[1]H. Ma

1] Key Laboratory of Particle and Radiation Imaging (Ministry of Education)
and Department of Engineering Physics, Tsinghua University, Beijing 100084,
China 2] Beijing Normal University, Beijing 100875, China

# Virtual segmentation of a small contact HPGe detector: inference of hit
positions of single-site events via pulse shape analysis

W.H. Dai<EMAIL_ADDRESS>Z. Zeng  L.T. Yang  Q. Yue  J.P. Cheng  [  [

###### Abstract

Exploring hit positions of recorded events can help to understand and suppress
backgrounds in rare event searching experiments. In this study, we virtually
segment a small contact P-type high purity germanium detector (HPGe) into two
layers. Single-site events (SSEs) in each layer are selected by an algorithm
based on two pulse shape parameters: the charge pulse drift time ($T_{Q}$) and
current pulse rise time ($T_{I}$). To determine the shapes and volumes of the
two layers, a Th-228 source is placed at top and side positions to irradiate
the detector. The double escape peak events from 2614.5 keV $\gamma$-ray are
selected as typical SSEs, their numbers in the two layers are used to
calculate the volumes and shapes of those layers. Considering the statistical
and systematic uncertainties, the inner layer volume is evaluated to be
47.2%$\pm$0.26(stat.)%$\pm$0.22(sys.)% of the total sensitive volume. We
extend our analysis for SSEs in 1400-2100 keV, the spectra of inner layer
events acquired from experimental data using the selection algorithm are in
good agreement with those from the simulation. For sources outside the HPGe
detector, the outer layer can act as a shielding for the inner layer.
Selecting the inner layer as the analysis volume can reduce the external
background in the signal region of Ge-76 neutrinoless double beta
(0$\nu\beta\beta$) decay. We use the Th-228 source to evaluate the background
suppression power of the virtual segmentation. After performing the single and
multi-site event discrimination, the event rate in the 0$\nu\beta\beta$ signal
region can be further suppressed by 12% by selecting the inner layer as the
analysis volume. The virtual segmentation could be used to efficiently
suppress surface background like electrons from Ar-42/K-42 decay in
0$\nu\beta\beta$ experiments using germanium detector immersed in liquid
argon.

###### keywords:

small contact HPGe, pulse shape analysis, detector segmentation

##  1 Introduction

Small contact high purity germanium (HPGe) detectors are widely used in
searching for rare events from physics beyond Standard Model, such as the
neutrinoless double beta ($0\nu\beta\beta$) decay and dark matter [4, 5, 6,
7]. Those searches need an extremely low background level in the signal region
to achieve sufficient sensitivity. The discrimination of background and signal
via pulse shape analysis is a powerful background suppression technology and
widely used in HPGe based experiments. [8, 9, 10, 11].

The energy depositions from $0\nu\beta\beta$ decay events and dark matter
interactions are typically within about a millimeter and are regarded as
single-site events (SSEs). Backgrounds can be single-site or multi-site events
(MSEs), depending on their origination. Small contact HPGe detectors, such as
point contact Ge (PCGe) and broad energy Ge (BEGe), have been demonstrated to
have SSE and MSE discrimination capability utilizing pulse shape analysis [3,
9, 10, 11]. After the SSE/MSE discrimination, signals are still mixed with
SSE-like backgrounds, such as single Compton scattering of incoming $\gamma$
or direct energy depositions from beta decay electrons penetrating the surface
layer of the detector. Signals are expected to have a uniform distribution in
the detector, while the backgrounds tend to be close to the detector surface.
Therefore, inference of the SSE position can help to understand and suppress
the SSE-like backgrounds.

Previous studies [12, 13, 14] have demonstrated that the charge collection
time in a small contact HPGe detector depends on the energy deposition
position. Past work [13] has shown that the rise time of the event pulse can
be used to estimate the distance of energy deposition from the contact in a
PCGe detector. Pulse shape simulation in [12] also showed that the signal
shape depends on the interaction position.

This work explores the position discrimination power of a small contact
$p$-type HPGe detector via pulse shape analysis. The detector is virtually
segmented into two layers, and single-site events with hit position in the
inner layer are identified. The shape and volume of the inner layer are
modeled, determined, and validated in a series of Th-228 irradiation
experiments. We also discuss the background suppression potential of this
method towards possible application in future $0\nu\beta\beta$ experiments.

Figure 1:  Schematic diagram of the DAQ system. Figure 2:  Experimental setup
at CJPL. Figure 3:  (a) an example of shaping amplifier pulse, the blue region
indicates the integral of the pulse after subtracting the baseline, and it is
used as the energy estimator; (b) an example of smoothed preamplifier pulse
and the extracted current pulse. Pulse time parameters $T_{Q}$, $T_{I}$, and
parameter ”A” in the A/E discriminator are also illustrated. The current pulse
is rescaled for demonstration.

##  2 Experimental setup

The detector used in this work is a small contact $p$-type HPGe detector
produced by ORTEC. The detector crystal has a height of 42.6 mm and a diameter
of 80.0 mm, and the thin $p$\+ contact is about 3.1 mm in diameter and is
implemented in a 1 mm deep hole on the bottom surface of the crystal. The
$n$\+ surface of the detector crystal, formed by the lithium diffusion,
contains an inactive layer and reduces the sensitive mass of the detector. The
thickness of the inactive layer is evaluated to be 0.87 mm in our previous
work [15]. Subtracting the inactive layer, the total sensitive mass of the
detector is 1.052 kg.

As shown in Fig.1, the data acquisition (DAQ) system is based on commercial
NIM/VME modules and crates. The detector is operated under 4500 V bias voltage
provided by a high voltage module. The output signal from the $p$\+ contact is
fed into an resistance-capacitance (RC) preamplifier. The RC-preamplifier
provides two identical output signals. One is loaded into a shaping amplifier
with a gain factor of 10 and shaping time of 6 $\mu$s. The output of the
shaping amplifier and the other output of the RC-preamplifier are fed into a
14-bit 100 MHz flash analog-to-digital convertor (FADC) for digitalization.
The digitalized waveforms are recorded by the DAQ software on a PC platform.

A detector scanning device is built in China Jinping Underground Laboratory
(CJPL) [16]. As shown in Fig.2, the detector and the liquid nitrogen (LN)
Dewar are installed with the scanning device. A Th-228 source with an activity
of 500 Bq is mounted on the source holder with a step motor controlling the
source position.

##  3 Pulse processing and event discrimination

###  3.1 Digital pulse processing

Typical pulses from the shaping amplifier and preamplifier are illustrated in
Fig.3. After subtracting the baseline, the integration of the shaping
amplifier pulse is used to estimate the event energy (as shown in Fig.3(a)).
Energy calibration is performed by the measured Th-228 spectrum with
characteristic $gamma$-ray peaks from decays of radionuclides in the Th-228
decay chain.

The pulses from the preamplifier are used to estimate the time features of the
event (as shown in Fig.3(b)). The charge drift time ($T_{Q}$) is defined as
the time between the moments when charge pulse reachs 0.2% and 10% of its
maximum amplitude. The current pulse is extracted from the charge pulse by a
moving average differential filter, and the current rise time ($T_{I}$) is the
time between the moments when the current pulse reachs 0.2% and 20% of its
maximum amplitude.

###  3.2 Single and multi-site event discrimination

The single/multi-site event discriminator (A/E) is defined as ratio of the
maximum amplitude of the current pulse (A) and the reconstructed energy (E).
It has been discussed in various literature [9, 11, 17, 18] that SSE tends to
have higher A/E value than MSE in a small contact HPGe detector. Therefore, we
apply a cut on A/E to select the SSEs. The acceptance region of the A/E cut is
determined by the double escape peak (DEP) events from a measured Th-228
spectrum. DEP events are typical SSEs and their A/E distribution is fitted by
a Gaussian function to determine the mean ($\mu_{SSE}$) and standard deviation
($\sigma_{SSE}$) of A/E parameter for SSEs. As shown in Fig.4, the cut
threshold is set to $\mu_{SSE}-5\sigma_{SSE}$, leading to about 80% survival
fraction of DEP events and 9% survival fraction of single escape peak events
(typical MSEs).

Figure 4:  A/E distributions of DEP and SEP events in Th-228 calibration data.
The dashed line is the A/E cut threshold ($\mu_{SSE}-5\sigma_{SSE}$). Figure
5:  Typical Th-228 spectra before and after the A/E cut. The characteristic
peaks from decay daughters of Th-228 (Tl-208, Bi-212) and other radionuclides
(K-40, and Bi-212) are labeled in the spectra. The double-escape peak (DEP) of
Tl-208 2614.5 keV $\gamma$-ray is marked in red.

Fig.5 shows typical Th-228 spectra before and after the A/E cut. Main
characteristic peaks from the Th-228 source and radionuclides in the
surrounding materials are labeled. The full-width-at-half-maximum (FWHM) of
the double escape peak (1592.5 keV) before (after) the A/E cut is $2.19\pm
0.05$ keV ($2.18\pm 0.03$ keV). The FWHM of the 2614.5 keV peak before (after)
the A/E cut is $2.51\pm 0.01$ keV ($2.46\pm 0.02$ keV). A slight improvement
in the energy resolution is observed after the A/E cut.

###  3.3 Linear and nonlinear event discrimination

The $T_{Q}$ and $T_{I}$ distribution of SSEs demonstrates two types of events:
events gathered in a rodlike region in Fig.6(a) are referred to as linear
events, and other events gathered in a cluster are referred to as nonlinear
events. As shown in Fig.6, the charge drift time ($T_{Q}$) and a linearity
index ($L$) are used to discriminate the linear and nonlinear events. The
linearity index is defined as:

$L=T_{I}-\left(k\times T_{Q}+b\right),$ (1)

where fit parameters $k$ and $b$ are calculated via fitting $T_{Q}$ and
$T_{I}$ of typical linear events with the function ($T_{I}=k\times T_{Q}+b$).
First, initial values of fit parameters ($k_{0}$ and $b_{0}$) are calculated
by fitting events with $T_{Q}$ and $T_{I}$ below 500 ns. Then events with
linearity $L=T_{I}-(k_{0}\times T_{Q}+b_{0})$ in [-50, 50] ns are fitted to
give the final value of $k$ and $b$. As shown in Fig.6(b), the distribution of
linearity index $L$ is fitted with two Gaussian functions corresponding to
linear and nonlinear events, respectively. The cut limit is set to
($\mu_{L,linear}-3\sigma_{L,linear}$), where $\mu_{L,linear}$ and
$\sigma_{L,linear}$ are the mean and standard deviation of $L$ distribution
for linear events. The distribution of $T_{Q}$ for nonlinear events selected
by linearity index $L$ is fitted with a Gaussian function, and the cut limit
is set to ($\mu_{T,linear}-3\sigma_{T,linear}$), where $\mu_{T,linear}$ and
$\sigma_{T,linear}$ are the mean and standard deviation of $T_{Q}$
distribution for nonlinear events as shown in Fig.6(c). The red dashed line in
Fig.6(a) shows the discrimination limit set by the linearity index $L$ and the
charge drift time $T_{Q}$.

Figure 6:  Discrimination of linear and nonlinear events. Data in the figure
are from DEP events (1592.5$\pm$5 keV, after A/E cut) in a Th-228 calibration
experiment (source placed at the center of detector top surface). (a)
Distribution of $T_{Q}$ and $T_{I}$. The blue dashed line is the fitted linear
function of $T_{Q}$ and $T_{I}$. Red dashed line is the cut limit for inner
layer events; (b) Histogram of event linearity index $L$, and the Gaussian fit
of linear (blue line) and nonlinear (red line) events; (c) $T_{Q}$ Histogram
for nonlinear events selected by $L$ cut in (b). The black dashed lines in (b)
and (c) are the cut limit for inner layer events.  Figure 7:  Pulse shape
simulation for SSEs in different positions of the detector. (a) Charge drift
time ($T_{Q}$) for SSE as a function of the interaction position; (b) Current
rise time ($T_{I}$) for SSEs as a function of the interaction position; (c)
Distribution of $T_{Q}$ and $T_{I}$ for pulses in (a) and (b), those events
are gathered in two clusters with a linear and nonlinear relationship between
$T_{Q}$ and $T_{I}$. Red crosses mark the positions of four selected SSEs.

##  4 Detector segmentation model

###  4.1 Demonstration of spatial distribution of linear and nonlinear events
via pulse shape simulation

We perform a pulse shape simulation (PSS) for the HPGe detector to demonstrate
the spatial distribution of the linear and nonlinear events. The electric
field and weight potential field in the detector are calculated using the
$mjd\\_fieldgen$ package [19], assuming a linear impurity profile in the
Z-direction with an impurity density of $3.7\times 10^{9}$ $\rm{cm^{3}}$ and
$8.0\times 10^{9}$ $\rm{cm^{3}}$ at the top and bottom surface of the crystal.
SSEs with 1 MeV energy deposition are placed at different positions in the
crystal. The corresponding charge pulses are calculated via the SAGE-PSS
package [20] and added with electric noise extracted from measured pulses.

Fig.7 demonstrates the $T_{Q}$ and $T_{I}$ as a function of the interaction
position. As shown in Fig.7(a) and (b), SSEs close to the $p$\+ contact have
shorter $T_{Q}$ and $T_{I}$. With the distance to contact increasing, the
$T_{Q}$ and $T_{I}$ of induced pulses increase simultaneously, for instance,
the SSE-3 and SSE-4. These events are typical linear events in Fig.7(c).
However, when SSEs near the top and side surfaces of the detector, their
$T_{Q}$ and $T_{I}$ are not sensitive to their positions. Those SSEs, such as
SSE-1 and SSE-2 are typical nonlinear events. It can be explained by the
Schockley-Ramo theory [21]: when SSEs deposit energy near the outer surface of
the detector, the induced charge and current pulses will not exceed the 0.2%
of their maximum amplitude as charge carriers drift in the weak electric and
weight potential field area near the surface. Thereby, the $T_{Q}$ and $T_{I}$
of those SSEs are not sensitive to the energy deposition position.

###  4.2 Parameterized segmentation model

According to the pulse shape simulation, the linearity between $T_{Q}$ and
$T_{I}$ of the SSE can be use to infer its hit position. We segment the
detector into two layers referring to the positions of linear and nonlinear
SSEs. The boundary between the two layers is related to the electric and
weight potential field of the detector. And due to the lack of precise
knowledge of the impurity profile within the Ge crystal, we can’t rely on the
PSS to calculate the shape of the two layers but take it as a reference.
Therefore, we take an empirical approach to build a segmentation model with 14
parameters to described the boundary.

As shown in Fig.8, the boundary of the inner layer is the linear connection of
8 spatial points. It is worth noting that the number of spatial points in the
model is arbitrary, and it will be demonstrated later that the 8 points model
is sufficient for this study. Table.1 lists the bound for each model
parameter. As the model only requires the two layers to be continuous, the
first spatial opint $(r_{1},z_{1})$ could be on the top surface or the central
axis. To determine the value of each model parameter, we design and conduct a
Th-228 scanning experiment.

Table 1: Bounds for segmentation model parameters, $R$ and $H$ are the radius and height of the Ge crystal. Parameter | Parameter bound
---|---
$(r_{1},z_{1})$ |  $r_{1}=0$, $0<z_{1}<H$
or $z_{1}=H$, $0<r_{1}<R$
$(r_{2},z_{2})$ |  $r_{1}\leq r_{2}$, $z_{2}\leq z_{1}$
$(r_{3},z_{3})$ |  $r_{2}\leq r_{3}$, $z_{3}\leq z_{2}$
$(r_{4},z_{4})$ |  $r_{3}\leq r_{4}\leq R$, $z_{4}\leq z_{3}$
$(r_{5},z_{5})$ |  $r_{5}\leq R$, $z_{5}\leq z_{4}$
$(r_{6},z_{6})$ |  $r_{6}\leq r_{5}$, $z_{6}\leq z_{5}$
$(r_{7},z_{7})$ |  $r_{7}\leq r_{6}$, $z_{7}\leq z_{6}$
$(r_{8},z_{8})$ |  $0\leq r_{8}\leq r_{7}$, $z_{8}=0$
|
Figure 8:  Parameterized segmentation model of the detector, where $H$ and $R$
are the height and radius of the crystal. The top spatial point
$(r_{1},z_{1})$ could be on the top surface $(z_{1}=H)$ or on the central axis
$(r_{1}=0)$ of the crystal. The green shadow region is the inner layer in the
segmentation model, and the gray shadow is the inactive layer in the $n$\+
surface. Figure 9:  Schematic of Th-228 source positions in calibration
experiments. The red points indicate the position of the Th-228 source. The
red, blue, and green dashed boxes mark the selected measurements for sub-
datasets in the uncertainty assessment. The Th-228 source is mounted on a
source holder. The carbon fiber vacuum cryostat and the copper crystal holder
are also shown.

##  5 Optimization of segmentation model parameters

###  5.1 Th-228 source scanning experiment

A Th-228 source is used to perform a scan of the detector top and side
surfaces at 19 different positions as shown in Fig.9. A background measurement
is also conducted for the detector.

Events in the DEP region (1592.5$\pm$5 keV) are selected as SSE candidates.
After removing MSEs by the A/E cut, the linear events in the remaining SSEs
are selected using the method in Sec 3.3. The ratio of linear events from the
Th-228 source ($R_{L,DEP}$) is then calculated by:

$R_{L,DEP}=\frac{N_{L,S}-N_{L,B}\cdot t_{S}/t_{B}}{N_{T,S}-N_{T,B}\cdot
t_{S}/t_{B}},$ (2)

where $N_{T,S}$ and $N_{T,B}$ are total numbers of selected single-site DEP
events in Th-228 and background measurements, respectively. $N_{L,S}$ and
$N_{L,B}$ are numbers of selected linear events. $t_{S}$ and $t_{B}$ are the
live time of source and background measurements. The uncertainty of
$R_{L,DEP}$ is calculated by propagating the Poisson uncertainties of event
counts in Th-228 and background measurement through Eq.(2). Fig.10 shows the
linear event ratio of SSEs in the DEP region as a function of Th-228 source
positions. The $R_{L,DEP}$ decreased from 33.3% to 24.0% as the source moved
from the top center to the edge of the detector. About 2.9% changes in
$R_{L,DEP}$ is observed when moving the source along the detector side
surface.

Figure 10:  Ratio of the linear event in selected DEP events as a function of
Th-228 source positions. Error bars indicate the 1$\sigma$ uncertainty.

###  5.2 Spatial distribution of DEP events

Figure 11:  $\delta_{D}$ histogram for simulated DEP events with the Th-228
source is placed at the center of the top detector surface.  Figure 12:
Spatial distribution of simulated SSEs in DEP region. (a) Th-228 source in the
center of the top surface; (b) Th-228 source on the side of the detector. The
labels of the color bar represent the distribution density (arbitrary unit).

As the linear events are located in the inner layer of the segmentation model,
the linear event ratio $R_{L,DEP}$ can be modeled by:

$R_{L,DEP}=\iint M(r,z\mid\theta)F_{DEP}(r,z)\cdot\rm{d}\it{r}\rm{d}\it{z},$
(3)

$M(r,z\mid\theta)=\left\\{\begin{matrix}1&(r,z)\in\rm inner\,layer\it\\\
0&(r,z)\in\rm outer\,layer\it\end{matrix}\right.,$ (4)

where $M(r,z\\!\mid\\!\theta)$ is the select function for the inner event
using the segmentation model, $\theta$ represents the model parameters in
Table.1, $F_{DEP}(r,z)$ is the spatial distribution of SSEs in the DEP region.
The energy deposition of $\gamma$ emitted by the Th-228 source is simulated by
Geant4 [22]. The energy depositions occured in the inactive layer of the
detector are not recorded in the simulation. The single-site events are
selected by the $\delta_{D}$ parameter. $\delta_{D}$ is the average distance
between the energy deposition points to the charge center of the event:

$\delta_{D}=\frac{1}{n}\sum_{i=0}^{n}\sqrt{(x_{i}-\hat{x})^{2}+(y_{i}-\hat{y})^{2}+(z_{i}-\hat{z})^{2}},$
(5)
$\hat{x}=\sum_{i=0}^{n}x_{i}\frac{E_{i}}{E_{tot}},\hat{y}=\sum_{i=0}^{n}y_{i}\frac{E_{i}}{E_{tot}},\hat{z}=\sum_{i=0}^{n}z_{i}\frac{E_{i}}{E_{tot}},$
(6)

where $n$ is the number of steps in one event, $(x_{i},y_{i},z_{i})$ and
$E_{i}$ are the hit position and energy deposition of the i-th step.
$(\hat{x},\hat{y},\hat{z})$ and $E_{tot}$ are the charge center and total
energy deposition of the event. Events with $\delta_{D}<\delta_{D,SSE}$ are
selected as SSEs, where $\delta_{D,SSE}$ is determined by matching the
survival fraction of DEP events in simulation with that of the A/E cut in the
experiment. Fig.11 demonstrates a typical $\delta_{D}$ distribution of
simulated DEP events when the Th-228 source is at the top center of the
detector. The charge center of the selected SSE is then used to simulate the
spatial distribution $F_{DEP}(r,z)$. Fig.12 shows the simulated $F_{DEP}(r,z)$
for the Th-228 source at two different positions.

###  5.3 Optimization of model parameters

As shown in Fig.12, the position of the Th-228 source affects the spatial
distribution of DEP events and therefore leads to different observed linear
event ratios in Fig.10. Thus, we use a minimum-$\chi^{2}$ method to calculate
the model parameters ($\theta$), in which $\chi^{2}$ is defined as:

$\chi^{2}=\sum_{k=1}^{19}\frac{\left(R_{k,exp}-\iint
M(r,z\mid\theta)F_{DEP}(r,z)\rm d\it r\rm d\it z\right)^{2}}{\sigma_{k}^{2}},$
(7)

where $R_{k,exp}$ is the measured linear event ratio for Th-228 source at
position $k$ ($k$=1,2,$\dots$19), $\sigma_{k}$ is the corresponding
uncertainty of $R_{k,exp}$. $F_{DEP,k}(r,z)$ is the simulated spatial
distribution of single-site DEP events for the Th-228 source at position $k$.
The minimalization of $\chi^{2}$ is implemented by the genetic algorithm using
a python-based calculation package Geatpy [23]. Fig.13 shows the optimized
results. The volume of the inner layer is 47.2% of the total sensitive volume
of the detector. The linear event ratios calculated by Eq.3 using the
optimized model parameters are shown in Fig.14. The fit result agrees well
with the measurements, the $p-value$ of the $\chi^{2}$ fit is 0.701.

##  6 Uncertainty assessment and model validation

Uncertainties of the shape and volume of the inner layer in the optimized
model mainly consist of three parts:

  * (1)

Uncertainty of the linear event ratio ($R_{L,DEP}$) propagated by the
$\chi^{2}$-method is evaluated using a toy Monte Carlo method. 3000 Monte
Carlo datasets are generated assuming a Gaussian distribution for the
$R_{L,DEP}$ with the mean and standard deviation equal to the measured value
and uncertainty, respectively. Model parameters are recalculated for each
dataset following the same analysis in Sec 5.3. The distribution of inner
layer shapes and volumes for the 3000 samples are illustrated in Fig.15. The
distribution of inner layer volume is fitted with a Gaussian function, and the
standard deviation, $\pm$0.26%, is adopted as the statistical uncertainty.

  * (2)

Systematic uncertainty due to the choice of dataset: we divide the measured
data in Fig.10 into three sub-datasets. Sub-dataset I and II each consists of
ten measured data (marked by red dashed boxes for sub-dataset I, and blue
dashed boxes for sub-dataset II in Fig.9). sub-dataset III consists of six
measured data (green dashed boxes in Fig.9). The fitting of model parameters
are performed in each sub-dataset, and the largest difference in inner layer
volume between all sub-datasets and the full dataset (Fig.16 (a)) is
$\pm$0.22% as a systematic uncertainty.

  * (3)

Systematic uncertainty due to the construction of the segmentation model: we
reconstruct the segmentation model using 6 spatial points (10 free parameters)
and 10 spatial points (18 free parameters) and calculate the model parameters
using the full dataset. Fig.16(b) shows the optimized results for the
reconstructed models. The overall shape and volume of the inner layer are
similar in the three models, and the largest difference in inner layer volume
is 0.02%, which is about 10 times smaller than the other two uncertainties and
thereby negligible. This indicates the 8-point segmentation model is
sufficient in this study.

Including the statistical and systematic uncertainties discussed above, the
volume of the inner layer is given as 47.2%$\pm$0.26%(stat.)$\pm$0.22%(sys.)

Figure 13:  Optimized result of the segmentation model, the volume of the
inner layer is 47.2% of the total sensitive volume.  Figure 14:  Linear event
ratio ($R_{L,DEP}$) calculated by Eq.(3) using the optimized model parameters
(red squares). The black dots and squares are the measured results from
Fig.10.  Figure 15:  (a) Inner layer shapes of the 3000 Monte Carlo datasets.
The green, yellow, and blue shadow bands are corresponding to 68%, 95%, and
99.7% quantiles, respectively. The gray shadow is the inactive layer on the
$n$\+ surface. (b) Distribution of inner layer volumes. The red line is the
fit of inner layer volumes using a Gaussian function, $\mu$ and $\sigma$ are
the mean and standard deviation, respectively.  Figure 16:  (a) Optimized
results using different datasets, full dataset (black line) consists of all
measured data, sub-dataset I, II, III are selected from the full dataset. (b)
Optimized results for three different models, the chi-square ($\chi^{2}$) and
$p$-$value$ are given to demonstrate the fit goodness of each model. The gray
shadow regions in both figures are the inactive layer on the detector $n$\+
surface.  Figure 17:  Comparison of simulation and experiment for Th-228
source placed on the side of the detector. (a) The linear event ratio as a
function of energy, The uncertainty band for simulation (the green shadow)
consists of uncertainty from the inner layer shape (68% quantile region in
Fig.15(a)) and statistical uncertainty in simulation. The normalized residuals
are shown in the bottom figure, (b) Measured and simulated spectra in
1400-2100 keV region.

The measured Th-228 spectra are compared to the simulated spectra to validate
the segmentation model. The energy depositions of the $\gamma$-rays emitted
from the Th-228 source are simulated via Geant4 and added with the energy
resolution of the detector. The SSEs are selected using the $\delta_{D}$
parameter defined in Sec 5.2, and the inner layer events are selected by their
charge center positions defined by Eq.(6). The measured background spectrum is
scaled by the live time of measurement and added to the simulated spectra.

Fig.17 compares the spectra and ratio of inner layer SSE events between
simulation and experimental results for one of Th-228 source measurements. The
gray band in Fig.17(a) is the statistic uncertainty of experiment data, the
green band is the combination of the statistic and systematic uncertainties in
the simulation. In this case, the systematic uncertainty is taken as the
discrepancy between linear event ratios corresponding to the innermost and
outmost shape of the 68% quantile of the inner layer (the green region in
fig.15(a)). Fig.17(b) is the comparison of measured and simulated spectra, it
demonstrates that the $\delta_{D}$ cut in the simulation is a good
approximation for the A/E cut, and the spectra of inner layer events also show
a good agreement between the simulation and measurement in the 1400-2100 keV
energy region.

##  7 Background suppression performance of virtual segmentation

In the search for Ge-76 $0\nu\beta\beta$ decay using HPGe detectors,
backgrounds, mostly $\gamma$-rays and electrons from outside the detector,
have to penetrate the outer layer of the detector to deposit their energy in
the inner layer. Thus, the outer layer in the virtual segmentation could act
as a shielding for the inner layer, and a lower background level of the inner
layer may improve the detection sensitivity.

We use the Th-228 scanning data to evaluate the background suppression power
of the virtual segmentation. The count rates in spectra are normalized to unit
sensitive mass to include the mass loss due to the analysis volume selection.
The masses of the detector are 1.052 kg and 0.496 kg for the total sensitive
volume and the inner layer, respectively. Fig.18 demonstrates spectra before
and after A/E cut and inner layer event selection when the Th-228 source is
placed on the side of the detector. First the whole detector is selected as
the analysis volume and the A/E cut is applied to removes multi-site events
(gray and blue regions in Fig.18). Then the inner layer of the virtual
segmentations is selected as the analysis volume, a further reduction on the
event rate is shown in Fig.18 (red region). It is expected that the SSEs
mostly come from the single Compton scattering of high energy $\gamma$-rays
emitted from the source and are clustered near the surface of the detector.
Thereby the inner layer has a lower background level in the detector.

Fig.19 shows the event rate in the $0\nu\beta\beta$ signal region (1900-2100
keV) as a function of the Th-228 source positions. The highest background
suppression power is achieved when the Th-228 source is at the side of the
detector. In this case, the A/E cut reduces the event rate by 62%, and the
virtual segmentation yeilds a further reduction of 12% on the basis of the A/E
cut.

Figure 18:  Measured spectra for the Th-228 source on the side surface of the
detector. cpkkd represents counts per kg per keV per day, $Q_{\beta\beta}$ is
the energy of Ge-76 $0\nu\beta\beta$ signal.  Figure 19:  Event rate in
$0\nu\beta\beta$ signal region (1900-2100 keV) as a function of Th-228 source
position. The left and right figures show the event rate for the Th-228 source
placed on the top and side surface of the detector, respectively.

In future $0\nu\beta\beta$ experiments using small contact HPGe detectors,
this method might be used to further suppress background in the signal region.
Especially for experiments using a liquid argon (LAr) veto system where the
HPGe detector is directly immersed in LAr, such as GERDA [1], the LEGEND [24],
and CDEX-300$\nu$ experiments [3]. The background from K-42 (daughter of
cosmogenic Ar-42 in LAr) beta-decay is mainly located in the surface of the
detector, therefore might be suppressed if the inner layer is selected as the
analysis volume. It should be noted that the balance between a lower
background and the loss in detector sensitive mass should be considered in the
the searching for the $0\nu\beta\beta$ signal.

Furthermore, the discrepancy between the inner and outer layer SSE spectrum
could be used to infer the location of the background source. A more precise
background model could be built by fitting the spectra of events in the inner
and the outer layer simultaneously.

##  8 Summary

In this study, we develop a virtual segmentation model for a small contact
HPGe detector and demonstrate its background suppression capability in the
Ge-76 $0\nu\beta\beta$ signal region. The HPGe detector is virtually segmented
into two layers, and a selection algorithm based on charge pulse drift time
($T_{Q}$) and current rise time ($T_{I}$) is established to identify the
position of the single-site event. The shape and volume of the inner layer in
the segmentation model are determined using the DEP events in a series of
Th-228 source calibration experiments. The volume of the inner layer is
evaluated to be 47.2%$\pm$0.26(stat.)%$\pm$0.22(sys.)% of the total sensitive
volume of the detector.

The background suppression power of the virtual segmentation in Ge-76
$0\nu\beta\beta$ signal region is evaluated by the Th-228 scanning data.
Choosing the inner layer as the analysis volume, a further 12% reduction of
background is achieved when the Th-228 source is on the side of the detector.
Other backgrounds in the $0\nu\beta\beta$ signal region, especially those
clustered on the surface of the detector, such as Ar-42 in future
$0\nu\beta\beta$ experiments, could also be reduced by the virtual
segmentation.

The principle of the virtual segmentation can be extended to other small
contact HPGe detectors, for instance, point-contact Ge (PCGe) and broad energy
Ge (BEGe) detectors.

## Acknowledgments

This work was supported by the National Key Research and Development Program
of China (Grant No. 2022YFA1604701) and the National Natural Science
Foundation of China (Grants No. 12175112). We would like to thank CJPL and its
staff for supporting this work. CJPL is jointly operated by Tsinghua
University and Yalong River Hydropower Development Company.

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# Mapping a new cluster of charmoniumlike structures at $e^{+}e^{-}$
collisions

Jun-Zhang Wang1,2<EMAIL_ADDRESS>Dian-Yong Chen3<EMAIL_ADDRESS>Xiang Liu1,2,4111Corresponding author<EMAIL_ADDRESS>Takayuki Matsuki5
<EMAIL_ADDRESS>1School of Physical Science and Technology, Lanzhou
University, Lanzhou 730000, China
2Research Center for Hadron and CSR Physics, Lanzhou University $\&$ Institute
of Modern Physics of CAS, Lanzhou 730000, China
3School of Physics, Southeast University, Nanjing 210094, China
4Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, and Frontiers Science Center for Rare Isotopes, Lanzhou
University, Lanzhou 730000, China
5Tokyo Kasei University, 1-18-1 Kaga, Itabashi, Tokyo 173-8602, Japan

###### Abstract

In this work, we find a Critical Energy induced Enhancement (CEE) mechanism
for the general three-body open-charm process at the $e^{+}e^{-}$ collisions,
which utilizes the peculiar kinematic behavior of the $e^{+}e^{-}$
annihilation process involving three-body final states. We present a general
analysis of a three-body process $e^{+}e^{-}\to BC\to B(C\to DE)$. When the
center-of-mass (CM) energy of the $e^{+}e^{-}$ collision satisfies a critical
relation $\sqrt{s}=m_{B}+m_{C}$, there clearly exists the reflection peak of
an intermediate $C$ state near the threshold of the invariant mass
distribution of $m_{BE}$ or $m_{BD}$, whose formation is very sensitive to the
CM energy. The reflection enhancement phenomenon induced at the critical
energy means that a new cluster of charmoniumlike structures can be
experimentally mapped. Taking an example of $e^{+}e^{-}\to
D_{s}^{*-}D_{s2}^{*}(2573)^{+}\to D_{s}^{*-}(D^{0}K^{+})$, we further
illustrate this novel phenomenon when $\sqrt{s}=4.680$ GeV. What is more
important is that a series of optimal CM energy points to search for new
charmoniumlike structures in three-body open-charm processes from $e^{+}e^{-}$
annihilation are suggested, which can be accessible at BESIII and further
BelleII as a new research topic.

Introduction.—Since 2003, dozens of charmoniumlike $XYZ$ states have been
observed with the accumulation of experimental data with high precision, which
form a new family in hadron spectroscopy. Study of charmoniumlike states is
constantly improving our knowledge of exotic hadronic matter and is deepening
our understanding of non-perturbative behavior of strong interaction (see
review articles Chen:2016qju ; Liu:2019zoy ; Guo:2017jvc ; Olsen:2017bmm ;
Brambilla:2019esw  for getting recent progress). Until now, the exploration of
charmoniumlike $XYZ$ state has become a frontier in science of matter.

At present, these reported charmoniumlike states can be grouped into five
clusters which involve several typical production processes like $B$ decay,
$e^{+}e^{-}$ collision, and $\gamma\gamma$ fusion Chen:2016qju ; Liu:2019zoy .
Among them, $e^{+}e^{-}$ collision is an ideal platform to capture
charmoniumlike states. For example, abundant $Y$ states Aubert:2005rm ;
Yuan:2007sj ; Wang:2007ea ; Aubert:2006ge ; Pakhlova:2008vn ; BESIII:2016adj ;
Ablikim:2016qzw  which result in ”Y problem” Ablikim:2019hff  can be directly
produced by $e^{+}e^{-}$ collisions, while two $X$ states Abe:2007jna ;
Abe:2007sya  are from double charmonium production in $e^{+}e^{-}$ collision.
Here, the Belle, BaBar and, running BESIII experiments based on $e^{+}e^{-}$
collision are the main force of finding out charmoniumlike states. Obviously,
$e^{+}e^{-}$ collision as an effective way to produce charmoniumlike states
has played a crucial role when constructing charmoniumlike state families in
the past 18 years. Standing at a new starting point of new ten years, we
believe that $e^{+}e^{-}$ collision will still be a rich mine of producing
charmoniumlike states.

In this letter, we find a Critical Energy induced Enhancement (CEE) mechanism
for the three-body open-charm processes from $e^{+}e^{-}$ annihilation, by
which we map a new cluster of charmoniumlike structures that can be largely
produced on the invariant mass distribution of the corresponding open-charm
channel by choosing a specific critical energy of positron-electron system.
Different from possible hidden-charm exotic hadronic resonances, these new
charmoniumlike structures with a non-resonant nature can be more easily
identified due to the existence of the unique CEE mechanism. In the
forthcoming years, BESIII and Belle II will have great ability and opportunity
to hunt for the predicted new cluster of charmoniumlike structures, for which
the concrete suggestion of searching for them will be given.

CEE mechanism.—In 2014, the BESIII collaboration reported two charged
charmoniumlike structures $Z_{c}(3885)^{\pm}$ and $Z_{c}(4025)^{\pm}$ in the
three-body open-charm processes $e^{+}e^{-}\to\pi^{\pm}(D\bar{D}^{*})^{\mp}$
Ablikim:2013xfr  and $e^{+}e^{-}\to\pi^{\pm}(D^{*}\bar{D}^{*})^{\mp}$
Ablikim:2013emm , respectively, whose peak positions are very close to the
production thresholds of respective open-charm channels $D^{(*)}\bar{D}^{*}$.
The two charged $Z_{c}$ structures are generally treated to be good candidates
of exotic hidden-charm tetraquark states due to the neutral property of a
charmonium meson Chen:2016qju ; Liu:2019zoy . In Ref. Wang:2020axi , the
Lanzhou group proposed a totally different opinion to understand the
structures of $Z_{c}(3885)^{\pm}$ and $Z_{c}(4025)^{\pm}$, both of which can
be uniformly decoded as the reflection structures of the $P$-wave charmed
meson $D_{1}(2420)$ involved in $e^{+}e^{-}\to D\bar{D}_{1}(2420)\to
D\bar{D}^{*}\pi$ and $e^{+}e^{-}\to D^{*}\bar{D}_{1}(2420)\to
D^{*}\bar{D}^{*}\pi$, respectively. Later, this unique reflection explanation
to $Z_{c}(3885)^{\pm}$ was further supported by the BESIII’s measurement on
the degree of asymmetry in the angular distribution of final states
Wang:2020axi . Building on previous studies, in this work, we find a CEE
mechanism existing in the general open-charm process with three-body final
states, which can widely induce this specific reflection peak phenomenon near
a threshold. Based on this mechanism, in addition to $Z_{c}(3885)^{\pm}$ and
$Z_{c}(4025)^{\pm}$, we can expect that more charmoniumlike $Z_{c}$ structures
can be produced from the $e^{+}e^{-}$ annihilation process by setting their
individual critical energy. And then a new cluster of charmoniumlike
structures can be established. In the following, we will illustrate it in
detail.

Focusing on the three-body open-charm channels from the $e^{+}e^{-}$
annihilation, $e^{+}e^{-}\to D^{(*)}\bar{D}_{J}^{**}\to
D^{(*)}(\bar{D}^{(*)}\pi)$, $e^{+}e^{-}\to D^{(*)}\bar{D}_{J}^{**}\to
D^{(*)}(\bar{D}^{(*)}_{s}K)$, $e^{+}e^{-}\to D_{s}^{(*)}\bar{D}_{sJ}^{**}\to
D_{s}^{(*)}(\bar{D}_{s}^{(*)}\eta)$, and $e^{+}e^{-}\to
D_{s}^{(*)}\bar{D}_{sJ}^{**}\to D_{s}^{(*)}(\bar{D}^{(*)}K)$, we start with a
general three-body process from the electron and positron annihilation,
$A(p_{1})\to BC\to B(p_{4})(C\to D(p_{3})E(p_{2}))$, whose schematic diagram
can be seen in Fig. 1. Here, $D_{(s)J}^{**}$ denotes the higher states in the
charmed/charmed-strange meson family. When choosing some special center-of-
mass (CM) energy of the $e^{+}e^{-}$ collision ($\sqrt{s}=m_{B}+m_{C}$), the
reflection signal of an intermediate $C$ clearly exists in the invariant mass
distribution of $m_{BE}$ or $m_{BD}$. Especially, this reflection is sensitive
to the width of $C$ if $C$ is a narrow state.

Figure 1:  The schematic diagram for a general three-body process from the
electron and positron annihilation. Here, $A$ may denote a vector charmonium
or charmoniumlike state with $J^{PC}=1^{--}$ coupling to a virtual photon.
$B$, $D$, and $E$ denote the particles involved in the final states, while $C$
is an intermediate state. $p_{1}$, $p_{2}$, $p_{3}$, and $p_{4}$ are defined
to be the four-momenta of initial state and three final states, respectively,
while $p_{C}=p_{1}-p_{4}$ corresponds to the propagation momentum of an
intermediate $C$.

As we all know, in the three-body decay process shown in Fig. 1, the pole
information of an intermediate $C$ can be reflected on the indirect invariant
mass spectrum of $m_{BE}$ or $m_{BD}$. However, a kinematical behavior of the
pole in indirect invariant mass spectrum is very different from that of direct
invariant mass spectrum. Here, we take $m_{BD}$ as an example. The pole
information of an intermediate resonance is fully determined by the propagator
$1/((p_{1}-p_{4})^{2}-m_{C}^{2}+im_{C}\Gamma_{C})$ and is not dependent on the
Lorentz structure of an interaction vertex. By solving the equation
$(p_{1}-p_{4})^{2}-m_{C}^{2}=0$, we can find a critical value
$\sqrt{s_{\textrm{critical}}}=m_{B}+m_{C}$, which is a critical energy point
of pole distributions of an intermediate state. In Fig. 2 , we show the
numerical contour plot of $(p_{1}-p_{4})^{2}-m_{C}^{2}$ dependent on the
invariant mass $m_{BD}$ and scattering angle $\theta_{2}$, where $\theta_{2}$
is the angle of three momentum of a particle $E$ to the $z$ axis in spherical
coordinates. We notice that there appears a continuous pole distribution in
the three-body phase space when the CM energy is larger than the critical
energy. When decreasing the CM energy close to the critical energy, the pole
curve gradually tends to a single pole point, which exactly corresponds to the
critical energy $\sqrt{s_{\textrm{critical}}}$. Of course, it is not
surprising that there is no pole structure in the CM energy region smaller
than $\sqrt{s_{\textrm{critical}}}$, where the intermediate state is always a
virtual state.

Therefore, this peculiar kinematic property near the critical energy point can
cause some fantastic phenomenon, i.e., the line shape of a reflection peak for
an intermediate $C$ is sensitive to the initial CM energy. In other word, a
reflection enhancement structure near threshold that corresponds to a single
pole can be largely induced at the CM energy point of
$\sqrt{s}=\sqrt{s_{\textrm{critical}}}$, which is main meaning of the CEE
mechanism. The position of this single pole can be determined analytically
with the following expression

$\displaystyle
m_{BD}^{pole}=\sqrt{\frac{(m_{C}^{2}-m_{E}^{2}+m_{D}^{2})m_{B}+(m_{D}^{2}+m_{B}^{2})m_{C}}{m_{C}}},$
(1)

which is useful for the concrete experimental analysis. By inputting the mass
of intermediate $C$, this pole position of a reflection peak structure can be
obtained.

Figure 2: The contour plot of $(p_{1}-p_{4})^{2}-m_{C}^{2}$ in the invariant
mass $m_{BD}$ and scattering angle $\theta_{2}$. Here, we assume the masses of
all final states to be 1 MeV and $m_{C}=3$ MeV. Diagrams ($a$)-($d$)
correspond to $\sqrt{s}-\sqrt{s_{\textrm{critical}}}=200,100,0,-100$ keV,
respectively.

Typical example.–To intuitively illustrate this novel phenomenon, we choose
$e^{+}e^{-}\to D_{s}^{*-}D_{s2}^{*}(2573)^{+}\to D_{s}^{*-}(D^{0}K^{+})$ as an
example, where the intermediate $D_{s2}^{*}(2573)^{+}$ has $J^{P}=2^{+}$ spin-
parity quantum number and is a narrow state Tanabashi:2018oca . Since the sum
of $m(D_{s2}^{*}(2573))=2.568$ GeV Tanabashi:2018oca ; Aaij:2014xza  and
$m(D_{s}^{*})=2.112$ GeV Tanabashi:2018oca  is precisely equal to 4.680 GeV,
we may set the CM energy of this discussed process to be $\sqrt{s}=4.68$ GeV.
Considering the existent charmoniumlike state $Y(4660)$ Wang:2007ea ;
Wang:2020prx , we may expect that $D_{s}^{*-}D_{s2}^{*}(2573)^{+}$ may couple
with this $Y(4660)$ via $S$-wave interaction for the discussed $e^{+}e^{-}\to
D_{s}^{*-}D_{s2}^{*}(2573)^{+}\to D_{s}^{*-}(D^{0}K^{+})$ process.
Additionally, $D_{s2}^{*}(2573)^{+}$ dominantly decays into $D^{0}K^{+}$
through the $D$-wave interaction Kubota:1994gn ; Song:2015nia ;
Godfrey:2015dva . In fact, this is the main reason why we would like to
firstly select this process as an example. Corresponding to Fig. 1, we
identify $A\equiv Y(4660)$, $B\equiv D_{s}^{*-}$, $C\equiv
D_{s2}^{*}(2573)^{+}$, $D\equiv D^{0}$, and $E\equiv K^{+}$.

In order to calculate this process, we adopt the effective Lagrangian
approach, where the related Lagrangians include Bauer:1975bv ; Bauer:1975bw ;
Liu:2020ruo

$\displaystyle\mathcal{L}_{\gamma Y}$ $\displaystyle=$
$\displaystyle\frac{-em_{Y}^{2}}{f_{Y}}Y_{\mu}A^{\mu},$
$\displaystyle\mathcal{L}_{D_{s2}{D}_{s}^{*}Y}$ $\displaystyle=$
$\displaystyle
g_{D_{s2}D_{s}^{*}Y}Y_{\mu}(D_{s2}^{\mu\nu{\dagger}}D^{*}_{s\nu}+D^{*{\dagger}}_{s\nu}D_{s2}^{\mu\nu}),$
$\displaystyle\mathcal{L}_{KD{D}_{s2}}$ $\displaystyle=$ $\displaystyle
g_{KDD_{s2}}(D_{s2}^{\mu\nu{\dagger}}(\partial_{\mu}\partial_{\nu}K)D+D^{{\dagger}}(\partial_{\mu}\partial_{\nu}K)D_{s2}^{\mu\nu}).$
(2)

Here, $A^{\mu}$, $D_{s2}^{\mu\nu}$, $K$ are photon, charmed-strange meson
$D_{s2}^{*}(2573)$, and kaon fields, respectively, and $g_{D_{s2}D_{s}^{*}Y}$
and $g_{KDD_{s2}}$ are the coupling constants. Thus, the general amplitude of
the $e^{+}(k_{1})e^{-}(k_{2})\to Y(p_{1})\to
D_{s}^{*-}(p_{4})D_{s2}^{*}(2573)^{+}\to D_{s}^{*-}(D^{0}(p_{3})K^{+}(p_{2}))$
channel can be written as

$\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle\bar{v}(k_{2})e\gamma_{\mu}u(k_{1})\frac{-g^{\mu\nu}em_{Y}^{2}}{sf_{Y}}\frac{-g_{\nu\rho}+p_{1\nu}p_{1\rho}/m_{Y}^{2}}{s-m_{Y}^{2}+im_{Y}\Gamma_{Y}}$
(3) $\displaystyle\times
g_{D_{s2}D_{s}^{*}Y}g_{KDD_{s2}}\epsilon_{D_{s}^{*}}^{\lambda*}\frac{-G_{\rho\lambda\alpha\beta}p_{2}^{\alpha}p_{2}^{\beta}}{(p_{1}-p_{4})^{2}-m_{D_{s2}}^{2}+im_{D_{s2}}\Gamma_{D_{s2}}},$

where
$G_{\rho\lambda\alpha\beta}=\frac{1}{2}(\tilde{g}_{\rho\alpha}\tilde{g}_{\lambda\beta}+\tilde{g}_{\rho\beta}\tilde{g}_{\lambda\alpha})-\frac{1}{3}\tilde{g}_{\rho\lambda}\tilde{g}_{\alpha\beta}$
with
$\tilde{g}_{\rho\alpha}=-g_{\rho\alpha}+(p_{1\rho}-p_{4\rho})(p_{1\alpha}-p_{4\alpha})/m_{D_{s2}}^{2}$
is the spin projection operator of tensor particle. With the above
preparation, the differential cross section vs. invariant mass
$m_{D_{s}^{*}D}$ or the recoil spectrum of $K^{+}$ can be calculated by
integrating over the phase space of three-body final states, i.e.,

$\displaystyle\frac{d\sigma}{dm_{D_{s}^{*}D}}=\frac{1}{(2\pi)^{5}}\frac{1}{32\sqrt{s}\sqrt{(k_{1}\cdot
k_{2})^{2}}}\overline{\left|\mathcal{M}\right|}^{2}\left|\bf{p_{2}}\right|\left|\bf{p_{3}}^{*}\right|d\Omega_{2}d\Omega_{3}^{*},$
(4)

where the overline above the amplitude stands for the average over the spin of
electron and positron and the sum over the spin of final states, and the
$\bf{p_{3}}^{*}$ and $\Omega_{3}^{*}$ are the three-momentum and solid angle
of $D^{0}$ meson in the center of mass frame of the $D_{s}^{*}D$ system.

Figure 3: The line shapes of a differential cross section for $e^{+}e^{-}\to
D_{s}^{*-}D_{s2}^{*}(2573)^{+}\to D_{s}^{*-}(D^{0}K^{+})$ vs. the invariant
mass $m_{D^{0}D_{s}^{*-}}$ with four CM energies $\sqrt{s}=4.630,4.680,4.690$,
and $4.730$ GeV. Here, the resonance parameter of $D_{s2}^{*}(2573)$ is fixed
as $m=2.568$ GeV and $\Gamma=16.9$ MeV Tanabashi:2018oca ; Aaij:2014xza . The
maximum of line shape is normalized to 1.  Figure 4: The relative differential
cross sections of reflection peaks on the invariant mass spectrum of
$D^{0}D_{s}^{*-}$ in $e^{+}e^{-}\to D_{s}^{*-}D_{s2}^{*}(2573)^{+}\to
D_{s}^{*-}(D^{0}K^{+})$ with five different CM energies. The maximum of the
reflection peak at $\sqrt{s}=4.680$ GeV is normalized to 1.

In Fig. 3, we test the sensitivity of the reflection line shape of
$D_{s2}^{*}(2573)$ near the critical CM energy by plotting a differential
cross section vs. the $m_{D^{0}D_{s}^{*-}}$ invariant mass with four CM
energies $\sqrt{s}=4.630,4.680,4.690$, and $4.730$ GeV. Here, $\sqrt{s}=4.680$
GeV is just the critical energy value for the discussed $e^{+}e^{-}\to
D_{s}^{*-}D_{s2}^{*}(2573)^{+}\to D_{s}^{*-}(D^{0}K^{+})$. Due to the peculiar
kinematic behavior, it can be found that the line shape of a reflection signal
is sensitive to the CM energy value around $4.680$ GeV. When $\sqrt{s}>4.680$
GeV, the line shape of the reflection does not look like a peak structure and
becomes more flat with increasing $\sqrt{s}$. When CM energy is exactly at
4680 MeV, the line shape of reflection represents a critical peak structure.
For another region of $\sqrt{s}<4.680$ GeV, although all reflection line
shapes look like a peak structure, their sensitivity is reflected on the
relative magnitude of events because of the off-shell suppression effect of
the pole. In Fig. 4, the relative differential cross sections based on five CM
energy points of $\sqrt{s}=4.680,4.675,4.670,4.650$, and 4.630 GeV are given.
It can be seen that the relative magnitude of differential cross sections is
vey sensitive to the CM energy. For example, there is about 40% suppression
for the tiny energy change of 5 MeV away from the critical energy.
Additionally, there is even an order of magnitude difference between
$\sqrt{s}=4.680$ and 4.650 GeV, which means that these reflection peaks that
are far from the critical energy are most likely invisible in experiments.
Hence, it can be seen that the reflective peak near a threshold is the most
obvious at critical position of $\sqrt{s}=\sqrt{s_{\textrm{critical}}}$, which
should be the best CM energy to induce the formation of the charmoniumlike
structures as a reflection nature.

Besides this, the reflection line shape is also sensitive to the width of an
intermediate state $D_{s2}^{*}(2573)$ as shown in Fig. 5, where all line
shapes have the same pole position at $m_{D^{0}D_{s}^{*-}}^{pole}=4.003$ GeV
when the CM energy is $\sqrt{s}=4.680$ GeV. In fact, until now the resonance
parameters of $D_{s2}^{*}(2573)$, especially its width, have not been well
established by directly analyzing the corresponding Breit-Wigner distribution
Tanabashi:2018oca . The measured width of $D_{s2}^{*}(2573)$ is from 2 to 33
MeV if considering experimental uncertainties Tanabashi:2018oca . Here, we
select four typical values of width as 6.0, 10.4, 16.9, and 27.1 MeV as an
example, which are from Ref. Song:2015nia , ARGUS Albrecht:1995qx , LHCb
Aaij:2014xza , and BABAR Aubert:2006mh , respectively. It can be found that
the larger width of intermediate $D_{s2}^{*}(2573)$ is also related to a wider
reflection peak structure, and its line shape is very sensitive to an
intermediate resonance width, which can be seen within the width change of
only several MeV.

Therefore, this critical energy induced enhancement mechanism existing in a
three-body open-charm process means that experimentalists can easily implement
to search for a critical reflection peak by scanning a series of the CM energy
points, which is expected to be somewhere in the middle of a series of scanned
CM energy points. At the same time, because of the existence of the CEE
mechanism, these induced reflection structures as a new cluster of
charmoniumlike structures can be more easily distinguished from the
charmoniumlike states with a resonant nature.

Figure 5: The dependence of the line shape of a differential cross section for $e^{+}e^{-}\to D_{s}^{*-}D_{s2}^{*}(2573)^{+}\to D_{s}^{*-}(D^{0}K^{+})$ vs. the invariant mass $m_{D^{0}D_{s}^{*-}}$ on the width of $D_{s2}^{*}(2573)^{+}$. Here, the CM energy is $\sqrt{s}=4.680$ GeV and four typical values of width of $D_{s2}^{*}(2573)^{+}$ are 6.0, 10.4, 16.9, 27.1 MeV, which are from Ref. Song:2015nia , ARGUS Albrecht:1995qx , LHCb Aaij:2014xza , and BABAR Aubert:2006mh , respectively. Here, The maximum of line shape is normalized to 1.  Table 1:  The proposal for inducing charmoniumlike structures through our proposed CEE mechanism. The last column shows the corresponding pole position of a reflective charmoniumlike peak in the invariant mass of $D_{(s)}^{(*)}D_{(s)}^{(*)}$.  $\sqrt{s_{\textrm{critical}}}~{}$222 Since the listed critical CM energies are only the reference values, experimentalists can plan to scan the CM energy points near the suggested reference value. | Recommended process | Involved state | $m_{D_{(s)}^{(*)}D_{(s)}^{(*)}}^{pole}$
---|---|---|---
4286 | $e^{+}e^{-}\to DD^{*}\pi$ |  $D_{1}(2420)$ ($1^{+}$) | 3890
4429 | $e^{+}e^{-}\to D_{s}D_{s}^{*}\pi$ |  $D_{s1}(2460)$ ($1^{+}$) | 4091
4430 | $e^{+}e^{-}\to D_{s}^{*}D_{s}\pi$ |  $D_{s0}(2317)$ ($0^{+}$) | 4092
4431 | $e^{+}e^{-}\to D^{*}D^{*}\pi$ |  $D_{1}(2420)$ ($1^{+}$) | 4035
4471 | $e^{+}e^{-}\to D^{*}D\pi$ |  $D_{2}^{*}(2460)$ ($2^{+}$) | 3910
4474 | $e^{+}e^{-}\to DD^{*}\pi$ |  $D^{*}(2600)$ ($1^{-}$) | 3906
4504 | $e^{+}e^{-}\to D_{s}D^{*}K$ |  $D_{s1}(2536)$ ($1^{+}$) | 3982
4572 | $e^{+}e^{-}\to D_{s}^{*}D_{s}^{*}\pi$ |  $D_{s1}(2460)$ ($1^{+}$) | 4234
4615 | $e^{+}e^{-}\to D_{s}D^{*}K$ |  $D_{s}(2646)$ ($0^{-}$)333They are theoretically predicted charmed/charmed-strange mesons that are still missing in experiments Song:2015nia ; Song:2015fha . | 3994
4617 | $e^{+}e^{-}\to DD^{*}\pi$ |  $D(2750)$ ($2^{-}$) | 3921
4619 | $e^{+}e^{-}\to D^{*}D^{*}\pi$ |  $D^{*}(2600)$ ($1^{-}$) | 4052
4647 | $e^{+}e^{-}\to D_{s}^{*}D^{*}K$ |  $D_{s1}(2536)$ ($1^{+}$) | 4125
4677 | $e^{+}e^{-}\to D_{s}DK$ |  $D^{*}_{s1}(2700)$ ($1^{-}$) | 3878
4750 | $e^{+}e^{-}\to DD^{*}\rho$ |  $D(2885)$ ($1^{+}$)22footnotemark: 2 | 3889
4758 | $e^{+}e^{-}\to D_{s}^{*}D^{*}K$ |  $D_{s}(2646)$ ($0^{-}$)22footnotemark: 2 | 4138
4762 | $e^{+}e^{-}\to D^{*}D^{*}\pi$ |  $D(2750)$ ($2^{-}$) | 4068
4820 | $e^{+}e^{-}\to D_{s}^{*}DK$ |  $D^{*}_{s1}(2700)$ ($1^{-}$) | 4023
4894 | $e^{+}e^{-}\to D^{*}D\rho$ |  $D(2884)$ ($2^{+}$)22footnotemark: 2 | 3914
4895 | $e^{+}e^{-}\to D^{*}D^{*}\rho$ |  $D(2885)$ ($1^{+}$)22footnotemark: 2 | 4034
5013 | $e^{+}e^{-}\to DD\rho$ |  $D(3148)$ ($2^{-}$)22footnotemark: 2 | 3812
5116 | $e^{+}e^{-}\to D_{s}^{*}D^{*}K^{*}$ |  $D_{s}(3004)$ ($2^{+}$)22footnotemark: 2 | 4138
5158 | $e^{+}e^{-}\to D^{*}D\rho$ |  $D(3148)$ ($2^{-}$)22footnotemark: 2 | 3960
5229 | $e^{+}e^{-}\to D_{s}DK^{*}$ |  $D_{s}(3260)$ ($2^{-}$)22footnotemark: 2 | 3924
5372 | $e^{+}e^{-}\to D_{s}^{*}DK^{*}$ |  $D_{s}(3260)$ ($2^{-}$)22footnotemark: 2 | 4070

Proposal for future experiments.–In the following, we list a series of three-
body open-charm processes from the positron-electron annihilation, where the
CM energies suitable for studying the charmoniumlike structures induced by the
CEE mechanism are given with the information of the involved intermediate
higher charmed/charmed-strange mesons. In a practical measurement, due to the
ensurance of the CEE mechanism, the reflection peak structure near a threshold
should be largely produced when the scanning energy point gradually close to
the critical position. Thus, for the listed three-body open charm processes,
experimentalists can plan a scanning interval in advance based on our
suggested CM energy point in Table 1.

The suggested critical CM energies in Table 1 can be directly obtained when we
determine the involved ground state charmed or charmed-strange meson $B$ and
the intermediate state $C$, which denotes a higher state of charmed/charmed-
strange meson families. These energy points can be covered by the future
BESIII and BelleII experiments, and even the discussing possible Super Tau-
Charm Factory in China, which is planned to operate in the CM energy of 2 to 7
GeV Luo:2019xqt . Here, we mention the theoretical reference masses
Song:2015nia ; Song:2015fha  to calculate the suggested CM energies for each
of the corresponding unobserved states $C$. Thus, we suggest that
experimentalists can plan to scan the CM energy points near the suggested
reference value. For the intermediate $C$ related to three-body open-charm
processes from the $e^{+}e^{-}$ annihilation, we select some relatively narrow
states for convenience of experimental measurement. Additionally, we give
approximate pole positions of the predicted charmoniumlike structures as a
reflection peak.

For the $e^{+}e^{-}$ collision, its CM energy can be changed according to the
corresponding physical aim. Obviously, our proposal can be an interesting
research topic achievable at BESIII and BelleII. We hope that this proposal
can stimulate experimentalists’ interest to apply this novel CEE mechanism to
detect more charmoniumlike $Z_{c}$ or $Z_{cs}$ structures. Of course, it is
worth mentioning that in a practical experimental measurement, the observation
of the critical peak structure in CEE mechanism is also dependent on the
coupling strength of reaction process. Therefore, based on this starting
point, further theoretical studies involving the suggested three-body open-
charm processes from the $e^{+}e^{-}$ annihilation are encouraged to consider
the detailed dynamics.

Summary.–We have proposed a Critical Energy induced Enhancement (CEE)
mechanism to detecting a new cluster of charmoniumlike structures, which is
based on a peculiar kinematic behavior of a three-body process from the
electron and positron annihilation. With $e^{+}e^{-}\to
D_{s}^{*-}D_{s2}^{*}(2573)^{+}\to D_{s}^{*-}(D^{0}K^{+})$ as an example, we
have presented the sensitivity of the reflective line shape from
$D_{s2}^{*}(2573)$ existing in the $m_{D_{s}^{*-}D^{0}}$ invariant mass
spectrum on the CM energy of the $e^{+}e^{-}$ system. When fixing a special CM
energy $\sqrt{s}=4.680$ GeV, which is just equal to
$m_{D_{s}^{*}}+m_{D_{s2}^{*}(2573)}$, an obvious reflection peak near the
threshold of $D_{s}^{*-}D^{0}$ can be produced. However, when the CM energy is
gradually larger than 4.680 GeV, the reflection peak will disappear. For
another case of $\sqrt{s}<4.680$ GeV, the events of reflection peak will be
obviously suppressed due to the missing of the single pole. Therefore, the
energy of $\sqrt{s}=m_{D_{s}^{*}}+m_{D_{s2}^{*}(2573)}$ is a special critical
point and there exist a CEE mechanism for the three-body open-charm process
from the $e^{+}e^{-}$ collisions. A direct application of this mechanism found
in this work is to study the new cluster of charmoniumlike $Z_{c}$ or $Z_{cs}$
structures. We have further suggested a series of three-body open-charm
processes from the electron and positron annihilation associated with the
corresponding suggested CM energies. We have also noticed that BESIII released
“White paper on the Future Physics Program of BESIII” recently Ablikim:2019hff
. Since the suggested CM energy points involved in three-body open-charm
processes can be covered by BEPCII, it is obvious that it is a good chance for
BESIII. The running BelleII is also an experimental facility based on the
electron and positron collision. By the initial state radiation technique,
BelleII has enough potential to measure the suggested three-body open-charm
processes. Experimentally studying three-body open-charm processes from
electron and positron annihilation with special CM energy points may form a
future research topic accessible at BESIII, BelleII and even Super Tau-Charm
factory Luo:2019xqt .

In fact, this method developed in the present work is not limited only to the
suggested three-body open-charm processes mentioned above. We may continue to
extend it to three-body open-strange processes and three-body open-bottom
processes from the electron and positron annihilation, which can be an
effective approach to construct new strangeoniumlike and bottomoniumlike
structures families, respectively.

## ACKNOWLEDGEMENTS

This work is partly supported by the China National Funds for Distinguished
Young Scientists under Grant No. 11825503, National Key Research and
Development Program of China under Contract No. 2020YFA0406400, the 111
Project under Grant No. B20063, the National Natural Science Foundation of
China under Grant Nos. 12047501 and 11775050, and the Fundamental Research
Funds for the Central Universities under Grant No. lzujbky-2020-it03.

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