# Bipedal Robot Running: Human-like Actuation Timing Using
Fast and Slow Adaptations

Yusuke Sakuraia , Tomoya Kamimuraa, Yuki Sakamotoa, Shohei Nishiia, Kodai
Satoa, Yuta Fujiwaraa, and Akihito Sanoa CONTACT Tomoya Kamimura. Email:
<EMAIL_ADDRESS>aDepartment of Electrical and Mechanical
Engineering, Nagoya Institute of Technology, Aichi, Japan

###### Abstract

We have been developing human-sized biped robots based on passive dynamic
mechanisms. In human locomotion, the muscles activate at the same rate
relative to the gait cycle during running. To achieve adaptive running for
robots, such characteristics should be reproduced to yield the desired effect,
In this study, we designed a central pattern generator (CPG) involving fast
and slow adaptation to achieve human-like running using a simple spring-mass
model and our developed bipedal robot, which is equipped with actuators that
imitate the human musculoskeletal system, Our results demonstrate that fast
and slow adaptations can reproduce human-like running with a constant rate of
muscle firing relative to the gait cycle. Furthermore, the results suggest
that the CPG contributes to the adjustment of the muscle activation timing in
human running.

###### keywords:

Humanoid and Bipedal Locomotion; Modeling and Simulating Humans; Passive
Walking; CPG

##  1 Introduction

We have been developing human-sized bipedal robots based on passive dynamic
mechanisms, which plays a significant role in human dynamical locomotion.
McGeer [1] developed a simple bipedal robot with passive legs attached to the
hip; this robot could walk stably without the need for any energy input other
than gravity by descending a slope. The aforementioned study indicated that
passive locomotion can play a significant role in gait. However, the dynamical
mechanisms under such locomotion is complex to fully understand sorely from
observation. To overcome the limitations of observational approach, several
researchers have investigated gait mechanisms using simple walking models [2,
3, 4, 5, 6, 7]. Bipedal locomotion consists of not only walking but also
running. Therefore, some researchers further developed simple running models.
For example, the spring-loaded inverted pendulum (SLIP) model, comprising a
point mass and prismatic massless spring, can effectively reproduce the
dynamics of running [8, 9, 10, 11, 12]. So far, to realize the running motion,
we developed a bipedal robot that utilizes bouncing rod dynamics [13].
Recently, we developed a bipedal robot with actuators that could imitate the
human musculoskeletal system, while utilizing the passive nature of the
dynamics of the robot [14].

Walking and running locomotion is executed through dynamic interactions
between the nervous system, body, and environment. Accordingly, bipedal robot
controllers have been proposed based on human neurophysiology. Reportedly, the
centers involved in the execution of patterned locomotion in humans, such as
walking and running, exist at the lower levels of the central nervous system
(e.g., the spinal cord and brainstem), which are simulated via central pattern
generators (CPGs) [15, 16, 17, 18].

CPGs employ a hierarchical network of rhythm generators that generate the
rhythm of walking and running and pattern formulators, which are responsible
for the timing of muscle activation. Rybak et al. [19] developed a model of
mammalian spinal cord circuits and investigated the firing patterns of muscle
activity. They incorporated CPGs obtained by observing muscle activity in cats
whose spinal cord was disconnected from the higher centers, and the model can
temporarily reset muscle activity upon sensory input from limbs. Through
analysis of human electromyograms, Ivarenko et al. [20] showed that five
signals are generated from the phase of CPGs which invokes leg muscle activity
to realize walking and running.

Several researchers have investigated the emergence of gait through the
interaction of CPGs, which are assigned to legs using dynamical models and
legged robots [21, 22, 23, 24, 25]. Aoi and Tsuchiya [26, 27] reported that
stable walking can be achieved in a bipedal compass model by adaptively
changing the gait period using a CPG that involves phase resetting, which is
inspired by the rhythm resetting of human walking.

Observably, humans fire their muscles at the same rate relative to the gait
cycle during running [28]. Thus, to achieve adaptive running using robots,
such a characteristic should be reproduced. When a CPG is used solely with
phase resetting for running control, although the overall period can be
adaptively varied, adjusting the timing of muscle firing is unachievable as
yet. Moreover, Fujiki et al. [29] report that applying slow adaptation to CPG
in addition to phase resetting for controlling a compact bipedal robot walking
on a split-belt, with different velocities of the left and right limbs, the
motion converges to a steady-state gait.

This study proposes an adaptive control system, which is based on the current
knowledge of human neurophysiology, to achieve a stable running condition in a
human-sized bipedal robot. We aimed to achieve adaptive running of the bipedal
robot by adjusting the actuation timings via fast and slow adaptations. We
designed a CPG-based controller to manifest adaptive behavior by providing
fast and slow adaptations. Specifically, in the rhythm generator, the phase
angular velocity is adjusted via fast adaptation (phase resetting) and slow
adaptation to the actual gait cycle. Furthermore, the pattern formulator
adjusts the muscle activation timing on the basis of the leg-state feedback to
yield a running pattern. We validated the proposed control system through
numerical simulations of a simple running model and experiments on a bipedal
robot.

##  2 Method

In this study, we propose a control system based on a CPG to achieve adaptive
actuation timings using fast and slow adaptation mechanisms. We designed a
rhythm generator that adapted to the gait cycle, which was determined by the
interaction between the robot body and environment, and a pattern formulator
that controlled the actuators based on the rhythm generator.

###  2.1 Bipedal robot based on passive dynamics

The human-sized bipedal robot, developed in our laboratory, is illustrated in
Fig. 1 (leg length: 0.76 [m]; body mass: 20 [kg]). The robot is composed of
passive elements such as springs and rubber bands. However, a minimum number
of actuators are implemented because the energy input is a prerequisite for
running on flat ground. In particular, the vastus muscles, which are involved
in knee extension, are represented by a wire connected to a pneumatic actuator
(CDQ2A63-75DZ, AirTAC). In addition, a brushless direct current (BLDC) motor
(RMD-X8 Pro, MyActuator) is used to actuate the hip-joint extension. To assist
in thigh extension, a rubber band is connected between the pelvis and thigh to
act as the thigh muscle. The touchdown timings are sensed using a pressure
sensor (FSR406, Interlink Electronics) on the sole. The robot is constrained
in the sagittal plane.

Figure 1: (a) Bipedal running robot. BLDC motors in hip joints actuate thigh
links. Muscle-tendon system represented by pneumatic actuators actuates knee
and foot links. (b) Schematics of robot. Links are connected with active
(solid red lines) or passive (dashed blue lines) wires and springs. (Colored
version is available online.)

###  2.2 Rhythm generator with fast and slow adaptations

We designed a rhythm generator involving fast and slow adaptations to adjust
its cycle period to the actual gait cycle. Moreover, this strategy guaranteed
that the left and right legs move in opposite phases. The rhythm generator was
constructed as coupled oscillators of the left and right legs, where the phase
oscillators ${\phi}_{\rm L}$ and ${\phi}_{\rm R}$ were defined for each leg as
shown in Fig.2.

Figure 2: Schematics of controller. Left and right neural oscillators (CPGs)
controls actuators according to their phases. Phases of CPGs are adjusted by
sensory feedback and inter-limb interaction.

We defined the $n$th half-period (the time from the landing of one leg till
the landing of the other leg) as $T_{n}$, and the half-period estimated by the
phase oscillator as $T_{n}^{\rm e}$. The basic angular velocity of the phase
oscillator was defined as $\pi/T_{n}^{\rm e}$. As explained later, we designed
the rhythm generator with slow adaptation to allow the estimated period
$T_{n}^{\rm e}$ to converge to the actual period $T_{n}$, which was determined
by the dynamical interaction between the body and the environment. Because the
left and right legs are designed to move in alternate phases, the target value
of the left-right phase difference is $\pi$. Furthermore, to adapt the
oscillator period to the gait cycle immediately, we introduced phase resetting
[15] at the moment of touchdown. Based on the design of the phase oscillator
proposed by Aoi et al. [27], the dynamics of each phase oscillator were
determined as follows:

$\displaystyle\dot{\phi}_{\rm R}$ $\displaystyle=\dfrac{\pi}{T_{n}^{\rm
e}}+\varepsilon\sin\\{(\phi_{\rm L}-\phi_{\rm R})-\pi\\}-\phi^{\rm td}_{\rm
R}\delta(t-t^{\rm td}_{\rm R}),$ (1a) $\displaystyle\dot{\phi}_{\rm L}$
$\displaystyle=\dfrac{\pi}{T_{n}^{\rm e}}+\varepsilon\sin\\{(\phi_{\rm
R}-\phi_{\rm L})-\pi\\}-\phi^{\rm td}_{\rm L}\delta(t-t^{\rm td}_{\rm L}),$
(1b)

where $\varepsilon$ denotes the gain of the phase difference, $\phi_{\rm
L}^{\rm td}$ and $\phi_{\rm R}^{\rm td}$ represent the phases of the left and
right oscillators at the touchdown moment, respectively, and $t_{\rm L}^{\rm
td}$ and $t_{\rm R}^{\rm td}$ symbolize the moments when the touchdown occurs.
The third term of each oscillator indicates the fast adaptation (phase reset)
at the touchdown moment. The estimated half-period $T_{n}^{\rm e}$ is designed
to converge to the actual half-period $T_{n}$ by the following update rule as
a slow adaptation by using the gain $K_{\rm p}$, $K_{\rm d}$, and the periodic
difference $\Delta T_{n}=T_{n}^{\rm e}-T_{n}$ as

$\displaystyle T_{n+1}^{\rm e}=T_{n}^{\rm e}-K_{\rm p}\Delta T_{n}-K_{\rm
d}\frac{\Delta T_{n}-\Delta T_{n-1}}{T_{n}}.$ (2)

Further, we defined the phase difference $\phi=\phi_{\rm R}-\phi_{\rm L}$ and
phase sum $\psi=\phi_{\rm R}+\phi_{\rm L}$. We assumed that running was
initiated upon touchdown of the right leg in our experiment. Therefore, an odd
number of $n$ indicates the touchdown of the right leg, and an even number of
$n$ indicates the touchdown of the left leg.

###  2.3 Pattern formulator for reproducing human-like actuation timing

To achieve human-like running, we designed a pattern formulator, which was
based on the muscle activity during human running [28]. Based on the phase
angles of the rhythm generator, the pattern formulator switches the control
laws of the hip and vastus pneumatic actuators.

Figure 3: Phases of rhythm generator are categorized into three phases:
stance, early swing, and late swing. Pattern formulator switches control laws
of actuators (hip joint motor and vastus pneumatic actuator) according to
phase angle $\phi_{i}$ ($i=\mathrm{L,R}$).

The phases are categorized into three ranges, as illustrated in Fig. 3. The
stance phase is defined as the phase range $0\leq\phi_{i}<2\pi\beta$ ($i={\rm
L,R}$), and the swing phase is defined as the phase range
$2\pi\beta\leq\phi_{i}<2\pi$, where $\beta$ indicates the duty rate, which is
determined using measured human data [28]. Furthermore, we categorized the
stance phase into two phases: the early swing phase
($2\pi\beta\leq\phi_{i}<2\pi\mu_{n}$) and the late swing phase
($2\pi\mu_{n}\leq\phi_{i}<2\pi$), where the rate $\mu_{n}$ is defined
hereafter.

The output torque of the hip motor is switched according to the phase. In the
stance phase, the leg exerts a force (i.e. kicks) on the ground because the
motor outputs a constant torque $\tau_{\rm c}$ while compensating for the
torque generated by the rubber bands. During the early swing phase, a rubber
band is used to swing the leg forward. In the late swing phase, the motor
outputs a torque $\tau_{\rm c}$ to retract the legs.

The pneumatic actuator switches the output according to the phase and state of
the leg. The thigh angle $\theta$ is defined as the current angle with respect
to the vertical downward direction of the robot, where the swing-forward
direction is positive. During the stance phase, the pneumatic actuator
tightens the vastus muscle wire. When the stance phase of the rhythm generator
terminates or the leg is lifted off the ground, the pneumatic actuator relaxes
the vastus muscle wires to prevent the robot from hitting the ground. When the
thigh angle $\theta$ exceeded a specific angle (we used 20 [deg] in the
experiments), the vastus muscle wire was slightly tightened in preparation for
the subsequent landing.

To maintain a steady running movement, a constant thigh-swing angle must be
maintained regardless of the gait cycle. To achieve this condition, we
introduced slow adaptation into the pattern formulator. Specifically,
adjusting the rate $\mu_{n}$ appropriately using the feedback of the thigh
swing angle is a prerequisite for properly adjusting the time between
touchdown and the beginning of the late swing phase. To follow the target
angle $\theta_{\rm d}^{\rm s}$, we measured the thigh angle $\theta^{\rm
s}_{n}$ at the beginning of the late swing phase and calculated $\mu_{n+1}$ by
using the gain $K_{\mu}$ as follows:

$\displaystyle\mu_{n+1}=\mu_{n}-K_{\mu}(\theta_{n}^{\rm s}-\theta_{\rm d}^{\rm
s}).$ (3)

Furthermore, in accordance with the relationship between the actual swing
angle $\theta_{n}^{\rm s}$ and target swing angle $\theta_{\rm d}^{\rm s}$,
the value of $K_{\mu}$ is switched as follows:

$\displaystyle K_{\mu}=\begin{cases}0.005,&(\theta_{n}^{\rm s}>\theta_{\rm
d}^{\rm s})\\\ 0.1.&(\theta_{n}^{\rm s}\leq\theta_{\rm d}^{\rm s})\end{cases}$
(4)

This condition yields a strong adjustment when the thigh-swing angle
$\theta_{n}$ is smaller than the target value $\theta_{\rm d}$. In the
experiment, we set $\theta^{\rm s}_{\rm d}=35$ [deg] based on the human
running and $K_{\rm p}=0.1$. Note that in human running [28], the hip joint
switches from extension to flexion when the leg angle to the vertical
direction is about 32 [deg].

###  2.4 Simple model

Before real robot experiments, we conducted numerical simulations to confirm
the validity of our controller, especially focusing on the slow adaptation of
the estimated half-period expressed by (2) using a simple running model. We
used a spring-loaded inverted pendulum (SLIP) model, which consists of a point
mass $m$ and a massless leg (Fig. 4). The leg is represented by a prismatic
spring $k$ and dumper $c$. The horizontal and vertical positions of the mass
are represented by $x$ and $y$, respectively. The gravitational acceleration
is $g$. When the leg is in the air, its length remains $l_{0}$ and its angle
keeps the touchdown angle $\gamma^{\rm td}$. When the tip of the leg reaches
the ground, it is constrained on the ground and behaves as a frictionless
joint. When the stance leg returns to its nominal length after compression,
the tip leaves the ground, and the leg angle immediately returns to the
touchdown angle. In the stance phase, the leg is actuated by the hip motor
constant torque $\tau_{\rm c}$.

Figure 4: Simple SLIP model with leg dumper and hip actuator.

The equation of motion is given by

$\displaystyle
M(\mbox{\boldmath$q$})\ddot{\mbox{\boldmath$q$}}+H(\mbox{\boldmath$q$},\dot{\mbox{\boldmath$q$}})+G(\mbox{\boldmath$q$})=\mbox{\boldmath$T$},$
(5)

where $\bm{q}=[x,y]^{\top}$. $M(\bm{q})$, $H(\bm{q},\dot{\bm{q}})$,
$G(\bm{q})$, and $\bm{T}$ represent inertia matrix, Coriolis and centrifugal
forces, conservative forces, and actuator input, respectively. According to
human parameter [30], we set $m=50$ [kg], $l_{0}=1$ [m], $k=8000$ [N/m],
$c=20$ [Ns/m], and $g=9.8$ [m/s2]. Furthermore, we heuristically set
$\varepsilon=4$, $\beta=0.6$, $\tau_{\rm c}=52$ [Nm], and $\gamma^{\rm
td}=\pi/6$ [rad]. Because our model involves a massless leg, control torque
terms related to leg inertia (Fig. 3) are omitted in the simulation. When the
step number is odd, the right leg CPG controls the leg, and when it is even,
the left CPG controls the leg.

##  3 Results

###  3.1 Simple model simulation

First of all, we verified the validity of fast and slow adaptation of the
rhythm generator using numerical simulation of the simple model. The
simulation results for $(K_{\rm p},K_{\rm d})=(0,0)$ (without feedback) and
$(K_{\rm p},K_{\rm d})=(0.8,0.1)$ (with feedback) are shown in Fig. 5.
Although the model fell down in 17 steps without feedback, it attracted to a
certain stable limit cycle which resulted into running over 3000 steps with
feedback.

Figure 5: Time profiles of $y$ and $\dot{x}$ of SLIP model. Left and right

figures indicate results for $(K_{\rm p},K_{\rm d})=(0,0)$ (without feedback)
and $(0.8,0.1)$ (with feedback), respectively. Model falls down around 5 [s]
without feedback.

The simulation results of step duration are shown in Fig. 6a, where the
initial estimated value is set as $T_{0}^{\rm e}=0.5$ [s]. Although the
estimated half-period $T_{n}^{\rm e}$ was not adjusted in the case without
feedback, $T_{n}^{\rm e}$ and actual half-period $T_{n}$ asymptotically
approached and converged to each other in the case with feedback.

Figure 6: Responses of CPG for $(K_{\rm p},K_{\rm d})=(0,0)$ (without
feedback) and $(0.8,0.1)$ (with feedback), respectively. (a) Responses of
$T_{n}$ (triangle) and $T_{n}^{\rm e}$ (circle) to step number $n$. (b)
Responses of phase difference $\phi_{N}^{-}$ (triangle) and phase sum
$\psi_{N}^{-}$ (circle) to stride number $N$.

Fig. 6b presents the response of phase difference $\phi_{N}^{-}$ and phase sum
$\psi_{N}^{-}$ to stride number $N$, where $N=(n+1)/2$ and $*_{n}^{-}$
indicates the phases immediately before the $n$th touchdown. Herein, we used
the stride number $N$ instead of the step number $n$ to focus solely on the
touchdown moments of the right leg. Notably, the stride refers to the footfall
number focusing only on the right leg. Although $\phi_{N}^{-}$ and
$\psi_{N}^{-}$ did not converge in the case without feedback, they converged
to $\pi$ and $3\pi$, respectively, in the case with feedback.

Therefore, by adjusting $T_{n}^{\rm e}$ using slow adaptation, the phase
difference approaches the opposite phase $\pi$ and the phase sum approaches
$3\pi$ as $N$ increases, indicating that the phase has been adjusted to
increase by exactly $2\pi$ during one cycle.

The aforementioned results demonstrate that the estimated half-period
$T_{n}^{\rm e}$ can be adapted to the actual period via slow adaptation while
guaranteeing the left-right opposite phase. Therefore, even if there remains a
mismatch between the actual period and the initial value of the estimated
half-period, muscle activation can be effectively performed at the same timing
like humans using the slow adaptation.

###  3.2 Robot experiments

####  3.2.1 Fast and slow adaptation of rhythm generator

To verify the effect of fast and slow adaptation of the estimated half-period,
we performed experiments on the robot for $K_{\rm p}=0$ and $0.1$ in (2),
respectively. For simplicity, we set $K_{\rm d}=0$ in the experiments. In
these experiments, we assumed the absence of a feedback of the thigh angle in
the pattern formulator, which was achieved by setting $K_{\mu}=0$. For the
rhythm generator, we set $\varepsilon=4$ and $T_{0}^{\rm e}=550$ [ms].

The value of $\varepsilon$ was set based on some preliminary experiments. When
$\varepsilon$ is set too small, the convergence of the relative phase $\phi$
is too slow, causing the robot’s left and right legs to be unable to move
alternately, ultimately leading to a fall.

For the pattern formulator, based on human parameter and several preliminary
experiments, we set $\beta=0.25$, $\mu=0.6$, and $\tau_{\rm c}=3$ [Nm]. Note
that the duty rate in human running is about 35 % and the hip extensor muscle
groups begin to contract after 50 % of the gait cycle [28]. The treadmill
speed during the experiment was set at 10 [km/h].

Fig. 7a presents a plot of the response of the estimated half-period
$T_{n}^{\rm e}$ to the number of steps $n$. For $K_{\rm p}=0$, $T_{n}^{\rm e}$
maintained its initial value of 550 [ms], regardless of the actual half-period
$T_{n}$ because the estimated half-period was not adjusted. In contrast, for
$K_{\rm p}=0.1$, the estimated half-period $T_{n}^{\rm e}$ and actual half-
period $T_{n}$ asymptotically converge to each other, same as the simple model
simulation (Fig. 6a). Note that, in both experiments, the fast adaptation
enabled running, even when there remained a mismatch between $T_{n}^{\rm e}$
and $T_{n}$.

Figure 7: Responses of rhythm generator. (a) Actual and estimated half-period
$T_{n}$ (triangle) and $T_{n}^{\rm e}$ (circle) to step number $n$. (b) Phase
difference $\phi_{N}^{-}$ (triangle) and phase sum $\psi_{N}^{-}$ (circle) at
touchdown moments of right leg to stride number $N$. Left and right figures
indicate $K_{\rm p}=0$ (without feedback) and $K_{\rm p}=0.1$ (with feedback),
respectively.

Fig. 7b depicts the change of phase difference $\phi_{N}^{-}$ and phase sum
$\psi_{N}^{-}$ at the moment immediately prior to the touchdown of the right
leg with respect to $N$. For both $K_{\rm p}=0$ and $0.1$, the phase
difference $\phi_{N}^{-}$ converges to $\pi$. On the other hand, while
$\psi_{N}^{-}$ remained at approximately $2.5\pi$ for $K_{\rm p}=0$, it
increased to approximately $3\pi$ for $K_{\rm p}=0.1$.

Although the robot achieved sustained running until $N=15$, it could not
achieve steady running and fell after running. The response of thigh angle
$\theta_{N}^{-}$ to $N$ is plotted in Fig. 8. Fig. 9 shows the snapshots of
the experiment for $13\leq N\leq 16$. Even when the slow adaptation in the
rhythm generator is activated, the robot cannot continue to run because the
swing angle of the thigh gradually decreases as $N$ increases, which resulted
in a stumble.

Figure 8: Response of swing angle of thigh $\theta^{-}_{N}$ at touchdown
moments of right leg to stride number $N$. $\theta^{-}_{N}$ gradually
decreased as $N$ increased

.

Figure 9: Snapshots of timings immediately before right leg touch down. The
touchdown angle gradually reduced and resulted in falling down.

####  3.2.2 Slow adaptation of pattern formulator

To elucidate the effect of the slow adaptation in the pattern formulator, we
conducted robot experiments with adaptation of the swing rate $\mu_{n}$, which
is determined using (3) and (4). From this experiment, we verified that a
longer period of sustained running can be achieved by adjusting $\mu_{n}$ on
the basis of the thigh-angle feedback in the pattern formulator.

Figure 10: Response of (a) $\mu_{N}$ and (b) thigh swing angle
$\theta^{-}_{N}$ to stride number $N$ for adaptive $\mu_{n}$.

Fig. 10a presents the response of $\mu_{N}$ to the stride number $N$.
$\mu_{N}$ decreases and converges to approximately 0.55 around $N=6$ to $12$.
When $N=12$, $\mu_{N}$ increases substantially to an approximate value of 0.80
and subsequently decreases again.

Fig. 10b shows the plot of the right-thigh angle $\theta_{N}^{-}$ with respect
to $N$. The swing angle decreases regardless of the value of $\mu_{N}$ until
$N=9$. In contrast, after $N=10$, although the swing angle continues to
decrease when $\mu_{N}$ is constant, it increases with $N$ when $\mu_{N}$ is
adaptive. Fig. 11 shows the snapshots for eight strides. Thanks to the
adaptive $\mu_{n}$, the thigh angle is kept almost constant compared to the
experiment with constant $\mu_{n}$ (Fig. 9), which contribute to keep running.
The obtained results reveal that the adaptive $\mu_{n}$ prevents the monotonic
decrease in the thigh swing angle, thereby resulting in continuous running of
more than $N=20$.

Figure 11: Snapshots of timings immediately before right leg touch down with
adaptive $\mu_{n}$. The touchdown angles are kept almost constant.

The response of $T_{n}^{\rm e}$ to $n$ is plotted in Fig. 12. When $\mu_{n}$
is adaptive, the decrement rate of $T_{n}^{\rm e}$ is suppressed, and
$T_{n}^{\rm e}$ converges to a constant value ($\approx 440$ [ms]) as $n$
increases.

Figure 12: Response of $T_{n}$ (triangle) and $T_{n}^{\rm e}$ (circle) to $n$
with adaptive $\mu_{n}$.

##  4 Discussion

###  4.1 Effect of estimated gait cycle adaptation in rhythm generator

In this study, to realize a bipedal robot capable of running with human-like
actuation timing, we designed a controller based on the CPG that could adapt
its gait cycle through slow adaptation, in addition to fast adaptation (phase
resetting) in the rhythm generator. Notably, the results validated that both
the simple model and the robot could achieve continuous running. Moreover,
owing to the slow adaptation of the rhythm generator, the estimated gait cycle
and the actual gait cycle asymptotically approached each other through slow
adaptation (Figs. 6a and 7), whereas the left and right legs were actuated in
alternate phases by employing the phase-difference compensation term.
Furthermore, fast adaptation allowed the robot to run even for a mismatch
between the estimated and actual gait cycles.

These results suggest that the fast and slow adaptation of the rhythm
generator plays an significant role to generate the adaptive locomotion. In
the real robot experiment, the pattern formulator actuated the pneumatic
actuators, which represent the vastus muscles of the robot, and the BLDC
motors, which represent the hip extensor muscle groups, according to the phase
angle of the rhythm generator. Therefore, the obtained results demonstrated
that the robot could reproduce human-like running with a constant rate of
muscle firing relative to the gait cycle, regardless of the actual running
cycle, owing to the fast and slow adaptation of the rhythm generator.

However, when we solely utilized adaptation in the rhythm generator, the
running motion could not be sustained for a long time. The thigh-swing angle
decreased as the number of steps increased, thereby leading to this inability,
as illustrated in Figs. 8 and 9. In particular, this result was presumably a
result of the constant $\mu_{n}$ used in the pattern formulator. When the
initial value of the estimated half-period $T_{n}^{\rm e}$ is larger than the
actual half-period $T_{n}$, $T_{n}^{\rm e}$ decreased as $n$ increased owing
to slow adaptation, where the phase angular velocity of the rhythm generator
increased. This resulted in a short stance phase, and the robot could not kick
the ground sufficiently. Moreover, the swing angle also decreased because the
duration of the early swing phase was reduced. In particular, the repetition
of such a sequence is expected to decrease the swing angle of the thighs. This
experimental result suggested that a further adaptation mechanism is required
to achieve continuous running.

###  4.2 Effect of $\mu_{n}$ adaptation in pattern formulator

We achieved the continuous running of our bipedal robot for approximately $20$
strides by using adaptive $\mu_{n}$ in the pattern formulator. The robot’s
fall was prevented by adjusting $\mu_{n}$ appropriately (Fig. 10a) when the
thigh swing angle temporarily decreased to an inappropriate value, as shown in
Figs. 10b and 11, thereby yielding the successful result. Furthermore,
$\mu_{n}$ converged to a certain value of approximately 0.55 for $N=6$ to
$11$, as depicted in Fig. 10a. This result suggests the existence of an
appropriate $\mu_{n}$ for running, generated by the dynamic interaction
between the body of the robot and the environment. Therefore, the obtained
results suggest that humans invariably activate their muscles at the same rate
with respect to the gait cycle using the appropriate muscle activation timing,
which is defined by the dynamic interaction between the human body and the
environment.

However, even when we used the pattern formulator with slow adaptation, the
robot could not maintain continuous running for a long period. We discovered
that the thigh angle at liftoff timing gradually decreased during running,
thereby causing difficulties for the robot to kick the ground. This trend
resulted in the robot’s inability to lift itself upward sufficiently, and the
robot consequently stumbled and fell onto the treadmill. As indicated by the
results in Fig. 10b, $\theta_{N}^{-}$ was maintained at approximately 50 [deg]
after $N=12$, following the considerable change in $\mu_{N}$. To achieve
longer periods of continuous running, the duty rate $\beta$ in the pattern
formulator should be accordingly adjusted, which will result in an appropriate
swing angle.

##  5 Conclusion

In this study, we designed a CPG-based controller involving fast and slow
adaptation and a constant muscle activation timing to achieve human-like
running motion of a bipedal robot.

Consequently, the CPG adapted to the actual period and achieved continuous
running for both the simple model and the actual bipedal robot. Furthermore,
the results suggested that the CPG contributed to adjusting the muscle firing
timing properly according to the state of the body, which could also result in
optimum timing for human running.

However, in the robot experiment, even upon the implementation of the CPG with
fast and slow adaptation, the swing angle of the thigh decreased as the number
of steps increased, and the robot was unable to achieve continuous running for
a long period. In future research, we will adjust the duty ratio $\beta$ on
the basis of the swing angle.

Moreover, we will aim to improve upon the developed running robot and develop
a walking robot to elucidate its relationship to adaptive human locomotion.
Thus, further studies in this research direction will lead to engineering
applications not only for improved bipedal robots but also for walking
assistive devices.

## Acknowledgement

This work was supported in part by JSPS KAKENHI Grant Number JP21K14104 and
JP22H01445.

## References

  * [1] McGeer T. Passive dynamic walking. The International Journal of Robotics Research. 1990;9:62–82.
  * [2] Garcia M, Chatterjee A, Ruina A, et al. The simplest walking model: stability, complexity, and scaling. Journal of Biomechanical Engineering. 1998;120:281–288.
  * [3] Goswami A, Thuilot B, Espiau B. A study of the passive gait of a compass-like biped robot: Symmetry and chaos. International Journal of Robotics Research. 1998;17:1282–1301.
  * [4] Collins S, Ruina A, Tedrake R, et al. Efficient bipedal robots based on passive-dynamic walkers. Science. 2005;307:1082–1085.
  * [5] Ikemata Y, Yasuhara K, Sano A, et al. A study of the leg-swing motion of passive walking. Proceedings - IEEE International Conference on Robotics and Automation. 2008;:1588–1593.
  * [6] Okamoto K, Aoi S, Obayashi I, et al. Disappearance of chaotic attractor of passive dynamic walking by stretch-bending deformation in basin of attraction. In: 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS); 2020.
  * [7] HKino, KSakata, MUemura, et al. Simulation verification for the robustness of passive compass gait with a joint stiffness adjustment. Advanced Robotics. 2019;33:1129–1143.
  * [8] Blickhan R. The spring-mass model for running and hopping. Journal of Biomechanics. 1989;22:1217–1227.
  * [9] Full R, Koditschek D. Templates and anchors: neuromechanical hypotheses of legged locomotion on land. J Exp Biol. 1999;202:3325–3332.
  * [10] Geyer H, Seyfarth A, Blickhan R. Spring-mass running: Simple approximate solution and application to gait stability. Journal of Theoretical Biology. 2005;232:315–328.
  * [11] Clark K, Weyand P. Are running speeds maximized with simple-spring stance mechanics? Journal of Applied Physiology. 2014;117:604–615.
  * [12] Gan Z, Yesilevskiy Y, Zaytsev P, et al. All common bipedal gaits emerge from a single passive model. Journal of the Royal Society Interface. 2018;15.
  * [13] Miyamoto H, Sano A, Ikemata Y, et al. A study of bouncing rod dynamics aiming at passive running. In: IEEE International Conference on Robotics and Automation, ICRA2010. IEEE; 2010. p. 3298–3303.
  * [14] Kamimura T, Sato K, Murayama D, et al. Dynamical effect of elastically supported wobbling mass on biped running. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, IROS2021; 2021. p. 4048–4055.
  * [15] Rybak IA, Shevtsova NA, Lafreniere-Roula M, et al. Modelling spinal circuitry involved in locomotor pattern generation: Insights from deletions during fictive locomotion. Journal of Physiology. 2006;577:617–639.
  * [16] Cappellini G, Ivanenko YP, Poppele RE, et al. Motor patterns in human walking and running. Journal of Neurophysiology. 2006;95:3426–3437.
  * [17] Ivanenko YP, Cappellini G, Dominici N, et al. Modular control of limb movements during human locomotion. Journal of Neuroscience. 2007;27:11149–11161.
  * [18] Molkov YI, Bacak BJ, Talpalar AE, et al. Mechanisms of left-right coordination in mammalian locomotor pattern generation circuits: A mathematical modeling view. PLoS Computational Biology. 2015;11(5, e1004270).
  * [19] IARybak, KStecina, NAShevtsova, et al. Modelling spinal circuitry involved in locomotor pattern generation: insights from the effects of afferent stimulation. Journal of Physiology. 2006;577:641–658.
  * [20] YPIvarenko, REPoppele, FLacquaniti. Motor control programs and walking. Neuroscientist. 2006;12:339–348.
  * [21] Taga G, Yamaguchi Y, Shimizu H. Self-organized control of bipedal locomotion by neural oscillators in unpredictable environment. Biological Cybernetics. 1991;65:147–159.
  * [22] Manoonpong P, Wörgötter F. Efference copies in neural control of dynamic biped walking. Robotics and Autonomous Systems. 2009;57:1140–1153.
  * [23] Fukuoka Y, Habu Y, Fukui T. Analysis of the gait generation principle by a simulated quadruped model with a cpg incorporating vestibular modulation. Biological Cybernetics. 2013;107:695–710.
  * [24] Aoi S, Manoonpong P, Ambe Y, et al. Adaptive control strategies for interlimb coordination in legged robots: A review. Frontiers in Neurorobotics. 2017;11:1–21.
  * [25] Fujiki S, Aoi S, Funato T, et al. Adaptive hindlimb split-belt treadmill walking in rats by controlling basic muscle activation patterns via phase resetting. Scientific Reports. 2018;8, 17341.
  * [26] Aoi S, Tsuchiya K. Stability analysis of a simple walking model driven by an oscillator with a phase reset using sensory feedback. IEEE Transactions on Robotics. 2006;22:391 – 397.
  * [27] Aoi S, Tsuchiya K. Self-stability of a simple walking model driven by a rhythmic signal. Nonlinear Dynamics. 2007;48:1–16.
  * [28] J Perry JB. Gait analysis: Normal and pathological function, second edition. Slack Incorporated, New Jersey; 2010.
  * [29] Fujiki S, Aoi S, Funato T, et al. Adaptation mechanism of interlimb coordination in human split-belt treadmill walking through learning of foot contact timing: A robotics study. Journal of the Royal Society Interface. 2015;12, 20150542.
  * [30] Geyer H, Seyfarth A, Blickhan R. Compliant leg behaviour explains basic dynamics of walking and running. Proceedings of the Royal Society B: Biological Sciences. 2006;273:2861–2867.


# Stabilisation of the Arrival Time of a Relativistic Electron Beam to the 50
fs Level

J. Roberts  John Adams Institute (JAI), University of Oxford, Denys Wilkinson
Building, Keble Road, Oxford, OX1 3RH, United Kingdom  The European
Organization for Nuclear Research (CERN), Geneva 23, CH-1211, Switzerland  P.
Skowronski  The European Organization for Nuclear Research (CERN), Geneva 23,
CH-1211, Switzerland  P.N. Burrows  John Adams Institute (JAI), University of
Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, United Kingdom
G.B. Christian  John Adams Institute (JAI), University of Oxford, Denys
Wilkinson Building, Keble Road, Oxford, OX1 3RH, United Kingdom  R. Corsini
The European Organization for Nuclear Research (CERN), Geneva 23, CH-1211,
Switzerland  A. Ghigo  Laboratori Nazionali di Frascati (LNFN), Via Enrico
Fermi, 40, 00044 Frascati RM, Italy  F. Marcellini  Laboratori Nazionali di
Frascati (LNFN), Via Enrico Fermi, 40, 00044 Frascati RM, Italy  C. Perry
John Adams Institute (JAI), University of Oxford, Denys Wilkinson Building,
Keble Road, Oxford, OX1 3RH, United Kingdom

###### Abstract

We report the results of a low-latency beam phase feed-forward system built to
stabilise the arrival time of a relativistic electron beam. The system was
operated at the Compact Linear Collider (CLIC) Test Facility (CTF3) at CERN
where the beam arrival time was stabilised to approximately 50 fs. The system
latency was $350$ ns and the correction bandwidth $>23$ MHz. The system meets
the requirements for CLIC.

High-energy linear electron-positron colliders have been proposed as next-
generation particle accelerators for exploring the subatomic world with
increased precision. They provide sensitivity to new physics processes, beyond
those described by the Standard Model (SM) of elementary particle
interactions, at mass scales that can exceed the reach of the CERN Large
Hadron Collider (LHC) CLIC and CLICdp collaborations (2016).

The Compact Linear Collider (CLIC) Aicheler  _et al._ (2012) is the most
technologically mature concept of a high-energy lepton collider for enabling
direct searches for new physics in the multi-TeV energy regime. It uses a
novel two beam acceleration concept to achieve a high accelerating gradient of
100 MV/m and centre-of-mass collision energies of up to 3 TeV. This energy
reach, combined with high-luminosity of the electron-positron collisions, will
also enable precise measurements of properties of the Higgs boson Abramowicz
_et al._ (2016) and the top quark, and provide sensitivity to beyond-SM
phenomena CLIC and CLICdp collaborations (2016).

The CLIC two-beam acceleration concept is shown schematically in Fig. 1. The
12 GHz RF power used to accelerate the colliding electron and positron beams
is extracted from high intensity ‘drive beams’. The drive beams are 2.4 GeV
electron beams, with an initial bunch frequency of 0.5 GHz, a pulse length of
$148~{}\mathrm{\mu s}$, and a pulse repetition rate of 50 Hz. The intensity of
the drive beams is increased by a factor 24 using a bunch recombination
process Aicheler  _et al._ (2012), thereby creating a series of 240 ns pulses
bunched at 12 GHz. Each 240 ns sub-pulse is directed into a ‘decelerator
sector’, in which the drive beam pulse is decelerated, producing 12 GHz RF
power which is transferred to the accelerating structures of the main beams.
Two drive beams with 25 decelerator sectors each are required for a 3 TeV
collider.

One of the major challenges is the synchronisation of the arrival of the drive
and main beams at the power-extraction and transfer structures to better than
50 fs rms. This requirement limits the luminosity loss, resulting from
subsequent energy errors of the main beams, to less than 1% of the design
value Schulte and Tomas (2009). Free-electron lasers (FELs) also demand a high
degree of beam arrival-time stability w.r.t. an externally-applied laser beam
for the purpose of seeding of lasing by the electron beam Savelyev  _et al._
(2017).

Figure 1:  Schematic of the CLIC drive-beam concept showing the electron
acceleration complex CLIC and CLICdp collaborations (2016).

We express the temporal stability of the drive beam in terms of phase
stability at the 12 GHz acceleration frequency. An arrival time jitter of 50
fs rms is equivalent to a phase jitter of $0.2^{\circ}$ at 12 GHz. In the CLIC
design the incoming drive-beam phase jitter cannot be guaranteed to be better
than $2^{\circ}$ Aicheler  _et al._ (2012). A mechanism to improve the phase
stability by an order of magnitude is therefore required. The correction must
be applied to the full drive beam pulse length and have a bandwidth exceeding
17.5 MHz. This bandwidth is derived from simulations of the system performance
whilst assuming a pessimistic frequency spectrum of the incoming phase errors
Gerbershagen  _et al._ (2015).

This is implemented via a ‘phase feed-forward’ (PFF) system which measures the
incoming beam phase and provides a correction to the same beam pulse after it
has traversed the turnaround loop (TA in Fig. 1). One PFF system will be
installed in each deceleration sector. The correction is provided by
electromagnetic kickers in a 4-bend chicane: bunches arriving early (late) in
time have their path through the chicane lengthened (shortened) respectively.
A particular challenge is that the PFF latency must be shorter than the beam
flight time of approximately. 250 ns around the turnaround loop.

We describe a prototype PFF system (Fig. 2) that implements this novel concept
at the CLIC Test Facility (CTF3) at CERN. CTF3 provides a 135 MeV electron
beam bunched at 3 GHz frequency with a beam-pulse length of 1.2 $\mathrm{\mu
s}$ and a pulse repetition rate of 0.8 Hz Aicheler  _et al._ (2012).

The incoming beam phase is measured in two upstream phase monitors
($M_{1},M_{2}$). While the beam transits the ‘turnaround loop’ a phase-
correction signal is evaluated and used to drive fast, high power amplifiers;
these drive two electromagnetic kickers ($\mathrm{K_{1},K_{2}}$) which are
used to alter the beam transit time in a four-bend chicane. A downstream phase
monitor ($M_{3}$) is used to measure the effect of the correction.

The beam time of flight between $M_{1}$ and $\mathrm{K_{1}}$ is around 380 ns.
The total cable delay for the PFF correction signals is shorter, around 250
ns. The correction in the chicane can therefore be applied to the entirety of
the beam pulse measured at the PFF input ($\phi_{1}$, the measured phase at
$M_{1}$), provided that the hardware latency is less than 130 ns. Significant
hardware challenges include the resolution and bandwidth of the phase
monitors, and the power, latency and bandwidth of the kicker amplifiers. A low
latency digitiser/feedforward controller is also required.

Figure 2: Schematic of the CTF3 PFF prototype, showing the phase monitors
($M_{1}$ , $M_{2}$ and $M_{3}$) and kickers (K1 and K2). The black box “PFF”
represents the calculation and output of the correction. Bunches arriving
early at $M_{1}$ are deflected on to longer trajectories in the chicane
(blue), and bunches arriving late on to shorter trajectories (red). Dashed
lines indicate beam lines that are not used.

The requirements of the CLIC system and their corresponding CTF3 values are
listed in Table 1. The main differences result from the different drive-beam
energies. Higher power amplifiers (500 kW rather than 20 kW) are required for
CLIC, which may be achieved by combining the output of multiple modules
similar to those built for CTF3. CLIC also requires a distributed timing
system to synchronise the phase of the drive and main beams along the 50 km
facility, which is not addressed here.

Table 1:  Requirements for the CLIC PFF system, and the respective CTF3 parameters; performance achieved with the prototype system is indicated by *. | CLIC | CTF3 |
---|---|---|---
Drive beam energy | 2400 | 135 | MeV
No. PFF systems | 50 | 1 |
Kickers per PFF chicane | 16 | 2 |
Power of kicker amplifiers | 500 | $\mathbf{20^{*}}$ | kW
Angular deflection per kicker | $\pm 94$ | $\mathbf{\pm 560^{*}}$ | $~{}\mathrm{\mu rad}$
Correction range | $\pm 10$ | $\mathbf{\pm 6^{*}}$ | ∘
Correction bandwidth | $>17.5$ | $\mathbf{>23^{*}}$ | MHz
Phase monitor resolution | $<0.14$ | $\mathbf{0.12^{*}}$ | ∘
Initial phase jitter | $2.0$ | $0.9$ | ∘
Corrected phase jitter | $0.2$ | $\mathbf{0.2^{*}}$ | ∘

The phase monitors Marcellini  _et al._ (2014) are cylindrical cavities with
an aperture of 23 mm and a length of 19 cm. Small ridges (’notch filters’) in
the cavity create an effective volume with a resonant frequency of 12 GHz. The
field induced by the beam traversing the cavity contains a beam-position-
independent monopole mode and an unwanted position-dependent dipole mode. The
effect of the latter is removed by summing the outputs from an opposing pair
of feedthroughs, on the top and bottom of the cavity, via a RF ‘hybrid’. To
extract the beam phase the output from each hybrid is mixed with a 12 GHz
reference signal derived from a 3 GHz source which is phase-locked to the CTF3
RF system and serves all three phase monitors. By comparing the signals from
$M_{1}$ and $M_{2}$ we have measured a phase resolution of $0.12^{\circ}$,
i.e. about 30 fs Roberts (2016).

The phase signals are digitised in the feedforward controller board Roberts
(2016), which is used to calculate and output the amplifier drive signals, and
to control the correction timing. It consists of nine 14-bit analogue to
digital converters clocked at 357 MHz, a field programmable gate array, and
four digital to analogue converters.

The kicker amplifiers Roberts (2016) consist of one central control module and
two drive and terminator modules (one per kicker). The control module
distributes power and input signals to the drive modules. The 20 kW drive
modules consist of low-voltage Si FETs driving high-voltage SiC FETs; an input
voltage range of $\pm 2$ V corresponds to an output range of $\pm 700$ V. The
response is linear to within 3% for input voltages between $\pm 1.2$ V, and
the output bandwidth is 47 MHz for small signal variations of up to 20% of the
maximum. For larger signal variations the bandwidth is slew-rate limited.

The two electromagnetic stripline kickers Ghigo  _et al._ (2011) are 1 m in
length and have an internal aperture of 40 mm between two strips placed along
their horizontal walls. They are designed to give a response within a few ns
of the input signal. Opposite polarity voltages of up to 700 V applied to the
strips at the downstream end horizontally deflect the 135 MeV beam by up to
560 $\mu$rad.

The measured total latency of the phase monitor signal processing, the
feedforward calculation, and amplifier response was approximately 100 ns.
Therefore the output from the controller was delayed by an additional 30 ns to
synchronise the correction at the kicker with the beam arrival Roberts (2016).

The PFF operation placed severe constraints on the setting of the magnetic
lattice in both the beamline between the upstream phase monitors and the
correction chicane, and in the chicane itself. The beam transfer matrix
coefficient $R_{52}$ between the two kickers characterises the change in path
length through the chicane relative to the deflection applied at the first
kicker. With an $R_{52}$ value of $0.74$ m/rad Roberts (2016) the expected
maximum path length change for operation of the PFF system, corresponding to
the maximum deflection of $\pm 560$ $\mu$rad from each kicker, is about $\pm
400~{}\mathrm{\mu m}$, equivalent to $\pm 6^{\circ}$ in phase. The chicane
magnets were also set so that PFF operation does not change the beam
trajectory at the exit of the chicane Roberts (2016).

A further challenge to PFF operation was obtaining a high correlation between
the upstream and uncorrected downstream phases measured at $M_{1}$ and $M_{3}$
respectively. The maximum measurable correlation depends on both the phase
monitor resolution and any additional phase jitter introduced in the beamlines
between $M_{1}$ and $M_{3}$. The monitor resolution of $0.12^{\circ}$ limits
the maximum upstream-downstream phase correlation to $98\%$ in typical
conditions, and places a theoretical limit of $0.17^{\circ}$ on the measurable
corrected downstream phase jitter. The dominant beam source of uncorrelated
downstream phase jitter arises from energy jitter that is transformed into
phase jitter in the beamlines between $M_{1}$ and $M_{3}$.

Figure 3: Measured downstream (red) and upstream (blue) phase jitter vs. TL1
$R_{56}$ value. Error bars show the statistical standard error on the measured
jitter values.

To first order the phase dependence on energy can be described via the beam
transfer matrix coefficient $R_{56}$: $\phi_{3}=\phi_{1}+\alpha R_{56}(\Delta
p/p)$ , where $\Delta p/p$ is the particle’s relative energy error, $\phi_{1}$
and $\phi_{3}$ are the phases measured at $M_{1}$ and $M_{3}$ respectively,
and the constant $\alpha=14400$ ${}^{\circ}/$m converts the units of $R_{56}$
from metres to degrees at 12 GHz ($360^{\circ}$ per 0.025 m).

The optimal condition is $R_{56}$ = 0. This was achieved by tuning the
$R_{56}$ value in the ‘TL1’ transfer line (Fig. 2) so as to compensate for
non-zero $R_{56}$ in the other beamline sections. With
$R_{56,\mathrm{TL1}}=10$ cm the downstream phase jitter is reduced to the same
level as the upstream jitter (Fig. 3). However, a large $R_{566}$ coefficient
(second-order phase dependence on energy) remained uncorrected. As a result,
drifts in beam energy lead to a degradation in upstream-downstream phase
correlation even after optimising the $R_{56}$ term. Drifts in the CTF3 RF
system, and the resulting changes in beam energy, therefore made it difficult
to maintain maximal upstream-downstream phase correlation for timescales
longer than 10 minutes. Optics for a future CLIC PFF system must zero both
$R_{56}$ and the higher order energy dependences.

The PFF system acts to remove the $M_{1}$ phase, multiplied by a ‘gain’
factor, from the phase at $M_{3}$. If the phases at $M_{3}$ and $M_{1}$ are
fully correlated, and the jitters are identical, the optimal system gain is
unity. In practice the gain is chosen to achieve optimal performance for real
beam conditions. A representative gain scan is shown in Fig. 4. The optimal
gain is typically in the range 0.9–1.3. Also shown in Fig. 4 is a prediction
of the corrected phase jitter at $M_{3}$, using a simple model including the
initial beam phase jitters at $M_{1}$ and $M_{3}$, the upstream-downstream
phase correlation, and the gain Roberts (2016). The model reproduces the data.

Figure 4: Measured corrected beam phase jitter at $M_{3}$ vs. PFF gain
(points). Error bars show the statistical standard error on the measured
jitter values. The expected performance is shown by the red shaded region (see
text).

The PFF system simultaneously corrects pulse-to-pulse phase jitter and phase
variations within the 1.2 $\mu s$ beam pulse at CTF3. Fig. 5 shows the effect
of the PFF system on the intra-pulse phase variations. The PFF system was
operated in interleaved mode, with the correction applied to alternating
pulses only. This allows the initial (‘PFF Off’) and corrected (‘PFF On’)
downstream phase at $M_{3}$ to be measured at the same time. The $M_{1}$ (PFF
input) phase is also shown for comparison.

It is an operational feature at CTF3 that there is a roughly parabolic phase
sag of $40^{\circ}$ along the pulse, resulting from the upstream RF pulse
compression scheme Aicheler  _et al._ (2012). Hence approximately a 440 ns
portion of the pulse is within the $\pm 6^{\circ}$ dynamic range of the PFF
system, and can be corrected to zero nominal phase. This time duration for the
full correction exceeds the CLIC drive-beam pulse length of 240ns and in any
case the CLIC design avoids such a large phase sag Aicheler  _et al._ (2012).
Vertical dashed lines in Fig. 5 mark the 440 ns portion of the pulse where
full correction is possible.

(a)

(b)

Figure 5: Correction of the pulse shape with the PFF system. Shown are: the
incoming phase ($\phi_{1}$) measured in $M_{1}$ (green), and the downstream
phase ($\phi_{3}$) measured in $M_{3}$ with PFF off (blue) and PFF on (red).
Each trace is the average over a 30 minute dataset. (a) The whole beam pulse.
Vertical dashed lines mark the time interval corresponding to the PFF dynamic
range. (b) The same data zoomed in to the central portion of the pulse. Shaded
areas represent the phase jitter at each sample point.

Within the range the PFF system flattens the phase, and almost all variations
are removed. The average intra-pulse phase variation (rms) over the dataset is
reduced from $0.960\pm 0.003^{\circ}$ (PFF off), to $0.285\pm 0.004^{\circ}$
(PFF on).

Figure 6: Amplitude of phase errors vs. frequency ($f$) in bins of 2.3 MHz
with the PFF system off (blue) and on (red), across a 30 minute dataset.

In order to meet CLIC requirements (Table 1) the PFF correction bandwidth
should be at least 17.5 MHz. A Fourier-Transform (FFT) method was used to
characterise the PFF on/off datasets. The FFT amplitude is shown vs. frequency
in Fig. 6. It can be seen that phase errors are corrected by up to a factor of
5 for frequencies up to 23 MHz, above which they are smaller than the monitor
resolution and not measurable. This is consistent with an expected system
bandwidth of around 30 MHz, and exceeds the CLIC requirement.

The effect of the PFF system on the pulse-to-pulse jitter, i.e. the jitter on
the mean phase of each beam pulse, is shown in Fig 7 for a dataset of around
ten minutes duration. The pulse-to-pulse phase jitter is reduced from $0.92\pm
0.04^{\circ}$ to $0.20\pm 0.01^{\circ}$, meeting CLIC-level phase stability.
The system acts to remove all correlations between the upstream and downstream
phase, reducing an initial correlation of $96\pm 2\%$ to $0\pm 7\%$ for this
dataset. Given the incoming upstream phase jitter and measured upstream-
downstream correlation, the performance is consistent with the theoretically
predicted correction of $0.26\pm 0.06^{\circ}$.

Figure 7: Distribution of the mean downstream phase with the PFF system off
(blue) and on (red).

The system was further tested by varying the incoming mean beam phase
systematically by around $\pm 3^{\circ}$ (Fig. 8). Variations of this
magnitude are comparable to the expected conditions in the CLIC design (Table
1). This is illustrated in Fig. 8. The system removed the induced phase
variations and achieved more than a factor-5 reduction in the downstream phase
jitter, correcting from $1.71\pm 0.07^{\circ}$ to $0.32\pm 0.01^{\circ}$.

Figure 8: Mean downstream phase vs. time with the PFF system off (blue) and on
(red) subject to large additional phase variations added to the incoming phase
(see text).

In summary, we have built, deployed and tested a prototype drive-beam phase
feedforward system for CLIC. The system incorporates purpose-built high-
resolution phase monitors, an advanced signal-processor and feedforward
controller, low-latency, high-power, high-bandwidth amplifiers, and
electromagnetic stripline kickers. The phase-monitor resolution was measured
to be $0.12^{\circ}\simeq$ 30 fs. The overall system latency, including the
hardware and signal transit times, was measured to be approximately 350 ns,
which is less than the beam time of flight between the input phase monitor and
the correction chicane. The system was used to stabilise the pulse-to-pulse
phase jitter to $0.20\pm 0.01^{\circ}\simeq$ 50 fs, and to simultaneously
correct intra-pulse phase variations at frequencies up to 23 MHz.

Our demonstration of a beam-based arrival-time stabilisation system with a
performance at the 50 fs level has potential application at other beamlines
where a high degree of beam arrival stability is required. For example, ‘pump-
probe’ experiments at FELs require laser/electron synchronisation ideally to
the few femtosecond level, see e.g. Savelyev  _et al._ (2017). The current
state-of-the-art in synchronisation at FELs is approximately 30 fs, using all-
optical techniques Schulz  _et al._ (2015). Our results are limited by the
beam arrival-time monitor resolution of approximately 30 fs. With higher
precision monitors (e.g. Löhl  _et al._ (2010)) 10 fs stabilisation could be
achieved with our technique. A key feature of our system is that it
incorporates a beam turnaround, which provides sufficient beam delay to allow
a feed-forward correction to be derived and applied with zero effective
latency. FEL designs based on energy-recovery linacs (see e.g. Sekutowicz  _et
al._ (2005); Kim  _et al._ (2008); Jackson  _et al._ (2016)) intrinsically
incorporate a beam turnaround section that would enable the deployment of a
high-performance system based on our technique.

###### Acknowledgements.

We acknowledge Alessandro Zolla and Giancarlo Sensolini (INFN Frascati) for
the mechanical design of the phase monitors and kickers, and Alexandra
Andersson, Luca Timeo and Stephane Rey (CERN) for their support on the phase
monitor electronics. We thank the operations team of CTF3 for their
outstanding support. This work was supported by the the UK Science and
Technology Facilities Council, and by the European Commission under the FP7
Research Infrastructures project Eu-CARD, grant agreement no. 227579.

## References

  * CLIC and CLICdp collaborations (2016) CLIC and CLICdp collaborations, “Updated baseline for a staged compact linear collider,” CERN-2016-004 (2016).
  * Aicheler  _et al._ (2012) M. Aicheler _et al._ , “CLIC conceptual design report,” CERN-2012-007 (2012).
  * Abramowicz  _et al._ (2016) H. Abramowicz _et al._ , Eur. Phys. J. C (Submitted), CLICdp-Pub-2016-001 (2016).
  * Schulte and Tomas (2009) D. Schulte and R. Tomas, in  _Proceedings of PAC09_ (Vancouver, Canada, 2009) pp. 3811–3813.
  * Savelyev  _et al._ (2017) E. Savelyev _et al._ , New. J. Phys 19, 043009 (2017).
  * Gerbershagen  _et al._ (2015) A. Gerbershagen _et al._ , Phys. Rev. STAB 18, 041003 (2015).
  * Marcellini  _et al._ (2014) F. Marcellini _et al._ , EuCARD-REP-2013-023 (2014).
  * Roberts (2016) J. Roberts, DPhil thesis, University of Oxford (2016).
  * Ghigo  _et al._ (2011) A. Ghigo _et al._ , in  _Proceedings of IPAC2011_ (San Sebastian, Spain, 2011) pp. 1000–1002.
  * Schulz  _et al._ (2015) S. Schulz _et al._ , Nat. Commun. 6, 5938 (2015).
  * Löhl  _et al._ (2010) F. Löhl _et al._ , Phys. Rev. Let. 104, 144801 (2010).
  * Sekutowicz  _et al._ (2005) J. Sekutowicz _et al._ , Phys. Rev. STAB 8, 010701 (2005).
  * Kim  _et al._ (2008) K. Kim _et al._ , Phys. Rev. Let. 100, 244802 (2008).
  * Jackson  _et al._ (2016) F. Jackson _et al._ , Phys. Rev. STAB 19, 120701 (2016).


# Electron-photon interaction in
de Sitter ambient space formalism

F. Jalilifard<EMAIL_ADDRESS>Department of Physics, Razi University,
Kermanshah, Iran  Y. Ahmadi<EMAIL_ADDRESS>Department of Physics, Razi
University, Kermanshah, Iran  M.V. Takook<EMAIL_ADDRESS>Department of
Physics, Razi University, Kermanshah, Iran

###### Abstract

The electron-photon interaction (the spinor-vector field interaction) in the
de Sitter ambient space formalism is investigated in the first (leading order)
approximation. The interaction Lagrangian and the scattering matrix are
presented. In this approximation, the scattering matrix can be written as a
expansion of the interaction Lagrangian. The tree-diagram of electron-photon
scattering amplitude is calculated. Finally the Minkowski limit is considered.

##  I Introduction

The effect of the gravitational field on the quantum fields is an interesting
problem for understanding the behavior of the quantum gravity in the linear
approximation. In this direction, the quantum field theory (QFT) in curved
space-time and the quantum black hole permit us to better understand the
quantum gravity effect. Therefore it is very important that the QFT in de
Sitter (dS) space-time is studied. The cosmic microwave background radiation
Nature , highly redshift observation of the Supernova Ia Perl ; Riess  and
galaxy clusters Henry 1 ; Henry 2 , indicate that our current universe in the
first approximation may be described by a dS space-time. The observational
data collected by BICEP2 BICEP2  may confirm that the early universe, in a
good approximation, is also the dS universe. Therefore, in this paper the
background curved space-time is chosen as the dS space-time.

The dS space-time can be considered as 4-dimensional hyperboloid embedded in
5-dimensional flat Minkowsi space-time, which is called the ambient space
formalism. Because of linearity of the action of dS group on the ambient space
formalism, unlike the intrinsic coordinates, working with this formalism leads
to simplicity of formulation and calculation of the Green functions. The
unitary irreducible representation of dS group and the analyticity of the
complexified dS space-time is a base of the rigorous mathematical construction
of QFT in dS space-time that have been studied in 77 ; Bros 1 ; brmo ; brmo03
; tak ; ta97 .

The unitary irreducible representation is used to consider the free field and
to construct the Hilbert space and Fock space. In Minkowski flat space-time,
the analyticity properties are the basis parts of the interaction fields
calculations, and the theory of interacting fields is formulated in terms of
global analyticity properties of the two-point functions. The analyticity of
the two-point function is a very important properties which permit us to
calculate the probability amplitude in the interaction case. These properties
for dS space-time was proved by Brose Bros 1 ; brmo ; brmo03 and in dS ambient
space formalism these are very similar to the Minkowski space-time. Thus it is
suitable to use this formalism to consider the interaction between the
different fields.

In this paper, the quantum electrodynamics in dS universe is discussed. In
this direction the electron-photon interaction (the spinor-vector fields
interaction) is studied. After fixing our notations in section II, which
include a description of the free spinor and vector field quantization in the
dS ambient space formalism, the interaction Lagrangian and the scattering
matrix are presented in section III. Section IV is devoted to the Compton
scattering. Finally, the Minkowski lime is presented in section V. We discuss
our results in section VI.

##  II Notations and terminology

The dS space-time can be considered as a four-dimensional hyperboloid embedded
in five-dimensional Minkowski space-time with the following equation:

$M_{H}=\left\\{x\in{\rm IR}^{5}|\;\;x\cdot
x=\eta_{\alpha\beta}x^{\alpha}x^{\beta}=-H^{-2}\right\\},\;\;\alpha,\beta=0,1,2,3,4\,.$

The metric is:

$ds^{2}=\eta_{\alpha\beta}dx^{\alpha}dx^{\beta}|_{x^{2}=-H^{-2}}=g_{\mu\nu}^{dS}dX^{\mu}dX^{\nu},\;\;\mu=0,1,2,3,$
(II.1)

where $\eta_{\alpha\beta}=$diag$(1,-1,-1,-1,-1)$, $H$ is Hubble parameter,
$X^{\mu}$ is dS intrinsic coordinates and $x^{\alpha}$ is the five-dimensional
dS ambient space formalism.

The dS group is defined by $SO(1,4)=\left\\{\Lambda\in GL(5,{\rm
IR})|\;\;\det\Lambda=1,\;\;\Lambda\eta\lambda^{t}=\eta\right\\},$ where
$\Lambda^{t}$ is the transpose of $\Lambda$. The action of dS group on the
ambient space coordinate $x^{\alpha}$ is linear and this leads to simplicity
of equation and calculation in this formalism 77 :

$x^{\prime\alpha}=\Lambda^{\alpha}_{\;\;\beta}x^{\beta},\;\;\;\Lambda\in
SO(1,4)\Longrightarrow x\cdot x=x^{\prime}\cdot x^{\prime}=-H^{-2}.$ (II.2)

In this formalism the five matrices $\gamma^{\alpha}$ are needed which satisfy
the following relation ta97 ; ta96 ; bagamota ; 77 :

$\gamma^{\alpha}\gamma^{\beta}+\gamma^{\beta}\gamma^{\alpha}=2\eta^{\alpha\beta}\;,\;\gamma^{\alpha\dagger}=\gamma^{0}\gamma^{\alpha}\gamma^{0}\,,$
(II.3)

and they may be chosen as:

$\gamma^{0}=\left(\begin{array}[]{clcr}I&\;\;0\\\ 0&-I\\\
\end{array}\right),\;\;\;\gamma^{4}=\left(\begin{array}[]{clcr}0&I\\\ -I&0\\\
\end{array}\right),\;$ $\gamma^{1}=\left(\begin{array}[]{clcr}0&i\sigma^{1}\\\
i\sigma^{1}&0\\\
\end{array}\right),\;\;\gamma^{2}=\left(\begin{array}[]{clcr}0&-i\sigma^{2}\\\
-i\sigma^{2}&0\\\
\end{array}\right),\;\;\gamma^{3}=\left(\begin{array}[]{clcr}0&i\sigma^{3}\\\
i\sigma^{3}&0\\\ \end{array}\right)\,,$ (II.4)

where $\sigma^{i}$ $(i=1,2,3)$ are the Pauli matrices. The $\gamma^{\alpha}$
matrices in dS ambient space formalism are different from Minkowski
$\gamma^{\prime\mu}$ matrices. In the null curvature limit the relation
between them are bagamota :

$\gamma^{\prime\mu}=\gamma^{\mu}\gamma^{4}.$ (II.5)

The dS Dirac first-order field equation is 77 ; ta97 :

$\left(\cancel{x}\;\cancel{\partial}^{\top}-2\pm
i\nu\right)\psi(x)=0,\;\;\;\cancel{x}=\gamma_{\alpha}x^{\alpha}=\gamma\cdot
x\,,$ (II.6)

where $\nu>0$ is related with dS mass parameter as
$m_{f,\nu}^{2}=H^{2}(\nu^{2}+2\pm i\nu)$ and
$\partial_{\alpha}^{\top}=\partial_{\alpha}+Hx_{\alpha}x.\partial$ is the
transverse derivative. The charged spinor field operator, which satisfy the
above field equation, is bagamota ; 77 :

$\psi(x)=\int_{S^{3}}d\mu(\xi)\sum_{\sigma}\left[a(\tilde{\xi},\sigma)(Hx.\xi)^{-2-i\nu}{\cal
U}(x,\xi,\sigma)+b^{{\dagger}}(\xi,\sigma)(Hx.\xi)^{-1+i\nu}{\cal
V}(x,\xi,\sigma)\right],$ (II.7)

where $\xi^{\alpha}=(\xi^{0},\vec{\xi},\xi^{4})\in C^{+}=\left\\{\xi\in{\rm
IR}^{5}|\;\;\xi\cdot\xi=0,\;\;\xi^{0}>0\right\\}$ is the transformed variable
of $x^{\alpha}$ in positive cone that correspond to energy-momentum four-
vector in Minkowski space-time.
$\tilde{\xi}^{\alpha}=(\xi^{0},-\vec{\xi},\xi^{4})$ and the
$\mathrm{d}\mu(\xi)$ is the $SO(4)$-invariant normalized volume. The explicit
form of $\cal{U}$ and $\cal{V}$ were presented in bagamota . The adjoint
spinor $\overline{\psi}(x)$ in ambient space formalism is define as
$\overline{\psi}(x)=\psi^{\dagger}(x)\gamma^{0}\gamma^{4}$ ta96 ; bagamota ;
ta97 ; 77 . The creation operators can be defined as 77 :

$a^{\dagger}(\xi,\sigma)\left|\Omega\right>\equiv\left|1_{\xi,\sigma}^{a}\right\rangle,\;\;b^{\dagger}(\xi,\sigma)\left|\Omega\right>\equiv\left|1_{\xi,\sigma}^{b}\right\rangle\,.$
(II.8)

The vacuum state $|\Omega>$ is invariant under the action of the UIR of the dS
group and it can be identified with the Bunch-Davies or Hawking-Ellis vacuum
state. Its norm can be fixed in the null curvature limit as
$\left<\Omega\right.\left|\Omega\right>=1$ brmo ; 77 . The anti-commutation
relations for creation and annihilation operators are:

$\left[a(\tilde{\xi}^{\prime},\sigma^{\prime}),a^{\dagger}(\xi,\sigma)\right]_{+}={\cal
N}_{p}(\xi,\sigma)\delta_{s^{3}}(\xi-\xi^{\prime})\delta_{\sigma\sigma^{\prime}}\,,$
$\left[b(\tilde{\xi}^{\prime},\sigma^{\prime}),b^{\dagger}(\xi,\sigma)\right]_{+}={\cal
N}_{p}(\xi,\sigma)\delta_{s^{3}}(\xi-\xi^{\prime})\delta_{\sigma\sigma^{\prime}},$
(II.9)

where ${\cal N}_{p}(\xi,\sigma)$ is the normalization constant.

Analytic field operator is define in complex dS space-time as 77 ; brmo :

$\psi(x)=\lim_{y\rightarrow 0}\psi(z)=\lim_{y\rightarrow 0}\psi(x+iy).$

The analytic two-point function of spinor field is:

$\left<\Omega\right|\psi_{i}(z_{1})\bar{\psi}_{j}(z_{2})\left|\Omega\right>=iS_{ij}(z_{1},z_{2})=\dfrac{c_{\frac{1}{2},\nu}}{2}\int_{S^{3}}d\mu(\xi)(z_{1}.\xi)^{-2-i\nu}(z_{2}.\xi)^{-2+i\nu}\left(\cancel{\xi}\gamma^{4}\right)_{ij},$
(II.10)

where
$c_{\frac{1}{2},\nu}=\frac{\nu(\nu^{2}+1)}{(2\pi)^{3}(e^{2\pi\nu}-1)}\,.$ The
analytic two-point function has been calculated in terms of generalized
Legendre function of first kind ta96 ; ta97 ; bagamota .

In the ambient space formalism one can write the massless vector field,
similar to other massless quantum field, in terms of the massless conformally
coupled scalar field. The field operator obtain in terms of annihilation $d$
and creation $d^{\dagger}$ operators as gagarota ; ta97 :

$A_{\alpha}(x)=\int_{B}d\mu({\xi})\sum_{n}\left[d({\tilde{\xi}},n)(Hx.\xi)^{-2}{\cal
E}_{2\alpha}(x,\xi,n)+d^{{\dagger}}(\xi,n)(Hx.\xi)^{-1}{\cal
E}_{1\alpha}(x,\xi,n)\right]\,,$ (II.11)

where $n=0,1,2,3$ are the polarization states and
$d\mu({\xi})_{B}=2\pi^{2}r^{3}drd\mu({\xi})$ is the Euclidean measure on the
unit ball $B$ tak ; ta97 . The commutation relation between creation and
annihilation operators is gata ; gagarota :

$\left[d(\tilde{\xi}^{\prime},n^{\prime}),d^{\dagger}(\xi,n)\right]_{-}={\cal
N}_{k}(\xi,n)\delta_{s^{3}}(\xi-\xi^{\prime})\delta_{nn^{\prime}}\,,$ (II.12)

and ${\cal N}_{k}(\xi,n)$ is also the normalization constant.

##  III Scattering matrix

In dS space-time, the dS-Dirac field equation is invariant under $U(1)$ global
symmetry:

$\psi(x)\rightarrow\psi^{\prime}(x)=e^{-i\epsilon}\psi(x),\;\;\;(-i\cancel{x}\;\cancel{\partial}^{\top}+2i+\nu)\psi^{\prime}(x)=0\,,$

where $\epsilon$ is a constant. This equation is not invariant under the local
$U(1)$ symmetry:

$\psi(x)\rightarrow\psi^{\prime}(x)=e^{-i\epsilon(x)}\psi(x),\;\;\;(-i\cancel{x}\;\cancel{\partial}^{\top}+2i+\nu)\psi^{\prime}(x)\neq
0.$

By changing the covariant derivative $\partial^{\top}_{\alpha}$ with a gauge
covariant derivative $D_{\alpha}$:

$D_{\alpha}=\partial^{\top}_{\alpha}+iqA_{\alpha},\;\;\;\;A_{\alpha}\rightarrow
A^{\prime}_{\alpha}=A_{\alpha}+\dfrac{1}{Q}\partial^{\top}_{\alpha}\epsilon(x).$

one can obtain dS-Dirac local gauge invariant equation agt2005 . In the free
field Lagrangian by applying this changing, the interaction Lagrangian can be
defined as agt2005 :

${\cal
L}_{0}=H\bar{\psi}(x)\gamma^{4}(-i\cancel{x}\;\cancel{\partial}^{\top}+2i+\nu)\psi(x),\;\;\;{\cal
L}_{int}=QH\bar{\psi}(x)\gamma^{4}\cancel{x}\cancel{A}(x)\psi(x).$

The $Q$ in the null curvature limit can be considered as the electric charge
$e$. As it is seen, the interaction Lagrangian is independent of derivative of
fields, therefor one can show easily that:

${\cal H}_{int}=-{\cal
L}_{int}=-QH\bar{\psi}(x)\gamma^{4}\cancel{x}\cancel{A}(x)\psi(x)\,.$ (III.13)

The interaction Lagrangian has the important role in the theoretical and
particle physics in Minkowski space-time for calculation of the probability
amplitude or the S-matrix elements, but in curved space-time one can not
define the asymptotic states ($t=-\infty$ and $t=+\infty$) for defining the
S-matrix elements. For defining the time evolution operator,
$|\alpha,t\rangle=U(t,t_{0})|\alpha,t_{0}\rangle$, at first, the time or
coordinate system must be defined. For simplicity one can choose the static
coordinate system:

$\left\\{\begin{array}[]{clcr}x^{0}&=\sqrt{H^{-2}-r^{2}}\sinh Ht_{s},\\\
x^{1}&=\sqrt{H^{-2}-r^{2}}\cosh Ht_{s},\\\ x^{2}&=r\cos\theta,\\\
x^{3}&=r\sin\theta\cos\phi,\\\ x^{4}&=r\sin\theta\sin\phi,\\\
\end{array}\right.$ (III.14)

where $-\infty<t_{s}<\infty\;,\;0\leq
r<H^{-1}\;,\;0\leq\theta\leq\pi\;,\;0\leq\phi<2\pi$. This coordinate system
does not cover all dS hyperboloid. In this coordinate system the metric is:

$ds^{2}=\left(1-r^{2}H^{2}\right)dt_{s}^{2}-\left(1-r^{2}H^{2}\right)^{-1}dr^{2}-r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (III.15)

In this coordinate system, the time evolution operator for a quantum dS black-
hole was considered (for more details see ta17 ). Here since the interaction
occur in the atomic dimension, we can use the flat limit approximation which
ignore the direct effect of curvature in this dimension but the indirect
effect exist. In this approximation the time evolution operator can be
expanded in terms of the Minkowski counterpart:

$U(t,t_{0})={U_{M}}(t,t_{0})+Hf(t,t_{0})+...\;,$

where in the null curvature limit $U_{M}$ is exactly the time evolution
operator in Minkowski space-time. The term $f(t,t_{0})$ is due to the direct
effect of curvature which is ignored in this calculation since the interaction
is in atomic level.

If we have the incoming free field state $|\phi_{in}\;,-\infty>$ which
interact with the interaction Lagrangian ${\cal L}_{int}$, and the outgoing
free field state $|\phi_{out}\;,+\infty>$, we can write
$|\phi_{out}\;,+\infty>$ in terms of $|\phi_{in}\;,-\infty>$ as:

$|\phi_{out}\;,+\infty>=U(\infty\;,-\infty)|\phi_{in}\;,-\infty>={\cal
S}|\phi_{in}\;,-\infty>.$ (III.16)

Therefore the scattering matrix may be presented in the following approximate
equation to calculate the indirect effect of curvature corrections, from the
Minkowski counterpart mandl shaw ; kaku :

${\cal S}\simeq\frac{(-i)^{\lambda}}{\lambda!}\sum_{\lambda=0}^{\infty}{\cal
S}^{(\lambda)},$

where

${\cal
S}^{(\lambda)}=\int\mathrm{d}\mu(x_{1})\int\mathrm{d}\mu(x_{2})...\int\mathrm{d}\mu(x_{\lambda})\;T[{\cal
H}_{int}(\phi(x_{1})){\cal H}_{int}(\phi(x_{2}))...{\cal
H}_{int}(\phi(x_{\lambda}))].$ (III.17)

The symbol $T$ is the time order product of operators. The probability
amplitude, Feynman amplitude, transition rate and other experimental quantity
can be written by calculating the ${\cal S}$ matrix elements. It is important
to note that in the null curvature limit the usual QED in Minkowski space-time
is obtained.

##  IV Compton scattering

In this section we will study the Compton scattering or the interaction
between the vector and spinor fields, in the tree level approximation. The
scattering matrix ${\cal S}^{(2)}$ will be calculated for spinor and vector
fields, which is very similar to the Minkowski QED. By inserting the
Lagrangian (III.13) in the equation (III.17) for $\lambda=2$ and applying
Wick’s theorem mandl shaw , the time order product of (III.17) can be
expressed in terms of the following normal order products:

$T\left[\overline{\psi}(x_{1})\gamma^{4}\cancel{x}_{1}\cancel{A}(x_{1})\psi(x_{1})\overline{\psi}(x_{2})\gamma^{4}\cancel{x}_{2}\cancel{A}(x_{2})\psi(x_{2})\right]=N\left[\overline{\psi}(x_{1})\gamma^{4}\cancel{x}_{1}\cancel{A}(x_{1})\psi(x_{1})\overline{\psi}(x_{2})\gamma^{4}\cancel{x}_{2}\cancel{A}(x_{2})\psi(x_{2})\right]\\\
\\\
+N\left[\underleftrightarrow{\overline{\psi}(x_{1})\gamma^{4}\cancel{x}_{1}\cancel{A}(x_{1})\psi(x_{1})\overline{\psi}(x_{2})\gamma^{4}\cancel{x}_{2}\cancel{A}(x_{2})\psi}(x_{2})\right]+N\left[\overline{\psi}(x_{1})\gamma^{4}\cancel{x}_{1}\cancel{A}(x_{1})\underleftrightarrow{\psi(x_{1})\overline{\psi}}(x_{2})\gamma^{4}\cancel{x}_{2}\cancel{A}(x_{2})\psi(x_{2})\right]\\\
\\\
+N\left[\overline{\psi}(x_{1})\gamma^{4}\cancel{x}_{1}\underleftrightarrow{\cancel{A}(x_{1})\psi(x_{1})\overline{\psi}(x_{2})\gamma^{4}\cancel{x}_{2}\cancel{A}}(x_{2})\psi(x_{2})\right]\\\
\\\
+N\left[\underleftrightarrow{\overline{\psi}(x_{1})\gamma^{4}\cancel{x}_{1}\underleftrightarrow{\cancel{A}(x_{1})\psi(x_{1})\overline{\psi}(x_{2})\gamma^{4}\cancel{x}_{2}\cancel{A}}(x_{2})\psi}(x_{2})\right]+N\left[\overline{\psi}(x_{1})\gamma^{4}\cancel{x}_{1}\underleftrightarrow{\cancel{A}(x_{1})\underleftrightarrow{\psi(x_{1})\overline{\psi}}(x_{2})\gamma^{4}\cancel{x}_{2}\cancel{A}}(x_{2})\psi(x_{2})\right]\\\
\\\
+N\left[\underleftrightarrow{\overline{\psi}(x_{1})\gamma^{4}\cancel{x}_{1}\cancel{A}(x_{1})\underleftrightarrow{\psi(x_{1})\overline{\psi}}(x_{2})\gamma^{4}\cancel{x}_{2}\cancel{A}(x_{2})\psi}(x_{2})\right]\\\
\\\
+N\left[\underleftrightarrow{\overline{\psi}(x_{1})\gamma^{4}\cancel{x}_{1}\underleftrightarrow{\cancel{A}(x_{1})\underleftrightarrow{\psi(x_{1})\overline{\psi}}(x_{2})\gamma^{4}\cancel{x}_{2}\cancel{A}}(x_{2})\psi}(x_{2})\right]\,,$

where $N[\cdots]$ is the normal order products and the symbol
$\leftrightarrow$ under the two fields represents the vacuum expectation value
of the time order product of the two fields. The second and third terms belong
to spinor-vector (electron-photon) scattering (Compton scattering) and forth
term belongs to spinor-spinor scattering. The spinor and vector fields
expansion (II.7) and (II.11) can be rewritten in the following forms:

$\psi(x)=\psi^{+}(x)+\psi^{-}(x),\;\;\overline{\psi}(x)=\overline{\psi}^{+}(x)+\overline{\psi}^{-}(x),\;\;A(x)=A^{+}(x)+A^{-}(x)\;.$
(IV.18)

Using the commutation and anti-commutation relations of creation and
annihilation operators (II.9) and (II.12), the act of
$\psi^{+},\overline{\psi}^{-}$ and $A^{+}$ on vacuum state are:

$\psi^{+}|\Omega>=0,\;\;\;\overline{\psi}^{-}|\Omega>=0,\;\;\;A^{+}|\Omega>=0.$

In the following equation:

$N\left[\overline{\psi}(x_{1})\gamma^{4}\cancel{x}_{1}\cancel{A}(x_{1})\underleftrightarrow{\psi(x_{1})\overline{\psi}}(x_{2})\gamma^{4}\cancel{x}_{2}\cancel{A}(x_{2})\psi(x_{2})\right],$
(IV.19)

the symbole $\underleftrightarrow{\psi(x_{1})\overline{\psi}}(x_{2})$ is the
spinor two-point function (II.10):

$\underleftrightarrow{\psi(x_{1})\overline{\psi}}(x_{2})\equiv\left<\Omega\left|\psi(x_{1})\overline{\psi}(x_{2})\right|\Omega\right>.$

In relation (IV.19) the $\gamma,\;\cancel{x}$ and $\psi$ are in the matrix
form, thus we write them in terms of their components. By inserting the
(IV.18) in (IV.19), using transversality condition, $x.A=0$, and some simple
calculation, one can see that the physical terms are as follows:

$\mbox{physical\;terems}=\overline{\psi}^{+}_{i}(x_{1})\cancel{A}^{-}_{jh}(x_{1})\cancel{A}^{+}_{lr}(x_{2})\psi^{+}_{r}(x_{2})+\overline{\psi}^{+}_{i}(x_{1})\cancel{A}^{-}_{lr}(x_{2})\cancel{A}^{+}_{jh}(x_{1})\psi^{+}_{r}(x_{2})$
$+\overline{\psi}^{-}_{i}(x_{1})\cancel{A}^{-}_{jh}(x_{1})\cancel{A}^{+}_{lr}(x_{2})\psi^{-}_{r}(x_{2})+\overline{\psi}^{-}_{i}(x_{1})\cancel{A}^{-}_{lr}(x_{2})\cancel{A}^{+}_{jh}(x_{1})\psi^{-}_{r}(x_{2})\,.$

The two first terms will display the electron-photon (spinor-vector fields)
scattering witch can be presented in figure 1:

$x_{1}$$x_{2}$$\overline{\psi}^{+}$$A^{-}$$A^{+}$$\psi^{+}$$x_{1}$$x_{2}$$\overline{\psi}^{+}$$A^{+}$$A^{-}$$\psi^{+}$$S_{F}(x_{1};x_{2})$$S_{F}(x_{1};x_{2})$
Figure 1: Electron-photon scattering tree diagram

By using the relation (II.10) the $\cal S$ matrix for Compton scattering in
ambient formalism of dS space in this approximation is obtained as:

${\cal
S}^{(2)}=-Q^{2}H^{2}\int\int\mathrm{d}\mu(x_{1})\mathrm{d}\mu(x_{2})\gamma^{4}_{ij}(\cancel{x}_{1}S(x_{1},x_{2})\gamma^{4}\cancel{x}_{2})_{hl}\;{\cal
M}_{ijhl},$

where

${\cal
M}_{ijhl}\equiv\overline{\psi}^{+}_{i}(x_{1})\cancel{A}^{-}_{jh}(x_{1})\cancel{A}^{+}_{lr}(x_{2})\psi^{+}_{r}(x_{2})+\overline{\psi}^{+}_{i}(x_{1})\cancel{A}^{-}_{lr}(x_{2})\cancel{A}^{+}_{jh}(x_{1})\psi^{+}_{r}(x_{2})\,.$

In this interaction the incoming state$|i>$ includes spinor and vector fields
and the outgoing state $|f>$ are another spinor and vector fields with
different momentums and polarizations. Thus for incoming and outgoing states
we have:

$|i>=|1_{p,\sigma}^{a},1_{k,n}^{d}>=a^{\dagger}_{\sigma}(\xi_{p})d^{\dagger}_{n}(\xi_{k})|\Omega>,\;\;<f|=<1_{p^{\prime},\sigma^{\prime}}^{a},1_{k^{\prime},n^{\prime}}^{d}|=<\Omega|d_{n^{\prime}}({\tilde{\xi}}^{\prime}_{k})a_{\sigma^{\prime}}({\tilde{\xi}}^{\prime}_{p})\,,$

then the ${\cal S}^{(2)}$ matrix element on these states is:

${\cal S}^{(2)}_{fi}=<f|{\cal
S}^{(2)}|i>=-Q^{2}\int\int\mathrm{d}\mu(x_{1})\mathrm{d}\mu(x_{2})\gamma^{4}_{ij}(\cancel{x}_{1}S(x_{1},x_{2})\gamma^{4}\cancel{x}_{2})_{hl}\;<f|{\cal
M}|i>,$

where

$<f|{\cal M}|i>={\cal N}_{p}{\cal N}_{p^{\prime}}{\cal N}_{k}{\cal
N}_{k^{\prime}}\left[(Hx_{1}.{\xi}^{\prime}_{p})^{-2+i\nu}(Hx_{2}.\xi_{p})^{-2-i\nu}\;\overline{\cal
U}_{i}({\xi}^{\prime}_{p},\sigma^{\prime}){\cal
U}_{r}({\xi}_{p},\sigma)\right]$
$\times\left[(Hx_{1}.{\xi}^{\prime}_{k})^{-2}(Hx_{2}.\xi_{k})^{-1}\cancel{{\cal
E}}_{1_{jh}}(x_{1},{\xi}^{\prime}_{k},n^{\prime})\cancel{{\cal
E}}_{2_{lr}}(x_{2},{\xi}_{k},n)\right.$
$\left.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+(Hx_{1}.\xi_{k})^{-1}(Hx_{2}.\xi^{\prime}_{k})^{-2}\cancel{{\cal
E}}_{1_{lr}}(x_{2},{\xi}^{\prime}_{k},n^{\prime})\cancel{{\cal
E}}_{2_{jh}}(x_{1},{\xi}_{k},n)\right].$

.

Hence one can write the scattering matrix element as:

${\cal S}^{(2)}_{fi}=-{\cal N}_{p}{\cal N}_{p^{\prime}}{\cal N}_{k}{\cal
N}_{k^{\prime}}Q^{2}\int\int\mathrm{d}\mu(x_{1})\mathrm{d}\mu(x_{2})\;\;\overline{\cal
U}_{i}(\xi^{\prime}_{p},\sigma^{\prime})\;\gamma^{4}_{ij}\;(\cancel{x}_{1}S(x_{1},x_{2})\gamma^{4}\cancel{x}_{2})_{hl}$
$\times\left[(Hx_{1}.{\xi}^{\prime}_{p})^{-2+i\nu}\;(Hx_{2}.\xi_{p})^{-2-i\nu}\right]$
$\times\left[(Hx_{1}.{\xi}^{\prime}_{k})^{-2}(Hx_{2}.\xi_{k})^{-1}\cancel{{\cal
E}}_{1_{jh}}(x_{1},{\xi}^{\prime}_{k},n^{\prime})\cancel{{\cal
E}}_{2_{lr}}(x_{2},{\xi}_{k},n)\right.+$
$\left.(Hx_{1}.\xi_{k})^{-1}(Hx_{2}.\xi^{\prime}_{k})^{-2}\cancel{{\cal
E}}_{1_{lr}}(x_{2},{\xi}^{\prime}_{k},n^{\prime})\cancel{{\cal
E}}_{2_{jh}}(x_{1},{\xi}_{k},n)\right]{\cal U}_{r}(\xi_{p},\sigma),$ (IV.20)

where ${\cal N}_{p}\;,\;{\cal N}_{p^{\prime}}\;,\;{\cal N}_{k}\;,\;{\cal
N}_{k^{\prime}}$ are normalization constant of incoming and outgoing spinor
and vector fields bagamota ; gagarota ; gata .

The integral (IV.20) is taken over the dS space-time and all elements are
known, it can be calculated by using the Bros-Fourier transformation. In
Minkowski flat space, the Fourier transforms of most functions are known, but
in dS space, apart from the scalar field which has been partly done by Bros et
al. brmo ; brmo03 , not much work has been done on this transformation.
Therefore, using numerical methods, one may find a solution for the integral
that determines the curvature effects in QED. It will be investigated in the
subsequent articles.

##  V Flat limit

The dS manifold is a four dimensional hyperboloid with radius $H^{-1}$ and for
larger radius the curvature of space-time decreases, so when $H\rightarrow 0$
the curvature of space-time disappears and space-time matches with the flat
Minkowski space-time. We can parametrize the $\xi_{p}$ in terms of the massive
particle four-momentum $p^{\mu}$, which is suitable for the null curvature
limit:

$\xi_{p}=\left(\xi^{0},\overrightarrow{\xi},\xi^{4}\right)=\left(\dfrac{p^{0}}{m}=\sqrt{\dfrac{\overrightarrow{p}^{2}}{m^{2}}+1},\dfrac{\overrightarrow{p}}{m},-1\right)\,.$

We can also express the dS point $x^{\alpha}$ in terms of the intrinsic
coordinates $X^{\mu}=\left(X^{0}=ct\;,\overrightarrow{X}\right)$ as:

$x^{\alpha}=\left(x^{0}=H^{-1}\sinh(HX^{0})\;,\overrightarrow{x}=H^{-1}\dfrac{\overrightarrow{X}}{\lVert\overrightarrow{X}\lVert}\cosh(HX^{0})\sinh(H\lVert\overrightarrow{X}\lVert)\;,x^{4}=H^{-1}\cosh(HX^{0})\cosh(H\lVert\overrightarrow{X}\lVert)\right),$

where
${\lVert\overrightarrow{X}\lVert}=(X_{1}^{2}+X_{2}^{2}+X_{3}^{2})^{\frac{1}{2}}$.
Therefor, it is straightforward to show that ta97 :

$\lim_{H\rightarrow
0}(x\cdot\xi_{p})^{-2-i\nu}=e^{-ip.X},\;\;\lim_{H\rightarrow
0}(x\cdot\xi_{p})^{-2+i\nu}=e^{ip.X},$ $\lim_{H\rightarrow
0}H\cancel{x}=\lim_{H\rightarrow
0}H\eta_{\alpha\beta}\gamma^{\alpha}x^{\beta}=-\gamma^{4},\;\;\;\lim_{H\rightarrow
0}\cancel{\xi}_{p}=\gamma^{\mu}p_{\mu}+\gamma^{4},$

where $\mu=0,1,2,3$ and $\alpha=\beta=0,1,2,3,4$. For the dS vector field we
can parametrize the $\xi$ in terms of the massless particle four-momentum
$k^{\mu}$ as:

$\xi_{k}^{\alpha}=\left(k^{0},\vec{k},H\right),\;\;k^{0}>0,\;\;\;\;(k^{0})^{2}-\vec{k}\cdot\vec{k}=H^{2}.$

For dS massless vector field one can showthat the polarization states are
gagarota ; ta97 :

$\lim_{H\rightarrow 0}{\cal
E}_{2\alpha}(x,\xi,n)=\epsilon_{\mu}^{(n)},\;\;\;\lim_{H\rightarrow 0}{\cal
E}_{1\alpha}(x,\xi,n)=\epsilon_{\mu}^{*(n)}\,,$

where

$\epsilon^{(n)}.k=0,\;\;\;\epsilon^{(n)}.\epsilon^{*(n^{\prime})}=\eta^{nn^{\prime}},\;\;\;\sum_{n=0}^{3}\dfrac{\epsilon_{\mu}^{(n)}\epsilon_{\nu}^{*(n)}}{\epsilon^{(n)}.\epsilon^{*(n)}}=\eta_{\mu\nu}\,.$

Two-point functions of spinor and vector fields at $H\rightarrow 0$ limit are
matched with Minkowski counterparts ta97 ; ta96 ; bagamota ; gagarota . Using
the definition $\overline{{\cal U}}={\cal U}^{\dagger}\gamma^{0}\gamma^{4}$
and (II.5), the null curvature limit of scattering matrix (IV.20) is:

$\lim_{H\rightarrow 0}{\cal S}^{(2)}_{fi}={\cal N}_{p}{\cal
N}_{p^{\prime}}{\cal N}_{k}{\cal
N}_{k^{\prime}}Q^{2}\dfrac{1}{(2\pi)^{4}}\int\mathrm{d}^{4}x_{1}\int\mathrm{d}^{4}x_{2}\int\mathrm{d}^{4}q$
$\times\\{e^{-ik^{\prime}.x_{1}}e^{ik.x_{2}}e^{iq.(x_{2}-x_{1})}e^{-ip^{\prime}.x_{1}}e^{ip.x_{2}}\;\;\overline{\cal
U}_{i}^{\prime}(p^{\prime},\sigma^{\prime})\;\gamma_{ij}^{4}\cancel{\epsilon}^{*(n^{\prime})}_{jh}(k^{\prime})\;\gamma_{ht}^{4}\;\dfrac{\left(q^{\mu}\gamma^{\prime}_{\mu}+m\right)_{tl}}{q^{2}-m^{2}+i\epsilon}\cancel{\epsilon}^{(n)}_{lr}(k){\cal
U}_{r}(p,\sigma)$
$+e^{ik.x_{1}}e^{-ik^{\prime}.x_{2}}e^{iq.(x_{2}-x_{1})}e^{-ip^{\prime}.x_{1}}e^{ip.x_{2}}\;\;\overline{\cal
U}_{i}^{\prime}(p^{\prime},\sigma^{\prime})\;\gamma_{ij}^{4}\cancel{\epsilon}^{(n)}_{jh}(k)\;\gamma_{ht}^{4}\;\dfrac{\left(q^{\mu}\gamma^{\prime}_{\mu}+m\right)_{tl}}{q^{2}-m^{2}+i\epsilon}\cancel{\epsilon}^{*(n^{\prime})}_{lr}(k^{\prime}){\cal
U}_{r}(p,\sigma)\\}.$

After a straightforward calculation, one obtained:

$\lim_{H\rightarrow 0}{\cal S}^{(2)}_{fi}={\cal N}_{p}{\cal
N}_{p^{\prime}}{\cal N}_{k}{\cal
N}_{k^{\prime}}(iQ)^{2}(2\pi)^{4}\;\;\delta(k+p-k^{\prime}-p^{\prime})$
$\times\\{\;{\cal
U}_{i}^{\dagger(\sigma^{\prime})}(p^{\prime})\;\gamma^{\prime
0}_{ij}\;\epsilon^{*(n^{\prime})}_{\tau}(k^{\prime})\gamma^{\prime\tau}_{jh}\;\dfrac{\left(k^{\mu}\gamma^{\prime}_{\mu}+p^{\mu}\gamma^{\prime}_{\mu}+m\right)_{hl}}{(k+p)^{2}-m^{2}+i\epsilon}\;\epsilon^{(n)}_{\nu}(k)\gamma^{\prime\nu}_{lr}\;{\cal
U}_{r}^{(\sigma)}(p)$ $+\;{\cal
U}_{i}^{\dagger(\sigma^{\prime})}(p^{\prime})\;\gamma^{\prime
0}_{ij}\;\epsilon^{(n)}_{\tau}(k)\gamma^{\prime\tau}_{jh}\;\dfrac{\left(k^{\mu}\gamma^{\prime}_{\mu}+p^{\mu}\gamma^{\prime}_{\mu}+m\right)_{hl}}{(k+p)^{2}-m^{2}+i\epsilon}\;\epsilon^{*(n^{\prime})}_{\nu}(k^{\prime})\gamma^{\prime\nu}_{lr}\;{\cal
U}_{r}^{(\sigma)}(p)\;\\}\,,$

which is exactly the Minkowski QED results.

##  VI Conclusion

The effect of gravitational field or curvature on the interaction of electron-
photon in dS ambient formalism is studied. The scattering matrix explicitly
includes the sentences that show the curvature effects. Since the Bros-Fourier
transform in dS space-time has not been defined explicitly, the integral
(IV.20) may not be calculated analytically for obtaining the scattering
matrix. This scattering matrix should be calculated by numerical methods,
which will be investigated in the next articles. In the null curvature limit
($H\rightarrow 0$) the Minkowski results are obtained. These calculations can
be the starting point for considering the interactions of the quantum
electron-photon fields with the classical gravitational field in dS ambient
space formalism, which may be examining with the experimental test.

Acknowledgments: The authors wish to express his particular thanks to M.
Amiri, M. Rastiveis, R. Raziani and S. Tehrani-Nasab for discussions.

## References

  * (1) P. de Bernardis, et al., A flat universe from high-resolution maps of the cosmic microwave background radiation, Nature 404, 955(2000).
  * (2) S. Perlmutter, et al., Measurement of $\Omega$ and $\Lambda$ from 42 high-redshift supernovae, Astrophys. J. 517, 565(1999).
  * (3) A.G. Riess, et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116, 1009(1998), [arXiv:9805201v1].
  * (4) J. P. Henry, U. G. Briel, and H. Bohringer, The evolution of galaxy clusters, Scientific American 279, 52(1998).
  * (5) J. P. Henry, Measuring cosmological parameters from the evolution of cluster $X-$ray temperatures, Astrophys. J. 534, 565(2000), [arXiv:0002365].
  * (6) P. A. R. Ade, et al., Detection of B-Mode polarization at degree angular scales by BICEP$2$, Phys. Rev. Lett. 112, 241101(2014), [arXiv:1403.3985v3].
  * (7) M.V. Takook , Thèse de l’université Paris VI (1997), Théorie quantique des champs pour des systèmes élémentaires “massifs” et de “masse nulle” sur l’espace- temps de de Sitter.
  * (8) M.V. Takook, Quantum Field Theory in de Sitter Universe: Ambient Space Formalism, [arXiv:1403.1204v5].
  * (9) B. Takahashi, Sur les représentations unitaires des groupes de Lorentz généralisés, Bull. Soc. Math. France 91, 289(1963).
  * (10) J. Bros, J.P. Gazeau, U. Moschella, Quantum field theory in the de Sitter universe, Phys. Rev. Lett. 73, 1746(1994).
  * (11) J. Bros, U. Moschella, Rev. Math. Phys. 8, 327 (1996), Two-point functions and quantum fields in de Sitter universe,[gr-qc/9511019].
  * (12) J. Bros, U. Moschella, Fourier analysis and holomorphic decomposition on the one-sheeted hyperboloid, [arXiv:math-ph/0311052v1].
  * (13) P. Bartesaghi, J.P. Gazeau, U. Moschella, M.V. Takook, Class. Quantum Gra. 18, 4373 (2001), Dirac fields and thermal effects in de Sitter universe.
  * (14) M.V. Takook, Proceedings of the Group 21, 15-20 july 1996, Goslar, Germany , Spin $\frac{1}{2}$ field theory in the de Sitter space-time, [arXiv:gr-qc/0005077].
  * (15) L.H. Thomas,  Ann. of Math. 42, 113 (1941), On unitary representations of the group of de Sitter space.
  * (16) T.D. Newton,  Ann. of Math. 51, 730 (1950), A note on the representations of the de Sitter group.
  * (17) J. Dixmier, Bull. Soc. Math. France 89, 9 (1961), Représentation intégrables du group de de Sitter.
  * (18) S. Helgason, Academi Press New York and London (1962), Differential Geometry and Symmetric Spaces.
  * (19) S. Helgason, American Mathematical Society Providence, Rhode Island (1994), Geometric Analysis on Symmetric Spaces.
  * (20) T. Garidi, J.P. Gazeau, S. Rouhani and M.V. Takook, J. Math. Phys. 49, 032501 (2008), Massless vector field in de Sitter universe, [ arXiv:gr-qc/0608004].
  * (21) J.P. Gazeau, M.V. Takook, J. Math. Phys. 41, 5920 (2000), Massive vector field in de Sitter space, [arXiv:gr-qc/9912080]; T. Garidi, J.P. Gazeau, M.V. Takook, ibid. 43, 6379 (2002), Comment on: Massive vector field in de Sitter space.
  * (22) S. Rouhani, M.V. Takook, Mod. Phys. Lett. A 20, 2387 (2005), Abelian gauge theory in de Sitter space, [arXiv:gr-qc/0502019 ].
  * (23) M.V. Takook, Quantum de Sitter-black hole in ambient space formalism, Lecture presented at 5th Workshop and Seminar on Topics in Theoretical Physics, Azarbaijan Shahid Madani University, 23-25 August (2017), [arXiv:1711.01067].
  * (24) F. Mandl, G. Shaw (2010) Quantum Field Theory, John Wiley Sons.
  * (25) M. Kaku (1993) Quantum Field Theory, Oxford university press.
  * (26) T. Garidi, J. P. Gazeau and M. V. Takook, J. Math.Phys. 44, 3838 (2003), Massive spin-2 field in de Sitter space, [arXiv:hep-th/0302022].