# A high precision calibration of the nonlinear energy response at Daya Bay

D. Adey  F. P. An  A. B. Balantekin  H. R. Band  M. Bishai  S. Blyth  D. Cao
G. F. Cao  J. Cao  J. F. Chang  Y. Chang  H. S. Chen  S. M. Chen  Y. Chen  Y.
X. Chen  J. Cheng  Z. K. Cheng  J. J. Cherwinka  M. C. Chu  A. Chukanov  J. P.
Cummings  N. Dash  F. S. Deng  Y. Y. Ding  M. V. Diwan  T. Dohnal  J. Dove  M.
Dvořák  D. A. Dwyer  M. Gonchar  G. H. Gong  H. Gong  W. Q. Gu  J. Y. Guo  L.
Guo  X. H. Guo  Y. H. Guo  Z. Guo  R. W. Hackenburg  S. Hans111Now at:
Department of Chemistry and Chemical Technology, Bronx Community College,
Bronx, New York 10453 M. He  K. M. Heeger  Y. K. Heng  A. Higuera  Y. K. Hor
Y. B. Hsiung  B. Z. Hu  J. R. Hu  T. Hu  Z. J. Hu  H. X. Huang  X. T. Huang
Y. B. Huang  P. Huber  D. E. Jaffe  K. L. Jen  S. Jetter  X. L. Ji  X. P. Ji
R. A. Johnson  D. Jones  L. Kang  S. H. Kettell  L. W. Koerner  S. Kohn  M.
Kramer  T. J. Langford  L. Lebanowski222Now at: University of Pennsylvania,
Department of Physics & Astronomy, 209 South 33rd Street, Philadelphia, PA
19104-6396 J. Lee  J. H. C. Lee  R. T. Lei  R. Leitner  J. K. C. Leung  C. Li
F. Li  H. L. Li  Q. J. Li  S. Li  S. C. Li  S. J. Li  W. D. Li  X. N. Li  X.
Q. Li  Y. F. Li  Z. B. Li  H. Liang  C. J. Lin  G. L. Lin  S. Lin  S. K. Lin
J. J. Ling  J. M. Link  L. Littenberg  B. R. Littlejohn  J. C. Liu  J. L. Liu
Y. Liu  Y. H. Liu  C. Lu  H. Q. Lu  J. S. Lu  K. B. Luk  X. B. Ma  X. Y. Ma
Y. Q. Ma  C. Marshall  D. A. Martinez Caicedo  K. T. McDonald  R. D. McKeown
I. Mitchell  L. Mora Lepin  J. Napolitano  D. Naumov  E. Naumova  J. P. Ochoa-
Ricoux  A. Olshevskiy  H.-R. Pan  J. Park  S. Patton  V. Pec  J. C. Peng  L.
Pinsky  C. S. J. Pun  F. Z. Qi  M. Qi  X. Qian  N. Raper  J. Ren  R. Rosero
B. Roskovec  X. C. Ruan  H. Steiner  J. L. Sun  K. Treskov  W.-H. Tse  C. E.
Tull  B. Viren  V. Vorobel  C. H. Wang  J. Wang  M. Wang  N. Y. Wang  R. G.
Wang  W. Wang  W. Wang  X. Wang  Y. Wang  Y. F. Wang  Z. Wang  Z. Wang  Z. M.
Wang  H. Y. Wei  L. H. Wei  L. J. Wen  K. Whisnant  C. G. White  H. L. H. Wong
S. C. F. Wong  E. Worcester  Q. Wu  W. J. Wu  D. M. Xia  Z. Z. Xing  J. L. Xu
T. Xue  C. G. Yang  L. Yang  M. S. Yang  Y. Z. Yang  M. Ye  M. Yeh  B. L.
Young  H. Z. Yu  Z. Y. Yu  B. B. Yue  S. Zeng  Y. Zeng  L. Zhan  C. Zhang  C.
C. Zhang  F. Y. Zhang  H. H. Zhang  J. W. Zhang  Q. M. Zhang  R. Zhang  X. F.
Zhang  X. T. Zhang  Y. M. Zhang  Y. M. Zhang  Y. X. Zhang  Y. Y. Zhang  Z. J.
Zhang  Z. P. Zhang  Z. Y. Zhang  J. Zhao  L. Zhou  H. L. Zhuang  J. H. Zou
Institute of High Energy Physics, Beijing  Institute of Modern Physics, East
China University of Science and Technology, Shanghai  University of Wisconsin,
Madison, Wisconsin 53706  Wright Laboratory and Department of Physics, Yale
University, New Haven, Connecticut 06520  Brookhaven National Laboratory,
Upton, New York 11973  Department of Physics, National Taiwan University,
Taipei  Nanjing University, Nanjing  National United University, Miao-Li
Department of Engineering Physics, Tsinghua University, Beijing  Shenzhen
University, Shenzhen  Sun Yat-Sen (Zhongshan) University, Guangzhou  North
China Electric Power University, Beijing  Chinese University of Hong Kong,
Hong Kong  Joint Institute for Nuclear Research, Dubna, Moscow Region  Siena
College, Loudonville, New York 12211  University of Science and Technology of
China, Hefei  Charles University, Faculty of Mathematics and Physics, Prague
Department of Physics, University of Illinois at Urbana-Champaign, Urbana,
Illinois 61801  Lawrence Berkeley National Laboratory, Berkeley, California
94720  Beijing Normal University, Beijing  Department of Nuclear Science and
Technology, School of Energy and Power Engineering, Xi’an Jiaotong University,
Xi’an  Department of Physics, University of Houston, Houston, Texas 77204
China Institute of Atomic Energy, Beijing  Shandong University, Jinan  Center
for Neutrino Physics, Virginia Tech, Blacksburg, Virginia 24061  Institute of
Physics, National Chiao-Tung University, Hsinchu  Department of Physics,
University of Cincinnati, Cincinnati, Ohio 45221  Department of Physics,
College of Science and Technology, Temple University, Philadelphia,
Pennsylvania 19122  Dongguan University of Technology, Dongguan  Department of
Physics, University of California, Berkeley, California 94720  Department of
Physics, The University of Hong Kong, Pokfulam, Hong Kong  School of Physics,
Nankai University, Tianjin  Department of Physics, Illinois Institute of
Technology, Chicago, Illinois 60616  Department of Physics and Astronomy,
Shanghai Jiao Tong University, Shanghai Laboratory for Particle Physics and
Cosmology, Shanghai  Joseph Henry Laboratories, Princeton University,
Princeton, New Jersey 08544  California Institute of Technology, Pasadena,
California 91125  College of William and Mary, Williamsburg, Virginia 23187
Instituto de Física, Pontificia Universidad Católica de Chile, Santiago
Department of Physics and Astronomy, University of California, Irvine,
California 92697  China General Nuclear Power Group, Shenzhen  College of
Electronic Science and Engineering, National University of Defense Technology,
Changsha  Iowa State University, Ames, Iowa 50011  Chongqing University,
Chongqing

###### Abstract

A high precision calibration of the nonlinearity in the energy response of the
Daya Bay Reactor Neutrino Experiment’s antineutrino detectors is presented in
detail. The energy nonlinearity originates from the particle-dependent light
yield of the scintillator and charge-dependent electronics response. The
nonlinearity model is constrained by $\gamma$ calibration points from deployed
and naturally occurring radioactive sources, the $\beta$ spectrum from 12B
decays, and a direct measurement of the electronics nonlinearity with a new
flash analog-to-digital converter readout system. Less than 0.5% uncertainty
in the energy nonlinearity calibration is achieved for positrons of kinetic
energies greater than 1 MeV.

###### keywords:

neutrino , liquid scintillator , energy calibration , Daya Bay

††journal: Nucl. Instr. Meth. A

##  1 Introduction

Liquid scintillator (LS), together with photosensors such as photomultiplier
tubes (PMTs), has been widely used for energy calorimetry. This kind of
detector plays an important role in neutrino physics, especially in reactor
$\overline{\nu}_{e}$ experiments, from the discovery of the neutrino at the
Savannah River Plant [1], to the measurement of the neutrino oscillation angle
$\theta_{12}$ at KamLAND [2], the observation of the neutrino oscillation
driven by $\theta_{13}$ at Daya Bay [3], RENO [4], and Double Chooz [5], and
the next generation experiment such as JUNO [6].

With reactor antineutrinos in the MeV energy range, these experiments are
sensitive to the $\overline{\nu}_{e}$ disappearance. The survival probability
is expressed as

$\begin{split}&P_{\rm
survival}=1-\cos^{4}\theta_{13}\sin^{2}2\theta_{12}\sin^{2}\Delta_{21}\\\
&-\sin^{2}2\theta_{13}(\cos^{2}\theta_{12}\sin^{2}\Delta_{31}+\sin^{2}\theta_{12}\sin^{2}\Delta_{32}),\end{split}$
(1)

where $\Delta_{ij}\simeq 1.267\Delta m^{2}_{ij}{\rm L}/{\rm E}_{\nu}$, L in
meters is the reactor-detector distance, Eν in MeV is the $\overline{\nu}_{e}$
energy, and $\Delta m^{2}_{ij}$ in ${\rm eV^{2}}$ is the mass-squared
difference between neutrinos of mass eigenstates $\nu_{i}$ and $\nu_{j}$.
Reactor antineutrinos are detected via the inverse beta decay reaction (IBD)
$\overline{\nu}_{e}$$+p\to e^{+}+n$. The $e^{+}$ and neutron form a prompt and
delayed coincident pair which efficiently discriminates $\overline{\nu}_{e}$
interactions from background. Convolving the $\overline{\nu}_{e}$ energy
spectrum with the IBD cross section [7] results in an expected spectrum which
rises from the 1.8 MeV interaction threshold, peaks at $\sim$4 MeV, and falls
to a very low rate above 8 MeV.

Based on the kinematics of IBD reactions, the neutron generally carries only a
small fraction of the initial $\overline{\nu}_{e}$ energy, O(10 keV). The
initial $\overline{\nu}_{e}$ energy can be estimated using ${\rm
E}_{\nu}\simeq{\rm E}_{\rm prompt}+0.78$ MeV, where Eprompt is the sum of the
positron kinetic plus annihilation energy. Since the observable in the
detector is the reconstructed positron energy E${}^{e^{+}}_{\mathrm{rec}}$,
the precise understanding of the detector energy response, which connects
Eprompt and E${}^{e^{+}}_{\mathrm{rec}}$, is critical for the measurement of
the oscillation parameters and for the absolute measurement of the reactor
$\overline{\nu}_{e}$ spectrum.

The energy response of LS is not linear with respect to the kinetic energies
of charged particles, primarily due to ionization quenching [8] and Cherenkov
radiation [9]. The former reduces the scintillation light yield at high
ionization density, such as protons, $\alpha$ particles and low energy
$e^{\pm}$, while the latter contributes additional photons if a particle’s
velocity is larger than the phase velocity of light in the LS. Further
nonlinearity can be introduced during the PMT charge estimation in the
electronics readout system.

In this paper, we present a precise energy nonlinearity calibration of the
Daya Bay antineutrino detectors (ADs). An energy nonlinearity model is built
accounting for the scintillation and electronics nonlinearity. The model is
constrained by ten $\gamma$ calibration points, the $\beta$ spectrum from 12B
decays, and a direct measurement of the electronics nonlinearity with a newly
installed flash analog-to-digital converter (FADC) readout system. Less than
0.5% uncertainty in the energy nonlinearity calibration is achieved for
positrons of kinetic energies greater than 1 MeV.

##  2 The Daya Bay experiment

The Daya Bay experiment consists of eight identically designed antineutrino
detectors (ADs). Two of them are deployed in each of two near halls (EH1 and
EH2) and four are installed in the far hall (EH3). To designate a particular
AD, EH1-AD1 means AD1 in EH1, for example. Each AD has three nested
cylindrical volumes separated by concentric acrylic vessels as shown in Fig.
1. The innermost volume is filled with 20 tons of gadolinium-loaded LS (Gd-
LS), serving as the primary $\overline{\nu}_{e}$ target. It is surrounded by
$\sim$22 tons of non-loaded LS to detect $\gamma$ rays escaping from the
target volume. The outermost volume is filled with mineral oil to shield the
LS from natural radioactivity. A total of 192 8-inch PMTs are installed on the
steel vessel (SS) to detect scintillation photons, giving about 160
photoelectrons per MeV of deposited energy (p.e./MeV). The ADs are immersed in
a water Cherenkov muon detector which functions both as a passive radiation
shield and as an active muon identification. Details of the detector system
are described in Refs. [10, 11].

Figure 1:  Schematic of a Daya Bay antineutrino detector. The three
cylindrical volumes, defined by three acrylic vessels, are filled with Gd-LS,
LS, and mineral oil from the innermost to the outermost. Three automatic
calibration units are installed on top of the detector to calibrate the PMTs
and detector energy response.

There are three automatic calibration units (ACUs) on the top of each AD to
calibrate the energy response along the vertical axes at detector center
(ACU-A), the edge of the Gd-LS volume (ACU-B), and the LS volume (ACU-C).
Regular calibration consists of the deployment of a 68Ge source, a 60Co
source, a 241Am-13C neutron source [12], or a light-emitting diode (LED). In
the summer of 2012, a special calibration deployed various radioactive sources
to calibrate the energy nonlinearity (137Cs, 54Mn, 40K, Pu-13C, 241Am-9Be).
Details of the calibration system are described in Ref. [13].

The radioactive sources are housed in a stainless steel capsule to shield the
accompanying $e^{-}$ or $e^{+}$ from the scintillator. The capsule is further
enclosed by acrylic, polytetrafluoroethylene (PTFE) with high reflectivity or
greenish PTFE as shown in Fig. 2. In the following the greenish PTFE will be
referred to as greenish teflon. There are weights on the top and bottom of the
source, made of stainless steel surrounded by acrylic, to maintain tension in
the deployment string. In January 2017, another special calibration campaign
was performed to better understand the optical shadowing of the source
enclosures and deployment system, which will be discussed in Sec. 4.1.

Figure 2:  Left: schematic for the radioactive sources, including a top and a
bottom weight (A and D) to maintain tension in the deployment string [13].
Right: sections of the 68Ge, 241Am-13C and 60Co sources. Dimensions are
millimeters.

The scintillation photons are detected by 192 PMTs (Hamamatsu R5912). The PMT
signals share the same cable with the high voltage power. A passive input to
an AC circuit is used to decouple the fast PMT signals, which are sent to the
frontend electronics (FEE) [14]. After an initial fast amplification, the
signal is fed to a pulse shaping circuit consisting of a differential CR and
four integrating RC circuits, and then amplified by a factor of ten. The peak
value of the shaped PMT signal is used as an estimate of the PMT charge
output, and it is sampled by a 40-MHz 12-bit ADC.

The energy reconstruction in data starts with a scale conversion from PMT
charge (${\rm Q}_{i}$) to number of photoelectrons, with a daily calibrated
conversion factor of roughly 19 ADC counts per single p.e. ($\overline{{\rm
Q}}^{\rm SPE}_{i}(t)$). The estimate of the total energy deposited by a
particle interaction, Erec, is proportional to the total number of
photoelectrons over the active PMTs ($\sum_{i}\frac{{\rm
Q}_{i}}{\overline{{\rm Q}}^{\rm SPE}_{i}(t)}$), as

${\rm E}_{\rm rec}=\left(\sum_{i}\frac{{\rm Q}_{i}}{\overline{{\rm Q}}^{\rm
SPE}_{i}(t)}\right)\cdot\frac{f_{\rm act}(t)}{N_{\rm PE}(t)}\cdot f_{\rm
pos}(\mathbf{r}_{\rm rec},t),$ (2)

where the correction $f_{\rm act}(t)$, defined as the ratio of total to active
AD PMTs, is to compensate for the reduced light when a PMT channel is
temporarily disabled333The average over all ADs of the number of disabled
channels is less than 0.5.. The time dependent light yield per MeV of
deposited energy, $N_{\rm PE}(t)$, is a scale conversion calibrated by the
neutron capture on Gd with a mean energy of 8.05 MeV. The time dependent
nonuniformity correction, $f_{\rm pos}(\mathbf{r}_{\rm rec},t)$, compensates
for the observed variation in total p.e. versus the estimated position
$\mathbf{r}_{\rm rec}$ in the AD. The light yield and nonuniformity correction
are determined by two independent methods: calibration sources and spallation
neutron captures. Details can be found in Ref. [15].

##  3 The energy nonlinearity model

In the $\overline{\nu}_{e}$ analysis of Daya Bay [15, 16, 17], the detector
energy response is analytically described in three steps connecting four
quantities: 1) the $e^{+}$ true energy ${\rm E}_{\rm prompt}$, which is the
sum of the positron kinetic plus annihilation energy, 2) the $e^{+}$ deposited
energy in the Gd-LS and LS Edep, 3) the visible energy proportional to the
total number of detectable photons Evis, and 4) the reconstructed energy Erec.
The first step is described by a matrix generated with the Geant4 [18] based
simulation and is reported in Ref. [16]. The nonlinear relationship between
Edep and Erec, expressed as $f_{\rm NL}$ = Erec/Edep, has two origins: in the
second step, the total detectable light is not linear with respect to the
total deposited energy, and in the third step, the total reconstructed PMT
charge is not linear to the total number of photoelectrons generated by the
PMT photocathodes.

To decouple the two origins, the visible energy Evis is introduced and the
nonlinearity expression is rewritten as $f_{\rm NL}=\frac{{\rm E}_{\rm
vis}}{{\rm E}_{\rm dep}}\times\frac{{\rm E}_{\rm rec}}{{\rm E}_{\rm vis}}$. In
this way the particle-dependent scintillation nonlinearity is described by
$\frac{{\rm E}_{\rm vis}}{{\rm E}_{\rm dep}}$, while the charge-dependent
electronics nonlinearity is described by $\frac{{\rm E}_{\rm rec}}{{\rm
E}_{\rm vis}}$. We will first discuss the modeling of scintillation
nonlinearity in Sec. 3.1 before introducing the direct measurement of the
electronics nonlinearity in Sec. 3.2.

It should be noted that the step-by-step strategy is feasible only in the case
of negligible coupling between energy leakage and energy nonlinearity, such as
the $\overline{\nu}_{e}$ analysis based on the neutron capture on gadolinium.
This approach cannot be used in an experiment with significant energy leakage.

###  3.1 Scintillation nonlinearity

The key particles involved in the $\overline{\nu}_{e}$ detection with LS are
$e^{-}$, $e^{+}$ and the gammas ($\gamma$s). The modeling of their
scintillation nonlinearity is discussed in this section.

####  3.1.1 Scintillation nonlinearity model of the $e^{-}$

##### Ionization quenching

When a particle deposits energy in the Gd-LS or LS, molecules of the solvent
(linear alkylbenzene in Daya Bay, referred to as LAB) are excited. The energy
is transferred to fluorescent molecules (3 g/L PPO) primarily via dipole-
dipole interactions [19]. Some fraction of energy is not transferred due to
quenching of the excitation by the high density of ionized and excited
molecules [8]. The consequence is that the scintillation light yield is not
proportional to the deposited energy for the highly ionizing particles such as
protons, $\alpha$s, and low energy $e^{\pm}$.

There are many measurements of the quenching effects in various kinds of the
organic scintillator [20, 21, 22, 23]. The Birks’ model, expressed as

${\rm E}_{\rm scint}~{}({\rm E}_{\rm dep},~{}kB)=\int_{0}^{{\rm E}_{\rm
dep}}\frac{dE}{1+kB\times\frac{dE}{dx}},$ (3)

is a popular way to describe quenching, where ${\rm E}_{\rm scint}$ is the
energy converted to scintillation photons, $kB$ is the Birks’ coefficient, and
$dE/dx$ is the stopping power.

Equation 3 can be calculated with either a numerical integral or a Geant4
simulation. The former requires knowing $dE/dx$ as a function of energy, which
is obtained from an ESTAR calculation [24] using the Daya Bay LS properties.
For the latter, the Birks’ model is implemented in each simulation step when
converting the deposited energy to scintillation photons.

A key configurable in the simulation is the production threshold of secondary
particles. For example, in the Daya Bay simulation the default threshold of
$e^{-}$ is 0.1 mm ($\sim$80 keV). If a secondary $e^{-}$, namely a $\delta$
electron, was generated with a kinetic energy less than $\sim$80 keV, it would
not be tracked by Geant4, so the energy loss due to the $\delta$ electron is
added to the ionization loss.

Thus there are several strategies to generate the quenching curves, which
describe the relationship between Edep and ${\rm E}_{\rm scint}$. Figure 3
shows four curves generated with different strategies: a numerical integral
with $dE/dx$ from ESTAR, a simulation with the default production threshold
0.1 mm, and two other simulations with 10 and 100 times lower production
thresholds, i.e., 0.01 mm and 0.001 mm, respectively. Even with the same $kB$,
the quenching factors from those methods do not agree. The lower the
production threshold, the more $\delta$ electrons are produced, resulting in
heavier quenching. This means that the $kB$ values reported by different
measurements cannot be compared without a unified simulation configuration, as
similarly reported in Ref. [21]. Figure 4 shows the quenching curves with
different $kB$ values. A larger $kB$ yields the larger quenching. It is
difficult to determine which calculation method is better, thus the numerical
integration method is adopted as the default and cross validations were
performed by simulated quenching curves with different thresholds.

Figure 3:  Relationship between the deposited energy Edep and the energy
converted to scintillation photons ${\rm E}_{\rm scint}$, derived from the
numerical calculation with the ESTAR $dE/dx$ (black), the Geant4-based Daya
Bay simulation with a 0.1 mm particle production threshold (red), the
simulations with 0.01 mm (green) and 0.001 mm (blue) production threshold. The
lower threshold yields more $\delta$ electrons, resulting in a heavier
quenching.  Figure 4:  Relationship between Edep and ${\rm E}_{\rm scint}$
with different $kB$ values, calculated with the numerical integral method. The
larger $kB$ yields the larger quenching.

##### Cherenkov radiation

Cherenkov photons are generated if a particle’s velocity is larger than the
phase velocity of light in the medium [9]. The intensity is calculated with
the Frank$-$Tamm formula [25], which is implemented in the simulation of
Geant4, generating photons with wavelengths as the blue chain line in Fig. 5.
The spectrum relies on the LS refractive index, thus the imperfect knowledge
of the index in the ultra-violet region introduces uncertainties in this.

Figure 5:  The wavelength p.d.f. of Cherenkov photons at generation (blue
chain line), the photons arriving at the PMT photocathode (red solid line),
and the ones converted to photoelectrons by the photocathode (black dashed
line). Most Cherenkov photons are absorbed and re-emitted with certain
efficiencies dependent on the photon wavelengths.

The LS has good transparency only for photons in a narrow range of
wavelengths, from 400 nm to 600 nm. Photons with other wavelengths would be
absorbed by the solvent or solute. Absorption by the latter can generate
another photon (re-emission) with a certain probability. The red solid line of
Fig. 5 shows the wavelengths of photons arriving at the PMT photocathode after
several meters propagation.

There is no conclusive measurement available for the LS absorption and re-
emission, especially in the ultra-violet and infrared region, thus the
detectable Cherenkov contribution heavily depends on the LS properties assumed
by the simulation. Given the fact that the wavelength spectrum of Cherenkov
photons has negligible dependence on the primary particle’s energy, in this
analysis we only use the shape of the Cherenkov contribution to the $e^{-}$
energies generated by the Geant4 simulation as shown in Fig. 6, and determine
the normalization by comparing the energy model predictions with the
calibration data.

Figure 6:  The Cherenkov contribution to the $e^{-}$ kinetic energies,
normalized at 1 MeV. The absolute contribution relies on the LS optical
properties assumed in the simulation, thus it is a free parameter in the fit
described in Sec. 5.

In summary, the scintillation nonlinearity of $e^{-}$ can be defined as

$\begin{split}&f_{\rm ScintNL}^{e-}({\rm E}_{\rm
dep},~{}kB,~{}k_{C},~{}A)\equiv\frac{{\rm E}_{\rm vis}}{{\rm E}_{\rm dep}}\\\
&=A\times\left(f_{q}({\rm E}_{\rm dep},k_{B})+k_{C}\times\frac{f_{C}({\rm
E}_{\rm dep})}{{\rm E}_{\rm dep}}\right),\end{split}$ (4)

where $f_{q}$ is the quenching curve with a given Birks coefficient $k_{B}$,
$k_{C}$ and $f_{C}$ are the normalization factor and the shape of Cherenkov
contribution (6), respectively, and $A$ accounts for the absolute energy scale
calibration with the neutron capture on gadolinium.

####  3.1.2 Scintillation nonlinearity model of the $\gamma$

The $\gamma$ carries zero charge, therefore it cannot directly excite the LS
molecules. Its deposited energy relies on three processes: the photoelectric
effect, Compton scattering, and $e^{+}$/$e^{-}$ pair production. In LS Compton
scattering is dominant for $\gamma$ energies above about 100 keV; otherwise
the photoelectric effect dominates. Pair production is a secondary process in
the energy region of the reactor $\overline{\nu}_{e}$ analysis (0 to 12 MeV),
since the nuclei in LS are light.

To determine the scintillation nonlinearity, the $\gamma$s are simulated in LS
with Geant4. The kinetic energies of induced $e^{\pm}$ for a range of initial
$\gamma$ energies are shown in Fig. 7. The distribution includes $e^{\pm}$
generated by the initial $\gamma$ and the secondary $\gamma$s from the
positron annihilation and bremsstrahlung radiation. The integral of each
distribution equals the average total number of $e^{\pm}$ generated by each
$\gamma$. Cross validations were done by varying the electromagnetic models to
EMPenelope and EMLivermore in Geant4; the differences are negligible.

Figure 7:  The kinetic energy distributions of induced $e^{\pm}$ for a variety
of initial $\gamma$ energies.

The expected Evis of a $\gamma$ is expressed as the weighted sum of the
$e^{\pm}$ visible energies as

${\rm E}^{\gamma}_{\rm vis}={\rm E}^{\gamma}_{\rm
dep}\times\frac{\int_{0}^{{\rm E}^{e^{\pm}}_{\rm max}}P({\rm
E}^{e^{\pm}})\times{\rm E}^{e^{\pm}}\times f_{\rm ScintNL}^{e-}({\rm
E}^{e^{\pm}})d{\rm E}^{e^{\pm}}}{\int_{0}^{{\rm E}^{e^{\pm}}_{\rm max}}P({\rm
E}^{e^{\pm}})\times{\rm E}^{e^{\pm}}d{\rm E}^{e^{\pm}}},$ (5)

where ${\rm E}^{e^{\pm}}_{\rm max}$ is the maximum kinetic energy of the
induced $e^{\pm}$, $P({\rm E}^{e^{\pm}})$ is the number of $e^{\pm}$ with the
kinetic energy ${\rm E}^{e^{\pm}}$, and $f_{\rm ScintNL}^{e-}$ is the $e^{-}$
scintillation nonlinearity. The denominator is a normalization accounting for
a $O(0.2\%)$ missing energy in the simulation due to the production
thresholds.

####  3.1.3 Scintillation nonlinearity model of the $e^{+}$

A positron deposits its kinetic energy in a similar way as $e^{-}$, via the
ionization and excitation in the LS molecules. Following the calculation
method in Ref. [26], the $dE/dx$ differences between $e^{-}$ and $e^{+}$ are
expected to yield a negligible ($<$0.05%) effect on scintillation
nonlinearity. Thus the difference is safely ignored. Most of the positrons
annihilate at rest, generating two 0.511 MeV $\gamma$s. The expected deposited
energy of $e^{+}$ is its kinetic energy ${\rm T}^{e+}$ plus the two $\gamma$s,
thus:

${\rm E}_{\rm dep}^{e+}={\rm T}^{e+}+2\times 0.511~{}{\rm MeV}~{}.$ (6)

There are two phenomena that could introduce a bias to Eq. 6: annihilation in
flight and formation of positronium. A small fraction of $e^{+}$ annihilates
during flight, generating two $\gamma$s with higher energies. Simulation
suggests that the fraction is about 1% per MeV of kinetic energy. If the
effect is included, the $e^{+}$ nonlinearity would change by less than 0.1%.
To simplify the nonlinearity model, we ignore the annihilation-in-flight
component in the model.

In LS, about half of the $e^{+}$ directly annihilates while the other half
would form positronium which has two spin states: singlet (p-Ps) and triplet
(o-Ps). The singlet has a life time of about 0.5 ns and decays to two 0.511
MeV $\gamma$s. The triplet has a life time of about 142 ns in vacuum and
decays to three $\gamma$s. In LS most of the o-Ps decays to p-Ps due to
chemical reactions, magnetic effects, or by positronium interactions with the
surrounding electrons. The measurements suggests the o-Ps’s life time is about
3.5 ns in LS [27], thus about 2% of o-Ps decay to three $\gamma$s and the
others to p-Ps. The total energy of the three $\gamma$s is 1.022 MeV but the
scintillation nonlinearity is slightly different from the two $\gamma$s case.
The phenomenon introduces a less than 0.05% variation to the energy
nonlinearity of $e^{+}$, which is also safely ignored in this analysis.

In summary, the expected visible energy of $e^{+}$ is expressed as

${\rm E}_{\rm vis}^{e+}={\rm T}^{e+}\times f_{\rm ScintNL}^{e-}({\rm
T}^{e+})+2\times{\rm E}^{\gamma}_{\rm vis}~{}(0.511~{}{\rm MeV}).$ (7)

###  3.2 Electronics nonlinearity

The electronics nonlinearity at Daya Bay is a combined effect from the time
distribution of detected light and the response of the readout electronics. In
this section, the origin and the direct measurement of the nonlinearity are
described in detail, and an improved energy reconstruction with the
electronics nonlinearity corrected at the single channel level is introduced.

####  3.2.1 Origin of the electronics nonlinearity

As mentioned in Sec. 2, the PMT signal is shaped by a CR-(RC)4 circuit. The
time constant of each RC and CR circuit is 25 ns, which means the effective
charge integral window is about 100 ns. PMT pulses within a 100 ns time window
will form one peak after shaping, and the peak value is used for the charge
estimate of these pulses. This charge reconstruction method works well if
there is only one pulse in the readout window, or multiple pulses being fully
overlapped, otherwise the charge is underestimated.

Figure 8 shows two simulated raw PMT waveforms (left) and the ones after
shaping (right). One waveform consists of two simultaneous pulses, and the
other one has two pulses separated by 30 ns. The total true charges of the two
waveforms are the same. The reconstructed charge is obtained with the peak
value of the waveform after shaping, as 7111 and 6753 in the right plot. For
the example with separated pulses, the reconstructed charge is underestimated
by 5%, and larger time separations lead to even larger underestimates of the
charge [28]. Since the LS of Daya Bay is found to have three timing
components, with decay constants of $\sim$5 ns, $\sim$30 ns, and $\sim$150 ns,
the interplay between the LS timing and the electronics response introduces a
10% nonlinearity in the energy region of interest (0 to 12 MeV).

Figure 8:  Left: simulated PMT waveforms based on the FADC measurements, with
two simultaneous pulses (black solid line) and two equal pulses separated by
30 ns (red dashed line). The total charges of the two waveforms are the same.
Right: waveforms after the CR-(RC)4 shaping circuit in the Daya Bay FEE. The
peak values are used to reconstruct the total charges, which is underestimated
for the case with two separated pulses [28].

####  3.2.2 Direct measurement of the electronics nonlinearity

To directly measure the electronics nonlinearity, a full FADC readout system
was installed on EH1-AD1 in Dec. 2015 and acquired data simultaneously with
the FEE readout system. PMT signals were split, then sent to the FEE and FADC
system after 1 and 4 times amplification, respectively. The two systems shared
the same trigger signal, and the FADC recorded PMT waveforms in a 1-$\mu$s
readout window at 1 GHz with a 10-bit resolution. Details of the FADC system
can be found in Ref. [28].

The difficulty in the FADC waveform charge reconstruction comes from the PMT
signal overshoot which is introduced by the AC circuit and has a $\sim$10%
amplitude to the main pulse, as shown in Fig. 8. The simple waveform integral
would result in a 10% nonlinearity in the estimated charge. A deconvolution
method is used to minimize the dependence on the pulse shape of a single
photoelectron, and to extract the integrated charge with minimum bias. An
electronics simulation is developed and is tuned to have a better than 1%
agreement with data. The nonlinearity in the reconstructed charge is estimated
to be less than 1% with the simulation, as shown in Fig. 9. Details of the
waveform reconstruction and simulation validation are also reported in Ref.
[28].

Figure 9:  Residual nonlinearity in the FADC waveform reconstruction of a
single channel, estimated with an electronics simulation tuned to data.

To compare the energies measured by the two electronics systems, similar
energy reconstruction strategies are applied, including the single p.e.
calibration, energy scale calibration with neutron capture on gadolinium and
the nonuniformity correction. Figure 10 shows the measured nonlinearity of the
FEE readout, which is defined as the ratio of reconstructed energies given by
the FEE and FADC readouts. A similar exponential shape is also found in the
Daya Bay simulation. The electronics nonlinearity is parameterized as

$\begin{split}&f_{\rm ElecNL}({\rm E}_{\rm
vis},~{}\alpha,~{}\tau)\equiv\frac{{\rm E}_{\rm rec}}{{\rm E}_{\rm vis}}\\\
&=1+\alpha\times\left(e^{-\frac{{\rm E}_{\rm
vis}}{\tau}}-e^{-\frac{8.05~{}{\rm MeV}}{\tau}}\right),\end{split}$ (8)

where $\alpha$ determines the amplitude of the nonlinearity, and $\tau$ sets
the energy dependence. The electronics nonlinearity is set to 1 at the
reference energy (8.05 MeV) of neutron capture on gadolinium. With the
parameterized electronics nonlinearity, the full nonlinearity model of
$e^{\pm}$ and $\gamma$ can be expressed as

$\begin{split}&f_{\rm FullNL}({\rm E}_{\rm
dep},~{}A,~{}kB,~{}k_{C},~{}\alpha,~{}\tau)=\frac{{\rm E}_{\rm rec}}{{\rm
E}_{\rm dep}}\\\ &=\frac{{\rm E}_{\rm vis}}{{\rm E}_{\rm dep}}\times f_{\rm
ElecNL}({\rm E}_{\rm vis})~{},\end{split}$ (9)

where the Evis of $e^{-}$, $e^{+}$, and $\gamma$ are given in Eq. 4, Eq. 7,
and Eq. 5, respectively.

To propagate the residual nonlinearity in the FADC’s single channel charge
reconstruction (Fig. 9) to the detector level, the reconstructed charge of
each FADC channel is corrected with this curve, then the energy reconstruction
strategy is applied. The comparison of the reconstructed energies between the
corrected FADC and FEE results in an about 0.2% variation shown as the red
squares in Fig. 10. The variation is assumed as the systematic uncertainty of
the measured electronics nonlinearity.

Figure 10:  Upper panel: the measured electronics nonlinearity (black dots) in
the Gd-LS volume, defined as the ratio of the energy given by the FEE system
(EFEE) to the one given by FADC system (EFADC). The red squares represent the
variation if the simulated residual nonlinearity in the FADC waveform charge
reconstruction in Fig. 9 is propagated to the full detector measurement. The
bottom panel shows the comparison between them.

####  3.2.3 Nonlinearity correction at the single channel level

The PMT charge pattern varies with the position of a particle interaction
within the detector, primarily from the geometric acceptance of the PMTs.
Convolving the nonlinearity in the FEE charge reconstruction, a position
dependence of the detector level electronics nonlinearity is expected. In data
the electronics nonlinearity of events in three cylinder or hollow cylinder
volumes with different radiuses and a 3-meters height is compared, as the
triangles in Fig. 11. An about 1% position dependence is found for events with
energies less than 1.5 MeV. The behavior is well reproduced in the simulation.

Figure 11:  Detector level electronics nonlinearity for the events in three
cylinder or hollow cylinder volumes, without (triangles) and with (dots) the
correction at channel level. The majority of the position dependence is
removed with the correction. See the text for detailed discussions.

To remove this effect, the FEE nonlinearity is further corrected at the
channel level, yielding a new reconstructed energy. The variation in the
nonlinearity of the 192 channels of EH1-AD1 is less than 1%, and a correction
curve is based on the average nonlinearity of them, as shown in Fig. 12. The
curve is assumed to be the same for all the channels of all ADs. The energy
reconstruction is modified as

${\rm E}_{\rm rec}=\left(\sum_{i}\frac{F_{\rm elec}({\rm
Q}_{i})}{\overline{{\rm Q}}^{\rm SPE}_{i}(t)}\right)\cdot\frac{f_{\rm
act}(t)}{N^{\prime}_{\rm PE}(t)}\cdot f^{\prime}_{\rm pos}(\mathbf{r}_{\rm
rec},t),$ (10)

where $F_{\rm elec}$ is the correction shown in Fig. 12. The detector energy
scale, $N^{\prime}_{\rm PE}$, increases by about 8% compared to the one
without correction. The nonuniformity correction, $f^{\prime}_{\rm pos}$, has
the same functional form as the one in Eq. 2 but with different coefficient
values. After the channel level correction, the new reconstructed energy is
also compared to the FADC energy as shown in Fig. 11. More than 80% of the
detector level electronics nonlinearity is removed, as well as majority of the
position dependence.

Figure 12:  The correction for the single channel nonlinearity, which is
defined as the ratio of the charge reconstructed by FEE to the one
reconstructed by FADC. The correction is derived from the average of the 192
channels of EH1-AD1.

In the following the reconstructed energy is without the single channel
correction unless otherwise specified, because it was used in the published
Daya Bay results [15, 16, 17]. The best-fit nonlinearity model of the energy
with the single channel correction will also be given in Sec. 5.

##  4 Inputs of the energy nonlinearity calibration

At Daya Bay, ten $\gamma$ calibration points, a continuous $\beta$ spectrum
from 12B decays, and the measured electronics nonlinearity are utilized to
determine the energy nonlinearity model. The electronics nonlinearity has been
described in Sec. 3.2. This section discusses the measurements, predictions,
and the systematic uncertainties of the others.

###  4.1 The $\gamma$ calibration points

The $\gamma$s come from the deployed 68Ge, 137Cs, 54Mn, 60Co, 40K sources, and
the naturally occurring 208Tl from the surrounding materials. For the deployed
sources, the accompanying $\beta$ particle is absorbed by the source
enclosure. There are also several $\gamma$s from the three neutron calibration
sources based on the ($\alpha,~{}n$) reaction of Pu-13C, 241Am-13C, and
241Am-9Be. The $\gamma$ energies and the enclosure materials are summarized in
Table 1.

####  4.1.1 The deployed $\gamma$ calibration sources

During physics data taking, the 68Ge, 60Co, and 241Am-13C sources are stored
in the ACUs and deployed weekly into the detector. The other sources were
utilized during special calibration campaigns in the summer of 2012 and
January 2017. The detector energy scale is calibrated with the 60Co source
before and after the deployment of any other sources, resulting in an energy
scale stability better than 0.1%.

The measured spectra of the deployed $\gamma$ calibration sources are obtained
by subtracting the spectra taken in the immediately preceding and following
runs without the source. Rejections of events from PMT flashers and muon vetos
follow the same methods used in the $\overline{\nu}_{e}$ analysis [15]. The
spectrum of a combined source of 137Cs and 60Co is shown in Fig. 13. The
shoulders (green) to the left from the peaks (red) are from the $\gamma$
energy loss in the source enclosure and they are well reproduced in the
simulation. In the fit, the full energy peak is modeled with a Gaussian
function and the shoulder is described by the simulated shape with a free
normalization. The simulation shows that such a procedure reduces the bias of
the peak energy to less than 0.2%. If the spectrum is fit with empirical
functions, which cannot describe the shoulder in a correct way, the peak could
have bias at a level of percent.

Figure 13:  An example of the fit to the 137Cs (left peak) and 60Co (right
peak) combined source. The shoulders in green are due to the $\gamma$ energy
loss in the source enclosure. The full energy peaks in red are modeled with
Gaussian functions.

####  4.1.2 The deployed neutron calibration sources

The $\gamma$s from the neutron calibration sources are accompanied by
neutrons, forming prompt and delayed signal coincidence pairs, which can be
used to select the $\gamma$. The 241Am-9Be source with the reaction of
9Be($\alpha,~{}n$)12C, produces a 4.44-MeV $\gamma$ from de-excitation of the
first excited state of 12C, while the 241Am-13C and Pu-13C sources with the
reaction 13C($\alpha,~{}n$)16O, can produce a 6.13-MeV $\gamma$ from the decay
of the second excited state of 16O. In the case of the 241Am-13C source a
1-$\mu$m gold foil was placed between the 241Am and 13C to absorb enough of
the $\alpha$ energy to inhibit the production of $\gamma$s from the 16O
excited states because the source is stored in the ACU during physics data
taking [12].

The energy deposited by the 4.4-MeV $\gamma$ is contaminated by the deposited
energy of recoil protons produced by the energetic ($>$1 MeV) neutrons
generated in the ($\alpha,~{}n$) reaction from the AmBe source. Similarly, the
4.44-MeV $\gamma$ from the excited 12C by high energy neutrons, is also
contaminated by the neutron scattering on protons before the excitation. For
this reason there is not a clean 4.44-MeV $\gamma$ available in the analysis.
On the other hand, the neutron accompanying the 6.13-MeV $\gamma$ from the de-
excitation of 16O has less than 100 keV kinetic energy. Considering the
quenching of recoil protons, the correction to Erec is only 0.5%. A 100%
relative uncertainty is assigned to this correction.

The neutron capture on gadolinium releases several $\gamma$s with a total
energy of either 7.94 MeV (for capture on 157Gd) or 8.54 MeV (on 155Gd). There
are several models and measurements to describe the $\gamma$ energy spectra,
such as the model in Geant4, the model based on the Nuclear Data Sheet, and
measurements by several groups [16, 29]. Large discrepancies are found among
these models and measurements. Detailed discussion can be found in Ref. [30].
In this analysis the gadolinium capture is not used.

A 241Am-13C source with a neutron emission rate of $\sim$59 s-1 was
temporarily installed on the lid of EH3-AD2 for more than 10 days in the
summer of 2012 [15]. The neutron captures on a nucleus (Fe, Cr, Ni) in the
steel are observed, as shown in Fig. 14. The spectrum is fitted with an
empirical background shape, and the third peak of the neutron capture on 56Fe
is used to constrain the energy nonlinearity in the high energy region. Since
the neutron capture concentrates at the top of the detector, the residual
nonuniformity is examined and found to be less than 0.5% with different vertex
cuts. A conservative 1% uncertainty is assigned to the Erec of this $\gamma$,
after considering the background modeling and residual nonuniformity.

Figure 14:  Spectrum of neutron capture on a nucleus in the stainless steel. The neutrons are from special calibration runs with a $\sim$59 Hz 241Am-13C source placed on the lid of EH3-AD2 in 2012. The subscripted in the legend is the true energy, and the corresponding reconstructed energy is also shown in the unit of MeV. The third peak in this plot is used in this study. Table 1: The $\gamma$ calibration points used in this analysis. A ${}^{\prime}+^{\prime}$ or ${}^{\prime}2\times^{\prime}$ in column 2 indicates two $\gamma$s are produced by the source. For sources with $\gamma$s with different branching ratios, the ratios are given in parentheses. An optical shadowing (OS) correction is applied to take into account absorption of optical photons by the source enclosures. The correction size is estimated with simulation and the corrected Erec is used in the nonlinearity model fit. Relative uncertainties of corrected Erec are uncorrelated among the $\gamma$ calibration points. Source |  Energy of $\gamma$(s) | Enclosure material |  Erec | OS correction |  Corrected Erec | Uncertainty
---|---|---|---|---|---|---
137Cs | 0.662 MeV | Teflon + SS | 0.642 MeV | 0.4% | 0.649 MeV | 0.5%
54Mn | 0.834 MeV | Teflon + SS | 0.827 MeV | 0.2% | 0.835 MeV | 0.5%
68Ge |  2$\times$0.511 MeV | Acrylic + SS | 0.958 MeV | 0.75% | 0.966 MeV | 0.5%
40K | 1.461 MeV | Teflon + SS | 1.475 MeV | 1.2% | 1.493 MeV | 0.5%
n-H | 2.223 MeV | Acrylic + SS | 2.292 MeV | 0.36% | 2.300 MeV | 0.5%
60Co | 1.173 + 1.332 MeV | Acrylic + SS | 2.474 MeV | 0.65% | 2.490 MeV | 0.5%
208Tl | 2.614 MeV | N/A | 2.701 MeV | N/A | 2.701 MeV | 0.5%
n-12C |  | 4.945 MeV (68%)
---
1.261 + 3.683 MeV (32%)
Acrylic + SS | 5.118 MeV |  $<$0.1% | 5.119 MeV | 0.8%
16O∗ | 6.130 MeV | Acrylic + SS | 6.312 MeV | 0.6% | 6.350 MeV | 0.8%
n-56Fe | 7.637 MeV | N/A | 7.940 MeV | N/A | 7.940 MeV | 1.4%

####  4.1.3 Correction for the optical shadowing of source enclosures

As discussed in Sec. 2, the calibration source enclosures and weights absorb
scintillation photons up to 1.5% depending on the $\gamma$ energy and the
surface reflectivity. The absorption fractions from the simulation are used to
correct the reconstructed energies as listed in Table 1. To anchor the
simulation, a special calibration was carried out in Jan. 2017. Several 60Co
sources, stored in a stainless steel chamber and with different enclosures
(PTFE of high reflectivity, acrylic, greenish teflon, acrylic without the
bottom weight) were deployed, absorbing $\leq$0.1%, 0.65%, 1.22%, and 0.45%
photons, respectively, as determined from the simulation. Comparison of the
data and simulation is shown in Fig. 15, and a better than 0.2% agreement is
found. When varying the electromagnetic models in the Geant4 simulation, a
0.2% variation is found in the absorption fractions of scintillation photons
at different $\gamma$ energies. Thus a conservative 0.3% uncertainty is
assigned to the correction.

Figure 15:  Results of special calibration utilizing 60Co sources stored in a
stainless steel chamber and enclosed by PTFE of high reflectivity, acrylic,
greenish teflon. A 60Co source enclosued by acrylic but without the bottom
weight is also deployed.

Table 1 summarizes the $\gamma$ calibration points used in this analysis. The
uncertainties in the table are the sum of the statistical uncertainty, the
optical shadowing correction uncertainty, and the one from spectrum fitting.
They are assumed to be uncorrelated among the calibration points. Large
uncertainties are assigned to the $\gamma$s from n-12C, 16O∗ and n-56Fe due to
the empirical background model. The 6.13-MeV $\gamma$ from 16O∗ de-excitation
has an additional 0.5% uncertainty because of the contamination due to prompt
neutron scattering. All the $\gamma$ calibration points except the n-56Fe have
a correlated 0.3% uncertainty taking into account the residual nonuniformity
between point-like $\gamma$ sources, which preferentially illuminate the
detector center, and IBD events over the full target volume.

Given the $\gamma$ energy, the predicted reconstructed energy Erec is
determined from Eq. 9. If there are multiple $\gamma$s in one event, such as
the 1.17-MeV and 1.33-MeV $\gamma$s of 60Co , Evis is the branching ratio
weighted sum of visible energies of the multiple $\gamma$s.

###  4.2 Continuous $\beta$ spectrum

####  4.2.1 The measured $\beta$ spectrum from 12B decays

The average $\beta$ energy and $Q$-value of 12B are about 6.4 MeV and 13.4
MeV, respectively. The $\beta$ spectrum provides a direct constraint on the
nonlinearity model of $e^{-}$, especially in the high energy region. An
enhanced 12B sample is selected in a (5 ms, 50 ms) time window after a
showering muon, while also requiring a reconstructed vertex in the Gd-LS
volume. Showering muons are selected by AD events with Erec larger than 2.5
GeV. The background is estimated with a (505 ms, 550 ms) window after the
muon. The start time of the signal window is chosen to eliminate the influence
of spallation neutron capture on 1H, and the end time of the signal window is
based on the 20.2 ms half-life of 12B.

Figure 16 shows the 12B signal and background spectra in EH1-AD1. The event
excess in 14 MeV to 18 MeV is from 12N decay, which is a $\beta^{+}$ isotope
with a 11 ms half-life and a $Q$-value of 17.3 MeV. The contribution from 12N
is estimated to be $(2.7\pm 0.8)$% and is included in the fitting of the
energy nonlinearity model. In the four ADs of EH1 and EH2, a total of about
470,000 events are observed in the 3.5 MeV to 18 MeV energy range and are used
in this analysis. The measured spectrum is provided in the Supplemental
Material [31].

Figure 16:  Beta spectrum of 12B in EH1-AD1 (blue dots) obtained by
subtracting backgrounds (black dash line) from an enhanced sample (red solid
line) with the (5 ms, 50 ms) window after a showering muon. The background is
estimated with the (505 ms, 550 ms) window after the muon.

####  4.2.2 Prediction of the 12B $\beta$ spectrum

Table 2 shows the primary decay channels of 12B [32]. The first channel is to
the ground state of 12C, the second to the first excited state of 12C* (4.44)
which releases a 4.44-MeV $\gamma$, and the third to the second excited state
of 12C*(7.65) which converts to three $\alpha$ particles. The other channels,
with less than 0.1% contribution, are safely ignored. The intensity of the
second channel is precisely measured with the $\beta-\gamma$ coincidence, and
that of the third channel is measured by detecting the unbound $\alpha$s. The
intensity of the first channel is obtained by subtracting all the other
observed decay channels from unity. Some measurements suggest the third
channel has a branching ratio larger than 0.54%. We inflate the uncertainty of
this branching ratio to 1%, to cover these discrepancies.

Table 2: Primary decay channels of 12B ($J^{\pi}=1^{+}$) [32]. |  Energy of 12C
---
excited states
Branching ratio (%) | $J^{\pi}$ |  E${}_{{\mathrm{max}}}^{\beta}$
Ground state |  98.300 $\pm$0.028 | $0^{+}$ | 13.37 MeV
4.44 MeV |  1.182$\pm$0.019 | $2^{+}$ | 8.93 MeV
7.65 MeV |  |  0.54$\pm$0.02
---
1.3$\pm$0.4
1.1$\pm$0.3
$0^{+}$ | 5.71 MeV

The $\beta$ decay of 12B is an allowed transition of the Gamow-Teller type.
The spectrum prediction follows the discussion in Refs. [33, 34]. We include
the shape corrections due to the screening effect, the electromagnetic and
weak interaction finite-size effect, and the weak magnetism effect. Since the
exchange of the virtual photon does not affect the $\beta$ shape and the
emitted real photon is detected in the LS detector, the radiative correction
is not included. The primary shape uncertainty comes from the weak magnetism
correction which is a linear correction with a coefficient of (0.48$\pm$0.24)
%/MeV, as shown in Fig. 17.

Figure 17:  Top panel: Predicted $\beta$ spectrum of 12B decaying to the
ground state of ${}^{12}C$ (blue solid line, nearly obscured by the red dashed
line), and the one if the weak magnetism (WM) correction is varied by
1$\sigma$ (red dashed line). Bottom panel: The ratio of the two predictions.

The predicted $\beta$ spectrum of each decay channel is calculated in 10 keV
bins and provided in the Supplemental Material [31]. The $e^{-}$ scintillation
nonlinearity $f_{\rm ScintNL}^{e-}$ in Eq. 4 is applied to each energy bin,
resulting in a visible energy spectrum. To the second channel, Evis of the
4.44-MeV $\gamma$ is added to the spectrum. To the third channel, a
0.5$\pm$0.1 MeV visible energy is added to take into account the subsequent
disintegration of 12C to 3 $\alpha$s. The electronics nonlinearity is applied
to the visible energy spectrum. Then the energy resolution function is
convolved. When summing over the three decay channels, the intensity of the
first channel is calculated by subtracting branching ratios of the second and
third channels from unity. Finally the summed spectrum is rebinned to 0.25
MeV/bin to compare with data.

Besides 12B, the $\beta$ spectra from 212Bi, 214Bi, and 208Tl could constrain
the energy nonlinearity of low energy electrons (less than 3 MeV). But most of
decay channels are first forbidden nonunique transitions. Prediction of the
$\beta$ spectra is difficult since the structure of the initial and final
nuclear states need to be considered. Generally the first forbidden nonunique
transitions use predictions of the corresponding allowed ones. The difficulty
is to verify how well the simplification works and how to assess uncertainties
in the prediction. Thus these three isotopes are not used in this analysis.

##  5 Results of the energy nonlinearity calibration

To determine the five free parameters of the energy nonlinearity model in Eq.
9, a $\chi^{2}$ function with nuisance parameters is constructed for a
combined fit on the ten $\gamma$ calibration points, the $\beta$ spectrum from
12B decays, and the directly measured electronics nonlinearity. This section
will introduce the definition of the $\chi^{2}$ function, and the fitted
results of the nonlinearity model.

###  5.1 Statistical methods

A $\chi^{2}$ test statistic with five free parameters is defined as

$\begin{split}\chi^{2}&(A,~{}kB,~{}k_{C},~{}\alpha,~{}\tau)=\chi^{2}_{\rm
stat}+\chi^{2}_{\rm syst}(\bm{\nu}),\\\ \chi^{2}_{\rm
stat}&=\sum_{i=1}^{58}\frac{(P^{B}_{i}-M^{B}_{i})^{2}}{(\sigma_{B}^{i})^{2}}+\sum_{i=1}^{10}\frac{(P^{\gamma}_{i}\times(1+\nu_{\gamma})-M^{\gamma}_{i})^{2}}{(\sigma_{\gamma}^{i})^{2}}\\\
&+\sum_{i=1}^{18}\frac{(P^{\rm elec}_{i}\times(1+\nu_{\rm elec}\times
a_{i})-M^{\rm elec}_{i})^{2}}{(\sigma_{\rm elec}^{i})^{2}},\\\ \chi^{2}_{\rm
syst}(\bm{\nu})&=\sum_{j=1}^{3}(\nu_{\rm
WM}^{j})^{2}+\sum_{j=2}^{3}\left(\frac{\nu^{j}_{\rm
BR}-\nu^{j~{}~{}\rm{Initial}}_{\rm BR}}{\sigma^{j}_{\rm BR}}\right)^{2}\\\
&+\frac{(\nu_{{\rm N12/B12}}-\nu^{\rm Initial}_{{\rm
N12/B12}})^{2}}{\sigma^{2}_{{\rm
N12/B12}}}+\frac{\nu^{2}_{\gamma}}{\sigma^{2}_{\gamma}}+\nu_{\rm
elec}^{2},\end{split}$ (11)

for comparison of the measurement ($M$) to the prediction ($P$). The
measurements consist of: 1) $M^{\gamma}_{i}$ ($\sigma_{\gamma}^{i}$), the ten
$\gamma$ calibration points and their uncorrelated uncertainties taken from
Table 1, 2) $M^{B}_{i}$ ($\sigma_{B}^{i}$), the reconstructed energy spectrum
of 12B in 58 energy bins and their statistical uncertainties as discussed in
Sec. 4.2, and 3) $M^{\rm elec}_{i}$ ($\sigma_{\rm elec}^{i}$), the measured
electronics nonlinearity in 18 energy bins and their statistical uncertainties
as shown in Fig. 10. The predicted values and uncertainties of these
quantities are described in Sec. 4.1, Sec. 4.2, and Eq. 8, respectively.

The $\chi^{2}_{\rm syst}(\bm{\nu})$ penalizes the total $\chi^{2}$ based on
deviations of the systematic nuisance parameters from their expected values.
The nuisance parameters consist of the correlated uncertainty among the
$\gamma$ rays ($\nu_{\gamma}$) except for the one from neutron capture on
56Fe, the uncertainty of 12N contribution in the 12B spectrum ($\nu_{{\rm
N12/B12}}$), the uncertainty of the weak magnetism correction in the 12B
prediction ($\nu_{\rm WM}^{j}$), the branching ratio uncertainties of the
second and third 12B decay channels ($\nu_{\rm BR}^{j}$), and the uncertainty
of the measured electronics nonlinearity due to the uncertainty of the single
channel’s charge reconstruction ($\nu_{\rm elec}$).

Because the branching ratio uncertainty of the 12B first decay channel is
fully correlated to those of the second and third channels, it is not
considered a nuisance parameter. The $\nu_{\rm elec}$ is deduced to follow a
standard Gaussian distribution by the coefficients $a_{i}$, which is the
1$\sigma$ variation of the electronics nonlinearity. The $\nu_{\rm WM}^{j}$ is
assumed to be uncorrelated among the three decay channels, and is also deduced
to follow a standard Gaussian distribution. Table 3 summarizes the free and
nuisance parameters, including their definitions, initial values and
uncertainties, and the values at the best-fit point.

Table 3: Summary of nuisance parameters in the study, including the initial values, estimated uncertainties, and the values at the best-fit point. The $\nu^{j}_{\rm WM}$ and $\nu_{\rm elec}$ have been deduced to follow the standard Gaussian distribution. Nuisance parameter | Definition | $\nu^{\rm Initial}$ | Estimated uncertainty | Best-fit value
---|---|---|---|---
$\nu_{\gamma}$ |  |  Correlated uncertainty among $\gamma$ points
---
accounting for the residual nonuniformity
0 | 0.003 |  3$\times 10^{-4}$
$\nu_{{\rm N12/B12}}$ |  |  Ratio of the 12N contribution
---
in the measured 12B spectrum
0.027 | 0.008 | 0.034
$\nu^{j}_{\rm WM}$ |  | Uncertainty of the weak magnetism
---
correction in 12B prediction
of the three decay channels
| 0
---
0
0
| 1
---
1
1
| -1.43
---
-0.02
-0.02
$\nu^{j}_{\rm BR}$ |  |  Uncertainty of the 2nd and 3rd decay
---
channels of 12B
| 0.012
---
0.005
|  2.2$\times 10^{-4}$
---
0.01
| 0.012
---
0.017
$\nu_{\rm elec}$ | Uncertainty of the measured electronics NL | 0 | 1 | -0.193

###  5.2 Results of the nonlinearity model calibration

The $\chi^{2}$ function in Eq. 11 is minimized, resulting in a
$\chi_{min}^{2}/$NDF = 70.6/(86-5) = 0.87. The nonlinearity model of $e^{+}$,
which is the prompt signal in the $\overline{\nu}_{e}$ detection, is derived
from the best-fit parameters, as shown in Fig. 18. The depicted uncertainty
band corresponds to the models consistent with the calibration data within 68%
C.L. (with a $\Delta\chi^{2}<5.89$ compared to $\chi_{min}^{2}$). A precision
better than 0.5% is achieved for prompt energies larger than 2 MeV. The
precision is limited by the systematic uncertainties associated with the
$\gamma$ rays at energies below 3 MeV, and by the 12B statistics at higher
energy. As discussed in Sec. 3.1.3, the in-flight annihilation and the 3
$\gamma$ decay from o-Ps have a $<$0.1% impact on the nonlinearity model. The
best-fit full nonlinearity and scintillation nonlinearity of $e^{-}$,
$e^{+}$and $\gamma$ are provided in the tabular form as Supplemental Materials
[31].

Figure 18:  Relationship between the reconstructed and true prompt energy, which is the sum of positron kinetic energy and two 0.511 MeV annihilation $\gamma$ rays. The previous model in Ref.[15] is shown for comparison (colour on-line). Table 4: Summary of free parameters in the $\chi^{2}$ minimization. The best-fit values and uncertainties are provided, and the correlation matrix is shown in Fig. 20. Free parameter | Definition | Best-fit value | Best-fit uncertainty
---|---|---|---
$A$ | Absolute energy scale | 1.013 | 0.006
$kB$ | Birks’ coefficient |  15.2$\times 10^{-3}$ g/cm2/MeV |  2.7$\times 10^{-3}$ g/cm2/MeV
$k_{C}$ | Absolute Cherenkov contribution | 0.019 | 0.009
$\alpha$ | Amplitude of the electronics nonlinearity | 0.104 | 0.002
$\tau$ | Decay constant of the electronics NL | 2.30 MeV | 0.06 MeV

The energy nonlinearity model agrees with the calibration data well, as shown
in Fig. 19. The values of best-fit parameter are provided in Table 4, and Fig.
20 shows the correlation matrix of the five parameters. The Birks’
coefficient, $kB$, has a large positive correlation with the absolute energy
scale $A$ and a negative correlation with the Cherenkov contribution $k_{C}$.
The correlation between the LS and electronics nonlinearity, $\alpha$ and
$\tau$, is weak due to the constraints from the directly measured electronics
nonlinearity. It is not practical to compare the best-fit values with those of
other experiments due to the dependence on simulation parameters.

Figure 19:  Top: comparison of the reconstructed 12B energy spectra between
the data and the prediction with best-fit nuisance parameters. Bottom: the
estimated LS nonlinearity (red line), and the measured from $\gamma$ rays. The
error bars of the 12B spectrum represents the statistical uncertainty, while
those of the $\gamma$ calibration points are taken from Table 1. Figure 20:
Correlation coefficients of the five parameters. The FADC measurement helps to
decouple the LS and electronics nonlinearity.

Cross validations were performed by removing the $\gamma$ calibration points
one by one, the measured electronics nonlinearity or the 12B from the fit.
Consistent results are found as shown in Fig. 21. Figure 21 is the
nonlinearity model without 12B, yielding three to eight times larger
uncertainties for prompt energies greater than 4 MeV. Although removing the
measured electronics nonlinearity (Fig. 21) has little impact on the model
precision, the correlation between the LS and electronics nonlinearity
increases by a factor of 5 to 10. The quenching curves produced by the
different methods in Sec. 3.1.1 are also tested in the fit, and the overall
behavior of the $e^{+}$ nonlinearity is unchanged at all energies within
0.05%.

Figure 21:  Energy nonlinearity models of positrons without constraints from
the 12B spectrum (a) or the measured electronics nonlinearity (b).

The energy nonlinearity model of $e^{+}$ for the reconstructed energy with the
electronics nonlinearity corrected at the single channel level is shown in
Fig. 22 and is provided as Supplemental Material [31]. The majority of
electronics nonlinearity has been removed. The scintillation nonlinearity
derived from the two reconstructed energies converges to 0.1%.

Figure 22:  Energy nonlinearity model of positrons with the electronics
nonlinearity corrected at the channel level. The majority of the electronics
nonlinearity is removed.

##  6 Summary

We present a precise energy calibration at Daya Bay. The nonlinearity in the
existing electronics readout system is measured with a full FADC readout
system. The measurements and systematic uncertainty estimates of the
calibration data, including $\gamma$ rays and the 12B spectrum, are described
in detail. Less than 0.5% uncertainty in the energy nonlinearity calibration
is achieved for positrons with kinetic energies greater than 1 MeV. There is
no significant deviation in the nonlinearity between detectors, so the model
is used for all eight ADs. A new reconstructed energy, with the electronics
nonlinearity corrected at the channel level, has been implemented in the Daya
Bay data processing.

##  7 Acknowledgements

Daya Bay is supported in part by the Ministry of Science and Technology of
China, the U.S. Department of Energy, the Chinese Academy of Sciences, the CAS
Center for Excellence in Particle Physics, the National Natural Science
Foundation of China, the Guangdong provincial government, the Shenzhen
municipal government, the China General Nuclear Power Group, Key Laboratory of
Particle and Radiation Imaging (Tsinghua University), the Ministry of
Education, Key Laboratory of Particle Physics and Particle Irradiation
(Shandong University), the Ministry of Education, Shanghai Laboratory for
Particle Physics and Cosmology, the Research Grants Council of the Hong Kong
Special Administrative Region of China, the University Development Fund of The
University of Hong Kong, the MOE program for Research of Excellence at
National Taiwan University, National Chiao-Tung University, and NSC fund
support from Taiwan, the U.S. National Science Foundation, the Alfred P. Sloan
Foundation, the Ministry of Education, Youth, and Sports of the Czech
Republic, the Charles University Research Centre UNCE, the Joint Institute of
Nuclear Research in Dubna, Russia, the National Commission of Scientific and
Technological Research of Chile, and the Tsinghua University Initiative
Scientific Research Program. We acknowledge Yellow River Engineering
Consulting Co., Ltd., and China Railway 15th Bureau Group Co., Ltd., for
building the underground laboratory. We are grateful for the ongoing
cooperation from the China General Nuclear Power Group and China Light and
Power Company.

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Yong-Hyeok Lee, Namhyun Cho

# PhonMatchNet: Phoneme-Guided Zero-Shot Keyword Spotting for User-Defined
Keywords

###### Abstract

This study presents a novel zero-shot user-defined keyword spotting model that
utilizes the audio-phoneme relationship of the keyword to improve performance.
Unlike the previous approach that estimates at utterance level, we use both
utterance and phoneme level information. Our proposed method comprises a two-
stream speech encoder architecture, self-attention-based pattern extractor,
and phoneme-level detection loss for high performance in various pronunciation
environments. Based on experimental results, our proposed model outperforms
the baseline model and achieves competitive performance compared with full-
shot keyword spotting models. Our proposed model significantly improves the
EER and AUC across all datasets, including familiar words, proper nouns, and
indistinguishable pronunciations, with an average relative improvement of 67%
and 80%, respectively. The implementation code of our proposed model is
available at https://github.com/ncsoft/PhonMatchNet.

Index Terms: keyword spotting, user-defined, zero-shot, open-vocabulary

##  1 Introduction

Keyword spotting (KWS) in speech processing enables the identification of
specific keywords in audio signals. Generally, conventional KWS methods
require a significant amount of labeled data for training, which is a
limitation in scenarios in which target keywords are rare, specialized, or
user-defined [1]. Recently, as user-friendly services, such as smart speakers,
AI assistants, and personalized digital humans have become widespread, the
demand for personalized technology that allows consumers to define their
keywords instead of using pre-determined keywords set by manufacturers has
increased.

Zero-shot KWS is key to addressing this challenge, enabling a model to detect
user-defined keywords with no prior training on those keywords. Most user-
defined KWS (UDKWS) models are implemented using the query-by-example (QbyE)
approach, which compares an input speech and a pre-registered speech in latent
space [2, 3, 4]. Despite the success of the QbyE approach, it is limited in
terms of performance and implementation because it compares the input speech
with the pre-enrolled speech.

Recently, a cross-modal correspondence detector (CMCD) that compares speech
and text and can omit speech registration when adding new keywords has been
proposed [5]. However, this method does not properly distinguish pairs of
similar pronunciations, as the similarity between speech and text is
calculated at the utterance level.

In this paper, we propose a novel zero-shot UDKWS model to address these
issues. Our proposed model includes a two-stream speech encoder with a pre-
trained speech embedder [6], self-attention-based pattern extractor [7], and
phoneme-level detection loss to achieve high performance in various
pronunciation environments. We conducted experiments on datasets containing
familiar words [8], proper nouns [9], and indistinguishable pronunciations [5]
to evaluate the effectiveness of our proposed model. Our proposed model
outperforms the baseline model and achieves competitive performance compared
with full-shot keyword spotting models. Particularly, our proposed model
achieved a significant improvement in the equal error rate (EER) and area
under the curve (AUC) across all datasets, with an average relative
improvement of 67% and 80%, respectively.

##  2 Related works

This section examines recent UDKWS methods and related studies for our
proposed method. The most recent model [5] showed degraded performance in
similar pronunciations that are difficult to distinguish owing to the
comparison of speech-text pairs at the utterance level. Furthermore, the audio
encoder cannot properly handle uncommon pronunciations in the training dataset
because it comprises only fully trainable modules. Thus, we employed a pre-
trained speech embedder and phoneme-level detection loss to overcome these
limitations.

Recently, various studies on non-semantic representation that allow a single
embedder model to be used across various target tasks, not only in speech
recognition [10] but also in other areas, such as keyword spotting [6],
speaker identification, and language identification [11] have been presented.
These studies improve models using techniques, such as self-supervised or
unsupervised learning, multi-task learning [12], and knowledge distillation
[13]. Our objective is to obtain embeddings that can show high performance in
general speech situations; therefore, we selected and froze the embedder model
[6] while considering the KWS performance and model size.

In various detection tasks using speech-text [14], visual-text [15], speech-
visual [16], and visual-only [17], temporal location information of targets
significantly assists in achieving high performance. However, providing the
correct labels for every timestamp in sequence data is challenging. Therefore,
connectionist temporal classification (CTC) loss [18], which does not require
explicit alignment information, is widely used. However, unlike speech
recognition in which speech and text labels always match, CTC loss is not
appropriate for UDKWS, where speech and text labels may not match. Therefore,
we propose a method that uses the sameness of speech and text pronunciation as
a label rather than the exact time information.

Figure 1:  Architecture of the proposed model. Boldface and red arrows denote
the unmatched phonemes, labels, and BCE losses. ``TConv'' is the abbreviation
of ``Transposed Convolution.''

##  3 Proposed method

Here, we describe our proposed model architecture and training criterion, as
shown in Figure 1. Our proposed model comprises three sub-modules: an audio
and text encoder with a pre-trained embedder, pattern extractor, and pattern
discriminator. We use two loss functions based on the cross-entropy loss in
the training criterion, which are computed at the utterance
($\mathcal{L}_{utt}$) and phoneme levels ($\mathcal{L}_{phon}$).

###  3.1 Model architecture

Audio Encoder. The audio encoder comprises two feature extractors: a pre-
trained speech embedder [6] with high performance in representing general
pronunciations and a fully trainable feature extractor, which learns
representations of special pronunciations, such as proper nouns. The pre-
trained speech embedder has $775\text{\,}\mathrm{ms}$ windows and computes
96-dimensional feature vectors every $80\text{\,}\mathrm{ms}$. We upsample the
feature and time dimensions using a 1-D transposed convolution with a kernel
size of 5 and stride of 4, along with fully connected layers. The fully
trainable feature extractor comprises two 1-D convolutions with a kernel size
of 3, batch normalization, and ReLU layers. The first convolution layer has a
stride of 2, whereas the others have a stride of 1. The 40-dimensional mel-
filterbank coefficients used as input are extracted every
$10\text{\,}\mathrm{ms}$ with a window of $25\text{\,}\mathrm{ms}$. Finally,
we compute the 128-dimensional feature vectors extracted every
$20\text{\,}\mathrm{ms}$ by adding the two feature vectors. We denote audio
embeddings as $\mathbf{E}^{a}\in\mathbb{R}^{T_{a}\times 128}$, where $T_{a}$
and 128 are the lengths of the audio and embedding dimension, respectively.

Text Encoder. The text encoder, similar to [5], includes a pre-trained
grapheme-to-phoneme (G2P) model111https://github.com/Kyubyong/g2p followed by
a fully connected layer and a ReLU activation function. We extract the G2P
embedding from the last hidden states of the encoder. We denote text
embeddings by $\mathbf{E}^{t}\in\mathbb{R}^{T_{t}\times 128}$, which is the
same as the audio encoder.

Pattern extractor. Considering the characteristics of the KWS task, which
requires maintaining fewer model parameters, our pattern extractor is based on
a self-attention [7, 15] rather than a cross-attention mechanism [16, 19]. As
in [20], during the fusion of multiple modalities, the self-attention method
does not require other modules, unlike other attention mechanisms. The matrix
of attention outputs for a set of queries $Q$ with keys and values packed into
matrices $K$ and $V$ is computed as

$\textrm{Attention}(Q,K,V)=\textrm{softmax}\left(\frac{QK^{T}}{\sqrt{d_{k}}}\right)V.$
(1)

Unimodal embeddings $\mathbf{E}^{a}$ and $\mathbf{E}^{t}$ are concatenated
along the time dimension to prepare the joint embeddings $\mathbf{E}^{j}$:

$\mathbf{E}^{c}=(\mathbf{E}^{a};\mathbf{E}^{t})\in\mathbb{R}^{(T_{a}+T_{t})\times
128}.$ (2)

The concatenated embedding $\mathbf{E}^{c}$ is calculated as joint embeddings
$\mathbf{E}^{j}$ using equation 1:

$\mathbf{E}^{j}=\textrm{Attention}(\mathbf{E}^{c},\mathbf{E}^{c},\mathbf{E}^{c})\in\mathbb{R}^{(T_{a}+T_{t})\times
128}.$ (3)

We used a lower triangular matrix as an attention mask to use only causal
information of the intra-modality.

Pattern discriminator. Our pattern discriminator determines two probabilities,
the match probability of audio and keywords and the audio and phoneme. To
detect an utterance-level matching, we use a single GRU layer with 128
dimensions, taking the joint embeddings $\mathbf{E}^{j}$ along the time
dimension as an input. The output of the last hidden state is fed into a fully
connected layer with a sigmoid function:

$P_{utt}=\sigma(\mathbf{W}^{u}\cdot\textrm{GRU}(\mathbf{E}^{j})+\mathbf{b}^{u})\in\mathbb{R}^{1\times
1}.$ (4)

Similarly, we extract only the phoneme sequences from the sequence of joint
embeddings $\mathbf{E}^{j}$ and feed this into the fully connected layer with
a sigmoid function to detect phoneme-level matching:

$P_{phon}=\sigma(\mathbf{W}^{p}\cdot\mathbf{E}^{j}_{\mathbf{s}}+\mathbf{b}^{p})\in\mathbb{R}^{T_{t}\times
1},$ (5)

where $\mathbf{W}$, $\mathbf{b}$, $\sigma$, and $\mathbf{s}$ denote the
trainable weights and biases of each fully connected layer, the sigmoid
function, and the frame index of $(T_{a},T_{a}+T_{t}]$, respectively.

Table 1: Performance of the baseline, proposed method, and ablations on various datasets. $\dagger$ refers to our implementation of H.-K. Shin et al. $\mathbf{G}$, $\mathbf{Q}$, $\mathbf{LP_{E}}$, and $\mathbf{LP_{H}}$ refer to our test sets from Section 4.1. w/o Phoneme-level Detection Loss represents the case of training using only utterance-level detection loss. w/o Self-attention-based Pattern Extractor means the pattern extractor is composed of cross-modality attention rather than self-attention. w/o Fully trainable Speech Encoder means the audio encoder comprises only a pre-trained speech embedder without convolution layers. We adjust the model parameters of the ablation studies to be the same as the proposed model. Method | Params | EER (%) $\downarrow$ | AUC (%) $\uparrow$
---|---|---|---
$\mathbf{G}$ | $\mathbf{Q}$ | $\mathbf{LP_{E}}$ | $\mathbf{LP_{H}}$ | $\mathbf{G}$ | $\mathbf{Q}$ | $\mathbf{LP_{E}}$ | $\mathbf{LP_{H}}$
† CMCD [5] | 653K | 31.94 | 26.09 | 10.48 | 29.34 | 73.86 | 82.42 | 95.63 | 77.60
Proposed | 655K | 6.77 | 4.75 | 2.80 | 18.82 | 98.11 | 98.90 | 99.29 | 88.52
w/o Phoneme-level Detection Loss | 655K | 7.52 | 6.05 | 3.39 | 19.65 | 97.45 | 98.45 | 99.30 | 88.04
w/o Self-attention-based Pattern Extractor | 11.32 | 18.24 | 3.47 | 20.51 | 95.73 | 90.14 | 99.26 | 87.49
w/o Fully trainable Speech Encoder | 14.34 | 22.90 | 5.13 | 25.77 | 93.91 | 84.19 | 98.71 | 82.53

###  3.2 Training criterion

Our training criterion ($\mathcal{L}_{total}$) comprises two binary-cross
entropy (BCE) losses: an utterance- ($\mathcal{L}_{utt}$) and a phoneme-level
detection losses ($\mathcal{L}_{phon}$),

$\mathcal{L}_{total}=\mathcal{L}_{utt}+\mathcal{L}_{phon}.$ (6)

Utterance-level detection loss. Similar to general KWS methods [21], the
utterance-level detection loss is used to train the similarity between speech
and keywords within a sample. The sample-level ground truth is used with
$P_{utt}$ from Equation 4 to calculate the BCE loss.

Phoneme-level detection loss. We propose the phoneme-level detection loss to
improve the performance of distinguishing similar pronunciations (e.g.,
``friend'' and ``trend'') without using speech and pronunciation alignment
information. The phoneme-level ground truth $\mathbf{y}$ is defined as 1 if
the phoneme sequence of the speech label is the same as that of the keyword
label and 0, otherwise.

$y_{p}=\begin{cases}1&\text{if $y^{t}_{p}=y^{s}_{p}$}\\\
0&\text{otherwise}\end{cases},$ (7)

where $y^{s}_{p}$, $y^{t}_{p}$, and $p$ denote the speech, keyword label, and
phoneme index from 1 to $T_{t}$. As with $\mathcal{L}_{utt}$, the phoneme-
level ground truth $\mathbf{y}$ and prediction $P_{phon}$ are used to
calculate the BCE loss.

##  4 Experiments

Here, we describe the experimental setup, including the datasets, evaluation
metrics, and implementation details for training and testing. Although most of
our experimental settings are similar to those in [5], we employ different
methods to create anchor-negative pairs in certain test sets. An example of
this is listed in Table 2 for clarity.

Table 2: Examples of anchor and negatives of each dataset Dataset | Anchor | Negatives
---|---|---
$\mathbf{G}$ | go |  | yes
---
no
up
down
left
right
on
off
stop
$\mathbf{Q}$ | hi galaxy |  | hey android
---
hey snapdragon
hi lumina

###  4.1 Datasets and metrics

We used three datasets: LibriPhrase [5], Google Speech Commands V1
($\mathbf{G}$) [8], and Qualcomm Keyword Speech dataset ($\mathbf{Q}$) [9],
for training and evaluation. In the training phase, we used the training set
of LibriPhrase and babble noise from the MS-SNSD dataset [22] for robust
detection. Detailed training conditions were similar to that of [5]. During
evaluation, we used the datasets $\mathbf{G}$, $\mathbf{Q}$, LibriPhrase-easy
($\mathbf{LP_{E}}$), and LibriPhrase-hard ($\mathbf{LP_{H}}$).
$\mathbf{LP_{E}}$ and $\mathbf{LP_{H}}$ are datasets reclassified as easy and
difficult to differentiate between anchor and negative pairs in the test sets
of LibriPhrase [5]. We evaluated our proposed models by measuring the EER and
AUC at the sample level on these datasets. Because the datasets $\mathbf{G}$
and $\mathbf{Q}$ do not provide negative pairs, we calculated the EER and AUC
by considering all keywords except the positive pairs as negative pairs among
the candidate keywords for each dataset. Table 2 provides examples of anchor
and negatives in $\mathbf{G}$ and $\mathbf{Q}$.

###  4.2 Implementation details

Our implementation was based on the TensorFlow library. The training criterion
was optimized using the Adam optimizer with the default parameters. The models
were trained for 100 epochs with a fixed learning rate of $10^{-3}$, and the
best model was selected based on performance on the test sets. For training,
we used a single V100 with a batch size of 2048 for a day.

##  5 Results

###  5.1 Comparison with baseline

We re-implemented the baseline model, CMCD, to compare its performance with
that of our proposed model. According to Table 1, our reconstructed CMCD
performed similarly in $\mathbf{LP_{E}}$ and $\mathbf{LP_{H}}$ but showed
decreased performance in $\mathbf{G}$ and $\mathbf{Q}$ datasets compared with
the results of the original study [5]. This can be attributed to the
difference in the method of generating anchor-negative pairs. Although the
method differs from [5], our method can measure the performance of these
models with increased accuracy because it computes one anchor and other
keywords. Our proposed model outperformed the baseline by a significant margin
in all datasets, including familiar words (e.g., ``yes'' and ``no''), proper
nouns (e.g., ``snapdragon'' and ``lumina''), and indistinguishable
pronunciations (e.g., ``friend'' and ``frind''). A relative improvement of 67%
in EER and 80% in AUC was observed on average across all the datasets.

###  5.2 Ablation Study

We evaluated the effectiveness of our proposed methods as shown in Table 1:
phoneme-level detection loss, self-attention-based modality fusion, and two-
stream speech encoder.

Effectiveness of Speech Encoder Components. Our speech encoder comprised fully
trainable and pre-trained modules. Comparing the first and last rows of Table
1, we observed a considerable improvement in EER and AUC when using only the
pre-trained speech embedder, except for the $\mathbf{Q}$ dataset. This could
be attributed to the limitation that the embedder model we used was trained on
general conversations from YouTube [6]. Therefore, we compensated for this by
adding fully trainable modules, which improved the performance on all test
sets, including the $\mathbf{Q}$ dataset, as shown in the fourth row.

Fusion Strategy for Audio-Text Representations. Comparing the third and fourth
rows of Table 1, the performance of cross-modality attention and that of self-
attention-based pattern extractor differed. Unlike the aforementioned
analysis, we observed performance improvements on all test sets, particularly
on the $\mathbf{Q}$ dataset. The keywords of $\mathbf{Q}$ comprise proper
nouns that are uncommon in typical training datasets. Unlike cross-modality
attention-based models, a self-attention-based pattern extractor concatenated
bi-modal data to produce outputs that contain all information in speech-to-
speech, text-to-speech, and text-to-text modalities. This permitted the
attention outputs, which are enhanced with speech information, to help
distinguish the pronunciation similarity of rare cases in the training
dataset.

Effectiveness of the Phoneme-level Detection Loss. We proposed the phoneme-
level detection loss to improve discrimination performance in speech-keyword
pairs with duplicate pronunciations. As shown in Table 1 and Figure 2, our
proposed loss function outperformed the utterance-level detection loss only,
particularly on the $\mathbf{LP_{H}}$ dataset and with low normalized
Levenshtein distance, indicating the effectiveness of our proposed loss in
discriminating speech-keyword pairs with similar pronunciations.

Figure 2: Evaluation results according to the normalized Levenshtein distance
in a LibriPhrase test set. The smaller the distance, the more similar the
pronunciation.

###  5.3 Performance analysis via keyword similarity

As shown in Figure 2, we examined the relationship between phonetic similarity
and model performance. We defined the similarity between speech-text label
pairs using the normalized Levenshtein distance [23] and calculated the mean
squared error (MSE) between our model predictions and the labels over the
similarity range greater than 0 and less than 1. Comparing our proposed model
to the baseline, we observed superior performances across all distances.
Notably, the error of the baseline model remained constant for normalized
Levenshtein distances below 0.45, whereas our proposed model exhibited a
linear improvement. Furthermore, we can attribute the performance improvements
in similar pronunciations to the effectiveness of our phoneme-level detection
loss.

Table 3: Comparison of conventional KWS models and the proposed model in the test set of Google Speech Commands V1-12. Model | Zero-shot | Accuracy $\uparrow$
---|---|---
DS-CNN [24] | ✗ | 95.4
Att-RNN [25] | 95.6
TC-ResNet [26] | 96.6
MHAtt-RNN [27] | 97.2
MatchBoxNet [28] | 97.5
KWT-3 [29] | 97.5
BC-ResNet-8 [30] |  | 98.0
Proposed | ✓ | 96.8

###  5.4 Comparison with Conventional Methods

We compared the performance of our proposed zero-shot KWS model with
conventional full-shot KWS models in Table 3. Because our model only
calculated the matching probability, we set a threshold of 0.8 to compute the
accuracy. Our proposed model outperformed some conventional models and
achieved competitive performance compared with the state-of-the-art model.
This result is noteworthy considering that we did not include the Google
Speech Command dataset when training our model.

##  6 Conclusions

This study presented a novel user-defined keyword spotting model that
leveraged the relationship between audio and phonemes of keywords. We
introduced a two-stream speech encoder containing pre-trained embedders, a
self-attention-based pattern extractor, and a phoneme-level detection loss to
improve the performance of distinguishing similar pronunciations. Our proposed
model outperformed the baseline model and achieved competitive performance
compared with full-shot KWS models. However, despite our various efforts,
relative performance improvements in challenging cases, such as
$\mathbf{LP_{H}}$ were lower compared with other test sets. Improving the
performance of highly similar pronunciations is our future research direction.

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