e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>e3e-mail<EMAIL_ADDRESS>
11institutetext: Physik Department E15, Technische Universität München, James-
Franck-Straße 1, 85748, Garching, Germany 22institutetext: Department of
Physics and Astronomy, University College London, Gower Street, London WC1E
6BT, UK

# Topologies of 76Ge double-beta decay events and calibration procedure biases

Tommaso Comellatoaddr1, e1  Matteo Agostiniaddr2, addr1, e2  Stefan
Schönertaddr1, e3

(Received: date / Accepted: date)

###### Abstract

The analysis of the time profile of electrical signals produced by energy
depositions in germanium detectors allows discrimination of events with
different topologies. This is especially relevant for experiments searching
for the neutrinoless double beta decay of 76Ge to distinguish the sought-after
signal from other background sources. The standard calibration procedures used
to tune the selection criteria for double-beta decay events use a 228Th
source, because it provides samples of signal-like events. These samples
exhibit energy spatial distributions with subtle different topologies compared
to neutrinoless double-beta decay events. In this work, we will characterize
these topological differences and, with the support of a 56Co source, evaluate
biases and precision of calibration techniques which use such event samples.
Our results will be particularly relevant for future experiments in which a
solid estimation of the efficiency is required.

††journal: Eur. Phys. J. C

##  1 Introduction

The search for neutrinoless double-beta decay ($0\nu\beta\beta$) is one of the
hottest topics in particle physics. Its discovery would unambiguously prove
that neutrinos are Majorana particles, i.e. they are their own antimatter
counterpart, and lepton number is not a global symmetry of the Standard Model
Agostini.2022 (1). $0\nu\beta\beta$ is not only predicted by our leading
theories of why neutrinos are massive particles, but also by those explaining
the matter-antimatter asymmetry of our universe rodejohann:neutrinos (2, 3,
4).

$0\nu\beta\beta$ is a nuclear transition in which two neutrons decay
simultaneously into two protons and two electrons, preserving the baryon
number but changing the lepton number by two units. Several isotopes have been
used to search for such transition, each offering different detection
techniques. Historically, 76Ge has always provided among the most stringent
constraints on the half-life of the process. Searches for $0\nu\beta\beta$ of
76Ge are carried out with high-purity germanium detectors built from material
isotopically enriched up to 92% in 76Ge. Such a detection concept offers
several advantages: Ge detectors have negligible radioactive internal
contaminations gerda:uraniumth (5), a per-mill energy resolution, and advanced
event reconstruction capabilities gerda.PSD.2022 (6). In addition, the high
density of germanium crystals ensures that the two electrons emitted in
$0\nu\beta\beta$ are absorbed within a few millimeters from the decay vertex,
generating well localized energy depositions which are fully contained within
the detector fiorini:0nbb (7). Ge detectors are also a well consolidated
technology, broadly used for gamma-ray spectroscopy and radioactivity
monitoring, and can be reliably produced in collaboration with industrial
partners.

$0\nu\beta\beta$ events in a Ge detector result in mono-energetic and well-
localized energy depositions. The whole decay energy is transferred to the two
emitted electrons, whose summed kinetic energy is equal to the 76Ge Q-value,
i.e. $Q_{\beta\beta}$$=$2039 keV. The two electrons are likely to share evenly
the decay energy, but extreme cases in which one electron takes most of the
energy are also possible. Electrons at these energies have an absorption
length of about a millimeter in Ge. However, secondary Bremsstrahlung photons
produced during the electron absorption can occasionally travel several
millimeters from their production vertex before interacting, producing
secondary energy deposition sites. Thus, the topology of $0\nu\beta\beta$
events is complex and its understanding is fundamental to develop techniques
for discriminating the sought-after signal from backgrounds such as gamma-rays
scattering multiple times within the detector.

In this manuscript, we first characterize the energy spatial distribution of
$0\nu\beta\beta$ and other classes of events occurring in Ge detectors. As all
events result in the generation of primary electrons/positrons, we first study
the energy deposition of single-electron events as a function of the electron
kinetic energy (Sec. 2.1), and then evaluate the difference between them and
more complicated classes of events (Sec. 2.2). In Sec. 3 we establish a
connection between the event spatial distributions and the estimator used in
$0\nu\beta\beta$ experiments to discriminate signal-like events from
background. In Sec. 3.3 we first review how such estimator is typically
calibrated, and then evaluate the biases and precision of such calibration
procedures on the $0\nu\beta\beta$-tagging efficiency. Finally, in Sec. 4, a
56Co source will allow for an experimental determination of these biases.

##  2 Characterization of the event spatial distribution

###  2.1 Absorption of electrons in Ge

(a) width=0.47

(b) width=0.47

Figure 1: Energy depositions of a $2$ MeV electron in germanium, through
collisional LABEL:sub@subfig:electronWalk_noGamma and collisional plus
radiative LABEL:sub@subfig:electronWalk_wGamma losses. The color code
represents the time of each energy deposition, which flows from blue to
yellow. The orange circle is centered around the center of energy and its
radius is the $R_{90}$ parameter described in the text. The path was simulated
in 3 dimensions, and is reported here in cylindrical coordinates with respect
to the starting point, which was taken as origin.

(a)

(b)

Figure 2: $R_{90}$ distributions from absorption of monoenergetic electrons in
germanium, and their dependence on energy.

Electrons lose energy mainly through collisional losses, i.e. ionization and
excitation. As the energy loss occurs via interaction with orbiting electrons,
a large fraction can be lost in a single collision. For this reason, the
electron trajectory follows a tortuous path, where the energy loss per
collision is inversely proportional to the electron kinetic energy. An example
of path for an electron depositing $2$ MeV through collisions in Ge is shown
in Fig. 1a. Every point in the plot represents a collision, in which an energy
proportional to the area of the marker is transferred to the medium. The color
code marks the time of each energy deposition, which flows from blue to
yellow, with a time scale which is of the order of a few ps. The electron in
Fig. 1a moves from left to right almost undisturbed, until its energy is
comparable to that of the orbiting electrons. This occurs after about $1$ mm,
when the electron begins a Brownian motion.

Electrons can also lose energy radiating Bremsstrahlung gammas, which
occasionally travel up to few centimeters in Ge, producing secondary
interaction sites. This is shown in Fig. 1b, where a gamma-ray with an energy
of about $200$ keV travels several millimeters upwards before undergoing
multiple Compton scatterings.

Therefore, according to the case, energy depositions from electrons with the
same energy can produce very different spatial distributions. To characterize
them we use the $R_{90}$ parameter. This is defined as the minimum radius of
the sphere which is centered in the energy weighted average of all energy
depositions and contains $90\%$ of the total deposited energy. As shown in
Fig. 1, in the case of energy loss through collisions (Fig. 1a) this parameter
is smaller than $1$ mm, while it can increase by an order of magnitude if a
gamma-ray with sufficient energy is also emitted (Fig. 1b).

We can thus use the $R_{90}$ parameter to study the size of the spatial
distributions of electrons depositing energy in germanium and estimate both
qualitatively and quantitatively how they are affected by the emission of
Bremsstrahlung gammas. To this purpose, we ran a set of Monte Carlo
simulations using the MaGe software framework MaGe (8) and the Geant4 toolkit
geant4 (9). The simulated setup consisted of an inverted coaxial Ge detector,
which is the state-of-the-art geometry developed for the future
$0\nu\beta\beta$-experiment Legend Legend.pCDR (10). The detector
specifications and a validation of the simulation are discussed in Ref.
comellato:modelingIC (11).

Fig. 2a shows the $R_{90}$ distribution for single-electron events with an
initial kinetic energy of $2$ MeV. The distribution is characterized by a
Gaussian peak at $0.6$ mm and a tail of events extending to higher values. The
events in the tail are those in which Bremsstrahlung gammas are produced, as
shown by the $R_{90}$ distribution of the subsets of events in which a gamma-
ray with energy higher than $50$ keV or $200$ keV is emitted.

The event topology is affected by the electron initial kinetic energy, both in
the collisional and in the radiative sector. The centroid of the gaussian peak
and the quantiles of the distributions as a function of the kinetic energy are
shown in Fig. 2b. As the scattering length of electrons increases with energy,
the spatial distributions get on average broader. This is captured by the
position of the $R_{90}$ peak, which shifts from a value of $0.2$ mm at $1$
MeV, to $0.8$ mm at $2.5$ MeV. Naturally, the variance $\sigma_{R_{90}}$ of
the peak also increases, as reported in Tab. 1. The higher the energy of the
electron, the larger is the probability of a high energetic gamma-ray
emission. As a consequence, the quantiles of the distributions increase with
different dependences, with the tail extending to larger and larger $R_{90}$
values.

To characterize the Bremsstrahlung radiative process, we show in Fig. 3 the
probability of producing photons with energy larger than $200$ keV111$200$ keV
is the energy threshold which corresponds to photon scattering length of
$\approx$ $10$ mm in germanium as a function of the energy of the primary
electron. This increases linearly from $1\%$ to $4.5\%$ for kinetic energies
from $1$ to $2.5$ MeV. With the help of the color scheme and the marker area,
both indicating the $90\%$ quantile of the Bremsstrahlung energy spectrum,
Fig. 3 also shows that, with increasing primary energy, Bremmstrahlung gammas
are also more energetic. This means that, with increasing energy, not only the
spatial distributions get on average broader, but the fraction of
exceptionally enlarged ones increases, as well. This is reflected in $R_{90}$
distributions as a shift of events from the peak to the tail, as shown in Tab.
1, under the _peak-to-tail_ label.222This value is calculated taking the ratio
of the integral outside and inside the main $R_{90}$ peak. The delimiting
point was arbitrarily taken $4\sigma$ away from the centroid of the peak.

Energy | _$R_{90}$ peak_ | $\sigma_{R_{90}}$ | _peak-to-tail_
---|---|---|---
$[MeV]$ | $[mm]$ | $[mm]$ | $[\%]$
1.0 | 0.25 | 0.07 | 4.7
1.5 | 0.43 | 0.13 | 5.4
2.0 | 0.63 | 0.18 | 6.5
2.5 | 0.84 | 0.24 | 6.5
Table 1: Summary of the parameters of interest for the description of spatial
distributions of electrons in germanium, and their evolution with energy.
Figure 3: Probability of emitting a photon with energy greater than 200 keV,
as a function of the energy of the primary electron. Both the marker size and
color indicate the 90% quantile of the gamma-rays’ energy spectrum.

###  2.2 Event samples of interest for $0\nu\beta\beta$ searches

The experimental signature of $0\nu\beta\beta$ is an energy deposition of 2
electrons sharing the full $Q$-value of the decay ($Q_{\beta\beta}$=$2039$
keV). They are likely to share evenly the decay energy, though very asymmetric
cases are also possible. Their spatial distributions will hence resemble those
of single-electron events with energy in the range $1-2$ MeV. Fig. 4 shows the
$R_{90}$ distribution for $0\nu\beta\beta$ and single-electron events with
initial energy equal to $Q_{\beta\beta}$. The distributions have similar
shape, with the Gaussian peaks slightly shifted due to the fact that not one,
but two electrons are present in $0\nu\beta\beta$ events.

Every technique aiming to select $0\nu\beta\beta$ events must be calibrated
with samples with a topology which is similar to $0\nu\beta\beta$. Samples of
$0\nu\beta\beta$-like events are typically produced by irradiating Ge
detectors with a 228Th calibration source, which provides a 2.6 MeV gamma-ray
due to the decay of 208Tl. The $2.6$ MeV gammas can interact through pair-
production, creating an electron and positron that share the whole energy
minus the mass energy of the electron-positron pair, thus $1.6$ MeV. When the
positron thermalizes, it annihilates with an orbiting electron, emitting two
secondary gamma-rays. If both escape the detector, then the pair-production
events have a topology similar to that of two-electron events with shared
energy of $1.6$ MeV. For this reason, as shown in Fig. 4, the shape of their
$R_{90}$ distribution is similar to that of $0\nu\beta\beta$ events. Due to
the different energy of the two processes, however, the Gaussian peak of pair-
production events is shifted to lower $R_{90}$ values.

A second group of $0\nu\beta\beta$-like events is created by the 2.6 MeV
gamma-rays scattering only once within the detector. This second samples is
composed of single-electron events with energy values between zero and the
Compton edge at $2382$ keV. This sample, however, contains also multiple
Compton-scattered events, which typically account for about 50% of the sample
size. Fig. 4 shows the $R_{90}$ distribution of events with an energy around
$Q_{\beta\beta}$ from the Compton continuum generated by the $2.6$ MeV gamma-
ray of 208Tl. Its Gaussian peak matches that from $0\nu\beta\beta$ events,
though multiple Compton-scattered events now create a higher tail at high
$R_{90}$values. On the other side, pair-production events can be effectively
selected applying a narrow energy cut around the double-escape peak at $1592$
keV, but will still contain multiple-Compton scattered events at the level of
a few percent.

Figure 4: Comparison of the $R_{90}$ distribution from $0\nu\beta\beta$ events
(filled histogram) with a single electron with initial energy equal to
$Q_{\beta\beta}$ (empty blue histogram). The yellow and green histograms
extend the comparison to the samples of interest for $0\nu\beta\beta$
searches, namely $e^{+}e^{-}$ events at $1.6$ MeV (yellow) and Compton-
scattered events around $Q_{\beta\beta}$ (green). Histograms are normalized on
the total number of events in each sample.

$0\nu\beta\beta$ experiments have been using these samples to calibrate their
tagging techniques and evaluate the signal detection efficiency. The
difference between calibration samples and the actual $0\nu\beta\beta$ signal
might lead to biases which are difficult to evaluate and are the focus of the
next sections.

##  3 $0\nu\beta\beta$-events identification techniques

(a)

(b)

Figure 5: $A/E$ distributions from absorption of monoenergetic electrons in
germanium, and their dependence on energy.

###  3.1 Signal-background discrimination estimators

Energy depositions in a germanium detector create charge carriers that drift
in the detector volume following the electric field lines. Their drift induces
a current, which for small anode detectors exhibits a characteristic peak
shape when the charge carriers are collected at the anode. As the shape and
number of peaks changes according to the event topology, the analysis of its
time profile is a powerful tool to discriminate among different event types.

The $A/E$ technique is a time profile analysis DusanBEGe (12) used by leading
$0\nu\beta\beta$ experiments gerda.PSD.2022 (6, 13) to discriminate events
characterized by multiple interaction sites, which are typical of external
gamma background, and keep point-like interactions, typical of the
$0\nu\beta\beta$ signal. It is based on a single parameter ($A/E$), which is
the ratio between the amplitude of the current peak ($A$) and the total
deposited energy ($E$). Whenever energy is deposited in multiple sites, the
current exhibits several peaks whose height $A_{i}$ is proportional to the
energy $E_{i}$ deposited in every site. Compared with the case in which the
full energy is deposited in a single location, multiple site events exhibit a
lower $A/E$ value, as every peak value $A_{i}$ gets divided by the full energy
$E$.

###  3.2 Reconstruction of the event topology

To investigate the relationship between $A/E$ and event topology, the Monte
Carlo simulations discussed in the previous section have been post-processed
with SigGen Radware (14), a software package able to simulate the Ge detector
physics and the electrical signals expected for each event. Then, the
generated signals have been convolved with typical electrostatic noise
recorded from an actual setup (resulting in an energy resolution $\sigma_{E}$
of $1.2$ keV at $Q_{\beta\beta}$), and finally analyzed to calculate the $A/E$
value. The $A/E$ distribution are shown in Fig. 5a and 5b. Similarly to the
$R_{90}$ distribution, the single-electron events without Bremmstrahlung are
reconstructed at similar $A/E$ values creating a Gaussian peak in the
distribution. Bremsstrahlung events populate a tail that extends to lower
$A/E$ values. Fig. 5b shows that the peak in the $A/E$ distribution decreases
linearly with energy by about $1\%$ per MeV. The quantiles of the distribution
decrease faster and non-linearly with the energy. This feature has been
identified during this work for the first time, and its has an important
impact on the $A/E$ analysis which will be highlighted in the following
sections.

The correlation between $R_{90}$ and $A/E$ is explicitly shown in Fig. 6,
where the two parameters are plotted one against the other for single-electron
events with energy between 1 and 2.5 MeV. The main feature is the densely
populated band with $A/E$ values between $0.99$ and $1$, and $R_{90}$ between
$0.2$ mm and $0.8$ mm, representing the correlation between the narrow peaks
of $R_{90}$ and $A/E$. As $R_{90}$ gets larger, the energy depositions tend to
be less localized, in turn lowering the $A/E$ value. Outside the band, we
observe events with $A/E>1$, which are known to be generated by interactions
in the volume surrounding the anode Agostini_SignalModeling (15), and
Bremsstrahlung events, populating the region of $A/E<0.99$ and $R_{90}$$>0.3$
mm. This is consistent with the results discussed in Ref. schweisshelm:thesis
(16).

All small anode detectors will show an inverse proportionality between $A/E$
and $R_{90}$. Our results are hence of general interest and apply to a large
group of detector geometries. They are however qualitative, as the actual
strength of the inverse proportionality depends on the detector geometry and
the resulting electric field.

Figure 6: Correlation between $A/E$ and $R_{90}$ for absorption of electrons
between 1 and 2.5 MeV.

###  3.3 Calibration of $0\nu\beta\beta$-events identification criteria

The selection criteria used to extract a data sample with enhanced fraction of
$0\nu\beta\beta$ events consist on an energy-dependent two-sided cut on the
$A/E$ parameter, selecting all events in the peak of the distribution and
rejecting those in the tails. The cut values are inferred from dedicated
calibration measurements creating samples of $0\nu\beta\beta$-like events. As
anticipated in Sec. 2.2, such events are typically obtained by irradiating Ge
detectors with a 228Th source. The 2.6 MeV gamma-rays from its decay create
two classes of $0\nu\beta\beta$-like events:

  * •

_Single Compton Scattering_ : events in which the gammas scatter only once in
the detector, before leaving the active volume. In such cases the energy
between zero and the Compton edge at $2.4$ MeV is transferred to a single
electron

  * •

_Pair-production with double escaping gammas_ ($e^{+}e^{-}$): events in which
the $2.6$ MeV gammas interact through pair-production inside the detector
creating an electron-positron pair. If the two photons from the annihilation
of the positron escape the detector, the energy deposition occurs only through
the electron and the positron, which deposit $2614-2\cdot 511=1592$ keV in the
detector

The first class of events provides $0\nu\beta\beta$-like samples at different
energies, up to the Compton edge, and can be used to study the energy
dependence of the $A/E$ parameter highlighted in Fig. 5b. The standard
calibration procedures performs this operation by tracking centroid and width
of the $A/E$ peak for single Compton-scattering samples at selected energies
between $1$ and $2.4$ MeV. The energy dependence thus obtained is then used as
a correction, such that the $A/E$ centroids align on a constant value and
exhibit the same width. This standardized approach was conceived without
noticing that the centroid and the quantiles in the $A/E$ distributions have
different energy dependencies DusanBEGe (12, 6). On the other hand,
$e^{+}e^{-}$ events occur at a fixed energy, and the contamination of single
and multiple Compton-scattered events can therefore be limited to a few
percents by applying a strict energy selection. For this reason, the sample of
$e^{+}e^{-}$ events is used as main proxy to tune and fix the efficiency of
the $0\nu\beta\beta$-selection. Historically, this has been done by setting a
lower threshold on their $A/E$ distribution, which keeps $90\%$ of the sample
events.

###  3.4 Estimation of the $0\nu\beta\beta$-tagging efficiency

The reconstruction of the event topology with the $A/E$ parameter depends on
several factors, like the detector geometry and, to a certain extent, the
interaction position. Indeed, it is well-known that small anode detectors
feature events with amplified $A/E$ for events occurring in the volume
surrounding the anode Agostini_SignalModeling (15). Therefore, $A/E$
distributions have features which not only depend on the starting kinetic
energy, but also on the event sample. A detailed modeling of the calibration
and $0\nu\beta\beta$ samples is thus mandatory for a precise estimation of the
tagging efficiency of $0\nu\beta\beta$-events.

The fact that the standard calibration procedure tracks the $A/E$ distribution
peak centroid (and not the desired quantile) results in an energy-decreasing
tagging efficiency for $0\nu\beta\beta$-like events. To estimate the bias at
$Q_{\beta\beta}$, we simulated $e^{+}e^{-}$ events occurring exactly at
$Q_{\beta\beta}$ and followed the standard calibration procedure with 228Th.
Fixing the tagging efficiency of $e^{+}e^{-}$ events at $1.6$ MeV to $90\%$
yields an efficiency of $(86.5\pm 0.4)$% at $Q_{\beta\beta}$. Fig. 7 shows the
$A/E$ distributions of the two $e^{+}e^{-}$ samples, after the standard
correction of the energy dependence. As an effect of the calibration
procedure, the two centroids are centered around the same value (which is
arbitrarily set to $1$). At $A/E$ values around $0.95$, the distribution of
$e^{+}e^{-}$ at $Q_{\beta\beta}$ shows an excess of events. As seen in Sec.
3.2, this is due to a higher production of secondary Bremsstrahlung gamma-
rays, which increases the background-like character of the sample and
therefore decreases its overall tagging efficiency.

Moreover, calibrating the $0\nu\beta\beta$-tagging efficiency on a different
sample brings an additional bias. Indeed, being 76Ge homogeneously
distributed, so will its decay products, while $e^{+}e^{-}$ events are more
likely to occur on detector lateral surfaces and corners, where the
probability for the annihilation photons to escape detection is maximal. To
evaluate the impact of this geometrical difference, we used a Monte Carlo
sample of $0\nu\beta\beta$-events and followed the standard calibration
procedure with 228Th. This yields a tagging efficiency of $(85.2\pm
0.4)\%$333The value here shows some tension with what we reported in
comellato:modelingIC (11), as it has been obtained with slightly different
analysis routines., i.e. a further $1\%$ reduction. Fig. 7 shows also the
$A/E$ distribution of the simulated $0\nu\beta\beta$ events in comparison to
$e^{+}e^{-}$ at 1.6 MeV and at $Q_{\beta\beta}$. It shows that this further
reduction is a complex balance of features in the whole $A/E$ spectrum. First,
$e^{+}e^{-}$ events exhibit a higher tail at very low $A/E$ values. Being
present for $e^{+}e^{-}$ events at 1.6 MeV as well, it suggests that its
origin is intrinsically related to the dynamics of pair-creation. On the high
$A/E$ side, being $0\nu\beta\beta$s homogeneously distributed, they probe the
volume region surrounding the anode more significantly, which gives signals
with amplified $A/E$.

Figure 7: $A/E$ distributions of pair production events and $0\nu\beta\beta$.

##  4 Energy dependent event discrimination with a 56Co source

In the previous sections, we have seen that A/E distributions shift and change
shape with energy. Also, we have seen that the samples used in the standard
calibration procedures, i.e. $e^{+}e^{-}$ and Compton-scattered events,
generate A/E distributions which actually differ from that expected from
$0\nu\beta\beta$. Before the present work, these facts were, to some extent,
already known. Pulse Shape Simulations (PSS) were used in the state-of-the-art
experiments to evaluate the systematic uncertainties due to these differences
gerda.PSD.2022 (6). With a thorough validation, the values from PSS could
eventually be used as a central value for the $0\nu\beta\beta$-tagging
efficiency. However this has been avoided so far as tens of even hundreds
detectors of different shapes are operated in the setup, making the tuning and
validation of their individual modeling extremely challenging. In this section
we study the option of a data-driven calibration of the energy dependence
avoiding the bias discussed in the previous section. As it’s the most
significant contribution to the overestimation of the $0\nu\beta\beta$-tagging
efficiency with a value of 4% out of a total of 5%, this would reduce the
systematic uncertainties to 1%, without requiring a detector-by-detector
modeling. This could be achieved exploiting the decays of a 56Co source.

The 56Co source used in this work has been custom-produced by the Jagiellonian
University in Krakow and its results have been cross-checked with post-
processed Monte Carlo simulations. Its energy spectrum is characterized by
several high-energy gamma-lines up to $3.6$ MeV. For this reason, it was early
recognized as a valuable source to calibrate germanium detectors barker1967
(17, 18, 19). In our work, we considered five gamma-rays with energy higher
than 2.5 MeV and branching ratio higher than 1%, as the probability of
creating electron-positron pairs for them is high enough to give samples of
$e^{+}e^{-}$ events with a small background contamination. The energies of the
considered $e^{+}e^{-}$ samples are listed in Tab. 2 next to their parent
gamma-rays. Lying in the range between 1.5 and 2.5 MeV, they offer the
opportunity to investigate the energy dependent tagging efficiency of
$0\nu\beta\beta$-like events with many data points and additionally to infer
its value directly at $Q_{\beta\beta}$.

|  |  | Data | Simulations
---|---|---|---|---
E (FEP) | E (DEP) | BR | SP | SP
$[keV]$ | $[keV]$ | % | % | %
2598.5 | 1576.5 | 17 | 88.8 | (5) | 88.6 | (4)
3009.6 | 1987.6 | 1 | 86.0 | (30) | 87.4 | (27)
3202.0 | 2180.0 | 3 | 84.6 | (11) | 85.9 | (10)
3253.5 | 2231.5 | 8 | 85.6 | (6) | 85.0 | (5)
3273.1 | 2251.1 | 2 | 83.9 | (18) | 85.4 | (16)
Table 2: Energies of the Double Escape Peaks (DEPs) used in our 56Co analysis
and respective Survival Probability (SP) after $A/E$ cut. Informations on the
energy and Branching Ratio (BR) of the relative Full Energy Peaks (FEPs) are
also given as reference. Figure 8: Drawing of the inverted coaxial detector
used for the experimental data with the 56Co source, inside the vacuum
cryostat provided by the vendor. Marked in yellow is the Boron-implanted p+
electrode and in blue the Lithium-diffused n+.

The detector used to record data with this source is shown in Fig. 8. It is a
$1.6$ kg inverted coaxial detector, which has been operated inside its
vendor’s vacuum cryostat (thus approaching the temperature of $77$ K) and
whose geometry is the one used as a reference for the simulations of the
present work. More details on the detector and on the validation of the
simulations can be found on our previous work comellato:modelingIC (11). The
source, which had an activity of $90$ kBq at the time of data taking, was
placed $20$ cm away from the cryostat, and the resulting energy spectrum is
shown in the top panel of Fig. 9, in the range where the $e^{+}e^{-}$ events
occur. The empty grey and filled blue histograms show the spectrum
respectively before and after the $A/E$ cut calibrated on 228Th. The ratio
between the two is shown in the middle panel, where the $6$ peaks from
$e^{+}e^{-}$ events clearly arise from the Compton continuum. The bottom panel
shows how all the events from the decays of 56Co populate the $A/E$ spectrum.
Its description is analogous to Fig. 6: the high density horizontal band
centered around $1$ represents all the energy depositions which occurr in a
single location, while the band below contains all the events which deposit
energy in more than one site, and the region where $A/E$ is higher than $1$
represents those which occur in the volume surrounding the anode. In this
spectrum, $e^{+}e^{-}$ and single Compton events are distributed around
$A/E=1$, while multiple Compton scatterings populate the region where $A/E<1$.

Figure 9: Region of interest of the energy spectrum (top) obtained from a 56Co
source. The middle panel shows the Survival Probability (SP) of every energy
bin after an $A/E$ cut calibrated on a 228Th source. The bottom panel
illustrates how the events in this energy range are distributed in the $A/E$
space. The position of the $e^{+}e^{-}$ events in the energy spectrum is
marked with a solid gray line.

The $e^{+}e^{-}$ tagging efficiency is extracted for experimental (simulated)
data following the standard calibration procedure with 228Th described in Sec.
3.3. This yields the results listed in the fourth (fifth) columns of Tab. 2,
and shown with the light blue squares (dark blue circles) in Fig. 10.

Figure 10: Survival probability of $e^{+}e^{-}$events from 56Co after $A/E$
cut. Light blue squares indicate results from simulated data, dark blue
circles from simulations.

The agreement between simulated and experimental data is within the
statistical uncertainties, which corroborates the results of our simulation
campaigns. In both datasets, the fraction of tagged events is systematically
lower than 90%, even at $1576$ keV. The tagging efficiency for this peak is
$1\%$ smaller than the value at $1592$ keV from the calibration on 228Th. This
comes from the complexity of the 56Co spectrum (which exhibits many features
in the $A/E$ space, see Fig. 9), which prevents the analysis from selecting a
completely pure sample of $e^{+}e^{-}$ events. For this reason, we take $1\%$
as systematic uncertainty of the method. Despite the small offset, the tagging
efficiency decreases as a function of energy. A linear interpolation of these
results yields a value of $(86.3\pm 0.4)\%$ at $Q_{\beta\beta}$ for
experimental data and $(86.2\pm 0.3)\%$ for simulations, which is in good
agreement with the results on the $e^{+}e^{-}$ events directly at
$Q_{\beta\beta}$ of Sec. 3.3.

##  5 Discussion and conclusions

In this paper we characterized the spatial distributions of electron energy
depositions in germanium and studied how they are affected by the initial
kinetic energy of the electron. We identified a connection between the spatial
distributions and the estimator used in $0\nu\beta\beta$ experiments to
discriminate single and multiple interaction sites, i.e. the $A/E$ parameter.
We used this information to review the standard calibration procedures with a
228Th source. In particular, we quantified that the standard calibration
procedures used so far by experiments result in two biases in the
determination of the $0\nu\beta\beta$-tagging efficiency. The first is an
energy-dependent overestimation of the efficiency at the level of a few
percents (4% for our simulated geometry). The second is a 1% contribution due
to using a different event sample than $0\nu\beta\beta$. Additionally, we
estimated the effect of different electronics noise levels on the
discrimination efficiency of $0\nu\beta\beta$ events. A total reduction of 3%
is obtained when an electronics noise at the level of the state-of-the-art
experiments (resulting in an energy resolution $\sigma_{E}$ of $1.5$ keV at
$Q_{\beta\beta}$) is used444Though an increase in the $0\nu\beta\beta$-tagging
efficiency is desirable, this comes at the expense of a higher increase in the
acceptance of background, as also stated in comellato:modelingIC (11). In
future experiments, which aims at intermediate noise levels between that of a
single detector and of present large-scale experiments, a
$0\nu\beta\beta$-tagging efficiency within $85\%$ and $87\%$ could therefore
be foreseen.

Though these biases are taken into account in the systematics uncertainties of
the state-of-the-art experiment, we showed that, with a 56Co source, these can
be strongly reduced from 5% to 1%. Furthermore, as 56Co provides samples of
almost pure $0\nu\beta\beta$-like events in a broad energy range, new
calibration routines could be developed for the correction of the $A/E$ energy
dependence, resulting in a bias-free tagging efficiency of signal-like events
at all energies. As three out of five $e^{+}e^{-}$ events of 56Co come from
very low branching ratio gamma-rays, a calibration with sufficient statistics
to observe the $2-4\%$ reduction in the tagging efficiency requires very long
acquisition times. For future-generation experiments, which are scheduled to
acquire data for a decade, the required counting statistics could be gained
with multiple or periodical calibration campaigns. Given 56Co’s half-life of
only $77.3$ days, this would require a regular production of radiation sources
and a calibration plan optimized for this application, which is beyond the
scope of this work.

###### Acknowledgements.

We are very grateful to Grzegorz Zuzel and the Jagiellonian University in
Krakow for providing the valuable 56Co source for this work. We are also
thankful to all the members of the Gerda and Legend collaborations, and in
particular to David Radford, for their valuable feedback. This work has been
supported in part by the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (Grant agreement No.
786430 - GemX), and by the Science and Technology Facilities Council, part of
U.K. Research and Innovation (Grant No. ST/T004169/1). M.A. acknowledges also
the support of the UCL Cosmoparticle Initiative.

## References

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# Hyper-Drive: Visible-Short Wave Infrared Hyperspectral Imaging Datasets for
Robots in Unstructured Environments

###### Abstract

Hyperspectral sensors have enjoyed widespread use in the realm of remote
sensing; however, they must be adapted to a format in which they can be
operated onboard mobile robots. In this work, we introduce a first-of-its-kind
system architecture with snapshot hyperspectral cameras and point
spectrometers to efficiently generate composite datacubes from a robotic base.
Our system collects and registers datacubes spanning the visible to shortwave
infrared (660-1700 nm) spectrum while simultaneously capturing the ambient
solar spectrum reflected off a white reference tile. We collect and
disseminate a large dataset of more than 500 labeled datacubes from on-road
and off-road terrain compliant with the ATLAS ontology to further the
integration and demonstration of hyperspectral imaging (HSI) as beneficial in
terrain class separability. Our analysis of this data demonstrates that HSI is
a significant opportunity to increase understanding of scene composition from
a robot-centric context. All code and data are open source online:
https://river-lab.github.io/hyper_drive_data

Index Terms—  hyperspectral imaging, robot spectroscopy, multi-modal sensing,
terrain segmentation

##  1 Introduction

Mirroring human-level terrain perception in robots is area of active research,
given its criticality in enabling actionable intelligence prior to traversing
the surface. Terrain can be best understood as a type of abstract material of
unspecified extent [1]. Unlike objects defined by regular geometric
properties, terrain challenges traditional perception systems because surface
materials vary widely in their size and shape. For instance, a farm field and
patch of soil have the same texture and micro appearance, but cover
drastically different geographic areas. Coarse labels like grass, soil, and
sand are useful in semantic segmentation, but intra-class differences affect
traversability. On road surfaces, these features might manifest as oil slicks,
standing water, or black ice.

Fig. 1: Hyper-Drive system mounted to off-road mobile robot, with sample data
representations of white reference target from a) the Visible to Near Infrared
(VNIR) hyperspectral camera b) Shortwave Infrared hyperspectral camera c) High
resolution RGB camera d) Combined point spectrometers.

Our approach to this problem builds on our previous work in terrain
classification with point-based spectroscopy and multi-modal sensing [2]. This
present work introduces hyperspectral imaging (HSI) to mobile robot terrain
understanding. In this research, we develop a system, HYPER DRIVE, shown in
Fig. 1, to capture datacubes from a moving platform with variable
illumination. Our work shows forward-facing HSI is a powerful tool for
robotics and is useful in a variety of terrain conditions. The contributions
of this paper are as follows.

  * •

Development of a sensing system to incorporate Visible to Short Wave Infrared
hyperspectral cameras onto a mobile robot with solar reference spectrum.

  * •

Open-source software framework and message types implemented for the Robot
Operating System [3]

  * •

Multi-modal dataset containing registered imaging products and ambient solar
spectra data for system.

##  2 RELATED WORK

Hyperspectral data acquired from moving vehicles is still in its infancy. The
recent innovation of snapshot hyperspectral cameras with fast integration
times has made it possible to capture images of a nonplanar scene, while
either the camera or objects are moving relative to each other. The absence of
artifacts like motion blurring is critical to obtain representative spectra
and distinct spatial characteristics.

###  2.1 Hyperspectral Terrain Datasets

There have been multiple efforts to develop vehicle-mounted hyperspectral
cameras to collect datacubes from off-road [4, 5], on-road [6, 7, 8]. Notably,
all the aforementioned examples make use of Visible-Near Infrared (VNIR)
cameras, which are well-suited to detect vegetative properties, but do not
have the same insight into the numerous absorption bands in the shortwave
infrared spectrum [9]. Unlike RGB image terrain segmentations which have
adopted standard label sets such as the widely-used KITTI classes [10], HSI
datasets are contextually driven with labels largely chosen by the operating
range of the camera. Examples of semantic label sets from the literature
include (dataset name bolded):

  * •

HSI-Drive: Road, road marks, vegetation, painted metal, sky,
concrete/stone/brick, pedestrian/cyclist, water, unpainted metal,
glass/transparent plastic [6].

  * •

Winkens et al. Drivable, rough, obstacle, sky[11]

  * •

HyKo Plastic, soil, paper, street, wood, chlorophyll, metal, sky [5].

  * •

Jakubczyk et al. Ground road, forest road, asphalt road, grass, forest [4].

  * •

Hyperspectral City V1.0: Car, human, road, traffic light, traffic sign, tree,
building, sky, object [7].

  * •

HSI Road: Road, background [12].

[6] contains annotations for the time of day and the time of year and
aggregations of classes into categories including drivable / non-drivable,
drivable / road markers / vehicles and drivable / road markers / vehicles /
pedestrians [6]. Similarly [5] included multi-modal sensors including a VNIR
spectrometer and Light Detection and Ranging (LIDAR) sensors in their HyKo
dataset, but did not elaborate on how the spectrometer could benefit the
system calibration [5]. HyKo also contains condensed annotations on
drivability classes (rough, sky, obstacle, drivable).

###  2.2 Spectral Informed Terrain Understanding

[11] shows the potential of HSI in terrain classification, even with simple
machine learning methods such as random forest, and later through manual
feature extraction [13].

[8] claims HSI overcomes challenges induced by object metamerism. RGB images
are empirically shown to have a lower degree of separability than HSI
datacubes. The authors exploit this information in a semantic segmentation
network, with a finetuning module leveraging 10 classes that combine semantic
purpose and material information.

[14] explores the use of RGB imagery alone to perform material segmentation
with transform-based neural networks. They also propose a dataset called
KITTI-Materials, containing 1000 frames with 20 different material categories
[14]. The same group also augmented RGB imagery with NIR and polarization
images to improve classification accuracy on materials such as metal and water
[15].

In our previous work, we demonstrated high accuracy in terrain classification
by measuring the spectral signatures of terrain near the contact point on
wheeled vehicles [2]. Combining feature-specific neural networks for IMU and
RGB image classification through a fusion network provided increased
performance. The results of [16] show that further fusing visual imagery and
NIR spectral signatures aids in the classification of urban road surface
types.

Fig. 2: Quantum efficiencies of each of the spectrum sensing devices used in
these experiments. Note: the Fabry–Pérot filters allow a spectral response
typically characterized by a primary transmission peak, followed by a smaller
secondary peak. This smaller feature is corrected for in the hypercube
demosaicing operation.

##  3 System Architecture

###  3.1 Imaging System

For this research we integrated two snapshot hyperspectral cameras onto a
reconfigurable platform. Both sensors are manufactured by IMEC. The VNIR
camera captures light from 660-900 in a 5$\times$5 Fabry–Pérot filters placed
in front of a silicon-based photodetector element. When corrected, these
filters produce a data cube with 24 spectral bands. Similarly, the IMEC SWIR
camera captures wavelength information from 1100-1700 nm using a 3$\times$3
filter array. These filters are placed on an uncooled Indium Gallium Arsenide
(InGaAs) photodetector. Unlike the VNIR camera, which has filters evenly
spaced across the sensitive spectral range, the SWIR camera concentrates its
bands in the lower range of the wavelength spectrum, where there are more
prevalent spectral absorption features. Combined, our hyperspectral sensing
solution features 33 channels, covering 1100 nm of the electromagnetic
spectrum. The cameras are housed inside a 3D printed Onyx housing weatherized
with epoxy. An uncoated Gorilla Glass shielding glass allows minimal
perturbation of the light entering the lens.

The hyperspectral cameras are coaligned with a 5 megapixel RGB machine vision
camera (Allied Vision). This system provides a high-resolution spatial
reference containing the hyperspectral cameras’ full field of view.

The hyperspectral cameras have a primary FOV of $25^{\circ}$. Each is
calibrated using a precision checkerboard target board. The individual bands
of the camera are then radially undistorted. The combined hyperspectral
datacube is generated by calculating projective transforms through the
checkerboard images, using bands that reveal the highest contrast between the
board’s squares. The combined datacube dimension is $1012\times 1666\times
33$; each cube is registered with the RGB camera.

###  3.2 Point Spectrometer

The protective water-proof computer housing also contains two point
spectrometers (Ibsen). The Pebble VIS-NIR, with a silicon detector, is
sensitive between 500 and 1100 nm with 256 spectral pixels; the Pebble NIR,
with an uncooled InGaAs detector, is sensitive from 950 to 1700 nm with 128
pixels. The two spectrometers have an overlap in the spectral signature range,
which is advantageous because of the decreased quantum efficiency of the VIS-
NIR spectrometer at wavelengths greater than 950 nm. As evidenced by Fig. 2,
we leverage the greater efficiency of the InGaAs detector, and truncate the
VIS-NIR device at wavelengths less than 950 nm. Together, the spectrometers
cover a greater spectral range than the hyperspectral cameras and with an
increased sensitivity. The InGaAs sensors are an order of magnitude less
sensitive than the silicon-based devices, resulting in integration times that
are appropriately larger.

The point spectrometers are coupled to low OH (Hydroxl) group fiber optic
cables (Thor Labs). This allows the spectrometers to remain inside the
weatherized housing, while still sensing light from the outside. The fiber
optic cables are connected to an SMA fitting, which holds the end ferrules
offset 4 cm above a 99% Spectralon white reference target (Labsphere). The
spectrometers measure a white reference signal data under the current
illumination conditions, both natural and artificial. This reference signal
will be used to dynamically generate reflectance calibrations in future work.

###  3.3 Messaging and Computation

Snapshot hyperspectral datacubes produced by the system are $\approx$20
megabytes. At its maximum operating rate, the system can generate nearly 1
gigabyte of data per second. To expedite data processing, the onboard compute
system leverages an Intel Core i7 processor with 3 terabytes of SSD storage.

The Robot Operating System [3] is used to synchronize data collection from all
the onboard systems. The three cameras are time-synchronized, so the RGB,
VNIR, and SWIR images all correspond to the same scene at a rate of 10 Hz. All
custom ROS compatible drivers, data structures, and algorithms are implemented
in Python and C++ and are included as an open-source resource in the project
repository. ROS enables time-synchronized data from the camera system and
point spectrometers to be acquired, even when device drivers operate at
different frequencies. Spectrometer messages contain time stamps, wavelength
values, and raw digital counts, in addition to optional metadata fields, such
as ambient humidity and device temperature. Hyperspectral data are transmitted
as flattened 1-D arrays, with the dimensions of the 3-D cube, as well as meta-
data fields for the central wavelengths, quantum efficiencies, and full-width
half-maximum values. These message structures are extensible to other
manufacturers, with examples available in the project repository.

Fig. 3: a) Warthog collection vehicle traversing grass environment with dense
surrounding vegetation. b) Ground-truth ATLAS segmentation labels overlayed on
high-resolution RGB image. c) Reduced order labels for drivability
assessments. Fig. 4: t-SNE two-dimensional embedding visualization of images
of selected dataset image from RGB data (left) and HSI data (right).

##  4 HYPER DRIVE DATASET

###  4.1 Dataset Construction

We conducted a field data collection at the autonomous systems track on the
campus of Olin College. The compute and sensor systems were mounted atop a
Warthog unmanned ground vehicle (Clearpath Robotics). The Warthog’s large
ground clearance and tire treads make it an ideal candidate for mobility in
both on-road and off-road terrain.

Data were collected over two days including late afternoon, during the setting
sun, early morning sunrise, and midday. This temporal variation increases the
variation of solar illumination and image intensity. The vehicle was driven on
4.0 kilometers of paved roads, hiking trails, and dirt roads. We downsampled
the data collection to 1 Hz to ensure scene differentiation between samples.
The published dataset contains the following structure:

  * •

ROS bag file containing compressed time synchronized datastream with all
following datatypes (.bag)

  * •

RGB image file registered to datacube (.png)

  * •

Hyperspectral datacube compressed in 3-D array (.npz)

  * •

White reference spectra from spectrometers (.npy)

  * •

Image segmentation masks (.png)

###  4.2 Annotations

Previous hyperspectral datasets suffer from a lack of consistent labels,
making it difficult to compare the performance of classification algorithms
between datasets. We address this problem by adopting the All-Terrain Labelset
for Autonomous System (ATLAS) [17]. ATLAS provides an extensible, hierarchical
ontology to generate fine-grain or coarse labels for off-road vehicle data.
Each datacube has a set of labels for the whole scene:  biome, time of day,
season, weather}. Additionally, instance labels mark the presence of specific
features in the image. At the highest level, these categorizations include:
{landscape, vegetation, animal, person, obstacle, atmospheric}. Each of these
labels can be decomposed into more specific classes or simplified to binary
labels such obstacle or landscape. Fig. 3 shows images from the dataset
overlayed with ATLAS labels.

As part of the initial data release, we provide 12,874 datacubes and RGB
images collected from the various data collections. 500 of these images have
been finely labeled with segmentation masks. To the best of the authors’
knowledge, this is the largest and most diverse vehicle-centric hyperspectral
dataset and the first to include shortwave infrared information. Table 1
contains an extracted breakdown of the full label information as a function of
the hierarchical classes.

Table 1: Class Structure Statistics in HYPER DRIVE Dataset Level #1 Label | Level # 2 Label | # Segments | # Images
---|---|---|---
Path | Dirt | 198 | 144
Rock/Gravel | 303 | 213
Paved | 143 | 116
Concrete | 117 | 92
Vegetation | Ground Cover | 806 | 464
Bush/Tree | 795 | 503
Leaves/Mulch | 233 | 158
Obstacle | Vehicle | 92 | 68
Infrastructure | 241 | 181
Road Signage | 127 | 98
Person | - | 15 | 15

##  5 DISCUSSION

As a motivating example for why this dataset is beneficial in robotic terrain
analysis, we consider the inter-class differences of the data. Fig. 4 shows a
t-SNE [18] separability analysis conducted on the data. t-SNE attempts to find
a two-dimensional embedding for the natively high-dimensional hyperspectral
datacube. The plot on the left shows the separability from the RGB color space
alone. The plot on the right is generated from the hyperspectral datacube.

From these embeddings, there is a clearer decision boundary between the
dominant classes in the HSI t-SNE embedding. We also observe that there are
more distinct groupings through the RGB data. There is still a significant
amount of overlap between the classes in this two-feature representation,
which is to be expected given the large number of classes (10) present in
these data. The RGB plot contains a significant number of outliers in the
clusters, especially in the “dirt” class. These tighter clusters suggest there
are natural distinctions amongst the classes that semantic segmentation
networks will exploit to generate more accurate classifications.

##  6 CONCLUSION

In this work, we presented a novel system architecture for collecting and
associating snapshot hyperspectral data from a moving vehicle. We release a
large dataset of VIS-SWIR images that encompass operating conditions as seen
from a mobile robot according to the ATLAS ontology. The project also contains
open-source software to integrate HSI into robotics applications. Future
iterations of this system will generate fully normalized datacubes without
imaging a camera white-reference image by predicting the unobserved white
reference from the spectrometers’ ambient spectra. We hope the introduction of
a ROS framework for hyperspectral data and dissemination of our dataset will
encourage further research on the applicability of HSI to other unstructured
environments.

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points represent intersections between $\mathcal{C}_{4,*}$ and the other
(geometrically irrelevant) components. For every $m\geq 5$, the inflection
polynomial $P^{\ell}_{m}$ factors as $P^{\ell}_{m}=x^{(-m)\text{ mod
}3}\cdot(4z-1)P^{\ell}_{m,*}$, where
$\mathcal{C}_{m,*}\subset\mathbb{A}^{2}_{x,z}$ cut out by $P^{\ell}_{m,*}$ is
irreducible. The affine curve $\mathcal{C}_{m,*}$ has associated Newton
polygons

$\begin{split}\text{New}_{p_{1}}(\mathcal{C}_{m,*})&=\text{Conv}(v_{1},v_{2},v_{3},\delta_{m\text{
mod }6\in\\{1,2,3\\}}v_{4},v_{5})\text{ and}\\\
\text{New}_{p_{j}}(\mathcal{C}_{m,*})&=\text{Conv}(v_{2},v_{3},\delta_{m\text{
mod }6\in\\{1,2,3\\}}v_{4},v_{6},v_{7},\delta_{2|(m-1)}v_{8})\end{split}$

where $v_{1}=(2m-(2m\text{ mod }3),0)$,
$v_{2}=(4m+6\lfloor\frac{m-4}{6}\rfloor+\varphi_{1}((m-4)\text{ mod }6,0)$,
$v_{3}=(0,\varphi_{2}(m))$, $v_{4}=(3,\varphi_{2}(m))$,
$v_{5}=(0,\lfloor\frac{2m}{3}\rfloor)$,
$v_{6}=(0,\lfloor\frac{m-1}{2}\rfloor)$, $v_{7}=(m-2,0)$, and
$v_{8}=(1,\frac{m-3}{2})$. Here $\varphi_{1}(0)=-4$, $\varphi_{1}(1)=-2$,
$\varphi_{1}(2)=0$, $\varphi_{1}(3)=-1$, $\varphi_{1}(4)=1$, and
$\varphi_{1}(5)=3$; while $\varphi_{2}(3)=2$, $\varphi_{2}(4)=2$,
$\varphi_{2}(5)=3$, $\varphi_{2}(6)=4$, and
$\varphi_{2}(m)=4+\lfloor\frac{m-7}{6}\rfloor\cdot 5+(m-1)\text{ mod }6$ for
every $m\geq 7$. In particular, the delta-invariants $\nu_{j}^{m,*}$ of
$\mathcal{C}_{m,*}$ in the singularities $p_{j}$ are given by

$\nu_{1}^{m,*}=\frac{3\lfloor 2/3m\rfloor\cdot(\lfloor
2/3m\rfloor-1)}{2}\text{ and
}\nu_{j}^{m,*}=\lfloor\frac{(m-3)^{2}}{4}\rfloor,j\geq 2$

and $\mathcal{C}_{m,*}$ is of geometric genus

$g(\mathcal{C}_{m,*})=3(\varphi_{2}(m))^{2}-3\varphi_{2}(m)-1-\nu_{1}^{m,*}-3\nu_{2}^{m,*}+3\delta_{m\text{
mod }6\in\\{1,2,3\\}}(\varphi_{2}(m)-1)$

whenever $m\geq 5$.

######  Remark 5.9.

_Conjecture 5.8, along with our results for Legendre, Weierstrass, and $D_{4}$
pencils, suggests that over fields of characteristic zero, there is a tight
relationship between singularities of the total space of a pencil of
superelliptic curves and singularities of the associated inflectionary curve.
The singularities of the total space of a pencil depend, in turn, on the
singularity types that arise in fibers. In the case of Legendre and
Weierstrass pencils of hyperelliptic curves, any fiber has at-worst a single
node and is (geometrically) irreducible; while $D_{4}$ and $D_{6}$ bielliptic
families include fibers with multiple nodes or simple cusps, and may be
reducible. In the final section below we initiate an investigation of
inflectionary varieties derived from the full two-dimensional family (27) of
bielliptic curves, in which case the interaction between the singular loci of
the family and of the associated inflectionary surfaces is more subtle._

###  5.2. Inflectionary surfaces from bielliptic curves

As explained in [2], Jung’s method for desingularizing a surface $X$ is in two
steps, the first of which is to realize $X$ as a branched cover of a plane $Y$
and compute the embedded desingularization of the discriminant curve of the
associated projection $\pi:X\rightarrow Y$.111111In the second step, $X$ is
replaced by its blown-up analogue $\widetilde{X}$ with smooth discriminant;
the singularities of $\widetilde{X}$ are then isolated, and may be resolved
via a deterministic combinatorial procedure. While computing
desingularizations of the inflectionary surfaces derived from the
superelliptic analogues $y^{n}=x^{6}-s_{1}x^{4}+s_{2}x^{2}-1$ of the
bielliptic surface (27) is itself a natural problem, we will not attack the
desingularization problem in full here.121212Kulikov [14] has solved the
analogue of this problem for two-dimensional families of plane curves subject
to a genericity hypothesis. Rather, we will focus on the structure of the
discriminant curves $\Delta^{\ell}_{m}$ associated with the projection of the
inflectionary surfaces $(P^{\ell}_{m}=0)$ derived from (27) to the
$(s_{1},s_{2})$-plane. Note that $\Delta^{\ell}_{m}$ always contains the
discriminant $\Delta$ of the bielliptic surface (27).

We begin by describing the stratification of the (reduced scheme associated
with the) discriminant of (27) according to the singularity configurations
along the curves it parameterizes. According to equation (28), the reduced
discriminant is a quartic curve
$\Delta_{*}\subset\mathbb{A}^{2}_{s_{1},s_{2}}$. In fact, it is easy to see
that $\Delta_{*}$ has nodes in the points $(3\zeta^{j},3\zeta^{-j})$, where
$\zeta$ is a cube root of unity, and that these nodes are permuted by a cyclic
$\mu_{3}$-automorphism of $\Delta_{*}$. In particular, $\Delta_{*}$ is of
geometric genus zero. We now apply a classical algorithm of Max Noether using
adjoint curves (see, e.g., [19, Ch. 4]), to compute a parameterization for
$\Delta_{*}$. More precisely, we single out adjoint conics through the
singularities $(3\zeta^{j},3\zeta^{-j})$ and the smooth point $(-1,-1)$ of
$\Delta_{*}$; there is a pencil of these, parameterized by

$a_{t}(s_{1},s_{2},z)=ts_{1}^{2}+s_{1}s_{2}+(3t-6)s_{1}z-(t-2)s_{2}^{2}-3ts_{2}z-9z^{2}$

where $t\in\mathbb{P}^{1}$. Solving the system of equations

$\text{res}_{s_{1}}(a_{t},\Delta_{*})=\text{res}_{s_{2}}(a_{t},\Delta_{*})=\text{res}_{z}(a_{t},\Delta_{*})=0$

in which “res” denotes the resultant, we deduce that the closure of
$\Delta_{*}$ in $\mathbb{P}^{2}_{s_{1},s_{2},z}$ is parameterized by

(32)
$[s_{1}(t):s_{2}(t):z(t))]=[(t-2)(3t^{3}-6t^{2}+12t-8):t(3t^{3}-12t^{2}+24t-16):t^{2}(t-2)^{2}].$

From (32), in turn, it is easy to identify those points of $\Delta_{*}$
corresponding to curves with singularities locally over $\overline{F}$ of the
form $y^{n}=x^{m}$ with $m\geq 3$; indeed, these are precisely the solutions
of

(33) $f(t,x)=D_{x}f(t,x)=D^{2}_{x}f(t,x)=0$

where $f(t,x)=x^{6}-\frac{s_{1}(t)}{z(t)}x^{4}+\frac{s_{2}(t)}{z(t)}x^{2}-1$.
The system (33) has eight solutions, divided into two groups of four each for
$t=1\pm\sqrt{\frac{-1}{3}}$. It is furthermore clear from the presentation
(26) that these are the only special configurations possible.

####  5.2.1. Further components of the inflectionary discriminant

To conclude, we describe the components of $\Delta^{\ell}_{m}$ for small
values of $m$.

Case: $m=3$. In this case, the reduced subscheme of the inflectionary
discriminant decomposes as
$\Delta^{\ell}_{3}=\Delta_{*}\cup\Delta^{\ell}_{3,1}\cup\Delta^{\ell}_{3,2}$,
where $\Delta^{\ell}_{3,1}$ and $\Delta^{\ell}_{3,2}$ have defining equations
$4s_{1}-s_{2}^{2}=0$ and

$-78125-118125s_{1}^{3}+756s_{1}^{6}+318750s_{1}s_{2}-31050s_{1}^{4}s_{2}+204375s_{1}^{2}s_{2}^{2}-189s_{1}^{5}s_{2}^{2}-337500s_{2}^{3}+500s_{1}^{3}s_{2}^{3}=0$

respectively. Clearly $\Delta^{\ell}_{3,1}$ is a smooth rational curve. On the
other hand, the Newton polygon of $\Delta^{\ell}_{3,2}$ has 10 interior
lattice points, while the closure of $\Delta^{\ell}_{3,2}$ in the toric
surface whose underlying polygon is $\text{New}(\Delta_{*})$ is singular in
precisely 9 points, all of which lie in the dense open lous
$\mathbb{A}^{2}_{s_{1},s_{2}}$. Closer inspection shows that each of these is
a node; so $\Delta_{3,2}^{\ell}$ is of geometric genus 1.

Case: $m=4$. The (reduced) inflectionary discriminant decomposes as
$\Delta^{\ell}_{4}=\Delta_{*}\cup\Delta^{\ell}_{4,1}\cup\Delta^{\ell}_{4,2}\cup\Delta^{\ell}_{4,3}\cup\Delta^{\ell}_{4,4}$,
where $\Delta^{\ell}_{4,1}$ and $\Delta^{\ell}_{4,2}$ are smooth rational
curves defined by $s_{1}^{2}-4s_{2}=0$ and $s_{2}^{2}-4s_{1}=0$, while
$\Delta^{\ell}_{4,3}$ and $\Delta^{\ell}_{4,4}$ are defined by

$\begin{split}&-1125+4s_{1}^{3}+110s_{1}s_{2}-s_{1}^{2}s_{2}^{2}+4s_{2}^{3}=0\text{
and }\\\
&20796875+3429000s_{1}^{3}+52272s_{1}^{6}-13942500s_{1}s_{2}-235440s_{1}^{4}s_{2}-571350s_{1}^{2}s_{2}^{2}+1512s_{1}^{5}s_{2}^{2}+3429000s_{2}^{3}\\\
&+6220s_{1}^{3}s_{2}^{3}-235440s_{1}s_{2}^{4}-3645s_{1}^{4}s_{2}^{4}+1512s_{1}^{2}s_{2}^{5}+52272s_{2}^{6}=0\end{split}$

respectively. Remarkably, $\Delta_{*}$ and $\Delta^{\ell}_{4,3}$ share the
same Newton polygon, namely

$\text{New}(\Delta^{\ell}_{4,3})=\text{New}(\Delta_{*})=\text{Conv}((0,0),(3,0),(2,2),(0,3)).$

In particular, $\Delta^{\ell}_{4,3}$ is of arithmetic genus 3. On the other
hand, closer inspection shows that $\Delta^{\ell}_{4,3}$ is singular in the
points $(-5\zeta,-5\zeta^{-1})\in\mathbb{A}^{2}_{s_{1},s_{2}}$, which are
permuted by a cyclic $\mu_{3}$-automorphism of $\Delta^{\ell}_{4,3}$; in
particular, $\Delta^{\ell}_{4,3}$ is itself a singular rational curve.
Likewise, we have $\text{New}(\Delta^{\ell}_{4,4})=2\text{New}(\Delta_{*})$;
as $\text{New}(\Delta^{\ell}_{4,4})$ contains 17 interior lattice points, it
follows that the arithmetic genus of the closure of $\Delta^{\ell}_{4,4}$ in
$\text{Tor}(\Delta_{*})$ is 17. On the other hand, (the closure of)
$\Delta^{\ell}_{4,4}$ is singular in 15 points of
$\mathbb{A}^{2}_{s_{1},s_{2}}$, each of which is a node; so
$\Delta^{\ell}_{4,4}$ is of geometric genus 2.

Case: $m=5$. The (reduced) inflectionary discriminant decomposes as
$\Delta^{\ell}_{5}=\Delta_{*}\cup\Delta^{\ell}_{5,1}\cup\Delta^{\ell}_{5,2}\cup\Delta^{\ell}_{5,3}$,
where $\Delta^{\ell}_{5,1}=\Delta^{\ell}_{4,1}$ and $\Delta^{\ell}_{5,2}$ is
the smooth rational curve defined by $-8+4s_{1}s_{2}-s_{2}^{3}=0$, while
$\Delta^{\ell}_{5,3}$ is defined by

$\begin{split}&-47148698016885339-1856430918636308s_{1}^{3}+913227582180384s_{1}^{6}+23648174414208s_{1}^{9}+140560353536s_{1}^{12}+711244800s_{1}^{15}\\\
&+33089373317849562s_{1}s_{2}-4358733253684512s_{1}^{4}s_{2}-300895277832256s_{1}^{7}s_{2}-4020413538816s_{1}^{10}s_{2}-3180234240s_{1}^{13}s_{2}\\\
&+209782436304845s_{1}^{2}s_{2}^{2}+619503583440656s_{1}^{5}s_{2}^{2}+23057043202656s_{1}^{8}s_{2}^{2}+118060510464s_{1}^{11}s_{2}^{2}-265317120s_{1}^{14}s_{2}^{2}-5091695995399316s_{2}^{3}\\\
&+1335207549416408s_{1}^{3}s_{2}^{3}+15852912617504s_{1}^{6}s_{2}^{3}-519443690880s_{1}^{9}s_{2}^{3}+1139913216s_{1}^{12}s_{2}^{3}-1226356470049056s_{1}s_{2}^{4}\\\
&-237101028007542s_{1}^{4}s_{2}^{4}-2958769613112s_{1}^{7}s_{2}^{4}-12989609248s_{1}^{10}s_{2}^{4}+17200512s_{1}^{13}s_{2}^{4}+55678112275728s_{1}^{2}s_{2}^{5}+16933654916732s_{1}^{5}s_{2}^{5}\\\
&+147246610976s_{1}^{8}s_{2}^{5}-65461824s_{1}^{1}1s_{2}^{5}+241313946391584s_{2}^{6}-23675574320096s_{1}^{3}s_{2}^{6}-579913339582s_{1}^{6}s_{2}^{6}-2298341488s_{1}^{9}s_{2}^{6}-3304800s_{1}^{12}s_{2}^{6}\\\
&+2280390203328s_{1}s_{2}^{7}+688503977416s_{1}^{4}s_{2}^{7}+33223815960s_{1}^{7}s_{2}^{7}-3479328s_{1}^{10}s_{2}^{7}+3980066847328s_{1}^{2}s_{2}^{8}-89661598176s_{1}^{5}s_{2}^{8}-881911439s_{1}^{8}s_{2}^{8}\\\
&+1118124s_{1}^{11}s_{2}^{8}-4300540393088s_{2}^{9}-196377843072s_{1}^{3}s_{2}^{9}+3550852496s_{1}^{6}s_{2}^{9}-425286s_{1}^{9}s_{2}^{9}+286292024832s_{1}s_{2}^{10}+6808519392s_{1}^{4}s_{2}^{10}\\\
&-30020384s_{1}^{7}s_{2}^{10}+81s_{1}^{10}s_{2}^{10}-30735987456s_{1}^{2}s_{2}^{11}-11464256s_{1}^{5}s_{2}^{11}-468s_{1}^{8}s_{2}^{11}+29621700864s_{2}^{12}+210366976s_{1}^{3}s_{2}^{12}-1376s_{1}^{6}s_{2}^{12}\\\
&+227179008s_{1}s_{2}^{13}+9600s_{1}^{4}s_{2}^{13}+5376s_{1}^{2}s_{2}^{14}-50176s_{2}^{15}=0.\end{split}$

Here $\text{New}(\Delta^{\ell}_{5,3})=5\text{New}(\Delta_{*})$, and as
$\text{New}(\Delta^{\ell}_{5,3})$ contains 131 interior lattice points, it
follows that the arithmetic genus of the closure of $\Delta^{\ell}_{5,3}$ in
$\text{Tor}(\Delta_{*})$ is 131. On the other hand, (the closure of)
$\Delta^{\ell}_{5,3}$ is singular in 105 points of
$\mathbb{A}^{2}_{s_{1},s_{2}}$, each of which is a node; so
$\Delta^{\ell}_{5,3}$ is of geometric genus 26.

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