∎

deceased Deceased addr29 Also at Coimbra Polytechnic - ISEC, 3030-199 Coimbra,
Portugal addr30 Also at Physikalisches Institut, Universität Heidelberg,
Heidelberg, Germany addr28 Also at INFN, Sez. di Ferrara and Dip. di Fisica e
Scienze della Terra, Università di Ferrara, via G. Saragat 1, Edificio C,
I-44122 Ferrara (FE), Italy

email1 e-mail<EMAIL_ADDRESS>(corresponding author) email2
e-mail<EMAIL_ADDRESS>(corresponding author) email3 e-mail:
<EMAIL_ADDRESS>(corresponding author) email4 e-mail:
<EMAIL_ADDRESS>(corresponding author) [$\ast$]emailxe e-mail:
<EMAIL_ADDRESS>
11institutetext: Physics Department, Columbia University, New York, NY 10027,
USA 22institutetext: Kamioka Observatory, Institute for Cosmic Ray Research,
and Kavli Institute for the Physics and Mathematics of the Universe (WPI),
University of Tokyo, Higashi-Mozumi, Kamioka, Hida, Gifu 506-1205, Japan
33institutetext: Department of Physics and Astronomy, University of Bologna
and INFN-Bologna, 40126 Bologna, Italy 44institutetext: LPNHE, Sorbonne
Université, CNRS/IN2P3, 75005 Paris, France 55institutetext: Institut für
Physik & Exzellenzcluster PRISMA+, Johannes Gutenberg-Universität Mainz, 55099
Mainz, Germany 66institutetext: Institut für Kernphysik, Westfälische
Wilhelms-Universität Münster, 48149 Münster, Germany 77institutetext: INAF-
Astrophysical Observatory of Torino, Department of Physics, University of
Torino and INFN-Torino, 10125 Torino, Italy 88institutetext: Nikhef and the
University of Amsterdam, Science Park, 1098XG Amsterdam, Netherlands
99institutetext: Oskar Klein Centre, Department of Physics, Stockholm
University, AlbaNova, Stockholm SE-10691, Sweden 1010institutetext: Department
of Physics & Kavli Institute for Cosmological Physics, University of Chicago,
Chicago, IL 60637, USA 1111institutetext: New York University Abu Dhabi -
Center for Astro, Particle and Planetary Physics, Abu Dhabi, United Arab
Emirates 1212institutetext: Physik-Institut, University of Zürich, 8057
Zürich, Switzerland 1313institutetext: Department of Physics and Astronomy,
Purdue University, West Lafayette, IN 47907, USA 1414institutetext: INFN-
Laboratori Nazionali del Gran Sasso and Gran Sasso Science Institute, 67100
L’Aquila, Italy 1515institutetext: Physikalisches Institut, Universität
Freiburg, 79104 Freiburg, Germany 1616institutetext: SUBATECH, IMT Atlantique,
CNRS/IN2P3, Université de Nantes, Nantes 44307, France 1717institutetext:
Department of Particle Physics and Astrophysics, Weizmann Institute of
Science, Rehovot 7610001, Israel 1818institutetext: Department of Physics &
Center for High Energy Physics, Tsinghua University, Beijing 100084, China
1919institutetext: LIBPhys, Department of Physics, University of Coimbra,
3004-516 Coimbra, Portugal 2020institutetext: Max-Planck-Institut für
Kernphysik, 69117 Heidelberg, Germany 2121institutetext: Institute for
Astroparticle Physics, Karlsruhe Institute of Technology, 76021 Karlsruhe,
Germany 2222institutetext: Department of Physics and Chemistry, University of
L’Aquila, 67100 L’Aquila, Italy 2323institutetext: Department of Physics and
Astronomy, Rice University, Houston, TX 77005, USA 2424institutetext:
Department of Physics “Ettore Pancini”, University of Napoli and INFN-Napoli,
80126 Napoli, Italy 2525institutetext: Kobayashi-Maskawa Institute for the
Origin of Particles and the Universe, and Institute for Space-Earth
Environmental Research, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi
464-8602, Japan 2626institutetext: Department of Physics, University of
California San Diego, La Jolla, CA 92093, USA 2727institutetext: Department of
Physics, Kobe University, Kobe, Hyogo 657-8501, Japan 2828institutetext:
Forschungsreaktor TRIGA Mainz, Johannes Gutenberg-Universität Mainz, 55099
Mainz, Germany

# Low-energy Calibration of XENON1T with an Internal ${}^{\textbf{37}}$Ar
Source

E. Aprileaddr3  K. Abeaddr23  F. Agostiniaddr0  S. Ahmed Maouloudaddr18  M.
Alfonsiaddr5  L. Althueseraddr7  B. Andrieuaddr18  E. Angelinoaddr14, email1
J. R. Angevaareaddr8  V. C. Antochiaddr12  D. Antón Martinaddr1  F.
Arneodoaddr9  L. Baudisaddr17  A. L. Baxteraddr10  L. Bellagambaaddr0  R.
Biondiaddr4  A. Bismarkaddr17  A. Brownaddr19  S. Bruenneraddr8  G.
Brunoaddr13  R. Budnikaddr16  T. K. Buiaddr23  C. Caiaddr27  C. Capelliaddr17
J. M. R. Cardosoaddr2  D. Cichonaddr6  A. P. Colijnaddr8  J. Conradaddr12  J.
J. Cuenca-Garcíaaddr17,addr26  J. P. Cussonneauaddr13,deceased  V.
D’Andreaaddr22,addr4,addr19  M. P. Decowskiaddr8  P. Di Gangiaddr0  S. Di
Pedeaddr8  S. Diglioaddr13  K. Eiteladdr26  A. Elykovaddr19  S. Farrelladdr11
A. D. Ferellaaddr22,addr4  C. Ferrariaddr4  H. Fischeraddr19  W.
Fulgioneaddr14,addr4  P. Gaemersaddr8  R. Gaioraddr18  A. Gallo Rossoaddr12
M. Gallowayaddr17  F. Gaoaddr27  R. Glade-Beuckeaddr19  L. Grandiaddr1  J.
Grigataddr19  M. Guidaaddr6  R. Hammannaddr6  A. Higueraaddr11  C. Hilsaddr5,
email2  L. Hoetzschaddr6  J. Howlettaddr3  M. Iacovacciaddr20  Y. Itowaddr21
J. Jakobaddr7  F. Joergaddr6  A. Joyaddr12  N. Katoaddr23  M. Karaaddr26  P.
Kavriginaddr16  S. Kazamaaddr21  M. Kobayashiaddr21  G. Koltmanaddr16  A.
Kopecaddr15  F. Kugeraddr19  H. Landsmanaddr16  R. F. Langaddr10  L.
Levinsonaddr16  I. Liaddr11  S. Liaddr10  S. Liangaddr11  S. Lindemannaddr19
M. Lindneraddr6  K. Liuaddr27  J. Loizeauaddr13  F. Lombardiaddr5  J.
Longaddr1  J. A. M. Lopesaddr2,addr29  Y. Maaddr15  C. Macolinoaddr22,addr4
J. Mahlstedtaddr12  A. Mancusoaddr0  L. Manentiaddr9  F. Marignettiaddr20  T.
Marrodán Undagoitiaaddr6  K. Martensaddr23  J. Masbouaddr13  D. Massonaddr19
E. Massonaddr18  S. Mastroianniaddr20  M. Messinaaddr4  K. Miuchiaddr24  K.
Mizukoshiaddr24  A. Molinarioaddr14, email3  S. Moriyamaaddr23  K. Moråaddr3
Y. Mosbacheraddr16  M. Murraaddr3  J. Mülleraddr19  K. Niaddr15  U.
Oberlackaddr5  B. Paetschaddr16  J. Palacioaddr6  R. Peresaddr17  C.
Petersaddr11  J. Pienaaraddr1  M. Pierreaddr13  V. Pizzellaaddr6  G.
Planteaddr3  J. Qiaddr15  J. Qinaddr10  D. Ramírez Garcíaaddr17  S.
Reichardaddr26  A. Rocchettiaddr19  N. Ruppaddr6  L. Sanchezaddr11  P.
Sanchez-Lucasaddr17  J. M. F. dos Santosaddr2  I. Sarnoffaddr9  G.
Sartorelliaddr0  J. Schreineraddr6  D. Schulteaddr7  P. Schulteaddr7  H.
Schulze Eißingaddr7  M. Schumannaddr19  L. Scotto Lavinaaddr18  M. Selviaddr0
F. Semeriaaddr0  P. Shaginaddr5  S. Shiaddr3  E. Shockleyaddr15, email4  M.
Silvaaddr2  H. Simgenaddr6  A. Takedaaddr23  P.-L. Tanaddr12  A.
Terliukaddr6,addr30  D. Thersaddr13  F. Toschiaddr19  G. Trincheroaddr14  C.
Tunnelladdr11  F. Tönniesaddr19  K. Valeriusaddr26  G. Voltaaddr17  C.
Weinheimeraddr7  M. Weissaddr16  D. Wenzaddr5  C. Wittwegaddr17  T. Wolfaddr6
D. Xuaddr3  Z. Xuaddr3  M. Yamashitaaddr23  L. Yangaddr15  J. Yeaddr3  L.
Yuanaddr1  G. Zavattiniaddr0,addr28  S. Zerboaddr3  M. Zhongaddr15  T.
Zhuaddr3 (XENON Collaborationemailxe) and C. GeppertaddrXX  J. RiemeraddrXX

(Received: date / Accepted: date)

###### Abstract

A low-energy electronic recoil calibration of XENON1T, a dual-phase xenon time
projection chamber, with an internal [37]Ar source was performed. This
calibration source features a 35-day half-life and provides two mono-energetic
lines at 2.82 keV and 0.27 keV. The photon yield and electron yield at 2.82
keV are measured to be ($32.3\,\pm\,0.3$) photons/keV and ($40.6\,\pm\,0.5$)
electrons/keV, respectively, in agreement with other measurements and with
NEST predictions. The electron yield at 0.27 keV is also measured and it is
($68.0^{+6.3}_{-3.7}$) electrons/keV. The [37]Ar calibration confirms that the
detector is well-understood in the energy region close to the detection
threshold, with the 2.82 keV line reconstructed at ($2.83\,\pm\,0.02$) keV,
which further validates the model used to interpret the low-energy electronic
recoil excess previously reported by XENON1T. The ability to efficiently
remove argon with cryogenic distillation after the calibration proves that
[37]Ar can be considered as a regular calibration source for multi-tonne xenon
detectors.

###### Keywords:

Dark Matter XENON1T Calibration [37]Ar Low-energy electronic recoil

††journal: Eur. Phys. J. C

##  1 Introduction

The XENON1T experiment, located underground at the INFN Laboratori Nazionali
del Gran Sasso (LNGS) in Italy, was designed for direct detection of dark
matter (DM) in the form of weakly interacting massive particles (WIMPs)
RefXE1TDMExp ; RefXE1TDMResSR1 . The detector was a liquid xenon dual-phase
time projection chamber (LXe TPC), hosting a liquid volume with a small layer
of gaseous xenon on top. It operated with a total of 3.2 t of ultra-pure
xenon. It was surrounded by a stainless steel tank filled with water which
operated as an active Cherenkov muon veto against cosmic radiation RefMuonVeto
. The cylindrical LXe TPC, with a height of 97 cm and a diameter of 96 cm,
contained 2 t of LXe as a target. It was instrumented with 248 photomultiplier
tubes (PMTs) arranged in two arrays at the top and bottom to efficiently
detect the signals RefPMTs . An electric drift field was applied between a
cathode at the bottom of the LXe TPC and a grounded electrode at the top
(named ”gate”). With a third electrode, the anode located in the gaseous
phase, an extraction field was applied.

In LXe, the total energy deposited by interacting particles is divided into
excitation, ionization and heat RefHitachi ; RefAprile2005 ; RefManzur ;
RefSorensen ; RefDahlThesis . The detectable energy is split between
excitation and ionization, causing these channels to be anti-correlated,
whereby the ratio of energy splitting is defined by the applied drift field.
Some of the freed electrons from ionization recombine with Xe ions. Excited Xe
atoms and Xe ions formed from recombination create Xe dimers which emit UV
scintillation light (S1). The remaining free electrons are drifted to the gate
by the applied field and extracted into the gas phase, generating a secondary
scintillation signal (S2) that is proportional to the number of extracted
electrons.

The light distribution pattern of the S2s on the top PMT array provides
$x$-$y$ coordinates ($x,y=0$ at the center of the TPC), while the time
difference between the S1 and S2 signals is used to measure the depth of the
interaction $z$ ($z=0$ at the gate). This allows for a full 3D position
reconstruction of the events. Furthermore, the ratio between the S2 and S1
signals makes it possible to discriminate between electronic recoils (ERs) and
nuclear recoils (NRs). ERs are induced by $\upbeta$-particles, $\upgamma$-rays
or neutrinos scattering off electrons, while NRs are induced by neutrons or
neutrinos scattering off nuclei. WIMPs are expected to induce NRs, while other
DM candidates can generate ERs. The difference in the S2/S1 ratio between ERs
and NRs is caused by the different ionization densities at the interaction
sites deposited by the two interaction types, thus resulting in different
excitation-to-ionization and recombination ratios in the LXe. For ERs, the
empirically-determined average energy required to produce a quantum
(scintillation photon or ionization electron) is independent from the energy
scale and includes the fraction of energy dissipated into heat, which is not
detectable in LXe. For NRs, a quenching factor is considered to account for
the larger fraction of energy lost into heat compared to the electronic
channel RefLenardo2015 .

In XENON1T, the unprecedentedly low background level of $(76\pm
2)\,\text{events}\,/(\text{t}\,\cdot\,\,\text{y}\,\cdot\,\text{keV})$ for ERs
between 1 keV and 30 keV opened up the possibility of investigating
alternative DM candidates and other physics beyond the Standard Model. A
significant excess with respect to background expectations was found in the
low-energy ER spectrum between 1 keV and 7 keV RefXE1TExcessER . The shape of
the excess statistically allowed several possible explanations: solar axions,
bosonic dark matter or neutrinos with enhanced magnetic moment. A trace amount
of tritium is also able to explain the excess, the data however is also
compatible with a null hypothesis. This is also valid for a possible
contamination with [37]Ar, but its concentration could be tightly constrained
and it was found to be negligible.

In XENONnT, the latest phase of the XENON project, no excess above background
has been found, excluding interpretations of the XENON1T excess as due to
physics beyond the Standard Model RefXEnTER . Despite this recent result, a
further investigation of the XENON1T excess is still valuable, and it requires
a robust description of the behaviour of the detector and validation of the
energy reconstruction in the region of interest.

It was important to calibrate the low-energy ER region with high statistics in
the whole TPC volume. Due to the size of the LXe TPC and the self-shielding
properties of LXe, external calibration sources were unable to probe the
internal regions of the detector. In XENON1T two internal calibration sources
were primarily used. [83m]Kr, a source with a two-step decay producing lines
at 32 keV and 9 keV, was used to study the spatial response of the detector
RefXE1TAnalysis1 . The $\upbeta$-decay of 212Pb (via injection of 220Rn) was
used to characterize the low-energy ER region, but produces a continuous
spectrum, which complicates studies of spatial dependence RefLangRn .
Therefore, a low-energy, uniformly-distributed, and mono-energetic calibration
source was desired to validate the detector response in the low energy region
where the XENON1T excess was observed.

[37]Ar decays by electron capture into 37Cl (Q-value of 813.9 keV) with half-
life of 35.01 days, emitting a mono-energetic neutrino RefTable37Ar . The
vacancy, which follows the electron capture from one of the shells (K-, L- or
M-shell) is filled by an electron rearrangement accompanied by a cascade
emission of x-rays and Auger-Meitner electrons. The overall energy deposits,
corresponding to the K-, L- and M-shell electron binding energies, result in
lines at 2.82 keV, 0.27 keV and 0.01 keV, with a branching ratio of 90.2%,
8.7% and 1.1%, respectively RefAr37energy .

The [37]Ar K- and L-shell lines allow to study the low-energy detector
response RefPixey ; RefXurich37Ar  , since they are well below the low-energy
peak of [83m]Kr at 9 keV and close to the 1 keV energy threshold of XENON1T.
In general, K-shell decays produce detectable S1s and S2s, while at L-shell
energy only S2s are expected. For this reason, two different analyses are
required and performed in the following.

This paper describes the calibration with [37]Ar performed for the first time
in XENON1T at the end of the last science run and it is organized as follows:
the [37]Ar source production, injection in the detector and removal after the
end of the calibration period are described in section 2. In section 3 the
standard analysis of [37]Ar K-shell signals, making use of both S1 and S2, is
presented. The impact of these results on the XENON1T low-energy ER excess is
discussed in section 4. Section 5 focuses on the S2-only analysis which
utilises only the ionization signal, lowering the energy threshold down to the
energy of the L-shell decays at 0.27 keV. Conclusions are reported in section
6.

##  2 Source production, injection and removal

###  2.1 Source production

The isotope [37]Ar is obtained by irradiation of argon enriched in 36Ar to
99.99 %, contained in a quartz glass container (ampule), with thermal
neutrons:

${}^{36}\text{Ar}(n,\gamma)^{37}\text{Ar}.$ (1)

The enrichment enhances the production of [37]Ar and reduces the production of
undesired isotopes. The sample was irradiated at the TRIGA Mark II reactor
located at the University of Mainz, which provides a flux of thermal neutrons
of $4.2\times 10^{12}\,\text{n}\,/\,(\text{cm}^{2}\,\cdot\,\text{s})$
111https://www.kernchemie.uni-mainz.de/reaktor/technische-daten/. The total
activity produced during the irradiation was (100 $\pm$ 10) kBq. This value
was calculated based on the neutron flux, the neutron capture cross-section of
36Ar and the irradiation time. The activity produced cannot be directly
measured as the low-energy electrons and x-rays do not penetrate the walls of
the quartz-ampule.

###  2.2 Dosing and insertion into XENON1T

A rate of $O$(10) Hz is needed in the detector to achieve a reasonable amount
of data, much lower than the overall activity produced during irradiation. A
dosing system is used to open the ampule with a guillotine mechanism. It is
also used to reduce the activity to a suitable calibration dosage and inject
the irradiated argon into the LXe TPC via the XENON1T purification system.
Besides the device to open the ampule and store the [37]Ar, the dosing system
consists of an arrangement of sections of well-defined volumes that are used
to successively dilute a small portion of the activity with xenon. The amount
of [37]Ar injected can be computed from the known volume sizes and the
starting activity. With an initial activity of about 100 kBq, the dosing
systems design enables doses of $O$(Bq) to be injected into the detector.

The evolution of the detector trigger rate is shown in Fig. 1, without any
data selection applied. The flat rate in the beginning is due to the [83m]Kr
source which was injected in parallel to allow simultaneous calibration with
both sources. This simultaneous calibration is possible due to the difference
in decay energy of both sources, which allows a clean separation of the
respective events. The [37]Ar calibration started with a computed dose of 2.2
Bq injected on October 22nd, 2018. The injection was immediately visible as an
increase in the event rate of the detector and is marked as day zero in Fig.
1. Two additional injections aimed at 8.0 Bq and 3.9 Bq, respectively, were
performed in the following days to ensure sufficient statistics for the later
analysis. The uncertainties on these values are estimated to be around 10%,
dominated by the initial uncertainty in the activity produced during ampule
irradiation. Inconsistencies between the expected doses and the rate increase
shown in Fig. 1 can be caused by unknown systematics from the dosing system
and the fact that different dosing procedures were tested. Injecting small
activities at first avoids an accidental overdose and allows verification that
the produced activity is in the expected order of magnitude. However a precise
knowledge of the absolute activity is not necessary to perform the detector
calibration.

Figure 1: Injection phase of the [37]Ar calibration with the activity increase
visible after each of the three injections with expected rates of 2.2, 8.0 and
3.9 Bq (Timestamps A, B and C, respectively), with estimated uncertainties of
10%. The [37]Ar injections were added on top of a continuous [83m]Kr injection
with a constant rate of 6 Bq. This allowed recording of [37]Ar and [83m]Kr
simultaneously for direct comparison of both sources at identical detector
conditions (yellow region). Each data point shows the time-averaged rate
during a one-hour long data set of all events detected in the complete TPC.
The rate drop at the end (red region) is caused by the decay of the [83m]Kr,
after the continuous supply of this source was stopped (D).

After data processing, an energy cut in the S2 vs S1 parameter space was
applied to select the 2.82 keV [37]Ar peak, together with a fiducial volume
cut to reduce background events. More details on the selection can be found in
Sec. 3.1. A detailed model RefXE1TOnlDist  was fitted to the data and yielded
injected activities of (2.6 $\pm$ 0.1) Bq, (4.3 $\pm$ 0.1) Bq and (2.1 $\pm$
0.1) Bq. Note that this model estimates the activity injected into the
purification system, while Fig. 1 only shows the rate increase caused by the
injection of argon inside the sensitive, liquid filled volume of XENON1T.

###  2.3 37Ar removal by cryogenic distillation

Figure 2: Time evolution of the [37]Ar K-shell event rate in one-day averages
during the distillation campaign which started 16 days after the first
injection. The red line presents a binned likelihood fit of an exponential
plus flat background. The background level is measured over the course of the
entire previous science run before the first injection, with correspondingly
small error, and is shown here at day -16 and with the dashed blue line. The
dashed gray line shows a projection of the [37]Ar rate without taking
background into account. The downward arrows represent days with no [37]Ar
events detected. The regions in the gray bands are excluded to avoid
contamination with background events caused by other calibrations at that
time. For comparison, the expected [37]Ar rate development based on natural
decay alone is also shown with a dashed-dotted gray line. The uncertainties
for the low rates are computed using Feldman-Cousins intervals RefFeldCousInt
. The lower panel shows the standardized residuals, computed by dividing the
difference between data and model by the $\upsigma$ of the data. The green and
yellow bands show the 1 and 2 $\upsigma$ deviation, respectively.

The [37]Ar half-life time of 35.01 days makes active removal of the isotope
after a calibration run necessary, since a reduction of at least four orders
of magnitude is required to return to normal background level. Due to the
higher volatility of argon compared to xenon, it accumulates in the gaseous
phase and makes removal via cryogenic distillation effective. Removal was
performed using the distillation column initially designed to remove 85Kr, one
of the major background sources of XENON1T RefXE1TOnlDist ; RefXE1TKrCryoDest
. Distillation was conducted online during normal detector operation which
allowed for continuous removal of [37]Ar from the gaseous phase.

The time evolution of the [37]Ar K-shell event rate during distillation is
shown in Fig. 2. With the same selection criteria and cuts used in the
injection phase, a simple exponential function, accounting for both [37]Ar
decay and removal by distillation, was used to fit the process. The removal
time constant found for the distillation is $\tau_{Ar,dist}=(1.97\,\pm\,0.10)$
days. This means a reduction of the [37]Ar activity by an order of magnitude
every (4.54$\,\pm\,$0.23) days. For this case, background level conditions
were reached again after 24 days of distillation when the [37]Ar activity has
been reduced by five orders of magnitude and can no longer be distinguished
from the background level measured before the first injection. For comparison,
a waiting time of approximately 570 days would have been necessary to reach
the same low [37]Ar activity level via natural decay. The last data point in
Fig. 2 exhibits a higher uncertainty because it refers to a day in which the
livetime was lower. This last measurement is at 1.1$\upsigma$ from the
expected background level. However, there are no data available after this
point to track the trend of the rate on longer time scales.

The difference for the time constant to the value of $\tau_{Ar}=1.7$ days
obtained in Ref. RefXE1TOnlDist  is caused by the extended dataset used, which
is taking two more weeks of distillation into account. Additional testing of
the detector was performed before the start of distillation. Due to this the
detector was transitioning into a new equilibrium. The resulting
instabilities, which cannot be modeled with this simple exponential function,
caused the large residuals shown in Ref. RefXE1TOnlDist .

Since it was possible to remove the source in a reasonable timescale, [37]Ar
should be considered as a regular calibration source for multi-tonne xenon
detectors equipped with on-site distillation capabilities.

##  3 S1-S2 analysis

In this analysis, [37]Ar K-shell events are considered where both signals, S1
and S2, are observed. The goal is to determine the absolute photon yield (PY,
ph/keV) and electron yield (EY, e-/keV) of the 2.82 keV energy deposition as
well as the linearity of the detector response down to $O$(keV) energies.

As mentioned in Sec. 1 the energy deposited by an ER is shared between the
scintillation and ionization channels as

$E=W(n_{\gamma}+n_{e}),$ (2)

where $W$ is the average energy required to produce one quantum (photon or
electron), while $n_{\gamma}$ and $n_{e}$ are the number of produced photons
and electrons, respectively. For consistency with other XENON data analyses,
$W=13.7\pm 0.2~{}\rm eV/quanta$ RefDahlThesis  is used here, although other
recent works suggest smaller values of $W=11.5^{+0.2}_{-0.3}~{}\rm eV/quanta$
RefExcEnBaudis ; RefExoExcEnAnton . The choice of a different $W$ value
impacts the determination of $n_{\gamma}$ and $n_{e}$ and the values of
detector dependent constants $g_{1}$ and $g_{2}$, which parametrize the
average size of the S1 and S2 signals per photon or electron, respectively,
but it does not affect the energy reconstruction itself. The S1 (S2) signals
in the XENON1T LXe TPC are proportional to the number of photons (electrons)
produced at the interaction site. Therefore, Eq. (2) can be converted to
observable quantities:

$\hat{E}=W\left(\frac{\mathrm{cS1}}{g_{1}}+\frac{\mathrm{cS2}}{g_{2}}\right),$
(3)

where $\hat{E}$ denotes reconstructed energy. The S1 and S2 signals are
corrected for position-dependent effects including light collection efficiency
(for both S1 and S2). Additionally S2 is corrected for the loss of electrons
due to attachment to electronegative impurities, which is a function of the
drift time and follows an exponential law, with the electron lifetime as a
decay parameter RefXE1TAnalysis1 . These corrections lead to the cS1 and cS2
variables seen in Eq. (3).

The $g_{1}$ and $g_{2}$ values, assuming a linear detector response, relate
the detected light (S1) and charge (S2) yields to the intrinsic PY and EY for
a given drift field:

$\displaystyle\mathrm{PY}(E)$ $\displaystyle=L_{y}(E)/g_{1},$ (4)
$\displaystyle\mathrm{EY}(E)$ $\displaystyle=Q_{y}(E)/g_{2},$ (5)

where $L_{y}$ ($Q_{y}$) is the light (charge) yield, i.e. the average cS1
(cS2) per unit energy $E$. The goal of this analysis is to first determine
$L_{y}$ and $Q_{y}$ of the [37]Ar K-shell, and then extract $g_{1}$ and
$g_{2}$ to calculate the PY and EY of LXe at the XENON1T drift field of $82$
V/cm RefXE1TAnalysis . The value of $g_{2}$ depends on how S2 is chosen. In
the S1-S2 analysis in this paper the corrected S2 signal from the bottom PMT-
array only is used.

###  3.1 Data selection

The data used in this analysis is from a period including roughly 3.2 days of
the injection phase as shown in Fig. 1.

The calibration sources can be distinguished in the S1-S2 parameter space. For
[37]Ar, all events with $0\,<\text{S1}<40~{}\text{photoelectrons (PE)}$ and
$500\,<\text{S2}<2500~{}\text{PE}$ are selected. A fiducial volume cut of
$-93\,<\,z\,<\,-9\,$ cm and radius $r<42$ cm is also applied for the main
results of this analysis, as described in Ref. RefXE1TExcessER .

The standard pre-selections of data are applied in order to exclude events
happening when the data acquisition was in a busy condition, which may result
in data loss, or when a single PMT detects a large fraction of the total
signal. Further data quality cuts are applied. These include selections to
ensure correct pairing of S1-S2 signals and to remove misclassified events
caused by single electron emission, events dominated by one PMT, or events
with an unreliable $x$-$y$ position reconstruction. A more complete list of
all applied selections with detailed explanations can be found in Ref.
RefXE1TAnalysis1 .

###  3.2 Data correction

Several corrections are applied to the data to account for non-uniform
detector response throughout the LXe TPC RefXE1TAnalysis1 . The S1 signal is
corrected to account for position-dependent light collection efficiency,
particularly as a function of depth ($z$) but also of $x$ and $y$. The S2
signal is corrected for both (a) charge loss as a function of $z$ due to
electronegative impurities and (b) light collection efficiency variation as a
function of $x$ and $y$, due to geometric effects on the S2 gain.

The corrections described above are determined from [83m]Kr calibration and
applied in all XENON1T analyses. Using the dataset collected with both [37]Ar
and [83m]Kr injected simultaneously, the same corrections are extracted using
[37]Ar and compared to the standard results from [83m]Kr, based on earlier
calibrations. The electron lifetime extracted from [37]Ar is approximately 10%
higher than the one from [83m]Kr. This discrepancy, which was also observed in
Ref. RefXE1TAnalysis1 , is attributed to a slight non-uniformity of the drift
field in the LXe TPC. Such an effect leads to position-dependent variations in
the light and charge yields.

Indeed, the non-uniform drift field produces variations in the number of
produced electrons and, consequently, the detected S2. This effect is not
separable from the variation of S2 due to electrons absorption by impurities.
Due to the different impact of drift field at different energies on charge
yield variations, the estimation of electron lifetime for various sources may
differ. Hence, in order to evaluate the true electron lifetime, energy-
dependent correction should be applied for specific calibration sources.

For the S1-S2 analysis described in this section, the electron lifetime based
on [83m]Kr is used in order to be consistent with other calibration points in
this science run (see Fig. 5) as well as other XENON1T analyses. While a
different choice of the electron lifetime impacts on the cS2 and $g_{2}$
values, the energy reconstructed with Eq. (3) is not affected.

Figure 3: Electron lifetime estimation from [37]Ar data. Without applying an
additional correction, the expected exponential decrease with drift time does
not well describe the S2 sizes near the top of the detector (low drift times).
The gray line shows the S2b (size of the S2 signal recorded in the bottom PMT
array) while the blue one shows the S2b after applying the afterpulses
correction. The red line is an exponential fit to the afterpulse corected data
to extract the electron lifetime. The region chosen for the exponential fit is
denoted in gray.

During the electron lifetime analysis, another effect was observed that
motivated an additional correction for [37]Ar data: in the top $\sim$20 cm of
the LXe TPC, the size of the observed S2 signals were on average a few percent
smaller than expected, as shown in Fig. 3. The source of this bias was
determined to be PMT afterpulsing. Afterpulses are due to residual gas
molecules in the PMT vacuum that get ionized by a photoelectron and then drift
to the photocathode generating a delayed signal. Afterpulses occur within a
few microseconds after the primary S2 peak and thus are typically merged
together with it RefPmtCharact . As the afterpulse size is directly
proportional to the primary S2 size, this results in a small but constant bias
that is absorbed into the value of $g_{2}$, which also applies to all
comparable signals. Due to the low energy of the [37]Ar interaction the energy
is deposited in a small volume, resulting in short S2. Additionally, at the
top of the LXe TPC, the broadening by diffusion is reduced, so many [37]Ar S2
signals occuring there were reconstructed separately from their afterpulses,
resulting in a relatively smaller S2 value. An additional correction is made
on these events that have secondary S2s within 5 $\mu$s of the primary and
with total area larger than 60 PE ($\sim$2 extracted electrons). For events
with this topology, the areas of the primary and secondary S2 signals are
summed up to a total S2 including the afterpulse area. This afterpulse
correction helps to ensure a linear energy scale across all energies and
across the full fiducial volume. The effect of this correction is shown in
Fig. 3.

###  3.3 Data analysis and results

Figure 4: Direct comparison of K-shell interactions at two different positions
in the TPC. The data is shown in the dataspace of drift-time and afterpulse
corrected S2 v. uncorrected S1. In the top and right panels are reported the
projections of the signals, with the black points for the data and the red
line for the fit. The dotted blue line indicates the S1 detection efficiency.
The upper plot shows events at the top and at the edge of the TPC, where the
S1 detection efficiency has the highest impact. The lower plot is taken from
central position at the bottom of the TPC, where the S1 is less affected by
the threshold.

After data selection and correction, the Ly and Qy of [37]Ar are determined
via fitting in the S1-S2 space. The event distribution for a mono-energetic
source is generally expected to follow a two-dimensional Gaussian
distribution. However, at low energies there are non-Gaussian effects to be
considered, particularly for S1 where the central limit theorem does not
apply, namely:

  1. 1.

with $O$(10) detected photons, the Gaussian approximation to the Poisson
distribution does not fully hold;

  2. 2.

the PMT single-PE response is not a Gaussian RefModelIndependentMethod ;

  3. 3.

for xenon scintillation light wavelength, PMTs
exhibit double-PE emission, a binomial process
with $p\sim 0.2$ RefDPE ; RefXE1TAnalysis .

Empirically, a skew Gaussian distribution was used for low-energy S1 signals,
as in Ref. RefEnRecTech ; RefZeplinSkew ; RefLuxSkew .

In addition to the non-Gaussian effects described above, the sharply-dropping
S1 detection efficiency for S1 $\lesssim$ 5 PE RefXE1TAnalysis1  is taken into
account, which has a position-dependent impact on the cS1 spectrum due to
spatial variations in the light collection efficiency. To include this effect,
cylindrical symmetry is assumed and the LXe TPC is divided into $r$-$z$ bins
of equal volumes. In each bin, a two-dimensional fit is performed in the space
of uncorrected S1 and corrected S2 (cS2b,AP, b: S2 signal detected on the
bottom PMT array; AP: afterpulse correction has been included). The full
model, including the detection efficiency and the signal shape of the S1
distribution is given by:

$f_{1}(S1)=\epsilon(S1)Ne^{\frac{-(S1-\xi)^{2}}{w^{2}}}\left(1+\mathrm{erf}(\alpha\frac{S1-\xi}{w\sqrt{2}})\right),$
(6)

where $\xi$ and $w$ are mean- and width-like parameters and $\alpha$ is
related to the skewness of the distribution, with $\alpha=0$ corresponding to
the standard Gaussian. The efficiency term $\epsilon(S1)$ is fixed (see Fig.
4) from dedicated analyses RefXE1TAnalysis1 , and $N$ is a normalization
constant.

The cS2 spectrum is modelled using a Gaussian, and the anti-correlation
between the S2 (Gaussian) mean and the S1 (skew Gaussian) mean is accounted
for in the full 2D fit. Two example fits are shown in Fig. 4, including the S1
and cS2b,AP projections. It can be noticed that the S1 at the top of the TPC
are much more affected by the detection efficiency, as they are generally
smaller. This is because they are mainly detected with the bottom PMT array
due to reflections at the liquid surface. A higher value of cS2b,AP at the
bottom of the TPC is also apparent. This is because the electron lifetime from
[83m]Kr, which is smaller than the one derived from [37]Ar, is used to correct
the signals. This leads to an over-correction of the [37]Ar signals at the
bottom of the TPC.

The mean of the underlying skew Gaussian in Eq. (6) is given by

$\mu=\xi+\sqrt{2/\pi}\frac{\alpha w}{\sqrt{1+\alpha^{2}}},$ (7)

and is taken as the mean S1 value in each $r$-$z$ bin. The S1 means are scaled
according to the light collection map derived using [83m]Kr calibration data
RefXE1TAnalysis1  and averaged to get an overall Ly to be compared with other
calibration points. Qy is determined by taking the sample mean of the fitted
cS2b,AP means. In the end, the light (charge) yields are found to be

$\displaystyle L_{y}(2.82~{}\mathrm{keV})$ $\displaystyle=(4.66\pm
0.01_{\mathrm{stat.}})~{}\mathrm{PE/keV},$ (8) $\displaystyle
Q_{y}(2.82~{}\mathrm{keV})$ $\displaystyle=(452.4\pm
0.1_{\mathrm{stat.}})~{}\mathrm{PE/keV}.$ (9)

The values of $g_{1}$ and $g_{2}$ can be obtained determining Ly and Qy for
other calibration points beyond [37]Ar. Eq. (3) can be converted to

$Q_{y}=\left(-\frac{g_{2}}{g_{1}}\cdot L_{y}+\frac{g_{2}}{W}\right).$ (10)

The additional calibration points used to determine $g_{1}$ and $g_{2}$
include the injected source [83m]Kr, as well as high-energy gamma lines from
material radioactivity and the activated xenon lines, 129mXe and 131mXe, which
are predominately produced during neutron calibration of the detector. Since
no threshold effects are applicable for these sources, the Ly and Qy are
determined by simple 2D (standard) Gaussian fits.

The Ly and Qy values for the various calibration sources are shown in Fig. 5,
where it can be seen that a linear energy response is observed over a wide
energy range, including the low-energy [37]Ar point.

Due to the high statistics for each calibration point, the uncertainty is
dominated by systematics. In order to evaluate them, the value of $\chi^{2}$
is set equal to the number of degrees of freedom. Assuming that every
calibration points is affected by the same systematics, they are estimated to
be around 0.6%, as the values that satisfy this condition. Then a simple
Monte-Carlo simulation is used to propagate these uncertainties on $g_{1}$ and
$g_{2}$. $L_{y}$ and $Q_{y}$, sampled many times from Gaussian distributions
which reflect such systematics, are fitted to evaluate the distribution of
$g_{1}$ and $g_{2}$, whose standard deviation is taken as uncertainty.
Potential systematic effects include small time dependencies not fully
accounted for as well as the small drift field inhomogeneity hypothesized to
explain the discrepancies in electron lifetime between different sources.

The linear fit to extract $g_{1}$ and $g_{2}$, taking into account these
systematic uncertainties, yields

$\displaystyle g_{1}$ $\displaystyle=(0.144\pm 0.001)~{}\mathrm{PE/ph},$ (11)
$\displaystyle g_{2}$ $\displaystyle=(11.1\pm 0.1)~{}\mathrm{PE/e}^{-}.$ (12)

Figure 5: Charge- vs. light-yield plot with linear fit to extract the detector
dependent light- and charge-gain constants $g_{1}$ and $g_{2}$. The black
points represent the average S1s and S2s for various mono-energetic peaks,
corrected for position-dependent effects and normalized to the energy. The red
line shows the linear fit of Eq. 10 to the data. The red band shows the
uncertainty on the fitted $g_{1}$ and $g_{2}$, dominated by systematics.

With $g_{1}$ and $g_{2}$ determined, the detector independent photon and
electron yields at the [37]Ar K-shell energy are

$\displaystyle\mathrm{PY}(2.82~{}\mathrm{keV})$ $\displaystyle=(32.3\pm
0.3)~{}\mathrm{ph/keV},$ (13) $\displaystyle\mathrm{EY}(2.82~{}\mathrm{keV})$
$\displaystyle=(40.6\pm 0.5)~{}\mathrm{e}^{-}\mathrm{/keV}.$ (14)

These values are shown in Fig. 11 and Fig. 12 at the end of the paper,
together with expected values from NEST (Noble Element Simulation Technique)
RefNestv2.3.6  and the results from other measurements at different energies
and drift fields. The photon yield is in disagreement ($>5\upsigma$) with the
NEST prediction at the same drift field, while it is in agreement within
2$\upsigma$ with measurements taken at same RefXurich37Ar  or similar energies
RefLuxTritium . A $>3\upsigma$ tension is also apparent with the values
reported by PIXeY RefPixey , which is not explained by the weak dependence on
the drift field (Fig. 12, left panel). The measured electron yield is
consistent with the predictions and other measurements at similar energies
within 2$\upsigma$.

Lastly, with the $g_{1}$ and $g_{2}$ values found above, the energy of any
event can be estimated using cS1 and cS2 signals. The reconstructed energy
spectrum of [37]Ar K-shell captures is shown in Fig. 6, along with 2 separate
fits: a standard Gaussian and a skew Gaussian. As expected, the standard
Gaussian does not describe the data as well as the skew Gaussian for this low-
energy source. The mean of the skew Gaussian is ($2.83\pm 0.02$) keV,
consistent with the literature value $2.82$ keV and with the uncertainty
dominated by the systematics of $g_{1}$ and $g_{2}$. The full-width half-
maximum of the K-shell peak is $(1.04\pm 0.01)$ keV.

Figure 6: Reconstructed energy spectrum of [37]Ar K-shell events using both
cS1 and cS2 signals. S2 correction include also afterpulses. The black points
show the data, the gray line is a Gaussian fit, and the red line is a skew
Gaussian fit, as described in the text. The dotted vertical line shows the
mean expected energy of the K-shell transition of 2.82 keV.

##  4 Implications for the XENON1T ER excess

The analysis reported here is relevant with respect to a previous result of
XENON1T in which an excess of low-energy ER events was reported
RefXE1TExcessER . As mentioned in Sec. 1, XENONnT data has shown no ER excess
RefXEnTER . However, it is still useful to reconsider the XENON1T ER data in
the context of the improved knowledge obtained from the [37]Ar calibration.
The running conditions of the detector at the time of [37]Ar calibration were
mostly the same as for the scientific run in which the excess was observed. In
particular drift field, temperature, pressure, charge yield and light yield
were stable. This makes it possible to directly compare the data from the two
periods.

The model used to fit the data is updated and compared to the one applied to
the results reported in Ref. RefXE1TExcessER . The improvements included in
the model are (a) the effect of PMT afterpulsing in low-energy S2s and (b) the
non-Gaussian response of the detector at low energies. A third effect is the
impact of field inhomogeneity leading to different electron lifetime values
for different sources, as mentioned in Sec. 3.2. However, the robust energy
reconstruction described in Sec. 4 suggests that this effect does not have a
large impact on mean reconstructed energy. Thus, its impact is implicitly
included in the empirical resolution model.

###  4.1 PMT afterpulsing

The impact of the afterpulse correction (see Sec. 3.2) in the top 20 cm of the
detector was investigated to cross-check the energy reconstruction as reported
in Ref. RefXE1TExcessER . The PMT afterpulsing rate was monitored throughout
the operation of XENON1T. There were PMTs that displayed large increases in
afterpulses consistent with xenon ions, suggesting these PMTs had small leaks
that degraded their performance over time. The dataset analyzed in Ref.
RefXE1TExcessER , called SR1, was recorded from February 2017 to February
2018, while the [37]Ar calibration was performed in October 2018. Therefore,
any bias from PMT afterpulsing is expected to be smaller in SR1 than in the
calibration reported in this work.

The afterpulse correction impacted 26% of events in the SR1 dataset with
reconstructed energy below 15 keV, almost entirely in the top half of the LXe
TPC, as expected. The S2 size for these events was increased on average by
3.3%, with a maximum change of 6.4%. This corresponds to the size of the
deficit shown in the residuals of Fig. 3.

###  4.2 Non-Gaussian detector response

In the SR1 analysis RefXE1TExcessER  a Gaussian response was assumed over the
full energy range for simplicity. However, as discussed in Ref. RefEnRecTech
and shown in this work, a skew Gaussian is a more realistic model at low
energies. As this change impacts mono-energetic spectra most, an improved peak
search is performed in the SR1 data with a new empirical resolution model.

The new empirical model is built using the calibration points of [37]Ar,
[83m]Kr, 131mXe, and 129mXe. A skew Gaussian model (Eq. (6)) is used for all
peaks instead of the typical Gaussian, and the skewness parameter ($\alpha$)
and width parameter ($w$) are then fit as a function of the energy. The energy
dependence of the skewness is modelled as a power law, while the width is
modelled in the same way as the standard deviation of a Gaussian in similar
contexts RefXE1TExcessER . Those empirical fits are given by

$\displaystyle\alpha(E)/E$ $\displaystyle=2.41(1)\cdot E^{-1.30(1)},$ (15)

$\displaystyle w(E)/E$ $\displaystyle=\frac{0.374(1)}{\sqrt{E}}+0.005(1).$
(16)

with $E$ expressed in keV.

Figure 7: Updated search for mono-energetic peaks in the XENON1T SR1 data.
Left panel: SR1 energy spectrum with energy reconstructed both with the
afterpulse correction (black data points) and without afterpulse correction
(gray data points). The red solid line shows the best fit to the black data
points for a mono-energetic peak with the updated skew Gaussian model. The red
dotted line shows the best fit to the gray data points for a mono-energetic
peak with the previously-used Gaussian model, as done in RefXE1TExcessER . The
dashed blue line shows the background-only fit to the black data points. Right
panel: Log-likelihood ratio for different mono-energetic peaks for both the
updated analysis using the skew Gaussian and including the afterpulse
correction (solid black) and the previous analysis with a standard Gaussian
without afterpulse correction (dotted gray).

###  4.3 Updated fit to SR1 ER data

A simplified version of the unbinned likelihood ratio analysis in Ref.
RefXE1TExcessER  is performed, using the same exposure, background models, and
search region, but without the partitioning of the dataset in time.

The result of the updated search for a mono-energetic peak is shown in Fig. 7.
The best-fit mean is $(2.4\pm 0.2)$ keV, to be compared with $(2.3\pm 0.2)$
keV from the original analysis RefXE1TExcessER . This indicates a small impact
of the updated analysis on the energy reconstruction. The likelihood ratio
curve for the peak mean is shown in Fig. 7 (right). The [37]Ar peak is
disfavoured at $\sim 2\sigma$ with the updated analysis, as it was already
found with the previous analysis RefXE1TExcessER . In conclusion, these
changes have no impact on the interpretation of the ER data in XENON1T.

##  5 S2-only analysis

Any S1s from the [37]Ar L-shell decay (0.27 keV) are expected to fall well
below the XENON1T energy threshold, which is determined by the 3-fold PMTs
coincidence. However, the S2s are still detectable on account of the electron
amplification in the gas phase. For this reason, the study of L-shell decay
can only be performed dropping the requirement of having an S1, with an
S2-only analysis. The XENON collaboration already made use of this technique
to search for light dark matter signals in XENON10, XENON100 and XENON1T
RefXE10S2only ; RefXE100S2only ; RefXE1TS2only ; RefXE1TSingleElectron . Fig.
8 shows the impact of dropping the S1 requirement, which results in a L-shell
peak with S2 below 1000 PE. The asymmetric shape of the peaks is due to
electron absorption during drift which here is not corrected. It results that
the fraction of K-shell events lost due to the S1 detection efficiency is
about 8%. The events visible at the L-shell position with S1 and S2 are the
results of a random S1 (e.g. dark counts) which is accidentally paired with a
S2 from L-shell decay.

While this approach allows a substantial improvement of the energy threshold,
it implies a worse energy resolution and no measurement of the depth of the
interaction using the drift time information. This in turn results in poorer
fiducialization and hence weaker background rejection.

The S2-only analysis is strictly necessary to study the L-shell, but it is
also applied to the K-shell region as a cross check with the results from the
standard S1-S2 analysis described in Sec. 3.

Figure 8: The [37]Ar uncorrected S2 spectrum with the standard requirement of
both S1 and S2 (red, hatched) and dropping the S1 requirement (blue).

###  5.1 Data selection

The S2-only analysis is performed on a 6.9 days dataset acquired in the stable
period after closing the krypton source and before distilling (day 7 to 14
after first injection).

The total S2 signal recorded by the top and bottom PMT-arrays is considered in
this analysis, differently from the S1-S2 analysis. It is not possible to
correct the S2 size depending on the interaction depth $z$. The S2s are only
corrected depending on their $x$-$y$ positions to account for position-
dependent light collection efficiency. Moreover, to be consistent with the
S1-S2 analysis, the afterpulse correction is applied, see Sec. 3.2.
Nevertheless, the relationship between S2 width and depth for interactions at
a given energy can be estimated with simulations. In this way, even for events
without S1, a $z$ coordinate can be statistically assigned depending on the
signal width even in the L-shell energy region.

The pre-selections of data already listed in Sec. 3.1 are applied and events
in which an S2 is recorded are selected. Events with S2 size lower than 90 PE
($\sim$ 3 extracted electrons) are excluded. Events greater than 5000 PE,
where [37]Ar signals are not expected, are also not used in this analysis.

The quality cuts applied are similar to those used in the search for light
dark matter RefXE1TS2only ; RefXE1TSingleElectron , but they are adapted to
the specific characteristics of [37]Ar signals, as described in the following:

  * –

Radial. Events reconstructed at a radius of $r\,>\,26.5\,\,$cm are removed.
This cut is tighter than the corresponding cut in the S1-S2 analysis. Since
$z$ is not reconstructed, field distortion corrections cannot be applied and,
hence, an inner region of the TPC where the field is more uniform must be
selected.

  * –

S2 Width. Events with an S2 width outside [400, 3000] ns are removed. Since
the width of S2 waveform is correlated with $z$, due to electrons diffusion
during drift, excluded events correspond to events at the top and the bottom
of the TPC.

  * –

S2 Area Fraction Top. On average $\sim$64% of S2 light is expected to be seen
by the top PMT array, given the detector geometry and the position where
signals are produced. A selection was applied to remove events where this
fraction is $>$69%, which are due to interactions in the gaseous xenon phase
above the usual secondary scintillation region (‘gas events’).

  * –

Pile-up. Two dedicated cuts are implemented against pile-up of randomly
emitted single-electron, which can be misidentified as S2s from real events.
First, events whose S2 hit-pattern on the top array indicate two or more
spatially separated small S2s which are reconstructed as a single event, as
determined by a likelihood test, are removed. Second, events with two or more
S2s or single electrons signals up to $\sim$1 ms before the largest S2 are not
taken into account. This additionally removes cluttered waveforms, where
reconstruction becomes difficult.

  * –

S2 Single Scatter. A selection is applied to remove multiple scatter events,
in which the secondary S2 has a hit-pattern compatible with that of a scatter
following a primary interaction .

  * –

Cathode. For events where multiple S1s are present, it is checked if one of
them paired with a main S2 indicates a reconstructed $z$ at the cathode
position. In this case the events are excluded.

The acceptances of the quality cuts and the trigger efficiency in a given $z$
slice ($-50\,<\,z\,<\,-30\,$ cm) are shown in Fig. 9, together with the signal
model. S2 Width and Cathode cuts have a $\sim$ 100% acceptance for the
complete S2 range in this $z$ slice, and they are not included in this plot.
The Radial cut features the lowest acceptance, but it is almost flat with
respect to S2 size. Due to the field distortion, events at the bottom of the
TPC are reconstructed more inward compared to events at the top, and are less
affected by Radial cut. This produce the step around 1000 PE in the transition
region, where L-shell events at the top and K-shell events at the bottom of
the TPC overlap.

The trigger efficiency, namely the ability to detect S2s depending on their
size, has the largest impact in the lowest S2 region. The background
contribution after the quality cuts are applied is negligible in the whole S2
size range.

Figure 9: Acceptances (fraction of [37]Ar events accepted) of the quality cuts
and trigger efficiency depending on S2 size for the slice
$-50\;\text{cm}\,<\,z\,<\,-30\,$ cm. The bands around the lines represent the
$\pm\,1\,\sigma\,$ uncertainties. The [37]Ar signal model (without acceptances
applied) is shown with a blue solid line. The number of extracted electrons
corresponding to S2 is reported on top axis.

###  5.2 Signal modeling

The expected signal from [37]Ar is evaluated following the same approach as in
Ref. RefXE1TS2only . Since the S2 signal is uncorrected, it is not possible to
use a simple Gaussian as it is done for the S1-S2 analysis. Instead, a
parametric model is considered which describes the response of the detector to
the [37]Ar energy deposition in the (S2, $z$) space. This model is built in
three steps.

The first step is the calculation of the rate of events $r_{p}(n_{e})$ with
$n_{e}$ produced electrons:

$r_{p}(n_{e})=\sum_{i=K,L}\,\,(\,R_{i}\times\,\mathrm{Binom}(n_{e}\,|\,N=(E_{i}/W),p_{i})\,)\,,$
(17)

where $R_{i}$ is the rate of K-shell ($i=K$) or L-shell ($i=L$) decays, $N$
the number of quanta produced, $E_{i}$ the energy of the decay, $W$ the
average energy per quanta and $p_{i}$ is the probability of producing an
electron, with $p_{i}\,=\,EY_{i}\,\cdot\,W$, where $EY_{i}$ is the electron
yield at the K-shell or L-shell energy.

The second step is the calculation of the rate $r_{d}$ of events with $k$
extracted electrons, which depends on $z$:

$r_{d}(k,z)=\sum_{n}\;r_{p}(n_{e})\cdot\mathrm{Binom}\left(k\,|\,n_{e},p=\eta
e^{-\frac{z}{v_{d}\tau}}\right)\,,$ (18)

where $\eta$ is the extraction efficiency, $v_{d}$ the electron drift velocity
and $\tau$ is the electron lifetime.

The third step is the calculation of the final rate of events $R(S2,z)$:

$\begin{split}R(S2,z)&=\sum_{k}\;r_{d}(k,z)\cdot\mathrm{Gauss}(S2\,|\,\mu(k)\,,\sigma(k))\,,\end{split}$
(19)

where $\mu(k)\,=\,b(k)\cdot(\mu_{\mathrm{SE}}\cdot k)$, $\mu_{\mathrm{SE}}$ is
the single electron gain (i.e. PE detected per extracted electron) and $b(k)$
is a reconstruction bias estimated from simulations, while

$\sigma(k)\,=\,\sqrt{\left(\,k\cdot\sigma^{2}_{\mathrm{SE}}\,+\,(\mu\cdot
s_{b}(k))^{2}\,+\,(c\cdot\mu)^{2}\right)}\,,$ (20)

where $\sqrt{k}\cdot\sigma_{\mathrm{SE}}$ is the resolution for $k$ electrons,
$s_{b}(k)$ is the gaussian smearing of $b(k)$ and $c\cdot\mu$ is the smearing
accounting for $x$-$y$ dependence of S2 ($c\,=\,0.05$ as estimated from
calibrations).

The parameters of the model have been set to the best-fit values from SR1
analysis in XENON1T RefXE1TAnalysis  with the exception of the single electron
gain $\mu_{\mathrm{SE}}\,=\,30.9\,~{}\mathrm{PE/e}^{-}$ which has been
calculated from the value of $g_{2}$ reported in Sec. 4. Five parameters,
namely the two rates ($R_{K}$, $R_{L}$), the two electron yields ($EY_{K}$,
$EY_{L}$) and the electron lifetime (for an independent estimate), are left
free to vary and are determined from the fit to the data.

The cut acceptances and trigger efficiency are applied to $R(S2,z)$ and then
the predicted signal is projected onto uncorrected S2 to compare with the
data.

###  5.3 Results

The model is fit to the data in the uncorrected S2 space with a binned maximum
likelihood method. The fit is restricted to events whose depth, estimated on
the basis of the model from S2 size and width , corresponds approximately to
$-85\,\text{ cm}\,<\,z\,<\,-12\,$ cm, in order to exclude the top and bottom
part of the LXe TPC. Moreover, it is performed on the central 95% region of
the two peaks, since the model cannot perfectly describe the outer tails due
to non accurate corrections of the field effects which also depend on $z$.

An extensive check of the events outside the central 95% fit region is
performed. The temporal evolution of these events and their uniform spatial
distribution inside the TPC are the same as for events in the central regions.
It is found that these events originate from [37]Ar decays but are poorly
reconstructed in $x$-$y$ and thus end up in the signal tail region. The
background contribution in the signal tails is also estimated to be
negligible.

The comparison of the best fit with the data is shown in Fig. 10. The
agreement is good for both L-shell and K-shell peaks
($\chi^{2}/\text{ndf}=135/111$). There is a slight mismatch for the L-shell
peak. This is a result of the electron lifetime, which is mainly driven by the
K-shell distribution and, hence, non-optimal for the L-shell region, at whose
energy less impact from field non-uniformity is expected. This value of the
electron lifetime agrees with that based on [37]Ar obtained with the standard
procedure (see 3.2), which only makes use of events from the K-shell.

Figure 10: Best-fit model (red line) of S2 spectrum compared with data (black
points). On top axis the number of extracted electrons corresponding to S2
size is reported. The gray-shaded regions are excluded from the fit, which is
performed in the central 95% region of the two peaks.

Systematic uncertainties on the values obtained from the fit are studied by
considering the variation of the parameters of the model inside their range of
uncertainty RefXE1TAnalysis . The main contributor to the systematic
uncertainties for the electron yields is the value of the single electron gain
$\mu_{\mathrm{SE}}$, which has an uncertainty of 6% given by the sum in
quadrature of the uncertainty on $g_{2}$ and of the reconstruction bias. This
occurs as a result of the anti-correlation between EY and the single electron
gain, for a given S2. Varying other parameters in the model has a minor impact
on the systematic uncertainties on the electron yield and on the branching
ratio between electron captures of the L- and K-shell (L/K ratio). This
results in a $\sim$7% systematic uncertainty on the electron yields and a
$\sim$4% systematic uncertainty on the L/K ratio.

The electron yield best fit values for the [37]Ar peaks are:

$\displaystyle\mathrm{EY}(0.27~{}\mathrm{keV})$
$\displaystyle=68.0^{+6.3}_{-3.7}~{}\mathrm{e}^{-}/\,\mathrm{keV}\,,$ (21)
$\displaystyle\mathrm{EY}(2.82~{}\mathrm{keV})$
$\displaystyle=40.0^{+2.7}_{-2.2}~{}\mathrm{e}^{-}/\,\mathrm{keV}\,.$ (22)

The electron yield at 2.82 keV is in agreement with the result from S1-S2
analysis, see Eq. (14). As shown in Fig. 11 and Fig. 12, the electron yield at
0.27 keV is compatible with both NEST predictions RefNestv2.3.6  and with the
extrapolations from the 220Rn calibration in XENON1T RefXE1TAnalysis . It is
also in agreement within 1$\upsigma$ with the measurement by PIXeY at a
slightly higher drift field (99 V/cm) RefPixey . It adds information in the
low-energy region ($<\,1$ keV), where only few measurements are available
RefPixey ; RefLux127Xe . It has to be noted that no subtraction of the
electrons produced directly by the decay and then drifted to the gas phase has
been applied, differently from what performed in Ref. ds50 . This allows a
direct comparison with all the other measurements reported in this work, in
which such a correction is also not applied.

The L/K ratio obtained from the best-fit model is

$\mathrm{L/K}\,=\,(10.11\pm 0.44)\%\,$ (23)

in agreement within 1$\upsigma$ with the expected value of 9.67% RefTable37Ar
; RefAr37energy  . This suggests that the S2 detection efficiency and cut
acceptances in the energy range between 0.27 keV and 2.82 keV are well
understood. Furthermore, the good description of the [37]Ar signal for both
peaks validates the modelling used for S2-only analysis in XENON1T
RefXE1TS2only .

Figure 11: Photon yield (_left panel_) and electron yield (_right panel_)
depending on energy for ER. The results obtained in this work for [37]Ar are
reported as red points, with vertical red bands highlighting the energy region
corresponding to [37]Ar decays. Are also reported the expected values from
NEST v2.3.6 RefNestv2.3.6  (black dashed line) and measurements from LUX 127Xe
RefLux127Xe  and CH3T RefLuxTritium , XELDA 127Xe RefXelda127Xe  , PIXeY
RefPixey  and Xurich II RefXurich37Ar  37Ar, XENON100 CH3T RefXenon100Tritium
, as well as the fit to the XENON1T calibration data from 220Rn (light blue
solid line, shaded region represents the 15%-85% credible region)
RefXE1TAnalysis . Figure 12: Photon yield (_left panel_) and electron yield
(_right panel_) at different drift fields for [37]Ar peaks obtained by PIXeY
RefPixey  and Xurich II RefXurich37Ar  compared to the results from this work
(red points). The vertical red bands indicate XENON1T drift field. The blue
triangles indicate yield at 2.82 keV while the green pentagons represent
electron yield at 0.27 keV. The expected values from NEST v2.3.6 RefNestv2.3.6
are shown with a black dashed line (2.82 keV) and a black dotted line (0.27
keV).

##  6 Conclusions

A calibration was performed with the low-energy lines of [37]Ar at 2.82 keV
(K-shell decay) and 0.27 keV (L-shell decay) in XENON1T. The mandatory removal
of this source after calibration via cryogenic distillation was successfully
demonstrated, reaching a reduction of activity by one order of magnitude every
4.5 days. This proves that [37]Ar can be used as a regular calibration source
for multi-tonne xenon detectors.

Two analyses of the [37]Ar data are performed, one using both S1 and S2
signals, the other only requiring the detection of the S2. At the XENON1T
drift field of 82 V/cm, the photon and electron yields at 2.82 keV are
$(32.3\pm 0.3)\,$photons/keV and $(40.6\pm 0.5)\,$electrons/keV, respectively.
The electron yield at 0.27 keV is $68.0^{+6.3}_{-3.7}\,$electrons/keV.

Since only a weak dependence from the drift field is expected at these
energies, these values can be compared with measurements taken at different
drift fields. As it is apparent from Fig. 11 and Fig. 12, there is in general
a good agreement with other measurements at the same or similar energies. In
particular agreement within 1$\upsigma$ is observed among XENON1T and LUX, the
two large detectors providing measurements at these energies. Some tension is
observed with the measurements from PIXeY at 2.82 keV, which is much larger
than the difference due to the higher drift field, whereas yields from Xurich
II are closer to the results of this work. Disagreement is also apparent for
the photon yield at 2.82 keV with respect to the NEST prediction at the same
drift field. The electron yield value at 0.27 keV is in agreement with
predictions and other measurements at the same or similar energies. This new
measurement adds relevant information in an energy region where only two other
experimental values are currently available.

The energy of the K-shell decay reconstructed with the combined energy scale
is (2.83$\pm$0.02) keV, in agreement with the expected value. The results from
the [37]Ar calibration suggests that XENON1T detector is well understood even
in the low-energy ER region close to the detection threshold.

The outcomes of this work are used to inform a re-analysis of XENON1T science
data in which a low-energy ER excess was previously reported. With the
additional correction to the data and the updated resolution model described
in this work, the main conclusions obtained in Ref. RefXE1TExcessER  remain
unaltered.

## Acknowledgements

We gratefully acknowledge support from the National Science Foundation, Swiss
National Science Foundation, German Ministry for Education and Research, Max
Planck Gesellschaft, Deutsche Forschungsgemeinschaft, Helmholtz Association,
Dutch Research Council (NWO), Weizmann Institute of Science, Israeli Science
Foundation, Binational Science Foundation, Fundacao para a Ciencia e a
Tecnologia, Région des Pays de la Loire, Knut and Alice Wallenberg Foundation,
Kavli Foundation, JSPS Kakenhi and JST FOREST Program in Japan, Tsinghua
University Initiative Scientific Research Program and Istituto Nazionale di
Fisica Nucleare. This project has received funding/support from the European
Union’s Horizon 2020 research and innovation programme under the Marie
Skłodowska-Curie grant agreement No 860881-HIDDeN. Data processing is
performed using infrastructures from the Open Science Grid, the European Grid
Initiative and the Dutch national e-infrastructure with the support of SURF
Cooperative. We are grateful to Laboratori Nazionali del Gran Sasso for
hosting and supporting the XENON project.

## References

  * (1) E. Aprile et al. (XENON Collaboration), The XENON1T dark matter experiment. Eur. Phys. J. C, 77, 881 (2017)
  * (2) E. Aprile et al. (XENON Collaboration), Dark matter search results from a one ton-year exposure of XENON1T. Phys. Rev. Lett. 121, 111302 (2018)
  * (3) E. Aprile et al. (XENON Collaboration), Conceptual design and simulation of a water Cherenkov muon veto for the XENON1T experiment. JINST, 9, P11006 (2014)
  * (4) E. Aprile et al. (XENON Collaboration), Lowering the radioactivity of the photomultiplier tubes for the XENON1T dark matter experiment. Eur. Phys. J. C, 75, 11, 546 (2015)
  * (5) A. Hitachi, Properties of Liquid Xenon Scintillation for Dark Matter Searches. Astropart. Phys., 24, 247 (2005)
  * (6) E. Aprile et al., Scintillation response of liquid xenon to low energy nuclear recoils. Phys. Rev. D 72, 072006 (2005)
  * (7) A. Manzur et al., Scintillation efficiency and ionization yield of liquid xenon for monoenergetic nuclear recoils down to 4 keV. Phys. Rev. C 81, 025808 (2010)
  * (8) P. Sorensen and C. E. Dahl, Nuclear recoil energy scale in liquid xenon with application to the direct detection of dark matter. Phys. Rev. D 83, 063501 (2011)
  * (9) C. E. Dahl, The physics of background discrimination in liquid xenon, and first results from Xenon10 in the hunt for WIMP dark matter. Ph.D. thesis, Princeton University, AAT-3364532 (2009)
  * (10) B. Lenardo et al., A Global Analysis of Light and Charge Yields in Liquid Xenon. IEEE Transactions on Nuclear Science, 62, 6 (2015)
  * (11) E. Aprile et al. (XENON Collaboration) and X. Mougeot, Excess electronic recoil events in XENON1T. Phys. Rev. D 102, 072004 (2020)
  * (12) E. Aprile et al. (XENON Collaboration), Search for New Physics in Electronic Recoil Data from XENONnT. Phys. Rev. Lett. 129, 161805 (2022)
  * (13) E. Aprile et al. (XENON Collaboration), XENON1T Dark Matter Data Analysis: Signal Reconstruction, Calibration and Event Selection. Phys. Rev. D 100, 052014 (2019)
  * (14) R. F. Lang, A. Brown, E. Brown, M. Cervantes, S. Macmullin, A 220Rn source for the calibration of low-background experiments. JINST, 11, P04004 (2016)
  * (15) M.-M. Bé et al., Table of Radionuclides, Monographie BIPM-5, volume 7. Bureau International des Poids et Mesures, Pavillon de Breteuil, F-92310 Sèvres, France (2013)
  * (16) V. P. Chechev, The evaluation of half-lives and other decay data used in nuclear astrophysics and cosmochronology. Phys. Atom. Nuclei, 74, 1713-1717 (2011)
  * (17) E. M. Boulton et al., Calibration of a two-phase xenon time projection chamber with a 37Ar source. JINST, 12, P08004 (2017)
  * (18) L. Baudis et al., The first dual-phase xenon TPC equipped with silicon photomultipliers and characterisation with 37Ar. Eur. Phys. J. C, 80, 477 (2020)
  * (19) E. Aprile et al. (XENON Collaboration), Application and modeling of an online distillation method to reduce krypton and argon in XENON1T. Prog. Theor. Exp. Phys., 5 (2022)
  * (20) E. Aprile et al. (XENON Collaboration), Removing krypton from xenon by cryogenic distillation to the ppq level. Eur. Phys. J. C, 77, 275 (2017)
  * (21) G. J. Feldman and R. D. Cousins, A Unified approach to the classical statistical analysis of small signals. Phys. Rev. D. 57, 3873 (1998)
  * (22) L. Baudis, P. Sanchez-Lucas, K. Thieme, A measurement of the mean electronic excitation energy of liquid xenon. Eur. Phys. J. C, 81, 1060 (2021)
  * (23) G. Anton et al. (EXO-200 Collaboration), Measurement of the scintillation and ionization response of liquid xenon at MeV energies in the EXO-200 experiment. Phys. Rev. C 101, 065501 (2020)
  * (24) E. Aprile et al. (XENON Collaboration), XENON1T Dark Matter Data Analysis: Signal & Background Models, and Statistical Inference. Phys. Rev. D 99, 112009 (2019)
  * (25) P. Barrow et al., Qualification tests of the R11410-21 photomultiplier tubes for the XENON1T detector. JINST, 12, P01024 (2017)
  * (26) R. Saldanha et al., Model independent approach to the single photoelectron calibration of photomultiplier tubes. Nuclear Instruments and Methods in Physics Research A, 863 (2017)
  * (27) C. H. Faham et al., Measurements of wavelength-dependent double photoelectron emission from single photons in VUV-sensitive photomultiplier tubes. JINST, 10, P09010 (2015)
  * (28) M. Szydagis et al., A Review of Basic Energy Reconstruction Techniques in Liquid Xenon and Argon Detectors for Dark Matter and Neutrino Physics Using NEST. Instruments, 5(1), 13 (2021)
  * (29) V. N. Lebedenko et al., Result from the First Science Run of the ZEPLIN-III Dark Matter Search Experiment. Phys. Rev. D 80, 052010 (2009)
  * (30) D. S. Akerib et al. (LUX Collaboration), Discrimination of electronic recoils from nuclear recoils in two-phase xenon time projection chambers. Phys. Rev. D 102, 112002 (2020)
  * (31) M. Szydagis et al., Noble Element Simulation Technique (v2.3.6) (2022). URL https://zenodo.org/record/6448408#.Y01Zo0xBw-U
  * (32) D. S. Akerib et al. (LUX Collaboration), Tritium calibration of the LUX dark matter experiment. Phys. Rev. D 93, 072009 (2016)
  * (33) J. Angle et al. (XENON10 Collaboration), Search for Light Dark Matter in XENON10 Data. Phys. Rev. Lett. 107, 051301 (2011)
  * (34) E. Aprile et al. (XENON Collaboration), Low-mass dark matter search using ionization signals in XENON100. Phys. Rev. D 94, 092001 (2016). Erratum (fixing LUX points) Phys. Rev. D 95, 059901 (2017)
  * (35) E. Aprile et al. (XENON Collaboration), Light dark matter search with ionization signals in XENON1T. Phys. Rev. Lett. 123, 251801 (2019)
  * (36) E. Aprile et al. (XENON Collaboration), Emission of single and few electrons in XENON1T and limits on light dark matter. Phys. Rev. D 123, 251801 (2022)
  * (37) D. S. Akerib et al. (LUX Collaboration), Ultra-Low Energy Calibration of LUX Detector using 127Xe Electron Capture. Phys. Rev. D 96, 112011 (2017)
  * (38) P. Agnes et al. (DarkSide Collaboration), Calibration of the liquid argon ionization response to low energy electronic and nuclear recoils with DarkSide-50. Phys. Rev. D 104, 082005 (2021)
  * (39) D. J. Temples et al., Measurement of charge and light yields for 127Xe L-shell electron captures in liquid xenon. Phys. Rev. D 104, 112001 (2021)
  * (40) E. Aprile et al. (XENON Collaboration), Signal Yields of keV Electronic Recoils and Their Discrimination from Nuclear Recoils in Liquid Xenon. Phys. Rev. D 97, 092007 (2018)