# Wire-Cell 3D Pattern Recognition Techniques for Neutrino Event
Reconstruction in Large LArTPCs:
Algorithm Description and Quantitative Evaluation with MicroBooNE Simulation

P. Abratenko  R. An  J. Anthony  L. Arellano  J. Asaadi  A. Ashkenazi  S.
Balasubramanian  B. Baller  C. Barnes  G. Barr  V. Basque  L. Bathe-Peters  O.
Benevides Rodrigues  S. Berkman  A. Bhanderi  A. Bhat  M. Bishai  A. Blake  T.
Bolton  J. Y. Book  L. Camilleri  D. Caratelli  I. Caro Terrazas  R. Castillo
Fernandez  F. Cavanna  G. Cerati  Y. Chen  D. Cianci  J. M. Conrad  M. Convery
L. Cooper-Troendle  J. I. Crespo-Anadón  M. Del Tutto  S. R. Dennis  P. Detje
A. Devitt  R. Diurba  R. Dorrill  K. Duffy  S. Dytman  B. Eberly  A. Ereditato
J. J. Evans  R. Fine  G. A. Fiorentini Aguirre  R. S. Fitzpatrick  B. T.
Fleming  N. Foppiani  D. Franco  A. P. Furmanski  D. Garcia-Gamez  S. Gardiner
G. Ge  S. Gollapinni  O. Goodwin  E. Gramellini  P. Green  H. Greenlee  W. Gu
R. Guenette  P. Guzowski  L. Hagaman  O. Hen  C. Hilgenberg  G. A. Horton-
Smith  A. Hourlier  R. Itay  C. James  X. Ji  L. Jiang  J. H. Jo  R. A.
Johnson  Y.-J. Jwa  D. Kalra  N. Kamp  N. Kaneshige  G. Karagiorgi  W. Ketchum
M. Kirby  T. Kobilarcik  I. Kreslo  R. LaZur  I. Lepetic  K. Li  Y. Li  K. Lin
B. R. Littlejohn  W. C. Louis  X. Luo  K. Manivannan  C. Mariani  D. Marsden
J. Marshall  D. A. Martinez Caicedo  K. Mason  A. Mastbaum  N. McConkey  V.
Meddage  T. Mettler  K. Miller  J. Mills  K. Mistry  A. Mogan  T. Mohayai  J.
Moon  M. Mooney  A. F. Moor  C. D. Moore  L. Mora Lepin  J. Mousseau  M.
Murphy  D. Naples  A. Navrer-Agasson  M. Nebot-Guinot  R. K. Neely  D. A.
Newmark  J. Nowak  M. Nunes  O. Palamara  V. Paolone  A. Papadopoulou  V.
Papavassiliou  S. F. Pate  N. Patel  A. Paudel  Z. Pavlovic  E. Piasetzky  I.
D. Ponce-Pinto  S. Prince  X. Qian  J. L. Raaf  V. Radeka  A. Rafique  M.
Reggiani-Guzzo  L. Ren  L. C. J. Rice  L. Rochester  J. Rodriguez Rondon  M.
Rosenberg  M. Ross-Lonergan  G. Scanavini  D. W. Schmitz  A. Schukraft  W.
Seligman  M. H. Shaevitz  R. Sharankova  J. Shi  J. Sinclair  A. Smith  E. L.
Snider  M. Soderberg  S. Söldner-Rembold  P. Spentzouris  J. Spitz  M.
Stancari  J. St. John  T. Strauss  K. Sutton  S. Sword-Fehlberg  A. M. Szelc
N. Tagg  W. Tang  K. Terao  C. Thorpe  D. Totani  M. Toups  Y.-T. Tsai  M. A.
Uchida  T. Usher  W. Van De Pontseele  B. Viren  M. Weber  H. Wei  Z. Williams
S. Wolbers  T. Wongjirad  M. Wospakrik  K. Wresilo  N. Wright  W. Wu  E.
Yandel  T. Yang  G. Yarbrough  L. E. Yates  H. W. Yu  G. P. Zeller  J. Zennamo
C. Zhang

###### Abstract

Wire-Cell is a 3D event reconstruction package for liquid argon time
projection chambers. Through geometry, time, and drifted charge from multiple
readout wire planes, 3D space points with associated charge are reconstructed
prior to the pattern recognition stage. Pattern recognition techniques,
including track trajectory and $dQ/dx$ (ionization charge per unit length)
fitting, 3D neutrino vertex fitting, track and shower separation, particle-
level clustering, and particle identification are then applied on these 3D
space points as well as the original 2D projection measurements. A deep neural
network is developed to enhance the reconstruction of the neutrino interaction
vertex. Compared to traditional algorithms, the deep neural network boosts the
vertex efficiency by a relative 30% for charged-current $\nu_{e}$
interactions. This pattern recognition achieves 80-90% reconstruction
efficiencies for primary leptons, after a 65.8% (72.9%) vertex efficiency for
charged-current $\nu_{e}$ ($\nu_{\mu}$) interactions. Based on the resulting
reconstructed particles and their kinematics, we also achieve 15-20% energy
reconstruction resolutions for charged-current neutrino interactions.

##  1 Introduction

The liquid argon time projection chamber (LArTPC) [1, 2, 3, 4] is a three-
dimensional tracking calorimeter widely used in neutrino physics [5, 6, 7, 8,
9]. Compared to water Cerenkov and liquid scintillator technologies, LArTPCs
are highly effective in differentiating electrons and photons in neutrino
interactions through identifying the gap between photon conversion vertex and
neutrino interaction vertex and measuring the $dE/dx$ (ionization energy loss
per unit length) in the first few centimeters of electron or photon
electromagnetic cascade (EM shower). This capability is advantageous for
detecting $\nu_{e}$ charged-current interactions with high signal efficiencies
and low background contamination. Utilizing this unique capability of LArTPCs,
the MicroBooNE experiment [7] aims to understand the nature of the low-energy
excess of $\nu_{e}$-like events observed in the MiniBooNE experiment [10] and
to measure neutrino-argon interaction cross sections in various inclusive and
exclusive channels.

Figure 1: Illustration of signal formation in the MicroBooNE three-plane
LArTPC, depicting the V plane (second induction plane) and Y plane (collection
plane) TPC wire signals on the right of the image [7].

Figure 1 illustrates the MicroBooNE LArTPC, which utilizes three readout wire
planes at the anode to obtain signals induced from drifting ionization
electrons produced from neutrino interactions within the active volume of the
detector. Drifting ionization electrons first pass by the two induction planes
(the “U plane” and “V plane”) and are finally collected by the collection
plane (the “Y plane”). Being one of the first large LArTPCs constructed,
MicroBooNE also bears important responsibilities in the research and
development of the LArTPC technology, which is key to many future experiments
[11, 12]. In particular, MicroBooNE pioneers the use of TPC cold electronics
[13], which significantly reduce electronic noise [14]. Paired with enhanced
TPC signal processing procedures [15, 16], the cold electronics lead to a much
improved performance of the induction anode wire planes. Good matching of
reconstructed ionization charge between the induction and collection wire
planes is demonstrated for the first time [15, 16]. The significantly enhanced
performance of the induction wire planes provides a solid foundation in
applying the Wire-Cell (a software package [17, 18]) reconstruction to the
MicroBooNE data.

Wire-Cell imaging [17, 18] is a 3D image reconstruction method developed by
the MicroBooNE collaboration. The main idea of Wire-Cell imaging is to
reconstruct 3D images in a tomographic way without involving heuristic
topological assumptions (e.g. track or shower) prior to the pattern
recognition stage. Using time and geometry information, Wire-Cell constructs
the 3D space points given the observed 2D images (wire vs. time) [17].
Afterwards, Wire-Cell utilizes the charge-matching constraint and the sparsity
condition to reduce ambiguities in 3D space points introduced by the wire
readout. The charge-matching constraint requires that the same amount of
reconstructed ionization charge should be seen by each (induction or
collection) wire plane locally. The distribution of ionization electrons in
space is expected to be sparse, typically occupying less than 10% of the local
bounding volume that contains the activities, for any physical signals [18].
This sparsity condition is applied through the compressed sensing technique
[19] via the coordinate descent algorithm [20]. The end results of Wire-Cell
imaging are 3D space points with associated charges. Figure 2a shows an
example of the Wire-Cell imaging result. We note that in dealing with
imperfect detector performance in MicroBooNE (e.g. dead channels) [14],
special algorithms, including two-wire-plane tiling and deghosting, are
necessary to retain the quality of 3D images. Here, “tiling” is a technique
that correlates 2D projections from multiple wire planes to obtain 3D
information; and “deghosting” is a technique that removes ghost energy
depositions caused by ambiguities in the 2D to 3D reconstruction [18].

In the MicroBooNE LArTPC, well-reconstructed 3D images form a good foundation
for the neutrino interaction selection [21, 22], which is a challenging task
for LArTPCs operating on the surface because of the numerous traversing
cosmic-ray muons. A set of clustering techniques on the 3D space points [18]
was developed to separate neutrino interactions from cosmic-ray muons. A many-
to-many TPC-charge to PMT-light matching algorithm [18] is applied to match
isolated TPC clusters to patterns of the PMT light signals, so that the
neutrino interaction candidates in coincidence with the Booster Neutrino Beam
(BNB) spill ($\sim$ 1.6 $\mu$s long) can be selected. Here, a “TPC cluster” is
defined to be a set of space points that are connected in space, given that
the LArTPC is a fully active tracking calorimeter. In addition, the links
between the 3D space points and their 2D projections are also kept. To further
reject cosmic muon backgrounds that occur in random coincidence with the BNB
spill, a selection based on the TPC geometry is applied to reject cosmic-ray
muons passing through the active TPC volume. The cosmic-ray muons that stop
inside the TPC active volume can be determined by identifying the direction of
the muon track. A 3D trajectory and $dQ/dx$ (ionization charge loss per unit
length) fitting procedure [22] was developed to identify Bragg peaks and
reject those stopped muons. Note that the measurable $dQ/dx$ is directly
related to the previously mentioned $dE/dx$ (energy loss per unit length).
These tools are used to form a generic neutrino selection, which aims at
rejecting cosmic-ray muon background while retaining high efficiencies for all
kinds of neutrino interactions. This generic neutrino selection achieves 80%
and 88% efficiencies for inclusive $\nu_{\mu}$ charged-current ($\nu_{\mu}$CC)
and $\nu_{e}$ charged-current ($\nu_{e}$CC) interactions with a 14.9% cosmic-
ray background contamination in the selected events [21]. The generic neutrino
selection serves as a good basis to further apply pattern recognitions and
identify various neutrino flavors with high efficiencies and purities. Figure
2b shows an example of the Wire-Cell generic neutrino selection result. In
addition to the removal of cosmic-ray activity, about 80% (90%) of the
neutrino charged-current interactions have greater than 80% (70%) of their
energy deposits reconstructed and included in the resulting 3D TPC clusters
[18]. This provides a solid foundation to the subsequent 3D pattern
recognition.

MicroBooNE data(run 5384, event 2561)

Figure 2: Example event displays of the Wire-Cell 3D imaging results before
(panel a) and after (panel b) generic neutrino selection. Most TPC activities
in panel a are identified as cosmic rays and removed in panel b. What remains
are likely associated with a neutrino interaction and will be the inputs for
the Wire-Cell pattern recognition algorithms described in this paper. Figure
3: Overview of the Wire-Cell pattern recognition procedure. The blue dashed
line indicates steps of the first iteration towards neutrino vertexing; the
orange dotted line indicates steps towards particle flow tree reconstruction
with a reconstructed neutrino vertex. Please note, intermediate objects
reconstructed in the first iteration, e.g. track segments and initial showers,
are also used in the second iteration, in addition to the reconstructed
neutrino vertex. More details can be found in the text.

This paper summarizes the pattern recognition techniques developed and applied
in Wire-Cell for the high-performance inclusive $\nu_{e}$CC and $\nu_{\mu}$CC
event selections [23, 24]. Both traditional algorithms (non-machine-learning
algorithms) and machine-learning algorithms are used in this pattern
recognition procedure. Some of the basic tools—the track trajectory and
$dQ/dx$ fitting, for example—are improved versions of the techniques developed
for generic neutrino detection [21, 22]. This fitting algorithm is expanded to
fit multiple tracks and multiple vertices rather than fitting a single track.
Figure 3 shows the overall flow of the Wire-Cell pattern recognition, which
consists of two iterations. In the first iteration, we focus on neutrino
vertex identification. An identified neutrino vertex location significantly
reduces the possible solution space for other pattern recognition tasks. Here,
solution space includes the origins and directions of particles as well as the
mother-daughter relationships between them. In the second iteration, full
particle flow trees are reconstructed based on the neutrino vertex locations
and other intermediate objects reconstructed in the first iteration, e.g.
track segments and initial showers. Here, a particle flow tree is a tree
structure that contains the mother-daughter relationship for all particles in
a set. At the beginning of the first iteration, vertices are defined by
searching for kinks and splits in the reconstructed 3D images (section 2.1).
With vertices determined, track segments between vertices are defined. A 3D
vertex fitting technique (section 2.2) can then be used to refine the 3D
vertex and connected track trajectories. Particle identification (PID) is
subsequently performed on segments using $dQ/dx$ and event topology
information (section 2.3). Event topology information is primarily targeted
toward electromagnetic (EM) shower identification (i.e. track/shower topology
separation). Using the PID information, directions of particles can be
determined in many cases. The neutrino interaction vertex is determined using
the reconstructed particle directions and later improved with a deep neutral
network (DNN, section 2.5). With the neutrino interaction vertex reconstructed
in the first iteration, EM showers can be fully clustered (section 3.1), and
often contain several separated sub-clusters. Finally, a particle flow tree is
reconstructed including the $\pi^{0}$s reconstructed from EM showers (section
3.2).

MicroBooNE data(run 5384, event 2561)

Figure 4:  Displays of the Wire-Cell pattern recognition results at different
stages. (a) In-beam candidate neutrino cluster selected by the generic
neutrino selection. The color scale represents the reconstructed charge
associated with each space point. (b) Identified tracks and EM showers, which
are displayed in blue and red, respectively. (c) Identified particles (or
track segments), which are displayed in different colors. (d) Fitted $dQ/dx$
associated with each piece ($\sim$6 mm) along the trajectories. The blue,
cyan, green, yellow, and red colors roughly correspond to 1/3, 1, 2, 3, and 4
times the $dQ/dx$ of a minimum ionizing particle (MIP). The short track
labeled as electron is likely to be a mis-reconstructed pion here. The EM
shower (listed as electron in the particle flow) is labeled as a $\gamma$,
given it is emitted by a proton track. It could be one of the two $\gamma$s
from a produced $\pi^{0}$. This $\gamma$ is likely converted to electron-
positron pair immediately. (e) Reconstructed particle flow starting from the
primary neutrino interaction vertex. The folder icon indicates the
corresponding particle has reconstructed daughter particle(s), while the file
icon indicates the corresponding particle has no reconstructed daughter
particle(s). Particles with check marks in the check-boxes are shown in panel
(e) with rainbow circles marking the starting positions and line segments
marking the starting and ending positions. A web based event display (BEE) for
this example can be found here:
https://www.phy.bnl.gov/twister/bee/set/uboone/reco/2021-09/lee/event/1/.

Figure 4 shows an example of the pattern recognition at different stages from
a neutrino interaction event in the MicroBooNE data. Colors in panels (d) and
(e) represent the magnitude of the reconstructed $dQ/dx$, which is especially
important for PID and determining the directions of particle trajectories. As
we currently do not differentiate particles and anti-particles, in the legend,
the charged particles can also refer to their antiparticles, e.g. $e^{-}$ can
refer to both $e^{-}$ or $e^{+}$, while the final state charged particles from
neutrino mode interactions are mostly negatively charged leptons and
positively charged pions.

The traditional algorithms in the Wire-Cell pattern recognition introduced in
this article are developed using hundreds of MicroBooNE BNB data events, while
the training of the DNN neutrino vertexing and the evaluation of the
performance of this pattern recognition are performed using high-statistics
MicroBooNE Monte-Carlo (MC) simulation events. E.g. in this paper, the DNN
neutrino vertexing model is trained with 48000 selected $\nu_{e}$ charged-
current events which leads to significant performance boost. The MicroBooNE MC
simulates neutrino interactions using a flux model following ref. [25] and
using the GENIE v3.0.6 neutrino event generator with a MicroBooNE tune [26,
27]. The particles produced in this neutrino interaction are processed through
a realistic detector response model which takes the energy depositions from a
Geant4 simulation [28], determines the ionization charge considering
recombination [29, 30], drifts them through the TPC electric field including
the space charge effect [31, 32], diffuses the charge [33], and calculates the
resulting signal on the wires using a sophisticated electric field response
[15]. This result is in good agreement between data and MC at a fundamental
level [16]. Moreover, the MicroBooNE MC simulation used in this analysis
adopts a scheme of overlaying the simulated neutrino interactions with the
dedicated off-beam data that is randomly triggered when no neutrino beam spill
is received, therefore the systematic uncertainties occurring in the
simulation of readout noise, time-invariant detector response, and cosmic-ray
backgrounds are avoided or largely reduced. Such MC simulation is referred to
as overlay MC. This paper is organized as follows. Section 2 describes
algorithms of the first iteration towards the identification of the neutrino
vertex. Follow up steps towards full particle flow tree reconstruction are
introduced in section 3. Neutrino energy reconstruction is described in
section 4. The conclusions are found in section 5.

##  2 Initial Iteration towards Neutrino Vertex Identification

A key step of the neutrino event reconstruction with LArTPCs is to identify
the neutrino interaction vertex location. In this section, we describe two
neutrino vertex identification algorithms. The first algorithm is a non-
machine-learning one, utilizing particle direction information and event
topology information. The second one is a machine-learning based algorithm,
utilizing a sparse regressional segmentation neural network. The final
neutrino vertex position is determined by combining results from both
algorithms. As described in section 1, to obtain the particle direction and
event topology information, dedicated algorithms for track segment finding,
$dQ/dx$ fitting and track shower separation are performed. These algorithms
are introduced in this section (sub-section 2.1, 2.2 and 2.3). Their results
are not only useful for the neutrino vertex identification but also the full
particle flow reconstruction which will be introduced in section 3.

###  2.1 Determination of Vertices and Track Segments

In Wire-Cell, a track segment is defined as a subset of a TPC cluster
including a set of 3D space points and associated 2D (projection) pixels with
deposited charge information. Based on a track segment, the best-fit 3D
trajectory points with about 6 mm spacing are obtained by minimizing a charge-
weighted distance. That list of trajectory points is defined as a track
trajectory. A key technique in determining vertices and track segments is the
multi-track trajectory and $dQ/dx$ fitting, which is an expansion of the
single-track trajectory and $dQ/dx$ fitting technique used in the generic
neutrino selection [22]. Figure 5 illustrates the relationships between key
concepts used in the multi-track trajectory and $dQ/dx$ fitting algorithm.

Figure 5:  This diagram illustrates relationships between key concepts used in
the multi-track trajectory and $dQ/dx$ fitting algorithm. More details about
the “3D Imaging”, the “Single-track trajectory and $dQ/dx$ fitting” and the
“3D vertex fitting” algorithms could be found in [18], [22] and section 2.2
respectively.

The single-track fitting technique development was inspired by the projection
matching algorithm [34]. First, an initial seed of the track trajectory is
obtained by constructing a Steiner-tree graph [35] from the 3D points in the
cluster and finding the shortest paths between extreme points. The Steiner-
tree ensures that points associated with the largest charges are included in
the initial seed. Then, the best-fit 3D trajectory points with about 6 mm
spacing are obtained by minimizing a charge-weighted distance, which is
constructed to compare the 2D measurements in time-versus-wire views from the
three wire planes to the predictions given a 3D trajectory point. A numerical
solver for large linear systems (BiCGSTAB [36]) is utilized to perform a
minimization. With the trajectory determined, the $dQ/dx$ associated with each
trajectory point can be obtained by minimizing the squared difference between
the reconstructed and predicted ionization charge. A parameterized model is
used to predict the measured charge taking into account the diffusion of
ionization electrons during transportation and the smearing of the charge
distribution in the signal processing. This two-step procedure is adopted to
avoid a nonlinear fitting process, ensuring the stability of the fit.
Regularization on smoothness is included to further improve the $dQ/dx$
fitting performance. More details about this single-track fitting technique
can be found in Ref. [22].

Moving towards the new multi-track fitting algorithm, major improvements
include:

  * •

The new algorithm handles simultaneous fitting of multiple track segments in
3D. In addition to the trajectory points that only belong to a single track
segment, a new type of data product, vertex, is added. A vertex can be
connected to one or more track segments.

  * •

The distance between two adjacent trajectory points is regulated to be as
close to 0.6 cm (twice the MicroBooNE TPC wire pitch) as possible, which makes
the $dQ/dx$ fitting easier. The more uniform step size also has advantages in
particle identification (PID) given the distribution of $dE/dx$ depending on
the step size ($dx$).

  * •

Grouping of the 2D pixels in the trajectory fitting is improved to better
handle multiple track segments at once. The determination of the 2D pixel
group for each track segment takes into account the associated 3D space points
with 2D pixels. The 3D space points are much easier to cluster with the
closest track trajectory, while the 2D pixels suffer more from ambiguities
because they are projections.

  * •

A dedicated fitting algorithm to determine the vertex position in 3D is
introduced. See more detailed discussions in section 2.2.

An iterative approach is used to determine track segments and vertices in a
TPC cluster, which allows one to systematically examine the TPC cluster given
its various possible topologies. This approach is described below:

  * •

Two extreme 3D space points (a pair of end points with the largest distance
between them) are found on the TPC cluster, and a single track trajectory is
fitted to connect the two extreme points.

  * •

An algorithm is run to search for kinks111sharp turns with local angle ¿ 25°
along the track trajectory from one end to the other. When found, the track
trajectory is broken at the kink vertex into separate segments, then the track
trajectory and $dQ/dx$ are fitted again. Segmenting the track helps prevent
track fitting from bypassing a kink by requiring the trajectory to pass
through the vertex connecting the segments. This process is repeated until no
more kinks are found.

  * •

Track segments associated with existing track trajectories are removed from
the TPC cluster and the remaining pieces are examined to search for new track
segments. The newly found track segments are connected back to the existing
track segments to seed new vertices. Track trajectory and $dQ/dx$ fitting is
repeated until no new track segments are found.

  * •

Additional empirical algorithms are used to merge very close vertices to each
other. For more details, please see Ref. [37].

The order of examinations (e.g. searching for kinks along a track or perform
new track fitting) is fixed given a certain input so that the results of
applying the aforementioned algorithms are reproducible. Figure 6 shows an
example of the multi-track trajectory and $dQ/dx$ fitting for the event shown
in figure 4.

MicroBooNE data (run 5384, event 2561)

Figure 6:  Example display of the results from multi-track trajectory and
$dQ/dx$ fitting. The three panels on the left show the reconstructed
ionization charge on different planes (two induction planes and one collection
plane) overlaid with the projection of the best-fit track trajectories. The
track trajectories are color coded and labeled with numbers of the same color.
The top right panel shows the 3D view of the best-fit track trajectories. The
bottom right panel shows the best-fit $dQ/dx$ for all track trajectories.
$dQ/dx$ distribution for each stopped track can be used to determine the
direction of the track according to the Bragg peak. The color coding can be
used to link the same trajectories in 2D and 3D views. Also using the color
coded numbers, the trajectories can be further linked to the $dQ/dx$ fitting
in the bottom right panel. The gray lines in the induction V view represent
non-functional channels. In the induction U view, short track number 6 is
highly compacted because of the nature of 2D projection.

###  2.2 3D Vertex Fitting

In order to improve the quality of the multi-track trajectory and $dQ/dx$
fitting (described in section 2.1), a dedicated algorithm is introduced to
better determine the position of the vertex when multiple track segments are
connected. Note, this vertex fitting algorithm is used to improve the
precision of all vertices, both primary and secondary. The neutrino vertex
(primary vertex) identification algorithms introduced later in section 2.4 and
2.5 are used to find the primary vertex.

Instead of operating on 2D projections, the vertex fitting procedure is
performed on the 3D space points. First, we denote the position of the vertex
to be determined as $\left(x,y,z\right)$. For the 3D space points associated
with each track segment $i$, the points that are within a certain range (with
a minimal distance of 1.5 cm and a maximal distance of 6 cm) are selected.
Requiring a minimal and maximal distance reduces the bias in the fit, as
points too close to the vertex can bias the fit due to incorrect associations
with track segments and points too far from the vertex can bias the fit for
tracks that change direction. A principle component analysis (PCA) is then
performed:

  * •

The center of the 3D points is $\left(x_{i0},y_{i0},z_{i0}\right)$. We further
define $\vec{r}_{i}=\left(x_{i0}-x,y_{i0}-y,z_{i0}-z\right)$.

  * •

The three eigenvectors of the PCA are denoted as $\vec{v}_{i1}$,
$\vec{v}_{i2}$, and $\vec{v}_{i3}$, with the eigenvalues as $\lambda_{i1}$,
$\lambda_{i2}$, and $\lambda_{i3}$, respectively. The distance from the vertex
position to the main axis (first principal component) of the 3D points can
thus be written as $\left(\vec{r}_{i}\cdot\vec{v}_{i2}\right)$ and
$\left(\vec{r}_{i}\cdot\vec{v}_{i3}\right)$ for the second and third principal
component directions respectively.

With these definitions, we can thus form a global test statistics to determine
the vertex position:

$T=\sum_{i}\left(\frac{\lambda_{i1}}{\lambda_{i2}}\left(\vec{r}_{i}\cdot\vec{v}_{i2}\right)^{2}+\frac{\lambda_{i1}}{\lambda_{i3}}\left(\vec{r}_{i}\cdot\vec{v}_{i3}\right)^{2}\right)+\lambda\cdot\left(\left(x-x_{org}\right)^{2}+\left(y-y_{org}\right)^{2}+\left(z-z_{org}\right)^{2}\right),$
(2.1)

where the first term in the summation minimizes normalized distance between
the vertex and the main axis of the 3D points from a track segment. The second
term in Eq. (2.1) minimizes the distance between the vertex position and its
initial value $\left(x_{org},y_{org},z_{org}\right)$ with a regularization
strength $\lambda$. The addition of this term is to avoid unrealistic fitting
results in certain situations (e.g. the track segments are all nearly parallel
to each other). The minimization of Eq. 2.1 can be solved with linear algebra,
which ensures the stability of the fitting procedure. Figure 7 shows the best-
determined track trajectories before and after the 3D vertex fitting. The
improvement close to the vertex is most prominent in the middle row plots.

MicroBooNE data (run 5384, event 2561)

Figure 7:  Best-determined track trajectories before (left) and after (right)
the 3D vertex fitting are displayed as lines on the reconstructed ionization
charge. The $z$-axis represents the amount of ionization charge. The
improvement of trajectory fitting near the vertex region (highlighted with
dashed red boxes) is most prominent in the middle row plots (induction V
plane). The original bias in the fitted track trajectory is the result of the
incorrect association of the 2D charge given that the initial track trajectory
is determined from the shortest path algorithm. The accurate vertex position
determination allows the shortest path algorithm to provide improved initial
track trajectories to determine the charge association for the fitting
algorithm.

###  2.3 Separation of Track and Shower Topology

We use three methods to differentiate tracks from electromagnetic (EM) showers
at different energies. They are: i) multiple Coulomb scattering measurements
to differentiate track-like electrons from other particles at low energy, ii)
the presence of additional isolated clusters in close proximity to track-like
electrons, also at low energy, and iii) the EM shower width perpendicular to
the trajectory direction for high energy electrons. We discuss each method in
more detail below.

At low energies ($\mathcal{O}(\text{10 MeV})$), the bremsstrahlung effect
becomes less important for electrons compared to the ionization effect. The
low-energy electrons could behave like tracks instead of electromagnetic
showers (i.e. multiple tracks including electron-positron pairs produced by
the bremsstrahlung photons). Nevertheless, the combination of an electron’s
small mass and ample interactions with atomic electrons in liquid argon means
low-energy electrons have more large-angle scattering than other particles. An
optimal way to differentiate low-energy electrons from low-energy muons,
pions, or protons in LArTPCs is to use the multiple Coulomb scattering (MCS)
method. For electrons, the MCS-derived momentum assuming a muon mass [38] will
be much lower than that reconstructed from the track residual range. We use a
simplified algorithm to differentiate a wiggled track from a straight track.
We divide each track segment into many sections with each section about 10 cm
long. For each track section, we calculate the straight-line distance between
the starting and ending points. Along its fitted track trajectory, the
trajectory length in this track section is also calculated. For a wiggled
track segment, the direct length after summing over all track sections is
expected to be much smaller than the trajectory length after summing over all
sections. For a straight track segment, on the other hand, the ratio of these
two lengths is close to one. 222Typically, a value of 0.9 or greater is
required empirically. Figure 8a shows such an example.

For low-energy electrons, in addition to the above mentioned length ratio, the
presence of nearby isolated clusters (i.e. bremsstrahlung photons emitted
during the development of EM shower) can also be used to identify an EM
shower. Figure 8b shows an identified EM shower in which case two isolated
clusters are associated with the wiggled track. The usage of isolated clusters
for EM shower identification requires an alignment of the isolated clusters to
the shower trunk (first few centimeters of the EM shower) direction. A good
neutrino vertex reconstruction plays an important role in determining the
shower trunk and its direction.

For high-energy electrons (see Fig. 8c), the resulting EM showers tend to
extend further in the projections perpendicular to the momentum direction
because a large number of secondary electrons and positrons are produced. The
measurement of the perpendicular spread of associated 3D space points
(reconstructed in the 3D imaging step) along the EM shower direction can be
used to identify an EM shower. The evolution of the width along the EM shower
direction (e.g. a gradual increase of the width) can also be used to determine
the starting point of the EM shower, which can be further used to determine
the direction of the EM shower using the stem of the EM shower (track-like
segment at the beginning of an EM shower). The direction of the EM shower is a
crucial input for neutrino vertex reconstruction. For any projective readout,
one expects ambiguities in the transverse plane (parallel to the readout
plane). The measurement of widths must take this into account. We define the
direction of the shower to be $\vec{r}_{mom}$ and the direction of drifting
electrons (perpendicular to the transverse plane) to be $\vec{r}_{drift}$. The
measurement of the widths is performed along the direction of
$\vec{r}_{mom}\times\left(\vec{r}_{mom}\times\vec{r}_{drift}\right)$, which is
perpendicular to both the momentum direction $\vec{r}_{mom}$ and the vector
representing the ambiguities $\vec{r}_{mom}\times\vec{r}_{drift}$.

MicroBooNE dataMicroBooNE dataMicroBooNE data

Figure 8: Example of various EM showers. (a) A very wiggled track. Color
represents reconstructed charge along the fitted track trajectories. The blue
(red) color indicates low (high) ionization charge. (b) A scattered track with
two isolated showers aligned with a straight line. Shower-like objects are
indicated by red color overlay; track-like objects are indicated by blue color
overlay. (c) A high-energy EM shower. Shower-like objects are indicated by red
color overlay; track-like objects are indicated by blue color overlay. Rainbow
circles indicate particle starting positions in all three panels.

###  2.4 Neutrino Vertex Identification with Traditional Techniques

The reconstructed particle direction provides important information for the
neutrino interaction vertex identification. In the primary neutrino
interaction vertex, the particles entering into the vertex would be the
neutrino (invisible to the detector) and the argon nuclei (also invisible to
the detector). Therefore, all tracks and EM showers would travel outward from
the primary neutrino interaction vertex. The situation is different for a
secondary interaction vertex in the main interaction cluster containing the
true neutrino interaction vertex. In this case, the particles entering into
the vertex would be a charged particle (e.g. produced by the primary neutrino
interaction) and an argon nuclei (invisible to the detector). The rest of
tracks and EM showers would travel outward from the secondary vertex. Given
each interaction always involves an argon nuclei, the chance to have two
visible particle tracks entering into a vertex is negligible. Therefore, the
determination of the track and EM shower directions is crucial. As described
in section 2.3, the gradual increase of the EM shower’s width can be used to
determine its direction. For tracks, the identification of a Bragg peak can be
used to determine the direction. Figure 9 shows the $dQ/dx$ distribution as a
function of the residual range (distance to the end point of a track). The
$dQ/dx$ rise can be used to mark the end point of a track. In addition, two
clearly separated bands, indicated by black and red lines, correspond to
protons and muons/pions. And this can be used to separate protons from
muons/pions.

MicroBooNE data

Figure 9:  $dQ/dx$ vs residual range from MicroBooNE data. Two clearly
separated bands, indicated by black and red lines, represent protons and
muons. The measurement of $dQ/dx$ near the end point of the track can be used
to determine the track direction and particle identification. The band around
$dQ/dx\sim$ 50k e/cm extending to zero residual range represents the muons
exiting the detector. The black and red curves are the predictions of most
probable $dQ/dx$ values for protons and muons, respectively, considering the
recombination effect as reported in Ref. [39]. The yellow curve indicates the
empirical cut to separate protons and muons/pions.

In addition to the track and EM shower’s directional information, the
following information is taken into consideration in determining the neutrino
vertex (primary vertex):

  * •

The position of the vertex. For example, the neutrino vertex in an interaction
cluster tends to be more upstream than any other activity near the neutrino
beam direction. On the other hand, a vertex at the detector boundary is less
likely to be the neutrino vertex.

  * •

The number of directly connected tracks and EM showers. Statistically, more
tracks and showers are connected to the neutrino vertex than secondary
vertices.

  * •

A set of connectivity rules for the activities (tracks or showers) near the
vertex. For example, there should not be two tracks going into the same
vertex. There should not be one EM shower going into a vertex and a track
coming out of the same vertex. In addition, the situation that one track comes
in and another track goes backward with a large angle is not preferred.

  * •

Some vertex candidates can be excluded by the local event topology. For
example, vertices inside an EM shower and vertices from a delta-ray emerging
from a straight muon track can be safely excluded. For details, please see
Ref. [40].

In practice, each piece of information is summarized into a score. The vertex
candidate with the highest score is determined to be the primary neutrino
vertex. Given this method is surpassed by the Deep-learning neutrino vertex
determination (discussed in Sec. 2.5), the technical details of the score
evaluation are skipped.

MicroBooNE dataMicroBooNE data

Figure 10:  Illustration of the primary neutrino vertex determination. (a) the
neutrino vertex is determined to be inside the main TPC cluster consisting of
multiple tracks. (b) the neutrino vertex is determined to be at a short proton
where two separated EM showers point back toward. This is a candidate
NC$\pi^{0}$ production event. The reconstructed particle flow information is
also displayed (Refer to the caption of figure 4 for more details).

Using the strategies above, one neutrino vertex candidate can be identified in
each TPC cluster. Given there could be multiple TPC clusters in each neutrino
event, another algorithm is used to select the primary interaction vertex
(neutrino vertex) from all vertex candidates, since the primary neutrino
vertex may be outside the main TPC cluster (the TPC cluster most closely
matched to the observed light pattern). Therefore, we utilize the position of
each candidate vertex and the size and direction of its associated TPC cluster
to select primary vertices among candidates in all TPC clusters. Figure 10a
shows an example case where the neutrino vertex is inside the main TPC cluster
consisting of multiple tracks. Figure 10b shows an example case where the
neutrino vertex is located on a short proton track with two separated EM
showers (reconstructed to be from a neutral pion decay) pointing back toward
it.

The performance of the primary neutrino vertex identification with these
traditional pattern recognition techniques has limitations. Among
$\nu_{\mu}$CC events ($\nu_{\mu}$ charged-current interactions), about 70% of
neutrino vertices are correctly determined, where a reconstructed vertex
position is defined to be correct if it lies within 1 cm of the true neutrino
vertex position. Among $\nu_{e}$CC events ($\nu_{e}$ charged-current
interactions), about 50% of neutrino interaction vertices are correctly
identified. We describe further improvements of the neutrino vertex
identification using deep learning techniques in section 2.5.

###  2.5 Neutrino Vertex Identification with a Regressional Segmentation
Network

While traditional pattern recognition techniques provide reasonable
performance in identifying the primary neutrino interaction vertex (section
2.4), an improved performance is desired to enhance the selection efficiency
of $\nu_{e}$ and $\nu_{\mu}$ charged-current interactions. We use a Deep
Neural Network (DNN) on the reconstructed 3D space points and vertex
candidates to further enhance the primary neutrino interaction vertex
identification performance.

This neutrino vertex identification problem can be considered as a special
case of an object detection task in computer vision. Object detection usually
takes 3 steps: region proposal, object type identification, and object
segmentation [41]. In the vertex identification case, we are targeting only
one type of object: the neutrino vertex and the region of the object, so we
focus on the region proposal step. We use a customized regressional
segmentation network to predict the distance map of each voxel (a 3D pixel) to
the neutrino vertex, which improves the training efficiency compared to only
predicting whether a voxel is a vertex or not. We use SparseConvNet [42, 43],
which makes the model training and inferencing (the process of using a trained
DNN model to make predictions against previously unseen data) much faster and
more scalable. Similar techniques were used in other MicroBooNE analyses [44].
The inferencing utilities are available in the Wire-Cell software package
[45].

Figure 11: Illustration of the Sparse U-Net network structure from
SparseConvNet [42, 43]. The bottom diagram shows a level 2 Sparse U-Net.

The key part of our regressional segmentation network is a Sparse U-Net [46]
which extracts a feature vector for each pixel or voxel. Figure 11 shows
details of a level 2 Sparse U-Net. One can add more levels by iteratively
inserting more “Concat-Join” blocks. We use level 5 in this paper. More
implementation details can be found in [47]. The SparseConvNet works on
sparsified 2D or 3D images, which consist of a list of pixels (2D) or voxels
(3D). The Wire-Cell reconstructed 3D points are placed into voxels, which is a
cube with 0.5 cm in each of three dimensions. If multiple 3D points fall into
the same voxel, the values are averaged. The SparseConvNet takes two tensors
as input. One is a coordinate tensor (integer type) with position information
for each voxel. The other one is a feature tensor (float type) with the charge
information. The output of the SparseConvNet is also a list of features for
each voxel, for now we extract only one feature which we call confidence value
related to the distance between a point and the truth vertex. The truth label
for this confidence value is calculated before voxelization, and the equation
used is:

$Conf_{\text{truth}}=\text{exp}\left(-\frac{\lVert\vec{x}-\vec{v}_{\text{truth}}\rVert^{2}}{2\sigma^{2}}\right),$
(2.2)

where $Conf_{\text{truth}}$ is the truth label of the confidence value;
$\vec{x}$ and $\vec{v}_{\text{truth}}$ are the reconstructed 3D charge and
truth vertex positions; $\sigma$ is a regularization parameter (1 cm is used).
Confidence values from multiple voxels (pixels) form a “Confidence Map” [48].
Figure 12 shows an example of the input (charge) and label before
voxelization.

Figure 12: Example of input (charge) and label (confidence value) images
before voxelization (showing a 2D projection). The color represents
reconstructed charge in the left panel and confidence value in the right
panel. From the right panel, we can see the confidence value becomes larger
when getting closer to the vertex as defined in Eq. 2.2.

The dataset used for training the network consists of 48k $\nu_{e}$CC events
simulated through the MicroBooNE simulation chain and Wire-Cell
reconstruction. The dataset for validating the network performance consists of
4k $\nu_{e}$CC events. The dataset for testing the network performance
consists of 4k $\nu_{e}$CC and 4k $\nu_{\mu}$CC events. The total dataset size
is chosen based on the available official MicroBooNE simulation production
sample size. Then we try to maximize the training set size while keeping
enough for validation and testing. The choice of focusing on the $\nu_{e}$CC
event topology is motivated by the need to improve $\nu_{e}$CC event
selection, which is much more challenging than the $\nu_{\mu}$CC event
selection. Nevertheless, the final testing results of the network show an
improvement over the performance of the traditional algorithm (described in
section 2.4) in both $\nu_{e}$CC and $\nu_{\mu}$CC events.

Machine learning model parameters are estimated with the help of loss
functions [49]. The loss function we used is:

$\mathcal{L}=\frac{1}{N_{voxel}}\sum^{N_{voxel}}\left\|Conf_{pred}-Conf_{truth}\right\|^{2}.$
(2.3)

The optimizer used is Adam [50]; the learning rate decays after each epoch as
$l_{0}\cdot exp(-n\cdot l_{d})$, where $l_{0}$ is $10^{-5}$ and $l_{d}$ is
0.05 and $n$ is the epoch number. Figure 13 shows the training progress, epoch
average loss, and hit rate as a function of epoch number for the $\nu_{e}$CC
training and validation samples. “Hit rate” here is defined as the probability
that the best predicted voxel is within 1 cm of the truth voxel (the voxel
closest to the truth vertex). From figure 13, we can see that: i) the solution
phase space is not fully covered by the training sample according to the gaps
between the train and validation curves (we also observed the gaps getting
smaller when increasing training samples); ii) there are signs of over-
training; and iii) validation performance stopped improving after epoch 24.
The results shown in this paper are based on the model at epoch 24.

MicroBooNE simulationMicroBooNE simulation

Figure 13: Left: epoch average loss as a function of epoch number for training
(red) and validation (blue) samples; right: probability that the best
predicted voxel is within 1 cm of the best truth voxel (voxel closest to the
truth vertex).

To accommodate existing Wire-Cell reconstruction/pattern recognition
algorithms, we select the neutrino vertex from the candidates provided by the
traditional pattern recognition algorithms. We call this vertex the hybrid
vertex. Figure 14 shows an event display to demonstrate finding the hybrid
vertex based on a DNN vertex. The decision proceeds as follows:

  * •

Find the voxel with the highest confidence value, which we call the “DNN
Vertex”.

  * •

Convert from voxel index back to continuous space coordinate.

  * •

From the list of traditional vertex candidates, find the closest one to this
DNN vertex in the coordinate space. (As described in section 2.4, the
traditional neutrino vertex is also selected from these traditional
candidates.)

  * •

Check if the distance between this traditional candidate and the DNN vertex is
smaller than a threshold, currently 2 cm. Note this “2 cm” threshold used in
the DNN-traditional vertex matching should not be confused with the “1 cm”
reco-truth matching threshold used in the vertex finding algorithm performance
evaluations.

  * •

If the previous distance check is good ( $<$ 2 cm) use this traditional
candidate as the “hybrid vertex”; if not, use the “reconstructed traditional
neutrino vertex” (section 2.4) as the “hybrid vertex”.

Figure 15 shows the performance of the trained model on the $\nu_{e}$CC test
samples. Compared to the performance of traditional pattern recognition
algorithm, the hybrid algorithm leads to a relative 30% improvement in
identifying the neutrino vertex. At low distance cut ($\sim$0.5 cm) between
the truth and the reconstructed neutrino vertex, the traditional algorithm
outperforms the DNN algorithm, because of the finite voxel size (0.5 cm in
each dimension) and the smearing of the reconstructed 3D space points. Figure
16 shows the performance of the current best model on the $\nu_{\mu}$CC test
samples. The hybrid algorithm leads to a relative 10% improvement in
identifying the neutrino vertex, which is achieved on the $\nu_{\mu}$CC
samples not used in the training. Furthermore, while the performance of the
hybrid algorithm is slightly worse than that of DNN for $\nu_{e}$CC events, it
is slightly better for $\nu_{\mu}$CC. This is because the performance of
traditional algorithm for $\nu_{e}$CC is worse than that for $\nu_{\mu}$CC. As
mentioned before, the hybrid vertex is used as the main result in the current
workflow to accommodate existing Wire-Cell reconstruction/pattern recognition
algorithms. In the future, it is likely we will revise the downstream
reconstructions directly based on the DNN vertexing results then we do not
need to use the “hybrid” one.

MicroBooNE simulation

Figure 14: An event display to demonstrate finding the hybrid vertex based on
a DNN vertex. Each filled circle represents a voxelized 3D charge projected on
the Y-Z plane. In this example, the hybrid vertex (indicated by the black
arrow) is different from the DNN one, which is better in this case.

MicroBooNE simulation

Figure 15: $\nu_{e}$CC neutrino vertex identification efficiency as a function
of the maximum distance between the reconstructed and the truth, i.e., the
numerator of the efficiency is defined as the number of events with a neutrino
vertex reconstructed within $X$ cm of the truth, where $X$ is the horizontal
coordinate of the plot. Three algorithms are compared: Traditional, DNN and
Hybrid.

MicroBooNE simulation

Figure 16: $\nu_{\mu}$CC neutrino vertex identification efficiency as a
function of the maximum distance between the reconstructed and the truth,
i.e., the numerator of the efficiency is defined as the number of events with
a neutrino vertex reconstructed within $X$ cm of the truth, where $X$ is the
horizontal coordinate of the plot. Three algorithms are compared: Traditional,
DNN and Hybrid.

###  2.6 Evaluation of the Neutrino Vertex Identification Performance

The efficiency of the neutrino vertex identification plays a crucial role in
$\nu_{\mu}$CC and $\nu_{e}$CC event selections. Therefore, we performed
evaluations of the neutrino, primary muon, and primary EM shower vertex
reconstruction. Pandora [51] is another state-of-the-art traditional pattern
recognition technique333Some new developments of Pandora reconstruction start
to involve deep-learning algorithms, but such developments were not included
in this comparison and are yet to be used in the MicroBooNE experiment. that
is also used in MicroBooNE. We compare the performance of the Wire-Cell vertex
identification with Pandora as a benchmark.

Two types of overlay MC samples are used to evaluate the neutrino vertexing
performance. BNB $\nu$ overlay MC, which simulates neutrino interactions based
on the BNB neutrino flux of which 99.5% are $\nu_{\mu}$’s, is used to evaluate
the performance for $\nu_{\mu}$CC and NC events. BNB intrinsic $\nu_{e}$
overlay MC, which is a $\nu_{e}$-enriched MC sample only simulating the
electron neutrino interactions based on the $\nu_{e}/\bar{\nu}_{e}$ component
in the BNB neutrino flux, is used to evaluate the performance for $\nu_{e}$CC
events.

The left panels of figure 17, 18, and 19 show distributions of the distance
between the reconstructed neutrino vertex and true neutrino vertex for
charged-current events in BNB $\nu$ overlay, neutral-current events in BNB
$\nu$ overlay, and charged-current events in BNB intrinsic $\nu_{e}$ overlay
samples, respectively. Percentages of events with a neutrino vertex
reconstructed within 1 cm of the truth for these three samples are 70.3%,
49.3% and 65.2%; within within 5 cm are 84.8%, 63.5% and 80.0%, respectively.
In the right panels of these figures we calculate the percentage of events
with the reco-truth vertex distance less than 1 cm.

\begin{overpic}[width=212.47617pt]{figs/pf_eval_2/nu_vtx_distance.png}
\put(42.0,64.0){\text{\small MicroBooNE simulation}} \end{overpic}

\begin{overpic}[width=212.47617pt]{figs/pf_eval_2/nu_vtx_eff.png}
\put(42.0,64.0){\text{\small MicroBooNE simulation}} \end{overpic}

Figure 17: Neutrino vertex reconstruction of the $\nu$ overlay sample,
charged-current interaction, reconstructed with Wire-Cell. Left: distance
between the reconstructed and truth neutrino vertex; Right: efficiency for
events with vertex reconstruction position within 1 cm of the truth, as a
function of truth neutrino energy. The error bars represent the statistical
uncertainties.

\begin{overpic}[width=212.47617pt]{figs/pf_eval_2/nuNC_vtx_distance.png}
\put(42.0,64.0){\text{\small MicroBooNE simulation}} \end{overpic}

\begin{overpic}[width=212.47617pt]{figs/pf_eval_2/nuNC_vtx_eff.png}
\put(42.0,64.0){\text{\small MicroBooNE simulation}} \end{overpic}

Figure 18: Neutrino vertex reconstruction of the $\nu$ overlay sample,
neutral-current interaction, reconstructed with Wire-Cell. Left: distance
between the reconstructed and truth neutrino vertex; Right: efficiency for
events with vertex reconstruction position within 1 cm of the truth, as a
function of truth neutrino energy. The error bars represent the statistical
uncertainties.

\begin{overpic}[width=212.47617pt]{figs/pf_eval_2/nue_vtx_distance.png}
\put(42.0,64.0){\text{\small MicroBooNE simulation}} \end{overpic}

\begin{overpic}[width=212.47617pt]{figs/pf_eval_2/nue_vtx_eff.png}
\put(42.0,64.0){\text{\small MicroBooNE simulation}} \end{overpic}

Figure 19: Neutrino vertex reconstruction of the intrinsic $\nu_{e}$ overlay sample, charged-current interaction, reconstructed with Wire-Cell. Left: distance between the reconstructed and truth neutrino vertex; Right: efficiency for events with vertex reconstruction position within 1 cm of the truth, as a function of truth neutrino energy. The error bars represent the statistical uncertainties. Table 1: The performance with the distance between reconstructed neutrino vertex and true neutrino vertex less than 1 cm. The performance are estimated with a BNB $\nu$ overlay sample, while charged-current $\nu_{e}$ event efficiency are estimated with a BNB intrinsic $\nu_{e}$ overlay sample.  | Pandora reconstruction | Wire-Cell Traditional | Wire-Cell DNN Hybrid
---|---|---|---
Charged-current $\nu_{\mu}$ | 70.3% | 67.1% | 72.9%
Neutral-current | 49.3% | 50.6% | 52.1%
Charged-current $\nu_{e}$ | 65.2% | 50.1% | 65.8%

Table 1 summarizes the performance comparison between Wire-Cell and Pandora
[51] reconstructions. In order to make a fair performance comparison focusing
on reconstruction algorithms, the same input events, with cosmic-rays cleaned
up and unresponsive detector areas recovered using the upstream Wire-Cell
imaging and clustering algorithms [18], are feed into both Wire-Cell and
Pandora reconstruction workflows. Except for the inputs, the Pandora
configuration is the same as used in Ref. [52]. For charged-current events in
the BNB $\nu$ overlay sample, Pandora reconstruction results in 70.3% good
reconstructed vertices (within 1 cm of the true vertex), and Wire-Cell input
plus Wire-Cell reconstruction leads to 67.1% (without DNN Vertexing) and 72.9%
(with DNN Vertexing) good reconstructed vertices. For the neutral-current
case, the Pandora reconstruction has 49.3% efficiency while the Wire-Cell
reconstruction has 50.6% (without DNN Vertexing) and 52.1% (with DNN
Vertexing). Neutrino vertex identification is generally harder for neutral-
current interactions due to the absence of primary charged particles
connecting to the neutrino interaction vertex and more complicated event
topology. When the BNB intrinsic $\nu_{e}$ sample is used, Pandora has 65.2%
efficiency while Wire-Cell has 50.1% (without DNN Vertexing) and 65.8% (with
DNN Vertexing) efficiency. The DNN Vertexing model is trained on a $\nu_{e}$
charged-current sample and boosts the Wire-Cell vertexing performance by about
30% for samples of the same type while it does not degrade the performance on
samples of other types. After using the DNN Vertexing, the Wire-Cell vertexing
performance is comparable with Pandora reconstruction for all samples tested.

Figure 20: 2D distribution of reconstructed primary shower vertex-true
neutrino vertex difference vs. reconstructed neutrino vertex-true neutrino
vertex of BNB intrinsic $\nu_{e}$ sample, with reconstructed with Pandora
(left) and with Wire-Cell (right). The two 2D histograms contain the same
number of events. For Wire-Cell, the distribution is more concentrated along
the diagonal line, which shows the reconstructed neutrino vertices and EM
shower vertices are better aligned. Red boxes in the lower left corner
indicates events with the position difference less than 1 cm. Note that the
z-axis is in log scale.

As noted earlier, the BNB intrinsic $\nu_{e}$ events can be used for the
shower vertex reconstruction study as there are only $\nu_{e}$CC interactions
in these events, which are supposed to have an EM shower attached to each
neutrino vertex. Figure 20 is a 2D distribution of distance between the
reconstructed primary shower vertex and the true neutrino vertex vs the
distance between the reconstructed neutrino vertex and the true neutrino
vertex. For the Pandora reconstruction, the percentage of events that have
both the reconstructed primary shower vertex and reconstructed neutrino vertex
within 1 cm of the true neutrino vertex is 30.8%, while the Wire-Cell
reconstruction achieves 62.0%. This is because that the hybrid Wire-Cell
neutrino vertex is selected from a list of candidates including the shower
vertices. So for $\nu_{e}$CC interactions, when the neutrino vertex is
identified correctly, there is a high chance that it is exactly the same
position of a shower vertex. While the Pandora reconstruction takes a
different approach that the shower vertices are reconstructed more
independently from the neutrino vertex, so that the analyzers would have
additional information to use. The better aligned neutrino and shower vertex
should enhance the capability to differentiate primary electrons from photons
and provide better a better capability in identifying the gap between the
photon EM shower vertex and the neutrino vertex. Furthermore, compared to the
65.8% efficiency in correctly identifying neutrino vertex, the lower
efficiency number when requiring both a reconstructed primary shower vertex
and a reconstructed neutrino vertex within 1 cm of the true neutrino vertex
essentially gives a measure of the reconstruction efficiency for the primary
EM shower.

In this section, we describe various pattern recognition techniques towards
neutrino vertex identification. 65.8% and 72.9% efficiencies have been
achieved in the Wire-Cell reconstruction for charged-current $\nu_{e}$ and
$\nu_{\mu}$ interactions respectively for the MicroBooNE experiment.

##  3 Particle Flow Tree Reconstruction after Neutrino Vertex Identification

In the first iteration of the Wire-Cell pattern recognition (section 2), we
identified the neutrino vertex as well as track segments and EM shower
segments. In this section, we describe the reconstruction of complete EM
showers and the particle flow tree including $\pi^{0}$ reconstruction.

###  3.1 Overall Electromagnetic Shower Clustering

As explained in section 1 and in ref. [22], “a TPC cluster”, which consists of
TPC activities that are connected, is used to organize (or divide) 3D and 2D
data. Within a TPC cluster, track segments are identified as outlined in
section 2.1. Such a data structure is not sufficient to describe
electromagnetic (EM) showers, which typically consist of many separated TPC
clusters. Therefore, once the primary neutrino vertex is identified (section
2.4), a dedicated EM shower clustering algorithm is applied. The clustering
algorithm considers directional and distance information. First, the EM shower
candidates within the TPC cluster that contains the primary neutrino vertex
are examined. Second, along the direction of each EM shower candidate, the
separated pieces within a certain distance (80 cm) and a certain angle (15
degrees) are clustered together. Finally, isolated EM shower candidates are
examined according to their distances to the primary neutrino vertex and the
same EM shower clustering algorithm is performed.

MicroBooNE data

Figure 21: Illustration of EM shower clustering. (a) shows the identified
track segments with different colors in different TPC clusters. (b) shows the
EM showers after clustering. The green lines illustrate the associations. The
red lines connect the start and end points of each EM shower. The numbers of
EM showers are also displayed with the particle flow information (Refer to the
caption of figure 4 for more details).

Figure 21 shows an example of the EM clustering algorithm. Here, the first,
second, third, and fifth EM showers are isolated EM showers. The fourth EM
shower is a Michel electron at the end of a stopped muon. The sixth EM shower
is connected to the primary neutrino vertex. It is expected to be the result
of a low-energy stopped charged pion, which is invisible because of the
detector spacial resolution.

###  3.2 Neutral Pion Reconstruction

With EM showers defined, one can reconstruct neutral pions which decay to two
photons with a $\sim$99% branching ratio. The reconstruction of neutral pions
is useful in validating the energy scale for the EM shower energy
reconstruction and is crucial for reconstructing the neutrino energy and in
selecting events with neutral pions in both charged-current and neutral-
current channels, which can be used to constrain backgrounds in single EM
shower searches, for example the $\nu_{e}$CC selection or a single photon
selection.

MicroBooNE dataMicroBooNE data

Figure 22: Examples of reconstructed neutral pions. (a): a neutral pion is
reconstructed at an invariant mass of 142 MeV/c2 without any TPC activity at
its generation location. (b): a neutral pion is reconstructed at an invariant
mass of 122 MeV/c2 from the primary neutrino vertex. Separate photons are
labeled with the corresponding numbers in the particle flow diagram. The color
coding in both (a) and (b) are to separate different reconstructed particles
similar as used in panel (c) of figure 4. The reconstructed particle flow
information is also displayed (Refer to the caption of figure 4 for more
details).

For a $\pi^{0}$ produced in a neutral current interaction (or outside the TPC
active volume), there may not be any other TPC information to identify the
$\pi^{0}$ vertex. In these cases, the reconstructed primary neutrino vertex is
erroneously connected to an EM shower. A dedicated algorithm is therefore
implemented to find a $\pi^{0}$ with a displaced vertex. For each EM shower, a
PCA is performed to find the EM shower’s primary axis (i.e. its direction).
Given any two EM showers with calculated primary axes, we find two points, one
on each of them. These points are the closest to the opposite primary axis.
The midpoint of the pair is then identified and labeled as the displaced
vertex. Once the displaced vertex is determined, the direction of each EM
shower is redefined with respect to that vertex for the invariant mass
calculation. Figure 22a shows such an example.

For each vertex with two or more associated EM showers, a $\pi^{0}$ is
reconstructed with two separate strategies. The first is to find the pair of
photons (above a certain energy threshold) that reconstruct as a $\pi^{0}$
with an invariant mass closest to the $\pi^{0}$ mass. The second is to find
the pair of photons that have the highest total energy. Both strategies are
implemented and applied to all events with two or more photons reconstructed
from the same vertex. The first strategy is used to construct the particle
flow and is generally good for rejecting backgrounds of $\nu_{e}$CC events, as
incorrectly-paired photons would generally result in an incorrect invariant
$\pi^{0}$ mass. The second strategy is used to select $\pi^{0}$ events, which
is useful in calibrating the electromagnetic energy reconstruction, given that
it has no bias in the invariant mass reconstruction. Figure 22 shows two
examples of $\pi^{0}$ reconstructions. In figure 22b the $\pi^{0}$ vertex is
also associated with other activities. Figure 23 shows the reconstructed
$\pi^{0}$ mass in simulation for the fully contained $\nu_{\mu}$CC $\pi^{0}$
selection. Here, the fully contained events are defined to be events with the
reconstructed TPC activity fully contained within the fiducial volume (3 cm
inside the effective TPC boundary [22], which is the corrected boundary that
takes a space charge effect [31, 53] into account). Compared to the earlier
MicroBooNE results [54], the resolution of the reconstructed $\pi^{0}$ mass is
improved by about 15%.

\begin{overpic}[width=265.37857pt]{figs/pf_eval_2/pi0_mass.png}
\put(52.0,64.0){\text{\small MicroBooNE simulation}} \end{overpic} Figure 23:
Reconstructed $\gamma\gamma$ invariant mass ($\sqrt{2E_{\gamma 1}E_{\gamma
2}\cdot(1-{\rm cos}\theta_{\gamma\gamma})}$) on simulated fully contained
charged-current $\nu_{\mu}\pi^{0}$ events. The tail at the high reconstructed
mass is the result of incorrect association of $\gamma$ candidates.

###  3.3 Reconstruction Performance of Primary Leptons

The reconstruction quality of the primary leptons (muons and electron EM
showers) is essential for both neutrino event flavor tagging and neutrino
energy reconstruction. Therefore, we evaluate the reconstruction efficiency
and angular resolution. The first step of the evaluation is to define truth-
reco matching. We select events with a good reconstructed neutrino vertex
(within 1 cm of the truth vertex), to decouple the evaluation of single
particle reconstruction and neutrino vertex reconstruction. Then within the
selected events, using the simulation truth information, we find all the
leading (with largest energy) primary (from neutrino interaction vertex) muons
or electrons originating inside a fiducial volume (20 cm from the active
volume boundary) as the truth target. The fiducial volume requirement ensures
that the particles deposit enough visible information for reconstruction in
the detector. With the particle flow information reconstructed from the
aforementioned pattern recognition algorithms, we find all the leading primary
muons or electrons. If those muons or electrons have a reconstructed vertex
within 1 cm of the truth counterpart, we count this as a truth-reco matching
in the following evaluations in this subsection.

Figure 24 shows the reconstruction efficiencies as a function of the truth
particle energy for leading primary muons and electrons. These efficiencies
are related to the performances of algorithms introduced in this section as
well as algorithms from section 2, i.e., the track segment finding, the
trajectory and $dQ/dx$ fitting, the track-shower separation, the PID and the
neutrino vertex identification. The denominators are the number of the truth
targets aforementioned and the numerators are the number of matched
reconstructed counterpart. The muon reconstruction efficiency rises to its
plateau at about 85% to 95% above 300 MeV. The electron reconstruction
efficiency plateaus at about 90% and slightly decreases above 1.5 GeV. The
slight reduction in efficiency at high energy is the result of incorrect
reconstruction for isochronous event topologies (close to parallel to the wire
planes), where a large amount of ionization electrons arrive at the anode
plane at the same time creating ambiguities in the projective wire-plane
readout.

\begin{overpic}[width=212.47617pt]{figs/pf_eval_2/single_muon_eff.png}
\put(42.0,64.0){\text{\small MicroBooNE simulation}} \end{overpic}

\begin{overpic}[width=212.47617pt]{figs/pf_eval_2/single_electron_eff.png}
\put(42.0,64.0){\text{\small MicroBooNE simulation}} \end{overpic}

Figure 24: Single particle reconstruction efficiency for the leading primary
muon (left) and electron (right). The x-axis is the truth particle energy.

Figure 25 shows the resolution of the reconstructed angle as a function of the
truth particle energy for leading primary muons and electrons. Here the
$\theta$ and $\phi$ refer to polar and azimuthal angles defined with respect
to the neutrino beam direction. The truth-reco matching is defined as before.
The angular resolutions for both muon and electron reconstruction are at the
0.1 rad (5.7 degrees) level.

\begin{overpic}[width=212.47617pt]{figs/pf_eval_2/muon_bias_dtheta.png}
\put(32.0,67.0){\text{\small MicroBooNE simulation}} \end{overpic}

\begin{overpic}[width=212.47617pt]{figs/pf_eval_2/muon_bias_dphi.png}
\put(32.0,67.0){\text{\small MicroBooNE simulation}} \end{overpic}

\begin{overpic}[width=212.47617pt]{figs/pf_eval_2/electron_bias_dtheta.png}
\put(32.0,67.0){\text{\small MicroBooNE simulation}} \end{overpic}

\begin{overpic}[width=212.47617pt]{figs/pf_eval_2/electron_bias_dphi.png}
\put(32.0,67.0){\text{\small MicroBooNE simulation}} \end{overpic}

Figure 25: Difference between reconstructed and truth track angle as a
function of truth particle energy for leading primary muons (top) and
electrons (bottom). The left panels are for polar angles and the right panels
are for azimuthal angles. Slice-wise mean and resolution values from Gaussian
fittings are overlaid. The slight bias in the polar angle is because of its
non-negative nature in its reconstruction.

##  4 Neutrino Energy Reconstruction

With a particle flow tree reconstructed from the neutrino vertex, the neutrino
energy can be calculated. There are, in general, three methods to compute the
deposited energy of a charged track or shower:

  * •

Range: the travel range is used to calculate the energy of track-like
particles if they stop inside the detector. The NIST PSTAR database [55] is
used to derive the relation between the range and the kinetic energy of
different particles with different masses.

  * •

Summation of $dE/dx$: the (visible) kinetic energy is estimated by summing up
$dE/dx$ for each piece ($\sim$6 mm) of the track. The energy loss per unit
length $dE/dx$ is converted from the ionization charge per unit length $dQ/dx$
considering the recombination effect when the ionization electrons are
produced. Note that an effective recombination model that takes into account
the overall normalization difference between MicroBooNE data and simulation
charge signals [39] is built by tuning the parameters of the modified box
model from ArgoNeuT [56]. This effective recombination model improves the
agreement with MicroBooNE data; however, it still underestimates the energy by
about 10%.

  * •

Overall charge-energy scaling: while the previous two methods are appropriate
for track-like objects, they are not suitable for EM showers because of the
difficulties in deriving $dQ/dx$ or range. After multiplying 23.6 eV per
ionization pair [57, 58], the energy of EM showers is estimated by applying
another overall scaling factor 2.50 ($=1/0.4$). This factor is derived from a
simulation that includes the bias in the reconstructed charge [18] and the
average recombination factor ($\approx$0.5) of an EM shower in argon. For
$\nu_{e}$CC events, the energy reconstruction of the primary electromagnetic
shower gives a 12% energy resolution with about 2% bias, indicating a good
performance of charge clustering for EM showers (described in section 3.1).

The three methods are combined in the neutrino energy estimation. For a
stopped track longer than 4 cm, range method is used. For short stopped tracks
($<$ 4 cm), the summation of $dE/dx$ method is used, given the range
measurement is expected to have a larger uncertainty since track trajectories
consist of points separated by 0.6 cm. For a long muon, where delta rays
become visible (multiple track segments are part of a reconstructed muon
track), we again use the summation of $dE/dx$ method, given the track length
of delta rays may bias the range calculation of the long muon. For EM showers,
the charge scaling method is used, since track trajectories are not well
defined in this case. The neutrino energy is estimated by summing up the
kinetic energy of all particles in the reconstructed particle flow tree. For
each muon or charged pion or electron, its mass is added to the energy
reconstruction. In addition, an average binding energy of 8.6 MeV per nucleon
in which case an argon-40 nucleus is completely disassembled is added for each
identified proton in the reconstructed particle flow. These protons may be
produced at the primary neutrino interaction or secondary interactions (e.g.
produced by a neutron).

\begin{overpic}[width=205.88908pt]{figs/dl_vtx_2/canv_nue_1.png}
\put(29.0,70.0){\text{\small MicroBooNE simulation}} \end{overpic}

\begin{overpic}[width=205.88908pt]{figs/dl_vtx_2/canv_nue_2.png}
\put(29.0,70.0){\text{\small MicroBooNE simulation}} \end{overpic}

\begin{overpic}[width=205.88908pt]{figs/dl_vtx_2/canv_nue_3.png}
\put(29.0,70.0){\text{\small MicroBooNE simulation}} \end{overpic}

Figure 26:  Energy reconstruction for selected fully contained $\nu_{e}$CC
candidates: reconstructed visible energy (top left), reconstructed neutrino
energy (top right), and ratio of reconstructed neutrino energy to truth
(bottom). In the bottom panel, the peak values and the corresponding
resolutions (asymmetric) for each true energy bin are plotted as well.

\begin{overpic}[width=205.88908pt]{figs/dl_vtx_2/canv_aa_1.png}
\put(29.0,70.0){\text{\small MicroBooNE simulation}} \end{overpic}

\begin{overpic}[width=205.88908pt]{figs/dl_vtx_2/canv_aa_2.png}
\put(29.0,70.0){\text{\small MicroBooNE simulation}} \end{overpic}

\begin{overpic}[width=205.88908pt]{figs/dl_vtx_2/canv_aa_3.png}
\put(29.0,70.0){\text{\small MicroBooNE simulation}} \end{overpic}

Figure 27:  Energy reconstruction for selected fully contained $\nu_{\mu}$CC
candidates: reconstructed visible energy (top left), reconstructed neutrino
energy (top right), and ratio of reconstructed neutrino energy to truth
(bottom). In the bottom panel, the peak values and the corresponding
resolutions (asymmetric) for each true energy bin are plotted as well.

Figures 26 and 27 summarize the neutrino energy reconstruction performance for
fully contained $\nu_{e}$CC and $\nu_{\mu}$CC interactions. The top left
panels show the reconstructed visible energy and the top right panels show the
reconstructed neutrino energy. We can see that the reconstructed neutrino
energy is closer to the diagonal line, meaning that the bias is largely
corrected from the reconstructed visible energy. The bottom panels show that
the reconstructed neutrino energy resolution is 10% to 15% for $\nu_{e}$CC
events and 15-20% for $\nu_{\mu}$CC events. At a truth neutrino energy of 800
MeV, the reconstructed neutrino energy resolution is 15% for $\nu_{e}$CC and
20% for $\nu_{\mu}$CC. The bias of the reconstructed neutrino energy relative
to the true neutrino energy is 7% and 10% on average for BNB $\nu_{e}$CC and
$\nu_{\mu}$CC interactions, respectively. The bias in the high energy region,
e.g. true neutrino energy greater than 1 GeV, is obvious for the selected
$\nu_{\mu}$CC sample. This is because the high energy $\nu_{\mu}$CC
interactions after the selection, in particular with the full containment
requirement, are more likely 1) to have high energy transfer (difference of
incoming neutrino energy and outgoing lepton energy) which results in a
sizable amount of invisible energy carried away by final state hadrons
especially neutrons or other neutral particles; and 2) to have high energy
muons (long muons with significant delta-rays), as discussed above, the
reconstructed energies of which have a relatively large bias because of the
usage of an imperfect calorimetric reconstruction method instead of the range
method.

In real data analysis [23, 24], the various methods used to reconstruct
neutrino energy are validated. The summation of $dE/dx$ method is validated by
comparing the $dQ/dx$ vs. residual range for the stopped muons and protons
between data and MC. The energy reconstruction of EM showers is validated by
comparing the reconstructed $\pi^{0}$ mass between data and MC. The
reconstructed hadronic energy has also been validated by comparing the lepton-
kinematics-constrained MC prediction and real data measurement, in which case
the systematic effect, in particular from the neutrino-argon interaction
modeling, is largely reduced.

##  5 Summary

To summarize, we describe a set of pattern recognition algorithms to
reconstruct neutrino events based on the Wire-Cell reconstruction paradigm.
These algorithms perform well on the tasks of neutrino vertex identification,
single particle reconstruction, and neutrino energy reconstruction, which are
essential to achieve high-performance inclusive $\nu_{e}$CC and $\nu_{\mu}$CC
event selections [23, 24]. Evaluated using the current state-of-the-art
MicroBooNE MC simulation, we achieve 65.8% and 72.9% neutrino vertex
identification efficiencies for $\nu_{e}$ and $\nu_{\mu}$ charged-current
interactions, respectively. These neutrino vertex identification efficiencies
are comparable to those achieved with the Pandora reconstruction paradigm
[51]. In addition, 80-90% single particle reconstruction efficiencies for
primary leptons and 15-20% reconstructed neutrino energy resolutions are
achieved for these charged-current neutrino interactions. The achievements
reported in this paper build a solid foundation for downstream physics
analyses, such as the search for new physics with electron neutrinos [23] and
measurement of charged-current muon neutrino interaction cross section on
Argon [24]. Regarding the performance comparison between the Wire-Cell
reconstruction and other reconstruction paradigms in MicroBooNE, more
information can be found in Ref. [59], which summarizes the searches for an
excess of electron neutrino interactions in MicroBooNE using multiple final
state topologies and multiple event reconstruction paradigms. In addition to
the search in the more demanding inclusive $\nu_{e}$CC channel using Wire-Cell
[23], searches in exclusive quasi-elastic $\nu_{e}$CC channel [60] and semi-
inclusive $\nu_{e}$CC channel without pions in the final state [52] are
performed with the Deep-Learning based reconstruction [61, 44, 62] and the
Pandora reconstruction [51], respectively. The current Wire-Cell pattern
recognition relies on many traditional (or non-Deep-Learning) algorithms, in
order to mitigate the risk of amplifying any discrepancies between simulation
and real data. As the simulation for LArTPCs becomes more mature, we are
expecting to adopt more Deep-Learning based algorithms in the Wire-Cell
reconstruction paradigm to further enhance the overall performance.

## Acknowledgments

This document was prepared by the MicroBooNE collaboration using the resources
of the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of
Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi
Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359.
MicroBooNE is supported by the following: the U.S. Department of Energy,
Office of Science, Offices of High Energy Physics and Nuclear Physics; the
U.S. National Science Foundation; the Swiss National Science Foundation; the
Science and Technology Facilities Council (STFC), part of the United Kingdom
Research and Innovation; the Royal Society (United Kingdom); and The European
Union’s Horizon 2020 Marie Sklodowska-Curie Actions. Additional support for
the laser calibration system and cosmic ray tagger was provided by the Albert
Einstein Center for Fundamental Physics, Bern, Switzerland. We also
acknowledge the contributions of technical and scientific staff to the design,
construction, and operation of the MicroBooNE detector as well as the
contributions of past collaborators to the development of MicroBooNE analyses,
without whom this work would not have been possible.

## References

  * [1] C. Rubbia.  The liquid argon time projection chamber: A new concept for neutrino detector.  CERN-EP/77-08, 1977.
  * [2] H. H. Chen, P. E. Condon, B. C. Barish, and F. J. Sciulli.  A Neutrino detector sensitive to rare processes. I. A Study of neutrino electron reactions.  FERMILAB-PROPOSAL-0496, 1976.
  * [3] W. J. Willis and V. Radeka.  Liquid argon ionization chambers as total absorption detector.  Nuclear Instruments and Methods, 120:221, 1974.
  * [4] D. R. Nygren.  The Time Projection Chamber: A New 4 pi Detector for Charged Particles.  eConf, C740805 (1974):58, PEP-0144.
  * [5] S. Amerio et al.  Design, construction and tests of the ICARUS T600 detector.  Nucl.Instrum.Meth., A527:329–410, 2004.
  * [6] C. Anderson et al.  First Measurements of Inclusive Muon Neutrino Charged Current Differential Cross Sections on Argon.  Phys. Rev. Lett., 108:161802, 2012, 1111.0103.
  * [7] R. Acciarri et al.  Design and Construction of the MicroBooNE Detector.  JINST, 12(02):P02017, 2017, 1612.05824.
  * [8] F. Cavanna, M. Kordosky, J. Raaf, and B. Rebel.  LArIAT: Liquid Argon In A Testbeam.  2014, 1406.5560.
  * [9] H. Berns et al.  The CAPTAIN Detector and Physics Program.  In Proceedings, 2013 Community Summer Study on the Future of U.S. Particle Physics: Snowmass on the Mississippi (CSS2013): Minneapolis, MN, USA, July 29-August 6, 2013, 2013, 1309.1740.
  * [10] A. A. Aguilar-Arevalo et al.  A Combined $\nu_{\mu}\rightarrow\nu_{e}$ and $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$ Oscillation Analysis of the MiniBooNE Excesses.  2012, 1207.4809.
  * [11] M. Antonello et al.  A Proposal for a Three Detector Short-Baseline Neutrino Oscillation Program in the Fermilab Booster Neutrino Beam.  arXiv:1503.01520, 2015, 1503.01520.
  * [12] Babak Abi et al.  Volume I. Introduction to DUNE.  JINST, 15(08):T08008, 2020, 2002.02967.
  * [13] Veljko Radeka et al.  Cold electronics for ’Giant’ Liquid Argon Time Projection Chambers.  J. Phys. Conf. Ser., 308:012021, 2011.
  * [14] R. Acciarri et al.  Noise Characterization and Filtering in the MicroBooNE Liquid Argon TPC.  JINST, 12(08):P08003, 2017, 1705.07341.
  * [15] C. Adams et al.  Ionization electron signal processing in single phase LArTPCs. Part I. Algorithm Description and quantitative evaluation with MicroBooNE simulation.  JINST, 13(07):P07006, 2018, 1802.08709.
  * [16] C. Adams et al.  Ionization electron signal processing in single phase LArTPCs. Part II. Data/simulation comparison and performance in MicroBooNE.  JINST, 13(07):P07007, 2018, 1804.02583.
  * [17] Xin Qian, Chao Zhang, Brett Viren, and Milind Diwan.  Three-dimensional Imaging for Large LArTPCs.  JINST, 13(05):P05032, 2018, 1803.04850.
  * [18] P. Abratenko et al.  Neutrino event selection in the MicroBooNE liquid argon time projection chamber using Wire-Cell 3D imaging, clustering, and charge-light matching.  JINST, 16(06):P06043, 2021, 2011.01375.
  * [19] E. J. Candès, J. K. Romberg, and T. Tao.  Stable signal recovery from incomplete and inaccurate measurements.  Communications on Pure and Applied Mathematics, 59:1207–1223, 2006, math/0503066.
  * [20] Hastie T Friedman J and Tibshirani R.  Journal of statistical software, 33:1, 2010.
  * [21] P. Abratenko et al.  High-performance Generic Neutrino Detection in a LArTPC near the Earth’s Surface with the MicroBooNE Detector.  12 2020, 2012.07928.
  * [22] P. Abratenko et al.  Cosmic Ray Background Rejection with Wire-Cell LArTPC Event Reconstruction in the MicroBooNE Detector.  Phys. Rev. Applied, 15(6):064071, 2021, 2101.05076.
  * [23] P. Abratenko et al.  Search for an anomalous excess of inclusive charged-current $\nu_{e}$ interactions in the MicroBooNE experiment using Wire-Cell reconstruction.  10 2021, 2110.13978.
  * [24] P. Abratenko et al.  First Measurement of Energy-dependent Inclusive Muon Neutrino Charged-Current Cross Sections on Argon with the MicroBooNE Detector.  10 2021, 2110.14023.
  * [25] A.A. Aguilar-Arevalo et al.  The Neutrino Flux prediction at MiniBooNE.  Phys. Rev. D, 79:072002, 2009, 0806.1449.
  * [26] Luis Alvarez-Ruso et al.  Recent highlights from GENIE v3.  6 2021, 2106.09381.
  * [27] P. Abratenko et al.  New Theory-driven GENIE Tune for MicroBooNE.  10 2021, 2110.14028.
  * [28] S. Agostinelli et al.  GEANT4–a simulation toolkit.  Nucl. Instrum. Meth. A, 506:250–303, 2003.
  * [29] Michal Jaskolski and Mariusz Wojcik.  Electron Recombination in Ionized Liquid Argon: A Computational Approach Based on Realistic Models of Electron Transport and Reactions.  J. Phys. Chem. A, 115:4317–4325, 2011.
  * [30] R. Acciarri et al.  A Study of Electron Recombination Using Highly Ionizing Particles in the ArgoNeuT Liquid Argon TPC.  JINST, 8:P08005, 2013, 1306.1712.
  * [31] C. Adams et al.  A Method to Determine the Electric Field of Liquid Argon Time Projection Chambers Using a UV Laser System and its Application in MicroBooNE.  JINST, 15(07):P07010, 2020.
  * [32] P. Abratenko et al.  Measurement of space charge effects in the MicroBooNE LArTPC using cosmic muons.  JINST, 15(12):P12037, 2020, 2008.09765.
  * [33] Yichen Li et al.  Measurement of Longitudinal Electron Diffusion in Liquid Argon.  Nucl. Instrum. Meth., A816:160–170, 2016, 1508.07059.
  * [34] M. Antonello et al.  Precise 3D track reconstruction algorithm for the ICARUS T600 liquid argon time projection chamber detector.  Adv. High Energy Phys., 2013:260820, 2013, 1210.5089.
  * [35] Steiner tree greedy algorithm.  http://paal.mimuw.edu.pl/docs/index.html, 2019.
  * [36] Biconjugate gradient stablized method (BiCGSTAB).  https://eigen.tuxfamily.org/dox/classEigen_1_1BiCGSTAB.html, 2019\.
  * [37] Code to merge close vertices: https://github.com/BNLIF/wire-cell-pid/blob/34c732755ee9c3426c2c25355404c5c28f0df3fe/src/NeutrinoID_proto_vertex.h#L2231.
  * [38] P. Abratenko et al.  Determination of muon momentum in the MicroBooNE LArTPC using an improved model of multiple Coulomb scattering.  JINST, 12(10):P10010, 2017, 1703.06187.
  * [39] C. Adams et al.  Calibration of the charge and energy loss per unit length of the MicroBooNE liquid argon time projection chamber using muons and protons.  JINST, 15(03):P03022, 2020, 1907.11736.
  * [40] Code to exclude certain vertex candidates according to the local event topology: https://github.com/BNLIF/wire-cell-pid/blob/34c732755ee9c3426c2c25355404c5c28f0df3fe/src/NeutrinoID_track_shower.h#L1356.
  * [41] A Gentle Introduction to Object Recognition With Deep Learning https://machinelearningmastery.com/object-recognition-with-deep-learning/.
  * [42] Benjamin Graham, Martin Engelcke, and Laurens van der Maaten.  3d semantic segmentation with submanifold sparse convolutional networks.  CVPR, 2018.
  * [43] Benjamin Graham and Laurens van der Maaten.  Submanifold sparse convolutional networks.  arXiv preprint arXiv:1706.01307, 2017.
  * [44] P. Abratenko et al.  Semantic segmentation with a sparse convolutional neural network for event reconstruction in MicroBooNE.  Phys. Rev. D, 103(5):052012, 2021, 2012.08513.
  * [45] Wire-Cell Prototype https://github.com/BNLIF/wire-cell.
  * [46] Olaf Ronneberger, Philipp Fischer, and Thomas Brox.  U-net: Convolutional networks for biomedical image segmentation.  CoRR, abs/1505.04597, 2015, 1505.04597.
  * [47] Wire-Cell Deep Learning Vertexing for MicroBooNE https://github.com/HaiwangYu/uboone-dl-vtx.
  * [48] Zhe Cao, Gines Hidalgo, Tomas Simon, Shih-En Wei, and Yaser Sheikh.  Openpose: Realtime multi-person 2d pose estimation using part affinity fields, 2018, 1812.08008.
  * [49] Sebastian Raschka and Vahid Mirjalili.  Python Machine Learning - Third Edition.  Packt, 2019.
  * [50] Diederik P. Kingma and Jimmy Ba.  Adam: A method for stochastic optimization, 2014, 1412.6980.
  * [51] R. Acciarri et al.  The Pandora multi-algorithm approach to automated pattern recognition of cosmic-ray muon and neutrino events in the MicroBooNE detector.  Eur. Phys. J. C, 78(1):82, 2018, 1708.03135.
  * [52] P. Abratenko et al.  Search for an anomalous excess of charged-current $\nu_{e}$ interactions without pions in the final state with the MicroBooNE experiment, 10 2021, 2110.14065.
  * [53] Michael Mooney.  The MicroBooNE Experiment and the Impact of Space Charge Effects.  In Proceedings, Meeting of the APS Division of Particles and Fields (DPF 2015): Ann Arbor, Michigan, USA, 4-8 Aug 2015, 2015, 1511.01563.
  * [54] C. Adams et al.  Reconstruction and Measurement of $\mathcal{O}$(100) MeV Energy Electromagnetic Activity from $\pi^{0}\rightarrow\gamma\gamma$ Decays in the MicroBooNE LArTPC.  JINST, 15(02):P02007, 2020, 1910.02166.
  * [55] PSTAR at NIST: https://physics.nist.gov/PhysRefData/Star/Text/PSTAR.html.
  * [56] R. Acciarri et al.  A Study of Electron Recombination Using Highly Ionizing Particles in the ArgoNeuT Liquid Argon TPC.  JINST, 8:P08005, 2013, 1306.1712.
  * [57] Eido Shibamura, Akira Hitachi, Tadayoshi Doke, Tan Takahashi, Shinzou Kubota, and Mitsuhiro Miyajima.  Drift velocities of electrons, saturation characteristics of ionization and W-values for conversion electrons in liquid argon, liquid argon-gas mixtures and liquid xenon.  Nucl. Instrum. Meth., 131:249–258, 1975.
  * [58] M. Miyajima, T. Takahashi, S. Konno, T. Hamada, S. Kubota, H. Shibamura, and T. Doke.  Average energy expended per ion pair in liquid argon.  Phys. Rev. A, 9:1438–1443, 1974.
  * [59] P. Abratenko et al.  Search for an Excess of Electron Neutrino Interactions in MicroBooNE Using Multiple Final State Topologies.  10 2021, 2110.14054.
  * [60] P. Abratenko et al.  Search for an anomalous excess of charged-current quasi-elastic $\nu_{e}$ interactions with the MicroBooNE experiment using Deep-Learning-based reconstruction.  10 2021, 2110.14080.
  * [61] C. Adams et al.  Deep neural network for pixel-level electromagnetic particle identification in the MicroBooNE liquid argon time projection chamber.  Phys. Rev. D, 99(9):092001, 2019, 1808.07269.
  * [62] P. Abratenko et al.  Convolutional neural network for multiple particle identification in the MicroBooNE liquid argon time projection chamber.  Phys. Rev. D, 103(9):092003, 2021, 2010.08653.