# Beamforming using Digital Piezoelectric MEMS Microphone Array

Ricky Leman, Ben Travaglione, and Melinda Hodkiewicz*
School of Engineering,
The University of Western Australia,
Perth, Australia
<EMAIL_ADDRESS>

###### Abstract

The recent explosion in low-cost, low-power wireless microcontrollers, coupled
with low-power, robust MEMS sensors has opened up the opportunity to create
new forms of low-cost Industrial Internet-of-Things (IIoT) devices for
condition monitoring. Piezoelectric MEMS microphones constructed with a
cantilever diaphragm are a potential solution against failure modes, such as
water and dust ingress, that have challenged the use of capacitive MEMS
microphones in industrial applications. In this paper, we couple a pair of
piezoelectric MEMS microphones to a COTS microcontroller to create a stand-
alone microphone array capable of discerning the direction of a noise source.
The microphone array is designed to acquire sound data without aliasing at
frequencies of 2000 Hz or below. Testing is conducted in an anechoic chamber.
We compare the performance of this microphone array to a simple idealized
theoretical model. The experimental results obtained in the anechoic chamber
compare well with the theoretical model. The work stands as a proof-of-
principle. By providing detailed information on how we coupled the sensors to
a COTS microcontroller, and the open-source code used to process the data, we
hope that others will be able to build upon this work by expanding on both the
number and type of sensors used.

_K_ eywords Beamforming $\cdot$ Commercial-Off-The-Shelf (COTS) $\cdot$
Microelectromechanical systems (MEMS) $\cdot$ Microphones $\cdot$ Microphone
arrays $\cdot$ Raspberry Pi $\cdot$ VM3000

##  1 Introduction

Acoustic analysis offers the ability to perform condition monitoring without
the need for surface contact on a machine. Sound emissions from machinery are
acquired by microphone arrays and processed using a signal processing
technique known as beamforming. This enables detection of the source location
of a desired signal amidst noise. This method of acoustic analysis has been
demonstrated in literature through underwater hydrophone applications [1] to
medical applications such as hearing aids [2], and in the resources industry
for leak detection [3] and fan bearing issues [4].

Use-cases of acoustic analysis conventionally utilise capacitive microphones
in their design. This is due to their high sensitivity, flat frequency
response and low noise level [5]. However, sensors used within heavy industry
are subject to high levels of dust exposure and water ingress. These are
environmental factors that have a known impact on the performance of
capacitive MEMS microphones [6, 7], rendering them potentially unsuitable for
chemical, mining or oil and gas applications. Piezoelectric MEMS microphones
provide a potential solution to the harsh environmental conditions on-site
within the resources industry. Their cantilever diaphragm removes the need for
a back-plate found in capacitive microphones, resulting in a waterproof and
dustproof design [8]. The VM3000 produced by Vesper Technologies is a
commercially-available digital piezoelectric MEMS microphone that has the
potential for industrial application under harsh environments. However, there
is a lack of published literature regarding the beamforming performance of the
VM3000s MEMS microphones. The aim of this paper is to describe the design and
assess the beamforming performance of a stand-alone digital microphone array
with Wi-Fi connectivity using the VM3000 piezoelectric MEMS microphones.

The paper is organised as follows. Section 2 explores past research on
condition monitoring using sound analysis, fundamental theory underpinning
microphone arrays, and a comparison between piezoelectric and capacitive MEMS
microphones. Section 3 outlines the engineering design process of the hardware
and software aspects of this work. Section 4 describes the build process of
the microphone array and Section 5 describes the method of testing to assess
beamforming performance. Section 6 discusses beamforming performance results
from testing, Section 7 concludes the paper with a recap of key takeaways and
outlines recommendations for future work.

##  2 Background

###  2.1 Condition Monitoring with Acoustic Analysis

Sound analysis for condition monitoring involves the acquisition of sound data
using microphones and subsequent signal conditioning where sound is amplified
and digitised. Sound data can then be converted into the frequency domain
through Fourier Transform methods for comparison against known sound
signatures. Several use-cases of this method have been demonstrated in the
monitoring of mechanical elements such as internal-combustion engine wear [9],
induction motor bearing faults [10], through to railway obstruction fault
detection applications [11]. These use cases demonstrated high fault
classification accuracy using standard vibration analysis methods as a
benchmark for comparison.

###  2.2 Microphone Arrays and Beamforming

Microphone array is a term used to denote a sound acquisition device that is
comprised of multiple microphones arranged in various geometrical layouts for
beamforming. Beamforming is a signal processing technique that enables
discrimination of different signals based on the physical location of sound
sources relative to a microphone array [12]. The combination and processing of
various microphone outputs allow received signals to be cleaned from
contaminating interference and ambient noise [13]. As a technology which
enables sound source localisation, it has seen commercial applications in
videoconferencing technology [14], communications infrastructure [15] and
hearing aids [2, 12]. The application of microphone arrays and beamforming as
a method for condition monitoring is not widely adopted in industry. However,
several papers evaluating its use-cases are available in literature [16, 17].

The geometrical design of a microphone array directly influences its
achievable beamforming resolution [18]. A long-standing issue with regularly
spaced microphones is spatial aliasing, where spatial sampling must occur at a
rate greater than half the wavelength of incoming sinusoids to produce an
unambiguous beam pattern [13]. This is achieved through an inter-microphone
spacing compliant with:

$d<\frac{\lambda_{\mathrm{min}}}{2}\;\mathrm{where}\;\lambda_{\mathrm{min}}=\frac{c}{f_{\mathrm{max}}}$
(1)

Where $\lambda_{\mathrm{min}}$ is the minimum wavelength of the signal of
interest, $c$ is the speed of sound (around 343 ms-1 in air) and
$f_{\mathrm{max}}$ is the maximum frequency of the signal of interest [12].

Spatial aliasing results in an inability to distinguish the direction of
arrival of a desired signal, producing ghost sources in beam patterns of
comparable amplitude to the true signal source direction [19]. As such, many
irregular array designs with unique inter-microphone spacing known as non-
redundant array designs have been explored in literature [18]. These designs,
however, are complex in comparison to the traditional uniform linear array
(ULA) design, the applications of which have been documented in several
publications [20, 21, 19].

There are numerous algorithms that can be implemented to achieve beamforming.
These algorithms can be categorised into time-domain beamforming such as
Delay-Sum beamforming or frequency-domain beamforming such as Discrete-Fourier
Transform (DFT) Single-Beam beamforming [12, 22]. Frequency-domain methods
offer improved computational efficiency, signal fidelity, and robustness
against two or more interfering sound sources in comparison to time-domain
methods [22, 23]. Though these are characteristics desirable for industrial
applications, frequency-domain methods are less intuitive and more complex in
their implementation. The processing of experimental data in this work has
utilised time-domain methods as an initial method for validating the
beamforming performance of the developed microphone array prototype.

The delay-and-sum (DS) beamformer is a standard time-domain technique that is
widely documented in literature [12, 24]. As illustrated in Figure 1 acquired
sound signals are time-shifted for each sensor within a ULA based on the time
difference of arrival between sensor $n$ and some reference sensor $M_{2}$.
The summation of these time-shifted signals results in an effective
amplification of the sound signal of interest and attenuation of unwanted
noise through constructive and destructive interference respectively [24].

Figure 1: Block diagram of the time-domain delay-and-sum beamforming method
which demonstrates time-shifting of sound data acquired by microphones in a
ULA and its subsequent amplified output. (adapted from [25])

This forms the output of the DS beamformer from which the ULA’s beam pattern
can be generated. The beam pattern illustrates the performance of the ULA DS
beamformer by visualising the estimated azimuth angle of a sound source
relative to the position of the ULA. The angle difference between the
estimated azimuth angle and the real azimuth angle provides a metric of
performance measure by which the ULA can be assessed.

A quantitative measure of beamforming performance for microphone arrays is its
array gain. The Signal-to-Noise ratio (SNR) of a microphone measures the ratio
between a reference signal level such as background noise to the noise level
of the microphone output signal [26]. By comparing the known SNR of a
microphone in isolation to the output of the same microphone in an array, the
array gain can be computed as follows [12]:

$\mathrm{Array\;Gain}=\frac{SNR_{Array}}{SNR_{Sensor}}$ (2)

Array gain quantifies the improvement in SNR between a reference sensor and
the array output. Multiple publications quantify the beamforming performance
of both digital and analog capacitive MEMS microphones using SNR gain.
Examples include the investigation of the SNR gain of a 52-microphone array
using digital output capacitive MEMS microphones [27] and a similar
investigation using analog output capacitive MEMS microphones [28]. However,
there is a lack of published literature on the beamforming performance of
digital output piezoelectric MEMS microphones.

###  2.3 Piezoelectric and Capacitive MEMS Microphones

Micro-electromechanical Systems (MEMS) refer to microscopic devices comprised
of moving mechanical elements and electrical components fabricated on a single
silicon chip. MEMS sensors offer the advantage of smaller size, lower weight,
less power demand, and greater versatility compared to their conventional
sensor counterparts [29]. In 2015 the global MEMS market was valued at USD
11.99 billion [30] and is projected to achieve a CAGR of 6.34% over the 2020 -
2025 period [31]. This growth is driven by the increasing adoption of MEMS in
consumer electronics devices, automotive and industrial applications, along
with continual developments within the Internet of Things (IoT) space [30].

Capacitive MEMS microphones are the current standard of microphones used in
beamforming. Their basic structure consists of an air-gap separated fixed
backplate in parallel with a flexible membrane that vibrates in response to
acoustic pressure. This produces a variation in the air gap, thereby altering
the parallel plate capacitance and producing a voltage that corresponds to the
incoming sound wave [32]. The ability of the membrane to flex freely is
critical to the performance of the device and can be compromised due to
particle ingress, water ingress or mechanical shock [6, 7]. If the membrane is
unable to flex freely, the microphone is no longer able to generate a
capacitance proportional to incoming sound waves, thereby reducing performance
or producing unusable data. The failure modes which can obstruct the motion of
the membrane are illustrated in Figure 2 which compares the structure of a
normal capacitive MEMS microphone against capacitive MEMS microphones under
different failure modes.

Figure 2: Failure modes of capacitive MEMS microphones: normal (upper left),
particle ingress (upper right), water ingress (lower left) and stiction
failure (lower right). (adapted from [7])

Piezoelectric MEMS microphones offer a potential solution to common failure
modes of their capacitive counterpart. The basic structure of the VM3000
comprises of a cantilever diaphragm that flexes against acoustic pressure and
generates a corresponding voltage through the piezoelectric effect [33]. The
piezoelectric cantilever plates remove the need for a backplate, thereby
producing a design that is innately more robust against particle ingress,
water ingress or mechanical shock. The lack of a backplate membrane ensures
either particles or water are unable to remain embedded within the structure
of the microphone, thereby ensuring performance is not inhibited even in the
presence of particle or water ingress. Figure 3 illustrates the cantilever
design of the VM3000 microphone. The microphone contains four cantilever
structures that act as sound transducers against incoming sound. The stress
generated by incoming sound is converted into a corresponding electrical
signal.

Figure 3: VM3000 cantilever piezoelectric diaphragms flexing in response to
incoming sound and generating stress within the plates which produces a
corresponding electrical signal (right). (adapted from [7, 8])

Digital microphones, such as the VM3000, contain an integrated analog-to-
digital converter (ADC) which enables digital signal processing to be
conducted without the need for an external ADC. This allows the VM3000s to be
interfaced directly with commercial off-the-shelf (COTS) hardware, such as the
Raspberry Pi, making the construction of a small-scale self-contained
microphone array capable of data acquisition, signal processing, and data
transmission feasible.

##  3 Hardware Selection and Design

To test the beamforming capabilities of the VM3000 digital piezoelectric MEMS
microphone, a microphone array prototype controlled through COTS hardware was
developed. The design method is split into hardware selection and design and
software development.

The focus of hardware development is to produce a prototype that is capable of
powering, interfacing with, and controlling the VM3000 microphones.

###  3.1 Digital MEMS Microphone

The Vesper VM3000 microphone is an omnidirectional, bottom-port, Pulse Density
Modulation (PDM) digital piezoelectric MEMS microphone. The PDM digital output
enables the multiplexing of two microphones on a single data line. In an
active state at 3.072MHz, the VM3000 consumes a maximum of 800$\mu$A and less
than 1$\mu$A in sleep mode. It has a flat frequency response over 500 to 2000
Hz and an ingress protection rating of IP57, certified for dust and water
ingress resistance [8].

###  3.2 Microcontroller

The Raspberry Pi 4 Model B is chosen as the microphone array COTS controller.
It is a 1.5GHz, quad-core CPU mini computer with wireless local area network
(LAN) and Bluetooth 5.0 capabilities. It is capable of reading PDM digital
data from two VM3000 MEMS microphones. It is space-efficient with dimensions
of 88mm x 58mm x 19.5mm and light-weight at 46g, characteristics which are
desirable for interfacing with small-scale MEMS devices [34]. Wi-Fi and
Bluetooth capabilities enable the prototype to be accessed remotely and
thereby operate as a stand-alone device.

###  3.3 Hardware Design

Design of the microphone array prototype PCB is performed using EAGLE and the
completed board is illustrated in Figure 4 and Figure 5. The PCB design
contains a LiPo 18650 battery cell and TPS81256 boost converter. These
components where to be utilised when the array was controlled by a Raspberry
Pi Zero, but became unnecessary once the Pi Zero was replaced with a Pi 4
Model B, as discussed in Section 5.1.2. The primary objective of this design
is to produce a microphone array prototype of minimal size to ensure
manufacturing costs remain low and portability is maintained. Inter-microphone
spacing is the key measure that determines the physical size of the microphone
array prototype PCB. A maximum inter-microphone spacing of 8.4 cm is chosen
through a compromise of the physical PCB dimensions, tolerable output signal
pitch during experimental testing, and the flattest frequency response range
of the VM3000 microphones (500 to 2000 Hz). By taking the upper bound of this
frequency range and using equation 1, the maximum inter-microphone spacing is
determined to be approximately 8.6 cm. Therefore the designed inter-microphone
spacing of 8.4 cm falls within the acceptable spacing distance, ensuring no
spatial aliasing occurs when recording signals within 500 to 2000 Hz.

Figure 4: PDM microphone array prototype layout produced from Autodesk EAGLE.
The array is designed for a LiPo 18650 battery cell, switch, TPS81256 boost
converter, two VM3000 MEMS microphones and a 40-pin GPIO header. Figure 5: Top
view of a completed, fully populated 2-element linear microphone array
prototype connected to a Raspberry Pi 4 Model B. (NB: The Pi 4 Model B is
unable to be powered from a single 18650 battery)

##  4 Software Design

The focus of software design is to develop a remote process for data
acquisition, data transfer, and signal processing using a laptop or equivalent
device. This involves the simultaneous control of lab equipment, to generate
test signals, and the Raspberry Pi on the microphone array prototype, for
sound recordings. A high-level overview of the software operation is as
follows:

  1. 1.

Import modules to be used in the script

  2. 2.

Establish connection with a Keysight 33500B Waveform Generator, either via
wireless local area network if the signal generator is Ethernet-enabled or via
direct USB connection with a laptop.

  3. 3.

Establish connection with Raspberry Pi on microphone array prototype via SSH
protocol.

  4. 4.

Load class of test functions which will control the test signal output, sound
recording duration of the microphone array prototype, and data transfer of
.WAV recordings to laptop or equivalent device.

  5. 5.

Perform time-domain delay-and-sum beamforming using the .WAV recordings and
using theoretical data.

  6. 6.

Visualise beam pattern of real data and theoretical data on the same plot.

##  5 Microphone Array Prototype Build

###  5.1 Hardware Implementation

####  5.1.1 Hardware Assembly

A combination of reflow soldering for SMD components and hand iron soldering
for through-hole components is used to assemble the microphone array
prototype. Due to the small footprint of the VM3000 MEMS microphones, TPS81256
boost converter, and other SMD components, a solder paste stencil are used to
lay solder paste accurately onto the pads of SMD components. These components
are mounted via reflow soldering using a hand-held heat gun. Through-hole
components such as the LiPo cell battery clips, switch and female 40-pin
general-purpose input/output (GPIO) header are assembled by hand using a
soldering iron. Since the female 40-pin GPIO sits on the underside of the
microphone array prototype, any Raspberry Pi with a male GPIO header will be
able to interface directly with the prototype.

####  5.1.2 Raspberry Pi Zero W and Raspberry Pi 4 Model B performance

Initial development was done using a Raspberry Pi Zero W. Interfacing the
microphone array prototype with the Raspberry Pi Zero W produced phantom noise
in sound recordings. Two separate builds of the microphone array prototype
were tested for different input signals. The phantom noise is present when the
microphone array is interfaced with the Raspberry Pi Zero W and powered using
either the 18650 battery cell or the micro USB power input. Sampling frequency
was adjusted to no effect. In all cases, a rhythmic, audible ticking noise in
both left and right channel data is audible when sound recordings are
replayed. Figure 6 illustrates this observable phantom noise through a plot of
a .WAV file from an ambient noise recording. The phantom noise is
characterised by periodic peaks within the .WAV file which is observable in
both left and right channel data. These peaks occur at the same points in time
and at the same frequency in both left and right channel data.

Figure 6: Phantom noise observable in the raw .WAV file through distinct,
periodic peaks during ambient noise recording when the microphone array
prototype is interfaced with the Raspberry Pi Zero W.

This phantom noise is no longer observable when the microphone array prototype
is interfaced with the Raspberry Pi 4 Model B with an official power supply.
Figure 7 illustrates an ambient noise recording when the microphone array
prototype is interfaced with a Raspberry Pi 4 Model B. The distinct periodic
peaks of the phantom noise are no longer observable within the .WAV file nor
is it audible when listening to replays of sound recordings.

It is likely that the phantom noise originates internally to the Raspberry Pi
Zero W. A possible cause is a lack of hardware processing power in the
Raspberry Pi Zero W for high frequency data sampling at 96 kHz. The Raspberry
Pi 4 Model B contains a quad-core CPU, 1.5GHz clock, and 8GB RAM whilst the
Raspberry Pi Zero W contains a single-core CPU, 1GHz clock, and 512MB RAM.
Since the Raspberry Pi Zero W is a single-core CPU device, it must share
processing power for PDM data with its other running processes. This inability
to multitask may result in some interaction between PDM recording and
background processes which corrupts the .WAV file data. Further work is
required to investigate this observation.

Figure 7: Phantom noise no longer observable in the raw .WAV file during
ambient noise recording as periodic peaks are no longer present when the
microphone array prototype is interfaced with the Raspberry Pi 4 Model B.

###  5.2 Software Implementation

####  5.2.1 Software Development

A central code base capable of controlling the Raspberry Pi 4 Model B,
commanding laboratory test equipment, transporting data remotely, and
conducting signal processing is developed specifically for this project by
Ricky Leman [35]. The Python implementation of the time-domain delay-sum
beamforming and visualisation of experimental data is developed by Dr. Ben
Travaglione. It is a simplistic implementation of the time-domain delay-sum
beamforming algorithm based on fundamental theoretical principles. It is
capable only of modelling the theoretical beam pattern of a single, perfect
sound source and a single receiver in infinite space. Jupyter Lab is used as
an open-source web-based tool for developing the project code. Python is
chosen as the programming language of choice due to its ease of implementation
and access to packages relevant to this work. The source code for this work is
available on the UWA System Health Lab GitHub [36].

Key Python packages that have been used include:

  1. 1.

vxi11 - This package supports the VXI-11 Ethernet instrument control protocol
for controlling VXI11 and LXI compatible instruments. It enables Ethernet
control of the Keysight 33500B Waveform Generator.

  2. 2.

pyvisa - This package enables control of measurement devices independently of
its interface. It enables USB control of the Keysight 33500B Waveform
Generator.

  3. 3.

paramiko - Paramiko is a Python implementation of the SSHv2 protocol which
provides client and server functionality. It enables remote SSH access to the
Raspberry Pi 4 Model B.

  4. 4.

scp - This package is used in conjunction with paramiko to send and receive
files via the scp1 protocol. It enables transfer of sound files locally stored
on the Raspberry Pi 4 Model B to the user’s device.

  5. 5.

bokeh - Bokeh enables the creation of interactive data visualisation on web
browsers. It enables raw .WAV file data to be visualised in an interactive
plot which dynamically updates based on changing inputs or user interaction.

####  5.2.2 Enabling PDM audio on Raspberry Pi Broadcom System on Chip

Interfacing PDM audio with the Raspberry Pi range involves altering the
Broadcom System on Chip (SoC) source code. The BCM2835(2012) and BCM2711(2020)
ARM Peripherals datasheets provide technical information regarding the PDM
input mode operation but do not provide detailed instructions on how to
activate the PDM input mode. In December 2020, an online review had yielded a
single set of high-level instructions on the Raspberry Pi forums and no formal
documented process. A reproducible method has been developed for this paper
and documented on The UWA System Health Lab GitHub site[35], as well being as
shared on public Raspberry Pi forums. This method has been tested and verified
to be successful on the Raspberry Pi 4 Model B. The method also enables PDM
audio on the Raspberry Pi Zero W, however there are unresolved recording
issues, as discussed in Section 5.1.2.

##  6 Experimental Method

The goal of this testing is to compare the accuracy of real beam patterns
created by sound data acquired using the VM3000 MEMS microphones against
theoretical beam patterns generated using idealised data. Differences between
peak angles and overall shape are the characteristics of interest as they
determine the beamforming accuracy of the microphone array prototype and
demonstrate the effects of sound interference.

###  6.1 Experiment Setup

A repeatable testing sequence is developed to ensure consistency in data
collection. The experiment setup consists of two distinct stages which are
illustrated in Figure 8. Testing is conducted in the UWA Anechoic chamber.

Figure 8: Two-stage experiment setup for beamforming testing of the microphone
array prototype. The microphone array records a 500 - 3000 Hz linear frequency
sweep generated by a Keysight 33500B Waveform Generator fed into a speaker.
The microphone array is rotated by $20\degree$ increments between $0\degree$
and $360\degree$. Time-Domain Delay-Sum beamforming is conducted on collected
data through Python to generate theoretical and real beampatterns.

The microphone array prototype is placed at a starting position perpendicular
to the speaker (array angle of $0\degree$) along the same horizontal plane. A
Python script running in a Jupyter Lab notebook is used to command a Keysight
33500B Waveform Generator to output a linear frequency sweep between 500 Hz to
3000 Hz through the speaker. This frequency range is chosen to observe the
beamforming performance for the operating design range of 500 to 2000 Hz and
to observe the effects of aliasing for frequencies greater than 2000 Hz. The
output signal amplitude is kept constant for all tests. The linear sweep
output is repeated for each $20\degree$ array angle rotation until 18 readings
corresponding to a $360\degree$ rotation is recorded. The angle bearing guide
used for this is illustrated in Figure 9 and was constructed manually using a
protractor. Experimental data is collected in .WAV format using the Linux
arecord function via the Raspberry Pi 4 Model B.

Figure 9: Angle bearing guide with increments of $20\degree$ used for rotating
the microphone array prototype about its inter-microphone centre for each
iteration of testing.

####  6.1.1 Signal Processing Stage

Python running on Jupyter Lab is used for signal processing and data
visualisation. Both experimental and theoretical data is processed using time-
domain delay-sum beamforming. Theoretical beam patterns are generated assuming
a completely anechoic environment free of reflections and a single, perfect
incoming sound signal. Real beam patterns are generated using experimental
data.

For assessing a single combination of array angle and frequency, a normalised
theoretical and real beam pattern is generated and overlayed on a single plot.
This allows for the comparison of peak angle and shape differences.

For assessing all combinations of array angles and frequencies, the area
difference between normalised theoretical and real beam patterns is computed
and plotted on a heatmap. The accuracy of the microphone array prototype
beamforming performance is quantified by calculating the Root Mean Square
Error (RMSE) between the real and theoretical beam patterns generated by the
dataset.

####  6.1.2 Anechoic environment setup

The test is conducted in the UWA Anechoic Chamber. This environment is chosen
to evaluate the microphone array prototype beamforming performance under
anechoic conditions, enabling like-for-like comparison with the theoretical
beam pattern data. The test environment is lined with sound-absorbing
materials such as curtains for the walls and foam padding for the floors
underneath the metal grating. These materials prevent sound reflections from
occurring. At the time of testing, a wooden wall had been installed for other
works within the anechoic chamber, introducing a surface for sound
reflections. The impact of this is minimised by placing the microphone array
prototype in a corner surrounded by sound-dampening surfaces.

##  7 Results and Discussion

Three trials of experimental data were gathered under the same test conditions
to minimise the impact of random errors in the anechoic chamber. To ensure the
effects of sound reflections remain consistent between trials, all test
apparatus such as the ladder and crates were kept in place between trials.
Only the microphone array prototype itself is rotated between each recording.
Sound data acquired by the microphone array prototype in an anechoic
environment demonstrates a beam pattern which agrees with the developed
theoretical model. Figure 10 illustrates the microphone array prototype’s beam
pattern at an array angle of $50\degree$ and frequency of 1650 Hz generated
using data collected in a single trial. The solid, blue line represents the
theoretical beam pattern generated using idealised sound data. The dotted,
cyan line represents the experimental beam pattern generated using collected
test data by the microphone array and the solid, black line represents the
source angle direction. No aliasing is present within the experimental beam
pattern, however, an approximately $10\degree$ mismatch in peak angles is
observable. This peak angle mismatch is likely to be resultant of surface
reflections by the wooden panel installed within the anechoic chamber.
Reflections from the microphone array hardware itself and the wooden panel
upon which it is resting may also be contributing factors.

Figure 10: Accurate beamforming performance as seen by a peak angle difference
of approximately $10\degree$ for data collected by the microphone array
prototype in an anechoic test environment at an array angle of $50\degree$ and
frequency of 1650 Hz.

Figure 11 compares the theoretical delay-sum waveform with the experimental
delay-sum waveform. Sound reflections and noise are observable within the
experimental delay-sum waveform signal characterised by perturbations within
the sinusoid. This suggests an imperfect anechoic environment as discussed and
is likely to be a large contributing factor to peak angle mismatches in
conjunction with the array’s physical structure.

Figure 11: Sample of time-domain delay-sum beamforming (cyan) compared with
the theoretical delay-sum beamforming (black dots). Differences in these
signals is likely due surface reflections in the UWA Anechoic Chamber.

The theoretical and experimental beam patterns change as a function of array
angle and frequency. As we are interested in both the peak angle difference
and shape difference between theoretical and experimental beam patterns,
finding the normalised area difference between the two beam patterns captures
both performance measures in a single value. A small normalised area
difference indicates the experimental beam pattern matches the shape of the
theoretical beam pattern well, therefore suggesting strong beamforming
performance. A large normalised area difference indicates the shape of the
experimental beam pattern is a mismatch with the theoretical beam pattern,
thereby suggesting poor beamforming performance.

Figure 12 visualises the normalised area difference between theoretical and
experimental beam patterns for all array angles and frequencies recorded. This
enables the beamforming performance of the microphone array prototype across
all combinations of array angles and frequencies to be observed. These values
have been mapped to a colour scale from purple to red. Purple represents the
smallest area difference and therefore the best beamforming performance.
Whereas red represents the largest area difference and hence the worst
beamforming performance.

Figure 12: Normalised area difference between theoretical and real beam
patterns for all array angle and frequency combinations for test data
collected in an anechoic environment. The heatmap shows agreement between
experimental and theoretical beam patterns for the majority of array angle and
frequency combinations, indicating accurate beamforming performance.

A strong match in shape between experimental and theoretical beam patterns is
observable, indicative of strong beamforming performance across all
combinations of array angles and frequencies tested for an anechoic
environment. The beamforming performance of the microphone array prototype is
quantified by computing the average RMSE between experimental and theoretical
beam patterns for each data set generated from each trial between the
frequency operating range of 500 - 2000 Hz. The average RMSE values for each
trial, and the overall average are shown in Table 1.

| Trial 1 | Trial 2 | Trial 3 | Overall
---|---|---|---|---
Average RMSE value | 12.52% | 10.33% | 10.45% | 11.10%
Table 1: Average RMSE values from each trial between the frequency operating
range of 500 - 2000 Hz.

The experimental beampattern generated by data collected from the microphone
array prototype produces an RMSE of 11.10 % when compared to theoretical beam
patterns.

When the signal of interest exceeds 2000 Hz, a decrease in beamforming
performance is observable. This observation agrees with theory as spatial
aliasing is expected to occur for signal frequencies that exceed the
microphone array prototype design frequency limit of 2000 Hz. However, for
signals below 2000 Hz, the heatmap indicates the majority of array angles and
frequencies produce experimental beam patterns which accurately match with
theoretical beam patterns. Hot spots in Figure 12 below 2000 Hz indicate a
deviation between theoretical and experimental beam patterns. These are likely
the result of sound reflections from the wooden wall interfering with the
signal of interest and also due to obstruction of sound by the asymmetric
microphone array prototype structure. Therefore, the developed microphone
array prototype exhibits a strong beamforming performance in anechoic
environments.

##  8 Conclusion

This paper demonstrates the design, build and test of a stand-alone digital
two-element microphone array prototype using the VM3000 MEMS microphone. The
prototype is tested in an anechoic environment. Time-domain delay-and-sum
beamforming is implemented using Python in a Jupyter notebook. Test results in
anechoic environments demonstrate that the microphone array prototype exhibits
accurate beamforming when subject to an environment with minimal surface
reflections. This suggest the VM3000 MEMS microphones are capable of
beamforming accurately under conditions of minimal sound reflections, however
further work is required to develop a microphone array using the VM3000 MEMS
microphones that is suitable for real-world applications.

The data presented in this study are available at [35].

## Acknowledgments

Melinda Hodkiewicz acknowledges support from the BHP Fellowship for
Engineering for Remote Operations which supports the UWA System Health Lab
work.

The authors declare no conflict of interest.

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# Measurements of Normalized Differential Cross Sections of Inclusive
$\pi^{0}$ and $K_{S}^{0}$ Production in $e^{+}e^{-}$ Annihilation at Energies
from 2.2324 to 3.6710 $\mathrm{GeV}$

M. Ablikim1, M. N. Achasov12,b, P. Adlarson72, M. Albrecht4, R. Aliberti33, A.
Amoroso71A,71C, M. R. An37, Q. An68,55, Y. Bai54, O. Bakina34, R. Baldini
Ferroli27A, I. Balossino28A, Y. Ban44,g, V. Batozskaya1,42, D. Becker33, K.
Begzsuren30, N. Berger33, M. Bertani27A, D. Bettoni28A, F. Bianchi71A,71C, E.
Bianco71A,71C, J. Bloms65, A. Bortone71A,71C, I. Boyko34, R. A. Briere5, A.
Brueggemann65, H. Cai73, X. Cai1,55, A. Calcaterra27A, G. F. Cao1,60, N.
Cao1,60, S. A. Cetin59A, J. F. Chang1,55, W. L. Chang1,60, G. R. Che41, G.
Chelkov34,a, C. Chen41, Chao Chen52, G. Chen1, H. S. Chen1,60, M. L. Chen1,55,
S. J. Chen40, S. M. Chen58, T. Chen1,60, X. R. Chen29,60, X. T. Chen1,60, Y.
B. Chen1,55, Z. J. Chen24,h, W. S. Cheng71C, S. K. Choi 52, X. Chu41, G.
Cibinetto28A, F. Cossio71C, J. J. Cui47, H. L. Dai1,55, J. P. Dai76, A.
Dbeyssi18, R. E. de Boer4, D. Dedovich34, Z. Y. Deng1, A. Denig33, I.
Denysenko34, M. Destefanis71A,71C, F. De Mori71A,71C, Y. Ding38, Y. Ding32, J.
Dong1,55, L. Y. Dong1,60, M. Y. Dong1,55,60, X. Dong73, S. X. Du78, Z. H.
Duan40, P. Egorov34,a, Y. L. Fan73, J. Fang1,55, S. S. Fang1,60, W. X. Fang1,
Y. Fang1, R. Farinelli28A, L. Fava71B,71C, F. Feldbauer4, G. Felici27A, C. Q.
Feng68,55, J. H. Feng56, K Fischer66, M. Fritsch4, C. Fritzsch65, C. D. Fu1,
H. Gao60, X. L. Gao68,m, Y. N. Gao44,g, Yang Gao68,55, S. Garbolino71C, I.
Garzia28A,28B, P. T. Ge73, Z. W. Ge40, C. Geng56, E. M. Gersabeck64, A
Gilman66, K. Goetzen13, L. Gong38, W. X. Gong1,55, W. Gradl33, M.
Greco71A,71C, L. M. Gu40, M. H. Gu1,55, Y. T. Gu15, C. Y Guan1,60, A. Q.
Guo29,60, L. B. Guo39, R. P. Guo46, Y. P. Guo11,f, A. Guskov34,a, W. Y. Han37,
X. Q. Hao19, F. A. Harris62, K. K. He52, K. L. He1,60, F. H. Heinsius4, C. H.
Heinz33, Y. K. Heng1,55,60, C. Herold57, G. Y. Hou1,60, Y. R. Hou60, Z. L.
Hou1, H. M. Hu1,60, J. F. Hu53,i, T. Hu1,55,60, Y. Hu1, G. S. Huang68,55, K.
X. Huang56, L. Q. Huang29,60, X. T. Huang47, Y. P. Huang1, Z. Huang44,g, T.
Hussain70, N. Hüsken26,33, W. Imoehl26, M. Irshad68,55, J. Jackson26, S.
Jaeger4, S. Janchiv30, E. Jang52, J. H. Jeong52, Q. Ji1, Q. P. Ji19, X. B.
Ji1,60, X. L. Ji1,55, Y. Y. Ji47, Z. K. Jia68,55, P. C. Jiang44,g, S. S.
Jiang37, X. S. Jiang1,55,60, Y. Jiang60, J. B. Jiao47, Z. Jiao22, S. Jin40, Y.
Jin63, M. Q. Jing1,60, T. Johansson72, S. Kabana31, N. Kalantar-Nayestanaki61,
X. L. Kang9, X. S. Kang38, R. Kappert61, M. Kavatsyuk61, B. C. Ke78, I. K.
Keshk4, A. Khoukaz65, R. Kiuchi1, R. Kliemt13, L. Koch35, O. B. Kolcu59A, B.
Kopf4, M. Kuemmel4, M. Kuessner4, A. Kupsc42,72, W. Kühn35, J. J. Lane64, J.
S. Lange35, P. Larin18, A. Lavania25, L. Lavezzi71A,71C, T. T. Lei68,k, Z. H.
Lei68,55, H. Leithoff33, M. Lellmann33, T. Lenz33, C. Li41, C. Li45, C. H.
Li37, Cheng Li68,55, D. M. Li78, F. Li1,55, G. Li1, H. Li49, H. Li68,55, H. B.
Li1,60, H. J. Li19, H. N. Li53,i, J. Q. Li4, J. S. Li56, J. W. Li47, Ke Li1,
L. J Li1,60, L. K. Li1, Lei Li3, M. H. Li41, P. R. Li36,j,k, S. X. Li11, S. Y.
Li58, T. Li47, W. D. Li1,60, W. G. Li1, X. H. Li68,55, X. L. Li47, Xiaoyu
Li1,60, Y. G. Li44,g, Z. X. Li15, Z. Y. Li56, C. Liang40, H. Liang32, H.
Liang1,60, H. Liang68,55, Y. F. Liang51, Y. T. Liang29,60, G. R. Liao14, L. Z.
Liao47, J. Libby25, A. Limphirat57, C. X. Lin56, D. X. Lin29,60, T. Lin1, B.
J. Liu1, C. Liu32, C. X. Liu1, D. Liu18,68, F. H. Liu50, Fang Liu1, Feng Liu6,
G. M. Liu53,i, H. Liu36,j,k, H. B. Liu15, H. M. Liu1,60, Huanhuan Liu1, Huihui
Liu20, J. B. Liu68,55, J. L. Liu69, J. Y. Liu1,60, K. Liu1, K. Y. Liu38, Ke
Liu21, L. Liu68,55, Lu Liu41, M. H. Liu11,f, P. L. Liu1, Q. Liu60, S. B.
Liu68,55, T. Liu11,f, W. K. Liu41, W. M. Liu68,55, X. Liu36,j,k, Y. Liu36,j,k,
Y. B. Liu41, Z. A. Liu1,55,60, Z. Q. Liu47, X. C. Lou1,55,60, F. X. Lu56, H.
J. Lu22, J. G. Lu1,55, X. L. Lu1, Y. Lu7, Y. P. Lu1,55, Z. H. Lu1,60, C. L.
Luo39, M. X. Luo77, T. Luo11,f, X. L. Luo1,55, X. R. Lyu60, Y. F. Lyu41, F. C.
Ma38, H. L. Ma1, L. L. Ma47, M. M. Ma1,60, Q. M. Ma1, R. Q. Ma1,60, R. T.
Ma60, X. Y. Ma1,55, Y. Ma44,g, F. E. Maas18, M. Maggiora71A,71C, S. Maldaner4,
S. Malde66, Q. A. Malik70, A. Mangoni27B, Y. J. Mao44,g, Z. P. Mao1, S.
Marcello71A,71C, Z. X. Meng63, J. G. Messchendorp13,61, G. Mezzadri28A, H.
Miao1,60, T. J. Min40, R. E. Mitchell26, X. H. Mo1,55,60, N. Yu. Muchnoi12,b,
Y. Nefedov34, F. Nerling18,d, I. B. Nikolaev12,b, Z. Ning1,55, S. Nisar10,l,
Y. Niu 47, S. L. Olsen60, Q. Ouyang1,55,60, S. Pacetti27B,27C, X. Pan11,f, Y.
Pan54, A. Pathak32, Y. P. Pei68,55, M. Pelizaeus4, H. P. Peng68,55, K.
Peters13,d, J. L. Ping39, R. G. Ping1,60, S. Plura33, S. Pogodin34, V.
Prasad68,55, F. Z. Qi1, H. Qi68,55, H. R. Qi58, M. Qi40, T. Y. Qi11,f, S.
Qian1,55, W. B. Qian60, Z. Qian56, C. F. Qiao60, J. J. Qin69, L. Q. Qin14, X.
P. Qin11,f, X. S. Qin47, Z. H. Qin1,55, J. F. Qiu1, S. Q. Qu58, K. H.
Rashid70, C. F. Redmer33, K. J. Ren37, A. Rivetti71C, V. Rodin61, M. Rolo71C,
G. Rong1,60, Ch. Rosner18, S. N. Ruan41, A. Sarantsev34,c, Y. Schelhaas33, C.
Schnier4, K. Schoenning72, M. Scodeggio28A,28B, K. Y. Shan11,f, W. Shan23, X.
Y. Shan68,55, J. F. Shangguan52, L. G. Shao1,60, M. Shao68,55, C. P. Shen11,f,
H. F. Shen1,60, W. H. Shen60, X. Y. Shen1,60, B. A. Shi60, H. C. Shi68,55, J.
Y. Shi1, Q. Q. Shi52, R. S. Shi1,60, X. Shi1,55, J. J. Song19, W. M. Song32,1,
Y. X. Song44,g, S. Sosio71A,71C, S. Spataro71A,71C, F. Stieler33, P. P. Su52,
Y. J. Su60, G. X. Sun1, H. Sun60, H. K. Sun1, J. F. Sun19, L. Sun73, S. S.
Sun1,60, T. Sun1,60, W. Y. Sun32, Y. J. Sun68,55, Y. Z. Sun1, Z. T. Sun47, Y.
H. Tan73, Y. X. Tan68,55, C. J. Tang51, G. Y. Tang1, J. Tang56, L. Y Tao69, Q.
T. Tao24,h, M. Tat66, J. X. Teng68,55, V. Thoren72, W. H. Tian49, Y.
Tian29,60, I. Uman59B, B. Wang68,55, B. Wang1, B. L. Wang60, C. W. Wang40, D.
Y. Wang44,g, F. Wang69, H. J. Wang36,j,k, H. P. Wang1,60, K. Wang1,55, L. L.
Wang1, M. Wang47, M. Z. Wang44,g, Meng Wang1,60, S. Wang11,f, S. Wang14, T.
Wang11,f, T. J. Wang41, W. Wang56, W. H. Wang73, W. P. Wang33,18,68,55, X.
Wang44,g, X. F. Wang36,j,k, X. L. Wang11,f, Y. Wang58, Y. D. Wang43, Y. F.
Wang1,55,60, Y. H. Wang45, Y. Q. Wang1, Yaqian Wang17,1, Z. Wang1,55, Z. Y.
Wang1,60, Ziyi Wang60, D. H. Wei14, F. Weidner65, S. P. Wen1, D. J. White64,
U. Wiedner4, G. Wilkinson66, M. Wolke72, L. Wollenberg4, J. F. Wu1,60, L. H.
Wu1, L. J. Wu1,60, X. Wu11,f, X. H. Wu32, Y. Wu68, Y. J Wu29, Z. Wu1,55, L.
Xia68,55, T. Xiang44,g, D. Xiao36,j,k, G. Y. Xiao40, H. Xiao11,f, S. Y. Xiao1,
Y. L. Xiao11,f, Z. J. Xiao39, C. Xie40, X. H. Xie44,g, Y. Xie47, Y. G.
Xie1,55, Y. H. Xie6, Z. P. Xie68,55, T. Y. Xing1,60, C. F. Xu1,60, C. J. Xu56,
G. F. Xu1, H. Y. Xu63, Q. J. Xu16, X. P. Xu52, Y. C. Xu75, Z. P. Xu40, F.
Yan11,f, L. Yan11,f, W. B. Yan68,55, W. C. Yan78, H. J. Yang48,e, H. L.
Yang32, H. X. Yang1, Tao Yang1, Y. F. Yang41, Y. X. Yang1,60, Yifan Yang1,60,
M. Ye1,55, M. H. Ye8, J. H. Yin1, Z. Y. You56, B. X. Yu1,55,60, C. X. Yu41, G.
Yu1,60, T. Yu69, X. D. Yu44,g, C. Z. Yuan1,60, L. Yuan2, S. C. Yuan1, X. Q.
Yuan1, Y. Yuan1,60, Z. Y. Yuan56, C. X. Yue37, A. A. Zafar70, F. R. Zeng47, X.
Zeng6, Y. Zeng24,h, X. Y. Zhai32, Y. H. Zhan56, A. Q. Zhang1,60, B. L.
Zhang1,60, B. X. Zhang1, D. H. Zhang41, G. Y. Zhang19, H. Zhang68, H. H.
Zhang56, H. H. Zhang32, H. Q. Zhang1,55,60, H. Y. Zhang1,55, J. L. Zhang74, J.
Q. Zhang39, J. W. Zhang1,55,60, J. X. Zhang36,j,k, J. Y. Zhang1, J. Z.
Zhang1,60, Jianyu Zhang1,60, Jiawei Zhang1,60, L. M. Zhang58, L. Q. Zhang56,
Lei Zhang40, P. Zhang1, Q. Y. Zhang37,78, Shuihan Zhang1,60, Shulei Zhang24,h,
X. D. Zhang43, X. M. Zhang1, X. Y. Zhang47, X. Y. Zhang52, Y. Zhang66, Y. T.
Zhang78, Y. H. Zhang1,55, Yan Zhang68,55, Yao Zhang1, Z. H. Zhang1, Z. L.
Zhang32, Z. Y. Zhang41, Z. Y. Zhang73, G. Zhao1, J. Zhao37, J. Y. Zhao1,60, J.
Z. Zhao1,55, Lei Zhao68,55, Ling Zhao1, M. G. Zhao41, S. J. Zhao78, Y. B.
Zhao1,55, Y. X. Zhao29,60, Z. G. Zhao68,55, A. Zhemchugov34,a, B. Zheng69, J.
P. Zheng1,55, Y. H. Zheng60, B. Zhong39, C. Zhong69, X. Zhong56, H. Zhou47, L.
P. Zhou1,60, X. Zhou73, X. K. Zhou60, X. R. Zhou68,55, X. Y. Zhou37, Y. Z.
Zhou11,f, J. Zhu41, K. Zhu1, K. J. Zhu1,55,60, L. X. Zhu60, S. H. Zhu67, S. Q.
Zhu40, T. J. Zhu74, W. J. Zhu11,f, Y. C. Zhu68,55, Z. A. Zhu1,60, J. H. Zou1,
J. Zu68,55 (BESIII Collaboration) 1 Institute of High Energy Physics, Beijing
100049, People’s Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Beijing Institute of Petrochemical Technology, Beijing 102617, People’s
Republic of China
4 Bochum Ruhr-University, D-44780 Bochum, Germany
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 Central South University, Changsha 410083, People’s Republic of China
8 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
9 China University of Geosciences, Wuhan 430074, People’s Republic of China
10 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind
Road, 54000 Lahore, Pakistan
11 Fudan University, Shanghai 200433, People’s Republic of China
12 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
13 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
14 Guangxi Normal University, Guilin 541004, People’s Republic of China
15 Guangxi University, Nanning 530004, People’s Republic of China
16 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
17 Hebei University, Baoding 071002, People’s Republic of China
18 Helmholtz Institute Mainz, Staudinger Weg 18, D-55099 Mainz, Germany
19 Henan Normal University, Xinxiang 453007, People’s Republic of China
20 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
21 Henan University of Technology, Zhengzhou 450001, People’s Republic of
China
22 Huangshan College, Huangshan 245000, People’s Republic of China
23 Hunan Normal University, Changsha 410081, People’s Republic of China
24 Hunan University, Changsha 410082, People’s Republic of China
25 Indian Institute of Technology Madras, Chennai 600036, India
26 Indiana University, Bloomington, Indiana 47405, USA
27 INFN Laboratori Nazionali di Frascati , (A)INFN Laboratori Nazionali di
Frascati, I-00044, Frascati, Italy; (B)INFN Sezione di Perugia, I-06100,
Perugia, Italy; (C)University of Perugia, I-06100, Perugia, Italy
28 INFN Sezione di Ferrara, (A)INFN Sezione di Ferrara, I-44122, Ferrara,
Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
29 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
30 Institute of Physics and Technology, Peace Avenue 54B, Ulaanbaatar 13330,
Mongolia
31 Instituto de Alta Investigacion, Universidad de Tarapaca, Casilla 7D,
Arica, Chile
32 Jilin University, Changchun 130012, People’s Republic of China
33 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
34 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
35 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
36 Lanzhou University, Lanzhou 730000, People’s Republic of China
37 Liaoning Normal University, Dalian 116029, People’s Republic of China
38 Liaoning University, Shenyang 110036, People’s Republic of China
39 Nanjing Normal University, Nanjing 210023, People’s Republic of China
40 Nanjing University, Nanjing 210093, People’s Republic of China
41 Nankai University, Tianjin 300071, People’s Republic of China
42 National Centre for Nuclear Research, Warsaw 02-093, Poland
43 North China Electric Power University, Beijing 102206, People’s Republic of
China
44 Peking University, Beijing 100871, People’s Republic of China
45 Qufu Normal University, Qufu 273165, People’s Republic of China
46 Shandong Normal University, Jinan 250014, People’s Republic of China
47 Shandong University, Jinan 250100, People’s Republic of China
48 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
49 Shanxi Normal University, Linfen 041004, People’s Republic of China
50 Shanxi University, Taiyuan 030006, People’s Republic of China
51 Sichuan University, Chengdu 610064, People’s Republic of China
52 Soochow University, Suzhou 215006, People’s Republic of China
53 South China Normal University, Guangzhou 510006, People’s Republic of China
54 Southeast University, Nanjing 211100, People’s Republic of China
55 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
56 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
57 Suranaree University of Technology, University Avenue 111, Nakhon
Ratchasima 30000, Thailand
58 Tsinghua University, Beijing 100084, People’s Republic of China
59 Turkish Accelerator Center Particle Factory Group, (A)Istinye University,
34010, Istanbul, Turkey; (B)Near East University, Nicosia, North Cyprus,
Mersin 10, Turkey
60 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
61 University of Groningen, NL-9747 AA Groningen, The Netherlands
62 University of Hawaii, Honolulu, Hawaii 96822, USA
63 University of Jinan, Jinan 250022, People’s Republic of China
64 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
65 University of Muenster, Wilhelm-Klemm-Strasse 9, 48149 Muenster, Germany
66 University of Oxford, Keble Road, Oxford OX13RH, United Kingdom
67 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
68 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
69 University of South China, Hengyang 421001, People’s Republic of China
70 University of the Punjab, Lahore-54590, Pakistan
71 University of Turin and INFN, (A)University of Turin, I-10125, Turin,
Italy; (B)University of Eastern Piedmont, I-15121, Alessandria, Italy;
(C)INFN, I-10125, Turin, Italy
72 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
73 Wuhan University, Wuhan 430072, People’s Republic of China
74 Xinyang Normal University, Xinyang 464000, People’s Republic of China
75 Yantai University, Yantai 264005, People’s Republic of China
76 Yunnan University, Kunming 650500, People’s Republic of China
77 Zhejiang University, Hangzhou 310027, People’s Republic of China
78 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
b Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
c Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia
d Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
e Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology,
Ministry of Education; Shanghai Key Laboratory for Particle Physics and
Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240,
People’s Republic of China
f Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and
Institute of Modern Physics, Fudan University, Shanghai 200443, People’s
Republic of China
g Also at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
h Also at School of Physics and Electronics, Hunan University, Changsha
410082, China
i Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China
j Also at Frontiers Science Center for Rare Isotopes, Lanzhou University,
Lanzhou 730000, People’s Republic of China
k Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou
730000, People’s Republic of China
l Also at the Department of Mathematical Sciences, IBA, Karachi , Pakistan
m Now at Zhejiang Jiaxing Digital City Laboratory Co., Ltd, Jiaxing 314051,
People’s Republic of China

###### Abstract

Based on electron positron collision data collected with the BESIII detector
operating at the BEPCII storage rings, the differential cross sections of
inclusive $\pi^{0}$ and $K_{S}^{0}$ production as a function of hadron
momentum, normalized by the total cross section of the $e^{+}e^{-}\to$ hadrons
process, are measured at six center-of-mass energies from 2.2324 to 3.6710
$\mathrm{GeV}$. Our results with a relative hadron energy coverage from 0.1 to
0.9 significantly deviate from several theoretical calculations based on
existing fragmentation functions.

Color confinement at long distance is one of the fundamental properties of
Quantum Chromodynamics (QCD), which is the underlying theory to describe the
strong interactions between quarks and gluons. Because of confinement, quarks
and gluons produced in hard scattering processes will ultimately become
colorless hadrons. The transition from quarks $q$, antiquarks $\bar{q}$ and
gluons $g$ to hadrons $h$, which occurs at a scale with a small momentum
transfer, must be treated non-perturbatively due to the large strong-coupling
constant $\alpha_{s}$ Schmelling:1994py ; dEnterria:2022hzv . Consequently,
the fragmentation function (FF) $D^{h}_{q,\bar{q},g}(z,\mu_{F})$ is used to
describe the non-perturbative long-distance behavior associated with the
hadronization process Metz:2016swz . Here, $\mu_{F}$ is the factorization
scale used to factorize the cross section in terms of the convolution of
perturbative hard-part coefficients and non-perturbative fragmentation
functions, and is usually set to be the center-of-mass (c.m.) energy
($\sqrt{s}$) in $e^{+}e^{-}$ experiments. The term $z\equiv
2\sqrt{p_{h}^{2}c^{2}+M^{2}_{h}c^{4}}/\sqrt{s}$ is the relative hadron energy,
where $p_{h}$ and $M_{h}$ are the momentum and mass of hadron $h$,
respectively. Single inclusive $e^{+}e^{-}$ annihilation, $e^{+}e^{-}\to h+X$,
where $h$ is an identified hadron and $X$ represents everything else, provides
a clean way to study FFs ParticleDataGroup:2020ssz . A typical experimental
observable is

$\frac{1}{\sigma(e^{+}e^{-}\rightarrow\textmd{hadrons})}\frac{\textmd{d}\sigma(e^{+}e^{-}\rightarrow
h+X)}{\textmd{d}p_{h}},$ (1)

where $\sigma(e^{+}e^{-}\rightarrow\textmd{hadrons})$ is the total cross
section for $e^{+}e^{-}$ annihilation to all possible hadronic final states
(referred to as inclusive hadronic events hereafter). At the leading order in
$\alpha_{s}$, the observable can be interpreted as
$\sum_{q}e_{q}^{2}[D_{q}^{h}(z,\sqrt{s})+D_{\bar{q}}^{h}(z,\sqrt{s})]$, where
$e_{q}$ is the fractional charge of the quark $q$. As summarized in Ref.
ParticleDataGroup:2020ssz , extensive high-precision measurements of this
observable have been made in $e^{+}e^{-}$ experiments in the c.m. energy range
above $10~{}\mathrm{GeV}$, but measurements in the region from 3.6 to 5.2
$\mathrm{GeV}$ have statistical uncertainties ranging from 20% to 50%
DASP:1978ftr ; PLUTO:1977qwj .

On the other hand, remarkable progress has been made in the experimental study
of nucleon structure by taking advantage of the semi-inclusive deep-inelastic
scattering (SIDIS) process. Due to its explicit dependence on both the
identified initial and final state hadrons, the SIDIS process plays a crucial
role in probing specific quark flavors. In particular, the FFs act as the
weights used to perform flavor separation in the initial state. For example,
precise kaon FFs might be the key to understand the puzzle of the strange-
quark polarization inside a longitudinally polarized proton Leader:2011tm ;
Leader:2014uua . The momentum-transfer $Q$ (equivalent to $\sqrt{s}$ in
$e^{+}e^{-}$ annihilation) for the existing SIDIS data from fixed-target
experiments like the ones at Jefferson Lab, COMPASS and HERMES Deur:2018roz
covers the range from 1 to 10 GeV, where only sparse single inclusive
$e^{+}e^{-}$ annihilation data exist. Moreover, the proposed Electron-Ion
Collider (EIC) and the Electron-Ion Collider in China (EicC) Accardi:2012qut ;
Anderle:2021wcy  would provide very high precision structure function
measurements, which, in order to determine precise parton distribution
functions (PDFs) for various quark flavors, would put unprecedented
requirements on the precision of FFs for the $Q$ value down to 1 GeV and
almost complete $z$ coverage from 0 to 1. Therefore, comprehensive
investigations on the single inclusive annihilation with identified pion and
kaon final states, especially in the region not well covered by previous data,
will provide valuable input for the nucleon structure study.

In this Letter, the processes $e^{+}e^{-}\to\pi^{0}/K_{S}^{0}+X$ are studied
at six c.m. energies from 2.2324 to $3.6710~{}\mathrm{GeV}$, with a $z$
coverage from 0.1 to 0.9. Data sets used in this research were collected with
the BESIII detector BESIII:2009fln  running at BEPCII. The detector has a
geometrical acceptance of 93% of the 4$\pi$ solid angle for the relatively
stable final state particles. It is based on a superconducting solenoid magnet
with a main drift chamber (MDC) as a central tracking system, plastic
scintillators as a time-of-flight system, CsI crystals as an electromagnetic
calorimeter (EMC) and resistive plate chambers as a muon system.

Monte Carlo (MC) simulations based on geant4 software GEANT4:2002zbu , which
includes the geometric description of the BESIII detector and implements the
interactions between the final-state particles and the detector, are used to
optimize the event selection criteria, estimate the number of residual
background events, and determine the correction factors accounting for the
efficiency loss and initial-state radiation (ISR) effects. The inclusive
hadronic events are simulated with the luarlw generator luarlwA ; luarlwB ;
BESIII:2021wib , where among others the signal processes
$e^{+}e^{-}\to\pi^{0}/K_{S}^{0}+X$ are contained. Background MC samples of the
processes $e^{+}e^{-}\to e^{+}e^{-}$, $\mu^{+}\mu^{-}$, and $\gamma\gamma$ are
generated by babayaga3.5 CarloniCalame:2000pz . At
$\sqrt{s}=3.6710~{}\mathrm{GeV}$, the $e^{+}e^{-}\to\tau^{+}\tau^{-}$ process
is simulated via the kkmc Jadach:1999vf  generator, in which the decay of
$\tau$ lepton is modeled by evtgen Lange:2001uf . The background events from
the two-photon processes are simulated by the corresponding dedicated MC
generators, as detailed in Ref. BESIII:2021wib . In addition, the beam-
associated events are estimated by a side-band method.

In this analysis, the inclusive hadronic events are selected first, from which
the $e^{+}e^{-}\to\pi^{0}/K_{S}^{0}+X$ events are identified by reconstructing
a $\pi^{0}$ or $K_{S}^{0}$ meson. We begin by removing the dominant background
processes, i.e., the $e^{+}e^{-}\to e^{+}e^{-}$ and
$e^{+}e^{-}\to\gamma\gamma$ events. In each remaining event, the good charged
hadronic tracks are identified with the same selection criteria as described
in Ref. BESIII:2021wib , and events with at least two good tracks are kept for
further analysis. For events with two or three good charged tracks, additional
requirements on the charged tracks and the showers in the EMC are implemented
to further suppress the background related to the quantum electrodynamics
process. Events with more than three good charged tracks are retained directly
without any additional requirements. The details of the selection of the
inclusive hadronic events are presented in Ref. BESIII:2021wib . The numbers
of inclusive hadronic ($N_{\textmd{had}}^{\textmd{tot}}$) events and residual
background ($N_{\textmd{bkg}}$) events, as well as the integrated luminosity
of each data sample, are summarized in Table 1.

Table 1: The integrated luminosities and the total observed hadronic and residual background events at various c.m. energy points. $\sqrt{s}$ (GeV) |  $\mathcal{L}$ ($\rm pb^{-1}$) | $N_{\textmd{had}}^{\textmd{tot}}$ | $N_{\textmd{bkg}}$
---|---|---|---
2.2324 | 2.645 | 83227 | 2041
2.4000 | 3.415 | 96627 | 2331
2.8000 | 3.753 | 83802 | 2075
3.0500 | 14.89 | 283822 | 7719
3.4000 | 1.733 | 32202 | 843
3.6710 | 4.628 | 75253 | 6461

A $\pi^{0}$ candidate is reconstructed via the decay $\pi^{0}\to\gamma\gamma$.
Photon candidates are identified using showers in the EMC. The deposited
energy of each shower must be greater than 25 $\mathrm{MeV}$ in the barrel
region ($|\cos\theta|<0.80$) and 50 $\mathrm{MeV}$ in the end-cap region
($0.86<|\cos\theta|<0.92$), where $\theta$ is the polar angle defined with
respect to the symmetry axis of the MDC. The shower is required to be
separated by more than 10 degrees from the closest charged track to eliminate
those produced by the charged particles. The difference between the EMC time
and the event start time has to be within $[0,700]$ ns to suppress electronic
noise and showers unrelated to the event. All combinations of two photons are
used to form $\pi^{0}$ candidates. To suppress background due to the
miscombination of photons, requirements are made on the polar angle of one
photon in the helicity frame of the $\pi^{0}$ candidate ($\theta_{\gamma}$).
For $\pi^{0}$ candidates with momentum less than $0.3~{}\mathrm{GeV}/c$,
$|\cos\theta_{\gamma}|$ is required to be less than 0.8, while for those with
momentum larger than $0.3~{}\mathrm{GeV}/c$, it must be less than 0.95. Each
$\pi^{0}$ candidate in an event is counted as a candidate of the separate
inclusive $\pi^{0}$ event, and the fraction of the observed hadronic events
containing more than one $\pi^{0}$ meson varies from 42% to 50% at the c.m.
energies from $\sqrt{s}=2.2324$ to $3.6710~{}\mathrm{GeV}$.

A $K_{S}^{0}$ candidate is formed by combining a pair of oppositely charged
tracks. The two tracks are required to satisfy $|\cos\theta|<0.93$. Due to the
relatively long lifetime of the $K_{S}^{0}$ meson, the distance of closest
approach of these charged tracks from the interaction point must be less than
30 cm along and 10 cm perpendicular to the beam direction. As a result of the
different requirements, the charged track here is not necessarily one of the
good charged tracks identified previously. To increase the number of events,
no particle identification is applied. Further, to select $K_{S}^{0}$ signal
events, the production and decay vertices are reconstructed, and the decay
length between these two vertices is required to be at least twice its
uncertainty. Each $K_{S}^{0}$ candidate in an event is counted as a candidate
of the separate inclusive $K_{S}^{0}$ event. In this measurement, fewer than
1% observed hadronic events contain more than one $K_{S}^{0}$ meson.

After imposing the above selection criteria, the residual contributions to the
mass spectra of the photon and charged-pion pairs, $M(\gamma\gamma)$ and
$M(\pi^{+}\pi^{-})$, from lepton-pair production, two-photon processes, and
beam-associated events are less than 0.1% in both signal processes. The
dominant background is caused by the miscombinations of the corresponding
daughter particles, which are reproduced by the inclusive hadronic MC samples
and well described by a polynomial.

The $M(\gamma\gamma)$ and $M(\pi^{+}\pi^{-})$ spectra are divided into the
momentum intervals with a step of $\Delta
p_{\pi^{0}/K_{S}^{0}}=0.1~{}\mathrm{GeV}/c$, which is 5 times larger than the
corresponding momentum resolutions. Unbinned maximum likelihood fits are
performed on the the $M(\gamma\gamma)$ and $M(\pi^{+}\pi^{-})$ spectra
obtained in each momentum interval to determine the corresponding numbers of
events, i.e., $N_{\pi^{0}}^{\textmd{obs}}$ and $N_{K_{S}^{0}}^{\textmd{obs}}$.
For the $e^{+}e^{-}\to\pi^{0}+X$ candidates, the signal is described by a
Crystal Ball function Skwarnicki:1986xj , while the background is
parameterized by a second-order Chebychev polynomial. For the $e^{+}e^{-}\to
K_{S}^{0}+X$ process, the signal is modeled by a double Gaussian function, and
the background is described by a first-order Chebychev polynomial. Figure 1
illustrates the fit results of the $\pi^{0}$ and $K_{S}^{0}$ candidates with
$p_{\gamma\gamma/\pi^{+}\pi^{-}}\in(0.4,0.5)~{}\mathrm{GeV}/c$ from the data
sample at $\sqrt{s}=2.8000~{}\mathrm{GeV}$. The summary of
$N_{\pi^{0}}^{\textmd{obs}}$ and $N_{K_{S}^{0}}^{\textmd{obs}}$ obtained in
each momentum range at each c.m. energy is given in the Supplemental Material
suppstuff .

\begin{overpic}[width=99.73074pt]{fit_gamgam2_bin4.eps}
\end{overpic}\begin{overpic}[width=99.73074pt]{fit_pipi2_bin4.eps}
\end{overpic}

Figure 1: The M($\gamma\gamma$) and M($\pi\pi$) distributions for $\pi^{0}$
(left) and $K_{S}^{0}$ (right) candidates, respectively, with
$p_{\gamma\gamma/\pi^{+}\pi^{-}}\in(0.4,0.5)~{}\mathrm{GeV}/c$ at
$\sqrt{s}=2.8000~{}\mathrm{GeV}$. The fit results are overlaid. The black
points with error bars are data. The black solid curves are the sum of fit
functions, while the red dashed and blue dotted curves represent the signal
and background, respectively. The pull variable $\chi$, defined as the
residual between the data and the total fit function, normalized by the
uncertainty of the data, is shown on the bottom of the figures.

In practice, the normalized differential cross section for the inclusive
production of any identified hadron, namely Eq. (1), is determined with

$\frac{N_{h}^{\textmd{obs}}}{N_{\textmd{had}}^{\textmd{obs}}}\frac{1}{\Delta
p_{h}}f_{h},$ (2)

where
$N_{\textmd{had}}^{\textmd{obs}}=N_{\textmd{had}}^{\textmd{tot}}-N_{\textmd{bkg}}$
is the net number of observed hadronic events in $e^{+}e^{-}$ annihilation at
a given c.m. energy, $N_{h}^{\textmd{obs}}$ is that of the
$e^{+}e^{-}\rightarrow h+X$ events within a specific momentum range $\Delta
p_{h}$, and $f_{h}$ is a correction factor which accounts for the effects
caused by the limited detector acceptance, the selection criteria, ISR, and
vacuum polarization. Since both $N_{\textmd{had}}^{\textmd{obs}}$ and
$N_{h}^{\textmd{obs}}$ are obtained from the same data sample, the integrated
luminosity usually used in the cross section measurement cancels.

In this analysis, the correction factor, $f_{h}$, is obtained from the signal
MC sample and is given by

$f_{h}=\frac{\bar{N}_{h}^{\textmd{tru}}(\textmd{off})}{\bar{N}_{\textmd{had}}^{\textmd{tru}}(\textmd{off})}\bigg{/}\frac{\bar{N}_{h}^{\textmd{obs}}(\textmd{on})}{\bar{N}_{\textmd{had}}^{\textmd{obs}}(\textmd{on})},$
(3)

where the variable “$\bar{N}$” denotes the numbers of events determined from
the signal MC sample, either at the detector observed level, similar to the
experimental data, with superscript “obs” or at the generation level with
superscript “tru”. The terms “on” and “off” in the parentheses indicate that
the corresponding quantities are obtained from signal MC samples with and
without simulating the ISR effects, respectively.
$\bar{N}_{h}^{\textmd{obs}}(\textmd{on})$ is determined by applying the same
fit procedure on the signal MC events as data. By definition, $f_{h}$
compensates the event lost caused by the limited acceptance of the BESIII
detector in the small polar angle region suppstuff . In this analysis, the
correction factor $f_{h}$ is by far dominated by the detection efficiency.
Calculated results of $f_{h}$ in different momentum ranges for $\pi^{0}$ and
$K_{S}^{0}$ are presented in the Supplemental Material suppstuff .

The systematic uncertainties of the normalized differential cross section
measurements mainly originate from the differences between the signal MC and
data samples, the reconstruction efficiencies of the $\pi^{0}$ and $K_{S}^{0}$
candidates, the fits to the events in the $M(\gamma\gamma)$ and
$M(\pi^{+}\pi^{-})$ bins, and the MC simulation model of the inclusive
hadronic events. To estimate the uncertainty caused by the imperfect
simulation of various kinematic variables of the signal events, all the
selection criteria are separately varied to be larger or smaller than their
nominal values by one time their resolutions, and the maximum changes of the
normalized differential cross sections are taken as the systematic
uncertainties. The uncertainty in the photon detection efficiency is estimated
to be 1% per photon BESIII:2018rdg , therefore 2% is taken as the systematic
uncertainty due to the $\pi^{0}$ reconstruction efficiency where the complete
correlation between the detection of the two photons is assumed. The momentum-
dependent systematic uncertainties due to the $K_{S}^{0}$ reconstruction
efficiency are obtained by applying a dedicated weighting procedure
BESIII:2015jmz , that incorporates weights due to the charged particle
tracking efficiency and the decay length requirement.

The uncertainties due to the fits on the events in the $M(\gamma\gamma)$ and
$M(\pi^{+}\pi^{-})$ intervals are examined by using alternative signal and
background shapes. The alternative signal shape of $e^{+}e^{-}\to\pi^{0}+X$ is
taken as the shape from the signal MC sample, while that for $e^{+}e^{-}\to
K_{S}^{0}+X$ is chosen as a single Gaussian function. The alternative
background shapes are obtained by varying the order of the Chebychev
polynomials. The relative differences from the original differential cross
sections are taken as the corresponding systematic uncertainties.

The dominant source of systematic uncertainty is the MC simulation model of
the inclusive hadronic events. The generation fractions of the exclusive
processes containing $\pi^{0}$ and $K_{S}^{0}$, which make up the inclusive
process, directly affect the correction factors $f_{\pi^{0}}$ and
$f_{K_{S}^{0}}$. To address the corresponding uncertainty, the hybrid
generator, which was developed in Ref. Ping:2016pms  and improved in Ref.
BESIII:2021wib , is used as an alternative MC model to generate the inclusive
hadronic events. In the hybrid generator, much knowledge of the allowed
exclusive processes in the c.m. energy region of BESIII has been incorporated,
including measured cross sections and production mechanisms. In addition, a
different simulation scheme of the ISR process is adopted BESIII:2021wib . The
changes of the correction factors $f_{\pi^{0}}$ and $f_{K_{S}^{0}}$ are
assigned as the systematic uncertainties. All these individual systematic
uncertainties are regarded as uncorrelated with each other and are summed in
quadrature.

\begin{overpic}[width=216.81pt]{TherExpPi0_17.eps} \end{overpic} Figure 2:
Normalized differential cross sections of the $e^{+}e^{-}\to\pi^{0}+X$
process. The points with error bars are the measured values, where the
uncertainties are the quadrature sum of the corresponding statistical and
systematic uncertainties. The bands or curves in red, green, blue, magenta,
and orange denote the NNFF, MAPFF, AKRS, ARS, and DSS calculations,
respectively, where only the former two cover $\pm 1\sigma$ limits. Normalized
differential cross sections as function of $z$ are shown in the Supplemental
Material suppstuff .  \begin{overpic}[width=216.81pt]{TherExpKS_20.eps}
\end{overpic} Figure 3: Normalized differential cross sections of the
$e^{+}e^{-}\to K_{S}^{0}+X$ process. The points with error bars are the
measured values, where the uncertainties are the quadrature sum of the
corresponding statistical and systematic uncertainties. The red band shows the
theoretical calculation from NNFF with $\pm 1\sigma$ limits and the orange
curve denotes the prediction of DDS. Normalized differential cross sections as
function of $z$ are shown in the Supplemental Material suppstuff

The normalized differential cross sections for the inclusive $\pi^{0}$ and
$K_{S}^{0}$ production in $e^{+}e^{-}$ annihilation at the six c.m. energies
are tabulated in the Supplemental Material suppstuff  and shown in Fig. 2 and
Fig. 3, respectively. Figure 2 also shows various theoretical predictions
extrapolated from different FFs determined from existing world data
ParticleDataGroup:2020ssz . The FFs are obtained with slightly different
assumptions and show the sensitivity of the prediction to assumptions about
the behavior at low-$z$ and different $\sqrt{s}$. ARS Anderle:2015lqa , AKRS
Anderle:2016czy  and NNFF1.0 Bertone:2017tyb  are all obtained from inclusive
annihilation data at NNLO accuracy. However, AKRS includes small-$z$
resummation, NNFF1.0 includes hadron-mass corrections and all of them have
different initial evolution scales and kinematics requirements on the data.
The MAPFF1.0 NLO study Khalek:2021gxf  contains low-$Q^{2}$ data from the
lepton-proton fixed-target experiments at HERMES HERMES:2012uyd  and COMPASS
COMPASS:2016xvm . The DSS NLO calculation deFlorian:2007aj  contains
low-$Q^{2}$ data from the lepton-proton fixed-target experiments at HERMES and
single inclusive production of proton-proton collisions. Figure 3 shows the
comparison of the normalized differential cross section of the inclusive
$K_{S}^{0}$ production with predictions using NNFF1.0 at NNLO precision and
DSS at NLO precision. In these comparisons, the disagreement is observed to
depend on both c.m. energy and hadron momentum. Here, the FFs are further away
from the kinematic region of the original data. One possible reason for these
discrepancies is that a calculation restricted to the leading twist may not be
sufficient at the BESIII energy scale. It may also be important to consider
quark mass and hadron mass correction effects Accardi:2014qda , and small-$z$
resummation effects Anderle:2016czy . Another problem may be with the
extrapolation of the FFs from existing $e^{+}e^{-}$ annihilation data at high
energy to the low-energy scale at BESIII. For instance, the predictions using
QCD-backward evolution for initial-state PDFs from high to low energies have
been found to deviate from the experimental measurements Caola:2010cy . Both
Fig. 2 and Fig. 3 show that BESIII data can be used to improve the fit
procedure to determine the FFs at the low energy scale. Also, the difference
between primary and secondary processes has to be taken into account as well.
Studies based on the signal MC samples show that the $\rho^{\pm}$ ($K^{*}$)
decay has contribution to the inclusive $\pi^{0}$ ($K_{S}^{0}$) production in
the c.m. energy region of this analysis. The results presented in this Letter
will be key to explore these possibilites as well as help to test collinear
perturbative QCD with data at the relatively low energy scale.

In summary, we have measured the normalized differential cross sections of the
$e^{+}e^{-}\to\pi^{0}/K_{S}^{0}+X$ processes, using data samples collected
from $\sqrt{s}=2.2324$ to $3.6710~{}\mathrm{GeV}$. The results obtained in
this work help to fill the region with $\sqrt{s}<10~{}\mathrm{GeV}$ where
precision $e^{+}e^{-}$ annihilation data have been rarely reported. The
results provide broad $z$ coverage from 0.1 to 0.9 with precision of around 3%
at $z\sim 0.4$. These results provide new ingredients for FF global data fits,
in which almost no single inclusive annihilation data measured in this special
energy region has been included.

The authors would like to thank Daniele Paolo Anderle and Hongxi Xing for
their valuable theoretical inputs. The BESIII Collaboration thanks the staff
of BEPCII, the IHEP computing center and the supercomputing center of USTC for
their strong support. This work is supported in part by National Key R&D
Program of China under Contracts Nos. 2020YFA0406300, 2020YFA0406400; National
Natural Science Foundation of China (NSFC) under Contracts Nos. 11635010,
11735014, 11835012, 11935015, 11935016, 11935018, 11961141012, 12022510,
12025502, 12035009, 12035013, 12192260, 12192261, 12192262, 12192263,
12192264, 12192265, 11335008, 11625523, 12035013, 11705192, 11950410506,
12061131003, 12105276, 12122509, 12205255; the Fundamental Research Funds for
the Central Universities, University of Science and Technology of China under
Contract No. WK2030000053; the Chinese Academy of Sciences (CAS) Large-Scale
Scientific Facility Program; Joint Large-Scale Scientific Facility Funds of
the NSFC and CAS under Contract No. U1832207, U1732263, U1832103, U2032111,
U2032105; the CAS Center for Excellence in Particle Physics (CCEPP); 100
Talents Program of CAS; China Postdoctoral Science Foundation under Contracts
No. 2019M662152, No. 2020T130636; The Institute of Nuclear and Particle
Physics (INPAC) and Shanghai Key Laboratory for Particle Physics and
Cosmology; ERC under Contract No. 758462; European Union’s Horizon 2020
research and innovation programme under Marie Sklodowska-Curie grant agreement
under Contract No. 894790; German Research Foundation DFG under Contracts Nos.
443159800, Collaborative Research Center CRC 1044, GRK 2149; Istituto
Nazionale di Fisica Nucleare, Italy; Ministry of Development of Turkey under
Contract No. DPT2006K-120470; National Science and Technology fund; National
Science Research and Innovation Fund (NSRF) via the Program Management Unit
for Human Resources & Institutional Development, Research and Innovation under
Contract No. B16F640076; Olle Engkvist Foundation under Contract No. 200-0605;
STFC (United Kingdom); Suranaree University of Technology (SUT), Thailand
Science Research and Innovation (TSRI), and National Science Research and
Innovation Fund (NSRF) under Contract No. 160355; The Royal Society, UK under
Contracts Nos. DH140054, DH160214; The Swedish Research Council; U. S.
Department of Energy under Contract No. DE-FG02-05ER41374.

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