A Hierarchical MPC Approach to Car-Following
via Linearly Constrained Quadratic Programming

This material is based upon work supported by the National Science Foundation under Grant Numbers CNS-1837244.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
This material is based upon work supported by the U.S. Department of Energy’s Office of Energy Efficiency and Renewable Energy (EERE) award number CID DE-EE0008872.
The views expressed herein do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

Fangyu Wu
Dept. of EECS

University of California, Berkeley

Berkeley, USA

Alexandre M. Bayen
Dept. of EECS

University of California, Berkeley

Berkeley, USA

Single-lane car-following is a fundamental task in autonomous driving.
A desirable car-following controller should keep a reasonable range of distances to the preceding vehicle and do so as smoothly as possible.
To achieve this, numerous control methods have been proposed:
some only rely on local sensing; others also make use of non-local downstream observations.
While local methods are capable of attenuating high-frequency velocity oscillation and are economical to compute,
non-local methods can dampen a wider spectrum of oscillatory traffic but incur a larger cost in computing.
In this article, we design a novel non-local tri-layer MPC controller that is capable of smoothing a wide range of oscillatory traffic and is amenable to real-time applications.
At the core of the controller design are 1) an accessible prediction method based on ETA estimation and 2) a robust, light-weight optimization procedure, designed specifically for handling various headway constraints.
Numerical simulations suggest that the proposed controller can simultaneously maintain a variable headway while driving with modest acceleration and is robust to imperfect traffic predictions.

Autonomous vehicles, predictive control for linear systems, hierarchical control.

§ INTRODUCTION

Despite significant progress in recent years, autonomous driving still remains a challenging problem.
Among many problems in this area, single-lane car-following is arguably a fundamental quest.

It is widely known that human car-following is inherently sub-optimal.

Suigiyama and Tadaki have demonstrated in [1, 2] that stop-and-go waves could emerge in a circular platoon of human vehicles without the presence of physical bottlenecks or lane changes.

To improve, various local controllers have been proposed [3, 4, 5, 6, 7].
For example, Luo has described a model predictive control (MPC) formulation for fuel-efficient, fixed-headway adaptive cruise control [5].
Kerner et. al. have presented a variable-headway controller that can attenuate propagation of stop-and-go waves [6].
In addition, Stern et. al. have demonstrated that a single vehicle controlled by a proportional-integral controller is able to dampen stop-and-go waves induced by human drivers in a circular platoon [7].

Beyond local methods, various approaches on non-local controllers have been proposed in an effort to attenuate a wider spectrum of oscillations, such as [8, 9, 10].
The key advantage of non-local controllers lies at the fact that with non-local knowledge of downstream traffic it can react in advance to dampen non-local oscillations.

For instance, [9, 10] introduced various variable-headway MPC methods to achieve smoother car-following by using information from downstream traffic lights.

A common feature of the non-local methods above is a bi-layer structure consisting of a prediction layer and a control layer.
To predict, these methods require dedicated sensing infrastructure and either vehicle-to-infrastructure [9, 10] or vehicle-to-vehicle [8] communication.
While non-local methods may have higher performance upper bound, they are also more prone to errors [11].
Yet, mitigating the impacts of erroneous predictions has been largely overlooked in previous works.

To address prediction errors, a common idea is to subdivide the control layer into a planning layer and a tracking layer.
First, the planning layer produces a reference trajectory using the forecast from the prediction layer.
Then, the tracking layer attempts to track this reference without violating certain performance constraints.
This approach has found many applications in transportation outside of car-following.
For example, [12, 13] have applied this idea for autonomous intersection coordination.

Adopting this idea of planning-then-tracking, we propose a novel hierarchical MPC controller, consisting of a prediction layer, a planning layer, and a tracking layer.
The core MPC formulation assumes the form of a linearly constrained quadratic program (LCQP).
Due to the tri-layer design, the controller is robust to prediction errors.
In contrast to [9, 14], the method does not require dedicated sensing and communication infrastructure for prediction.
Rather, it only uses an estimated time of arrival (ETA) estimator, which is commonly accessible from mainstream map service providers and needs only cellular network.
In addition, compared to [9, 10], our method admits a more flexible formulation of headway constraints and is better understood when it comes to theoretical properties such as optimality and feasibility.

We regard the primary contribution of the paper as a novel tri-layer MPC design, along with a comprehensive numerical study that demonstrates its robustness to prediction errors.
At the code of the MPC design are:
1) A simple, accessible prediction method using an ETA estimator and
2) A robust, light-weight LCQP formulation.

We state the problem of the two-vehicle car-following in Section <ref>.
Next, we formally define and analyze the tri-layer hierarchical MPC control scheme in Section <ref>.
We evaluate the performance of our MPC controller via numerical simulations in Section <ref> and discuss the numerical results in Section <ref>.
Lastly, we conclude our findings in Section <ref>.

§ PROBLEM STATEMENT

Consider a two-vehicle setup, where a considered vehicle follows a preceding vehicle that travels along a path of length $L$.
Denote the initial and final times of the trip as $t_{0}$ and $t_{f}$, respectively.

Let the state of a vehicle be $\vec{x} \coloneqq [s \quad v]^{\top}$, where $s$ and $v$ represent the position and speed of a vehicle, respectively.
Define $a$ as the acceleration of a vehicle.
Let $\ell$ be the length of a vehicle.
We use superscript to identify to which vehicle a variable belongs, where the considered vehicle is indexed as one and the preceding vehicle zero.
Thus, $\vec{x}^{1}$ refers to the state of the considered vehicle.
Let the initial position of the considered vehicle be $w_{0}$, that is, $s^{1}(t_{0}) \coloneqq w_{0}$.
Let the initial position of the preceding vehicle be $w_{1}$, that is, $s^{0}(t_{0}) \coloneqq w_{1}$.
Let $w_{f} \coloneqq w_{1} + L$.
Then, $s^{0}(t_{f}) \coloneqq w_{f}$.
The two-vehicle scenario is illustrated in Figure <ref>.

[xscale=-1] (considered_vehicle) at (0.75, 0.5) ;
[xscale=-1] (preceding_vehicle) at (3.5, 0.5) ;
[->, >=latex, line width=0.35mm] (0, 0) to (8, 0) node[right] $s$;
(1.5, 0.1) to (1.5, 0) node[below] $w_{0}$;
(4.25, 0.1) to (4.25, 0) node[below] $w_{1}$;
(5, 0.1) to (5, 0) node[below] $w_{2}$;
(5.75, 0) node[below] $\dots$;
(6.5, 0.1) to (6.5, 0) node[below] $w_{f-1}$;
(7.25, 0.1) to (7.25, 0) node[below] $w_{f}$;
(0.75, 0.9) node[above] $\vec{x}^{1}, a^{1}, \ell^{1}$;
(0.75, 1.5) node[above] Consid. Veh.;
(3.5, 0.9) node[above] $\vec{x}^{0}, a^{0}, \ell^{0}$;
(3.5, 1.5) node[above] Preced. Veh.;
[decoration=brace, decorate] (4.25, 0.2) – node[above=3pt] $\Delta{s}$ (5, 0.2);

[decoration=brace,mirror,raise=5pt, decorate] (4.25, -0.25) – node[below=6pt] $L$ (7.25, -0.25);
[->, >=latex, line width=0.2mm, dotted] (1.5, 0.5) to node[above]RADAR (2.75, 0.5);
[->, >=latex, line width=0.2mm, dotted] (-0.75, 1.25) |- (0, 0.5);
(network) at (-0.75, 1.5) ;
(-0.375, 0.5) node[above] ETA;

Scenario of the two-vehicle car-following problem.
At $t_{0}$, the considered vehicle is at $w_{0}$ and the preceding vehicle at $w_{1}$.
The preceding vehicle travels through a set of way points $(w_{1}, w_{2}, \dots, w_{f})$, where each pair of consecutive way points are spaced $\Delta{s}$ apart, barring boundary condition.

For the considered vehicle to properly follow the preceding vehicle, we impose the following five requirements.
First and foremost, to prevent collision, we constrain the space headway between the two vehicles to be greater than some minimal headway $h_{\min}(t): \mathbb{R} \rightarrow \mathbb{R}_{\geq 0}$.
Next, to follow the preceding vehicle, we constrain the space headway to be smaller than some maximum headway $h_{\max}(t): \mathbb{R} \rightarrow \mathbb{R}_{\geq 0}$.
The third constraint requires the speed of the considered vehicle to fall within some speed limits $[v_{\min}, v_{\max}]$.
Next, we constrain the acceleration of the considered vehicle to be bounded within some range $[a_{\min}, a_{\max}]$.
Last but not least, among all trajectories that satisfies the constraints above, we select one that is the smoothest, as measured by $\ell^{2}$-norm of the acceleration.

We summarize the above specifications in Problem <ref>.

Provided $s^{0}(t)$ for $t \in [t_{0}, t_{f}]$ with some finite $t_{f}$, determine a trajectory of least acceleration in $\ell^{2}$-norm for the considered vehicle $s^{1}(t)$ for $t \in [t_{0}, t_{f}]$ such that 1) $\min_{\forall t} s^{0}(t) - s^{1}(t) \geq h_{\min}(t)$, 2) $\max_{\forall t} s^{0}(t) - s^{1}(t) \leq h_{\max}(t)$, 3) $v^{1}(t) \in [v_{\min}, v_{\max}]$ for all $t$, and 4) $a^{1}(t) \in [a_{\min}, a_{\max}]$ for all $t$.

With the problem stated above, we present the proposed controller in the next section.

§ OPTIMAL CAR-FOLLOWING CONTROLLER

In this section, we present a hierarchical MPC scheme consisting of 1) a prediction layer, 2) a planning layer, and 3) a tracking layer.

The layered controller design is illustrated in Figure <ref>.
We present the details of the three layers below.

[draw, minimum width=4cm, minimum height=0.5cm, anchor=south west, text width=4cm, align=center] (predictor) at (0, 0) Prediction Layer;
[draw, minimum width=4cm, minimum height=0.5cm, anchor=south west, text width=4cm, align=center] (planner) at (0, -1.1) Planning Layer;
[draw, minimum width=4cm, minimum height=0.5cm, anchor=south west, text width=4cm, align=center] (controller) at (0, -2.2) Tracking Layer;
[draw, minimum width=4cm, minimum height=0.5cm, anchor=south west, text width=4cm, align=center] (plant) at (0, -3.3) Plant;
[->, >=latex] (2, 0) to node[left] Predicted traj. $\hat{s}^{0}(t)$ (2, -0.55);
[->, >=latex] (2, -1.1) to node[left] Planned accel. $\check{a}^{1}(t)$ (2, -1.65);
[->, >=latex] (2, -2.2) to node[left] Actual accel. $a^{1}(t)$ (2, -2.75);
[->, >=latex] (2, -3.3) to (2, -4) to (5, -4) to node[right, text width=0.5cm] $\vec{x}^{0}(t)$ $\vec{x}^{1}(t)$ $a^{0}(t)$ (10 Hz) (5, -1.9) to (4.25, -1.9);
[->, >=latex] (5, -1.95) to node[right, text width=1.5cm] Updated predictor state (1 Hz) (5, 0.275) to (4.25, 0.275);

Diagram of the MPC controller design.

Prediction layer and planning layer are invoked every 1 second.
Tracking layer is triggered every 0.1 second.

§.§ Prediction layer

The prediction layer predicts the trajectory of the preceding vehicle based on an ETA estimator.
Because the ETA estimator uses downstream information, it renders the controller non-local.

Define a way point every $\Delta{s}$ from $s^{0}(t)$ to $s^{0}(t) + l \cdot \Delta{s}$ at some time $t \in [t_{0}, t_{f}]$ and for some spatial receding horizon $l \cdot \Delta{s} > 0$.
Non-integer $l$ indicates that the spatial horizon cannot be evenly divided by $\Delta{s}$.
Collect all way points into a sequence, we have $\mathcal{W}(t) \coloneqq (w_{1}(t), w_{2}(t), \dots, w_{l+1}(t)) = ( s^{0}(t) + i \cdot \Delta{s} \mid i = 0, 1, \dots, l )$.
For convenience, we assume $w_{l+1} \leq w_{f}$.

Define $\widehat{\mathcal{T}}^{0}(t)$ as the corresponding ETA set for the preceding vehicle, that is, $\widehat{\mathcal{T}}^{0}(t) \coloneqq (t_{1}^{0}(t), \hat{t}_{2}^{0}(t), \dots, \hat{t}_{l+1}^{0}(t))$,

where $t_{i}^{0}, \hat{t}_{i}^{0}$ are the true and estimated arrival times of the preceding vehicle at the way point $w_{i} \in \mathcal{W}(t)$.

For the problem to be well-posed, we require both true and estimated arrival time sequences to be strictly increasing.

We assume there exists an ETA estimator[Common ETA service providers include Google Maps, Waze, INRIX, and Garmin.] $f: \mathcal{W}(t) \rightarrow \widehat{\mathcal{T}}^{0}(t)$ that estimates the arrival time of the preceding vehicle at each way point in $\mathcal{W}(t)$.
To model prediction errors, we assume that the relative estimation error is uniformly distributed with an average at one and a radius of $\sigma$, that is,

\begin{equation*}
\frac{\hat{t}_{i+1}^{0} - \hat{t}_{i}^{0}}{t_{i+1}^{0} - t_{i}^{0}} \sim \mathcal{U}(1-\sigma, 1+\sigma), \quad i = 1, 2, \dots, l,
\end{equation*}

where $\hat{t}_{1}^{0} = t_{1}^{0}$.
We use $f_{\sigma}$ to denote that an ETA estimator $f$ has an error radius of $\sigma$.

With $\mathcal{W}(t)$ and $\widehat{\mathcal{T}}^{0}(t) = f_{\sigma}(\mathcal{W}(t))$, we can then generate a predicted trajectory for the preceding vehicle $\hat{s}^{0} : [t_{1}^{0}, \hat{t}_{l}^{0}] \rightarrow [s^{0}(t), s^{0}(t) + l \cdot \Delta{s}]$ using a standard interpolation method.

The above receding horizon design is illustrated in the $s$ and $\hat{t}^{0}$ axes of Figure <ref>.

[->, >=latex, line width=0.35mm] (0, 0.6) to (7, 0.6) node[right] $s$;
(0, 0.7) to (0, 0.6) node[below] $w_{1}$;
(1.25, 0.7) to (1.25, 0.6) node[below] $w_{2}$;
(2.5, 0.7) to (2.5, 0.6) node[below] $w_{3}$;
(3.75, 0.7) to (3.75, 0.6) node[below] $w_{4}$;
(5.125, 0.6) node[below] $\dots$;
(6.5, 0.7) to (6.5, 0.6) node[below] $w_{l+1}$;

[->, >=latex, line width=0.35mm] (0, 0) to (7, 0) node[right] $\hat{t}^{0}$;
(0, 0.1) to (0, 0) node[below] $t_{1}^{0}$;
(1.75, 0.1) to (1.75, 0) node[below] $\hat{t}_{2}^{0}$;
(3.00, 0.1) to (3.00, 0) node[below] $\hat{t}_{3}^{0}$;
(4.00, 0.1) to (4.00, 0) node[below] $\hat{t}_{4}^{0}$;
(5.25, 0) node[below] $\dots$;
(6.5, 0.1) to (6.5, 0) node[below] $\hat{t}_{l+1}^{0}$;

[->, >=latex, line width=0.35mm] (0, -0.7) to (5.5, -0.7) node[right] $t^{p}$;
(0, -0.6) to (0, -0.7) node[below] $t^{p}_{0}$;
(0.75, -0.6) to (0.75, -0.7) node[below] $t^{p}_{1}$;
(1.5, -0.6) to (1.5, -0.7) node[below] $t^{p}_{2}$;
(2.25, -0.6) to (2.25, -0.7) node[below] $t^{p}_{3}$;
(3, -0.6) to (3, -0.7) node[below] $t^{p}_{4}$;
(4, -0.7) node[below] $\dots$;
(5, -0.6) to (5, -0.7) node[below] $t^{p}_{m}$;

[->, >=latex, line width=0.35mm] (0, -1.4) to (3.7, -1.4) node[right] $t^{c}$;
(0, -1.3) to (0, -1.4) node[below] $t^{c}_{0}$;
(0.2, -1.3) to (0.2, -1.4) node[below] ;
(0.4, -1.3) to (0.4, -1.4) node[below] $t^{c}_{1}$;
(0.6, -1.3) to (0.6, -1.4) node[below] ;
(0.8, -1.3) to (0.8, -1.4) node[below] $t^{c}_{3}$;
(1.0, -1.3) to (1.0, -1.4) node[below] ;
(1.2, -1.3) to (1.2, -1.4) node[below] $t^{c}_{5}$;
(1.4, -1.3) to (1.4, -1.4) node[below] ;
(1.6, -1.3) to (1.6, -1.4) node[below] $t^{c}_{7}$;
(1.8, -1.3) to (1.8, -1.4) node[below] ;
(2.0, -1.3) to (2.0, -1.4) node[below] $t^{c}_{9}$;
(2.6, -1.4) node[below] $\dots$;
(3.2, -1.3) to (3.2, -1.4) node[below] $t^{c}_{n}$;

Specification of the receding-horizon control scheme.
The prediction layer has a fixed spatial horizon and a variable temporal horizon.
The planning layer has a temporal horizon $m$.
The tracking layer has a fixed temporal horizon $n$.

§.§ Planning layer

The planning layer plans a reference trajectory for the considered vehicle based on the predicted trajectory of the preceding vehicle.

With $\hat{s}^{0}(t)$ predicted by $f_{\sigma}$, we can derive the following four trajectories:
1) $\hat{s}_{t}^{-}(t) \coloneqq \hat{s}^{0}(t - \Delta{t}^{-})$: predicted envelope of some minimal time headway $\Delta{t}^{-}$,
2) $\hat{s}_{t}^{+}(t) \coloneqq \hat{s}^{0}(t - \Delta{t}^{+})$: predicted envelope of some maximal time headway $\Delta{t}^{+}$,
3) $\hat{s}_{s}^{-}(t) \coloneqq \hat{s}^{0}(t) - \Delta{s}^{-}$: predicted envelope of some minimal space headway $\Delta{s}^{-}$, and
4) $\hat{s}_{s}^{+}(t) \coloneqq \hat{s}^{0}(t) - \Delta{s}^{+}$: predicted envelope of some maximal space headway $\Delta{s}^{+}$.
The above four trajectories are illustrated in Figure <ref>.

While many possible designs are possible, in this study we select the minimal and maximal headway envelopes as follows

\begin{equation}\begin{aligned}
\hat{s}_{\min}\left( t \mid \mathcal{W}, \widehat{\mathcal{T}}^{0} \right) &\coloneqq \max\left( \min\left( \hat{s}_{s}^{-}(t), \hat{s}_{t}^{-}(t) \right), \hat{s}_{s}^{+}(t) \right),\\
\hat{s}_{\max}\left( t \mid \mathcal{W}, \widehat{\mathcal{T}}^{0} \right) &\coloneqq \min\left( \max\left( \hat{s}_{s}^{+}(t), \hat{s}_{t}^{+}(t) \right), \hat{s}_{s}^{-}(t) \right).
\end{aligned}
\label{eq:headway_constraints}
\end{equation}

The area between $\hat{s}_{\min}(t)$ and $\hat{s}_{\max}(t)$ is shaded in green in Figure <ref>.
Consequently, the minimum and maximum headways are

\begin{equation}\begin{aligned}
h_{\min}(t) &= \hat{s}^{0}(t) - \hat{s}_{\min}\left( t \mid \mathcal{W}, \widehat{\mathcal{T}}^{0} \right),\\
h_{\max}(t) &= \hat{s}^{0}(t) - \hat{s}_{\max}\left( t \mid \mathcal{W}, \widehat{\mathcal{T}}^{0} \right).\\
\end{aligned}\end{equation}

The objective of the design in (<ref>) is to generate smooth car-following without falling far behind nor getting too close.
Compared to [9, 10], our formulation is stricter, which leads to a tighter yet still safe and smooth car-following.

axis lines=middle,
axis line style=-Latex,
legend style=at=(1.5,0.125),anchor=south east,
legend style=draw=none,font=,
+[no marks,domain=5:65,samples=100,black,thick] x + 15;
+[mark=square*,domain=5:65,samples=2,red,thick] x + 10;
+[mark=triangle*,domain=15:75,samples=2,pink,thick,name path=lower] (x-10) + 15;
+[mark=diamond*,domain=5:65,samples=2,cyan,thick,name path=upper] x - 2.5;
+[mark=*,domain=35:95,samples=2,blue,thick] (x-30) + 15;
[mark=none,black,dotted] coordinates (5,0) (5,20);
[mark=none,black,dotted] coordinates (15,0) (15,20);
[mark=none,black,dotted] coordinates (35,0) (35,20);
[mark=none,black,dotted] coordinates (0,80) (65,80);
[mark=none,black,dotted] coordinates (0,75) (65,75);
[mark=none,black,dotted] coordinates (0,62.5) (65,62.5);
[<->] (5,3) – node[above] $\Delta{t}^{-}$ (15,3);
[<->] (5,15) – node[above] $\Delta{t}^{+}$ (35,15);
[<->] (3,75) – node[right] $\Delta{s}^{-}$ (3,80);
[<->] (20,62.5) – node[right] $\Delta{s}^{+}$ (20,80);
$\hat{s}^{0}(t)$, $\hat{s}^{-}_{s}(t)$, $\hat{s}^{-}_{t}(t)$, $\hat{s}^{+}_{s}(t)$, $\hat{s}^{+}_{t}(t)$;
[olive,opacity=0.25] fill between[of=lower and upper, soft clip=domain=15:65];

Headway envelope design.
$\hat{s}^{0}(\cdot)$ is the predicted trajectory of the proceeding vehicle.
$\hat{s}^{-}_{s}(\cdot), \hat{s}^{+}_{s}(\cdot)$ are estimated minimum and maximum space headway envelopes, respectively.
$\hat{s}^{-}_{t}(\cdot), \hat{s}^{+}_{t}(\cdot)$ are estimated minimum and maximum time headway envelopes, respectively.
The shaded region represents the feasible set in space and time.

Now, let $m$ be the temporal planning horizon and $\Delta{t}_{p}$ be the temporal planning resolution.
For all $i_{p} = 0, \dots, m$, denote $t_{i_{p}}^{p} \coloneqq t^{0}_{1} + i_{p}\Delta{t}_{p}$.
Non-integer $m$ indicates that $\hat{t}_{m}^{0} - t^{0}_{1}$ cannot be evenly divided by $\Delta{t}_{p}$.

Define the planned states $\widecheck{\mathcal{X}}^{1p}$ and the planned accelerations $\widecheck{\mathcal{U}}^{1p}$ of the considered vehicle as follows:

\begin{equation}
\begin{aligned}
\widecheck{\mathcal{X}}_{m}^{1p} &\coloneqq \{\check{\vec{x}}^{1}(t_{0}^{p}), \check{\vec{x}}^{1}(t_{1}^{p}), \dots, \check{\vec{x}}^{1}(t_{m}^{p}) )\},\\
\widecheck{\mathcal{U}}_{m-1}^{1p} &\coloneqq \{\check{a}^{1}(t_{0}^{p}), \check{a}^{1}(t_{1}^{p}), \dots, \check{a}^{1}(t_{m-1}^{p})\}.
\end{aligned}
\label{eq:decision_variables_planning}
\end{equation}

Here, we introduce the check accent $\check{x}$ to denote that a variable $x$ is a planned decision variable.

With everything defined above, we plan for the states and accelerations of the considered vehicle via the following LCQP:

\beginmini!|s|[2]
.75!${\displaystyle \alpha \sum_{i_{p}=0}^{m-1} \left(\check{a}^{1}(t_{i_{p}}^{p})\right)^{2} + \beta \sum_{j_{p}=1}^{m} \left( \xi_{j_{p}}^{p} \right)^{2} + \gamma \sum_{j_{p}=1}^{m} \left( \zeta_{j_{p}}^{p} \right)^{2}}$

x̌⃗̌^1(t_0^p) = x⃗^1_0
A_i_p^p ·x̌⃗̌^1(t_i_p^p) +
B_i_p^p ·ǎ^1(t_i_p^p)
0 ≤ξ_j_p^p,   š^1(t_j_p) - ŝ_min( t_j_p |𝒲, 𝒯^0 ) ≤ξ_j_p^p
0 ≤ζ_j_p^p,   -š^1(t_j_p) + ŝ_max( t_j_p |𝒲, 𝒯^0 ) ≤ζ_j_p^p
v_min ≤v̌^1(t_j_p^p)≤v_max
a_min ≤ǎ^1(t_i_p^p)≤a_max,
\endmini!

\begin{equation*}\begin{aligned}
X &\coloneqq \left\{ \widecheck{\mathcal{X}}_{m}^{1p},\widecheck{\mathcal{U}}_{m-1}^{1p}, \{ \xi_{j_{p}}^{p} \}_{1}^{m}, \{ \zeta_{j_{p}}^{p} \}_{1}^{m} \right\},\\
\mat{A}_{i_{p}}^{p} &\coloneqq e^{\mat{A}(t_{i_{p}+1}^{p} - t_{i_{p}}^{p})}, \quad \mat{A} = \left[[0 \; 0]^{\top} \: [1 \; 0]^{\top}\right],\\
\mat{B}_{i_{p}}^{p} &\coloneqq \int^{t_{i_{p}+1}^{p}}_{t_{i_{p}}^{p}} e^{\mat{A}(t_{i_{p}+1}^{p} - \tau)} d\tau \cdot \mat{B}, \quad \mat{B} = [0 \; 1]^{\top},
\end{aligned}\end{equation*}

for $i_{p} = 0, \dots, m-1$ and $j_{p} = 1, \dots, m$.
The receding horizon design is illustrated in the $t^{p}$ axis of Figure <ref>.
We have $\mathcal{W}, \widehat{\mathcal{T}}^{0}$, $\alpha, \beta, \gamma$, and $\vec{x}_{0}^{1}$ as the inputs to the optimization program, where $\alpha, \beta, \gamma \in \mathbb{R}_{>0}, \alpha + \beta + \gamma = 1$, and $\vec{x}_{0}^{1}$ is the initial state of the considered vehicle.

In (<ref>), the first term penalizes large accelerations, while the second and third term, coupled with (<ref>) and (<ref>), regulate the vehicle inside the admissible space headway ranges.
Constraints (<ref>) and (<ref>) impose kinematic constraints using zero-order hold.
Lastly, constraints (<ref>) and (<ref>) impose speed and acceleration limits.

If the feasible set is nonempty, problem (<ref>) has unique global optimum.

Because $\alpha, \beta, \gamma > 0$, the objective function (<ref>) is strongly convex.
By construction, the feasible set is a polyhedron, which is also convex.
Therefore, a direct application of Lemma 8.2 and Theorem 8.6 in [15] reveals that if the feasible set is nonempty, problem (<ref>) has an unique global optimum.

For some initial time $t_{0}$, if the feasible set of problem (<ref>) is nonempty, then its corresponding receding-horizon policy has a solution for all $t > t_{0}$, that is, it is persistently feasible.

To show that problem (<ref>) is persistently feasible, it suffices to show that it has a solution at every future time step.
Because the feasible set is initially nonempty, the problem has a solution at $t_{0}$.
For every other time step $t > t_{0}$, it is easy to verify that zero acceleration is a feasible point due to the fact that constraints (<ref>) and (<ref>) are soft.
Therefore, it follows that problem (<ref>), when repeatedly applied in receding horizon, always stays feasible.

When the initial velocity is too high or too low, the feasible set of (<ref>) will become empty.
For example, when the initial velocity is greater than $v_{\max}+a_{\max}\Delta{t}^{p}$, there will be no acceleration within the actuation limits that can steer the velocity below $v_{\max}$ in the next time step.
To prevent any chance of infeasibility, we can replace the hard constraints on velocity with suitable soft constraints.

Because the constraints on minimum headway are soft, it is possible for the considered vehicle to violate constraint (<ref>).
Therefore, it is theoretically permissible for the preceding vehicle to collide with the preceding vehicle.
Note that this is chosen by design because the hard constraint version of (<ref>) can never be guaranteed in practice due to irregularity of human driving and inevitable errors in prediction.
Therefore, the presented soft constraints in (<ref>) should be viewed as a “best-effort” attempt to respect the minimum headway requirement.

Furthermore, we briefly comment on how constraints (<ref>) and (<ref>) can be extended to account for cut-ins and cut-outs.
For example, a possible design is illustrated in Figure <ref>.
Under this design, the considered vehicle will not overreact when a discontinuous change occurs in the position of the preceding vehicle.
Rather, it gradually recovers to a comfortable headway over time.
With the help a switching condition, this policy can be combined with that of (<ref>) to form a hybrid MPC controller to perform car-following with cut-ins and cut-outs.

axis lines=middle,
axis line style=-Latex,
legend style=at=(1,0.125),anchor=south east,
+[no marks,domain=5:25,samples=100,black,thick] x + 40;
+[no marks,domain=25:85,samples=100,black,thick] x + 20;
[mark=none,black,thick] coordinates (25,65) (25,45);
[mark=*,red,thick,name path=lower] coordinates (5,20) (25,40) (85,80);
[mark=triangle*,blue,thick,name path=upper] coordinates (5,5) (25,25) (85,65);
[olive,opacity=0.25] fill between[of=lower and upper, soft clip=domain=5:85];
[<->] (5,20) – node[right] $\Delta{s}^{*}$ (5,45);
[<->] (85,80) – node[right] $\Delta{s}^{*}$ (85,105);
[-latex] (25,110) node[above]Cut-In to (25,65);

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[mark=none,black,thick] coordinates (25,65) (25,85);
[<->] (5,20) – node[right] $\Delta{s}^{*}$ (5,45);
[<->] (85,120) – node[right] $\Delta{s}^{*}$ (85,145);
[-latex] (25,110) node[above]Cut-Out to (25,85);
$\hat{s}^{0}(t)$, $\hat{s}_{\min}(t)$, $\hat{s}_{\max}(t)$;

Headway envelope design to handle cut-ins and cut-outs.
Left design is for cut-ins and right for cut-outs.
The proposed designs aim to to keep a comfortable headway $\Delta{s}^{*}$ without overreaction.

§.§ Tracking layer

To handle prediction errors, an additional tracking layer is introduced to track the planned accelerations and to guard the considered vehicle from imminent collisions.

Concretely, the layer takes the following form, which is a modified version of (<ref>).
\beginmini!|s|[2]
λ∑_i_c=0^n-1 (ǎ^1(t_i_c^c) - a̅^1(t_i_c^c))^2 + μ∑_j_c=1^n ( ξ_j_c^c )^2

x̅⃗̅^1(t_0^c) = x⃗^1_0
A_i_c^c ·x̅⃗̅^1(t_i_c^c) +
B_i_c^c ·a̅^1(t_i_c^c)
0 ≤ξ_j_c^c,   s^1(t_j_c^c) - s̃_min( t_j_c^c |a^0(t_0^c) ) ≤ξ_j_c^c
v_min ≤v̅^1(t_j_c^c)≤v_max
a_min ≤a̅^1(t_i_c^c)≤a_max,
\endmini!

\begin{equation*}\begin{aligned}
t_{i_{c}}^{c} &\coloneqq t + i_{c}\Delta{t}^{c},\quad Z \coloneqq \left\{ \overline{\mathcal{X}}_{n}^{1c},\overline{\mathcal{U}}_{n-1}^{1c}, \{ \xi_{j_{c}}^{c} \}_{1}^{n} \right\},\\
\mat{A}_{i_{c}}^{c} &\coloneqq e^{\mat{A}(t_{i_{c}+1}^{c} - t_{i_{c}}^{c})}, \quad \mat{B}_{i_{c}}^{c} \coloneqq \int^{t_{i_{c}+1}^{c}}_{t_{i_{c}}^{c}} e^{\mat{A}(t_{i_{c}+1}^{c} - \tau)} d\tau \cdot \mat{B},
\end{aligned}\end{equation*}

for some horizon $n$, resolution $\Delta{t}^{c}$, $i_{c} = 0, \dots, n-1$, and $j_{c} = 1, \dots, n$.

$\overline{\mathcal{X}}_{n}^{1c},\overline{\mathcal{U}}_{n-1}^{1c}$ are defined similar to those in (<ref>).
Note that here we use accent $\bar{x}$ to indicate that a variable $x$ is an internal variable of the tracking controller.
The receding horizon design is illustrated in the $t^{c}$ axis of Figure <ref>.
Unlike the planning layer, the minimum headway envelope $\tilde{s}_{\min}(\cdot \mid a^{0}(\cdot))$ is generated by assuming that the preceding vehicle accelerates constantly from $t_{0}^{c}$ to $t_{n}^{c}$.
Similar to the planning layer once again, we have $a^{0}(t_{0})$, $\lambda, \mu$, and $\vec{x}_{0}^{1}$ as inputs to the optimization program, where $\lambda, \mu \in \mathbb{R}_{>0}, \lambda + \mu = 1$.

The modified LCQP in the tracking layer in (<ref>) is structurally identical to (<ref>) except for:
1) we remove the soft constraint on maximum headway;
2) we adopt a shorter planning horizon $n$ and a smaller temporal resolution $\Delta{t}^{c}$; and
3) we predict the trajectory of the preceding vehicle by assuming that it keeps its current acceleration for the entire tracking horizon.

Note that the last change above is made to ensure safety.
Because constraints (<ref>) only takes vehicular RADAR measurements, which are generally highly reliable, the collision avoidance of the car-following is mostly decoupled from the ETA estimation.
Hence, a bad prediction from the ETA estimator cannot severely impact the safety of the vehicle.

Because the optimization problem (<ref>) may be viewed as a special case of problem (<ref>) where $\gamma = 0$, it has the same properties as those highlighted in Propositions <ref>,<ref> and Remarks <ref>,<ref>.

We assume that we can directly control the acceleration of the vehicle.
Consequently, dynamic constraints (<ref>) are purely kinematic.

Lastly, we briefly describe how our proposed method can be extended for multi-vehicle platooning.
For example, consider a platoon of vehicles with indices $0, 1, \dots, k$, where index $0$ indicates the preceding human vehicle and indices $1, \dots, k$ indicate the $k$ following automated vehicles.
As before, we can use an ETA estimator to generate a prediction for vehicle zero.
For all the vehicles that follow, we can substitute the predicted trajectories of the preceding vehicles by the corresponding planned trajectories.
That is, for $i = 1, 2, \dots, k$, we can use $\check{a}^{i-1}(t)$ to compute $\check{s}^{i-1}(t)$ and approximate $\hat{s}^{i-1}(t)$ with $\check{s}^{i-1}(t)$.
Integrating the above prediction method with the planning and tracking methods presented in this paper, we have consequently produced a complete platoon controller design.

§ NUMERICAL SIMULATION

To evaluate the proposed controller, we test its performance through an array of numerical experiments, each equipped with an ETA estimator of an unique combination of spatial resolution $\Delta{s}$ and noise level $\sigma$.

For all simulations, we emulate the preceding vehicle by replaying a recorded drive from [16].
The velocity and acceleration of the recorded drive are shown in the second and third rows of Figure <ref>.

Spatial horizon of the ETA estimators are fixed to $l \cdot \Delta{s} = \SI{3000}{\meter}$.
We run an array of simulations by varying the spatial resolution $\Delta{s}$ between $\SI{10}{\meter}$ and $\SI{500}{\meter}$ and noise level $\sigma$ between $0.01$ and $0.25$.
To establish a performance baseline, we run an additional simulation with an intelligent driver model (IDM) controller [17].
Parameters of the IDM are chosen to roughly match that of the MPC controller: $a = 1.5, b = 3, \delta = 4, s_{0} = 3.5, \ell = 4.65$.
Likewise, to establish a performance upper bound, we run one more simulation with a controller that can perfectly foresee the trajectory of the preceding vehicle.
We call this controller the oracle controller.

Shared parameters of the planning and tracking layers are: $v_{\min} = 0, v_{\max} = \SI{35}{\meter\per\second}, a_{\min} = \SI{-1.5}{\meter\per\second\squared}, a_{\max} = \SI{3}{\meter\per\second\squared}, \Delta{s}^{-} = \SI{5}{\meter}, \Delta{s}^{+} = \SI{100}{\meter}, \Delta{t}^{-} = \SI{0.6}{\second}$, and $\Delta{t}^{+} = \SI{3.0}{\second}$.
Unique parameters of the planning layer are: $\alpha = 0.2, \beta = 0.7, \gamma = 0.1, \Delta{t}^{p} = \SI{1}{\second}$, and $m = 60$.
Unique parameters of the tracking layer are: $\lambda = 0.1, \mu = 0.9, \Delta{t}^{c} = \SI{0.1}{\second}$, and $n = 30$.

To evaluate performance, we use two measures $e$ and $f$ to quantify tracking error and fuel saving, respectively.
The measure $e$ is defined to be the standard deviation of $\check{v}^{1*} - \bar{v}^{1}$ for all sampled time instants, where $\check{v}^{1*}$ denotes the planned velocity under perfect prediction.
The measure $f$ is defined to be the ratio of the fuel consumption of the baseline IDM controller to that of the MPC controller, using the energy model described in [18].

trim axis right,
axis line style=-Latex,
ylabel=$s - \hat{s}^{0}$ (m),
legend style=draw=none,legend columns=-1,at=(0.5,1.6),anchor=north,
enlarge x limits=false,
mark repeat=10,
[no marks,thick] table[x=t, y expr=1-1,col sep=comma] data/7050_solution.csv;
[mark=*,olive,thick] table[x=t, y expr=4-1,col sep=comma] data/7050_solution.csv;
[mark=square*,red,name path=lower] table[x=t, y expr=7-1,col sep=comma] data/7050_solution.csv;
[mark=triangle*,blue,name path=upper] table[x=t, y expr=8-1,col sep=comma] data/7050_solution.csv;
[olive,opacity=0.25] fill between[of=upper and lower];

axis line style=-Latex,
ylabel=$v$ (m/s),
legend pos=outer north east,
legend style=draw=none,legend columns=-1,at=(0.5,1.6),anchor=north,
enlarge x limits=false,
mark repeat=10,
[no marks,thick] table[x=t, y expr=2,col sep=comma] data/7050_solution.csv;
[mark=*,olive,thick] table[x=t, y expr=5,col sep=comma] data/7050_solution.csv;
[mark=square*,red,name path=lower,samples=100] table[x=t, y=v_min,col sep=comma] data/7050_solution.csv;
[mark=triangle*,blue,name path=upper,samples=100] table[x=t, y=v_max,col sep=comma] data/7050_solution.csv;
[olive,opacity=0.25] fill between[of=upper and lower];

axis line style=-Latex,
xlabel=$t$ (s),
ylabel=$a$ (m/s$^2$),
legend style=draw=none,legend columns=-1,at=(0.5,1.6),anchor=north,
enlarge x limits=false,
mark repeat=10,
[no marks,thick] table[x=t, y expr=3,col sep=comma] data/7050_solution.csv;
[mark=*,olive,thick] table[x=t, y expr=6,col sep=comma] data/7050_solution.csv;
[mark=square*,red,name path=lower,samples=100] table[x=t, y=a_min,col sep=comma] data/7050_solution.csv;
[mark=triangle*,blue,name path=upper,samples=100] table[x=t, y=a_max,col sep=comma] data/7050_solution.csv;
[olive,opacity=0.25] fill between[of=upper and lower];

Demonstration of the planning layer.
The oracle controller assumes perfect prediction over the entire planning horizon.

colorbar style=
ylabel=$e$ (m/s),
yticklabel style=
width=0.1*/pgfplots/parent axis width
xtick = 0.01,0.05,0.1,0.15,0.2,0.25,
xticklabels = 0.01,0.05,0.1,0.15,0.2,0.25,
x tick label style=rotate=60,anchor=east,
xlabel=$\sigma$ (s),
ytick = 10,100,200,300,400,500,
3[surf, shader=flat] table[x index=0,y index=1,z index=2, col sep=comma] data/error_stdev.txt;

colorbar style=
ylabel=$f$ (%),
yticklabel style=
width=0.1*/pgfplots/parent axis width
xlabel=$\sigma$ (s),
ylabel=$\Delta{s}$ (m),
xtick = 0.01,0.05,0.1,0.15,0.2,0.25,
xticklabels = 0.01,0.05,0.1,0.15,0.2,0.25,
x tick label style=rotate=60,anchor=east,
xlabel=$\sigma$ (s),
ytick = 10,100,200,300,400,500,
3[surf, shader=flat] table[x index=0,y index=1,z index=2, col sep=comma] data/fuel_reduction.txt;

[id=FW]Performance heat maps of the MPC controllers.
The top heat map measures tracking error $e$, while the bottom heat map measures fuel saving $f$.

Position, velocity, and acceleration of the oracle controller are shown in Figure <ref>.
As the oracle vehicle has demonstrated in Figure <ref>, the optimal behavior is to avoid accelerating or decelerating unless there is an imminent violation of at least one of the two headway constraints.

Tracking error and fuel saving are visualized in the top and bottom heat maps in Figure <ref>.
As expected, tracking error $e$ is proportional to spatial resolution $\Delta{s}$ and noise level $\sigma$.
Likewise, fuel saving $f$ is inversely proportional to spatial resolution $\Delta{s}$ and noise level $\sigma$.
Note that both performance measures platoon near the origin, indicating that the MPC controller is robust to prediction errors and especially so to that induced by $\sigma$.

Lastly, we comment on the running times of the planning and tracking layers.
The optimization solver is part of a python package called CVXOPT [19].
Benchmarked on an Intel i7-6700K CPU, the average running time of the planning layer is about 0.12 s and of the tracking layer about 0.04 s.
From Figure <ref>, we find that both times fit comfortably into their allocated time budgets.

§ DISCUSSIONS

In this section, we discuss two of the most consequential features in the design of the controller, namely, 1) the space between the maximum and minimum headway constraints and 2) the resolution and noise level of the ETA estimators.

It is clear that the larger the space between the maximum and minimum headway constraints, the smoother the car-following could be.
For example, if the considered car is allowed to be arbitrarily far behind the preceding vehicle without incurring any penalties, it can simply wait for the preceding vehicle to exit the road and then accelerate to a very small constant velocity to complete the trip.
Nevertheless, we know from common sense that such behavior is not acceptable in most of the real-world applications.

On the contrary, when the space between the two headway constraints is small, the MPC controller will become sensitive to prediction errors.
Consider a situation where the allowable headway gap is thin and the prediction errors are significant.
Because the considered vehicle is already close to the boundaries of the headway constraints, an error in prediction could easily mislead the planning layer to falsely believe that it will soon violate one of the headway constraints.
In an effort to steer away from imminent constraint violation, the planning layer overreacts, leading to undesirably large acceleration or deceleration.

To resolve the above problem, one could modify the headway constraints proposed in (<ref>).
One possible solution is to enforce a minimum space gap between the maximum and minimum headway envelopes and to add a small penalty to encourage the considered vehicle to drive at the center of the allowable headway constraints.

Last but not least, we comment on how to choose an ETA estimator.
As illustrated in Figure <ref>, the performance of the controller changes significantly with the resolution and noise level of the ETA estimator.
In practice, one may first define a desirable tracking error $e_{\textnormal{des}}$ and a desirable fuel saving $f_{\textnormal{des}}$.
With $e_{\textnormal{des}}$ and $f_{\textnormal{des}}$ defined, one can then draw two level sets in the two heat maps of Figure <ref>.
Any ETA estimator with $e$ and $f$ that simultaneously fall within the two level sets may be deemed sufficient to meet the design specifications.

§ CONCLUSIONS

In this article, we propose a hierarchical MPC control scheme based on a LCQP.
We show via simulations that the controller can achieve smooth and tight car-following and is robust to prediction errors.
Additional constructions can be added to further enhance its robustness to erroneous forecasts.
Possible future works include field tests with various ETA estimators, extension to handle cut-ins and cut-outs, and modification for platooning.

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June 18, 2022 OU-HET-1108

KYUSHU-HET-237

Bhabha scattering

in the gauge-Higgs unification

Shuichiro Funatsu1, Hisaki Hatanaka2, Yutaka Hosotani3, Yuta Orikasa4 and
Naoki Yamatsu5 1Institute for Promotion of Higher Education, Kobe University,
Kobe 657-0011, Japan

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

We examine effects of $Z^{\prime}$ bosons in gauge-Higgs unification (GHU)
models at $\mathrm{e}^{+}\mathrm{e}^{-}\to\mathrm{e}^{+}\mathrm{e}^{-}$ Bhabha
scatterings. We evaluate differential cross sections in Bhabha scatterings
including $Z^{\prime}$ bosons in two types of
$\mathrm{SO}(5)\times\mathrm{U}(1)\times\mathrm{SU}(3)$ GHU models. We find
that deviations of differential cross sections in the GHU models from those in
the SM can be seen at $\sqrt{s}=250\,\mathrm{GeV}$. With
$80\,\%$-longitudinally polarized electron and $30\,\%$-longitudinally
polarized positron beams, the left-right asymmetries in the GHU A- and
B-models are resolved at more than $3\,\sigma$ at $L_{\rm
int}=250\,\mathrm{fb}^{-1}$. We also show that Bhabha scattering with
scattering angle less than 100 mrad can be safely used as luminosity
measurements at $\mathrm{e}^{+}\mathrm{e}^{-}$ colliders since the effects of
$Z^{\prime}$ bosons are well suppressed for small scattering angle. We propose
a new observable which can be measured at future TeV-scale
$\mathrm{e}^{+}\mathrm{e}^{-}$ linear colliders.

##  1 Introduction

With the establishment of the standard model (SM) by the discovery of the
Higgs boson, searching for physics beyond the SM (BSM) and understanding the
electroweak phase transition have become main topics in particle physics. As
one scenario of BSM, the gauge-Higgs unification (GHU) scenario has been
studied [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. In GHU, the Higgs
boson is a part of the extra-dimensional component of gauge potentials,
appearing as a fluctuation mode of an Aharonov-Bohm (AB) phase $\theta_{H}$ in
the fifth dimension. Many GHU models are proposed for electroweak unification
[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35], and GHU models for grand unification are also proposed [36, 37, 38, 39,
40, 41, 42, 43, 44, 45]. Among them, two types of
$\mathrm{SU}(3)_{C}\times\mathrm{SO}(5)\times\mathrm{U}(1)_{X}$ GHU models,
the A-model[16, 17, 18, 19, 20, 21, 22] and B-model[23, 24, 25, 26, 27, 28],
in warped spacetime have been extensively studied. At low energies below the
electroweak scale the mass spectrum and gauge and Higgs couplings of SM
particles are nearly the same as in the SM.

Couplings of the first Kaluza-Klein (KK) neutral gauge bosons to right-handed
SM fermions are large in the A-model, whereas those to left-handed SM fermions
become large in the B-model. In proton-proton collisions KK bosons of photons
and $Z$ bosons appear as huge broad resonances of $Z^{\prime}$ bosons in the
Drell-Yan process, and can be seen in current and future hadron collider
experiments [19, 20, 28]. The KK-excited states of the $W$ boson are also seen
as resonances of $W^{\prime}$ bosons. In the A-model the couplings of the
first KK $W$ boson to the SM fermions are small. In the B-model the couplings
of the first KK $W$ boson to right-handed fermions are negligible, while the
couplings to left-handed fermions are much larger than the $W$ boson
couplings. Therefore at the LHC the first KK $W$ boson appears as a narrow
resonance of $W^{\prime}$ boson in the A-model, but appears as a broad
resonance in the B-model[20, 28].

In $\mathrm{e}^{+}\mathrm{e}^{-}$ collider experiments, effects of GHU can be
examined by exploring interference effects among photon, $Z$ boson and
$Z^{\prime}$ bosons. In the previous papers we have studied effects of new
physics (NP) on such observable quantities as cross section, forward-backward
asymmetry and left-right asymmetry in $\mathrm{e}^{+}\mathrm{e}^{-}\to
f\bar{f}$ ($f\neq\mathrm{e}$) with polarized and unpolarized
$\mathrm{e}^{+}\mathrm{e}^{-}$ beams [21, 25, 30, 31, 49, 50, 51]. In Ref.
[21] we compared such observable quantities of GHU with those of the SM in LEP
experiments at $\sqrt{s}=M_{Z}$, and LEP2 experiments for
$130\text{GeV}\leq\sqrt{s}\leq 207\,\text{GeV}$ [52, 53]. In Refs. [21, 25] we
also gave predictions of several signals of $Z^{\prime}$ bosons in GHU in
$\mathrm{e}^{+}\mathrm{e}^{-}$ collider experiments designed for future with
collision energies $\sqrt{s}\geq 250\text{ GeV}$ with polarized electron and
positron beams. In the $\mathrm{e}^{+}\mathrm{e}^{-}\to f\bar{f}$
($f\neq\mathrm{e}$) modes, the deviations of total cross sections become large
for right-polarized electrons in the A-model, whereas in the B-model the
deviations are large for left-polarized electrons.

Deviations from the SM can be seen in the Higgs couplings as well. $HWW$,
$HZZ$ and Yukawa couplings deviate from those in the SM in a universal
manner[16, 17, 32]. They are suppressed by a common factor

$\displaystyle\frac{g_{HWW}^{\text{GHU}}}{g_{HWW}^{{\rm
SM}}},\,\frac{g_{HZZ}^{\rm\text{GHU}}}{g_{HZZ}^{{\rm
SM}}}\simeq\cos(\theta_{H})$ (1)

for $W$ and $Z$ bosons, and

$\displaystyle\frac{y_{\bar{f}f}^{\text{GHU}}}{y_{\bar{f}f}^{{\rm
SM}}}\simeq\begin{cases}\cos(\theta_{H})&\text{A-model
\cite[cite]{[\@@bibref{}{Hosotani:2008by}{}{}]}}\\\
\cos^{2}(\theta_{H}/2)&\text{B-model
\cite[cite]{[\@@bibref{}{Funatsu:2019fry}{}{}]}}\end{cases}$ (2)

for SM fermions $f$. In the analysis of both $Z^{\prime}$ and $W^{\prime}$
bosons in hadron colliders [19, 20, 28], it is found that the AB phase is
constrained as $\theta_{H}\lesssim 0.1$. For $\theta_{H}=\mathcal{O}(0.1)$,
probable values in the model, the deviation of the couplings amounts to
$(1-\cos\theta_{H})=\mathcal{O}(0.005)$, and is small. At the International
Linear Collider (ILC) at $\sqrt{s}=250\text{GeV}$, the $ZZH$ coupling can be
measured in the $0.6\,\%$ accuracy with $2\text{ ab}^{-1}$ data [47]. Since
the masses and Higgs couplings of the SM fields in the GHU models are very
close to those in the SM, the electroweak phase transition (EWPT) occurs at
$T_{C}\sim 160\text{GeV}$ and appears very weak first order[22, 27] in both A-
and B-models, which is very similar to EWPT in the SM[48].

In this paper we study effects of $Z^{\prime}$ bosons in GHU models on the
$\mathrm{e}^{+}\mathrm{e}^{-}\to\mathrm{e}^{+}\mathrm{e}^{-}$ Bhabha
scattering. Measurements of Bhabha scattering at linear colliders have
contributed to the establishment of the SM[56, 57, 58]. Bhabha scattering is
also useful to explore NP [59, 60]. Unlike $\mathrm{e}^{+}\mathrm{e}^{-}\to
f\bar{f}$ ($f\neq\mathrm{e}$) scattering, in Bhabha scattering not only
$s$-channel but also $t$-channel contributions enter the process. Since the
$t$-channel contribution of photon exchange dominates in forward scattering
amplitudes, the cross section becomes very large for small scattering angles,
which improves the statistics of experiments. It will be seen below that
effects of $Z^{\prime}$ bosons on cross sections can be measured with large
significances.

Bhabha scattering at very small scattering angles is used for the
determination of the luminosity of the $\mathrm{e}^{+}\mathrm{e}^{-}$ beams.
Since cross sections of all other scattering processes depend on the
luminosity, one needs to know whether or not effects of $Z^{\prime}$ bosons on
the $\mathrm{e}^{+}\mathrm{e}^{-}\to\mathrm{e}^{+}\mathrm{e}^{-}$ cross
section are sufficiently small. Forward-backward asymmetry $A_{FB}$ of the
cross section in Bhabha scattering is no longer a good quantity for searching
NP, since the backward scattering cross section is much smaller than the
forward scattering cross section. We will propose a new quantity $A_{X}$ to
measure with polarized $\mathrm{e}^{+}\mathrm{e}^{-}$ beams, which can be used
for seeing NP effects instead of $A_{FB}$.

In Section 2 we briefly review the GHU A- and B-models and discuss the
$\mathrm{e}^{+}\mathrm{e}^{-}\to\mathrm{e}^{+}\mathrm{e}^{-}$ scattering in
both the SM and GHU models. In Section 3, we show the formulas of
$\mathrm{e}^{+}\mathrm{e}^{-}\to\mathrm{e}^{+}\mathrm{e}^{-}$ scattering cross
sections for longitudinally polarized $\mathrm{e}^{\pm}$ beams, and
numerically evaluate the effects of $Z^{\prime}$ bosons in GHU models on
differential cross sections and left-right asymmetries. We also show that
effects of $Z^{\prime}$ bosons on the cross section are very small at the very
small scattering angle.

##  2 Gauge-Higgs unification

In GHU A- and B-models the electroweak $\mathrm{SU}(2)\times\mathrm{U}(1)$
symmetry is embedded in $\mathrm{SO}(5)\times\mathrm{U}(1)_{X}$ symmetry in
the Randall-Sundrum warped space [46], whose metric is given by

$\displaystyle ds^{2}$
$\displaystyle=\frac{1}{z^{2}}\left[\eta_{\mu\nu}dx^{\mu}dx^{\nu}+\frac{dz^{2}}{k^{2}}\right],\quad
1\leq z\leq z_{L}=e^{kL}$ (3)

where $\eta_{\mu\nu}=\operatorname{diag}(-1,+1,+1,+1)$ and $k$ is the AdS-
curvature. We refer two 4D hyperplanes at $z=1$ and $z=z_{L}$ as the UV and IR
branes, respectively. The $\mathrm{SO}(5)$ symmetry is broken to
$\mathrm{SO}(4)\simeq\mathrm{SU}(2)_{L}\times\mathrm{SU}(2)_{R}$ by the
boundary conditions at $z=1,\,z_{L}$ and the
$\mathrm{SU}(2)_{R}\times\mathrm{U}(1)_{X}$ symmetry is broken to
$\mathrm{U}(1)_{Y}$ by a scalar field localized on the UV brane.
$\mathrm{SU}(2)_{L}\times\mathrm{U}(1)_{Y}$ symmetry is broken to the
electromagnetic $\mathrm{U}(1)_{\rm EM}$ symmetry by the VEV of the
$z$-component gauge fields of $\mathrm{SO}(5)/\mathrm{SO}(4)$. The VEV is
related to the gauge-invariant AB phase $\theta_{H}$.

Table 1: The fermion content in the first generation of the lepton sector is shown. In the A-model bulk fermions are introduced in the vector representation of $\mathrm{SO}(5)$, which are decomposed of $\mathrm{SO}(4)$-vector and singlet. Zero modes of fermions appear in the left-handed components of $\mathrm{SO}(4)$-vectors and in the right-handed components of $\mathrm{SO}(4)$-singlets. The extra zero modes of $L_{j}$ ($j=1,2,3$) couple to brane fermions at the UV brane to become massive[18]. In the B-model bulk fermions are introduced in the spinor representation of $\mathrm{SO}(5)$. Zero modes appear in the left-handed components of $\mathrm{SU}(2)_{L}$ doublet $(\nu_{\mathrm{e}},\mathrm{e})$ and in the right-handed components of $\mathrm{SU}(2)_{R}$ doublet $(\nu^{\prime}_{\mathrm{e}},\mathrm{e}^{\prime})$ [23].  | A-model | B-model
---|---|---
Bulk fermion | $\begin{matrix}\left(\begin{pmatrix}\nu_{\mathrm{e}}&L_{1X}\\\ \mathrm{e}&L_{1Y}\end{pmatrix},\,\mathrm{e}^{\prime}\right),\\\ \left(\begin{pmatrix}L_{2X}&L_{3X}\\\ L_{2Y}&L_{3Y}\end{pmatrix},\,\nu^{\prime}_{\mathrm{e}}\right)\end{matrix}$ |  $\begin{pmatrix}\nu_{\mathrm{e}}\\\ \mathrm{e}\\\ \nu^{\prime}_{\mathrm{e}}\\\ \mathrm{e}^{\prime}\end{pmatrix}$

The difference between the A- and B-models lies in the content of fermions as
tabulated in Table 1. In the A-model, quarks and leptons in the SM are
embedded in the $\mathrm{SO}(5)$-vector representation. In the B-model quarks
and leptons are embedded in the $\mathrm{SO}(5)$-spinor, vector and singlet
representations, which are naturally derived from spinor and vector multiplets
in the $\mathrm{SO}(11)$ gauge-Higgs grand unification[37, 38]. We also note
that the bulk mass parameter for the bulk electron field, $c_{\mathrm{e}}$, is
positive in the A-model [18] whereas $c_{\mathrm{e}}$ in the B-model has to be
negative[23]. In the B-model positive $c_{\mathrm{e}}$ leads to an exotic
light neutrino state and therefore negative $c_{\mathrm{e}}$ must be chosen.
Zero modes of fermion fields with positive bulk mass parameters are localized
near the UV brane whereas zero modes of fermion fields with negative bulk mass
parameters are localized near the IR brane.

Interactions of the electron and gauge bosons are given by

$\displaystyle\int
d^{4}x\int_{1}^{z_{L}}\frac{dz}{k}\biggl{\\{}\bar{\check{\Psi}}\gamma^{\mu}(\partial_{\mu}-igA_{\mu})\check{\Psi}\biggr{\\}}$
(4)

where $A_{\mu}(x,z)$ is a four-dimensional component of the 5D gauge field and
$\Psi(x,z)=z^{2}\check{\Psi}$ is the 5D electron field. The electron
corresponds to the zero mode of $\check{\Psi}(x,z)$. In the A-model the left-
handed electron is localized in the vicinity of the UV brane and the right-
handed component is localized near the IR brane. In the B-model the right-
handed electron is localized in the vicinity of the UV brane and the left-
handed component is localized near the IR brane. $A_{\mu}(x,z)$ has a KK
expansion which contains the photon, $Z$ boson and their KK excited modes. The
wave function of the photon is constant in the fifth dimension coordinate $z$.
The wave function of the $Z$ boson is almost constant in $z$, but has
nontrivial dependence near the IR brane. Couplings of the electron to the $Z$
boson are very close to those in the SM. Wave functions of the first KK-
excited modes of gauge bosons are localized near the IR brane so that the
first KK-excited gauge bosons couple strongly with fermions localized near the
IR brane. In the A-model right-handed electrons have large couplings to the
first KK-excited gauge bosons whereas in the B-model left-handed electrons
couple strongly to the first KK-excited gauge bosons.

In Tables 2 and 3, parameters and couplings in the A- and B-models are
tabulated. Here, model parameters ($\theta_{H},m_{{\rm KK}}$ and $z_{L}$),
masses, widths and couplings of $Z^{\prime}$-bosons are referred from Refs.
[26, 25].

Table 2: Parameters in GHU models. $c_{\mathrm{e}}$ in the rightest column is the bulk parameter for the electron field. Model | $\theta_{H}$ | $m_{{\rm KK}}$ | $z_{L}$ | $m_{\gamma^{(1)}}$ | $\Gamma_{\gamma^{(1)}}$ | $m_{Z^{(1)}}$ | $\Gamma_{Z^{(1)}}$ | $m_{Z_{R}^{(1)}}$ | $\Gamma_{Z_{R}^{(1)}}$ | $c_{\mathrm{e}}$
---|---|---|---|---|---|---|---|---|---|---
| [rad] | [TeV] |  | [TeV] | [TeV] | [TeV] | [TeV] | [TeV] | [TeV] |
A | $0.08$ | $9.54$ | $1.01\times 10^{4}$ | $7.86$ | $0.99$ | $7.86$ | $0.53$ | $7.31$ | $1.01$ | 2.0342
B | $0.10$ | $13.0$ | $3.87\times 10^{11}$ | $10.2$ | $3.25$ | $10.2$ | $7.84$ | $9.95$ | $0.816$ | $-1.0067$

The big difference in the magnitude of $z_{L}$ in the A- and B-models
originates from the formulas of top-quark mass. In the A-model $m_{\rm
top}^{\rm A}\simeq(m_{{\rm KK}}/(\sqrt{2}\pi))\sqrt{1-4c_{\rm
top}^{2}}\sin\theta_{H}$ [18] whereas in the B-model $m_{\rm top}^{\rm
B}\simeq(m_{{\rm KK}}/\pi)\sqrt{1-4c_{\rm top}^{2}}\sin\frac{1}{2}\theta_{H}$
[23]. In both models the $W$ boson mass is given by $m_{W}\simeq m_{{\rm
KK}}/(\pi\sqrt{kL})\sin\theta_{H}$ so that the lower bound of $z_{L}$ becomes
$z_{L}\gtrsim 8\times 10^{3}$ in the A-model and $z_{L}\gtrsim 7\times 10^{7}$
in the B-model. In Table 2 the bulk mass parameter for the electron field
$c_{\mathrm{e}}$ is given. As explained before, $c_{\mathrm{e}}$ is positive
in the A-model whereas $c_{\mathrm{e}}$ is negative in the B-model.

Table 3:  Left-handed and right-handed couplings of the electron, $\ell_{V},r_{V}$ ($V=Z,Z^{(1)}$, $Z_{R}^{(1)}$ and $\gamma^{(1)}$), in units of $g_{w}\equiv g_{A}/\sqrt{L}$ (see text). Ratios of $e$ and $g_{w}$ are shown in the second column. Model | $(e/g_{w})^{2}$ | $\ell_{Z}$ | $r_{Z}$ | $\ell_{Z^{(1)}}$ | $r_{Z^{(1)}}$ | $\ell_{Z_{R}^{(1)}}$ | $r_{Z_{R}^{(1)}}$ | $\ell_{\gamma^{(1)}}$ | $r_{\gamma^{(1)}}$ |
---|---|---|---|---|---|---|---|---|---|---
A | $0.2312$ | $-0.3066$ | $0.2638$ | $0.1195$ | $0.9986$ | $0.0000$ | $-1.3762$ | $0.1879$ | $-1.8171$ |
B | $0.2306$ | $-0.3058$ | $0.2629$ | $-1.7621$ | $-0.0584$ | $-1.0444$ | $0.0000$ | $-2.7587$ | $0.1071$ |

In Table 3, the left- and right-handed electron couplings to $Z^{\prime}$
bosons, $r_{V},\ell_{V}$ ($V=Z,Z^{(1)},Z_{R}^{(1)},\gamma^{(1)}$), are
tabulated. In the table $g_{w}\equiv g_{A}/\sqrt{L}$ is the 4D gauge coupling
of the $\mathrm{SO}(5)$ where $g_{A}$ is the 5D $\mathrm{SO}(5)$ coupling. In
terms of $g_{A}$ and the 5D $\mathrm{U}(1)_{X}$ coupling $g_{B}$, a mixing
parameter is defined as [19, 25]

$\displaystyle e/g_{w}=\sin\theta_{W}^{0}$
$\displaystyle\equiv\frac{s_{\phi}}{\sqrt{1+s_{\phi}^{2}}},\quad
s_{\phi}\equiv g_{B}/\sqrt{g_{A}^{2}+g_{B}^{2}}.$ (5)

The value of $\sin\theta_{W}^{0}$ is determined so as to reproduce the
experimental value of the forward-backward asymmetry in
$\mathrm{e}^{+}\mathrm{e}^{-}\to\mu^{+}\mu^{-}$ scattering at the $Z$-pole. In
the A-model electron’s right-handed couplings to $Z^{\prime}$-bosons are
larger than left-handed couplings. In the B-model electron’s left-handed
couplings to $Z^{\prime}$-bosons are larger than right-handed couplings.

##  3 Bhabha scattering in $\mathrm{e}^{+}\mathrm{e}^{-}$ colliders

We consider the $\mathrm{e}^{+}\mathrm{e}^{-}\to\mathrm{e}^{+}\mathrm{e}^{-}$
scattering in the center-of-mass frame. In this frame, the Mandelstam
variables $(s,t,u)$ are given by

$\displaystyle s$ $\displaystyle=4E^{2},$ $\displaystyle t$
$\displaystyle=-\frac{s}{2}(1-\cos\theta)=-s\sin^{2}\frac{\theta}{2},$
$\displaystyle u$
$\displaystyle=-\frac{s}{2}(1+\cos\theta)=-s\cos^{2}\frac{\theta}{2}$ (6)

where $E$ is the energy of initial electron and positron, and $\theta$ is the
scattering angle of the electron. Since the
$\mathrm{e}^{+}\mathrm{e}^{-}\to\mathrm{e}^{+}\mathrm{e}^{-}$ scattering
process consists both $s-$ and $t-$channel processes, the scattering amplitude
is written in terms of the following six building blocks:

$\displaystyle S_{LL}=S_{LL}(s)$
$\displaystyle\equiv\sum_{i}\frac{\ell_{V_{i}}^{2}}{s-M_{V_{i}}^{2}+iM_{V_{i}}\Gamma_{V_{i}}},$
$\displaystyle S_{RR}=S_{RR}(s)$
$\displaystyle\equiv\sum_{i}\frac{r_{V_{i}}^{2}}{s-M_{V_{i}}^{2}+iM_{V_{i}}\Gamma_{V_{i}}},$
$\displaystyle S_{LR}=S_{LR}(s)$
$\displaystyle\equiv\sum_{i}\frac{\ell_{V_{i}}r_{V_{i}}}{s-M_{V_{i}}^{2}+iM_{V_{i}}\Gamma_{V_{i}}},$
$\displaystyle T_{LL}=T_{LL}(s,\theta)$
$\displaystyle\equiv\sum_{i}\frac{\ell_{V_{i}}^{2}}{t-M_{V_{i}}^{2}+iM_{V_{i}}\Gamma_{V_{i}}},$
$\displaystyle T_{RR}=T_{RR}(s,\theta)$
$\displaystyle\equiv\sum_{i}\frac{r_{V_{i}}^{2}}{t-M_{V_{i}}^{2}+iM_{V_{i}}\Gamma_{V_{i}}},$
$\displaystyle T_{LR}=T_{LR}(s,\theta)$
$\displaystyle\equiv\sum_{i}\frac{\ell_{V_{i}}r_{V_{i}}}{t-M_{V_{i}}^{2}+iM_{V_{i}}\Gamma_{V_{i}}},$
(7)

where $M_{V_{i}}$ and $\Gamma_{V_{i}}$ are the mass and width of the vector
boson $V_{i}$. $\ell_{V_{i}}$ and $r_{V_{i}}$ are left- and right-handed
couplings of electrons to the vector boson $V_{i}$ ($V_{0}=\gamma$,
$V_{1}=Z$), respectively. In particular, we have
$\ell_{\gamma}=r_{\gamma}=Q_{\mathrm{e}}e$, $Q_{\mathrm{e}}=-1$,
$\ell_{Z}=\frac{e}{\sin\theta_{W}^{0}\cos\theta_{W}^{0}}[I_{\mathrm{e}}^{3}-Q_{e}\sin^{2}\theta_{W}^{0}]$,
$r_{Z}=\frac{e}{\sin\theta_{W}^{0}\cos\theta_{W}^{0}}[-Q_{e}\sin^{2}\theta_{W}^{0}]$,
$I^{3}_{\mathrm{e}}=-\frac{1}{2}$ in the SM. Here $e$, $I_{\mathrm{e}}^{3}$
and $\theta_{W}^{0}$ are the electromagnetic coupling, weak isospin of the
electron and the bare Weinberg angle defined in (5), respectively.

When initial state electrons and positrons are longitudinally polarized, the
differential cross section is given by

$\displaystyle\frac{d\sigma}{d\cos\theta}(P_{\mathrm{e}^{-}},P_{\mathrm{e}^{+}})$
$\displaystyle=\frac{1}{4}\biggl{\\{}(1+P_{\mathrm{e}^{-}})(1+P_{\mathrm{e}^{+}})\frac{d\sigma_{\mathrm{e}^{-}_{R}\mathrm{e}^{+}_{R}}}{d\cos\theta}+(1-P_{\mathrm{e}^{-}})(1-P_{\mathrm{e}^{+}})\frac{d\sigma_{\mathrm{e}^{-}_{L}\mathrm{e}^{+}_{L}}}{d\cos\theta}$
$\displaystyle\qquad+(1+P_{\mathrm{e}^{-}})(1-P_{\mathrm{e}^{+}})\frac{d\sigma_{\mathrm{e}^{-}_{R}\mathrm{e}^{+}_{L}}}{d\cos\theta}+(1-P_{\mathrm{e}^{-}})(1+P_{\mathrm{e}^{+}})\frac{d\sigma_{\mathrm{e}^{-}_{L}\mathrm{e}^{+}_{R}}}{d\cos\theta}\biggr{\\}},$
(8)

where $P_{\mathrm{e}^{-}}$ and $P_{\mathrm{e}^{+}}$ are the polarization of
the electron and positron beam, respectively. $P_{\mathrm{e}^{-}}=+1$
($P_{\mathrm{e}^{+}}=+1$) denotes purely right-handed electrons (positrons)
[54, 55]. $\sigma_{\mathrm{e}_{X}^{-}\mathrm{e}_{Y}^{+}}$ ($X,Y=L,R$) denotes
the cross section for left-handed or right-handed electron and positron. When
the electron mass is neglected, these cross sections are given by

$\displaystyle\frac{d\sigma_{\mathrm{e}^{-}_{L}\mathrm{e}^{+}_{R}}}{d\cos\theta}$
$\displaystyle=\frac{1}{8\pi
s}\left(u^{2}|S_{LL}+T_{LL}|^{2}+t^{2}|S_{LR}|^{2}\right),$
$\displaystyle\frac{d\sigma_{\mathrm{e}^{-}_{R}\mathrm{e}^{+}_{L}}}{d\cos\theta}$
$\displaystyle=\frac{1}{8\pi
s}\left(u^{2}|S_{RR}+T_{RR}|^{2}+t^{2}|S_{LR}|^{2}\right),$
$\displaystyle\frac{d\sigma_{\mathrm{e}^{-}_{L}\mathrm{e}^{+}_{L}}}{d\cos\theta}$
$\displaystyle=\frac{d\sigma_{\mathrm{e}^{-}_{R}\mathrm{e}^{+}_{R}}}{d\cos\theta}=\frac{1}{8\pi
s}\cdot\left(s^{2}|T_{LR}|^{2}\right).$ (9)

When $s,t\ll M_{Z}^{2}$, the cross section is approximated by the one at the
QED level, where we obtain $S_{LL}=S_{RR}=S_{LR}=e^{2}/s$ and
$T_{LL}=T_{RR}=T_{LR}=e^{2}/t$, and

$\displaystyle\frac{d\sigma_{\rm QED}}{d\cos\theta}(P_{e^{-}},P_{e^{+}})$
$\displaystyle=\frac{e^{4}}{16\pi
s}\left\\{(1-P_{\mathrm{e}^{-}}P_{\mathrm{e}^{+}})\frac{t^{4}+u^{4}}{s^{2}t^{2}}+(1+P_{\mathrm{e}^{-}}P_{\mathrm{e}^{+}})\frac{s^{2}}{t^{2}}\right\\}.$
(10)

For unpolarized electron or positron beams, the above expression reduces to a
familiar formula

$\displaystyle\frac{d\sigma^{\rm unpolarized}_{\rm QED}}{d\cos\theta}$
$\displaystyle=\frac{e^{4}}{16\pi s}\frac{s^{4}+t^{4}+u^{4}}{s^{2}t^{2}}.$
(11)

We also note that in terms of building blocks (7) we can write down components
of s-, t-channels, and interference terms as

$\displaystyle\frac{d\sigma}{d\cos\theta}$
$\displaystyle=\frac{d\sigma^{\text{s-channel}}}{d\cos\theta}+\frac{d\sigma^{\text{t-channel}}}{d\cos\theta}+\frac{d\sigma^{\text{interference}}}{d\cos\theta},$
(12)

where each component is given by

$\displaystyle\frac{d\sigma^{\text{s-channel}}}{d\cos\theta}(P_{\mathrm{e}^{-}},P_{\mathrm{e}^{+}})$
$\displaystyle=\frac{1}{32\pi
s}\biggl{\\{}(1+P_{\mathrm{e}^{-}})(1-P_{\mathrm{e}^{+}})\left[u^{2}|S_{RR}|^{2}+t^{2}|S_{LR}|^{2}\right]$
$\displaystyle\qquad+(1-P_{\mathrm{e}^{-}})(1+P_{\mathrm{e}^{+}})\left[u^{2}|S_{LL}|^{2}+t^{2}|S_{LR}|^{2}\right]\biggr{\\}},$
$\displaystyle\frac{d\sigma^{\text{t-channel}}}{d\cos\theta}(P_{\mathrm{e}^{-}},P_{\mathrm{e}^{+}})$
$\displaystyle=\frac{1}{32\pi
s}\biggl{\\{}(1+P_{\mathrm{e}^{-}})(1+P_{\mathrm{e}^{+}})s^{2}|T_{LR}|^{2}+(1-P_{\mathrm{e}^{-}})(1-P_{\mathrm{e}^{+}})s^{2}|T_{LR}|^{2}$
$\displaystyle\qquad+(1+P_{\mathrm{e}^{-}})(1-P_{\mathrm{e}^{+}})u^{2}|T_{RR}|^{2}+(1-P_{\mathrm{e}^{-}})(1+P_{\mathrm{e}^{+}})u^{2}|T_{LL}|^{2}\biggr{\\}},$
$\displaystyle\frac{d\sigma^{\text{interference}}}{d\cos\theta}(P_{\mathrm{e}^{-}},P_{\mathrm{e}^{+}})$
$\displaystyle=\frac{1}{16\pi
s}u^{2}\biggl{\\{}(1+P_{\mathrm{e}^{-}})(1-P_{\mathrm{e}^{+}})\mathrm{Re}(S_{RR}T_{RR}^{*})$
$\displaystyle\qquad+(1-P_{\mathrm{e}^{-}})(1+P_{\mathrm{e}^{+}})\mathrm{Re}(S_{LL}T_{LL}^{*})\biggr{\\}}.$
(13)

When initial electrons and/or positrons are longitudinally polarized, one can
measure left-right asymmetries. The left-right asymmetry of polarized cross
sections is given by

$\displaystyle A_{\text{LR}}(P_{-},P_{+})$
$\displaystyle\equiv\frac{\sigma(P_{\mathrm{e}^{-}}=-P_{-},P_{\mathrm{e}^{+}}=-P_{+})-\sigma(P_{\mathrm{e}^{-}}=+P_{-},P_{\mathrm{e}^{+}}=+P_{+})}{\sigma(P_{\mathrm{e}^{-}}=-P_{-},P_{\mathrm{e}^{+}}=-P_{+})+\sigma(P_{\mathrm{e}^{-}}=+P_{-},P_{\mathrm{e}^{+}}=+P_{+})}$
$\displaystyle=(P_{-}-P_{+})\cdot\frac{\sigma_{\mathrm{e}^{-}_{L}\mathrm{e}^{+}_{R}}-\sigma_{\mathrm{e}^{-}_{R}\mathrm{e}^{+}_{L}}}{(1+P_{-}P_{+})(\sigma_{\mathrm{e}^{-}_{L}\mathrm{e}^{+}_{L}}+\sigma_{\mathrm{e}^{-}_{R}\mathrm{e}^{+}_{R}})+(1-P_{-}P_{+})(\sigma_{\mathrm{e}^{-}_{L}\mathrm{e}^{+}_{R}}+\sigma_{\mathrm{e}^{-}_{R}\mathrm{e}^{+}_{L}})},$
$\displaystyle 1\geq P_{-}\geq 0,\quad 1\geq P_{+}\geq-1,$ (14)

where the cross section in a given bin $[\theta_{1},\theta_{2}]$ is given by
$\sigma\equiv\int_{\cos\theta_{1}}^{\cos\theta_{2}}\frac{d\sigma}{d\cos\theta}d\cos\theta$.
We have used
$\sigma_{\mathrm{e}^{-}_{L}\mathrm{e}^{+}_{L}}=\sigma_{\mathrm{e}^{-}_{R}\mathrm{e}^{+}_{R}}$
because
$\frac{d\sigma_{\mathrm{e}^{-}_{L}\mathrm{e}^{+}_{L}}}{d\cos\theta}-\frac{d\sigma_{\mathrm{e}^{-}_{R}\mathrm{e}^{+}_{R}}}{d\cos\theta}=0$.
We can also define the left-right asymmetry of the differential cross section
as

$\displaystyle A_{\text{LR}}(P_{-},P_{+},\cos\theta)$
$\displaystyle\equiv\frac{\dfrac{d\sigma}{d\cos\theta}(P_{\mathrm{e}^{-}}=-P_{-},P_{\mathrm{e}^{+}}=-P_{+})-\dfrac{d\sigma}{d\cos\theta}(P_{\mathrm{e}^{-}}=+P_{-},P_{\mathrm{e}^{+}}=+P_{+})}{\dfrac{d\sigma}{d\cos\theta}(P_{\mathrm{e}^{-}}=-P_{-},P_{\mathrm{e}^{+}}=-P_{+})+\dfrac{d\sigma}{d\cos\theta}(P_{\mathrm{e}^{-}}=+P_{-},P_{\mathrm{e}^{+}}=+P_{+})}$
$\displaystyle=\frac{(P_{-}+P_{+})\left(\dfrac{d\sigma_{\mathrm{e}^{-}_{L}\mathrm{e}^{+}_{L}}}{d\cos\theta}-\dfrac{d\sigma_{\mathrm{e}^{-}_{R}\mathrm{e}^{+}_{R}}}{d\cos\theta}\right)+(P_{-}-P_{+})\left(\dfrac{d\sigma_{\mathrm{e}^{-}_{L}\mathrm{e}^{+}_{R}}}{d\cos\theta}-\dfrac{d\sigma_{\mathrm{e}^{-}_{R}\mathrm{e}^{+}_{L}}}{d\cos\theta}\right)}{(1+P_{-}P_{+})\left(\dfrac{d\sigma_{\mathrm{e}^{-}_{L}\mathrm{e}^{+}_{L}}}{d\cos\theta}+\dfrac{d\sigma_{\mathrm{e}^{-}_{R}\mathrm{e}^{+}_{R}}}{d\cos\theta}\right)+(1-P_{-}P_{+})\left(\dfrac{d\sigma_{\mathrm{e}^{-}_{L}\mathrm{e}^{+}_{R}}}{d\cos\theta}+\dfrac{d\sigma_{\mathrm{e}^{-}_{R}\mathrm{e}^{+}_{L}}}{d\cos\theta}\right)}$
$\displaystyle=(P_{-}-P_{+})\cdot\frac{\Sigma_{LR-
RL}}{(1+P_{-}P_{+})\Sigma_{LL+RR}+(1-P_{-}P_{+})\Sigma_{LR+RL}},$ (15)

where we have used
$d\sigma_{\mathrm{e}^{-}_{L}\mathrm{e}^{+}_{L}}/d\cos\theta=d\sigma_{\mathrm{e}^{-}_{R}\mathrm{e}^{+}_{R}}/d\cos\theta$
and defined

$\displaystyle\Sigma_{LL+RR}$ $\displaystyle\equiv 2s^{2}|T_{LR}|^{2},$
$\displaystyle\Sigma_{LR+RL}$ $\displaystyle\equiv
u^{2}(|S_{LL}+T_{LL}|^{2}+|S_{RR}+T_{RR}|^{2})+2t^{2}|S_{LR}|^{2},$
$\displaystyle\Sigma_{LR-RL}$ $\displaystyle\equiv
u^{2}(|S_{LL}+T_{LL}|^{2}-|S_{RR}+T_{RR}|^{2}).$ (16)

In linear colliders we can measure the cross sections for
$(P_{\mathrm{e}^{-}},P_{\mathrm{e}^{+}})=(P_{-},P_{+})$, $(P_{-},-P_{+})$,
$(-P_{-},P_{+})$ and $(-P_{-},-P_{+})$. Combining these quantities, one can
make a new observable quantity which does not depend on the value of
$P_{\pm}$. In $\mathrm{e}^{+}\mathrm{e}^{-}\to\mathrm{e}^{+}\mathrm{e}^{-}$
scatterings we have $A_{LR}(P_{-},+P_{+},\cos\theta)$ and
$A_{LR}(P_{-},-P_{+},\cos\theta)$ as independent observables and one may
define the following non-trivial quantity:

$\displaystyle
A_{X}(\cos\theta)\equiv\frac{\Sigma_{LL+RR}-\Sigma_{LR+RL}}{\Sigma_{LL+RR}+\Sigma_{LR+RL}}$
$\displaystyle\quad=\frac{1}{P_{-}P_{+}}\cdot\dfrac{(P_{-}-P_{+})A_{\text{LR}}(P_{-},-P_{+},\cos\theta)-(P_{-}+P_{+})A_{\text{LR}}(P_{-},+P_{+}\cos\theta)}{(P_{-}-P_{+})A_{\text{LR}}(P_{-},-P_{+},\cos\theta)+(P_{-}+P_{+})A_{\text{LR}}(P_{-},+P_{+},\cos\theta)}.$
(17)

where the second equality holds only when $P_{\pm}\neq 0$ and
$|P_{+}|\neq|P_{-}|$. As is evident in the first line of (17),
$A_{X}(\cos\theta)$ is independent of the magnitudes of polarization
$P_{\pm}$. This quantity may be used to explore NP beyond the SM as discussed
below.

Since $\mathrm{e}^{+}\mathrm{e}^{-}\to\mathrm{e}^{+}\mathrm{e}^{-}$ scattering
contains $t$-channel processes, forward scatterings dominate. Therefore unlike
the $\mathrm{e}^{+}\mathrm{e}^{-}\to f\bar{f}$ ($f\neq\mathrm{e}^{-}$)
scattering, the forward-backward asymmetry of
$\mathrm{e}^{+}\mathrm{e}^{-}\to\mathrm{e}^{+}\mathrm{e}^{-}$ scattering is a
less-meaningful quantity.

We note that all of the above formulas can be applied to
$\ell^{+}\ell^{-}\to\ell^{+}\ell^{-}$ ($\ell=\mu,\tau$) scatterings.

##  4 Numerical Study

In the followings we calculate
$\mathrm{e}^{+}\mathrm{e}^{-}\to\mathrm{e}^{+}\mathrm{e}^{-}$ cross sections
both in the SM and GHU models, and we evaluate effects of $Z^{\prime}$ bosons
in GHU models on observables given in the previous section. As benchmark
points, we have chosen typical parameters of the A- and B-models in Tables 2
and 3. For experimental parameters we choose $\sqrt{s}=250$ GeV and
$L_{\mathrm{int}}=250$ fb-1 as typical value of linear
$\mathrm{e}^{+}\mathrm{e}^{-}$ colliders like ILC[61]. We also choose
$L_{\mathrm{int}}=2$ ab-1, which will be achieved at circular
$\mathrm{e}^{+}\mathrm{e}^{-}$ colliders like FCC-ee[62] and CEPC[63]. For the
new asymmetry $A_{X}(\cos\theta)$ in (17), we consider $\sqrt{s}=3$ TeV for
future linear colliders like CLIC[64]. As for the longitudinal polarization,
we set the parameter ranges $-0.8\leq P_{\mathrm{e}^{-}}\leq+0.8$ and
$-0.3\leq P_{\mathrm{e}^{+}}\leq 0.3$, which can be achieved at ILC[61].

In Figure 1 $\mathrm{e}^{+}\mathrm{e}^{-}\to\mathrm{e}^{+}\mathrm{e}^{-}$
differential cross sections in the SM are plotted. In the forward-scattering
region ($\cos\theta>0$), the magnitudes of cross sections of $t$-channel and
the interference parts are much larger than those of the $s$-channel part.

Figure 1: Differential cross sections for unpolarized
$\mathrm{e}^{+}\mathrm{e}^{-}$ initial states in the SM. (a) Log-scale plot
with $-0.9\leq\cos\theta\leq 0.9$. (b) Linear plot with
$-0.7\leq\cos\theta\leq 0.7$. In both plots, red-solid lines indicate the
total of $s$-, $t$-channels and interferences. The $s$-channel and $t$-channel
contributions are drawn as blue-dashed and purple-dotted lines, respectively.
In (b), the negative contribution from interferences is plotted with the
green-dashed line.

Figure 2:  Deviations of differential cross sections of GHU from those in the
SM, $\frac{d\sigma^{\text{GHU}}}{d\cos\theta}/\frac{d\sigma^{{\rm
SM}}}{d\cos\theta}-1$, for unpolarized $\mathrm{e}^{+}\mathrm{e}^{-}$ beams
are plotted. The left plot is for the A-model and the right plot is for the
B-model. In each plot, the red-solid curve represents the deviation of the sum
of all the components of the differential cross section. Blue-dashed, purple-
dotted and green dot-dashed curves correspond to the deviations of $s$-,
$t$-channels and interference components of differential cross sections,
respectively. Error-bars are estimated for $L_{\mathrm{int}}=250\text{
fb}^{-1}$.

Figure 3: Deviations of differential cross sections of GHU models from the SM
for polarized $\mathrm{e}^{+}\mathrm{e}^{-}$ beams. GHU-A [(a) and (b)] and
GHU-B [(c) and (d)]. (a) and (c) are for $(P_{e^{-}},P_{e^{+}})=(-0.8,+0.3)$.
(b) and (d) are for$(P_{e^{-}},P_{e^{+}})=(+0.8,-0.3)$. Meanings of the curves
and error-bars are the same as in Figure 2.

In Figures 2 and 3, the differences of differential cross sections of the GHU
from the SM for unpolarized and polarized beams are plotted, respectively. In
the figures, differences of $s$-channel, $t$-channel and interference
contributions are also plotted. In the $s$-channel, the NP effects contribute
destructively in the forward scattering. On the other hand, in the $t$-channel
NP effects contribute constructively. Since the cross section is dominated by
$t$-channel, the total of $s$-, $t$\- and interference channels increases due
to the NP effects.

In the A-model $Z^{\prime}$ bosons have larger couplings to right-handed
electrons than to left-handed electrons. Therefore the cross section of the
$\mathrm{e}_{R}^{-}\mathrm{e}_{L}^{+}$ initial states becomes larger than that
of $\mathrm{e}_{L}^{-}\mathrm{e}_{R}^{+}$. On the other hand, in the B-model
$Z^{\prime}$ bosons have larger couplings to left-handed electrons than to
right-handed electrons, and the cross section of
$\mathrm{e}_{L}^{-}\mathrm{e}_{R}^{+}$ initial states becomes larger.

NP effects become smaller when $\theta$ becomes smaller. The statistical
uncertainty, however, also becomes small since the cross section becomes very
large. Therefore deviations of the cross section relative to statistical
uncertainties may become large.

For unpolarized $\mathrm{e}^{+}\mathrm{e}^{-}$ beams (Figure 3), the new
physics effect in both models tends to enhance the cross section at forward
scattering with almost the same magnitude. In the B-model the suppression of
NP effects due to larger $Z^{\prime}$ masses is compensated by larger
couplings of $Z^{\prime}$ bosons than the couplings in the A-model. The
enhancement of the differential cross section due to the NP effects at
$\cos\theta\sim 0.3$ is around $1\,\%$.

For polarized beams deviations can be much larger. In the A-model, electrons
have large right-handed couplings to $Z^{\prime}$ bosons and for right-handed
polarized beam relative deviations of the cross-section become as much as
$2\,\%$ [Figure 3-(b)], whereas for right-handed beams relative deviations are
around $0.1\,\%$ [Figure 3-(a)]. Contrarily, in the B-model a left-handed
electron has large couplings to $Z^{\prime}$ bosons. Therefore in the B-model
deviations become large for left-polarized beam [Figures 3-(c) and (d)]. In
Figure 2, we have also shown statistical 1 $\sigma$ relative errors at
$L_{\mathrm{int}}=250$ fb-1 for bins
$[\cos\theta_{0}-0.05,\cos\theta_{0}+0.05]$
($\cos\theta_{0}=-0.90,-0.80,\dots,+0.90$) as vertical bars. In each bin, the
observed number of events and statistical uncertainty are given by $N$ and
$\sqrt{N}$, respectively. Therefore relative error of the cross section is
estimated as the inverse of the square of the number of events $N$.

Figure 4: Estimated significances on differential cross sections of GHU models
with unpolarized $\mathrm{e}^{+}\mathrm{e}^{-}$ beam with the integrated
luminosity $L_{\mathrm{int}}=2\text{ ab}^{-1}$. A statistical significance of
0.1 % non-statistical error is plotted with a purple dot-dashed line.

In Figure 4, the statistical significances in the GHU models are plotted. An
estimated significance of the deviation of the cross section in a bin is given
by

$\displaystyle\frac{|N_{\text{GHU}}-N_{{\rm SM}}|}{N_{{\rm
SM}}}\bigg{/}\frac{\sqrt{N_{{\rm SM}}}}{N_{{\rm
SM}}}=\frac{|N_{\text{GHU}}-N_{{\rm SM}}|}{\sqrt{N_{{\rm SM}}}},$ (18)

where $N_{\text{GHU}}$ and $N_{{\rm SM}}$ are observed numbers of events in a
bin. In Figure 4, significances are larger than 5 $\sigma$ for
$\cos\theta\gtrsim 0.1$. Significances are very large for forward scatterings,
but are very small for backward scatterings. In Figure 4, relative 0.1 %
errors are also shown. Errors due to the NP effects become very small and are
around $0.1\,\%$ for $\cos\theta\simeq 0.9$. A similar analysis has been given
in Ref.[59].

For small scattering angles $\theta$, the scattering amplitude is dominated by
$t$-channel contributions which are constructed with the blocks $T_{LL}$,
$T_{RR}$ and $T_{LR}$. When $|t|\simeq s\theta^{2}/4\ll
M_{Z}^{2},M_{Z^{\prime}}^{2}$, we can approximate the SM and NP contributions
to the block $T_{LL}$ in the scattering amplitude as

$\displaystyle T_{LL}^{{\rm NP}}$
$\displaystyle\equiv\sum_{Z^{\prime}}\frac{\ell_{Z^{\prime}}^{2}}{t-M_{Z^{\prime}}^{2}+iM_{Z^{\prime}}\Gamma_{Z^{\prime}}}\simeq\sum_{Z^{\prime}}\frac{\ell_{Z^{\prime}}^{2}}{-M_{Z^{\prime}}^{2}+iM_{Z^{\prime}}\Gamma_{Z^{\prime}}}.$
(19)

When $s\theta^{2}/4\ll M_{Z}^{2}$, $T_{LL}$ is dominated by the QED part
$T_{LL}^{\rm QED}\simeq-4e^{2}/(\theta^{2}s)$ and the NP effects are estimated
as

$\displaystyle\frac{T_{LL}^{{\rm NP}}}{T_{LL}^{\rm QED}}$
$\displaystyle\simeq\theta^{2}\sum_{Z^{\prime}}\frac{(\ell_{Z^{\prime}}^{2}/4e^{2})s}{M_{Z^{\prime}}^{2}-iM_{Z^{\prime}}\Gamma_{Z^{\prime}}}=\theta^{2}\cdot\mathcal{O}(s/M_{Z^{\prime}}^{2}),$
(20)

and similar analysis is applied to $T_{LR}$ and $T_{RR}$. Consequently, this
correction arises not only in amplitudes but also in differential cross
sections. For $\sqrt{s}=250\,\text{GeV}$ and $\theta\lesssim 300\text{ mrad}$,
the QED $t$-channel contribution dominates and corrections due to $Z^{\prime}$
bosons are suppressed by a factor $\theta^{2}s/M_{Z^{\prime}}^{2}$. In Figure
5, deviations of differential cross sections of GHU from the SM for
$\theta<300\text{ mrad}$ are plotted. Deviations of cross sections from the SM
are proportional to the square of $\theta$ and become smaller than $0.1\,\%$
when $\theta<250\text{ mrad}$. The measurement of Bhabha scatterings at small
scattering angle is used for the determination of the luminosity of
$\mathrm{e}^{+}\mathrm{e}^{-}$ collision and uncertainties of the luminosity
should be smaller than $0.1\,\%$. In GHU models the NP effects on such
uncertainty are well suppressed when $\theta\lesssim 100\text{ mrad}$. At ILC,
the luminosity calorimeter in ILD (SiD) operates between 43 and 68 (40 and 90)
mrad [61], where the influence of the $Z^{\prime}$ bosons is below $0.1\,\%$.

Figure 5: Deviations of differential cross sections of GHU models from the SM
in the forward-scattering region ($\theta\leq 300$ mrad).

When the initial electron and positron beams are longitudinally polarized, the
left-right asymmetry $A_{\text{LR}}$ (14) can be measured. In Figure 6, the
left-right asymmetries of the SM and GHU models are plotted. The measured
asymmetries become larger when $|P_{\mathrm{e}^{-}}-P_{\mathrm{e}^{+}}|$ are
larger. As seen in Figure 2, in the A-model cross section of
$\mathrm{e}_{R}^{-}\mathrm{e}_{L}^{+}$ initial states becomes large whereas in
the B-model cross section of $\mathrm{e}_{L}^{-}\mathrm{e}_{R}^{+}$ initial
states is enhanced due to the large left-handed $Z^{\prime}$ couplings.
Therefore $A_{\text{LR}}$ of B-models are larger than the SM, whereas
$A_{\text{LR}}$ of A-models are smaller. Since the $A_{\text{LR}}$ is
proportional to $|P_{\mathrm{e}^{-}}-P_{\mathrm{e}^{+}}|$, the asymmetries in
Figure 6-(a) are almost twice as large as in (b).

Figure 6: Left-right asymmetries. In both plots, blue-dotdashed, and red-
dashed and black-solid curves correspond to the GHU-A, GHU-B and the SM,
respectively. Error-bars are indicated for
$L_{\mathrm{int}}=250\,\mathrm{fb}^{-1}$ in each polarization. (a) Asymmetries
for $(P_{\mathrm{e}^{-}},P_{\mathrm{e}^{+}})=(\mp 0.8,\pm 0.3)$. (b)
Asymmetries for $(P_{\mathrm{e}^{-}},P_{\mathrm{e}^{+}})=(\mp 0.8,\mp 0.3)$.

In Figure 6, an asymmetry $A_{\text{LR}}$ in a bin and the statistic error
$\Delta A_{\text{LR}}$ are also shown. Here

$\displaystyle A_{\text{LR}}$ $\displaystyle=\frac{N_{L}-N_{R}}{N_{L}+N_{R}},$
$\displaystyle\Delta A_{\text{LR}}$
$\displaystyle=\sqrt{\frac{2(N_{L}^{2}+N_{R}^{2})}{(N_{L}+N_{R})^{3}}}$ (21)

with $N_{L}$ and $N_{R}$ being observed number of events for the left-handed
($P_{\mathrm{e}^{-}}<0$) and right-handed ($P_{\mathrm{e}^{-}}>0$) electron
beams, respectively. For small scattering angle $\cos\theta\gtrsim 0.8$, both
$A_{\text{LR}}^{\text{GHU}}$ and $A_{\text{LR}}^{{\rm SM}}$ become close to
each other.

To see how NP effects against statistical uncertainty grow for small $\theta$,
we plotted in Figure 7 the averages and statistical significances of left-
right asymmetries in GHU models in each bin which are estimated as

$\displaystyle\frac{A_{LR}^{\text{GHU}}-A_{LR}^{{\rm SM}}}{\Delta
A_{\text{LR}}}.$ (22)

For the forward scattering with $\cos\theta\gtrsim 0.2$, the deviations are
bigger than several times of standard deviations.

Figure 7: Estimated statistical significance on left-right asymmetries in GHU
models. The blue-dotdashed, red-dashed and black-dotted lines indicate GHU-A,
GHU-B and 0.1 % non-statistical errors, respectively.

In Figure 7, the significance is larger than $5\,\sigma$ for
$\cos\theta\gtrsim 0.2$. Both models are well distinguished from the SM. Using
the magnitude and sign of deviations, the A-model and B-model can be
distinguished.

In Figure 8, the asymmetry $A_{X}$ defined in Eq. (17) is plotted for
$\sqrt{s}=250\,\text{GeV}$ and $\sqrt{s}=3\,\text{TeV}$. At
$\sqrt{s}=250\,\text{GeV}$, the NP effect on $A_{X}$ is very small. For
$\sqrt{s}=3\,\text{TeV}$, the asymmetry $A_{X}$ of the SM and GHU models is
clearly different and may be discriminated experimentally. In the present
analysis of NP effects, only first KK excited states of neutral bosons are
taken into account. At $\sqrt{s}\sim 3\,\text{TeV}$, effects of second KK
modes on $A_{X}$ are estimated as a few percent. These effects are much
smaller than the effects of the first KK modes.

Figure 8: Plot of an asymmetry $A_{X}$ in Eq. (17). The left plot (a) is for
$\sqrt{s}=250\,\text{GeV}$ with $L_{\mathrm{int}}=250\,\mathrm{fb}^{-1}$ for
each set of ($P_{\mathrm{e}^{-}},P_{\mathrm{e}^{+}}$). The right plot (b) is
for $\sqrt{s}=3\,\text{TeV}$ with $L_{\mathrm{int}}=3\,\mathrm{ab}^{-1}$ for
each set of ($P_{\mathrm{e}^{-}},P_{\mathrm{e}^{+}}$). The red-dashed, blue-
dotdashed and black-solid curves correspond to the A-model, B-model and the
SM, respectively. Error-bars are indicated for
$L_{\mathrm{int}}=250\,\mathrm{fb}^{-1}$ in (a) and
$L_{\mathrm{int}}=3\,\mathrm{ab}^{-1}$ in (b) in each set of
($P_{\mathrm{e}^{-}},P_{\mathrm{e}^{+}}$).

##  5 Summary

In this paper we examined the effects of $Z^{\prime}$ bosons in the gauge-
Higgs unification (GHU) models in the
$\mathrm{e}^{+}\mathrm{e}^{-}\to\mathrm{e}^{+}\mathrm{e}^{-}$ (Bhabha)
scatterings. We first formulated differential cross sections in Bhabha
scattering including $Z^{\prime}$ bosons. We then numerically evaluated the
deviations of differential cross sections in the two
$\mathrm{SO}(5)\times\mathrm{U}(1)\times\mathrm{SU}(3)$ GHU models (the A- and
B-models) at $\sqrt{s}=250\,\text{GeV}$. We found that at
$L_{\mathrm{int}}=2\,\mathrm{ab}^{-1}$ with unpolarized
$\mathrm{e}^{+}\mathrm{e}^{-}$ beams, the deviation due to $Z^{\prime}$ bosons
in the GHU models from the SM can be clearly seen. We also found that for
$80\,\%$-longitudinally polarized electron and $30\,\%$-polarized positron
beams, deviations of the differential cross sections from the SM become as
large as a few percent for $\cos\theta\sim 0.2$, and that the A-model and the
B-model are well distinguished at more than $3\,\sigma$ significance at
$L_{\mathrm{int}}=250\,\mathrm{fb}^{-1}$. We also checked the effects of
$Z^{\prime}$ bosons are negligible for the scattering angle smaller than $100$
mrad at $\sqrt{s}=250\,\text{GeV}$. Therefore Bhabha scatterings at very small
$\theta$ can be safely used as the measurements of the luminosity in the
$\mathrm{e}^{+}\mathrm{e}^{-}$ collisions. Finally we introduced the new
observable $A_{X}$. We numerically evaluated it at $\sqrt{s}=250\,\text{GeV}$
and $3\,\text{TeV}$. Effects of the GHU models on $A_{X}$ can be seen at
future TeV-scale $\mathrm{e}^{+}\mathrm{e}^{-}$ colliders.

In this paper the effects of $Z^{\prime}$ bosons are calculated at the Born
level. Higher-order QED effects should be taken into account for more precise
evaluation [65, 66].

## Acknowledgements

This work was supported in part by European Regional Development Fund-Project
Engineering Applications of Microworld Physics (No.
CZ.02.1.01/0.0/0.0/16_019/0000766) (Y.O.), by the National Natural Science
Foundation of China (Grant Nos. 11775092, 11675061, 11521064, 11435003 and
11947213) (S.F.), by the International Postdoctoral Exchange Fellowship
Program (IPEFP) (S.F.), and by Japan Society for the Promotion of Science,
Grants-in-Aid for Scientific Research, Nos. JP19K03873 (Y.H.) and JP18H05543
(N.Y.).

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