# CHIPS Event Reconstruction and Design Optimisation

A. Blake  S. Germani  Y. B. Pan  A. J. Perch  M. M. Pfützner  J. Thomas  L. H.
Whitehead111Now at CERN, Geneva, Switzerland Department of Physics and
Astronomy, University College London, Gower Street, London, WC1E 6BT, United
Kingdom  Department of Physics, University of Wisconsin, Madison, WI 53706,
United States of America  Physics Department, Lancaster University, Lancaster,
LA1 4YB, United Kingdom

###### Abstract

The CHIPS experiment will comprise a $10\,$kton water Cherenkov detector in an
open mine pit in northern Minnesota, USA. The detector has been simulated
using a full GEANT4 simulation and a series of event reconstruction algorithms
have been developed to exploit the charge and time information from all of the
PMTs. A comparison of simulated CCQE $\nu_{\mu}$ and $\nu_{e}$ interactions
using 10 inch and 3 inch PMTs is presented, alongside a comparison of 10% and
6% photocathode coverage for 3 inch PMTs. The studies demonstrate that the
required selection efficiency and purity of charged-current $\nu_{e}$
interactions can be achieved using a photocathode coverage of 6% with 3 inch
PMTs. Finally, a dedicated $\pi^{0}$ fitter is shown to successfully
reconstruct a clean sample of $\pi^{0}$ mesons despite the low 6% photocathode
coverage with 3 inch PMTs.

###### keywords:

Neutrino Oscillation, CHIPS, Event Reconstruction

##  1 Introduction

It has been conclusively shown over the last 20 years that neutrinos undergo
oscillations from one flavour to another, and the field has now entered a
precision measurement era. The oscillations arise because the three weak
interaction eigenstates are not the same as the three mass eigenstates but are
formed from a linear combination, as defined by the PMNS matrix [1, 2, 3]. The
PMNS matrix is commonly expressed in terms of four parameters: three mixing
angles $\theta_{12}$, $\theta_{23}$ and $\theta_{13}$; and the $CP$-violating
phase $\delta_{CP}$. The oscillations themselves are driven by the mass-
squared differences between the mass eigenstates $\Delta m^{2}_{ij}\equiv
m^{2}_{i}-m^{2}_{j}$ with $i>j$, giving two independent mass-squared
splittings.

With the measurement of a non-zero value of $\theta_{13}$ by reactor
antineutrino experiments[4, 5, 6], the main focus of the neutrino oscillation
community has moved to resolving the mass hierarchy ambiguity and measuring
the value of the CP-violating phase, $\delta_{CP}$. Long-baseline neutrino
oscillation experiments can probe the mass hierarchy and $\delta_{CP}$ by
looking for $\nu_{\mu}$ $\rightarrow$$\nu_{e}$ transitions. The $\nu_{e}$
appearance probability, taking into account the MSW matter effect [7, 8],
expanded to second order in $\alpha\equiv\Delta m^{2}_{21}/\Delta m^{2}_{32}$
can be written as [9]

$\displaystyle
P(\nu_{\mu}\rightarrow\nu_{e})\approx\sin^{2}\theta_{23}\sin^{2}$
$\displaystyle
2\theta_{13}\frac{\sin^{2}\Delta(1-A)}{(1-A)^{2}}+\alpha\tilde{J}\cos(\Delta\pm\delta_{CP})\frac{\sin\Delta
A}{A}\frac{\sin\Delta(1-A)}{(1-A)}$
$\displaystyle+\alpha^{2}\cos^{2}\theta_{23}\sin^{2}2\theta_{12}\frac{\sin^{2}\Delta
A}{A^{2}},$ (1)

with $\Delta=\Delta m^{2}_{31}L/4E$, $\tilde{J}=\cos\theta_{13}\sin
2\theta_{13}\sin 2\theta_{12}\sin 2\theta_{23}$, and
$A=2\sqrt{2}G_{F}n_{e}E/\Delta m^{2}_{31}$, where $G_{F}$ is the Fermi weak
coupling constant and $n_{e}$ is the electron density. The sign of $A$ is
positive for neutrinos and negative for antineutrinos. Similarly, the sign in
front of $\delta_{CP}$ in the second term is positive for neutrinos and
negative for antineutrinos. Equation 1 shows that the
$\nu_{\mu}$$\rightarrow$$\nu_{e}$ probability has sensitivity to the mass
hierarchy through the matter effect parameter $A$ and to the $CP$-violating
phase through the second term.

Water Cherenkov (WC) neutrino detectors infer the presence of a neutrino
interaction by measuring the Cherenkov light [10] emitted by charged particles
resulting from the interaction. Cherenkov radiation is emitted by all
electrically charged particles travelling faster than the local speed of light
in a dielectric medium. The charged particles emit a cone of radiation with an
opening angle $\theta_{C}$ defined as

$\cos\theta_{C}=\frac{1}{\beta n},$ (2)

where $n$ is the refractive index of the material and $\beta$ is the particle
speed divided by the speed of light in the vacuum. Inside a detector, this
cone of light projects onto the wall and forms a ring shape.

The study of neutrino oscillations generally requires clean samples of
charged-current (CC) interactions for different neutrino flavours. Water
Cherenkov detectors are well suited to distinguishing CC $\nu_{\mu}$ and
$\nu_{e}$ interactions, particularly at low neutrino energies when the track
multiplicity is lower. These CC interactions proceed by the exchange of a $W$
boson and are of the form $\nu_{l}+X\rightarrow l^{-}+X^{\prime}$, where the
flavour of the final-state lepton, $l$, is used to identify the flavour of the
neutrino. Electrons initiate an electromagnetic cascade meaning that the
produced Cherenkov radiation is actually the sum of the individual Cherenkov
cones from all of the electrons and positrons in the cascade, and hence the
ring of light detected is wider and fuzzier than the corresponding ring from a
muon. There are three main types of CC interaction, ranging from those with
minimal hadronic activity to those with a high energy transfer to the hadronic
system: quasi-elastic (QE), resonant (RES) and deep inelastic scattering
(DIS).

The primary background to the CC $\nu_{e}$ appearance search, once the CC
$\nu_{\mu}$ interactions have been identified, comes from the production of
$\pi^{0}$ mesons in neutral-current (NC) interactions:
$\nu+X\rightarrow\nu+X^{\prime}+\pi^{0}$. The $\pi^{0}$ meson decays into a
pair of photons with a probability of 98.8% [11], and can mimic an electron
signal in two ways: firstly, when the $\pi^{0}$ has a high boost the decay
photons are emitted almost colinearly and produce closely overlapping rings of
light on the detector wall; and secondly, if one photon is not detected then
the single remaining photon ring is indistinguishable from that of an
electron, since the photon will also initiate an electromagnetic shower. The
first of these two possibilities is reducible in the event that the two
closely overlapping rings can be reconstructed and shown to give a better fit
than a single electron ring.

##  2 The CHIPS Experiment

The CHerenkov detectors In mine PitS (CHIPS) experiment plans to deploy a WC
neutrino detector into a large body of fresh water, such as a flooded mine
pit. The main goal of the project is to construct a 10 kton water Cherenkov
detector at a dramatically reduced cost compared to previous detectors through
novel design choices. The water surrounding the detector will provide the
structural support and sufficient overburden, and means that no large-scale
civil engineering is required, as would be necessary for building a detector
in an underground cavern. A demonstration of this ability would pave the way
towards future affordable mega-ton scale neutrino detectors.

CHIPS will only study neutrino interactions from accelerator neutrino beams.
This means that the design requirements with regard to photodetectors are
different from previous and existing large WC neutrino detectors such as
Super-Kamiokande [12]. These experiments have wide physics programs including
low energy physics arising from solar neutrinos and proton decay searches. The
energy threshold of water Cherenkov detectors is primarily dictated by the
photocathode coverage, and the cost of photodetectors, such as photomultiplier
tubes (PMTs), is a major driver to the total cost of the experiment.
Considering only accelerator beam neutrinos enables the photocathode coverage
to be reduced since the energy threshold can be much larger than for solar
neutrino studies, for example.

The CHIPS project began with the deployment of CHIPS-M, a small prototype
detector 3.3 m tall and 3 m in diameter, into a water-filled, dis-used mine
pit in northern Minnesota in the summer of 2014. The mine pit is located
approximately 700 km from the NuMI [13] beam target at Fermilab and 7 mrad
off-axis. CHIPS-M was then subsequently refurbished and redeployed in the same
location in the summer of 2015. CHIPS-M was instrumented with five 10 inch
PMTs and after refurbishment, an additional 32 3 inch PMTs. The next step will
be the building of a $10\,$kton module called CHIPS-10, which is planned for
deployment in the summer of 2018. This document details simulation and event
reconstruction studies to benchmark the performance of different detector
geometry options.

##  3 Simulation

A GEANT4 [14] simulation has been written to simulate the CHIPS detectors. It
was originally based on the WCSim package [15] developed for the proposed T2K
2km and LBNE WC detectors, but has been overhauled and almost entirely
rewritten to better suit the requirements for CHIPS. The main design goal of
the simulation was to produce a versatile package to allow many different
geometry options to be considered without the need for recompilation.

The simulation builds an n-sided, regular polygonal prism detector geometry
consisting of the top and bottom end caps separated by the main barrel region.
The dimensions of the geometry are stored in a configuration file and are
loaded at run time. Each of the sides of the n-sided barrel form a separate
_region_ and each of the caps can be divided into regions defined by the angle
subtended. A concentric veto volume is included in order to study the
efficiency of rejecting cosmic-ray muons and the effect of cosmic-ray muons
overlapping with a beam neutrino interaction.

The base unit of the geometry is known as the _unit cell_. Each region of the
detector can have a uniquely defined unit cell that contains a pattern of
PMTs. PMTs can be placed such that they face outwards in order to look at the
veto region as well as inwards to observe the main detector volume. This
pattern can consist of any number of PMTs of different sizes and types that
are defined in a second configuration file. The geometry is then built by
tiling each region with its corresponding unit cell, and the unit cells are
scaled in size to produce the required level of photocathode coverage for the
region. The unit cell represents what would be a plane of PMTs in the real
detector. Figure 1 shows a cartoon of the inner detector with example regions
and unit cells displayed.

Figure 1: An illustration of an eight-sided inner detector showing regions
defined on the cap and barrel. One of the barrel regions has been tiled with
unit cells, and an example unit cell is shown to contain six PMTs. The region
on the top cap is defined by the opening angle.

A simple dynode chain simulation was written to convert the number of photons
incident on the PMT photocathode returned by GEANT4 into a digitised number of
photoelectrons. The simulation assumes a total gain of $1\times 10^{7}$ across
ten dynode chains, the relative gains of which are taken from the base of the
10 inch Hamamatsu PMT developed for the IceCube experiment [16], the PMT used
in the first deployment of the CHIPS-M prototype. The simulation also accounts
for non-linearity at high numbers of incident photons resulting from the
electron cloud in the dynode chain effectively shielding some of the
accelerating electric field.

###  3.1 CHIPS-10 Geometries

The CHIPS-10 detector is simulated as a 20-sided prism with a height of
$20\,$m and a $12.5\,$m radius. The two different sizes of PMT considered here
are $10\,$inch and $3\,$inch. The properties of the 3 inch PMT, other than the
actual diameter, were set to be identical to the 10 inch PMT in order to be
able to perform a direct comparison of the effect of the PMT size. Both 10
inch and 3 inch tubes were assumed to have high quantum efficiency, peaking at
34% at $380\,$nm. These parameters can be trivially changed in a configuration
file once an actual PMT has been selected for the experiment. The three
geometry options considered are:

  * 1.

Uniform 10% coverage of 10 inch PMTs

  * 2.

Uniform 10% coverage of 3 inch PMTs

  * 3.

Uniform 6% coverage of 3 inch PMTs.

In order to obtain the same photocathode coverage using 3 inch PMTs and 10
inch PMTs, the 3 inch PMT geometry has a factor 11 more PMTs based on the
ratio of the areas of the PMTs.

The attenuation length of the water as a function of wavelength is
parametrised from $200-800\,$nm using data from measurements of pure water
[17, 18, 19] and uniformly scaled down to account for impurities such that it
peaks at $50\,$m at $405\,$nm. A recent dedicated study showed that the
attenuation length attainable using just filtration and UV-sterilisation of
the water from the mine pit is up to 100 m [20], meaning this could be a
slightly conservative choice.

###  3.2 Event Generation

The generation of beam neutrino events is performed by the GENIE [21] package
using as input the predicted NuMI [13] beam spectrum at the CHIPS detector
location, 700 km from the target and 7 mrad off-axis. The predicted spectrum
is that of the $\nu_{\mu}$ flux, shown in Fig. 2, which is then oscillated to
provide the $\nu_{e}$ flux using the values $\Delta m^{2}_{32}=2.39\times
10^{-3}$eV2, $\sin^{2}\theta_{23}=0.43$, $\sin^{2}2\theta_{13}=0.0945$,
$\delta_{CP}=0$, and the solar scale oscillation terms are neglected. Cosmic-
ray muons are generated with the CRY [22] package, producing as output a
series of cosmic-ray muons with their positions spread over a square kilometre
at the Earth’s surface surrounding the detector. The interactions from both
generators are converted into the same file format and used as input to the
detector simulation.

Figure 2: The simulated NuMI muon neutrino flux at a distance of 700 km and 7
mrad off-axis.

The beam interaction or cosmic-ray muon files are passed into the detector
simulation to produce events. It is also possible to use a combination of
particles with beam and cosmic origins in order to produce beam events with
cosmic-ray muons overlaid to study how the reconstruction software can cope
with overlapping events. The beam events are assigned a random time within the
NuMI beam spill window of approximately $10\,\mu$s. The full simulated events
are then produced with the GEANT4 simulation described previously.

##  4 Reconstruction

The first steps of the reconstruction are employed in order to seed the tracks
for the full likelihood fit that forms the bulk of the reconstruction method.
The seeding algorithms are used to provide the main fitter with a good
starting point from which to begin the minimisation routines, and to allow it
to efficiently find the optimal set of parameters that describe the
interaction.

###  4.1 Track seeding

The first stage in seeding the tracks is to try to find clearly separated
energy deposits in space and time. The digitised hits from each PMT are sorted
into time order and sliced up when gaps are found in time between sequential
hits. Each time slice is then considered in turn to look for distinct spatial
collections of PMT hits. A charge cut is applied at this stage in order to
prevent low charge hits joining up the different regions222These hits are only
removed from the seeding process, all hits are considered in the full
likelihood fit.. Hits are then clustered together beginning with the highest
charged hit, and clustering continues until no further clusters can be formed.
The slicing algorithm then returns a series of slices that consist of a
collection of associated PMT hits.

A series of vertexing algorithms are applied individually to the slices in
order to produce estimates of the interaction vertex position, vertex time and
track direction. The slices are then passed through a Hough transform
algorithm to further refine the track direction values, and to look for other
sub-dominant directions that could indicate that the energy deposit was caused
by multiple particles. The output from the Hough transform, and hence the
seeding algorithms, is a list of seeds where each seed consists of a vertex
position, vertex time and track direction. The particle energy is not
estimated in the seeding algorithms.

The outputs from the Hough transform for all slices are sorted by their peak
height in the Hough-transform space, where a large height corresponds to a
stronger seed. The number of tracks, $n$, defined before the fit, are then set
with initial parameters from the $n$ best seeds.

####  4.1.1 Cosmic-ray muon track seeding

The detector geometry includes a veto volume in order to determine whether a
beam interaction also contains a cosmic-ray muon track. Traditionally veto
volumes are used to define some dead time to the detector in order to avoid
contamination of beam interactions with the cosmic-ray muons. However, in
order to minimise the detector dead time, an algorithm has been developed to
exploit the information from the veto region to tag and seed the cosmic-ray
muon tracks. This process requires the identification of the cosmic-ray muon
entry and exit point, and as such, only works for through-going cosmic-ray
muons.

The algorithm begins by attempting to cluster the veto PMT hits. If two (or
more) clusters can be formed, the largest two are taken and the entry point is
assigned as the position of the highest charge hit in the upper cluster, and
the exit point to the highest charge hit in the lower one. If only a single
cluster is found, then the highest charge hits in the top and bottom halves of
the detector are taken as the entry and exit points. The track vertex is then
simply assigned as the entry point, the track direction from the vector
joining the entry and exit points, and the time from the hit time of the hit
with the highest charge at the entry point.

If a good cosmic-ray muon seed is obtained it is used to form a shadowed
region inside the inner detector. All those inner detector hits that fall
within the Cherenkov cone defined by the cosmic-ray muon seed are masked out
of the standard seeding algorithm outlined previously, such that the beam
interaction can also be accurately seeded. Figure 3 shows an event display of
an example output from seeding a cosmic-ray muon overlaid on to a beam
$\nu_{e}$ CCQE interaction.

Figure 3: An event display of a simulated CCQE $\nu_{e}$ interaction with an
overlaid cosmic-ray muon. The top panel shows the unrolled barrel section of
the detector and the top and bottom caps are shown in the bottom left and
bottom right panels, respectively. The colour shows each inner PMT that
received an energy deposit and represents the collected charge, limited to a
maximum of 100 p.e. for clarity. The two rings show projections of the true
tracks onto the detector wall. The figure was produced immediately after the
seeding algorithm, meaning no further reconstruction was performed, and hence
no attempt to reconstruct the particle energies was made.

###  4.2 Likelihood track-fitting method

The reconstruction algorithms for CHIPS are based on calculating the
likelihood to observe energy deposits of a given charge and time for a
particular particle track hypothesis. The algorithms were written from
scratch, with the charge prediction based upon the likelihood method developed
for the MiniBooNE experiment [23, 24]. The method is relavitely simple in
theory: given a hypothesised track or tracks, the number of photoelectrons and
time at which the first photoelectron is detected for each PMT are predicted.
The measured hit charges and times can then be compared to these predictions
in order to calculate the likelihood that the given track hypothesis was
responsible for the measured signals. The parameters that define the track
hypothesis are then varied until the negative logarithm of the likelihood is
minimised, giving the best-fit track hypothesis to the measurements.

The method uses information from all PMTs, whether there was a registered
charge deposit or not. The method can be extended in order to perform multiple
track fits and this is a vital part of identifying the $\pi^{0}$ decays that
form a background to the $\nu_{e}$ appearance search. The method is also very
modular such that only one of the components depends on the detector geometry.
This means it is easy to test different detector geometry configurations
without having to recalculate many of the parts that go into the likelihood.

The basic form of the likelihood consists of two components:

$\mathcal{L}(\bm{x})=\mathcal{L}_{\textrm{unhit}}(\bm{x})\mathcal{L}_{\textrm{hit}}(\bm{x})=\prod_{\textrm{unhit}}P_{\textrm{unhit}}(\bm{x})\prod_{\textrm{hit}}P_{\textrm{charge}}(\bm{x})P_{\textrm{time}}(\bm{x}),$
(3)

where $\bm{x}$ is the track hypothesis. The first term calculates the
likelihood that a given track $\bm{x}$ will not predict a hit in those PMTs
without registered hits. The second term calculates the likelihood that the
same track hypothesis _will_ cause a hit PMT to register a given charge at a
given time. The track hypothesis $\bm{x}$ consists of the following
parameters:

  * 1.

The track vertex position $(x_{0},y_{0},z_{0})$ and time $t_{0}$.

  * 2.

The initial track direction $(d_{x},d_{y},d_{z})$.

  * 3.

The initial kinetic energy of the particle.

  * 4.

The particle type - muon, electron or photon.

The charge prediction calculation is identical for both the PMTs with and
without registered charge deposits, and is discussed in Section 4.2.1. The
PMTs with registered hits also must have a hit time prediction which is
detailed in Section 4.2.2.

####  4.2.1 Charge component

There are three main steps in order to calculate the charge component of the
likelihood:

  1. 1.

Predict the mean number of photons incident on the PMT photocathode.

  2. 2.

Convert the number of photons into the output digitised charge from the PMT.

  3. 3.

Calculate the likelihood to see the measured energy deposit given the
predicted charge.

The formalism to perform the first step was described in detail by the
MiniBooNE collaboration [23, 24]. Adaptations to the method have been made to
account for the differences between the experiments, such as the cylindrical
geometry in CHIPS as opposed to the spherical symmetry of the MiniBooNE
detector, and the lack of scintillation light from water compared to mineral
oil. Considerable detail of the adaptations and the specific implementation of
the algorithms is given in Ref. [25].

An extended track, where light is emitted at each point along the length, can
be considered as a series of point sources. Each of these sources produces a
certain number of photons, $\mu_{i}$ that reach the PMT. These contributions
are summed over the track to give the total number of photons as:

$\mu=\sum_{i}\mu_{i}=\sum_{i}\Phi_{i}T(R_{i})\epsilon(\psi_{i})\frac{\Omega(R_{i})}{4\pi}.$
(4)

The components of $\mu_{i}$ are as follows: $\Phi_{i}$ is the number of
photons emitted in step $i$ in the direction of the PMT; $T(R_{i})$ is the
transmission function and parametrises the probability that a photon will
survive after traversing a distance $R_{i}$, the distance from the point
source $i$ to the PMT; $\epsilon(\psi_{i})$ is the angular efficiency of the
PMT as a function of the angle of the incident photon, where $\psi_{i}=0$
along the PMT normal direction; and $\frac{\Omega(R_{i})}{4\pi}$ is the
fraction of the total solid angle subtended by the PMT as viewed from the
point source $i$.

To first order, only $\Phi_{i}$ depends on the particle type and energy and
can be written as

$\Phi_{i}=\Phi(E)\rho(E,s_{i})g(E,s_{i},\cos{\theta(s_{i})}),$ (5)

where $s_{i}$ is the distance along the track and $\theta(s_{i})$ is the angle
of the emitted photon from the track direction, illustrated in Fig. 4.
$\Phi(E)$ is the total number of photons emitted by a particle of energy $E$.
The function $\rho(E,s_{i})$ gives the fraction of all photons emitted in step
$i$, and $g(E,s_{i},\cos{\theta(s_{i})})$ gives the number of photons from
step $i$ emitted in the angular range
$\cos\theta\rightarrow\cos\theta+d(\cos\theta)$, and normalised such that the
sum over $\cos\theta$, multiplying by the bin width, for a given $s$ is equal
to one. The energy dependence of the emission profiles $\rho(E,s_{i})$ and
$g(E,s_{i},\cos{\theta(s_{i})})$ is handled by binning in energy. This means
that for each particle type and energy bin, two emission profiles are created,
$\rho(s_{i})$ and $g(s_{i},\cos{\theta(s_{i})})$, using large samples of Monte
Carlo simulated events.

Figure 4: The definition of the distance along the track $s$, the angle of the
emitted photon $\theta$ and the distance travelled by the photon $R$.

Combining Eqs. 4 and 5 and noting that since the track vertex, track direction
and PMT position are fixed for a given hypothesis, both $\psi_{i}$ and $R_{i}$
depend solely on $s_{i}$, gives

$\mu=\Phi(E)\sum_{i}T(s_{i})\epsilon(s_{i})\frac{\Omega(s_{i})}{4\pi}\rho(E,s_{i})g(E,s_{i},\cos{\theta(s_{i}))}.$
(6)

In calculating the likelihood, Eq. 6 must be evaluated for each PMT and for
each iteration of the track hypothesis, which would prove to be very
computationally intensive. In order to combat this, as described in Refs. [23,
24], Eq. 6 can be rewritten as

$\mu=\Phi(E)\sum_{i}J(s_{i})\rho(E,s_{i})g(E,s_{i},\cos{\theta(s_{i}))}.$ (7)

with $J(s_{i})=T(s_{i})\epsilon(s_{i})\frac{\Omega(s_{i})}{4\pi}$. Since
$J(s)$ is a slowly varying function, it can be expanded quadradically in terms
of $s$: $J(s)\approx j_{0}+j_{1}s+j_{2}s^{2}$. The coefficients of the
expansion are calculated using three values of $s$ and solved by matrix
inversion, rather than actually performing a fit. The three values are chosen
to cover a large portion of the track length, but differ depending on whether
the track is contained inside the detector. For contained tracks, the points
are at: $10\,$cm, the distance at which $75\%$ of photons have been emitted
and half this distance. For exiting tracks, the three points are at: $10\,$cm,
and $40\%$ and $80\%$ of the distance to the exiting point. Writing
$\xi(E,s_{i},\cos{\theta(s_{i})})=\rho(E,s_{i})g(E,s_{i},\cos{\theta(s_{i}))}$,
Eq. 7 becomes

$\mu=\Phi(E)\left(j_{0}\sum_{i}\xi(E,s_{i},\cos{\theta(s_{i})})+j_{1}\sum_{i}s_{i}\xi(E,s_{i},\cos{\theta(s_{i})})+j_{2}\sum_{i}s^{2}_{i}\xi(E,s_{i},\cos{\theta(s_{i})})\right).$
(8)

The summations in Eq. 8 depend only on the emission profiles and can hence be
calculated in advance for each particle type and stored in a look-up table as
a function of the track energy, distance from the vertex to the PMT $R_{0}$,
and the angle between the direction from the vertex to the PMT and the
particle direction $\cos\theta_{0}$. This means that the computationally
intensive part is performed once and then the values are just used by the
fitter from the look-up table. It is important that there is a smooth
behaviour as a function of energy from the look-up table, hence cubic splines
are used to interpolate between energy bins. For a given
$\left(R_{0},\cos\theta_{0}\right)$ bin the spline is loaded and evaluated at
the desired energy.

The next step is to convert the predicted number of photons $\mu$ into a
number of photoelectrons $\mu_{pe}$. This is performed by applying the quantum
efficiency of the PMT, the values of which are parametrised as a function of
photon wavelength in the simulation. The quantum efficiency is averaged over
the wavelengths of light expected at the PMT, since the wavelength spectrum
changes as a function of the distance travelled by the photons.

The final step is to calculate the likelihood of the measured PMT charge given
the predicted number of photoelectrons for each PMT $p$. This is done by
considering Poisson statistics with $\lambda_{p}=\mu_{pe,p}$ for each PMT by
comparing to the measured charge $q_{p}$ in the following way:

$\mathcal{L}_{\textrm{charge}}=\prod_{\textrm{hit}}P_{\textrm{charge}}=\prod_{p}\frac{\lambda_{p}^{q_{p}}e^{-\lambda_{p}}}{q_{p}!},$
(9)

or in $-2\ln\mathcal{L}$ form, noting also that the Gamma function
$\Gamma(n+1)=n!$ is used to replace the factorial since $q_{p}$ is not
constrained to be an integer,

$-2\ln\mathcal{L}_{\textrm{charge}}=-2\sum_{p}q_{p}\ln\lambda_{p}-\lambda_{p}-\ln\Gamma(q_{p}+1).$
(10)

In the case that the PMT had no measured charge deposit, i.e. $q_{p}=0$, the
contribution to the likelihood simplifies to $2\lambda_{p}$.

####  4.2.2 Time component

The treatment of the time component of the fit differs substantially from the
method used by MiniBooNE, and is hence described here in detail. A further
fully detailed description can be found in Ref. [25]. The method described
here tries to predict the time at which each PMT will receive a hit by
calculating the trajectories of the photons emitted along the length of the
track. The large size of CHIPS-10 compared to MiniBooNE means that there is a
much larger spread in PMT hit times within an event, and hence the time of the
hits provides a lot of information. The time difference between the first and
last hits, not including scattered light, is of the order of $50\,$ns for
CHIPS-10, which when compared with typical PMT timing resolutions of 1 or 2
ns, shows the potential power of the timing information.

The predicted hit time $t_{PMT}$ of a photon incident on a PMT at position
$\bm{x_{PMT}}$ emitted by a particle with a vertex four-vector
$(\bm{x_{0}},t_{0})$ and momentum unit vector $\bm{\hat{p}}$, travelling
through a medium with refractive index $n$, can be written as the sum of three
terms: the vertex time, the time taken until the photon emission and the
travel time of the photon. This gives

$t_{PMT}=t_{0}+\frac{s}{v}+\left(\frac{n}{c}\times|\bm{x_{PMT}}-(\bm{x_{0}}+s\bm{\hat{p}})|\right),$
(11)

where $s$ is the distance along the track at which the photon was emitted and
$v$ is the speed of the charged particle. The effective speeds of the muons
and electromagnetic showers were found to be $0.94c$ and $0.81c$ from the
simulation, respectively. These effective speeds account for the fact that the
emission of photons does not occur along a perfectly straight line. It is
clear that if the particle only emits photons at a single angle then there is
only one value of $s$ that will cause a hit on a given PMT. In reality,
however, the photons are emitted at a range of angles, and this range is much
larger for electrons than muons since the photons are emitted by all of the
electrons and positrons in the electromagnetic shower. There are hence
different combinations of $s$ and the distance travelled by the photon that
can cause a hit on a given PMT.

The time prediction proceeds by stepping along the length of the track in
units of $25\,$cm. At each step $\Delta s$, the goal is to calculate the time
that a hit would occur on each PMT, and a corresponding weight that describes
how likely it is to be hit at that time, given the distribution of light
emitted. The time of the hit is simply given by Eq. 11. The weight must be
calculated from emission profiles, similar to those described in the
calculation of the charge prediction. The profiles $\rho_{t}(E,s_{i})$ and
$g_{t}(E,s_{i},\cos\theta_{i})$, where $\rho_{t}(E,s_{i})$ gives the fraction
of all photons emitted in step $i$, and $g_{t}(E,s_{i},\cos{\theta(s_{i})})$
gives the number of photons from step $i$ emitted in the angular range
$\cos\theta\rightarrow\cos\theta+d(\cos\theta)$, are normalised differently to
their charge equivalents such that the weight, equal to the fraction of the
total number of photons emitted towards the PMT in the $\Delta s$ step, is
given by:

$w_{i}=\rho_{t}(s_{i})g_{t}(s_{i},\cos{\theta_{i}})\Delta
s_{i}\Delta{\cos{\theta_{i}}}.$ (12)

Hence, for each PMT, a distribution of hit times and their corresponding
weights is obtained after stepping along the entire length of the track. The
predicted hit time is then taken as the weighted mean of this array of points
calculated in the following way:

$\bar{t}_{PMT}=\left(\displaystyle\sum\limits_{i}{t_{i}w_{i}}\right)/\left(\displaystyle\sum\limits_{i}{w_{i}}\right)$
(13)

The weighted RMS also provides important information for calculating the
likelihood. This width $\sigma_{source}$, describes the intrinsic uncertainty
in the hit time due to the emission of photons by the charged particle, and is
defined as:

$\sigma^{2}_{source}=\left(\displaystyle\sum\limits_{i}(t_{i}-\bar{t}_{PMT})^{2}w_{i}\right)/\left(\displaystyle\sum\limits_{i}w_{i}\right).$
(14)

In the event that only one step of $\Delta s$ provides a predicted hit time,
the width is set to be $\sigma_{source}=0.1\,$ns in order for later
calculations to succeed.

The mean and RMS of the predicted time, along with the probability weights,
are interpolated as a function of energy. In particular, Catmull-Rom [26]
splines were used such that both the values and their derivatives vary
smoothly between the energy bins. They are also very computationally fast as
they rely only on the nearest four points to construct the spline.

There is one further complication in the calculation when considering PMTs hit
by multiple photons; the hit time recorded by the PMT is the time of the first
hit, not the mean of all the photons. Assuming that the distribution of photon
arrival times is Gaussian with mean $\bar{t}_{PMT}$ and width
$\sigma_{source}$, then the larger the number of photons considered, the
earlier the first hit time should fall within that distribution. Order
statistics gives the probability density functions $P_{k}(t)$ for the
$k^{\textrm{th}}$ smallest value obtained from $n$ values drawn from a
probability distribution $f(t)$. In this case the minimum value corresponds to
$k=1$ such that

$P(t_{min})=n\left(1-F(t)\right)^{(n-1)}f(t),$ (15)

where $t_{min}$ is the earliest time, $F(t)$ is the cumulative distribution
function of $f(t)$, and $n$ is the number of photons detected. For a Gaussian
$f(t)$ Eq. 15 becomes:

$P(t_{min})=\frac{n}{{2^{n}}\sigma\sqrt{2}}\left(1-\textrm{erf}{\left(\frac{t-\mu}{\sqrt{2}\sigma}\right)}\right)^{n-1}\exp{\left(-\frac{(t-\mu)^{2}}{2\sigma^{2}}\right)}.$
(16)

The probability for the PMT to be hit at the predicted time is equal to the
overlap between the $P(t_{min})$ distribution and a Gaussian distribution,
$f_{PMT}$, with a mean set to the measured PMT hit time and the width equal to
the PMT time resolution. Mathematically, this overlap area is described as

$P_{\textrm{time}}=\int^{\infty}_{-\infty}\textrm{min}(P(t),f_{PMT}(t))dt$
(17)

It is faster to calculate this value by finding the points where the functions
cross, and using the exact integrals of the two functions to compute the area.

###  4.3 Single Track Fits

In the case of a single track, the best track seed is taken from the seeding
algorithms described in Section 4.1. The minimisation covers a complicated
parameter space with non-trivial correlations. In particular there is a three-
way correlation between the track vertex position along the track direction,
the vertex time and the energy, such that a movement in the vertex position
can be compensated by altering the vertex time and energy to give a similar
fit result. The minimisation routine proceeds in the following steps:

  1. 1.

Fit the time using the time component of the fit only.

  2. 2.

Fit the energy using both charge and time components.

  3. 3.

Fit the energy, track vertex along the track direction, and the time, and then
re-fit the track direction. This step is then repeated once more.

  4. 4.

Fit all track parameters using charge and time components.

  5. 5.

Fit the energy and track vertex along the track direction using charge only.

  6. 6.

Fit the energy with charge and time components.

  7. 7.

Fit the time with the charge and time components.

The track parameters are varied during the minimisation process until the
minimum combined likelihood from the charge and time components of the fit is
obtained. The values of the parameters at the minimum value of the negative
log-likelihood are the best estimate of the true values. Figure 5 shows
example single track fits of a CCQE $\nu_{\mu}$ interaction under the muon
hypothesis (top), and a CCQE $\nu_{e}$ interaction under the electron
hypothesis (bottom), showing good agreement between the reconstructed and true
parameter values.

Figure 5: The top (bottom) event display shows the result of a single track
fit to a simulated CCQE $\nu_{\mu}$ ($\nu_{e}$) interaction under the muon
(electron) hypothesis using a 6% coverage of 3 inch PMTs. The magenta ring
shows the best-fit from the reconstruction algorithms.

###  4.4 Multiple Track Fits

The reconstruction method is relatively straightforward to extend to a
multiple track hypothesis. The predicted charge for each PMT simply becomes
the sum of the predicted charges from each of the $j$ tracks:

$\mu_{total}=\sum_{j}\mu_{j}$ (18)

For the time component, the contribution to the time likelihood for each PMT
from each of the tracks, $\mathcal{L}^{t}_{j}$, is weighted by the predicted
charge in the PMT:

$\mathcal{L}^{t}_{total}=\frac{\sum_{j}\mu_{j}\mathcal{L}^{t}_{j}}{\sum_{j}\mu_{j}}$
(19)

Seeding multiple track events can be challenging, and Section 4.1 outlined the
standard track seeding method and the cosmic-ray muon track seeding. A special
procedure is used in the case of seeding tracks for the $\pi^{0}$ fits.

####  4.4.1 $\pi^{0}$ Fits

One of the main challenges in identifying $\nu_{e}$ interactions is to
efficiently and accurately separate them from background events containing
$\pi^{0}$ mesons. Unless good seeds are found for the two decay photon tracks
it is hard to prevent the fit from collapsing down into an effective single
track fit. The procedure begins by performing a single-track electron fit in
order to provide a baseline likelihood to which the two-track fit can be
compared. Two algorithms are then applied in order to obtain seeds for the two
$\pi^{0}$ decay photon tracks.

Firstly, the two best track seeds are then taken from the output of the
standard seeding algorithm described in Section 4.1. A common vertex is formed
by extrapolating the tracks backwards to a point of closest approach, and the
initial energies of the tracks are set to be in the ratio of the height of the
peaks in the Hough transform under the condition of the invariant mass of the
$\pi^{0}$. This vertex, the track directions and energies are then used to
create two photon hypothesis tracks to represent the $\pi^{0}$ and the
likelihood is calculated. If this likelihood is better than the single
electron likelihood then this seed is passed to the fitter, otherwise a second
attempt to seed a $\pi^{0}$ is made using a method similar to the procedure
used by MiniBooNE [23, 24], outlined below.

The PMT hit positions are projected onto a plane containing the single
electron vertex position and perpendicular to the electron direction. The
major and minor axes of the covariance ellipse of the hits are formed.
Rotations are then performed on the electron track to represent the first of
the two photons: 0, $\pm\frac{\pi}{20}$, $\pm$0.45 and $\pm\frac{\pi}{5}$
radians about the major axis; and $\pm\frac{\pi}{20}$ radians about the minor
axis. The second photon track direction is obtained through a grid search
between $\pm\frac{\pi}{8}$ in both the $\theta$ and $\phi$ directions with
respect to the first photon direction. The range of the grid search was chosen
to concentrate on the small separations where the Hough transform does not
perform as well. The two configurations giving the best likelihood when the
second photon has greater than or less than half the energy of the first
photon are passed on to the fitter. Both of these are then minimised in the
fitter, and the best likelihood is returned.

##  5 Comparison of PMT options

A study was performed to benchmark the reconstruction software performance for
the three geometry options discussed in Sec. 3.1. These options are 10 inch
PMTs at 10% coverage, and 3 inch PMTs at both 10% and 6% coverage. Samples of
CCQE $\nu_{\mu}$ and CCQE $\nu_{e}$ interactions were produced using the
simulation, with energy spectra following those expected from the NuMI beam.
The events were reconstructed using a single track fit with the track
hypothesis set to match the flavour of the leading charged lepton. Following
the reconstruction, a preselection was applied to select events that should be
well reconstructed:

  * 1.

Track does not exit the detector

  * 2.

Number of hit PMTs greater than 50

  * 3.

Vertex position at least 1 m from the detector wall

  * 4.

Track energy in the range $550\,$MeV to $4950\,$MeV (this is the range covered
by the generated emission profiles).

The difference between the reconstructed and simulated distributions for the
vertex position, the vertex time, track direction and track energy are shown
for CCQE $\nu_{e}$ and CCQE $\nu_{\mu}$ interactions comparing 10 inch and 3
inch PMTs, both at 10% coverage, in Figs. 6 and 7, respectively. Similar
comparisons are shown in Figs. 8 and 9 for the 3 inch PMTs with 10% and 6%
photocathode coverage for CCQE $\nu_{e}$ and CCQE $\nu_{\mu}$ interactions,
respectively. A summary of the data displayed in the figures is given in Table
1. The tables show the resolutions for each of the variables for the three
different geometry options, where the resolution for time and energy is given
by the width of the Gaussian fit, and by the region containing 68% of entries
for the vertex position and direction since they are non-Gaussian. A small
bias in the time and energy variables can be seen, and further studies are on-
going to investigate the cause of these shifts. Since the shifts only change
the mean of the distributions a correction function could be developed to
account for the effect. The results show that for a fixed photocathode
coverage of 10% the 3 inch PMTs outperform the 10 inch PMTs. This is expected
as the more numerous 3 inch PMTs provide extra position, timing and shape
information. Furthermore, it shows only a small degradation in performance for
the 6% coverage case, which in terms of PMT cost represents a $40\%$ saving.
This 6% coverage with 3 inch PMTs is the default design for CHIPS-10, and the
performance shown suggests that this is a realistic option to use. The studies
detailed in the following sections were hence performed using the 3 inch PMTs
with 6% coverage geometry option.

|
---|---
|
Figure 6: A comparison of the reconstruction performance for a single track fit to CCQE $\nu_{e}$ interactions for 10 inch and 3 inch PMTs for a photocathode coverage of 10%. The distributions show (clockwise from top left): the distance of the reconstructed vertex from the simulated vertex, the difference between the reconstructed and simulated values for the vertex time, the difference between the reconstructed and simulated values for the electron energy and the angle between the reconstructed and simulated track directions. |
---|---
|
Figure 7: A comparison of the reconstruction performance for a single track fit to CCQE $\nu_{\mu}$ interactions for 10 inch and 3 inch PMTs for a photocathode coverage of 10%. The distributions show (clockwise from top left): the distance of the reconstructed vertex from the simulated vertex, the difference between the reconstructed and simulated values for the vertex time, the difference between the reconstructed and simulated values for the muon energy, and the angle between the reconstructed and simulated track directions. |
---|---
|
Figure 8: A comparison of the reconstruction performance for a single track fit to CCQE $\nu_{e}$ interactions for 10% and 6% photocathode coverage of 3 inch PMTs. The distributions show (clockwise from top left): the distance of the reconstructed vertex from the simulated vertex, the difference between the reconstructed and simulated values for the vertex time, the difference between the reconstructed and simulated values for the electron energy, and the angle between the reconstructed and simulated track directions. |
---|---
|
Figure 9: A comparison of the reconstruction performance for a single track fit to CCQE $\nu_{\mu}$ interactions for 10% and 6% photocathode coverage of 3 inch PMTs. The distributions show (clockwise from top left): the distance of the reconstructed vertex from the simulated vertex, the difference between the reconstructed and simulated values for the vertex time, the difference between the reconstructed and simulated values for the muon energy, and the angle between the reconstructed and simulated track directions. Sample | Geometry | Reconstruction Resolution
---|---|---
Vertex Position (cm) | Vertex Time (ns) | Direction (∘) | Energy (MeV)
CCQE $\nu_{e}$ | 10 inch, 10% | 35 | 0.9 | 2.1 | 208
3 inch, 10% | 35 | 0.84 | 1.9 | 210
3 inch, 6% | 38 | 0.89 | 2.1 | 211
CCQE $\nu_{\mu}$ | 10 inch, 10% | 47 | 1.35 | 2.6 | 113
3 inch, 10% | 44 | 1.14 | 2.7 | 110
3 inch, 6% | 51 | 1.28 | 3.0 | 113
Table 1: The resolutions of various parameters from a single electron (muon)
track fit to a sample of CCQE $\nu_{e}$ ($\nu_{\mu}$) interactions with
energies following those expected from the NuMI beam. The resolutions were
calculated from the width of the distributions shown in Figures 6 \- 9.

##  6 Particle Identification

The fact that the method is based on likelihoods naturally allows for the use
of likelihood ratios as particle identification (PID) variables. However, it
was found that other variables in addition to the likelihoods have some power
to separate event types. In addition, there is more power to be extracted from
the likelihoods by considering the charge and time components separately, as
opposed to simply adding the two components together.

For this reason, two artificial neutral networks (ANNs) were developed using
the TMVA package [27] within the ROOT framework [28]. The first of the
networks is used to perform a separation of electron-like and muon-like
interactions and is henceforth referred to as $\textrm{ANN}_{e\mu}$. The
second network, called $\textrm{ANN}_{NC}$, is designed to to discriminate
between electron-like and NC interactions. The list of variables used in the
networks is as follows:

  * 1.

$\Delta\ln\mathcal{L}$ between muon and electron hypotheses for time and
charge components

  * 2.

$N_{\textrm{hits}}$, the number of hit PMTs

  * 3.

$\frac{\Delta\ln\mathcal{L}_{\textrm{charge}}}{N_{\textrm{hits}}}$

  * 4.

Fraction of hits outside, within, and inside the central hole of the ring for
both muon and electron hypotheses

  * 5.

Fraction of predicted charge outside the ring for both muon and electron
hypotheses

  * 6.

Ratio of the total predicted charge to the total measured charge for both muon
and electron hypotheses

  * 7.

The ratio of the reconstructed energy to the total measured charge

  * 8.

$\textrm{ANN}_{NC}$ only: reconstructed track direction under the electron
hypothesis

  * 9.

$\textrm{ANN}_{NC}$ only: fraction of hits in the downstream half of the
detector

The two $\textrm{ANN}_{NC}$ only variables reflect the fact that NC events are
typically less collimated along the neutrino beam axis, but do not provide any
information for the other ANN. The two networks each produce a single output
parameter that can be used to discriminate between the different hypotheses.
These outputs are referred to as $a_{e\mu}$ and $a_{NC}$ for
$\textrm{ANN}_{e\mu}$ and $\textrm{ANN}_{NC}$, respectively. The variables are
distributed in such a way that a value close to one suggests that the event is
signal-like, where the signal in both cases is electron-like, and a value
close to zero suggests the event is background-like.

A sample of CCQE $\nu_{e}$ and CCQE $\nu_{\mu}$ interactions was used to train
the $\textrm{ANN}_{e\mu}$ classifier so that the ANN was trained on the
cleanest subset of CC interactions. Similarly, the $\textrm{ANN}_{NC}$
classifier was trained on a sample of CCQE $\nu_{e}$ and NC interactions. The
same preselection described in Section 5 was applied to both the training and
testing samples. An additional requirement on the $\textrm{ANN}_{NC}$ training
samples was that the output variable from $\textrm{ANN}_{e\mu}$, $a_{e\mu}$ is
greater than 0.8. This additional requirement was to ensure that the
$\textrm{ANN}_{NC}$ network did not try to predominantly remove CC
$\nu_{\mu}$-like events.

###  6.1 Particle identification performance

In order to study the PID performance, samples of CC $\nu_{e}$, CC $\nu_{\mu}$
and NC $\nu$ interactions were processed through the reconstruction algorithm
twice, each time using a single track fit. On the first pass they were all
fitted under the electron track hypothesis, and on the second pass they were
fitted assuming the muon hypothesis. Fitting each event under both hypotheses
provides the basis to distinguish which hypothesis is the more likely.

The value $\epsilon\mathbb{P}$, where $\epsilon$ is the CC $\nu_{e}$ selection
efficiency and $\mathbb{P}$ is the CC $\nu_{e}$ selection purity, was
calculated over a grid of cut values on the outputs of the
$\textrm{ANN}_{e\mu}$ and $\textrm{ANN}_{NC}$ classifiers and is shown in Fig.
10. The cut values that maximise the figure of merit are shown as the green
star such that the requirements for an event to be selected as a $\nu_{e}$
candidate event are $a_{e\mu}>0.9$ and $a_{NC}>0.75$.

Figure 10: The value of CCQE $\nu_{e}$ efficiency $\times$ purity (colour
axis) as a function of the cut value on the output variables of
$\textrm{ANN}_{e\mu}$ and $\textrm{ANN}_{NC}$. The green star shows the
optimised cut position that maximises the figure of merit, with those events
falling in the upper right region selected as CC $\nu_{e}$ candidates.

The ANNs were trained to look for CCQE $\nu_{e}$ interactions, but in the
CHIPS neutrino spectrum CCQE interactions only make up approximately 20% of
the total CC sample for both $\nu_{\mu}$ and $\nu_{e}$ flavours. It is
therefore important to test the performance of these ANNs on the total
expected event samples that also include CC RES and CC DIS interactions, in
addition to the CCQE and NC events. Figure 11 shows the distribution of
$a_{e\mu}$ (left) and $a_{NC}$ (right), including the preselection and a cut
on the other ANN, for CC $\nu_{\mu}$, CC $\nu_{e}$ (and the CCQE subset) and
NC interactions. The relative number of events in each sample is set to match
the predictions for the beam composition at the CHIPS-10 detector. A
comparison of the shapes of the two ANN output variables shows that it is
relatively simple to separate the CC $\nu_{\mu}$ and CC $\nu_{e}$
interactions, but the NC background is more difficult to remove. It also shows
that whilst the networks were trained on CCQE interactions, even a single ring
fit can do well at identifying more complex CC $\nu_{e}$ interactions in order
to perform a counting experiment.

Figure 11: The output distribution from each of the ANNs, shown after the
application of the preselection and the optimised cut on the other ANN output,
for CC $\nu_{e}$, CCQE $\nu_{e}$, CC $\nu_{\mu}$ and NC interactions. The cut
on the other ANN was applied in order to show more clearly the events that
each network was designed to separate. The black line and arrow shows the
optimised cut value on the displayed network output variable used to selected
$\nu_{e}$ events.

A summary of the selection efficiency and purity is shown in Fig. 12 for the
requirement that each event must have $a_{e\mu}>0.9$ and $a_{NC}>0.75$ in
addition to passing the preselection. In particular, the efficiency is plotted
for the signal CC $\nu_{e}$ interactions, and also for the CCQE $\nu_{e}$
subsample. Additionally, the efficiencies for the CC $\nu_{\mu}$ and NC
backgrounds are also shown. This shows that even performing a simple single
track fit, a sample of CC $\nu_{e}$ interactions can be selected with
approximately 30% efficiency and 58% purity. The background rejection for CC
$\nu_{\mu}$ and NC interactions is 99.6% and 98.3%, respectively. The
comparison of the CCQE $\nu_{e}$ efficiency and the CC $\nu_{e}$ efficiency,
shown in black and blue respectively, shows that the selectors still work well
even for the more complex event topologies.

Figure 12: The efficiency and purity of various components with a selection
requiring $a_{e\mu}>0.9$ and $a_{NC}>0.75$ in addition to the preselection,
shown as a function of the reconstructed energy under the electron hypothesis.

####  6.1.1 Fitting $\pi^{0}$ events

A sample of simulated $\pi^{0}$ mesons were obtained by extracting them from
the standard GENIE NC interactions to ensure a representative energy
distribution, and were then passed through the simulation and reconstruction
chain. The events were fit under the $\pi^{0}$ hypothesis without constraining
the photon energies and produced the invariant mass distribution shown in Fig.
13, with the requirement that each reconstructed photon had an energy greater
than $150\,$MeV. A clear peak can be seen in the distribution, and a Gaussian
fit yields a mean of $130\pm 34\,$MeV in excellent agreement with the
$\pi^{0}$ mass $m_{\pi^{0}}=134.98\,$MeV [11]. It shows that the fitter is
capable of reconstructing $\pi^{0}$ mesons of the energy expected in the
CHIPS-10 detector even with only a 6% coverage of 3 inch PMTs.

The $\pi^{0}$ fitter will be integrated with the rest of the PID algorithms in
order to help further reduce the NC background to the $\nu_{e}$ appearance
search.

Figure 13: The invariant mass of the two photon tracks reconstructed using the
two track $\pi^{0}$ fit for all events where the two decay photons were
reconstructed with energies greater than 150 MeV.

##  7 Conclusions

A series of event reconstruction algorithms have been developed for the CHIPS
experiment, using charge and time information from all PMTs. The charge
component was based on the method developed by MiniBooNE, whereas the time
component was developed from first principles. The software was used to
perform a comparison between three geometry options and showed that a very low
6% photocathode coverage of 3 inch PMTs is a plausible option for CHIPS-10.
The low photocathode coverage of small PMTs is a vital part of minimising the
cost of the experiment and hence the demonstrated ability to successfully
reconstruct neutrino interactions under these conditions is a very important
result.

A simple single track fitting technique under both the electron and muon
hypothesis was tested on a representative sample of events predicted for
CHIPS-10 as exposed by the NuMI beam. Signal CC $\nu_{e}$ events were selected
with 30% efficiency and 58% purity. The particle identification was performed
using two neural networks: $\textrm{ANN}_{e\mu}$, trained to separate CCQE
$\nu_{e}$ events from CCQE $\nu_{\mu}$ interactions; and $\textrm{ANN}_{NC}$
designed to separate CCQE $\nu_{e}$ interactions from NC interactions. The
selection efficiency and background rejection compare well to the values
assumed in the CHIPS Letter of Intent [29].

The reconstruction software and particle identification capability will be
further studied in the future in order to optimise the physics capability of
the detector. Firstly, a $\pi^{0}$ fitter has been demonstrated to work on a
sample of $\pi^{0}$ interactions with representative energies of those
expected to be produced in NC interactions in the NuMI beam. The $\pi^{0}$
fitter will be integrated with the two PID ANNs in order to minimise the
background arising from NC interactions as well as those from CC $\nu_{\mu}$
events. Secondly, a technique has been developed in order to fit cosmic-ray
muon interactions that overlap with the beam interactions. Further studies
will determine the effects of the cosmic-ray muons on the reconstruction of
the beam interaction and determine the level of degradation of the various
reconstruction resolutions. Finally, additional methods of performing particle
identification will be investigated. For example, a Convolutional Neural
Network based event classifier using minimal event reconstruction has been
shown to give a 30% improvement in the $\nu_{e}$ event selection efficiency on
the NO$\nu$A experiment [30]. This technique draws from image feature
recognition, which should also be applicable to a water Cherenkov detector.

##  8 Acknowledgments

This work was supported in the UK by the Leverhulme Trust.

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