# An open dataset for research on audio
field recording archives: freefield1010

Dan Stowell and Mark D. Plumbley
Centre for Digital Music, Queen Mary University of London
<EMAIL_ADDRESS>
###### Abstract

We introduce a free and open dataset of 7690 audio clips sampled from the
field-recording tag in the Freesound audio archive. The dataset is designed
for use in research related to data mining in audio archives of field
recordings / soundscapes. Audio is standardised, and audio and metadata are
Creative Commons licensed. We describe the data preparation process,
characterise the dataset descriptively, and illustrate its use through an
auto-tagging experiment.

##  1 Introduction

Digital sound archives hold vast resources of material, including speech,
music, and naturalistic and ethnographic field recordings (Ranft, 2004).
However, there are still many challenges in organising and searching in these
archives. In recent decades, research fields such as automatic speech
recognition (ASR) and music information retrieval (MIR) have developed
automatic methods for labelling and transcribing specific types of sound. Yet
even if we put speech and music to one side, we still have a large and
valuable range of recorded sound, and there has been relatively little work in
organising and searching this non-speech-non-music audio. Such research may
come under the umbrella of “computational auditory scene analysis” (CASA)
(Wang and Brown, 2006). Tasks have begun to be addressed in recent years such
as automatically labelling the type of audio scene, or automatically detecting
and labelling the events within the audio scene (Giannoulis et al., 2013, and
citations therein).

Important for research development is the existence of standard datasets that
can be independently reused by researchers. This has motivated many community
efforts in ASR and in MIR. In our own recent work we created a set of “audio
scene” recordings as part of the IEEE AASP “D-CASE” challenge, consisting of
30-second binaural recordings made by three recordists in locations around
London (Giannoulis et al., 2013). Such focussed datasets are valuable for
developing algorithms for the specific tasks considered, but they are quite
different from most audio archives, being tightly calibrated in their
production and of moderate size.

In order for datasets to be relevant to applications in sound archives, they
need to be large enough that they (a) reflect the diversity of content in
audio archives and (b) give some indication of scalability issues for analysis
and for visualisation/navigation. However, large datasets are expensive and
time-consuming to record from scratch, while most existing archives cannot be
freely redistributed due to the associated copyright and licensing terms. It
is illustrative that the creators of a recent large pop-music dataset (the
Million Song Dataset) worked around this issue by distributing not the audio,
but some pre-computed features derived from the audio (Bertin-Mahieux et al.,
2011). This enabled them to distribute a large open dataset of relevance to
pop music archives, but as a consequence it restricted the types of analysis
possible: researchers are constrained to using the specific pre-computed
features they provided.

For our purposes, a notable initiative is the Freesound
archive,111http://freesound.org/ which hosts extensive holdings of
crowdsourced audio recordings, reusable under Creative Commons and public
domain licences. Established in 2005, it holds more than 160,000 sounds from
thousands of users around the world. It contains a wide range of sound types,
including field recordings, recordings from contact mics and hydrophones, and
synthetic sounds.

Freesound is already usable for various research purposes, having open
licensing conditions and an easy-to-use application programming interface
(API). However, it is a large crowdsourced and continually-updated collection,
not a fixed dataset. Files are continually added and removed, and metadata
changed; it is too large to easily redistribute among researchers; various
licences are used, not all mutually compatible; and files are in various
formats (e.g. WAV/AIFF/ MP3/Ogg, number of channels, sample-rate, duration)
which can be inconvenient for those developing algorithms. Also, a
crowdsourced archive such as Freesound is typically more heterogeneous than
traditional archives such as the British Library Sound Archive, whose curation
involves manual attention to file formats and metadata (Ranft, 2004).

We therefore chose to compile a free and open dataset of a wide range of
sounds, standardised and curated from a fraction of the extensive holdings of
the Freesound archive. This dataset is intended to be of use to researchers
developing methods for working with field recordings in audio archives. In the
following we describe how we designed and prepared the data. We then describe
how anyone can access the data and work with it, and illustrate with an
automatic recognition experiment to infer the presence of tags.

##  2 Dataset

In order to maximise the potential usefulness of the dataset, before
preparation we considered the following design criteria:

Content: Firstly we aimed to reflect the content that a general audio archive
might collect. Given the range of archive policies, from crowdsourced to
strictly curated, we opted for a middle way, by using Freesound contributions
but only those under the field-recording tag. Initial inspection of the
various tags in Freesound determined that this tag was mostly free of
interpretation issues, unlike for example ambience which Freesound users often
use for ambient field recordings but also for synthesised atmospheric
soundtrack sounds. On the other hand, we decided not to manually curate the
collection to a specific definition of “field recording”, in part because such
curation is difficult to apply to material from unknown third parties, and
also because audio archive collections are rarely so narrowly construed.

Licensing: Freesound contains material under various licences. Most common are
the Creative Commons CC-BY licence and the “CC0” public domain dedication,
though there is a small proportion of older material under other licences such
as Creative Commons “sampling” licences which are not compatible with CC-BY.
We wished to be able to apply a single open licence to the overall dataset, so
we restricted ourselves to CC-BY and CC0 material, which means the overall
dataset can be published under CC-BY. We also needed to respect the
attribution requirements in the CC-BY source material, which we implemented by
ensuring we stored the author metadata with each file as well as a URL to link
back to the original source.

Size and duration: We aimed to produce a dataset of manageable storage size,
for ease of redistribution, yet suitably diverse. We also wished to produce
sound excerpts of a standard and relatively short duration, so that they could
be used in listening tests without risking listener fatigue, and so that
automatic tests could run efficiently. These motivations led us to settle on a
fixed ten-second duration for each excerpt.

File formats: We also aimed to use a standardised file format. Freesound
allows users to upload sounds of any sample rate, any number of channels, and
in various file formats (uncompressed and lossy-compressed). We chose not to
use audio coming from lossy compressed file formats, in case of artifacts
introduced by the codecs. We also chose to convert all downloaded sounds to
standard CD-quality mono WAV files. We considered standardising on 24-bit
and/or 96 kHz as recommended in the IASA “TC04” archiving standard (IASA
Technical Committee, 2009). However, in our experience 16-bit PCM has been
more widely compatible than 24-bit PCM: the latter is not well handled by some
older versions of Matlab and some Python audio libraries, although this
situation is improving. Also the majority of the original downloaded audio was
in 44.1 kHz. We therefore settled on 16-bit 44.1 kHz. In the crowdsourced
Freesound archive, amplitude levels are uncontrolled, which may be problematic
for listening tests. We therefore chose to amplitude-normalise each excerpt in
our dataset.

Dataset partitioning: In data mining and machine-learning experiments, it is
useful to have a dataset partitioned into separate subsets—e.g. one for
training and another for testing (Witten and Frank, 2005, Chapter 5). To
facilitate this we chose to partition our data into ten equally-sized subsets.
Partitioning can be purely random or can be stratified: for example, if the
data was intended for an experiment detecting the presence of geotags, then
the partitions could be arranged such that each partition had an equal mix of
geotagged and non-geotagged data (Witten and Frank, 2005, Chapter 5). However,
our dataset is intended for various purposes and not for a single specific
experiment, so we opted for the simple random partitioning. Researchers can
choose to use these subsets for comparability with others, or to perform their
own partitioning.

###  2.1 Preparation

We first obtained specific permission from Freesound to perform our relatively
large-scale data download. We created a Python script based on the official
freesound-python code, which we used to download the files from
Freesound.222https://github.com/danstowell/freesound-python/tree/tagsearch The
script was run to download all files matching all of the following criteria:

  * •

Tagged field-recording (which contained 17807 sounds in total)

  * •

Length 10 seconds or greater

  * •

Audio file format WAV

  * •

Published under either the CC-BY licence or CC0

  * •

Audio with 1 or 2 channels

  * •

Audio sample format one of: pcm16, pcm24, pcm32

Each file was saved along with its metadata in JSON
format.333http://tools.ietf.org/html/rfc4627 A small number of files (44)
failed to download completely; these were detected by using the sox command-
line audio tool (v 14.3.2) to attempt WAV decoding, and deleting the files
which reported end-of-file errors or similar. The script was run in July 2013,
taking about a week to download 328 GB of material.

We then prepared a 10-second standardised excerpt from each audio file, taken
from the middle of the audio recording, and used sox to convert it to a
standardised file format: WAV, single-channel, sample rate 44.1 kHz, 16-bit
PCM, amplitude normalised to $-2$ dB (empirically selected as the maximum gain
before clipping).

We inspected the excerpts for any further issues, by listening to all
10-second extracts. We found a few (seven) which were pure silence with DC
offset. While this may sometimes be valid audio when considered in context, it
led to normalisation issues, and could be problematic for some applications;
we decided to remove these files.

The above procedure resulted in 7690 sound excerpts with accompanying
metadata. We placed the exerpts into 10 separate partitions, where the
allocation was by pseudorandom shuffle initialised with a fixed seed value for
repeatability (given in the Python script referenced above). Each of the 10
partitions has about 128 minutes of audio; the dataset totals over 21 hours of
audio.

Figure 1: Density plot of all geolocation tags in the dataset. Source code for
this plot is in Appendix A.

Each file is associated with various metadata (author, date, licence etc.)
including a median of 7 tags per file (range 1–68). Around 40% of the files
come with geolocation metadata. Figure 1 shows a density plot for all
geolocations in the dataset. It indicates a broad geographic spread, although
with much the strongest density in Europe (perhaps understandable given that
Freesound is a European project), and a relative lack of sounds tagged from
Africa and Russia.

###  2.2 Availability

The full dataset is available online, hosted on the Internet
Archive,444http://archive.org/details/freefield1010 and our institutional
repository555http://c4dm.eecs.qmul.ac.uk/rdr/handle/123456789/35, under an
overall CC-BY licence. For portability, each of the ten subsets of the data is
presented as a single zip file which can fit on a data CD (around 570 MB per
subset). We list MD5 checksums of these zip files so that their integrity can
be verified.

The metadata stored alongside the audio is easy to work with. Appendix A gives
a simple Python code example for working with the JSON metadata, in this case
the code used to generate Figure 1.

##  3 A classification experiment

To demonstrate the dataset in use, we conducted an experiment using a binary
classification paradigm using audio content analysis to infer the
presence/absence of particular tags (sometimes called auto-tagging (Ellis et
al., 2011)). For this we used the simple baseline classifier smacpy presented
in Giannoulis et al. (2013), which provides a standard implementation of a
common approach based on Gaussian mixture modelling of Mel frequency cepstral
coefficients (MFCCs).

Freesound uses a free-tagging system, with users able to associate an
arbitrary number of tags with an audio file, and also able to create new tags.
We thus manually chose a small selection of tags which appear relatively
often—birdsong, city, people, nature, train, voice, water—to use for this
experiment, creating a binary classification task for each one. We note that
we might expect the tags to have a varying directness of connection with the
audio content: we would expect recordings tagged voice to feature human voice
sounds relatively prominently, whereas recordings tagged city might contain
many sounds in ensemble, with perhaps no sonic component always present.

We further added two pseudo-tags to the study, based on other metadata
attributes: an indicator of whether or not the item comes with geolocation
data (__geotagged), and an indicator of whether the item is CC-BY licensed
(__ccby), as opposed to the only other licence present, public domain. These
pseudo-tags in principle have no direct connection to the audio content,
although there may be correlations due to circumstantial effects (for example,
geolocation might more often be stored when recording outdoor scenes). We
therefore expected only mild if any ability to predict these metadata
attributes from audio.

Table 1: Prevalence of selected tags in freefield1010. Tag | Num tagged | Proportion (%)
---|---|---
birdsong | 198 | 2.6
city | 562 | 7.3
nature | 905 | 11.8
people | 321 | 4.1
train | 411 | 5.3
voice | 556 | 7.2
water | 707 | 9.2
__geotagged | 3058 | 39.8
__ccby | 6111 | 79.5

Table 1 provides a summary of the prevalence of the selected tags in the
freefield1010dataset. It is important to note that, for the “true” tags
especially, each tag is present in only a minority of the items; the ratio of
positive to negative instances is highly skewed. This has consequences for how
we evaluate automatic classification: rather than using raw accuracy, which
fails to account for this skew, we use the area under the curve (AUC)
statistic derived from a receiver operating characteristic (ROC) curve
(Provost et al., 1998; Fawcett, 2006).

Our experiment proceeded as follows: for each of the selected tags and pseudo-
tags, we performed a ten-fold cross-validation experiment using the folds
defined by the ten subsets of freefield1010. This means that for each fold, we
used nine of the ten subsets as training data for the classifier, where the
presence/absence of the tag was the binary attribute to be learnt, and then
tested the classifier using the audio from the one remaining subset. For each
such run, we calculated the numbers of correct and incorrect decisions, and
used this to calculate the AUC statistic (as in Fawcett (2006)). Source code
for this experiment is available
online.666https://github.com/danstowell/smacpy/tree/freefield1010

Figure 2: Results for automatic inference of tag presence/ absence. Plot shows
the mean and the 95% confidence interval of the AUC score for each tag (and
pseudo-tag) studied, across ten-fold crossvalidation.

Results in Figure 2 show that the tags can be automatically inferred from
audio with varying degrees of reliability. The best result is for birdsong at
82% AUC, while the weakest result for a true tag is city at 63%. Note that the
standard interpretation of an AUC value is that it tells us the probability
that the algorithm will rank a random positive instance higher than a random
negative instance (Fawcett, 2006); chance performance is always 50% for the
AUC statistic. Results for the two pseudo-tags attain around 58% AUC: above
chance, but still very weak. It indicates that there is some mild difference
in audio content between the positive and negative instances, but much less
than for the true tags.

The classification performance for true tags appears to show some connection
with the “directness” issue raised above: the tags which yield weakest
recognition performance (city, people, nature) can be said to have an indirect
connection with the audio content. However these results are illustrative
only, using a baseline classifier rather than a leading-edge algorithm. The
strongest performance (82%) is still much lower than is desirable for a binary
classifier deployed in a live system.

The 95% confidence intervals (error bars) in Figure 2 are relatively small and
well-separated. This illustrates that the dataset is of sufficient size to
make inferences about the relative predictability of these tags from the audio
content, and also that there is relative consistency among the ten folds of
the data.

##  4 Conclusions

In this paper we have described the preparation of a free and open audio
dataset, designed primarily for use in research on data mining of audio
archives. The dataset is derived from a subset of the Freesound archive, but
fixed and standardised so as to facilitate reproducible research. In an auto-
tagging experiment we demonstrated that the dataset can be used to probe
issues such as the differential predictability of tags from audio. We hope
that the dataset will prove useful to others.

## Acknowledgments

We wish to thank the Freesound developers and maintainers at the Music
Technology Group in Universitat Pompeu Fabra, for running the excellent
Freesound archive and for allowing us API access to the data.

DS & MP are supported by an EPSRC Leadership Fellowship EP/G007144/1.

## References

  * Bertin-Mahieux et al. [2011] T. Bertin-Mahieux, D. P. W. Ellis, B. Whitman, and P. Lamere.  The Million Song Dataset.  In _Proceedings of the 12th International Conference on Music Information Retrieval (ISMIR-11)_ , pages 591–596, Miami, FL, USA, Oct 2011.
  * Ellis et al. [2011] K. Ellis, E. Coviello, and G. Lanckriet.  Semantic annotation and retrieval of music using a bag of systems representation.  In _Proceedings of the 12th International Conference on Music Information Retrieval (ISMIR-11)_ , pages 723–728, Miami, FL, USA, Oct 2011.
  * Fawcett [2006] T. Fawcett.  An introduction to ROC analysis.  _Pattern recognition letters_ , 27(8):861–874, 2006.  doi: 10.1016/j.patrec.2005.10.010.
  * Giannoulis et al. [2013] D. Giannoulis, D. Stowell, E. Benetos, M. Rossignol, M. Lagrange, and M. D. Plumbley.  A database and challenge for acoustic scene classification and event detection.  In _Proceedings of the European Signal Processing Conference (EUSIPCO 2013)_ , 2013.
  * IASA Technical Committee [2009] IASA Technical Committee.  _Guidelines on the Production and Preservation of Digital Audio Objects_.  Number IASA-TC04. International Association of Sound and Audiovisual Archives, Aarhus, Denmark, second edition, 2009.  URL http://www.iasa-web.org/tc04/audio-preservation.
  * Provost et al. [1998] F. J. Provost, T. Fawcett, and R. Kohavi.  The case against accuracy estimation for comparing induction algorithms.  In _Proceedings of the International Conference on Machine Learning (ICML)_ , pages 445–453, 1998.
  * Ranft [2004] R. Ranft.  Natural sound archives: Past, present and future.  _Anais da Academia Brasileira de Ciências_ , 76(2):456–460, 2004.  doi: 10.1590/S0001-37652004000200041.
  * Wang and Brown [2006] D. L. Wang and G. J. Brown, editors.  _Computational Auditory Scene Analysis: Principles, Algorithms, and Applications_.  IEEE Press, New York, 2006.
  * Witten and Frank [2005] I. H. Witten and E. Frank.  _Data Mining: Practical Machine Learning Tools and Techniques_.  Morgan Kaufmann, San Francisco, CA, USA, 2nd edition, 2005.

## Appendix A: Code example

The following Python code was used to generate Figure 1 (using Python 2.7).

⬇

import glob, json

import matplotlib.pyplot as plt

import matplotlib.cm as cm

ffpath = ’./freefield1010’

outfile = ’plots/plotgeo.pdf’

lats = [], lons = []

for onepath in glob.iglob(’%s/*/*.json’ % ffpath):

jsonfile = open(onepath, ’r’)

jsondata = json.load(jsonfile)

jsonfile.close()

geo = jsondata.get(u’geotag’)

if geo != None:

lats.append(float(geo[u’lat’]))

lons.append(float(geo[u’lon’]))

# plot

plt.figure()

plt.hexbin(lons, lats, gridsize=25, bins=’log’, cmap=cm.binary)

plt.xlabel(’Longitude’)

plt.ylabel(’Latitude’)

cb = plt.colorbar()

cb.set_label(’log10(N)’)

plt.savefig(outfile, papertype=’A4’, format=’pdf’)


aainstitutetext: Institute for Particle Physics Phenomenology, Department of
Physics, Durham University, Durham, DH1 3LE, UKbbinstitutetext: Department of
Physics and McDonnell Center for the Space Sciences, Washington University,
St. Louis, MO 63130, USAccinstitutetext: Theoretical Physics Department, Fermi
National Accelerator Laboratory, Batavia, IL 60510, USA

# Impact of Improved Energy Resolution on DUNE sensitivity to Neutrino Non-
Standard Interactions

Sabya Sachi Chatterjee<EMAIL_ADDRESS>b P. S. Bhupal Dev
<EMAIL_ADDRESS>c Pedro A. N. Machado<EMAIL_ADDRESS>
###### Abstract

The full physics potential of the next-generation Deep Underground Neutrino
Experiment (DUNE) is still being explored. In particular, there have been some
recent studies on the possibility of improving DUNE’s neutrino energy
reconstruction. The main motivation is that a better determination of the
neutrino energy in an event-by-event basis will translate into an improved
measurement of the Dirac $CP$ phase and other neutrino oscillation parameters.
To further motivate studies and improvements on the neutrino energy
reconstruction, we evaluate the impact of energy resolution at DUNE on an
illustrative new physics scenario, viz. non-standard interactions (NSI) of
neutrinos with matter. We show that a better energy resolution in comparison
to the ones given in the DUNE conceptual and technical design reports may
significantly enhance the experimental sensitivity to NSI, particularly when
degeneracies are present. While a better reconstruction of the first
oscillation peak helps disentangling standard $CP$ effects from those coming
from NSIs, we find that the second oscillation peak also plays a nontrivial
role in improving DUNE’s sensitivity.

###### Keywords:

Neutrino Oscillation, Long-baseline, NSI, DUNE

††preprint: FERMILAB-PUB-21-260-T, IPPP/20/107††arxiv: 2106.04597

##  1 Introduction

Decades of solar, atmospheric, accelerator and reactor neutrino experiments
have firmly established the phenomenon of neutrino oscillations among the
three flavors Zyla:2020zbs . After the recent discovery of the relatively
large reactor mixing angle $\theta_{13}$, the three-neutrino oscillation
paradigm has entered a new precision era, where the known oscillation
parameters are being measured with an ever-increasing accuracy. At the same
time, several short, medium and long-baseline neutrino oscillation
experiments, either running or in the pipeline, are poised to resolve the sub-
dominant effects in oscillation data sensitive to the currently unknown
oscillation parameters, namely the Dirac $CP$ phase $\delta_{\rm CP}$, the
sign of the atmospheric neutrino mass-squared difference $\Delta m^{2}_{32}$,
and the octant of the atmospheric mixing angle $\theta_{23}$.

Unraveling the neutrino properties within the three-neutrino framework has
been a great success so far, when analyzing either a single experiment or the
entirety of current neutrino oscillation data in terms of global fits
Capozzi:2018ubv ; deSalas:2020pgw ; Esteban:2020cvm . Nevertheless, the quest
for new physics beyond the Standard Model (SM) in the neutrino sector remains
remarkably vibrant. Indeed, the phenomenon of neutrino oscillations itself
suggests nonzero neutrino masses, which requires some beyond the SM (BSM)
physics. Therefore, exploiting the full potential of the current and next-
generation neutrino experiments to probe BSM physics is an important ongoing
research topic.

One interesting model-independent framework for parameterizing BSM physics in
the neutrino sector is the so-called non-standard interactions (NSI)
Wolfenstein:1977ue ; for reviews on various aspects of NSI, see Refs.
Ohlsson:2012kf ; Miranda:2015dra ; Farzan:2017xzy ; Dev:2019anc . The NSI
framework is an effective field theory below the electroweak scale that may
encode the oscillation phenomenology of an entire class of new physics
scenarios; see e.g. Refs Farzan:2015doa ; Farzan:2015hkd ; Farzan:2016wym ;
Forero:2016ghr ; Babu:2017olk ; Dey:2018yht ; Babu:2019mfe  for specific
ultraviolet (UV)-complete models of NSI. The presence of NSI in either
neutrino production, detection or propagation through matter can crucially
affect the interpretation of the experimental data Esteban:2019lfo ;
Coloma:2019mbs . At the very least, it could serve as a foil for the three-
neutrino oscillation scheme Miranda:2004nb ; Girardi:2014kca ; Liao:2016hsa ;
Masud:2016bvp ; deGouvea:2016pom ; Masud:2016nuj ; Agarwalla:2016fkh ;
Deepthi:2016erc ; Deepthi:2017gxg ; Flores:2018kwk ; Hyde:2018tqt ;
Masud:2018pig ; Capozzi:2019iqn ; Yasuda:2020cff ; Esteban:2020itz ;
Agarwalla:2021twp . Therefore, a better understanding of the NSI effects is
essential for the success of the future precision neutrino experiments in
answering the open questions in neutrino physics.

Since neutral-current interactions affect neutrino propagation coherently,
long-baseline neutrino experiments with a well-understood beam and trajectory
are an ideal place to probe matter NSI effects. Numerous studies have been
performed to this effect exploring the NSI prospects in the long-baseline
experiments; see e.g. Refs. Ota:2001pw ; Blennow:2007pu ; Kopp:2007ne ;
Blennow:2008ym ; Kopp:2010qt ; Adhikari:2012vc ; Friedland:2012tq ;
Coelho:2012bp ; Rahman:2015vqa ; deGouvea:2015ndi ; Coloma:2015kiu ;
Forero:2016cmb ; Huitu:2016bmb ; Bakhti:2016prn ; Soumya:2016enw ;
Blennow:2016etl ; Liao:2016bgf ; Fukasawa:2016lew ; Liao:2016orc ;
Chatterjee:2018dyd ; Han:2019zkz ; Denton:2020uda ; Chatterjee:2020kkm ;
Bakhti:2020fde ; Giarnetti:2021  and references therein. In fact, a mismatch
in the determination of the $\delta_{\rm CP}$ extracted from the latest data
of the two running long-baseline neutrino experiments, NO$\nu$A
NOVA_talk_nu2020  and T2K T2K_talk_nu2020 , might already be hinting towards
some possible indications of nonzero NSI Denton:2020uda ; Chatterjee:2020kkm .

The next-generation long-baseline experiment DUNE Acciarri:2015uup ;
Abi:2020evt , with its high-intensity neutrino beam, huge statistics, wide-
band spectrum and improved systematic uncertainties, is in an excellent
position to probe matter NSI deGouvea:2015ndi ; Coloma:2015kiu ;
Blennow:2016etl ; Liao:2016orc  and BSM physics involving neutrinos in general
Abi:2020kei . At present, the full physics potential of liquid argon time
projection chamber (LArTPC) detectors is still being explored. Several
developments and novel proposals have been put forward recently to bolster the
capabilities of these detectors, such as sub-MeV ionization energy detection
in ArgoNeuT Acciarri:2018myr  and other LArTPC detectors Castiglioni:2020tsu ,
sub-GeV study of atmospheric neutrinos Kelly:2019itm , and novel $\nu_{\tau}$
detection strategies Conrad:2010mh ; deGouvea:2019ozk ; Machado:2020yxl  at
DUNE. These studies are aimed at further enhancing the DUNE sensitivity to BSM
physics.

Of particular interest for the present paper is the possibility of having an
improved reconstruction of the neutrino energy by collecting all ionization
energy and/or identifying each particle in an event-by-event basis
DeRomeri:2016qwo ; Friedland:2018vry . These studies show that the current
energy resolution as reported in the original DUNE Conceptual Design Report
(CDR) Acciarri:2015uup  and the DUNE Technical Design Report (TDR) Abi:2020qib
can be considerably improved by up to a factor of 2 to 3. It has also been
shown that the better energy resolution will contribute to a more precise
determination of the Dirac $CP$ phase DeRomeri:2016qwo .

In this work, we will show for the first time that not only the determination
of the standard oscillation parameters will benefit from an improved energy
resolution, but also DUNE’s sensitivity to new physics. We will take NSI as a
general example of new physics and will analyze how the experimental
sensitivity changes when going from the CDR and TDR resolutions to an improved
neutrino energy resolution scenario. For the improved case, we will take the
best energy reconstruction results of Ref. Friedland:2018vry , which leverages
particle identification and the detection of all ionization energy, including
de-excitation gammas from argon nuclei. Our findings show that the better
neutrino energy resolution significantly boosts the DUNE sensitivity to NSI,
improving its constraints by up to 25-30% for several NSI parameters. The
improved energy resolution leads to a better reconstruction of both first and
second oscillation peaks. While the first peak comprises most of the
statistical power in DUNE, we find that the second peak provides a non-
negligible contribution to the sensitivity, helping better disentangle
standard $CP$ effects from those coming from NSIs.111 For discussions related
to the relevance of DUNE’s second oscillation peak, see e.g. Ref.
DeRomeri:2016qwo ; Rout:2020emr  in the context of standard oscillation
physics and Ref. Huber:2010dx  in the context of NSI. These findings hold true
for single NSI, with or without associated $CP$ phases, and also when multiple
NSI parameters are considered at the same time.

The rest of the paper is organized as follows: In Sec. 2, we briefly review
the theoretical framework of NSIs. In Sec. 3, we review the DUNE experimental
setup. In Sec. 4, we give our simulation details. In Sec. 5, we discuss the
improved energy resolution. Our main numerical results are presented in Sec.
6. In Sec. 7, we present a qualitative discussion of our results. Our
conclusions are given in Sec. 8.

##  2 Theoretical framework

Non-standard interactions, or NSIs, first recognized in Ref.
Wolfenstein:1977ue , is an effective field theory below the weak scale,
without explicit $SU(2)_{L}$ invariance, that encompasses four-fermion
operators involving at least one neutrino field. Logically, we can divide NSIs
into two categories: charged-current (CC) operators, in which the leptonic
current has a neutrino and a charged lepton; and neutral-current (NC) NSI
where the leptonic current has two neutrino fields. While CC-NSIs may affect
neutrino production and detection Grossman:1995wx , NC-NSIs induce non-trivial
matter effects and modify the dynamics of flavor conversion in matter
Wolfenstein:1977ue . Long-baseline experiments like DUNE are strongly affected
by matter effects, and thus are expected to have exceptional sensitivity to
NC-NSIs.

While the ultraviolet completion of NC-NSIs can arise either via heavy
mediators (see e.g. Refs. Biggio:2009nt ; Forero:2016ghr ; Babu:2019mfe ) or
light mediators Farzan:2015doa ; Farzan:2015hkd ; Babu:2017olk ; Datta:2017pfz
, we adopt an agnostic approach here and analyze the low energy phenomenology
irrespective of model building considerations. To be more precise, NC-NSI can
be described by dimension-six operators as,

$\mathcal{L}_{\mathrm{NC\mbox{-}NSI}}\;=\;-2\sqrt{2}G_{F}\varepsilon_{\alpha\beta}^{fC}\bigl{(}\overline{\nu}_{\alpha}\gamma^{\mu}P_{L}\nu_{\beta}\bigr{)}\bigl{(}\overline{f}\gamma_{\mu}P_{C}f\bigr{)}\;,$
(1)

where $\alpha,\beta=e,\mu,\tau$ denote the neutrino flavors, $f=e,u,d$
indicate the matter fermions, $C=L,R$ corresponds to the chirality of the
fermionic $f$-$f$ current, and $\varepsilon_{\alpha\beta}^{fC}$ are the
strengths of the NSI. The hermiticity of the Lagrangian requires

$\varepsilon_{\beta\alpha}^{fC}\;=\;\left(\varepsilon_{\alpha\beta}^{fC}\right)^{*}\;.$
(2)

It is worth mentioning that the diagonal ($\alpha=\beta$) NSIs are always real
whereas the non-diagonal ($\alpha\neq\beta$) NSIs are in general complex as
they always appear along with their associated $CP$-phases.

For neutrino propagation in the Earth, the potential induced by matter is the
sum of the potentials induced by electrons, protons and neutrons. Since only
the vector part of the current is relevant for coherent forward scattering, it
is convenient to define

$\varepsilon_{\alpha\beta}\;\equiv\;\sum_{f=e,u,d}\varepsilon_{\alpha\beta}^{f}\dfrac{N_{f}}{N_{e}}\;\equiv\;\sum_{f=e,u,d}\left(\varepsilon_{\alpha\beta}^{fL}+\varepsilon_{\alpha\beta}^{fR}\right)\dfrac{N_{f}}{N_{e}}\;,$
(3)

where $N_{f}$ is the number density of fermion $f$. For the crust of the
Earth, which is what long baseline neutrinos travel through, we can assume
neutral and isoscalar matter, implying $N_{n}=N_{p}=N_{e}$, in which case
$N_{u}=N_{d}=3N_{e}$. Therefore,

$\varepsilon_{\alpha\beta}\,\simeq\,\varepsilon_{\alpha\beta}^{e}+3\,\varepsilon_{\alpha\beta}^{u}+3\,\varepsilon_{\alpha\beta}^{d}\;.$
(4)

The presence of NSI modifies the effective Hamiltonian of neutrino propagation
in matter, which in the flavor basis becomes

$H\;=\;U\begin{bmatrix}0&0&0\\\ 0&k_{21}&0\\\
0&0&k_{31}\end{bmatrix}U^{\dagger}+V_{\mathrm{CC}}\begin{bmatrix}1+\varepsilon_{ee}&\varepsilon_{e\mu}&\varepsilon_{e\tau}\\\
\varepsilon_{e\mu}^{*}&\varepsilon_{\mu\mu}&\varepsilon_{\mu\tau}\\\
\varepsilon_{e\tau}^{*}&\varepsilon_{\mu\tau}^{*}&\varepsilon_{\tau\tau}\end{bmatrix}\,,$
(5)

where $U$ is the PMNS matrix, which, in its standard parameterization, depends
on three mixing angles ($\theta_{12},\,\theta_{13},\,\rm{and,}\,\theta_{23}$)
and one $CP$-phase ($\delta_{\rm CP}$). The quantities $k_{21}\equiv\Delta
m^{2}_{21}/2E$ and $k_{31}\equiv\Delta m^{2}_{31}/2E$ represent the solar and
atmospheric wavenumbers, where $\Delta m^{2}_{ij}\equiv m^{2}_{i}-m^{2}_{j}$
and $E$ is the neutrino energy. $V_{\mathrm{CC}}$ is the CC matter potential,

$V_{\mathrm{CC}}\;=\;\sqrt{2}G_{F}N_{e}\;\simeq\;7.6\,Y_{e}\times
10^{-14}\bigg{[}\dfrac{\rho}{\mathrm{g/cm^{3}}}\bigg{]}\,\mathrm{eV}\,,$ (6)

where $Y_{e}=N_{e}/(N_{p}+N_{n})\simeq 0.5$ is the relative electron number
density in Earth’s crust and $\rho$ is the Earth matter density, which for
DUNE we take as $\rho=2.848$ g/cm3.

For convenience we introduce the dimensionless quantity
$\hat{v}=V_{\mathrm{CC}}/k_{31}$, which gauges the sensitivity to matter
effects. Its absolute value

$|\hat{v}|\;=\;\bigg{|}\frac{V_{\mathrm{CC}}}{k_{31}}\bigg{|}\;\simeq\;8.8\times
10^{-2}\bigg{[}\frac{E}{\mathrm{GeV}}\bigg{]}\;,$ (7)

appears in the analytical expressions of the $\nu_{\mu}\to\nu_{e}$ appearance
probability and $\nu_{\mu}\to\nu_{\mu}$ survival probability, as we will see
below.

Let us now discuss the appearance and survival probabilities relevant for the
long-baseline experiment DUNE. In the presence of NSI, one can realize that
the mixing angle $\theta_{13}$, the parameter $\hat{v}$ at DUNE and non-
diagonal NSIs $|\varepsilon_{\alpha\beta}|$ are small. While the first two
have similar size $\eta\sim 0.1$, where $\eta$ is simply a small expansion
parameter, for neutrino energies of a couple of GeV or so, the latter cannot
be much larger than that. Therefore we can perform an expansion on those
parameters considering them to be roughly of the same magnitude
$\mathcal{O}(\eta)$. We can also define $\alpha\equiv\Delta m^{2}_{21}/\Delta
m^{2}_{31}=\pm 0.03$, which would then be $\mathcal{O}(\eta^{2})$. Note that
at DUNE $\hat{v}$ can be relatively large, particularly for the high energy
tail of the spectrum, and thus the expansions that we will perform on small
$\hat{v}$ may break down if $|\varepsilon_{\alpha\beta}|$ is not much smaller
than 1. Nevertheless it is still useful to do the expansion in order to
understand the role of energy resolution in the search for NSIs. Our numerical
analysis is, however, exact and does not rely on any such expansions.

Let us discuss the appearance and disappearance oscillation probabilities up
to the third order in $\eta$ given in Refs. Kikuchi:2008vq ; Liao:2016hsa ;
Liao:2016orc . We will write the NSI parameters in general as
$\varepsilon_{\alpha\beta}=|\varepsilon_{\alpha\beta}|e^{i\phi_{\alpha\beta}}$
for non-diagonal NSIs ($\alpha\neq\beta$) while for the diagonal NSI couplings
we simply have $\varepsilon_{\alpha\alpha}$ as a real parameter (its sign is
physical). First, we start with the appearance probability,

$\displaystyle P\left(\nu_{\mu}\rightarrow\nu_{e}\right)\,\simeq\,$
$\displaystyle\,\sin^{2}2\theta_{13}s_{23}^{2}f^{2}+2\alpha s_{13}\sin
2\theta_{12}\sin 2\theta_{23}fg\cos(\Delta+\delta_{\rm CP})$
$\displaystyle+8\hat{v}s_{13}s_{23}\bigg{\\{}|\varepsilon_{e\mu}|\left[s_{23}^{2}f^{2}\cos(\phi_{e\mu}+\delta_{\rm
CP})+c_{23}^{2}fg\cos(\Delta+\delta_{\rm CP}+\phi_{e\mu})\right]$
$\displaystyle\qquad+|\varepsilon_{e\tau}|s_{23}c_{23}\left[f^{2}\cos(\phi_{e\tau}+\delta_{\rm
CP})-fg\cos(\Delta+\delta_{\rm CP}+\phi_{e\tau})\right]\bigg{\\}},$ (8)

where $\Delta\equiv\Delta m^{2}_{31}L/4E$ is the atmospheric oscillating
frequency, $L$ being the baseline. For compactness, we have used the notation
$s_{ij}\equiv\sin\theta_{ij}$, $c_{ij}\equiv\cos\theta_{ij}$, and,

$\displaystyle
f\equiv\frac{\sin\left\\{[(1-\hat{v}(1+\varepsilon_{ee})]\Delta\right\\}}{1-\hat{v}(1+\varepsilon_{ee})}\,,\qquad
g\equiv\frac{\sin[\hat{v}(1+\varepsilon_{ee})\Delta]}{\hat{v}(1+\varepsilon_{ee})}.$
(9)

The first line of Eq. (2) is the standard approximated formula if
$\varepsilon_{ee}\to 0$, see e.g. Refs. Arafune:1997hd ; Freund:2001pn ;
Akhmedov:2004ny , while the second and third lines are the modifications
induced by NSIs. Notice that for small oscillation phase $\Delta$, both $f$
and $g$ are linear in $\Delta$. The leading term in $|\varepsilon_{e\tau}|$
tends to cancel for small $\Delta$, and thus we expect this NSI parameter to
not strongly affect the high energy part of DUNE’s far-detector spectrum.
Therefore, the impact of $|\varepsilon_{e\tau}|$ may be relatively more
pronounced in the second oscillation maximum. The observability of the second
oscillation maximum is directly related to the precision of DUNE’s energy
reconstruction. We will see later that an improved energy resolution will
significantly enhance the sensitivity to $|\varepsilon_{e\tau}|$, and that
both first and second oscillation peaks contribute appreciably for this
improvement. This effect is not present for $|\varepsilon_{e\mu}|$, and we
will confirm that with our numerical analysis.

The $\nu_{\mu}\to\nu_{\mu}$ disappearance oscillation probability is affected
by a different set of NSI parameters:

$\displaystyle P\left(\nu_{\mu}\rightarrow\nu_{\mu}\right)\,\simeq\,$
$\displaystyle\,1-\sin^{2}2\theta_{23}\sin^{2}\Delta+\alpha
c_{12}^{2}\sin^{2}2\theta_{23}\Delta\sin
2\Delta-4s_{23}^{4}s_{13}^{2}\frac{\sin^{2}[(1-\hat{v})\Delta]}{(1-\hat{v})^{2}}$
$\displaystyle-\frac{\sin^{2}2\theta_{23}s_{13}^{2}}{(1-\hat{v})^{2}}\biggl{\\{}\hat{v}\Delta\sin
2\Delta+\sin\big{[}(1-\hat{v})\Delta\big{]}\sin\big{[}(1+\hat{v})\Delta\big{]}\biggr{\\}}$
$\displaystyle-2\hat{v}|\varepsilon_{\mu\tau}|\cos\phi_{\mu\tau}\biggl{(}\sin^{3}2\theta_{23}\Delta\sin
2\Delta+2\sin 2\theta_{23}\cos^{2}2\theta_{23}\sin^{2}\Delta\biggr{)}$
$\displaystyle+\left[\hat{v}\sin^{2}2\theta_{23}\cos
2\theta_{23}\left(\varepsilon_{\mu\mu}-\varepsilon_{\tau\tau}\right)-\frac{\hat{v}^{2}}{2}\sin^{4}2\theta_{23}\left(\varepsilon_{\mu\mu}-\varepsilon_{\tau\tau}\right)^{2}\right]$
$\displaystyle\qquad\times(\Delta\sin 2\Delta-2\sin^{2}\Delta)\,.$ (10)

The first two lines describe standard oscillations, while the effect of NSIs
is encoded in the last 2 lines. The leading term on $\varepsilon_{\mu\tau}$ is
also multiplied by $\cos\phi_{\mu\tau}$, see the third line in the equation
above. Thus, if one marginalizes over the phase when presenting the
sensitivity to this NSI parameter, the result will typically be weak for DUNE
as the leading term can always be put to zero. In experiments where matter
effects are more relevant like IceCube, $\hat{v}\gtrsim 1$, the expansion
performed here breaks down and the impact of the $\phi_{\mu\tau}$ phase
becomes less pronounced, see e.g. Ref. IceCube:2021abg . It is also worth
mentioning that the second term in the parenthesis is proportional to
$\cos^{2}2\theta_{23}$, which may be fairly suppressed as $\theta_{23}$
approaches maximality.

The leading $\varepsilon_{\mu\mu}-\varepsilon_{\tau\tau}$ term is suppressed
by $\cos^{2}2\theta_{23}$, and thus sensitivity to this parameter should
present a strong dependence on $\theta_{23}$. Moreover, both leading and next-
to-leading terms cancel out for small $\Delta$, and so we anticipate an
important role in the experimental sensitivity to this parameter to be played
by the second oscillation and the energy resolution as a consequence. Our
numerical simulations will confirm this expectation as well. Also, note that
the first term in parenthesis in the last line of Eq. (2) is linear in
$\varepsilon_{\mu\mu}-\varepsilon_{\tau\tau}$, whereas the second term is
quadratic. This is the reason for the difference in the DUNE sensitivities to
$\varepsilon_{\mu\mu}$ and $\varepsilon_{\tau\tau}$, when each NSI is
considered at a time, as we will see later.

In the expressions above, for normal mass ordering, the sign of $\Delta$,
$\alpha$ and $\hat{v}$ are all positive, while for inverted ordering they are
all negative. Analogous expressions to those in Eqs. (2) and (2) for
$P\left(\nu_{\mu}\rightarrow\nu_{e}\right)$ and
$P\left(\nu_{\mu}\rightarrow\nu_{\mu}\right)$ can be obtained for
antineutrinos by flipping the sign of all $CP$ phases and $\hat{v}$. For
concreteness, we will present all our numerical results for normal ordering
(NO) only.

##  3 Experimental setup

DUNE is a future long-baseline accelerator neutrino experiment, where
neutrinos are expected to travel a distance of 1300 km from the source at
Fermilab to the far-detector placed deep underground at the Sanford Lab in
South Dakota. In order to estimate the sensitivity of the experiment to NSIs,
we adopt three benchmark experimental configurations for DUNE: the Conceptual
Design Report configuration (CDR) Acciarri:2015uup ; Alion:2016uaj ; the more
recent Technical Design Report configuration (TDR) Abi:2020qib ; Abi:2021arg ;
and a third which is the TDR configuration with improved energy resolution
based on the findings of Ref. Friedland:2018vry , as we will discuss later,
which we refer to as the Best Reconstruction (Best Rec.).

To perform the experimental simulations we have used the GLoBES package
Huber:2004ka ; Huber:2007ji  together with the NSI tool from Ref. Kopp_NSI .
In all configurations we have assumed a 40 kton LArTPC detector. The CDR beam
configuration assumes an 80 GeV proton beam with 1.07 MW beam power resulting
in $1.47\times 10^{21}$ POT/year, while the TDR uses a 120 GeV proton beam
with 1.2 MW beam power resulting in $1.1\times 10^{21}$ POT/year. We have also
assumed equal time of 3.5 years running in each forward and reverse horn
configurations (that is, equal neutrino and antineutrino runtime), which
results in 300 and 336 kton-MW-year exposures for the CDR and TDR setups,
respectively. Incidentally, the TDR staged 7-year running gives the same 336
kton-MW-year exposure Abi:2020qib . For the CDR and TDR simulations, specific
details on systematic errors and efficiencies can be found in Refs.
Alion:2016uaj ; Abi:2021arg .

##  4 Simulation details

In order to quantify the statistical sensitivity of all the numerical
simulations performed in this work, we use the built-in $\chi^{2}$ function in
GLoBES, which incorporates systematic uncertainties via pull parameters and
penalty terms (see Ref. Huber:2007ji  for details). The total $\chi^{2}$ is
the sum of all four channels that can be studied at DUNE: muon disappearance
and electron appearance, in both forward and reverse horn polarity (i.e.
neutrino and antineutrino modes). We marginalize over SM and new physics
parameters whenever specified to obtain the minimum $\Delta\chi^{2}$.

In all our numerical simulations we have used a line-averaged constant Earth
matter density $\rho$ = 2.848 gm/$\rm{cm}^{3}$ for DUNE following the PREM
stacey:1977 ; PREM:1981  profile. Unless otherwise stated, we have used the
following benchmark values of the standard three-neutrino oscillation
parameters: $\Delta m_{21}^{2}=7.5\times 10^{-5}\,\rm eV^{2}$, $\Delta
m_{31}^{2}=2.51\times 10^{-3}\,\rm eV^{2}$, $\sin^{2}\theta_{12}=0.310$,
$\sin^{2}\theta_{13}=0.022$, and $\sin^{2}\theta_{23}=0.56$. Moreover,
throughout this paper for simplicity we have assumed normal mass ordering
(NO). Note that our benchmark parameters closely agree with the current global
fit results Capozzi:2018ubv ; deSalas:2020pgw ; Esteban:2020cvm . As the solar
mixing parameters have very small impact on DUNE, we have fixed $\theta_{12}$
and $\Delta m_{21}^{2}$ to their above mentioned values. Atmospheric
parameters $\Delta m^{2}_{31}$, $\theta_{23}$ and $\delta_{\rm CP}$ are always
allowed to vary freely as their measurements are among the main goals of DUNE.
We adopt a 3.7% uncertainty on the reactor angle, which is better constrained
at experiments like Daya Bay Adey:2018zwh  and RENO Bak:2018ydk . Finally, we
assume a 5% uncertainty on the average matter density, slightly larger than
the 2% uncertainty assumed in the TDR. Whenever appropriate, we also
marginalize over some new physics parameters.

##  5 Improved energy resolution

Charged particles traversing LArTPC ionize the liquid argon, leading to free
electrons which drift against the electric field and are collected by wires.
The charge deposit is used to reconstruct objects, such as tracks or showers,
which are then used for particle identification. As the amount of energy
deposited depends on the particle momentum and mass, the identification,
together with the overall energy deposited in the detector via ionization is
used to reconstruct the energy of each individual particle. With those
energies and momenta, the incoming neutrino energy is inferred. This is to say
that the reconstruction of the incoming neutrino energy, a crucial step to
understand neutrino oscillations, is not a simple business.

Recently, there has been some re-evaluation of the best resolution that could
be achieved by the DUNE experiment DeRomeri:2016qwo ; Friedland:2018vry . In
particular, Ref. Friedland:2018vry  studies the best case scenario, regarding
incoming neutrino energy reconstruction, if the detector is able to identify
all particles in an event and if the small but frequent energy deposits from
recoiled neutrons are also accounted for. The authors find that a significant
enhancement, up to a factor of 4, in the reconstruction with respect to the
CDR values could be achieved. 222Note that in Ref. Friedland:2018vry , besides
the “best reconstruction,” another possibility is also studied for
reconstructing the neutrino energy, that is, adding all calorimetric energy of
the hadronic system without relying on individual particle identification. It
is observed that this method leads to a neutrino energy resolution comparable
to the one in DUNE’s TDR, as so we do not present this “Charge” case
separately here.

In this paper, we are interested in understanding how much an improved energy
resolution would enhance DUNE’s sensitivity to new physics, particularly NSIs.
In Fig. 1 we present the energy resolution for neutrinos (left panel) and
antineutrinos (right panel) as a function of the true neutrino energy for the
CDR (red line, worst resolution), and for the TDR (green line). We also show
the best case scenario for the energy resolution obtained in Ref.
Friedland:2018vry  (black line, labeled “Best Rec.”) as well as a simple
parameterization we perform to reproduce the best case results (blue line,
labeled “Our fit”). Note that from Fig. 2 onwards we use our parameterization
of the energy resolution in the Best Reconstruction case.

Our parameterization of the Best Reconstruction case is obtained by using an
energy resolution function
$R(E,E_{r})=e^{-(E-E_{r})^{2}/2\sigma^{2}}/\sigma\sqrt{2\pi}$, where $E$ is
the true neutrino energy, $E_{r}$ is the reconstructed energy, and the energy
resolution $\sigma$ is given by

$\displaystyle\sigma(E)/{\rm GeV}=\alpha.(E/{\rm GeV})+\beta.\sqrt{E/{\rm
GeV}}+\gamma\,,$ (11)

where, $(\alpha,\,\beta,\,\gamma)$ are the parameters for the fit:
$(0.045,\,0.001,\,0.048)$ for neutrinos and $(0.026,\,0.001,\,0.085)$ for
antineutrinos (cf. Fig. 1). We assume the same energy resolution for the
appearance and disappearance modes. To simulate the neutral current, $\nu_{e}$
contamination, misidentified muon, and $\nu_{\mu}\to\nu_{\tau}$ backgrounds we
have used the same energy resolution migration matrices as provided in Refs.
Alion:2016uaj  (CDR) and Abi:2021arg  (TDR and Best Reconstruction).

Figure 1: Energy resolution as a function of true neutrino energy. Left
(right) panel corresponds to neutrinos (antineutrinos). The red and green
curves depict the energy resolutions from DUNE’s CDR and TDR, respectively,
while the best reconstruction scenario from Ref. (Friedland:2018vry, ) is
shown in black. Our fit to the best reconstruction scenario is given by the
blue curve. The numbers $\alpha$, $\beta$, and $\gamma$ in each panel
correspond to the best fit values used in Eq. (11).

As we can see from Fig. 1, the energy resolution is the worst for the CDR
configuration, somewhat better for the TDR and improves considerably in the
Best Reconstruction case. Quantitatively, for neutrinos at 1 GeV we have about
25%, 19% and 9% energy resolution, respectively. For reference, near 0.9 GeV,
an energy resolution of about 20% would completely wash out the second
oscillation maximum.

A concrete example illustrating the importance of the energy resolution for
DUNE is presented in Fig, 2, where we show, as a function of the true
$\delta_{\rm CP}$ phase, the $CP$ violation sensitivity (left panel) 333The
$CP$ violation sensitivity is defined as the statistical significance at which
one can reject the test hypothesis of no $CP$ violation i.e., $\delta_{\rm
CP}(\rm{test})=0,\rm{or}\pm\pi$. and the precision on $\delta_{\rm CP}$
achievable at DUNE at 1$\sigma$ ($\Delta\chi^{2}=1$, right panel) for 3.5
years in each neutrino and antineutrino mode. For both panels, the red, the
green, and the blue lines correspond to the CDR, TDR, and the Best
Reconstruction benchmarks. As discussed above we marginalize over all relevant
oscillation parameters except the solar ones, as well as over the matter
density. As we can see, the effect of a better energy resolution is non-
negligible, substantially enhancing the $CP$ precision, in particular for
maximal $CP$ violation. Quantitatively, for $\delta_{\rm
CP}(\rm{true})=-90^{\circ}$, the $CP$ precision is $\sim 24^{\circ}$ for the
CDR, $\sim 21^{\circ}$ for the TDR, and $\sim 17^{\circ}$ for the Best
Reconstruction case. Similar findings have been reported previously in Ref.
DeRomeri:2016qwo .

It is important to mention that the $CP$ precision obtained here with DUNE’s
GLoBES simulation files Abi:2021arg  does not match the official DUNE version
of the plot given in Ref. Abi:2020qib . The main reason seems to be the
treatment of systematics related to the flux and/or cross section shape
uncertainties. Compared to a full-fledged Monte Carlo simulation, the GLoBES
treatment of spectral uncertainties in Ref. Abi:2021arg  is somewhat
simplified.

Figure 2: Left panel: $CP$ violation discovery ($\delta_{\rm
CP}(\textrm{test})\neq 0,\pm\pi$) potential of DUNE with different assumptions
on the DUNE energy resolution. Right Panel: 1$\sigma$ ($\Delta\chi^{2}=1$)
uncertainties on the true value of $\delta_{\rm CP}$ determined by DUNE. Red,
green and blue curves represent the CDR, TDR and Best Reconstruction setup
respectively. We have assumed normal ordering (NO) in the analysis for both
plots.

##  6 Numerical Results

In this section we discuss the main results we have obtained from our
numerical simulations regarding the impact of the energy resolution on the
sensitivity to NSIs.

Let us start with Fig. 3, where we have adopted a simplified framework in
which all NSI parameters $\varepsilon_{\alpha\beta}$ are taken to be real
parameters and we take one at a time when analyzing DUNE’s sensitivity. Here,
and throughout this paper, we will always assume standard three neutrino
oscillations as the true hypothesis and test nonzero NSI against it, that is
our $\Delta\chi^{2}=\chi_{\rm NSI}^{2}-\chi_{\rm SM}^{2}$. For concreteness we
have assumed $\delta_{\rm CP}(\textrm{true})\,=\,-90^{\circ}$, which is close
to the current preferred value from global fits Capozzi:2018ubv ;
deSalas:2020pgw ; Esteban:2020cvm . We marginalize on the two mixing angles
$\theta_{13}$ and $\theta_{23}$, the standard $CP$ phase $\delta_{\rm CP}$,
the mass-squared splitting $\Delta m^{2}_{31}$, and the matter density, as
discussed in the previous section. The CDR, TDR and Best Reconstruction
scenarios are shown in red, green and blue, respectively. The upper panels
show the sensitivity for the diagonal NSI parameters whereas the lower panel
shows the sensitivity for the non-diagonal NSI parameters, which are assumed
real here.

Figure 3: One-dimensional projections of the DUNE sensitivity to the diagonal NSI parameters $\varepsilon_{ee},\varepsilon_{\mu\mu},\varepsilon_{\tau\tau}$ in the upper panel and non-diagonal $\varepsilon_{e\mu},\varepsilon_{e\tau},\varepsilon_{\mu\tau}$ in the lower panel. The red line represents the result with “CDR” configuration, black line is for the “TDR” configuration, and the blue line corresponds to the “Best Reconstruction” configuration. Normal mass ordering is assumed in the data as well as in the theory. We have considered $\delta_{\rm CP}(\rm{true})=-90^{\circ}$ and the non-diagonal NSI parameters as real. NSI Parameter | CDR | TDR | Best Rec.
---|---|---|---
$\varepsilon_{ee}$ | $[-0.249,\,+0.552]$ | $[-0.256,\,+0.399]$ | $[-0.246,\,+0.360]$
$\varepsilon_{\mu\mu}$ | $[-0.415,\,-0.240]$, | $[-0.445,\,+0.549]$ | $[-0.416,\,-0.335],$
$[-0.214,\,0.232]$, | $[-0.117,\,0.128],\,$
$[0.289,\,0.522]$ | $[0.393,\,0.520]$
$\varepsilon_{\tau\tau}$ | $[-0.550,\,-0.357]$, | $[-0.214,\,+0.164]$ | $[-0.126,\,+0.112]$
$[-0.200,\,+0.154]$
$\varepsilon_{e\mu}$ | $[-0.046,\,+0.043]$ | $[-0.047,\,+0.045]$ | $[-0.045,\,+0.043]$
$\varepsilon_{e\tau}$ | $[-0.038,\,+0.041]$ | $[-0.039,\,+0.041]$ | $[-0.033,\,+0.033]$
$\varepsilon_{\mu\tau}$ | $[-0.018,\,+0.018]$ | $[-0.021,\,+0.021]$ | $[-0.015,\,+0.015]$
Table 1: DUNE sensitivities on the NSI parameters at $2\sigma$ confidence
level (C.L.) with 1 degree of freedom (d.o.f.) (i.e. $\Delta\chi^{2}=4$),
assuming normal mass ordering and $\delta_{\rm CP}(\rm true)=-90^{\circ}$ (cf.
Fig. 3). Left, middle, and right columns correspond to CDR, TDR, and Best
Reconstruction cases respectively. Here the non-diagonal NSI parameters have
been assumed real.

We can clearly see that, while the CDR and TDR cases yield comparable results,
an improved energy reconstruction significantly enhances DUNE’s sensitivity to
NSIs, in particular for $\varepsilon_{\mu\mu}$, $\varepsilon_{\tau\tau}$, and
$\varepsilon_{e\tau}$, as well as $\varepsilon_{\mu\tau}$ to a certain extent.
There is some impact in $\varepsilon_{ee}$, though somewhat small, and almost
no effect on $\varepsilon_{e\mu}$. Numbers comparing the allowed regions at
$2\sigma$ ($\Delta\chi^{2}=4$) for all NSI parameters and for the three cases
can be found in Table 1. Note that there are existing constraints from several
experiments on all NSI parameters, see e.g. Ref. Farzan:2017xzy .

Until now we have treated the non-diagonal NSI parameters as real parameters.
However it might be interesting to see how the bound changes when we treat
these parameters as complex. As we discussed before, the NSI $CP$ phases may
significantly impact the sensitivity to $\varepsilon_{\mu\tau}$, as the first
order contribution of this NSI parameter to the $\nu_{\mu}$ disappearance
probability goes like $|\varepsilon_{\mu\tau}|\cos\phi_{\mu\tau}$, see Eq.
(2).

We show DUNE’s sensitivity to complex non-diagonal NSI parameters in Fig. 4.
While the procedure adopted here is the same as in the previous figure, we
have additionally marginalized over the corresponding NSI $CP$ phases. As
discussed, the sensitivity to $|\varepsilon_{\mu\tau}|$ greatly diminishes
(right panel), while a smaller but still significant effect can be seen for
$|\varepsilon_{e\tau}|$ (middle panel). $|\varepsilon_{e\mu}|$ is largely
unaffected by the presence of a nonzero phase. Even accounting for the $CP$
phases, the improvement due to a better energy resolution remains for
$\varepsilon_{e\tau}$, as we can see in the middle panel of Fig. 4.
Quantitatively, we found the upper bounds on $|\varepsilon_{e\tau}|$ at
2$\sigma$ ($\Delta\chi^{2}=4$) to be 0.12, 0.10 and 0.08 for the CDR, TDR and
Best Reconstruction cases, respectively, revealing a possible 20% improvement
between the last two. In Table 2, we quote the $2\sigma$ sensitivity on the
non-diagonal, complex NSI parameters for all three scenarios. We highlight
that DUNE will significantly improve current sensitivities to
$\varepsilon_{e\mu}$ and $\varepsilon_{e\tau}$ (for both real and complex NSI
cases). For the $\varepsilon_{\mu\tau}$, DUNE could compete with the IceCube
constraint if the NSI is real. The presence of a nonzero $\phi_{\mu\tau}$
considerably worsens DUNE sensitivity, but not IceCube’s IceCube:2021abg .

Figure 4: Same as in Fig. 3 for the non-diagonal NSI parameters $\varepsilon_{e\mu}$, $\varepsilon_{e\tau}$, and $\varepsilon_{\mu\tau}$, but now assuming them to be complex and marginalizing over their phases.  NSI Parameter | CDR | TDR | Best Rec.
---|---|---|---
$|\varepsilon_{e\mu}|$ | $\leq 0.048$ | $\leq 0.052$ | $\leq 0.047$
$|\varepsilon_{e\tau}|$ | $\leq 0.123$ | $\leq 0.096$ | $\leq 0.085$
$|\varepsilon_{\mu\tau}|$ | $\leq 0.191$ | $\leq 0.196$ | $\leq 0.189$
Table 2: Same as in Table 1, but for the magnitude of the non-diagonal NSI
parameters which are taken to be complex here (cf. Fig. 4).

To show how an improved energy resolution can affect the sensitivity to NSIs
for different true values of the $CP$ violation phase $\delta_{\rm CP}$, we
present Figs. 5 and 6. The simulation details are the same as used before. The
contours in each panel is shown for $2\sigma$ C.L. with 1 d.o.f. (i.e.
$\Delta\chi^{2}=4$). The CDR, TDR and Best Reconstruction cases are
represented by red, green and blue contours respectively. Fig. 5 shows the
allowed regions for each NSI parameter
($\varepsilon_{ee},\varepsilon_{e\mu},\varepsilon_{e\tau},\varepsilon_{\mu\tau},\varepsilon_{\mu\mu},\,\textrm{and}\,\varepsilon_{\tau\tau}$),
taking one NSI at a time and assuming all NSI parameters to be real. The null
hypothesis corresponds to no NSI and a given value of $\delta_{\rm
CP}(\textrm{true})$ as indicated on the $y$-axis. Therefore, this figure
should be understood as the allowed region of each NSI parameter for a given
input value of $\delta_{\rm CP}$. The variation among CDR, TDR and Best
Reconstruction shows the role of the energy resolution. If we take
$\varepsilon_{ee}$ as an example, we can see that the energy resolution can
significantly improve DUNE’s sensitivity to this NSI, except for $\delta_{\rm
CP}$(true)$\sim 80^{\circ},-110^{\circ}$. It is evident from this figure that
the better energy resolution in DUNE plays a significant role in improving the
constraints on most of the NSI parameters. Also note that some spurious
degeneracies in the $\varepsilon_{\tau\tau}$ case are only present for the CDR
and TDR resolutions, while completely disappear for the Best Reconstruction
scenario.

Figure 5: Projected allowed regions in the plane spanned by the $\delta_{\rm
CP}$ (true) and the NSI parameters at $2\sigma$ C.L. ($\Delta\chi^{2}\,=\,4$).
The red contours represent the DUNE sensitivity for the CDR configuration,
green contours correspond to the TDR setup, and the blue contours are for the
Best Reconstruction case. Normal mass ordering is assumed to be fixed in the
data as well as in theory. We have considered the NSI parameters to be real.

Figure 6: Same as in Fig. 5 for the non-diagonal NSI parameters, but now
assuming them to be complex.

In the previous discussion we focused on real NSI parameters. Now, we present
in Fig. 6 the allowed region for the absolute value of non-diagonal NSI
parameters, marginalizing over their corresponding $CP$ phase, as a function
of the true value of $\delta_{\rm CP}$. All analysis details are the same as
before. As expected, the allowed regions are larger when one consider complex
NSI parameters, particularly for $\varepsilon_{\mu\tau}$ due to the
$\cos\phi_{\mu\tau}$ factor in the leading-order contribution of this NSI to
the $\nu_{\mu}$ survival probability, see Eq. (2). We also observe that the
energy resolution plays an important role in the sensitivity to
$\varepsilon_{e\tau}$ even in the presence of the NSI $CP$ phase.

Figure 7: $2\sigma$ C.L. (2 d.o.f.) sensitivity regions spanned by the
different sets of NSI parameters for three different cases CDR, TDR, and Best
Reconstruction respectively. We have assumed $\delta_{\rm CP}(\rm
true)=-90^{\circ}$ and normal ordering is assumed in both true and test
hypotheses of the analysis. The benchmark point marked in the upper left panel
will be used later in Sec. 7.

Now that we have established the important role of the energy resolution in
constraining certain NSI parameters, it is important to assess how robust
these conclusions are if we do not assume one NSI parameter at a time. In
general, UV complete models will predict a combination of NSI parameters, with
possible correlations stemming from the new particles and interactions that
generate the NSI at a higher scale Farzan:2015doa ; Farzan:2015hkd ;
Farzan:2016wym ; Forero:2016ghr ; Babu:2017olk ; Dey:2018yht ; Babu:2019mfe .
Although a comprehensive study of specific models is beyond the scope of this
paper, we can redo our analysis taking into account two NSI parameters at a
time, instead of individual $\varepsilon_{\alpha\beta}$. Note that further
redoing the analysis with more than two NSI parameters will not be very
illuminating for our purposes.

In Fig. 7 we show the allowed regions at $2\sigma$ C.L. with 2 d.o.f.
($\Delta\chi^{2}=6.18$) for different pairs of NSI parameters. We consider
$\varepsilon_{ee}$, $\varepsilon_{\mu\mu}$, $\varepsilon_{\tau\tau}$, and
$\varepsilon_{e\tau}$, as those are the parameters that profit the most from
an improved energy resolution (cf. Figs. 3 and 4). As before, we indicate the
CDR, TDR and Best Reconstruction configurations by the red, green and blue
shaded regions, respectively. In order to draw the two-dimensional contours we
have marginalized away the two mixing angles $\theta_{13}$ and $\theta_{23}$,
the mass splitting $\Delta m^{2}_{31}$, the $CP$ phase $\delta_{\rm CP}$, and
the matter density $\rho$, within the uncertainties discussed in Sec. 4.
Moreover the nonstandard $CP$ phases for non-diagonal NSIs have also been
marginalized away. We have fixed $\delta_{\rm CP}$(true) at $-90^{\circ}$ as
well as normal ordering for the mass spectrum. We have considered a few
combinations of two NSI parameters at a time keeping others fixed to zero. As
expected (see e.g. Refs. Coloma:2016gei ; Liao:2016hsa ), we observe
significant degeneracies between NSI parameters, particularly for the pairs
containing $(\varepsilon_{ee},\,|\varepsilon_{e\tau}|)$. We also see here a
substantial impact of the improved energy resolution in several of these NSI
combinations. In the upper left panel of Fig. 7 we can see that the allowed
region in the Best Reconstruction scenario in the
$(\varepsilon_{ee},\,|\varepsilon_{e\tau}|)$ plane for positive
$\varepsilon_{ee}$ is reduced to about half of the allowed region in the CDR
case. Moreover, the allowed regions in both $\varepsilon_{\mu\mu}$ and
$\epsilon_{\tau\tau}$ versus other parameters are connected for the CDR and
TDR cases, whereas three disconnected islands appear for the Best
Reconstruction case.

Figure 8: Allowed regions in the $\varepsilon_{e\tau}$ versus $\Delta
m^{2}_{31}$ (left) and $\varepsilon_{e\tau}$ versus $\delta_{\rm CP}$ (right)
planes for the DUNE CDR, TDR and Best Reconstruction cases (red, green, and
blue contours, respectively) at $2\sigma$ C.L. ($\Delta\chi^{2}=6.18$). The
assumed true value is indicated as a magenta star while the black triangle on
the right panel is a NSI benchmark, see text.

##  7 Discussion

In this section we discuss the results obtained previously and the role of the
energy resolution in a qualitative manner. First, we present Fig. 8, in which
we show an illustrative example of DUNE’s sensitivity to NSI for CDR, TDR and
Best Reconstruction (red, green and blue, resp.) in the planes of
$(\varepsilon_{e\tau},\Delta m^{2}_{31})$ (left panel) and
$(\varepsilon_{e\tau},\delta_{\rm CP})$ (right panel) at $2\sigma$ C.L. for 2
d.o.f. ($\Delta\chi^{2}\,=\,6.18$). The magenta star in each panel represents
the choice of true parameters where we have assumed no NSI. Note that we have
marginalized away the NSI $CP$ phase $\phi_{e\tau}$ in both panels.

The improvement on the measurement of $\Delta m^{2}_{31}$ due to a better
energy resolution is obvious: a more precise determination of the minimum of
oscillation allows for a better measurement of the oscillation frequency and
therefore of the mass splitting. In fact, the better energy resolution makes
the first oscillation minimum in the disappearance channel more prominent and
helps resolving the second oscillation minimum, around 0.8 GeV. Nevertheless,
the right panel is more interesting for us, as we see a degeneracy between
$\delta_{\rm CP}$ and $\varepsilon_{e\tau}$. To understand the role of the
energy resolution, let us take the NSI benchmark point denoted by the black
triangle in the right panel, which is allowed by the CDR but excluded by the
TDR or Best Reconstruction simulations. The relevant parameters for this
benchmark point are $\delta_{\rm CP}\,=\,-35^{\circ}$,
$|\varepsilon_{e\tau}|\,=\,0.16$, and $\phi_{e\tau}\,=\,10^{\circ}$ for the
CDR, and $\phi_{e\tau}\,=\,15^{\circ}$ for the TDR and Best Reconstruction
cases. The other standard oscillation parameters were chosen to be the best
fit values in the analysis.

Figure 9: Upper left (right) panel presents the $\nu_{\mu}\rightarrow\nu_{e}$
($\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$) appearance event spectra as a
function of reconstructed energy. Red, green and blue lines correspond to the
CDR, TDR, and Best Reconstruction configurations, respectively. Solid lines
represent the standard three-neutrino spectra with $\delta_{\rm
CP}\,=\,-90^{\circ}$ and dashed lines represent the spectra with NSI in the
$e-\tau$ sector. In case of NSI, we have taken the best fit parameters
corresponding to the black triangle in Fig. 8, i.e., $\delta_{\rm
CP}\,=\,-35^{\circ}$, $|\varepsilon_{e\tau}|\,=\,0.16$, and
$\phi_{e\tau}\,=\,10^{\circ}$ for the CDR, and $\phi_{e\tau}\,=\,15^{\circ}$
for the TDR and Best Reconstruction cases. The best fit values of
$\theta_{13}$, $\theta_{23}$, and $\Delta m_{31}^{2}$ have also been chosen
from the simulation. Lower left (right) panel represents the absolute
difference in the upper left (right) appearance events between the standard
and the NSI scenarios divided by the statistical uncertainty in each bin. The
color coding is the same as in the upper panels.

In Fig. 9 we show the appearance spectra for the neutrino mode (upper left
panel) and antineutrino mode (upper right panel) for the CDR, TDR and Best
Reconstruction (red, green and blue, respectively), for the assumed true
parameters (magenta star in Fig. 8 right panel, solid lines in Fig. 9) and NSI
benchmark point (black triangle in Fig. 8 right panel, dashed lines in Fig.
9). First we can understand why it is hard to disentangle this NSI effect: the
spectra themselves look very similar between the standard model assumed true
values and the NSI benchmark. This is particularly true for the first maximum.
The second maximum shows a larger deviation, but with lower statistics. To
better see the differences between the standard and NSI cases, we show in the
lower panels the absolute difference on the number of events divided by the
statistical uncertainty in each bin, i.e. $\Delta=|N_{\rm SM}-N_{\rm
NSI}|/\sqrt{N_{\rm SM}}$. This statistical figure of merit should be viewed
only as a crude approximation for the statistical relevance of each bin, but
it serves to demonstrate the relevance of the first and second oscillation
maxima in a qualitative level. As we can see, a better energy resolution not
only leads to a slight increase in the statistical power around the first
oscillation maximum, but also on the second maximum. This draws us to the
conclusion that the improved energy resolution enhances the sensitivity not
only due to a better reconstruction of the first maximum: the role of the
second maximum is indeed very much significant. These considerations apply to
both neutrino and antineutrino modes.

To be more explicit, we compare in Fig. 10 the appearance spectra between the
standard case (solid lines) and the NSI case (dashed lines) corresponding to a
point in parameter space which is ruled out by the Best Reconstruction but not
by the TDR configuration (see the black star in Fig. 7 upper left panel).
Upper left (right) panel in Fig. 10 presents the $\nu_{\mu}\rightarrow\nu_{e}$
($\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$) appearance event spectra as a
function of reconstructed energy. Green and blue lines correspond to the TDR
and Best Reconstruction configurations respectively. Solid lines represent the
standard three-neutrino spectra with $\delta_{\rm CP}\,=\,-90^{\circ}$ and
dashed lines represent the spectra with NSI in the $e-e$ and $e-\tau$ sector
simultaneously. In case of NSI, we have taken the best fit parameters
corresponding to the black star in the upper left panel of Fig. 7, i.e.,
$\delta_{\rm CP}\,=\,-22^{\circ}$, $\varepsilon_{ee}\,=\,2.70$,
$|\varepsilon_{e\tau}|\,=\,0.65$, and $\phi_{e\tau}\,=\,-177^{\circ}$ for the
TDR, and $\delta_{\rm CP}\,=\,-23^{\circ}$, $\varepsilon_{ee}\,=\,2.70$,
$|\varepsilon_{e\tau}|\,=\,0.65$, and $\phi_{e\tau}\,=\,-170^{\circ}$ for the
Best Reconstruction case respectively. The best fit values of $\theta_{13}$,
$\theta_{23}$, and $\Delta m_{31}^{2}$ have also been chosen from the
simulation for both the standard and the NSI scenarios respectively. In the
lower panels we represent the absolute difference $\Delta$ in the appearance
events between the standard and the NSI scenarios divided by the statistical
uncertainty in each bin. The color coding is the same as in the upper panels.
Here, the effect of the improved energy resolution is very clear: both first
and second maxima become statistically more powerful in rejecting this NSI
hypothesis. Although the second maximum has less events and less bins, the
relative change in the number of events is significantly larger than in the
first maximum. This evidences that both oscillation maxima can contribute
comparably to a better sensitivity, if they are resolved more precisely.

Figure 10: Upper left (right) panel presents the $\nu_{\mu}\rightarrow\nu_{e}$
($\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$) appearance event spectra as a
function of reconstructed energy. Green and blue lines correspond to the TDR
and Best Reconstruction configurations respectively. Solid lines represent the
standard three-neutrino spectra with $\delta_{\rm CP}\,=\,-90^{\circ}$ and
dashed lines represent the spectra with NSIs in the $e-e$ and $e-\tau$ sectors
simultaneously. In case of NSI, we have taken the best fit parameters
corresponding to the black star in the upper left panel of Fig. 7, i.e.,
$\delta_{\rm CP}\,=\,-22^{\circ}$, $\varepsilon_{ee}\,=\,2.70$,
$|\varepsilon_{e\tau}|\,=\,0.65$, and $\phi_{e\tau}\,=\,-177^{\circ}$ for the
TDR, and $\delta_{\rm CP}\,=\,-23^{\circ}$, $\varepsilon_{ee}\,=\,2.70$,
$|\varepsilon_{e\tau}|\,=\,0.65$, and $\phi_{e\tau}\,=\,-170^{\circ}$ for the
Best Reconstruction case. The best fit values of $\theta_{13}$, $\theta_{23}$,
and $\Delta m_{31}^{2}$ have also been chosen from the simulation for both the
standard and the NSI scenarios. Lower left (right) panel represents the
absolute difference in the upper left (right) appearance events between the
standard and the NSI scenarios divided by the statistical uncertainty in each
bin. The color coding is the same as in the upper panels.

##  8 Conclusions

In this work we have investigated the impact of improved energy resolution on
DUNE sensitivity to new physics. As an illustrative example, we have shown how
a better energy resolution could lead to improved constraints on neutrino non-
standard interactions with matter. We have found that a better energy
resolution would help to disentangle degeneracies among effects arising from
standard three-neutrino oscillations, such as the usual $CP$ violation, and
those stemming from NSIs. In fact, by studying in more detail certain
benchmark points, we have identified that the improved sensitivity to NSIs
comes not only from the first oscillation peak, but the second maximum also
contributes significantly to the experimental sensitivity as long as it can be
appropriately resolved. Therefore, a better neutrino energy determination
allows the experiment to further leverage its broad-band beam in BSM physics
searches.

More specifically, we have found that potentially significant sensitivity
improvements could be achieved for $\varepsilon_{ee}$, $\varepsilon_{\mu\mu}$,
$\varepsilon_{\tau\tau}$, and $\varepsilon_{e\tau}$, particularly when the NSI
$CP$ violating phase is considered for $\varepsilon_{e\tau}$. We have also
found that certain degeneracies could be better resolved with an improved
energy resolution. For any combination of two of the aforementioned NSI
parameters, we find a significant reduction in the allowed regions, when going
from the CDR case to the TDR and especially to the Best Reconstruction
scenario (see Fig. 7). Finally, as a by-product of our analysis, we have also
shown how the $CP$ violation sensitivity and the $CP$ phase precision would
improve with the neutrino energy resolution (see Fig. 2). We hope our analysis
will further motivate the pursuit of an enhanced neutrino energy
reconstruction in liquid argon time projection chambers.

### Acknowledgments

We would like to thank Shirley Li for useful discussions on the DUNE energy
resolution, Chris Marshall and Elizabeth Worcester for discussions and
clarifications regarding the GLoBES simulation for DUNE TDR, and Sanjib Kumar
Agarwalla for some earlier discussions on NSI at DUNE. Fermilab is operated by
the Fermi Research Alliance, LLC under contract No. DE-AC02-07CH11359 with the
United States Department of Energy. This project has received support from the
European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie grant agreement No 860881-HIDDeN. The work of BD is
supported in part by the US Department of Energy under Grant No. DE-SC0017987,
by the Neutrino Theory Network Program, and by a Fermilab Intensity Frontier
Fellowship. BD and SSC would like to thank the Fermilab Theory Group for local
hospitality during a summer visit in 2019, where this project was initiated.

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