#  Snowmass Neutrino Frontier:
NF01 Topical Group Report
Three-Flavor Neutrino Oscillations  Submitted to the Proceedings of the US
Community Study on the Future of Particle Physics (Snowmass 2021)

Peter B. Denton Topical Group Conveners High Energy Theory Group, Physics
Department, Brookhaven National Laboratory, Upton, NY 11973, USA  Megan Friend
Mark D. Messier  Hirohisa A. Tanaka  Sebastian Böser  University of Mainz,
Germany  João A. B. Coelho  Université de Paris, CNRS, Astroparticule et
Cosmologie, F-75013 Paris, France  Mathieu Perrin-Terrin  Aix Marseille Univ,
CNRS/IN2P3, CPPM, Marseille, France  Tom Stuttard  Niels Bohr Institute,
University of Copenhagen, Denmark

(November 30, 2022)

This is the report from the Snowmass NF01 topical group and colleagues on the
current status and expected future progress to understand the three-flavor
neutrino oscillation picture.

###### Contents

  1. Executive Summary
  2. 1 Introduction and Current Three-Flavor Status
    1. 1.1 Neutrino Oscillations in Particle Physics
    2. 1.2 Current Knowns and Known Unknowns in Neutrino Oscillations
    3. 1.3 Neutrino oscillations and the Previous P5
  3. 2 Three-Flavor Oscillation Theory
    1. 2.1 Neutrino Oscillation Probabilities
    2. 2.2 Matter Effect
    3. 2.3 Role of Each Oscillation Parameter
    4. 2.4 Flavor Model Predictions and Desired Precision
  4. 3 Three-Flavor Neutrino Oscillation Facilities
    1. 3.1 JUNO
    2. 3.2 Fermilab/SURF Program
      1. 3.2.1 NOvA
      2. 3.2.2 DUNE
    3. 3.3 J-PARC/Kamioka Program
      1. 3.3.1 Super-Kamiokande (SK)
      2. 3.3.2 Tokai to Kamioka (T2K)
      3. 3.3.3 Hyper-Kamiokande (HK)
    4. 3.4 South Pole: IceCube/DeepCore
      1. 3.4.1 Neutrino oscillation physics at IceCube
      2. 3.4.2 Neutrino oscillation measurements
      3. 3.4.3 Next-generation oscillation physics with the IceCube Upgrade
      4. 3.4.4 Neutrino mass ordering
    5. 3.5 KM3NeT
      1. 3.5.1 Scientific Status and Prospects
      2. 3.5.2 Technical Status and Schedule
    6. 3.6 Expected Sensitivity Milestones
  5. 4 Three Flavor Oscillation Supporting Program
    1. 4.1 Experiments
    2. 4.2 Oscillation Theory
    3. 4.3 Joint Fits
      1. 4.3.1 T2K + SK
      2. 4.3.2 T2K + NOvA
      3. 4.3.3 JUNO + NOvA + T2K
      4. 4.3.4 DUNE + HK
      5. 4.3.5 JUNO + ORCA
      6. 4.3.6 IceCube/DeepCore + JUNO
  6. 5 Possible Upgrades to Planned Experiments
    1. 5.1 DUNE
      1. 5.1.1 Realizing the P5 DUNE
      2. 5.1.2 Near Detector
      3. 5.1.3 Third and Fourth Modules
      4. 5.1.4 Accelerator Upgrades
    2. 5.2 Hyper-K
      1. 5.2.1 HK Near Detector
      2. 5.2.2 HK-Gd
      3. 5.2.3 Detector in Korea
      4. 5.2.4 J-PARC Accelerator Upgrades
  7. 6 Other Probes of the Oscillation Parameters
    1. 6.1 Galactic Supernova
    2. 6.2 Astrophysical Neutrinos
    3. 6.3 Absolute Mass-Scale Based Probes
  8. 7 Possible Future Experiments
    1. 7.1 DUNE to THEIA
    2. 7.2 ESSnuSB
    3. 7.3 ICAL at INO
    4. 7.4 Neutrinos from muons
    5. 7.5 Protvino to Orca: P2O
  9. 8 Conclusions
  10. A LOIs tagged in NF01, “Neutrino Oscillations”
  11. B Snowmass whitepapers related to NF01

## Executive Summary

The discovery of neutrino oscillations in 1998 and 2002 added at least seven
new parameters to our model of particle physics, and oscillation experiments
can probe six of them. To date, three of those parameters are fairly well
measured: the reactor mixing angle $\theta_{13}$, the solar mixing angle
$\theta_{12}$, and the solar mass splitting $\Delta m^{2}_{21}$, although
there is only one good measurement of the last parameter. Of the remaining
three oscillation parameters, we have some information on two of them: we know
the absolute value of the atmospheric mass splitting $\Delta m^{2}_{31}$
fairly well, but we do not know its sign, and we know that the atmospheric
mixing angle $\theta_{23}$ is close to maximal $\sim 45^{\circ}$, but we do
not know how close, nor on which side of maximal it is. Finally, the sixth
parameter is the complex phase $\delta$ related to charge-parity (CP)
violation, which is largely unconstrained.

Determining these remaining three unknowns, the sign of $\Delta m^{2}_{31}$,
the octant of $\theta_{23}$, and the value of the complex phase $\delta$, is
of the utmost priority for particle physics. In addition to the absolute
neutrino mass scale which can be probed with cosmological data sets, they
represent the only known unknown parameters in our picture of particle
physics. It is our job as physicists to determine the parameters of our model.
The values of these parameters have important implications in many other areas
of particle physics and cosmology, as well as providing insights into the
flavor puzzle.

To measure these parameters, a mature experimental program is underway with
some experiments running now and others under construction. In the current
generation we have NOvA, T2K, and Super-Kamiokande (SK) which each have some
sensitivity to the three remaining unknowns, but are unlikely to get to the
required statistical thresholds. Next generation experiments, notably DUNE and
Hyper-Kamiokande (HK) are expected to get to the desired thresholds to answer
all three oscillation unknowns. Additional important oscillation results will
come from JUNO, IceCube, and KM3NeT. This broad experimental program reflects
the fact that there are many inter-connected parameters in the three-flavor
oscillation picture that need to be simultaneously disentangled and
independently confirmed to ensure that we truly understand these parameters.

To achieve these ambitious goals, DUNE and HK will need to become the most
sophisticated neutrino experiments constructed to date. Each requires
extremely powerful neutrino beams, as many measurements are statistics
limited. Each will require a very sophisticated near detector facility to
measure that beam, as well as to constrain neutrino interactions and detector
modeling uncertainties, which are notoriously difficult in the energy ranges
needed for oscillations. To augment the near detectors, additional
measurements and theory work are crucial to understand the interactions
properly, see NF06 [1]. Finally, large highly sophisticated far detectors are
required to be able to reconstruct the events in a large enough volume to
accumulate enough statistics. DUNE will use liquid argon time-projection
chamber (LArTPC) technology most recently demonstrated with MicroBooNE.
LArTPCs provide unparalleled event reconstruction capabilities and can be
scaled to large enough size to accumulate the necessary statistics. HK will
expand upon the success of SK’s large water Cherenkov tank and build a new
larger tank using improved photosensor technology.

It is fully expected that with the combination of experiments described above,
a clear picture of three-flavor neutrino oscillations should emerge, or, if
there is new physics in neutrino oscillations (see NF02 [2] and NF03 [3]),
that should fall into stark contrast in coming years.

##  1 Introduction and Current Three-Flavor Status

###  1.1 Neutrino Oscillations in Particle Physics

The Standard Model has been highly successful, but we know it cannot be a
complete fundamental physics theory. The Standard Model does not provide a
source of dark matter and dark energy whose imprints on the universe we see in
the night sky through our telescopes. The Standard Model does not explain the
vast difference in strength between the electric, weak, and strong forces and
gravity. The Standard Model does not explain the hierarchy of masses of quarks
and leptons. The Standard Model does not explain why charge-parity (CP) is
sometimes violated and sometimes conserved. And the Standard Model does not
explain the mechanism behind neutrino masses, nor why they are hierarchically
different from the other fermions. Of these four, the effects of neutrino
masses are the only sign of physics outside the Standard Model which, to date,
we can manipulate in our laboratories.

While there are various simple ways to introduce neutrino masses to the
Standard Model, these approaches leave many questions unanswered. They do not
account for the unusual smallness of the neutrino masses relative to their
charged partners; they do not explain why neutrino mixing is large, whereas
mixing in other sectors of the Standard Model is small. They don’t predict
patterns or symmetries in the masses or answer a fundamental question about
the differences between neutrinos, antineutrinos, and lepton charge
conservation.

Since the discovery of neutrino oscillations in 1998, a global program has
been developed to explore neutrino masses and their mixing using a wide
variety of natural and artificial sources. This program has narrowed the
allowed ranges of neutrino mass splittings and shown that all neutrino mixing
is relatively, perhaps surprisingly, large. However, many questions remain. Do
neutrinos follow a “normal hierarchy” that would associate most of the
electron flavor with the lightest states or an “inverted hierarchy” where the
electron flavor is mainly in the heavier states? Do neutrino masses and mixing
possess new symmetries? Do neutrino oscillations violate charge-parity? If, as
many think, the lightness of neutrino masses is associated with physics
approaching the scale of grand unification, will their oscillations contain
echos of new physics at these energy scales?

The answers to these questions will demand levels of precision not yet seen in
neutrino physics. They will place new demands on the experimental program:
larger, more precise detectors, a more accurate understanding of the
mechanisms which produce neutrinos, and the physics behind the interactions
which experiments use to detect them.

This report documents the current and future program to measure neutrino
oscillations to precisely elucidate the now standard
Pontecorvo–Maki–Nakagawa–Sakata (PMNS) framework of neutrino masses and
mixings. Precise measurements of the PMNS framework will answer questions
about new symmetries in neutrino mixing, enable tests of predictions of
neutrino mass models, and resolve the neutrino mass hierarchy. These
measurements will establish a baseline understanding of neutrino oscillations
upon which theorists and experimenters will build searches for additional
physics beyond the Standard Model associated with neutrinos.

###  1.2 Current Knowns and Known Unknowns in Neutrino Oscillations

Given many oscillation experiments over the last several decades, a clear
picture of the overall framework of three-flavor oscillations has emerged. The
details are discussed in more detail in section 2.3 below, but generally we
have two mass-squared differences: $\Delta m^{2}_{21}\sim+7.5\times 10^{-5}$
eV2 and $\Delta m^{2}_{31}\sim\pm 2.5\times 10^{-3}$ eV2. We know two of the
mixing angles fairly well: $\theta_{13}\sim 9^{\circ}$ and $\theta_{12}\sim
34^{\circ}$. The third mixing angle is close to maximal, but somewhat
uncertain, $\theta_{23}\sim 45^{\circ}$, and the complex phase $\delta$ is
largely unconstrained.

The three primary goals of the coming years of neutrino oscillations are to 1)
determine the sign of $\Delta m^{2}_{31}$, known as the atmospheric mass
ordering question, 2) determine whether $\theta_{23}$ is more than or less
than $45^{\circ}$ and how close to maximal it is, known as the octant
question, and 3) measure $\delta$ and determine if $\sin\delta=0$, and thus CP
conservation, can be excluded or not.

While some of these parameters can be assessed at an individual experiment
(with input from orthogonal experiments), some parameters will first be
measured as a result of global analyses of all relevant neutrino data, known
as global fits. There are three primary global fit groups at the moment, all
based in Europe [4, 5, 6]. While they generally agree on some parameters like
$\theta_{13}$, there are some differences among them for $\theta_{12}$,
$\Delta m^{2}_{21}$ and $\theta_{23}$.

###  1.3 Neutrino oscillations and the Previous P5

The study of neutrino oscillations featured prominently in the previous P5
report [7] which identified “Pursue the physics associated with neutrino mass”
as one of five key science drivers for U.S. high energy physics. P5 identified
the three key questions for neutrino physics as “What is origin of neutrino
mass?”, “How are the neutrino masses ordered?”, and “Do neutrinos and
antineutrinos oscillate differently?”.

In the spirit of its Recommendation 1, “Pursue the most important
opportunities wherever they are, and host unique, world-class facilities that
engage the global scientific community”, P5 recommended an ambitious neutrino
oscillation program to be based at the proton source at Fermilab
(Recommendations 12 and 13). On this point, P5 was very specific, recommending
the formation of an international collaboration to build a neutrino beam
capable of delivering 1.2 MW with the possibility of upgrades to multi-
megawatt power. This beam is to target an underground liquid argon detector of
40 kt fiducial mass located at a baseline of 1350 km in Lead, SD. The program,
known as the Deep Underground Neutrino Experiment (DUNE) is expected to begin
data taking in 2029 and deliver 3$\sigma$ sensitivity for 75% of $\delta_{\rm
CP}$ values by 2035. Recommendation 13 concluded that this experiment “is the
highest priority large project in its timeframe.”.

To support this program, P5 recommended that Fermilab proceed with
construction of PIP-II (Recommendation 14) to achieve multi-megawatt beam
powers and set goals of reaching an exposure of 600 kt $\cdot$ MW $\cdot$ yr,
and precise control of systematic uncertainties: $\pm 1$% signal uncertainty,
and $\pm 5$% background uncertainty for electron neutrino appearance in
neutrino and antineutrino beams. This control of systematic uncertainties is
unprecedented in long baseline physics and places ambitious requirements on
the experiment near detector suite and may also necessitate ancillary
measurements of neutrino cross-sections and the production of hadrons which
produce neutrino fluxes. These ancillary measurements align with P5’s
Recommendation 4; “Maintain a program of projects of all scales, from the
largest international projects to mid- and small-scale projects.”

This P5 vision for building a U.S. based long baseline program built on PIP-II
and LBNF came with some choices. Recommendation 15 of the 2014 P5 report
advised against new experimental projects on the NuMI beamline. The committee
also recommended termination of U.S. participation in the MICE program of
neutrino factory R&D and recommended against advancing a proposal (NuSTORM) to
construct a muon storage ring to conduct short baseline measurements and
exercise ideas for a future neutrino factory. The committee endorsed continued
development of the idea to use high intensity cyclotrons to measure CP
violation in the neutrino sector (IsoDAR and Daedalus) and encouraged further
exploration of the potential to measure neutrino oscillations at the South
Pole using the IceCube neutrino telescope.

##  2 Three-Flavor Oscillation Theory

In this section we briefly review the standard paradigm of oscillation theory
in vacuum and in matter and the role of each of the six oscillation
parameters. Notably, there are three major current unknowns that the community
is attempting to determine while simultaneously making precision measurements
of all the oscillation parameters. Additional discussion of the role of theory
supporting neutrino oscillations can be found in 4.2.

###  2.1 Neutrino Oscillation Probabilities

The discovery of neutrino oscillations, coming from Super-Kamiokande [8] and
SNO [9] data confirmed the existence of neutrino oscillations at high
$>5\sigma$ significance independent of theoretical flux predictions. This
immediately added at least seven new parameters to our model of particle
physics: three masses and four parameters related to the mixing between the
weak interaction basis and the mass basis. Moreover, since there is not a
single obvious minimal mechanism for neutrino mass generation, there are
likely additional parameters and particles waiting to be discovered at scales
other than the neutrino scale $\sim 0.01$ eV. Nonetheless the neutrino
oscillation phenomenon is sensitive to six of the seven parameters and
measuring these parameters has been the focus of an intense global effort over
the last several decades and is the focus of this report.

Neutrinos are produced in flavor states which are linear combinations of the
mass states. Since neutrinos propagate in the mass states (i.e. the mass
states are the eigenstates of the Hamiltonian), quantum mechanical
interference happens during propagation. This interference is related to the
difference in the momenta or energy of the states and is proportional to
$m_{i}^{2}-m_{j}^{2}\equiv\Delta m^{2}_{ij}$. Given the known values of the
$\Delta m^{2}$’s and the conveniently available neutrino energies, it is
somewhat a coincidence that it is feasible to see neutrino oscillations on
human scales. Atmospheric neutrinos experience their oscillations across a
range of energies easily accessible for a range of distances smaller than the
diameter of the Earth. Reactor neutrinos experience their oscillations over
baselines of $\sim 1$ km or $\sim 30$ km which make for relatively easy
placement of detectors.

The probability that a neutrino that begins as flavor $\alpha$ is detected in
a charged-current (CC) interaction as flavor $\beta$ is

$P_{\alpha\beta}=\delta_{\alpha\beta}-4\sum_{i>j}\Re(U_{\alpha i}^{*}U_{\beta
i}U_{\alpha j}U_{\beta j}^{*})\sin^{2}\left(\frac{\Delta
m^{2}_{ij}L}{4E}\right)\pm 8J\prod_{i>j}\sin\left(\frac{\Delta
m^{2}_{ij}L}{4E}\right)\,,$ (1)

where $L$ is the distance traveled, $E$ is the neutrino energy,
$J\equiv\Im(U_{\alpha i}^{*}U_{\beta i}U_{\alpha j}U_{\beta j}^{*})$ is the
Jarlskog coefficient governing CP violation [10], the upper (lower) sign is
for (anti-)neutrinos for $P_{\mu e}$ and is cyclic in flavor, and $U$ is the
PMNS matrix [11, 12] given by

$|\nu_{i}\rangle=U_{\alpha i}|\nu_{\alpha}\rangle\,.$ (2)

There are many valid parameterizations of the mixing matrix [13] and the one
that is phenomenologically useful and standard in the field is

$\displaystyle U$ $\displaystyle=\begin{pmatrix}1\\\ &c_{23}&s_{23}\\\
&-s_{23}&c_{23}\end{pmatrix}\begin{pmatrix}c_{13}&&s_{13}e^{-i\delta}\\\ &1\\\
-s_{13}e^{i\delta}&&c_{13}\end{pmatrix}\begin{pmatrix}c_{12}&s_{12}\\\
-s_{12}&c_{12}\\\ &&1\end{pmatrix}$ (3)
$\displaystyle=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{13}s_{23}e^{i\delta}&c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}&c_{13}s_{23}\\\
s_{12}s_{23}-c_{12}s_{13}c_{23}e^{i\delta}&-c_{12}s_{23}-s_{12}s_{13}c_{23}e^{i\delta}&c_{13}c_{23}\end{pmatrix}$
(4)

where $c_{ij}$, $s_{ij}$ are $\cos\theta_{ij}$, $\sin\theta_{ij}$
respectively. If neutrinos have a Majorana mass term then there are two
additional complex phases, but they are not measurable in neutrino oscillation
experiments as the effect is suppressed by $\mathcal{O}((m/E)^{2})\lesssim
10^{-14}$ or smaller for all neutrino oscillation experiments.

###  2.2 Matter Effect

The above discussion of neutrino oscillations is valid only in the vacuum or
sufficiently low densities such as the Earth’s atmosphere. In other
environments, the presence of electrons modifies the propagation of the states
and, notably, does so in a basis that is not the usual propagation basis which
is the mass basis [14]. This modifies the oscillation probabilities in a
significant way that is relevant in a number of environments. In fact, for
neutrinos produced in the Sun, it is the dominant effect. It is because of
this fact that the solar mass ordering was determined: that is, that $\Delta
m^{2}_{21}$ is positive111It is useful to define the numbering of the mass
states by $|U_{e1}|^{2}>|U_{e2}|^{2}>|U_{e3}|^{2}$ since those elements of the
mixing matrix are all fairly well measured. If one adopts a different
definition then one could instead reframe the solar mass ordering measurement
as a measurement that $\theta_{12}<45^{\circ}$ in the usual parameterization
of the mixing matrix..

The same matter effect can be used to determine the atmospheric mass ordering
which is the strategy DUNE will leverage. In the normal mass ordering DUNE
will see a relatively large number of $\nu_{e}$’s and a relatively small
number of $\bar{\nu}_{e}$’s and vice-versa in the inverted ordering. While
writing the neutrino oscillation probabilities for the different channels in
vacuum as a function of the six underlying oscillation parameters is
relatively straightforward, doing so in the presence of matter is
significantly more complicated, even if the matter density is essentially
constant [15, 16, 17, 18, 19]. As such numerous approximations have been made
to better understand the role of the matter effect on long-baseline neutrino
oscillations [20].

In the Sun the matter effect is fairly large relative to the mass splittings
leading to an approximate alignment between the $\nu_{e}$ state and the second
eigenstate of the propagation basis for the higher energy 8B neutrinos [21].
Then, since the density profile in the Sun changes sufficiently slowly towards
the surface [22], the neutrinos are emitted in a dominantly $\nu_{2}$ state
with subleading $\nu_{1}$ and $\nu_{3}$ contributions. En route to the Earth,
the neutrinos decohere [23] and the probability for detection is
$P_{e\beta}=\sum_{i}|\hat{U}_{ei}|^{2}|U_{\beta i}|^{2}$ where $\hat{U}$ is
the matrix that diagonalizes the Hamiltonian at densities of the production
region. In addition, solar neutrinos detected at night pass through the Earth
and the decohered states begin oscillations again leading to the day-night
effect for which SK has reported preliminary evidence for [24] and DUNE and HK
may be able to measure well [25, 26, 27].

The matter effect has only been significantly detected in the Sun by a
comparison of solar neutrino data and reactor neutrino data. Expected
improvements on the matter effect in the Earth will come from nighttime solar
neutrinos at DUNE and HK, long-baseline accelerator neutrinos at DUNE [28],
and low-energy atmospheric neutrinos at DUNE [29, 30].

###  2.3 Role of Each Oscillation Parameter

Due to the rich phenomenology of neutrino oscillation physics, the human and
planet sized oscillation lengths, and wide range in available neutrino
energies, the interplay of the six oscillation parameters is quite
complicated. The history of their measurements is shown in Fig. 1. The six
oscillation parameters affect oscillations in particular ways discussed here.

Figure 1: The six oscillation parameters listed in the order they were first
measured, and the evolution of our understanding of them. Data comes from [8,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40].

##### $\boldsymbol{\Delta m^{2}_{31}}$

Neutrino oscillation experiments are only sensitive to the difference of mass
squareds since neutrino oscillations occur from the relative accumulated
phase. Given three neutrinos, there are thus two degrees-of-freedom often
quantified as $\Delta m^{2}_{31}$ and $\Delta m^{2}_{21}$; $\Delta m^{2}_{32}$
then follows in a straightforward fashion as $\Delta m^{2}_{32}=\Delta
m^{2}_{31}-\Delta m^{2}_{21}$. $\Delta m^{2}_{31}$ is known as the atmospheric
mass squared difference and is known to be $\sim\pm 2.5\times 10^{-3}$ eV2. An
oscillation maximum or minimum happens when $\Delta m^{2}L/(4E)\simeq n\pi/2$
where $n$ is some integer; there are additional corrections due to the other
$\Delta m^{2}$’s as well as the matter effect. These $\Delta m^{2}$’s are
sometimes referred to as frequencies since they dictate when the probability
for a neutrino to interact with a certain charged lepton is a maximum or a
minimum. Most experiments are probing the first oscillation maximum or
minimum. This value for $\Delta m^{2}_{31}$ corresponds to oscillations at a
baseline of 500 m for 1 MeV neutrinos (e.g. reactor neutrinos) or 500 km for 1
GeV neutrinos (e.g. accelerator or atmospheric neutrinos). There is an
impressive level of agreement on $|\Delta m^{2}_{31}|$ among a total of seven
experiments, reactor, accelerator, and atmospheric, spanning about four orders
of magnitude in energy and baseline [41, 42, 43, 44, 45, 46, 47, 48]. The sign
of $\Delta m^{2}_{31}$ is undetermined which is one of the three major known
unknowns in neutrino oscillations; some hints currently suggest that it is
positive, although the significance is at the $\sim 1.5-3\sigma$ level from
global analyses [4, 5, 6]. Determining the sign of $\Delta m^{2}_{31}$ has
important implications for neutrinoless double beta decay, cosmological
measurements of the sum of the neutrino masses, kinematic end point mass
measurements, supernova neutrinos, and detections of the cosmic neutrino
background.

##### $\boldsymbol{\Delta m^{2}_{21}}$

The other $\Delta m^{2}$ is known as the solar $\Delta m^{2}$ and is $\Delta
m^{2}_{21}\sim+7.5\times 10^{-5}$ eV2. This corresponds to oscillations at 17
km for 1 MeV neutrinos. The absolute value of $\Delta m^{2}_{21}$ was
determined by KamLAND [49] and will be further measured by JUNO at high
precision [50]. $\Delta m^{2}_{21}$ can also be determined with solar
neutrinos, although with generally worse resolution. Nonetheless, the matter
effect in the Sun combined with the fact that the disappearance probability
decreases with energy from $\sim 0.55$ at $E\lesssim 1$ MeV for pp neutrinos
[51, 52, 53] to $\sim 0.3$ at $E\gtrsim 5$ MeV for 8B neutrinos [54, 55, 53,
24] indicates that $\Delta m^{2}_{21}$ is positive. In addition, the Earth’s
matter effect for nighttime solar neutrinos provides some information about
$\Delta m^{2}_{21}$. Extracting the complex phase $\delta$ in long-baseline
accelerator neutrinos depends on the precision of this parameter.

##### $\boldsymbol{\theta_{23}}$

The atmospheric mixing angle is $\theta_{23}$ and is known to be close to
maximal: $\sim 45^{\circ}$. Thus the $\nu_{\mu}$ disappearance probability at
the first oscillation minimum is close to zero. This parameter is currently
measured dominantly from $\nu_{\mu}$ disappearance measurements at MINOS [43],
T2K [44], NOvA [45], IceCube [47, 48], and SK [46]. These measurements
constrain $\sin^{2}(2\theta_{23})$ which seems to be close to 1, and thus can
only tell how close to 45∘ $\theta_{23}$ is, but not whether it is preferred
to be below 45∘ or above; this is known as the octant problem and is one of
the three big known unknowns in neutrino oscillations. The octant question
tells us whether the mass state that is least $\nu_{e}$ (typically defined as
$\nu_{3}$) is more $\nu_{\mu}$ (upper octant) or more $\nu_{\tau}$ (lower
octant). Determining the octant likely requires an appearance measurement and
is a goal of current accelerator experiments NOvA and T2K and the upcoming
experiments DUNE and HK. Existing data is inconclusive on the octant question.

##### $\boldsymbol{\theta_{12}}$

The solar mixing angle is $\theta_{12}$ and is $\sim 34^{\circ}$. This
indicates that in long-baseline reactor neutrinos at the $\Delta m^{2}_{21}$
oscillation minimum that the $\nu_{e}\to\nu_{e}$ probability drops to $\sim
15\%$. This parameter has been fairly well measured by KamLAND [49] and will
be well measured by JUNO [50], both with reactor neutrinos. The solar mixing
angle can also be measured with solar neutrinos as well as nighttime solar
neutrinos with comparable precision as the reactor experiments.

##### $\boldsymbol{\theta_{13}}$

The third and final mixing angle measured, $\theta_{13}$ is known as the
reactor mixing angle. Based on theoretical predictions, it was anticipated by
many to be quite small or zero. $\theta_{13}$ plays a role in both $\Delta
m^{2}_{21}$ and $\Delta m^{2}_{31}$ measurements. For example, in solar
neutrinos there is a contribution from $\theta_{13}$ since some of the
$\nu_{e}$’s escape the Sun as $\nu_{3}$ for all solar neutrino energies and
this fraction is $s_{13}^{2}$. It can also be measured in long-baseline
accelerator experiments and it plays a role in CP violation, as do all of the
other parameters, see below. Finally, the best way to probe $\theta_{13}$ is
with reactor neutrinos at a baseline of $\sim 1$ km where Daya Bay [41], RENO
[42], and Double Chooz [56] all agree in their measurements of
$\theta_{13}\sim 8.5^{\circ}$ with excellent precision.

##### $\boldsymbol{\delta}$

There is one guaranteed complex phase in the mixing matrix known as $\delta$.
Additional phases may be physical if lepton number is violated e.g. by the
presence of a Majorana mass term, although the effect of these phases on
neutrino oscillations is completely negligible. In addition, $\delta$ is only
physical if all three neutrino masses are different and all three mixing
angles are non-zero; the discovery of non-zero $\theta_{13}$ was the final
piece to open the door to a physical $\delta$. Since this phase is complex it
governs the difference between neutrinos and antineutrinos and thus dictates
the amount of CP violation in the lepton mixing matrix. The simplest
measurement of $\delta$ results from comparing appearance measurements of
neutrinos and antineutrinos which is sensitive to $\sin\delta$, although this
channel provides no information about the sign of $\cos\delta$; this is the
scenario for T2K and HK to a good approximation. Meanwhile, $\cos\delta$ can
be determined either in the presence of matter (this is DUNE’s approach) or by
an extremely careful analysis of both $\nu_{e}$ and $\nu_{\mu}$ (or
$\nu_{\tau}$) disappearance data which is likely not feasible in upcoming
experiments. Currently T2K shows a mild preference for $\delta\sim
270^{\circ}$ [44] while NOvA, which has comparable sensitivity, is compatible
with most values of $\delta$ [45]. Determining CP violation in the lepton
sector has important implications for understanding CP violation in the larger
context where we see that the weak interaction violates CP but the strong
interaction doesn’t and the quark mass matrix violates CP but only a little
bit. The Jarlskog invariant, which usefully quantifies the “amount” of CP
violation [10], for the quark matrix is $J_{\rm CKM}=+3\times 10^{-4}J_{\max}$
while for leptons it could be much larger, $|J_{\rm PMNS}|<0.34J_{\max}$ where
$J_{\max}\equiv\frac{1}{6\sqrt{3}}\approx 0.096$. Understanding this mystery
of CP violation is a top priority in particle physics.

The three unknowns presented here: the atmospheric mass ordering, the octant
of $\theta_{23}$, and the value of $\delta$ represent three of the four known
unknowns in particle physics with the absolute neutrino mass scale being the
fourth parameter. We show the preferred values of the parameters in the lepton
mixing matrix and the quark mixing matrix in table 1.

Table 1: The preferred values of the parameters governing the lepton and quark mixing matrices in the usual parameterization. The quark parameters are quite well measured while $\theta_{23}$ for the leptons has modest uncertainties and $\delta$ for the leptons is largely undetermined. Note that the masses for all of the quarks are extremely well known, the masses for neutrinos are largely unconstrained. | $\theta_{23}$ | $\theta_{13}$ | $\theta_{12}$ | $\delta$
---|---|---|---|---
Leptons | $\sim 45^{\circ}$ | $8.5^{\circ}$ | $34^{\circ}$ | ?
Quarks | $2.4^{\circ}$ | $0.20^{\circ}$ | $13^{\circ}$ | $69^{\circ}$

###  2.4 Flavor Model Predictions and Desired Precision

Understanding the flavor structure of particle physics: why the fermion masses
are distributed as they are and why their mixing with the weak interaction has
the structure it has, is one of the biggest open questions in understanding
our model of particle physics. Neutrino oscillation measurements of the
remaining mixing unknowns, the atmospheric mass ordering, the octant of
$\theta_{23}$ and the value of $\delta$, will provide a wealth of information
to enhance our knowledge of the three family structure of particle physics.
Before these measurements are made it is important to consider the model
predictions which can, in turn, motivate a desired level of precision for
these parameters.

Different flavor structures make different predictions, notably in terms of
the complex phase. Many of these predictions are functions of $\cos\delta$.
For example, various models based on discrete symmetries are of the form
$U_{\rm PMNS}=U_{e}^{\dagger}U_{\nu}$ where $U_{e}$ has a (12) rotation
inspired by the Cabibbo angle and $U_{\nu}$ contains two discrete rotations
that are often fixed such that the $\sin^{2}\theta_{ij}$ form a ratio of small
integers. Given the existing oscillation constraints one can then extract a
range of expected values for e.g. $\cos\delta$ shown in Fig. 2. In this we see
that, for example, some models such as Golden Ratio (GR) 1 and 2 predict quite
different values of $\cos\delta$ but similar values of $|\cos\delta|$
emphasizing the need to measure the sign of $\cos\delta$. Being able to
differentiate among these models should be a goal for the upcoming
measurements of $\delta$. In addition, these models predict specific
correlations of the values of the other mixing parameters, $\theta_{12}$,
$\theta_{13}$, and $\theta_{23}$ that require more precise measurements to
discriminate among the models. Thus improving the precision of all mixing
parameters, not just $\delta$ and $\theta_{23}$, is very important to add
clarity to the flavor puzzle, see e.g. [57] for a recent overview.

Figure 2: Predicted ranges of $\cos\delta$ given existing known oscillation
parameters under several example model predictions. Figure from [58].

##  3 Three-Flavor Neutrino Oscillation Facilities

###  3.1 JUNO

The JUNO experiment is the successor to the very successful program of
experiments which used nuclear reactors as a source for studies of
$\bar{\nu}_{e}\rightarrow\bar{\nu}_{e}$ oscillations. Nuclear reactors are an
abundant source of $\bar{\nu}_{e}$ with energies below 8 MeV. When measured at
distances of 1 – 2 km, reactor neutrino spectra have provided precise
measurements of $\sin^{2}2\theta_{13}$ and $\Delta m^{2}_{31}$ and the near
detectors used in these experiments have made high statistics measurements of
reactor neutrino fluxes [59, 60, 61].

Figure 3:  Left: A schematic of the JUNO detector. Right: Predicted
$\bar{\nu}_{e}$ spectra for the JUNO experiment showing the dependence on
neutrino oscillations parameters.

JUNO (Jiangmen Underground Neutrino Observatory) is constructing a 35 m
diameter detector containing 20 kt of liquid scintillator 700 m underground in
Kaiping, South China at a distance of 53 km from the reactor cores of the
Yangjiang and Taishan nuclear power plants. This $\simeq 25\times$ increase in
baseline over the Daya Bay experiment will give JUNO unique sensitivity to the
solar oscillation parameters $\Delta m^{2}_{21}$, and $\sin^{2}2\theta_{12}$
among the next generation experiments. JUNO projects that measurements of
$\sin^{2}2\theta_{12}$, $\Delta m^{2}_{21}$, and $\Delta m^{2}_{32}$ with
reach $\simeq 1\%$ precision in six years of data taking.

JUNO will instrument its volume with 43,000 photomultipliers to reach 75%
photo-cathode coverage yielding an expected energy resolution of $3\%$. This
will allow JUNO to provide information on the atmospheric mass ordering at up
to the $3~{}\sigma$ level by observing the phase of the two “fast”
oscillations from $\Delta m^{2}_{31}$ and $\Delta m^{2}_{32}$ in the recoil
electron energy spectrum, see Fig. 3.

Construction of JUNO began in 2015 and construction of the detector started in
2022. JUNO expects to complete the detector in 2023.

###  3.2 Fermilab/SURF Program

####  3.2.1 NOvA

The NOvA experiment uses the Neutrinos from the Main Injector (NuMI) beam,
originally constructed for the MINOS experiment [62] to measure
$\nu_{\mu}\rightarrow\nu_{\mu}$, $\nu_{\mu}\rightarrow\nu_{e}$,
$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{\mu}$,
$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$, in a relatively narrow-band beam
peaked at 2 GeV. The neutrino beam passes through a near detector located 1 km
from the proton target and then through a second 14,000 ton detector located
810 km away in Northern Minnesota. The detectors employ liquid scintillator
segmented into 4 cm $\times$ 6 cm $\times$ 15 m cells by a reflective PVC
structure. NOvA’s segmentation provides for 3.5% muon energy resolution
coupled with a 25% hadronic energy resolution giving an overall neutrino
energy resolution of 8–10% for $\nu_{\mu}$-charged current events crucial for
measurements of $\nu_{\mu}\rightarrow\nu_{\mu}$ oscillations which determine
the parameters $|\Delta m^{2}_{32}|$ and $\sin^{2}2\theta_{23}$. The use of
low $Z$ materials allows NOvA to efficiently separate electrons from photons
which is crucial for measurements of electron neutrino appearance which are
used for the measurement of the $\theta_{23}$ octant, atmospheric mass
ordering, and $\delta_{\rm CP}$. Due to the long baseline, $L=810$ km, the
choice of mass ordering modifies the $\nu_{\mu}\rightarrow\nu_{e}$ and
$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$ oscillation probabilities by $\pm
20$%. Of planned long-baseline accelerator experiments, NOvA’s baseline will
remain the longest until the DUNE experiment begins operations.

Figure 4:  The $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ neutrino spectra (left
panels) and $\nu_{\mu}\rightarrow\nu_{e}$ and
$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$ (right) spectra recorded by the NOvA
far detector. [45].  Figure 5:  Left: The best fit parameters for $\Delta
m^{2}_{32}$ and $\sin^{2}\theta_{23}$ from the NOvA experiment and others.
Right: The best-fit values of $\sin^{2}\theta_{23}$ and $\delta_{\rm CP}$ to
NOvA data for the normal neutrino mass ordering (a), and the inverted neutrino
mass ordering (b). Also shown are recent best-fits to T2K data [63].

NOvA has been taking data since 2014 releasing new data on standard three-
flavor oscillations each year since its first results in 2016. During that
time the accelerator power to NuMI has steadily increased throughout NOvA’s
run. Typical running today averages 650 kW to 730 kW depending on other beam
consumers at Fermilab. An hourly peak power of 893 kW has been demonstrated
exceeding the design power of 700 kW. In total, NOvA has recorded $13.6\times
10^{20}$ protons-on-target (POT) in neutrino beam and $12.5\times 10^{20}$ POT
in anti-neutrino beam, yielding 214 $\nu_{\mu}\rightarrow\nu_{\mu}$ events, 59
$\nu_{\mu}\rightarrow\nu_{e}$ events, 103
$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{\mu}$, and, 19
$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$ events in the far detector. The
spectra of these events (Fig. 4) give $\Delta m^{2}_{32}=(2.41\pm 0.07)\times
10^{-3}~{}{\rm eV}^{2}$, $\sin^{2}\theta_{23}=0.57^{+0.03}_{-0.04}$, and
$\delta_{\rm CP}=0.82^{+0.27}_{-0.87}$. NOvA has not observed a large
asymmetry in the rates of $\nu_{\mu}\rightarrow\nu_{e}$ vs.
$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$ oscillations and hence does not
favor large CP violation and has excluded parameter combinations in the
inverted mass ordering which would produce a large
$\nu_{\mu}\rightarrow\nu_{e}$ vs. $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$
asymmetry (Fig. 5). NOvA’s observation is different from T2K which has seen a
large asymmetry in these rates and the two experiments are in slight tension.
NOvA’s modest asymmetry also has not produced a clear resolution of the
atmospheric mass ordering; NOvA data favors the normal ordering at 1 $\sigma$.

NOvA’s statistical uncertainties are roughly 2$\times$ larger than its
systematic uncertainties and the experiment will remain statistics limited
until the projected end of its run in 2026 when a shutdown of NuMI becomes
necessary for PIP-II construction. The NuMI beam intensity is projected to
increase to 900 kW by 2024 allowing NOvA to double its current data set prior
to the planned end of its run. This final data set would have 3 $\sigma$
sensitivity to resolve the neutrino mass ordering for $\simeq$50% of
$\delta_{\rm CP}$ values and up to $>4~{}\sigma$ sensitivity for the most
favorable parameters. The experiment projects 2 $\sigma$ sensitivity to CP
violation for $\simeq$20% of $\delta_{\rm CP}$ values.

####  3.2.2 DUNE

The DUNE experiment is the next step in the evolution of the Fermilab program.
The DUNE collaboration was formed to realize the 2014 P5 vision of a best-in-
class long-baseline experiment based at Fermilab.

For the experiment, a new neutrino beamline is under construction directed
from Fermilab to the Sanford Underground Research Facility (SURF) in South
Dakota at a distance of 1285 km. This beamline will use protons from an
upgraded Main Injector fed from the PIP-II accelerator. DUNE will use these
protons to make an intense and $\simeq 90\%$ pure beam of muon neutrinos and
antineutrinos ranging in energy from 0.5 GeV up to 10 GeV as part of the near
site of its Long-Baseline Neutrino Facility (LBNF) which will house both the
neutrino beam and near detector for the experiment. The combination of a very
long baseline and wide-spectrum beam are unique features of the DUNE beamline.
Construction of the PIP-II source is underway, as is the construction of LBNF.

Figure 6:  Overview of the DUNE experiment. The Long-Baseline Neutrino
Facility will house the beamline and near detector caverns at Fermilab. At the
far site, $L=1285$ km away, DUNE will construct four liquid argon time
projection chambers.

To receive this beam, DUNE intends to construct its far detector in four 18 m
$\times$ 19 m $\times$ 66 m modules to be constructed and deployed in stages.
Excavation of the caverns to house these modules is underway. Design of two of
these modules are in advanced stages, and construction is underway. These
modules will instrument a fiducial mass of 10 or more kilotons of liquid argon
using a time projection chamber capable of $\simeq 5$ mm ($\simeq X_{0}/30$)
spatial resolution and waveguides to collect scintillation light produced by
the liquid argon. This large size and high resolution will allow for detailed
reconstriction of the interactions of neutrinos with $\mathcal{O}(1$ GeV$)$ of
energy and a very efficient tag of the incident neutrino flavor through
separation of muons from pions and electrons from photons.

The DUNE TPCs build on considerable experience with liquid argon gained
through the ICARUS [64] and MicroBooNE [65] experiments. The first of these
modules will use a so-called single-phase technology. The ionization electrons
will be drifted horizontally to three layers of wires to read their induction
and collection signals. For the second module, the design will exploit the
long drift lengths demonstrated and drift electrons vertically. This dual-
phase, vertical drift technology offers several potential simplifications to
the construction of the detector and an increase in its fiducial fraction.
Plans for the third and fourth modules are still taking shape [66].

The collaboration has tested the horizontal drift technology in a large-scale
prototype built and operated in a test beam at CERN [67] and a run using the
vertical drift technology is in progress. Figure 7 shows some results
published from the horizontal prototype run.

Figure 7:  A sample of data from the ProtoDUNE run of the horizontal, single-
phase, prototype detector. Top left: A 3D reconstruction of a beam muon
passing through the detector. Bottom left: The measured muon $dE/dx$ profile
for muons stopping in the detector. Top right: A 6 GeV electron shower. Bottom
right: A candidate conversion of a charged to neutral pion.

To fully exploit the potential of the detectors and beam, DUNE will require
several advances in the understanding of its neutrino flux, the properties of
neutrino interactions, and the efficiency of the detector performance. To meet
these goals, DUNE will construct a suite of near detectors [68]. The
downstream detector, System for on-Axis Neutrino Detection, (SAND) will use a
straw tube tracker placed inside a magnet and calorimeter reused from the KLOE
experiment. The goal is to measure the momentum spectrum and charge-sign of
muons produced by muon neutrino interactions to ensure that the neutrino flux
is stable and well understood in the on-axis location. Upstream, a LArTPC
system consisting of an array of modular LArTPCs optimized for the high
multiplicity environment (ND-LAr) and a magnetized iron scintllator muon
spectrometer (TMS) downstream to analyze muons which exit the back of ND-LAr,
will enable the experiment to measure and control uncertainties in the flux,
cross section, and detector response in order to predict the expected far
detector spectrum [68]. The ability to perform this extrapolation precisely is
essential for extracting the oscillating parameters, which also depends on
obtaining measurements on a near detector which is functionally the same in
fundamental aspects as the far detector.

While the SAND detector remains in place, the ND-LAr+TMS system will be
installed on rails allowing them to move to different off-axis positions in
the DUNE beam. As the neutrino spectrum peaks at a different and lower
energies at each position, the ensemble forms a complete set of neutrino
spectra which allow for the synthesis of oscillated spectra as the far
detector. This data-driven approach, known as DUNE-PRISM [68] to the near to
far extrapolation is an essential piece of the DUNE program to mitigate
systematic uncertainties. To reach the P5 goals of controlling signal and
background uncertainties to 1% and 5% respectively, the collaboration plans to
replace this spectrometer with gaseous argon TPC. This low-density TPC will
allow for much more detailed reconstruction of neutrino event topologies in
argon enabling systematic uncertainties to be controlled to a precision
matched to the ultimate exposure of the DUNE far detector [69].

Figure 8:  The layout of the DUNE near detector is shown here, with the front
two components in their most off-axis configuration. Note, the “day one” DUNE
near detector replaces the ND-GAr detector with a temporary muon spectrometer.
[68] Figure 9:  Left: The oscillated muon neutrino (top) and muon antineutrino
(bottom) spectra predicted at the DUNE far detector Right: The electron
neutrino (top) and anti-electron (bottom) appearance spectra at the DUNE far
detector. The calculations assume currently best-fit PMNS parameters. The
spectra are normalized to a combined 336 kt-MW-year exposure divided equally
between neutrino and antineutrino beam running. [70]

DUNE will make precise measurements of the effects of oscillations on the muon
neutrino and antineutrino spectra at its far detector and the resulting
appearance of electron neutrinos and antineutrinos (Fig. 9). These
measurements will enable precise determinations of the probabilies of
$\nu_{\mu}\rightarrow\nu_{\mu}$, $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{\mu}$,
$\nu_{\mu}\rightarrow\nu_{e}$, and, $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$
oscillations and the PMNS parameters $\Delta m^{2}_{32}$, $\theta_{23}$,
$\theta_{13}$, and $\delta_{\rm CP}$. DUNE’s long-baseline provides excellent
sensitivity to the matter effect on neutrino oscillations and hence strong
resolving power for the neutrino mass ordering. Figure 10 shows the early
sensitivity of the DUNE experiment to the mass-ordering and CP violation. This
early data ($\simeq$100 kt-MW-yr exposure) can provide $5~{}\sigma$ resolution
of the atmospheric mass ordering and $3~{}\sigma$ sensitivity to CP violation
if it is maximal. Figure 11 shows the ultimate precision of the experiment.
The high precision of these standard three-flavor measurements in a wide-band
beam will provide a platform for searches of additional physics beyond the
standard model that might impact neutrino oscillations.

Figure 10:  The early DUNE sensitivity to the neutrino mass ordering (left
panels) and CP violation (right panels) calculated assuming the neutrino mass
ordering is normal (top panels) and inverted (bottom panels). The two curves
show the median sensitivity for 66 and 100 kt-MW-year exposures; the bands
indicate the $1\sigma$ range of these curves based on variations in
oscillation parameter assumptions, statistics, and systematics. The dotted
lines indicate where the experiment reaches $5\sigma$ resolution of the
neutrino mass ordering and $3~{}\sigma$ observation of CP violation.  Figure
11:  90% confidence intervals for $\sin^{2}2\theta_{13}$ – $\delta_{\rm CP}$
(left), and $\sin^{2}\theta_{23}$ – $\Delta m^{2}_{32}$ (right) after seven,
ten, and fifteen years of staged DUNE operations. Yellow regions indicate
recent global fits from the NuFIT 4.0 group [71, 72].

In addition to the long-baseline neutrino physics program DUNE will collect
40,000 atmospheric neutrino interactions [73] of which 800 will be
$\nu_{\tau}$ charged-current interactions. This sample, measured with the
unprecedented detail afforded by the liquid argon TPC will present several
opportunities for a new level of precision in the study of
$\nu_{\mu}\rightarrow\nu_{\tau}$ oscillations [74].

Finally, DUNE will also measure solar neutrinos providing considerably
improved solar measurement of the solar oscillation parameters $\theta_{12}$
and $\Delta m^{2}_{21}$ [25, 75], although they will not be competitive with
the precision of JUNO.

###  3.3 J-PARC/Kamioka Program

The J-PARC/Kamioka program consists of a neutrino beam produced at the Japan
Proton Accelerator Research Complex (J-PARC) in Tokai village, Ibaraki
prefecture, Japan. A suite of near detectors which measure the neutrino beam
flux and neutrino interactions is also located in Tokai. A large-scale water
Cerenkov detector operated as part of the Kamioka Observatory, is installed
295 km away from the beam neutrino source, near Kamioka town, Gifu Prefecture,
Japan, and acts as a stand-alone neutrino (and proton decay) detector, as well
as a far detector for the J-PARC neutrino beam.

####  3.3.1 Super-Kamiokande (SK)

The Super-Kamiokande (SK) detector is a massive water Cherenkov detector
installed 1,000 m underground in the Mozumi Mine in western Japan [76]. The
detector contains 50 kt of ultra-pure water instrumented by $>$10,000 50 cm
inward facing Photo Multiplier Tubes (PMTs). An optically separated, highly
instrumented outer detector is used to veto particles with initial interaction
points outside of the tank volume. The SK detector has excellent PID
performance between showering (electron-like) and non-showering (muon-like)
particles at high energies, with a $\sim$1% mis-PID rate at 1 GeV. SK has been
operating with continual improvements since 1996.

SK was initially conceived as a massive, highly instrumented, water Cherenkov
detector, designed to resolve the atmospheric neutrino anomaly, resolve the
solar neutrino puzzle, and search for nucleon decay. SK also serves as the T2K
far detector, as described in Sec. 3.3.2. The discussion in this SK text will
focus on the SK sensitivity to three-flavor-oscillations via atmospheric
neutrino oscillations as T2K is a separate collaboration.

SK recently entered a new phase of operation following upgrade work in 2018.
The detector was refurbished and the water purification system was upgraded to
allow operation with dissolved gadolinium sulfate, Gd2(SO4)3, 0.2% by mass.
First operation of this new phase of running, called SK-Gd, commenced in 2020
with 0.02% loading. Additional loading is planned for the near future. Neutron
capture on gadolinium will improve differentiation between neutrinos and
antineutrinos, as well as improving energy reconstruction using neutron
counting. These improvements enhance nearly every aspect of physics
measurements at SK.

Recent SK atmospheric oscillation fit results are shown in Fig. 5. These
results slightly prefer the normal mass ordering, and the oscillation
parameter best fit values are
$\sin^{2}\theta_{23}=0.588^{+0.031}_{-0.064}(0.575^{+0.036}_{-0.073})$,
$|\Delta m^{2}_{32,31}|=2.50^{+0.13}_{-0.20}(2.50^{+0.08}_{-0.37})\times
10^{-3}$ eV2, and $\delta_{CP}=4.18^{+1.41}_{-1.61}(4.18^{+1.52}_{-1.66})$
assuming the normal (inverted) ordering [77]. The precision of these
measurements continues to improve as SK continues to collect data. The
expected SK sensitivity to reject the wrong mass ordering is given in Fig. 12.

Figure 12: SK expected sensitivity to the normal mass ordering as a function
of livetime assuming $\sin^{2}\theta_{23}=0.5$. Gray and blue lines show the
sensitivity of the atmospheric neutrino sample reconstructed with the latest
fit algorithm using the conventional SK Fiducial Volume (FV) and expanded FV,
respectively. Orange lines denote the sensitivity when events are
reconstructed with an older fit algorithm using the conventional FV.
Reproduced from [46].

SK data also delivers data sets that are necessary for understanding water
Cherenkov reconstruction algorithms, their efficiencies, and related
systematic uncertainties, all of which are essential for precision T2K
measurements, discussed in the following section. Atmospheric neutrino
interactions are also used to validate neutrino interaction simulations on
water nuclei. In addition, cosmic ray data is critical for tuning the detector
simulation. The atmospheric neutrino single-$\pi^{0}$ sample is the most
important in-situ energy scale calibration in the T2K energy range, and is
also a valuable sample for studying and improving software algorithms.

####  3.3.2 Tokai to Kamioka (T2K)

The Tokai to Kamioka (T2K) experiment is a long-baseline neutrino oscillation
experiment [78] which utilizes a high-intensity, highly pure muon or antimuon
neutrino beam produced by 30 GeV protons from the Main Ring (MR) accelerator
at J-PARC. The neutrino flux and neutrino interactions are measured near the
source by a suite of near detectors installed 280 m downstream of the neutrino
production target. The far detector is the SK detector, which sits 295 km away
from the neutrino source and 2.5$\degree$ off-axis from the neutrino beam
center. T2K hosts a broad physics program of precision neutrino oscillation
measurements, neutrino cross section measurements, as well as searches for
exotic physics.

The T2K experiment began commissioning in 2009 and physics data taking in
2010. In 2013, T2K discovered $\nu_{\mu}\rightarrow\nu_{e}$ appearance [79]
and in 2019, T2K presented the most stringent constraints on CPV in the lepton
sector [80], as shown in Fig. 13. Recent T2K measurements of the three-flavor
oscillation parameters are shown in Fig. 5 and can be summarized as
$\sin^{2}\theta_{23}=0.53_{-0.04}^{+0.03}$, $\Delta m^{2}_{32}=(2.45\pm
0.07)\times 10^{-3}$ eV2, and
$\delta=-1.89^{+0.70}_{-0.58}(-1.38^{+0.48}_{-0.54})$ with a 3$\sigma$
confidence interval of [-3.41, -0.03] assuming the normal (inverted) mass
ordering [44, 80].

The J-PARC MR proton beam, which has stably operated at 515 kW, is used to
produce the neutrino beam for T2K. A MR magnet power supply upgrade in
2021/2022 will decrease the time between proton beam spills from 2.48 s to
1.32 s, increasing the beam power to $>$700 kW. Additional RF upgrades and
machine development should allow the beam power to reach $>$1 MW by 2025. A
series of three electromagnetic horns focus hadrons outgoing from the neutrino
production target, allowing for either a neutrino-enhanced or an antineutrino-
enhanced beam. Upgrades to the horns and the horn power supplies are also
taking place in 2021/2022. These upgrades will allow the T2K experiment to
increase the horn current from $\pm$250 kA to $\pm$320 kA, increasing the
neutrino beam flux right-sign component (e.g. muon neutrinos in a neutrino-
enhanced beam) by $\sim$10% and decreasing the wrong-sign component (e.g. muon
antineutrinos in a neutrino-enhanced beam) by $\sim$5% at the flux peak.

As of 2020, T2K oscillation analyses incorporate the charged pion yields
measured by the NA61/SHINE experiment using a T2K-replica target [81] to
constrain the beam flux prediction, and this analysis will be expanded to
include more species and higher statistics in the near future [82]. These
improvements have reduced the T2K flux uncertainty from $\sim$10% to $\sim$5%
near the flux peak prior to a near detector constraint. Future hadron
production data at lower energies and high-statistics replica-target data will
reduce these uncertainties further, and improvements in proton and tertiary
beam monitoring and analysis are also being developed. These efforts to
improve flux uncertainties are useful not only to T2K, but rather are relevant
to the full future global program of accelerator-based neutrino beams.

Multiple near detectors sample the neutrino beam at different locations 280 m
from the neutrino production target. T2K’s near detector suite includes an on-
axis detector colinear with the neutrino beam axis (INGRID), and two
magnetized tracking off-axis detectors at different positions transverse to
the beam (ND280 at 2.5$\degree$ and WAGASCI+BabyMIND at 1.5$\degree$). The
different positions in the beam result in three distinct energy spectra for
use in physics analyses. A new, upgraded ND280 detector, with a fully active
target (SuperFGD), surrounded by new horizontal TPCs, is being installed in
2022 [83]. This detector will have significantly increased acceptance to
particles emitted at high and backward angles, and to low energy protons. The
improved capabilities of this detector will help to reduce systematic
uncertainties due to the neutrino interaction model to an unprecedented level.

The SK far detector has also been undergoing upgrades since 2018, as discussed
in Sec. 3.3.1.

T2K expects to take data with the upgraded beam and detectors until the start
of Hyper-Kamiokande, aiming to collect data corresponding to $10\times
10^{21}$ protons-on-target (POT). The physics program will continue to pursue
$3~{}\sigma$ observation of CPV in neutrinos and enhanced sensitivity to the
mass ordering and atmospheric parameters [84, 85]. The expected sensitivity of
T2K to resolve CPV is given in Fig. 13, and the expected sensitivity to the
atmospheric mixing parameters is given in Fig. 14. Future T2K running will be
used to make further improvements to the J-PARC neutrino beam, and future data
will be used to develop analysis techniques and approaches for precision
oscillation measurements, especially including the use of upgraded, high
performance near detectors. T2K’s program is therefore essential to support
the next generation of long-baseline neutrino experiments.

Figure 13: T2K 2020 $\delta_{CP}$ measurement (left) and expected sensitivity
to resolving $\sin\delta_{CP}\neq 0$ plotted as a function of T2K POT assuming
$\delta_{CP}=-\pi/2$, normal MO, $\Delta m^{2}_{32}=2.4\times 10^{-3}$ eV2,
and various true values of $\sin^{2}\theta_{23}$ (right). Reproduced from [86]
and [84].

Figure 14: T2K expected sensitivity to $\sin^{2}\theta_{23}$ (left) and
$\Delta m^{2}_{32}$ (right) plotted as a function of T2K POT assuming
$\delta_{CP}=-\pi/2$, normal MO, $\Delta m^{2}_{32}=2.4\times 10^{-3}$ eV2,
and $\sin^{2}\theta_{23}=0.5$. Reproduced from [84].

####  3.3.3 Hyper-Kamiokande (HK)

As the next-generation underground water Cherenkov detector in Japan, Hyper-
Kamiokande (HK) [27] builds on the highly successful Super-Kamiokande and T2K
experiments. The HK experiment plans to utilize the upgraded J-PARC MR 1.3 MW
proton beam, the upgraded 280 m near detector suite, a new intermediate
detector to be installed $\sim$1 km away from the neutrino production target,
and a new far detector 295 km away from the neutrino source. The 260 kton
Hyper-Kamiokande far detector has an 8.4 times larger fiducial volume than SK
and is expected to acquire an unprecedented exposure of 3.8 Mton-year over a
period of 20 years of operation. The following also assumes collecting
$2.7\times 10^{21}$ POT per year, running at a ratio of 1 to 3 neutrino-
enhanced to antineutrino-enhanced beam. Civil construction towards the HK tank
is underway now. Detector installation and commissioning is planned for
2025/2026, and datataking is scheduled to start in 2027.

The planned HK near detector suite will include INGRID, WAGASCI+BabyMIND, and
the upgraded ND280 inherited from T2K222An MOU related to property transfer is
under discussion now.. Additional upgrades to ND280 specific for HK are also
under consideration. A new Intermediate Water Cherenkov Detector (IWCD) is
proposed to be newly constructed $\sim$1 km from the neutrino production
target. This kiloton-scale water Cherenkov detector can be moved vertically to
measure the neutrino beam intensity and energy spectra at different off-axis
angles, yielding different true neutrino energy spectra. Precision neutrino
cross section measurements, as well as measurements of the neutrino flux and
intrinsic electron neutrino backgrounds will be performed using this suite of
near detectors.

The HK detector will be located 8 km south of the SK detector site in a cavern
with a rock overburden of 650 m. Like SK for T2K, the HK detector will sit
2.5$\degree$ off of the center of the J-PARC neutrino beam axis. The detector
tank will be 71 m in height and 68 m in diameter, with an expected fiducial
volume of 188 kt. Newly developed, high sensitivity 50 cm “Box & Line” PMTs
will provide 20% photo-coverage for the HK inner detector, while additional
photo-coverage using multi-PMT modules is also planned. Like SK, a highly-
instrumented outer detector will be used to veto incoming particles.

HK, like SK, will be sensitive to various physics processes, including nucleon
decay, solar neutrinos, supernova burst neutrinos, relic supernova neutrinos,
atmospheric neutrinos, and accelerator neutrinos. Sensitivities to the last
two are relevant to the subject of this report and are detailed here.

Long-baseline neutrinos from the J-PARC accelerator in particular give
sensitivity to the oscillation parameters $\sin^{2}\theta_{23}$, $\Delta
m^{2}_{32}$, and $\delta_{CP}$. The HK sensitivity to excluding
$\sin\delta_{CP}=0$ assuming a known mass ordering for different possible true
values $\delta_{CP}$ is shown in Fig. 15. As can be seen in this figure, the
measurement of $\delta_{CP}$ is particularly sensitive to the systematic error
on the $\nu_{e}/\bar{\nu}_{e}$ cross section ratio. Precision measurements by
the near detector suite to constrain this error are essential.

Figure 15: HK expected sensitivity to exclude $\sin\delta_{CP}=0$ plotted as a
function of the true value of $\delta_{CP}$ (top left) and the percent of true
$\delta_{CP}$ values for which $\sin\delta_{CP}=0$ can be excluded plotted as
a function of years of operation (top right). HK expected precision on
$\delta_{CP}$ plotted as a function of year (bottom left) and sensitivity to
exclude $\sin\delta_{CP}=0$ plotted as a function year in the case of true
$\delta_{CP}=-\pi/2$ and $-\pi/4$ (bottom right). Plots all compare different
systematic error assumptions and assume that the MO is known. From [87].

Atmospheric neutrinos span a wide range of energies and path lengths, unlike
the narrow-band accelerator neutrino beam, and matter effects for atmospheric
neutrinos that travel through the earth give HK sensitivity to the neutrino
mass ordering. Simultaneously fitting HK beam and atmospheric neutrinos gives
enhanced sensitivity to oscillation parameters, especially in the case that
the mass ordering is unknown, as shown in Fig. 16.

Figure 16:  The HK sensitivity to the mass ordering plotted as a function of
time for a combination of beam and atmospheric neutrinos (left) reproduced
from [27] and the HK sensitivity to exclude $\sin\delta_{CP}=0$ plotted as a
function of the true value of $\delta_{CP}$ assuming the mass ordering is
unknown (right). A combined fit of HK beam and atmospheric neutrinos
significantly enhances the HK sensitivity to $\delta_{CP}$.

Hyper-Kamiokande also has sensitivity to measure the values of $\Delta
m^{2}_{32}$ and $\sin^{2}\theta_{23}$. HK can determine the $\theta_{23}$
octant at 3$\sigma$ significance if the true value of $\sin^{2}\theta_{23}$ is
between 0.47 and 0.55 [87].

In order to reach this unprecedented precision in the oscillation parameter
measurements, systematic errors must be controlled to unprecedented levels.
Highly precise detector calibration, both before detector installation and in-
situ, is necessary. Both HK internal measurements using near detectors,
control samples, etc, and external measurements to constrain neutrino
interactions and fluxes are also essential. Precisely determining the
$\nu_{e}/\bar{\nu}_{e}$ cross section ratio will be key to a precision
measurement of $\delta_{CP}$, but other unexpected and/or incorrectly modeled
systematics could also have a major impact on the HK precision. Global effort
towards understanding these potential “unknown-unknowns” will be important.

###  3.4 South Pole: IceCube/DeepCore

####  3.4.1 Neutrino oscillation physics at IceCube

The IceCube Neutrino Observatory [88] at the South Pole instruments a cubic km
of clear glacial ice 1.5-2.5 km below the surface, using 5160 Digital Optical
Modules (DOMs) to detect Cherenkov light resulting from neutrino interactions
in the ice. The observatory detects vast numbers of GeV-TeV atmospheric
neutrinos as well as a high energy (up to PeV) flux of astrophysical origin,
with an overall neutrino detection rate of $\sim$10 mHz. A more densely
instrumented region known as DeepCore in the clear ice 2.1 km below the
surface greatly improves the detector’s low energy performance down between
5-100 GeV [89].

Figure 17: Left: $\nu_{\mu}\rightarrow\nu_{\mu}$ survival probability in the
energy range of interest for IceCube-DeepCore and the IceCube Upgrade,
including effects of Earth matter. Both the disappearance of $\nu_{\mu}$ and
the corresponding appearance of $\nu_{\tau}$ are observed by IceCube. Right:
The most recent IceCube oscillation measurement [90] and sensitivity of an
ongoing analysis, alongside results from other experiments [44, 45, 91, 62].

IceCube-DeepCore observes strong atmospheric neutrino oscillations in
DeepCore’s energy range for neutrinos crossing the Earth before detection,
primarily in the $\nu_{\mu}\rightarrow\nu_{\mu}$ and
$\nu_{\mu}\rightarrow\nu_{\tau}$ channels (Fig. 17). At higher energies
($\gtrsim$100 GeV) no oscillations are expected in the standard three flavor
paradigm as the baseline required exceeds the Earth’s diameter. With a large
range of energies, baselines, flavors and matter profiles encompassed by its
dataset, IceCube is sensitive to broad range of standard oscillations –
$\nu_{\mu}$ disappearance, $\nu_{\tau}$ appearance, and the mass ordering.

The strengths of the IceCube-DeepCore oscillation program include:

  * •

High statistics: The size of the DeepCore sub-detector (10 Mton) coupled with
the copious natural atmospheric flux affords very high detection rates,
yielding $>$200,000 neutrinos in the current all-flavor 9-year data sample;
orders of magnitude more events than current accelerator experiments.

  * •

$\nu_{\tau}$ detection: Charged-current $\nu_{\tau}$ interactions are
difficult to access at many other neutrino experiments due to the kinematic
suppression resulting from the large $\tau$ lepton mass. DeepCore is uniquely
able to detect large numbers of $\nu_{\tau}$ in the
$\nu_{\mu}\rightarrow\nu_{\tau}$ appearance channel at $\mathcal{O}$(10-20)
GeV, with $>$9,000 $\nu_{\tau,\rm{CC}}$ events expected in the current 9-year
dataset. While Deepcore’s particle identification for $\nu_{\tau}$ does not
reach the levels of e.g. DONuT [92] or OPERA [93], Deepcore’s existing data
set already far exceeds the $\sim$300 $\nu_{\tau}$ detected to date by all
other neutrino experiments combined [92, 93, 94, 95].

  * •

Complementarity: DeepCore probes comparable $L/E$ oscillations to long
baseline accelerator experiments, but at an order of magnitude higher energy
(in the deep inelastic scattering regime) and with very different systematic
uncertainties, and is thus highly complementary to other global measurements.
Moreover, the flavor composition of the astrophysical neutrino flux above 100
TeV also constrains neutrino mixing [96] (see Sec. 6), providing an additional
level of complementary.

####  3.4.2 Neutrino oscillation measurements

IceCube measures the atmospheric mass splitting, $\Delta m^{2}_{32}$, and
mixing angle, $\theta_{23}$, via the simultaneous measurement of $\nu_{\mu}$
disappearance and corresponding $\nu_{\tau}$ appearance, with a background of
largely unoscillated $\nu_{e}$. The dataset is binned in three dimensions:
reconstructed neutrino energy, cosine of the reconstructed zenith angle (a
proxy for baseline), and particle ID (PID, a proxy for flavor), with
oscillations producing 3D deformations in these dimensions that cannot be
easily mimicked by systematic uncertainties, providing robust measurements.
Neutrino properties are reconstructed using a maximum likelihood fit utilizing
tabulated expectations of the detector response to light sources at a given
location. PID utilizes boosted decision tree (BDT) ensembles to predict
whether an event resulted from a $\nu_{\mu,\rm{CC}}$ interaction, largely
based on the presence of track-like light emission from the secondary $\mu$
produced [97]. All other neutrino interaction types exhibit more spherical,
cascade-like emission profiles.

In addition to upgoing events, the data samples used for these measurements
include neutrinos from above the detector which travel only
$\mathcal{O}$(10-100 km) and thus provide unoscillated neutrino events that
can constrain systematic uncertainties related to the neutrino flux cross-
sections and detector effects, in essence somewhat like a near detector in
long baseline accelerator experiments. Overwhelming backgrounds of atmospheric
muons and random noise coincidences at trigger level are suppressed by seven
orders of magnitude via multiple layers of event selection, ultimately
resulting in $\sim$1 mHz neutrino rates in current GeV-scale oscillation
analyses, with vanishing noise and $<$1% atmospheric muon contamination.

Fig. 17 shows the most recent IceCube oscillation measurement using 8 years of
data, which uses a sub-sample (known as the verification sample) of
$\sim$23,000 predominantly track-like events with minimally scattered light,
making it extremely robust against uncertainties in modeling the optical
properties of the ice [90]. Fig. 17 also shows the expected sensitivity of a
9-year high-statistics IceCube measurement that is currently underway, which
features $>$200,000 neutrinos and improved reconstruction and background
rejection methods. The verification sample result is already approaching the
precision of the latest long baseline accelerator measurements (NO$\nu$A and
T2K), whilst the high-statistics analysis is expected to achieve precision
commensurate with current world-leading results.

IceCube also detects $\nu_{\tau}$ appearance in atmospheric neutrinos
providing an additional test of the oscillation parameters.

####  3.4.3 Next-generation oscillation physics with the IceCube Upgrade

Figure 18: Event displays of a 30 GeV $\nu_{\mu,\rm{CC}}$ event in DeepCore
(left) and the Upgrade (right). White (yellow) spheres represent optical
modules (photon hits) respectively, with the neutrino direction in red.

In 2025-26, a dense new sub-array of 7 strings featuring $\sim$700 multi-PMT
optical modules, spaced 3 m apart vertically, will be deployed in IceCube,
vastly improving the photocathode density in a 2 Mton fiducial volume within
the already dense 10 Mton DeepCore volume. Known as the IceCube Upgrade [98],
this next-generation detector has the following advantages:

Figure 19: Initial (conservative) estimate of oscillation sensitivity with the
IceCube Upgrade [98], alongside other experiments [44, 45, 91, 62]. The 2018
IceCube fit value is assumed [47].

  * •

Lower energy threshold: The denser instrumentation will allow the Upgrade to
measure atmospheric neutrinos down to $\sim$1 GeV, and more generally increase
the $<$10 GeV neutrino rate by an order of magnitude compared to DeepCore
[98]. This allows the Upgrade to probe higher order oscillation bands, and
significantly increases the fraction of detected neutrinos that have
oscillated.

  * •

Reconstruction: Denser instrumentation will also result in major improvements
in neutrino energy and direction resolution, by a factor 2-4 at the
$\mathcal{O}$(10-20) GeV energies relevant for current DeepCore oscillation
measurements (see Fig. 18 for an example of the increased light detection)
[98, 99]. The neutrino rates at these energies will also increase by a factor
$\sim$2-4, depending on fiducial volume.

  * •

Calibration: A plethora of new and densely deployed calibration devices will
dramatically improve detector and ice property calibration, also allowing re-
calibration of more than a decade of existing IceCube data.

This precision neutrino physics detector will facilitate the significant
reduction in systematic uncertainties required to fully exploit IceCube’s huge
statistical power. Initial studies of the oscillation physics potential of the
Upgrade are highly conservative, using only the 2 Mton Upgrade fiducial volume
(e.g. excluding the remaining 8 Mton of DeepCore), simplified reconstruction,
PID and background rejection methods (not yet exploiting the segmented new
optical sensors or recent advances in DeepCore analyses), and neglecting both
improvements in detector calibration and the existing 10+ years of DeepCore
data. Nonetheless, even these conservative estimates indicate that the Upgrade
can achieve $\Delta m^{2}_{23}$ precision exceeding current long baseline
accelerator results [44, 45] (see Fig. 19) and $\lesssim$6% $\nu_{\tau}$
normalization precision by the end of this decade [98].

####  3.4.4 Neutrino mass ordering

A measurement of the matter effect is required to determine the mass
orderings. For atmospheric neutrino experiments such as IceCube however,
matter effects in the Earth’s core ($\cos\theta<-0.8$ in Fig. 17) create
additional potential that provides some information.

While neutrino telescopes like IceCube cannot distinguish between $\nu$ and
$\bar{\nu}$ on an per-event level, two effects – an energy-dependent asymmetry
in atmospheric flux of $\nu$ and $\bar{\nu}$ of up to 25% [100] and a
$\nu$-nucleon cross-section that is about twice as large for $\nu$ as
$\bar{\nu}$ [101] – combine into a net signature in the $\sim
2-15\,\mathrm{GeV}$ energy range for core-crossing events that IceCube is
sensitive to.

An analysis of 3 years of IceCube-DeepCore data yields a weak sensitivity to
the mass ordering of $0.45-0.65\sigma$ in the preferred range for the mixing
angle $0.45<\sin^{2}(\theta_{23})<0.55$, with an insignificant preference for
the normal ordering [102]. In contrast, the vastly improved low energy
performance of the IceCube Upgrade dramatically increases the sensitivity in
the normal ordering, with an initial study predicting that
$3.8\sigma(1.8\sigma)$ can be reached in case of true normal (inverted) mass
ordering after 6 years of operation.

Additionally, studies are underway to evaluate the Upgrade’s
$\nu_{\mu}/\bar{\nu}_{\mu}$ separation capabilities using a combination of
reconstructed inelasticity in charged-current interactions and Michel electron
tagging, which could offer significant gains in mass ordering sensitivity.

###  3.5 KM3NeT

KM3NeT is an international research project building a series of neutrino
telescopes in the depths of the Mediterranean Sea. The main goals of KM3NeT
are twofold: measuring the neutrino mass ordering; and identifying high-energy
neutrino sources in the Universe.

The KM3NeT detectors consist of 3D arrays of detection units (DUs) each
comprising 18 digital optical modules (DOM) made of 31 photo-multiplier tubes
(PMTs). Various neutrino energy ranges can be accessed with this technology by
simply adjusting the spacing between lines and between DOMs. Two detectors are
under construction: one offshore Toulon (France) and one offshore Capo Passero
(Italy). The former is called ORCA (Oscillation Research with Cosmics in the
Abyss) and features a dense DOM geometry optimised to study neutrino
oscillations in the GeV energy range with atmospheric neutrinos. The latter is
called ARCA (Astronomy Research with Cosmics in the Abyss) and features a
sparser DOM geometry adapted to the study of high-energy neutrino sources in
the Universe.

Beyond the two main objectives mentioned above, the detectors offer a wide
range of scientific opportunities not only in fundamental physics but also in
astrophysics, earth and sea sciences. The full description of the physics
program can be found in the Letter of Intent [103].

####  3.5.1 Scientific Status and Prospects

Neutrino oscillation with the first KM3NeT/ORCA data The modular design of
KM3NeT allows physics analyses to be performed during detector construction.
An oscillation measurement using the first data collected with 6 DUs during
355 days ($\sim$0.3 Mton-year) was presented in 2021 [104].

Fig. 20 shows the distribution of events as a function of the baseline (L) to
energy (E) ratio, relative to the expectation without oscillations. A clear
depletion of events is visible as L/E increases, indicating a characteristic
oscillation pattern. The data excludes the no-oscillation hypothesis with a
confidence level of 5.9$\sigma$. The atmospheric neutrino oscillation
parameters are measured to be $\Delta{m^{2}_{31}}=1.95^{+0.24}_{-0.22}\times
10^{-3}\ \mbox{eV}^{2}$ and $\sin^{2}\theta_{23}=0.50\pm 0.10$. The allowed
90% CL region is reported in Fig. 20 and is compatible with other experiments.

Figure 20: Left: L/E distribution of the expected number of events relative to
the no-oscillation hypothesis for the data collected with the KM3NeT/ORCA
first six DUs (black) and the fit distribution (light-blue). The expected
number of events assuming the no-oscillation hypothesis (purple) and the
global best fit oscillation parameters [71] (red) are also shown for
comparison. Right: Contour at 90% CL of the oscillation parameters obtained
with the data collected with the KM3NeT/ORCA first six DUs (red). Contours of
other experiments [47, 77, 80, 105, 106] have been added for comparison
purposes as well as the global best fit value [71].

Sensitivity studies to neutrino oscillation with the complete detector Upon
completion (115 DUs), KM3NeT/ORCA will have an instrumented mass of 7 Mton. In
the following, we present results in terms of Mton-years, with typical
benchmarks 7 and 21 Mton-year (i.e. 1 and 3 years of operation of the full
detector). These results are based on a recent study [107] reporting the
sensitivity to neutrino oscillation measurements updated with respect to the
Letter of Intent [103] to include improvements in detector modeling, trigger,
reconstruction and particle identification methods.

##### Mass ordering

The sensitivity to the atmospheric mass ordering with an exposure of 21 Mton-
years is reported as a function of $\theta_{23}$ for both orderings in Fig.
21. Assuming the best estimates for $\theta_{23}$ from [71], the mass ordering
sensitivity is 4.4$\sigma$ if the true ordering is normal and 2.3$\sigma$ if
it is inverted. Fig. 21 shows the sensitivity for both orderings as a function
of data taking time with a complete detector. The mass ordering can be
determined at 3$\sigma$ level with 9 Mton-years exposure if the true ordering
is normal, and with 35 Mton-years exposure if it is inverted.

Figure 21: Left: Sensitivity to the mass ordering for an exposure of 21 Mton-
years as a function of the true $\theta_{23}$ value, for both normal (red) and
inverted (blue) orderings under three assumptions for the $\delta$ value: the
world best fit points for NO and IO reported in [71] (solid line), $None$
(dotted line) or $None$ (dashed line). The coloured shaded areas represent the
range of significance values that would occur with 68% probability according
to the Asimov approach [108]. Right: Sensitivity to the mass ordering as a
function of data taking time with a 7 Mton detector for both normal (red) and
inverted (blue) orderings and assuming the oscillation parameters reported in
[71].

##### Oscillation Parameter Measurements

We also present the expected sensitivity of KM3NeT/ORCA to the atmospheric
oscillation parameters. Fig. 22 shows the expected contours in the $\Delta
m^{2}_{32}-\sin^{2}\theta_{23}$ plane and illustrates the precision that
KM3NeT/ORCA will be able to provide with 21 Mton-years exposure compared to
the current knowledge.

Figure 22: Expected precision of $\Delta m^{2}_{32}$ and $\theta_{23}$ for
both NO (left) and IO (right) with 3 years of KM3NeT/ORCA data taking with the
complete detector (21 Mton-years) at 90% confidence level (red) overlaid with
existing results from other experiments [47, 77, 80, 105, 106] and the
oscillation parameters reported in [71] (black cross).

#####  $\nu_{\tau}$ appearance

KM3NeT/ORCA will also be able to detect $\nu_{\tau}$ appearance at $>5\sigma$
within one year of running which will also contribute to the oscillation
results.

####  3.5.2 Technical Status and Schedule

Since November 2021, ten DUs are operational at the KM3NeT/ORCA site. Before
that, a configuration with 6 DUs was operated for 22 months continuously,
between January 2020 and November 2021, corresponding to a total exposure of
approximately 0.6 Mton-years. As of February 2022, a total exposure of
approximately 0.8 Mton-years has been collected.

The experience gained in assembling and deploying the first ten KM3NeT/ORCA
DUs indicates the full ORCA detector (115 DUs) can be completed by the end of
2025, as long as no limitations are imposed due to availability of funding.
Since data taking proceeds during construction, under this schedule a total of
14.5 Mton-years would be accumulated by the time the detector is complete,
assuming a constant DU production rate and a data-taking efficiency of 95%, in
line with the performance achieved with the first 6 DUs.

At the best-fit oscillation parameters reported in [71], a 3$\sigma$
measurement would then be possible by 2026 in the NO and by 2030 in the IO. In
the NO scenario, a 5$\sigma$ determination could be achieved by 2028 if
$\theta_{23}>$$$. These projections assume the detector instrumented mass
grows linearly with the number of operational DUs.

###  3.6 Expected Sensitivity Milestones

Fig. 23 shows the current best estimate of the construction and data taking
timelines of experiments in the current program.

We estimate the year at which experiments will reach major milestones
($3\sigma$ and $5\sigma$ or the best achievable) for the three known unknowns
in neutrino oscillations in table 2. Note that the results show optimistic and
conservative results which take into account both staging (detector
construction, accelerator construction, and reactor power construction) as
well as the oscillation parameters, some of which can significantly enhance
the sensitivities.

Table 2: The year individual next-generation experiments are currently expected to reach key milestones for the three known unknowns in neutrino oscillations. Optimistic vs. conservative includes both staging effects as well as effects from the true oscillation parameters. Combined fits will be significantly more powerful than the naive sum, especially in the case of the atmospheric mass ordering. Experiments estimate their sensitivities somewhat differently. Additional time for analysis is not included. |  | Atmospheric Mass Ordering |  $\theta_{23}$ Octant |  $\sin\delta\neq 0$ for 50% of $\delta$
---|---|---|---|---
JUNO | Optimistic | 2030: $3\sigma$ | - | -
Conservative | 2030: $2.5\sigma$ | - | -
DUNE | Optimistic | 2030: $5\sigma$ | 2036: $3\sigma$, 2040: $5\sigma$ | 2035: $3\sigma$, 2039: $5\sigma$
Conservative | 2032: $5\sigma$ | 2040: $2\sigma$ | 2037: $3\sigma$
HK | Optimistic | 2033: $5\sigma$ | 2033: $5\sigma$ | 2029: $3\sigma$, 2032: $5\sigma$
Conservative | 2032: $3\sigma$ | 2034: $3\sigma$ | 2029: $3\sigma$, 2037: $5\sigma$
IceCube | Optimistic | 2030: $3\sigma$, 2033: $4\sigma$ | - | -
Conservative | 2033: $2\sigma$ | - | -
KM3NeT | Optimistic | 2026: $3\sigma$, 2029 $5\sigma$ | - | -
Conservative | 2030: $3\sigma$, 2032: $4\sigma$ | - | -

The choice of assumed true oscillation parameters can affect the sensitivity
quite a bit. The KM3NeT [107] and IceCube [109] mass ordering sensitivities
both assume the best fit oscillation parameters from [71], notably the upper
octant while we note that one of the three global fits, [4], prefers the lower
octant which would significantly reduce their sensitivity. The HK sensitivity
to the mass ordering [110] includes atmospheric and accelerator neutrinos, is
largely independent of the true mass ordering, and is better in the upper
octant than the lower. For the octant fiducial values of
$\sin^{2}\theta_{23}=0.45,0.55$ are taken; the impact of $\delta$ is small and
median values are used. For CPV improved systematics of $\nu_{e}$ cross
section uncertainties are assumed, along with the upper octant and the normal
mass ordering both of which are assumed to be known. A start date of 2027 is
assumed for HK. The DUNE sensitivity to $\delta$ [111] assumes that the
accelerator starts at 1.2 MW and only the currently funded baseline program,
while the optimistic numbers show the improvement if the full scope is
achieved after six years. The DUNE sensitivity to the mass ordering also
assumes 1.2 MW initially, that the true ordering is inverted, and is valid for
any value of $\delta$.

Finally, we want to draw special attention to the fact that combined fits,
either by the experiments or by theorists, can perform significantly better
than the naive comparison. Comparing $\nu_{e}\to\nu_{e}$ measurements with
$\nu_{\mu}\to\nu_{\mu}$ measurements provides a powerful way to probe the
atmospheric mass ordering since the dominant $\Delta m^{2}$ measured in each
differs in different directions depending on the mass ordering [112]. The
details of how, when, and which experiments are to be combined is subtle and
will depend on exactly when each experiment has reported data in the coming
years. Further, it should be noted that as precision in the knowledge of
oscillation parameters improves, searches for additional physics may likely
require measurements over a wide energy range and at the longest baselines;
this is, e.g. the approach of the DUNE experiment.

Figure 23: Assumptions about the construction timeline for future neutrino
projects.

##  4 Three Flavor Oscillation Supporting Program

###  4.1 Experiments

Auxiliary measurements, particularly hadron production measurements to
constrain neutrino fluxes and flux systematic errors, as well as precision
measurements of cross sections to constrain interaction systematic errors, are
necessary for precision three-flavor oscillation measurements. Reports from
NF06 (Neutrino Interaction Cross Sections) [1] and NF09 (Artificial Neutrino
Sources) [113] detail some of these essential measurements.

The CERN Neutrino Platform is another important supporting program towards
precision three-flavor oscillation measurements. From the CERN website: “The
CERN Neutrino Platform is CERN’s undertaking to foster and contribute to
fundamental research in neutrino physics at particle accelerators worldwide,
as recommended by the 2013 European Strategy for Particle Physics. It includes
the provision of a facility at CERN to allow the global community of neutrino
experts to develop and prototype the next generation of neutrino detectors.
The CERN Neutrino Platform is CERN’s main contribution to a globally
coordinated programme of neutrino research.”

###  4.2 Oscillation Theory

Understanding the physics behind three flavor oscillation has taken a
convoluted path with missteps on the way. The role of resonances in solar
neutrinos is now understood [114] as is the probability to jump from one
eigenstate to another [22, 115, 116]. The three-flavor oscillation probability
in sufficiently uniform matter density, relevant in the Earth for $\nu_{e}$
appearance experiments, was initially thought to be fairly intractable [16]
but has since been increasingly well understood [17, 18, 20, 19]. Nonetheless,
in order to perform statistically robust estimates of confidence intervals
[117] a very large number of pseudo-experiments must be thrown and calculating
the oscillation probabilities for each set of the oscillation parameters is
often the limiting factor. To this end, some initial studies of the
computational speed of different expressions has been carried out in the
context of long-baseline accelerator neutrino experiments [20]. Preliminary
implementations suggest a speed up of 3-10 is realistic, suggesting hat there
may be more progress possible. Atmospheric and nighttime solar neutrinos are
even more computationally expensive due to both the range of zenith angles
(baselines) as well as the shell structure of the Earth. As HK, DUNE, IceCube,
and KM3NeT look to considerably increase the precision of these channels,
additional improvements in computational methods would also assist these
calculations.

###  4.3 Joint Fits

Joint fits of data from different experiments give enhanced sensitvities to
oscillation parameters by breaking degeneracies. A clear example can be found
in Fig. 25, where a combination of JUNO and IceCube data under a joint
oscillation parameter hypothesis, beyond a simple statistical sum, gives a
clear enhancement to the mass ordering sensitivity.

When experiments use similar neutrino sources, energy scales, detectors, etc.,
it is extremely important to correctly take into account correlations between
systematic errors. Especially, incorrect/inconsistent treatment of the
neutrino interaction model has a significant impact on resulting joint fits.

Producing joint fits in a timely manner is important for quickly realizing
measurements of various oscillation parameters. Even in the case of highly
different energy scales, clarity in how data inputs are used and how fits are
performed is important.

Several currently running experiments are working on dedicated joint fits
including consistent treatment of inputs and correct treatment of correlations
between systematics. In other cases, joint fits by phenomenologists are
proposed.

####  4.3.1 T2K + SK

An MOU between the T2K and SK collaborations has been established and a joint
working group has been formed to produce joint fits of T2K long-baseline and
SK atmospheric neutrino oscillation data. These joint fits use a unified set
of inputs and properly take into account correlations in systematics between
the two experiments. In particular, significant correlations between
experiments are expected from the neutrino interaction model, and these
correlations are especially important given that the two experiments use the
same detector.

With a joint fit, sensitivity to oscillation parameters, in particular the
mass ordering, $\theta_{23}$, and $\delta_{CP}$, is enhanced beyond the naive
sum of $\chi^{2}$ values published by the two experiments separately. First
joint fit results are expected to be released publicly in 2022.

####  4.3.2 T2K + NOvA

An MOU between the T2K and NOvA collaborations has been established and a
joint working group has been formed to produce joint fits of T2K and NOvA
oscillation data. Combinations of experimental data from different baselines
and neutrino energies resolves degeneracies in oscillation parameters in a
complementary way, which provides improved sensitivity to the physics and/or
physics reach outside the parameter space accessible by each experiment. In
particular, a joint fit of T2K and NOvA is sensitive to the (unknown) mass
ordering, $\theta_{23}$, $\theta_{13}$, and $\delta_{CP}$. As significant
correlations between experiments are expected from the neutrino interaction
model, direct collaboration between the experiments is necessary.

####  4.3.3 JUNO + NOvA + T2K

A joint fit of JUNO + NOvA + T2K data could provide a first $\geq 5\sigma$
measurement of the mass ordering, as proposed by phenomenologists in [118].

####  4.3.4 DUNE + HK

There is interest in exploiting the complementarity of the HK and DUNE
experiments. This complementarity is similar to the one of the T2K and the
NOvA experiments: different baselines, beam energies (narrow-band vs. wide-
band), detector technology, detection mechanism and detector size. A joint fit
of the data from both experiments including common inputs and correct
correlations would be beneficial following several years of data-taking by
both experiments. Setting up analysis frameworks with this in mind now would
greatly benefit the future generation of analyzers. It is imperative that the
lessons learned in the NOvA+T2K working group extends to each experiment’s
successor.

####  4.3.5 JUNO + ORCA

The sensitivity of a combined analysis of the JUNO and KM3NeT/ORCA experiments
to determine the neutrino mass ordering was performed [119]. This combination
is particularly interesting as it significantly boosts the potential of either
detector, beyond simply adding their neutrino mass ordering sensitivities, by
removing a degeneracy in the determination of $\Delta m_{31}^{2}$ between the
two experiments when assuming the wrong ordering. A $5\sigma$ determination of
the neutrino mass ordering is expected after 6 years of joint data taking for
any value of the oscillation parameters. As shown in Fig. 24, this sensitivity
would be achieved after only 2 years of joint data taking assuming the current
global best-fit values for those parameters for normal ordering.

Figure 24: NMO sensitivity as a function of time for only JUNO (red), only
KM3NeT/ORCA (blue), and the combination of JUNO and KM3NeT/ORCA (green),
considering 2 (solid) or 4 (dashed) Taishan NPP reactors, corresponding
respectively to 8 or 10 reactor cores at 53 km from JUNO for NO (left) and IO
(right).

####  4.3.6 IceCube/DeepCore + JUNO

Figure 25: The sensitivity to the mass ordering for the IceCube Upgrade and
JUNO, both individually and in a combined analysis [109].

Similarly to ORCA + JUNO, IceCube/DeepCore data can also be advantageously
combined with JUNO. This can be seen in the mass ordering analyses with both
the IceCube Upgrade and with future JUNO data [120], which is sensitive to the
mass ordering via sub-leading survival probability terms arising from the
difference between $\Delta m^{2}_{31}$ and $\Delta m^{2}_{32}$ for
$\bar{\nu}_{e}$ [121]. When analyzing IceCube Upgrade data in combination with
JUNO data under a joint oscillation parameter hypothesis, the sensitivity to
the mass ordering improves significantly beyond the pure statistical
combination of independent measurements [109], primarily due to the
correlation of $\Delta m^{2}_{31}$ with the mass ordering hypothesis in both
experiments. Such a combined analysis has an expected $6.5\sigma(5.1\sigma)$
mass ordering sensitivity for the normal (inverted) ordering after 6 years of
data taking of each fully constructed and powered experiment, giving the
possibility of a conclusive determination of the mass ordering in the early
2030s shown in Fig. 25. Similar results would be expected for the combination
of any measurements of $\Delta m^{2}_{31}$ from $\nu_{e}$ disappearance and
$\nu_{\mu}$ disappearance.

##  5 Possible Upgrades to Planned Experiments

###  5.1 DUNE

####  5.1.1 Realizing the P5 DUNE

DUNE Phase I consists of two far detector modules, totaling 35 kton, a beam
capable of 1.2 MW of proton delivery, and a suite of near detectors adequate
for the initial physics goals of resolving the mass ordering and beginning the
search for CP violation with 3$\sigma$ sensitivity. To fully realize the
neutrino program imagined by the 2014 P5 panel, DUNE Phase II will upgrade
these three pieces of the experiment [122]. Two additional modules will be
added to the far detector, the beam intensity will be upgraded to 2.4 MW, and
the near detector will be augmented to gain additional precision in neutrino
cross-section measurements with limit the neutrino oscillation measurements.
As illustrated in Fig. 26, each of these three improvements are required to
achieve 5$\sigma$ sensitivity to CP violation and to realize the full
oscillation measurement program.

Figure 26:  The significance in units of $\sigma$ with which the DUNE
experiment could observe CP violation for at least 50% of the possible
$\delta_{\rm CP}$ values as a function of running time. In each of the three
panels the lower green curve shows the evolution of the DUNE Phase-I
experiment and the upper red curve shows the evolution for the DUNE Phase-II
experiment. The intermediate cyan curves forego one of the three pieces of the
Phase-II program; modules 3 and 4 at the far detector on the left, the 2.4 MW
beam power upgrade in the center, and the near detector upgrades on the right.

####  5.1.2 Near Detector

The Phase-I DUNE near detector will have a liquid argon TPC system with a muon
spectrometer, ND-LAr + TMS. These sub-detectors will move off axis as part of
PRISM to constrain flux and cross-section uncertainties. These components are
sufficient for the DUNE Phase-I goals to resolve the neutrino mass hierarchy
and to reach 3$\sigma$ sensitivity to CP violation. However, to reach
5$\sigma$ sensitivity across a wide range of $\delta$ values, improved
knowledge of neutrino-Ar interactions will be required. In DUNE Phase-II, TMS
is planned to be replaced with a magnetized high-pressure gaseous argon TPC.
This detector, “ND-GAr” [69], will enable a precise measurement of the momenta
of muons produces in the ND-LAr and will additionally allow for high
resolution reconstruction of the vertex activity produced by neutrino
interactions on the low-density gaseous argon target. The improvement in
neutrino cross-section knowledge will enable the search for CP violation to be
extended to 5$\sigma$.

####  5.1.3 Third and Fourth Modules

During DUNE Phase-I two drifts sized to house the four DUNE liquid argon
modules will be excavated and two modules will be installed; one per drift
which allows for outfitting, installation, and operations to proceed in
parallel. DUNE requires two additional detector modules to realize 5$\sigma$
sensitivity to CP violation, to search for physics beyond the standard model
in neutrino oscillations, and to fully realize the non-accelerator physics
measurement program of solar neutrinos, proton decay, and atmospheric
neutrinos. Modules three and four will need to be at least as capable of
executing the DUNE accelerator physics program as modules one and two but need
not be identical to these initial modules and could implement improved
technologies or modifications to the detector design targeted at specific
topics in the DUNE physics program. Possibilities include several ideas for
novel TPC readout technologies [123, 124, 125, 126], introduction of Xe for
neutrinoless double beta decay [127], and deployment of liquid scintillator
[128]. The DUNE collaboration has hosted one workshop focused on these ideas
in 2019 [66] and another is planned in November 2022.

####  5.1.4 Accelerator Upgrades

The Main Injector at Fermilab recently (May 2022) set a new record power of
0.9 MW of protons delivered to the NOvA experiment at 120 GeV. The PIP-II
upgrades [129] to the Fermilab proton source are now underway and when
completed later this decade will enable delivery of 1.2 MW for the DUNE
experiment. Beam delivery beyond 1.2 MW is limited by the Booster ring which
is entering its 50th year of service. Fermilab is investigating two options
[130] for future investments to realize the 2.4 MW beam power envisioned by
the 2014 P5 for DUNE. Both options increase the PIP-II linac energy to 2 GeV
and call for the construction of a new rapid-cycling synchrotron. The “Initial
Configuration Document 2”, ICD-2 [131], option is likely the most cost
effective route to 2.4 MW power for the neutrino program. The “Booster Science
Replacement”, BSR [132], could deliver of 2.4 MW for DUNE while also
increasing the power available to experiments at 8 GeV to 750 kW over the 170
kW possible with ICD-2.

###  5.2 Hyper-K

####  5.2.1 HK Near Detector

Upgrades to the T2K near detector ND280 are underway now, as discussed in Sec.
3.3.2. Measurements made by the upgraded detector will be used to understand
the detector performance in the coming few years. Further detector upgrades or
configuration changes towards dedicated precision measurements for the HK
experiment are now under consideration, and near-future “ND280 Upgrade”
measurements will inform design decisions for a future HK “ND280++” project.

####  5.2.2 HK-Gd

As discussed in Sec. 3.3.1, the SK detector has recently been upgraded to SK-
Gd, giving SK enhanced access to various new physics processes. Similarly, Gd
doping of the HK detector is under consideration as a possible upgrade of the
HK detector.

The main motivator towards HK-Gd is a significant enhancement in the
sensitivity to Diffuse Supernova Neutrino Backgrounds (DSNB), since HK-Gd will
be able to significantly reduce spallation backgrounds at $E<20$ MeV, measure
the DSNB spectrum at $E>10$ MeV with 100s of events, and study the supernova
rate down to $Z\sim 1$, see e.g. [133] for a recent DSNB review. HK-Gd could
also offer direct benefits to the HK atmospheric and long-baseline
measurements by allowing for enhanced neutrino-antineutrino separation and
improving the detector energy reconstruction via neutron counting. These
improvements can directly impact the HK precision of three-flavor
oscillations.

####  5.2.3 Detector in Korea

A second HK tank located at the Mt. Bisul site in Korea, $\sim$1100 km away
and 1.3$\degree$ off-axis from the J-PARC neutrino beam source, is proposed to
serve as a far detector for T2HKK (Tokai to Hyper-K Korea). This second tank
would enhance HK sensitivities by doubling the detector volume, but would also
give access to the second oscillation maximum for long-baseline neutrinos,
allowing for significantly enhanced sensitivity to oscillation parameters
beyond a simple increase in statistics [134]. Differences in tank geometry,
off-axis angle, and overburden between the detectors in Japan and Korea would
further break degeneracies and enhance sensitivities when fitting data from
both detectors.

Design of the Korea Neutrino Observatory (KNO), which will serve as the T2HKK
far detector tank, is now underway.

####  5.2.4 J-PARC Accelerator Upgrades

Currently planned upgrades to the J-PARC MR are aiming to achieve 1.3 MW at
this point, but further future upgrades beyond that could achieve higher beam
powers. For example, installation of a new 8 GeV booster ring at J-PARC for
injection into the MR could allow the MR beam power to be increased beyond 3.2
MW.

Several main components of the J-PARC neutrino beamline infrastructure
(secondary beamline shielding, decay volume, and hadron absorber) are no
longer human accessible due to irradiation, and will never be accessed again.
Those components were all designed to be able to accept a 3$\sim$4 MW proton
beam from the J-PARC MR.

Construction of a new Continuous Wave (CW), 10 MW beam has also been
considered at the KEK site in Tsukuba, Ibaraki prefecture, Japan. This new
accelerator would use a series of 4 CW linear accelerators installed in the
tunnel currently in use for KEKB. One major challenge for this design would be
development of a CW horn to focus secondary particles produced by a CW beam.
Another challenge is that neutrino beamline and near detector infrastructure
do not exist at the KEK site now, meaning that construction of a new neutrino
beamline and near detector suite would be necessary.

##  6 Other Probes of the Oscillation Parameters

Beyond the traditional probes there are various other ways to learn about the
oscillation parameters in astrophysical and nuclear environments.

###  6.1 Galactic Supernova

When the next supernova goes off in our galaxy, a wealth of information will
be immediately acquired about astrophysics, astroparticle physics, and
particle physics. Since the mass ordering affects the flavor transitions in
the supernova, a measurement of the neutrino signal will provide crucial
evidence about the mass ordering; either as the first measurement or as a
confirmation of terrestrial results. For a typical supernova distance of $\sim
10$ kpc one can expect that one of the current or upcoming experiments will
have between $2-6\sigma$ sensitivity to the mass ordering subject to possibly
additional systematic uncertainties [135]. Note that the dependence of the
diffuse supernova neutrino background on the mass ordering is likely too small
to measure even with many years of HK [136].

###  6.2 Astrophysical Neutrinos

IceCube has detected high energy astrophysical neutrinos [137] from beyond our
galaxy [138, 139]. These neutrinos are expected to be some combination of
$\nu_{e}$ and $\nu_{\mu}$ at the source and by the time they reach the Earth
they have decohered with probabilities
$P_{\alpha\beta}=\sum_{i=1}^{3}|U_{\alpha i}|^{2}|U_{\beta i}|^{2}$, thus
measurements of the flavor ratio detected at the Earth provides information
about the mixing matrix. Existing constraints are not sensitive [140, 141] but
it is conceivable that with IceCube’s upgrade they could provide some
information on the mixing parameters [142].

###  6.3 Absolute Mass-Scale Based Probes

There are various other means of probing the six oscillation parameters. In
particular, measurements of the absolute mass scale can provide information
about the mass ordering in some cases.

  * •

Kinematic end-point experiments such as KATRIN, ECHo, HOLMES, and Project-8
[143, 144, 145, 146] are sensitive to $\sqrt{\sum_{i}|U_{ei}|^{2}m_{i}^{2}}$
which is $\gtrsim 10$ meV in the NO and $\gtrsim 50$ meV in the IO, although
these experiments may not have sensitivity to the mass ordering before
oscillation experiments do.

  * •

Cosmological measurements of the cosmic microwave background temperature and
polarization information, baryon acoustic oscillations, and local distance
ladder measurements lead to an estimate that $\sum_{i}m_{i}<90$ meV at 90% CL
which mildly disfavors the inverted ordering over the normal ordering [147]
since $\sum_{i}m_{i}\gtrsim 60$ meV in the NO and $\gtrsim 110$ meV in the IO;
although these results depend on one’s choice of prior of the absolute
neutrino mass scale [135, 148]. Significant improvements are expected to reach
the $\sigma(\sum m_{\nu})\sim 0.04$ eV level with upcoming data from DESI and
VRO, see the CF7 report [149], which should be sufficient to test the results
of local oscillation data in the early universe at high significance,
depending on the true values.

  * •

If lepton number is violated via an effective operator related to neutrino
mass, then we expect neutrino-less double beta decay to occur proportional to
$m_{\beta\beta}=|\sum_{i}U_{ei}^{2}m_{i}|$ which could be as low as zero in
the NO but is expected to be $\gtrsim 1$ meV in the IO, thus a detection below
1 meV would imply the mass ordering is normal. The latest data from KamLAND-
Zen disfavors some fraction of the inverted hierarchy for favorable nuclear
matrix element calculations which are fairly uncertain [150].

  * •

Finally, a measurement of the cosmic neutrino background is sensitive, in
principle, to a combination of the absolute mass scale, whether neutrinos are
Majorana or Dirac, and the mass ordering [151, 152].

Among these non-oscillation measurements, only the cosmological sum of the
neutrino masses is likely to be sensitive to the atmospheric mass ordering
within the next decade.

##  7 Possible Future Experiments

###  7.1 DUNE to THEIA

THEIA is a proposal to build a large mass detector ($\gg 10$ kilotons) using
water-based liquid scintillator. Although it is not strictly an upgrade to the
DUNE experiment, if it were built at SURF as a fourth DUNE module, THEIA could
add to the long baseline mission of DUNE. Scintillator has enhanced detection
efficiency at MeV energies and THEIA would primarily be focused on the
measurement of solar neutrinos and supernova neutrinos. However, using a
combination of scintillation light and Cherenkov light, event reconstruction
of GeV neutrino interactions is possible. A 17-kt THEIA module would have
$>5\sigma$ sensitivity to the neutrino mass ordering after seven years of
running at SURF, and could observe CP violation at $>2\sigma$ for 50% of
$\delta_{\rm CP}$ values after seven years of running [128].

###  7.2 ESSnuSB

The European Spallation Source neutrino Super-Beam (ESSnuSB) is a proposed
long-baseline accelerator neutrino experiment in Sweden designed for measuring
CP violation using the existing ESS facility [153, 154, 155]. Two proposed
baselines of 360 km and 540 km are being considered for a neutrino beam peaked
at $\sim 0.3$ GeV targeting the second oscillation maximum. The experiment
imagines a later water Cherenkov far detector similar to HyperK or the MEMPHYS
project [156] with an assumed detector mass of 538 kt, roughly the size of two
HyperK tanks, and with 40% PMT coverage. The accelerator facility, under
construction for other uses, could be upgraded to 5 MW with $2.7\times
10^{23}$ POT/year with a proton energy of 2.5 GeV. With a preliminary
systematics analysis, ESSnuSB can achieve 5 $\sigma$ discovery of CPV for 75%
of $\delta$ values in about eight years and can make a high precision
measurement of $\delta$ as well.

###  7.3 ICAL at INO

The ICAL detector at the India-based Neutrino Observatory (INO) is a proposed
magnetized iron calorimeter to measure atmospheric neutrinos [157]. The
detector would be 50 kt and instrument with resistive plate chambers. The
detector would also be inside a 1.5 T field allowing for charge identification
of muons produced in atmospheric $\nu_{\mu}$ CC interactions. This would allow
for a measurement of the mass ordering at 3 $\sigma$ with 10 years go data. An
85 t mini-ICAL was commissioned in 2018 including a muon veto. In addition to
the mass ordering, ICAL at INO is also well suited for various new physics
studies such as Lorentz invariance violation [158].

###  7.4 Neutrinos from muons

A neutrino factory is a class of proposed experiments using neutrinos from
muon storage rings, see e.g. [159]. While the experimental challenges are
considerable, they build on an effort within the energy frontier to study
electroweak physics with a muon collider [160, 161] and could be a natural
intermediate step on the way. The advantages of such a machine are extremely
high neutrino intensities with an extremely well characterized beam coming
from muon decay which contains equal numbers of muon neutrinos and electron
neutrinos providing a unique flavor combination in the flux allowing for
measurements of up to six out of the nine oscillation channels. Such a
proposal would likely lead to the highest precision measurement of the three-
flavor oscillation parameters as well numerous additional oscillation and BSM
searches, although the details depend strongly on many design choices which
have to be made. See also nuSTORM which uses as similar approach and would
assist the accelerator oscillation program by measuring neutrino cross
sections and possibly provide a first step towards a neutrino factory [162].

In China, the MuOn-decay MEdium-baseline NeuTrino beam facility (MOMENT) is
another proposed high intensity neutrino beam [163, 164]. A 15 MW 1.5 GeV
proton beam would lead to a beam of neutrinos from muon decays of 200-300 MeV
targeting a baseline of 150 km. The far detector could be Gd-doped water
Cherenkov or other designs. Such an experiment is expected to be sensitive to
$\delta$ at the level of 8∘-18∘ in 10 years [165].

###  7.5 Protvino to Orca: P2O

Studies were made on the possibilities to use KM3NeT/ORCA as far detector for
a long baseline neutrino experiment with a neutrino beam made at the U70
accelerator complex in Protvino (Russia). It has been shown that P2O offers
sensitivity to $\delta$ comparable to or beyond that of DUNE or HK [166, 167,
168], reaching about $\sigma(\delta)\sim 6^{\circ}-8^{\circ}$ for all values
of $\delta$ including tagging of the initial state given 10 years of 450 kW
beam.

At modest beam power, it appears that the technology being developed for the
HL-LHC would be sufficient to instrument the beam line with silicon trackers.
Those trackers would allow to reconstruct the kinematics of each and all
neutrinos produced by the $\pi^{+}\to\mu^{+}\nu_{\mu}$ decays with energy
resolutions better than 1%. Using time and angular coincidence each neutrino
observed in KM3NeT/ORCA could be individually matched with the one
reconstructed kinematically. Such an association would remove most of the
systematic uncertainties and would allow to reach energy resolutions
inaccessible with conventional neutrino detectors.

##  8 Conclusions

Since the discovery of neutrino oscillations almost 25 years ago, there has
been a massive global effort to measure the six oscillation parameters. This
effort has been extremely successful. Since then, three of the parameters have
become fairly well measured: $\theta_{13}$, $\theta_{12}$, and $\Delta
m^{2}_{21}$, we have partial information about two parameters: $\theta_{23}$
and $\Delta m^{2}_{31}$, and we are just starting to be sensitive to the final
parameter: $\delta$. As the next generation experiments aim to measure the
final parameters, we expect the three-flavor oscillation scenario to either
fall into stark relief, or holes will emerge pointing us towards new physics.

With the support of the last P5 report, the US is becoming a world leader in
measuring and understanding these elusive particles. In particular, the report
set the stage to build the Deep Underground Neutrino Experiment (DUNE) which
will be the most sophisticated neutrino oscillation experiment ever
constructed. This will build on the US’s success of long-baseline accelerator
experiments MINOS and NOvA and the LArTPC detectors of MicroBooNE and the rest
of the short-baseline neutrino program. DUNE will provide competitive
measurements of all six oscillation parameters and will likely be world
leading in several of them.

In Japan, building of the success of T2K and SuperK, HyperK will be built
which will also have sensitivity to the majority of the oscillation
parameters. In China, building on the success of Daya Bay, JUNO will have
excellent sensitivity to the solar oscillation parameters and some sensitivity
to the atmospheric mass ordering. At the south pole and in the Mediterranean
IceCube and KM3NeT respectively will have very good sensitivity to the
atmospheric oscillation parameters. In addition, oscillation parameters can be
somewhat probed in astrophysical environments in very orthogonal ways.
Finally, there exist several ideas for neutrino oscillation experiments beyond
even those under construction now to further refine and improve our
understanding neutrino physics.

In addition to the impressive strengths of individual experiments, the
combined fit of several or all of them will be extraordinarily powerful. Such
combined fits and joint fits as well as the initial experimental analyses are
crucial to support to ensure a timely presentation of results.

If the proposed experiments are fully constructed in a timely fashion, the
atmospheric mass ordering will be robustly tested in multiple different
oscillation scenarios. We will also have excellent sensitivity to determining
the octant of $\theta_{23}$ values of $\theta_{23}$ fairly close to maximal.
Finally, we will have the capability to make two truly independent
measurements of $\delta$ and, depending on the true parameters, discover CP
violation at high significance $>5$ $\sigma$.

In order to fully realize the goals of understanding the neutrino sector, the
community needs to continue the development and construction of existing
experiments. This includes high power accelerator facilities, large state-of-
the-art far detectors, and comprehensive near detectors for the long-baseline
experiments. To augment these challenging measurements, smaller experiments to
improve our understanding of detector technologies and neutrino interactions
will also be necessary. These experiments will also need the personnel to
carry out the analyses in a timely fashion to ensure that the data collected
by the machines is properly analyzed. Finally, as this program is measuring
unknown parameters of nature which are extremely difficult to carry out and
sensitive to involved systematics including neutrino cross sections, two or
more distinct measurements of the same quantities are required to have a firm
understanding of the oscillation parameters.

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