# Extracting low energy signals from raw LArTPC waveforms using deep learning
techniques — A proof of concept

Lorenzo Uboldi  David Ruth  Michael Andrews  Michael H. L. S. Wang
<EMAIL_ADDRESS>Hans-Joachim Wenzel  Wanwei Wu  Tingjun Yang  CERN, The
European Organization for Nuclear Research, 1211 Meyrin, Switzerland  Niles
North High School, Skokie, IL 60077, USA  Carnegie Mellon University,
Pittsburgh, PA 15213, USA  Fermi National Accelerator Laboratory, Batavia, IL
60510, USA

###### Abstract

We investigate the feasibility of using deep learning techniques, in the form
of a one-dimensional convolutional neural network (1D-CNN), for the extraction
of signals from the raw waveforms produced by the individual channels of
liquid argon time projection chamber (LArTPC) detectors. A minimal generic
LArTPC detector model is developed to generate realistic noise and signal
waveforms used to train and test the 1D-CNN, and evaluate its performance on
low-level signals. We demonstrate that our approach overcomes the inherent
shortcomings of traditional cut-based methods by extending sensitivity to
signals with ADC values below their imposed thresholds. This approach exhibits
great promise in enhancing the capabilities of future generation neutrino
experiments like DUNE to carry out their low-energy neutrino physics programs.

###### keywords:

Low-energy neutrinos , LArTPC , Triggering , Signal processing , Machine
learning , Convolutional neural networks

##  1 Introduction

The liquid argon time projection chamber (LArTPC) has been successfully
deployed in a number of recent and currently running neutrino experiments and
is the technology of choice for massive, next-generation neutrino experiments
like the Deep Underground Neutrino Experiment (DUNE) [1]. Born out of
combining the novel three-dimensional imaging capabilities of the time
projection chamber (TPC) [2] with the unique properties of liquefied noble
gases like liquid argon (LAr), it represents the modern, electronic equivalent
of the bubble chamber [3].

Unlike the latter, LArTPC detectors are “always-on” devices, continuously
detecting and acquiring signals induced by ionization charges on wire planes
at the end of the drift path. Furthermore, the electronic readout of these
signals from multiple wire planes with different angular orientations that
provide 2 spatial coordinates, together with a third determined from drift
times, enables the automated reconstruction of detailed topologies, while
simultaneously performing calorimetry from the integrated charge.

Alongside the unique benefits provided by the TPC are the excellent
characteristics of LAr which include high ionization and scintillation yields.
The ability to transport electrons efficiently across long drift distances due
to the vanishing electronegativity and low dispersion in LAr permits the high
spatial resolution possible with finely segmented wire planes. This allows
very large detectors like the DUNE far detectors [1] to be built. Finally, the
high density of LAr makes it an ideal neutrino detector due to the low
neutrino-nucleon interaction cross sections.

The powerful capabilities of the LArTPC make it an excellent choice for DUNE’s
long baseline physics program with goals that include determining the neutrino
mass hierarchy, observing CP violation in the lepton sector, and making
precise measurements of the oscillation parameters using the wide-band beam
from Fermilab [4, 5]. Performing these measurements with a LArTPC will not be
too challenging due to the relatively high incident neutrino energies, ranging
from hundreds of MeVs to a few GeVs, and the reduced background levels from
cosmogenic and atmospheric sources made possible by the deep underground
location of the far detectors.

Beyond its long-baseline program, DUNE’s physics goals also include the
detection of neutrinos from core-collapse supernovas, searches for nucleon
decay, studies of solar neutrinos, and atmospheric neutrino oscillation
studies to supplement the long-baseline measurements [6, 7, 8]. Of these, the
solar and core-collapse supernova neutrinos involve low energy neutrinos in
the 1 MeV (solar) to 10 MeV (supernova) range. Ionization from the products of
their interactions in the LAr can induce signals that are close to the noise
threshold, making them challenging to detect. This is further exacerbated by
the conventional approach of applying minimum ADC threshold cuts to
discriminate signal waveforms from noise which results in poor low-energy
efficiency.

|  |  |
---|---|---|---
(a) | (b) | (c) | (d)
Figure 1: Plots (a) and (b) show simulated wire waveforms for the
$\tensor[^{39}]{\mathrm{Ar}}{}$ sample in the induction and collection planes,
respectively. Plots (c) and (d) show simulated wire waveforms for the
isotropic electron sample in the induction and collection planes,
respectively. The first row in each plot shows the pure signal waveform,
followed by the noise waveform in the second row, and the sum of these two
components in the third row. Details on the simulation of these waveforms is
described in Section 2 of the text.

In this paper, we take a deep learning (DL) approach to address the drawbacks
of conventional threshold-based methods and to optimize the efficiency to low-
energy neutrinos. We develop deep learning techniques and apply them to the
raw waveforms from individual LArTPC wires, to detect the presence of a signal
and narrow down its location in the full waveform in terms of a region of
interest (ROI). While DL methods have been applied to LArTPC data, they tend
to be applied at later stages in offline reconstruction and on 2D “images”
based on wire plane views. To our knowledge, this is the first attempt to
apply such methods directly to the raw waveforms associated with single LArTPC
wires. This implies potentially promising applications of this method in low-
level filtering and triggering in online DAQ systems.

In the discussion that follows, we begin by describing a minimalist but
realistic _toy_ LArTPC detector that we created and used to develop and test
our deep learning models. This is followed by a detailed description of the
neural network architecture. The model’s performance on training and
validation sets is presented. It is then tested on an independent sample to
determine various performance metrics including the detection efficiency as
functions of energy deposited and ADC counts produced. The main purpose of
this paper is to demonstrate a proof-of-principle for using deep learning
networks in extracting signals from raw LArTPC waveforms. After successfully
accomplishing this, we conclude by summarizing the results and discussing
future work and application of the method.

##  2 A minimal toy LArTPC detector

To test the idea of using DL-based methods for extracting signals from raw
LArTPC waveforms, we first developed a software simulation of a minimal _toy_
LArTPC detector. It consists of a LAr volume defined by a rectangular prism
measuring $50\times 50\times 180$ cm3, with the long dimension oriented along
the $z$ axis and the two shorter ones along the $x$ and $y$ axes, centered at
$(x,y)=(25,0)$. There are two instrumented anode wire planes lying in the
$y-z$ plane consisting of an induction plane at $x=0.075$ cm and a collection
plane at $x=-0.075$ cm. Both planes consist of 280 equally spaced parallel
wires with a pitch of 0.25 cm, and whose axes are oriented at $+60$°and
$-60$°with respect to the $y$ axis for the induction and collection planes,
respectively. A uniform electric field is oriented along the $x$ axis with a
field strength of 500 V/cm in the main drift region.

The propagation of particles through the LAr volume was simulated using the
Geant4 simulation toolkit [9, 10] with a step limit of 100 microns through the
_LArG4_ package in the LArSoft framework [11]. If the particle’s interaction
with the medium in a given step led to energy deposition, the amount of this
energy and the position of the interaction was recorded in a
_SimEnergyDeposit_ object. This information from the Geant4 simulation was
passed on to the next stage which simulated the propagation of electrons to
the readout planes and the response of the detector that ultimately produced
the digitized raw wire waveforms from the LArTPC electronics.

Figure 2: Architecture of the one-dimensional convolutional neural network
used to recognize signals in LArTPC waveforms.

The detector response was simulated by first calculating the initial number of
ionization electrons resulting from the energy deposition, based on the
ionization work function in LAr
($\nicefrac{{\mathrm{\\#\,electrons}}}{{\mathrm{GeV}}}=\nicefrac{{1}}{{2.36\times
10^{-8}\mathrm{GeV}}}$) [12]. Electron-ion recombination effects were taken
into account by multiplying this initial number by a factor determined from
the _box model_ of recombination [13] modified to extend into the low $dE/dx$
region [14]. Diffusion effects were then simulated by first splitting up the
total number of electrons after recombination into clusters of a given size.
The initial drift time for all clusters was calculated from the drift velocity
and the distance between the original energy deposition and the wire plane.
Diffusion in the longitudinal direction was simulated by smearing this drift
time according to a Gaussian distribution for each cluster. Diffusion in the
transverse direction was simulated by smearing the transverse positions of the
original energy deposition by a Gaussian distribution for each cluster,
providing their spatial coordinates at the wire planes. These smeared
coordinates were used to determine the wire closest to a cluster in a plane,
on which an induced signal would be simulated. The amount of energy and number
of electrons for that cluster were recorded in a _SimChannels_ object, and
were identified by the channel number corresponding to the wire. Only channels
associated with clusters were stored in the object, and the cluster
information was saved in the time bin (TDC) corresponding to its drift time.
The waveforms from all LArTPC wires were assumed to be digitized at a rate of
5.05 MHz (198 ns/sample), starting from the instant energy was deposited by
the Geant4 simulated tracks (assumed to all occur simultaneously), and lasting
for a duration corresponding to 2,048 samples.

To simulate the waveforms produced by the LArTPC wires, we modeled the field
response of the wire planes, as a function of $j$’th TDC bin, with a quadratic
($\propto j^{2}$) for the collection plane, and with an asymmetric sinusoid,
where the amplitude of the negative-going half cycle was 10% larger, for the
induction plane. The electronics response was modeled using the
parameterization described in Reference [15]. For each LArTPC wire, the charge
in each TDC bin, determined from the total number of ionization electrons
stored in the _SimChannels_ object for that bin, was convoluted with the field
response function for the appropriate wire plane and the electronics response
in order to produce what we will refer to as the _pure signal waveform_. Noise
was modeled by parameterizing the modulus $r$ of its complex frequency
components according to:

$\displaystyle r=|re^{-i\phi}|=p_{0}\times
e^{-\frac{1}{2}\left(\frac{f-p_{1}}{p_{2}}\right)^{2}}\times
e^{-\frac{1}{2}\left(\frac{f}{p_{3}}\right)^{p_{4}}}+p_{5}+e^{-p_{6}(f-p_{7})}$
(1)

where $f$ represents the midpoint of the $j$’th frequency bin in kHz and the
parameters $p_{k}$ are given by $p_{0}=4450$, $p_{1}=-530$, $p_{2}=280$,
$p_{3}=110$, $p_{4}=-0.85$, $p_{5}=18$, $p_{6}=0.064$, and $p_{7}=74$. The
modulus $r$ was fluctuated according to a modified Poisson distribution of the
form $P(x)=e^{-\mu}\mu^{x}/(x-1)!$ with $\mu=0.28$, while the phase $\phi$ was
generated uniformly from $0\rightarrow 2\pi$. The parameters used in this
model were chosen to generate noise very similar to that observed in actual
detectors like that described in Reference [16].

The waveform produced by transforming this randomly generated noise into the
time domain is what we will refer to as the _noise waveform_ , which was
generated uniquely for each channel and for every event. The pure signal
waveform was added to the noise waveform to produce the simulated LArTPC
waveform for a wire if it was associated with a _SimChannels_ object;
otherwise, only the noise waveform was used. From here on, we will refer to
the simulated waveform formed from the sum of a pure signal and a noise
waveform simply as a _signal waveform_. Examples of these simulated waveforms
are shown in Figure 1 for the generated samples used in training and testing
the DL model described below and in evaluating its performance.

##  3 Deep learning approach to LArTPC waveform recognition

|  |  |
---|---|---|---
(a) | (b) | (c) | (d)
Figure 3: Plots (a) and (b) show how accuracy and loss evolve as a function of
training epoch for the induction plane. Plots (c) and (d) shows the evolution
of the same quantities as a function of training epoch for the collection
plane.

###  3.1 1D-CNN model architecture

The architecture of the neural network we developed for detecting signal
waveforms in a LArTPC is the one-dimensional convolutional neural network
(1D-CNN) shown in Figure 2. Starting on the left is the waveform presented to
the input of the network, which consists of three one-dimensional
convolutional layers (Conv1Ds), all of which use rectified linear unit (ReLU)
activation functions, defined by $f(x)=\max(0,x)$. Each Conv1D is immediately
followed by a pooling layer that reduces the size of the input feature map.
The first two are maximum pooling layers with pool sizes of 2, and the last is
a global pooling layer. There are 16, 32, and 64 filters or kernels in the
first, second, and third Conv1D layers, respectively. The kernel sizes in this
sequence of three layers are 3, 5, and 9, respectively, with stride lengths of
2 for the first two and 1 for the last. Two dropout layers are used, one with
a dropout fraction of 0.1 after the second maximum pooling layer, and the
other with a dropout fraction of 0.2 after the global pooling layer. By
randomly ignoring a fraction of their inputs, dropout layers help prevent the
formation of co-adaptations between layers, which do not generalize well to
unseen data and could therefore cause overfitting [17]. The outputs of the
global pooling layer terminate into a dense layer with a single node that is
activated by a sigmoid function, defined by $f(x)=1/(1+e^{-x})$. This function
yields an output bounded between 0 and 1, which can be conveniently
interpreted as the probability that the waveform contains a signal or not.
This network has a total of 21,217 trainable parameters.

###  3.2 Training the model

To create the simulated sample of digitized LArTPC waveforms used to train the
model, radiological events from the $\beta$ decay of the
$\tensor[^{39}]{\mathrm{Ar}}{}$ nuclide contaminating LAr were first
generated. The interactions of the particles produced in these events with the
LAr volume were then simulated using the Geant4 toolkit, followed by the
detector response simulation, both of which are described in Section 2. Next,
all the particles in an event that deposited energy in the LAr, leading to a
detectable signal from at least one wire, were then identified. This was
ensured by requiring the maximum ADC value of the pure signal component of the
digitized wire waveform to be $>3$ ADC counts. In addition, the minimum energy
of the parent ionizing particle was required to be $\geq 50$ keV, and the
maximum number of ionization electrons associated with the signal was required
to be $<11,000$. For each particle identified above, we randomly selected a
single wire channel among all those with signals associated with this
particle, satisfying the requirements above. This was done in order to
minimize possible correlations between signals from neighboring wires
originating from the same particle, which could impact the training process
negatively. Since there can be multiple signal contributions from different
particles in a given channel, only the largest contribution, based on the
energy deposited, was selected. A cutout of the full waveform consisting of
200 time bins, with the selected signal region randomly positioned within it,
was then created. The training set used in the procedure described below
consisted of such 200-tick waveforms.

The model was trained separately for the induction and collection planes of
the LArTPC described in Section 2. The total number of samples used to train
the model was $\approx$2.88 M ($\approx$3.30 M) for the induction (collection)
plane. A separate validation sample of $\approx$721 K ($\approx$824 K) in the
induction (collection) plane was not used to train the model directly, but to
monitor its performance in the course of training. Both training and
validation samples were split roughly equally between signal waveforms and
noise waveforms. Prior to feeding the waveforms to the model, the mean
$\bar{x}$ and standard deviation $\sigma$ of all ADC values over all waveforms
in the sample were first computed. Each waveform, identified by the index $i$,
was then _standardized_ by scaling its ADC values $x_{i}$ by a factor
$s_{i}=(x_{i}-\bar{x})/\sigma$. To fit for the optimal model weights in the
training process, we made use of the _Adam_ (adaptive moment estimation)
optimization algorithm [18]. This is an extension of the mini-batch stochastic
gradient method that uses per-parameter learning rates, whose values are
adapted based on how quickly the weights have been changing. The weights were
determined and updated iteratively using random batches of 2,048 waveforms
(_batch size_), in which a full pass over the entire sample was completed in
one _epoch_. A total of 8 (12) training epochs were performed for the
induction (collection) plane. We used the binary cross-entropy loss function,
calculated according to
$L=-\frac{1}{N}\sum_{i=1}^{N}y_{i}\cdot\log(p_{i})+(1-y_{i})\cdot\log(1-p_{i})$,
where the index $i$ runs over the number of observations $N$. $p_{i}$ is the
model output representing the predicted probability for observation $i$ to
contain a signal, and $y_{i}$ is the correct label for that observation (1 for
signal and 0 otherwise).

The evolution of the accuracy and the loss as a function of training epoch is
shown separately in Figure 3 for the induction (blue) and collection (green)
planes. We define
$accuracy=\frac{\mathrm{TP}+\mathrm{TN}}{\mathrm{TP}+\mathrm{FN}+\mathrm{TN}+\mathrm{FP}}$,
where TP, TN, FP, and FN are the number of true positives, true negatives,
false positives, and false negatives, respectively. Loss is as defined above.
For each plane, the accuracy and loss curves are shown for both the training
(solid) and validation (dashed) samples. In all cases, the validation curves
follow the general trend of the training curves, reassuring us that
overfitting is not an issue. The values for TN, FP, FN, and TP (elements of
the $2\times 2$ confusion matrix) at the end of the final training epoch are
shown for the training and validation samples in the first two rows of Table
1.

| Induction Plane | Collection Plane
---|---|---
Sample | TN | FP | FN | TP | TN | FP | FN | TP
training | 0.477 | 0.023 | 0.052 | 0.448 | 0.483 | 0.017 | 0.028 | 0.472
validation | 0.476 | 0.024 | 0.052 | 0.448 | 0.482 | 0.017 | 0.028 | 0.473
testing | 0.476 | 0.024 | 0.052 | 0.448 | 0.483 | 0.017 | 0.028 | 0.472
Table 1: The table above shows the elements of the confusion matrix when
applying the fully trained 1D-CNN model on the training, validation, and
testing samples. Results are shown for both planes.

###  3.3 Verifying with independent test set

When the training and validation samples were created, a separate and
independent test sample of $\tensor[^{39}]{\mathrm{Ar}}{}$ waveforms was also
generated. This served as an unbiased sample that was not used to train the
sample and did not influence the hyperparameter choices for the model. The
size of this sample was $\approx$899 K ($\approx$1.29 M) for the induction
(collection) plane, consisting of roughly equal portions of signal and noise
waveforms. We applied the fully trained model described in Section 3.2 to the
test sample after scaling its ADC values with the same standardization
parameters used for the training sample. The results, in terms of the elements
of the confusion matrix, are shown in the last row of Table 1. From these
numbers, we calculate _precision_ $=p=\mathrm{\frac{TP}{TP+FP}}$, _negative
predictive value_ $=npv=\mathrm{\frac{TN}{TN+FN}}$, _recall_
$=r=\mathrm{\frac{TP}{TP+FN}}$, _specificity_ $=s=\mathrm{\frac{TN}{TN+FP}}$,
and accuracy as defined previously. The values for these metrics for the test
sample are shown in Table 2. In Figure 4, the true positive rate is plotted
against the false positive rate to show the receiver operating characteristic
curves (ROCs) from applying the fully trained model on the test sample for
each plane. The diagonal dash-dotted line in each plot represents the
reference case when a model has absolutely no ability to tell two classes
apart, in which case the area under the ROC curve (AUC) is 0.5. In comparison,
our model is able to achieve AUC=0.97 on the induction plane and AUC=0.99 on
the collection plane, indicating useful discriminating power between signal
and noise waveforms.

Metric | Induction Plane | Collection Plane
---|---|---
precision | 0.950 | 0.965
negative predictive value | 0.901 | 0.946
recall | 0.896 | 0.944
specificity | 0.953 | 0.966
accuracy | 0.924 | 0.955
Table 2: The table above summarizes the performance of the fully trained 1D-CNN model on the test sample. |
---|---
(a) | (b)
Figure 4: Receiver operating characteristic curves (ROCs) and the associated
areas under the ROC curves (AUCs) for the independent test set are shown for
the induction plane in (a) and the collection plane in (b). Shown for
reference in each plot is the dash-dotted line representing the case when
there is absolutely no discriminating power.

##  4 Region of interest finding and model performance

###  4.1 Extending the model to localize signals

The results presented in Section 3.3 demonstrate that our 1D-CNN model can
make useful predictions about whether 200-tick waveform snippets contain
signals or not. This section describes how we extended this capability to the
localization of signals within the full waveforms in terms of ROIs. This was
done in a straightforward manner by simply scanning a window across the entire
waveform starting from the leftmost edge and shifting it repeatedly to the
right with some finite stride length until it reached the rightmost edge. By
performing an inference on the portion of the waveform contained within the
window at each step, the signal region could effectively be localized. In our
implementation in this paper, the full 2,048-tick simulated LArTPC waveform
was subdivided into 14 overlapping 200-tick windows, beginning with the first,
whose left edge was aligned with the start of the waveform. Each of the 12
subsequent windows after the first was offset from the previous one by a
stride length of 150 ticks, while the last window (13th after the first) was
offset from the preceding one by 48 ticks, so that its right edge aligned with
the last tick of the full waveform. An inference was then performed on each of
these 14 windows to identify the ROI/s within the full waveform. Examples of
such ROIs are shown in Figure 5.

|
---|---
(a) | (b)
Figure 5: Two examples each of simulated full (2,048-tick) waveforms are shown
for the (a) induction and the (b) collection planes. _Regions-of-interest_
(ROIs) that localize the signals within the full waveforms, using the method
described in Section 4.1, are indicated by the rectangular pulses drawn in
red.

###  4.2 Evaluating efficiency with single electrons

|
---|---
(a) | (b)
Figure 6: Efficiencies of the 1D-CNN ROI finder as a function of
$ADC_{\mathrm{pk}}$ determined using the isotropic electron sample for the (a)
induction and (b) collection planes. Results are shown in each case for the
traditional _over-ADC-threshold_ and the 1D-CNN based approaches. The average
energy deposited in MeV and the average number of electrons associated with
the signal, for each $ADC_{\mathrm{pk}}$ bin, are shown below each plot.

To further quantify the performance and capabilities of our 1D-CNN model, we
generated a single electron sample with electron momenta ranging from 1 MeV to
1 GeV. The initial position of each electron was generated uniformly within a
rectangular prism that centered on the LAr volume and which measures $30\times
30\times 90$ cm3, with the long dimension oriented along the $z$ axis and the
two shorter ones along the $x$ and $y$ axes. The initial direction of each
electron was generated pseudo-isotropically with angles $\theta_{xz}$ and
$\varphi_{yz}$ distributed uniformly within $\pm 180\degree$ and $\pm
90\degree$, respectively. After this, the electrons were propagated through
the detector using Geant4 and the detector response simulated as described in
Section 2, to produce the raw LArTPC wire waveforms. The fully trained 1D-CNN,
described in Section 3.2, was then applied to each waveform to find the signal
ROI as detailed in Section 4.1.

The signal detection efficiency for the isotropic electron sample is presented
here as a function of the peak ADC value associated with the pure signal
component of a digitized waveform, which shall be referred to hereon as
$ADC_{\mathrm{pk}}$. We choose to work with ADC values because they are
ultimately what the 1D-CNN directly “sees”. This avoids the need to convert
from some other quantity like energy to ADC values, which would be experiment
specific.

To calculate efficiency, two histograms are created to represent the numerator
($H_{\mathrm{N}}$) and denominator ($H_{\mathrm{D}}$) of the ratio. Each
histogram has 12 bins representing $ADC_{\mathrm{pk}}$ ranging from 1 to 12.
If a signal is present within the full 2,048-tick waveform, an entry is made
in the $H_{\mathrm{D}}$ bin corresponding to the $ADC_{\mathrm{pk}}$
associated with the signal. In case multiple signals are present within the
full waveform, only the one with the largest $ADC_{\mathrm{pk}}$ is selected
for calculating efficiency. Following the procedure described in Section 4.1,
the 1D-CNN is used to scan the entire waveform for ROIs. If the signal lies
within an ROI identified by the 1D-CNN, an entry is also made in the same bin
of $H_{\mathrm{N}}$. Once the two histograms are filled, their ratio is taken
to yield the efficiency as a function of $ADC_{\mathrm{pk}}$. This is done
separately for the induction and collection planes, and the results are shown
in red in Figure 6. The average energy deposited and average number of
ionization electrons associated with the signal, for each bin of
$ADC_{\mathrm{pk}}$, are also shown below each plot. The CNN output is
required to be $>0.91$ ($>0.95$) in order to classify a waveform as a signal
in the induction (collection) plane.

Also shown for comparison in each plot are the efficiencies (in blue) for the
traditional _over-ADC-threshold_ approach. In this case, signal waveforms are
discriminated from noise by requiring the measured ADC values in a region
containing the signal to be above noise threshold. The mean ADC value of the
pure noise waveforms generated as described in Section 2 is 1.77, with a
standard deviation of 1.05. In these plots, we require
$\left|\mathrm{ADC}\right|>6$ counts for the induction plane and
$\mathrm{ADC}>6$ counts for the collection plane. These correspond to $\approx
4$ standard deviations above the noise average. Such a requirement achieves a
background rejection rate of 0.926 (0.962) on the induction (collection)
plane, which closely matches the rate of 0.931 (0.966) from the CNN-based
method when requiring the output to be $>0.91$ ($>0.95$).

The fact that the efficiency plots for the _over-ADC-threshold_ method do not
exhibit a sharp cutoff at $ADC_{\mathrm{pk}}=6$ might seem counter-intuitive
at first. However, this is to be expected because $ADC_{\mathrm{pk}}$
represents the ADC value associated with the pure signal prior to any noise
fluctuations. A signal with $ADC_{\mathrm{pk}}\leq 6$ could satisfy the
requirement of $\mathrm{ADC}>6$ if it fluctuates above this threshold after
the addition of noise. To demonstrate more clearly how the 1D-CNN is able to
detect signals in regions inaccessible to the traditional approach, the 1D-CNN
efficiencies are shown as a function of the actual ADC value produced by the
detector ($ADC_{\mathrm{pk}}+\mathrm{noise}$) in Figure 7. The blue hatched
region represents the region above the ADC cut that is accessible to the
_over-ADC-threshold_ method.

It is not difficult to see from Figure 6 that the CNN-based method outperforms
the traditional approach throughout the range from $ADC_{\mathrm{pk}}=2$ to 8,
after which both achieve essentially 100% efficiency. The former begins to
achieve better than 90% efficiency at $ADC_{\mathrm{pk}}=6$, while the latter
only reaches this level at around $ADC_{\mathrm{pk}}=8$.

|
---|---
(a) | (b)
Figure 7: Efficiencies of the 1D-CNN ROI finder for the (a) induction and (b)
collection planes as a function of the actual ADC value produced by the
detector, which is the sum $ADC_{\mathrm{pk}}+\mathrm{noise}$. The blue
hatched region represents the efficiency of the _over-ADC-threshold_ method.

##  5 Conclusion

In this work, we have successfully demonstrated that deep learning methods can
be applied directly to the raw wire waveforms produced by the individual
channels in a LArTPC detector in order to discriminate signal waveforms from
background. This was achieved using a 1D-CNN which was implemented in a way
that allowed the ROI of the signal to be identified within the full waveform.
The discriminating power of CNNs derives from their ability to learn and
detect subtle features that distinguish signal from background such as, but
not limited to, shape characteristics. Because of this, they are not
constrained by user defined cuts imposed in traditional threshold based signal
ROI finders and can maintain signal sensitivity in energy regions inaccessible
to such methods. The implications are significant for the rich low-energy
physics program of future neutrino experiments like DUNE.

A major advantage of our approach over other ML-based ones, is its use of a
simple neural network architecture consisting of a mere $\approx$20K trainable
parameters compared with the millions found on typical designs. The low
resource utilization makes it feasible for deployment near the detector front
end or in edge applications, as part of an online trigger system or for
implementing an intelligent zero-suppression filter. Such applications of this
1D-CNN at the very early stages of the DAQ system can help achieve optimal
signal efficiency and background rejection that will surely benefit and
complement downstream selection algorithms, including more complex ML-based
ones.

This paper set out to establish a proof-of-concept and provide a detailed
description of the methodology, and it has succeeded in this task. In
subsequent and related papers, this method will be used in actual
applications, including those involving real experimental data [16].

##  6 Acknowledgements

We wish to thank Brian Nord and Gabe Perdue of Fermilab for useful discussions
on machine learning techniques. This manuscript has been authored by Fermi
Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S.
Department of Energy, Office of Science, Office of High Energy Physics. This
work has received funding from the European Union’s Horizon 2020 Research and
Innovation Programme under grant agreement number 858199, “INTENSE”.

## References

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# Neutrino quantum kinetics in two spatial dimensions

Marie Cornelius ID Shashank Shalgar ID and Irene Tamborra ID

###### Abstract

Our understanding of neutrino flavor conversion in the innermost regions of
core-collapse supernovae and neutron star mergers is mostly limited to
spherically symmetric configurations that facilitate the numerical solution of
the quantum kinetic equations. In this paper, we simulate neutrino quantum
kinetics within a $(2+1+1)$ dimensional setup: we model the flavor evolution
during neutrino decoupling from matter in two spatial dimensions, one neutrino
momentum variable, and time; taking into account non-forward neutral current
and charged current collisions of neutrinos with the matter background, as
well as neutrino advection. In order to mimic fluctuations in the neutrino
emission and matter background, and explore their effect on the flavor
evolution, we introduce perturbations in the collision term as well as in the
vacuum term of the Hamiltonian. Because of such perturbations, the initial
symmetry of the neutrino field across the simulation annulus is broken and
flavor conversion is qualitatively affected, with regions of larger flavor
conversion alternating across the simulation annulus. In addition, neutrino
advection is responsible for spreading flavor waves across neighboring spatial
regions. Although based on a simplified setup, our findings highlight the
importance of modeling neutrino quantum kinetics in multi-dimensions to assess
the impact of neutrinos on the physics of compact astrophysical sources and
nucleosynthesis.

##  1 Introduction

Neutrinos play a fundamental role in core-collapse supernovae and neutron star
mergers, despite their weakly interacting nature [1, 2, 3, 4, 5]. In the core
of these sources, the neutrino number density is so large that the coherent
forward scattering of neutrinos off each other significantly affects the
flavor evolution and makes the neutrino equations of motion non-linear [6, 2,
7, 8, 9, 10].

In order to model the rich flavor phenomenology resulting from flavor
conversion in neutrino-dense astrophysical sources and grasp its consequences
on the source physics and multi-messenger observables [11, 12, 13, 14, 15, 16,
17, 18], a quantum-kinetic treatment of the Boltzmann neutrino transport is
necessary [9, 10, 3, 19, 20, 21]. This is a computationally challenging task
due to the fact that the characteristic quantities entering the neutrino
quantum kinetic equations (QKEs) change over several orders of magnitude;
additionally the QKEs should be solved in seven dimensions (i.e., time, three
spatial coordinates and three neutrino momentum coordinates). As a
consequence, this problem can only be tackled relying on simplified setups,
imposing symmetry assumptions on the system to reduce its dimensionality [22,
23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35].

Crossings in the electron lepton number (ELN) angular distribution of
neutrinos can develop as neutrinos decouple from matter, induce flavor
instabilities [36, 37, 38], and eventually fast flavor conversion [39]. The
latter is characterized by a frequency directly proportional to the number
density of neutrinos and could take place in the absence of neutrino mass
difference. The advection of neutrinos may be responsible for smearing the ELN
structures responsible for flavor conversion, unless such structures are self-
sustained in time [26, 40]. In addition, non-forward collisions of neutrinos
with the background medium can dynamically affect the flavor evolution [41,
42, 28, 43, 44] or lead to the development of collisional flavor instabilities
[45, 46, 47, 48, 49], whose relevance in the core of astrophysical sources is
yet to be understood [50, 51, 32].

Understanding the quasi-steady state flavor configuration resulting from fast
flavor conversion is an active field of research [22, 23, 24, 13, 50, 52, 25,
26, 32, 27, 28, 53, 54, 29, 35, 55, 56, 30]. While a robust criterion to
predict the quasi-steady state flavor configuration is still missing, it is
clear that the boundary conditions assumed for the QKE simulations as well as
the shape of the ELN distribution could dramatically affect the final flavor
configuration [25, 22, 23, 32]. On the other hand, it remains to be clarified
how the quasi-steady state configuration could change, when relaxing the
symmetry assumptions imposed on the system. Preliminary work shows that
increasing the dimensionality of the simulation box [35, 26] or breaking the
azimuthal symmetry [57] can lead to efficient spreading of flavor waves across
neutrino angular modes and spatial regions.

In this paper, we explore the flavor conversion phenomenology in two spatial
dimensions, moving beyond the assumption of spherical symmetry previously
employed [22, 23, 32, 25]. Our goal is to investigate how the flavor
conversion physics changes in the presence of multiple spatial dimensions.
Introducing perturbations across the simulation annulus, we find that the
quasi-steady state flavor configuration changes qualitatively, breaking the
initial symmetry of the system. In addition, the dynamical impact of neutrino
advection and collisions spreads flavor instabilities across neighboring
regions in the simulation shell.

Our work is organized as follows. Section 2 presents the QKEs in two spatial
dimensions as well as our simulation setup. Section 3 focuses on our findings
on the quasi-steady state flavor configuration in two spatial dimensions in
the absence of perturbations. We then investigate how the flavor structure
changes as a result of flavor conversion when the vacuum Hamiltonian or the
collision term are perturbed in Sec. 4. We summarize and discuss our results
in Sec. 5. Additionally, Appendix A outlines the derivation of the advection
term in two spatial dimensions, and Appendix B presents the solution of the
QKEs in two spatial dimensions for a different collision term (i.e. different
ELN configuration) to validate the robustness of our findings.

##  2 Neutrino quantum kinetic equations

In this section, we introduce the neutrino QKEs. We also describe the employed
simulation setup.

###  2.1 Neutrino equations of motion

In order to solve the neutrino QKEs in two spatial dimensions, we rely on a
shell characterized by the radius $r$ and the polar angle $\Theta$, as
depicted in Fig. 1. For each point in the simulation annulus, determined by
$(r,\Theta)$, the neutrino direction of propagation is given by the angle
$\theta$. Hence, the neutrino field depends on the coordinates
$(r,\Theta,\theta)$, with the radius varying between $r_{\min}=15$ km and
$r_{\max}=30$ km. As for the angles, we consider $\theta\in[0,2\pi)$ and
$\Theta\in[0,2\pi)$, with periodic boundaries in $\Theta$ (which means that
neutrinos move across $\Theta=0$ and $2\pi$). This choice of the simulation
setup is motivated by the fact that, allowing for $\Theta\in[0,2\pi)$,
(anti)neutrinos can freely move around the annulus and we investigate an
intermediate configuration with respect to a 3D one; this approach allows for
a breaking of the initially spherically symmetric configuration without
introducing the dependence on the azimuthal angles.

Figure 1: Sketch of the shell adopted to solve the QKEs numerically. A point
in the simulation annulus is characterized by the radius $r$, varying between
the innermost (outermost) radius $r_{\min}$ ($r_{\max}$), and the polar angle
$\Theta$. The neutrino direction of propagation is defined by $\theta$.

For simplicity, we focus on two neutrino flavors $\nu_{e}$ and $\nu_{x}$, with
$x=\mu$ and $\tau$. We consider $2\times 2$ density matrices, whose diagonal
terms $\rho_{ii}$ (with $i=e,x$) represent the neutrino occupation numbers and
are normalized such that
$\rho_{ee}(r_{\min},\Theta,\theta)=\rho_{xx}(r_{\min},\Theta,\theta)=\bar{\rho}_{xx}(r_{\min},\Theta,\theta)=1$
and $\bar{\rho}_{ee}(r_{\min},\Theta,\theta)=1.2$. The coherence between the
flavors is described by the off-diagonal terms $\rho_{ij}$, with $i\neq j$.
The correspondent equations of motion for neutrinos and antineutrinos,
respectively, are:

$\displaystyle i\left(\frac{\partial}{\partial
t}+\vec{v}\cdot\vec{\nabla}\right)\rho(r,\Theta,\theta,t)$
$\displaystyle=\left[H,\rho(r,\Theta,\theta,t)\right]+i\mathcal{C}\ ,$ (2.1)
$\displaystyle i\left(\frac{\partial}{\partial
t}+\vec{v}\cdot\vec{\nabla}\right)\bar{\rho}(r,\Theta,\theta,t)$
$\displaystyle=\left[\bar{H},\bar{\rho}(r,\Theta,\theta,t)\right]+i\bar{\mathcal{C}}\
,$ (2.2)

where we have neglected general relativistic corrections [13]. The advective
term on the left-hand side of the equations of motion is derived in Appendix A
and is given by

$\vec{v}\cdot\vec{\nabla}\rho(r,\Theta,\theta)=\cos\theta\frac{\partial\rho}{\partial
r}+\frac{\sin\theta}{r}\frac{\partial\rho}{\partial\Theta}-\frac{\sin\theta}{r}\frac{\partial\rho}{\partial\theta}\
.$ (2.3)

The flavor conversion physics is taken into account through the commutator of
the Hamiltonian and the density matrix on the right-hand side of the equations
of motion. The Hamiltonian consists of the vacuum, matter, and self-
interaction terms. For simplicity, we neglect the matter term, as it
effectively suppresses the vacuum mixing angle, and use a smaller effective
vacuum mixing angle instead. Hence,

$H=H_{\rm{vac}}+H_{\nu\nu}\ \mathrm{and}\ \bar{H}=-{H}_{\rm{vac}}+H_{\nu\nu}\
.$ (2.4)

The vacuum term of the Hamiltonian is defined as

$H_{\rm{vac}}=\frac{\omega}{2}\begin{pmatrix}-\cos 2\vartheta_{\rm{V}}&\sin
2\vartheta_{\rm{V}}\\\ \sin 2\vartheta_{\rm{V}}&\cos
2\vartheta_{\rm{V}}\end{pmatrix}\ ,$ (2.5)

where $\vartheta_{\rm{V}}=10^{-3}$ is the effective vacuum mixing angle,
$\omega=\Delta m^{2}/2E$ is the vacuum frequency ($\Delta m^{2}=2.5\times
10^{-3}~{}\rm{eV}^{2}$ is the mass difference squared and $E=20$ MeV is the
neutrino energy; note that we consider mono-energetic neutrinos for the sake
of simplicity). The self-interaction Hamiltonian is

$H_{\nu\nu}=~{}\mu_{0}\int_{0}^{2\pi}\left[\rho(r,\Theta,\theta^{\prime})-\bar{\rho}(r,\Theta,\theta^{\prime})\right]\times\left(1-\cos\theta\cos\theta^{\prime}-\sin\theta\sin\theta^{\prime}\right)d\theta^{\prime}\
,$ (2.6)

where $\mu_{0}=10^{4}~{}\rm{km}^{-1}$ is the strength of the neutrino-neutrino
interaction at $r_{\rm min}$ (the neutrino self-interaction strength then
decreases with radius). We note that, since we solve the neutrino QKEs in 2D,
$H_{\nu\nu}$ carries an additional dependence on
$\sin\theta\sin\theta^{\prime}$, which is not present in 1D and contributes to
affect the flavor evolution.

We use a simplified model for non-forward collisions of neutrinos off matter,
taking into account emission, absorption, and direction-changing scatterings.
It is given by
$\mathcal{C}(r,\theta)=\mathcal{C}_{\rm{emission}}+\mathcal{C}_{\rm{absorb}}+\mathcal{C}_{\rm{dir-
ch}}$, where [23]:

$\displaystyle\mathcal{C}_{\text{emission}}^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda_{\text{emission}}^{i}(r)}\ ,$ (2.7)
$\displaystyle\mathcal{C}_{\text{absorb}}^{i}$ $\displaystyle=$
$\displaystyle-\frac{1}{\lambda_{\text{absorb}}^{i}(r)}\rho_{ii}(\theta)\ ,$
(2.8) $\displaystyle\mathcal{C}_{\text{dir-ch}}^{i}$ $\displaystyle=$
$\displaystyle-\frac{2\pi}{\lambda_{\text{dir-
ch}}^{i}(r)}\rho_{ii}(\theta)+\int_{0}^{2\pi}\frac{1}{\lambda_{\text{dir-
ch}}^{i}(r)}\rho_{ii}\left(\theta^{\prime}\right)d\theta^{\prime}\ .$ (2.9)

Here, $\lambda^{i}(r)$ are the flavor-dependent mean free paths. It should be
noted that the direction changing term $\mathcal{C}_{\text{dir-ch}}^{i}$, by
definition, conserves the number of particles. However, this is not the case
for the emission and absorption terms. For the modeling of the collision term,
we follow Case C of Ref. [23], with the mean free paths for each flavor being
summarized in Table 1. With this choice of the collision term, we have
isotropic angular distributions for all flavors at $r_{\rm min}$, then
neutrinos gradually decouple from matter as $r$ increases in a flavor-
dependent fashion and ELN crossings arise (see Sec. 3).

Table 1: Mean free paths of emission, absorption, and direction-changing interactions for each flavor as a function of radius, with $\xi(r)=\exp(15-r)$. The modeling of the collision term follows Case C of Ref. [23]. | $\nu_{e}$ | $\bar{\nu}_{{e}}$ | $\nu_{x},~{}\bar{\nu}_{x}$
---|---|---|---
$\lambda_{\text{emission}_{\phantom{l}}}^{i}$[km] | $1/[50\xi(r)]$ | $1/[30\xi(r)]$ | $1/[10\xi(r)]$
$\lambda_{\text{absorb }_{\phantom{l}}}^{i}$[km] | $1/[50\xi(r)]$ | $1/[25\xi(r)]$ | $1/[10\xi(r)]$
$\lambda_{\text{dir-ch }_{\phantom{l}}}^{i}$[km] | $1/[50\xi(r)]$ | $1/[25\xi(r)]$ | $1/[12.5\xi(r)]$

###  2.2 Simulation setup

The QKEs are solved relying on an adaptive multi-step Adams-Bashforth-Moulton
method with adaptive step size for the time variable; the spatial and momentum
derivatives are instead computed using the central difference method. We use a
uniform grid of $75$ bins for radius, angle $\Theta$, and momentum angle
$\theta$ (we have tested the convergence of our simulations in the
correspondent 1D model—results not shown here).

We aim to compute the quasi-steady state configuration that is achieved by the
system in the presence of neutrino flavor conversion. In order to efficiently
tackle this issue, we follow the approach presented in Refs. [23, 22] and
first solve Eqs. 2.1 and 2.2 for $H=\bar{H}=0$ for $t=100~{}\mu$s to compute
the “classical steady state” solution. This solution is then used as input to
solve Eqs. 2.1 and 2.2 when the physics of neutrino flavor conversion is taken
into account 111It should be noted that the approach introduced in Refs. [23,
22] is employed for numerical efficiency, but the quasi-steady state
configuration is independent of the initial configuration used in the
simulation. This method has the added benefit of highlighting the effect of
flavor evolution on neutrino transport.. The “quasi-steady state”
configuration is computed solving Eqs. 2.1 and 2.2 for $t=50~{}\mu$s. This
represents the time it takes for neutrinos to cross the simulation annulus and
we have tested that an approximately steady state configuration has been
reached (results not shown here).

In order to compute the classical steady-state solution (as well as the quasi-
steady state one), we have optimized the choice of the boundary conditions for
what concerns $r_{\rm min}$ and $r_{\rm max}$ such that neutrinos of all
flavors have isotropic angular distributions at $r_{\rm min}$ and are fully
decoupled from matter with a negligible backward flux at $r_{\rm max}$ [23,
22]. Through this approach, ELN crossings also appear within the simulation
annulus when neutrinos decouple, as illustrated in the next section.

##  3 Flavor evolution in two dimensions

In this section, we present the quasi-steady state neutrino flavor
configuration in the absence of $\Theta$-dependent perturbations in the vacuum
Hamiltonian and neutrino backgrounds. We first focus on this configuration
since it allows us to recover a quasi-steady state configuration equivalent to
the one obtained in a simulation shell with one spatial dimension. This is due
to the symmetry across $\Theta$, which is preserved by flavor conversion to a
good approximation.

Case 1NP (no perturbations)

Figure 2: Quasi-steady state flavor configuration obtained for the QKE
solution without perturbations (Case 1NP; evolved for $t=50~{}\mu$s). Left
panel: Contour plot of the difference of $dn_{\nu_{e}}/d\theta$ between the
classical (c) and the quantum (q) solutions in the plane spanned by $\theta$
and $r$. The blue regions indicate a deficit of electron neutrinos due to
flavor conversion. Middle and right panels: Radial evolution of the neutrino
number densities with (solid) and without (dashed) flavor conversion. The
middle panel shows the number density of $\nu_{e}$ (red) and $\nu_{x}$ (blue),
and the right panel shows the same for $\bar{\nu}_{e}$ (teal) and
$\bar{\nu}_{x}$ (orange). The relative flavor content changes due to flavor
conversion as a function of radius, but not of $\Theta$.

Figure 2 shows the quasi-steady state solution obtained by solving Eqs. 2.1
and 2.2 with the collision term defined as in Table 1 (Case 1NP, no
perturbations). The contour plot on the left depicts the difference in the
steady state configuration of the differential electron neutrino number
density ($dn_{\nu_{e}}/d\theta$), between the classical solution (c) and the
quantum solution (q) in the plane spanned by the momentum angle $\theta$ and
the radius $r$. We define the neutrino number density as

$n_{\nu_{i}}(r,\Theta)=\frac{\mu_{0}}{\sqrt{2}G_{F}}\int_{0}^{2\pi}\rho_{ii}(r,\Theta,\theta)d\theta\
,$ (3.1)

with $G_{F}$ being the Fermi constant. Note that we do not consider the number
density as the number of particles per unit area, as expected within a 2D
geometry, instead choose to use the effective number of particles per unit
volume at the location $(r,\Theta)$ of a 3D equivalent system (this is done
rescaling by $\mu_{0}$). The plots in Fig. 2 have been obtained for
$\Theta=0^{\circ}$, however the QKE solution is identical for any $\Theta$.

The colored regions of the contour plot on the left panel of Fig. 2 show where
neutrinos have changed flavor compared to the classical solution: blue (red)
represents a deficit (surplus) of $\nu_{e}$ in the quantum solution with
respect to the classical one. At radii close to $r_{\rm min}$, neutrinos are
trapped inside the neutrinosphere. As expected, we find no flavor conversions
at such radii since the angular distributions of neutrinos and antineutrinos
are both approximately isotropic. As the neutrinos move to larger radii, they
start to decouple from the matter background and free-stream; this results in
a forward-peaking of their angular distributions [58, 59]. The electron
antineutrinos decouple at smaller radii than the electron neutrinos since they
have a larger mean free path (see Table 1). Consequently, an ELN crossing
develops which triggers flavor instabilities. This is visible through the blue
band arising at $\simeq 20$ km and spreading towards larger radii in the
forward direction (at $\theta=0$ and $\theta=2\pi$).

The QKE solution is independent of $\Theta$, hence Case 1NP is the analog of
Case C presented in Ref. [23] (cf. their Fig. 7) because
${\partial\rho}/{\partial\Theta}=0$ in Eq. 2.3; however, note that
$H_{\nu\nu}$ (Eq. 2.6) acquires an additional dependence on
$\sin\theta\sin\theta^{\prime}$ which is not present the correspondent 1D
solution. It can be seen from the contour plot that the quasi-steady state
configuration has a left-right asymmetry around $\theta=\pi$. This is due to
the fact that the self-interaction Hamiltonian includes a numerical integral
over $\sin\theta^{\prime}$ whose floating-point error provides a seed for the
spontaneous breaking of the symmetry around $\theta=\pi$.

The middle and right panels of Fig. 2 show the radial profiles of the neutrino
number densities for $e$ and $x$ neutrinos with and without flavor conversion,
and the same for antineutrinos, for $\Theta=0^{\circ}$. As neutrinos decouple
from matter and ELN crossings form, flavor conversion is triggered. Because of
flavor conversion, the electron (anti)neutrino distribution approaches the one
of the $x$ flavor. However, flavor equipartition is not achieved for neutrinos
nor antineutrinos for this ELN configuration.

To better understand how neutrino flavor conversion evolves with time as a
function of radius and momentum angle $\theta$, we rely on the Fourier
spectrum in the momentum angle ($k_{\theta}$) for the diagonal element of the
density matrix ($\rho_{ii}$):

$\hat{F}_{i}(t,k_{\theta},r)=\frac{1}{2\pi}\int_{0}^{2\pi}\rho_{ii}(t,r,\Theta,\theta)e^{-ik_{\theta}\theta}d\theta\
,$ (3.2)

which is independent of $\Theta$. The relative contribution of the Fourier
modes gives an indication of how the angular scales of the solution develop,
allowing for a comparison across different radii.

Case 1NP (no perturbations)

Figure 3: Fourier spectrum for the $\nu_{e}$ quasi-steady state flavor
configuration obtained for the standard configuration without perturbations
(Case 1NP). The Fourier spectrum for different simulation times ($t=0$, $2$,
and $50~{}\mu$s) is shown from left to right, respectively. For each time, the
Fourier spectrum is plotted for five selected radii ($r=17$, $19$, $21$, $23$,
and $25$ km). For $r\gtrsim 20$ km, where neutrinos change flavor (see Fig.
2), the contribution of the higher order Fourier modes increases for $t=2$ and
$50~{}\mu$s. Below $20$ km, lower order modes dominate.

Figure 3 shows the absolute value of the Fourier spectrum for the quasi-steady
state solution of Case 1NP for $\nu_{e}$ for five selected representative
radii ($r=17$ km and $19$ km are below the region where flavor conversion
happens, while the other three selected radii correspond to a radial region
where flavor conversion takes place). The left panel shows the Fourier
spectrum of the quantum solution for $t=0~{}\mu$s (i.e. when there is no
flavor conversion). The Fourier spectrum peaks at low order Fourier modes
$k_{\theta}$ at small radii. For larger radii, the higher order modes are
populated. This is due to the increasing forward-peaking of the angular
distributions as a function of radius. The middle and right panels of Fig. 3
show the Fourier spectra for $t=2~{}\mu$s and $t=50~{}\mu$s, respectively. At
small radii, the Fourier spectrum is dominated by lower order modes
$k_{\theta}$. As $r$ increases, higher order Fourier modes contribute to the
Fourier spectrum, since neutrinos experience flavor conversion, with flavor
waves cascading down to smaller angular scales. Comparing these two panels,
one can see that the Fourier spectra at the selected radii are qualitatively
comparable.

##  4 Flavor evolution in two dimensions with perturbations

The neutrino and matter backgrounds are not homogeneous in the core of a
compact astrophysical source [60, 61, 62, 63, 64, 65]. In this section, we
investigate the impact of such fluctuations on the flavor conversion physics
solving the QKEs in the presence of $\Theta$-dependent perturbations in the
vacuum term of the Hamiltonian and in the neutrino background.

###  4.1 Flavor evolution in two dimensions with perturbations in the
Hamiltonian

Case 1HP (perturbations in vacuum Hamiltonian)

Figure 4: Quasi-steady state flavor configuration obtained in the presence of
perturbations in the vacuum Hamiltonian (Case 1HP, cf. Eq. 4.1; evolved for
$t=50~{}\mu$s). Top panels: Contour plots of the difference in
$dn_{\nu_{e}}/d\theta$ without flavor conversion and with flavor conversion in
the plane spanned by the momentum angle $\theta$ and radius. Each panel
represents the quasi-steady state configuration for a selected angle $\Theta$,
as indicated in the inset in the upper left corner of the figure
($\Theta_{1}=60^{\circ}$, $\Theta_{2}=156^{\circ}$, and
$\Theta_{3}=252^{\circ}$). The $\Theta$-dependence of the perturbations in the
vacuum Hamiltonian is illustrated in the inset in the upper right corner. The
Hamiltonian perturbations are responsible for the small scale structure
differences across $\Theta$, while the overall flavor conversion trend is
comparable for the different $\Theta$’s. Bottom panels: Radial profiles of the
neutrino number densities of $\nu_{e}$ (red) and $\nu_{x}$ (blue) with (solid)
and without flavor conversions (dashed) for the three selected $\Theta$
directions. While the classical steady state configurations are independent of
$\Theta$, $\Theta$-dependent structures develop because of flavor conversion.

We introduce a $\Theta$-dependent perturbation in the vacuum term of the
Hamiltonian (Case 1HP):

$H_{\rm{vac},\,p}=H_{\rm{vac}}(1+0.1\cos{\Theta})\ ;$ (4.1)

the perturbations $H_{\rm{vac},\,p}$ affect all flavors. The evolution of
$H_{\rm{vac},\,p}$ as a function of $\Theta$ is illustrated in the inset in
the top right corner of Fig. 4.

Figure 4 shows the quasi-steady state solution for Case 1HP. The contour plots
in the top row show the difference in $dn_{\nu_{e}}/d\theta$ without and with
flavor conversion in the plane spanned by the radius and momentum angle
$\theta$, for three different $\Theta$ angles ($\Theta_{1}$, $\Theta_{2}$, and
$\Theta_{3}$) from left to right, respectively. The three selected $\Theta$
angles are marked in the inset on the top left corner of the figure to guide
the eye. The bottom panels of Fig. 4 represent the corresponding radial
evolution of the neutrino number densities for $\nu_{e}$ and $\nu_{x}$ for the
classical solution and quantum solutions.

Comparing Fig. 4 (Case 1HP) with Fig. 2 (Case 1NP), one can see that the
perturbations introduced in the Hamiltonian break the symmetry across the
simulation annulus, leading to the development of small scale structures and
different radial evolution of $n_{\nu_{i}}(r,\Theta)$. Because of neutrino
advection and collision, the impact of perturbations is not limited to the
specific $\Theta$ where they are introduced, rather it spreads across neighbor
cells, in agreement with the findings of Refs. [26, 40, 27, 50].

The impact of the perturbations introduced in Eq. 4.1 is investigated in
Appendix B for another set of neutrino angular distributions (cf. Fig. 9 vs.
10). With respect to the example presented in this section, we find a
qualitatively similar trend for what concerns the changes induced by flavor
conversion across $\Theta$.

###  4.2 Flavor evolution in two dimensions with perturbations in the
collision term

While Case 1HP focuses on flavor-independent perturbations, in this section we
consider perturbations in the emission term of electron neutrinos and
antineutrinos (with no perturbations in the vacuum term of the Hamiltonian),
i.e. we perturb the system in a flavor-dependent fashion. For electron
neutrinos:

$\mathcal{C}_{\rm{emission},\,p}^{e}=\mathcal{C}_{\rm{emission}}^{e}[1+0.05\cos({4\Theta-\pi/3})]\
;$ (4.2)

while for electron antineutrinos, we consider

$\bar{\mathcal{C}}_{\rm{emission},\,p}^{e}=\bar{\mathcal{C}}_{\rm{emission}}^{e}\left[1+0.05\cos({4\Theta+2\pi/3})\right]\
.$ (4.3)

The functional form of the perturbations for $\nu_{e}$ and $\bar{\nu}_{e}$ is
illustrated in the inset in the top right corner of Fig. 5.

Case 1CP (perturbations in collision term)

Figure 5: Same as Fig. 4, but for the configuration obtained introducing
perturbations in the collision term (Case 1CP, Eqs. 4.2 and 4.3; evolved for
$t=50~{}\mu$s). From left to right, each panel refers to the angle $\Theta$
marked in the inset in the upper left corner ($\Theta_{1}=36^{\circ}$,
$\Theta_{2}=151.2^{\circ}$, and $\Theta_{3}=285.6^{\circ}$). The quasi-steady
state configuration varies according to $\Theta$ because of the perturbations
that affect the flavor conversion physics. The perturbations in the collision
term are modeled as indicated in the inset in the top right corner of the
figure for $\nu_{e}$ (red) and $\bar{\nu}_{e}$ (teal). For $\Theta_{1}$, the
quasi-steady state configuration resembles the one found for Case 1NP with a
larger region of flavor conversion between $20$ and $23$ km. For $\Theta_{2}$,
a smaller region of flavor conversion appears between $20$ and $21$ km, and
for $\Theta_{3}$ the variation of $\nu_{e}$ due to flavor conversion is
negligible for $r\simeq 21$ km. These differences are also visible in the
radial evolution of the neutrino number densities plotted in the bottom
panels.

Case 1CP (perturbations in collision term)

Figure 6: Angular distributions of the ELN number density for the classical
steady state of Case 1CP as a function of $\theta$ for three different radii:
$r=19.1$ km (dotted, violet), $r=22.1$ km (solid, pink), and $r=25.1$ km
(dashed, indigo). Each panel refers to a specific angle $\Theta$, as indicated
in the inset in the top left corner. The loci of ELN crossings are marked by a
gray horizontal line to guide the eye. The inset in the top right corner of
the figure illustrates the modeling of the perturbation as a function of
$\Theta$. Because of the different ELN configuration as a function of $\Theta$
induced by the perturbation of the collision term, the quasi-steady state
neutrino number densities plotted in Fig. 5 exhibit a strong dependence on
$\Theta$.

Figure 5 shows the quasi-steady state configuration for Case 1CP for three
selected angles $\Theta_{1}$, $\Theta_{2}$, and $\Theta_{3}$ (cf. the inset in
the top left corner of the figure), respectively chosen to represent a part of
the perturbation spectrum where the $\nu_{e}$ and $\bar{\nu}_{e}$ emissions
are comparable, a part where the $\nu_{e}$ emission dominates, and where the
$\bar{\nu}_{e}$ emission dominates. The top panels display the difference in
the (quasi-)steady state configurations for the classical and quantum
solutions. For $\Theta_{1}$, a configuration similar to Case 1NP is found,
with a broad region of flavor conversion between $20$ and $23$ km. For
$\Theta_{2}$, a less extended region of flavor conversion appears (between
$20$ and $21$ km), and for $\Theta_{3}$ the classical and quantum solutions
are comparable in the proximity of $r\simeq 21$ km (cf. white band). The
significant differences, as a function of $\Theta$, obtained in the flavor
configuration after flavor conversion imply that flavor waves have not spread
enough to homogenize the neutrino distribution across the simulation annulus.
Yet, one can see that flavor structures spread across neighbor bins in the
simulation annulus. This trend is also evident from the radial profiles of the
neutrino number densities in the bottom panels of Fig. 5: for $\Theta_{1}$,
$\nu_{e}^{q}$ and $\nu_{x}^{q}$ fluctuate close to each other, whereas for
$\Theta_{3}$ they are far from each other. For $\Theta_{2}$ and $\Theta_{3}$,
the perturbation is such that the radial distribution achieved in the
classical steady state is different between $\nu_{e}$ and $\nu_{x}$ at small
radii.

To get a better understanding of the $\Theta$-dependence of the flavor
evolution, Fig. 6 shows the ELN angular distributions of the classical steady
state for three selected radii. At $r=19.1$ km, the angular distributions of
$\nu_{e}$ and $\bar{\nu}_{e}$ are isotropic and do not cross, hence no fast
flavor instability is expected. At $r=22.1$ km and $r=25.1$ km, two ELN
crossings appear. The difference in the location of the ELN crossings for
different $\Theta$ directions are responsible for the development of flavor
instabilities in different spatial regions, as shown in the top panels of Fig.
5.

Appendix B provides another example of the quasi-steady state configuration
achieved in the presence of perturbations in the collision term for a
different flavor configuration (cf. Fig. 11). Similar conclusions are achieved
also in this case for what concerns the impact of perturbations on the quasi-
steady state configuration across $\Theta$ directions. In fact, the difference
in the location and extension of flavor conversion is more pronounced in this
case, with the initial $\theta$ angular distributions exhibiting no ELN
crossings for selected $\Theta$’s, but the spread of flavor waves contributes
to the development of flavor conversion (cf. Fig. 12).

##  5 Discussion and outlook

We have presented the solution of the QKEs in $(2+1+1)$ dimensions, including
two spatial dimensions, one neutrino momentum angle (focusing on mono-
energetic neutrinos), and time. The neutrino angular distributions are
computed through neutral and charged current interaction terms entering the
collisional kernel in the QKEs. We have tracked the neutrino flavor evolution
during neutrino decoupling from matter (with isotropic angular distributions
at the innermost radius, that consistently evolve becoming forward peaked as
the radius increases). As already pointed out in Refs. [26, 40, 32, 41, 50,
27, 48, 66, 19, 28, 67, 68], the non-trivial interplay among advection,
collisions, and flavor conversion shapes the quasi-steady state configuration
achieved by neutrinos.

Figure 7: Difference in the $\nu_{e}$ number density of the steady state
configurations obtained without flavor conversion (c) and with flavor
conversion (q) for Cases NP (left, without perturbations), PH (middle, with
perturbations in the vacuum Hamiltonian), and PC (right, with perturbations in
the collision term). The top (bottom) panels refer to Case 1 (Case 2; cf.
Appendix B). The perturbations in the Hamiltonian or in the collision term
contributes to break the symmetry in $\Theta$, giving rise to
$\Theta$-dependent quasi-steady state configurations, with flavor waves
marginally spreading across neighbor $\Theta$ regions.

Figure 7 summarizes our findings for the $\nu_{e}$ quasi-steady state
configurations for Case 1 (top panels) and Case 2 (bottom panels, presented in
Appendix B and obtained for a collision term leading to different
$\bar{\nu}_{e}$ emission with respect to Case 1). At the innermost radii, the
angular distributions for all flavors are isotropic, and flavor conversion is
triggered above $\simeq 20$ km. In the absence of any perturbations (left
panels) and despite the difference in the neutrino self-interaction
Hamiltonian between the 2D and 1D simulation setups, our findings are
qualitatively comparable with Cases C and B presented in Ref. [23] (except for
a small left-right asymmetry in $\theta$ appearing in 2D). This implies that
the quasi-steady state flavor configuration is independent of $\Theta$ to a
good approximation.

In order to mimic fluctuations occurring in the neutrino and matter
backgrounds, we introduce perturbations in the vacuum term of the Hamiltonian
(affecting all flavors equally) or in the collision term (in the emission term
of $\nu_{e}$ and $\bar{\nu}_{e}$). In both scenarios, the presence of
perturbations contributes to break the symmetry in $\Theta$, giving rise to
$\Theta$-dependent quasi-steady state configurations.

When all flavors are perturbed in the same way in $H_{\rm{vac}}$ (Figs. 4, 10
and middle panels of Fig. 7), the ELN crossings in the classical steady state
configuration occur at the same momentum angle $\theta$, for all $\Theta$.
Hence, we roughly obtain the same regions of flavor transformation on the
contour plots as in the non-perturbed case (cf. left panels of Fig. 7), except
for small fluctuations. For Case 1HP, the difference between the flavor
configurations obtained for different $\Theta$’s is more pronounced than for
Case 2HP.

Introducing flavor-dependent perturbations in the collision term (Figs. 5 and
11 and right panels of Fig. 7), the symmetry is clearly broken in $\Theta$
because the ELN crossings occur at different $\theta$’s for each $\Theta$.
Notably, because of neutrino advection, flavor waves spread through their
neighbor angular regions, remaining localized in specific ranges of $\Theta$.
This trend is more pronounced in Case 2CP, where there is no crossing for some
$\Theta$’s in the initial angular distribution, but advection favors the
spread of the regions affected by flavor conversion (Fig. 12). These results
suggest a larger variation of the flavor conversion physics across different
$\Theta$ regions when the number of emitted neutrino flavors is not
homogeneous.

In none of the explored ELN configurations, flavor conversion leads to flavor
equipartion, despite Refs. [27, 52, 28, 53, 54, 29, 69, 35, 34, 55, 56, 30,
50] conclude that this might be a generic flavor outcome in 1D. In this
respect, our findings are in agreement with the 1D results of Refs. [23, 22],
the achievement of flavor equipartition being dependent on the choice of the
simulation boundary conditions and the shape of the ELN distributions [25].

Earlier work attempted to explore the neutrino flavor conversion physics in
multi-dimensions [40, 26, 35]. Here, for the first time, we consistently
evolve the (anti)neutrino angular distributions through the collision term,
tracking neutrino decoupling from matter. Also, we do not rely on the
employment of seeds in the off-diagonal terms of the density matrix to aid the
growth of flavor instabilities, but evolve the QKEs including the vacuum term
in the Hamiltonian. Our main findings are in general agreement with the ones
reported in previous multi-D work within simplified setups, concerning the
fact that advection allows for the spreading of flavor structures within small
spatial regions. However, our dynamical modeling of the ELN distributions does
not lead to flavor equilibration as an outcome of flavor conversion. Future
work should also focus on the interplay between the increased dimensionality
of the simulation shell and the inclusion of the azimuthal angle in the
neutrino distribution—the inclusion of the latter has been shown to lead to
spontaneous symmetry breaking [57].

In summary, this work highlights the non-trivial interplay among neutrino
flavor conversion, collisions, and advection in multi-dimensions, showing that
flavor conversion can spread across neighbor regions of the simulation shell
through neutrino advection. Our findings suggest that it is important to
assess how the quasi-steady state flavor configuration changes when relaxing
symmetry assumptions in the solution of the QKEs in order to work out robust
criteria to forecast the flavor configuration resulting from flavor
conversion, and its consequent impact on the physics of compact astrophysical
sources and related multi-messenger observables.

## Acknowledgments

We acknowledge useful discussions with Manuel Goimil García. This project has
received support from the Villum Foundation (Project No. 13164), the Danmarks
Frie Forskningsfond (Project No. 8049-00038B), the European Union (ERC, ANET,
Project No. 101087058), and the Deutsche Forschungsgemeinschaft through
Sonderforschungbereich SFB 1258 “Neutrinos and Dark Matter in Astro- and
Particle Physics” (NDM). Views and opinions expressed are those of the authors
only and do not necessarily reflect those of the European Union or the
European Research Council. Neither the European Union nor the granting
authority can be held responsible for them.

##  Appendix A Neutrino advection term in two spatial dimensions

Figure 8: Sketch of the simulation annulus (cf. Fig. 1). A neutrino traveling
with velocity $\vec{v}$ in the simulation annulus is described by the radius
$r$, angle $\Theta$, and momentum angle $\theta$. The minimum distance between
the neutrino trajectory and the center is given by $r_{0}$. The velocity
vector can be decomposed into the radial and angular components $v_{r}$ and
$v_{\Theta}$.

In this appendix, we derive the expression for the advection term
($\vec{v}\cdot\vec{\nabla}\rho(r,\Theta,\theta)$) in Eqs. 2.1 and 2.2. The del
operator in two dimensions and polar coordinates is

$\vec{\nabla}=\frac{\partial}{\partial
r}\hat{r}+\frac{1}{r}\frac{\partial}{\partial\Theta}\hat{\Theta}\ ;$ (A.1)

while the velocity vector is defined as

$\vec{v}=v_{r}\hat{r}+v_{\Theta}\hat{\Theta}=\frac{\partial r}{\partial
t}\hat{r}+r\frac{\partial\Theta}{\partial t}\hat{\Theta}\ .$ (A.2)

The velocity components can be expressed in terms of the neutrino momentum
angle $\theta$ considering that $\partial r/\partial
t=|\vec{v}|\cos\theta=\cos\theta$ and $\partial\Theta/\partial
t=|\vec{v}|\sin\theta/r=\sin\theta/r$, since $|\vec{v}|\approx c=1$ (see Fig.
8). Hence

$\vec{v}\cdot\vec{\nabla}=\cos{\theta}\frac{\partial}{\partial
r}+\frac{\sin{\theta}}{r}\frac{\partial}{\partial\Theta}\ .$ (A.3)

Using the expressions above, the advection term in Eqs. 2.1 and 2.2 becomes

$\begin{split}\vec{v}\cdot\vec{\nabla}\rho(r,\Theta,\theta)=&\cos{\theta}\frac{\partial\rho(r,\Theta,\theta)}{\partial
r}+\frac{\sin{\theta}}{r}\frac{\partial\rho(r,\Theta,\theta)}{\partial\Theta}\\\
=&\cos{\theta}\left(\frac{\partial\rho(r,\Theta,\theta)}{\partial
r}+\derivative{\theta}{r}\frac{\partial\rho(r,\Theta,\theta)}{\partial\theta}\right)+\frac{\sin{\theta}}{r}\frac{\partial\rho(r,\Theta,\theta)}{\partial\Theta}\
.\end{split}$ (A.4)

Using the setup illustrated in Fig. 8 to express the neutrino momentum angle
as a function of the radius, we have:

$\cos\theta(r)=\frac{\sqrt{r^{2}-r_{0}^{2}}}{r}\ ,$ (A.5)

with $r_{0}$ being the the minimum distance to the neutrino trajectory. As a
consequence, the radial derivative of $\cos\theta$ is:

$\derivative{\cos\theta}{r}=\frac{r_{0}^{2}}{r^{2}\sqrt{r^{2}-r_{0}^{2}}}=\frac{r^{2}\sin^{2}{\theta}}{r^{2}\sqrt{r^{2}-r^{2}\sin^{2}\theta}}=\frac{\sin^{2}\theta}{r\cos\theta}\
,$ (A.6)

such that

$\derivative{\cos\theta}{r}=-\frac{\sin\theta}{r\cos\theta}\ .$ (A.7)

Inserting Eq. A.7 in Eq. A.4, we obtain

$\begin{split}\vec{v}\cdot\vec{\nabla}\rho(r,\Theta,\theta)=\cos\theta\partialderivative{\rho(r,\Theta,\theta)}{r}+\frac{\sin\theta}{r}\partialderivative{\rho(r,\Theta,\theta)}{\Theta}-\frac{\sin\theta}{r}\partialderivative{\rho(r,\Theta,\theta)}{\theta}\
.\end{split}$ (A.8)

##  Appendix B Flavor evolution in two dimensions with perturbations: another
electron neutrino lepton number angular configuration

In order to assess the robustness of our findings for what concerns the role
of the perturbations on the quasi-steady state flavor configurations, in this
appendix, we investigate the flavor conversion physics for another set of
neutrino angular distributions. To this purpose, we consider the mean free
paths defined as in Table 2, following Case B of Ref. [23].

Table 2: Mean free paths of emission, absorption, and direction-changing interactions for each flavor as a function of radius. This model for the collision term follows Case B of Ref. [23]. | $\nu_{e}$ | $\bar{\nu}_{{e}}$ | $\nu_{x},~{}\bar{\nu}_{x}$
---|---|---|---
$\lambda_{\text{emission}_{\phantom{l}}}^{i}$[km] | $1/[50\xi(r)]$ | $1/[26\xi(r)]$ | $1/[10\xi(r)]$
$\lambda_{\text{absorb }_{\phantom{l}}}^{i}$[km] | $1/[50\xi(r)]$ | $1/[25\xi(r)]$ | $1/[10\xi(r)]$
$\lambda_{\text{dir-ch }_{\phantom{l}}}^{i}$[km] | $1/[50\xi(r)]$ | $1/[25\xi(r)]$ | $1/[12.5\xi(r)]$

Figure 9 shows the corresponding quasi-steady state configuration obtained in
the absence of perturbations (Case 2NP). Due to the absence of
$\Theta$-dependent perturbations, the quasi-steady configuration of Case 2NP
corresponds to the one obtained for Case B in Ref. [23] (cf. their Fig. 7)
except for the additional term present in the neutrino self-interaction
Hamiltonian in our 2D framework and responsible for small variations in the
quasi-steady state number densities. The ELN radial evolution in the classical
steady state is such that there are no flavor instabilities developing in a
narrow band between $21$ and $22$ km, compared to Case 1NP where flavor
instabilities occur at all radii above $20$ km.

Case 2NP (no perturbations)

Figure 9: Same as Fig. 2, but for the collision term modeled as in Table 2
(Case 2NP; evolved for $t=50~{}\mu$s). The ELN radial evolution allows for
negligible flavor conversion between $21$ and $22$ km.

Figure 9 shows the quasi-steady state solution for Case 2HP. Consistently with
the findings of Case 1HP, despite the different ELN configuration, we find
that perturbations are responsible for small changes in the quasi-steady state
configuration (cf. Fig. 9 for the case without any perturbations).

Case 2HP (perturbations in vacuum Hamiltonian)

Figure 10: Same as Fig. 4, but for the collision term modeled as in Table 2
(Case 2HP; evolved for $t=50~{}\mu$s). Similar to Case 1HP, we find that
perturbations in the vacuum term of the Hamiltonian induce small changes in
the quasi-steady state configuration.

Figure 11 is the analog of Fig. 5, but it has been obtained for the collision
term modeled as in Table 2 (Case 2CP). Similar to Case 1CP, we find that the
perturbations in the collision term introduce $\Theta$-dependent fluctuations
in the $\nu_{e}$ and $\bar{\nu}_{e}$ local distribution densities with
$\Theta$-dependent implications on the flavor conversion physics.

Case 2CP (perturbations in collision term)

Figure 11: Same as Fig. 5, but for the collision term modeled as in Table 2
(Case 2CP; evolved for $t=50~{}\mu$s). The perturbations in the collision term
are responsible for the development of $\Theta$-dependent small-scale
structures in the neutrino ensemble.

It is interesting to note that, in Case 2CP shown (Fig. 11), the regions of
flavor conversion do not match the peaks in the $\nu_{e}$ emission, rather the
ones in the $\bar{\nu}_{e}$ emission. This can be explained by inspecting Fig.
12 which shows the angular distributions: in the locations with more
$\bar{\nu}_{e}$, the ELN crossings form at two angles for larger radii
($\Theta_{2}$), and in the sites with more $\nu_{e}$’s, the ELN distribution
does not show any crossings at the selected radii ($\Theta_{3}$). Furthermore,
for $r=25.1$ km at $\Theta_{3}$ there are no ELN crossings, but when comparing
to the contour plot of $\Theta_{3}$ in Fig. 11, flavor conversion has
developed at $r=25.1$ km, suggesting that neutrino advection contributes to
the spreading of flavor waves.

Case 2CP (perturbations in collision term)

Figure 12: Same as Fig. 6, but for the collision term modeled as in Table 2
(Case 2CP).

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