# Decomposition by Approximation with Pulse Waves Allowing Further Research on
Sources of Voltage Fluctuations

Piotr Kuwałek, _Member, IEEE_ Manuscript received 11 January 2023; revised 14
March 2023 and 13 April 2023; accepted 26 May 2023. This work was funded by
National Science Centre, Poland – 2021/41/N/ST7/00397. For the purpose of Open
Access, the author has applied a CC–BY public copyright licence to any Author
Accepted Manuscript (AAM) version arising from this submission. Piotr Kuwałek
is with the Institute of Electrical Engineering and Electronics, Faculty of
Control, Robotics and Electrical Engineering, Poznan University of Technology,
Poznan, Poland, (e-mail: piotr.kuwalek@put.poznan.pl).

###### Abstract

Voltage fluctuations are common disturbances in power grids. Initially, it is
necessary to selectively identify individual sources of voltage fluctuations
to take actions to minimize the effects of voltage fluctuations. Selective
identification of disturbing loads is possible by using a signal chain
consisting of demodulation, decomposition, and assessment of the propagation
of component signals. The accuracy of such an approach is closely related to
the applied decomposition method. The paper presents a new method for
decomposition by approximation with pulse waves. The proposed method allows
for an correct identification of selected parameters, that is, the frequency
of changes in the operating state of individual sources of voltage
fluctuations and the amplitude of voltage changes caused by them. The article
presents results from numerical simulation studies and laboratory experimental
studies, based on which the estimation errors of the indicated parameters were
determined by the proposed decomposition method and other empirical
decomposition methods available in the literature. The real states that occur
in power grids were recreated in the research. The metrological interpretation
of the results obtained from the numerical simulation and experimental
research is discussed.

###### Index Terms:

approximation, decomposition, demodulation, power quality, voltage
fluctuation, voltage fluctuations indices, voltage variation.

††publicationid: pubid:

©2023 IEEE. Personal use of this material is permitted. Permission from IEEE
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lists, or reuse of any copyrighted component of this work in other works. DOI:
10.1109/TIE.2023.3283698

##  I Introduction

Electrical energy is one of the basic energy sources used to supply devices.
Regardless of power requirements and load characteristics, it is necessary to
supply energy with appropriate parameters. This forces power suppliers to
distribute energy of a certain quality, allowing for the efficient use of
supplied devices. This requirement is defined by the appropriate criteria for
parameters determining power quality [1] (e.g., rms value of voltage $U$,
fundamental frequency of voltage $f_{c}$, total harmonic distortion of voltage
$\textnormal{THD}U$, short–term/long–term flicker indicator $P_{st}/P_{lt}$).
The technical and commercial conditions force the development and use of
different methods of power quality evaluation [2, 3, 4, 5]. Common reasons for
the deterioration of power quality are voltage fluctuations [6]. This
phenomenon can cause flicker for different types of light sources (e.g.
incandescent lamps, fluorescent lamps, LEDs) [7] and can disturb the operation
and reduce the life of other loads (e.g. induction motors [8]). Depending on
the cause, voltage fluctuations occur in power grids from the LV network to
the HV network. The causes (sources) of voltage fluctuations are, for example,
operations of a specific load (e.g. arc furnace, machine controlling
technological process) or their groups (in this case, the resultant frequency
$f_{m}$ of changes in the operating state of a group of loads can be higher
than the power frequency $f_{c}$) [9, 10, 11], changes in the network topology
and impedance of the supply circuit. The effects of operations of disturbing
loads are changes in the current, which are the direct cause of voltage
fluctuations. Voltage fluctuations can also come from other circuits (usually
voltage fluctuations propagate from circuit with higher rated voltage to
circuit with lower rated voltage). It is worth noting that now also power
electronic devices can be a source of voltage fluctuations (even if there is
high–frequency switching of the order of $\rm\,kHz$), the number of which is
significantly increasing (e.g. due to the increase in the number of new
installations of renewable energy sources in the power grid) [12]. Therefore,
there is a need to identify the occurring sources of voltage fluctuations, in
order to indicate the supply point of individual sources of voltage
fluctuations, and then to take actions to minimize the effects of voltage
fluctuations.

In the process of identification of sources of voltage fluctuations,
information is obtained on the frequency $f_{m_{i}}$ of changes in the
operating state of individual $i$–th sources of voltage fluctuations in the
power grid (a feature dependent on disturbing loads) and on the amplitude
$k_{m_{i}}$ of voltage changes caused by them (a feature dependent on
disturbing loads and their supply circuit). The accurate estimation of
indicated parameters ($f_{m_{i}}$,$k_{m_{i}}$) supports the diagnostic of
voltage fluctuations [13, 11], and in particular allows for the selective
localization of sources of voltage fluctuations in the power grid based on
simultaneous series of measurements at particular points in the power grid
[14]. Currently, there are not too many methods in the literature that allow
for the achievement of indicated goal (even in a limited way). Unfortunately,
the methods available in the literature:

  * •

limit the identification of disturbing loads to those whose frequency
$f_{m_{i}}$ of operating state changes is lower than the power frequency
$f_{c}$ [15, 16, 17, 18]; or

  * •

can cause incorrect estimation of indicated parameters
($f_{m_{i}}$,$k_{m_{i}}$) due to limited possibilities of the decomposition
method used (in some cases) [19, 18]; or

  * •

have limited diagnostic capabilities (e.g. lack of automation of the
identification process, additional expert knowledge required, the need for
iterative procedures in which the dominant disturbing load is identified and
the effect of changes in its operating state is eliminated) [15, 20, 21, 18].

The paper presents a new proprietary method of decomposition by approximation
with pulse waves (DAPW), the description of which is presented in Section II.
The presented method allows for the accurate identification of selected
parameters, i.e. the frequency $f_{m_{i}}$ of changes in the operating state
of individual sources of voltage fluctuations and the amplitude $k_{m_{i}}$ of
voltage changes caused by them, which allows for automatic selective
identification and localization of many sources of voltage fluctuations based
on simultaneous series of measurements at particular points of the power grid
(without additional expert knowledge). Section III presents the results of
numerical simulation studies and laboratory experimental studies, on the basis
of which were determined the estimation errors ($\delta f_{m_{i}}$,$\delta
k_{m_{i}}$) of indicated parameters ($f_{m_{i}c}$,$k_{m_{i}c}$) estimated by
the proposed decomposition method and by other empirical decomposition methods
available in the literature, which have not yet been used for this purpose
other than Enhanced Empirical Wavelet Transform (EEWT). The presented research
results show that the proposed approach is characterized by the smallest
estimation errors ($\delta f_{m_{i}}$,$\delta k_{m_{i}}$) of the indicated
parameters, thanks to which the use of the proposed method in the process of
selective identification of voltage fluctuations sources based on a
demodulation with a carrier signal estimation, allows for the accurate
estimation of selected features of disturbing loads in the power grid
(including power electronic devices).

##  II Proposed Approach

A simplified diagram of the proposed proprietary decomposition approach using
approximation by pulse waves is shown in Fig. 1 (the colors of graphic
representation for the individual steps of the proposed approach are
maintained). The idea of the proposed approach is that the amplitude
modulating signal of supply voltage in the power grid is approximated by a
series of pulse waves. The selection of basis functions in the form of pulse
waves results from the fact that probably most present sources of voltage
fluctuations cause step (rectangular) voltage changes (chaotic sources causing
irregular voltage changes currently have mostly separate supply circuits or
their effects are already minimized by appropriate equipment). In the proposed
decomposition process, information about the actual shape of voltage changes
caused by individual disturbing loads is lost. Nevertheless, the assumption of
the pulse waves as the basis functions allows for the accurate extraction of
the two most important parameters ($f_{m_{i}}$,$k_{m_{i}}$) used in the
process of indication of the supply points of individual sources of voltage
fluctuations. The amplitude $k_{m_{i}c}$ of the estimated $i$–th pulse wave
corresponds to the maximum voltage change $k_{m_{i}}$ caused by the operation
of the $i$–th disturbing load, and the fundamental frequency $f_{m_{i}c}$ of
the estimated $i$–th pulse wave corresponds to the average value of frequency
$f_{m_{i}}$ of the quasi–periodic changes in the operating state of $i$–th
disturbing load.

Figure 1: The simplified block diagram of the proposed approach

In the first stage of the proposed approach, the fundamental frequencies
$f_{m_{i}c}$ for particular $i$–th pulse waves are estimated. For this
purpose, for the input signal $u_{IN}\left(t_{k}\right)$ (in terms of
diagnostics of voltage fluctuations, the input signal
$u_{IN}\left(t_{k}\right)$ is the estimated amplitude modulating signal
$u_{\textnormal{mod}}\left(t_{k}\right)$, obtained using the demodulation with
estimation of carrier signal), the autocovariance function
$u_{\textnormal{acov}}\left(t_{k}\right)$ for individual samples $t_{k}$ is
determined (sample interval $\Delta t=1/f_{s}$, where $f_{s}$ is sampling
rate) according to the relation [22]:

$\displaystyle
u_{\textnormal{acov}}\left(t_{k}\right)=\frac{1}{L}\sum_{i=1}^{L-k}\Bigg{\\{}$
$\displaystyle\left(u_{IN}\left(t_{i}\right)-\frac{1}{L}\sum_{j=1}^{L}u_{IN}\left(t_{j}\right)\right)$
(1)
$\displaystyle\cdot\left(u_{IN}\left(t_{i+k}\right)-\frac{1}{L}\sum_{j=1}^{L}u_{IN}\left(t_{j}\right)\right)\Bigg{\\}}\,,$

where $L$ is the number of samples in the measurement window. For the obtained
autocovariance function, the signal spectrum is determined using fast Fourier
transform (FFT) (if the number of samples does not allow for the
implementation of the Cooley–Tukey algorithm, discrete Fourier transform (DFT)
algorithm is used instead of FFT). In order to estimate the fundamental
frequencies $f_{m_{i}}$ for $N$ pulse waves, the local maxima
$\hat{f}_{m_{j}c}$ of the autocovariance function spectrum are determined and
$N$ maxima with highest values are selected, except that those maxima are
rejected that meet the relation:

$\displaystyle\forall_{j,k\in 1,...,N_{j}:\left(j\neq
k\right)}\hat{f}_{m_{j}c}\cong n\hat{f}_{m_{k}c}\,,$ (2)

where: $n\in\left\\{2,3,4,5\right\\}$, $N_{j}$ is the number of determined
local maxima of the spectrum of autocovariance function
$u_{\textnormal{acov}}\left(t_{k}\right)$. The approximately equal in
dependence (2) is assumed to be equal with the resolution $\Delta f_{mc}$:

$\displaystyle\Delta f_{mc}=\frac{10f_{s}}{L}.$ (3)

An example of procedure in the first stage is shown in Fig. 2.

(a) recreated amplitude modulating signal
$u_{\textnormal{mod}c}\left(t_{k}\right)$

(b) autocovariance function $u_{\textnormal{acov}}\left(t_{k}\right)$ of the
recreated signal $u_{\textnormal{mod}c}\left(t_{k}\right)$

(c) spectrum $u_{\textnormal{acov}}\left(f\right)$ of the determined function
$u_{\textnormal{acov}}\left(t_{k}\right)$ – selection of appropriate
$f_{m_{i}c}$ values

Figure 2: The example of procedure for estimation of frequencies $f_{m_{i}c}$
of basis pulse waves, where $N$=2

In the second stage, initial phases $\varphi_{m_{i}c}$ are estimated for
individual component signals with frequencies $f_{m_{i}c}$ determined in the
first stage. In order to determine the initial phase $\varphi_{m_{i}c}$ for
the $i$–th pulse wave with fundamental frequency $f_{m_{i}c}$, the
cross–correlation function $u_{\textnormal{xcorr}}\left(t_{k}\right)$ between
the input signal $u_{IN}\left(t_{k}\right)$ =
$u_{\textnormal{mod}c}\left(t_{k}\right)$ and the normalized square wave with
frequency $f_{m_{i}c}$ is determined for individual samples $t_{k}$ according
to the dependence [22]:

$\displaystyle
u_{\textnormal{xcorr}}\left(t_{k}\right)=\frac{1}{L}\sum_{i=1}^{L-k}\left\\{u_{IN}\left(t_{i}\right)\cdot\text{sign}\left(\sin\left(2\pi
f_{m_{i}c}t_{i+k}\right)\right)\right\\}.$ (4)

In the next step, based on the determined cross–correlation function
$u_{\textnormal{xcorr}}\left(t_{k}\right)$, the global maximum
$t_{k\textnormal{max}}$ is determined. The value of global maximum
$t_{k\textnormal{max}}$ is the delay time expressed in seconds. In order to
calculate the initial phase $\varphi_{m_{i}c}$ in radians based on the delay
time $t_{k\textnormal{max}}$, the following relationship should be used:

$\displaystyle\varphi_{m_{i}c}=\mod\left(2\pi
t_{k\textnormal{max}}f_{m_{i}c},2\pi\right)\left[\text{rad}\right].$ (5)

An example of procedure in the second stage is shown in Fig. 3.

(a) cross–correlation function $u_{\textnormal{xcorr}}\left(t_{k}\right)$ for
input signal $u_{IN}\left(t_{k}\right)$ =
$u_{\textnormal{mod}c}\left(t_{k}\right)$ and square wave with $f_{m1c}$=0.1
$\rm\,Hz$

(b) cross–correlation function $u_{\textnormal{xcorr}}\left(t_{k}\right)$ for
input signal $u_{IN}\left(t_{k}\right)$ =
$u_{\textnormal{mod}c}\left(t_{k}\right)$ and square wave with $f_{m2c}$=8.25
$\rm\,Hz$

Figure 3: The example of procedure for estimation of initial phases
$\varphi_{m_{i}c}$ of basis pulse waves, where $N$=2

In the third stage, the duty cycles $\delta_{m_{i}c}$ are estimated for
individual pulse waves. For this purpose, for each $i$–th basis pulse waves, a
set of pulse waves is created with an amplitude equal to half of the
peak–to–peak value of the input signal $u_{IN}\left(t_{k}\right)$ =
$u_{\textnormal{mod}c}\left(t_{k}\right)$, with the fundamental frequency
$f_{m_{i}c}$, with the initial phase $\varphi_{m_{i}c}$, and with the duty
cycle $\left[\Delta\delta,2\Delta\delta,...,1-\Delta\delta\right]$, where
$\Delta\delta$ is the adopted resolution with which the unknown value of the
duty cycle of a particular basis pulse wave is estimated. The resolution
$\Delta\delta$=0.01 is arbitrarily assumed in the research. As the estimated
value of the duty cycle $\delta_{m_{i}c}$ of $i$–th basis pulse wave, the
value of $n\Delta\delta$ $\left(n\in N\land
n:n\Delta\delta\in\left(0;1\right)\right)$ is assumed, for which the largest
value of the global maxima is obtained for individual cross–covariance
functions $u_{\textnormal{xcov}}\left(t_{k}\right)$ between input signal
$u_{IN}\left(t_{k}\right)$ = $u_{\textnormal{mod}c}\left(t_{k}\right)$ and a
properly created set of pulse waves. To calculate the cross–covariance between
any two signals $u_{1}\left(t\right)$ and $u_{2}\left(t\right)$ for individual
samples $t_{k}$ is used the relationship [22]:

$\displaystyle
u_{\textnormal{xcov}}\left(t_{k}\right)=\frac{1}{L}\sum_{i=1}^{L-k}\Bigg{\\{}$
$\displaystyle\left(u_{1}\left(t_{i}\right)-\frac{1}{L}\sum_{j=1}^{L}u_{1}\left(t_{j}\right)\right)$
(6)
$\displaystyle\cdot\left(u_{2}\left(t_{i+k}\right)-\frac{1}{L}\sum_{j=1}^{L}u_{2}\left(t_{j}\right)\right)\Bigg{\\}}.$

An example of procedure in the third stage is shown in Fig. 4.

Figure 4: The example of procedure for estimation of duty cycle
$\delta_{m_{1}c}$ of basis pulse wave, where $N$=2, $f_{m_{1}c}$=0.1$\rm\,Hz$,
$\varphi_{m_{1}c}$=0 $\rm\,rad$

In the last stage, the amplitudes $k_{m_{i}c}$ are estimated for individual
$i$–th basis pulse waves. For this purpose, the spectra are determined using
the FFT algorithm (if the number of samples does not allow for the
implementation of Cooley–Tukey algorithm, DFT is used instead of FFT) for the
input signal $u_{IN}\left(t_{k}\right)$ =
$u_{\textnormal{mod}c}\left(t_{k}\right)$ and for a set of pulse waves with
the fundamental frequency $f_{m_{i}c}$, initial phase $\varphi_{m_{i}c}$, duty
cycle $\delta_{m_{i}c}$, and with amplitude $\left[\Delta k,2\Delta
k,...,R\Delta k\right]$, where $\Delta k$ is the adopted resolution with which
the unknown value of the amplitude of a particular basis pulse wave is
estimated, and $R$ is such a natural number that the value of $R\Delta k$ is
equal to the peak–to–peak value of the input signal $u_{IN}\left(t_{k}\right)$
= $u_{\textnormal{mod}c}\left(t_{k}\right)$. As an estimator of the unknown
amplitude $k_{m_{i}c}$ for the $i$–th basis pulse wave is assumed the value
for which the energy value for the neighborhood of the global maximum of the
spectrum from the set of pulse waves and the energy value for the
corresponding bandwidth of the input signal $u_{IN}\left(t_{k}\right)$ =
$u_{\textnormal{mod}c}\left(t_{k}\right)$ is comparable. An example of
procedure in the fourth stage is shown in Fig. 5.

Figure 5: The example of procedure for estimation of amplitude $k_{m_{1}c}$ of
basis pulse wave, where $N$=2, $f_{m_{1}c}$=0.1$\rm\,Hz$, $\varphi_{m_{1}c}$=0
$\rm\,rad$, $\delta_{m_{1}c}$=0.50

##  III Research Results and Discussion

###  III-A Test Signal

For the verification of the proposed approach, as the test signal is selected
signal described by the relationship:

$\displaystyle
u_{\textnormal{test}}\left(t_{k}\right)=u_{c}\left(t_{k}\right)\cdot\left(1+u_{\textnormal{mod}}\left(t_{k}\right)\right).$
(7)

The relationship described by (7) defines amplitude modulation (AM) without
attenuated carrier wave and is a properly representation of voltage
fluctuations that occur in the real stiff power grid [13, 11, 14], i.e., where
the power frequency has negligible deviations (otherwise it is necessary to
recreate voltage fluctuations as amplitude–phase/frequency modulation
(AM–PM/FM) [23]). The carrier signal $u_{c}\left(t_{k}\right)$ in (7)
describes the supply voltage before voltage fluctuations occur. In order to
recreate the states occurring in the real power grid, the “clipped cosine”
type signal is adopted as a carrier signal $u_{c}\left(t_{k}\right)$, which is
associated with the effect of input stages of switching power supplies.
Therefore, the carrier signal $u_{c}\left(t_{k}\right)$ is given by:

$\displaystyle
u_{c}\left(t_{k}\right)=\left\\{\begin{matrix}{k_{U}}{m_{c}}&\text{if}&\cos\left(2\pi{f_{c}}t_{k}\right)>{m_{c}}\\\
{k_{U}}\cos\left(2\pi{f_{c}}t_{k}\right)&\text{if}&-{m_{c}}\geq\cos\left(2\pi{f_{c}}t_{k}\right)\leq{m_{c}}\\\
-{k_{U}}{m_{c}}&\text{if}&\cos\left(2\pi{f_{c}}t_{k}\right)<-{m_{c}}\\\
\end{matrix}\right.,$ (8)

where $f_{c}$ is the carrier frequency ($f_{c}=50$$\rm\,Hz$ was adopted in the
research); $k_{U}$ is a scaling value which, for a given value of $m_{c}$,
allows conversion the rms value of test signal to the rated value
($U_{c}$=230$\rm\,V$ was adopted in the research, i.e., the rated rms value in
the LV network in Europe was adopted in the research); $m_{c}$ defines the
clipping level and is given by:

$\displaystyle m_{c}=\frac{M_{c}}{M}\leq 1,$ (9)

where: $M_{c}$ and $M$ are amplitudes after and before clipping, respectively.
The value of $m_{c}$=0.8 is adopted in the research, which corresponds to the
occurrence of voltage distortion at the level of the limit of acceptable
distortion in the LV network [1]. An example of the adopted carrier signal
$u_{c}\left(t_{k}\right)$ is shown in Fig. 6.

Figure 6: The exemplary waveform of “clipped cosine” for $m_{c}=0.8$ and
$k_{U}\approx 361.6$

The modulating signal $u_{\textnormal{mod}}\left(t_{k}\right)$ in (7) is a
signal associated with the resultant operation of $N$ disturbing loads. In the
research, the modulating signal $u_{\textnormal{mod}}\left(t_{k}\right)$ was
adopted as the sum of $N$ asymmetric rectangular signals
$u_{\textnormal{mod}_{i}}\left(t_{k}\right)$ (a signal associated with step
voltage changes, which are probably the most common in the power grid) with a
duty cycle $\delta_{m_{i}}$:

$\displaystyle\delta_{m_{i}}=\frac{t_{{ON}_{i}}}{T_{m_{i}}},$ (10)

associated with individual $i$–th disturbing loads, and noise
$u_{\textnormal{noise}}\left(t_{k}\right)$ with uniform distribution and
standard deviation equal to $10^{-5}$. Therefore, the modulating signal
$u_{\textnormal{mod}}\left(t_{k}\right)$ is given by:

$\displaystyle
u_{\textnormal{mod}}\left(t_{k}\right)=\sum_{i=1}^{N}u_{\textnormal{mod}_{i}}\left(t_{k}\right)+u_{\textnormal{noise}}\left(t_{k}\right),$
(11) $\displaystyle
u_{\textnormal{mod}_{i}}\left(t_{k}\right)=\left\\{\begin{matrix}k_{m_{i}}&\text{if}&lT_{m_{i}}<t_{k}<lT_{m_{i}}+t_{ON}\\\
-k_{m_{i}}&\text{if}&lT_{m_{i}}+t_{ON}\leq
t_{k}\leq\left(l+1\right)T_{m_{i}}\\\ \end{matrix}\right.,$ (12)

where $l\in\mathbb{Z}$. The exemplary component
$u_{\textnormal{mod}_{i}}\left(t_{k}\right)$ of the modulating signal
$u_{\textnormal{mod}}\left(t_{k}\right)$ is shown in Fig. 7. Fig. 8 shows an
exemplary resultant modulating signal
$u_{\textnormal{mod}}\left(t_{k}\right)$, assuming the occurrence of three
disturbing loads ($N$=3).

Figure 7: The example of adopted component of modulating signal Figure 8: The
example of adopted resultant modulating signal associated with the operation
of three dominant sources of voltage fluctuations

###  III-B Numerical Simulation Studies

In numerical simulation studies, a set of 9000 test signals was generated in
accordance with the relationship defined by (7), with the following
parameters:

  * •

the amplitude $k_{m_{i}}$ of the $i$–th component
$u_{\textnormal{mod}_{i}}\left(t_{k}\right)$ of modulating signal
$u_{\textnormal{mod}}\left(t_{k}\right)$ selected randomly from the range
$\left[5\cdot 10^{-4};2.5\ 10^{-2}\right]$, which corresponds to the amplitude
modulation depth $0pt$ in the set $\left[0.1;5\right]\%$;

  * •

the fundamental frequency $f_{m_{i}}$ of the $i$–th component
$u_{\textnormal{mod}_{i}}\left(t_{k}\right)$ of modulating signal
$u_{\textnormal{mod}}\left(t_{k}\right)$ selected randomly from the set
$\left[0.1;150\right]$$\rm\,Hz$ (150$\rm\,Hz$ corresponds to 3$f_{c}$ for
$f_{c}$=50$\rm\,Hz$ [9, 10]);

  * •

the duty cycle $\delta_{m_{i}}$ of the $i$–th component
$u_{\textnormal{mod}_{i}}\left(t_{k}\right)$ of modulating signal
$u_{\textnormal{mod}}\left(t_{k}\right)$ selected randomly from the set
$\left\\{0.1,0.2,...,0.9\right\\}$;

  * •

an initial phase $\varphi_{m_{i}}$ of the $i$–th component
$u_{\textnormal{mod}_{i}}\left(t_{k}\right)$ of modulating signal
$u_{\textnormal{mod}}\left(t_{k}\right)$ selected randomly from the set
$\left[0;2\pi\right)$$\rm\,rad$;

  * •

the number $N$ of components $u_{\textnormal{mod}_{i}}\left(t_{k}\right)$ of
modulating signal $u_{\textnormal{mod}}\left(t_{k}\right)$ belonging to the
set $\left\\{2,3,4\right\\}$.

In the process of generation of test signals, cases in which fundamental
frequencies $f_{m_{i}}$ of individual component signals are equal were
avoided. The sampling rate $f_{s}$ of individual test signals is
20$\rm\,kSa/s$, and the time duration of measurement window is 1$\rm\,min$.
According to [11], this is an enough time for correct extraction of slow
voltage changes of 0.1$\rm\,Hz$. Numerical simulation studies were carried out
in the MATLAB software on a computer with an Intel Core i5-1035G1 processor
with a clock rate of 3.6$\rm\,GHz$ and 16$\rm\,GB$ of RAM.

###  III-C Laboratory Experimental Studies

In laboratory experimental studies, a set of 4500 test signals was generated
in accordance with the relationship defined by (7), with the following
parameters:

  * •

the amplitude $k_{m_{i}}$ of the $i$–th component
$u_{\textnormal{mod}_{i}}\left(t_{k}\right)$ of modulating signal
$u_{\textnormal{mod}}\left(t_{k}\right)$ selected randomly from the range
$\left[5\cdot 10^{-4};2.5\cdot 10^{-2}\right]$, which corresponds to the
amplitude modulation depth $0pt$ in the set $\left[0.1;5\right]\%$;

  * •

the fundamental frequency $f_{m_{i}}$ of the $i$–th component
$u_{\textnormal{mod}_{i}}\left(t_{k}\right)$ of modulating signal
$u_{\textnormal{mod}}\left(t_{k}\right)$ selected randomly from the set
$\left[0.1;150\right]$$\rm\,Hz$ (150$\rm\,Hz$ corresponds to 3$f_{c}$ for
$f_{c}$=50$\rm\,Hz$ [9, 10]);

  * •

the duty cycle $\delta_{m_{i}}$ of the $i$–th component
$u_{\textnormal{mod}_{i}}\left(t_{k}\right)$ of modulating signal
$u_{\textnormal{mod}}\left(t_{k}\right)$ equal to 0.5;

  * •

an initial phase $\varphi_{m_{i}}$ of the $i$–th component
$u_{\textnormal{mod}_{i}}\left(t_{k}\right)$ of modulating signal
$u_{\textnormal{mod}}\left(t_{k}\right)$ equal to 0$\rm\,rad$;

  * •

the number $N$ of components $u_{\textnormal{mod}_{i}}\left(t_{k}\right)$ of
modulating signal $u_{\textnormal{mod}}\left(t_{k}\right)$ belonging to the
set $\left\\{2,3,4\right\\}$.

Fundamental frequencies $f_{m_{i}}$ of the $i$–th components
$u_{\textnormal{mod}_{i}}\left(t_{k}\right)$ of modulating signal
$u_{\textnormal{mod}}\left(t_{k}\right)$ were selected randomly, but the
following conditions were maintained:

  * •

the fundamental frequency $f_{m_{i}}$ expressed in$\rm\,cpm$ is a natural
number;

  * •

least common multiple for $f_{m_{i}}$ where $i$=1,..,$N$ and for $f_{c}$
expressed in$\rm\,cpm$ is different from 1.

In addition, the standard deviation of noise
$u_{\textnormal{noise}}\left(t_{k}\right)$ was adopted to be zero. Such
conditions allowed for the correct generation of test signals on the
laboratory setup, the block diagram of which is shown in Fig. 9. The photo of
the laboratory setup is shown in Fig. 10. In the process of generation of test
signals, cases in which the fundamental frequencies $f_{m_{i}}$ of individual
component signals $u_{\textnormal{mod}_{i}}\left(t_{k}\right)$ are equal were
avoided. The sampling rate $f_{s}$ of individual test signals is
50$\rm\,kSa/s$, and the time duration is 2$\rm\,min$ (in the research, the
first minute of the recorded signal was rejected due to the possible
occurrence of transient states related to the imperfection of the laboratory
equipment used).

Figure 9: The block diagram of the laboratory setup, where: AWG is an
arbitrary waveform generator, PA is a power amplifier, VM is a voltmeter, AP
is an active differential probe Figure 10: The photo of the laboratory setup

###  III-D Research Results

In the first step, each test signal from the generated set was demodulated
using the amplitude demodulation with a carrier signal estimation [24]. Then,
the individual recreated modulating signals were decomposed by: the proposed
decomposition method described in Section II, i.e., decomposition by
approximation with pulse waves (DAPW), empirical wavelet transform (EWT) [25],
enhanced empirical wavelet transform (EEWT) [26], empirical mode decomposition
(EMD) [27], variational mode decomposition (VMD) [28], multidimensional
variational mode decomposition (MVMD) [29], empirical Fourier decomposition
(EFD) [30]. For each component signal obtained from the decomposition process,
the fundamental frequency $f_{m_{i}c}$ of $i$–th component signal and its
amplitude $k_{m_{i}c}$ in the considered 1$\rm\,min$ measurement window were
determined using particular methods of decomposition.

To determine the fundamental frequency $f_{m_{i}c}$ of $i$–th component signal
obtained from decomposition methods other than DAPW, the autocorrelation
function was used with a window function that allows determining the
fundamental frequency of any function [31]. In this research, the window
function was assumed to be a moving average function. The selection of the
proposed method for estimation of the fundamental frequency is due to the
generation of a quasi–periodic signal as a result of the decomposition
process. In addition, some types of decomposition showed numerous undulations
around the zero value, which prevented the use of basic algorithm of frequency
detection, i.e., the zero crossing detector.

The median value of the local maxima of the absolute value of component signal
$u_{\textnormal{mod}_{i}}\left(t_{k}\right)$ (obtained from decomposition
methods other than DAPW) after subtracted its average value was used to
determine the amplitude $k_{m_{i}c}$.

On the basis of estimated values of $f_{m_{i}c}$ and $k_{m_{i}c}$, the values
of relative estimation errors of indicated parameters were determined
according to the relationship:

$\displaystyle\delta
f_{m_{i}}=\frac{\left|f_{m_{i}c}-f_{m_{i}}\right|}{f_{m_{i}}},$ (13)
$\displaystyle\delta
k_{m_{i}}=\frac{\left|k_{m_{i}c}-k_{m_{i}}\right|}{k_{m_{i}}}.$ (14)

For the analysis of accuracy of decomposition process, only errors in the
estimation of amplitudes $k_{m_{i}c}$ and frequencies $f_{m_{i}c}$ of
component signals are focused, because these parameters allow for selective
identification of individual disturbing loads in the power grid [14, 11]. It
is worth noting that the determined relative errors include error caused by
the decomposition method used and error caused by the demodulation [24] used
in the process of recreation of amplitude modulating signal. This approach is
important from the point of view of the process of selective identification of
sources of voltage fluctuations in the power grid [14, 11], because it is
necessary to extract the $f_{m_{i}c}$ and $k_{m_{i}c}$ values by the proposed
signal chain “demodulation – decomposition – statistic assessment of
propagation of voltage fluctuations” in [14], with the smallest errors $\delta
f_{m_{i}}$ and $\delta k_{m_{i}}$.

(a) $N$=2

(b) $N$=3

(c) $N$=4

Figure 11: The distribution of errors $\delta f_{m_{i}}$ for selected
decomposition methods - numerical simulation studies

(a) $N$=2

(b) $N$=3

(c) $N$=4

Figure 12: The distribution of errors $\delta k_{m_{i}}$ for selected
decomposition methods - numerical simulation studies

Fig. 11 and Fig. 12 show the statistical assessment of determined errors
$\delta f_{m_{i}}$ and $\delta k_{m_{i}}$ in the form of “box–plots” for
results obtained from numerical simulation studies.

Fig. 13 and Fig. 14 show the statistical assessment of determined errors
$\delta f_{m_{i}}$ and $\delta k_{m_{i}}$ in the form of “box–plots” for
results obtained from laboratory experimental studies.

(a) $N$=2

(b) $N$=3

(c) $N$=4

Figure 13: The distribution of errors $\delta f_{m_{i}}$ for selected
decomposition methods - laboratory experimental studies

(a) $N$=2

(b) $N$=3

(c) $N$=4

Figure 14: The distribution of errors $\delta k_{m_{i}}$ for selected
decomposition methods - laboratory experimental studies

Analyzing the distribution of errors presented in Figs. 11–14, it can be seen
that for the proposed approach, the smallest errors $\delta f_{m_{i}}$ and
$\delta k_{m_{i}}$ are obtained. Figs. 11 and 13 show that the average and
median values of frequency estimation errors $\delta f_{m_{i}}$ of individual
components are close to zero for the proposed approach. The smallest errors
$\delta f_{m_{i}}$ obtained support the process of voltage fluctuations
diagnostics focused on identification and initial recognition of disturbing
loads. Figs. 12 and 14 show that the mean and median values of errors in the
estimation of amplitudes $\delta k_{m_{i}}$ of individual components are
greater than for $\delta f_{m_{i}}$. However, errors $\delta k_{m_{i}}$ are
acceptable, because for the correct localization (indication of the power
supply point) of individual disturbing loads, it is necessary to estimate the
amplitudes of individual component signals at individual points of the power
grid while maintaining a constant value of relative error. Errors for
decomposition methods other than the proposed method are related to the fact
that they are based on decomposition into AM–FM modulated sinusoidal signals.
However, voltage fluctuations of this type rarely occur in practice, so the
diagnostic utility of methods other than the proposed one is lower.

It is worth noting that the proposed approach is based on functions that can
be calculated according to the Cooley–Tukey algorithm when the number of
samples in the measurement window is a power of 2. As a consequence, the
complexity of the proposed solution is $\textbf{O}(n)$ [32]. Other
decomposition methods that were used for comparison in this paper have a level
of complexity of at least of $\textbf{O}(n^{2})$, which significantly extends
the calculation time when considering longer measurement windows used to
consider slow voltage changes (e.g., time duration of measurement window equal
to 10$\rm\,min$). For short time duration of windows, the time consumption of
selected decomposition methods is comparable. However, it is worth noting that
in order to correct recreation of the modulating signal with a frequency of up
to 3$f_{c}$, it is necessary to use a sampling rate greater than
12$\rm\,kSa/s$, which in turn results in a large size of the input signals in
the decomposition process for long measurement windows.

##  IV Conclusion

Diagnostics of voltage fluctuations focused on selective identification and
localization of sources of voltage fluctuations in the power grid, which
change their operating state with a frequency of up to 3$f_{c}$, where $f_{c}$
is the power frequency, requires decomposition method with the smallest
estimation errors $\delta f_{m_{i}}$ and $\delta k_{m_{i}}$ in the signal
chain: “demodulation – decomposition – statistical assessment of propagation”.
Therefore, the paper presents a new method of decomposition by approximation
with pulse basis functions. Numerical simulation studies and laboratory
experimental studies were carried out for the verification of accuracy of the
proposed approach in its target application for the purposes of diagnostics of
voltage fluctuations. The proposed approach is compared with other empirical
decomposition methods available in the literature, intended for use for
non–stationary and noisy signals, i.e., signals that are associated with real
sources of voltage fluctuations, which can have a random nature. The analysis
focused on step changes in voltage, considering the asymmetric operation of
disturbing loads. Considering this type of modulating signal components is
related to the fact that probably most sources of voltage fluctuations cause
step voltage changes. The presented research results indicate that only the
proposed approach allows for the correct identification of the frequency of
individual disturbing loads in the power grid, which change their operating
state with a frequency of up to 3$f_{c}$. In addition, the proposed approach
is characterized by low computational complexity in the case of selection of
the appropriate number of samples compared to other considered methods of
empirical decomposition, which supports the process of implementation of the
proposed approach in measuring and recording equipment with limited
performance.

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# J-PARC hadron physics and future possibilities on color transparency

S. Kumano  KEK Theory Center, IPNS, KEK, Oho 1-1, Tsukuba, Ibaraki, 305-0801,
Japan  J-PARC Branch, KEK Theory Center, IPNS, KEK,
and Theory Group, Particle and Nuclear Physics Division, J-PARC Center,
Shirakata 203-1, Tokai, Ibaraki, 319-1106, Japan

(March 31, 2022)

###### Abstract

The J-PARC is a hadron-accelerator facility to provide secondary beams of
kaons, pions, neutrinos, muons, and the others together with the primary
proton beam for investigating a wide range of science projects. High-energy
hadron physics can be studied by using high-momentum beams of unseparated
hadrons, which are essentially pions, and also primary protons. In this
report, possible experiments are explained on color transparency and
generalized parton distributions (GPDs). These projects are complementary to
lepton-scattering experiments at JLab, COMPASS/AMBER, and future electron-ion
colliders. Because of hadron-beam energies up to 30 GeV, the J-PARC is a
unique facility to investigate the transition region from the hadron degrees
of freedom to the quark-gluon one. It is suitable for finding mechanisms of
the color transparency. Such color-transparency studies are also valuable for
clarifying factorization of hadron-production processes in extracting the GPDs
from actual measurements. These studies will lead to the understanding of
basic high-energy hadron interactions in nuclear medium and to clarifications
on the origins of hadron spins, masses, and internal pressure mechanisms.

Keyword: QCD, Quark, Gluon, Color transparency, High-energy hadron reactions,
J-PARC

††preprint: KEK-TH-2388, J-PARC-TH-0265

##  I Introduction

The Japan Proton Accelerator Research Complex (J-PARC) is a multi-purpose
facility ranging from life sciences to nuclear and particle physics J-PARC-web
. By using secondary beams of kaons, pions, neutrinos, muons, and the others
as well as the primary proton beam, a wide range of experiments are possible.
A well-known particle physics experiment is on neutrino oscillations, and
there are also promising projects, including future ones, to probe physics
beyond the standard model such as by lepton-flavor violation, $g-2$, rare kaon
decays, and the neutron electric-dipole moment. It is a flagship facility not
only in particle physics but also in hadron physics.

The J-PARC hadron physics, so far, focuses on low-energy hadron and nuclear
physics with strangeness. It is a unique project in extending our knowledge of
standard nuclear physics by exploring the new flavor degrees of freedom,
strangeness. The recent highlights are discoveries on (1) event excess
observed by the missing spectrum of ${}^{12}C(K^{-},p)$ toward understanding
kaonic- and hyper-nuclei, (2) the $\bar{K}NN$ bound state in
${}^{3}He(K^{-},\Lambda p)n$, (3) $\Xi^{-}$–${}^{14}N$ bound system with the
$\Xi^{-}$ binding energy of 1.27$\pm$0.21 MeV and subsequent findings on other
bound states J-PARC-recent-highlights . Such strangeness nuclear-physics
studies will be continued at J-PARC, and we will expand our knowledge of the
current nuclear physics to the new realm. These investigations will lead to
the understanding of the equation of state for neutron-star matter, which is a
developing field with recent progress on astrophysical observations including
gravitational waves. Right now, the hadron-hall extension is under
consideration, and charmed hadron physics will be also an interesting topic at
J-PARC.

In addition, high-momentum beamlines with the primary 30-GeV protons and
unseparated hadrons, mainly pions, are now available up to about 20-GeV/$c$
momentum. These hadron beam energies are not very high, which indicates a
unique opportunity to investigate the intermediate region from hadron degrees
of freedom (d.o.f.) to the quark and gluon d.o.f. described by perturbative
QCD. In this sense, the J-PARC is a similar facility to the Thomas Jefferson
National Accelerator Facility (JLab); however, it is complementary to JLab
projects because different aspects can be investigated by hadron reactions.

Such complementarity is clear in the studies of the color transparency and the
generalized parton distributions (GPDs). The color transparency means that a
hadron is expected to pass freely through the nuclear medium at large momentum
transfer. There was a mysterious experimental result from the BNL-AVA
collaboration in 2004 that the transparency increases with the beam momentum
as expected; however, it started to drop at about 10 GeV BNL-EVA-2004 . It is
inconsistent with the color transparency prediction, which cannot be explained
at this stage. Since the BNL measurement is not understood easily, it should
be confirmed by independent experiments and the J-PARC is the ideal facility
for this experiment. There was a new color-transparency measurement from the
JLab that the transparency does not appear even up to $Q^{2}=$14 GeV2 in 2021
jlab-trans-2021 . Together with this new development, a possible J-PARC
experiment becomes more valuable for finding the mechanism. It is a timely
project by considering the new experimental development at JLab in connection
with the peculiar BNL result and because the high-momentum beam line is now
ready at J-PARC.

On the other hand, the GPDs can be investigated in the same processes at
hadron accelerator facilities KSS-GPD-2009 ; J-PARC-GPD-2016 ; J-PARC-GPD-LoI
. In particular, the color transparency argument is essential for
factorization of the processes in extracting the GPDs from experimental
measurements Collins:1997hv . The GPDs are valuable quantities for
understanding the origins of hadron mass and spin compositions GPD-review .
They contain information on form factors and parton distribution functions
(PDFs), and their second moments are related to quark orbital-angular-momentum
contributions to the nucleon spin. Furthermore, they are related to
gravitational form factors of hadrons as shown, for example, in Ref. KSS-
grav-2018 , they are also important for finding mechanisms of hadron mass and
internal pressure generations. Therefore, the GPDs will play a crucial role in
future developments of hadron physics. We believe that their studies will lead
to the understandings on the origins of nucleon spin and mass in terms of
quarks and gluons. In this report, we explain possible high-energy hadron-
structure physics especially related to color transparency, which is the main
topic of this workshop, along with possible projects on the GPDs.

##  II Transition from hadron to quark-gluon degrees of freedom and color
transparency

###  II.1 Hard exclusive reactions and constituent counting rule

Figure 1:  Hard-gluon exchange process in an exclusive hadron reaction.

Figure 2:  Transition from hadron d.o.f. to quark-gluon d.o.f. and constituent
counting rule in $\gamma+p\to\pi^{+}+n$.

Let us consider a high-energy exclusive reaction. The cross section for a hard
exclusive hadron reaction $a+b\to c+d$ is generally expressed by the partonic
scattering term $H_{ab\to cd}$ and lightcone amplitudes $\phi_{h}\
(h=a,\,b,\,c,\,d)$, and its matrix element is given by kks-2013 ; kk-2014 ;
cks-2016

$\displaystyle M_{ab\to
cd}=\int[dx_{a}]\,[dx_{b}]\,[dx_{c}]\,[dx_{d}]\,\phi_{c}([x_{c}])\,\phi_{d}([x_{d}])$
$\displaystyle\times H_{ab\to
cd}([x_{a}],[x_{b}],[x_{c}],[x_{d}],Q^{2})\,\phi_{a}([x_{a}])\,\phi_{b}([x_{b}]).$
(1)

Here, $[x_{h}]$ is a set of the lightcone momentum fractions
$x_{i}=p_{i}^{+}/p_{h}^{+}$ for the parton $i$ and the hadron $h$. The matrix
element or the cross section is described by the hard part $H_{ab\to cd}$
calculated in perturbative QCD and the soft one given by the distribution
amplitudes. A typical hard exclusive process is shown in Fig. 2. The exclusive
reaction is possible if the hard momentum is shared among the constituents,
which should be close to each other (“small hadron component”) to form a
hadron in the final state. Assigning the hard gluon and quark propagators as
well as external quark factors, we obtain the cross section expressed by the
Mandelstam variables $s$ and $t$ as kks-2013 ; kk-2014 ; cks-2016 ; ss-2011 ;
abrs-2021

$\displaystyle\frac{d\sigma_{ab\to cd}}{dt}=\frac{1}{s^{\,n-2}}\,f_{ab\to
cd}(t/s),$ (2)

where $f_{ab\to cd}(t/s)$ is a function which depends on the scattering angle,
and $n$ is the total number of constituents ($n=n_{a}+n_{b}+n_{c}+n_{d}$)
involved in the reaction.

This constituent-counting rule was confirmed by high-energy exclusive
reactions. For example, the cross-section ($s^{7}d\sigma/dt$) data are shown
for $\gamma+p\to\pi^{+}+n$ JLab-counting-2003  as the function of the center-
of-mass energy $\sqrt{s}$ in Fig. 2. At low energies, it indicates bumpy
resonance structure, and it becomes constant at high energies, as predicted by
the constituent counting rule. Therefore, hadrons and nuclei should be
described by quark and gluon degrees of freedom in high-energy reactions.
However, the transition energy depends on the reaction process and it is not
easily determined. There are two major reasons for this issue. One is that
there are too many processes like Fig. 2, and the other is that the
distribution amplitudes are not well understood except for the pion one.

###  II.2 Color transparency

Hadron interactions in nuclear medium are valuable for understanding dynamical
properties of QCD and for application to high-energy hadron reactions. As seen
in the constituent-counting rule in the exclusive hadron reactions, the
internal constituents should stick together to form a hadron. Namely, a small
hadron component dominates the reaction at a large-momentum transfer. The
small-size hadron is expected to pass through the nuclear medium without
interactions. This phenomena is called color transparency, and the nuclear
transparency $T=\sigma_{A}/(A\sigma_{N})$ is used for showing this quantity.
Here, $\sigma_{A}$ and $\sigma_{N}$ are nuclear and nucleon cross sections,
respectively, and $A$ is the mass number of the nucleus. The nuclear
transparencies are measured by the reactions $eA\rightarrow e^{\prime}p(A-1)$
and $pA\rightarrow 2p(A-1)$ at electron and proton accelerator facilities. The
color transparency involves both the energy scale at which a hadron is formed
in a point-like configuration as discussed in Sec. 2.1, and how it then
evolves through the medium. For example, $A(p,2p)$ measurements were done at
about 90∘ in the $pp$ center-of-mass frame in the BNL-AGS experiment BNL-
EVA-2004 , so that the large-momentum-transfer mechanism of Sec. 2.1 could be
applied. On the other hand, the expansion of the wave packet is controlled by
the proton momentum in the final state as its expansion is characterized by
the coherence length which is proportional to the proton momentum 2010-ks .
The color transparency contains these two mechanisms. The $(e,e^{\prime}p)A$
reaction is illustrated in Fig. 3 together with our expectation on the nuclear
transparency. At low energies, hadrons interact strongly with nucleons in the
nucleus, so that the nuclear transparency is small. As the energy increases,
it becomes larger and $T$ should become one in the high-energy limit according
to the color transparency.

Figure 3: Color transparency in high-energy electron scattering
$(e,e^{\prime}p)A$. Figure 4: BNL experiment on nuclear transparency by
$pA\rightarrow 2p(A-1)$.

The BNL-AGS data BNL-EVA-2004  are shown in Fig. 4 for the nuclear
transparency in the reaction $pA\rightarrow 2p(A-1)$ as the function of the
beam momentum. As the hard scale of the reaction, the proton-beam energy in
this case, becomes larger, the transparency is expected to increase. In fact,
the transparency increases as the momentum becomes larger up to $p\sim 10$
GeV/$c$; however, it drops at $p>10$ GeV/$c$. This result seems to be in
contradiction to the color transparency expectation and it is not easily
explained theoretically. Since the J-PARC has the 30-GeV primary-proton-beam
energy, the larger momentum region can be measured and the transparency
phenomena shown in Figs. 3 and 4 should be tested. Therefore, the J-PARC
provides a valuable opportunity for investigating basic hadron interactions in
the nuclear medium as predicted by perturbative QCD.

Next, we proposed a new $2\to 3$ reaction process for investigating the color
transparency phenomena by using the high-momentum pion beam at J-PARC and
COMPASS /AMBER 2010-ks . In particular, we studied hard branching $2\to 3$
processes $\pi^{-}A\to\pi^{-}\pi^{+}A^{*}$ as illustrated in Fig. 6. The basic
reaction process $\pi^{-}N\to\pi^{-}\pi^{+}N^{\prime}$ is shown in Fig. 6,
where the kinematics of the upper hard-reaction part $\pi h\to\pi\pi$ is
controlled by the lower part $N\to hN^{\prime}$. This hard branching $2\to 3$
processes with nuclei provide an effective way to determine the momentum
transfers needed for effects of point-like configurations to dominate large-
angle $2\to 2$ processes.

Figure 5: $\pi^{-}A\to\pi^{-}\pi^{+}A^{*}$ for color transparency.

Figure 6: $\pi^{-}N\to\pi^{-}\pi^{+}p$ process.

Figure 7: Color transparency in $2\to 3$ process
$\pi^{-}A\to\pi^{-}\pi^{+}A^{*}$.

Calculated nuclear transparencies are shown in Fig. 7 as the functions of the
pion-beam energy for three different nuclei 2010-ks . The $2\to 3$ reaction
allows the effects of the transverse size of configurations to be decoupled
from effects of the space-time evolution of these configurations, which is
different from past $2\to 2$ methods. Namely, it is a new method for probing
the dynamics of large angle hadron-hadron scattering using the color-
transparency phenomenon which is free from the limitations imposed by the
expansion effects of the point-like configurations. The beam energies are in
the 20-200 GeV range, namely from the J-PARC to COMPASS/AMBER energies, and
they are appropriate for these $2\to 3$ reaction studies.

##  III Possible J-PARC project on generalized parton distributions

###  III.1 J-PARC hadron facility

The J-PARC provides the most intense proton beam in the energy region above
multi-GeV, and it is a multi-purpose facility from nuclear and particle
physics, material and life sciences to industrial applications. Nuclear and
particle physics experiments are done by secondary beams (pions, kaons,
neutrinos, muons, and antiprotons) as well as the primary 30-GeV proton beam.

Figure 8: J-PARC hadron hall.

Hadron-physics projects are done in the hadron experimental hall of Fig. 8.
There are also particle-physics projects on lepton-flavor violation and rare
kaon decays in this hadron hall. The K1.8 indicates kaons with momentum around
1.8 GeV/c and this beamline is used for strangeness nuclear physics such as
strangeness $-$2 hypernuclei with $\Xi^{-}$ by the $(K^{-},K^{+})$ reaction.
The K1.1 beamline is intended for low-momentum stopped kaon experiments such
as the studies of kaonic nuclei. The high-momentum beamline indicates for 30
GeV protons and unseparated hadrons, which are essentially pions with up to
about 20 GeV/$c$ momentum. Using these high-momentum protons and pions, we can
investigate interesting high-energy hadron physics topics, particularly in the
transition region from hadron to quark-gluon degrees of freedom, as explained
in this paper.

###  III.2 Generalized parton distributions

The studies of the color transparency are closely related to the GPD
measurements in connection with factorization and reaction mechanisms. The
spacelike GPDs have been measured in deeply virtual Compton scattering and
meson productions at charged-lepton accelerator facilities. On the other hand,
the timelike GPDs are measured in two-photon processes of $e^{+}e^{-}$
reactions, for example, at the KEK-B facility KSS-grav-2018 . Furthermore, a
GPD measurement is also considered at the Fermilab neutrino facility GPD-
neutrino . In addition to these projects, it is possible to investigate the
GPDs at hadron accelerator facilities such as the J-PARC.

Figure 9: GPD kinematics.

The quark GPD is given by the amplitude to extract a quark from a hadron and
then to insert it into the hadron at a different spacetime point as shown in
Fig. 9. The generalized scaling variable $x$ and a skewdness parameter $\xi$
are defined by the lightcone momenta $k^{+}$ and $P^{+}$ as $x=k^{+}/P^{+}$
and $\xi=-\Delta^{+}/(2P^{+})$. Here, $k$ is the average momentum of the
quarks, and $\Delta^{\mu}$ is the four momentum transfer $\Delta=p^{\prime}-p$
for the hadron or the quark. There is another variable $t$ defined by
$t=\Delta^{2}$. The variable $x$ indicates the momentum fraction carried by a
quark in the hadron, and the skewdness parameter $\xi$ indicates the momentum-
transfer fraction. The GPDs are expressed by these three variables ($x$,
$\xi$, $t$). The GPDs of the nucleon, $H^{q}$, $E^{q}$, $\tilde{H}^{q}$, and
$\tilde{E}^{q}$ are defined by the matrix elements of nonlocal vector and
axial-vector operators as

$\displaystyle\int\frac{dy^{-}}{4\pi}$ $\displaystyle
e^{ixP^{+}y^{-}}\left<p^{\prime}\left|\bar{q}(-y/2)\gamma^{+}q(y/2)\right|p\right>_{y^{+}=\vec{y}_{\perp}=0}$
$\displaystyle=\frac{1}{2P^{+}}\bar{u}(p^{\prime})\left[H^{q}(x,\xi,t)\gamma^{+}+E^{q}(x,\xi,t)\frac{i\sigma^{+\alpha}\Delta_{\alpha}}{2m_{N}}\right]u(p),$
(3) $\displaystyle\int\frac{dy^{-}}{4\pi}$ $\displaystyle
e^{ixP^{+}y^{-}}\left<p^{\prime}\left|\bar{q}(-y/2)\gamma^{+}\gamma_{5}q(y/2)\right|p\right>_{y^{+}=\vec{y}_{\perp}=0}$
$\displaystyle=\frac{1}{2P^{+}}\bar{u}(p^{\prime})\left[\tilde{H}^{q}(x,\xi,t)\gamma^{+}\gamma_{5}+\tilde{E}^{q}(x,\xi,t)\frac{\gamma_{5}\Delta^{+}}{2m_{N}}\right]u(p).$
(4)

These GPDs are very useful quantities. In the forward limit, the GPDs become
the unpolarized and longitudinally-polarized PDFs. Their first moments become
the corresponding form factors, and the second moment is the quark
contribution to the nucleon spin: $J_{q}=\int
dx\,x\,[H^{q}(x,\xi,t=0)+E^{q}(x,\xi,t=0)]/2=\Delta q^{+}/2+L_{q}$. Here,
$L_{q}$ is a quark orbital-angular-momentum contribution to the nucleon spin.
Because the quark-spin contribution $\Delta q^{+}$ is known from experimental
measurements, it is possible to determine $L_{q}$ from the GPDs. In addition,
the GPDs can be used for extracting gravitational form factors of hadrons KSS-
grav-2018 , so that they are also valuable for finding the origin of hadron
masses and investigating internal pressure phenomena of hadrons.

###  III.3 GPDs in the ERBL region by $\bm{2\to 3}$ process $\bm{NN\to N\pi
B}$

The $2\to 3$ reaction processes $NN\to N\pi B$ can be used for probing the
GPDs in a unique kinematical region, which is called the ERBL (Efremov-
Radyushkin-Brodsky-Lepage) region shown in Fig. 11. The GPDs have three
kinematical regions, (I) $-1<x<-\xi$, (II) $-\xi<x<\xi$, (III) $\xi<x<1$. The
region II is called the ERBL region and the process corresponds to a quark
emission of momentum fraction $x+\xi$ with an antiquark emission of momentum
fraction $\xi-x$. It is a quark-antiquark (meson) distribution amplitude. The
regions I and III are called DGLAP (Dokshitzer-Gribov-Lipatov-Altarelli-
Parisi) regions, and the process III (I) corresponds to a quark (antiquark)
emission of momentum fraction $x+\xi$ ($\xi-x$) with a quark (antiquark)
absorption of momentum fraction $x-\xi$ ($-\xi-x$). It is a quark (antiquark)
distribution amplitude.

Figure 10: DGLAP and ERBL regions of GPDs.

Figure 11: Typical $2\to 3$ process $pp\to p\pi B$ and GPD in the ERBL region.

Figure 12: Cross sections for $pp\to p\pi B$.

For investigating the ERBL region, we proposed to use the hadronic $2\to 3$
process $pp\to p\pi B$ KSS-GPD-2009 , where the baryon $B$ could be a nucleon
or $\Delta$, in Fig. 11. If the pion and the final proton have large and
nearly opposite transverse momenta and a large invariant energy, an
intermediate exchange could be considered as a $q\bar{q}$ state. This
$q\bar{q}$ attached to the nucleon is expressed by the GPDs in the ERBL
region. This experiment is possible at J-PARC, and it is complementary to
other GPD measurements because of this uniqueness on the ERBL kinematics. The
$pp\to p\pi^{+}B$ ($B=\Delta^{0},\,n$) cross sections are shown in Fig. 12 for
the proton beam energy of 30 and 50 GeV by considering the J-PARC experiment
KSS-GPD-2009 . In the hard kinematical region,
$s^{\prime},|t^{\prime}|,|u^{\prime}|\gg m_{N}^{2}$ with
$t^{\prime}/s^{\prime}={\rm const}$, the cross section could be factorized
into the GPD part and the hadron ($q\bar{q}$)-proton scattering part as ${\cal
M}_{NN\to N\pi B}={\cal M}_{N\to hB}\,{\cal M}_{hN\to\pi N}$, where $h$
indicates the intermediate $q\bar{q}$ hadronic state. Here, the transverse
sizes of the projectile and two outgoing hadrons near the interaction point
are given by $\sim 1/\sqrt{|t^{\prime}|}$, so that the color transparency
arguments are essential in proving the factorization for an exclusive hadron
production Collins:1997hv . Actually, this factorization was proved for the
deeply virtual meson-production process in Ref. Collins:1997hv , so that we
need such a study for the current hadronic reaction. Depending on the final
state $\Delta^{0}$ or $n$ and the proton-beam energy, the cross section and
the kinematical range vary. It is a unique opportunity to investigate the ERBL
region of the GPDs.

###  III.4 GPDs by exclusive Drell-Yan process
$\bm{\pi^{-}p\to\mu^{+}\mu^{-}B}$

Figure 13: Exclusive Drell-Yan process.

The GPDs can be investigated by the exclusive Drell-Yan process
$\pi^{-}p\to\mu^{+}\mu^{-}B$ at J-PARC BDP-2001 ; J-PARC-GPD-2016 , and the
experimental proposal is under preparation now J-PARC-GPD-LoI . This part is
discussed by Wen-Chen Chang at this workshop, so that the detailed should be
found in his contribution Wen-Chen-Chang-2022 .

The unseparated hadron beam, which is essentially the pion beam, is available
at the high-momentum beamline of J-PARC. The beam momentum is up to about 20
GeV/$c$. A typical process is shown in Fig. 13 with the GPDs. The nucleonic
GPDs are investigated if $B$ is the neutron, whereas the transition GPDs are
studied for $B\neq N$. In addition to the GPDs, the pion distribution
amplitude is involved in the reaction; however, it could be taken from other
theoretical and experimental studies pion-distribution . Therefore, the GPDs
should be extracted from experimental measurements. For such studies, one
should note that possibly a large contribution could come from another non-
factorized process, so called the Feynman mechanism, and it needs to be taken
into account properly Tanaka-Feynman-2017 .

Figure 14:  Detector design for the J-PARC GPD experiment J-PARC-GPD-2016 ;
APS .

Figure 15:  Monte-Carlo simulation for the missing-mass ($M_{X}$) spectra of
the $\mu^{+}\mu^{-}$ events at $p_{\pi}=15$ GeV/$c$ J-PARC-GPD-2016 ; APS .

A new proposal will be submitted soon for this project J-PARC-GPD-LoI  in
collaboration with the approved E50 experiment on charmed baryons J-PARC-E50 .
In this experiment, the muon identification system is added to the E50
spectrometer as shown in Fig. 15. The details of the detectors are explained
in Refs. J-PARC-GPD-2016 ; J-PARC-GPD-LoI . The exclusive Drell-Yan process
can be identified by the missing-mass technique. The actual Monte-Carlo
simulation is shown in Fig. 15 as the function of the missing mass $M_{X}$ for
the pion momentum $p_{\pi}=15$ GeV/$c$. It is clear that the exclusive Drell-
Yan process is identified in comparison with other contributions from the
inclusive Drell-Yan, $J/\psi$ production, and random-background processes.
This measurement probes the GPDs in the region $0.1<x<0.3$ by the timelike
photon, which is comlementary to the JLab meson-production experiment with the
spacelike photon. In addition, pion-to-nucleon transition distribution
amplitudes can be investigated by backward charmonium production in pion-
nucleon collisions TDA-2017 . Strangeness-hadron production experiments are
possible at J-PARC as mentioned in Ref. Wen-Chen-Chang-2022 , so that the
discussed topics in this paper could be investigated including strangeness.
Furthermore, we expect that a separated high-momentum kaon beam will become
available in future j-parc-extension . The color-transparency and the $2\to 3$
GPD (Fig. 12) experiments are still under consideration at J-PARC. Hopefully,
they will be realized in addition to the exclusive Drell-Yan experiment.

##  IV Summary

The J-PARC is the most intense proton-accelerator facility above the multi-GeV
region. By using the high-momentum beams, it is possible to have unique
experiments on the color transparency and the GPDs. Especially, it should be
the key experiment to solve the mysterious BNL-AVA result on the color
transparency. In addition, the process $\pi^{-}A\to\pi^{-}\pi^{+}A^{*}$ ($2\to
3$ reaction) provides a unique opportunity to separate the transverse size
configurations from the space-time evolution, in contract to the past $2\to 2$
studies. Next, the J-PARC GPD experiment by the exclusive Drell-Yan processes
is a complementary project to the JLab one because of the smaller-$x$
kinematical region and the timelike photon instead of the spacelike one. We
also discussed another possible GPD project at J-PARC by the reaction $pp\to
p\pi B$ to investigate the ERBL region of the GPDs. All these projects are
valuable for the development of high-energy hadron physics along with other
experimental projects at JLab, Fermilab, KEKB, NICA, FAIR, LHCspin, EIC, EicC,
and the others. From these measurements, we understand basic mechanisms of
high-energy hadron reactions in nuclear medium. Furthermore, the GPD studies
will lead to clarifications on the origins of hadron spins, masses, and
internal pressures.

## Acknowledgments

S. Kumano was partially supported by Japan Society for the Promotion of
Science (JSPS) Grants-in-Aid for Scientific Research (KAKENHI) Grant Number
19K03830. The author thanks T. Takahashi for suggestions on the J-PARC
experiments. He also thanks the American Physical Society for the copyright
permission to use Figures 14 and 15 from Ref. J-PARC-GPD-2016 .

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#  New CP phase of $B_{s}-\bar{B}_{s}$ mixing on ${\rm T}$ violation in
$B_{d(s)}\to K^{*}(\phi)\ell^{+}\ell^{-}$

Chuan-Hung Chen1,2111Email<EMAIL_ADDRESS>Chao-Qiang
Geng3222Email<EMAIL_ADDRESS>and Lin
<EMAIL_ADDRESS>1Department of Physics, National Cheng-
Kung University, Tainan 701, Taiwan
2National Center for Theoretical Sciences, Hsinchu 300, Taiwan
3Department of Physics, National Tsing-Hua University, Hsinchu 300, Taiwan

###### Abstract

A large CP-violating phase uncovered recently by CDF and D$\O$ collaborations
in the time-dependent CP asymmetry (CPA) of the $B_{s}\to J/\Psi\phi$ decay
clearly indicates that a non-Cabibbo-Kobayashi-Maskawa (CKM) phase has to be
brought into the $b\to s$ transition. We find that the model with $SU(2)_{L}$
singlet exotic quarks can not only provide the new phase induced from the
Z-mediated flavor changing neutral current (FCNC) at tree level, but also
strongly relate the $B_{s}-\bar{B}_{s}$ mixing, $B_{q}\to
V_{q}\ell^{+}\ell^{-}$ ($V_{d[s]}=K^{*}[\phi])$ and $B_{s}\to\mu^{+}\mu^{-}$
together. In particular, we show that the new CP phase can be unambiguously
exposed by the large statistical significances of $T$ violating observables in
$B_{q}\to V_{q}\ell^{+}\ell^{-}$, while the branching ratio of
$B_{s}\to\mu^{+}\mu^{-}$ can be enhanced to be $O(10^{-8})$.

CP violation (CPV) has been one of the most mysterious phenomena in high
energy physics since it was discovered in the K system CP . At B factories,
BABAR and BELLE have observed both the mixing-induced time-dependent CP
asymmetry (CPA) in the $B_{d}$ oscillation through the golden mode of $B\to
J/\Psi K_{S}$ and the direct CPAs in exclusive $B\to\pi\pi$ and $B\to\pi K$
decays HFAG , where the former is dictated by the $b\to d$ transition while
the latter $b\to s$. Although three generations of the standard model (SM) can
provide a unique CP violating phase in the Cabibbo-Kobayashi-Maskawa (CKM)
matrix CKM  to interpret the observed CPAs, it does not provide a solution to
understand the matter-antimatter asymmetry in the Universe and it does not
stop people to search for a new CPV source.

Nevertheless, it is very difficult to unfold a new CPV phase in the direct
CPAs of the nonleptonic exclusive decays due to the inevitable large
uncertainty of nonperturbative QCD effects. Hence, the best environment to
look for the new phase is that the QCD effects are less involved while the SM
contributions are highly suppressed. Now, the dawn to see the new effects
could be in the $B_{s}$ system. By the $B_{s}$ production in Tevatron Run II,
besides CDF and $D{\O}$ observations on the $B_{s}$ oscillation of $\Delta
m_{s}=17.77\pm 0.10\pm 0.07$ ps-1 CDF  and $\Delta m_{s}=18.56\pm 0.87$ ps-1
D0 , respectively, and the large direct CPA of $0.39\pm 0.17$ for $B_{s}\to
K^{-}\pi^{+}$ BsCP , an unexpected large CPV phase has been detected in the
mixing-induced CPA for $B_{s}\to J/\Psi\phi$.

To explain the new phase, we write the transition matrix element for
$\bar{B}_{s}\to B_{s}$ as

$\displaystyle M^{s}_{12}$ $\displaystyle=$ $\displaystyle A^{\rm
SM}_{12}e^{-2i\beta_{s}}+A^{\rm NP}_{12}e^{2i(\theta^{\rm NP}_{s}-\beta_{s})}$
(1)

where $\beta_{s}\equiv arg(-V_{ts}V^{*}_{tb}/V_{cs}V^{*}_{cb})$ is the CPV
phase in the SM and $\theta^{NP}_{s}$ is the new CPV phase in some extension
of the SM. Here, the convention has been chosen to be the same as that in Ref.
UTfit . Due to $\Delta\Gamma_{s}\ll\Delta m_{s}$ in the B-system PDG06 , the
time-dependent CPA could be simplified to be

$\displaystyle-S_{J/\Psi\phi}$ $\displaystyle\simeq$ $\displaystyle{\rm
Im}\left(\sqrt{\frac{M^{s^{*}}_{12}}{M^{s}_{12}}}\right)=\sin(2\beta_{s}-\phi^{\rm
NP}_{s})\,,$ $\displaystyle\phi^{NP}_{s}$ $\displaystyle=$
$\displaystyle\arctan\left(\frac{r\sin 2\theta^{\rm NP}_{s}}{1-r\cos
2\theta^{\rm NP}_{s}}\right)$ (2)

with $r=A^{NP}_{12}/A^{\rm SM}_{12}$. By adopting Wolfenstein parametrization
Wolfenstein  of the CKM matrix up to $O(\lambda^{4})$, in which
$V_{tb}=1-A^{2}\lambda^{4}/2$,
$V_{ts}=-A\lambda^{2}+1/2A\lambda^{4}\left(1-2(\rho+i\eta)\right)$,
$V_{cb}=A\lambda^{2}$ and $V_{cs}=1-\lambda^{2}/2-1/8\lambda^{4}(1+4A^{2})$
Buras , one can easily find

$\displaystyle\beta_{s}\approx\lambda^{2}\eta\approx 0.019\,,$ (3)

where $\lambda=0.2272$ and $\eta=0.359$ PDG06  have been used. Clearly, the
mixing-induced CPA of $B_{s}$ in the SM is only few percent. Astonishingly,
the nonvanished CP phase measured by CDF CDFCP  and D$\O$ D0CP  is

$\displaystyle\phi_{s}=2\beta_{s}-\phi^{NP}_{s}=\left\\{\begin{array}[]{c}[0.24,\,1.36]\cup[1.78,\,2.82]\
(\textrm{CDF})\\\ 0.57^{+0.30+0.02}_{-0.24-0.07}\hskip
48.36958pt(\textrm{D$\O$})\end{array}\right.$ (6)

at the $68\%$ confidence level (CL), while the allowed range at the $90\%$ CL
by D$\O$ is given to be $\phi_{s}\in[-0.06,\,1.20]$. Recently, the UTfit
collaboration has combined all available information in the $B_{s}$ system and
concluded that the CPV phase of the $B_{s}$ mixing amplitude deviates more
than $3\sigma$ from the SM value UTfit . To explain the large CPV phase in the
$b\to s$ transition, several new physics models have been proposed models . In
this paper, we explore the effects of the large phase in the decays of
$B_{q}\to V_{q}\ell^{+}\ell^{-}$ ($V_{d(s)}=K^{*}(\phi)$), corresponding to
$b\to s\ell^{+}\ell^{-}$ at the quark level. In particular, we will show that
the large CPV phase can be directly probed by measuring T-odd observables in
the decays.

To comprehend the beauty of using T violation to probe the CPV phase, we
briefly summarize the characters of CP-odd and T-odd observables. In a decay
process, the direct CPA or CP-odd observable is defined by ${\cal
A}_{CP}\equiv(\bar{\Gamma}-\Gamma)/(\bar{\Gamma}+\Gamma)$, where $\Gamma$
($\bar{\Gamma}$) is the partial decay rate of the (CP-conjugate) process. As a
result, ${\cal A}_{CP}\propto\sin\theta_{w}\sin\theta_{st}$ with
$\theta_{w(st)}$ being the weak (strong) phase. Clearly, to have a nonvanished
CPA, both phases are needed. The efficiency on the CPA is mainly dictated by
the uncertain calculations of the strong phase. Another way to probe the CPV
phase is through the spin-momentum triple correlation, such as
$\vec{s}\cdot(\vec{p}_{i}\times\vec{p}_{j})$ Garisto-CGL ; geng-BRD ;
CG_NPB636  for a three-body decay, where $\vec{s}$ is the spin carried by one
of outgoing particles and $\vec{p}_{i}$ and $\vec{p}_{j}$ denote any two
independent momenta. The triple correlation is a T-odd observable since it
changes sign under the time reversal (${\rm T}$) transformation of
$t\rightarrow-t$. We note that the T transformation defined here is different
from the real time-reversal transformation which also contains the interchange
of initial and final states. By the CPT invariant theorem, T violation (TV)
implies CPV. Therefore, the study of the T-odd observable can help us to
understand the origin of CPV. Intriguingly, the T-odd triple correlation is
proportional to $\sin(\theta_{w}+\theta_{st})$, which indicates that the
strong phase is not necessary to achieve a nonzero T-odd observable. It has
been shown in Ref. CG_NPB636  that T-violating effects in the exclusive $b\to
s\ell^{+}\ell^{-}$ processes are sensitive to new physics with small QCD
uncertainties, which could provide a good place to directly observe the new
phase revealed by CDF and D$\O$.

The transition amplitudes for $B_{q}\to V_{q}\ell^{+}\ell^{-}$
($\ell=e,\,\mu$) are given by CG_NPB636 ; CG_PRD66

$\displaystyle{\cal M}_{V_{q}}^{(\lambda)}$ $\displaystyle=$
$\displaystyle-\frac{G_{F}\alpha\lambda_{t}}{2\sqrt{2}\pi}\left\\{{\cal
M}_{1\mu}^{(\lambda)}L^{\mu}+{\cal M}_{2\mu}^{(\lambda)}L^{5\mu}\right\\}\,,$
$\displaystyle{\cal M}_{a\mu}^{(\lambda)}$ $\displaystyle=$ $\displaystyle
if_{1}\varepsilon_{\mu\nu\alpha\beta}\epsilon^{*\nu}(\lambda)P^{\alpha}q^{\beta}+f_{2}\epsilon_{\mu}^{*}(\lambda)+f_{3}\epsilon^{*}\cdot
qP_{\mu}\,,$

where $\lambda_{t}=V_{tb}V_{ts}^{*}$, $L_{\mu}=\bar{\ell}\gamma_{\mu}\ell$,
$L^{5}_{\mu}=\bar{\ell}\gamma_{\mu}\gamma_{5}\ell$, $P=p_{B}+p_{V}$,
$q=p_{B}-p_{V}$, $a=1\,(2)$ while $f_{i}=h_{i}$ $(g_{i})$ and

$\displaystyle h_{1}$ $\displaystyle=$ $\displaystyle{C^{\rm eff}_{9}V\over
m_{B}+m_{V}}+\frac{2m_{b}}{q^{2}}C_{7}T_{1}\,,$ $\displaystyle h_{2}$
$\displaystyle=$ $\displaystyle-\frac{1}{2}(m_{B}+m_{V})C^{\rm
eff}_{9}A_{1}-\frac{1}{2}\frac{2m_{b}}{q^{2}}P\cdot qC_{7}T_{2}\,,$
$\displaystyle h_{3}$ $\displaystyle=$ $\displaystyle{C^{\rm
eff}_{9}A_{2}\over
m_{B}+m_{V}}+\frac{2m_{b}}{q^{2}}C_{7}\Big{(}T_{2}(q^{2})+\frac{q^{2}}{P\cdot
q}T_{3}\Big{)}\,,$ $\displaystyle g_{i}$ $\displaystyle=$ $\displaystyle
h_{i}|_{C^{\rm eff}_{9}\to C_{10},C_{7}=0}\,,\ \ (i=1,2,3)\,.$ (7)

Here, $m_{B}(m_{V})\equiv m_{B_{q}}(m_{V_{q}})$, $C^{\rm eff}_{9}$, $C_{7}$
and $C_{10}$ are the Wilson coefficients BBL  and the definitions of the form
factors in Eq. (7) can be found in Ref. CG_NPB636 . Furthermore, to obtain
T-odd terms, the polarizations of $V_{q}$ should be kept in the averaged
squared-amplitude. To achieve the requirement, we have to consider the decay
chain $B_{q}\to V_{q}(\to P_{1}P_{2})\ell^{+}\ell^{-}$ in which $P_{1}P_{2}$
is $K\pi(KK)$ as $V_{q}=K^{*}(\phi)$. Consequently, the differential decay
rate associated with these terms is given by

$\displaystyle\frac{d\Gamma}{d\cos\theta_{K}d\cos\theta_{\ell}d\phi
dq^{2}}=\frac{3\alpha^{2}G_{F}^{2}\left|\lambda_{t}\right|^{2}\left|\vec{p}\right|}{2^{14}\pi^{6}m_{B}^{2}}$
$\displaystyle\times\\{4\cos^{2}\theta_{K}\sin^{2}\theta_{\ell}\sum_{i=1,2}|{\cal
M}_{i}^{0}|^{2}+\sin^{2}\theta_{K}(1+\cos^{2}\theta_{\ell})$
$\displaystyle\sum_{i=1,2}\left(|{\cal M}_{i}^{+}|^{2}+|{\cal
M}_{i}^{-}|^{2}\right)-\sin 2\theta_{K}\sin 2\theta_{\ell}\sin\phi$
$\displaystyle\sum_{i=1,2}{Im}\left({\cal M}_{i}^{+}-{\cal
M}_{i}^{-}\right){\cal
M}_{i}^{0*}-2\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\sin 2\phi$
$\displaystyle\sum_{i=1,2}{Im}\left({\cal M}_{i}^{+}{\cal
M}_{i}^{-*}\right)+2\sin 2\theta_{K}\sin\theta_{\ell}\sin\phi(Im{\cal
M}_{1}^{0}$ $\displaystyle({\cal M}_{2}^{+*}+{\cal M}_{2}^{-*})-Im({\cal
M}_{1}^{+}+{\cal M}_{1}^{-}){\cal M}_{2}^{0*})+\cdots]\\}\,,$ (8)

where $\theta_{\ell}(\theta_{K})$ is the polar angle of the lepton (K-meson)
in the $q^{2}$ ($V_{q}$) rest frame,
$|\vec{p}|=[((m_{B}^{2}+m_{V}^{2}-q^{2})/(2m_{B}))^{2}-m_{V}^{2}]^{1/2}$ and
${\cal M}_{i}^{0}$ and ${\cal M}_{i}^{\pm}$ denote the longitudinal and
transverse polarizations of $V_{q}$ with their explicit expressions being

$\displaystyle{\cal M}_{a}^{0}$ $\displaystyle=$
$\displaystyle\sqrt{q^{2}}\left(\frac{E_{V}}{m_{V}}f_{2}+2\sqrt{q^{2}}\frac{\left|\vec{p}_{V}\right|^{2}}{m_{V}}f_{3}\right),$
$\displaystyle{\cal M}_{a}^{\pm}$ $\displaystyle=$
$\displaystyle\sqrt{q^{2}}\left(\pm
2\left|\vec{p}_{V}\right|\sqrt{q^{2}}f_{1}+f_{2}\right)\,,$ (9)

respectively. To shorten the expressions, we have only presented the relevant
pieces in Eq. (8), where the three imaginary terms denote the T-violating
effects. The whole expression for differential decay rate could refer to Refs.
CG_NPB636 ; BHP .

From Eqs. (7), (8) and (9), it is easy to show that the large contributions to
the T-violating effects arise from $Im{\cal M}_{1}^{0}({\cal M}_{2}^{+*}+{\cal
M}_{2}^{-*})-Im({\cal M}_{1}^{+}+{\cal M}_{1}^{-}){\cal M}_{2}^{0*}$. To
explore the effects, we examine the T-odd observable, defined by
$\left\langle{\cal O}_{T}\right\rangle=\int{\cal O}_{T}d\Gamma$ where ${\cal
O}_{T}$ is a T-odd five-momentum correlation, given by

${\cal
O}_{T}=\frac{\vec{p}_{B}\cdot\vec{p}_{K}}{\left|\vec{p}_{B}\right|\left|\vec{p}_{K}\right|}\frac{\vec{p}_{B}\cdot(\vec{p}_{K}\times\vec{p}_{\ell^{+}})}{\left|\vec{p}_{B}\right|\left|\vec{p}_{K}\right|\omega_{\ell^{+}}}\,$
(10)

with $\omega_{\ell^{+}}=q\cdot p_{\ell^{+}}/\sqrt{q^{2}}$. In the $V_{q}$ rest
frame, we note that ${\cal
O}_{T}=\cos\theta_{K}\sin\theta_{K}\sin\theta_{\ell}\sin\phi$. To signal the
nonvanished CPV phase, we employ the statistical significance of the
observable, defined by

$\varepsilon_{T}(q^{2})={\frac{\int{\cal O}_{T}d\Gamma}{\sqrt{(\int
d\Gamma)(\int{\cal O}^{2}_{T}d\Gamma)}}}\,.$ (11)

Integrating all relevant angles in Eq. (11), we obtain

$\displaystyle\varepsilon_{T}(q^{2})$ $\displaystyle\simeq$
$\displaystyle\frac{0.76}{\sqrt{{\cal D}_{1}{\cal D}_{2}}}[Im{\cal
M}_{1}^{0}({\cal M}_{2}^{+*}+{\cal M}_{2}^{-*})-$ $\displaystyle Im({\cal
M}_{1}^{+}+{\cal M}_{1}^{-}){\cal M}_{2}^{0*}]\,,$ $\displaystyle{\cal D}_{a}$
$\displaystyle=$ $\displaystyle\sum_{i=1,2}\left[\left|{\cal
M}_{i}^{0}\right|^{2}+\frac{1}{a}\left(\left|{\cal
M}_{i}^{+}\right|^{2}+\left|{\cal M}_{i}^{-}\right|^{2}\right)\right].\,\ $
(12)

To observe the effect at $n\sigma$ level, the required number of B mesons is
$N_{B}=n^{2}/(Br\cdot\varepsilon^{2}_{T})$.

We now use the clue of the current data to illustrate our model-independent
analysis. Although the principle of the minimal flavor violation (MFV) MFV
could be as a dogma to rule the new source of CPV MFVCP , to focus on the
criterion of the minimal extension of the SM, we employ the vector-like-quark
model (VQM), in which the vector-like quarks (VQs) are $SU(2)_{L}$ singlet
exotic quarks, as the ones naturally realized in $E_{6}$ models E_6 .
Explicitly, we include $SU(2)_{L}$ singlet VQs in the SM, where the right-
handed component is the same as the right-handed down-quark. Since the singlet
quarks do not couple to W-bosons directly, one of the fascinating characters
of the model is that the corresponding CKM matrix is not a unitary matrix. By
introducing flavor mixing matrices to diagonalize the $4\times 4$ down-type
quark mass matrix, we will display that the model interestingly leads to
Z-mediated flavor changing neutral currents (FCNCs) at tree level ZFCNC ,
which clearly have significant impacts on the $B_{s}-\bar{B}_{s}$ mixing,
$B_{q}\to V_{q}\ell^{+}\ell^{-}$ and $B_{s}\to\mu^{+}\mu^{-}$ processes. We
note that the non-unitary CKM matrix could result in new contributions to the
processes from box and penguin diagrams. However, to simply illustrate the new
phase, only the Z-mediated effects at tree level are considered here, while
those from the box and penguin diagrams could be referred to Ref. Aguilar-
Saavedra .

In the mass eigenstates, the coupling of the $Z$-boson to fermions is written
by

$\displaystyle{\cal L}_{Z}$ $\displaystyle=$
$\displaystyle-\frac{gc^{f}_{L}}{2\cos\theta_{W}}\bar{F}\gamma^{\mu}\left(V^{L}_{F}X_{F}V^{L^{\dagger}}_{F}\right)P_{L}FZ_{\mu}\,,$
$\displaystyle X_{Q}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}\begin{array}[]{c}\openone_{3\times
3}\\\ \end{array}&|&{\bf 0}_{3\times 1}\\\ -\ -\ -&-&\ -\\\ {\bf 0}_{1\times
3}&|&\xi_{Q}\\\ \end{array}\right]\,,\ \ \ X_{\ell}=\openone_{3\times 3}\,,$
(17)

where $g$ is the coupling constant of $SU(2)_{L}$, $\theta_{W}$ is the
Weinberg’s angle, $P_{R(L)}=(1\pm\gamma_{5})/2$,
$F^{T}=(q_{1},q_{2},q_{3},q_{4})$ and $(e,\mu,\tau)$ represent quarks (Qs) and
leptons $(\ell s)$, $c^{f}_{L}$ is defined as $c^{f}_{L}=c^{f}_{V}+c^{f}_{A}$
with

$\displaystyle c^{f}_{V}$ $\displaystyle=$ $\displaystyle
T^{3}_{f}-2\sin^{2}\theta_{W}Q_{f}\,,\ \ \ c^{f}_{A}=T^{3}_{f}$ (18)

in which $T^{3}_{f}$ and $Q_{f}$ are the third component of the weak isospin
and the electric charge of the particle, respectively, and
$\xi_{f}=-2\sin^{2}\theta_{W}Q_{f}/c^{f}_{L}$. Here, as the quantum number of
the new right-handed VQ is the same as the right-handed down-quark, the tree
FNCNs only occur in the left-handed quarks. Accordingly, the interaction for
$b$-$s$-$Z$ is given by

$\displaystyle{\cal L}_{b\to
s}=\frac{gc^{D}_{L}\lambda_{23}}{2\cos\theta_{W}}\bar{s}\gamma^{\mu}P_{L}bZ_{\mu}+H.c.$
(19)

with
$\lambda_{23}=(1-\xi_{D})(V^{L}_{D})_{24}(V^{L}_{D})^{*}_{34}\equiv|\lambda_{23}|\exp[i(\theta^{\rm
NP}_{s}-\beta_{s})]$. By Eqs. (1) and (19), the transition matrix element for
the $\Delta B=2$ process is obtained as

$\displaystyle A^{\rm NP}_{12}$ $\displaystyle=$
$\displaystyle\frac{G_{F}\left(\lambda_{23}\right)^{2}}{3\sqrt{2}}m_{B_{s}}f^{2}_{B_{s}}\hat{B_{s}}\,.$
(20)

As a result, the $B_{s}-\bar{B}_{s}$ mixing in the VQM is

$\displaystyle\Delta m_{s}=\Delta m^{\rm SM}_{s}\left(1+2r\cos\theta^{\rm
NP}_{s}+|r|^{2}\right)^{1/2}\,.$ (21)

From the above equation, it is clear that a large new CPV phase can have a
significant influence on the $B_{s}-\bar{B}_{s}$ mixing. By combining the SM
and Z-mediated FCNC, the branching ratio (BR) of $B_{s}\to\mu^{+}\mu^{-}$ is
found to be

$\displaystyle{\cal
B}_{\ell^{+}\ell^{-}}=\tau_{B_{s}}\frac{G^{2}_{F}}{16\pi}m_{B_{s}}f^{2}_{B_{s}}m^{2}_{\ell}\left(1-\frac{4m^{2}_{\ell}}{m^{2}_{B_{s}}}\right)^{1/2}\left(|\Re|^{2}+|\Im|^{2}\right),$
$\displaystyle\Re$ $\displaystyle=$
$\displaystyle-\frac{|\lambda_{t}|\alpha}{\pi\sin^{2}\theta_{W}}Y(x_{t})+|\lambda_{23}|c^{D}_{L}\cos\theta^{\rm
NP}_{s},$ $\displaystyle\Im$ $\displaystyle=$
$\displaystyle|\lambda_{23}|c^{D}_{L}\sin\theta^{\rm NP}_{s}.$ (22)

We can also obtain the effects of the Z-mediated FCNCs on $B_{q}\to
V_{q}\ell^{+}\ell^{-}$ decays by utilizing the replacement:

$\displaystyle C^{\rm eff}_{9}[V,A_{1(2)}]$ $\displaystyle\to$
$\displaystyle\left(C^{\rm
eff}_{9}+\frac{4\lambda_{23}}{\alpha\lambda_{t}}c^{D}_{L}c^{\ell}_{V}\right)[V,A_{1(2)}],$
$\displaystyle C_{10}[V,A_{1(2)}]$ $\displaystyle\to$
$\displaystyle\left(C_{10}-\frac{4\lambda_{23}}{\alpha\lambda_{t}}c^{D}_{L}c^{\ell}_{A}\right)[V,A_{1(2)}].$
(23)

We note that in the following numerical analysis we will concentrate on
$B_{d}\to K^{*}\ell^{+}\ell^{-}$ as they have already been observed. However,
the same discussions can be easily applied to $B_{s}\to\phi\ell^{+}\ell^{-}$.

In the Z-mediated $b\to s$ transition, the magnitude $|\lambda_{23}|$ and the
phase $\theta^{\rm NP}_{s}$ can be determined by the observed $B_{s}$ mixing
and BR of $B_{d}\to K^{*0}\mu^{+}\mu^{-}$. We adopt the average value of
$\Delta m_{s}=18.17\pm 0.86$ ps-1 CDF ; D0  and ${\cal B}(B_{d}\to
K^{*0}\mu^{+}\mu^{-})=(1.22^{+0.38}_{-0.32})\times 10^{-6}$ PDG06  as the
inputs, while the SM results are taken to be $\Delta m^{\rm SM}_{s}=19.3\pm
6.7$ ps-1 LN , ${\cal B}(B_{d}\to K^{*0}\mu^{+}\mu^{-})_{\rm SM}=1.3\times
10^{-6}$ CGA21  and ${\cal B}(B_{s}\to\mu^{+}\mu^{-})_{\rm SM}\approx
0.33\times 10^{-8}$ with $V_{ts}=-0.04$ and $f_{B_{s}}=0.23$ GeV. In order to
reveal the strong correlations among $\Delta m_{s}$, ${\cal B}(B_{d}\to
K^{*0}\ell^{+}\ell^{-})$ and ${\cal B}(B_{s}\to\mu^{+}\mu^{-})$ influenced by
the same parameters, we present ${\cal B}(B_{s}\to\mu^{+}\mu^{-})$ versus
$\Delta m_{s}$ [${\cal B}(B_{d}\to K^{*0}\ell^{+}\ell^{-})$] in Fig.
1(a)[(b)]. From the figures, we see clearly that the Z-mediated effects could
enhance the BR of $B_{s}\to\mu^{+}\mu^{-}$ to be $O(10^{-8})$, which is close
to the current upper limit of $4.7\times 10^{-8}$ CDFBsmumu .

Figure 1: (a)[(b)] ${\cal B}(B_{s}\to\mu^{+}\mu^{-})$ versus $\Delta m_{s}$
[${\cal B}(B_{d}\to K^{*0}\ell^{+}\ell^{-})$].

With the constrained values of $\lambda_{23}$ and $\theta^{\rm NP}_{s}$, in
Fig. 2(a) we show the allowed $\phi_{s}$ given by Eq. (6). It is wroth
mentioning that ${\cal B}(B_{s}\to K^{*0}\ell^{+}\ell^{-})$ excludes
$\phi_{s}$ larger than 0.2 rad. Furthermore, we present the contributions of
the new CP violating source to the T-odd observable of Eq. (12) in Fig. 2(b).
Intriguingly, the new phase can lead to a large statistical significance of
the T-odd observable in $B_{q}\to V_{q}\ell^{+}\ell^{-}$.

In summary, motivated by the large CP phase found by CDF and D$\O$ in the
$B_{s}-\bar{B}_{s}$ mixing, we have investigated the $SU(2)_{L}$ singlet VQM.
This model can not only provide the large phase through the Z-mediated FCNCs
at tree level, but also strongly relate $\Delta m_{s}$, $B_{q}\to
V_{q}\ell^{+}\ell^{-}$ and $B_{s}\to\mu^{+}\mu^{-}$ processes. In particular,
we have shown that the new CP phase can be unambiguously exposed by the large
statistical significances of the T-violating observables in $B_{q}\to
V_{q}\ell^{+}\ell^{-}$ ($V_{q}=K^{*},\,\phi)$. In addition, we have found that
${\cal B}(B_{s}\to\mu^{+}\mu^{-})$ can be enhanced as large as $O(10^{-8})$.
Finally, we remark that the T-violating effects in $B_{q}\to
V_{q}\ell^{+}\ell^{-}$ as well as the result on ${\cal
B}(B_{s}\to\mu^{+}\mu^{-})$ are accessible at future super-B factories, such
as the SuperKEKB SuperKEKB  and LHCb LHCb . For example, 4400 events/year for
$B\to K^{*}\ell^{+}\ell^{-}$ decays will be produced at the LHCb,
corresponding to the accuracy of the T-odd observable being around percent
level.

Figure 2: (a) Correlation of $\phi_{s}=2\beta_{s}-\phi^{\rm NP}_{s}$ and
$\Delta m_{s}$. (b) Statistical significance $\varepsilon_{T}$ of $B_{d}\to
K^{*0}\mu^{+}\mu^{-}$ as a function of $q^{2}$.

###### Acknowledgements.

This work is supported in part by the National Science Council of R.O.C. under
Grant #s: NSC 97-2112-M-006 -001-MY3 and NSC-95-2112-M-007-059-MY3.

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