# Quality Assurance Test of Silicon Photomultipliers and Electronic Boards for
STAR Event Plane Detector

Ming Shao  Yitao Wu  Zheng Liang  Kaifeng Shen  Zebo Tang  M.A. Lisa  R. Reed
G. Visser  Yongjie Sun  Yi Zhou  Jian Zhou  Guofeng Song  Dongdong Hu  Xu Wang
Xinjian Wang

###### Abstract

The event plane detector (EPD), installed in the Solenoid Tracker at the
Relativistic Heavy-Ion Collider located at the Brookhaven National Laboratory
is a plastic scintillator-based device that measures the reaction centrality
and event plane in the forward region of the relativistic heavy-ion
collisions. We used silicon photomultiplier (SiPM) arrays to detect the
photons produced in the scintillator via the fiber connection. Signals from
the SiPM arrays were amplified by the front-end electronic (FEE) board, and
sent to  the analog-to-digital converter (ADC) boards for further processing
via the receiver (RX) board. The full EPD system consisted of 24  super-
sectors (SSs); each SS was equipped with two SiPM boards, two FEE boards and
two RX boards, and they corresponded to 744 readout channels. All these boards
were mass produced at the University of Science and Technology of China, with
a dedicated quality assurance (QA) procedures applied to  identify any
problems before deployment. This article describes the details of the QA
method and the related test system. The QA test results are presented along
with the discussions.

##  1 Introduction

One of the main physics aims of the Relativistic Heavy-Ion Collider (RHIC) at
Brookhaven National Lab (BNL) is to discover and study the  properties of
quark-gluon plasma (QGP), which is a state of strongly acting matter that
exists only under extreme pressure and energy density [1, 2]. The phase
structure, especially the phase boundary and the anticipated critical end-
point (CEP), of the nuclear matter  is of particular interest to researchers.
During the beam energy scan (BES) program at RHIC there have been hints of a
CEP and first-order phase transition at center-of-mass energies ranging from 5
GeV to 30 GeV [3]. However, many key measurements for locating the CEP and
determining the phase transition were limited by the event plane resolution,
statistics, and centrality determination by the time projection chamber (TPC)
only, which may have caused unwanted auto-correlation in the analyses.
Therefore, in phase II of the BES program  (BES-II), it  was proposed to take
data with increased statistics and upgraded detectors in the forward region,
which would help to further investigate the nuclear phase structure.  To
explore the rich science of BES-II, the Soleniod Tracker at RHIC (STAR) was
upgraded with the event plane detector (EPD), which is a highly segmented
device with broad coverage at forward rapidity. The EPD allows for the precise
determination of event planes through measuring the hit positions of minimum-
ionizing particles (MIP), and provides a measure of collision centrality in a
kinematic range that avoids autocorrelations [4].

The EPD [5] is made up of plastic scintillator with wavelength shifting fibers
and SiPM for readout. It consists of two wheels that extend from a radius of
4.6 cm to 91 cm and are placed at $z=\pm 375$ cm relative to the center of
STAR, which corresponds to $5.1<|\eta|<2.1$. Each wheel is made up of 12
wedge-shaped super-sectors (SS) that consists of 2 rows of 15 tiles and a
single innermost tile, which gives a total of 31 tiles per SS, as depicted in
Figure 1. Thus there are 372 channels on each side and a total of 744
channels. The segmentation, size, and shape of the EPD design maximized the
event plane resolution, centrality estimation and flow harmonic measurements
[5, 6]. Embedded in the scintillator are the wavelength shifting (WLS) fibers
that guide the light from each tile up to a connector at the outer edge of the
SS. These WLS fibers were then coupled with two bundles of clear optical
fibers which were finally air-coupled to arrays of silicon photomultipliers
(SiPMs) that converted and amplified the photons generated by the charged
particle crossing the EPD.

Figure 1: EPD layout.

The construction of EPD began in 2016. A quarter of one EPD disc was installed
in STAR and commissioned in 2017. After verifying the main performance
characteristics, the full production of EPD was launched in early 2017 and
completed by the end of that year. During this campaign, the heavy-ion group
of the University of Science and Technology of China (USTC) had taken on the
task of producing all the front-end boards, including the SiPM boards, the
front-end electronics (FEE) boards and the receiver (RX) boards. To ensure the
high-quality functionality and safe operation of these boards, we had
developed a dedicated test stand and quality assurance (QA) procedure, which
consisted of a test platform, a graphical user interface (GUI) controller and
a web-based database. In section 2, we will give a brief introduction to these
electronic boards. And in sections 3 and 4, we will present the details of the
QA process and the test results of the EPD mass-production.

##  2 Electronics Boards

For each charged particle passing through the sensitive layer (plastic
scintillator) of the EPD, the scintillation light produced in the scintillator
will be detected by the SiPM arrays via a fiber connection. The output signals
from the SiPMs are then processed by the front-end electronic (FEE) board,
which is closely connected to the SiPM board. After pulse shaping, the signals
are sent to the analog-to-digital converter (ADC) boards for further
processing and analysis via the receiver (RX) board that matches the
electrical properties at the front and rear ends.

Hamamatsu MPPC (SiPM) S13360-1325PE was chosen as the photon sensor because of
its high quantum efficiency (QE) in the blue-green light wavelength range, low
noise rate, good linearity, intrinsic fast rise time and modest cost. The
sensitive area 1.3 mm $\times$ 1.3 mm fitted the size of clear fiber (1.1 mm
in diameter) quite well. The relatively small 25 $\rm\mu$m pixel pitch ensured
a linearity of up to approximately 1000 photons, corresponding to
approximately 10 minimum-ionizing particles (MIPs) hitting one EPD tile that
is considered the upper limit of the normal EPD operation [6].

The fiber-SiPM optical coupling was achieved by a properly designed connector,
spacer and SiPM assembly board, as illustrated in Figure 3. Each SiPM board is
2 cm $\times$ 5 cm in size, populated with 16 SiPMs (see also Figure 3), and
connected to a 16-fiber bundle. The spacer block is between the flat-faced
fiber bundle connector and the SiPM board, which relieved the SiPMs from the
stress of joining. No coupling silicone oil or grease was used, and the fibers
were air-coupled to the SiPMs to ensure long-term stability. A thermal sensor
was installed to the SiPM board to monitor the local temperature of the SiPMs,
which was used to compensate the bias voltage of the SiPM.

The SiPM board was connected to the FEE board using gold fingers, as shown in
Figure 3. In the lower-left corner of the plot a SiPM board was directly
plugged into the housing connector of the FEE board. The FEE board provides
SiPM with an adjustable bias voltage (typically approximately 56 V) and reads
back the output signal from SiPM, and the local temperature. The SiPM signal
was amplified by a "regulated common base" amplifier with low input impedance,
which maintained a more constant voltage on the SiPM for more constant gain
and gathered the charge out of the SiPM more quickly. After the first stage
amplifier the signal was shaped by an L-C shaper with a peaking time of 16 ns,
and then further amplified to drive the output. The bias voltage can be set
individually for each SiPM, with compensations from the temperature feedback
obtained from a single temperature sensor (negative temperature coefficient
(NTC) thermistor) on the SiPM board with an adjustable compensation slope. The
individual SiPM currents can be monitored with a 10 nA resolution and a range
of 41 uA. Meanwhile, the FEE board is capable of adjusting the baseline
offsets of the output signals within a certain range, which is required by the
ADC board used in STAR. By setting a specific digital-analog converter (DAC)
on the board to values between -128 and 127, the FEE circuit can tune the
current level to 0, which means a pedestal voltages 0 $\rm mV$ on 50
$\rm\Omega$ load at the RX board output. The SiPM board temperature can also
be monitored. Via on-board digital-to-analog converters (DACs) and ADCs, the
bias voltage and dark current are set and monitored remotely for each channel.

Control communications were performed by the I2C 111A multi-master, multi-
slave, single-ended, serial computer bus invented by Philips Semiconductor
(now NXP Semiconductors) bus, with 12 FEE boards on a single bus in the
installed system at STAR. For the test stand, a single FEE was controlled by
the I2C from the test computer and connected by means of a LinkUSB
222Copyright by iButtonLink LLC., East Troy, Wi USA interface under the
1-Wire333A trademark of Dallas Semiconductor Corp, Dallas, Texas, USA. bus
protocol. The ouput signals from the FEE board were sent in differential mode
to the RX board, which converted the differential signal to unipolar signal,
required to interface with the standard STAR ADC system. Differential signals
were used on the long cables between the FEE and RX boards because the cables
and connectors were compact, and helped avoid difficulties with the
interference and ground loops. The RX board outputs were connected to the ADC
boards by short coaxial cables.

Figure 2: Schematic of SiPM and fiber coupling

Figure 3: Top: SiPM card with gold finger, to be inserted in the FEE board,
Bottom: SiPM spacer

Figure 4: Left: FEE board; Right: RX board

##  3 Quality Assurance

The SiPMs, FEEs and RX boards play a critical role in converting photons from
the mounted SSs into clean electrical pulses for downstream digital
conversion. During the production, all these items were produced or
manufactured by different companies: SiPMs by Hamamatsu photonics, FEE boards
by E-TekNet Inc., RX and SiPM boards by Fastprint Circuit Tech. Co. Ltd, and
SiPM assembly by Edadoc Co. Ltd.444Their websites are listed as below,
Hamamatsu: https://www.hamamatsu.com/, E-TekNet: https://www.e-teckoffer.com,
Fastprint: www.chinafastprint.com, Edadoc: http://www.edadoc.com/. Besides
some basic quality checks were provided by the vendors, and additional checks
were required to ensure the quality of these products.

Given the channel count and EPD production schedule (accelerated to make full
use of the RHIC beamtime), approximately 1000 channels of the electronics
chain needed to be produced and thoroughly tested in only four months. Once
the most important aspects of the QA were understood, we needed to perform the
QA tests on complete SiPM-FEE-RX chains, which focused on the individual
components only if a problem was found.

First, we needed to ensure that SiPM could properly receive light from the
clear fiber, which meant the air coupling between the fiber and SiPM needed to
be correct. This could be achieved by controlling the position of SiPM on the
board and by controlling its height relative to the board surface. To ensure
uniform light coupling and to reduce the stress, the SiPMs were machine
soldered onto the board with 0.05mm tolerance (horizontal and vertical). Each
SiPM board was individually examined under a microscope555 Optical Gaging
Products (ogp) StarLiteTM 200. to check if there were any flux residues or
dust around the soldering point or if there were any visible scratches on the
surface of SiPMs. Most importantly, all the 16 SiPMs needed to be soldered
firmly and at the right positions; the height and parallelism of each SiPM was
measured and controlled, as shown in Figure 5. The microscope is able to
measure the distance between different points. Practically we chose 3 couples
of points along the edge of SiPM and the PCB surface respectively, and the
microscope gave the distances between each couple of points. The average of
the measurements is taken as the height of SiPM, the resolution of which is
about 10$\mu$m and is enough to guarantee the light tightness of the fiber-
SiPM coupling.

Figure 5: Left: SiPM assembled properly. Right: SiPM assembled badly. We
choose three points on the SiPM surface and the average distances to the
surface of PCB board as the measured height.

Second, each SiPM needs to function properly with the expected
characteristics, such as gain, noise, etc.

This preliminarily requires the FEE board to correctly communicate with the
data acquisition (DAQ) devices, provide HV for SiPM with controllable value,
and transmit signals from SiPM and process it as desired. After inserting the
SiPM board to the FEE board, we set different bias to measure dark current
through current monitor on FEE board and acquire noise spectrum of SiPM from
oscilloscope, in order to check the basic connection and functions of the
SiPM.

Gain is one of the most important parameters of SiPM and is influenced by the
bias HV and temperature. It is primarily determined by the overvoltage, which
is defined as the quantity of supplied bias voltage beyond the breakdown
voltage, $V_{br}$. The breakdown voltage is the bias voltage at which the
electric field strength in the depletion region of the SiPM is sufficient to
create a Geiger discharge. Also, $V_{br}$ should also be the bias voltage at
which the electrons in the SiPM stop multiplication during the avalanche
procedure. Thus, the overvoltage determines how much charge is produced during
an avalanche. The signal charge is calculated by the simple formula
$Q=C\cdot\Delta V$, where $C$ is the capacitance of SiPM and $\Delta V$ is the
overvoltage. The gain of SiPM, which is defined as the output charge of a
single photoelectron (SPE), increases linearly with the bias voltage under a
fixed temperature [7].

We have tried two methods to determine $V_{br}$. One was to measure the
voltage-current (U-I) curve; another was to directly acquire the Gain-$\Delta
V$ relationship. In the UI curve, where U is the supplied voltage, and I is
the dark current of SiPM, the current rises rapidly at $V_{br}$. Thus, the UI
curve, as shown in Figure 6, can be used to measure $V_{br}$. However,
different theories will give different values for $V_{br}$ [8]. Because of the
relatively low precision (0.01 $\mu$A) of the current monitor on the FEE
board, we choose the inflection point of UI curve as breakdown threshold, the
corresponding abscissa of the intersection point is determined as breakdown
voltage, $V_{br}$, and calculate $V_{op}=V_{br}+5$ V.

For a Gain-$\Delta V$ line, the $V_{br}$ can be measured by finding the
intersection point with $Gain=0$ line. Figure 7a shows a charge spectrum of
SiPM at 38${}^{\circ}C$, measured by a charge-to-digital converter (QDC) (CAEN
V965A, 25fC/ch). A multi-Gaussian fit was applied to the data, wherein the
first peak was the pedestal, and the other peak represented single and
multiple photoelectrons. The distance between the adjacent two peaks is the
gain of one photoelectron. Figure 7b depicts how the extracted gain of SiPM
increases linearly with the bias voltage, as expected. The bias voltage at
which the fit line crosses the X-axis is $V_{br}$.

Results from both the methods provide similar $V_{br}$ values. Although the
value from the U-I curve has a lower resolution than that from the
Gain-$\Delta V$ line, the U-I curve is much easier to measure, and the
resolution is sufficient for our QA. Thus, we finally choose to measure the UI
curve to get $V_{br}$ for each SiPM.

(a) U-I curve in the Cartesian coordinate system. (b) U-I curve in the
Logarithm coordinate system.

Figure 6: U-I Curve

(a) Charge spectrum of SiPM

(b) Gain versus bias voltage fit with a linear function

Figure 7: Breakdown voltage measured via gain variations. See text for
details.

$V_{br}$ of SiPM usually increases for higher temperature [9]. Therefore the
environment temperature will also influence the gain of SiPM. The thermistor
on the SiPM board can sense the local temperature change, which is used (by
the FEE) to compensate for the gain loss accordingly by adjusting the bias
voltage. To keep the SiPM operation stable at different temperature
conditions, the effect of gain compensation needs to be tested. The expected
compensation slope provided by the vendor is approximately 54 $\rm
mV/^{\circ}C$, as shown in Figure 8a. Compared with the measurement without
temperature compensation, the dependence of the gain on the temperature was
greatly suppressed, which also demonstrates the need for such corrections. It
was also found that the gain variation with temperature in Figure 8a had not
been fully corrected; this was because of the temperature dependence of the
input voltage of the common-base preamplifier. In our test, we calculated the
correct compensation slope parameter as 62.5 $\rm mV/^{\circ}C$, the
compensation effect under which is shown in Figure 8b. In our QA procedure the
temperature compensation parameter 62.5 $\rm mV/^{\circ}C$ was used and under
this condition, the operation voltage of each SiPM needed to fall into the 50
V to 60 V interval.

(a) No compensation versus incomplete compensation

(b) Incomplete compensation versus complete compensation

Figure 8: Temperature compensation

Energy resolution is another important characteristic of SiPM, especially for
the counting of photons. However the light yield is expected to be
approximately 100 photoelectrons for a MIP hitting the EPD, and the
fluctuation is huge (Landau fluctuation). Therefore, energy resolution is not
a crucial parameter in our research. The typical oscilloscope signal (in the
trace mode) for a single photoelectron (SPE) is shown in Figure 9a. Signals
for multiple photons are also shown, and the splitting of waveforms is
evident. By fitting the charge spectrum measured by the QDC, as in Figure 7a,
the energy resolution (or the signal-to-noise ratio) can be quantitatively
determined. The charge spectrum can also be obtained by directly analyzing the
recorded waveforms. To save on measurement time, we use a 16-channel
digitizer, instead of the oscilloscope, to record the signal waveform. Results
from both the oscilloscope and the digitizer are shown in Figure 9. Although
the resolution of SPE is a little poorer with the digitizer than the
oscilloscope, it is sufficient for our QA. We define the SPE energy resolution
as $R_{\rm{SPE}}=\sigma_{\rm{SPE}}/(\mu_{\rm{SPE}}-\mu_{\rm{PED}})$, where
$\sigma_{\rm{SPE}}$ is the Gaussian fit $\sigma$ of SPE; $\mu_{SPE}$ and
${\mu_{PED}}$ are the mean values of the SPE and pedestal peaks respectively.
Note that the resolutions of SPE and pedestal are similar, which shows an
unstable baseline as the main source of fluctuation. For our purpose $\sim
5\%$ contamination of the SPE peak from pedestal is a satisfactory criteria
which conservatively imply $R\lesssim 0.4$($\sim 2.5\sigma$ separation between
the two Gaussian peaks).

(a) Single photon signal from the oscilloscope

(b) Single photon signal from the digitizer; a small slope of the baseline is
visible in the waveform

(c) Spectrum from the oscilloscope

(d) Spectrum from the digitizer, with the background subtracted

Figure 9: Single photon signal

Besides maintaining the proper operation of SiPM, the FEE board can also
provide adjustable pedestals for each channel. In our QA, the range of the
pedestal needs to contain a zero value.

The last QA item is to check if unwanted noise was introduced into the signal
after being shaped and filtered by the electronic boards at the front end. By
analyzing the frequency spectrum of the signals and noises, we can check
whether any particular noise patterns appear. When measuring the noise
spectrum of a set of validated reference boards (SiPM board, FEE board and RX
board), the typical frequency spectrum is smoothly continuous (see Figure
10c), without any notable sharp peaks or spikes. Both the oscilloscope and
digitizer were used to acquire the noise spectrum. From Figure 10a, we can see
that the frequency spectrum measured by the oscilloscope shows clear spikes at
certain frequencies. However, these spikes can be seen as features of our
oscilloscope, because they exist even without any inputs, as in Figure 10b.
The measurement done by using the CAEN N6742 digitizer is shown in Figure 10c.
Although the result obtained from the digitizer is less accurate than that
obtained from the oscilloscope, it does not show any specific noise patterns.
We choose the digitizer in our QA to check the noise frequency spectrum, in
which no obvious peaks or differences from the reference spectrum are allowed.

(a) Electronic noise in oscilloscope

(b) Oscilloscope noise with no input signal

(c) Electronic noise in digitizer

Figure 10: Comparison of electronic noise between digitizer & oscilloscope

Note that the timing performance is not included in the QA procedure. The
production QA only detects production problems; therefore, most kinds of
defects (e.g., bad solder joint, wrong component value, etc.) would already
have been flagged by the tests conducted. Meanwhile, the timing performance of
the EPD is limited by the scintillator and light collection than by the
electronics.

Figure 11: System Flow Chart

Figure 12: Light source system

(a) Waveforms obtainded while measuring the pedestal. When the bias was set to
46V, DAC at 127, 0 and -127, the waveforms showed the pedestal adjustable
range.

(b) Pedestal vs. DAC setting. A similar tendency and almost the same slope can
be seen from the Figure.

Figure 13: DAC measurement. Figure 14: U-I curves of 16 SiPMs on one board

##  4 QA Batch Test and Result

To accomplish the QA task in time for approximately 1000 channels of SiPMs and
the corresponding 60 (approx.) SiPM boards, FEE boards and RX boards, we
established a dedicated batch test setup, by which we could test 16 channels
quickly in one run. Figure 12 shows the flowchart of this system. A waveform
generator (LeCroy ArbStudio 1104) provided the trigger for the DAQ system and
drove the light-emitting diode (LED) light source. The peak emission
wavelength of the LED was approximately 470-480 nm, which matched the maximum
detection efficiency of the S13360-1325PE SiPM. A highly transparent hexagonal
quartz prism was used to diffuse the LED light injected from one end. A bundle
of 24 clear fibers receive the relatively uniform light output from the other
end of the quartz prism, with the optical coupling via the silicon oil.
Sixteen out of the 24 fibers from the inner part of the bundle were chosen to
illuminate the SiPMs under the test, with a thin air gap between them. The
LED, the quartz diffuser and the fibers were placed in a light-tight blackened
Aluminum box, whose photo and inner structure are shown in Figure 12. The bias
HV for SiPM/FEE was provided by the CAEN module N1470A, which was set at -90
V. Two Keithley 2231A-30-3 DC power suppliers offered required low voltages
(LV) for the FEE ($\rm\mp 6V$) and RX ($\rm\mp 6.5V$) boards. Despite their
relatively low accuracy the CAEN N6742 digitizer was chosen over the
oscilloscope to record the signal waveform. The main reason was that the
digitizer could measure 16 channels simultaneously, which was four times
faster than the oscilloscope and important for our QA task in the limited
time. Throughout the batch test, the measurements were performed at the
laboratory for the photoelectric device study, with controlled temperature of
approximately $\rm 25^{\circ}C$ and a relative humidity of approximately $\rm
50\%$.

All the QA tests described in the previous section were applied in a
sequential mode. The following steps were proformed:

(1) Before doing any other QA tests, we used a microscope to visually inspect
the mechanical aspects of the SiPMs, and check whether their positions on the
board fitted the spacer well. The height of SiPMs should not exceed the
thickness of the spacer (approx. 2 mm).

(2) Insert the SiPM Board to the FEE board, set bias HV at 40 V to simply
check communication of each part and functionality of the system.

(3) We first set the bias voltage at 46 V to ensure no SiPM signal would be
produced. Under this voltage, the pedestal offsets and noise spectrum could be
measured. We recorded 3000 waveforms with the offset DAC values setting at
-128, 0 and 127 respectively. These waveforms showed how the offsets changed
with the DAC settings. Figure 13a shows the baseline level of the waveform,
which depicts a small but none-zero slope. However this slope only affects the
determination of the pedestal slightly, and is negligible because the range of
the offsets is much larger. The dependence on the DAC setting of all 16
channels of the FEE board under test is shown in Figure 13b. Obviously, all
the 16 channels show similar pedestal-DAC relations, all of which contain the
pedestal 0, as required. Also, by performing a fast Fourier transform (FFT) on
the recorded waveforms, we can obtain the average noise frequency spectra, as
shown in Figure 10c. Different DAC settings do not affect the spectra because
they only change the baseline level.

(4) After measuring the noise spectrum, we scanned the bias voltage from 50.0
V to 63.0 V in steps of 0.5 V to obtain the U-I curves of SiPMs. A set of
typical U-I curves measured for 16 channels of one SiPM board are shown in
Figure 14. All the U-I curves exhibit consistent shapes, with only modest
differences on $V_{br}$. As discussed in section 3, we take the bias voltage
at $I=0.03\mu A$ as $V_{op}$, and require this voltage to be within the range
of 50.0 V to 60.0 V.

(5) We finally set the SiPM bias voltage to 60 V to analyze the signal
characteristics. In fact, 60 V is not the nominal $V_{op}$ for the SiPMs used
by EPD, but rather it is the maximum bias voltage permitted. However running
SiPMs under this voltage for a short period of time in the QA test should not
be a problem, and a high gain at 60 V also eases the analysis of the test
data. The charge of a signal is determined by integrating the waveform over
time, with the baseline subtracted. The effect of the slope of the baseline is
reduced by fitting the baseline with a linear function. Once the charge
distributions are obtained, they are fitted by a multi-Gaussian function to
resolve the position and resolution ($\sigma$) of the pedestal and multi-
photoelectron peaks. A gain greater than $5\times 10^{5}$ and an SPE energy
resolution less than $40\%$ are required. A typical measurement of the signal
waveform and the charge distribution are shown in Figure 9.

Figure 15: GUI control panel

Figure 16: Database GUI

To standardize and speedup the QA procedure, we developed a semi-automatic DAQ
program using a GUI control panel. The controller of this system is a C++
script program, which is based on the ROOT package. It was developed to
encapsulate the drivers of the FEE and the digitizer, and to provide a
graphical interface for easy use. As shown in Figure 16, there are five
sections in the GUI, which contains a user interface of the test information,
hardware settings, and action buttons.

A database was established to save the QA test data. Based on the QA criteria,
the qualities of the SiPMs and the FEE boards were examined, and the QA
results were stored. To organize the data in a manner that allows us to store,
query, sort, and manipulate the data, we developed a simple database called
DetectorDB, which was launched on our website. The DetectorDB provides a web
viewer to browse the records and test results in the database. It consists of
a main table (bootstrap-table), an image viewer (baguetteBox.js), and a ROOT
file browser (jsroot). An example of a page of the database is shown in Figure
16. For more details about the implementation, one may visit the corresponding
repository on GitHub666https://github.com/yatowoo/DetectorDB.

Figure 17: Operating Voltage by using the inflection point of UI curve; Blue:
Data from Hamamatsu in which $V_{op}$ is defined as $V_{br}+5V$, Red: Data
from U-I curve.

Figure 18: Integrated plot for the charge spectra of all SiPMs with bias
voltage at 60V.

Figure 19: All SiPMs’ gain measured by digitizer.

Figure 20: All SiPMs’ SPE resolution $R_{SPE}$.

Table 1: QA Batch Test Items and Results Items |  Methods |  Criteria |  Result
---|---|---|---
SiPM assembly |  Visual inspection with a microscope |  Height $\in$ [0.83, 0.93]mm, soldered properly, no visible pollution |  1 SiPM (out of 960) was soldered badly. The SiPM board containing this SiPM was marked out.
FEE communication |  Send/receive command/data via DAQ |  Can read serial NO., set HV/offset |  Only 1 FEE board was unable to connect to DAQ777Has been repaired after sending to BNL.
RX board signal |  Feed output signal to oscilloscope, measure noise spectrum |  No unreasonable peak in the noise spectrum |  Only 1 channel in one RX board was unable to output the signal
SiPM operating voltage |  Measure the U-I curve |  $V_{op}\in(50V,60V)$, $I_{leak}<0.03\mu A$ |  All SiPMs pass the test
Single photon charge resolution |  Collect waveforms by the digitizer and translate into the charge spectrum off-line |  $R<0.4$ |  All SiPMs and electronic devices passed the test
Signal offset |  Collect waveforms by the digitizer and measure the offset off-line |  Offset can be adjusted to 0 |  All electronic devices passed the test
SiPM and FEE noise |  Collect waveforms by the digitizer, do FFT off-line |  No unreasonable peak in the FFT spectrum |  All SiPMs and FEE boards pass test

All the QA items, criteria and statistical results are presented in Table 1.
On testing all the 60 SiPM boards and 960 SiPM channels, only one SiPM was not
soldered properly, and the related SiPM module was rejected by the QA test.
One FEE board was unable to connect to the DAQ, while one RX board contains a
channel with no signal output. Very consistent performance was shown by 57 out
of the 60 electronic suite of SiPM-FEE-RX boards, and these boards passed the
test. Since STAR only requires 52 boards including 4 for 2 extra super-
sectors, there are still enough good boards (57) to fulfill the standard of
failure rate. However, the problematic boards have been fixed by the BNL
electronics group after further investigations.

As a key parameter, we focused on the distribution of the operating voltages
and compared them to the distribution of the breakdown voltages measured by
the manufacturer (see Figure 18). Figure 18 illustrates an integrated 2D
histogram depicting the signals of all SiPMs at 60V, which is also the minimum
value to cover the distribution of $V_{op}$ (see 18). Most channels have at
least three peaks that can be easily distinguished by eye. The two regimes
with an overall shift, which are divided by the boundary between B30 and B31,
should be caused by an adjustment of LED amplitude. It was proposed to reduce
the luminous flux and to extract higher value at the first three peaks. Beyond
all that, we found some boards failing the test because of fault soldered
joint (B17), connection error (B18) and noisy response (B47). Figure 20 and
Figure 20 are gains and $R_{SPE}$ results that calculated from charge spectra
(see Section 3). Due to the excessive operation voltage, the SiPMs show the
atypically high gain around $7\\--8\times 10^{6}$ rather than the nominal
$0.5\\--1\times 10^{6}$. Nevertheless, the distribution of single photon
charge resolution displays an excellent performance at range from 0.05 to
0.25, which is much better than the criteria of 0.4.

##  5 Summary

The production and installation of the EPD sub-system into the STAR experiment
is an important upgrade for the heavy-ion physics program at RHIC. To meet the
scheduled BES-II program, we need to efficiently manage and reliably excute
the mass production of all EPD components, including the detectors, the
readout electronics, and the integration and commissioning of the whole
system. The heavy-ion group at USTC has taken the task of producing all the
FEE and the related QA testing to ensure the proper functionalities of the
photon sensors (SiPMs) and the three electronic boards. Based on the
experience learned from a previous study of the EPD prototypes [6], a
dedicated QA procedure was established and applied to all SiPMs and electronic
boards, up to $\sim 1000$ channels. The QA test results revealed that our mass
production and quality control action were very successful, with more than
$98\%$ of these channels passing the QA requirements. Very few channels faced
any problems.

The full production of EPD was launched in the early of 2017 and completed by
the end of that year. From the 2018 to 2019 operations at RHIC, the EPD sub-
system was continuously optimized and adjusted. Most of the EPD components,
including the FEEs, worked well and remained stable. The results of the data
analysis prove that the EPD performance fulfilled all the technical and
physical requirements [5]. The EPD continues to run along with the STAR
experiment.

## References

  * [1] J. D. Bjorken, _Highly Relativistic Nucleus-Nucleus Collisions: The Central Rapidity Region_ , Phys. Rev. D, 27, (1983), 140-151. doi:10.1103/PhysRevD.27.140.
  * [2] J. Adams, et al., _Experimental and theoretical challenges in the search for the quark gluon plasma: The STAR Collaboration’s critical assessment of the evidence from RHIC collisions_ , Nucl. Phys. A 757, (2005), 102-183. arXiv:nucl-ex/0501009, doi:10.1016/j.nuclphysa.2005.03.085.
  * [3] Aprahamian, Ani, et al., _Reaching for the horizon: The 2015 long range plan for nuclear science_ , (2015).
  * [4] Chi YANG, and STAR Collaboration, _The STAR beam energy scan phase II physics and upgrades._ , Nuclear Physics A, 967, (2017), 800-803.
  * [5] Joseph Adams, Annika Ewigleben, Sierra Garrett, etal, _The STAR Event Plane Detector_ , Nucl. Instrum. Meth. A, 968, (2020), 163970. DOI:10.1016/j.nima.2020.163970
  * [6] Alex Schmah, Rosi Reed, Mike Lisa, for the STAR collaboration, _An Event Plane Detector for STAR_ , STAR Note 0666 (2016), https://drupal.star.bnl.gov/STAR/starnotes/public/sn0666
  * [7] P. Buzhan, et al, _The Advanced Study of Silicon Photomultiplier_ , Advanced Technology and Particle Physics, (Nov. 2002), 717-728.
  * [8] Ferenc Nagy, et al, _A model based DC analysis of SiPM breakdown voltages_ , Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 849, (2017), 55-59.
  * [9] M.Ramilli, et al, _Characterization of SiPM: Temperature dependencies_ , 2008 IEEE Nuclear Science Symposium Conference Record, (2018), 2467-2470.


\mathit{UPol}(\mathit{BPol}(\mathcal{G}^{+}))&=&\Delta_{2}(<,+1,\mathbbm{P}_{\mathcal{G}})&=&\textup{FO}^{2}(<,+1,\mathbbm{P}_{\mathcal{G}})&=&\textup{TLX}(\mathcal{G}).\end{array}$

The proofs of the missing inclusions are based on Theorem 6.5, the algebraic
characterization of
$\mathit{Pol}(\mathit{BPol}(\mathcal{C}))\cap\mathit{co\textup{-}}\\!\mathit{Pol}(\mathit{BPol}(\mathcal{C}))=\Delta_{2}(\mathbbm{I}_{\mathcal{C}})$.
We use Proposition 84 to prove that it is satisfied by the class
$\textup{TL}(\mathcal{C})$ when $\mathcal{C}$ is a prevariety of group
languages $\mathcal{G}$ or its well-suited extension $\mathcal{G}^{+}$.
Actually, we simplify the generic characterization presented in Theorem 6.5
and present specialized characterizations for these two special cases. In
particular, this yields generic characterizations of
$\textup{FO}^{2}(<,\mathbbm{P}_{\mathcal{G}})$ and
$\textup{FO}^{2}(<,+1,\mathbbm{P}_{\mathcal{G}})$. They generalize well-known
results for particular instances of these logical classes. Let us point out
that here, we directly characterize the languages of these classes (using a
property of their syntactic morphism) rather than characterizing the morphisms
associated to the class. This is because in this case, we do need to use the
characterizations as subresults.

###  10.1 Group languages

We first consider the classes $\mathit{UPol}(\mathit{BPol}(\mathcal{G}))$
where $\mathcal{G}$ is a prevariety of group languages. The algebraic
characterization is based on the class of finite monoids $\mathbf{DA}$, whose
connection with unambiguous polynomial closure [35] and two-variable first-
order logic [39] is well known. We use a definition based on an equation (see
[38] for details). A finite monoid $N$ belongs to $\mathbf{DA}$ when it
satisfies the following equation:

$(st)^{\omega}=(st)^{\omega}t(st)^{\omega}\quad\text{for every $s,t\in N$}.$
(15)

We are now ready to present the generic characterization for classes of the
form $\mathit{UPol}(\mathit{BPol}(\mathcal{G}))$.

######  Theorem 10.1.

Let $\mathcal{G}$ be a prevariety of group languages and let $L$ be a regular
language. The following properties are equivalent:

  1. 1.

$L\in\mathit{UPol}(\mathit{BPol}(\mathcal{G}))$.

  2. 2.

$L\in\mathit{Pol}(\mathit{BPol}(\mathcal{G}))\cap\mathit{co\textup{-}}\\!\mathit{Pol}(\mathit{BPol}(\mathcal{G}))$.

  3. 3.

$L\in\Delta_{2}(<,\mathbbm{P}_{\mathcal{G}})$.

  4. 4.

$L\in\textup{FO}^{2}(<,\mathbbm{P}_{\mathcal{G}})$.

  5. 5.

$L\in\textup{TL}(\mathcal{G})$.

  6. 6.

The $\mathcal{G}$-kernel of the syntactic morphism of $L$ belongs to
$\mathbf{DA}$.

######  Proof 10.2.

By Lemma 13, we have
$\Delta_{2}(<,\mathbbm{P}_{\mathcal{G}})=\Delta_{2}(\mathbbm{I}_{\mathcal{G}})$.
Therefore, $(2)\Leftrightarrow(3)$ follows from Theorem 4.7. Moreover,
$(1)\Leftrightarrow(2)$ is given by Theorem 4.14. Since
$\textup{FO}^{2}(<,\mathbbm{P}_{\mathcal{G}})=\textup{FO}^{2}(\mathbbm{I}_{\mathcal{G}})$
by Lemma 13, the equivalence $(4)\Leftrightarrow(5)$ is immediate from Theorem
9.8. Finally, the implication $(1)\Rightarrow(5)$ follows from Proposition 86.
We now prove independently that $(6)\Rightarrow(2)$ and $(5)\Rightarrow(6)$,
which completes the argument. Let $\alpha:A^{*}\to M$ be the syntactic
morphism of $L$ and let $N\subseteq M$ be its $\mathcal{G}$-kernel.

Let us start with the implication $(6)\Rightarrow(2)$. We assume that $N$
belongs to $\mathbf{DA}$ and show that
$L\in\mathit{Pol}(\mathit{BPol}(\mathcal{G}))\cap\mathit{co\textup{-}}\\!\mathit{Pol}(\mathit{BPol}(\mathcal{G}))$.
Since $L$ is recognized by its syntactic morphism $\alpha$, it suffices to
show that $\alpha$ is a
$(\mathit{Pol}(\mathit{BPol}(\mathcal{G}))\cap\mathit{co\textup{-}}\\!\mathit{Pol}(\mathit{BPol}(\mathcal{G})))$-morphism.
By Theorem 6.5,this boils down to proving that for every $s,t\in M$ and $e\in
E(M)$ such that $(e,s)\in M^{2}$ is a $\mathcal{G}$-pair, we have
$(eset)^{\omega+1}=(eset)^{\omega}et(eset)^{\omega}$. By Lemma 39, we have
$es\in N$ and $et(eset)^{2\omega-1}\in N$. Since $N$ belongs to $\mathbf{DA}$,
it follows from (15) that,

$(eset(eset)^{2\omega-1})^{\omega}=(eset(eset)^{2\omega-1})^{\omega}et(eset)^{2\omega-1}(eset(eset)^{2\omega-1})^{\omega}.$

This exactly says that $(eset)^{\omega}=(eset)^{\omega}et(eset)^{3\omega-1}$.
It now suffices to multiply by $eset$ on the right to get
$(eset)^{\omega+1}=(eset)^{\omega}et(eset)^{\omega}$, as desired.

Finally, we prove that $(5)\Rightarrow(6)$. We assume that
$L\in\textup{TL}(\mathcal{G})$ and show that $N\in\mathbf{DA}$, _i.e._ , that
$N$ satisfies (15). Let $s,t\in N$. We prove that
$(st)^{\omega}=(st)^{\omega}t(st)^{\omega}$. By hypothesis, $L$ is defined by
a formula $\varphi_{L}$ of $\textup{TL}[\mathcal{G}]$. Let $k\in{\rm Nature}$
be the rank of $\varphi_{L}$. Fact 9.2 yields a $\mathcal{G}$-morphism
$\eta:A^{*}\to G$ such that $\varphi_{L}$ is a $\textup{TL}[\eta]$ formula.
Moreover, since $s,t$ belong to the $\mathcal{G}$-kernel $N$ of $\alpha$ and
$\eta$ is a $\mathcal{G}$-morphism, Lemma 33 yields $u,v\in A^{*}$ such that
$\eta(u)=\eta(v)=1_{G}$, $\alpha(u)=s$ and $\alpha(v)=t$. Hence, since
$\varphi_{L}$ is an $\eta$-formula of rank $k$, it follows from Lemma 83 and
Proposition 84 (applied for $z=\varepsilon$ which also maps to $1_{G}$) that
$x(uv)^{k}(uv)^{k}y\in L\Leftrightarrow x(uv)^{k}v(uv)^{k}y\in L$ for all
$x,y\in A^{*}$. In other words, $(uv)^{k}(uv)^{k}$ and $(uv)^{k}v(uv)^{k}$ are
equivalent for the syntactic congruence of $L$. Since $\alpha$ is the
syntactic morphism of $L$, we get
$\alpha((uv)^{k}(uv)^{k})=\alpha((uv)^{k}v(uv)^{k})$. By definition of $u$ and
$v$, it follows that $(st)^{k}(st)^{k}=(st)^{k}t(st)^{k}$. It now suffices to
multiply by enough copies of $st$ on the left and on the right to obtain
$(st)^{\omega}=(st)^{\omega}t(st)^{\omega}$, as desired.

Given as input a regular language $L$, one can compute its syntactic morphism.
Moreover, by Lemma 32, the $\mathcal{G}$-kernel of this morphism can be
computed when $\mathcal{G}$-separation is decidable. It is then simple to
decide whether it belongs to $\mathbf{DA}$: this boils down to checking
whether (15) holds by testing all possible combinations for $s$ and $t$. By
Theorem 10.1, this decides if $L\in\mathit{UPol}(\mathit{BPol}(\mathcal{G}))$.
Altogether, we obtain the following corollary.

######  Corollary 87.

Let $\mathcal{G}$ be a prevariety of group languages with decidable
separation. Then, membership is decidable for the class
$\textup{FO}^{2}(<,\mathbbm{P}_{\mathcal{G}})=\textup{TL}(\mathcal{G})=\Delta_{2}(<,\mathbbm{P}_{\mathcal{G}})=\mathit{UPol}(\mathit{BPol}(\mathcal{G}))$.

Recall that separation is decidable for the four standard prevarieties of
group languages ST, MOD, AMT and GR that we presented in Section 2.3.
Therefore, Corollary 87 applies in these cases. In particular, we get the
decidability of membership for the logical classes $\textup{FO}^{2}(<)$,
$\textup{FO}^{2}(<,\mathit{MOD})$, $\textup{FO}^{2}(<,\mathit{AMOD})$ and
$\textup{FO}^{2}(<,\mathbbm{P}_{\textup{GR}})$.

Again, Theorem 10.1 generalizes classic results. The most prominent one is for
$\mathcal{G}=\textup{ST}$. Since $\textup{ST}=\\{\emptyset,A^{*}\\}$, it is
easy to check that the ST-kernel of a surjective morphism $\alpha:A^{*}\to M$
is the whole monoid $M$. Also by Lemma 26, we have
$\mathit{UPol}(\mathit{BPol}(\textup{ST}))=\mathit{UPol}(\and)$. Therefore, in
this case, Theorem 10.1 yields that
$\mathit{UPol}(\and)=\Delta_{2}(<)=\textup{FO}^{2}(<)=\textup{F}+\textup{P}$
and that a regular language belongs to this class if and only if its syntactic
monoid belongs to $\mathbf{DA}$. Historically, this was proved by combining
several independent results. First, the correspondence between
$\mathit{UPol}(\and)$ and $\mathbf{DA}$ is due to Schützenberger [35]. Then,
it was shown by Pin and Weil [20] that $\mathit{UPol}(\and)=\Delta_{2}(<)$. As
we explained in the previous section, that
$\textup{FO}^{2}(<)=\textup{F}+\textup{P}$ was proved by Etessami, Vardi and
Wilke [9]. Finally, the correspondence between $\textup{FO}^{2}(<)$ and
$\mathbf{DA}$ is due to Thérien and Wilke [39].

Another well-known application of Theorem 10.1 is the case where
$\mathcal{G}=\textup{MOD}$. In particular, the theorem implies that
$\Delta_{2}(<,MOD)=\textup{FO}^{2}(<,\mathit{MOD})$ and that a regular
language belongs to this class if and only if the MOD-kernel of its syntactic
morphism belongs to $\mathbf{DA}$. The correspondence between
$\textup{FO}^{2}(<,\mathit{MOD})$ and the membership of the MOD-kernel of the
syntactic morphism to $\mathbf{DA}$ is due to Dartois and Paperman [7], while
the equality $\Delta_{2}(<,MOD)=\textup{FO}^{2}(<,\mathit{MOD})$ is due to
Kufleitner and Walter [13].

######  Remark 88.

While membership for $\mathit{UPol}(\mathit{BPol}(\mathcal{G}))$ boils down to
$\mathcal{G}$-separation, not much is known about _separation_ and _covering_
for $\mathit{UPol}(\mathit{BPol}(\mathcal{G}))$. The only class of this kind
for which separation and covering are known to be decidable is
$\mathit{UPol}(\mathit{BPol}(\textup{ST}))=\textup{FO}^{2}(<)$. This is the
class $\mathit{UPol}(\and)$ by Lemma 26. Since and is a _finite_ class,
$\mathit{UPol}(\and)$-covering is decidable by Corollary 68.

###  10.2 Well-suited extensions

We now consider classes of the form
$\mathit{UPol}(\mathit{BPol}(\mathcal{G}^{+}))$, where $\mathcal{G}^{+}$ is
the well-suited extension of an arbitrary prevariety of group languages
$\mathcal{G}$. In this case, the characterization is based on the class of
finite semigroups $\mathbf{LDA}$. Again, we use a definition which is based on
an equation. A finite semigroup $S$ belongs to $\mathbf{LDA}$ when it
satisfies the following equation:

$(esete)^{\omega}=(esete)^{\omega}ete(esete)^{\omega}\quad\text{for every
$s,t\in S$ and $e\in E(S)$}.$ (16)

We now present the generic statement characterization of
$\mathit{UPol}(\mathit{BPol}(\mathcal{G}^{+}))$.

######  Theorem 10.3.

Let $\mathcal{G}$ be a prevariety of group languages and let $L$ be a regular
language. The following properties are equivalent:

  1. 1.

$L\in\mathit{UPol}(\mathit{BPol}(\mathcal{G}^{+}))$.

  2. 2.

$L\in\mathit{Pol}(\mathit{BPol}(\mathcal{G}^{+}))\cap\mathit{co\textup{-}}\\!\mathit{Pol}(\mathit{BPol}(\mathcal{G}^{+}))$.

  3. 3.

$L\in\Delta_{2}(<,+1,\mathbbm{P}_{\mathcal{G}})$.

  4. 4.

$L\in\textup{FO}^{2}(<,+1,\mathbbm{P}_{\mathcal{G}})$.

  5. 5.

$L\in\textup{TLX}(\mathcal{G})$.

  6. 6.

The strict $\mathcal{G}$-kernel of the syntactic morphism of $L$ belongs to
$\mathbf{LDA}$.

######  Proof 10.4.

In this case as well, we already proved most of the implications. By Lemma 13,
we have
$\Delta_{2}(<,+1,\mathbbm{P}_{\mathcal{G}})=\Delta_{2}(\mathbbm{I}_{\mathcal{G}^{+}})$.
Hence, $(2)\Leftrightarrow(3)$ follows from Theorem 4.7. Moreover,
$(1)\Leftrightarrow(2)$ is given by Theorem 4.14. Finally, since
$\textup{FO}^{2}(<,+1,\mathbbm{P}_{\mathcal{G}})=\textup{FO}^{2}(\mathbbm{I}_{\mathcal{G}^{+}})$
by Lemma 13 and $\textup{TLX}(\mathcal{G})=\textup{TL}(\mathcal{G}^{+})$ by
Lemma 80, the equivalence $(4)\Leftrightarrow(5)$ is immediate from Theorem
9.8. Finally, the implication $(1)\Rightarrow(5)$ follows from Proposition 86.
We now prove independently that $(6)\Rightarrow(2)$ and $(5)\Rightarrow(6)$ to
complete the proof. We let $\alpha:A^{*}\to M$ be the syntactic morphism of
$L$. Let also $N$ be the $\mathcal{G}$-kernel of $\alpha$, and $S$ be its
strict $\mathcal{G}$-kernel. Recall from Section 5 that we have
$S=N\cap\alpha(A^{+})$ by definition.

We start with the implication $(6)\Rightarrow(2)$. Assume that $S$ belongs to
$\mathbf{LDA}$. We have to show that
$L\in\mathit{Pol}(\mathit{BPol}(\mathcal{G}^{+}))\cap\mathit{co\textup{-}}\\!\mathit{Pol}(\mathit{BPol}(\mathcal{G}^{+}))$.
Since $L$ is recognized by its syntactic morphism $\alpha$, it suffices to
show that $\alpha$ is a
$(\mathit{Pol}(\mathit{BPol}(\mathcal{G}^{+}))\cap\mathit{co\textup{-}}\\!\mathit{Pol}(\mathit{BPol}(\mathcal{G}^{+})))$-morphism.
By Theorem 6.5, this boils down to proving that for every $s,t\in M$ and $e\in
E(M)$ such that $(e,s)\in M^{2}$ is a $\mathcal{G}^{+}$-pair, we have
$(eset)^{\omega+1}=(eset)^{\omega}et(eset)^{\omega}$. By Fact 5.4, we have
$e\in N$. We consider two cases depending on whether $e\in S$ or $e\in
N\setminus S$. Assume first that $e\in N\setminus S$. Since
$S=N\cap\alpha(A^{+})$, this implies that $e=1_{M}$ and that
$\alpha^{-1}(1_{M})=\\{\varepsilon\\}$. Since $(1_{M},s)$ is a
$\mathcal{G}^{+}$-pair and $\\{\varepsilon\\}\in\mathcal{G}^{+}$, it follows
that $s=1_{M}$ as well. Hence,
$(eset)^{\omega+1}=t^{\omega+1}=(eset)^{\omega}et(eset)^{\omega}$ follows
directly. We now consider the case where $e\in S$. By Lemma 39, we have $es$
and $et(eset)^{2\omega-1}\in N$. Since $e\in S$, this implies that $es\in S$
and $et(eset)^{2\omega-1}\in S$. Since $S$ belongs to $\mathbf{LDA}$, we get
from Equation (16) that,

$(eeseet(eset)^{2\omega-1}e)^{\omega}=(eeseet(eset)^{2\omega-1}e)^{\omega}eet(eset)^{2\omega-1}e(eeseet(eset)^{2\omega-1}e)^{\omega}.$

Since $e$ is idempotent, this simplifies into
$(eset)^{\omega}e=(eset)^{\omega}et(eset)^{3\omega-1}e$. It now suffices to
multiply by $set$ on the right to get the desired equality
$(eset)^{\omega+1}=(eset)^{\omega}et(eset)^{\omega}$.

We finally turn to the implication $(5)\Rightarrow(6)$. Assume that
$L\in\textup{TLX}(\mathcal{G})=\textup{TL}(\mathcal{G}^{+})$. We show that $S$
belongs to $\mathbf{LDA}$, _i.e._ , that $S$ satisfies (16). Let $s,t\in S$
and $e\in E(S)$. We prove that
$(esete)^{\omega}=(esete)^{\omega}ete(esete)^{\omega}$. By hypothesis, $L$ is
defined by a formula $\varphi_{L}$ of $\textup{TL}[\mathcal{G}^{+}]$. Let
$k\in{\rm Nature}$ be the rank of $\varphi$. Fact 9.2 yields a
$\mathcal{G}^{+}$-morphism $\eta:A^{*}\to Q$ such that $\varphi_{L}$ is a
$\textup{TL}[\eta]$ formula. It follows from Lemma 10 that $G=\eta(A^{+})$ is
a group in $N$ and that the morphism $\beta:A^{*}\to G$ defined by
$\beta(w)=\eta(w)$ for every $w\in A^{+}$ is a $\mathcal{G}$-morphism. Using
the hypothesis that $s,t,e\in S$, we prove that there exist $u,v,z\in A^{*}$
such that $\alpha(u)=s$, $\alpha(v)=t$, $\alpha(z)=e$ and
$\eta(u)=\eta(v)=\eta(z)=1_{G}$. By symmetry, we only prove the existence of
$u\in A^{*}$. There are two cases, depending on whether $s=1_{M}$ or not. If
$s=1_{M}$, then we have $1_{M}\in S\subseteq\alpha(A^{+})$. Thus, there exists
$u^{\prime}\in A^{+}$ such that $\alpha(u^{\prime})=1_{M}$. We let
$u=(u^{\prime})^{p}$ for $p=\omega(G)$. Clearly, $\alpha(u)=1_{M}=s$ and
$\eta(u)=(\eta(u^{\prime}))^{p}=1_{G}$ since $G$ is a group. Assume now that
$s\neq 1_{M}$. We have $s\in S\subseteq N$. Since $\beta:A^{*}\to G$ is a
$\mathcal{G}$-morphism, it follows from Lemma 33 that there exists $u\in
A^{*}$ such that $\alpha(u)=s$ and $\beta(u)=1_{G}$. Since $s\neq 1_{M}$, we
have $u\in A^{+}$, which also implies that $\eta(u)=\beta(u)=1_{G}$, as
desired.

We now use $u,v,z$ (satisfying $\alpha(u)=s$, $\alpha(v)=t$, $\alpha(z)=e$ and
$\eta(u)=\eta(v)=\eta(z)=1_{G}$) to conclude the proof. Since $\varphi_{L}$ is
an $\eta$-formula of rank $k$ and $1_{G}\in E(N)$, it follows from Lemma 83
and Proposition 84 that for all $x,y\in A^{*}$, we have,

$x(z^{k}uz^{2k}vz^{k})^{k}(z^{k}uz^{2k}vz^{k})^{k}y\in L\Leftrightarrow
x(z^{k}uz^{2k}vz^{k})^{k}z^{k}vz^{k}(z^{k}uz^{2k}vz^{k})^{k}y\in L.$

Hence, $(z^{k}uz^{2k}vz^{k})^{k}(z^{k}uz^{2k}vz^{k})^{k}$ and
$(z^{k}uz^{2k}vz^{k})^{k}z^{k}vz^{k}(z^{k}uz^{2k}vz^{k})^{k}$ are equivalent
for the syntactic congruence of $L$. Since $\alpha$ is the syntactic morphism
of $L$, these words have the same image under $\alpha$. This yields
$(esete)^{k}(esete)^{k}=(esete)^{k}ete(esete)^{k}$ by definition of $u,v$ and
$z$. It now suffices to multiply by enough copies of $esete$ on the left and
on the right to obtain $(esete)^{\omega}=(esete)^{\omega}ete(esete)^{\omega}$,
as desired.

Given as input a regular language $L$, one can compute its syntactic morphism.
Moreover, by Lemma 32, the strict $\mathcal{G}$-kernel of this morphism can be
computed as soon as $\mathcal{G}$-separation is decidable. It is then simple
to decide whether it belongs to $\mathbf{LDA}$: this boils down to checking if
(16) holds by testing all possible combinations for $s$, $t$ and $e$. By
Theorem 10.3, this decides whether
$L\in\mathit{UPol}(\mathit{BPol}(\mathcal{G}^{+}))$. We state this in the
following corollary.

######  Corollary 89.

Let $\mathcal{G}$ be a prevariety of group languages with decidable
separation. Then, membership is decidable for
$\textup{FO}^{2}(<,+1,\mathbbm{P}_{\mathcal{G}})=\textup{TLX}(\mathcal{G})=\Delta_{2}(<,+1,\mathbbm{P}_{\mathcal{G}})=\\!\mathit{UPol}(\mathit{BPol}(\mathcal{G}^{+}))$.

Recall again that separation is decidable for the prevarieties of group
languages ST, MOD, AMT and GR, so that Corollary 89 applies in these cases:
membership is decidable for the logical classes $\textup{FO}^{2}(<,+1)$,
$\textup{FO}^{2}(<,+1,\mathit{MOD})$, $\textup{FO}^{2}(<,+1,\mathit{AMOD})$
and $\textup{FO}^{2}(<,+1,\mathbbm{P}_{\textup{GR}})$.

Here again, Theorem 10.3 generalizes known results in the particular case
$\mathcal{G}=\textup{ST}$. Since $\textup{ST}=\\{\emptyset,A^{*}\\}$, it is
straightforward to verify that the strict ST-kernel of a morphism
$\alpha:A^{*}\to M$ is the semigroup $\alpha(A^{+})$. When $\alpha$ is the
syntactic morphism of a language $L$, this object is called the _syntactic
semigroup_ of $L$. Thus, Theorem 10.3 yields that
$\Delta_{2}(<,+1)=\textup{FO}^{2}(<,+1)=\textup{F}+\textup{X}+\textup{P}+\textup{Y}$
and that a regular language belongs to this class if and only if its syntactic
semigroup belongs to $\mathbf{LDA}$. As seen in the previous section, the
equality $\textup{FO}^{2}(<,+1)=\textup{F}+\textup{X}+\textup{P}+\textup{Y}$
is due to Etessami, Vardi and Wilke [9]. The remaining correspondences are due
to Thérien and Wilke [39]. In particular, the connection with $\mathbf{LDA}$
is established by relying on results of Almeida [1], based on a complex
algebraic framework by Tilson [42], which involves categories and wreath
products of finite semigroups. We bypass this intricate machinery here.

######  Remark 90.

In this case as well, there exists no generic result concerning separation and
covering for classes of the form
$\mathit{UPol}(\mathit{BPol}(\mathcal{G}^{+}))$. However, both problems are
known to be decidable in _two_ particular cases: $\mathcal{G}=\textup{ST}$ and
$\mathcal{G}=\textup{MOD}$. In view of Theorem 10.3, this corresponds to the
logical classes $\textup{FO}^{2}(<,+1)$ and
$\textup{FO}^{2}(<,+1,\mathit{MOD})$. The decidability of covering for these
two classes can be obtained by applying generic transfer results of [25] and
[23] to the simpler class $\textup{FO}^{2}(<)$ (as we mentioned above,
decidability of covering for this class was first proved in [32]). The
techniques involved in these results are orthogonal and independent from the
ones of the present paper.

##  11 Conclusion

We presented a generic algebraic characterization of unambiguous polynomial
closure and used it to prove that when $\mathcal{C}$ is a prevariety,
membership for $\mathit{UPol}(\mathcal{C})$ reduces to membership for
$\mathcal{C}$. An interesting byproduct of the proof is that
$\mathit{UPol}(\mathcal{C})=\mathit{APol}(\mathcal{C})=\mathit{Pol}(\mathcal{C})\cap\mathit{co\textup{-}}\\!\mathit{Pol}(\mathcal{C})$
in that case. Moreover, we showed that when $\mathcal{C}$ is a finite
prevariety, covering and separation are decidable for
$\mathit{UPol}(\mathcal{C})$. This completes similar results of [31] for
$\mathit{Pol}(\mathcal{C})$ and $\mathit{Bool}(\mathit{Pol}(\mathcal{C}))$ and
of [22] for $\mathit{Pol}(\mathit{Bool}(\mathit{Pol}(\mathcal{C})))$. Finally,
we presented logical characterizations of the classes built with unambiguous
polynomial closure. In particular, we proved that if the input class
$\mathcal{G}$ is a prevariety of _group languages_ , then
$\mathit{UPol}(\mathit{BPol}(\mathcal{G}))=\Delta_{2}(<,\mathbbm{P}_{\mathcal{G}})=\textup{FO}^{2}(<,\mathbbm{P}_{\mathcal{G}})=\textup{TL}(\mathcal{G})$
and
$\mathit{UPol}(\mathit{BPol}(\mathcal{G}^{+}))=\Delta_{2}(<,+1,\mathbbm{P}_{\mathcal{G}})=\textup{FO}^{2}(<,+1,\mathbbm{P}_{\mathcal{G}})=\textup{TLX}(\mathcal{G})$.
This generalizes earlier results corresponding to particular examples of
prevarieties of group languages $\mathcal{G}$.

A natural follow-up question is whether our result for separation and covering
can be pushed further to encompass more inputs than the finite classes. In
particular, in view of the above logical characterizations, it would be
interesting to look at both problems for classes of the form
$\mathit{UPol}(\mathit{BPol}(\mathcal{G}))$ and
$\mathit{UPol}(\mathit{BPol}(\mathcal{G}^{+}))$ when $\mathcal{G}$ is a
prevariety of group languages. While this is difficult, there are known
results of this kind; in particular, it is known [30] that separation and
covering are decidable for $\mathit{Pol}(\mathit{BPol}(\mathcal{G}))$ as soon
as separation is decidable for $\mathcal{G}$.

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