††thanks: Supported by the Strategic Priority Research Program A-JUNO, High
Energy Physics Experiment and Detector R&D and National Natural Science
Foundation of China

# Design & Optimization of the HV divider for JUNO 20-inch PMT

Feng-Jiao Luo  University of South China, School of Nuclear Science and
Technology, Hengyang 421001, China  Zhi-Min Wang<EMAIL_ADDRESS>Institute
of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
State Key Laboratory of Particle Detection and Electronic, Beijing 100049,
China  An-Bo Yang  Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China  Yue-Kun Heng  Institute of High Energy
Physics, Chinese Academy of Sciences, Beijing 100049, China  State Key
Laboratory of Particle Detection and Electronic, Beijing 100049, China  Zhong-
Hua Qin  Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China  State Key Laboratory of Particle Detection and
Electronic, Beijing 100049, China  Mei-Hang Xu  Institute of High Energy
Physics, Chinese Academy of Sciences, Beijing 100049, China  State Key
Laboratory of Particle Detection and Electronic, Beijing 100049, China  Sen
Qian  Institute of High Energy Physics, Chinese Academy of Sciences, Beijing
100049, China  State Key Laboratory of Particle Detection and Electronic,
Beijing 100049, China  Shu-Lin Liu  Yi-Fang Wang  Institute of High Energy
Physics, Chinese Academy of Sciences, Beijing 100049, China  State Key
Laboratory of Particle Detection and Electronic, Beijing 100049, China  Wei
Wang  Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen
University, Zhuhai, 519082, China  Alexander Olshevskiy  Joint Institute for
Nuclear Research, Dubna, 141980, Russia  Guo-Rui Huang  Zhen Jin  Ling Ren
Xing-Chao Wang  Shu-Guang Si  Jian-Ning Sun  Northern Night Vision Technology
Co., LTD, 211100, Nanjing, China

###### Abstract

The Jiangmen Underground Observatory (JUNO) is a 20-kton liquid scintillator
detector that employs 20,000 20-inch photomultiplier tubes (PMTs) as photon
sensors, with 5,000 dynode-PMTs from HAMAMATSU Photonics K.K. (HPK), and
15,000 MCP-PMTs from North Night Vision Technology (NNVT) installed in pure
water. JUNO aims to provide long-lasting and the best performance operation by
utilizing a high-transparency liquid scintillator, high detection efficiency
PMTs, and specially designed electronics including water-proof potting for the
high voltage (HV) dividers of PMTs. In this paper, we present a summary of the
design and optimization of HV dividers for both types of 20-inch PMTs, which
includes collection efficiency, charge resolution, HV divider current, pulse
shape, and maximum amplitude restriction. We have developed and finalized four
schemes of the HV divider for different scenarios, including the final version
selected by JUNO. All 20,000 20-inch PMTs have successfully undergone
production and burning tests.

JUNO, 20-inch PMT, HV divider, MCP-PMT, dynode-PMT

##  I Introduction

The Jiangmen Underground Neutrino Observatory (JUNO) is a multipurpose
neutrino experiment that has been designed to achieve a wide range of
objectives. Its primary goal is to determine the ordering of neutrino masses
at the 3-4 sigma level in six years of data taking and to measure neutrino
oscillation parameters by detecting reactor neutrinos with an accuracy of less
than 1%. JUNO is also capable of observing supernova neutrinos and studying
atmospheric, solar, and geo-neutrinos, in addition to performing searches for
exotic phenomena ref8 . The JUNO experiment is located in Kaiping, Jiangmen,
Guangdong province of China. It is situated about 53 km away from the
Yangjiang and Taishan nuclear power plants, with a planned total thermal power
of 36 GWth. To further suppress muon-induced backgrounds, the detector will be
located deep underground with a 700 m rock overburden ref9 . Fig. 1 shows the
schematic of the JUNO detector.

The detector system of JUNO consists of several sub-detectors, which include
the central detector (CD) CD-ref10 , water pool (WP, a water Cerenkov
detector) ref11 , and top tracker (TT) ref12 . The CD is an acrylic vessel
with a diameter of 35.4 m filled with 20-kton 1iquid scintillator, which uses
17,612 20-inch PMTs ref13 ; ref14 ; ref15  and 25,600 3-inch PMTs as photon
sensors JUNO:3inchPMT:CAO2021165347 ; ref16 ; ref17 . The water Cerenkov
detector is a cylindrical pool equipped with 2,400 20-inch PMTs filled with
35-kton of ultra-pure water as a muon veto and radioactive shielding. The top
tracker detector will use to tag muon tracks precisely.

Figure 1: Schematic view of the JUNO detector JUNO:physics:2022PrPNP.12303927J

To reach the expected sensitivity of neutrino mass ordering, the energy
resolution of JUNO has to be better than 3% at 1 MeV, which requires a
scintillation light collection of at least 1,200 photoelectrons(pes)/MeV ref20
. To obtain a good energy resolution, the PMTs need to have high photon
detection efficiency (PDE) and good performance. The 20-inch PMT system of
JUNO JUNO:20inchPMT:2022hlz ; ref18 ; ref19  includes 5,000 dynode PMTs
R12860-50 from Hamamatsu Photonics K.K. (HPK) and 15,012 micro-channel plate
PMTs (MCP-PMTs) from North Night Vision Technology Co., Ltd. (NNVT). The
3-inch dynode PMTs are from Hainan Zhuanchuang Photonics Technology Co., Ltd.
(HZC). The PMT are commonly used in nuclear physics, particle physics, and
astrophysics, such as Super-Kamiokande (Super-K) ref1 , Daya Bay ref2-dyb ,
Borexino ref3 , Chooz ref4 , Double Chooz ref5 , and China’s Large High
Altitude Air Shower Observatory (LHAASO) ref7 , and so on. The performances of
PMTs are not only related to the processing technology of PMTs themselves but
also the design of the High Voltage (HV) divider (HV divider or base). The
PMTs of JUNO will be submerged in pure water, whereas the PMTs with the HV
divider should be sealed and water-proof, and have good stability and
lifetime. Furthermore, the HV divider of the newly developed 20-inch MCP-PMT
is one of the key features differentiating it from traditional dynode-PMTs.

This paper focuses on the R&D, optimization, production, and quality control
of HV dividers for JUNO 20-inch PMTs. The paper is structured as follows: In
Sec.II, the requirements and pre-design of the HV divider for JUNO 20-inch
PMTs are briefly introduced. In Sec.III, the optimization of the HV divider is
discussed, including the collection efficiency of MCP-PMT, charge resolution
of MCP-PMT, HV divider current, pulse shape, and restriction on maximum
amplitude. In Sec.IV, the final designs of four options, including JUNO
selection, is presented. In Sec.V, the burning test and quality control are
described. In Sec.VI, a summary is provided.

##  II Requirements & Pre-design of HV divider for JUNO 20-inch PMT

###  II.1 Requirements

To achieve high energy resolution, as little light as possible needs to be
lost, so 20012 LPMT will be used. Consequently, a list of PMT performance
criteria has been required for all the selected PMTs PMT-selection-
WEN2019162766 ; JUNO:20inchPMT:2022hlz . For this purpose, JUNO has selected
two types of 20-inch PMTs: the traditional dynode-PMT and the innovative
micro-channel plate (MCP) PMT, as depicted in Figure 2.

Figure 2: Physical view of JUNO 20-inch PMTs. (a) 20-inch dynode-PMT; (b)
20-inch MCP-PMT.

In addition to the performances, the system lifetime is also required for the
planned running period, and the voltage divider cannot be disassembled once
the PMT waterproof is completed. The HV divider of the PMT is a critical
component that directly affects its function, performance, and reliability.
Therefore, the design of the HV divider is crucial to ensuring the performance
and lifetime of JUNO. The JUNO collaboration has established strict
requirements for the PMT HV divider, as shown in Table 1.

Table 1: Key parameters of 20-inch PMT HV divider required by JUNO Parameters | Requirements
---|---
Current of HV divider | Dynode-PMT: < 300 µA
| MCP-PMT: < 300 µA
Gain | $10^{7}$
Dynamic Range | 0-4000 pe
Linearity | 1000 pe@$\pm$10%
Rise time of waveform | > 3 ns
Overshoot in amplitude | < 1%
Life time | > 20 years
Failure rate in 6 years | < 0.5%

###  II.2 Pre-Design

A high voltage from 500 to 3000 V is applied usually across the cathode (K)
and anode (A) of a PMT, with a proper voltage gradient among the photoelectron
focusing electrode, and dynodes or MCPs. In practice, the inter-stage voltage
for each electrode/MCP is provided by voltage-dividing resistors or Zener
diodes connected between the anode and cathode ref21 , naming it as an HV
divider (or base). For JUNO, a resistor chain is suggested for voltage-
dividing taking into account the reliability and simplicity of electronic
components.

Table 2 shows the suggested voltage ratio of the two types of PMTs from the
vendors. Further optimization of the design to meet the requirements of JUNO
is needed, including optimizing the voltage ratio, pulse shape, and working
direct current (DC) to limit power consumption, and implementing the suggested
final design. Additionally, for the newly developed 20” MCP-PMTs, the
characteristics of the MCPs needed to be further considered.

Table 2: The HV ratio of 20-inch dynode- and MCP-PMT provided by the vendors.

dynode-PMT | K | Dy1 | F | Dy2 | Dy3 | Dy4 | Dy5 | Dy6 | Dy7 | Dy8 | Dy9 | Dy10 | A
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Ratio | 11.3 | 0.6 | 3.4 | 3.7 | 3 | 2 | 1 | 1 | 1 | 1 | 1 | 1
Capacitor (nF) |  |  |  |  |  |  |  | 10 | 10 | 10 | 10 |

MCP-PMT | K | F1 | F2 | MCP1a | MCP1b | MCP2a | MCP2b | A
---|---|---|---|---|---|---|---|---
Ratio | 7 | 1 | 0 | 10 | 0.68 | 10 | 1
Capacitor (nF) |  |  |  |  |  | 10 | 10

Positive high voltage is favored in most large-scale neutrino experiments for
lower noise, lower cost, and easier installation, such as SNOSNO-
JILLINGS1996421 , Double CHOOZref22 ; ref23 , etc. PMT with positive high
voltage will work with only one readout cable and a de-coupler between HV and
signal. However, due to the existence of the de-coupling capacitor, the output
pulse of PMT will suffer from an overshoot, which is coupled to the primary
pulse, and will generate problems to charge measurements and system
triggering, as observed by Double Chooz ref24 , KamLAND ref25 , SNOref26 ,
Borexino ref27  and Daya Bayref28 . According to the requirements of JUNO, the
overshoot, and small divider-current is part of the key points taken into
account in our design for the 20-inch PMTs. We have finished a detailed study
on the overshoot of PMT output to a controlled contribution. More details
could refer to ref15 ; ref29 . At the same time, the DC is further required by
the power consumption, heating effect in the detector, and reliability.

The typical waveforms of the two types of PMTs can be found in Fig. 3 with the
pre-designed HV dividers of the dynode-PMT and MCP-PMT. The rise time and fall
time of MCP-PMT do not meet the JUNO requirements. The MCP-PMT waveforms
exhibit two peaks and an uneven falling edge, which can impact charge
measurement and waveform de-convolution. In the subsequent sections, we will
describe our optimization of the HV divider to meet JUNO’s requirements for
the relevant features.

Figure 3: Typical single photoelectron (SPE) waveforms of 20-inch PMTs with
pre-designed HV dividers before optimization. (a) Dynode-PMT; (b) MCP-PMT.

##  III Optimization

###  III.1 Collection efficiency of MCP-PMT

The high photon detection efficiency (PDE) of the PMT is a critical
requirement for JUNO. PDE is determined by photon transmission, quantum
efficiency (QE), and collection efficiency (CE) of the PMT. The collection
efficiency is influenced by the shape of the photocathode, the design of the
focusing electrode, and the electric field distribution. Without an
appropriate applied voltage between the cathode and the first multiplier,
photoelectrons (pe) cannot be efficiently collected or amplified, thus
affecting the collection efficiency and charge amplification process. In this
section, we will focus on optimizing the first several stages of the JUNO 20”
MCP-PMT.

For an MCP-PMT, the applied voltages between the photocathode, the focusing
stage, and the first MCP will affect the internal electric field distribution
and the process of photoelectron collection or amplification. Efficient
collection of photoelectrons by the first MCP results in a good collection
efficiency for the PMT. However, the collection efficiency of PMT cannot be
measured directly. A measurement system of relative detection efficiency is
set up, as shown in Fig. 4relative-eff-anbo , where a 20-inch dynode PMT is
used as a reference. The collection efficiency of photoelectrons will saturate
when the collection field is strong enough haiqiong-relative-collection-
efficiency . The following tests have satisfied this condition.

The ratio of K-F1 (cathode to focus stage 1) and F1-F2 (focus stage 1 to stage
2) (in Table 2) has an impact on the collection efficiency of the MCP-PMT. To
investigate this, we surveyed the voltage ratio between K-F1 and F1-F2 with
the system, applying a strong enough field to collect all photoelectrons. The
relative detection efficiency of the MCP-PMT with different ratios
(K-F1)/(F1-F2) from 0.2 to 5.0 was measured, as shown in Fig.5. The total
applied voltage between the cathode (K) and focus stage 2 (F2) was kept
constant. The measured charge of a PMT in pe was calculated based on two
methods: Poisson distribution (mu=-ln(1-N_sig/N_total))JUNO-PMT-gain-
Zhang2021GainAC ; JUNO:20inchPMT:2022hlz  and PMT gain (pe=(S-P)/($e$×Gain)),
where S and P (in pC) are the peak locations of the pedestal and signal on the
charge spectrum, respectively, of an electron of the MCP-PMT and the reference
PMT, and e is the charge of an electron. The output charge relationship
between the MCP-PMT and the reference PMT, including different light
intensities, was fitted by a linear curve relative-eff-anbo  for each ratio of
(K-F1)/(F1-F2), and the results of different ratios were compared. It is
important to note that the y-axis in Fig.5 is a normalized value relative to
the measurement of the ratio of (K-F1)/(F1-F2) = 1 of each curve itself.

The design of MCP-PMT with respect to the electric field between K-F1 and
F1-F2 is crucial. If the electric field of K-F1 or F1-F2 is too high, the
photoelectrons will be easily collected and absorbed by F1 or F2 itself,
rather than the first MCP for further amplification. This will result in a
reduced effective collection efficiency. Only when the ratio between K-F1 and
F1-F2 is around one, can the photoelectrons be effectively collected by the
first MCP for the following multiplication. With this setting, the field
distribution between K, F1, and F2 is optimal, and the photoelectrons can be
better collected by the first MCP. For the JUNO MCP-PMT, a voltage ratio of
K-F1 to F1-F2 of one is suggested.

Figure 4: Diagram of the measurement system for the relative detection
efficiency. A light source with diffuser ball is located between the two 20”
PMTs.

Figure 5: Relative detection efficiency relative to the reference PMT from
different ranges of light intensity and a survey on different voltage ratios
of the K-F1 and F1-F2 for the MCP-PMT. It is calculated from the relative
output charge of the MCP-PMT and the reference dynode-PMT in pe, which is
related to the solid angle, PDE of each individual PMT. The y-axis is a
further normalized value of each curve with the ratio of (K-F1)/(F1-F2)

= 1.

###  III.2 Charge resolution of MCP-PMT

To achieve better amplification and charge resolution, MCP-PMTs use two
internal MCPs in parallel ref30 . It has been confirmed that the charge
resolution of the MCP-PMT output can be improved with an inverted voltage in
the gap between the two MCPs (the gap stage MCP1b-MCP2a as shown Fig.8 between
MCPs) ref31 ; ref32 ; ref33 . The inverted voltage reduces the spread of
electrons arriving at the second MCP by rejecting those with lower kinetic
energies in the periphery and only allowing electrons with higher kinetic
energies to pass to MCP2. This feature affects both the amplification gain and
the charge resolution of the MCP-PMT.

Figure 6: (a) Gain versus voltage of MCP-PMT (serial No. 16105) with internal
normal and inverted gap voltage (black: inverted gap voltage; red: normal gap
voltage). (b) Charge resolution (calculated from a Gaussion fitting to the SPE
peak) versus voltage of MCP-PMT (serial No. 16105) with internal normal and
inverted gap voltage; (c) SPE spectrum of MCP-PMT with internal normal and
inverted gap voltage with a same peak gainJUNO-PMT-gain-Zhang2021GainAC .

We conducted an evaluation of the effects of the inverted gap voltage on the
PMT gain and charge resolution of several PMT samples, using a Gaussian
fitting to the spectra SPE(Single Photoelectron) peak. The SPE spectra of the
MCP-PMT with both the inverted and normal gap voltage are shown in Fig.6. As
expected, the PMT gain reduced from the normal gap voltage to the inverted gap
voltage due to fewer electrons reaching the second MCP. A higher voltage was
required in total to achieve the same gain for the configuration of the
inverted gap voltage. However, the charge resolution of the MCP-PMT with an
inverted gap voltage is better than that with the normal gap voltage for the
same applied voltage or the same gain, which is consistent with our
expectations. It is important to note that the higher the working voltage of a
PMT, the larger the dark noise rate. Further checks were conducted with
different inverted gap voltages of MCP-PMT in detail on the gain, dark noise,
charge resolution of SPE, and peak-to-valley ratio of MCP-PMT, as shown in
Table 3 and Fig.7. Based on our findings, an inverted gap voltage between the
two MCPs of the JUNO MCP-PMT HV divider is suggested, as shown in Fig.8. This
voltage should be higher than 30 V, which can be satisfied by most of the
working voltage of MCP-PMTs with a gain of 1$\times 10^{7}$.

Figure 7: (a) Peak-to-Valley ratio versus inverted gap voltage; (b) Charge
resolution versus inverted gap voltage.  Table 3: Gain and dark noise rate of
the MCP-PMT with different inverted gap voltage.

Gap Voltage | Voltage Value | HV | Dark Rate
---|---|---|---
|  | (gain@$10^{7}$) | (gain@$10^{7}$)
Inverted | 20 V | 1560 V | 15 kHz
| 30 V | 1560 V | 14 kHz
| 40 V | 1650 V | 17 kHz
| 50 V | 1675V | <20kHz
| 60 V | 1675 V | 17 kHz
Normal | 36 V | 1535 V | 14 kHz

Figure 8: Schema of the HV divider of MCP-PMT with inverted gap voltage
between MCPs

###  III.3 HV divider direct current

For JUNO, the high voltage for each PMT will be provided by a high voltage
unit (HVU) produced by the HVSYS companyHVU-ru . However, the maximum current
of the designed HVU is less than 300 µA, which will limit the working DC of
the PMT or even further lower it, taking into account the reliability. It is
important to note that the higher the working current, the lower the
reliability.

For a dynode-PMT, the resistance between the electrodes is virtually infinite,
so the resistance of the HV divider chain is almost decoupled from the PMT
itself within a large range, allowing for a constant voltage division
ratioref21 . Based on our analysis, we conclude that the working DC of the HV
divider should be less than 1/3 of the maximum DC of the HVU, taking into
consideration the reliability, linearitydiru-PMT-linearity , and component
selection.

For an MCP-PMT, the finite body resistance of the MCP is an important feature
that distinguishes it from the dynode-PMT. The MCP and the HV divider resistor
chain will influence each other, which will further affect the performance of
the MCP-PMT. The effective resistance ($R$) of the MCP with a designed HV
divider can be calculated using Formula 3-1.

$R=\frac{R_{divider}\times R_{body}}{R_{divider}+R_{body}},$ (3-1)

The $R_{divider}$ represents the resistance of R4 and R6 in Fig.8, which is in
parallel with the MCPs, while the $R_{body}$ represents the body resistance of
the MCP. When the $R_{divider}$ is much smaller than the $R_{body}$, the
effective resistance $R$ is approximately equal to the $R_{divider}$, and the
divider ratio of the HV divider will not be influenced by the $R_{body}$. The
mean value of $R_{body}$ is around 20 M$\Omega$ for the 15,000 JUNO MCP-PMTs.
Taking into consideration the possible options of the HV divider DC of around
180 µA (High DC) and 70 µA (Low DC), we compared the peak-to-valley ratio and
charge resolution, as shown in Fig.6 and Fig.7. The charge distribution of SPE
of high DC is slightly narrower than that of low DC.

We also measured the non-linearity of the MCP-PMT with different DCs, as shown
in Fig.6. When the intensity is small (at a single photoelectron level), the
measured non-linearity of PMT response is deeply affected by the system noise
and statistical fluctuations. When the intensity is less than 1000 pe (with a
frequency of 200 Hz), there is no obvious difference in non-linearity between
the DCs. The peak-to-valley ratio of MCP-PMT and the charge resolution with
High DC are better than that of Low DC. When the MCP-PMT is working at the
same gain, the working voltage at high DC is lower than that of low DC.
However, there is no obvious effect on timing performance among different DCs.

In summary, when the MCP-PMT is working at high DC (180 µA@1800 V), its peak-
to-valley ratio and charge resolution are better than those of low DC, and the
required voltage to achieve the same gain is lower. Based on the requirements
for the performance of MCP-PMT, we advise a divider DC of 180 µA, which has
been adopted by JUNO for a balance of performance and acceptable reliability.

Figure 9: (a) The SPE spectrum of MCP-PMT with divider current of 70 µA and
180 µA; (b) The non-linearity of MCP-PMT with different DC.

Figure 10: (a) The working voltage distribution of the MCP-PMT at a gain of
$10^{7}$; (b) The gain distribution of MCP-PMT with the same voltage; (c)
Peak-to-valley ratio; (d) Energy resolution; (e) Rise time; (f) Fall time.

###  III.4 Pulse shape

In a large-scale high-precision neutrino experiment, the quality of the PMT
output pulse is one of the key factors in achieving accurate measurements. In
the circuit shown in Fig. 8, resistors Ra1, Ra2, R8, R9, and R12 effectively
reduce the ringing of the output waveform by forming an internal filter. Fig.
11 shows the waveforms of the MCP-PMT with different configurations, and the
ringing is efficiently reduced when the resistors Ra1=Ra2=R8=R9=50 $\Omega$.
In Fig. 11, the red line represents the output waveform of a single
photoelectron, and the blue square waveform is the trigger gate synchronized
to the signal. The resistors will also affect the rise time, fall time, and
amplitude of the SPE. The performances of MCP-PMT are measured as shown in
Fig. 12c. The larger the resistance, the smaller the amplitude of the output
waveform. Finally, we suggest using Ra1=Ra2=R8=R9=50 $\Omega$ for the JUNO
MCP-PMT.

Figure 11: The SPE waveforms of MCP-PMT with different configurations for
ringing: X-axis: time, 50ns/div; Y-axis: amplitude, 4mV/div. (a) Ra1= Ra2= R8=
R9= 0 $\Omega$; (b) Ra1= Ra2= R8= R9= 50 $\Omega$;

Figure 12: (a) Rise time of SPE versus resistors. (b) Fall time of SPE versus
resistors. (c) Amplitude of SPE versus resistors.

In JUNO, a Flash ADC with 1 GSample/s is used to sample the waveform of the
PMT, and the rise time of the raw waveform of MCP-PMT is about 1-2 ns, which
is determined by the structure of MCP-PMT. The distance between the MCPs and
the thickness of the MCP itself are very small order of few millimeter, and
the amplification process of electrons is very quickref34 . This is much
faster compared to the rise time (5 ns) of Hamamatsu dynode-PMT and the
bandwidth of the sampling electronics. As a result, only 1-2 points can be
sampled on the rising edge of the pulse, which can lead to information loss.
Therefore, it is prescribed to slow down the rise time of MCP-PMT for JUNO. A
specialized filter circuit (a capacitor, Ca) for MCP-PMT is designed to slow
down its rise time, as shown in Fig.13, where Ca and R12 are composed of an RC
filter ($\tau$ is around 2 ns, the high-frequency part is filtered out by the
RC circuit), where the exact value will be confirmed in upcoming measurements
because of distributed factors. Without the capacitor of Ca, the waveform of
SPE will always have two peaks, as seen in Fig.3, which is due to the
impedance mismatching among the internal structure of MCP-PMT. With the Ca,
the waveform is smooth and the rising edge is slowed down, and the fall time
of MCP-PMT is also affected to around 25 ns, which is related to the value of
R12.

Figure 13: Optimized output circuit for MCP-PMT

We surveyed the value of R12 while keeping the other values constant. The
value of R12 affects the fall time, amplitude, and ringing of the waveform by
forming another RC circuit. The smaller the R12, the faster the fall time and
the greater the amplitude. Additionally, the ringing of the waveform is
related to the R12, with smaller R12 values leading to larger ringing.When R12
is equal to 20 $\Omega$, the ringing is acceptable for JUNO, as shown in
Fig.14. However, to further reduce the ringing, we advise using a value of R12
of 30 $\Omega$. After optimizing the rise time of the waveform, the fall time
is also changed.

The features of MCP-PMT with different configurations are measured and
summarized in Table 4. The rise time of the waveform can be slowed down to
less than 3 ns when the Ca is set to 22 pF or 30 pF, where the high-frequency
part of the waveform is filtered by the RC circuit. Fig.14 shows the waveform
at Ca=0 pF and Ca=22 pF when keeping the R12 unchanged. Fig.14 also shows the
waveform of SPE after applying the Fourier transforms with different Ca
values. From the spectrum of the waveform in the frequency domain, we observe
that the greater the Ca, the larger the waveform distortion.

In summary, based on the rise time, the shape of the waveform, and the optimal
filtering of high-frequency components, we suggest using Ca=22 pF and R12=30
$\Omega$ as the optimal choice for JUNO MCP-PMT.

Figure 14: Waveform of SPE at different Ca. (a) Ca=0 pF; (b) Ca=22 pF; (c)
Spectrum of waveform of SPE under different Ca in frequency domain: the x-axis
is the frequency of the spectrum in MHz, and the y-axis is the power. Table 4:
The parameters of MCP-PMT with different Ca

| Ca | Gain | Voltage | PV | Charge | Rise | Fall | FWHM | SPE
---|---|---|---|---|---|---|---|---|---
| pF | $10^{7}$ | V | ratio | Res. % | time/ns | time/ns | ns | Amp./mV
PMT1 | 0 | 1.17 | 1770 | 6.7 | 21 | 1.9 | 28.5 | 10.9 | 6.7
| 10 | 1.17 | 1770 | 7.0 | 22 | 2.5 | 28.9 | 11.0 | 6.7
| 22 | 1.17 | 1770 | 6.3 | 23 | 3.3 | 28.1 | 12.2 | 6.4
| 30 | 1.17 | 1770 | 7.4 | 21 | 3.3 | 29.8 | 12.1 | 6.3
PMT2 | 10 | 1.17 | 1705 | 4.6 | 32 | 2.5 | 27.3 | 10.6 | 7.2
| 22nF | 1.17 | 1705 | 4.9 | 34 | 3.3 | 26.0 | 10.6 | 7.0
| 30 | 1.17 | 1705 | 4.3 | 31 | 3.3 | 26.4 | 11.1 | 6.8

###  III.5 Restrict the maximum amplitude

The JUNO detector was established with a rock overburden of 700 meters
underground to reduce the backgrounds related to cosmic rays. According to
simulations of the actual terrain, there will be around 3 Hz muons passing
through the central detector, and the average energy of muons is 215 GeV.
Muons reaching the CD will be detected and seen by the PMTs. The output
strength of high-energy muons passing through the CD will be very large, with
thousands to tens of thousands of photoelectrons generated in each PMT. The
amplitude of the PMT pulse can reach tens of volts ref6  if there is no
saturation or limitation. For a JUNO MCP-PMT, the maximum output amplitude can
reach around 30 to 50 Vdiru-PMT-linearity , while the maximum output amplitude
of HPK PMT is around 7-8 V with a load of 50 $\Omega$. The designed dynamic
range of JUNO electronics is from 0-8 V with a load of 50 $\Omega$. Working
under such a large dynamic range to cover from a few millivolts to several
tens of volts with high accuracy is challenging and potentially dangerous for
the electronics system.

To limit the maximum output amplitude of the MCP-PMT and meet JUNO
requirements, we considered a limitation circuit to restrict the largest
amplitude while maintaining an acceptable waveform frequency and reliability.
In the circuit design, a diode is suggested as a protection and restriction
device, as shown in Fig.15.

Figure 15: Schematic diagram of MCP-PMT protection circuit

Table 5 shows the output amplitude of several tested diodes for the MCP-PMT.
Without a protective diode, the maximum output of the MCP-PMT is around 28 V
(200 Hz). With diode protection, the maximum output of the MCP-PMT is limited,
which is related to the cut-off voltage of the tested diode. In Table 5, the
amplitude of the PMT output begins to decrease when the PMT output voltage
exceeds the cut-off voltage of the diode. According to the frequency domain of
the waveform, the bandwidth of the PMT waveform is less than 500 MHz.
According to the RF interface parameters, as shown in Table 6, when the diode
capacitance is less than 3.5 pF, there is no effect on the signal. Figure 16
shows the timing parameters of SPE under four different diodes in Table 5.

Table 5: MCP-PMT output amplitude with different diodes

Type | Max output | Specification | Capacitors (pF)
---|---|---|---
| amp. (V) | Clam voltage (V) | typical (max.)
No diode |  28 | - | -
NXP-ESD-10V |  17 |  10 | 1.55 (1.75)
NXP-TVS-5v |  7.5 |  5 | 152 (200)
ESD08V32D-LC |  15 |  8 | 1.2

Table 6: RF interface requirements (Reference criterion: 50 $\Omega$ system,
port voltage standing wave ratio degraded from 1 to 5)

Frequency | Cp | Frequency | Cp
---|---|---|---
| (Appro. value) |  | (Appro. value)
100 Mhz | $\leq{16\,pF}$ | 1.8 GHz |  $\leq{0.9\,pF}$ (1.75)
450 MHz | $\leq{3.5\,pF}$ | 2.1 GHz |  $\leq{0.9\,pF}$ (200)
800 MHz | $\leq{2\,pF}$ | 3 GHz | $\leq{0.5\,pF}$
1 GHz | $\leq{1.5\,pF}$ | 5 GHz | $\leq{0.33\,pF}$

Figure 16: Timing performance of MCP-PMT versus the incident photoelectrons of
MCP-PMT with different diodes. (a) Rise Time; (b) Fall Time; (c) FWHM; (d)
TTS.

When using the NXP-ESD-10V or ESD08V32D-LC diode models, the time features of
the MCP-PMT waveform are not affected because the capacitance of these two
diodes is small, less than 3.5 pF. Therefore, the NXP-ESD-10V and ESD08V32D-LC
diode models are the optimal choices for limiting the maximum output signal
amplitude of the MCP-PMT while maintaining the time features of the waveform.

##  IV Final designs

Several versions of the HV dividers have been designed to match the
requirements of different options of JUNO electronics for R$\&$D and testing
purposes.

###  IV.1 Design I: w/ restriction on Maximum amplitude

As shown in the previous sections, the maximum amplitude of the dynode-PMT is
around 7-8 V, but the maximum amplitude of the MCP-PMT is around 30-50 V. In
the early stage of JUNO electronics design, the maximum amplitude needed to be
lower than the 8 V required by the electronics interface. Therefore, we used a
diode after the decoupling capacitor to limit the output amplitude and the
maximum amplitude of the PMT, which was limited to 8 V without affecting the
signal bandwidth. The optimization of the HV divider with the scheme of
limiting the maximum amplitude of the PMT has also been finalized, as shown in
Fig. 17.

Figure 17: HV divider with maximum amplitude limitation. (a) MCP-PMT; (b)
Dynode-PMT

###  IV.2 Design II: Two outputs scheme

During the R$\&$D of JUNO, another option was required to provide two
identical outputs from the HV divider to the electronics JUNO-BX-
BELLATO2021164600 . The two-output option, based on design I, was designed in
the HV divider for overshoot suppression by the following electronics. The
final design with the two-output scheme is shown in Fig. 18.

Figure 18: Anode readout with two outputs and maximum output amplitude
limitation. (a) MCP-PMT; (b) Dynode-PMT

###  IV.3 Design III: BX2 scheme

The BX2 scheme was another early option for the JUNO design. In the BX2
scheme, as shown in Fig. 19, the HV divider and the decoupling capacitor
circuit (Splitter) of the PMT, as well as the high voltage and Front End Unit
(FEU), are integrated. The final design of the HV divider with the BX2 scheme
is shown in Fig. 20, where it is only integrated with the decoupling capacitor
and the amplitude limitation diode has been removed.With the BX scheme, there
are many interfaces, such as the interface between the electronics and PMT
waterproof potting, cables, etc. In addition, the power of each channel with
the BX2 scheme is 17.5 W, and there are a total of 20,000 channels.
Furthermore, it needs to deal with a more complicated reliability problem.

Figure 19: JUNO electronic pre-scheme of BX2

Figure 20:  HV divider of the BX2 scheme. (a) MCP-PMT; (b) Dynode-PMT

####  IV.3.1 For PMT acceptance test

A special version of the PMT HV divider of the BX2 schema (only the HV divider
as in Fig.20) with a decoupling capacitor of 10 nF is required to support the
acceptance testing of the bare PMTs, which is designed as a plug-gable version
to test the bare PMTs JUNO:20inchPMT:2022hlz . The products used for bare PMT
testing are shown in Fig. 21.

Figure 21: Plug-gable HV divider for bare PMT testing. Left for HPK PMTs;
right for NNVT PMTs.

###  IV.4 Design IV: 1F3 scheme

To avoid the disadvantages of the BX scheme and decouple the electronics and
PMTs, a new scheme for the JUNO electronics system has been proposed, named
1F3 (one electronics box for three PMT channels)
JUNO:physics:2022PrPNP.12303927J . In the 1F3 scheme, the HV divider of the
PMT is separated from the splitter, FEC, and ADC, as compared to the BX
scheme. The challenge of this design is the impedance matching between the PMT
and the electronics Liu_2023_1F3 . The circuit for the 1F3 scheme is shown in
Fig. 22.

Figure 22: Final design of HV divider of 1F3 scheme. (a) MCP-PMT; (b) Dynode-
PMT

In the 1F3 scheme, a 2 m long high-voltage coaxial cable is used between the
PMT and the electronic underwater box. The impedance of the cable is required
to be 50 $\Omega$, but the HV connector is not easy to match the requirement.
When the impedance cannot match perfectly, the PMT output waveform will show
some ringing, reflection, and a larger overshoot. According to the study of
the MCP-PMT, R12 will affect the ringing. When the R12 value is smaller, the
ringing is larger, the SPE amplitude is larger, and the fall time is faster.
R12=5 $\Omega$ is recommended, and the ringing of the waveform of MCP-PMT is
very small, as shown in Fig. 23.

Figure 23: The waveform of SPE of MCP-PMT, R12=5 $\Omega$

####  IV.4.1 Final design for JUNO

For JUNO, the 1F3 scheme was adopted to match the final electronics, and it
was used as the final design of the HV dividers for the 20-inch PMT, as shown
in Fig. 22. The SPE waveforms of dynode-PMT and MCP-PMT are shown in Fig. 24.

Figure 24: SPE waveforms. (a) Dynode-PMT; (b) MCP-PMT

In the JUNO, the PCB design of the HV divider with good reliability is also
one of the key factors. Table 7 shows the design requirements for the PCB.
Fig. 25 shows the PCB design for the JUNO PMT HV divider, where the resistors
and capacitors on the PCB have been selected from Vishay’s wire-wound
components for reliability. The capacitors were selected after a pre-sample
aging test from eight types of candidates.

Table 7: The requirements of PCB design

Width of PCB wring | 0.5 mm (Common wiring)
---|---
| 0.7 mm
| (Wiring between
| high voltage and ground)
Through hole size | Actual pin diameter+0.2 mm
Pad size | 2-2.5 times diameter of PMT Pin
Minimum distance | >1.6 kV/mm
design of | (40 V/mil)
breakdown/solder joint |

Figure 25: The PCB of JUNO PMT divider; (a) Dynode-PMT; (b) MCP-PMT

##  V Quality control & burning test

We have finalized the design, optimization, and production of the JUNO HV
divider. The performance acceptance testing of 20,000 20-inch PMTs has
finished in early 2021, and the final production of the HV divider for JUNO
has also been completed.

###  V.1 Reliability

For the JUNO, the HV divider is required to have a lifetime of over 20 years.
With the selected components, a reliability expectation has been forecasted
for the lifetime and possible failure rate of the HV dividers GJBZ-299C-2006 .
This has been done with different component standards and temperatures, and
the results with commercial components at 20∘C are shown in Fig. 26. The
capacitors contribute the maximum according to the model, even though the
workload of each component in power and voltage is designed to be less than
1/2 of their nominal value. The final failure rate of each HV divider is less
than 50 FIT with commercial components, which already meets the requirements
of JUNO.

Figure 26: Components of HV divider and reliability calculation. (a) resistors
and capacitors; (b) reliability calculation of each components for HPK dynode-
PMT; (c) reliability calculation of each components for NNVT MCP-PMT.

###  V.2 Soldering

In order to meet the requirements on the PMT flasher, special testing on the
flasher of the HV divider has been conducted PMT-flasher-Yang_2020 . Smooth
soldering and special cleanliness are required for the flasher, and the
soldering quality is further checked with a special procedure and pre-samples,
as shown in Fig. 27. The soldering, production, and following burning test of
PCBs of HV dividers for 20,000 20-inch PMTs have been entrusted to Tianjin
Centre Advanced Tech, CO, LTD tjcentre . This ensures a high level of quality
control and reliability for the production of the HV dividers. The soldering
quality, temperature underworking, and maximum voltage of 6000 V were applied
to four random samples, and all of them passed the test, as shown in Fig. 27.
This indicates that the production process for the HV dividers is reliable and
meets the design requirements.

Figure 27: HV divider soldering quality control. (a) X-ray check; (b)
soldering quality; (c) temperature monitoring; (d) smooth soldering shape.

###  V.3 Burning test

After soldering and cleaning the PCBs, a burning test is carried out for each
HV divider piece for 12 days $\times$ 24 hours at a high voltage workload of
3000,V and a temperature of 75∘C to reach 90% strength, as shown in Fig. 28.
In total, 21,000 PCBs of HV dividers are subjected to aging experiments, and
their aging performances are monitored by the current of the voltage divider.
During the test, only four 362 k$\Omega$ resistors of dynode-PMT were found to
be broken, and the failure ratio was approximately 0.000067. The burning test
effectively screens out defective components. The failed resistors were
further checked by the vendor and another standalone analysis company, and the
conclusion was that the failure was due to a possible random voltage surge.
Even considering these kinds of failures, the system still meets the
requirements of JUNO.

Figure 28: Burning test of the HV divider PCBs. (a) a sample of the produced
divider; (b) The incubator and assembled samples of the burning test; (c) The
burning strength calculation with different conditions;

##  VI Summary

In summary, we have successfully designed, optimized, and produced the 20-inch
PMT HV divider for the JUNO. The HV divider is designed with low current and
small overshoot and ringing for the HPK dynode-PMT. We also designed and
developed the HV divider for the new large-area 20-inch MCP-PMT, with
optimized HV ratio and DC to improve its collection efficiency, P/V, and
charge resolution. The overshoot and ringing of the 20-inch MCP-PMTs have been
effectively controlled, and the rise time and fall time have been improved to
meet the JUNO requirements. Four schema designs of HV dividers for different
options have been designed and optimized.The mass production of HV dividers
for 20,000 JUNO 20-inch PMTs has been completed with good reliability after a
burning test. The design, optimization, PCB design and production, soldering
and aging test, and quality control of the HV divider have been presented in
this article, which can provide valuable experiences for other photodetector
experiments.

## Acknowledgments

This work was supported by the National Natural Science Foundation (NSFC) of
China No. 11875282, the Strategic Priority Research Program of the Chinese
Academy of Sciences, Grant No. XDA10011100, the CAS Center for Excellence in
Particle Physics and the Research Foundation of Education Bureau of Hunan
Province, China (Grant No. 22A0281).

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# Mass Composition of UHECRs from $X_{\rm max}$ Distributions Recorded by the
Pierre Auger and Telescope Array Observatories

Nicusor Arsene<EMAIL_ADDRESS>Institute of Space Science,
P.O.Box MG-23, Ro 077125 Bucharest-Magurele, Romania

###### Abstract

In this paper we infer the mass composition of the ultra high energy cosmic
rays (UHECRs) from measurements of $X_{\rm max}$ distributions recorded at the
Pierre Auger (2014) and Telescope Array (TA) (2016) Observatories, by fitting
them with all possible combinations of Monte Carlo (MC) templates from a large
set of primary species (p, He, C, N, O, Ne, Si and Fe), as predicted by EPOS-
LHC, QGSJETII-04 and Sibyll 2.1 hadronic interaction models. We use the
individual fractions of nuclei reconstructed from one experiment in each
energy interval to build equivalent MC $X_{\rm max}$ distributions, which we
compare with the experimental $X_{\rm max}$ distributions of the other
experiment, applying different statistical tests of compatibility. The results
obtained from both experiments confirm that the mass composition of the UHECRs
is dominated ($\gtrsim$$70\%$) by protons and He nuclei in the energy range
investigated $\lg E(\rm eV)$ = [17.8–19.3] (Auger) and $\lg E\rm(eV)$ =
[18.2–19.0] (TA). The indirect comparisons between the $X_{\rm max}$
distributions recorded by the two experiments show that the degree of
compatibility of the two datasets is good, even excellent in some high energy
intervals, especially above the ankle ($\lg E(\rm eV)\sim 18.7$). However, our
study reveals that, at low energies, further effort in data analysis is
required in order to harmonize the results of the two experiments.

##  I Introduction

The UHECRs ($E>10^{18}$ eV) are the most energetic particles in the universe
produced by the most energetic astrophysical objects. Their origin and
acceleration mechanisms are still unknown despite the huge efforts in the last
decades of the entire astrophysics community Mollerach and Roulet (2018). A
realistic description of such a complex process of production and transport
through the intergalactic medium requires accurate information about the
extra-galactic magnetic fields Harari  _et al._ (2016); Mollerach and Roulet
(2020), interaction with the cosmic microwave background (CMB) and the extra-
galactic background light (EBL), distribution and type of sources which
accelerate these particles Thoudam  _et al._ (2016); Aloisio (2017); De Marco
and Stanev (2005); Berezinsky  _et al._ (2006); Aloisio  _et al._ (2012) and,
last but not least, accurate information about their energy spectrum Abu-
Zayyad  _et al._ (2015); Aab  _et al._ (2020), mass composition Abbasi  _et
al._ (2019); Yushkov (2020); Aab  _et al._ (2017a); Glushkov and Saburov
(2019) and arrival directions Aab  _et al._ (2017b, 2018) measured at ground
based cosmic ray experiments. These former properties are indirectly
reconstructed, making use of different parameters of extensive air showers
(EAS) recorded by the cosmic ray experiments such as the Pierre Auger
Observatory (Auger) Aab  _et al._ (2015) and the Telescope Array (TA) Tokuno
_et al._ (2012).

The mass composition of the primary UHECRs is still a matter of debate even if
it was extensively studied by combining different complementary techniques Aab
_et al._ (2014a); Abbasi  _et al._ (2018); Aab  _et al._ (2014b, 2016);
Knurenko and Petrov (2020). The most reliable parameter from the EAS used to
infer the mass composition proved to be $X_{\rm max}$, the atmospheric depth
where the energy deposit profile of the secondary particles reaches its
maximumGaisser and Hillas (1977). This parameter is related to the primary
particle mass as $\langle X_{\rm max}\rangle\propto-\ln A$.

The measurements of mass composition reported by the Auger Aab  _et al._
(2014a); Bellido (2018) and TAAbbasi  _et al._ (2015) experiments, on the
basis of the first two moments of the $X_{\rm max}$ distributions ($\langle
X_{\rm max}\rangle$ and $\sigma_{X_{\rm max}}$) are not in very good agreement
on the entire energy range. While the TA results suggest only a light
composition above $E>10^{18.2}$ eV, the Auger results clearly indicate a
transition to a heavier component starting from $E\sim 10^{18.5}$ eV and
becoming increasingly heavier at the highest energies.

On the other hand, it was shown that using only the limited information given
by the first two moments of a $X_{\rm max}$ distribution may lead, in very
particular cases, to a misinterpretation of the mass composition, since
different mixes of primary particles can reproduce exactly the same $\langle
X_{\rm max}\rangle$ and $\sigma_{X_{\rm max}}$ values. To avoid such
situations, a method was proposed, which uses the entire shape of each $X_{\rm
max}$ distribution by fitting them with Monte Carlo (MC) templates for four
fixed primary species (p, He, N and Fe) obtaining in this way information
about fractions of individual nuclei Aab  _et al._ (2014c). Following this
approach, the measurements of $X_{\rm max}$ distributions recorded by Auger
and TA experiments were indirectly compared de Souza (2018) concluding that
both measurements are compatible in the limits of statistical and systematical
uncertainties.

In a recent paper Arsene and Sima (2020), we showed that by fitting the
$X_{\rm max}$ distributions only with four fixed elements (p, He, N and Fe),
an artificial worsening of the fit quality can be induced and the
reconstructed fractions might be biased as a consequence of a high abundance
of some intermediate elements (e.g., Ne/Si) not included in the fitting
procedure. We argued that a more appropriate approach is to fit the observed
$X_{\rm max}$ distributions with all possible combinations of elements from a
larger set of primaries (p, He, C, N, O, Ne, Si and Fe) obtaining the “best
combination” of elements that best describe the data.

In this work, we perform an indirect comparison between $X_{\rm max}$
distributions measured by Auger (2014) Aab  _et al._ (2014a) and TA (2016)
Abbasi  _et al._ (2018) experiments following the approach proposed in Arsene
and Sima (2020).

In Section II, we describe the simulation procedure for obtaining the MC
templates for a large set of primary species (p, He, C, N, O, Ne, Si and Fe),
taking into account the detector effects (acceptance and resolution) of both
experiments, employing different hadronic interaction models. In Section III,
we obtain the fractions of individual nuclei which best describe the $X_{\rm
max}$ distributions measured by Auger and TA experiments on the entire energy
range $\lg E(\rm eV)$ = [17.8–19.3] (Auger) and $\lg E\rm(eV)$ = [18.2–19.0]
(TA) following two methods. The first method reconstructs the fractions of
individual nuclei of the “best combination” of elements which best describe
the observed $X_{\rm max}$ distributions. With the second method we extract
the average of the reconstructed fractions of each species from all possible
combinations of fitting elements whose fit quality was higher than a threshold
value, obtaining in this way the evolution of the abundances of all primary
species as a function of energy.

In Section IV, we present the compatibility of the measurements of the two
experiments on the basis of three statistical tests: p-value as goodness-of-
fit, Kolmogorov–Smirnov ($KS$) and Anderson–Darling ($AD$). Section V
concludes the paper.

##  II Monte Carlo templates

The MC templates ($X_{\rm max}$ distributions or Probability Density Functions
(PDFs) of $X_{\rm max}$) for 8 primary species (p, He, C, N, O, Ne, Si and Fe)
for each energy interval of $0.1$ in $\lg E(\rm eV)$, employing three hadronic
interaction models EPOS-LHC Werner  _et al._ (2006), QGSJETII-04 Ostapchenko
(2006) and Sibyll 2.1 Ahn  _et al._ (2009) were generated using CONEX v4r37
simulation code Pierog  _et al._ (2006); Bergmann  _et al._ (2007). Due to the
experimental limitations, the MC templates for the TA experiment were
simulated for only eight energy intervals in the range $\lg E(\rm eV)$ =
[18.2–19.0] and 15 energy intervals for Auger $\lg E(\rm eV)$ = [17.8–19.3].
The zenith angle of the showers were isotropically sampled in the interval
$\theta=[0^{\circ}$–$60^{\circ}]$ ensuring an isotropic flux on flat surface
$dN/dcos(\theta)\sim cos(\theta)$. The statistics of each $X_{\rm max}$
distribution are of the order of $10^{4}$–$10^{5}$ events, with a larger
number of simulations for proton induced showers in comparison with the
heavier nuclei.

A PDF of $X_{\rm max}$ for a primary nuclear species in a given energy
interval consists of a binned $X_{\rm max}$ distribution in the range [0–2000]
g/cm2 with a bin width $=20$ g/cm2 for the Auger case, while for the TA
experiment the $X_{\rm max}$ distributions are binned in the range [500–1300]
g/cm2 with a bin width $=40$ g/cm2.

To account for detector effects, the true $X_{\rm max}$ values calculated with
CONEX were modified in accordance with the acceptance and resolution of each
experiment. In the case of Auger, the true $X_{\rm max}$ values were modified
by using Equations (7) and (8) from Aab  _et al._ (2014a), while for the TA
case the true MC templates were modified in accordance with the biases
(reconstruction + acceptance) and resolutions computed in Abbasi  _et al._
(2018) in Table 1 and Figure 9 of that paper, considering full detector
simulations for p, He, N and Fe for the QGSJETII-04 model. For the
intermediate elements, the bias and resolution of $X_{\rm max}$ are obtained
using a 2nd degree polynomial interpolation. It is worth mentioning that the
possible uncertainties on the bias and resolution of the intermediate
elements, artificially introduced by this interpolation, would be much smaller
than the experimental resolution of the $X_{\rm max}$ parameter. The exact
values of the biases and resolutions used to construct the MC templates for
the TA MC templates are listed in Table 1. For EPOS-LHC and Sibyll 2.1 models
we considered the same values of biases and resolutions as computed for the
QGSJETII-04 model. Again, we have to stress that this approximation cannot
affect the results significantly, since the possible deviations from the true
values of bias and resolution for EPOS-LHC and Sibyll 2.1 cannot exceed more
than few g/cm2. As stated in Abbasi  _et al._ (2018), the bias is larger for
deeply penetrating showers, for example, protons. As an example, the
difference between the average $<X_{\rm max}>$ induced by protons predicted by
Sibyll 2.1 and QGSJETII-04 in the energy interval $\lg E(\rm eV)$ =
[18.2–18.3] is about $2.4$ g/cm2, so the bias of the reconstructed $X_{\rm
max}$ for protons predicted by these models should be roughly the same.

Table 1: The values of the biases and resolution in units of [g/cm2] used to build MC templates for the case of TA data in each energy interval for 8 primary species. | p | He | C | N | O | Ne | Si | Fe
---|---|---|---|---|---|---|---|---
lgE (eV) |  | bias
---
| res
---
| bias
---
| res
---
| bias
---
| res
---
| bias
---
| res
---
| bias
---
| res
---
| bias
---
| res
---
| bias
---
| res
---
| bias
---
| res
---
[18.2 - 18.3] | -6.17 | 17.2 | -5.62 | 15.7 | -4.04 | 14.43 | -3.7 | 14.2 | -3.41 | 13.75 | -2.87 | 13.18 | -2.03 | 12.39 | -1.78 | 13.2
[18.3 - 18.4] | -6.21 | 17.2 | -5.56 | 15.7 | -4.04 | 14.43 | -3.72 | 14.2 | -3.41 | 13.75 | -2.87 | 13.18 | -2.03 | 12.39 | -1.78 | 13.2
[18.4 - 18.5] | -11.07 | 17.2 | -8.65 | 15.7 | -3.27 | 14.43 | -2.17 | 14.2 | -1.17 | 13.75 | -0.55 | 13.18 | -0.81 | 12.39 | -1.54 | 13.2
[18.5 - 18.6] | -9.22 | 17.2 | -7.68 | 15.7 | -4.23 | 14.43 | -3.51 | 14.2 | -2.85 | 13.75 | -1.7 | 13.18 | -0.1 | 12.39 | -1.8 | 13.2
[18.6 - 18.7] | -9.31 | 17.2 | -7.13 | 15.7 | -2.33 | 14.43 | -1.37 | 14.2 | -0.49 | 13.75 | -0.99 | 13.18 | -0.83 | 12.39 | -2.34 | 13.2
[18.7 - 18.8] | -7.75 | 17.2 | -6.4 | 15.7 | -3.42 | 14.43 | -2.81 | 14.2 | -2.27 | 13.75 | -1.34 | 13.18 | -0.17 | 12.39 | -3.18 | 13.2
[18.8 - 18.9] | -7.75 | 17.2 | -6.4 | 15.7 | -3.42 | 14.43 | -2.81 | 14.2 | -2.27 | 13.75 | -1.34 | 13.18 | -0.17 | 12.39 | -3.18 | 13.2
[18.9 - 19.0] | -7.17 | 17.2 | -6.09 | 15.7 | -3.66 | 14.43 | -3.16 | 14.2 | -2.7 | 13.75 | -1.92 | 13.18 | -0.87 | 12.39 | -2.63 | 13.2
Figure 1: PDFs of $X_{\rm max}$ for proton and iron induced showers as
predicted by QGSJETII-04 and EPOS-LHC hadronic interaction models in the
energy interval $\lg E(\rm eV)=[18.2-18.3]$ for Auger (left) and TA (right)
experiments. Note the difference in the bin width.

An example of PDFs of $X_{\rm max}$ is given in Figure 1 for proton and iron
induced showers in the energy interval $\lg E(\rm eV)$ = [18.2–18.3] employing
QGSJETII-04 and EPOS-LHC hadronic interaction models and taking into account
the specific detector effects of each experiment.

In the next Section we will use these PDFs to fit the experimental $X_{\rm
max}$ distributions measured by Auger and TA experiments following a binned
maximum-likelihood procedure.

##  III Fitting fractions of Auger and TA $X_{\rm max}$ distributions

The experimental $X_{\rm max}$ distributions in each energy interval recorded
by Auger and TA experiments are fitted with all possible combinations of
primary elements (p, He, C, N, O, Ne, Si and Fe) following a binned maximum-
likelihood procedure. Thus, we obtain the fractions of individual nuclei that
best describe the observed $X_{\rm max}$ distributions.

Figure 2: Fitted fractions of individual nuclei in each energy interval
obtained with method #1 (left) and method #2 (right) employing the EPOS-LHC
hadronic interaction model. Full orange circles stand for TA (2016) data while
the black open circles represent the Auger (2014) data. The p-value parameters
are displayed on the bottom panels as one single value per energy interval
(left) and a range of p-values represented with arrows (right) (see text).

In this fitting procedure the minimizing quantity is $-\ln L$, which is
defined as:

$-\ln L=\sum_{i}y_{i}-n_{i}+n_{i}\ln(n_{i}/y_{i}),$ (1)

where $n_{i}$ represents the measured counts in the “$i$”-th bin of the
experimental $X_{max}$ distribution and $y_{i}$ is the MC prediction for that
bin Baker and Cousins (1984) given by $y_{i}=\sum_{j}[c_{j}f_{j}(i)]$ where
$j=$ p, He, C, N, O, Ne, Si and Fe, $f_{j}(i)$ is an MC template and $c_{j}$
is the fraction of the $j$ component.

We quantify the goodness of fit using the p-value parameter defined as:

$\textit{p-value}=1-\Gamma\left(\frac{ndf}{2},\frac{\chi^{2}}{2}\right),$ (2)

representing the probability of obtaining a worse fit than the one observed
even if the distribution predicted by the fit results is correct. Here, $ndf$
represents the number of degrees of freedom computed as the number of bins
(including empty bins) of each $X_{\rm max}$ distribution minus the number of
elements considered in the fit, $\chi^{2}$ is the sum of the square of
residuals using the parameters computed by the maximum-likelihood procedure
and $\Gamma$ is the normalized upper incomplete Gamma function. The results of
the fitted fractions of $X_{\rm max}$ distributions measured by Auger and TA
are presented following two methods. In method #1 we obtained the “best
combination” of elements from all possible combinations that best describe the
experimental $X_{\rm max}$ distributions based on the highest p-valuemax, in
each energy interval. The results are displayed in the left plots of Figures
2–4 for EPOS-LHC, QGSJETII-04 and Sibyll 2.1 hadronic interaction models. The
error bars of each point were computed using the MINOS technique based on
$\Delta L=1/2$ rule Eadie (1971). The reconstructed fractions obtained for
both experiments clearly show that the mass composition of primary UHECRs is
dominated by light elements (p and He), which present a modulation of their
abundances as a function of energy, but keeping the sum ($f_{p}+f_{He}$)
roughly constant on the entire energy spectrum. This feature is predicted by
all hadronic interaction models. An interesting aspect is the presence of Fe
nuclei in a quite high abundance ($\sim$$20\%$ predicted by EPOS-LHC model) on
the entire energy spectrum of TA data which, at least qualitatively, seems to
contradict the results of mass composition obtained on the basis of the first
two moments of the $X_{\rm max}$ distributions Abbasi  _et al._ (2015).

Figure 3: Fitted fractions of individual nuclei in each energy interval
obtained with method #1 (left) and method #2 (right) employing the QGSJETII-04
hadronic interaction model. Full orange circles stand for TA (2016) data while
the black open circles represent the Auger (2014) data. The p-value parameters
are displayed on the bottom panels as one single value per energy interval
(left) and a range of p-values represented with arrows (right) (see text).
Figure 4: Fitted fractions of individual nuclei in each energy interval
obtained with method #1 (left) and method #2 (right) employing the Sibyll 2.1
hadronic interaction model. Full orange circles stand for TA (2016) data while
the black open circles represent the Auger (2014) data. The p-value parameters
are displayed on the bottom panels as one single value per energy interval
(left) and a range of p-values represented with arrows (right) (see text).

In method #2 we extract the average of the reconstructed fractions of each
species from all possible combinations of fitting elements whose goodness of
fit parameter p-value was in the range [0.5 $\cdot$ p-valuemax, p-valuemax].
For example, if the combination of the elements p + N + Fe and p + He + Si or
simply p + He, and so on, give a good fit quality, then the average fraction
of protons is the mean of the proton fractions reconstructed from all these
combinations. In this case, the error bar associated to one species is
computed as the square root of the sum of the squared uncertainties of the
species obtained from different combinations of fitting elements whose
goodness of fit was in the range mentioned above. In this way, we have an
overview of the evolution of all fractions of individual nuclei as a function
of primary energy. The results are displayed on the right plots of Figures 2–4
for the three hadronic interaction models. Both methods give similar results
on the entire energy range. It is worth mentioning that the shortcoming of
method #2 is that the sum of all reconstructed fractions in a given energy
interval will be biased (¿1) as a consequence of the limitations of the
fitting procedure when using extreme combinations of elements. This bias is
evaluated for each energy interval as the sum of all reconstructed fractions,
$\sum_{i=p}^{Fe}f_{i}$, for all hadronic interaction models and is represented
in Figure 5.

Figure 5: The sum of all reconstructed fractions obtained with method #2 (see
text) as a function of energy for EPOS-LHC (up), QGSJETII-04 (middle) and
Sibyll 2.1 (bottom) hadronic interaction models.

As can be seen, the obtained values are not too far from $1$ for almost the
entire energy spectrum of Auger data, while for TA we observe larger biases in
many energy intervals, mainly due to the lower statistics in the experimental
$X_{\rm max}$ distributions. Even so, we consider that these biases are not
significant compared to the systematic and statistical uncertainties of the
experimental $X_{\rm max}$ distributions.

In the next Section, we will perform an indirect comparison between the
$X_{\rm max}$ distributions recorded by Auger and TA experiments using the
individual fractions of nuclei reconstructed in this Section III.

##  IV Auger vs. TA mass composition compatibility

The measurements of $X_{\rm max}$ distributions obtained by both experiments
cannot be directly compared because they use different approaches when
analyzing the recorded events Yushkov  _et al._ (2019). This is the reason we
produced two sets of MC predictions of PDFs of $X_{\rm max}$ taking into
account the specific resolution and acceptance of each experiment.

We test the compatibility of the measurements of $X_{\rm max}$ distributions
recorded by both experiments following two approaches on the common energy
range $\lg E(\rm eV)=$ [18.2–19.0]. In the first approach, we translate the
individual fractions of each primary species reconstructed from TA data in
equivalent PDFs of $X_{\rm max}$ predicted for Auger (PDFs of $X_{\rm
max}^{TA\rightarrow Auger}(i)=\sum_{j}c_{j}^{TA}f_{j}^{Auger}(i)$) and we
perform the comparison PDFs of $X_{\rm max}^{TA\rightarrow Auger}$ vs. Auger
data, while in the second approach we translate the individual fractions of
each primary species reconstructed from Auger data in equivalent PDFs of
$X_{\rm max}$ predicted for TA (PDFs of $X_{\rm max}^{Auger\rightarrow
TA}(i)=\sum_{j}c_{j}^{Auger}f_{j}^{TA}(i)$) and the comparison is performed
between PDFs of $X_{\rm max}^{Auger\rightarrow TA}$ vs. TA data.

In each approach, we use the fitting fractions obtained following method #1
and #2 (see Section III). Each comparison is performed for all hadronic
interaction models and is characterized by three statistical tests: p-value as
goodness of fit, $KS$ and $AD$. The p-value parameter is obtained by fitting
Auger data with PDFs of $X_{\rm max}^{TA\rightarrow Auger}$ and TA data with
PDFs of $X_{\rm max}^{Auger\rightarrow TA}$ following the same binned maximum-
likelihood procedure. Both $KS$ and $AD$ tests calculate the probability that
two distributions come from the same parent distribution. The $KS$ test is one
of the most used compatibility procedure with an improved response around the
peak of the distributions. On the other hand, the $AD$ test is optimized to
probe the compatibility with a higher accuracy on the tails of the
distributions which, in our cases, are dominated by protons and He nuclei.

In Figure 7 we present the comparison between Auger data and PDFs of $X_{\rm
max}^{TA\rightarrow Auger}$ for the energy interval $\lg E(\rm eV)=$
[18.2–18.3] considering the individual fitting fractions obtained from TA data
by the two methods (see Section III) for all three hadronic interaction
models. The statistics of the $X_{\rm max}$ distribution ($N=1952$) as well as
the p-value, $KS$ and $AD$ parameters are displayed on the plots. We chose to
give this example because it is the energy interval with the highest
statistics of events recorded by both experiments.

Figure 7 presents the same analysis as in Figure 7 but in the energy bin $\lg
E(\rm eV)$ = [18.6–18.7] in which the number of events recorded by Auger is
$N=575$. In this case, the compatibility of the two datasets is much better,
reaching $AD=0.35$ for the QGSJetII-04 model. It is important to note that the
highest degree of compatibility is obtained with the $AD$ test, which is more
sensitive on the tails of the distributions dominated by protons and He
nuclei. This is an indication that the measurements of both experiments are
more compatible with respect to the light component (p and He) of primary
UHECRs.

In Figures 9 and 9 we present the comparison between TA data and the fitted
fractions reconstructed from Auger data (second approach) for the energy bins
$\lg E(\rm eV)$ = [18.2–18.3] and $\lg E(\rm eV)$ = [18.6–18.7], respectively.

As in the first approach, the two experiments show that the degree of
compatibility of the two datasets is good, reaching excellent agreement in
some high energy intervals, especially when using the fractions reconstructed
with method #2 (e.g., p-value$=0.83$, $KS=0.94$ and $AD=0.86$) for $\lg E(\rm
eV)$ = [18.6–18.7] considering QGSJetII-04 model. At lower energies, our study
shows that the reconstruction methods need to be refined in order to achieve a
better compatibility between the two sets of measurements.

The complete set of probabilities computed with the three statistical tests is
listed in Tables 3 and 2 for the first approach and second approach,
respectively.

Figure 6: The comparison between Auger data (black points) and PDFs of $X_{\rm
max}^{TA\rightarrow Auger}$ in the energy interval $\lg E(\rm eV)=[18.2-18.3]$
using the TA fractions obtained with method #1 (top) and method #2 (bottom)
for EPOS-LHC (left), QGSJETII-04 (middle) and Sibyll 2.1 (right) hadronic
interaction models.

Figure 7: The comparison between Auger data (black points) and PDFs of $X_{\rm
max}^{TA\rightarrow Auger}$ in the energy interval $\lg E(\rm eV)=[18.6-18.7]$
using the TA fractions obtained with method #1 (top) and method #2 (bottom)
for EPOS-LHC (left), QGSJETII-04 (middle) and Sibyll 2.1 (right) hadronic
interaction models.

Figure 8: The comparison between TA data (orange points) and PDFs of $X_{\rm
max}^{Auger\rightarrow TA}$ in the energy interval $\lg E(\rm eV)=[18.2-18.3]$
using the Auger fractions obtained with method #1 (top) and method #2 (bottom)
for EPOS-LHC (left), QGSJETII-04 (middle) and Sibyll 2.1 (right) hadronic
interaction models.

Figure 9: The comparison between TA data (orange points) and PDFs of $X_{\rm max}^{Auger\rightarrow TA}$ in the energy interval $\lg E(\rm eV)=[18.6-18.7]$ using the Auger fractions obtained with method #1 (top) and method #2 (bottom) for EPOS-LHC (left), QGSJETII-04 (middle) and Sibyll 2.1 (right) hadronic interaction models. Table 2: The probabilities computed for the tree statistical tests of the comparison TA data vs. $X_{\rm max}^{Auger\rightarrow TA}$ considering the Auger fractions obtained using method #1 (left) and #2 (right). model | lg$(E/\rm eV)$ | $p-value$ | $KS$ | $AD$ | $p-value$ | $KS$ | $AD$
---|---|---|---|---|---|---|---
| $[18.2-18.3]$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$
| $[18.3-18.4]$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$
| $[18.4-18.5]$ | $2.0\times 10^{-5}$ | $8.5\times 10^{-5}$ | $1.3\times 10^{-3}$ | $<10^{-5}$ | $7.5\times 10^{-2}$ | $4.2\times 10^{-2}$
EPOS-LHC | $[18.5-18.6]$ | $2.7\times 10^{-4}$ | $2.1\times 10^{-3}$ | $4.4\times 10^{-2}$ | $2.8\times 10^{-5}$ | $8.0\times 10^{-2}$ | $4.8\times 10^{-2}$
| $[18.6-18.7]$ | $7.6\times 10^{-3}$ | $2.2\times 10^{-2}$ | $1.7\times 10^{-1}$ | $9.0\times 10^{-5}$ | $3.6\times 10^{-2}$ | $1.9\times 10^{-1}$
| $[18.7-18.8]$ | $2.1\times 10^{-5}$ | $1.8\times 10^{-2}$ | $1.1\times 10^{-2}$ | $<10^{-5}$ | $7.5\times 10^{-2}$ | $2.5\times 10^{-2}$
| $[18.8-18.9]$ | $8.5\times 10^{-2}$ | $1.5\times 10^{-1}$ | $4.4\times 10^{-1}$ | $8.5\times 10^{-4}$ | $2.7\times 10^{-1}$ | $4.8\times 10^{-1}$
| $[18.9-19.0]$ | $6.3\times 10^{-1}$ | $9.0\times 10^{-1}$ | $7.7\times 10^{-1}$ | $8.2\times 10^{-2}$ | $9.6\times 10^{-1}$ | $7.6\times 10^{-1}$
| $[18.2-18.3]$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$
| $[18.3-18.4]$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$
| $[18.4-18.5]$ | $<10^{-5}$ | $3.2\times 10^{-2}$ | $4.3\times 10^{-3}$ | $<10^{-5}$ | $3.2\times 10^{-2}$ | $4.3\times 10^{-3}$
QGSJETII-04 | $[18.5-18.6]$ | $4.0\times 10^{-5}$ | $4.4\times 10^{-3}$ | $2.0\times 10^{-3}$ | $<10^{-5}$ | $4.7\times 10^{-2}$ | $6.7\times 10^{-3}$
| $[18.6-18.7]$ | $8.3\times 10^{-1}$ | $9.4\times 10^{-1}$ | $8.6\times 10^{-1}$ | $2.2\times 10^{-1}$ | $9.4\times 10^{-1}$ | $8.6\times 10^{-1}$
| $[18.7-18.8]$ | $4.4\times 10^{-5}$ | $<10^{-5}$ | $5.7\times 10^{-4}$ | $4.0\times 10^{-4}$ | $5.6\times 10^{-1}$ | $5.5\times 10^{-1}$
| $[18.8-18.9]$ | $7.9\times 10^{-2}$ | $7.4\times 10^{-1}$ | $3.7\times 10^{-1}$ | $8.7\times 10^{-4}$ | $7.4\times 10^{-1}$ | $3.7\times 10^{-1}$
| $[18.9-19.0]$ | $9.0\times 10^{-1}$ | $1.0$ | $1.0$ | $2.5\times 10^{-1}$ | $1.0$ | $1.0$
| $[18.2-18.3]$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $1.4\times 10^{-3}$ | $7.5\times 10^{-4}$ | $3.0\times 10^{-4}$
| $[18.3-18.4]$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$
| $[18.4-18.5]$ | $3.5\times 10^{-3}$ | $3.9\times 10^{-3}$ | $1.9\times 10^{-2}$ | $1.5\times 10^{-3}$ | $9.0\times 10^{-2}$ | $1.6\times 10^{-1}$
Sibyll 2.1 | $[18.5-18.6]$ | $4.5\times 10^{-2}$ | $4.9\times 10^{-2}$ | $1.5\times 10^{-1}$ | $4.5\times 10^{-2}$ | $4.9\times 10^{-2}$ | $1.5\times 10^{-1}$
| $[18.6-18.7]$ | $2.9\times 10^{-1}$ | $2.8\times 10^{-1}$ | $5.1\times 10^{-1}$ | $3.6\times 10^{-2}$ | $4.6\times 10^{-1}$ | $6.4\times 10^{-1}$
| $[18.7-18.8]$ | $2.8\times 10^{-3}$ | $9.9\times 10^{-2}$ | $7.9\times 10^{-2}$ | $1.5\times 10^{-5}$ | $9.9\times 10^{-2}$ | $7.9\times 10^{-2}$
| $[18.8-18.9]$ | $4.8\times 10^{-2}$ | $4.2\times 10^{-1}$ | $4.5\times 10^{-1}$ | $4.0\times 10^{-4}$ | $4.2\times 10^{-1}$ | $4.5\times 10^{-1}$
| $[18.9-19.0]$ | $8.2\times 10^{-1}$ | $9.9\times 10^{-1}$ | $7.7\times 10^{-1}$ | $1.9\times 10^{-1}$ | $1.0$ | $1.0$
Table 3: The probabilities computed for the tree statistical tests of the comparison Auger data vs. $X_{\rm max}^{TA\rightarrow Auger}$ considering the TA fractions obtained using method #1 (left) and #2 (right). model | lg$(E/\rm eV)$ | $p-value$ | $KS$ | $AD$ | $p-value$ | $KS$ | $AD$
---|---|---|---|---|---|---|---
| $[18.2-18.3]$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$
| $[18.3-18.4]$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$
| $[18.4-18.5]$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$
EPOS-LHC | $[18.5-18.6]$ | $<10^{-5}$ | $<10^{-5}$ | $1.0\times 10^{-3}$ | $<10^{-5}$ | $<10^{-5}$ | $1.9\times 10^{-4}$
| $[18.6-18.7]$ | $<10^{-5}$ | $1.4\times 10^{-2}$ | $3.9\times 10^{-2}$ | $<10^{-5}$ | $8.4\times 10^{-3}$ | $3.1\times 10^{-2}$
| $[18.7-18.8]$ | $<10^{-5}$ | $<10^{-5}$ | $8.5\times 10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$
| $[18.8-18.9]$ | $1.1\times 10^{-2}$ | $7.7\times 10^{-3}$ | $2.4\times 10^{-2}$ | $1.4\times 10^{-4}$ | $6.8\times 10^{-2}$ | $1.3\times 10^{-1}$
| $[18.9-19.0]$ | $3.8\times 10^{-1}$ | $5.2\times 10^{-2}$ | $2.5\times 10^{-1}$ | $5.2\times 10^{-4}$ | $5.4\times 10^{-2}$ | $1.9\times 10^{-1}$
| $[18.2-18.3]$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$
| $[18.3-18.4]$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$
| $[18.4-18.5]$ | $<10^{-5}$ | $2.1\times 10^{-4}$ | $3.6\times 10^{-5}$ | $<10^{-5}$ | $8.2\times 10^{-5}$ | $1.2\times 10^{-5}$
QGSJETII-04 | $[18.5-18.6]$ | $<10^{-5}$ | $1.1\times 10^{-2}$ | $2.5\times 10^{-2}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$
| $[18.6-18.7]$ | $2.5\times 10^{-4}$ | $3.5\times 10^{-1}$ | $3.6\times 10^{-1}$ | $<10^{-5}$ | $5.2\times 10^{-2}$ | $7.2\times 10^{-2}$
| $[18.7-18.8]$ | $<10^{-5}$ | $6.1\times 10^{-5}$ | $6.3\times 10^{-4}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$
| $[18.8-18.9]$ | $<10^{-5}$ | $<10^{-5}$ | $2.1\times 10^{-4}$ | $<10^{-5}$ | $6.1\times 10^{-4}$ | $8.9\times 10^{-3}$
| $[18.9-19.0]$ | $7.9\times 10^{-2}$ | $1.6\times 10^{-2}$ | $8.1\times 10^{-2}$ | $4.1\times 10^{-4}$ | $1.7\times 10^{-2}$ | $8.6\times 10^{-2}$
| $[18.2-18.3]$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$
| $[18.3-18.4]$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$
| $[18.4-18.5]$ | $<10^{-5}$ | $8.4\times 10^{-4}$ | $1.4\times 10^{-4}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$
Sibyll 2.1 | $[18.5-18.6]$ | $<10^{-5}$ | $6.8\times 10^{-5}$ | $1.6\times 10^{-3}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$
| $[18.6-18.7]$ | $<10^{-5}$ | $2.7\times 10^{-2}$ | $4.7\times 10^{-2}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$
| $[18.7-18.8]$ | $<10^{-5}$ | $5.5\times 10^{-5}$ | $5.9\times 10^{-4}$ | $<10^{-5}$ | $<10^{-5}$ | $<10^{-5}$
| $[18.8-18.9]$ | $5.4\times 10^{-5}$ | $<10^{-5}$ | $1.8\times 10^{-4}$ | $<10^{-5}$ | $<10^{-5}$ | $2.4\times 10^{-4}$
| $[18.9-19.0]$ | $3.8\times 10^{-1}$ | $1.5\times 10^{-2}$ | $2.9\times 10^{-2}$ | $1.7\times 10^{-3}$ | $1.4\times 10^{-2}$ | $7.9\times 10^{-2}$

##  V Discussions and Conclusions

In this paper, we present an alternative approach to inferring the mass
composition of the primary UHECRs using the available $X_{\rm max}$
distributions recorded by Auger (2014) and TA (2016) experiments, by
comparisons with MC templates predicted by EPOS-LHC, QGSJETII-04 and Sibyll
2.1 hadronic interaction models.

A common general remark is that the measurements of both experiments suggest
that the mass composition of primary UHECRs is dominated ($\gtrsim$$70\%$) by
protons and He nuclei, which present a modulation of their abundances as a
function of energy but keeping the sum ($f_{p}+f_{He}$) roughly constant on
the entire energy spectrum. This conclusion holds for all the three hadronic
interaction models. An interesting aspect is the presence of Fe nuclei in a
quite high abundance ($\sim$$20\%$) in TA data on the entire energy spectrum
$\lg E(\rm eV)=$ [18.2–19.0] as predicted by the EPOS-LHC hadronic interaction
model.

The general conclusion is that, using the approaches proposed in this paper,
the degree of compatibility of the two datasets is good, reaching excellent
agreement in some high energy intervals above the ankle, as computed by the
three statistical tests (e.g., p-value$=0.83$, $KS=0.94$ and $AD=0.86$ for
$\lg E(\rm eV)=$ [18.6–18.7] considering the QGSJetII-04 model). The best
agreement is achieved when the indirect comparison is performed using the
reconstructed fractions of nuclei from Auger data to build the equivalent
distributions, PDFs of $X_{\rm max}^{Auger\rightarrow TA}$, which are compared
with TA data. However, our study reveals that, at low energies, further effort
in terms of data analysis is required in order to harmonize the results of the
two experiments, taking into account that the models are not able to
accurately describe the data in some energy intervals.

We believe that the current approach could be very useful in future studies on
mass composition, especially to crosscheck the measurements of $X_{\rm max}$
between Auger and TA once the statistics of the recorded events will increase.

### Acknowledgments

I would like to thank Octavian Sima and Ionel Lazanu for many useful
discussions and suggestions. This work was supported by a grant of the
Romanian Ministry of Education and Research, CNCS - UEFISCDI, project number
PN-III-P1-1.1-PD-2019-0178, within PNCDI III.

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