ABSTRACT
Title of dissertation: A Search for the Neutrinoless Double Beta
Decay of Xenon-136 with Improved
Sensitivity from Denoising
Clayton G. Davis, Doctor of Philosophy, 2014
Dissertation directed by: Professor Carter Hall
Department of Physics
The EXO-200 detector is designed to search for the neutrinoless double beta
decay of 136Xe. ??0? decay, if it occurs in nature, would demonstrate the fun-
damental nature of neutrino mass; set the mass scale of the neutrino sector; and
demonstrate lepton number non-conservation. Since the ??0? decay produces a mo-
noenergetic peak, the energy resolution of the detector is of fundamental importance
for the sensitivity of the experiment.
The present work describes a new analysis technique which improves the energy
resolution of EXO-200 through a combination of waveform denoising and weighting
of waveform components based on their expected signal-to-noise ratio. With this
method, the energy resolution of the detector is improved by 21% and the expected
background in the 2? region of interest is reduced by 32%. Applying this technique
to 99.8 kg*years of exposure collected by EXO-200 between October 5, 2011 and
September 1, 2013, we find no statistically significant evidence for the presence of
??0? in the data. We set a half-life limit T1/2 > 1.1 ? 1025 years at 90% confidence.
We also describe further improvements which could impact the energy resolution of
EXO-200, and consider implications for the planned nEXO experiment.
A Search for the Neutrinoless Double Beta Decay of Xenon-136 with
Improved Sensitivity from Denoising
by
Clayton G. Davis
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2014
Advisory Committee:
Professor Carter Hall, Chair/Advisor
Professor Radu Balan, Dean?s Representative
Professor Elizabeth Beise
Professor Rabindra Mohapatra
Professor Peter Shawhan
c? Copyright by
Clayton G. Davis
2014
Dedication
For an endless supply of coffee, patience, support, and love, this work is dedi-
cated to my wife Laura.
ii
Acknowledgments
Without the help of my colleagues, advisor, friends, family, and my wife, this
thesis and the work it embodies would never have happened. Thanks go to them
and to the EXO-200 collaboration, which has provided a supportive and stimulating
environment for my work and learning over the past four years.
Particular thanks go to: David Auty, Phil Barbeau, Giorgio Gratta, Steve
Herrin, Sam Homiller, Mike Jewell, Tessa Johnson, Tony Johnson, Caio Licciardi,
Mike Marino, Dave Moore, Russell Neilson, Igor Ostrovskiy, J.J. Russell, Simon
Slutsky, Erica Smith, Kevin O?Sullivan, Tony Waite, Josiah Walton, Liangjian Wen,
Liang Yang, Yung-Ruey Yen, and of course my advisor Carter Hall. All of you have
taught me to be the physicist I am today. Denoising has been a tremendously
collaborative effort, and I am grateful that my last project with this collaboration
has provided the chance to work with so many of you so closely.
My advisor Carter Hall has given me the freedom and support to try all kinds
of crazy things in the EXO analysis, and I am grateful to him for his open-minded
insight throughout. He has also been enormously helpful to me in finishing this
dissertation on a very condensed schedule, taking much time from his own life to
improve the quality of this work and its presentation; it could not have been what
it is without so much of his help and guidance.
Finally, my friends, family, and wife have been extremely supportive of me
throughout my time in graduate school. You have my deepest gratitude for every-
thing you have done over the past five years and before. Thank you all.
iii
Contents
List of Figures vii
List of Abbreviations xvii
1 Introduction 1
2 Theory of Neutrinoless Double-Beta Decay 4
2.1 Two-Neutrino Double-Beta Decay . . . . . . . . . . . . . . . . . . . . 5
2.2 Neutrinoless Double-Beta Decay . . . . . . . . . . . . . . . . . . . . . 8
2.3 Double-beta Decay Nuclear Matrix Calculations . . . . . . . . . . . . 15
2.4 Neutrino Flavor Physics . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Particle Physics Constraints . . . . . . . . . . . . . . . . . . . . . . . 23
2.6 Nuclear Physics Constraints from ??0? . . . . . . . . . . . . . . . . . 27
3 The EXO-200 Detector 34
3.1 Overview of the EXO-200 Detector . . . . . . . . . . . . . . . . . . . 34
3.2 Backgrounds to ??0? Decay . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Passive Background Rejection . . . . . . . . . . . . . . . . . . . . . . 47
3.4 Active Background Rejection . . . . . . . . . . . . . . . . . . . . . . 57
3.5 Pulse Amplification and Waveform Readout . . . . . . . . . . . . . . 59
3.6 Calibration Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4 Denoising Theory and Implementation 69
4.1 What is Denoising? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Notational Conventions and External Input . . . . . . . . . . . . . . 77
4.3 APD Noise Model: the Photon Perspective . . . . . . . . . . . . . . . 82
4.4 Bridge between Photons and Pulses . . . . . . . . . . . . . . . . . . . 89
4.5 Derivation of an Optimal Energy Estimator . . . . . . . . . . . . . . 96
4.6 Matrix Formulation of Denoising . . . . . . . . . . . . . . . . . . . . 104
4.7 Preconditioning of the Matrix Formulation . . . . . . . . . . . . . . . 110
4.8 Matrix Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.9 Computational Considerations . . . . . . . . . . . . . . . . . . . . . . 115
4.10 Denoising in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
iv
4.11 Future Extensions to Denoising . . . . . . . . . . . . . . . . . . . . . 124
4.11.1 Anticorrelated Cluster Energies . . . . . . . . . . . . . . . . . 124
4.11.2 Wire Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5 Electronic Noise Measurements 129
5.1 Mathematical Framework for Noise Correlations . . . . . . . . . . . . 129
5.2 Time Windows of Constant Noise . . . . . . . . . . . . . . . . . . . . 135
5.3 Algorithm for Measuring Noise . . . . . . . . . . . . . . . . . . . . . 146
5.4 Future Directions with Noise Measurement . . . . . . . . . . . . . . . 151
6 The Lightmap 156
6.1 A History of EXO-200 Lightmaps . . . . . . . . . . . . . . . . . . . . 157
6.2 Four-Dimensional Lightmap . . . . . . . . . . . . . . . . . . . . . . . 160
6.3 Algorithm Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.3.1 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.3.2 Function Parametrization . . . . . . . . . . . . . . . . . . . . 166
6.3.3 Convergence and Error Estimation . . . . . . . . . . . . . . . 170
6.4 Implementation for this Analysis . . . . . . . . . . . . . . . . . . . . 173
6.5 Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7 EXO-200 Analysis and Results from Denoising 183
7.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.1.1 Simulation of Particles using GEANT . . . . . . . . . . . . . . 184
7.1.2 Digitization of Waveforms . . . . . . . . . . . . . . . . . . . . 188
7.2 Cluster Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.2.1 Pulse Finding . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
7.2.2 Pulse Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7.2.3 Clustering Pulses into Deposit Sites . . . . . . . . . . . . . . . 205
7.3 Energy Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
7.3.1 Charge Corrections . . . . . . . . . . . . . . . . . . . . . . . . 209
7.3.2 Light Corrections . . . . . . . . . . . . . . . . . . . . . . . . . 213
7.3.3 Rotated Energy Calibration . . . . . . . . . . . . . . . . . . . 216
7.3.4 Measurement of Rotated Energy Resolution . . . . . . . . . . 220
7.3.5 Cross-Checks on the Energy Calibration . . . . . . . . . . . . 224
7.4 Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
7.5 Results and Physics Reach . . . . . . . . . . . . . . . . . . . . . . . . 243
7.6 Comparison to Results without Denoising . . . . . . . . . . . . . . . 248
7.6.1 Comparison of Energy Resolution and Calibration . . . . . . . 248
7.6.2 Comparison of Physics Results . . . . . . . . . . . . . . . . . . 257
7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
8 Conclusions and Future Plans 275
v
Bibliography 282
vi
List of Figures
2.1 Feynman diagram for ??2? decay. The reaction products are equiv-
alent to two ? decays in succession, but this reaction can sometimes
occur even if a single ? decay would be energetically forbidden. Fig-
ure from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Energy diagram of isotopes with atomic mass A = 136. The energies
?E are the binding energies of the atom, compared to the bare masses
of the same nuclei. Values are from [3]. . . . . . . . . . . . . . . . . . 7
2.3 Feynman diagram for ??0? decay. A virtual neutrino mediates the
exchange. This is only possible if ?R can flip its handedness to ?L,
and the interaction that induces this parity change also generates
neutrino mass. Figure from [1]. . . . . . . . . . . . . . . . . . . . . . 9
2.4 Even without any assumptions about the mechanism which leads to
??0? decay, we can use that process to generate an effective neutrino
mass as a higher-order process. Figure from [1]. . . . . . . . . . . . . 10
2.5 A selection of ??0? half-life limits versus the publication year of the
limit. Colors indicate which isotope is under study. Open circles
indicate experiments which have not yet concluded data-taking. Data
is from [12?68]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6 Nuclear matrix element calculations for a variety of isotopes. The
behavior is relatively stable from isotope to isotope due to the short-
range interaction for ??0? decay. The shaded regions represent ranges
based on known biases of the various methods, as suggested by [72].
Figure from [72]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.7 The electron spectrum of Tritium (3H) ? decay. The endpoint con-
tains only a small fraction of the total statistics. Figure from [82]. . . 26
2.8 The relationship between the effective Majorana mass ?m??? and
other fundamental neutrino quantities: the lightest neutrino mass
eigenstate mmin, the sum of mass eigenstates M =
?mi, and the ef-
fective single-beta-decay neutrino mass ?m??. Black (magenta) lines
indicate the allowed region for the inverted (normal) hierarchy; red
(blue) hatches indicate uncertainty for the inverted (normal) hierar-
chy due to the unknown CP-violating and Majorana phases ?, ?1,
and ?2 [86]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
vii
3.1 A two-dimensional energy spectrum of scintillation and ionization
from a testbed liquid xenon experiment under an electric field of 4
kV/cm. The spectrum is from a 207Bi source with dominant gamma
lines at 570 and 1064 keV. Figure reproduced from [97]. . . . . . . . . 35
3.2 Schematic of the inner EXO-200 TPC. Figure reproduced from [96]. . 37
3.3 Anode collection wires (u-wires) and induction wires (v-wires) from
EXO-200. (1) and (5) indicate the wire support frame; (2) indi-
cates the wires themselves, constructed as gangs of three wires; (3)
illustrates the attachment between wires and the support frame; (4)
shows the return cables from the wires to the data acquisition system.
Figure reproduced from [96]. . . . . . . . . . . . . . . . . . . . . . . . 39
3.4 Representative 238U spectra as the energy resolution of the detec-
tor changes. Red lines indicate the 2? region of interest around the
Q-value. Normalization is arbitrary, but all spectra have the same
normalization. Betas and gammas are assumed to have the same
calibration here; alternative beta scales are considered in section 7.4. 44
3.5 Representative 232Th spectra as the energy resolution of the detec-
tor changes. Red lines indicate the 2? region of interest around the
Q-value. Normalization is arbitrary, but all spectra have the same
normalization. Betas and gammas are assumed to have the same
calibration here; alternative beta scales are considered in section 7.4. 46
3.6 Relative change in background rates expected in our 2? region of
interest as a function of the energy resolution. Rates are normalized
to one at a Q-value energy resolution of 1.6% ?/E, the design goal
of EXO-200. The three backgrounds shown, from 232Th, 238U, and
137Xe, are currently our dominant backgrounds. Betas and gammas
are assumed to have the same calibration here; alternative beta scales
are considered in section 7.4. . . . . . . . . . . . . . . . . . . . . . . . 48
3.7 X-ray attenuation lengths in xenon. Compton scattering, pair pro-
duction, and photoelectric absorption are shown, along with their
combined attenuation length; coherent (Rayleigh) scattering is omit-
ted because it produces no observable energy deposit. The Q-value of
136Xe ??0? decay is indicated with a dashed black line. The vertical
axis, in cm2/g, can be multiplied by the density of the xenon to derive
an attenuation factor per unit length. Data from [104]. . . . . . . . . 50
3.8 Wire triplet, read as one channel. Figure from [96]. . . . . . . . . . . 52
3.9 APDs ganged together (bottom right). Wiring to the front-end elec-
tronics are visible as yellow ?tape.? Figure from [96]. . . . . . . . . . 53
3.10 A cutaway schematic image of the TPC and surrounding materials
in the cleanroom. HFE-7000 refrigerant is contained between the
cryostat and the LXe vessel. Figure from [96]. . . . . . . . . . . . . . 54
3.11 Muon flux as a function of depth, in meters water equivalent. The
WIPP site is indicated in relation to other underground science facil-
ities. Figure from [107]. . . . . . . . . . . . . . . . . . . . . . . . . . . 56
viii
3.12 An overview of the EXO-200 electronics and readout subsystems.
The area inside the dashed line indicates cold subsystems inside the
cryostat or TPC. Figure from [96]. . . . . . . . . . . . . . . . . . . . 60
3.13 A cross-sectional schematic of an APD is shown in (a). The electric
field as a function of depth is shown in (b). Figure from [108]. . . . . 61
3.14 A guide tube permits gamma sources to be inserted to known loca-
tions near the TPC. Green dots indicate a few of the locations where
the source may conventionally be placed. Figure from [96]. . . . . . . 65
4.1 Detector energy resolution (without denoising) is strongly correlated
with noise observed on the APDs. Noise data provided by Josiah
Walton. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Coherent and incoherent noise power spectra for a sample set of APD
channels without pulse shaping. The coherent noise power spectrum
is measured by adding together waveforms from different channels and
computing a power spectrum; the incoherent noise power spectrum is
measured by computing power spectra for each channel individually
and adding them together [114]. . . . . . . . . . . . . . . . . . . . . . 72
4.3 APD waveforms from a single event. No energy deposit occured dur-
ing this event, so the waveforms represent pure electronic noise. The
horizontal axis indicates time, and the vertical axis indicates chan-
nel number; colors ranging from blue to red indicate the baseline-
subtracted waveform magnitude. Vertical streaks are indicative of
correlations in noise across channels. . . . . . . . . . . . . . . . . . . 73
4.4 Shaped and unshaped APD waveforms. The normalization is shown
to make the peak of the shaped waveform have a magnitude of one,
and the time axis is shifted so that the unshaped waveform is a step
function centered at t = 0. . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5 It is known that the parameters used for denoising in this analysis
are not optimal. Here we use a scaling factor to adjust the parameter
controlling the importance of Poisson noise. Data from run 3516 and
4544 are shown on the left and right, respectively; top plots show
single-site resolution, bottom plots show multi-site resolution. We
can see that an improvement of roughly 0.05 percentage points at
2615 keV could be achieved by moving from the expected optimum.
It is not currently understood why scaling factors above the expected
optimum yield further improvements to resolution. . . . . . . . . . . . 123
5.1 Sum noise for all APD channels measured from charge injection runs,
with environmental changes indicated which may correlate with ob-
served changes in noise. Data provided by Josiah Walton. . . . . . . . 136
5.2 Sum noise for each APD plane measured from charge injection runs,
with environmental changes indicated which may correlate with ob-
served changes in noise. Data provided by Josiah Walton. . . . . . . . 138
ix
5.3 Sum noise for each APD electronics board measured from charge
injection runs, with environmental changes indicated which may cor-
relate with observed changes in noise. Data provided by Josiah Walton.139
5.4 Sum noise for each APD pie (6-7 channels) measured from charge
injection runs, with environmental changes indicated which may cor-
relate with observed changes in noise. Data provided by Josiah Walton.140
5.5 Time trend of
?
N?R192[385]N?R193[385]
?
(black),
?
N?R192[385]N? I193[385]
?
(green), and
?
N? I192[385]N? I193[385]
?
(red) corresponding to the corre-
lations between channels 192 and 193 at 188 kHz. Blue lines indicate
tentative noise windows. This frequency is chosen because it is as-
sociated with a peak in the noise power spectrum, and the pair of
channels is selected as a representative example for seeking changes
in noise behavior. A restart of the front-end power supply at run
2700 is coincident with the reduction in correlated noise shown here,
but the reason for this effect is not understood [128]. . . . . . . . . . 141
5.6 Time trend of
?
N?R202[383]N?R203[383]
?
(black),
?
N?R202[383]N? I203[383]
?
(green), and
?
N? I202[383]N? I203[383]
?
(red) corresponding to the corre-
lations between channels 202 and 203 at 187 kHz. Blue lines indicate
tentative noise windows. This frequency is chosen because it is as-
sociated with a peak in the noise power spectrum, and the pair of
channels is selected as a representative example for seeking changes
in noise behavior. Installation of electronics board cooling fans is co-
incident with the reduction of noise around run 2900, with different
installation times for different channels, but the reason this particular
noise frequency is so strongly affected is not understood [128]. . . . . 142
5.7 Cumulative low-background livetime collected in EXO-200. Figure
provided by David Auty. . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.8 The amplitude of the TEM injected pulses (?glitch? pulses) over time
for channel 158. Figure provided by Sam Homiller. . . . . . . . . . . 152
6.1 Lightmap position-dependence R(~x) for selected APD gangs. . . . . . 175
6.2 Lightmap position-dependence R(~x) for selected APD gangs. Here
extreme anode positions are omitted to permit better contrast for
the lightmap in the fiducial volume. . . . . . . . . . . . . . . . . . . . 176
6.3 Functions S(t) for selected channels. . . . . . . . . . . . . . . . . . . 177
6.4 Functions S(t) for selected channels. . . . . . . . . . . . . . . . . . . 178
6.5 Functions S(t) for selected channels. . . . . . . . . . . . . . . . . . . 179
7.1 The GEANT simulation includes large-scale features of the EXO-200
detector, including the outer and inner lead wall (black), outer and
inner cryostat and TPC legs (red), and the TPC itself (brown). Com-
ponents are assembled from simple geometrical shapes, and distant
objects are only described coarsely [111]. . . . . . . . . . . . . . . . . 185
x
7.2 To ensure computational time is only spent on important details,
detector components which are close to the liquid xenon are simulated
in GEANT with greater accuracy than distant objects. The TPC
contains the xenon, so it is simulated in detail [111]. . . . . . . . . . . 186
7.3 Energy spectra from 238U in the TPC vessel for single-site (red) and
multi-site (black) events. No energy smearing is performed; peak
widths are due to energy loss outside of the liquid xenon [111]. . . . . 189
7.4 Energy spectra from 232Th in the TPC vessel for single-site (red) and
multi-site (black) events. No energy smearing is performed; peak
widths are due to energy loss outside of the liquid xenon [111]. . . . . 190
7.5 The wire planes are modeled in only two dimensions; charge drifts
along the field lines, which are arranged to terminate only on the
u-wires [111]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.6 Weight potential of a u-wire channel consisting of three ganged wires.
Electric field lines are superimposed [111]. . . . . . . . . . . . . . . . 194
7.7 Weight potential of a v-wire channel consisting of three ganged wires.
Electric field lines are superimposed [111]. . . . . . . . . . . . . . . . 195
7.8 Comparison between simulated and observed waveforms on a u-wire
(left) and v-wire (right) from 228Th sources. The events are chosen
to have similar energies so that the magnitudes match [111]. . . . . . 196
7.9 Comparison between simulated and observed energy spectra (a) and
standoff distance (b) in single-site 226Ra events from a source located
at position S5 [89]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
7.10 A u-wire waveform (left) and the output from operation of the matched
filter (right). The red line on the right indicates our filtered pulse
threshold; the matched filter output exceeds the threshold, so this
waveform is determined to contain a pulse [112]. . . . . . . . . . . . . 201
7.11 A u-wire waveform composed of two pulses near in time is shown (top
left); the matched filter (top right) correctly detects the presence of
a pulse, but does not detect the presence of two distinct pulses. At
bottom left, the waveform is unshaped; at bottom right the wave-
form is reshaped with shorter differentiation times, leading to easier
detection [112]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7.12 Fits to data for a u-wire (top left), v-wire (top right), and summed-
APD waveform (bottom). The red model line indicates the time
extent of the fit window [112]. . . . . . . . . . . . . . . . . . . . . . . 204
7.13 The purity charge correction is measured by fitting the 2615-keV 208Tl
gamma line as a function of Z-position [140]. . . . . . . . . . . . . . . 211
7.14 The expected grid correction of a u-wire as a function of Z-position.
In this plot, Z = 6 mm represents the position of the u-wire plane
and Z = 12 mm is the position of the v-wire plane [110]. . . . . . . . 211
7.15 After applying all charge corrections, we find that the 2615-keV 208Tl
ionization peak is observed at the expected location for single-wire
events; however, for two-wire events a residual bias is observed which
also exhibits Z-dependence [140]. . . . . . . . . . . . . . . . . . . . . 212
xi
7.16 A residual calibration offset and Z-dependent behavior is observed
in the denoised scintillation measurements. The top plots show the
relative peak position of the scintillation-only 2615-keV 208Tl peak
(one-wire and two-wire) and average alpha energy from 222Rn versus
Z (top left) and R (top right). The bottom plot shows the absolute
denoised scintillation energy from the 2615-keV 208Tl peak for one
and two wires, with the measured correction function (equation 7.7)
overlaid [140]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
7.17 Corrected scintillation versus charge energy are shown for a thorium
run. The projection angle for the rotated energy measurement is
illustrated at the 2615-keV peak; the projection onto a calibrated axis
is shown. The dotted red line indicates the diagonal cut described in
section 7.4. Figure provided by Liangjian Wen. . . . . . . . . . . . . 218
7.18 Charge-only, scintillation-only, and rotated energy spectra are shown
from a thorium source run; the significant improvement in energy res-
olution with the rotated energy measurement is apparent, as are low-
energy features which are washed out in the lower-resolution spectra.
Figure provided by Liangjian Wen. . . . . . . . . . . . . . . . . . . . 219
7.19 The time-averaged resolution functions (left) and relative resolution
functions (right) for single-site (red) and multi-site (purple) denoised
data. Shaded bands indicate uncertainties. Data provided by Caio
Licciardi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
7.20 A typical monoenergetic gamma line, where the horizontal axis in-
dicates ionization energy and the vertical axis indicates scintillation
energy. We note that the ionization-only and scintillation-only reso-
lutions are dominated by the length and angle of the island, which
are dominated by xenon physics and the choice of electric field rather
than the accuracy of the readouts in those channels. . . . . . . . . . . 223
7.21 Scintillation-only resolution over time, including: single-site denoised
(blue), multi-site denoised (red), single-site undenoised (black solid),
multi-site undenoised (hollow square). Although denoising does show
some improvement in the scintillation-only resolution, it is quite mod-
est; this is because the scintillation-only resolution is dominated by
fluctuations in light yield. [140]. . . . . . . . . . . . . . . . . . . . . . 223
7.22 The single-site (left) and multi-site (right) gamma line at 2.2 MeV
from neutron capture on hydrogen can be used as a low-background
cross-check on the resolution calibration. Here, data which is coin-
cident with a muon veto is shown to include the expected neutron
capture peak [140]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7.23 The low-background potassium peak at 1461 keV is used as a cross-
check on the single-site (left) and multi-site (right) energy calibra-
tions. Fits also include the nearby 1332-keV cobalt peak [140]. . . . . 226
xii
7.24 The residuals between calibrated and true peak positions for all sources
is compared. Calibrations were obtained from the 137Cs, 60Co, and
232Th peaks at 662, 1173, 1332, and 2615 keV. The peak from 226Ra
at 2448 keV is also shown, but was not used for calibration. Low-
background calibration lines are neutron capture on hydrogen (2200
keV) and 40K (1461 keV). Figure from [140]. . . . . . . . . . . . . . . 227
7.25 Time-dependence of the peak positions from all calibration cam-
paigns. Error bars, when not visible, are smaller than the circle.
Top row is the 60Co peak at 1173 keV; middle row is the 60Co peak at
1332 keV; bottom row is the 137Cs peak at 662 keV. Thorium is not
included because it is used to measure the time-dependence of the
peak positions, so it is necessarily calibrated to a time-independent
value. Figure from [140]. . . . . . . . . . . . . . . . . . . . . . . . . . 228
7.26 Time-dependence of the energy resolution at the 208Tl 2615 keV
gamma line. Single-site (blue) and multi-site (red) energy resolu-
tions are shown. The time-averaged energy resolutions (1.47% ?/E
for single-site, 1.59% ?/E for multi-site, both at 2615 keV) are shown
in blue and red dashed lines, respectively. Data provided by Liangjian
Wen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
7.27 Here the X-Y orthogonal coordinates are shown along with the U-V
coordinates that run orthogonal to the wire planes. The X-Y fiducial
cuts applied to the data (dashed hexagon) do not include the en-
tire active volume (solid hexagon); for the ??0? search we find that
aggressive fiducial cuts optimize our sensitivity, so very little active
xenon is left unused. . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
7.28 Vetoing events coincident with muons can reduce our 1? and 2? event
counts. Figure provided by David Auty. . . . . . . . . . . . . . . . . 233
7.29 The fit provides constraints on the beta-scale. Figure provided by
Liangjian Wen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
7.30 Energy spectra with best-fit PDFs for single-site (top) and multi-
site (bottom). Residuals between data and the combined PDFs are
shown below the spectra. The last bin of the spectra is an overflow
bin. The 2? region of interest around the Q-value is shown on the
single-site spectrum. Insets for single-site and multi-site zoom around
the Q-value. The simultaneous standoff-distance fit is not shown [89]. 241
7.31 The positions of events in the 1? region of interest (red) and wider
energy range 2325? 2550 keV (black) are shown projected onto their
X-Y (left) and R2-Z (right) coordinates. Lines indicate the fiducial
volume. Figure provided by Dave Moore. . . . . . . . . . . . . . . . . 242
7.32 Profile scan of negative log-likelihood as a function of the number of
fit ??0? counts [89]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
xiii
7.33 Assuming that the background measurements of EXO-200 from this
analysis are accurate, toy datasets are simulated. The top plot shows
the probability distribution of our 90% confidence limit; we find that
our median upper limit is 14.04 counts attributed to ??0?, compared
to our observed upper limit of 23.92 counts. The bottom plot shows
the probability distribution of our ability to reject the null hypothe-
sis (no ??0?) based on the negative log-likelihood; we find that the
present dataset rejects the null hypothesis with probability less than
90%. Figures provided by Ryan Killick. . . . . . . . . . . . . . . . . . 244
7.34 ??0? constraints are shown from 136Xe (horizontal) and 76Ge (ver-
tical). The four diagonal lines represent four recent matrix element
computations [71,145?147] and one recent phase-space factor [94] re-
lating the two half-lives; tick marks along the diagonal represent the
corresponding mass ?m???. 90% confidence limits and median 90%
sensitivities are shown for the KamLand-Zen [13], GERDA [15], and
EXO-200 experiments [89]; the claimed 2006 discovery by Klapdor-
Kleingrothaus and Krivosheina is shown as a 1? confidence inter-
val [148]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
7.35 A comparison of the spectra before and after denoising for a set of
representative thorium source runs (4758 and 4766). The thickness
of the lines indicate Poisson error bars on the number of counts in a
bin; smoothing is applied to the spectra to permit easier comparison
of peaks. Important gamma lines are indicated. Around 500-600 keV
threshold effects become significant, making comparison difficult. . . . 249
7.36 A comparison of the spectra before and after denoising for a repre-
sentative cobalt source run (4787 and 4788). The thickness of the
lines indicate Poisson error bars on the number of counts in a bin;
smoothing is applied to the spectra to permit easier comparison of
peaks. The two gamma lines of 60Co are indicated. . . . . . . . . . . 250
7.37 A comparison of the spectra before and after denoising for a represen-
tative cesium source run (4777-4781). The thickness of the lines indi-
cate Poisson error bars on the number of counts in a bin; smoothing
is applied to the spectra to permit easier comparison of peaks. The
gamma line of 137Cs is indicated. Threshold effects are significant
at this energy and the calibration is extrapolated from higher-energy
sources, but the spectra illustrate the fact that resolution improves
dramatically at low energy. . . . . . . . . . . . . . . . . . . . . . . . . 251
7.38 The time-averaged resolution functions (top) and relative resolution
functions (bottom) are compared for denoised and undenoised data.
Bands indicate uncertainties. Data provided by Caio Licciardi. . . . . 252
7.39 Time dependence of the single-site energy resolution at 2615 keV with
denoised (blue) and undenoised (red) scintillation. Time-averaged
energy resolutions at this energy (1.47% denoised, 1.86% undenoised)
are overlaid as dashed lines. Data provided by Mike Marino. . . . . . 255
xiv
7.40 The non-linearity of the calibration is shown here for denoised (red)
and undenoised (black) data in single-site (left) and multi-site (right).
Rrec is defined as the ratio between the observed peak position and
the observed peak position of the 2615 keV 232Th peak, and Rreal is
defined as the true ratio between the peak position and 2615 keV;
the vertical axis shows the relative difference between Rrec and Rreal,
which reflects how much non-linear correction the calibration must
perform. The cesium, cobalt, and thorium peaks (662, 1173, 1332,
and 2615 keV) are used to obtain a calibration; the radium line at
2448 keV and the neutron capture line from hydrogen at 2200 keV
are shown as cross-checks [140]. . . . . . . . . . . . . . . . . . . . . . 256
7.41 Denoised (red) and undenoised (blue) profile likelihood curves for ??0?.262
7.42 Denoised (red) and undenoised (blue) data counts are shown in 30
keV bins. The total best-fit PDF, where the ??0? contribution is
fixed to zero, is overlaid for denoised and undenoised data. Dashed
lines indicate the 2? region of interest. . . . . . . . . . . . . . . . . . 264
7.43 Denoised (red) and undenoised (blue) single-site low-background data
and pdfs in the energy window from 1200 to 1800 keV. The thick lines
indicate the total single-site pdfs; thin lines indicate contributions
from 40K. Both fits are constrained to have no contribution from ??0?.265
7.44 Denoised (red) and undenoised (blue) multi-site low-background data
and pdfs in the energy window from 1200 to 1800 keV. The thick lines
indicate the total multi-site pdfs; thin lines indicate contributions
from 40K. Both fits are constrained to have no contribution from ??0?.266
7.45 Denoised (red) and undenoised (blue) multi-site low-background data
and pdfs in the energy window from 1000 to 1600 keV. The thick lines
indicate the total multi-site pdfs; thin lines indicate contributions
from 60Co. Both fits are constrained to have no contribution from
??0?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
7.46 Denoised (red) and undenoised (blue) single-site low-background data
and pdfs in the energy window from 2000 to 2600 keV. The thick
lines indicate the total single-site pdfs; thin lines indicate contribu-
tions from isotopes in the 238U decay chain. Dashed vertical lines
indicate the 2? region of interest around the Q-value. Both fits are
constrained to have no contribution from ??0?. The uranium decay
chain contributes 10.0 (8.9) mean expected counts in the 2? region
of interest for denoised (undenoised) data. . . . . . . . . . . . . . . . 268
7.47 Denoised (red) and undenoised (blue) multi-site low-background data
and pdfs in the energy window from 1600 to 2800 keV. The thick lines
indicate the total multi-site pdfs; thin lines indicate contributions
from isotopes in the 238U decay chain. Both fits are constrained to
have no contribution from ??0?. . . . . . . . . . . . . . . . . . . . . . 269
xv
7.48 Denoised (red) and undenoised (blue) single-site low-background data
and pdfs in the energy window from 2200 to 2800 keV. The thick
lines indicate the total single-site pdfs; thin lines indicate contribu-
tions from isotopes in the 232Th decay chain. Dashed vertical lines
indicate the 2? region of interest around the Q-value. Both fits are
constrained to have no contribution from ??0?. The thorium decay
chain contributes 18.7 (34.5) mean expected counts in the 2? region
of interest for denoised (undenoised) data. . . . . . . . . . . . . . . . 270
7.49 Denoised (red) and undenoised (blue) multi-site low-background data
and pdfs in the energy window from 2200 to 2800 keV. The thick lines
indicate the total multi-site pdfs; thin lines indicate contributions
from isotopes in the 232Th decay chain. Both fits are constrained to
have no contribution from ??0?. . . . . . . . . . . . . . . . . . . . . . 271
7.50 Denoised (red) and undenoised (blue) single-site PDF contributions
from 137Xe, where we constrain ??0? to have no decays in both fits. . 272
8.1 NEST simulation software has been used to estimate how the ro-
tated energy resolution at 2615 keV (vertical axis) depends on the
scintillation-only resolution at 2615 keV (horizontal axis); this theo-
retical estimate is shown in blue, as applicable to the EXO-200 de-
tector with fixed electric field. Black points indicate measurements
from thorium source runs without denoising; red points indicate mea-
surements from thorium runs with denoising. Figure provided by
Liangjian Wen using NEST software [118]. . . . . . . . . . . . . . . . 276
xvi
List of Abbreviations
EXO Enriched Xenon Observatory
EXO-200 Current (200-kg) iteration of the EXO family of detectors
nEXO Planned next-generation EXO detector
APD Avalanche Photo-diode
ADC Analog-Digital Converter, or one of its discrete output units
DAQ Data Acquisition System
TPC Time Projection Chamber
WIPP Waste Isolation Pilot Plant
PDF Probability Distribution Function
ROI Region Of Interest (around 2457 keV)
??0? Neutrinoless double-beta decay
??2? Two-neutrino double-beta decay
FEC Front-End Card
QRPA Quasi-Random Phase Approximation
VUV Vacuum Ultra-Violet
xvii
Chapter 1: Introduction
The EXO-200 detector is designed to search for a hypothetical decay mode of 136Xe
whereby two electrons and no neutrinos are emitted from the nucleus. This neutri-
noless double-beta (??0?) decay, if it occurs, would have profound implications for
nuclear and particle physics: it would set the absolute mass scale of the neutrino
sector, provide clues to the mechanism which generates mass in the neutrino sector,
and give the first observation of non-conservation of total lepton number.
EXO-200 contains roughly 110 kg of liquid xenon in its active volume. The
xenon used by EXO is enriched to 80.6% in xenon-136, resulting in approximately
3.9? 1026 atoms of 136Xe which can be monitored. The observation of ??0? decay
would present itself as a peak in the energy spectrum at a Q-value of 2458 keV, so
EXO-200 is designed to subject such a peak to as little background as possible
Background is controlled primarily by reducing the level of radioactivity of
materials in and around the liquid xenon. All materials used in the construction
of the EXO-200 detector were carefully screened to insure low concentration of
radioactive isotopes, making EXO-200 one of the lowest-background detectors in
the world.
1
After construction, background reduction must be performed by discrimination
between ??0? and other processes. This can be performed primarily by exploiting
the event topology and by refining the energy measurements. The event topology of
??0? consists of a highly localized energy deposit due to the short interaction length
of ? particles; in contrast, backgrounds will often produce diffuse energy deposits
which can be rejected as candidate ??0? events.
Perhaps the most conceptually straightforward method of background reduc-
tion, however, is by precise energy estimation. An improved energy resolution as-
sures us that ??0? candidate events must come from a narrower region of the energy
spectrum and allows us to reject events outside of that region of interest (ROI). The
EXO-200 detector observes energy in two forms, ionization (charge) and scintillation
(light), and both are necessary to achieve excellent energy resolution. The scintil-
lation measurement has a lower accuracy, so it is the limiting factor to the energy
resolution of EXO-200.
In this work, we describe a new technique for improving the accuracy of the
scintillation measurements of the EXO-200 detector through a detailed offline wave-
form analysis. This technique, identified throughout this work as ?denoising?, con-
sists of understanding the signal-to-noise ratio of the different components of our
scintillation measurement. The signal-to-noise content of our scintillation measure-
ments depends on both the proximity of a light sensor to the energy deposit and on
the spectral shape of light pulses compared to waveform noise. An active noise re-
duction program is included in this effort, which in turn improves the signal-to-noise
content of the scintillation measurements. An overall improvement in the EXO-200
2
energy resolution of more that 20% is achieved at the ??0? Q-value.
We provide an overview of the theoretical motivations for the search for ??0?
decay in chapter 2. Chapter 3 describes the design of the EXO-200 detector, the
expected background, and some of the techniques it is capable of using to reduce
those backgrounds. Chapter 4 derives the mathematical framework of denoising and
some practical considerations of its application; chapters 5 and 6 describe the mea-
surements of electronic noise and light yield which are critical inputs to denoising.
In chapter 7 we describe the components and results of the EXO-200 double beta
decay search, including comparisons between results with and without denoising in
sections 7.3.3 and 7.6. Conclusions and future outlook are contained in chapter 8.
3
Chapter 2: Theory of Neutrinoless Double-
Beta Decay
Toward the end of the 1990s neutrinos were conclusively proven to possess a non-zero
mass. Subsequent effort has been directed toward understanding the absolute scale
of neutrino mass spectrum and the nature of the interaction which causes it. Neu-
trinoless double-beta (??0?) decay, would provide a window onto both questions.
This chapter presents the theoretical motivations for searching for ??0? decay. Sec-
tion 2.1 describes the closely-related two-neutrino double-beta (??2?) decay which
is allowed in the standard model. Section 2.2 defines ??0? decay and reasons for
hypothesizing its occurrence. Considerations and challenges in nuclear physics are
described in section 2.3. The parametrization of the neutrino sector is provided in
section 2.4. Section 2.5 summarizes constraints on neutrino masses which can be ob-
tained from observations of single-? decay and cosmology, and section 2.6 describes
in detail the considerations and challenges involved in searching for ??0? decay. The
reader will obtain a general understanding of the state of the field before continuing
on to the details of the EXO-200 detector in chapter 3.
4
??
?
n
n p
p
e
e
?
W
W
Figure 2.1: Feynman diagram for ??2? decay. The reaction products are
equivalent to two ? decays in succession, but this reaction can sometimes
occur even if a single ? decay would be energetically forbidden. Figure
from [1].
2.1 Two-Neutrino Double-Beta Decay
Standard-model two-neutrino double beta (??2?) decay is the result of the particle
interaction
2d? 2u+ 2e? + 2??e (2.1)
mediated by W?-exchange, as depicted in Figure 2.1. It is effectively the simulta-
neous occurrence of two beta (?) decays from the same nucleus.
Because ??2? decay is a second-order weak interaction, it has a remarkably
slow rate compared to most ? decay processes. Although many nuclei are expected
to decay by ??2?, the process is thoroughly masked by conventional ? decay in
most of them. In most cases, we can only hope to detect ??2? decay in isotopes
where ? decay is forbidden or highly suppressed.
5
An example of an isotope for which ? decay is highly suppressed is 4820Ca. The
ground state of 4820Ca has zero units of angular momentum, whereas its single-? decay
daughter product 4821Sc has six units of total angular momentum in its ground state,
and is thus highly suppressed by angular momentum conservation. In contrast, 4822Ti
has zero units of total angular momentum, making ??2? decay of 4820Ca permitted
by angular momentum and energy considerations, and as a result the ??2? decay
mode of 4820Ca dominates [2].
In the most promising ??2? candidates, single ? decay is forbidden by energy
conservation. It is well-known that nuclei minimize their energy by arranging similar
nucleons to have overlapping wavefunctions [2]. Thus, isotopes with an even number
of protons and an even number of neutrons will have less nucleon-pairing potential
energy than an isotope with either an odd number of protons or an odd number
of neutrons, which in turn will have less nucleon-pairing potential energy than an
isotope with an odd number of protons and an odd number of neutrons.
The effect is illustrated by the nuclear energy level diagram shown in figure 2.2
for the A = 136 isobar. Xenon, barium, cerium, and neodymium are even-even iso-
topes, and have systematically lower energies than the odd-odd isotopes iodine,
cesium, lanthanum, and praseodymium. For this particular isobar, we can see that
xenon is energetically forbidden from single-? decaying to cesium because the odd-
odd isotope of cesium has slightly more potential energy than the even-even iso-
tope of xenon. As a result, the primary mode of decay of xenon-136 will be ??2?
decay to barium-136. Similarly, cerium-136 can undergo double-electron capture
(ECEC), electron capture with positron emission (EC?+), or double-positron emis-
6
Figure 2.2: Energy diagram of isotopes with atomic mass A = 136. The
energies ?E are the binding energies of the atom, compared to the bare
masses of the same nuclei. Values are from [3].
7
sion (?+?+); however, in practice the expected rates for these decays will be lower
than the rates for ??2? decay, so we will not consider them further in this work.
2.2 Neutrinoless Double-Beta Decay
The detection and study of ??2? decay provides an opportunity to test a class
of nuclear matrix element computations; however, the decay does not violate any
fundamental symmetries and its existence is, in this sense, a mundane prediction of
the Standard Model. The primary appeal of isotopes which undergo ??2? decay is
the opportunity these isotopes provide to probe the nature of neutrinos through the
related neutrinoless decay.
It had been suggested as early as 1937 that neutrinos could possess mass
through a neutrino-antineutrino interaction, provided that the neutrino is its own
antiparticle [4]. The theorized Majorana interaction comes from Lagrangian terms
of the form (for each of three neutrino eigenstates)
LMaj = ?
mL
2
(
?cL?L + ?L?cL
)
? mR2
(
?cR?R + ?R?cR
)
,
(2.2)
where the superscript-c represents charge conjugation. (This is the reason that
Majorana mass terms are only possible for a chargeless lepton.) The masses mL
and mR may be chosen independently; since there has never been an observation of
right-handed neutrinos or left-handed anti-neutrinos, it is possible that mR = 0 and
that the fields ?R and ?cR do not exist in nature [1].
8
??
n
n p
p
e
e
W
W
x
Figure 2.3: Feynman diagram for ??0? decay. A virtual neutrino me-
diates the exchange. This is only possible if ?R can flip its handedness
to ?L, and the interaction that induces this parity change also generates
neutrino mass. Figure from [1].
If neutrinos do have Majorana mass interactions, then any isotope that un-
dergoes ??2? decay can also undergo the related process 2d ? 2u + 2e?, depicted
in Figure 2.3, in which the two outgoing neutrinos in ??2? decay are replaced by
one virtual neutrino. This process is called neutrinoless double-beta (??0?) decay.
We can interpret this as a mixing interaction between a left-handed neutrino and a
right-handed antineutrino; this is only possible if neutrinos are Majorana particles.
The tree-level diagram for ??0? has one additional interaction vertex com-
pared to ??2? decay, and as a result we would expect it to occur at an even slower
rate. However, the more immediate consequence of ??0? decay is that lepton num-
ber conservation is violated. The lepton number changes by two units corresponding
to the creation of two leptons with no balancing anti-leptons. Numerous theories
have suggested other plausible modes of lepton number non-conservation [5, 6], but
none have yet reported a positive result. In the conventional Standard Model with
9
(?)R ?L??(0?)
p p
_
nnW W
ee
Figure 2.4: Even without any assumptions about the mechanism which
leads to ??0? decay, we can use that process to generate an effective
neutrino mass as a higher-order process. Figure from [1].
massless neutrinos, lepton number conservation is an accidental symmetry [7], but
in a model with massive neutrinos there may no longer be any reason a priori to
expect conservation of lepton number.
It is worth noting that the interaction term above assumes no mediating par-
ticles in the mass mechanism, whereas it is possible that ??0? could be mediated by
some higher-order interaction terms. However, if ??0? decay is observed, it leads
very generally to a conclusion that neutrinos have an effective Majorana mass inter-
action [8]. We can see this by embedding the ??0? process into a higher-order one,
as shown in figure 2.4. Regardless of the details of how ??0? occurs, its existence
would necessarily generate an effective neutrino mass as a higher-order process.
We have described so far only the theorized Majorana interaction of equa-
tion 2.2. However, it is also possible that neutrinos could have a Dirac mass term
analogous to the other fermions. The full set of neutrino mass terms in the La-
10
grangian would then be:
LMaj+Dirac = ?
mL
2
(
?cL?L + ?L?cL
)
? mR2
(
?cR?R + ?R?cR
)
?mD
(
?L?R + ?R?L
)
,
(2.3)
where mD is the new Dirac mass term. The three flavors observed in nature would
result in three sets of these terms, one for each flavor, and nine possible mass terms
in total [1].
For simplicity, we consider now the mass terms for a one-flavor system. We
can rearrange the Lagrangian terms as:
LMaj+Dirac = ?
1
2
(
(nL)cMnL + (nL)MncL
)
,where (2.4a)
M =
?
?
?
?
mL mD
mD mR
?
?
?
? (2.4b)
nL =
?
?
?
?
?L
(?R)c
?
?
?
? . (2.4c)
The matrix M can, in all cases, be diagonalized by some unitary matrix, which
means that there is always a basis in which two interaction terms are sufficient. In
nearly all cases the eigenvalues will be distinct values, so the two eigenstates are
non-degenerate. Both eigenstates in this case are their own antiparticles, so they
are Majorana. The only exceptions to this are when mL = mR and mD = 0, or
11
when mL = mR = 0. Both of those special cases lead to degenerate eigenspaces; in
the former, the system is clearly purely Majorana from the beginning, and in the
latter the system is purely Dirac. Thus we can see that even though it is possible
to include a Dirac neutrino mass term, neutrinos will be Majorana particles unless
mL = mR = 0. Similar results hold in the three-flavor case [1].
We have spoken of the possibility that the Majorana mass term comes from
a tree-level Lagrangian term. However, this would be a surprising result. The
Majorana fields would not have the same quantum number under the SU(2)L ?
SU(2)R symmetry, meaning that either neutrinos would need to choose between the
Majorana and Dirac terms or some larger symmetry would need to take the place
of SU(2)L ? SU(2)R. It is viewed as more likely that neutrino mass is generated
through an effective interaction. A natural mechanism to generate neutrino mass
called the see-saw mechanism was developed around 1980 [9, 10]. In this scheme
we presume that mR  mD  mL. The Dirac term could come from the Higgs
mechanism, so we expect mD to occupy the 1 MeV energy range typical of other
leptons and quarks. The large mass mR of the right-handed neutrino ?R is meant
to explain the absence of ?R and ?cR from all observations. When we take mL = 0
12
and diagonalize the single-flavor neutrino mass matrix M we obtain:
M =
?
?
?
?
0 mD
mD mR
?
?
?
? (2.5)
?
?
?
?
?
i mD/mR
?imD/mR 1
?
?
?
?
?
?
?
?
?m2D/mR 0
0 mR
?
?
?
?
?
?
?
?
?i imD/mR
mD/mR 1
?
?
?
? . (2.6)
In other words, if we presume that there is a Dirac neutrino mass similar to the
masses of other leptons and a right-handed Majorana neutrino mass mR  mD, then
in another basis a small left-handed Majorana neutrino mass m2D/mR is generated.
This is widely considered to be the most natural mechanism for explaining the
lightness of the neutrinos observed in nature [1, 11].
A selection of ??0? half-life limits are shown in figure 2.5 for the 76Ge, 100Mo,
130Te, and 136Xe isotopes with the publication year of the result. Although some
results did exist before 1980, after the see-saw mechanism described in [9, 10] was
appreciated experimental interest in neutrinoless double-beta decay flourished, and
we can see that for the last thirty years there has been steady progress in improving
experimental sensitivity for many isotopes. The discovery of oscillation of atmo-
spheric neutrinos ( [69]) verified that neutrinos do have finite mass of some sort,
adding to the motivation to search for the ??0? process. The following sections
describe computational and experimental considerations associated with searches
for ??0? decay.
13
Pu
bli
ca
tio
n Y
ea
r
19
70
19
80
19
90
20
00
20
10
 Limit (years) 1/2 T
21
102
2
10
23
10
24
10
25
10
26
10
Ge
76
Mo
10
0 T
e
13
0 X
e
13
6
Figure 2.5: A selection of ??0? half-life limits versus the publication
year of the limit. Colors indicate which isotope is under study. Open
circles indicate experiments which have not yet concluded data-taking.
Data is from [12?68].
14
2.3 Double-beta Decay Nuclear Matrix Calcula-
tions
If the dominant mechanism of ??0? decay is a tree level neutrino mass term as
shown in figure 2.3, then the rate of ??0? decay will also reflect the magnitude
of the neutrino mass parameters. To specify this relation precisely we need an
understanding of the nuclear physics of the decaying isotope. This section will
identify the relevant quantities which must be computed and provide a survey of
the computational approaches to estimating them.
For a tree-level Majorana interaction we can write the partial half-life T 0?1/2 of
an isotope which undergoes ??0? decay as
[
T 0?1/2
]?1 = G0?(Q??, Z) |M0? |2 ?m???2 , (2.7)
where G0?(Q??, Z) is a phase-space factor coming from the range of possible output
states, M0? is a nuclear matrix element, and ?m??? is the effective ??0? neutrino
mass which will be defined in section 2.6.
The two factors G0?(Q??, Z) and M0? may be computed by a variety of meth-
ods, and it is tempting to draw nuclear matrix elements and phase factors from
different publications. However, this must be done carefully: scaling factors can be
absorbed from one into the other, and if nuclear matrix elements and phase space
factors are calculated using different conventions, the result of combining them will
15
not be correct.
For example, in early calculations, the nuclear matrix element and phase-space
factor were generally computed in units of fm?1 and yrs?1fm2, respectively. Starting
in the mid-1980s common practice shifted to multiply the nuclear matrix element
by the nuclear radius R0 ? A3, where A is the number of nucleons, and divide
the phase-space factor by R20, making the nuclear matrix element unitless and the
phase-space factor have units of yrs?1. With this convention care must be taken
that the nuclear radius used in both calculations is the same, whereas for many years
different authors might choose R0 = 1.1A3 fm or R0 = 1.2A3 fm without specifying
that choice [70]. Modern practice is now that the value of R0 is explicitly specified
in any matrix element or phase-space factor calculations. Another convention is
whether the fourth power of the axial vector current g4A should be included with
the phase space factor, nuclear matrix element, or separately as its own factor of
equation 2.7, and again one must be careful to understand the chosen conventions
before combining results from different sources [71].
The phase-space factor G0?(Q??, Z) accounts for the phase space of the final
state of ??0? decay. This includes two outgoing electrons and the final state nu-
cleus. The mass of the outgoing nucleus is always much larger than the masses of
the neutrinos, so nearly all momentum will be carried by the two electrons. Each
contributes a phase space integral of the form
? pmax
0 d ? ppE, which individually con-
tribute a factor proportional to p3max. Considering the system in the rest frame of
the initial state nucleus, the sum of the two electron momenta are constrained to be
equal to zero, so the combined phase space integral for both electrons is proportional
16
to
? pmax
0
dp1p1E1
? pmax
0
dp2p2E2?(p2 ? p1) ? p5max. (2.8)
As a result, the phase space factor G0?(Q??, Z) of ??0? decay will be proportional
to Q5, where Q is the total energy of the decay [11]. The strong dependence of
G0?(Q??, Z) on Q means that isotopes which have high Q-values will be expected
to have shorter ??0? half-lives than isotopes with low Q-values, which is one reason
that most ??0? searches focus on high-Q isotopes.
Another important contribution to the phase space factor come from the elec-
tric potential of the atomic nucleus and its atomic electrons. The potential V (r)
experienced by the escaping electron as a function radius is approximated by: [71]
V (r) = ?Z?h?c ?
?
????
????
1/r r ? R0
(3? (r/R0)2) /2R0 r < R0,
(2.9)
where Z is the nuclear charge, ? is the fine structure constant, h? is Planck?s constant,
c is the speed of light, and R0 is the radius of the nucleus. Higher-order corrections
include the change in nuclear charge due to the decay and the density of the electron
cloud (which includes angular asymmetries). A modern treatment can be found
in [71].
The nuclear matrix element M0? describes the transition rate from the initial
to the final nuclear state of the decay process. The transition may be treated as a
17
 76 82 96 100 128 130 136 150  A0
1
2
3
4
5
6
7
8 GCMIBMISMQRPA(J)QRPA(T)
M
0
?
Figure 2.6: Nuclear matrix element calculations for a variety of isotopes.
The behavior is relatively stable from isotope to isotope due to the short-
range interaction for ??0? decay. The shaded regions represent ranges
based on known biases of the various methods, as suggested by [72].
Figure from [72].
18
two-step process:
(Z,A)? (Z + 1, A) + e? + ??e
?e + (Z + 1, A)? (Z + 2, A) + e?, (2.10)
where the intermediate state is a virtual state because it does not conserve energy.
Because the two converted protons must be near each other within the nucleus for
the neutrino interaction to occur, the M0? is not very sensitive to the variations in
nuclear structure between elements; this is in contrast to the transition probability
for ??2?, M2? , for which the neutrons may be well-separated within the nucleus [73].
This similarity can be observed in figure 2.6.
To compute the nuclear matrix elements M0? , two main approaches exist: the
quasi-random phase approximation (QRPA) and the nuclear shell model. In the
simpler random phase approximation (RPA), the transition from initial to interme-
diate state and from intermediate to final state are produced by operators of the
form p+n, where p+ represents the proton creation operator and n represents the
neutron annihilation operator. The goal is to make these operators more bosonic
(and hence more amenable to treatment in a statistical fashion), and so these oper-
ators are diagonalized in a new basis which acts on pairs of nuclei and obeys bosonic
commutation rules. QRPA is similar to RPA, but additionally takes into account
the preference of like nuclei to pair together. This is accomplished at the cost of
breaking nucleon number conservation in the operator, but nucleon number is still
preserved on average, and the modification can have a significant impact on the
19
result [1].
The other common approach to nuclear matrix calculations, the nuclear shell
model, attempts to capture more fully the dynamics of a nuclear system by including
the full nucleon state space and using nucleon-pair (or higher-order) interactions
which can be measured empirically from small nuclei. The disadvantage compared
to QRPA is its enormous computational demands which make a full shell model
treatment of relevant ??0? nuclei impossible at this point. However, it is possible
to perform nuclear shell model calculations with a severely truncated state space,
and for nuclei whose shape is close to spherical the results can be reasonable. As
computational power increases it is likely that the shell model will overtake QRPA
methods, but for now only a few large-scale shell model calculations of ??0? matrix
elements have been undertaken [1].
One important component of both the shell model and QRPA techniques is
feedback from experimental data. This can serve two purposes. The first is val-
idation: because both approaches make significant approximations, the results of
those approximations must generally be tested empirically to ensure they do not
have adverse consequences for the accuracy of the result. Observables such as nu-
clear energy levels and emission spectra are commonplace; only recently, however,
have precision decay rates for ??2? decay begun to appear in the literature [74].
Although there are many differences between the calculations of M2? and M0? , still
it is currently the best source of validation for M0? calculations available, so this can
help researchers to improve their understanding of the appropriate approximations
for double-beta decay matrix elements.
20
The other benefit of experimental data is that it can help to constrain in-
put parameters to the calculations. In QRPA, a parameter called gpp controls the
strength of phonon-phonon interactions and needs to be fixed from experimental
data; the double-beta matrix elements depend strongly on the value of this param-
eter, and precision observations of ??2? can be used to constrain its value [1, 73].
In the shell model, the extreme truncation of the single-particle state space results
in a need to renormalize the axial vector current gA, which requires a related ob-
servable to control that renormalization; precision observations of ??2? can be an
appropriate method to constrain that value as well. One disadvantage to the use of
??2? decay rates to constrain parameters which are input to calculations is that the
same observations can no longer be used to validate the approximations for double-
beta decays [73]. A description of efforts to obtain new experimental data useful for
nuclear theory can be found in [75].
When the results from modern shell model and QRPA calculations are com-
pared, it is found that the results can differ by as much as a factor of 2-3 [1].
Provided that there is no systematic effect which impacts both methods similarly,
this permits us to understand in a rough way the level of accuracy provided by these
calculations. Progress in reducing these differences continues primarily by increas-
ing the number of single-particle states that can be included, but it is clear that for
the near term the search for ??0? will only be able to set rough limits on Majorana
neutrino mass ?m??? of equation 2.7.
21
2.4 Neutrino Flavor Physics
Neutrinos are known to exist in three flavors, or eigenstates which are diagonal with
respect to the lepton interaction terms of the Standard Model. These flavors are
?e, ??, and ?? ; they interact, respectively, with e, ?, and ? leptons. We also expect
there to be a basis in which the neutrino mass matrix is diagonalized, and these two
bases are not the same. We call the mass eigenstates ?1, ?2, and ?3 with respective
masses m1, m2, and m3. The unitary operator which translates between the two
bases is specified by:
?
?
?
?
?
?
?
?
?e
??
??
?
?
?
?
?
?
?
?
= U
?
?
?
?
?
?
?
?
?1
?2
?3
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
Ue1 Ue2 Ue3
U?1 U?2 U?3
U?1 U?2 U?3
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?1
?2
?3
?
?
?
?
?
?
?
?
. (2.11)
These formalities are uninteresting if the neutrino sector is massless, since the
mass eigenstates are degenerate. However, in the case where neutrinos are massive,
we can see that neutrinos which are created in one flavor eigenstate will oscillate
between the flavor eigenstates with a period which depends on the differences be-
tween masses, as is typical in an N-state quantum system. In the neutrino sector,
we can more specifically state that the probability for a transition from flavor ? to
flavor ? will be: [76]
P?? = ??? ? 4
2?
i=1
3?
j=i+1
Re
[
U?iU??iU??jU?j
]
sin2
([
m2i ?m2j
]
L
4E
)
(2.12)
22
where L is the distance (or time, in c = 1 units) between neutrino source and
destination, and E is the relativistic energy of the emitted neutrinos.
The transition probability is sensitive to the masses of the neutrinos, but
only in the form
?
?m2i ?m2j
?
?. These measurements have now been performed in
a variety of neutrino oscillation experiments, and the best current constraints are
|m21 ?m22| = (7.50? 0.20) ? 10?5eV2 [77] and |m22 ?m23| = 2.32+0.12?0.08 ? 10?3eV2 [78].
However, oscillation experiments cannot constrain the overall mass scale of
neutrinos; they can only set a conservative lower limit that max(m2,m3) ? 0.048
eV if we assume one of m2 or m3 is zero. Furthermore, they do not establish the
sign of the difference. We can see that m1 and m2 are fairly close together, and
m3 is significantly different; but we cannot see whether m3 is larger or smaller
than the other two masses. We refer to the situation with m1 ' m2  m3 as
the normal hierarchy, and m3  m1 ' m2 as the inverted hierarchy; the regime
where m1 ' m2 ' m3  |m22 ?m23| is called the degenerate region. Distinguishing
between these three situations is one of the significant open questions in neutrino
physics because of its impact on observable quantities.
2.5 Particle Physics Constraints
To produce more detailed constraints on neutrino physics, it is generally useful to
provide a parametrization of the mixing matrix U from equation 2.11. The standard
23
parametrization is:
U =
?
?
?
?
?
?
?
?
Ue1 Ue2 Ue3
U?1 U?2 U?3
U?1 U?2 U?3
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
1 0 0
0 cos(?23) sin(?23)
0 ?sin(?23) cos(?23)
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
cos(?13) 0 sin(?13)e?i?
0 1 0
?sin(?13)ei? 0 cos(?13)
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
cos(?12) sin(?12) 0
?sin(?12) cos(?12) 0
0 0 1
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
ei?1/2 0 0
0 ei?2/2 0
0 0 1
?
?
?
?
?
?
?
?
.
(2.13)
Out of the six parameters used in defining this matrix, the only ones which have
been measured are the three mixing angles sin2(2?12) = 0.857+0.023?0.025 [77], sin2(2?13) =
0.089? 0.010(stat)? 0.005(sys) [79], and sin2(2?23) > 0.95 [80]. The Dirac phase ?
is in principle observable from oscillation experiments, but no current experiments
have achieved the sensitivity necessary to accomplish this. The Majorana phases ?1
and ?2 cannot be extracted from neutrino oscillations [1].
The sum of the three mass eigenstates, M =
?
mi, can be constrained by
cosmological observations. This constraint, like all cosmologically-based constraints,
is model-dependent; it relies on the expectation that low-mass, hot forms of dark
matter similar to neutrinos promote the formation of large-scale structures in the
early universe by allowing extremely remote regions of matter to remain in thermal
24
equilibrium. Recent results from Planck combined with WMAP and baryon acoustic
oscillations have restricted M < 0.230 eV with 95% confidence [81]. Considering
the assertion in section 2.4 that the heaviest neutrino must have a mass no less
than 0.048 eV, we can see that this cosmological constraint pushes M to within a
factor of five of its lower limit. Taken at face value, the Planck measurement is the
strongest existing constraint on the absolute mass scale of neutrinos.
Closer to home, the mass of neutrinos is also reflected in ? decay, d? u+e?+
??e. The total energy of the daughter products is known, and is shared between the
electron and antineutrino; the minimum energy of the antineutrino is its rest mass,
so by searching for the maximum energy of the electron we can simultaneously
measure the rest mass of the neutrino. The electron anti-neutrino emitted from
beta decay is a mixture of all three mass eigenstates; since no current or planned
experiment has sufficiently good energy resolution to resolve the separate endpoints
from the three neutrino masses, we can instead write an effective rest mass of an
electron antineutrino as: [1]
?m??2 =
?
i
m2i |Uei|2 . (2.14)
To measure ?m?? we must observe the electron energy spectrum of beta decays
at the endpoint, which is complicated by the fact that this portion of the electron
spectrum contains only a small fraction of the total electron statistics. Tritium
(3H) is commonly used for these experiments because it has a medium-length half-
life of 12.3 years and an extremely low ? decay endpoint energy of 18.6 keV, which
25
Figure 2.7: The electron spectrum of Tritium (3H) ? decay. The
endpoint contains only a small fraction of the total statistics. Figure
from [82].
maximizes the relative shift in endpoint energy due to ?m??. Figure 2.7 shows that
for ?m?? = 1 eV in Tritium, experiments must observe a shift in the spectrum which
affects only about one decay in 5 ? 1012, making this level of sensitivity extremely
difficult to achieve. The best existing limit from ? decay is ?m?? < 2.05 eV, with
95% confidence, from the Troitsk experiment which ran from 1994 to 2004 [83]. The
KATRIN experiment hopes to achieve a sensitivity of 0.2 eV, and is expected to
begin taking data in 2015 [84,85].
26
2.6 Nuclear Physics Constraints from ??0?
As stated in equation 2.7, it is possible to relate the rate of ??0? decay to the
effective Majorana neutrino mass ?m??? by:
[
T 0?1/2
]?1 = G0?(Q??, Z) |M0? |2 ?m???2 , (2.15)
where the phase-space factor G0?(Q??, Z) and nuclear matrix element M0? have
been described in section 2.3. We can now proceed to relate the effective Majorana
neutrino mass to the parametrization of section 2.4. We follow this relation with a
discussion of the considerations which affect a ??0? decay search and a summary
of the current state of the field.
The effective Majorana neutrino mass ?m??? comes from a combination of the
three mass eigenvalues described in section 2.4. It takes the form:
?m??? =
?
?
?
?
?
?
k
mkU2ek
?
?
?
?
?
. (2.16)
Unlike ?m??, which is an incoherent sum of strictly positive terms in equation 2.14,
we can see that ?m??? is a coherent sum of terms, each of which may have arbitrary
complex phase which may increase or decrease the result of equation 2.16. In other
words it is possible, even if neutrinos do have Majorana mass, for ?m??? to be
arbitrarily small if U is tuned to produce cancellations between terms, specifically
by tuning the Majorana phases and selecting the normal rather than inverted mass
27
hierarchy [86].
It is possible to relate the observables ?m???, ?m??, and M within the model
for U of equation 2.13. These relations are shown in figure 2.8; we recall from sec-
tion 2.5 that beta spectrum measurements constrain ?m?? < 2.05 eV and cosmolog-
ical observations constrain M < 0.230 eV, both with 90% confidence. The strongest
constraints on ?m??? from ??0? searches place ?m??? < 0.15 ? 0.4 eV, depending
on the choice of matrix element calculations. We can see that if the cosmological
limits are to be trusted, they provide the strongest constraints on the neutrino mass
parameters; however, all three observables are complementary, and the wide range
of experimental approaches means that systematic effects are unlikely to be shared
by all three methods. The figure shows the relation between these parameters in
blue for the normal hierarchy and red for the inverted hierarchy; we note that it
is only for the normal hierarchy at equation 2.16 can lead to an extremely small
?m???. In the case of the inverted hierarchy we can see ?m??? > 0.013 eV, only an
order of magnitude lower than the current limits.
The sensitivity of an experiment for measuring T 0?1/2 can be described by the
approximate formula: [1]
T 0?1/2(n?) =
4.16 ? 1026yrs
n?
( a
W
)
?
Mt
b?E , (2.17)
where M is the mass of material, a is the isotopic enrichment, W is the molecular
mass of the material in atomic units, and t is the livetime of the experiment;  is
the signal efficiency, b is the background rate (in counts per kg keV year, or some
28
Figure 2.8: The relationship between the effective Majorana mass ?m??? and other
fundamental neutrino quantities: the lightest neutrino mass eigenstate mmin, the
sum of mass eigenstates M =
?
mi, and the effective single-beta-decay neutrino
mass ?m??. Black (magenta) lines indicate the allowed region for the inverted (nor-
mal) hierarchy; red (blue) hatches indicate uncertainty for the inverted (normal)
hierarchy due to the unknown CP-violating and Majorana phases ?, ?1, and ?2 [86].
29
similar units), and ?E is the energy resolution of the detector at the Q-value; and
n? is the desired confidence limit, in sigmas, where the standard 90% confidence
limit will require n? = 1.64. The scaling of this equation is most accurate when the
energy resolution is much smaller than the Q-value and the background is uniformly
distributed in energy; however, when combined with phase factor and nuclear matrix
element estimates it roughly allows us to compare the sensitivity of different ??0?
experiments.
According to equation 2.17 we should prefer experiments for which:
? A large quantity of highly-enriched isotope can be obtained.
? Signal detection is highly efficient.
? Background contamination around the Q-value is small, and does not scale
with detector mass.
? Good energy resolution can be achieved.
The leading experiments which are planned or currently searching for ??0?
are as follows:
1. The leading search for 76Ge comes from the GERDA experiment at the Gran
Sasso Laboratory in Italy. GERDA consists of an array of cryogenic germa-
nium detectors with charge readout. 76Ge has a Q-value of only 2039 keV,
giving it a lower phase factor than other popular materials. It also is expen-
sive to grow large uniform crystals of germanium; this means that it is difficult
for a germanium experiment to take advantage of self-shielding from external
30
radioactive backgrounds. However, its strongest advantage is its excellent en-
ergy resolution: GERDA has achieved an energy resolution at its Q-value of
1.1-1.7 keV (?) with its newer crystals [15].
2. The best results with 130Te have come from the CUORICINO experiment
which ran at the Gran Sasso Laboratory in Italy. CUORICINO was a bolo-
metric experiment: it cooled tellurium crystals to extremely low temperatures
where the heat capacity becomes small, so that a decay inside the tellurium
creates a measurable change in temperature. Similarly to 76Ge, it is expensive
to grow large crystals of tellurium, so most tellurium experiments use an ar-
ray of detectors. 130Te has a Q-value of 2528 keV, somewhat larger than that
of 76Ge, but the energy resolution of the enriched crystals was 2.1-10.6 keV
(?), depending on the crystal. CUORICINO stopped running in 2008 [20];
planned experiments in 130Te include CUORE [87] and SNO+ [88], both of
which anticipate data-taking beginning in 2015.
3. The two leading experiments in 136Xe are KamLAND-Zen, located at the
Kamioka Observatory, and EXO-200, located in the WIPP facility. 136Xe also
has a Q-value of 2458 keV, giving it a higher phase-space factor than 76Ge
but not 130Te. KamLAND-Zen dissolves its xenon in a liquid scintillator and
observes only scintillation energy; it achieves an energy resolution of 103 keV
(?), modest compared to the resolution achieved in tellurium and germanium,
but its monolithic nature allows it to keep backgrounds low [13]. EXO-200 is
a pure liquid xenon detector which observes both ionization and scintillation.
31
Isotope T 0?1/2 (years) M0? G0? (yrs?1) ?m??? (eV) Collaboration
48Ca 5.8 ? 1022 2.28 2.48 ? 10?14 3.7 ELEGANT IV [90]
76Ge 2.1 ? 1025 5.98 2.36 ? 10?15 0.24 GERDA [15]
82Se 3.6 ? 1023 4.84 1.02 ? 10?14 1.1 NEMO-3 [91]
96Zr 9.2 ? 1021 2.89 2.06 ? 10?14 8.0 NEMO-3 [92]
100Mo 1.1 ? 1024 4.31 1.59 ? 10?14 0.56 NEMO-3 [27]
116Cd 1.6 ? 1022 3.16 1.67 ? 10?14 6.1 NEMO-3 [91]
130Te 3.0 ? 1024 4.47 1.42 ? 10?14 0.34 Cuoricino [21]
136Xe 1.1 ? 1025 3.67 1.46 ? 10?14 0.22 EXO-200 [89]
136Xe 1.9 ? 1025 3.67 1.46 ? 10?14 0.16 KamLAND-Zen [13]
150Nd 1.8 ? 1022 2.74 6.30 ? 10?14 3.4 NEMO-3 [93]
Table 2.1: A listing of the strongest available ??0? limits; all half-life and mass
limits are quoted at 90% confidence. Limits on ?m??? are obtained using phase space
factors from [94] and matrix elements from [95], both chosen for the completeness
of their tabulations. These sources explicitly factor out gA; we use gA = 1.269. For
136Xe, both Kamland-Zen?s published results and the EXO-200 results described in
this work are included in the table.
This, combined with analysis improvements described in the present work,
allows it to achieve a somewhat better energy resolution of 38 keV (?) [89].
Table 2.1 tabulates the most current T 0?1/2 limits in all ??0? isotopes for which
??0? limits have been published. Representative limits on ?m??? are also included;
these come from one particular set of phase space factors and nuclear matrix element
calculations performed using the interacting-boson model; this pair of sources for
calculations is made because they include tabulations for a broad range of isotopes
and permit comparisons across all available half-life limits [94, 95]. However, one
should bear in mind that errors in the matrix elements propagate to errors in ?m???
of as much as a factor of two for each isotope. We can see that although there are
a favored set of isotopes, active and successful programs exist in a wide range of
isotopes, and no one isotope is ideal in all respects.
As the table indicates, 136Xe has provided some of the strongest constraints
32
on ?m??? in spite of its relatively modest energy resolution. The present work
demonstrates significant improvements to the energy resolution observed by the
EXO-200 detector in 136Xe which have been achieved through offline denoising of the
scintillation signals. This denoising technique is applied to data from the detector,
and results in a stronger limit on T 0?1/2 from EXO-200 than could be obtained from
the same data without denoising. Chapter 7 presents recent results from EXO-200
which benefit from this denoising technique, as well as increased livetime and other
improvements.
33
Chapter 3: The EXO-200 Detector
This chapter describes the physical apparatus of the EXO-200 detector. Section 3.1
gives a broad overview of the detector. Section 3.2 identifies the dominant back-
grounds for ??0? decay, and sections 3.3 and 3.4 describe methods used to mitigate
these backgrounds. The pulse and waveform readout subsystems are described in
section 3.5, where discussion of the scintillation readout will be particularly im-
portant for subsequent chapters. We conclude with a description of the calibration
systems in section 3.6. Throughout, the reader is referred to the detailed description
in [96] for more information.
3.1 Overview of the EXO-200 Detector
The EXO-200 detector is a cryogenic experiment containing 175 kg of liquid xenon
enriched to 80.6% in 136Xe. Of those 175 kg of liquid xenon, 110 kg are contained
within the ?active? volume where the detector is fully sensitive to deposited energy
from ??0? decay [96], and 94.7 kg are contained within the fiducial volume where
the detector?s response is well-understood. This implies that 3.39 ? 1026 atoms 136Xe
which are for the search [89].
34
Figure 3.1: A two-dimensional energy spectrum of scintillation and ion-
ization from a testbed liquid xenon experiment under an electric field of
4 kV/cm. The spectrum is from a 207Bi source with dominant gamma
lines at 570 and 1064 keV. Figure reproduced from [97].
35
Energy deposits in xenon can be measured primarily in two ways: vacuum
ultraviolet (VUV) scintillation photons are emitted from the de-excitation of xenon
eximers, and xenon atoms are ionized to produce free electrons. It is well-known [97]
that liquid noble element calorimeters show significant fluctuations in their separate
production of scintillation photons and free electrons, but that these separate quan-
tities are strongly anticorrelated. As a result, it is possible to achieve far better
energy resolution if both light and charge are independently measured than if only
one is detected. Figure 3.1 illustrates this phenomenon in a testbed liquid xenon
experiment, where it is apparent that using light and charge simultaneously lets us
observe narrower gamma lines than either would individually; the same is also true
of beta and double-beta lines such as the ??0? decay we seek.
To observe free electrons we must impose an electric field on the xenon which
drifts the electrons onto a collection anode. The EXO-200 detector is shaped as
a cylinder, and the required electric field is produced by placing a cathode grid
in the center of that cylinder and anode wires along each of the two endcaps of
the detector; such a detector is called a time projection chamber, or TPC. The
EXO TPC is shown in figure 3.2. Charge is collected on the anode wires and light
is collected by avalanche photo-diodes (see section 3.5) mounted on the endcaps
behind the anode wires; the time difference between the two signals indicates the
initial position of the charge along the direction of the field. The anode wires are
thin, so the u- and v-wire planes together have 91.8% optical transparency which
does not interfere significantly with APD light collection [96].
The cathode is maintained at a voltage of ?8 kV below the anode wire voltage,
36
Figure 3.2: Schematic of the inner EXO-200 TPC. Figure reproduced
from [96].
37
which is at virtual ground. Field shaping rings encircle the charge drift region to
ensure that electric field lines are parallel and the magnitude of the electric field
is constant in the bulk volume of xenon. (Near the edges, non-uniformities in the
electric field are expected to exist; these are a continuing topic of research.) The
cathode and APDs are separated by 20.4 cm. The bulk volume of xenon has an
electric field of 374 V/cm.
Electrons in liquid xenon drift at a velocity of 1.71 mm/?s under an electric
field of 374 V/cm. The maximum drift distance from the cathode to the anode is
198.4 mm, resulting in a maximum electron drift time of 116 ?s. Free electrons are
not absorbed by xenon ? as for other noble elements, xenon has a low electronega-
tivity, so in pure xenon electrons drift unimpeded indefinitely. The EXO-200 liquid
xenon has small quantities of electronegative impurities which could include oxy-
gen or nitrogen. The exact nature of these impurities is unknown. To minimize
the concentration of these impurities, the xenon is constantly circulated through a
hot zirconium getter which extracts chemically active molecules and permits noble
elements to pass through [96].
It is necessary to associate charge pulses with their corresponding scintillation
pulses because they are separated by an unknown drift time of up to 116 ?s. This
is easily done provided the typical time between events is much longer than 116
?s, and the time difference serves as our measurement of the position of the energy
deposit along the direction of the electric field. Our 1 ?s sampling rate permits us
to reconstruct this position coordinate with an accuracy of 0.42 mm [74].
At the anode, there are two sets of parallel wire planes as shown in figure 3.3.
38
Figure 3.3: Anode collection wires (u-wires) and induction wires (v-
wires) from EXO-200. (1) and (5) indicate the wire support frame; (2)
indicates the wires themselves, constructed as gangs of three wires; (3)
illustrates the attachment between wires and the support frame; (4)
shows the return cables from the wires to the data acquisition system.
Figure reproduced from [96].
39
The first (closer to the cathode) parallel wire plane is called the ?v-wire? plane,
and the second plane is called the ?u-wire? plane. The voltages of the two wire
planes are set so that no drift field lines terminate on the v-wires. Instead, all drift
field lines penetrate through the v-wire plane and terminate on the u-wires. When
charge is deposited in the liquid xenon and drifts toward the anode, it induces a
bipolar current on the v-wires as it passes by, and then produces a unipolar current
on the u-wires as it is collected. See section 3.5 for details on the wire readout and
section 7.1.2 for illustrations of the expected pulse shapes.
The u- and v-wires are rotated sixty degrees relative to each other. Figure 3.3
indicates the orientation of the u-wires and v-wires, and figure 7.27 illustrates from
top-down perspective the u-v coordinate system defined by the wires alongside the
x-y coordinate system used in analysis. By detecting which wires observe a current
pulse in both the u-wire and v-wire channels, it is possible to identify the two-
dimensional location on the anode where the charge arrived. Channels in each
plane are 9 mm wide; by taking advantage of signal sharing which sometimes occurs
between u-wires and always occurs between v-wires, we are able to achieve position
accuracy of 2.4 mm perpendicular to the u-wires and 1.2 mm perpendicular to the
v-wires [74].
Energy which appears to come from a single location is called a charge deposit
?cluster? or ?site?, and events are classified according to the number of sites they
produce (their ?multiplicity?) as either single-site or multi-site. We will see in
section 3.4 that this provides a powerful tool for background characterization and
rejection.
40
This section has described how energy deposits in the liquid xenon are trans-
formed into scintillation and charge, which are then observed by APDs and anode
wires, respectively. These observations are sufficient for us to reconstruct the posi-
tions and magnitudes of energy deposits in the liquid xenon in all three coordinates.
3.2 Backgrounds to ??0? Decay
136Xe has a relatively high Q-value compared to 76Ge; in the neutrinoless mode the
two emitted electrons will share 2456.7 keV [89]. This means that the sensitivity of
T 0?1/2 to the mass of the neutrino is good in 136Xe compared to 76Ge, as was described
in section 2.6; it also means that the energy spectrum around our Q-value will
naturally be relatively free from most sources of background radiation. This section
will identify the types of background which can be expected to affect our ??0?
search; subsequent sections will describe methods of mitigating those backgrounds.
It is common for alpha decays to have energies well in excess of our Q-value.
As a result, we might expect all alpha decays to be possible backgrounds to ??0?.
However, alpha particles are stopped rapidly by even a small quantity of shielding,
and as a result the only alpha decays which can be observed in the detector are those
from sources which are dissolved into the xenon. Radon is the only alpha-emitting
noble element, and in EXO-200 only 222Rn is sufficiently long-lived to diffuse into
the xenon; as a result, we only expect 222Rn and its daughter products to contribute
a significant quantity of alpha decays to the detector.
Cosmic rays produce high-energy muons; these are another source of high-
41
energy backgrounds. Observations using the EXO-200 detector have indicated that
the vertical flux of muons ?v is
?v =
(
4.01? 0.04(stat)+0.04?0.05(sys)
)
? 10?7 Hzcm2sr (3.1)
which implies a depth of 1481+8?6 meters water-equivalent [98]. The depth and muon
flux of WIPP relative to other underground facilities can be seen in figure 3.11.
Muons which pass through the TPC will produce a streak of energy in the detector
rather than discrete clusters; generally they will have sufficient energy to pass fully
through the detector. We can expect that such an event will look substantially
different from a ??0? event, and generally will deposit substantially more energy
than our Q-value as well.
More interesting are the associated by-products from the nearby passage of
such a high-energy particle through the EXO-200 shielding or the surrounding salt,
called spallation products. Although high-energy muons can produce a range of
fission products, the most significant spallation product of muons will be neutrons
which can diffuse into the detector and activate materials there. 134Xe and 136Xe are
both present in significant quantities in the TPC, and by activation can be converted
into the radioactive isotopes 135Xe and 137Xe respectively. 135Xe has an endpoint
decay energy of 1165 keV which is too small for it to be a background to ??0?
decay, but 137Xe undergoes ? decay with an endpoint energy of 4173 keV. We will
see that indeed 137Xe is a significant source of background in our detector [89,99].
Some sources of background are intrinsic to the ??0? search. Any isotope
42
which can undergo ??0? decay can also undergo ??2? decay, and the endpoint of
the 136Xe ??2? spectrum is necessarily at our Q-value. The only means of reducing
background from ??2? is to improve the energy resolution of the detector so that
fewer such decays can mimic ??0? decay. Fortunately with the energy resolution
exhibited by the EXO-200 detector, ??2? is subdominant to other backgrounds by
many orders of magnitude [89].
Similarly, it is hypothetically possible for neutrino absorption to stimulate a
single-? decay which would otherwise have been energetically forbidden; that isotope
may then undergo a second single-? decay with a Q-value greater than the Q-value
of the ??0? decay, acting as a background for our search. For example, in 136Xe the
reaction 136Xe + ?e ?136 Cs + gammas can occur; 136Cs will then undergo single-?
decay with an endpoint energy of 2548 keV and half-life of 13.6 days. This energy is
above the Q-value for 136Xe ??0? decay. However, this process is expected to occur
at quite a low rate, and estimates indicate that it will be a negligible background
until 136Xe sensitivities reach T 0?1/2 ? 1027 years [100,101].
A gamma background which is particularly detrimental to EXO-200 comes
from 214Bi, a member of the 226Ra decay chain. 214Bi emits a gamma at 2448 keV
with a branching ratio of 1.548%, which with our energy resolution is indistinguish-
able from our Q-value. 226Ra has a half-life of 1600 years and generally will be
supported by 230Th (75, 000 years), 234U (250, 000 years), and ultimately by 238U
(4.5 ? 109 years) which has a primordial abundance in the Earth?s crust and most
natural materials [102]. Figure 3.4 shows the 238U energy spectrum as the energy
resolution improves; we can see that even with excellent resolution the 2448 keV
43
10-2
10-1
100
R
at
e 
(a
.u
.) 2% ?/E at 2457 keV
10-2
10-1
100
R
at
e 
(a
.u
.) 1.6% ?/E at 2457 keV
10-2
10-1
100
R
at
e 
(a
.u
.) 1.2% ?/E at 2457 keV
10-2
10-1
100
R
at
e 
(a
.u
.) 0.8% ?/E at 2457 keV
2200 2300 2400 2500 2600 2700
Energy (keV)
10-2
10-1
100
R
at
e 
(a
.u
.) 0.4% ?/E at 2457 keV
Figure 3.4: Representative 238U spectra as the energy resolution of the
detector changes. Red lines indicate the 2? region of interest around the
Q-value. Normalization is arbitrary, but all spectra have the same nor-
malization. Betas and gammas are assumed to have the same calibration
here; alternative beta scales are considered in section 7.4.
44
gamma line will be a problematic ??0? background.
Most other backgrounds to ??0? will be gamma decays from 208Tl, a member
of the 232Th decay chain. 208Tl is produced from 232Th with a 35.94% branching
ratio and emits a gamma at 2614.5 keV with a 99.74% branching ratio; with our Q-
value of 2456.7 keV, the two energies are separated by 157.8 keV [102]. The energy
resolution (in ?/E at our Q-value) of EXO-200 has historically been 1.5?2% at these
energies [89], meaning that separation between the central values of the gamma lines
will be 3 ? 4?. (If betas and gammas have different calibrations then this number
may vary; see section 7.4 for more information on the beta scale.) 232Th has a half-
life of 14 billion years and a significant abundance in most natural materials; we
expect it to contribute a significant fraction of our radioactive backgrounds, and as
a result some events from the 2615 keV 208Tl line will leak across those 3?4? and act
as backgrounds. Improving the energy resolution to 1.5% will reduce contamination
from this gamma line.
The Compton edge is a well-known feature of gamma scattering spectra; it
originates from gammas which enter a calorimeter, scatter once, and escape from
the detector. The maximum energy which a gamma of incident energy Einc can
deposit in a single scatter is: [103]
Edep =
2E2inc
mec2 + 2Einc
. (3.2)
When the incident gamma has an energy of 2615 keV, the Compton edge will lie at
2382 keV, or only 74.5 keV below our Q-value. If our energy resolution is around
45
10-2
10-1
100
R
at
e 
(a
.u
.) 2% ?/E at 2457 keV
10-2
10-1
100
R
at
e 
(a
.u
.) 1.6% ?/E at 2457 keV
10-2
10-1
100
R
at
e 
(a
.u
.) 1.2% ?/E at 2457 keV
10-2
10-1
100
R
at
e 
(a
.u
.) 0.8% ?/E at 2457 keV
2200 2300 2400 2500 2600 2700
Energy (keV)
10-2
10-1
100
R
at
e 
(a
.u
.) 0.4% ?/E at 2457 keV
Figure 3.5: Representative 232Th spectra as the energy resolution of the
detector changes. Red lines indicate the 2? region of interest around the
Q-value. Normalization is arbitrary, but all spectra have the same nor-
malization. Betas and gammas are assumed to have the same calibration
here; alternative beta scales are considered in section 7.4.
46
1.5 ? 2% ?/E at these energies, this separation is 1.5 ? 2?. At energy resolutions
which are better than about 1.4%, we find that the Compton edge begins to con-
tribute more background to our 2? region of interest than the gamma line at 2615
keV; we can see this phenomenon from the energy spectra of figure 3.5.
Figure 3.6 summarizes the background rates from our three dominant back-
grounds, 232Th, 238U, and 137Xe, as they depend on the energy resolution of the
detector. We can see that the background rates of 232Th and 238U do not vary
smoothly with resolution due to the presence of gamma lines; by contrast, the spec-
trum of 137Xe is smooth and its background rates decrease linearly with resolution.
The EXO-200 detector has a time-averaged resolution of 1.94% ?/E without the
denoising technique described in this work, and 1.53% ?/E with denoising; we can
see that this results in a significant reduction of backgrounds from 232Th. How-
ever, improvements in resolution will not decrease the backgrounds from 238U unless
resolutions below 0.5% ?/E are reached, which is not feasible with the current de-
tector. The following sections will identify some of the mechanisms beyond energy
resolution which are used to further minimize these backgrounds.
3.3 Passive Background Rejection
We can distinguish between two classes of background rejection methods: passive
methods, in which backgrounds are reduced by reducing the amount of background
reaching the liquid xenon, and active methods, in which backgrounds are observed
by the detector but discriminated from ??0? signal based on identifying character-
47
0.0 0.5 1.0 1.5 2.0 2.5
Resolution (?/E, %) at 2457 keV
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Ba
ck
gr
ou
nd
 R
ate
s i
n 2
? 
Re
gi
on
 of
 In
ter
es
t (
ar
bi
tra
ry
 un
its
) Th-232
Xe-137
U-238
Figure 3.6: Relative change in background rates expected in our 2?
region of interest as a function of the energy resolution. Rates are nor-
malized to one at a Q-value energy resolution of 1.6% ?/E, the design
goal of EXO-200. The three backgrounds shown, from 232Th, 238U, and
137Xe, are currently our dominant backgrounds. Betas and gammas are
assumed to have the same calibration here; alternative beta scales are
considered in section 7.4.
48
istics. In this section, passive methods will be described; section 3.4 will describe
the active methods of background rejection. The passive methods we will iden-
tify are exploiting the self-shielding properties of xenon, screening the materials for
radioactivity, and putting the experiment deep underground.
The simplest method of reducing the presence of backgrounds is by exploiting
the self-shielding properties of xenon. The xenon itself is extremely pure due to
the ease of chemical purification. As a result most backgrounds will be external to
the xenon. Since external gammas are weakly attenuated by dense materials such
as liquid xenon, which has a density of 3.0305 ? 0.0077 g/cm3 [74], we can expect
that the xenon near the center of the detector will be exposed to less background
than the xenon near the TPC walls. This is one of the primary advantages of xenon
as a source: it is possible to construct a large monolithic detector, maximizing the
quantity of xenon which is shielded from the walls.
Figure 3.7 shows the attenuation lengths of gammas at a range of energies
in xenon. Photoabsorption and pair production both convert the gamma entirely
into short-ranged electron and positron carriers; Compton (incoherent) scattering
results in a fractional energy deposition, with the rest remaining in the gamma which
rebounds and continues on a deflected path. Enriched liquid xenon has a density
of 3.0305 ? 0.0077 g/cm3 [74], which means that the minimal attenuation factor is
0.108/cm at an energy of 4.34 MeV. Unfortunately, this is similar to our Q-value at
2456.7 keV, where the attenuation factor is still only 0.116/cm, which means that
self-shielding is minimally effective for ??0?. Nevertheless, there is some reduction
in backgrounds deeper in the interior of the xenon, and lower-energy gammas are
49
10-2 10-1 100 101 102
Gamma Energy (MeV)
10-5
10-4
10-3
10-2
10-1
100
101
102
103
A
tte
nu
at
io
n 
in
 X
en
on
 (
cm
2
/g
)
Compton Scattering
Photoelectric Absorption
Pair Production in Nuclear Field
Total Attenuation (w/o Coherent Scattering)
Figure 3.7: X-ray attenuation lengths in xenon. Compton scattering,
pair production, and photoelectric absorption are shown, along with
their combined attenuation length; coherent (Rayleigh) scattering is
omitted because it produces no observable energy deposit. The Q-value
of 136Xe ??0? decay is indicated with a dashed black line. The vertical
axis, in cm2/g, can be multiplied by the density of the xenon to derive
an attenuation factor per unit length. Data from [104].
50
attenuated more effectively [104].
To reduce the quantity of background around the detector, all materials near
the xenon were carefully screened for radioactive contamination [105]. Backgrounds
from 40K, 232Th, and 238U were cataloged for all materials which have unshielded
line-of-sight access to the detector system. Requirements for 238U are particularly
stringent because its daughter products include the 214Bi background which cannot
be resolved from ??0? decay. For the copper of the TPC vessel and the lead shielding
around the detector apparatus, samples from a range of companies and mines were
tested to locate an optimal choice of material source. For the wires which were
placed into the TPC, a range of manufacturing methods were tested; improvements
to a photo-etching scheme were identified which led to a reduction in background
from these materials [96]. This thorough material screening research is one of the
distinctive processes which has enabled EXO-200 to achieve its background goals.
Details of the quantification of and constraints on material radioactivity can be
found in [105].
Beyond selecting extremely clean materials, it is possible to reduce back-
grounds by minimizing the mass of these external materials. Most notably, the
EXO-200 copper TPC vessel is, in most places, only 1.37 mm thick. To maintain
structural integrity, supporting structures are welded to the TPC where needed.
The total mass of copper is less than 30 kg [96].
Some materials were dispensed with entirely. Typically, silicon APDs are
encapsulated with ceramic to isolate them from water contamination and provide
electrical insulation. However, this ceramic material would have contributed back-
51
Figure 3.8: Wire triplet, read as one channel. Figure from [96].
grounds. Instead, the APDs were delivered ?bare,? without any encapsulation, and
protected from water by storing them in a dry-nitrogen container. Liquid xenon
itself serves as an electrical insulator, ensuring the APDs would function properly
during detector operations [106].
The cabling of the wires and APDs was also done with great care. The elec-
trical amplifiers and digitizers contain many high-background plastics, metals, and
other complex materials, so they were placed outside the lead shielding rather than
close to the TPC. Wires connect the sensors in the TPC to these electronics; or-
dinarily such long wires would need to be shielded by coaxial cabling to minimize
cross-talk noise. However, the coaxial cabling was expected to contribute a high
52
Figure 3.9: APDs ganged together (bottom right). Wiring to the front-
end electronics are visible as yellow ?tape.? Figure from [96].
quantity of radioactive background, and was omitted; instead wires are partially
electrically shielded by surrounding them with inactive wires which reduce cross-
talk noise. Furthermore, the total number of wires needed was reduced by ganging
together triplets of u- and v-wires and groups of six or seven APDs into single chan-
nels. This reduced the fineness of event information available in analysis, but also
reduced the quantity and complexity of material placed near the detector. The wire
gangs are shown in figure 3.8; APD gangs cabling are visible in figure 3.9 [96].
The TPC is shielded by nested layers of clean material designed to prevent
backgrounds from reaching the xenon. In the innermost layer, HFE-7000 refrigerant
(see figure 3.10) is used to maintain the detector at liquid xenon temperatures and
to shield from external gammas. However, the HFE-7000 has a significant quantity
of hydrogen, which also makes it an excellent stopping agent for thermal neutrons.
53
VETO PANELS
DOUBLE-WALLED 
CRYOSTAT
LXe VESSEL
LEAD SHIELDING
JACK AND FOOT
VACUUM PUMPS
FRONT END 
ELECTRONICS
HV FILTER AND 
FEEDTHROUGH
Figure 3.10: A cutaway schematic image of the TPC and surrounding
materials in the cleanroom. HFE-7000 refrigerant is contained between
the cryostat and the LXe vessel. Figure from [96].
54
The refrigerant layer is more than 50 cm thick, so most external neutrons thermalize,
absorb onto hydrogen, and emit a 2.2 MeV gamma rather than reaching the TPC
and producing 137Xe with its more harmful 4173 keV beta endpoint energy [96].
The following layer of shielding is copper cryostat, then lead, shown in fig-
ure 3.10. This is a common approach for low-background experiments due to the
high density of lead, which enables it to stop gammas in a short distance. Lead
bricks surround the detector on all sides; the bricks were designed with an interlock-
ing shape to ensure no seams between bricks left a line-of-sight path from external
sources into the TPC. In the front of the detector, where plumbing must enter and
exit the detector through the first lead wall, a second lead wall is assembled with
the purpose of blocking any line-of-sight paths into the TPC through the plumbing
gaps of the first wall [96].
Outside of the lead shielding, restrictions on materials are less stringent; how-
ever, control over the presence of materials is still desired. The entire apparatus is
contained inside a class-1000 cleanroom facility. The entire facility is located in the
Waste Isolation Pilot Plant (WIPP) facility, a Department of Energy waste reposi-
tory located in a salt formation near Carlsbad, NM. The overburden of the facility
is 1481+8?6 meters water-equivalent [98]. Figure 3.11 shows the reduction in muon
flux due to depth at the WIPP site and a selection of other underground science
facilities.
These passive approaches have all been demonstrated to reduce the rate of
backgrounds depositing energy in the liquid xenon. The following section will de-
scribe an active set of background-rejection approaches.
55
Figure 3.11: Muon flux as a function of depth, in meters water equiva-
lent. The WIPP site is indicated in relation to other underground science
facilities. Figure from [107].
56
3.4 Active Background Rejection
This section will describe the ?active? forms of background reduction employed by
the EXO-200 detector, in which events are observed but discriminated from ??0?
events based on their characteristics. We neglect the energy resolution as a back-
ground rejection tool here, since section 3.2 has already described how the impact
of certain backgrounds may be reduced by improvement to the energy resolution.
A number of cosmogenic backgrounds have been described: neutrons and fis-
sion products can be produced from muons penetrating down to the depth of the
WIPP facility. To reject many of these events, it is sufficient to detect the passage of
a muon anywhere near the detector. This is accomplished by a set of plastic scintil-
lator veto panels surrounding the cleanroom on four of six sides. Muons are detected
by coincident PMT hits on opposite sides of a single panel, with (96.0? 0.5)% effi-
ciency for detecting muons which traverse the TPC [96]. Further reduction in the
impact of muon-induced backgrounds is achieved by searching for muon tracks ob-
served within the TPC itself, and also by applying a one-second coincident event
cut as a catch-all for spallation products of muons which may decay multiple times
in rapid succession.
Alpha decays within the xenon are another possible source of high-energy back-
grounds. However, it is possible to discriminate alpha decays from beta and gamma
decays based on the properties of their energy measurement. Alphas produce a far
more dense energy deposit in liquid xenon than betas and gammas; the dense col-
57
lection of free electrons and ionized xenon has a higher rate of charge recombination
than is observed with betas and gammas, which translates into a higher ratio of
scintillation to ionization for alphas than for betas and gammas. This feature can
be used to eliminate essentially all alpha backgrounds.
Discriminating between gamma, beta, and double-beta decays is more difficult.
Betas from external backgrounds will not penetrate into the xenon due to their
short attenuation lengths, so only dissolved beta emitters are backgrounds to ??0?
decay; this constrains the number of plausible beta decays that must be considered.
Gammas generally deposit their energy by ionizing atomic electrons of xenon, so in
this way their behavior closely mimics that of betas. However, most gammas within
our energy window (and particularly gammas near our Q-value) will be attenuated
by Compton (incoherent) scattering, as can be seen in figure 3.7. By this process
the gamma will deposit energy in more than one discrete location, which beta and
double-beta decay will not do. We are generally able to resolve the positions of
these energy deposits and classify the event as multi-site. More than half of external
gamma backgrounds are classified as multi-site, whereas the fraction of ??0? events
classified as multi-site due to bremsstrahlung is only 5% [74].
Finally, the liquid xenon itself is continuously purified and quite low in back-
grounds, so most radioactive backgrounds in EXO-200 are external. We have noted
that 94.7 kg of liquid xenon are treated as fiducial; the remaining quantity of roughly
15 kg of xenon which is in the active volume of the detector is used as an active veto
shield, where any interaction which deposits energy in this volume disqualifies the
entire event from use as a ??0? candidate. Moreover, the fits which will be described
58
in section 7.4 take into account the depth into the detector of an energetic deposit,
giving events near the exterior greater likelihood of being an external background
than events deep inside the xenon. These strategies give us some discriminating
power against external backgrounds.
These are some of the basic techniques employed to reduce backgrounds in
offline analysis, and the detector has been designed with the goal of facilitating
their application. The EXO-200 detector currently has two and a half years of
data with enriched xenon, and although improvements to the passive methods of
background reduction cannot be applied to this existing dataset, it may be hoped
that these active methods may be improved or extended through further analysis.
These improvements could have the potential to reduce backgrounds below their
current levels in offline analysis, making them a particularly attractive target now
that EXO-200 has collected a significant dataset.
3.5 Pulse Amplification and Waveform Readout
This section will provide a brief overview of the EXO electronics and waveform read-
out. Particular attention is paid to subsystems which are relevant to the denoising
algorithm described in chapter 4, including the APDs and front-end electronics. A
schematic of the electronics and readout subsystems can be seen in figure 3.12.
The APDs are constructed from a silicon semiconductor which is doped and
biased so that: [108]
1. Photons arriving at the active surface of the APD can deposit their energy by
59
Figure 3.12: An overview of the EXO-200 electronics and readout sub-
systems. The area inside the dashed line indicates cold subsystems inside
the cryostat or TPC. Figure from [96].
60
Figure 3.13: A cross-sectional schematic of an APD is shown in (a). The
electric field as a function of depth is shown in (b). Figure from [108].
61
generating an electron-hole pair in the silicon semiconductor.
2. The electron-hole pair is drifted apart by an electric field. Here the electric
field is low and the silicon is doped to be of p-type (electron-poor), ensuring a
similar number of electron-hole pairs are produced by all photons before any
amplification can occur.
3. The electron enters a high-field n-type (electron-rich) region of silicon, and
liberates additional electron-hole pairs in an avalanche effect.
4. Electrons reach another p-type region of silicon, and then are collected on a
cathode as an output current pulse.
These phases of the APD are illustrated in figure 3.13.
851 APDs were specially made for EXO-200; from those, 468 were selected
for use based on their favorable noise and gain properties [106]. For each APD, the
relative quantum efficiency and the bias voltage needed to ensure the product of
gain and quantum efficiency equals one hundred were both measured. It is impos-
sible to make the gain of all APDs perfectly uniform because of limitations in the
configurability of the APD bias voltages in EXO-200: APDs are grouped into twelve
pies, and the APD bias voltage within a pie is controlled by a single parameter [96].
To mitigate this effect, APDs were assigned to gangs and pies so that their products
of gain and quantum efficiency could be set to one hundred as uniformly as pos-
sible [109]. We note that the current operating conditions of EXO-200 establish a
higher operating gain of around 200? 300. This may have the side effect of making
the product of gain and quantum efficiency less uniform in each APD gang.
62
In the u-wires, charge is collected and delivered directly to the electronics as
current. In the v-wires, no net charge is delivered, but transient currents are induced
in such a way that the v-wire remains at a constant voltage; these transient currents
produce a bipolar pulse, such that the integral (before shaping) is zero. We note
that the pulse shape depends on the drift path of ionization in the liquid xenon ?
both sets of wires deliver a current signal which is induced from approaching charge
before the charge deposits on the u-wires, and the path of the electrons can affect
the time development and, in the case of the v-wires, the magnitude of that induced
current. Furthermore, ions (being much more massive than electrons) drift much
more slowly through liquid xenon. As a result, the pulse magnitude will depend on
the point of origin of the electrons because from that location the ions will hold a
net charge on the u- and v-wires which is released only on much longer timescales
than we can observe with our electronics [110,111].
The front-end electronics, located between the two lead walls in front of the
detector at room temperature, consist of four stages: [96,112]
1. A transimpedance amplifier which converts the current signals from the de-
tector to voltage signals. This stage applies amplification and a differentiator
with a long time constant (roughly 60?s for wires and 300?s for APDs).
2. Two differentiators and two integrators, to improve signal-to-noise ratio for
real-time triggering. For the u-wires, each integrator has a time constant of
1.5?s and each differentiator has a time constant of 40?s; for the APDs and
v-wires, each integrator has a time constant of 3?s and each differentiator has
63
a time constant of 10?s. Some signal amplification is applied at this stage as
well.
3. An analog-to-digital converter (ADC) converts voltage pulses into 12-bit dig-
ital waveforms. The full 12-bit scale is equivalent to 2.5V. Digitization is
performed at a rate of 1MHz.
4. A triggering module reads the digitized waveforms and issues a trigger to the
data acquisition (DAQ) when signals exceed programmable thresholds. The
triggering module can also accept external requests for a ?solicited? trigger.
In most cases, when a trigger is issued, the DAQ will write out 2048 samples (just
over 2 ms) from all waveforms, centered on the first sample responsible for the
trigger. In cases where the data rate is high, waveforms can be truncated to a
length less than 2048 samples.
3.6 Calibration Systems
Energy resolution has been discussed as a general feature of the detector which
can have a significant impact on the reduction of backgrounds. A significant factor
in the resolution is the quality of the detector?s absolute energy scale calibration.
This is particularly important because of the long lifetime of the experiment. If
the calibration of the detector drifts over time, and this drift is not corrected, this
will be perceived in the cumulative low-background spectrum as a smearing out of
energy peaks, worsening the effective resolution. This section will describe some
64
Figure 3.14: A guide tube permits gamma sources to be inserted to
known locations near the TPC. Green dots indicate a few of the locations
where the source may conventionally be placed. Figure from [96].
of the techniques used to produce an accurate time-dependent calibration for the
energy scale.
The most critical type of calibration comes from external gamma sources. The
EXO experiment was constructed with a guide tube passing through the HFE-7000
refrigerant and wrapping loosely around the TPC, as shown in figure 3.14. Currently
available to the experiment are sources containing 228Th (primary gamma line at
2615 keV), 60Co (simultaneous emission of gammas at 1173 and 1332 keV), and
137Cs (gamma line at 662 keV); additionally, a new 226Ra source became available
in the summer of 2013 (numerous gammas, including 2448 keV). The 228Th and
65
60Co sources are available in weak and strong strengths, where the strong strength
is chosen to produce a data rate that is still low enough that charge-light associa-
tion can usually be done unambiguously. Together these sources allow us to observe
the detector response to external gammas with known energies, permitting energy
calibrations, energy resolution measurements, and verification of Monte Carlo sim-
ulations.
All of these sources have been used periodically (roughly every three months)
in calibration campaigns intended to permit a full characterization of the detector
response across a wide energy range at a particular moment. Additionally, the 228Th
source has been deployed regularly, typically three times every week for 2?3 hours,
for the entire history of the dataset. The richness of the accumulated 228Th source
dataset, along with the benefit that it contains a relatively clean high-energy gamma
line, make it particularly convenient to work with. It will play a prominent role in
the generation of our lightmap described in chapter 6.
Other calibrations are also performed on the detector, and may in some in-
stances supplement the information provided by the source data. To perform direct
calibrations of the APDs, one APD mount contains a light diffuser rather than an
APD device; the diffuser can be illuminated by a laser through optical fibers [96].
The APDs can be calibrated by pulsing the laser while varying the voltages of the
APDs to adjust their gain, and in this way it is possible to measure the gain of the
APDs in situ. A laser system was available at the beginning of the dataset used in
this work; however, there were certain deficiencies to the laser system which made
its data difficult to use, and as a result a new and more accurate laser system was
66
commissioned in August 2012. Laser data in some form has been collected roughly
once per week for much of the time containing this dataset. Composing the laser
data into a coherent history of the APD gains is an ongoing project, but it has been
possible to use APD gain information from specific moments to inform the APD
noise model described in section 4.3.
Just as it is possible and desirable to learn the gain of the APDs separately
from other components using laser runs, it is also desirable to learn the relative
gain of the electronics amplifiers separately from other detector function. This is
accomplished using ?internal calibrations?: simultaneously with software triggers
solicited by the data acquisition, a switched capacitor injects charge into the input
of the electronics cards of individual or groups of channels [113]. The resulting
waveform is then read normally, and the size of the output pulse reflects the gain
of the electronics. This system is employed approximately daily on all channels; it
provides an excellent measure of relative changes in the electronics gains over time.
However, the absolute gains cannot be extracted accurately from this data due to
variances in the capacitors, and must be obtained by other means.
To measure the absolute gains of the electronics, the front-end cards must be
physically removed from their enclosures; when this is done, it is possible to use
a more accurately calibrated capacitor to make an absolute gain measurement in
the same style as the internal calibrations. These ?external calibrations? can be
performed in two flavors: manual, in which a capacitor calibrated to 1% is used to
inject charge, and automatic, in which the calibration is automatically performed
at a much faster rate but accuracy is limited to 2 ? 3%. Due to the inconvenience
67
of removing electronics cards from their enclosures, this process has only been per-
formed once in February 2011. Both styles of external calibration were performed
on the u-wire channels. Less demanding needs for resolution on the v-wires and
APD channels were foreseen, so only the less accurate automatic calibration was
performed on those channels. The data from these runs has not been fully ana-
lyzed, but may be revived in future analyses to permit a better calibrations of the
subsequent internal calibrations which were taken. A few electronics boards have
been swapped since February 2011; channels on those boards will no longer have
an absolute calibration available after the board swap, so it may be advisable to
perform an additional external calibration [110].
3.7 Summary
This chapter has described some of the key characteristics of the construction and op-
eration of the EXO-200 detector. Background reduction has been a theme through-
out ? EXO-200 has employed both passive and active methods to reject background
and improve its sensitivity to ??0? decay. Energy resolution has also been described
as a key element in background rejection because the 2? region of interest shrinks
linearly with energy resolution, so with sufficiently precise energy measurements the
rates of most backgrounds will decrease. Chapter 4 will discusses a new scheme to
extract more accurate energy measurements from the APD channels, thereby im-
proving our overall energy resolution and the sensitivity of our ??0? half-life limit.
68
Chapter 4: Denoising Theory and Imple-
mentation
We have shown in section 3.2 that if it is possible to improve the energy resolution
of EXO-200 then backgrounds, particularly from 232Th and 137Xe, can be reduced.
The energy resolution of EXO-200 is limited by its scintillation energy resolution, so
any attempt to improve the energy resolution should focus on improved extraction
of APD pulse magnitudes. In this chapter we present a new technique for extracting
optimal scintillation energy measurements based on an improved model of the pulses
and noise on APD waveforms. The estimator derived in this chapter is optimal
partly due to its ability to operate in the presence of noise; we therefore describe it
throughout as a ?denoising? technique, though at no point are denoised waveforms
produced.
We begin with a general overview in section 4.1 of what qualitative methods
of noise reduction a denoising technique may hope to apply and why they are neces-
sary for our APDs. Notational conventions and some crucial input parameters are
listed in section 4.2. Section 4.3 presents a detailed physics model of the fluctua-
69
tions in observed pulse magnitude, and section 4.4 connects that physics model to
observable output of the APD channels. The optimal energy estimator is presented
in section 4.5 with a derivation. Sections 4.6-4.9 translate the optimal energy esti-
mator into a usable form and describe the computational steps needed to apply it to
data. Differences between the optimal version of denoising presented in this chap-
ter and the version used for the present physics analysis are shown in section 4.10.
We finish in section 4.11 by identifying some extensions to the denoising technique
presented here which may prove interesting for future analyses.
4.1 What is Denoising?
Before the investigations described in this work, scintillation energy was not mea-
sured from individual APD channels. Instead, two summed waveforms were con-
structed: a sum of all waveforms from APDs on the North plane, and a sum of
all waveforms from APDs on the South plane. This simplified the construction of
a position-dependent correction function, or lightmap. (For more details on why
this is so, see section 6.1.) However, when we sum waveforms together we discard
information; this section shows that the information which is thrown away is in fact
critical to improving our scintillation energy measurements.
Energy resolution is a critical factor in the strength of the EXO-200 exper-
iment. Figure 4.1 shows the energy resolution trend which was observed before
denoising was developed. We can see that there was a long period of worsening res-
olution from March 2012 to February 2013, when the single-site energy resolution
70
Dec 2011 Feb 2012 Apr 2012 Jun 2012 Aug 2012 Oct 2012 Dec 2012 Feb 2013 Apr 2013 Jun 20131.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
Res
olu
tion
 (%
 ?/
E)
SS Resolution without denoising
MS Resolution without denoising
50
60
70
80
90
100
110
120
RM
S N
ois
e (A
DC
 co
unt
s)
Summed APD Noise
Figure 4.1: Detector energy resolution (without denoising) is strongly
correlated with noise observed on the APDs. Noise data provided by
Josiah Walton.
changed from 1.4% to 1.9% ?/E at our Q-value, followed by a period of improve-
ment to 1.6% in July 2013. There has always been interest in ways to improve the
energy resolution, but the upward trend gave greater urgency to understand the
observed fluctuations, reverse them, and find a way to counter the effect in offline
analysis for existing data.
A search began for environmental changes which could be correlated to the
worsening energy resolution. In addition to the energy resolution trend, figure 4.1
overlays a trend of the root-mean-square noise on the sum of APD waveforms, and
we can immediately see that the overall energy resolution is closely tied to electronic
noise on the summed APD channels.
The design goal of EXO-200 was for individual APD channels to have root-
mean-square noise levels of 2000 electrons, and this goal was met [114]. However,
71
Figure 4.2: Coherent and incoherent noise power spectra for a sample set
of APD channels without pulse shaping. The coherent noise power spec-
trum is measured by adding together waveforms from different channels
and computing a power spectrum; the incoherent noise power spectrum
is measured by computing power spectra for each channel individually
and adding them together [114].
72
Figure 4.3: APD waveforms from a single event. No energy deposit oc-
cured during this event, so the waveforms represent pure electronic noise.
The horizontal axis indicates time, and the vertical axis indicates channel
number; colors ranging from blue to red indicate the baseline-subtracted
waveform magnitude. Vertical streaks are indicative of correlations in
noise across channels.
rather than observing the noise on summed APD channels increase proportionally to
the square root of the number of channels, the summed APD noise is roughly two to
three times higher than projected. Figure 4.2 compares the power spectrum of the
summed APD waveforms to the sum of power spectra of individual APD waveforms;
the power spectrum of the sum is 2-3 times higher than the sum of power spectra.
These observations indicate that the phases of the noise frequency components on
different channels are positively correlated, leading to a worse impact on energy
resolution than the same level of uncorrelated noise would give.
73
Electronics Board APD Channels
0 158-163, 176-182
3 189-195, 202-207
4 208-213, 220-225
5 196-201, 214-219
6 170-175, 183-188
9 152-157, 164-169
Table 4.1: Electronics boards and their corresponding channels, as defined during
the time range from February 2012 to this writing. Although electronics board
numbers are different at other times, their corresponding channel numbers do not
change.
The correlations in noise across channels are visible even by eye. Figure 4.3
shows a single noise event from the detector taken on January 30, 2013, when noise
levels were near their highest. The horizontal axis represents time, the vertical
axis represents channel number, and color indicates the magnitude of the baseline-
subtracted waveform. If noise on different channels were uncorrelated, we should
see no relation between the noise on different horizontal rows (channels) of the plot.
Instead, vertical streaks can be seen which indicate noise which is in phase across
multiple channels. Table 4.1 identifies the channel numbers corresponding to each
electronics board, and we can see that the correlations are strongest for channels on
the same board.
The observation that noise is correlated across channels has provided an im-
portant clue to the origin of the noise, and investigations led to the discovery that
switching power supplies and a switched voltage regulator introduce noise into the
electronics; these are magnified by a ground loop in the electronics and a pre-
amplifier design which is not optimized to reject noise in its power supply. Efforts
are underway to address these issues in hardware [115,116].
74
The observation that coherent APD electronic noise is the limiting factor in
our energy resolution also implies that we can mitigate this noise in offline analysis.
Although the correlations in noise initially worsen our energy resolution due to their
linear scaling with the number of channels, knowledge of these correlations permits
us to take advantage of the redundancy in the noise and reduce its overall magnitude.
By observing that the majority of our APD noise is correlated, we have learned that
we can eliminate the majority of our APD noise in offline analysis and improve the
quality of data which has already been taken. Efforts to denoise the data in this
way have been made by many members of the EXO collaboration; here we describe
the success of one of these denoising approaches.
Although the previous discussion emphasizes the electronic noise which induces
time variations in our energy resolution, we note that this is not the only source of
noise in APD waveforms. The types of noise which are considered in the present
work are:
? Electronic noise. More generally, we consider collectively additive noise which
is uncorrelated with pulses. APD dark current is another possible source of
such noise.
? Fluctuations in photon collection efficiency.
? Fluctuations in APD gain.
The last two are sometimes collectively described as Poissonian noise because they
each introduce a variance term which is roughly linear in the energy of the event.
75
We note that the electronic noise scales linearly with the number of channels
used, in the case where it is dominated by coherent noise; by contrast, Poissonian
noise decreases proportionally to the square root of the number of channels used,
in the case where all channels are expected to have similar mean photon collection
and gain. Thus, there are competing forces to form our energy estimates from a
small number of channels to minimize the electronic noise and from a large number
of channels to minimize the Poissonian noise. Finding the optimal balance between
these two extremes is one of the challenges of denoising.
The energy estimator which we derive in this chapter is an optimal linear
estimator, and we do not attempt to control how it produces its estimates. However,
we still can identify a number of qualitative approaches to reducing the noise levels
in the scintillation channel. We refer to ?passive? approaches as those in which
components of our waveforms are weighted based on their relative signal-to-noise
content. ?Active? approaches, by contrast, are classified as those which attempt
to improve the signal-to-noise content of the waveforms. We identify the following
three types of denoising:
Frequency weighting
On a given channel waveform, weight more heavily the frequency components
which contain larger signal-to-noise ratio. This passive denoising scheme re-
quires knowledge of the power spectra of the pulse signal and the noise.
Channel weighting
Different channels may have differing levels of noise, so some may generally
76
have lower signal-to-noise ratios. More importantly, though, the pulse magni-
tude on a given channel depends strongly on the proximity of the APD gang to
the source of scintillation within the detector. This passive denoising scheme
therefore consists of weighting more heavily the channels which have larger
expected pulse magnitudes, as determined on an event-by-event basis.
Noise cancellation
This active form of denoising consists of using correlations between noise on
multiple channels to produce a better estimate of the noise component of wave-
forms than each waveform taken independently could provide. To accomplish
this, we require detailed information about the pairwise noise correlations
across channels at each frequency.
Section 4.2 specifies the mathematical framework for denoising. This frame-
work is general enough to include all of the features described in this section.
4.2 Notational Conventions and External Input
In this section, we first establish a set of notational conventions which clarify the
mathematics of this chapter. We then describe the overarching framework in which
denoising takes place, taking care that the framework is general enough to include
all of the qualitative features described in section 4.1.
Throughout this chapter we use the following notational conventions:
? i, j represents indices over APD channels (or equivalently, APD gangs).
77
? a, b, c represents indices of scintillation clusters in an event; a scintillation
cluster is a set of simultaneous energy deposits which may occur at multiple
locations.
? ? represents the time indices of a discrete-time waveform; ta represents the
calendar time of a scintillation deposit a.
? f , g represents the frequency indices of Fourier-transformed waveforms.
? ?ij is the Kronecker delta, equal to 1 if i = j and 0 otherwise.
? For a waveform ?[? ], we represent the discrete Fourier transform of that wave-
form with ??[f ], where the particular convention used to evaluate the Fourier
transform is not significant.
? For a Fourier-transformed waveform ??[f ], we denote the real and imaginary
parts of that waveform by ??R[f ] and ??I [f ], respectively.
? For an unknown parameter p, the symbol p? identifies an estimator for p.
? For an expression f(?) containing random variables, ?f(?)? is the expectation
value of f(?). Here the expectation value is meant in a frequentist sense: if we
could repeat the experiment of generating the same number of initial photons
in the same places, ?f(?)? would be the average value of f(?) from many trials
of this hypothetical experiment. The expectation value of f(?) conditional on
the value of some random variable X is written ?f(?)|X?.
? When a bare index appears on both sides of an equation, the equation holds
78
for all possible values of that index; for example, fi(?) = gi(?) would mean that
for all possible indices i the equation holds.
? Einstein notation is not used; all summations are written explicitly.
? All energies are in units of 2615 keV (the energy of the 208Tl gamma line). This
energy scale is natural for us because the lightmap of chapter 6 is measured
with events from that gamma line.
We describe the data as a collection of discretely sampled waveforms, Xi[? ].
We assume that all pulse times and shapes are already known, and only their mag-
nitudes need to be extracted; so we can model the waveforms by
Xi[? ] =
?
a
MiaYia[? ] +Ni[? ] + bi, (4.1)
where Yia[? ] is the template function of the pulse caused by scintillation cluster a on
channel i, Mia is the unknown magnitude of that pulse, and Ni[? ] and bi represent
the electronic noise and baseline, respectively, of the channel.
To break the degeneracy between M and Y we must fix the magnitude of the
template function Y . We choose to require that the function Y has a baseline of
zero and a peak magnitude of one, as illustrated in figure 4.4.
The magnitude of a pulse is a random variable which depends on both the
energy of the scintillation cluster and on random fluctuations detailed in section 4.3.
We assume that the expected magnitude ?Mia? of a pulse on channel i from a single-
site 2615-keV deposit is known, and is characterized as a function Li(~xa, ta) of the
79
Figure 4.4: Shaped and unshaped APD waveforms. The normalization
is shown to make the peak of the shaped waveform have a magnitude
of one, and the time axis is shifted so that the unshaped waveform is a
step function centered at t = 0.
80
deposit position ~xa and calendar time ta. We can thereby characterize the expected
yields Mia from a scintillation cluster with energy Ea (in units of 2615 keV) as:
?Mia? = Li(~xa, ta)Ea. (4.2)
The exact methods for measuring Li(~x, t) is described in chapter 6.
In cases where a multi-site scintillation cluster deposits energy in multiple
locations, we use the charge information to estimate the energy deposited in each
location; the estimated lightmap yield LMSi from a multi-site scintillation cluster is
a weighted sum:
LMSi (~x1, . . . , ~xnmax , t) =
?
nEchargen Li(~xn, t)
?
nE
charge
n
, (4.3)
where the index n ranges over the charge clusters associated with this scintillation
cluster. For notational simplicity, we still write the expected yield on a channel i
from a multi-site scintillation cluster as Li(~x, t); in practice it is always clear how
to compute the expected yields for any scintillation cluster from this function.
The electronic noise correlations are simplest in frequency space, so we take
the Fourier transform of equation 4.1 and drop the zero-frequency component (which
only serves as a measure of the baseline bi) to obtain
X?i[f ] =
?
a
MiaY?ia[f ] + N?i[f ]. (4.4)
We characterize the electronic noise by its second moments. It is computationally
81
useful to decompose complex-valued numbers into their rectangular coordinates, so
our characterization of the electronic noise should include the expectation values of:
?
N?Ri [f ]N?Rj [f ]
?
(4.5a)
?
N?Ri [f ]N? Ij [f ]
?
(4.5b)
?
N? Ii [f ]N?Rj [f ]
?
(4.5c)
?
N? Ii [f ]N? Ij [f ]
?
, (4.5d)
for all pairs i, j of APD channels and all non-zero frequency components f . Corre-
lations between noise in different frequencies,
?
N?i[f ]N?j[g]
?
where f 6= g, are always
equal to zero. The measurement of these correlations is described in chapter 5; this
chapter simply presumes that they are known.
4.3 APD Noise Model: the Photon Perspective
In this section we describe a model for the Poissonian noise which comes from
fluctuations in photon collection and APD gain. The model is constructed from
a physical understanding of the APDs, and includes a step-by-step breakdown of
where noise is introduced at each stage of the light collection process. For reference,
the reader is referred to section 3.5 which describes the readout process, including
the operation of APDs.
We define P (0)a to be the initial number of photons generated by scintillation
82
cluster a; this parameter is an unknown non-random parameter. Subsequent steps in
the readout process result in P (n)ia quanta (photons, electron-hole pairs, or electrons)
on channel i, where n indicates the step of the readout process; this quantity is a
function of P (0)a and also incorporates the randomness of photon collection and gain
fluctuations, so P (n)ia is a random parameter whose distribution reflects the unknown
non-random parameter P (0)a . We now describe the readout stages.
First, each APD gang i has some number P (1)ia of photons which reach its active
surface. The mean fraction of photons reaching a particular gang i from position
~xa is written fi(~xa) and is assumed to be known; however, the initial paths of
the optical photons emitted from the source, their trajectories through the xenon,
and their success in reflecting off of teflon surfaces are all random, so P (1)ia is a
Poisson-distributed random variable with mean fi(~xa)P (0)a . A typical ??0? event
may deposit up to ? 1000 photons on a nearby APD gang, resulting in roughly 3%
uncertainty from photon statistics; for a deposit in the bulk of the xenon, individual
APD gangs may collect only one-tenth of this number of photons, resulting in closer
to 10% uncertainty due to photon statistics on the nearest APD gang. Thus, we
find that the noise on a single APD channel due to photon collection fluctuations
may be quite significant.
Additionally, the numbers of photons reaching different gangs are not uncor-
related; since a photon which reaches gang i cannot deposit on a different gang j,
P (1)ia and P
(1)
ja are anticorrelated for different gangs i 6= j. This process is described
by a multinomial distribution. Explicitly, the expectation values of our multinomial
83
distribution are known to be: [117]
?
P (1)ia
?
= fi(~xa)P (0)a (4.6a)
?
P (1)ia P
(1)
jb
?
=
?
P (1)ia
??
P (1)jb
?
+ [fi(~xa)?ij ? fi(~xa)fj(~xa)]P (0)a ?ab (4.6b)
(As a detail, we should note that the multinomial distribution is an incorrect
model in one important respect: a photon may reflect off of teflon one or more times
on its way to the APD. Photons incident on the teflon may be absorbed, which would
lead to a reflection efficiency which is less than one. Furthermore, it is possible
that teflon is be mildly fluorescent at the wavelength of our scintillation, leading
to an increase in reflection efficiency. Combining the variances which come from
each segment on a photon?s trajectory and the variances from fluctuations in the
reflection efficiency of teflon can be done with the conditional variance formula [117],
but we currently have no estimate of the number of reflections a photon undergoes.
The consequence is that we probably mis-estimate the variance of P (1)ia .)
Next, photons which arrive at the active surface of an APD must convert to
electron-hole pairs in the silicon semiconductor layer of the APD; see section 3.5 for
details. The energy required to produce one electron-hole pair in silicon is 3.66 eV,
so each incident photon produces roughly 1.9 electron-hole pairs. We define P (2)ia to
be the number of electron-hole pairs actually produced from P (1)ia incident photons.
The Fano factor for electrons produced in silicon is roughly 0.1, meaning that in
addition to the uncertainty in P (1)ia which we have already characterized, there is an
additional uncorrelated variance in P (2)ia equal to 0.1
?
P (2)ia
?
[106]. The correlations
84
in the parameters P (2)ia are therefore:
?
P (2)ia
?
?
?P (1)ia
?
= 1.9 ? P (1)ia (4.7a)
?
P (2)ia P
(2)
jb
?
?
?P (1)ia , P
(1)
jb
?
= (1.9)2 ? P (1)ia P (1)jb + 0.1 ? 1.9 ? P
(1)
ia ?ij?ab (4.7b)
Electron-hole pairs are then amplified by an avalanche process inside the
APDs. The magnitude of this gain is APD-dependent, generally on the order of
200 ? 300, and can be identified by a time-dependent quantity GDi (t) (where D
stands for diode); we call the number of output electrons P (3)ia . In addition to am-
plification of existing noise in P (2)ia , two additional source of noise are introduced:
? First, there is statistical variance in the amplification experienced by each
electron due to the randomness of the avalanche process. This variance is
dependent on many factors, including the gain, and scales linearly with the
number of electrons; we define the variance on the gain experienced by a single
electron on channel i as ?2Gi(t). From [106], ?2Gi(t) is approximately equal to
(
GDi (t)
)2 when GDi (t) = 100; as a result, we approximate
?2Gi(t) ?
(
GDi (t)
)2 . (4.8)
? Second, there are gain non-uniformities in the diode volume, which contribute
variance proportional to the square of the number of initial electron-hole pairs;
we identify the proportionality constant as ?2NU , which may depend on time
and APD gang. The magnitude of this uncertainty is not well-known, but
85
may be significant. In particular, the ganging together of APDs means that
a single channel is a sum of pulses which have been exposed to six or seven
different gain factors from the different APDs in that gang; the diameter of
an APD gang is 58 mm [96], and gain non-uniformity is expected to be a
significant factor when photons deposit preferentially on one portion of that
area. Our current code base treats ?2NU as zero; however, future analyses will
likely attempt to estimate it, and the derivation which follows retains it as an
aid to that anticipated work.
These fluctuations are not correlated across channels, so their variances are multi-
plied by ?ij?ab to ensure they only contribute to variance terms and not covariance
terms. The correlations in the parameters P (3)ia are therefore:
?
P (3)ia
?
?
?P (2)ia
?
= GDi (ta)P
(2)
ia (4.9a)
?
P (3)ia P
(3)
jb
?
?
?P (2)ia , P
(2)
jb
?
= GDi (ta)GDj (tb)P
(2)
ia P
(2)
jb +
[
P (2)ia ?2Gi(ta) +
(
P (2)ia
)2
?2NU
]
?ij?ab
(4.9b)
Finally, there is amplification GEi (t) associated with the electronics of the
APDs which is dependent on time and channel. This includes preamplifier gain,
shaper gain, gain associated with the shaping times, and conversion from voltage to
ADC counts. We assume that this gain does not suffer any fluctuations, so no new
variance is produced. The output from this amplification is a waveform in ADC
counts, so we can link it to the magnitude variable Mia of equation 4.4 by stating
86
that:
?
Mia
?
?
?P (3)ia
?
= GEi (ta)P
(3)
ia (4.10a)
?
MiaMjb
?
?
?P (3)ia , P
(3)
jb
?
= GEi (ta)GEj (tb)P
(3)
ia P
(3)
jb . (4.10b)
Although we assume that no noise correlated with pulses is introduced during this
stage, there is electronic noise Ni[? ] introduced at this stage which has been de-
scribed above and is uncorrelated with the pulses. Additionally, the APDs can
contribute noise in the form of a dark current which is uncorrelated with pulses;
this is inseparable from the electronic noise, and so we absorb it into our description
of Ni[? ].
Equations 4.6, 4.7, 4.9, and 4.10 collectively chart the mean and variance
of the APD pulse magnitudes at each step of readout and amplification. We can
eliminate the intermediate terms P (1)ia , P
(2)
ia , and P
(3)
ia by substitution and obtain a
full description for the first and second moments of the pulse magnitudes Mia, Mjb
depending only on the unknown numbers of emitted photons P (0)a , P (0)b and not on
87
any random variables:
?Mia? = 1.9 ?Gi(ta)fi(~xa)P (0)a (4.11a)
?MiaMjb? = (1.9)2 ?Gi(ta)Gj(tb)fi(~xa)fj(~xb)P (0)a P (0)b
? (1.9)2 ?Gi(ta)Gj(ta)fi(~xa)fj(~xa)P (0)a ?ab (4.11b)
+ 1.9 ?
(
GEi (ta)
)2 fi(~xa)P (0)a
[
[
(0.1 + 1.9) + 1.9 ? fi(~x1)
(
P (0)a ? 1
)]
?2NU
+ (1.9 + 0.1)
(
GDi (ta)
)2 + ?2Gi(ta)
]
?ij?ab,
where for notational convenience we have defined Gi(t) = GEi (t)GDi (t) as the product
of the electronic and APD gains.
A simple substitution is possible by the property of conditional expectations,
which states that the expectation value of a random value X is the same as the result
from taking an expectation value of X conditional on Y followed by an expectation
value of the result: [117]
?X? = ??X|Y ?? . (4.12)
The choice throughout to convey second moments of random variables with expecta-
tion values, as ?XY ?, rather than with covariances as cov(X, Y ), is for the simplicity
this provides; had we chosen to use covariances, combining these separate results
would have required the use of the conditional covariance formula: [117]
cov(X, Y ) = ?cov(X, Y |Z)?+ cov (?X|Z? , ?Y |Z?) . (4.13)
88
So, simplicity has been gained by avoiding a description of the model in terms of
covariances.
Equations 4.11 fully characterize the first and second moments of the random
variables Mia, which are the magnitudes of the observed pulses. This is a key step in
building a full model of the APD noise. However, the expressions for these first and
second moments are in terms of some parameters which are not directly observable.
In particular, we do not have the ability to directly measure the function fi(~x) which
describes the average fraction of photons which arrive on channel i from an initial
position ~x. In section 4.4 we restate equations 4.11 in terms of quantities which are
directly observable so that it is usable in practice.
4.4 Bridge between Photons and Pulses
We have characterized in equation 4.11 the first and second moments of the random
variables Mia. However, that characterization is phrased in terms of quantities which
are not observable, so its practical utility is limited. In this section we transform
equation 4.11 into a form which is usable in practice, and identify where we obtain
the key parameters which it contains.
The main part of equation 4.11 which we are unable to access experimentally is
fi(~x), the mean fraction of photons originating from position ~x which arrive at APD
channel i. However, we do have observations of a related quantity: in equation 4.2
we make use of a lightmap function Li(~x, t) which identifies the average magnitude
(in ADC counts) of a pulse on channel i produced from a 2615-keV energy deposit
89
at position ~x and calendar time t (where 2615 keV is selected because we measure
it from the 2615-keV 208Tl gamma line of our thorium source calibrations). Chap-
ter 6 describes how this lightmap function is measured from data; here we simply
note that, because Li(~x, t) relates deposited energy to measured pulse sizes without
reference to the number of photons internally created, it is possible to measure it
directly from data.
Additionally, we must relate the quantity of energy deposited to the number
of photons produced. This is not directly measurable in the EXO-200 detector, so
we must rely on simulation to estimate it; however, in section 4.5 we find that this
relation has a very small impact on our energy estimator, so it need not be precise.
Simulations using NEST [118] which have been performed within the EXO group
by Liangjian Wen at our bulk electric field of 376 V/cm indicate that we can expect
82, 000 photons from a 2615-keV gamma deposit. We identify this ratio between
photons generated and energy deposited with the parameter c ? 82, 000, and express
the average relation between energy and photon yield in EXO-200 by
P (0)a = c ? Ea, (4.14)
where Ea is the energy of scintillation cluster a in units of 2615 keV.
Equation 4.14 is treated here as exact, but this may not be so. It is likely that
photon yield does not scale linearly with energy; additionally, there are fluctuations
in the number of photons produced for a particular energy deposit, and we are not
treating P (0)a as a random variable depending on E. The essential point here is that
90
the scintillation ?energy? which we measure with the APD waveforms is a direct
reflection of the number of photons P (0)a , and only indirectly reflects the true scin-
tillation cluster energy Ea. Similarly, the ionization energy measurement which is
made with the u-wires is a direct reflection of the number of free electrons gener-
ated, and only indirectly reflects the true deposit energy. The purpose of the energy
calibration, described in section 7.3, is to combine our estimates of scintillation en-
ergy (proportional to the number of photons) and ionization energy (proportional
to the number of free electrons) into a single estimate of the true deposited energy,
taking into account any non-linearities in either component. We find it convenient
to report the scintillation energy as P (0)a /c rather than the true number of photons
P (0)a , so the reported scintillation energy is roughly comparable to the true amount
of deposited energy; but our goal is still only to measure (a quantity proportional
to) the number of photons created. In the derivations of this chapter we refer to Ea
as the energy of scintillation cluster a, but it is the scintillation energy, not the true
deposited energy, which we mean by this variable.
We can now combine equations 4.2, 4.11a, and 4.14 to relate fi(~x) and Li(~x, t):
Li(~x, t) = 1.9 ? cfi(~x)Gi(t). (4.15)
We use this relation and equation 4.14 to eliminate fi(~x) from equation 4.11 and
91
restate the moments of the pulse magnitudes Mia as:
?Mia? = Li(~xa, ta)Ea (4.16a)
?MiaMjb? = Li(~xa, ta)Lj(~xb, tb)EaEb
? Li(~xa, ta)Lj(~xa, ta)Ea?ab/c (4.16b)
+ Li(~xa, ta)Ea
[
[(0.1 + 1.9)Gi(ta) + Li(~xa, ta) (Ea ? 1/c)]
?2NU
(GDi (ta))
2
+ (0.1 + 1.9)Gi(ta) +
GEi (ta)
GDi (ta)
?2Gi(ta)
]
?ij?ab.
To simplify the notation, let us define a new function:
q(i, a) := Li(~xa, ta)Ea
[
[(0.1 + 1.9)Gi(ta) + Li(~xa, ta) (Ea ? 1/c)]
?2NU
(GDi (ta))
2
+ (0.1 + 1.9)Gi(ta) +
GEi (ta)
GDi (ta)
?2Gi(ta)
]
(4.17)
which we note should be strictly positive if measured correctly (because all gains
and lightmap values should be strictly positive, and all variances are non-negative).
Using this shorthand, we re-write the expectation values of equation 4.16 in the
more compact form:
?Mia? = Li(~xa, ta)Ea (4.18a)
?MiaMjb? = Li(~xa, ta)Lj(~xb, tb)EaEb
? Li(~xa, ta)Lj(~xa, ta)Ea?ab/c (4.18b)
+ q(i, a)?ij?ab.
92
We will see in equation 6.1 that the lightmap Li(~x, t) is treated as a separable
function, Li(~x, t) = Ri(~x)Si(t). The right hand side of equation 4.15 has factors
which depend strictly on either position or time, but not both; so we can separate
the position-dependent and time-dependent factor from both sides and state:
Ri(~x) ? fi(~x) (4.19a)
Si(t) ? Gi(t). (4.19b)
So, we can use the lightmap to understand the time-dependent behavior of the
product of electronic and APD gain, Gi(t), for each channel; but we cannot use it to
set the absolute scale, nor can we use it to understand separately GEi (t) and GDi (t).
Measurements of GEi (t) are possible from internal charge injection runs, as
described in section 3.6. However, using these runs in a uniform way is difficult.
Internal charge injection runs are only useful for understanding relative changes in
the electronic gain over time, and an external charge injection run is needed to set
an absolute scale for the gain. The only APD external charge injection runs were
taken in February 2011, and have not been analyzed since that time; also, since
that time some APD electronics boards have been replaced, resetting the scale of
the internal charge injection runs. For the present analysis, the charge injection
runs are not used to fix GEi (t).
However, we can still derive an estimate for the electronic gain which is suf-
ficient for the present analysis. The electronic gain comes entirely from known
electronic components; based on these electronic components, the preamplifier has
93
a gain of 1/(5 pF), shapers have a combined 21.2, and the digitizer digitizes 12
bits (4096 ADC counts) for a 2.5 volt pulse. Combining these factors, we use a
time-independent estimate:
GEi (t) = 1.1 ? 10?3 ADC/e?. (4.20)
Future work should include a more careful study of the charge injection runs to
improve this estimate, but we use this value to perform the present analysis. A
consequence of this choice is that all time-dependence of Si(t) must be absorbed
into GDi (t), so we can strengthen equation 4.19b to:
Si(t) ? GDi (t). (4.21)
We also have independent measurements of the APD gains GDi (t) available
from laser calibration runs. These special runs allow a laser to shine into the detector
from a fixed point and with a stable amplitude while the bias voltages on the APDs
are varied from an effective unity gain to our standard voltage settings. Using these
measurements, we are able to measure GDi (t) at weekly intervals from September
2012 to the present time. Before September 2012, some less-reliable laser data is
available, but results from that data are not readily available as of this writing.
It would be possible, and should be a goal for future improvements, to make
use of this full range of laser data. However, the laser data provides a less uniform
history of APD gains over the full data-taking window of EXO-200 from September
94
2011 to November 2013, and it is much easier to track time-dependent behavior from
thorium source data which have been collected regularly throughout that period. As
a result, a compromise is used to characterize GDi (t). One particular laser run, run
4540 (taken on December 13, 2012), is used to fix GDi (t4540), and equation 4.21 is
used to extrapolated using thorium source data:
GDi (t) ? GDi (t4540) ? Si(t)/Si(t4540). (4.22)
This assumption makes use of the approximation that GEi (t) is roughly constant in
time, which is probably only accurate to one significant figure; therefore when an
electronics change is made to a channel, we can expect that the accuracy of GDi (t) is
no better than one significant figure. These results mean that we can also estimate
with the same level of accuracy:
fi(~x) ?
Si(t4540)
GDi (t4540)
? Ri(~x)1.9 ? cGEi (t)
. (4.23)
The set of inputs needed to use equations 4.18 are Li(~x, t), GDi (t), GEi (t), ?2NU ,
?2Gi(t), and c. The lightmap Li(~x, t) is measured in chapter 6. Our approximations
for GDi (t) and GEi (t) have been described in equations 4.22 and 4.20, respectively.
c ? 82, 000 was identified for equation 4.14. We have stated that ?2NU is set equal to
zero for this analysis, and is retained in this work to facilitate future investigations.
Equation 4.8 has estimated ?2Gi(t) ?
(
GDi (t)
)2.
Thus, we have specified fully the parameters needed as inputs to equations 4.18.
95
We now have at our disposal a full model for the Poissonian noise on APD chan-
nels. We are finally prepared with all tools necessary to derive an optimal energy
estimator, and this is done in section 4.5.
4.5 Derivation of an Optimal Energy Estimator
It is now possible to specify the optimization criteria for generating an energy esti-
mate from the APD waveforms. We wish to identify an energy estimator which is
unbiased and has a minimal expected error. For the problem to remain tractable, we
demand that the estimator be linear. Furthermore, although the Fourier-transformed
waveforms X?i[f ] are complex-valued, we require that the energy estimate be strictly
real-valued. This section describes the optimization criteria and constraints for the
energy estimator, and then proceeds to derive an optimal estimator satisfying those
constraints.
We recall from equation 4.4 our model for the APD waveforms:
X?i[f ] =
?
a
MiaY?ia[f ] + N?i[f ], (4.24)
where Mia and N?i[f ] are random variables with known probability distributions, we
choose to drop the zero-frequency component, and the probability distribution of
Mia reflects the energy parameter which we intend to estimate. We therefore take
96
the energy estimator E?a for the energy of scintillation cluster a to be of the form:
E?a =
?
if
Aia[f ]X?Ri [f ] +Bia[f ]X?Ii [f ], (4.25)
where we recall from section 4.2 that ?R and ?I are the real and imaginary parts,
respectively, of a complex-valued parameter. This expression is one form for the
most general real-valued linear functional on Xi[f ]. The goal of denoising is therefore
reduced to identifying the optimal parameters Aia[f ] and Bia[f ] for this estimator.
The mean squared error 2a in the energy estimate E?a of Ea is defined by:
2a =
?(
E?a ? Ea
)2
?
. (4.26)
Our goal is to minimize 2a under the constraint of no bias, ie. that:
?
E?a ? Ea
?
= 0 (4.27)
or, by substitution of equation 4.24 and applying our knowledge of the expectation
value of Mia from equation 4.18a and our knowledge that the noise terms have mean
zero, the no-bias constraint is equivalent to:
?
ifb
[
Aia[f ]Y? Rib [f ] +Bia[f ]Y? Iib[f ]
]
Li(~xb, tb)Eb = Ea. (4.28)
This constraint holds for all values of the scintillation cluster indices a and b, in-
dicating that even in the presence of multiple scintillation clusters we still demand
97
that the energy estimators are unbiased.
However, we find that it is necessary to specify a slightly stronger constraint.
In particular, it is desirable to ensure that the constraints are as independent of
energy as possible to reduce the need to input an estimated energy into our energy
estimator; therefore we freely employ the stronger constraint
?
if
[
Aia[f ]Y? Rib [f ] +Bia[f ]Y? Iib[f ]
]
Li(~xb, tb) = ?ab for all b, (4.29)
which implies the earlier forms and leads to advantageous cancellations of terms.
Conceptually, this constraint is equivalent to saying that estimates of the energy Ea
of a scintillation cluster a should be unbiased and independent of any other energies
Eb which are present on the same waveform.
We now proceed with the optimization. We start by expanding 2a:
2a =
?(
E?a ? Ea
)2
?
(4.30)
=
?
E?2a
?
? Ea
?
E?a
?
?
?
E?a ? Ea
?
Ea (4.31)
98
We employ the constraint of equation 4.27 to simplify the second term of this ex-
pansion, and eliminate the third altogether; we then proceed:
2a =
?
E?2a
?
? E2a (4.32)
=
?(
?
if
[
Aia[f ]X?Ri [f ] +Bia[f ]X?Ii [f ]
]
)2?
? E2a (4.33)
=
?(
?
if
[
Aia[f ]N?Ri [f ] +Bia[f ]N? Ii [f ]
]
+
?
ifb
[
Aia[f ]Y? Rib [f ] +Bia[f ]Y? Iib[f ]
]
Mib
)2?
? E2a
(4.34)
The noise N? and pulse magnitude Mia are uncorrelated, so multiplicative cross-terms
have an expectation value of zero:
2a =
?(
?
if
[
Aia[f ]N?Ri [f ] +Bia[f ]N? Ii [f ]
]
)2?
+
?(
?
ifb
[
Aia[f ]Y? Rib [f ] +Bia[f ]Y? Iib[f ]
]
Mib
)2?
? E2a (4.35)
99
Additionally, electronic noise cross-terms between different frequencies evaluate to
zero:
2a =
?
?
ijf
[
Aia[f ]N?Ri [f ] +Bia[f ]N? Ii [f ]
] [
Aja[f ]N?Rj [f ] +Bja[f ]N? Ij [f ]
]
?
+
?(
?
ifb
[
Aia[f ]Y? Rib [f ] +Bia[f ]Y? Iib[f ]
]
Mib
)2?
? E2a (4.36)
=
?
ijf
[
Aia[f ]Aja[f ]
?
N?Ri [f ]N?Rj [f ]
?
+ Aia[f ]Bja[f ]
?
N?Ri [f ]N? Ij [f ]
?
+Bia[f ]Aja[f ]
?
N? Ii [f ]N?Rj [f ]
?
+Bia[f ]Bja[f ]
?
N? Ii [f ]N? Ij [f ]
?]
+
?
ifb
jgc
[
Aia[f ]Y? Rib [f ] +Bia[f ]Y? Iib[f ]
] [
Aja[g]Y? Rjc [g] +Bja[g]Y? Ijc[g]
]
?MibMjc? ? E2a
(4.37)
100
We now expand ?MibMjc? using equation 4.18 and take advantage of the stronger
form of our constraint 4.29 to simplify the expression:
2a =
?
ijf
[
Aia[f ]Aja[f ]
?
N?Ri [f ]N?Rj [f ]
?
+ Aia[f ]Bja[f ]
?
N?Ri [f ]N? Ij [f ]
?
+Bia[f ]Aja[f ]
?
N? Ii [f ]N?Rj [f ]
?
+Bia[f ]Bja[f ]
?
N? Ii [f ]N? Ij [f ]
?]
+
(
?
ifb
[
Aia[f ]Y? Rib [f ] +Bia[f ]Y? Iib[f ]
]
Li(~xb, tb)Eb
)2
? E2a
?
?
b
(
?
if
[
Aia[f ]Y? Rib [f ] +Bia[f ]Y? Iib[f ]
]
Li(~xb, tb)
)2
Eb
c
+
?
ifgb
[
Aia[f ]Y? Rib [f ] +Bia[f ]Y? Iib[f ]
] [
Aia[g]Y? Rib [g] +Bia[g]Y? Iib[g]
]
q(i, b) (4.38)
=
?
ijf
[
Aia[f ]Aja[f ]
?
N?Ri [f ]N?Rj [f ]
?
+ Aia[f ]Bja[f ]
?
N?Ri [f ]N? Ij [f ]
?
+Bia[f ]Aja[f ]
?
N? Ii [f ]N?Rj [f ]
?
+Bia[f ]Bja[f ]
?
N? Ii [f ]N? Ij [f ]
?]
+
?
ib
(
?
f
[
Aia[f ]Y? Rib [f ] +Bia[f ]Y? Iib[f ]
]
)2
q(i, b)
? Eac (4.39)
We are now in a position to evaluate the partial derivatives of 2a with respect
101
to the parameters Aia[f ] and Bia[f ]. They are:
?2a
?Aia[f ]
= 2
?
j
[
Aja[f ]
?
N?Ri [f ]N?Rj [f ]
?
+Bja[f ]
?
N?Ri [f ]N? Ij [f ]
?]
+ 2
?
gb
[
Aia[g]Y? Rib [g] +Bia[g]Y? Iib[g]
]
q(i, b)Y? Rib [f ] (4.40a)
?2a
?Bia[f ]
= 2
?
j
[
Aja[f ]
?
N? Ii [f ]N?Rj [f ]
?
+Bja[f ]
?
N? Ii [f ]N? Ij [f ]
?]
+ 2
?
gb
[
Aia[g]Y? Rib [g] +Bia[g]Y? Iib[g]
]
q(i, b)Y? Iib[f ] (4.40b)
We use Lagrange?s method to minimize 2a while satisfying our constraints; we
define
Cab =
?
if
[
Aia[f ]Y? Rib [f ] +Bia[f ]Y? Iib[f ]
]
Li(~xb, tb), (4.41)
where the indices of Cab are ordered, and restate the constraints of equation 4.29 as
Cab = ?ab. (4.42)
Then, the partial derivatives of these constrained expressions are:
?Cbc
?Aia[f ]
= Y? Ric [f ]Li(~xc, tc)?ab (4.43a)
?Cbc
?Bia[f ]
= Y? Iic[f ]Li(~xc, tc)?ab (4.43b)
Denoting the set of Lagrange multipliers for 2a with ?ab, where the indices are
ordered, and allowing these parameters to absorb constant factors, we can at last
102
identify the full set of linear equations describing the optimal energy estimator E?a:
?
j
[
Aja[f ]
?
N?Ri [f ]N?Rj [f ]
?
+Bja[f ]
?
N?Ri [f ]N? Ij [f ]
?]
+
?
gb
[
Aia[g]Y? Rib [g] +Bia[g]Y? Iib[g]
]
q(i, b)Y? Rib [f ]
+
?
b
?abY? Rib [f ]Li(~xb, tb) = 0 for each i, f (4.44a)
?
j
[
Aja[f ]
?
N? Ii [f ]N?Rj [f ]
?
+Bja[f ]
?
N? Ii [f ]N? Ij [f ]
?]
+
?
gb
[
Aia[g]Y? Rib [g] +Bia[g]Y? Iib[g]
]
q(i, b)Y? Iib[f ]
+
?
b
?abY? Iib[f ]Li(~xb, tb) = 0 for each i, f (4.44b)
?
if
[
Aia[f ]Y? Rib [f ] +Bia[f ]Y? Iib[f ]
]
Li(~xb, tb) = ?ab for each b (4.44c)
It is important to note here that energies Eb, which we are intending to mea-
sure, do in fact enter into this linear set of equations. At first this would seem to
demonstrate that the equations here are impossible to apply, but in fact they remind
us that our Poissonian noise is dependent on the energy. Statistical fluctuations in
the number of photons observed on each APD, for instance, scale like the square
root of the number of photons emitted. This is in contrast to the electronic noise,
which is not correlated with the energies.
We can see, then, that some estimate of energy is necessary to know the relative
importance of different forms of noise. But the constraint equation ensures that our
estimate is unbiased regardless of what energy estimates are fed into the system of
103
equations, so we can be confident that it is only a rough estimate of the energy
scale which is needed, and not worry that the optimization is biased to estimate
energies similar to the estimates we feed in. For this purpose, it is sufficient to use
the charge-only energy as an estimate of the scintillation-only energy.
The optimal energy estimator is now fully specified by the parameters Aia[f ]
and Bia[f ] which solve equations 4.44, and the theory of denoising is fully specified.
We can see that the optimal estimator varies from event to event, reflecting the facts
that we should weight channels differently according to the event position and that
the balance of electronic to Poissonian noise changes based on the estimated energy
of an event, so we need to solve this system of equations separately for each event
in the detector. Sections 4.6-4.9 consider the computational aspects of solving for
this energy estimator.
4.6 Matrix Formulation of Denoising
We intend to solve the system of equations 4.44 numerically, and note that it is
linear in the unknown parameters Aia[f ] and Bia[f ]. It is most convenient if we can
restate the system of equations as a matrix equation before attempting to solve it.
This section specifies a matrix formulation of the system of equations.
First, we must choose an ordering of the unknowns; the best ordering is the
one which groups nonzero entries into blocks, since this allows us to make more
104
efficient use of matrix software libraries. We define:
~Aa =
(
A1a[1] B1a[1] A2a[1] B2a[1] ? ? ? Bimaxa[1] A1a[2] ? ? ?
? ? ? A1a[fmax] A2a[fmax] ? ? ? Aimaxa[fmax]
)
(4.45)
so that the entries alternate between A and B, iterating quickly through channels
and more slowly through frequencies. Note that only the A terms are included for
the maximum frequency; this is because Xi[t] is real-valued, so X?i[0] and X?i[fmax]
are real-valued as well. We also find it convenient to define the matrix:
A =
?
?
?
?
?
?
?
?
... ...
~A1 ? ? ? ~Aamax
... ...
?
?
?
?
?
?
?
?
(4.46)
which includes the parameters needed for each of the estimators E?a corresponding
to each of the scintillation clusters a.
We can then define the electronic noise blocks as:
Nf =
?
????
????
COV
(
N?R1 [f ], N? I1 [f ], N?R2 [f ], . . . , N? Iimax [f ]
)
for f < fmax
COV
(
N?R1 [f ], N?R2 [f ], . . . , N?Rimax [f ]
)
for f = fmax
(4.47)
105
where COV specifies the covariance matrix of an ordered list of random variables,
COV (x1, x2, . . . , xn) =
?
?
?
?
?
?
?
?
?
?
?
?
cov(x1, x1) cov(x1, x2) ? ? ? cov(x1, xn)
cov(x2, x1) cov(x2, x2) ? ? ? cov(x2, xn)
... ... . . . ...
cov(xn, x1) cov(xn, x2) ? ? ? cov(xn, xn)
?
?
?
?
?
?
?
?
?
?
?
?
. (4.48)
Note that because our noise variables N?Ri [f ] and N? Ii [f ] have mean zero (for f 6= 0),
we can move smoothly between expectation values of products and covariances:
cov(N?Ri [f ], N?Rj [f ]) =
?
N?Ri [f ]N?Rj [f ]
?
(4.49a)
cov(N?Ri [f ], N? Ij [f ]) =
?
N?Ri [f ]N? Ij [f ]
?
(4.49b)
cov(N? Ii [f ], N?Rj [f ]) =
?
N? Ii [f ]N?Rj [f ]
?
(4.49c)
cov(N? Ii [f ], N? Ij [f ]) =
?
N? Ii [f ]N? Ij [f ]
?
. (4.49d)
We pack together the blocks of equation 4.47 into a sparse noise matrix:
N =
?
?
?
?
?
?
?
?
?
?
?
?
N1 0 . . . 0
0 N2 . . . 0
... ... . . . ...
0 0 . . . Nfmax
?
?
?
?
?
?
?
?
?
?
?
?
. (4.50)
In a similar way, we can define the other noise terms in terms of matrix opera-
tions. We find that it is possible to describe the Poissonian noise terms as a product
106
of two matrices, P = P1P2. First we define the matrix P1, which steps horizontally
through the APD channels and scintillation clusters. (It does not matter what or-
dering is used provided P2 follows the same one.) For a particular choice of indices
j and b, the corresponding column of P1 is:
P1(column j, b) =
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
Y? Rjb [1]?1j
Y? Ijb[1]?1j
Y? Rjb [1]?2j
...
Y? Ijb[1]?imaxj
Y? Rjb [2]?1j
...
Y? Rjb [fmax]?1j
Y? Rjb [fmax]?2j
...
Y? Rjb [fmax]?imaxj
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
q(j, b). (4.51)
Note that in each column, only a small subset of the entries are nonzero.
This means that P2 should be a matrix with rows that step through the APD
channels and scintillation clusters as well (and with the same ordering as P1). For
107
a particular choice of indices j, b, the corresponding row of P2 is:
P2(row j, b) =
(
Y? Rjb [1]?1j Y? Ijb[1]?1j Y? Rjb [1]?2j ? ? ? Y? Ijb[1]?imaxj Y? Rjb [2]?1j ? ? ?
? ? ? Y? Rjb [fmax]?1j Y? Rjb [fmax]?2j ? ? ? Y? Rjb [fmax]?imaxj
)
?
q(j, b).
(4.52)
Since we use the product P = P1P2, it clearly must be the case that the ordering
of columns in P1 is the same as the ordering of rows in P2. Because in practice
there are always be more APD channels than scintillation clusters, it is generally
advantageous to iterate through j fastest, and step through b more slowly. We can
also see that P1 = P2>, so
P = P2>P2. (4.53)
The constraint equations are represented by a matrix C with rows that step
through the scintillation clusters; for a particular choice of b the corresponding row
of C is:
C(row b) =
(
Y? R1b [1]L1(~xb, tb) Y? I1b[1]L1(~xb, tb) Y? R2b [1]L2(~xb, tb)
? ? ? Y? Iimaxb[1]Limax(~xb, tb) Y? R1b [2]L1(~xb, tb)
? ? ? Y? R1b [fmax]L1(~xb, tb) Y? R2b [fmax]L2(~xb, tb)
? ? ? Y? Rimaxb[fmax]Limax(~xb, tb)
)
.
(4.54)
Finally it is possible to restate the full linear equation specified above in matrix
form, where we can simultaneously include the equations for all of the estimators
108
E?a: ?
?
?
?
N + P C>
C 0
?
?
?
?A =
?
?
?
?
0
I
?
?
?
? . (4.55)
The matrix N is the same as the covariance matrix of the electronic noise,
so it must be symmetric and positive-semidefinite. Similarly, we have seen that
P = P2>P2, implying that P must be symmetric and positive-semidefinite. This
means that there is a Cholesky decomposition N + P = LL>, where L is a lower-
triangular matrix, and that N + P is symmetric and positive semi-definite. This
implies that the full matrix equation is symmetric; however, it is insufficient for
showing that the full system is positive semi-definite, so we are not able to make
use of specialized solver algorithms which exploit positive semi-definiteness.
We also note the important observation that this matrix is sparse. Given
roughly 1000 frequency components in our Fourier-transformed waveforms, only
0.1% of the entries of N are non-zero. And with 70 APD channels and only a small
number of scintillation clusters we wish to denoise, the combined number of non-
zero entries in P1 and P2 is easily an order of magnitude less than the number of
non-zero entries in N. As a result, we can conclude that this matrix is extremely
sparse, and should ensure that whatever method we use to solve these equations
takes advantage of this sparsity.
The resulting system of equations 4.55 fully specifies the optimal energy es-
timators for all scintillation clusters in an event. In sections 4.7-4.9 we discuss
computational aspects of the efficient solution of that system of equations.
109
4.7 Preconditioning of the Matrix Formulation
We have specified scintillation energy estimators with matrix equation 4.55 which
we now wish to solve. However, it is generally recommended that large matrix
equations should not be solved directly. Instead, it is recommended that the matrix
should be preconditioned. This means that if a matrix equation A~x = ~b should
be solved, one should first find a matrix A? which is approximately equal to A
and which is easily invertible. Finding the right balance between the quality of the
approximation A? ? A and the difficulty of computing the inverse (A?)?1 is more
art than science, but this section presents a preconditioning matrix which seems to
strike a good balance for this particular system.
The matrix equation we wish to solve takes the form:
?
?
?
?
N + P C>
C 0
?
?
?
?A =
?
?
?
?
0
I
?
?
?
? (4.56)
Let us define D = diag(N) and approximate
?
?
?
?
N + P C>
C 0
?
?
?
? ?
?
?
?
?
D C>
C 0
?
?
?
? (4.57)
This approximate form is easy to invert (assuming every frequency component of
every channel has non-zero noise and every constraint is satisfiable, which is true
in practice), and can be used to precondition our problem for better numerical
110
behavior.
We can further factor this approximate form of the matrix:
?
?
?
?
D C>
C 0
?
?
?
? =
?
?
?
?
D1/2 0
CD?1/2 H
?
?
?
?
?
?
?
?
D1/2 D?1/2C>
0 ?H>
?
?
?
? (4.58)
=
?
?
?
?
D1/2 0
0 I
?
?
?
?
?
?
?
?
I 0
CD?1/2 H
?
?
?
?
?
?
?
?
I D?1/2C>
0 ?H>
?
?
?
?
?
?
?
?
D1/2 0
0 I
?
?
?
? (4.59)
where H is a lower-diagonal square matrix defined uniquely as the Cholesky de-
composition HH> = CD?1C>. (The solution is guaranteed to be real because D
is positive-semidefinite in all cases, and positive-definite in practice ? it consists of
noise variance terms, which are strictly positive in practice.) Note that CD?1C>
has dimensions equal to the number of scintillation clusters in an event, which in
practice is small enough that we can compute its Cholesky decomposition HH>
directly.
The two diagonal factors are easily inverted; we further define
K1 =
?
?
?
?
I 0
CD?1/2 H
?
?
?
? K
?1
1 =
?
?
?
?
I 0
?H?1CD?1/2 H?1
?
?
?
? (4.60a)
K2 =
?
?
?
?
I D?1/2C>
0 ?H>
?
?
?
? K
?1
2 =
?
?
?
?
I D?1/2C>H?>
0 ?H?>
?
?
?
? , (4.60b)
where M?> represents the inverse of the transpose of M. We then follow the
standard prescription for preconditioning a linear system by replacing our original
111
matrix equation with the new form:
K?11
?
?
?
?
D?1/2 0
0 I
?
?
?
?
?
?
?
?
N + P C>
C 0
?
?
?
?
?
?
?
?
D?1/2 0
0 I
?
?
?
?K
?1
2 A? = K?11
?
?
?
?
D?1/2 0
0 I
?
?
?
?
?
?
?
?
0
I
?
?
?
?
(4.61a)
A? = K2
?
?
?
?
D1/2 0
0 I
?
?
?
?A (4.61b)
which reduces to
K?11
?
?
?
?
?
?
?
?
D?1/2ND?1/2 0
0 0
?
?
?
?+
?
?
?
?
D?1/2PD?1/2 D?1/2C>
CD?1/2 0
?
?
?
?
?
?
?
?K
?1
2 A? =
?
?
?
?
0
H?1
?
?
?
?
(4.62a)
A =
?
?
?
?
D?1/2 0
0 I
?
?
?
?K
?1
2 A? (4.62b)
This preconditioned set of equations is numerically quite stable. It is also
possible to provide an excellent initial guess A?0 by taking the approximate form of
the matrix to be exact, yielding
A?0 =
?
?
?
?
0
H?1
?
?
?
? (4.63)
Based on these various advantages, it is this system which we attempt to solve rather
than the original form. Below we define matrices M and B for convenience, and
112
summarize the system to be solved:
M = K?11
?
?
?
?
?
?
?
?
D?1/2ND?1/2 0
0 0
?
?
?
?+
?
?
?
?
D?1/2PD?1/2 D?1/2C>
CD?1/2 0
?
?
?
?
?
?
?
?K
?1
2 (4.64a)
B = A?0 =
?
?
?
?
0
H?1
?
?
?
? (4.64b)
MA? = B (4.64c)
A =
?
?
?
?
D?1/2 0
0 I
?
?
?
?K
?1
2 A? (4.64d)
4.8 Matrix Solver
The matrix equation we have derived is sparse: most of its entries are equal to zero.
The challenge to solving this system is therefore to find an algorithm which takes
advantage of that sparsity to speed up computations. Although there are some di-
rect methods to solving sparse matrix equations by finding a decomposition which
preserves the sparsity structure, we find that these are not very effective for our
particular sparsity structure. Instead, we choose to adopt an iterative solver which
works by repeated multiplication of vectors by our matrix; then we can take advan-
tage of the sparsity of our matrix simply by writing a matrix-vector multiplication
routine which is efficient. Section 4.9 describes how we make our matrix-vector
multiplication routine efficient; this section describes the iterative matrix solver al-
gorithm we have chosen.
113
The matrix A? for which we attempt to solve has as many columns as there
are scintillation clusters we wish to denoise within an event; often there is only
one scintillation cluster and A? is simply a column vector. However, there certainly
are many events with multiple scintillation clusters; we also discuss extended ap-
plications of denoising where we may treat events as having additional pulses in
section 4.11.1. In these cases the number of columns may be larger. In such cases
it would be possible to solve for each column of A? independently, but more efficient
to solve the entire system simultaneously. In this way, information obtained from
multiplying the matrix by one column may be exploited to solve the other columns
as well, effectively multiplying the benefit from each matrix-multiplication by the
number of columns of A?.
Matrix-solving algorithms which handle multiple vectors simultaneously are
commonly referred to as block algorithms. We use the Block-BiCGSTAB algo-
rithm [119], an iterative method based on the popular stabilized biconjugate-gradient
(BiCGSTAB) method. BiCGSTAB and Block-BiCGSTAB are notable for being able
to deal with arbitrary matrices rather than requiring that the matrix be positive-
definite (which our matrix is not). The algorithm is reproduced in the form used
by our code as Algorithm 1. This algorithm only requires that we are able to left-
multiply vectors and matrices by M quickly, which makes it possible to exploit the
sparsity of M.
Since each column of A? corresponds to one scintillation cluster, the norm of
the corresponding column of R tells us how well we have identified the optimal
magnitude estimator for that scintillation cluster; thus, to test for termination we
114
Algorithm 1 Block-BiCGSTAB Algorithm
1: A?? A?0
2: R? B?MA?
3: S? R
4: R?0 ? R
5: loop
6: V?MS
7: Solve (R?>0 V)? = (R?>0 R) for ?.
8: R? R?V?
9: T?MR
10: ? ? ?T,S?F/?T,T?F , where ??, ??F is the Frobenius matrix norm.
11: A?? A? + S? + ?R
12: R? R? ?T
13: R currently equals B?MA?
14: if all columns of R have sufficiently small magnitudes then
15: return A?
16: end if
17: Solve (R?>0 V)? = ?(R?>0 T) for ?.
18: S? R + (S? ?V)?
19: end loop
should evaluate the norm of each column of R, and while any of those norms are
above some threshold, the solver must be permitted to continue. Because our reso-
lution is only on the order of 1%, solutions generally need not be very accurate; in
practice, we establish reasonable thresholds by plotting resolution against threshold
for a small subset of data, and observing when the resolution stops improving.
4.9 Computational Considerations
Section 4.8 describes the matrix solver algorithm which we have chosen; as we
describe there, the key to its efficiency is that we use an efficient matrix-vector
multiplication routine. In this section we describe the tricks used to ensure our
matrix-vector multiplication takes full advantage of any special structure in our
115
problem.
The majority of non-zero matrix entries comes from the electronic noise matrix
N, making it the computational bottleneck of the algorithm. N is block-diagonal,
as seen in equation 4.50; if we assume roughly 70 APD channels, then the last block
of N has 702 non-zero entries and all other blocks have 4 ? 702 non-zero entries.
With waveforms having 2048 samples, and ignoring the baseline component, the
problem of multiplying a vector by the electronic noise terms thus consists of 1024
matrix-matrix multiplications, with each of our 1024 left sub-matrices Nf containing
roughly 20, 000 non-zero terms and each right matrix consisting of O(1) columns and
roughly 150 rows. (The number of columns in the right matrix is equal to the number
of scintillation clusters, which may be sometimes 3?5 or more; however, it is always
much smaller than 150, and describing it as O(1) captures this fact.)
The key points here which create an opportunity for optimization are that:
? The electronic noise terms generally do not change event-by-event. They are
treated as constant across many runs.
? It is possible to multiply two N ? N matrices together much faster than it
is possible to perform N multiplications of an N ? N matrix with N N ? 1
matrices. This can be achieved with a combination of algorithms faster than
naive O(N3) speed and exploitation of low-level computer hardware features
such as the CPU cache [120,121].
What we wish to do, then, is reorganize the solver algorithm so that whenever
multiplication by the electronic noise matrix is required, rather than performing
116
that multiplication on a ?skinny? 150 ? O(1) matrix, we pack together many such
?skinny? matrices into a single matrix with many columns. Multiplication can then
be performed in bulk; and individual columns from the result can be extracted and
used as before.
Additionally, it is important that matrix multiplication be made as fast as
possible. It has long been known that matrix multiplication provides significant op-
portunities for low-level optimizations [121]. For example, most of the time taken by
naive matrix multiplication is spent fetching and writing data to and from RAM. Sig-
nificant speedups can be achieved by minimizing the number of CPU cache misses,
which can be accomplished by operating on matrices in blocks with size chosen so
they fit entirely in the cache. Multiplication instructions can also often be packed
into vector instructions such as SSE or AVX. For extremely large matrices, there
are even algorithms which require fewer than O(N3) multiplications, and these can
sometimes be beneficial [121].
Optimization of matrix-matrix multiplication is a large field, but fortunately
there are a number of well-tuned software libraries implementing matrix multiplica-
tion well-tuned to specific machines. These libraries generally provide the standard
Basic Linear Algebra Subroutines (BLAS) interface, making them interchangeable
with ease. In this instance, Intel?s MKL library has been used; benefits to this
implementation are its availability on scientific computing platforms and its ability
to adapt to heterogeneous compute clusters by detecting the architecture of the
machine on which it is run [122].
Finally, implementation at the National Energy Research Scientific Computing
117
Center (NERSC) has the interesting feature that NERSC?s computing systems are
designed for multi-core processing. The Hopper and Edison computing systems
at NERSC allocate whole nodes, each of which contains 24 cores. Although it is
possible to simply run 24 independent processes on each node, memory on NERSC
machines is highly constrained; memory use can be reduced by sharing certain large
data structures across multiple cores on a node. Additionally, as more events are
packed together for electronic-noise multiplication, greater savings can be realized;
so it is better to handle many events in coordination. To this purpose, a multi-
threaded version of the program has been developed to exploit portions of the code
which are conveniently distributed. Because NERSC nodes have a Non-Uniform
Memory Access (NUMA) architecture, processes are constrained to only run on
cores with a similar memory access pattern; on Hopper this results in four 6-threaded
processes per node, while on Edison this results in two 12-threaded processes per
node.
As a result, and because the bulk of our computational time is spent multi-
plying by N, we pursue the following strategies to improve code performance:
1. One denoising process runs on each NUMA node of a NERSC machine. Pro-
cesses should be run in ?strict memory mode,? where a process only has access
to memory resources on its own NUMA node.
2. Within a process, algorithm 1 should be performed in parallel on all available
cores to maximize CPU utilization.
3. When a thread encounters the need for a matrix multiplication in lines 2, 6,
118
or 9, the thread should not multiply by N. Instead, it should push the left
hand side into new columns of a matrix RHS of ?thin? matrices needing to
be multiplied by N.
4. When enough columns have accumulated in RHS, collect the running threads;
these threads should each perform some fraction of the matrix multiplications
between blocks of N and the corresponding rows of RHS. When this is
complete, the threads should resume their processing of algorithm 1 in parallel
on the available pool of events.
5. Every matrix multiplication should be performed by a direct call to the In-
tel MKL library. To facilitate this process, N should be stored in a format
that ensures its blocks are accessed in a contiguous segment of memory. Fur-
thermore, because there is ample opportunity for higher-level parallelization,
at no time should a multithreaded BLAS operation be invoked; this reduces
computational overhead of parallelization.
These represent the most significant set of computational optimizations. How-
ever, there are also smaller optimizations which can still produce non-negligible
speedups. The Block-BiCGSTAB algorithm (in the case of only one scintillation
cluster in an event) requires two matrix-vector multiplications, but it also requires a
number of vector-vector operations, and the cumulative cost of these vector-vector
operations can become non-negligible. One obvious shortcut is to reuse our compu-
tation of R?>0 V from line 7 of Algorithm 1 to save work in line 17.
Another speedup for our vector-vector operations consists of combining mul-
119
tiple operations into one step. We see on line 7 that both R?>0 V and R?>0 R are com-
puted. As described earlier in the context of multiplication of vectors by electronic
noise, it is possible to take advantage of caching and speed up vector operations
by packing multiple vectors together into a more square-shaped matrix. In this
case, rather than computing these two products separately, we can pack V and R
together and compute
R?>0
(
V R
)
(4.65)
as one step. The rest of the algorithm is not affected by the fact that these two
matrices are stored contiguously in memory.
This section has described the optimizations which are available to us to accel-
erate matrix-vector operations. However, even with the application of these tricks,
the amount of computation required to denoise our full dataset is quite large. Each
event in our dataset requires us to solve a matrix equation where the matrix has
roughly twenty million non-zero entries, and our full dataset contains hundreds of
millions of events. Denoising our full dataset currently takes a total of roughly fifty
thousand core-hours on the NERSC computing system, making it manageable but
not light on resources. Bottlenecks to the code remain, including latency in data
access and some inefficient use of multiple cores; improving the speed of denoising
remains a high priority and continues to be investigated.
120
4.10 Denoising in Practice
The preceding description of denoising has been intended to convey our fullest cur-
rent understanding of denoising as it should be performed. However, due to the
computational intensity of this algorithm it is not feasible to conduct routine data
processing of the full EXO-200 dataset whenever updates to the code base are imple-
mented. As a result, the most recent denoised data available from EXO-200 demon-
strates certain deviations from the description above. It is known, and has been
demonstrated, that correcting some of these flaws can lead to further improvements
in resolution, but no physics analysis will be performed with those improvements
until the computational cost of denoising is again deemed worthwhile.
In this section we outline the places where the denoising used for the current
physics analysis differs from the description above. Differences in implementation
also exist, but are not described here. The differences of denoising are:
? The Poisson noise coming from sources internal to the APDs was not fully
understood when denoising was first implemented, and as a result its magni-
tude is significantly underestimated. Specifically, in place of equations 4.7b
and 4.9b we take
?
P (3)ia P
(3)
jb
?
= (1.9)2 ?GDi (ta)GDj (tb)
?
P (1)ia P
(1)
jb
?
. (4.66)
? We have described with equation 4.20 that the electronic gain is treated as
having a constant value of 1.1 ? 10?3 ADC counts per electron emitted from
121
the APDs. The wires use a similar set of electronics; however, their gain is
nominally set to a different value of roughly 6.3 ?10?4 ADC counts per electron
collected on the wires. In the EXO code the two numbers are mistakenly
reversed. As a result, GEi (t) was underestimated by almost a factor of two;
the effect of this is to overestimate by almost a factor of two the number of
photons being deposited on APDs, which again causes us to underestimate
the Poisson noise introduced within the APDs.
The effect of both of these issues is to underestimate the importance of Poisson
noise, effectively fooling denoising into believing its photon statistics are greater than
they truly are; this, in turn, leads it to attach too much weight to too few channels in
its energy estimator. Thus, the resolution is negatively impacted by these mistakes;
the effect of these two issues can be summarized in one scaling factor which controls
the importance of Poisson noise. In figure 4.5 sample runs are denoised with different
values of this scaling factor, including the value of 0.6 which was used for the present
analysis and 1.6 which we would expect to be the optimal choice. It appears that
an improvement in resolution of roughly 0.05 percentage points at 2615 keV will be
gained when thes issues are addressed.
However, neither of these mistakes have any impact on the correctness of the
constraints. As a result, the denoised energy measurements should still be unbiased
even though they are non-optimal. Results from the use of this data will be described
in chapter 7.
122
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Gain Scaling Factor1.15
1.20
1.25
1.30
1.35
1.40
SS Re
soluti
on, ?/
E (%)
 at 26
15 ke
V
This Analysis
Theoretical Optimum
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Gain Scaling Factor1.35
1.40
1.45
1.50
1.55
1.60
SS Re
soluti
on, ?/
E (%)
 at 26
15 ke
V
This Analysis
Theoretical Optimum
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Gain Scaling Factor1.40
1.45
1.50
1.55
1.60
1.65
MS Re
soluti
on, ?/
E (%)
 at 26
15 ke
V
This Analysis
Theoretical Optimum
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Gain Scaling Factor1.55
1.60
1.65
1.70
1.75
1.80
1.85
1.90
MS Re
soluti
on, ?/
E (%)
 at 26
15 ke
V
This Analysis
Theoretical Optimum
Figure 4.5: It is known that the parameters used for denoising in this
analysis are not optimal. Here we use a scaling factor to adjust the
parameter controlling the importance of Poisson noise. Data from run
3516 and 4544 are shown on the left and right, respectively; top plots
show single-site resolution, bottom plots show multi-site resolution. We
can see that an improvement of roughly 0.05 percentage points at 2615
keV could be achieved by moving from the expected optimum. It is not
currently understood why scaling factors above the expected optimum
yield further improvements to resolution.
123
4.11 Future Extensions to Denoising
We have shown that denoising is capable of producing optimal estimates of the
total energy emitted in scintillation by a particular event. However, this algorithm
can be applied to a more general range of problems in the EXO-200 detector with
only minor modifications. Here we describe possible extensions of the denoising
technique to produce anticorrelated energies of individual clusters and to produce
denoised wire waveforms.
4.11.1 Anticorrelated Cluster Energies
One important generalization is in our definition of scintillation clusters. In the body
of this chapter, we have defined a scintillation cluster to include all energy deposits
which occur simultaneously in time. By requiring that different scintillation clusters
occur at different times, we ensure that the pulse template Y?ia[f ] has a different
shape for each scintillation cluster a, so we can disentangle waveform contributions
from each energy parameter Ea and estimate all of them separately.
Historically, time was the only observable available for this sort of discrimina-
tion. However, now that we use all APD channels independently, we can use the
pattern of pulse magnitude for different channels as a clue to the position of the
source of scintillation photons. Even if two scintillation deposits occur simultane-
ously in time it is possible to estimate their separate energies Ea based on their
positions ~xa and our knowledge, from the APD lightmap, of which channels collect
124
the strongest pulse from those positions. Thus, it is possible for the energy estima-
tors described in this chapter to assign scintillation energies to individual clusters,
which would lead to anticorrelated energies of individual clusters with better energy
resolution than charge alone can currently provide.
In principle, this is achievable ? in the extreme case, when two clusters are at
opposite ends of the detector we can expect them to produce light on an entirely
different set of channels, and it should be quite easy to separate pulses produced
by each. However, charge clusters may be quite close together. For denoising to
measure the scintillation from two simultaneous charge clusters solely based on their
relative position, they must be spatially separate enough that their respective light
yields on nearby APD channels are significantly different. As a result, denoising
can only measure scintillation for individual charge clusters if it first performs a re-
clustering process in which clusters occurring near each other are artificially merged
together for the purpose of scintillation measurement.
Anticorrelated cluster energies would not have a significant impact on single-
site data, so it would have minimal impact on our primary goal of searching for
neutrinoless double-beta decay. However, it could have a significant impact on a
search for excited-state two-neutrino double-beta decay by improving the resolution
of gamma lines associated with barium de-excitation. Barabash states in [123] that
the dominant mode of excited-state 2??? decay is to the 0+1 state, which in 136Ba
would lead to the emission of a 760 keV and 819 keV gamma in coincidence; if
we could produce anticorrelated energy measurements for individual clusters, the
sensitivity to these gamma lines could be significantly enhanced.
125
There are also potential benefits to our understanding of the detector itself.
Currently the only mono-energetic beta calibration available for EXO is the double-
escape peak, formed when a gamma particle enters the detector, pair-produces an
electron and positron, and the positron interacts with an electron and emits gammas
which travel sufficiently far from the initial interaction site so that the initial electron
interactions and gamma interactions can be clearly separated. Currently we have the
ability to separately measure charge yield from the gammas and the initial electron
in some of these events, allowing us to understand the different charge yields from
gammas and betas. However, we have no similar capabilities for light yield because
the scintillation energy of the whole event is measured together. If we could separate
the light yield from these spatially separated clusters, we could use these events to
understand both the light and charge yield for different types of interactions.
Another type of detector study which might benefit from anticorrelated cluster
energies is the Compton telescope. This is a technique that combines our knowledge
of cluster energies and scattering angles, along with the Compton scattering for-
mula [124], and allows us to produce estimates of the origin of the incident gamma.
Such a technique permits us to study the sources of our backgrounds, informing
future attempts to reduce them by indicating which materials had the most signif-
icant negative effect. The accuracy with which we locate the origin of the incident
gammas is strongly related to the accuracy of our measurements of the energies of
the individual clusters, so it is expected that improving the accuracy of our cluster
energies would have a positive impact on our ability to locate the backgrounds of
our experiment.
126
The last benefit to this approach is more technical: evaluating the lightmap
for a multi-site event requires an energy-weighted combination of the values of the
lightmap at individual cluster locations, as shown in equation 4.3. This process
is complicated and error-prone, and can be avoided entirely when we treat each
scintillation cluster as having a single well-defined position.
4.11.2 Wire Denoising
The current EXO-200 resolution is limited by the scintillation resolution of the
APDs. As a result, little benefit is anticipated from denoising the wires. However,
it is possible that a future detector could reduce the scintillation noise to a point
where resolution of the wires is non-negligible; alternatively, a future detector could
have noise which is correlated between wires and APDs, making it possible to use
noise information from the wires to improve our scintillation resolution even though
no direct benefit is expected to the wire resolution itself.
In any case, it is possible to apply equations 4.44 to a system with a combina-
tion of wires and APDs without significant adjustment. The template functions Y [t]
will of course be significantly different, and must include a time offset to account for
the different time of arrival of the electrons and photons. The ?lightmap? of charge
will be far less diffuse than for APDs because charge from any location generally
drifts directly to only a small number of wires along externally imposed field lines.
And finally, because electron statistics should be negligible and pulse amplification
should not be a noisy process, for wire channels we would modify equation 4.17 to
127
state that q(iwire, a) = 0. In this sense, denoising of wires consists of minimizing
the variance in the energy estimator due to electronic noise, and using the wires to
improve our denoising of APDs consists of making use of correlations between wire
and APD noise without expecting any correlations between APD and wire pulses,
all while ensuring that the energy estimates are unbiased by retaining the same
constraints.
4.12 Summary
We have shown in this chapter that a technique exists to produce an optimal lin-
ear energy estimate for the scintillation clusters in EXO-200 and is fully described
by equations 4.64. It requires detailed knowledge of the origins of noise in APD
waveforms and a full lightmap describing the scintillation yield on each channel as
a function of cluster position. Chapters 5 and 6 describe aspects of how we measure
electronic noise and the APD lightmap, both of which have proven to be critical
inputs to denoising. We will show in section 7.6 that the result is a significant im-
provement in the energy resolution of EXO-200 and a corresponding increase in its
sensitivity to ??0? decay.
128
Chapter 5: Electronic Noise Measurements
As described in chapter 4, a detailed noise model is required to perform our denois-
ing algorithm. Equation 4.5 specifies that the correlations in electronic noise are
taken as inputs to denoising; in this chapter we describe the measurement of those
correlations in EXO-200. Section 5.1 specifies the desired measurement; section 5.2
identifies the time-dependent behavior of noise; and section 5.3 describes the algo-
rithm to measure noise from data. We conclude in section 5.4 with some possible
future work to improve the quality of the noise measurements and their use in other
aspects of the EXO-200 analysis.
5.1 Mathematical Framework for Noise Correla-
tions
A waveform on channel i with no pulse on it consists entirely of a noise function
Ni[? ]. The noise is a random function: its value at each time ? is a random variable.
Our goal is to describe the joint probability distribution of those random variables.
Noise on different channels is correlated, so our joint probability distribution should
129
describe not only the noise for all time samples ? on a particular channel i, but also
the relation between noise on any distinct pair of channels i and j.
We can guarantee that the noise is stationary (meaning that the distribution
does not change over time, ie. due to a drifting baseline) because the waveforms
are all subject to shaping in the front-end electronics which removes low-frequency
noise components, as described in section 3.5. It is conventional to study station-
ary noise in Fourier space. This may lead to sharper features because noise often
originates from environmental factors that demonstrate periodic behavior. In EXO-
200, examples of possible sources of periodic waveform noise include acoustic noise
from the cleanroom, mechanical vibrations of the various wires and cables in the
TPC, or switching noise from the digital power supplies. The Fourier transform can
also lead to a simpler characterization (at lowest order) of noise correlations: in a
steady-state environment the second-order correlations between noise components
at different frequencies is always zero because the inner product of two sinusoidal
functions with different frequencies is always zero.
We write the discrete Fourier transform of Ni[? ] as N?i[f ], consistent with the
notation described in section 4.2. Although the specific choice of convention will not
matter for any of the analysis in this work, for completeness we specify explicitly
the definition of the discrete Fourier transform and its inverse as
N?i[f ] =
T?1?
?=0
Ni[? ]e?2pii?f/T (5.1)
Ni[? ] =
1
T ?
T?1?
f=0
N?i[f ]e2pii?f/T , (5.2)
130
where T is the (unitless) number of samples in the time domain waveform and
the roman-font i is
?
?1. Both of these waveforms are assumed to be periodic, so
we store N [? ] with ? ? [0, T ) and N? [f ] with f ?
[
0, bT2 c
]
, where b c indicates that
rounding is performed downward. This set of conventions matches the conventions of
the real-to-complex discrete Fourier transform implemented by the popular FFTW
library, which is available on all major computational platforms [125].
Our sampling frequency of 1 MHz means that we can associate N?i[f ] with
noise at a frequency of f/T MHz (where we again recall that T and f are unitless).
The accuracy of this association is dependent on the accuracy of the nominal 1 MHz
sampling. This is controlled by a nominal 80 MHz oscillator in the master TEM unit;
preliminary investigations indicate that this frequency changes over time, and may
deviate from a true 80 MHz period by as much as ten parts per million [113, 126].
Since we apply low-pass filters with a combined effective cutoff frequency around
167 kHz, as described in section 3.5, this is expected to have a negligible effect on
our noise measurements.
To quantify the second moments of the noise, we have seen in equation 4.44
that it will be useful to measure expectation values of the real-valued quantities for
each pair of channels i, j and each frequency index f :
?
N?Ri [f ]N?Rj [f ]
?
(5.3a)
?
N?Ri [f ]N? Ij [f ]
?
(5.3b)
?
N? Ii [f ]N? Ij [f ]
?
, (5.3c)
131
where N?R and N? I denote the real and imaginary parts of the noise in Fourier space,
respectively.
We can see that equations 5.3 possess some symmetries which may reduce
the size of a file and permit the exploitation of faster matrix operations. First,
equations 5.3a and 5.3c are symmetric under exchange of channel indices i and j, so
we can store these values on disk more compactly by only storing the expectation
values where i ? j.
A second symmetry comes from the observation that for most frequencies the
phase is random because a waveform trigger time is uncorrelated with the phase of
any noise frequency. More specifically, we can decompose N? [f ] into a real-valued
amplitude A[f ] and phase ?[f ] by
N? [f ] = A[f ]e?[f ]i. (5.4)
The phase and amplitude cannot be correlated because the phase depends solely on
where we choose to sample a waveform from the detector, whereas the amplitude
at a particular frequency is constant in time. By the Nyquist-Shannon sampling
theorem (see for example [127]) we can reconstruct this function unambiguously for
all frequencies less than half of the sampling rate; in our case, we can reconstruct
perfectly all parameters for f < T/2. This means that in a waveform with odd-
valued T we can perfectly reconstruct all components, and in a waveform with
even-valued T we can perfectly reconstruct all components except f = T/2.
For the frequency f = T/2 in an even-length time-domain waveform, we cope
132
with the ambiguous reconstruction of A[T/2] and ?[T/2] by asserting that ?[T/2] is
zero, so that the last Fourier coefficient is strictly real-valued; this comes automat-
ically from equation 5.1. Additionally, for the zero-frequency component ?[0] = 0
is enforced by the fact that N [? ] is real-valued. In all other frequency components,
however, there is a unique choice of ?[f ] ? [0, 2pi), and it can have no directional
preference because the time of the event trigger has no preference. Put another
way, for some fixed time ?0 it is equally likely that we sample a waveform starting
at ?0, ?0 + 1, ..., each of which would result in a different phase ?[f ]. This can also
be viewed as a symmetry of the noise: if we translate the time of the waveform we
sample, it has the effect of rotating the phase ?[f ]. All expectation values should
be invariant under such an action.
We can then expand equation 5.3a, taking (as discussed earlier) only the phase
to be random:
?
N?Ri [f ]N?Rj [f ]
?
= ?Ai[f ]Aj[f ]cos (?i[f ]) cos (?j[f ])? (5.5)
= ?Ai[f ]Aj[f ]? ?
?cos (?i[f ] + ?j[f ]) + cos (?i[f ]? ?j[f ])?
2 . (5.6)
133
We can see that ?i[f ]+?j[f ] is not invariant under time translation, because a small
shift in time ?t will decrease the value of a phase ?i[f ] by ?2pif?t/T ; as a result,
we can assert that the left term must have an expectation value equal to zero. On
the other hand, ?i[f ] ? ?j[f ] is invariant under time translation, so that term can
survive:
?
N?Ri [f ]N?Rj [f ]
?
= ?Ai[f ]Aj[f ]?2 ? ?cos (?i[f ]? ?j[f ])? . (5.7)
Similarly for the other expectation values:
?
N?Ri [f ]N? Ij [f ]
?
= ?Ai[f ]Aj[f ]?2 ? ?sin (?j[f ]? ?i[f ])? (5.8)
?
N? Ii [f ]N? Ij [f ]
?
= ?Ai[f ]Aj[f ]?2 ? ?cos (?i[f ]? ?j[f ])? . (5.9)
We can see from equations 5.7-5.9 that for frequencies other than 0 and T/2,
the following identities hold:
?
N?Ri [f ]N?Rj [f ]
?
=
?
N? Ii [f ]N? Ij [f ]
?
(5.10)
?
N?Ri [f ]N? Ij [f ]
?
= ?
?
N?Rj [f ]N? Ii [f ]
?
. (5.11)
Taking advantage of these symmetries, we can compute that for 71 channels
and waveforms containing 2048 samples we need to store roughly five million in-
dependent values to characterize the noise correlations; this results in a file of size
roughly 40 megabytes for each snapshot of the noise correlations. We show in sec-
134
tion 5.2 that only a small number of snapshots appear to be necessary, so this can
be a manageable dataset.
5.2 Time Windows of Constant Noise
Section 5.1 has described the noise correlation information which denoising requires
and demonstrated that a snapshot of the noise will require roughly 40 MB of data.
Although EXO-200 has a significant amount of noise information available and is
capable of producing a detailed history of the noise variations in time, taking such
an approach would quickly produce an unmanageable quantity of data. This section
explains how the noise behavior appears to be stable for long periods of time, per-
mitting us to create a coarse-grained history of the noise without losing significant
accuracy.
The approach to identifying these constant-noise time windows is two-fold.
Firstly, we identify certain environmental changes which are likely to have a signif-
icant impact on the noise observed on the APDs. Since the times at which these
changes occur are generally known precisely (and usually fall between runs), we can
place time boundaries accurately when an environmental change is traced as the
origin of a change.
Secondly, we develop a set of parameters which can easily be viewed in plots
versus time. These trend plots are then reviewed qualitatively by collaboration
members, and stepwise changes in any of these parameters can be interpreted as a
change in detector noise at the time of the step. Although sometimes it is necessary
135
Dec 2
011
Mar 
2012 Jun 2
012
Sep 2
012
Dec 2
012
Mar 
2013 Jun 2
013
Sep 2
013
Date
50
60
70
80
90
100
110
120
RM
S [A
DC
]
FEC PS Restart
Ebox Fan1
Ebox Fan2
APD 01 -> APD 09
Util. Power Outage
Util. Power Outage
APD 09 Trouble
APD 09 Restarted
Ventilation Shutdown
Surf. Power Outage
Ventilation Turn-on
Power Switch EBox Filters Removed
Temp. Excursion Start
Temp. Excursion EndAll APDs
Figure 5.1: Sum noise for all APD channels measured from charge in-
jection runs, with environmental changes indicated which may correlate
with observed changes in noise. Data provided by Josiah Walton.
to guess the precise time when a change in noise occurred, when possible we review
the environmental conditions of the detector in more detail and search for possible
causes for the change in noise which would permit us to pinpoint the time of the
change.
One source of noise information is provided from charge injection runs. These
runs have been taken daily for the entire history of our dataset, and are designed to
inject a known pulse half-way through the 2048-sample waveform. Since the pulse
time is known, it is also generally true that the pretrace has no pulse on it and can
be viewed as a pure noise sample. It is possible by coincidence for a low-background
event to deposit energy which is observed as a pulse in the pretrace, but this is
extremely rare.
136
Runs Dates Comments
2401-2423 9/28/2011-9/30/2011 APDs biased to special ?9-28-11? settings.
2424-2690 9/30/2011-11/2/2011 FEC voltage adjustment on 11/2/2011.
2691-2852 11/2/2011-11/28/2011 Cooling fan installed on Ebox 1,
11/28/2011.
2853-2891 11/28/2011-12/4/2011 Cooling fan installed on Ebox 2, 12/4/2011.
2892-3117 12/4/2011-1/13/2012 Power outage, no data 1/13-1/19.
3118-3326 1/13/2012-2/23/2012 During runs 3314-3320 (2/22) APD channel
163 was disconnected. During runs 3324-
3331 (2/23) front-end cards were swapped.
3327-3700 2/23/2012-5/11/2012 Brief power outage on 5/11.
3701-3933 5/11/2012-7/10/2012 Possibly associated with a TEM tempera-
ture spike on the morning of 7/10; cause
not known.
3934-4003 7/10/2012-7/24/2012 Possibly associated with a new TEM mod-
ule on 7/24.
4004-4126 7/24/2012-8/28/2012 Power outage, no data 8/28-8/30, messy re-
covery of APD01.
4127-4589 8/28/2012-12/27/2012 Pump reset on 12/27, but it is not clear this
is the cause of the short spike in APD noise.
4590-4609 12/27/2012-1/1/2013 As mysteriously as the noise came on 12/27,
it disappeared sometime between 1/1 and
1/2.
4609-4779 1/1/2013-2/20/2013 Not clear that anything happened on 2/20;
we should review this boundary to ensure it
is significant.
4780-5061 2/20/2013-5/14/2013 Noise on TPC2 changed sometime around
5/14-5/20. Thermal stores stopped cooling
on the evening of 5/17, but it it unclear how
this would impact the APDs.
5062-5197 5/14/2013-6/7/2013 Modifications to electronics boards on 6/7.
5198-5590 6/7/2013-8/31/2013 Temperature excursions in the cleanroom
around 8/31 permanently impacted the
APD behavior.
5591-5892 8/31/2013-11/11/2013 Differential-pressure excursion on 11/11.
Table 5.1: Recommended noise windows, based on the current understanding of
changes in noise and their possible causes. This is not the same as the noise windows
actually used in the present analysis; for those, please see table 5.2.
137
Dec 2
011
Mar 
2012 Jun 2
012
Sep 2
012
Dec 2
012
Mar 
2013 Jun 2
013
Sep 2
013
Date
30
40
50
60
70
80
RM
S [A
DC
]
FEC PS Restart
Ebox Fan1
Ebox Fan2
APD 01 -> APD 09
Util. Power Outage
Util. Power Outage
APD 09 Trouble
APD 09 Restarted
Ventilation Shutdown
Surf. Power Outage
Ventilation Turn-on
Power Switch EBox Filters Removed
Temp. Excursion Start
Temp. Excursion EndAPD Plane 1APD Plane 2
Figure 5.2: Sum noise for each APD plane measured from charge in-
jection runs, with environmental changes indicated which may correlate
with observed changes in noise. Data provided by Josiah Walton.
Runs Dates
2401-2423 9/28/2011-9/30/2011
2424-2690 9/30/2011-11/2/2011
2691-2852 11/2/2011-11/28/2011
2853-2891 11/28/2011-12/4/2011
2892-3117 12/4/2011-1/13/2012
3118-3326 1/13/2012-2/23/2012
3327-3700 2/23/2012-5/11/2012
3701-3949 5/11/2012-7/12/2012
3950-4140 7/12/2012-9/2/2012
4141-4579 9/2/2012-12/24/2012
4580-4779 12/24/2012-2/20/2013
4780-5197 2/20/2013-6/7/2013
5198-5590 6/7/2013-8/31/2013
5591-5892 8/31/2013-11/11/2013
Table 5.2: Noise windows used for the current analysis. For the noise windows
recommended for future analyses and a more detailed explanation of the causes of
changes in noise behavior, see table 5.1.
138
Dec 2
011
Mar 
2012 Jun 2
012
Sep 2
012
Dec 2
012
Mar 
2013 Jun 2
013
Sep 2
013
Date
0
10
20
30
40
50
60
RM
S [A
DC
]
FEC PS Restart
Ebox Fan1
Ebox Fan2
APD 01 -> APD 09
Util. Power Outage
Util. Power Outage
APD 09 Trouble
APD 09 Restarted
Ventilation Shutdown
Surf. Power Outage
Ventilation Turn-on
Power Switch EBox Filters Removed
Temp. Excursion Start
Temp. Excursion End
APD 00 RMS
APD 03 RMS
APD 04 RMS
APD 05 RMS
APD 06 RMS
APD 09 RMS
Figure 5.3: Sum noise for each APD electronics board measured from
charge injection runs, with environmental changes indicated which may
correlate with observed changes in noise. Data provided by Josiah Wal-
ton.
139
Dec 2
011
Mar 
2012 Jun 2
012
Sep 2
012
Dec 2
012
Mar 
2013 Jun 2
013
Sep 2
013
Date
5
10
15
20
25
30
35
RM
S [A
DC
]
FEC PS Restart
Ebox Fan1
Ebox Fan2
APD 01 -> APD 09
Util. Power Outage
Util. Power Outage
APD 09 Trouble
APD 09 Restarted
Ventilation Shutdown
Surf. Power Outage
Ventilation Turn-on
Power Switch EBox Filters Removed
Temp. Excursion Start
Temp. Excursion End
Pie 1
Pie 2
Pie 3
Pie 4
Pie 5
Pie 6
Pie 7
Pie 8
Pie 9
Pie 10
Pie 11
Pie 12
Figure 5.4: Sum noise for each APD pie (6-7 channels) measured from
charge injection runs, with environmental changes indicated which may
correlate with observed changes in noise. Data provided by Josiah Wal-
ton.
140
10/11 12/11 3/12 7/12 12/12 4/13 
~190kHz 
Figure 5.5: Time trend of
?
N?R192[385]N?R193[385]
?
(black),
?
N?R192[385]N? I193[385]
?
(green), and
?
N? I192[385]N? I193[385]
?
(red) cor-
responding to the correlations between channels 192 and 193 at 188
kHz. Blue lines indicate tentative noise windows. This frequency is
chosen because it is associated with a peak in the noise power spectrum,
and the pair of channels is selected as a representative example for
seeking changes in noise behavior. A restart of the front-end power
supply at run 2700 is coincident with the reduction in correlated noise
shown here, but the reason for this effect is not understood [128].
141
10/11 12/11 3/12 7/12 12/12 4/13 
~190kHz 
Figure 5.6: Time trend of
?
N?R202[383]N?R203[383]
?
(black),
?
N?R202[383]N? I203[383]
?
(green), and
?
N? I202[383]N? I203[383]
?
(red) cor-
responding to the correlations between channels 202 and 203 at 187
kHz. Blue lines indicate tentative noise windows. This frequency is
chosen because it is associated with a peak in the noise power spectrum,
and the pair of channels is selected as a representative example for
seeking changes in noise behavior. Installation of electronics board
cooling fans is coincident with the reduction of noise around run
2900, with different installation times for different channels, but the
reason this particular noise frequency is so strongly affected is not
understood [128].
142
The approach described in [129] is to use the pretrace samples between 200
and 800 ?s from charge injection runs as noise samples. In that reference, subset
of waveforms ? APD pies, electronics boards, planes, or all APD channels together
? are summed together for each event, and the root-mean-square variation in the
summed waveform is averaged over the samples between 200 and 800 ?s and over all
events in the charge injection run. Each charge injection run contains 13,200 events,
all of which can be used to improve the quality of this average root-mean-square
measurement of noise on that subset of channels.
Figures 5.1, 5.2, 5.3, and 5.4 show the noise for all APDs, APD planes, APD
electronics boards, and APD pies respectively, with notable environmental changes
overlaid to demonstrate their correlation with changes in noise. These trending
plots demonstrate that although the noise undergoes day-to-day fluctuations (pos-
sibly due to insufficient sample statistics in the charge injection runs) and some
gradual changes over time, the dominant effects are from stepwise changes which
are generally correlated with a change in the detector environment.
Another approach to understanding the behavior of noise with respect to time
is to compute noise correlations
?
N?Ri [f ]N?Rj [f ]
?
,
?
N?Ri [f ]N? Ij [f ]
?
, and
?
N? Ii [f ]N? Ij [f ]
?
for some particular choice of channels i and j and frequency f , and identify changes
in those particular values over time. The algorithm for computing these noise cor-
relations will be described fully in section 5.3; the philosophy is simply that it is
difficult to visualize changes in a dataset of five million values, but easy to visualize
changes in some very small subset of those values. Changes in the small subset are
expected to indicate changes in overall noise behavior; by sampling enough choices
143
of i, j, and f we can hope to identify all times when the noise behavior changed.
To guide this search, we focus our attention on known peaks in the noise
power spectrum; furthermore, we focus on frequencies which have been observed
to produce noise that is highly correlated across channels. One example of such
a noise peak is around 190 kHz, and examples of the trending of noise correlation
values are shown in figures 5.5 and 5.6. There we can see that the changes in
noise correlations for a particular frequency and channel pair can be dramatic. This
method of understanding our noise seems to be a powerful and sensitive approach
to complement studies based on root-mean-square noise measurements described
earlier.
The currently recommended noise run windows are listed in table 5.1. An
attempt has also been made to identify the reasons for changes in noise behavior,
but in some cases there is no clear change to the detector that is correlated with the
change in noise. Future work may include combining all sources of noise trending
information to obtain a more precise understanding of exactly when the behavior
change occurred.
The run windows recommended in table 5.1 are recommended for future work;
however, they differ in some instances from the run windows used for the present
analysis. These are listed in table 5.2. In some instances the change in run range
is minor and comes from a closer examination of exactly when the noise behavior
changed; in others, more careful analysis revealed additional stepwise changes to
the noise which had not previously been observed. The energy resolution achieved
will be shown in section 7.3.4 to be fairly stable in time, so it is not believed that
144
the present analysis was significantly impacted by these slight variations in choice
of noise run windows, but this assertion has not been quantified.
Future work can continue to improve the choice of noise run windows. One
detail which is neglected in the present analysis is the selection of runs to be used
within a run range. In the present analysis, all low-background runs which were
used in the final dataset are also used for noise measurements. However, this may
not always be ideal.
A specific example comes from runs 3321-3323, which occur after channel 163
was disconnected but before electronics boards were replaced. It is not advisable in
this case for so few runs to form their own noise window because the quantity of
statistics would be insufficient, but retaining them in the noise window formed by
runs 3118-3326 could bias our estimates of noise on channel 163. Instead, those runs
should not have been used to measure noise at all; this will be corrected in future
work.
Additional studies may be necessary if denoising is extended to include the
wire channels. The wire noise is generally more stable than the APD noise [129].
However, there are exceptions to this observation; one known exception comes from
the Spring of 2012, when u-wire channel 16 was mistakenly dropped from data
acquisition. If data prior to run 2464 is used in a future analysis, the change in
u-wire electronics in early Fall 2011 would also need to be accounted for.
The most significant future work, though, simply consists of more careful study
to characterize exactly when the noise behavior changed and whether slow changes
in noise behavior may warrant the creation of additional time windows. These
145
Da
te
01
/12
04
/12
07
/12
10
/12
12
/12
04
/13
07
/13
Cumulative livetime (days)
0
10
0
20
0
30
0
40
0
50
0
60
0
go
lde
n l
ive
tim
e
no
t g
old
en
 liv
eti
me
Figure 5.7: Cumulative low-background livetime collected in EXO-200.
Figure provided by David Auty.
investigations will feed into further improvements to denoising, but they may also
inform a better understanding of the noise in our detector. The current body of work
already constitutes a rich set of studies performed by many collaboration members,
and has enabled the use of a manageable set of noise files to characterize the whole
history of our detector.
5.3 Algorithm for Measuring Noise
For each of the noise windows specified in table 5.2, it is necessary to locate a pool
of ?clean? events when no pulse occurred; these will be interpreted as samples of
pure noise. This section will describe the selection cuts for clean events, followed by
the data format currently used to store noise correlation information.
146
Although EXO-200 does occasionally take noise data, which consists of rapid
requests for waveforms without any corresponding physics trigger, these runs have
not been taken consistently throughout the EXO-200 dataset and will not be usable
for measuring noise that changes over time. However, during all low-background
data-taking it has been standard practice to force one trigger every ten seconds.
These solicited triggers, though infrequent, have cumulatively amounted to an enor-
mous quantity of data which is expected to be purely noise. Figure 5.7 shows the
cumulative low-background livetime which has been collected, and we can see that
it is approximately evenly distributed across the dataset and, at a rate of 0.1 Hz,
millions of solicited triggers have been taken during the livetime of the detector.
The cuts we will make to select clean events are:
? Only golden-quality low-background runs (those deemed usable for final fits)
will be used for noise measurements, to reduce the inclusion of anomalously
noisy runs.
? Events which occur during a ?bad? time (times identified as anomalous within
a run, eg. due to loud noises) are excluded. Also, events occurring during
the clean room siren are explicitly excluded, even though these times should
generally also be marked as bad.
? Only solicited-trigger events are used; events where a physics trigger fired may
contain a pulse, regardless of what is found by reconstruction.
? Truncated waveforms are not used, because they cannot provide the same set
of noise frequencies as a full 2048-sample waveform.
147
? Waveforms which saturate in any channel are not used.
? Waveforms which are flagged as a known type of sporadic noise are excluded.
? Events on which any u-wire or APD pulse was found by reconstruction are
excluded.
From the set of acceptable events produced in this way, calculation of the
noise correlation values is done in exactly the way implied by the definition of our
noise correlation terms. The expectation values are computed as averages over the
set of events, so for example
?
N?Ri [f ]N?Rj [f ]
?
is evaluated for a particular pair of
channels i, j and frequency index f by computing N?Ri [f ]N?Rj [f ] for all acceptable
events, and then using the average of these values. The symmetries of equations 5.10
and 5.11 are not currently used, but future versions of the code will exploit these
symmetries to effectively double the number of independent measurements of each
noise correlation.
The data are currently stored in a binary file as a sequence of eight-byte
floating point numbers. Binary portability has not been a serious concern because
all machines on which the files are used have a similar architecture (little-endian,
IEEE floating point format). To allow studies of u-wire denoising, correlations are
computed between all u-wire and APD channels, so the ordered list of channels in
the file is
[0, 1, 2, ..., 36, 37, 76, 77, ..., 112, 113, 152, 153, ..., 225]. (5.12)
No channel mapping is included in the file, so the set of included channels must be
148
fixed regardless of the addition or subtraction of channels in the data (specifically,
channel 163 and channel 16, both of which are only present in some of our data,
and channels 178, 191, and 205, which were never active channels); these channels
are included in the noise file, with noise fixed to zero where appropriate, and it is
assumed that downstream software will be intelligent enough to use only those noise
values which are meaningful.
The file stores noise correlations as a sequence of matrices, with one matrix
for each frequency component except the baseline component f = 0. We recall from
equation 4.48 that the covariance matrix of a list of random variables is defined as:
COV (x1, x2, . . . , xn) =
?
?
?
?
?
?
?
?
?
?
?
?
cov(x1, x1) cov(x1, x2) ? ? ? cov(x1, xn)
cov(x2, x1) cov(x2, x2) ? ? ? cov(x2, xn)
... ... . . . ...
cov(xn, x1) cov(xn, x2) ? ? ? cov(xn, xn)
?
?
?
?
?
?
?
?
?
?
?
?
. (5.13)
We recall that when ?x? = ?y? = 0, the covariance of random variables x and y
will be simply ?xy?; for us, all noise coefficients with f > 0 have mean zero. For
frequencies 0 < f < 1025, we define an ordering for the electronic noise coefficients
~?N [f ] =
(
N?R0 [f ], N?R1 [f ], ..., N?R225[f ], N? I0 [f ], ..., N? I225[f ]
)
(5.14)
where the channel index ranges over the channels listed in equation 5.12. For f =
149
1025 the imaginary components are all identically zero, so we define
~?N [1025] =
(
N?R0 [1025], N?R1 [1025], ..., N?R225[1025], N? I0 [1025], ..., N? I225[1025]
)
. (5.15)
Note that this is not the same as the index ordering used for computations in
denoising and identified in equation 4.47; since the on-disk format outlined here
will need to be read into memory anyway, and since the list of channels on disk is
different from the list of channels used for denoising, this is not a technical challenge
but only a point which requires careful programming.
The binary-format noise file is then the sequence COV
(
~?N [1]
)
, COV
(
~?N [2]
)
,
..., COV
(
~?N [1025]
)
, where each covariance matrix in turn is serialized to disk. No
current analysis requires baseline information, so the f = 0 covariance matrix is
omitted. Covariance matrices are symmetric, so the matrix serialization can be
viewed equally well as column-major or row-major. The matrices are all currently
stored in unpacked format, meaning that no symmetries are currently exploited; fu-
ture work should include specifying a more efficient packing of the noise coefficients
which takes advantage of the symmetries described in section 5.1.
Besides the anticipation of a change in file format in future versions of the
code, there are other improvements to this algorithm which should be pursued in
the future. Some applications have been considered which would require knowledge
of the noise on all channels, so the noise on v-wire channels should be recorded
as well. As another means of improving the usability of these files for other ap-
plications, an independent interface to the file format should be designed. This
150
could perhaps be written as a wrapper around some matrix serialization library,
which could simplify the conversion between packed (for storage) and unpacked (for
computation) symmetric matrix formats.
The highest priority, of course, will continue to be improving the quality of the
noise correlations which are calculated. Future topics for improvement are detailed
in the next section.
5.4 Future Directions with Noise Measurement
Although the noise correlations described in this chapter have been sufficient for
the present analysis, and there are no current indications that they are the limiting
factor on denoising, still there are a number of possible improvements and extensions
which could be investigated in future work. Here we describe possible improvements
to the quality of the noise correlations, followed by possible applications outside of
denoising which may benefit from the use of noise correlations described in this
chapter.
The general algorithm for creating noise correlations is fairly straightforward:
identify time windows of constant noise behavior, select events within a noise win-
dow which appear to be clean representations of noise, and measure the average
noise correlations (pairwise products) within those clean events. However, there are
potential improvements in all of these steps which may be subjects for future study.
For identifying the time windows, the current analysis is quite detailed in its
attempt to correlate dramatic changes in noise behavior to known changes to the
151
Run Number
2000 2500 3000 3500 4000 4500 5000 5500
Av
er
ag
e 
Gl
itc
h P
ea
k V
alu
e 
(AD
C U
nit
s)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Plot of Trigger Glitch Peak vs. Run Number on Channel 158
Figure 5.8: The amplitude of the TEM injected pulses (?glitch? pulses)
over time for channel 158. Figure provided by Sam Homiller.
detector. However, we have only begun to understand the more gradual changes
to noise; one can imagine slow changes to noise on the order of weeks or months,
or daily variations due to varying temperature, and neither would be well-captured
by the current methods of identifying noise windows. Combining charge injection
noise and low-background noise studies should allow us to form a high-resolution
map of the noise behavior over time, and it should be possible for us to understand
exactly how stable the detector noise is over all time scales. This is perhaps the
most pressing need for future noise analysis.
The selection of clean events is also a topic requiring further study. Although
solicited triggers are treated as clean events, it has long been known that the TEM
152
can inject pulses into events during solicited triggers; the mechanism by which it
does so is unknown. This injected pulse is correlated with the trigger time, meaning
that it has a non-random phase; Sam Homiller has made a preliminary study to
separate the injected pulses from noise by exploiting this property of non-random
phase, and the amplitudes of these pulses over time for a particular channel are
shown in figure 5.8. In the present analysis these injected pulses are assumed to be
small and ignored. However, the tools to subtract the mean injected pulses have
already been developed, and we should investigate:
? Do injected pulses occur with all trigger types, or only solicited triggers?
? Does subtraction of injected pulses improve the quality of the noise correlations
or the results from denoising?
Finally, section 5.1 described how we neglect the possible aliasing of extremely
low-frequency noise. To a first approximation this would simply be observed as
a drift in the baseline of waveforms, which has no significant effect on the offline
analysis (though it may have an impact on the online triggering threshold). However,
a more accurate treatment may observe that low-frequency noise is aliased into the
low Fourier components with f = 0, 1, 2, ... in a way which does not accurately
reflect the true noise in the detector. Such aliasing would perhaps lead to an event-
by-event bias in reconstructed peak magnitudes, both from standard reconstruction
and denoising. It may be possible to estimate the contribution from aliasing by
looking for periodicity in the low frequency Fourier components event-by-event;
noise runs (such as run 4796, a ten-minute 50-Hz solicited trigger run taken under
153
normal running conditions) would be a particularly good place to look for such an
effect because of the higher rate of usable noise events in that data. One approach
to combining many events into a single estimate of noise frequency components that
extends to lower frequency could be to use a non-equispaced Fourier transform [130].
Another extension of the current work with noise correlations, related to the
question of whether aliasing is a significant effect, is to study whether we can interpo-
late from the noise correlations of a 2048-sample waveform what the expected noise
correlations for a shorter (truncated) waveform would be. Generally, it is expected
that the longer a waveform is, the more information it will provide; so it should be
true that a 2048-sample noise waveform provides more information than a truncated
waveform provides. We would like to be able to downsample the noise correlations
from a 2048-sample waveform to the expected noise correlations for a truncated
waveform on this premise. However, if aliasing is a significant factor then it may
be that the low-frequency aliasing observed in a 2048-sample waveform is different
from the aliasing observed in a truncated waveform, and it may be that a sim-
ple interpolation of the 2048-sample noise correlations is not an accurate reflection
of the noise observed on truncated waveforms. (Conversely, it may be that trun-
cated waveforms are more vulnerable to low-frequency aliasing than 2048-sample
waveforms; in this case it may be that the downsampled noise from a 2048-sample
waveform is an accurate reflection of noise on the truncated waveform, but that
the Fourier transform of the shorter waveform still exhibits too much aliasing to be
usable.) Understanding the degree to which we can understand noise on truncated
waveforms and evade low-frequency aliasing issues for truncated waveforms would
154
be a key step to understanding whether denoising can be extended to operate on
such events.
It may also be hoped that these noise correlations will find uses beyond denois-
ing and other signal processing applications. One example which is currently under
consideration is for simulated noise. It is a general observation within EXO-200 that
the noise model employed in our simulations is simple, ignoring all correlations and
all noise frequency peaks which have been observed in data. For many purposes
this is sufficient, but to fully model our energy threshold a more accurate model
is needed. Preliminary investigations have focused on sampling noise directly from
the observed waveforms and using these noise samples, as described in section 7.1.2.
However, it would also be possible to use the noise correlations described in this
chapter to represent our most complete understanding of the noise in the detector,
and simulate noise which matches those noise correlations. This can be accom-
plished by computing the Cholesky decomposition of the noise correlations, viewed
as a covariance matrix, as described in [131].
This chapter has described the measurement of electronic noise correlations
from low-background data, with significant input from charge injection runs as well.
Noise is one of the central features of the EXO-200 analysis, and understanding it
in detail is a critical component to an effective denoising strategy. Thus, the result
of this chapter can be understood as one of the two key inputs to denoising. The
other, the APD lightmap, is described in chapter 6.
155
Chapter 6: The Lightmap
EXO-200 uses a lightmap to characterize the expected scintillation pulse magnitudes
from scintillation clusters with a known position and energy. Chapter 7 has shown
that the lightmap is a critical input to the denoising algorithm. In this chapter, we
develop from a simpler type of lightmap used in prior analysis to a fully detailed
individual-APD lightmap which characterizes scintillation pulse magnitudes on ev-
ery channel as they depend on scintillation cluster position. All of this is done in
a time-dependent fashion, so the main result of this chapter will be, for each APD
channel, a four-dimensional function of scintillation cluster position and time that
yields the expected pulse magnitudes. In section 6.1 we describe the lightmap which
existed prior to this work, its accomplishments, and its issues which made an up-
grade necessary. Section 6.2 will present details of the production of the lightmap
from empirical data; deviations from that algorithm for the present analysis are
presented in section 6.4. Visualizations of the result are provided in section 6.5.
156
6.1 A History of EXO-200 Lightmaps
The EXO-200 detector has roughly 450 APDs ganged into 74 data channels of five
to seven APDs each. Three of the APD ganged channels were disabled due to noisy
components before physics data collection began; a fourth channel was disabled in
February 2012 due to increasing noise. The APDs are set into the two endcaps of
the cylindrical EXO-200 detector. To improve light collection, teflon sheets cover
the inside of the detector and reflect light back into the liquid xenon rather than
allowing it to be absorbed by the vessel walls.
Scintillation is not collected uniformly by all APDs. Given the same amount of
energy deposited into the detector, APDs channels nearest to the deposited energy
show significantly larger pulses than APD channels far from the deposit. To accu-
rately measure the true scintillation energy, it is critical to map out the response of
the APDs as a function of deposit position ~x.
Furthermore, the APDs and their front-end electronics show time-dependent
changes. Gains drift in each APD channel at a variety rates, and stepwise changes
occur when the electronics are changed or channels are disabled. This has happened
on multiple occasions during the course of the experiment. This means that, in
addition to mapping the response of the APDs as a function of deposit position ~x,
we must also map it as a function of time. We call this the lightmap.
Earlier lightmaps were derived from periodic source calibration campaigns
collected over one or more days. The ?strong? thorium-228 source (34.04 kBq, or
157
0.9201 ?Ci, on September 1, 2009 [132]) was used, and an on-site expert would
position the source in a wide range of locations around the detector. The 2615-keV
single-site gamma line of the thallium-208 (a daughter product of thorium-228, see
figure 7.4) allows the pulse magnitude from a single-position mono-energetic deposit
in the detector to be determined in offline analysis.
Even with such a significant quantity of source data, statistics are generally
insufficient to fully characterize the lightmap with this method. The source guide
tube shown in figure 3.14 does not approach near to every region of the detector;
as a result, some regions are difficult to illuminate with the 2615 keV gamma line.
Additionally, APD pulses on individual channels may be quite small for cluster
positions which are not quite near to the APDs, compounding the problem of low
statistics. To simplify the problem, pulses on the waveforms of each endcap can
be summed together (without gain corrections) such that the 70 or 71 active APD
channels can be treated as merely two APD sum channels. This effectively increases
the size of the pulses and the signal-to-noise ratio of these waveforms. It also makes
the spatial dependence of the response smoother, so that a coarser binning of the
position coordinates is sufficient to characterize the response. The resulting lightmap
was characterized in [98] and used in [14,74] to perform a precision measurement of
the rate of ??2? decay and a search for ??0? decay in 136Xe.
Although this lightmap permitted an energy resolution of 1.67% at our Q-value
to be achieved in [14], it is an incomplete characterization of the APD response. For
example, for ??0? studies we place a fiducial volume cut as close to the edges of
the detector as possible, and in certain regions of the detector the scintillation pulse
158
Nominal Source Position Deployment Position Triggers
P2 nz S2 6207613
P4 px S5 85931937
P2 pz S8 5395820
P4 py S11 6775279
P4 ny S17 5482649
Other Other 1773128
Table 6.1: The total number of thorium triggers taken at each source position;
data include only runs which were used to generate the lightmap. Nominal source
position follows a code where the number is 2 for anode runs and 4 for cathode
runs; the second part of the code indicates the non-zero coordinate, eg. ?px? means
positive-x direction, ?nz? means negative-z direction. A diagram of the source tube
is shown in figure 3.14.
is highly concentrated on a small number of channels. Summing together multiple
channels is, in this case, a lossy form of compression of the data, and it is possible
to extract a better energy measurement if it is avoided.
The present chapter addresses the challenges inherent with the formulation
of an individual-APD lightmap. We do this primarily by a significant increase in
available statistics: whereas the older lightmap used thorium data from a specific
calibration campaign lasting for only a few days, we combine all thorium data that
has ever been collected by the EXO-200 detector between October 5, 2011 and June
24, 2013. The total number of triggers included in this dataset are listed in table 6.1.
This vast extension of the available data raises the challenge that APD behavior is
time-dependent, so pulse magnitudes may not be constant throughout the dataset;
we account for this with a parametrization of the lightmap which permits us to
simultaneously extract this time-dependent behavior and the position-dependent
light yield that is independent of time. The result is a complete characterization of
light yield in EXO-200.
159
The denoising algorithm described in chapter 4 describes one analysis by which
we can exploit knowledge of the behavior of individual APD channels; other analyses
include reducing our scintillation thresholds by allowing us to search for pulses
in subsets of the APDs rather than only in sums across entire endcaps. Thus,
characterizing the APD yield on individual APD-gang channels is a key component
of denoising and may enable additional future studies which would benefit from
individual APD channel information. Section 6.2 describes the method used to
extract an individual-APD lightmap.
6.2 Four-Dimensional Lightmap
As described above, in constructing a channel-by-channel lightmap we face two
conflicting needs: we must use as much data as possible to handle the rapid spatial
variation and smaller pulses which are expected, but if too much data is included
then we run the risk of combining data taken when an APD or the electronics had a
different gain. If we truly wish to use all available data, then we must simultaneously
understand the full time-dependence of the gain. In other words, rather than forming
a small number of independently-measured three-dimensional lightmaps, we instead
measure a four-dimensional lightmap L(~x, t). Since we use thorium-228 source data
to generate the lightmap, we require that our lightmap should predict the magnitude
of the pulses on each APD gang induced by a single-site deposit of energy 2615 keV.
Figure 4.4 shows our definition of a unit-magnitude pulse.
This may at first seem infeasible. After all, adding an extra time dimension
160
to the arguments of the lightmap is equivalent to generating a new lightmap for
every time bin, which is exactly the situation we wish to avoid. But we can make
a simplifying assumption that the lightmap is separable. In a physical sense, we
assume that:
1. From a given position ~x, photons deposit on each APD channel at a constant
rate.
2. Each APD channel amplifies and shapes its pulse with a gain which may vary
in time, but does not depend on the point of origin of the photons.
So, we demand that the lightmap have the much simpler form
L(~x, t) = R(~x)S(t) (6.1)
Although this is a reasonable approximation, there are details which are omited
by this model. Since the electric field and the reflectivity of detector surfaces are
constant in time, assumption 1 is likely to be valid.
However, assumption 2 may not hold in certain cases. For example, each
channel draws its pulse from multiple APDs, each of which has an independent
time-varying gain. Photons from a deposit may preferentially sample the gain from
the closest APD within a channel, so deposits in different locations may track more
closely the gain of the closest APD within a channel. A study of the impact of this
effect is a topic for future study. The analysis presented here will treat it as a small
effect.
161
One simple scheme to find R and S iteratively is described in algorithm 2.
Step 1 of that algorithm is to select 2615-keV single-site events; it is understood
that some compton-scattered events will inevitably leak into our selection window,
but we use the best available energy measurements to minimize that likelihood.
As analysis improvements continue to improve the energy resolution, the number
of selected compton-scattered events will become smaller and the quality of the
lightmap will improve. The rest of the algorithm describes how we can efficiently
converge on estimates of R(~x) and S(t) through an iterative approach. We do not
prove that the algorithm is guaranteed to converge; however, in practice S(t) is
almost constant, resulting in fairly rapid convergence.
Algorithm 2 Generating a Lightmap
1: Tabulate, using the best energy measurement available, a list of single-site 2615-
keV events from thorium-228 source data.
2: Set S(t) = 1 for all APD channels.
3: repeat
4: For each APD channel, estimate R(~x) = L(~x, t)/S(t) from the set of events
tabulated in line 1.
5: For each APD channel, estimate S(t) = L(~x, t)/R(~x) from the set of events
tabulated in line 1.
6: until convergence is reached.
6.3 Algorithm Details
The previous section outlines a general algorithm for computing a four-dimensional
channel-by-channel lightmap from source data. In this section we specify the im-
plementation choices which are made in the code currently in use by EXO-200. In
attempt to encourage future experimentation, alternative options will also be listed
162
in some detail, along with some motivations for these alternatives. Time has not
permitted testing of all of these options, but it is hoped that they will be explored
in the future.
6.3.1 Event Selection
The single-site thorium-228 spectrum is well-peaked at 2615 keV, making it an
excellent monoenergetic calibration line. Our challenge is to select truly 2615-keV
events as efficiently as possible, while simultaneously avoiding near-2615-keV events
which may leak into the dataset. Any events which are not truly 2615-keV events will
reduce the accuracy of our lightmap, so by ensuring that we have a clean sample
of truly 2615-keV events we reduce the impact of input energy resolution on the
accuracy of the lightmap.
This is inherently an iterative process. The first lightmap is constructed from
events selected based on an ionization-only spectrum because no useful position-
corrected scintillation measurement yet exists. The resolution of the ionization-only
spectrum is relatively poor, roughly 3.5% (sigma) of the energy, as described in [98],
and compromises are made to keep the Compton-shoulder leakage to acceptable lev-
els. [98] required events to have ionization between +0.33? and +3? of the ionization
peak.
Such a strongly asymmetric cut is chosen to avoid leakage from compton shoul-
der events. However, because of the anticorrelation between scintillation and ion-
ization, events whose ionization fluctuates high preferrentially selects events whose
163
scintillation fluctuates low, introducing a bias in the pulse magnitude. Addition-
ally, the cut accepts less than 37% of good events, which is a substantial loss of
event statistics. This has a significant impact in certain low-statistics regions of the
detector.
The current work benefits from this existing position-corrected scintillation
measurement of [98], leading to a rotated energy spectrum with resolution of roughly
1.8%. As a result, it is possible to accept more events while still keeping Compton
shoulder leakage small. We currently accept events within 2? of the peak, with
better than 95% acceptance and only small leakage. An additional benefit is that
the improved energy resolution allows us to use a symmetric acceptance region,
avoiding the implicit bias introduced from the earlier asymmetric cut window.
Preliminary investigations have been conducted to see the impact of using
the denoised scintillation measurements which are the subject of this work in the
event selection itself. Presently the improvement in resolution (to 1.53%) has not
demonstrated any significant improvement in the quality of the next iteration of the
lightmap beyond what we can achieve with 1.8% resolution, suggesting that energy
resolution is no longer an important factor in event selection. This is a topic of
continuing study.
Beyond the cuts described above (single-site within an appropriate energy
window), other basic cuts are applied to ensure only well-behaved scintillation pulses
are selected (where we emphasize that all scintillation cuts are with reference to the
old reconstruction scale, which was not normalized):
164
? Charge and light must individually lie within a reasonable range: charge must
lie between 1 and 5000 uncalibrated keV, and light must lie between 1 and
15000 counts in the old scintillation reconstruction scale.
? The charge-to-light ratio must not identify an event as an alpha decay: the
light counts L and charge counts C must obey the relation L < 3.405 ? C +
2600.67.
? The scintillation time must be well-separated from any other scintillation clus-
ters in same event frame: a scintillation cluster is only acceptable if no other
scintillation clusters occur within 220 ?s before or 180 ?s after it.
? All charge clusters which are assigned to this scintillation cluster must be
assigned unambiguously: for each charge cluster assigned to this scintillation
cluster, there can be no other scintillation cluster occurring within 140 ?s
before or 5 ?s after it.
? All three position coordinates of the charge deposit must be reconstructed.
(No fiducial cut is placed, since that would restrict the volume where the
lightmap is specified.)
Many of these cuts are probably unnecessary. As the energy resolution has improved,
the chances of Compton-shoulder contamination have decreased. Note that our
??0? decay search analysis only accepts events with a single scintillation cluster,
a requirement which would carry a high cost in lost statistics for determination of
the lightmap. (Strong thorium runs (34.04 kBq, or 0.9201 ?Ci, on September 1,
165
2009 [132]) produce a significant rate of events with multiple scintillation clusters in
a single 2 ms event frame. These statistics are quite valuable, particularly because
they are the primary source of events in some poorly illuminated regions of the
detector.)
6.3.2 Function Parametrization
Because we are attempting to empirically measure the functions R(~x) and S(t) from
a finite dataset, we must specify some more limited form for them to take. All current
approaches to describing R(~x) first bin the detector volume into three-dimensional
voxels, and then define R(~x) to interpolate linearly between known values at the
center of the voxels. The size of these voxels must be chosen with some care; if
they are too small, then low per-voxel statistics cause the statistical errors on the
pulse magnitude to dominate, whereas if the voxels are too large then the spatial
variation of the lightmap is not fully captured.
In the current lightmap, the detector is binned into 1 cm3 voxels; the detector
is easily contained within a box with sides 40 cm long, leading to 64, 000 voxels (of
which roughly 20% lie outside of the detector and will be empty). The choice of
voxel size was made based on the size of an individual APD, which is roughly 2.5
cm in diameter. Near the anode we would like for the size of a voxel to be much
smaller than 2.5 cm. When 1 cm3 voxels are used, much of the detector has sufficient
statistics per voxel; however, there are some regions of the detector with fewer than
ten hits per voxel, indicating that R(~x) may have quite significant statistical error
166
in these regions.
It is possible to justify a choice of larger voxels. The APDs lie at ?204 mm
from the cathode, which means that there is more than 2 cm between our fiducial
volume and the APDs. At this distance, the dependence of the lightmap on x and y
is expected to be much slower than at the APD plane, indicating that perhaps 2 cm
binning in x and y may be sufficient. Additionally, z-dependence of the lightmap is
expected to be fairly smooth throughout the detector. Since we interpolate linearly
between voxels, it may be possible to use a z-binning much coarser than 1 cm. This
is a topic for future investigation.
It is also worth mentioning that alternative voxel geometries have been tried in
the past. The older APD-plane lightmap [98] used a cylindrical-coordinate binning.
Binning in r was selected to make the bin volumes roughly constant along the r axis,
binning in the angular coordinate was uniform, and binning in z was chosen to be
coarser near the anodes and finer near the cathode to reflect faster variation there.
In all cases the binning was coarser than with the current cubic voxels being used.
Our finer binning is made possible by the larger quantity of available statistics from
combining the full set of thorium source data, and is justified by the potential for
the yield on a single APD gang to vary more rapidly than the yield averaged across
an entire APD plane.
It is well-known [133] that when it is necessary to estimate a multivariate
function from limited statistics, a choice of binning can have a significant impact on
the result. It is preferable to use an unbinned technique such as kernel regression.
In particular, our data density is highly non-uniform, and it should be possible to
167
measure the lightmap with high fidelity in regions of high statistics, while smoothly
transitioning to a coarser interpolation in regions of low statistics to minimize the
impact of uncertainty from individual hits. State-of-the-art solutions to this problem
would rely on locally adaptive kernel regression; see [133] for a detailed explanation
of the related issues in non-parametric multivariate regression. Attempting to use a
locally adaptive kernel regression for R(~x) should be considered a highly appealing
extension to the algorithm described here for generating a lightmap.
The parametrization of S(t) presents a very different set of choices. Thorium
data is taken in bursts, with the time between mostly filled by low-background
runtime. When we choose to treat S(t) as a smoothly varying function, it becomes
critical to interpolate properly ? after all, the low-background runtime is the critical
part of the experiment. (If we only produce a lightmap accurate during source
calibration runtime, we will measure an energy resolution from source data which is
not borne out in the low-background data, so we should in fact be able to give some
guarantee that S(t) is almost as accurate during the low-background runtime as
during the source runtime.) Fortunately, it is generally true that the time variation
of the APD response is quite slow; exceptions are generally due to changes in the
electronics which occur at well-specified times.
Currently each source run is treated as a burst of data taken at the midpoint of
the run, and S(t) is measured at that point only from the data of that run; then S(t)
is linearly interpolated between those points. In principle it is possible that between
two source runs an electronics change may have been performed, which would mean
that a better interpolation would be to assume S(t) is flat except for a step at the
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time of the electronics change; in practice, though, EXO-200 has generally taken
a source run immediately before and after an electronics change, so no high-value
data is taken during that interval and this detail can be ignored.
Another concern with this method is the treatment of short source runs. If
a run is too short, then the measurement of S(t) coming from that run may have
significant errors. We currently mitigate this issue by entirely throwing out all
data from runs with fewer than 100 usable events. We justify this approach by
claiming that even though those events might in principle have contributed some
useful information on R(~x), without a good measurement of the relevant S(t) it is
impossible to use that data.
In the future, it would be useful if instead we performed smoothed interpola-
tions between electronics changes. This could be done in the same style as EXO?s
lifetime corrections, using polynomial fits, where the degree of the polynomial could
be determined by eye. A candidate set of time intervals when S(t) might be treated
as constant could be the same as the set of constant noise windows described in
chapter 5. It would be possible to check whether this time binning is fine enough
by looking at the thorium scintillation peak position versus time in denoised data
? if it appears to drift in time within a window where S(t) is treated as constant,
then it is likely that S(t) needs to be binned more finely.
Alternatively, often there is a long string of consecutive source runs which
should be combined into one higher-statistics dataset. This process must be done
by hand, and has not been performed for the current analysis, but it could benefit
the accuracy of the S(t) function and also recover some statistics in cases where the
169
individual runs might be too short for inclusion in the lightmap dataset.
The choice of binning or parametrization forR(~x) and S(t) can have a profound
impact on the accuracy of the lightmap, and the current analysis has only skimmed
the surface of the various options. It is hoped that further work on the lightmap
will include significant investigation in these topics.
6.3.3 Convergence and Error Estimation
Our treatment of the convergence of algorithm 2 and the resulting statistical errors
in L(~x, t) is rudimentary. Cronvergence is checked not by analysis of the lightmap
itself, but by verifying that the energy resolution from denoising is not improved by
further iterations through algorithm 2. Lightmap errors do not directly enter into
the calculations of denoising (which treats the lightmap as perfectly known), so they
are not estimated at all in the current analysis.
As a proxy for estimation of the statistical errors in the lightmap, we in-
stead study the number of hits observed in each position voxel of the detector (see
section 6.5), and presume that the most significant source of error comes from low-
statistics regions of the detector. This information can motivate future data-taking
campaigns to probe the light response in those regions of the detector and reduce
these uncertainties. However, no explicit estimation of the lightmap uncertainty has
been attempted.
One difficulty with estimating the errors in the lightmap function L(~x, t) comes
from the correlation between errors in R(~x) and S(t). If one of those two compo-
170
nents were known perfectly, then we could treat the fit uncertainties of each hit as
independent errors, and propagate those errors into an uncertainty for each voxel
of R(~x) (if S(t) is assumed to be perfectly known) or S(t) (if R(~x) is assumed to
be perfectly known). But on the contrary, all of the measurements of R(~x) are
correlated with all of the measurements of S(t), meaning that even the errors of
different voxels of R(~x) or different times in S(t) are correlated with each other.
Fully characterizing these errors would require a significant effort and computation.
Although a full estimate of lightmap errors with correlations would probably
be computationally quite intensive to produce, it might be possible to estimate the
independent errors of each voxel of the position lightmap R(~x) or each independent
run making up S(t) by treating the other function as perfectly known, as described
above. This would generally produce an underestimate of the uncertainty in each,
but the estimate might still give some benefit.
To produce a more accurate estimate of errors, it would probably be necessary
to do a simultaneous fit by varying both R(~x) and S(t) together. In the current
scheme, R(~x) contains far more complexity than S(t) with roughly 50, 000 non-
empty voxels, so for each APD gang we would need to simultaneously vary roughly
50, 000 parameters to obtain the optimal lightmap. It is exactly this time-intensive
process which was avoided by choosing the iterative approach for measuring the
lightmap; however, it would only need to be performed occasionally when deriving a
new lightmap, so it is not infeasible to imagine attempting this project. The prospect
of feeding in a high-quality guess obtained from the iterative method presents an
additional significant time-saver. This method would not fully account for the cor-
171
relations between errors of the different voxels of R(~x) or runs in S(t), but it would
come closer than the naive method described earlier.
However, it is also possible that identifying the error of each voxel or run
is heading down the wrong path. As described earlier, it is likely that the opti-
mal (lowest-error) method for estimating the lightmap will be an unbinned method
such as a locally-adaptive kernel regression method. Error estimation in kernel re-
gression presents significant additional challenges compared to errors from binned
parametrizations. At present, I am not aware of any simple scheme to manage this
difficulty, meaning that there may be a paradoxical tradeoff that using the best
method for minimizing lightmap errors simultaneously makes those errors infeasible
to estimate.
Note that under the iterative method, the most naive method for estimating
errors is not valid. It might appear that when S(t) is initialized to a constant
value of 1, we could also assign to it some constant error. Then when we compute
R(~x), we could propagate independent errors from the fit uncertainties of pulses
and from S(t); and when we compute S(t) we could propagate independent errors
from the fit uncertainties and from R(~x); and so forth. However, such a scheme
provides no compensating feedback mechanism to force the iterated errors to a
reasonable or accurate value, so there is no reason to believe the errors from such
a system (if, that is, they converge at all). This difficulty may underscore the
fundamental challenge associated with measuring the lightmap errors within our
scheme ? iterative methods are well-suited to solving a system, but when correlated
errors are mistreated as uncorrelated an iterative method can easily magnify the
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impact of that mistreatment.
On the topic of convergence, it has been mentioned already that by starting
with a generally accurate initial guess for S(t) as constant, the iterative solution
method tends to converge rapidly. As a result, and because iterations take only a
few hours to perform on a single machine, we currently perform three iterations and
claim that convergence is approximately reached. Ideally, we would require that
none of the position voxels or runs change value within an iteration by more than
some fraction of their estimated errors; but in the absence of estimated errors, this
is of course impossible.
It would be possible, in a conservative approach, to require each value to con-
verge to some small fraction of an ADC unit, ensuring that the convergence is better
than our ability to measure pulses. Such a requirement might force us to compute
more iterations than are truly warranted by our lightmap errors, but given the
reasonable speed of each iteration, such a method still might not be unreasonable.
6.4 Implementation for this Analysis
The physics analysis which will be described in this paper includes data from EXO
runs 2464 to 5597, which were taken between October 5, 2011 and September 1,
2013. However, at the time when the initial denoising processing was begun, the
tentative range of runs to be used only extended up to 5367 on June 24, 2013. As
a result, calibrated data at that point was only available up to June 24, 2013, and
the lightmap had only been generated using the same set of thorium source data.
173
When the run range for the present analysis was extended, a significant portion
of the dataset had already been denoised with the lightmap based only on data
extending to June 24, 2013. Although the possibility of creating a new lightmap
was considered, this would have required a re-analysis and re-verification of the
existing denoised dataset. As a result, the existing lightmap continued to be used.
The functions S(t), which had only been directly measured up to June 24, 2013,
were assumed to remain constant between June 24 and September 1, 2013; no known
changes to the APDs occurred during this time, nor did any known environmental
factors change, so this assumption was considered acceptable. Future lightmaps of
course will make use of a more complete dataset rather than relying on extrapolation.
Furthermore, one bug was discovered in the lightmap which was used for de-
noising. A set of radium-226 source data was taken before the corresponding iden-
tifier could be set in data files, and as a result they were provisionally labeled as
thorium runs. These runs were mistakenly incorporated into the generated lightmap,
and events from the 2204-keV gamma line of the bismuth-214 daughter product of
radium-226 were selected and handled as though they were legitimate 2615-keV
events.
It was possible to partially remedy this after the fact: since currently there
is a one-to-one relationship between individual source runs and their corresponding
points in the functions S(t), it was possible to artificially erase the data points of
S(t) originating from radium runs. It is not so easy to remove the impact they may
have had on R(~x). However, the radium runs were taken at a location which is
extremely well-populated by thorium data throughout the life of the detector; since
174
Figure 6.1: Lightmap position-dependence R(~x) for selected APD gangs.
the quantity of radium data is small by comparison, the effect on R(~x) is presumed
to be quite small. Subsequent studies using a correctly-generated lightmap on small
subsets of data indicate that the effect is indeed negligible, as expected.
6.5 Visualization
Although it is not strictly necessary to be able to visually inspect the lightmap for
it to be useful, nevertheless it is reassuring to see that the lightmap is qualitatively
similar to intuitive expectations of how light should propagate through the detector.
We have chosen to store the position-dependent R(~x) for each APD gang as a set
of ROOT TH3D objects, and fortunately ROOT provides a number of excellent
175
Figure 6.2: Lightmap position-dependence R(~x) for selected APD gangs.
Here extreme anode positions are omitted to permit better contrast for
the lightmap in the fiducial volume.
176
Figure 6.3: Functions S(t) for selected channels.
177
Figure 6.4: Functions S(t) for selected channels.
178
Figure 6.5: Functions S(t) for selected channels.
179
plotting features suitable for a three-dimensional dataset. In figure 6.1, it is possible
to view the values of R(~x) for a sample of gangs on top of each other. Larger boxes
and denser color indicates a higher yield on the APD gang in question, and it is
immediately apparent that events near the anodes produce light which is highly
concentrated on a single APD gang.
Figure 6.1 shows the highest yield on the very boundary of the detector, well
outside of our fiducial volume; to permit us to more easily view contrast inside of
the detector, figure 6.2 shows the same map while omitting the most extreme bin
near either anode. Viewing this map, we can see more interesting characteristics of
the lightmap:
? Events near the anodes show a high pulse concentration on the one gang
nearest to their position; however, even deep into the detector near the cathode
it is still possible to see the higher concentration of pulse magnitude on the
gang directly aligned with the event.
? We can also see, from gang 201 (green) in this visualization, that events can
produce significant pulse magnitudes on APD gangs which which they are not
directly aligned (in the Z direction); yield on gang 201 can be seen to decrease
smoothly in all directions.
? APD gangs in the corners of the detector, such as gang 195 (red), are not
effective at measuring light from events which are far away; even directly
above gang 195, it is clear that gang 198 is more effective at collecting light
farther away than about five to ten centimeters. This can be attributed to the
180
reflection of photons by teflon, which may enhance the light yield on gangs
which are not hidden in corners.
Figures 6.3, 6.4, and 6.5 show the functions S(t) for the same sample of gangs.
The vertical scale can be treated as having arbitrary units; we note that all of
these functions lie roughly around 1, which is attributed to the initial placement of
S(t) = 1 in algorithm 2. We draw the following primary observations from these
plots:
? Many of the gangs show their values of S(t) decreasing rapidly up to around
February 2012; some of the gangs also show a sharp decrease in value at that
point. The decrease in gain corresponds with observations which were made
at the time, leading to a decision to replace electronics on some APD channels
on February 23, 2012. Thus, we do see ?real? features from these plots.
? The functions S(t) are otherwise dominated by jitter between points, indi-
cating that we do not collect enough statistics from each run to sufficiently
constrain S(t). This is taken as the strongest evidence that we should use a
coarser time binning for S(t), as described in section 6.3.2. Preliminary work
has been performed to do this with more recent lightmaps, but no studies have
evaluated the impact on energy resolution.
? Individual points on these functions may spike by as much as 40%. These
points have been investigated, and their cause is not understood. These jumps,
along with the overall jitter, will certainly be reduced by the use of a coarser
time binning.
181
6.6 Summary
In this chapter we have described the generation of an individual-APD lightmap
which characterizes the expected pulse magnitude on each APD channel as a func-
tion of scintillation origin, calendar time, and the quantity of energy. The two key
steps to this process are the use of the full dataset for extra statistics and the as-
sumption that the lightmap is separable between position and time coordinates, as
shown in equation 6.1. Previous lightmaps could only characterize the light yield
on large sums of waveforms; by providing a full lightmap for every channel, we en-
able the denoising algorithm of chapter 4 to make use of channel-by-channel pulse
and noise information, so this lightmap is a key component of denoising and the
improvement in resolution presented in section 7.6.
182
Chapter 7: EXO-200 Analysis and Results
from Denoising
The dataset used for the present work extends from October 5, 2011 to September
1, 2013 with a cumulative livetime of 477.60?0.01 days [89]. This chapter describes
the analysis which has been applied to this livetime. Sections 7.1-7.4 describe the
basic elements of the EXO-200 analysis. Section 7.5 presents the results from this
set of data. In section 7.6 we compare the results obtained using the denoising
scheme of chapter 4 to the results which would have been obtained without that
algorithm applied and demonstrate that denoising has contributed to the strength
of our physics reach.
7.1 Simulation
Here we describe the simulation of ionizing events in the EXO-200 detector. We
begin by describing the framework for simulating the deposition of energy from
primary decay particles into the liquid xenon and surrounding materials. From
there, we continue to describe the simplified electric field model which permits us
183
to model the trajectory of ionization drifting towards the anode wires.
7.1.1 Simulation of Particles using GEANT
To simulate the deposition of energy from primary decay particles, version 4.9.3p02
of the GEANT software package is used [134,135]. This package includes a database
containing attenuation properties of many common materials as well as detailed
decay modes for most radioactive isotopes.
A model of the EXO-200 detector is simulated in GEANT with a collection
of simple geometric volumes which can be composed to form more complicated
structures. Simulation speed decreases as more volumes are generated, and we
expect that it is unimportant to simulate fine geometrical details far away from the
liquid xenon. We attempt to find a balance between accurate modeling of detailed
features and simplification of distant features [111]. In figure 7.1 we see the full
geometry described in GEANT, where distant objects are constructed from a small
number of geometric pieces. Figure 7.2 shows how the TPC is described in GEANT,
and we can see the level of detail is much greater.
To simulate particles, GEANT uses a Monte Carlo method. This means that
it models decay processes as a sequence of samples from a probability distribution
which are chosen using a pseudorandom number generator. The energy and direction
of the emitted particle, its interaction locations and quantity of deposited energy,
and the formation of secondary particles are all generated randomly as choices from
a set of options with known probabilities. It is possible to override these probability
184
Figure 7.1: The GEANT simulation includes large-scale features of the
EXO-200 detector, including the outer and inner lead wall (black), outer
and inner cryostat and TPC legs (red), and the TPC itself (brown).
Components are assembled from simple geometrical shapes, and distant
objects are only described coarsely [111].
185
Figure 7.2: To ensure computational time is only spent on important
details, detector components which are close to the liquid xenon are
simulated in GEANT with greater accuracy than distant objects. The
TPC contains the xenon, so it is simulated in detail [111].
186
distributions when necessary; for example, EXO-200 uses its own double-beta and
forbidden beta decay spectra rather than the defaults provided by GEANT. Gener-
ally, though, the existing options in GEANT are suitable for simulating nuclear and
electromagnetic processes [111].
Only energy which is deposited in the liquid xenon is observable. When pri-
mary decays are simulated far from the detector, it may be that most of those
simulated events deposit no energy in the liquid xenon and are not observable; the
simulation continues running until a sufficient number of simulated events deposit
energy in the liquid xenon. This means that sources which are farther from the
liquid xenon require significantly more computational time to accumulate a usable
number of statistics. We find that sources outside of the HFE-7000 are subject to
4.5 attenuation lengths before reaching the liquid xenon, and events reaching the
liquid xenon from the inner cryostat can only be simulated at 0.01 Hz/core by this
approach [111].
To improve this rate, importance sampling is employed for distant sources.
This approach consists of the following techniques to maximize the impact of limited
computational time: [111,136]
? Low-energy beta and alpha particles outside of the TPC may be ?killed?, or
prematurely eliminated from the simulation based on our knowledge that their
attenuation length is quite short.
? The detector is surrounded by importance sampling ?boundaries?: when a
particle passes into a boundary it may be cloned (with a user-selected prob-
187
ability), where the rate of cloning is tracked by a corresponding decrease in
particle ?weight.?
? To avoid biasing the spectrum, it is also necessary to kill particles which
pass out of a boundary with the same probability, and increase their weight
accordingly.
Implicitly, importance sampling simulates the properties of particles reaching the
outermost boundary; then draws samples from that distribution and simulates the
properties of particles reaching the next boundary; and so forth, amplifying the
impact of statistics at each stage. This also means that statistical errors in the sim-
ulated particle distributions at outer boundaries are magnified by this amplification
process. For far backgrounds, though, an approach such as this is essential: simula-
tion speeds from the inner cryostat are increased from 0.01 Hz/core to a few Hz/core,
and simulations from outside the lead wall which would be wholely infeasible are
made possible [111].
The result of these GEANT simulations is a measurement of the efficiency with
which events from various sources reach the liquid xenon, and also of the energy and
position spectra of energy deposits from these sources. Representative spectra of our
primary backgrounds, 238U and 232Th, are shown in figures 7.3 and 7.4 respectively.
7.1.2 Digitization of Waveforms
After energy deposits are simulated using GEANT, it is necessary to model the
conversion of those energy deposits into collected photons and electrons, and then
188
Figure 7.3: Energy spectra from 238U in the TPC vessel for single-site
(red) and multi-site (black) events. No energy smearing is performed;
peak widths are due to energy loss outside of the liquid xenon [111].
189
Figure 7.4: Energy spectra from 232Th in the TPC vessel for single-site
(red) and multi-site (black) events. No energy smearing is performed;
peak widths are due to energy loss outside of the liquid xenon [111].
190
the generation of digitized waveforms resembling the waveforms which are collected
in real data.
To generate a scintillation pulse, a purely empirical model is used to estimate
the relative pulse magnitudes on the north and south APD planes. This model only
takes into account Z-dependence of the light collection, and does not incorporate
the different light yields expected on each individual APD channel. This model is
therefore rather crude, and can only be used as a rough check on the pulse-finding
efficiency for scintillation as a function of energy. Attempts to track optical photons
in the TPC have met with only partial success which is insufficient to justify their
significant computational cost. Thus, the EXO simulations are unable to model
most aspects of scintillation pulses. Section 7.4 will describe the methods used to
cope with this aspect of simulation.
Simulation of charge pulses is better-understood. The detector?s electric field
is modeled in a two-dimensional geometry illustrated in figure 7.5 where, rather than
one-dimensional wires arranged in two-dimensional planes, we have wire ?points?
at fixed voltage which are grouped in a one-dimensional pattern. The v-wires are
treated as stacked directly on top of the u-wires; it is impossible in two dimensions to
model the true orientation of the v-wires which is rotated relative to the u-wires, but
this has generally proven to be a minor detail for us. Only one TPC half is modeled
because of the approximate mirror symmetry of the detector across the cathode.
We assume that the two-dimensional geometry is periodic along the direction of the
wire ?plane?, which permits us to always locate drifting charge above the central
wire. Only a few wires to either side of the center are simulated because each wire
191
Figure 7.5: The wire planes are modeled in only two dimensions; charge
drifts along the field lines, which are arranged to terminate only on the
u-wires [111].
192
acts as a shield between wires to its left and right, so the electrostatic effect of any
one wire will be not extend beyond a few wire spacings [111].
Electrostatic effects are simulated using the ANSYS Maxwell field simulator.
To simulate the electric fields in the detector, wires are treated as circles with a radius
matching the approximate radii of the physical wires, with a constant voltage on
their surfaces. The APD plane and cathode plane are treated as constant-voltage
boundaries, and a periodic boundary condition is established on the two remaining
boundaries of the model geometry.
These electric fields can be used to trace the paths followed by charge deposits
in the detector. Charge deposits are drifted in small steps based on the direction
of the electric field. The speed of drift is taken from external measurements rather
than from the magnitude of the electric field; in most of the bulk of xenon, the
electric field and drift velocity are treated as constant, but near the u-wire plane the
drift velocity is increased slightly to account for higher electric fields experienced in
that region of the detector. Charge attenuation due to finite purity can be modeled
at this step, but generally is treated as infinite here; charge diffusion effects are
ignored.
We have discussed in section 3.5 that charge is induced on the u-wires and
v-wires; this means that we must record the amount of charge induced at each step
along the drift path of the charge deposit. This is done based on the Shockley-Ramo
Theorem, which states that the change in induced charge ?qi on an electrode i is
equal to:
?qi = Q?Wi, (7.1)
193
Figure 7.6: Weight potential of a u-wire channel consisting of three
ganged wires. Electric field lines are superimposed [111].
194
Figure 7.7: Weight potential of a v-wire channel consisting of three
ganged wires. Electric field lines are superimposed [111].
195
s]?
Tim
e 
[
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95
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Pulse height [ADC units]
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ta
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nte
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Tim
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Pulse height [ADC units]
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5005010
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Figure 7.8: Comparison between simulated and observed waveforms on
a u-wire (left) and v-wire (right) from 228Th sources. The events are
chosen to have similar energies so that the magnitudes match [111].
where Q is the total drifting charge and Wi is the weighting potential of electrode
i, defined as the potential which would be induced in our geometry if the potential
on electrode i were set to one and the potential on all other boundaries were set to
zero [137, 138]. Figures 7.6 and 7.7 illustrate the weighting potentials of a u-wire
and v-wire channel, respectively.
Finally, the functions of integrated charge versus time must be converted to
shaped digitized waveforms, and noise must be added. The shaping and gain am-
plification is performed to match the electronics described in section 3.5. To ensure
accurate time shaping, the functions are all sampled at a bandwidth of 10 MHz, and
then downsampled after shaping to the nominal 1 MHz. Digitization is performed
by converting voltages into units of ADC counts, then truncating to an integer value.
To model saturation effects, this number is pulled into the integer range [0, 4096).
We have the ability to add electronic noise to the pulses by extracting a set of
representative noise waveforms from real data. These are taken from a large number
196
of solicited triggers which have been checked for the absence of a coincident event.
To increase the number of noise waveforms available, the noise waveforms sampled
from the detector are spliced together, and simulated events may draw their noise
from any subrange of the spliced-together noise waveform. This method provides
our most accurate noise model because it can include all noise frequency peaks and
channel correlations without requiring a full understanding of those features.
However, current analyses do not employ this method; instead, we simulate
noise which is white (has a flat Fourier spectrum) before shaping, and then shaped
to have a spectrum which roughly resembles the one observed in the detector. Fig-
ure 7.8 compares waveforms observed from a sample event in data and simulation,
and demonstrates that at high energies excellent agreement is achieved [111].
In this way, we are able to generate simulated data which in most respects
resembles real data collected from the detector. The EXO-200 simulation has shown
remarkable agreement with the detector in the properties of energy and cluster
location, as illustrated in figure 7.9. A significant effort has been made to achieve
excellent agreement between simulation and data, and the result is that we can
claim a strong understanding of the behavior of the EXO-200 detector.
7.2 Cluster Reconstruction
The first stage of data processing involves locating candidate pulses in the waveforms
and fitting for the magnitude and time of any candidate pulses which are found.
The methods of accomplishing this are described in sections 7.2.1 and 7.2.2. We
197
1000 1500 2000 2500 3000
Energy (keV)
10-6
10-5
10-4
10-3
10-2
10-1
No
rm
al
ize
d 
Co
un
ts
 (a
.u
.) (a)
0 40 80 120 160
Standoff Distance (mm)
0.0
0.1
0.2
No
rm
al
ize
d 
Co
un
ts
 (a
.u
.)
(b)
Figure 7.9: Comparison between simulated and observed energy spectra
(a) and standoff distance (b) in single-site 226Ra events from a source
located at position S5 [89].
198
will complete our description of cluster reconstruction with a brief explanation of
the approach to the clustering of waveform pulses into three-dimensional clusters in
section 7.2.3.
7.2.1 Pulse Finding
Pulse-finding proceeds in two steps. First, on u-wire, v-wire, and summed APD
waveforms a matched filter is applied to do a preliminary search on all channels.
Then, a secondary search is performed on the u-wire waveforms to improve sensi-
tivity to multiple pulses near in time.
The matched filter algorithm was first described by D.O. North in 1943 [139].
It attempts to decide between the null hypothesis that a waveform X[? ] consists of
only noise and an alternative hypotheses that the waveform contains a pulse and
noise: ?
????
????
H0 : X[? ] = n[? ]
H1 : X[? ] = s[? ] + n[? ].
(7.2)
To discriminate between these hypotheses optimally, we search for a linear functional
L0 which will act on X[? ] and maximize the expected signal-to-noise ratio,
SNR = (L0 {s[? ]})
2
?
(L0 {n[? ]})2
? . (7.3)
In other words, if a waveform is composed of only noise, the functional should result
in a small value; however, if the waveform contains a pulse then the functional
should result in a large value.
199
We omit the derivation, which is a standard result, and simply state that the
linear functional which maximizes the expected signal-to-noise ratio can be expressed
as: [139]
L0 {X[? ]} = F?1
{ F {X} [f ]F {s}? [f ]
?F {n} [f ]F {n}? [f ]?
}
[0], (7.4)
where F represents the Fourier transform and F?1 represents the inverse Fourier
transform. (The ? = 0 index in this test statistic indicates that here we are testing
for a pulse at only one instant in time.) This expression has an added benefit that if
instead we?d like to test the hypothesis that the translated pulse s[???] is contained
in the waveform, substitution lets us show that we can do so with the related test
statistic:
L? {X[? ]} = F?1
{ F {X} [f ]F {s}? [f ]
?F {n} [f ]F {n}? [f ]?
}
[?]. (7.5)
Finally, we can define a test statistic T for the waveform X[? ] to have a pulse
anywhere with:
T = max
?
L? {X[? ]} . (7.6)
The values L? for all ? can be computed efficiently using the fast Fourier transform.
The time of the pulse is guessed as the value of ? which led to the maximum of
L? {X[? ]}.
The first phase of pulse-finding performs a search using the matched filter on:
? All u-wire channels.
? All v-wire channels.
200
s)?t (0 500 1000 1500 2000
ADC
 Units
s)?t (0 500 1000 1500 2000
ADC
 Units
Figure 7.10: A u-wire waveform (left) and the output from operation of
the matched filter (right). The red line on the right indicates our filtered
pulse threshold; the matched filter output exceeds the threshold, so this
waveform is determined to contain a pulse [112].
? The sum of all APD channels in the North plane.
? The sum of all APD channels in the South plane.
An example application of the matched filter to a u-wire waveform is shown in
figure 7.10.
The matched filter has proven to be an excellent metric for deciding whether
a waveform has a pulse on it or not. However, it is not as effective at distinguishing
between the cases where one or multiple pulses occur on a waveform. We would
particularly like to be able to classify u-wire waveforms by the number of distinct
pulses they contain because this can help us distinguish between single-site and
multi-site events, which can reduce our background as described in section 3.4. The
matched filter output function is designed to have a tall peak in the presence of a
pulse, but it may also be a broad peak which is difficult to resolve as the sum of two
distinct pulse contributions.
201
s)?t (1000 1050 1100 1150 1200
ADC
 (a.u.)
s)?t (800 1000 1200 1400
ADC
 (a.u.)
s)?t (1000 1050 1100 1150 1200
ADC
 (a.u.)
s)?t (1000 1050 1100 1150 1200
ADC
 (a.u.)
Figure 7.11: A u-wire waveform composed of two pulses near in time
is shown (top left); the matched filter (top right) correctly detects the
presence of a pulse, but does not detect the presence of two distinct
pulses. At bottom left, the waveform is unshaped; at bottom right the
waveform is reshaped with shorter differentiation times, leading to easier
detection [112].
202
To improve sensitivity to multiple pulses in u-wire waveforms, a second pass
is performed on u-wire channels using a multiple-signal finder. This scheme consists
of first unshaping the waveform offline to remove the effect of the shapers, and
then reshaping the waveform using shorter differentiation times than the hardware
shapers apply. This is a type of high-pass filter; in a multiple-pulse waveform, it can
have the effect of damping the first pulse faster to reduce its overlap in time with the
second pulse. It is then possible to search for pulses using a simple threshold which
is not as sensitive to small pulses as the matched filter, but is capable of detecting
additional pulses to complement the matched filter. This process is illustrated in
figure 7.11, and is the last procedure for finding pulses [112].
7.2.2 Pulse Fitting
After finding pulses, it is necessary to perform a fit to the expected pulse shape.
Fits are performed allowing both the pulse magnitude and time to float freely, where
only the initial guesses for these parameters come from the finding step. The metric
for fits is a simple chi-square between waveform data and the expected pulse shape,
where error on each point is estimated by the root-mean-square noise of the channel.
To minimize the impact of pulse templates whose shape may not perfectly
reflect the shaping times in hardware, and also to reduce the effect which waveform
noise may have on our fit, we do not fit the entire waveform to our pulse model.
Instead, we use only an 80?s fit window in the case of v-wires and APDs, and a
180?s fit window in the case of u-wires. The fit windows are illustrated with example
203
ADC
 (a.u.)
 
s)?Time (1000 1200 1400 1600
ADC
 (a.u.)
s)?Time (1000 1100 1200 1300
950 1000 1050 1100 1150 310?
ADC
 (a.u.)
s)?Time (950 1000 1050 1100 1150
Figure 7.12: Fits to data for a u-wire (top left), v-wire (top right), and
summed-APD waveform (bottom). The red model line indicates the
time extent of the fit window [112].
204
events in figure 7.12.
As pulse times and magnitudes are extracted, it may be discovered at this
phase of processing that two pulses on the same waveform which were separately
reported by the finding phase are extremely close together in time. In this case, we
can generally conclude that the finding phase inadvertently reported the same pulse
multiple times; the multiple candidate pulses should then be merged together, and
the waveform refit. The specific criteria for merging or removing candidate pulses
are: [112]
? If two pulses occur within 10 ?s of each other and one pulse has a fit magnitude
error of at least 15 ADC counts, the two pulses are merged.
? If any pulse has a magnitude to magnitude-error ratio of less than 6.0 for u-
wires, 5.0 for summed APD waveforms, or 3.0 for v-wires, then the smallest
such pulse is removed.
? If any pulse has a magnitude of less than 5 ADC counts, then the smallest
such pulse is removed.
Each time a single pulse is removed or a pair of pulses is merged, the fit is re-run
and we search again for any pulses warranting removal. In this way, we attempt to
ensure at the pulse-fitting stage that only appropriate fit results are retained.
7.2.3 Clustering Pulses into Deposit Sites
The result of pulse finding and fitting is a list of pulse magnitudes and times for
each channel. It remains for reconstruction to combine this information into three-
205
dimensional clusters, where each cluster should have a three-dimensional position,
time, and estimates for charge and light pulse magnitudes.
We begin by bundling groups of similar u-wire, v-wire, and APD pulses. Multi-
ple u-wire pulses may be bundled together when they occur on neighboring channels
close together in time. This may represent a cluster which occurred halfway between
two u-wire channels and was split, with some charge being collected by each u-wire
channel; it may also represent a combination of charge being collected on one chan-
nel and an induced charge pulse on a nearby u-wire channel due to the proximity
of the charge drift path to that neighboring channel. V-wire pulses are bundled
together based on a combination of pulse magnitude and time to account for the
observation that the passage of a single charge cloud will induce pulses on many
nearby v-wires. APD pulses are bundled based on time only, particularly to ensure
that when a pulse is observed on both the North and South APD planes, the pulses
are interpreted as originating from the same event in the detector.
We then attempt to join u-wire pulse bundles and APD pulse bundles into two-
dimensional charge clusters, identified by their U-Z positions and the time of the
energy deposit. The algorithm to do this is simple: for each u-wire pulse bundle, we
join it with the most recent APD pulse bundle in the event. The maximum difference
in time is the maximum drift time of charge clusters in the detector, 116?s. To allow
for fit errors in the time parameters of the APD and u-wire pulses, an additional
3?s allowance is added on either end of the permitted time difference; thus, an APD
and u-wire pulse bundle may be joined if the APD occurred between 119?s before
and 3?s after the u-wire collection time, and from these candidates the latest APD
206
pulse bundle is selected for each u-wire pulse bundle. The cluster?s charge energy
is the sum of the energies attributed to all of the u-wire pulses in the u-wire pulse
bundle; no attempt is made to measure the cluster?s scintillation energy.
Finally, we cluster v-wire pulse bundles with two-dimensional charge clusters
to form three-dimensional clusters possessing X-Y-Z positions, deposit time, and
charge energy. The only information taken from the v-wire pulse bundle is the third
position coordinate; it is not used to adjust our estimates of energy. The challenge
to this portion of clustering is that charge clusters may arrive at the same u-wire or
v-wire at the same time, yet from two different locations. For example, it may be
that what appeared to be one u-wire pulse bundle in fact comes from two charge
clusters with common U-Z positions and different V-positions. So, in addition to
searching for the most likely associations between u-wire/APD clusters and v-wire
pulse bundles, we must consider the possibility of splitting a two-dimensional cluster
or v-wire pulse bundle into pieces before performing this association.
To do this, every possible combination of u-wire/APD cluster and v-wire pulse
bundle is considered, along with every possible choice of split for clusters or pulse
bundles. A likelihood is assigned to each possible combination; this likelihood takes
into account:
? The ratio of v-wire and u-wire pulse magnitudes. These pulse magnitudes are
expected to be linearly related because they both are observations of the same
charge drift cloud, and deviations from that linear relation are penalized by a
decreased likelihood.
207
? The time difference between the v-wire and u-wire pulses. There is an expected
drift time between the v-wires and u-wires which is measured from data; if
the observed time difference deviates from this expectation, the likelihood is
decreased to reflect a preference for a more realistic pairing.
? The two-dimensional position which the pairing implies. Due to the geometry
of the detector, not all pairs of u-wire and v-wire can be hit by the same
charge cluster. When a u-wire and v-wire lie in different halves of the TPC, it
is of course impossible for the same charge cloud to induce pulses on both. In
some cases a u-wire and v-wire may almost overlap, and it may be considered
possible yet unlikely that the same charge cloud could induce pulses on both;
these instances are penalized as well.
Further details of the clustering criteria may be found in [112].
This section has described the situation where the pulses of an event may
be reconstructed as a set of three-dimensional clusters with well-defined position,
time, and charge energy. Although every effort is made to accomplish this whenever
possible, in practice there are events for which some clusters may not be fully re-
constructible. Primarily this occurs when charge deposits on a u-wire near the edge
of the detector where no v-wires exist; because a cluster falls below the threshold
for v-wire pulse-finding, or the event falls below the scintillation threshold; because
the pulse finder reports a false positive, yielding a non-physical pulse; or because
an event has extremely high multiplicity and it is not possible to select a preferred
clustering. Such events are difficult to use in higher-level analyses and will generally
208
be removed, with an associated energy-dependent efficiency loss of 9.1% [89].
7.3 Energy Corrections
Each cluster which is found can be assigned a preliminary energy estimate based
on the fit magnitudes of its pulses. However, to achieve the best possible energy
resolution it is necessary to make an assortment of energy corrections. These correc-
tions are identified in this section. We first describe the corrections which are made
to charge energies on a cluster-by-cluster basis, and then the corrections made to
the denoised scintillation energy. Section 7.3.3 describes the calibration of a rotated
energy measurement using charge and light together, and section 7.3.4 describes the
measurement of the energy resolution of this rotated energy measurement.
7.3.1 Charge Corrections
The preliminary charge cluster energy measurements come from the sum of the
u-wire pulse magnitudes described in section 7.2.2. These magnitudes reflect the
quantity of charge which is collected by the u-wire.
However, each u-wire channel has its own set of electronics, so pulse magni-
tudes must be adjusted by a gain correction which depends on the channel. These
gains are extracted from source data by selecting pair production events from the
thorium source, in which a 2615-keV gamma pair-produces an electron and positron,
and the positron subsequently annihilates with another electron to emit two 511-
keV gammas. The high multiplicity of these events makes it possible to select them
209
accurately, and the process guarantees that a single 1593-keV cluster at the site
of the pair production must be truly single-site because it originates from a single
electron-positron pair. These properties make it ideal for producing an accurate
u-wire gain measurement. Using the resulting gain corrections, individual u-wire
pulse magnitudes are corrected based on their gain, and the charge cluster energy
measurements are corrected according to the corrections on their bundled u-wire
pulses [110].
There are also two Z-dependent energy corrections which must be applied to
clusters. The first accounts for charge attenuation due to the imperfect purity of the
xenon, as discussed in section 3.1. As electrons drift, some will attach to electroneg-
ative impurities such as oxygen or nitrogen; the exact nature of the electronegative
impurities in EXO-200 are not known. The level of impurities in the xenon are
time-dependent due to the strong effects which the xenon pump speed and periodic
injections of additional xenon may have.
To extract the purity correction, the dataset is divided into time windows
when the pump speed was constant and no feeds occurred; the purity is assumed to
be constant during these time intervals. It is then possible to combine all thorium
source data during these time windows and fit for the 2615-keV single-site 208Tl
gamma line in Z-bins ranging from the North to the South anode. An illustration
of the ionization peak position versus Z is shown in figure 7.13. Charge attenuation
is exponential with a typical e-fold attenuation occurring in 4 ? 5 ms drift time;
the purity correction to charge energy is typically 2? 3% for charge deposits at the
cathode [140].
210
Figure 7.13: The purity charge correction is measured by fitting the
2615-keV 208Tl gamma line as a function of Z-position [140].
Figure 7.14: The expected grid correction of a u-wire as a function of
Z-position. In this plot, Z = 6 mm represents the position of the u-wire
plane and Z = 12 mm is the position of the v-wire plane [110].
211
Figure 7.15: After applying all charge corrections, we find that the 2615-
keV 208Tl ionization peak is observed at the expected location for single-
wire events; however, for two-wire events a residual bias is observed
which also exhibits Z-dependence [140].
The other Z-dependent charge correction is from the shielding grid. Section 3.5
described the process by which both electron clouds and ionized xenon will induce
pulses on the wires. The electron cloud drifts rapidly, and its pulse is observable;
the ionized xenon will drift quite slowly, and will not produce an observable pulse
in our electronics. However, the ionized xenon will counteract the induced current
pulse from the drifting electrons by holding some electrons on the wire; the degree to
which they are able to influence the observed current pulse depends on the weight
potential of the wire evaluated at the position of the xenon ions. The simulated
weight potential of a u-wire is shown as a function of Z in figure 7.14, and we can
see that inside the v-wire plane this correction is expected to remain below 1% [110].
212
These are all of the known corrections to the charge energy. However, we
find that after applying them, there remains a residual z-dependence in the charge
calibration. Figure 7.15 shows that although the 2615-keV 208Tl ionization peak has
been properly corrected for clusters which deposit on only one u-wire, for clusters
which are split between two u-wires there is a residual offset in peak position of
roughly 1% and an additional z-dependent effect of up to roughly 0.3%. These effects
are not presently understood; however, they are believed to induce a Z-dependent
bias effect in the lightmap which is used for scintillation denoising, which in turn
is believed to cause the scintillation z-bias described in section 7.3.2 [140]. Active
work is underway to understand this z-bias in charge and develop an appropriate
correction.
7.3.2 Light Corrections
Although scintillation measurements are made from fitting the magnitudes of the
scintillation pulses found on the summed North and South APD planes in recon-
struction, as described in section 7.2.2, these scintillation measurements are super-
seded by the denoised scintillation energies estimated using the denoising method of
chapter 4. This subsection will describe evaluations of the success of that denoising
effort and subsequent corrections made to the denoised scintillation energy.
When the denoising algorithm operates properly, it should be constrained to
produce scintillation energy measurements calibrated in keV. This calibration is
accomplished by the lightmap, which retains information about the magnitude of
213
Figure 7.16: A residual calibration offset and Z-dependent behavior is
observed in the denoised scintillation measurements. The top plots show
the relative peak position of the scintillation-only 2615-keV 208Tl peak
(one-wire and two-wire) and average alpha energy from 222Rn versus Z
(top left) and R (top right). The bottom plot shows the absolute de-
noised scintillation energy from the 2615-keV 208Tl peak for one and
two wires, with the measured correction function (equation 7.7) over-
laid [140].
214
pulses produced by 2615-keV energy deposits. However, in practice it is seen that
the denoised scintillation measurements are not properly calibrated; this can be
seen in figure 7.16, where it is clear that events of all types (alpha decays in low-
background data and both one-wire and two-wire single-site gamma decays from
an external source) measure a scintillation energy which is systematically too low
and displays significant Z-dependence. The thorium data, which provides the best
measurement statistics, shows a mis-calibration which ranges from 2.5% near the
anodes to 6.5% at the cathode [140].
The cause for the discrepancy in scintillation energy measurements is not cur-
rently known. One hypothesis is that the discrepancy in two-wire charge cluster
measurements described in section 7.3.1 may lead to inaccurate event selection in
the generation of the lightmap described in section 6.3.1; poor event selection would
lead to a systematic error in the lightmap which could depend on Z, and would be
capable of creating the observed bias. Attempts to address this Z-bias within the
generation of the lightmap or investigate alternative explanations for its origin have
so far been unsuccessful, but these investigations continue, and it is hoped that this
issue can be fixed for the next analysis.
For the present analysis an empirical correction of the discrepancy is applied.
The scintillation correction function takes the form:
Ecorr = Emeas ?
?
????
????
[
(c?) + (a?) ? |Z|b?
]?1 if Z < 0
[
(c+) + (a+) ? |Z|b+
]?1 if Z > 0,
(7.7)
215
where we measure from data c? = 0.9380, c+ = 0.9355, b? = 1.71, b+ = 2.00,
a? = 0.69, a+ = 1.3, and Z is in units of meters. The bottom plot of figure 7.16
shows this empirical correction overlaid on one-wire and two-wire thorium source
data, demonstrating that the function agrees well with the observed Z-dependent
offset [140].
7.3.3 Rotated Energy Calibration
From the corrected scintillation and charge energy measurements, it is necessary to
form a calibrated rotated energy measurement which combines them. This com-
bination takes advantage of anticorrelation between charge and light to achieve an
optimal energy resolution. To accomplish these measurements, it is necessary to
use multiple calibration points; we use the 2615-keV gamma line from regularly-
taken thorium source runs to measure time-dependent calibration parameters and
the 662-keV cesium and 1173-keV and 1332-keV cobalt gamma lines from source
runs to characterize the energy dependence of calibrations. The linear combination
of scintillation and charge is selected to optimize the energy resolution of the tho-
rium line; the precise measurement of energy resolution is described in section 7.3.4,
but for this purpose we simply seek to make energy peaks as narrow as possible.
The calibrated rotated energy E is computed by a calibration function of the
216
form:
Erot = ES ? sin(?(t)) + EC ? cos(?(t)) (7.8a)
Eratio =
Erot
Ethrot(t)
(7.8b)
E = (Ethorium ? Ebias) ?
(
p0 + p1Eratio + p2E2ratio
)
, (7.8c)
where EC and ES are the measured charge and scintillation energies (with cor-
rections applied as described in sections 7.3.1 and 7.3.2), ?(t) is a time-dependent
rotation angle measured to optimize the fractional energy resolution of the thorium
gamma line, Ethrot(t) is the time-dependent location of the thorium peak in the spec-
trum of Erot, Ethorium is the true thorium peak position equal to 2615 keV, and
Ebias, p0, p1, and p2 are time-independent calibration parameters. Entirely separate
calibration parameters are obtained for single-site and multi-site events.
The result of this calibration for a particular thorium source calibration run
is shown in figure 7.17. We can see here that a projection is performed from the
two-dimensional charge-light spectrum along an axis that minimizes the width of
the thorium gamma line. Figure 7.18 demonstrates the spectra we obtain from
projections only in scintillation, only in ionization, and using the optimal linear
combination we have described; the improvement in resolution from using an ap-
propriate linear combination of scintillation and ionization is apparent, as the peak
at 2615 keV becomes sharper and additional low-energy features of the spectrum
become apparent which were washed out with worse resolution. The next section
will describe the process of measuring the energy resolution.
217
Ion
iza
tio
n e
ne
rgy
 (ke
V)
50
0
10
00
15
00
20
00
25
00
30
00
35
00
Scintillation energy (keV) 5
00
10
00
15
00
20
00
25
00
30
00
35
00
Ro
tat
ed 
ene
rgy
 (ke
V)
10
001
20
014
001
60
018
002
00
022
002
40
026
002
80
0
Figure 7.17: Corrected scintillation versus charge energy are shown for a
thorium run. The projection angle for the rotated energy measurement
is illustrated at the 2615-keV peak; the projection onto a calibrated axis
is shown. The dotted red line indicates the diagonal cut described in
section 7.4. Figure provided by Liangjian Wen.
218
Figure 7.18: Charge-only, scintillation-only, and rotated energy spectra
are shown from a thorium source run; the significant improvement in
energy resolution with the rotated energy measurement is apparent, as
are low-energy features which are washed out in the lower-resolution
spectra. Figure provided by Liangjian Wen.
219
7.3.4 Measurement of Rotated Energy Resolution
The previous section has described the calibration of the rotated energy spectrum.
In this section, we describe the measurement of energy resolution with this rotated
energy measurement.
The resolution is assumed to be energy-dependent; if a monoenergetic deposit
is made in the liquid xenon, we hypothesize that the observed spectrum should be
Gaussian and define the energy resolution at that energy to be the value of ?. We
can parametrize the dependence of ? on energy as:
?(E) =
?
p20E + p21 + p22E2, (7.9)
where typically we interpret the parameters p1 to come from electronic noise (which
contributes a smearing independent of energy), p0 to come from statistical fluc-
tuations in the number of photons or electrons observed (which follows a Poisson
distribution in the number of photons), and p2 to come from mis-calibration of
components of the detector (whose effect increases linearly with the size of the
pulse) [141]. We also often find it useful to refer to the relative or fractional energy
resolution, defined as:
?(E)/E =
?
p20/E + p21/E2 + p22. (7.10)
The resolutions of single-site and multi-site events differ, so there are separate res-
220
Parameter SS MS Units
p0 0.628 0.602
?
keV
p1 20.8 25.8 keV
p2 0.0011 0.0040 1
Table 7.1: Time-averaged resolution parameters for the single-site and multi-site
denoised dataset [142].
olution functions measured for both classes of events.
The resolution functions are measured using only the cobalt and thorium cal-
ibration gamma lines (at 1173, 1332, and 2615 keV). The resolution parameters
are time-dependent; average resolution parameters are computed by weighting the
resolutions observed in different time windows by the fraction of EXO-200 low-
background data taken within that window. The resulting resolution parameters
are listed in table 7.1. Figure 7.19 shows the resulting resolution functions. We can
see that at our typical energies of ? 2500 keV the p0 and p1 terms dominate the
resolution function of equation 7.9, and this manifests itself as a concave-downward
shape in the absolute resolution function. The shape of these resolution curves can
be interpreted as meaning that the electronic noise and Poisson fluctuations domi-
nate our resolution; this may not be surprising, since we recall from section 4.10 that
Poisson fluctuations were underweighted in the version of denoising implemented for
this analysis. Our interpretation is only suggestive because the curvature is slight
and the energy-dependent calibration effects cannot be ruled out, but it will be inter-
esting to see how these terms change when future work addresses the underweighting
of Poisson fluctuations.
It may be tempting, given that the topic of this work is a system for denois-
221
En
erg
y (k
eV)
10
00
15
00
20
00
25
00
30
00
35
00
Energy Resolution (keV) 253035404550
SS
 De
no
ise
d
MS
 De
no
ise
d
En
erg
y (k
eV)
10
00
15
00
20
00
25
00
30
00
35
00
/E) ? Energy Resolution (% 11.522.533.5
SS
 De
no
ise
d
MS
 De
no
ise
d
Figure 7.19: The time-averaged resolution functions (left) and relative
resolution functions (right) for single-site (red) and multi-site (purple)
denoised data. Shaded bands indicate uncertainties. Data provided by
Caio Licciardi.
222
Figure 7.20: A typical monoenergetic gamma line, where the horizontal
axis indicates ionization energy and the vertical axis indicates scintil-
lation energy. We note that the ionization-only and scintillation-only
resolutions are dominated by the length and angle of the island, which
are dominated by xenon physics and the choice of electric field rather
than the accuracy of the readouts in those channels.
20
11
Oc
t 0
2
20
12
Jan
 01
20
12
Ap
r 0
2
20
12
Jul
 02
20
12
Oc
t 0
1
20
12
De
c 3
1
20
13
Ap
r 0
2
20
13
Jul
 02
20
13
Oc
t 0
1
Scintillation Energy resolution
0.0
5
0.0
550.0
6
0.0
650.0
7
0.0
750.0
8
0.0
850.0
9
0.0
950.1
SS
, d
en
ois
ed
 (T
h)
M
S, 
de
no
ise
d (T
h)
Figure 7.21: Scintillation-only resolution over time, including: single-site
denoised (blue), multi-site denoised (red), single-site undenoised (black
solid), multi-site undenoised (hollow square). Although denoising does
show some improvement in the scintillation-only resolution, it is quite
modest; this is because the scintillation-only resolution is dominated by
fluctuations in light yield. [140].
223
ing scintillation waveforms, to prefer to focus on the scintillation-only resolution.
However, the scintillation-only resolution is in fact dominated not by the accuracy
of the readout system but by fluctuations in the fraction of deposited energy which
is released in the scintillation channel. Figure 7.20 shows a schematic of a typical
monoenergetic gamma line; the length and angle of the island represents the anti-
correlation between light and charge. We can see that although the charge and light
yield are highly correlated, individually they are also subject to significant fluctu-
ations. A scintillation-only energy measurement is fundamentally limited by these
fluctuations in light yield, and only to second order is it affected by the accuracy
of our event-by-event measurements of light yield. A comparison between the de-
noised and undenoised scintillation-only resolution can be seen in figure 7.21, and
as expected the improvement from denoising is quite modest. Considering that the
scintillation-only energy measurement is not used in the final analysis and does not
provide a window onto the performance of denoising, we will not consider it further
in this work.
7.3.5 Cross-Checks on the Energy Calibration
The energy calibration is extracted from source data; the large quantity of source
data leads to small statistical errors on the calibration constants. However, the
calibration constants are time-dependent and it is valuable to have cross-checks
verifying that the time-dependent nature of the calibration is accounted for properly.
To cross-check the time-dependence of the calibrations derived from sources,
224
Rota
ted E
nergy
1900
2000
2100
2200
2300
2400
Events / ( 20 ) 02468101214161820SS
 
14?
Mean
 =  2
215  
11?
Sigm
a =  3
9  0
.28?
coe
ff =  
0.24 
SS
Rota
ted E
nergy
1950
2000
2050
2100
2150
2200
2250
2300
2350
Events / ( 20 ) 02468101214161820MS
 
5.9?
Mean
 =  2
219.3
  4.4?
Sigm
a =  2
7.7 
 
0.050?
coe
ff =  
0.098
 
MS
Figure 7.22: The single-site (left) and multi-site (right) gamma line at 2.2
MeV from neutron capture on hydrogen can be used as a low-background
cross-check on the resolution calibration. Here, data which is coincident
with a muon veto is shown to include the expected neutron capture
peak [140].
we use gamma lines in the low-background data. These can verify that gamma
lines from a wide range of times are combined to produce coherent peaks in our
cumulative energy spectrum; they can also verify that the time-averaged resolutions
extracted from source data are accurate because the resolution extracted from low-
background peaks is inherently averaged over the duration of the data run. Two
energy lines from the low-background spectrum have been identified as particularly
useful.
One gamma line in the low-background data comes from neutron capture on
hydrogen in the HFE-7000 refrigerant. Such events are generally vetoed by the
coincidence cuts, primarily the veto panels, so their impact on the low-background
data is suppressed; conversely, it is possible to study only events which occur in
coincidence with muon detection by the veto panels, thereby enriching our dataset
in neutron capture spectra. Neutron capture on hydrogen emits a 2.2 MeV gamma;
225
Rota
ted E
nergy
 
(keV)
1100
1200
1300
1400
1500
1600
1700
1800
1900
Events / ( 10 keV ) 02004006008001000SS r
otate
d spe
ctrum
 (Run2
abc)
 
2.3?
Mean
 =  1
458.0
  1.9?
Sigm
a =  3
9.9 
 
0.000
14
?
coe
ff =  
0.004
68 
 
0.000
25
?
coe
ff2 =
  0.99
372 
 
0.030?
erfc_
coef
f =  0
.494 
 
5.3?
p0 = 
 343.
1  133?
p1 = 
 120 
SS ro
tated
 spec
trum
 (Run2
abc)
Rota
ted E
nergy
 
(keV)
1100
1200
1300
1400
1500
1600
1700
1800
1900
Events / ( 10 keV ) 050100150200250300350400450MS 
rotat
ed sp
ectru
m (Ru
n2abc
)
 
0.87?
Mean
 =  1
454.2
7  0.7
1
?
Sigm
a =  3
3.19 
 
0.003
4
?
coe
ff =  
0.282
8  
0.004
2
?
coe
ff2 =
  0.68
31 
 
0.011?
erfc_
coef
f =  0
.303 
 
1422?
p0 = 
 2000
 
 
5.2?
p1 = 
 6.4 
MS r
otate
d spe
ctrum
 (Run2
abc)
Figure 7.23: The low-background potassium peak at 1461 keV is used
as a cross-check on the single-site (left) and multi-site (right) energy
calibrations. Fits also include the nearby 1332-keV cobalt peak [140].
figure 7.22 shows the data which is produced by this source with a fit to a Gaussian
function and simple background model. [99,140].
The other gamma line which has been used as a cross-check is the 1461-keV
40K gamma line. 40K is expected to occur in the copper of the TPC at some low level,
and this gamma line is readily visible in the multi-site spectrum of low-background
data. The improved energy resolution of denoised data allows us to also identify the
40K single-site peak over the ??2? background, which was not previously possible.
These peaks are shown in figure 7.23.
The result from combining all of these calibration points is shown in figure 7.24.
We can see that all sources agree well with the calibration, indicating that the cali-
bration derived from thorium and cobalt extends well to the other calibration sources
and captures time-dependent calibration information well. (With 14 independent
measurements under consideration, the 2? discrepancy of the 662 keV cesium multi-
site peak position should not be considered significant.)
226
En
erg
y (k
eV
)
50
0
10
00
15
00
20
00
25
00
Residual (%)
-
1
-
0.500.51
SS
Ca
lib
rat
ion
 So
urc
e
Lo
w 
ba
ck
gro
un
d d
ata
En
erg
y (k
eV
)
50
0
10
00
15
00
20
00
25
00
Residual (%)
-
1
-
0.500.51
M
S
Figure 7.24: The residuals between calibrated and true peak positions
for all sources is compared. Calibrations were obtained from the 137Cs,
60Co, and 232Th peaks at 662, 1173, 1332, and 2615 keV. The peak
from 226Ra at 2448 keV is also shown, but was not used for calibration.
Low-background calibration lines are neutron capture on hydrogen (2200
keV) and 40K (1461 keV). Figure from [140].
227
Co1_
ss
Entr
ies 
 
5
Mean
 y  
 
 
1170
RMS
 y 
 
 
2.086
Cam
paign
 Inde
x
1
2
3
4
5
6
7
8
Energy (keV) 116011701180Co Peak
1 SS
Co1_
ms
Entr
ies 
 
5
Mean
 y  
 
 
1168
RMS
 y 
 
 
1.878
Cam
paign
 Inde
x
1
2
3
4
5
6
7
8
Energy (keV) 116011701180Co Peak
1 MS
Co2_
ss
Entr
ies 
 
5
Mean
 y  
 
 
1330
RMS
 y 
 
 
2.207
Cam
paign
 Inde
x
1
2
3
4
5
6
7
8
Energy (keV) 132013301340Co Peak
2 SS
Co2_
ms
Entr
ies 
 
5
Mean
 y  
 
 
1327
RMS
 y 
 
 
1.898
Cam
paign
 Inde
x
1
2
3
4
5
6
7
8
Energy (keV) 132013301340Co Peak
2 MS
Cs_s
s
Entr
ies 
 
5
Mean
 y  
 
657.2
RMS
 y 
 
 
1.848
Cam
paign
 Inde
x
1
2
3
4
5
6
7
8
Energy (keV) 650660670Cs Peak
 SS
Cs_m
s
Entr
ies 
 
5
Mean
 y  
 
651.2
RMS
 y 
 
 
1.376
Cam
paign
 Inde
x
1
2
3
4
5
6
7
8
Energy (keV) 650660670Cs Peak
 MS
Figure 7.25: Time-dependence of the peak positions from all calibration
campaigns. Error bars, when not visible, are smaller than the circle.
Top row is the 60Co peak at 1173 keV; middle row is the 60Co peak at
1332 keV; bottom row is the 137Cs peak at 662 keV. Thorium is not
included because it is used to measure the time-dependence of the peak
positions, so it is necessarily calibrated to a time-independent value.
Figure from [140].
228
20
11
Oc
t 0
2
20
12
Ja
n 0
1
20
12
Ap
r 0
1
20
12
Ju
l 0
1
20
12
Oc
t 0
1
20
12
De
c 3
1
20
13
Ap
r 0
1
20
13
Ju
l 0
2
20
13
Oc
t 0
1
/E) ? Rotated Energy Resolution (% 
1.21.31.41.51.61.71.8
SS
, d
en
ois
ed
 (T
h)
M
S, 
de
no
ise
d (T
h)
Figure 7.26: Time-dependence of the energy resolution at the 208Tl 2615
keV gamma line. Single-site (blue) and multi-site (red) energy resolu-
tions are shown. The time-averaged energy resolutions (1.47% ?/E for
single-site, 1.59% ?/E for multi-site, both at 2615 keV) are shown in blue
and red dashed lines, respectively. Data provided by Liangjian Wen.
We also use the calibration source peak positions to check that the calibrated
peak positions remains constant over time. This is automatic for the thorium peak
position at 2615 keV because that peak is used to measure the time-dependence
of the calibration. However, if we see that the cesium calibrated peak position
changes over time, for example, then this would indicate that the time-dependent
calibration parameters ?(t) and Ethrot(t) of equation 7.8 are inadequate to describe
the time-dependent behavior of the detector. The peak positions of cesium and
cobalt are shown in figure 7.25. There we can see that indeed the peak positions of
these sources do vary by more than their statistical error bars. In this analysis, the
variations in peak position are treated as variations in the spectral shapes of sources
and incorporated as a systematic effect [140,143].
229
Lastly, it is necessary to study the energy resolution as a time-dependent quan-
tity. This is only done with the thorium source because of the regularity of thorium
runs that have been taken. The time-dependent resolution observed from thorium
calibration runs is shown in figure 7.26. We can see that there are statistically sig-
nificant variations in resolution over time; these, combined with the time-varying
peak positions shown in figure 7.25, lead to a weakened time-averaged energy res-
olution which accounts for the smearing of low-background peaks over the span of
our dataset [142].
The cross-checks described here identify features of the energy response which
are not fully calibrated; these are treated as smearing effects over time and lead to
a weakening of our time-averaged energy resolution. Future work will be targeted
toward understanding the peak position variations with event position shown in
figure 7.16 and with time shown in figures 7.25 and 7.26. Resolving these issues
should lead to further improvements in the time-averaged resolution of EXO-200 in
later analyses.
7.4 Fitting
After the energy calibrations have been performed, it is possible to produce a set of
data with calibrated energy measurements. This section describes the selection of
usable events from this dataset, the creation of corresponding background probabil-
ity distribution functions (PDFs), the fitting of data to extract the best-fit number
of ??0? events in the dataset, and the association of errors with that number. Re-
230
sults from the current analysis are demonstrated along the way, leading to a limit
for ??0? decay.
It is necessary to perform a number of cuts on the dataset before it is usable.
These cuts are intended to reject events which may not agree with the model used
to generate background PDFs. These cuts are: [143]
? A fiducial volume cut to remove events with any energy deposits outside of
a pre-defined volume. We choose to define a fiducial volume defined in Z by
10mm < |Z| < 182mm and in the X-Y plane as the area within a hexagon
with an apothem of 162 mm, as illustrated in figure 7.27. These cuts allow
us to ignore regions near the detector walls or v-wire plane where the elec-
tric fields may not behave as expected. They also ensure that beta or alpha
backgrounds from the walls of the detector do not penetrate into our fiducial
volume, allowing us to simplify the set of backgrounds modeled and included
in fits.
? A ?diagonal? cut to remove events with excessive scintillation relative to ion-
ization. Alpha decays produce a light-to-charge ratio much higher than beta
and gamma decays, as described in section 3.4, so excluding events with ex-
cessive light permits us to neglect alpha-decay backgrounds dissolved in the
xenon. Events with excessive light can also be caused by decays outside of
our active xenon, where charge is not collected efficiently; these events will be
difficult to calibrate, and should not be included. Both are excluded by a cut
on the light-to-charge ratio, as shown in figure 7.17.
231
18
3 m
m
to
tefl
on
refl
ect
or
162 mm (fiducial Xe)
171 mm (active Xe)
X
Y
U V
 Z out
of page
Figure 7.27: Here the X-Y orthogonal coordinates are shown along with
the U-V coordinates that run orthogonal to the wire planes. The X-
Y fiducial cuts applied to the data (dashed hexagon) do not include
the entire active volume (solid hexagon); for the ??0? search we find
that aggressive fiducial cuts optimize our sensitivity, so very little active
xenon is left unused.
232
Figure 7.28: Vetoing events coincident with muons can reduce our 1?
and 2? event counts. Figure provided by David Auty.
233
? A coincidence cut to remove events occurring near in time to a passing muon.
Muons and their spallation products can produce a wide range of backgrounds
which is difficult to model; most of these backgrounds are quite short-lived, so a
coincidence cut can provide an effective means of reducing them. Currently we
cut all events occurring within one minute of a muon observed passing directly
through the TPC and within 25 ms after a muon observed passing through
the veto panels. Furthermore, events which occur within one second of each
other are both cut based on the expectation that the events were probably
correlated and therefore more likely to be some form of background. Fig-
ure 7.28 demonstrates the significant reduction in data rate which is achieved
by applying this coincidence cut.
The fiducial volume identified above corresponds to a fiducial 136Xe mass of 76.3 kg;
for our livetime of 477.60? 0.01 days identified at the beginning of the chapter, the
dataset has a combined 136Xe exposure of 99.8 kg-years [89].
Probability distribution functions (PDFs) must also be formed from Monte
Carlo data which reflects the spectrum expected from different sources. We have
discussed the simulation of these datasets in section 7.1, and noted that the simu-
lations are sufficiently realistic that they are reconstructed in exactly the same way
as data. However, we cannot hope to measure energy in simulation and data the
same way because the scintillation pulse amplitudes are not simulated realistically.
Instead, energy is taken directly from simulation and smeared by the energy
resolution function which has been measured from data. Smearing is performed by
234
first producing a binned perfect-resolution PDF which models perfect energy reso-
lution, and then looping over the bins of the perfect-resolution PDF and smearing
each bin by a Gaussian function with width equal to the absolute resolution at that
energy [143].
Fits are then performed using the profile-likelihood method, as described
in [144]. An energy window of 980-9800 keV is used; this window is chosen to
avoid lower-energy parts of the spectrum where pdf shape agreement is poor [89].
Each data point is assigned a likelihood PDF(x) equal to the evaluation of the nor-
malized PDF at that point; the test statistic of interest is the negative log-likelihood
(NLL), defined by:
NLL = ?
?
i
ln [PDF(xi)] + constraints. (7.11)
Constraints are Gaussian penalty terms applied to parameters whose value is con-
strained by independent studies external to the fit. The best fit is the one which
minimizes the NLL. Fit errors on parameters are extracted by observing how the
NLL changes when those parameters are forced away from their best-fit values; 1?
limits are placed where the NLL changes by 0.5 units from its best-fit value, and
90% limits are placed where the NLL changes by 1.35 units from its best-fit value. In
cases where the best-fit value of a parameter is near its boundary, these errors may
not provide a good estimate of the true confidence interval or limits, so when this
situation occurs for a parameter of significant interest (specifically, the rate of ??0?,
which fits near to zero) we can verify that the confidence limit is not overstated using
235
toy Monte Carlo studies [143].
The fits make use of three observables. The first, event energy, has been
discussed at length in section 7.3.3. The second is the event classification as single-
site or multi-site; we define an event as single-site if reconstruction located only one
cluster, and that cluster deposited charge on no more than two u-wire channels [143].
This choice of definition ensures that a dense cloud of charge which approaches the
anode at a midpoint between two u-wire channels and gets split apart will still be
counted as single-site.
The third observable is called standoff distance, and it attempts to capture
the nearness of an event to the TPC walls. Most backgrounds are external, and
can be expected to deposit energy preferentially near the walls of the TPC; by
contrast, ??0? and ??2? decay come from the xenon, and should be distributed
uniformly through the detector. We define the standoff distance to be the shortest
distance between some deposit cluster and either the v-wire planes or the teflon walls.
Ideally, it would be possible to capture finer position information by using all three
position coordinates as observables; however, in practice it is difficult to construct a
PDF with sufficient statistics in so many dimensions, so the standoff observable has
been constructed to capture the most interesting distances in a single observable.
The exact definition is not too important provided it is modeled properly by the
simulation; figure 7.9 demonstrates that standoff distance is simulated well for all
but the smallest values (closest to the TPC walls or anode), and this is accounted
for as a shape systematic [143].
Fit parameters include the number of counts observed from each PDF and the
236
fraction of counts from each PDF which populate the single-site spectrum; the latter
is constrained based on the agreement between simulated and observed single-site
fraction from source runs, but is allowed to float within those uncertainties. The
PDFs which are included are:
? 136Xe ??2? decay.
? 136Xe ??0? decay.
? 232Th from the TPC vessel.
? 232Th from distant sources (HFE-7000 refrigerant or cryostat).
? 238U from the TPC vessel.
? 214Bi from the TPC cathode (238U chain).
? 214Bi from the air gap inside the lead wall (238U chain).
? 214Pb dissolved in active xenon (238U chain).
? 222Rn dissolved in inactive xenon (238U chain).
? 60Co from the TPC vessel.
? 40K from the TPC vessel.
? 65Zn from the TPC vessel.
? 54Mn from the TPC vessel.
? 137Xe from neutron capture on 136Xe.
237
? 135Xe from neutron capture on 134Xe.
? Various neutron captures on xenon, copper, and hydrogen.
Similar PDFs from different locations are included because the sharpness and rela-
tive intensity of peaks can be affected by intervening material. The rate of 214Bi from
the air gap is constrained by measurements of radon levels in air, and the rates of
214Bi from the TPC cathode, 214Pb dissolved in active xenon, and 222Rn dissolved in
inactive xenon are jointly constrained by independent searches for 214Bi-214Po rates.
The overall rate of neutron captures is allowed to float, but the relative intensities of
the captures on xenon, copper, and hydrogen are constrained by simulation, and the
single-site fraction is fixed because there are not expected to be sufficient statistics
in the single-site spectrum to constrain it [143].
In addition to the magnitude and single-site fraction parameters of the fit,
there are a few global parameters which are also allowed to float. There is a nor-
malization constant which is allowed to float within constraints and accounts for
uncertainty in the fiducial volume or detection efficiency. Constraints on this term
come from comparisons between the predicted and observed event rate from sources.
Denoising does not currently operate on events whose waveforms are shorter than
the standard length of 2048?s; since these occur more in source data than low-
background data, an additional correction to the simulated event rate must be gen-
erated, and denoising may in this way worsen the event rate agreement in source
data even though it is not expected to show a significant effect in low-background
data. Additionally, the normalization constants for gamma and beta particles are
238
Beta
 Scal
e
0.99
0.99
5
1
1.00
5
 NLL ? 00.511.522.533.544.55
Figure 7.29: The fit provides constraints on the beta-scale. Figure pro-
vided by Liangjian Wen.
allowed to float separately because their reconstruction efficiency may be different.
Furthermore, it is expected that betas and gammas may need to be calibrated
separately. To a first approximation, gammas deposit energy in the detector by
exciting electrons, so these processes should be quite similar. However, to account
for possible differences, we define a beta-scale ? which quantifies a linear calibration
offset between beta and gamma deposits:
E? = ?E?. (7.12)
The beta scale is permitted to float, and is constrained by the fit; figure 7.29 shows
the NLL profile with various choices of beta scale, and demonstrates that it is
possible to constrain the beta scale to a value close to 1 using only the spectral
239
information in the fit. Uncertainty on the beta scale is incorporated by permitting it
to float freely. We note that currently the beta scale is implemented by shifting beta-
like PDF components (??0?, ??2?, 135Xe, 137Xe) relative to the other components
which are gamma-like; this ignores the detail that some events may contain clusters
generated through both beta and gamma interactions, such as pair production from
energetic gammas. These issues will be addressed in a future analysis, but for now
the beta scale is assigned only to decays whose primary particle is a beta [143].
When all of this is done, it is possible to obtain a fit to the low-background
spectrum; the maximum-likelihood fit is shown in figure 7.30. The best-fit ex-
pected background in the 2? single-site energy window around our Q-value is 31.1?
1.8(stat) ? 3.3(sys) counts, corresponding to a background rate of (1.7 ? 0.2) ?
10?3keV?1kg1yr1. The backgrounds in this 2? energy window come from the tho-
rium decay chain (51%), uranium decay chain (26%), and 137Xe (23%), with other
contributions negligible. The observed event count in the 2? single-site energy win-
dow is 39 counts [89].
(We remind the reader at this point that the quoted limit on ??0? is not
simply based on this comparison of counts, but based on the full set of energy and
standoff-distance spectral information. Representative position information can be
seen in figure 7.31. However, the simple comparison of counts can be instructive as
an easier-to-understand approximation to the following limits.)
The likelihood profile for the total number of ??0? counts occurring in our
detector during the lifetime of the experiment is shown in figure 7.32. Based on the
profile, we can see that a 1? confidence interval for this parameter would exclude
240
-1
0
1
2
3
4
log
10(C
oun
ts/1
4 k
eV)
Energy (keV) 6
3
0
3
6
Res
id. 
(?)
Energy (keV)-1
0
1
2
3
4
log
10(C
oun
ts/1
4 k
eV)
Data
Best Fit
Rn
LXe bgd
n-capture
232Th (far)
Vessel
0???
2???
1000 1500 2000 2500 3000 3500 4000Energy (keV)
6
3
0
3
6
Res
id. 
(?)
2250 2300 2350 2400 2450 2500 2550 2600Energy (keV)
0
1
2
3
4
5
6
7
Cou
nts
(a) SS
2100 2200 2300 2400 2500 2600 2700Energy (keV)
0
10
20
30
40
50
Cou
nts
(b) MS
Figure 7.30: Energy spectra with best-fit PDFs for single-site (top) and
multi-site (bottom). Residuals between data and the combined PDFs are
shown below the spectra. The last bin of the spectra is an overflow bin.
The 2? region of interest around the Q-value is shown on the single-site
spectrum. Insets for single-site and multi-site zoom around the Q-value.
The simultaneous standoff-distance fit is not shown [89].
241
Figure 7.31: The positions of events in the 1? region of interest (red)
and wider energy range 2325 ? 2550 keV (black) are shown projected
onto their X-Y (left) and R2-Z (right) coordinates. Lines indicate the
fiducial volume. Figure provided by Dave Moore.
0 5 10 15 20 25 300??? counts0.0
0.5
1.0
1.5
2.0
2.5
3.0
?
1?
90%
Figure 7.32: Profile scan of negative log-likelihood as a function of the
number of fit ??0? counts [89].
242
the null hypothesis of zero ??0? counts; however, only 1? exclusion of the null
hypothesis is far too weak to make an inference of a non-zero signal. Instead, we
follow the conventions of the field and quote a 90%-confidence upper limit of 24
total counts from ??0?. This corresponds to a 90%-confidence lower limit on the
half-life of 136Xe decay by the ??0? mode of 1.1 ? 1025 years [89].
7.5 Results and Physics Reach
We have stated the half-life limit on ??0? decay of 136Xe, which is the main result
of the present analysis. We shall now continue to put this result in context with the
expected sensitivity of the experiment and compare it to the limits and sensitivities
achieved by other leading ??0? experiments.
To place the limit derived with the current dataset in context with the overall
strength of the experiment, we generate a number of toy Monte Carlo datasets. We
assume that our best-fit backgrounds are fully accurate and that the null hypothesis
(no ??0?) is true; we then use this best-fit background as our PDF and generate
toy datasets. These can be used to study the ?typical? results our detector would
have produced if the same experiment could be repeated many times.
The resulting distributions are shown in figure 7.33. The top plot shows the
distribution of 90%-confidence upper limits on the number of ??0? counts; we can
see that typical datasets from the EXO-200 experiment result in limits ranging from
8 to 22 counts. The median limit, is 14 counts; this value is called the median
sensitivity, and it illustrates the overall strength of an experiment. We contrast this
243
 
Co
unt
s (9
0%
 CL
)
?0??
Up
pe
r L
imi
t o
n 
0
10
20
30
40
50
60
70
80
0
10
0
20
0
30
0
40
0
50
0
60
0
Up
pe
r lim
it fr
om
 da
ta,
 23
.92
 co
un
ts
Me
dia
n u
pp
er 
lim
it, 1
4.0
4 c
ou
nts
68
% 
Int
erv
al a
rou
nd
 M
ed
ian
 
NL
L
?
 
-
2
0
2
4
6
8
10
1102 10
3
10
4
10
p-v
alu
e =
 0.
10
8
?
Sig
nifi
can
ce 
= 1
.23
9
 
NL
L f
rom
 da
ta
?
Figure 7.33: Assuming that the background measurements of EXO-200
from this analysis are accurate, toy datasets are simulated. The top
plot shows the probability distribution of our 90% confidence limit; we
find that our median upper limit is 14.04 counts attributed to ??0?,
compared to our observed upper limit of 23.92 counts. The bottom
plot shows the probability distribution of our ability to reject the null
hypothesis (no ??0?) based on the negative log-likelihood; we find that
the present dataset rejects the null hypothesis with probability less than
90%. Figures provided by Ryan Killick.
244
with the observed limit of 24 counts; as is apparent from the distribution, this limit
is consistent with an unlucky upward fluctuation in the backgrounds observed by
EXO-200.
We saw in the NLL profile on the total number of ??0? counts in figure 7.32
that the NLL for the hypothesis of no ??0? counts is about 1.1 higher than for the
best fit; this quantity reflects how well we can reject the null hypothesis in favor of
the best fit. To put this in context, we also observe the distribution of this value from
toy Monte Carlo datasets in the bottom plot of figure 7.33. Here, we see that 10.8%
of identical experiments will yield an NLL profile which is at least 1.1 worse for no
??0? than for the best fits even though no ??0? is simulated. We can conclude
that this difference in NLL is not strong enough to reject the null hypothesis with
high confidence.
The wide spread of limits produced in toy Monte Carlo studies illustrates the
significant role that luck plays in the limits set by low-statistics experiments such
as EXO-200. To compare EXO-200 to other experiments, it is useful to compare
both the limits and the median sensitivities (median 90% confidence limits). This
is done in figure 7.34. Vertical lines correspond to the reported median sensitivities
of the two leading 136Xe experiments, KamLAND-Zen and EXO-200; right-pointing
arrows indicate the actual limits set by each. For both experiments we can see
that statistical fluctuations in the background account for factor-of-two differences
between median sensitivity and observed limits; EXO-200 claims to be the most sen-
sitive 136Xe experiment, but KamLAND-Zen currently reports the strongest limit.
Both experiments intend to continue running, and EXO-200 hopes that their next
245
1024 1025 1026
T1/2 
136Xe (yr)
1024
1025
1026
T 1
/2
 76
G
e 
(y
r)
0.5
0.4
0.3
0.2
GC
M
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
NS
M
0.7
0.6
0.5
0.4
0.3
0.2
IB
M-
2
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
RQ
RP
A
KK&K 68% CL
EX
O
-2
00
 S
en
si
tiv
ity
K
am
LA
N
D
-Z
en
 S
en
si
tiv
ity
GERDA Sensitivity
G
ER
D
A
 L
im
it
EXO-200 Limit
KamLAND-Zen Limit
Figure 7.34: ??0? constraints are shown from 136Xe (horizontal) and
76Ge (vertical). The four diagonal lines represent four recent matrix ele-
ment computations [71, 145?147] and one recent phase-space factor [94]
relating the two half-lives; tick marks along the diagonal represent the
corresponding mass ?m???. 90% confidence limits and median 90% sensi-
tivities are shown for the KamLand-Zen [13], GERDA [15], and EXO-200
experiments [89]; the claimed 2006 discovery by Klapdor-Kleingrothaus
and Krivosheina is shown as a 1? confidence interval [148].
246
dataset is atypical in the other way.
Nuclear matrix elements have large uncertainties, as described in 2.3. Fig-
ure 7.34 illustrates four diagonal lines corresponding to four different modern nuclear
matrix calculations, and uses these values to relate the half-lives of 136Xe and 76Ge.
The current median sensitivity and limit on the ??0? half-life of 76Ge from GERDA
are shown as a horizontal line and vertical arrow, respectively. This is contrasted
with the 2006 claimed discovery by Klapdor-Kleingrothaus and Krivosheina of ??0?
decay, which is not excluded by the more recent GERDA limit.
Furthermore, the nuclear matrix elements permit us to translate half-life lower
limits into upper limits on the effective Majorana neutrino mass ?m???. The relation
is indicated, for each choice of nuclear matrix elements, by tick marks on the cor-
responding diagonal line. We can see that the GERDA and EXO-200 experiments
are of similar strengths, depending most on the choice nuclear matrix elements;
KamLAND-Zen is comparable in strength to GERDA for the RQRPA nuclear ma-
trix elements, and stronger for all other choices in this plot. The 90%-confidence
mass limit quoted by EXO-200 is ?m??? < 190? 450 meV, depending on the choice
of matrix elements. Recalling figure 2.8, this corresponds to a 90%-confidence limit
on the mass of the lightest neutrino eigenstate of mmin < 0.691.63 eV under the
most conservative choice of Majorana phases [89].
247
Parameter Undenoised DenoisedSS MS SS MS
p0 0 0 0.628 0.602
p1 39.8 43.0 20.8 25.8
p2 0.0107 0.0096 0.0011 0.0040
% ?/E at Q-value 1.94 2.00 1.53 1.65
Table 7.2: Time-averaged resolution parameters for denoised and undenoised data.
For undenoised data, measurements were consistent with p0 = 0, so this parame-
ter was not used in the analysis of undenoised data. The resulting time-averaged
resolutions at our Q-value are also shown [142].
7.6 Comparison to Results without Denoising
The results presented in section 7.5 are obtained using the most up-to-date analysis
techniques adopted by the EXO-200 collaboration. These include the denoising
technique described in chapter 4. In this section we compare physics results with
and without denoising and demonstrate the degree to which resolution gains from
denoising strengthen the EXO-200 result. We begin with a comparison of the energy
measurements in section 7.6.1, and then proceed to a comparison of the physics
results in section 7.6.2.
7.6.1 Comparison of Energy Resolution and Calibration
The process of denoising visibly sharpens source spectra, leading to easier extraction
of peak positions and widths. Single-site source spectra for representative runs are
shown in figures 7.35, 7.36, 7.37, all of which show significantly sharper peaks after
denoising. The rest of this section will contrast the calibrations of energy and
resolution for the two datasets.
248
En
erg
y (k
eV)
0
50
0
10
00
15
00
20
00
25
00
Rate (arbitrary units)
0
50
0
10
00
15
00
20
00
25
00
30
00
35
00
40
00
45
00
51
1 58
372
7 86
1 1
07
9
10
94
16
21
26
15
De
no
ise
d T
h D
ata
Un
de
no
ise
d T
h D
ata
Figure 7.35: A comparison of the spectra before and after denoising for a
set of representative thorium source runs (4758 and 4766). The thickness
of the lines indicate Poisson error bars on the number of counts in a bin;
smoothing is applied to the spectra to permit easier comparison of peaks.
Important gamma lines are indicated. Around 500-600 keV threshold
effects become significant, making comparison difficult.
249
En
erg
y (k
eV)
80
0
90
0
10
00
11
00
12
00
13
00
14
00
15
00
Rate (arbitrary units)
0
20
0
40
0
60
0
80
0
10
00
12
00
14
00
16
00
18
00
20
00
22
00
24
00
11
73
13
32
De
no
ise
d C
o D
ata
Un
de
no
ise
d C
o D
ata
Figure 7.36: A comparison of the spectra before and after denoising for
a representative cobalt source run (4787 and 4788). The thickness of
the lines indicate Poisson error bars on the number of counts in a bin;
smoothing is applied to the spectra to permit easier comparison of peaks.
The two gamma lines of 60Co are indicated.
250
En
erg
y (k
eV)
40
0
45
0
50
0
55
0
60
0
65
0
70
0
75
0
80
0
Rate (arbitrary units)
0
20
00
40
00
60
00
80
00
10
00
0
66
2
De
no
ise
d C
s D
ata
Un
de
no
ise
d C
s D
ata
Figure 7.37: A comparison of the spectra before and after denoising for a
representative cesium source run (4777-4781). The thickness of the lines
indicate Poisson error bars on the number of counts in a bin; smoothing
is applied to the spectra to permit easier comparison of peaks. The
gamma line of 137Cs is indicated. Threshold effects are significant at this
energy and the calibration is extrapolated from higher-energy sources,
but the spectra illustrate the fact that resolution improves dramatically
at low energy.
251
En
erg
y (k
eV)
10
00
15
00
20
00
25
00
30
00
35
00
Energy Resolution (keV) 2
530354045505560
SS
 De
no
ise
d
MS
 De
no
ise
d
SS
 Un
de
no
ise
d
MS
 Un
de
no
ise
d
En
erg
y (k
eV)
10
00
15
00
20
00
25
00
30
00
35
00
/E) ? Energy Resolution (% 
1
1.522.533.544.5
SS
 De
no
ise
d
MS
 De
no
ise
d
SS
 Un
de
no
ise
d
MS
 Un
de
no
ise
d
Figure 7.38: The time-averaged resolution functions (top) and relative
resolution functions (bottom) are compared for denoised and undenoised
data. Bands indicate uncertainties. Data provided by Caio Licciardi.
252
The energy resolution parameters with and without denoising, using the parametriza-
tion defined in equation 7.9, are listed in table 7.2. The undenoised resolution func-
tions have fixed p0 = 0 because fits with undenoised data led to resolution functions
consistent with zero. The resolution functions are compared in figure 7.38. The
resolution at our Q-value is also indicated in table 7.2; in single-site data we can
see that implementing denoising improves our energy resolution at the Q-value from
1.94% to 1.53%.
Recalling the interpretation of the resolution parameters which was described
in section 7.3.4 leads to a number of suggestive observations. First, p1 has decreased
by a factor of 1.9 in single-site data and 1.7 in multi-site data when we move from
undenoised to denoised resolutions. The parameter p1 is associated with electronic
noise, so we can interpret this to mean that we have indeed reduced the impact of
electronic noise after denoising.
Secondly, p2 has decreased by a factor of 2.4 in the multi-site data and a
factor of 9.7 for single-site data. The parameter p2 is associated with improper gain
calibrations; the old method of APD reconstruction handled only a simple sum of
APD waveforms with no APD channel gain corrections, whereas the lightmap used
with denoising has detailed time-dependent gain information for every channel built
into it. Thus, the reduction in p2 may be a reflection of the impact of improved
APD gain information on our energy resolution.
And finally, p0 is observed to increase from a value consistent with zero to the
dominant component of the denoised resolution at our Q-value. The parameter p0
is associated with Poisson fluctuations in the number of photons detected. In the
253
old method of APD reconstruction we always used a sum of all APD waveforms for
scintillation measurements; implicitly, we were always using the full light collecting
efficiency of the EXO-200 detector. By contrast, denoising is designed with the
flexibility to weight fewer APD channels more heavily, effectively sacrificing light
collection for the benefit of a reduction in electronic noise. It is likely that p0 is
zero with undenoised data because Poisson fluctuations in the total light collec-
tion of EXO-200 are small, whereas denoising uses a sufficiently small number of
APD channels that Poisson fluctuations become important. Moreover, we recall in
section 4.10 that the current analysis mistakenly underestimates Poissonian noise
produced inside the APDs, and this would manifest itself by subjecting the scintil-
lation estimate to even more statistical fluctuations than would be ideal. We can
imagine that when this error is corrected in future analyses the p0 parameter may
respond by decreasing, possibly at the expense of a larger p0 value as more channels
are used.
It should be noted that although these interpretations of p0, p1, and p2 are
highly suggestive, they depend on the assumption that scintillation energy resolution
continues to be the dominant factor in our rotated energy resolution at all energy
scales. As an alternative, it has been suggested that the rotation angle may have
an energy dependence which is not described by our calibrations [110]; failure to
include this dependence may be reflected by an energy dependence in the resolution
function which does not agree with the interpretation we provide here. Also, the
error bands in figure 7.38 are still somewhat broad, and more accurate resolution
measurements from a wider range of energies would be useful to constrain the values
254
Nov 20
11
Jan 20
12
Mar 20
12
May 20
12Jul 201
2
Sep 20
12
Nov 20
12
Jan 20
13
Mar 20
13
May 20
13Jul 201
3
1.4
1.6
1.8
2.0
2.2
Res
olut
ion
(?/E
)(%
)
208Tl res. (with denoising)
208Tl res. (without denoising)
Figure 7.39: Time dependence of the single-site energy resolution at
2615 keV with denoised (blue) and undenoised (red) scintillation. Time-
averaged energy resolutions at this energy (1.47% denoised, 1.86% un-
denoised) are overlaid as dashed lines. Data provided by Mike Marino.
of these parameters better. Further investigation will be required to verify that the
scintillation energy measurement accuracy is still the dominant contributor to energy
resolution at all energy scales.
We also find that the time-dependence of the energy resolution has been dra-
matically affected by denoising. Figure 7.39 compares the single-site energy reso-
lutions at the 208Tl 2615 keV gamma line with denoised and undenoised data, as
255
realR
0.2
0.4
0.6
0.8
1
1.2
real ) / R real  - R rec (R -0.0
3
-
0.025-0.02
-
0.015-0.01
-
0.00500.005
un-
deno
ised
deno
ised
SS
realR
0.2
0.4
0.6
0.8
1
1.2
real ) / R real  - R rec (R -0.0
3
-
0.025-0.02
-
0.015-0.01
-
0.00500.005
un-
deno
ised
deno
ised
MS
Figure 7.40: The non-linearity of the calibration is shown here for de-
noised (red) and undenoised (black) data in single-site (left) and multi-
site (right). Rrec is defined as the ratio between the observed peak
position and the observed peak position of the 2615 keV 232Th peak,
and Rreal is defined as the true ratio between the peak position and
2615 keV; the vertical axis shows the relative difference between Rrec
and Rreal, which reflects how much non-linear correction the calibration
must perform. The cesium, cobalt, and thorium peaks (662, 1173, 1332,
and 2615 keV) are used to obtain a calibration; the radium line at 2448
keV and the neutron capture line from hydrogen at 2200 keV are shown
as cross-checks [140].
measured from 232Th source calibration data. The denoised energy resolution varies
by as much as 0.3 percentage points, but the undenoised energy resolution varies by
as much as 0.6 percentage points. We saw in figure 4.1 that the energy resolution
of the detector was strongly correlated with APD noise, but after denoising this
effect is significantly reduced. In particular, the period of particularly poor energy
resolution November 2012 and May 2013 is no longer exceptional, indicating that
this poor energy resolution came from noise which denoising can reduce to negligible
levels [89].
One concern with the denoised energy calibration comes from the observed
256
non-linearity of the observation, shown in figure 7.40. The ratio of reconstructed
peak positions become farther from what we expect with denoised energy, indicating
more significant contributions from the Ebias, p0, and p2 parameters of the calibration
defined in equation 7.8. This is not in itself a problem ? it is the job of calibration
to correct for such things ? but as the calibration changes more rapidly with energy
it becomes important for us to constrain it with more calibration points at a wide
range of energies. Starting in June 2013 we have a 226Ra source which contains many
gamma lines, including its primary line at 2448 keV which serves as an excellent
constraint for the energy of our Q-value. For existing data, there is some hope of
using the lower-energy gamma lines of the 228Th source, visible in figure 7.18, as a
new set of calibration points which is accessible due to the improved resolution of
the denoised dataset.
7.6.2 Comparison of Physics Results
We now proceed to compare the physics results obtained with denoising to those
which would have been obtained without denoising. We recall from figure 3.6 that
improvement of the energy resolution will have the strongest effect on our 232Th
backgrounds, and will have a smaller (but not negligible) effect on our 137Xe back-
grounds. Here we will show changes as they impact our 2? region of interest, and
will conclude with a comparison of the profile likelihood scans obtained from full
fits to the spectrum.
We begin by listing the events which occur in the denoised single-site spectrum
257
in the energy window from 2325 to 2550 keV. These are provided in table 7.3, which
also includes the corresponding undenoised energy and the change in energy due to
denoising. The table indicates whether the event would fall within the 2? region of
interest in both the denoised and undenoised analysis.
Table 7.3: All single-site events which fall into the denoised energy window between
2325 and 2550 keV. This fully includes the 2? window around the Q-value; we
indicate whether the event falls into that 2? window. The correponding undenoised
energy is also indicated, along with whether that undenoised energy falls into the
undenoised 2? region of interest.
Run# Event#
Denoised Undenoised
?E (keV)
Energy (keV) In ROI? Energy (keV) In ROI?
2669 3889 2327.4 No 2322.6 No 4.8
2782 2237 2464.2 Yes 2560.3 No -96.0
2831 4864 2488.5 Yes 2245.3 No 243.2
3155 2053 2524.6 Yes 2541.6 Yes -17.0
3155 6861 2448.5 Yes 2403.4 Yes 45.1
3255 8687 2360.4 No 2353.4 No 7.0
3321 8688 2332.4 No 2315.2 No 17.2
3461 845 2357.7 No 2336.6 No 21.1
3573 1958 2406.1 Yes 2421.4 Yes -15.2
3657 3515 2417.0 Yes 2399.1 Yes 17.8
3740 11325 2405.7 Yes 2324.8 No 80.9
3912 10712 2391.3 Yes 2399.8 Yes -8.5
Continued on next page
258
Table 7.3 ? continued from previous page
Run# Event#
Denoised Undenoised
?E (keV)
Energy (keV) In ROI? Energy (keV) In ROI?
3933 3459 2441.4 Yes 2507.3 Yes -65.9
3964 9703 2459.2 Yes 2464.8 Yes -5.6
3972 754 2509.6 Yes 2470.9 Yes 38.8
3994 12296 2412.7 Yes 2367.1 Yes 45.5
4030 5037 2502.1 Yes 2482.8 Yes 19.3
4032 2727 2426.4 Yes 2349.3 No 77.1
4065 2350 2428.8 Yes 2442.8 Yes -14.0
4068 8841 2543.4 No 2640.4 No -97.0
4068 9897 2423.9 Yes 2453.5 Yes -29.7
4078 2595 2370.3 No 2372.4 Yes -2.0
4229 9176 2541.3 No 2519.5 Yes 21.9
4258 13386 2340.7 No 2338.8 No 2.0
4288 1334 2333.9 No 2340.6 No -6.7
4288 11179 2466.0 Yes 2494.1 Yes -28.1
4288 13040 2418.6 Yes 2486.3 Yes -67.7
4306 10543 2544.3 No 2574.1 No -29.7
4373 3036 2349.4 No 2395.3 Yes -45.8
4378 11150 2492.3 Yes 2494.2 Yes -1.9
Continued on next page
259
Table 7.3 ? continued from previous page
Run# Event#
Denoised Undenoised
?E (keV)
Energy (keV) In ROI? Energy (keV) In ROI?
4402 10759 2393.0 Yes 2398.0 Yes -5.1
4433 16390 2529.9 No 2494.1 Yes 35.8
4459 29435 2529.3 No 2472.2 Yes 57.1
4475 11191 2454.5 Yes 2521.3 Yes -66.8
4490 4613 2441.5 Yes 2481.4 Yes -40.0
4507 6129 2435.6 Yes 2412.4 Yes 23.2
4672 1891 2383.5 Yes 2418.5 Yes -35.0
4705 5799 2404.6 Yes 2465.7 Yes -61.1
4712 5996 2466.0 Yes 2439.5 Yes 26.5
4733 3595 2364.9 No 2381.2 Yes -16.2
4809 3937 2427.7 Yes 2429.1 Yes -1.4
4860 5170 2513.8 Yes 2596.3 No -82.5
4882 310 2474.2 Yes 2450.2 Yes 24.1
4899 9833 2342.4 No 2323.7 No 18.7
4903 7366 2472.6 Yes 2456.7 Yes 15.9
4918 4313 2377.1 No 2458.2 Yes -81.1
4957 8387 2398.6 Yes 2384.9 Yes 13.7
4978 6796 2521.1 Yes 2518.8 Yes 2.3
Continued on next page
260
Table 7.3 ? continued from previous page
Run# Event#
Denoised Undenoised
?E (keV)
Energy (keV) In ROI? Energy (keV) In ROI?
5010 2201 2365.3 No 2349.5 No 15.7
5082 9766 2517.7 Yes 2477.1 Yes 40.6
5097 7670 2472.1 Yes 2441.0 Yes 31.1
5105 15012 2548.5 No 2559.8 No -11.3
5124 7626 2480.4 Yes 2411.8 Yes 68.6
5305 3256 2406.0 Yes 2393.2 Yes 12.8
5372 11466 2513.7 Yes 2557.9 No -44.2
5380 4415 2458.2 Yes 2478.9 Yes -20.7
5409 38 2405.1 Yes 2456.9 Yes -51.8
To compare the limits set with denoised and undenoised data, figure 7.41
includes the profile likelihood curves derived from both datasets. The beta scale is
included as a systematic here, broadening both curves. The 90% confidence limit is
the value of ??0? at which the curve has a value of 1.35; for the denoised data we
can set a 90% confidence limit of 24 counts, whereas with the undenoised data we
set a 90% confidence limit of 27 counts. This means denoising has given us an 11%
stronger half-life limit than the undenoised dataset could have provided; or, since
the neutrino mass ?m??? is proportional to the square root of half-life, denoising
261
0 5 10 15 20 25 30 35
??0? counts
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Ne
gat
ive
 Lo
g-L
ike
liho
od
1?
90%
Denoised
Undenoised
Figure 7.41: Denoised (red) and undenoised (blue) profile likelihood
curves for ??0?.
262
gives us a 6% stronger limit on the neutrino mass.
Another way to compare the denoised and undenoised datasets is by studying
their expected number of background counts. We choose to present the best fits
where ??0? is fixed at zero, rather than the best fits overall. We do this because
we are claiming in section 7.4 that the result is a limit on ??0? decay, not an
observation; any differences in the best-fit number of ??0? counts should therefore
be viewed as statistical fluctuations rather than meaningful results, and we do not
wish to allow them to skew our comparison.
Figure 7.42 shows the denoised and undenoised data and total best-fit PDFs
with the ??0? contribution fixed to zero. We can see that a significant portion
of the expected background reduction in undenoised data comes from pulling the
region of interest away from the 208Tl peak. The total expected background in the
2? region of interest is 37.4 counts in the denoised fit, as opposed to 54.8 counts in
the undenoised fit.
We also show a number of features in the single-site and multi-site low-
background spectra and contrast the data, total pdfs, and pdf contributions from
individual components. Figures 7.43 and 7.44 illustrate the 40K gamma line at 1461
keV. Figure 7.45 shows the multi-site contribution from 60Co with gamma lines at
1173 and 1332 keV. Figures 7.46 and 7.47 show contributions from daughter prod-
ucts of 238U. Figures 7.48 and 7.49 show contributions from daughter products of
232Th. The change in contribution from 137Xe is shown in figure 7.50.
The observation that all of these features become sharper gives us confidence
that the resolution improvements observed in source data are also seen in the low-
263
2200 2300 2400 2500 2600 2700 2800Energy (keV)
0
5
10
15
20
25
30
35
40
Co
un
ts/3
0 k
eV
Denoised (37.4 exp. counts in ROI)
Undenoised (54.8 exp. counts in ROI)
Denoised (39 counts in ROI)
Undenoised (51 counts in ROI)
Figure 7.42: Denoised (red) and undenoised (blue) data counts are shown
in 30 keV bins. The total best-fit PDF, where the ??0? contribution is
fixed to zero, is overlaid for denoised and undenoised data. Dashed lines
indicate the 2? region of interest.
264
1200 1300 1400 1500 1600 1700 1800Energy (keV)
0
500
1000
1500
2000
2500
3000
Co
un
ts/3
0 k
eV
Denoised Total
Undenoised Total
Denoised 40K
Undenoised 40K
Denoised Data
Undenoised Data
Figure 7.43: Denoised (red) and undenoised (blue) single-site low-
background data and pdfs in the energy window from 1200 to 1800 keV.
The thick lines indicate the total single-site pdfs; thin lines indicate con-
tributions from 40K. Both fits are constrained to have no contribution
from ??0?.
265
1200 1300 1400 1500 1600 1700 1800Energy (keV)
0
200
400
600
800
1000
1200
1400
Co
un
ts/3
0 k
eV
Denoised Total
Undenoised Total
Denoised 40K
Undenoised 40K
Denoised Data
Undenoised Data
Figure 7.44: Denoised (red) and undenoised (blue) multi-site low-
background data and pdfs in the energy window from 1200 to 1800 keV.
The thick lines indicate the total multi-site pdfs; thin lines indicate con-
tributions from 40K. Both fits are constrained to have no contribution
from ??0?.
266
1000 1100 1200 1300 1400 1500 1600Energy (keV)
0
200
400
600
800
1000
1200
1400
Co
un
ts/3
0 k
eV
Denoised Total
Undenoised Total
Denoised 60Co
Undenoised 60Co
Denoised Data
Undenoised Data
Figure 7.45: Denoised (red) and undenoised (blue) multi-site low-
background data and pdfs in the energy window from 1000 to 1600 keV.
The thick lines indicate the total multi-site pdfs; thin lines indicate con-
tributions from 60Co. Both fits are constrained to have no contribution
from ??0?.
267
2000 2100 2200 2300 2400 2500 2600Energy (keV)
0
20
40
60
80
100
120
Co
un
ts/3
0 k
eV
Denoised Total
Undenoised Total
Denoised 238U chain
Undenoised 238U chain
Denoised Data
Undenoised Data
Figure 7.46: Denoised (red) and undenoised (blue) single-site low-
background data and pdfs in the energy window from 2000 to 2600 keV.
The thick lines indicate the total single-site pdfs; thin lines indicate con-
tributions from isotopes in the 238U decay chain. Dashed vertical lines
indicate the 2? region of interest around the Q-value. Both fits are con-
strained to have no contribution from ??0?. The uranium decay chain
contributes 10.0 (8.9) mean expected counts in the 2? region of interest
for denoised (undenoised) data.
268
1600 1800 2000 2200 2400 2600 2800Energy (keV)
0
50
100
150
200
Co
un
ts/3
0 k
eV
Denoised Total
Undenoised Total
Denoised 238U chain
Undenoised 238U chain
Denoised Data
Undenoised Data
Figure 7.47: Denoised (red) and undenoised (blue) multi-site low-
background data and pdfs in the energy window from 1600 to 2800
keV. The thick lines indicate the total multi-site pdfs; thin lines in-
dicate contributions from isotopes in the 238U decay chain. Both fits are
constrained to have no contribution from ??0?.
269
2200 2300 2400 2500 2600 2700 2800Energy (keV)
0
5
10
15
20
25
30
35
40
Co
un
ts/3
0 k
eV
Denoised Total
Undenoised Total
Denoised 232Th chain
Undenoised 232Th chain
Denoised Data
Undenoised Data
Figure 7.48: Denoised (red) and undenoised (blue) single-site low-
background data and pdfs in the energy window from 2200 to 2800 keV.
The thick lines indicate the total single-site pdfs; thin lines indicate con-
tributions from isotopes in the 232Th decay chain. Dashed vertical lines
indicate the 2? region of interest around the Q-value. Both fits are con-
strained to have no contribution from ??0?. The thorium decay chain
contributes 18.7 (34.5) mean expected counts in the 2? region of interest
for denoised (undenoised) data.
270
2200 2300 2400 2500 2600 2700 2800Energy (keV)
0
20
40
60
80
100
120
Co
un
ts/3
0 k
eV
Denoised Total
Undenoised Total
Denoised 232Th chain
Undenoised 232Th chain
Denoised Data
Undenoised Data
Figure 7.49: Denoised (red) and undenoised (blue) multi-site low-
background data and pdfs in the energy window from 2200 to 2800
keV. The thick lines indicate the total multi-site pdfs; thin lines in-
dicate contributions from isotopes in the 232Th decay chain. Both fits
are constrained to have no contribution from ??0?.
271
2200 2300 2400 2500 2600 2700 2800Energy (keV)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
13
7 Xe
 Ba
ckg
rou
nd
 as
sum
ing
 no
 ??
0?
 (C
oun
ts/3
0 k
eV
) Denoised (7.8 exp. counts in 2? ROI)Undenoised (10.3 exp. counts in 2? ROI)
Figure 7.50: Denoised (red) and undenoised (blue) single-site PDF con-
tributions from 137Xe, where we constrain ??0? to have no decays in
both fits.
272
background spectrum; they also improve the ability of the fit to reduce correlations
between different background components. These fits also permit us to gauge the
change in expected background due to each background component. In particular,
we see that background in the 2? region of interest due to thorium decreases dramat-
ically after denoising, as expected. Background due to 137Xe decreases moderately.
We expect no change in the expected background due to the uranium chain; in fact
the expected number of background counts due to uranium increases slightly, but
this is due to a shift in the best-fit rate of uranium components to 864 expected
counts over the entire fit window, compared to only 724 in the undenoised dataset.
The uranium decay chain, thorium decay chain, and 137Xe are our dominant back-
grounds in both the denoised and undenoised data.
The mean expected number of background counts drawn from these fit results
give some indication of how easily we could discover ??0? decay with the denoised
and undenoised analysis because we expect background fluctuations to be similar to
the square root of the number of expected counts, and to discover ??0? decay it must
produce more counts than can be attributed to a typical background fluctuation.
The number of expected background counts in denoised data is 32% lower than in
undenoised data, so the number of ??0? counts required for a discovery of similar
strength is
?
32% = 17% lower for denoised data than undenoised data. Since the
neutrino mass ?m??? is proportional to the square root of the half-life T 0?1/2, we can
expect that denoising typically permits us to discover neutrino masses 9% lower
than without denoising.
273
7.7 Summary
We have seen that denoising now occupies a central place in the EXO-200 analysis,
providing one of the two energy quantities (charge and light) which together deter-
mine the size of the region of interest we must use. Denoising has been shown to
produce a 21% improvements in single-site energy resolution at our Q-value and a
32% reduction in the mean expected backgrounds in our 2? region of interest. Even
with denoising, the ??0? half-life limit we obtain is less than our mean sensitivity
would lead us to hope; however, this fluctuation is not unreasonable and is a natural
part of low-background physics research. The upgrade of the present analysis using
denoising has led to exciting new investigations, and we can already see that our
improved resolution has exposed new avenues for improvements in our calibration
techniques. In chapter 8 we conclude with a reminder of the accomplishments of
the present work and a look ahead to more anticipated upgrades to the EXO-200
system.
274
Chapter 8: Conclusions and Future Plans
This work has presented a wide-ranging upgrade to the scintillation waveform anal-
ysis of EXO-200. The result has been a 21% improvement in the single-site energy
resolution at the Q-value, from 1.94% to 1.53% ?/E. We strengthen the half-life
limit of ??0? decay for this analysis by 11%, corresponding to a 6% stronger limit
on the Majorana neutrino mass. Separate background fits to the denoised and un-
denoised data indicate that the 2? region of interest background is reduced 32% by
denoising, corresponding to a mean sensitivity which is strengthened by 17% and a
mean sensitivity for the Majorana mass which is 9% lower than could be achieved
without denoising.
One question which must be asked is whether there is any room for further
improvement in the energy resolution. Figure 8.1 shows simulations of the rela-
tionship between scintillation-only resolution and rotated energy resolution. Points
indicate the resolutions measured from denoised and non-denoised data; we can see
that the model agrees well with both sets of data. Based on these simulations, it
is clear that further improvements to the scintillation resolution will indeed have a
significant impact on the overall resolution of EXO-200; projections from figure 3.6
275
 
[ke
V]
ph
oto
n s
tat
ist
ics
 + 
no
ise
?
Sc
int
illa
tio
n a
bso
lut
e r
eso
lut
ion
, 
20
40
60
80
10
0
12
0
14
0
/E ? Rotated resolution, 0.
00
5
0.0
1
0.0
150.0
2
0.0
25
EX
O-
20
0 T
ho
riu
m
 
so
ur
ce
 
Da
ta 
(no
n-
de
no
ise
d)
EX
O-
20
0 T
ho
riu
m
 
so
ur
ce
 
Da
ta 
(de
no
ise
d)
NE
ST
 
M
C (
10
% 
L.E
, 
re
ali
sti
c 
U-
w
ire
 
no
ise
 
lev
el)
Figure 8.1: NEST simulation software has been used to estimate how
the rotated energy resolution at 2615 keV (vertical axis) depends on the
scintillation-only resolution at 2615 keV (horizontal axis); this theoret-
ical estimate is shown in blue, as applicable to the EXO-200 detector
with fixed electric field. Black points indicate measurements from tho-
rium source runs without denoising; red points indicate measurements
from thorium runs with denoising. Figure provided by Liangjian Wen
using NEST software [118].
276
indicate that 232Th and 137Xe backgrounds can continue to be significantly reduced
by resolution improvements, so these attempts are indeed worthwhile.
We have noted in section 4.10 that the parameters used for this denoising are
not optimal, and that preliminary studies indicate roughly 0.05 percentage points
can be gained in overall energy resolution at 2615 keV with an up-to-date set of de-
noising parameters. This issue is straightforward to address, and the next processing
of our dataset will incorporate it.
Beyond this improvement, we have described in section 7.3.2 that the denoised
scintillation peak position displays a large position dependence which cannot be
fully calibrated in downstream analysis. The effect of this feature is to dilute our
resolution improvements. Although the cause of this position dependence is not
fully understood, we believe that it originates in the lightmap because the lightmap
is the input to denoising which encodes position-dependent yield information; one
preliminary theory is that the discrepancy between 1-wire and 2-wire charge peak
positions described in section 7.3.1 leads to a bias in event selection for the lightmap.
We believe that this issue can be addressed with further investigation and will lead
to additional significant improvements in resolution.
The most exciting improvement in resolution may come not from offline anal-
ysis but from planned electronics upgrades. There are indications that the source
of electronic noise in the APDs is now understood and can be fixed in hardware in
the near future, with an expectation of energy resolutions below 1% after all up-
grades [116]. Energy resolution improvements on this scale would have a significant
impact on backgrounds and the sensitivity of EXO-200 to ??0? decay.
277
One may wonder, after a hardware upgrade which reduces electronic noise on
the APDs, whether denoising will still be a necessary component of the analysis. It
may indeed be the case that the full denoising scheme is not needed for lower-noise
waveforms. However, certain components of this analysis will still be critical to
good resolution. In particular, the techniques developed in this work to measure an
APD-by-APD lightmap will still be needed to relate pulse magnitudes to deposit
energies; we have seen hints in section 7.6.1 that application of channel-dependent
gain corrections reduces their impact on energy resolution by a factor of 9.7 in
single-site data, and if true then this will be a critical component of any scintillation
measurements regardless of changes in electronic noise.
Beyond energy resolution upgrades to EXO-200, the techniques described in
this work have led to the most complete understanding to date of what limits the
energy resolution of EXO-200. This understanding is timely because the successor
experiment to EXO-200, called nEXO, is currently in design stages of development.
Options for nEXO which are currently being considered include the types of light
sensors to use and how much to gang sensors together; studies are currently under-
way to understand exactly how design decisions have impacted EXO-200 scintillation
energy resolution, and the results will provide useful feedback to the nEXO design
process.
Additionally, other aspects of the EXO-200 analysis besides energy resolution
may benefit from the progress discussed here. It may be possible that our improved
understanding of scintillation enables us to enlarge the EXO-200 dataset. Currently,
data taken between May 2011 and October 2011 is not used in the ??0? search
278
because it was prior to a u-wire electronics upgrade and an increase in the APD
bias voltages. Studies which used that data were primarily charge-driven and only
used scintillation to measure Z-positions of deposits. However, it is possible that
denoising will permit us to make use of the weaker scintillation pulses from that
dataset and achieve an acceptable energy resolution; the result could be months of
added livetime for future studies.
Our deeper understanding of the scintillation pulses from an individual-APD
lightmap also has the potential to improve our pulse-finding threshold. The EXO-
200 energy threshold of 980 keV is limited by scintillation-finding. A lower energy
threshold would not directly impact a ??0? search, but would allow backgrounds
to be better-constrained by fits. Furthermore, there are low-energy physics searches
like 134Xe ??2? decay and 136Xe ??0?? Majoron decay searches which have not been
described in this work but which would benefit strongly from a lower energy thresh-
old [98]. Currently all scintillation pulses are found on summed APD waveforms;
our detailed APD lightmap should permit us to make use of individual-APD infor-
mation, which may permit better noise rejection and lead to a lower pulse-finding
threshold.
Another set of alternative physics searches which could be improved using tech-
niques from this work are excited-state decay searches. We expect 136Xe to have a
??2? decay mode to the second excited state of 136Ba, followed by the prompt emis-
sion of two de-excitation gammas. Currently the EXO-200 analysis only produces
scintillation measurements for all simultaneous charge deposits together, making it
impossible to generate anticorrelated energy estimates for single energy deposits.
279
However, a natural extension of denoising would permit us to extract separately the
scintillation energies of each deposit site, using knowledge of which channels collected
photons to assign energy appropriately. By enabling anticorrelated individual-site
energy measurements, we could improve our ability to identify the clusters produced
by each of the de-excitation gammas from an excited-state decay.
Finally, there is a possibility, remote but tantalizing, that a variation of these
methods could lead to improved discriminating power between beta and gamma
deposits. The leading edge of u-wire pulses includes information about the size of
the charge cloud: a diffuse charge cloud penetrates the v-wire shielding grid slowly,
leading to a slowly-rising leading edge to the u-wire pulse, whereas a more pointlike
charge cloud penetrates the v-wire shielding grid in a shorter span of time and leads
to a sharply-rising leading edge to the u-wire pulse, so the risetime of a pulse may
provide information about the size of its charge cloud. The leading edge of the u-
wire pulse is quite short, contained in only a few samples, and current studies have
shown no significant improvement in discriminating power when it is used; however,
simulations hint that it should be possible to use this information. One plan for
improving the quality of this discriminator is to denoise the u-wire waveforms so that
the samples from the leading edge of the pulse are less noisy and provide a better
measure of the risetime of the pulse. Improved ability to discriminate betas from
gammas would lead to fewer backgrounds for ??0? decay, improving our sensitivity
further.
These are some of the many ways in which the denoising techniques of this
work may be applied. In all of these applications, the broader message we take away
280
from denoising is that having a complete model of a detector?s signals is a powerful
thing. We have constructed a full description of the APD noise correlation behavior
and a complete characterization of how photons generate pulses on waveforms and
with what fluctuations. Neither tool was available previously, and using them we
have been able to transform the challenge of improving energy resolution into a
purely mathematical optimization. We have confidence that these same tools will
yield benefits in many EXO-200 analyses to come.
281
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