. In _Computer Science Logic_ , Michael Kaminski and Simone Martini (Eds.). Springer Berlin Heidelberg, 385–400.
  * Chellas (1968) B.F. Chellas. 1968\.  _The Logical Form of Imperatives_.  Department of Philosophy, Stanford University.
  * Clarke et al. (1999) Edmund M. Clarke, Orna Grumberg, and Doron A. Peled. 1999\.  _Model Checking_.  MIT Press, Cambridge, Massachusetts.
  * Gabbay et al. (2013) Dov Gabbay, John Horty, and Xavier Parent (Eds.). 2013\.  _Handbook of deontic logic and normative systems_.  College Publications.
  * Gerdes and Thornton (2015) J. Christian Gerdes and Sarah M. Thornton. 2015.  _Implementable Ethics for Autonomous Vehicles_.  Springer Berlin Heidelberg, Berlin, Heidelberg, 87–102.
  * Giordano et al. (2013) Laura Giordano, Alberto Martelli, and Daniele Theseider Dupré. 2013\.  Temporal Deontic Action Logic for the Verification of Compliance to Norms in ASP. In _Proc. of the 14th Intl. Conf. on Artificial Intelligence and Law_ (Rome, Italy) _(ICAIL ’13)_. ACM, New York, NY, USA, 53–62.
  * Girault (2004) Alain Girault. 2004\.  A hybrid controller for autonomous vehicles driving on automated highways.  _Transportation Research Part C: Emerging Technologies_ 12, 6 (2004), 421 – 452.  https://doi.org/10.1016/j.trc.2004.07.008
  * Hilpinen and McNamara (2013) Risto Hilpinen and Paul McNamara. 2013.  Deontic Logic: A historical survey and introduction.
  * Horty (2001) John Horty. 2001\.  _Agency and Deontic Logic_.  Cambridge University Press.
  * Horty and Belnap (1995) John F. Horty and Nuel Belnap. 1995.  The deliberative stit: A study of action, omission, ability, and obligation.  _J. Philos. Logic_ (1995), 583–644.
  * Jha et al. (2019) Susmit Jha, Tuhin Sahai, Vasumathi Raman, Alessandro Pinto, and Michael Francis. 2019.  Explaining AI Decisions Using Efficient Methods for Learning Sparse Boolean Formulae.  _J Autom Reasoning_ (2019), 1055––1075.
  * Kulicki et al. (2018) Piotr Kulicki, Robert Trypuz, and Michael P. Musielewicz. 2018\.  Towards a Formal Ethics for Autonomous Cars. In _Deontic Logic and Normative Systems - 14th International Conference, DEON 2018, Utrecht, The Netherlands, July 3-6, 2018_ , Jan M. Broersen, Cleo Condoravdi, Nair Shyam, and Gabriella Pigozzi (Eds.). College Publications, 193–209.
  * Lin et al. (2014) Patrick Lin, Keith Abney, and George A. Bekey (Eds.). 2014\.  _Robot Ethics: The Ethical and Social Implications of Robotics_.  The MIT Press.
  * Lokhorst ([n.d.]) Gert-Jan Lokhorst. [n.d.].  Mally’s Deontic Logic.  _The Stanford Encyclopedia of Philosophy (Summer 2019 Edition)_ ([n. d.]).  https://plato.stanford.edu/archives/sum2019/entries/mally-deontic/.
  * Lomuscio et al. (2017) Alessio Lomuscio, Hongyang Qu, and Franco Raimondi. 2017\.  MCMAS: an open-source model checker for the verification of multi-agent systems.  _Intl. Jrnl. on Software Tools for Technology Transfer_ 19, 1 (01 Feb 2017), 9–30.
  * Mally (1926) Ernst Mally. 1926\.  _Grundgesetze des Sollens. Elemente der Logik des Willens_.
  * Manna and Pnueli (1992) Zohar Manna and Amir Pnueli. 1992.  _The Temporal Logic of Reactive and Concurrent Systems — Specification_.  Springer.
  * McNamara (2018) Paul McNamara. 2018\.  Deontic Logic.  _The Stanford Encyclopedia of Philosophy_ (Fall 2018).
  * Murakami (2004) Yuko Murakami. 2004\.  Utilitarian Deontic Logic. In _in ‘Proceedings of the Fifth International Conference on Advances in Modal Logic (AiML 2004_. 288–302.
  * Pnueli (1977) Amir Pnueli. 1977\.  The Temporal Logic of Programs. In _Proceedings of the 18th IEEE Symposium Foundations of Computer Science_. 46–57.
  * Prisacariu and Schneider (2012) Cristian Prisacariu and Gerardo Schneider. 2012.  A dynamic deontic logic for complex contracts.  _The Journal of Logic and Algebraic Programming_ 81, 4 (2012), 458 – 490.  Special Issue: NWPT 2009.
  * Puterman (1994) Martin L Puterman. 1994\.  _Markov decision processes : discrete stochastic dynamic programming_.  Wiley, New York.
  * Raimondi and Lomuscio (2004) Franco Raimondi and Alessio Lomuscio. 2004.  Automatic Verification of Deontic Interpreted Systems by Model Checking via OBDD’s. In _Procs. of the 16th European Conf. on Artificial Intelligence_.
  * Rizaldi and Althoff (2015) A. Rizaldi and M. Althoff. 2015.  Formalising Traffic Rules for Accountability of Autonomous Vehicles. In _2015 IEEE 18th International Conference on Intelligent Transportation Systems_. 1658–1665.
  * Shalev-Shwartz et al. (2018) Shai Shalev-Shwartz, Shaked Shammah, and Amnon Shashua. 2018\.  On a Formal Model of Safe and Scalable Self-driving Cars. (October 2018).  arXiv:1708.06374v6.
  * Shea-Blymyer and Abbas (2020) Colin Shea-Blymyer and Houssam Abbas. 2020.  A Deontic Logic Analysis of Autonomous Systems’ Safety. In _Proceedings of the 23rd International Conference on Hybrid Systems: Computation and Control_ (Sydney, New South Wales, Australia) _(HSCC ’20)_. Association for Computing Machinery, New York, NY, USA, Article 26, 11 pages.  https://doi.org/10.1145/3365365.3382203
  * Thekkilakattil and Dodig-Crnkovic (2015) A. Thekkilakattil and G. Dodig-Crnkovic. 2015.  Ethics Aspects of Embedded and Cyber-Physical Systems. In _2015 IEEE 39th Annual Computer Software and Applications Conference_ , Vol. 2. 39–44.
  * von Wright (1951) Georg H. von Wright. 1951\.  Deontic Logic.  _Mind_ 60, 237 (January 1951).
  * Wiewiora (2010) Eric Wiewiora. 2010\.  _Reward Shaping_.  Springer US, Boston, MA, 863–865.  https://doi.org/10.1007/978-0-387-30164-8_731