The KPIPE detector was originally envisioned to be paired with the 3 GeV, 730 kW (currently, with 1 MW planned) J-PARC Spallation Neutron Source. Aside from the primary proton energy, which is above the kaon production threshold, and the high power, this source is particularly attractive because the beam timing structure, two $\sim$80 ns pulses separated by 540 ns at 25 Hz, provides an extremely low duty factor ($4\times 10^{-6}$), essential for cosmic background rejection. The drawback of this source, however, is that the 3 GeV primary proton energy, while above threshold, is somewhat lower than optimal for charged kaon production per unit power: at 3 GeV, the MARS15 software package <cit.> predicts 0.007 KDAR $\nu_\mu$/POT. With an increase in proton energy to 8 GeV, for example, the production rate increases by a factor of 10 to 0.07 KDAR $\nu_\mu$/POT. Spatial and facility issues, especially in consideration of the existing materials-science-focused beamlines and experiments, also means that optimal detector placement, with KPIPE calling for a 120 m long detector with closest distance of 32 m from the neutrino source, is challenging.

The future Fermilab particle accelerator complex <cit.>, including PIP-II <cit.> and the RCS upgrade <cit.>, can provide an optimal beam-dump/stopped-kaon neutrino source for KPIPE, in terms of beam energy (8 GeV), beam timing ($\sim10^{-5}$ duty factor), and spatial considerations. Using the detector and Fermilab-accelerator assumptions shown in Table <ref>, and scaling based on the detailed study in Ref. <cit.>, we expect KPIPE could achieve the sensitivity to short-baseline $\nu_\mu$ disappearance shown in Figure <ref>. As can be seen, this sensitivity surpasses, and is highly complementary to, SBN (6 years) at $\Delta m^2>10$ eV$^2$ for both scenarios considered and $\Delta m^2>1$ eV$^2$ for the RCS upgrade era case.

Experimental assumptions
Detector length      120 m

Active detector radius     1.45 m

Closest distance to source      32 m

Liquid scintillator density      0.863 g/cm$^3$

Active detector mass     684 tons

Primary proton energy     8 GeV

Target material     Hg or W

KDAR $\nu_\mu$ yield (MARS15)       0.07 $\nu_\mu$/POT

$\nu_\mu$ CC $\sigma$ @ 236 MeV (NuWro)       $1.3\times10^{-39}~\mathrm{cm}^2/\mathrm{neutron}$

KDAR signal efficiency        77%

Vertex resolution        80 cm

Light yield     4500 photons/MeV

Uptime (5 years)     5000 hours/year

$\nu_{\mu}$ creation point uncertainty        25 cm
PIP-II era assumptions
Proton rate (0.08 MW)     1.0 $\times 10^{21}$ POT/year

Beam duty factor     $1.6\times10^{-5}$

Cosmic ray background rate        110 Hz

Raw KDAR CC event rate      $2.7\times10^4$ events/year
RCS upgrade era assumptions
Proton rate (1.2 MW)     1.5 $\times 10^{22}$ POT/year

Beam duty factor     $5.3\times10^{-5}$

Cosmic ray background rate       360 Hz

Raw KDAR CC event rate      $4.0\times10^5$ events/year
Summary of the relevant KPIPE experimental parameter assumptions.

The 90% CL sensitivities of the KPIPE at Fermilab scenarios considered here, in both the PIP-II and RCS upgrade eras. For reference, we also show the expected 90% CL SBN sensitivity (6 years) <cit.>, existing 90% CL MiniBooNE+SciBooNE limit <cit.>, and 99% allowed region from the Collin et al. global fit <cit.>.

§.§.§ IsoDAR

Through tracing $\bar \nu_e$ disappearance continuously across $L/E$ from 1 to $\sim 10$ m/MeV, the IsoDAR (Isotope Decay At Rest) experiment uniquely addresses the fundamental question raised by this white paper:  What, if any, new physics phenomenology underlies the short-baseline anomalies?  Despite the enormous consequences if new physics is the cause, the question has been unanswered for more than 20 years.  Incremental improvements on our present approaches are likely to yield more of the same confusing results.  IsoDAR represents an entirely new approach–the experiment makes use of a flux from $^8$Li $\beta$ decay, produced through a 60 MeV proton beam that is
targeted on $^9$Be to yield neutrons that enter
a surrounding  isotopically-pure $^7$Li sleeve and capture.  When this source is paired with the 2.3 kton Yemilab liquid scintillator detector (LSC), approximately 1.6 million inverse beta decay (IBD) events can be reconstructed in 5 years of running.  The high statistics, relatively high energy, $E$, of $\bar \nu_e$ from $^8$Li decay, and the ideal matching baseline, $L$, due to the size of the LSC,
gives unprecedented capability to study the $L/E$ dependence of short-baseline disappearance in an agnostic manner, determining
its cause
without design assumptions that bias toward specific underlying physics models.  Fig. <ref> illustrates the power of IsoDAR to resolve various popular proposals for the source of the effect, with a 3+1 model at top left;  3+1 with nonzero quantum mechanical wave packet effects at top right;  introduction of additional sterile neutrinos, in this case 3+2 at bottom left; and introduction of new interactions, in this case 3+1+decay, at bottom right.    These examples show that IsoDAR can clearly elucidate the underlying oscillation-related phenomenology of the electron-flavor short-baseline anomalies, even in the case of physics that produces very complex waveforms.

The IsoDAR@Yemilab capability to trace $\bar \nu_e$ disappearance versus $L/E$ for IBD interactions.  Top left
and right present a 3+1 example without and with wavepacket effects described in Ref. <cit.>.
Bottom left and right are 3+2 and 3+1+decay models for the global best fit points in Ref. <cit.>.  Orange is the true underlying model.   Points represent the measurement capability.  See text for further discussion.

IsoDAR has received preliminary approval to run at Yemilab in the configuration shown in Fig. <ref>.
Ref. <cit.> provides an overview  of the technology and installation-plan.  In Fig. <ref>, the cyclotron that drives the flux production is located at the far right.  This

novel 5 mA
H$_2^+$ ion accelerator, producing 10 mA of 60 MeV protons, yields an order of magnitude higher proton beam current than on-market cyclotrons at similar energies.   Since the 2013 Snowmass Study, cyclotron development has culminated in the design described in Ref. <cit.>, which presents start-to-end simulations and prototypes of components now under test <cit.>.   As seen in Fig. <ref>, the proton beam is transported from right to left and then bent through 180$^\circ$ to the target surrounded by the sleeve, hence fast neutrons are directed away from the LSC detector, shown in green.  Substantial R&D and engineering have established successful target, sleeve and shielding designs <cit.>.

Layout of IsoDAR@Yemilab in the Yemilab caverns.   The excavation of the IsoDAR cavern complex is complete.

IsoDAR@Yemilab is designed to address issues that have arisen during studies of reactor and MegaCurie source experiments.  The $\bar \nu_e$ flux is generated from a single, well understood isotope, avoiding issues faced by reactor flux modeling.   The $\bar \nu_e$ energy range is from about 3 to 13 MeV, with peak at $\sim 6$ MeV, well beyond environmental backgrounds and backgrounds from neutron capture.   The source creation region is compact ($\sim$41 cm at 1$\sigma$) and isotropic.  The experiment has the capability of event-by-event reconstruction, with prompt (e$^+$) energy resolution of 2.3% and vertex resolution of 4 cm at 8 MeV <cit.>, in contrast to MegaCurie (MCi) source experiments that do not reconstruct event kinematics. Use of the well-known IBD cross section is also an advantage over the gallium experiments.  Lastly, and  importantly, the size of the LSC detector, when combined with the energy range of the source, leads to the wide $L/E$ range, allowing precision reconstruction of the oscillation wave across many cycles.

A simple 3+1 model is traditionally used for cross-comparison of experimental reach. The reach of IsoDAR in $\sin^2 2\theta_{ee}$ at 95% CL is presented in Tab. <ref>, column 2 for a range of $\Delta m^2$.   For comparison, column 3 shows the combined limits at 95% CL from Prospect, RENO and Daya Bay <cit.>. The IsoDAR mixing angle reach is $\times 4$ ($\times 35$) that of the reactor limits at 1 (8) eV$^2$. For allowed region comparisons for Neutrino-4 <cit.> and the Gallium experiments <cit.>, we use the $2\sigma$ lower edge in $\sin^2 2\theta_{ee}$.   In the Neutrino-4 case, the allowed region is narrow in $\Delta m^2$ and does not coincide with 8 eV$^2$, so we present the mixing angle reach for the best fit mass splitting of 7.3 eV$^2$.

$\Delta m^2$     IsoDAR@Yemilab     Combined Reactor <cit.>     Neutrino-4 <cit.>      Gallium <cit.>

Sensitivity     Limits     Allowed     Allowed
1 eV$^2$     0.004     0.016     N/A     0.28

2 eV$^2$     0.004     0.07     N/A      N/A

4 eV$^2$     0.005     0.13     N/A     0.27

8 (7.3) eV$^2$     0.008      0.28     (0.12)     0.28

Quantitative comparison of the low $\sin^22\theta_{ee}$ 2$\sigma$ reach of electron-flavor experiments in the $\Delta m^2$ range of interest.   IsoDAR sensitivity is based on assumptions in Ref. <cit.>.   Combined reactor limits are from PROSPECT, NEOS, and Daya Bay. N/A indicated $\Delta m^2$ is not within 95% CL allowed region. The Neutrino-4 2$\sigma$ reach is quoted at 7.3 eV$^2$.

As a result of the novel design, IsoDAR@Yemilab design is able to elucidate $\bar \nu_e$ disappearance across $L/E$ of 1 to $\sim$10 m/MeV without guidance from any phenomenological model.    Fig. <ref> illustrates the complex oscillation waves that are able to be differentiated in 5 years of running, with efficiency included.   The upper plots illustrate the oscillation wave without (left) and with (right) wavepacket effects as discussed in Ref. <cit.>, at a point where the combined reactor limit and gallium allowed region overlap assuming the wavepacket model. Comparison of the two plots shows that the distinctive damping due to wavepacket effects can be observed given IsoDAR's high statistics.
The lower plots present 3+2 and 3+1+decay models evaluated at the best fit points from Ref. <cit.>. The value of the high statistics and excellent reconstruction of IsoDAR is particularly emphasized by the 3+2 case, where the second predicted modulation is clear due to the capability of using very fine binning.   The orange line indicates the true underlying distribution, while the points with error bars present the expected measurements, illustrating the loss of information from finite statistics and bin sizes.   The 3+1 (top) and 3+2 points (bottom right) also include detector energy and position smearing.

The experiment will also collect $\times 4$ the world's sample $\bar \nu_e$-electron elastic scattering events in 5 years which may help further decipher new physics, depending on the source.   In fact, IsoDAR@Yemilab has an extensive discovery-level physics program beyond searching for the short-baseline anomalies, including an order of magnitude improvement in NSI searches through elastic scattering from electrons <cit.>, unique neutrino-based searches for Z$^\prime$ signatures <cit.>, and exotic non-neutrino searches, such as for neutrons shining through walls <cit.>.   As such, IsoDAR represents a leap forward for electron-flavor neutrino experiments.

§.§ Decay-in-Flight Accelerator Experiments

§.§.§ Short-Baseline Experiments

The Fermilab SBN Program

The Short-Baseline Neutrino (SBN) program consists of three LArTPC detectors located along the BNB at Fermilab: the MicroBooNE detector, which completed operations in 2021; the ICARUS detector, which began operations in the BNB in 2021; and the upcoming Short-Baseline Near Detector (SBND), which is expected to begin operations in 2023. This program represents an exciting opportunity for a multi-baseline search for light sterile neutrino oscillations in multiple exclusive or inclusive oscillation channels, and a test of the 3+1 light sterile neutrino oscillation interpretation of past experimental anomalies at $\ge 5\sigma$ <cit.>. In particular, $\nu_\mu$ CC measurements across the three detectors will probe $\nu_\mu \rightarrow \nu_{\not{\mu}}$ oscillations with world-leading sensitivity as shown in Fig. <ref> (right); $\nu_e$ CC measurements will probe $\nu_\mu\rightarrow\nu_e$ and/or $\nu_e\rightarrow\nu_{\not{e}}$ oscillations with sensitivity as shown in Fig. <ref> (left) and <ref>. Additionally, NC-based oscillation searches have been proposed, e.g. <cit.>, with unique sensitivity to $U_{si}$ under a 3+$N$ model, as well as potentially $U_{\tau i}$ (for $i>3$) when combined with $\nu_e$ and $\nu_\mu$ CC-based appearance and disappearance searches.

SBN light sterile neutrino sensitivities in the $\nu_\mu\to\nu_e$ appearance channel (left) and $\nu_\mu\to\nu_\mu$ disappearance channel (right) according to the SBN proposal <cit.>. The $3\sigma$ ($5\sigma$) sensitivities are given by the solid (dotted) red curves.  The LSND 90% C.L. (99% C.L.) allowed region is shown as shaded blue (grey) <cit.>.  The global $3\sigma$ $\nu_e$ ($\nu_{mu}$) appearance (disappearance) regions from Ref. <cit.> are shown by the shaded red region (black line), and the global best fit regions from Ref. <cit.> are shown in green.

Due to the large intrinsic $\nu_e$ statistics at SBND, SBN is also sensitive to $\nu_e$ disappearance, probing $\sin^22\theta_{ee}$ at high $\Delta m^2 \ge 0.2$ eV$^2$.  This provides a complementary probe of oscillations  traditionally probed using reactor antineutrinos at a much lower (MeV) energy scale. The figure is from <cit.>, and assumes $\theta_{24}$=0.

The power of a multi-baseline and multi-channel search has been shown to be advantageous not only for 3+1 searches, but for 3+$N$ searches more generally. For example, Ref. <cit.> has found that SBN is capable of ruling out 85%, 95% and 55% of the 99%-globally-allowed parameter space region[Global allowed regions as of 2018.] of 3+1, 3+2, and 3+3 light sterile neutrino oscillation parameters at 5$\sigma$ CL, assuming a null observation, particularly when appearance and disappearance effects are studied simultaneously (including correlations). This is illustrated in Fig. <ref>, for the 3+3 scenario. Additionally, it has been pointed out that within the context of 3+$N$ oscillations with N$>1$, SBN offers an opportunity for measuring potential CP violation in the leptonic sector, particularly if future antineutrino beam running is possible with SBN. In particular, if antineutrino exposure is considered, for maximal values of the (3+2) CP violating phase $\phi_{54}$, SBN could be the first experiment to directly observe $\sim 2\sigma$ hints of CP violation associated with an extended leptonic sector. This is illustrated in Fig. <ref>, for the 3+2 scenario.

SBN's coverage of globally-allowed 3+3 light sterile neutrino oscillation parameters, defined as the fraction of 99%-CL-globally-allowed parameter space that can be ruled out by SBN at a given CL indicated by the $x$-axis, assuming a null observation. The figure is from <cit.>. Coverage of 3+1 and 3+2 globally-allowed parameter space is provided in <cit.>.

The significance at which SBN can observe CP violation in the (3+2) sterile neutrino scenario, as a function of true CP violating phase $\phi_{54}$, for two injected signals corresponding to the global (3+2) best fit point (red lines) as well as the point with largest total allowed mixings (blue lines), for a variety of POT in neutrino and antineutrino running modes at SBN. The figure is from <cit.>.

Beyond searches for physics associated with eV-scale sterile neutrinos, SBN's main physics goals include detailed studies of neutrino–argon interactions at the GeV energy scale, enabled by millions
of neutrino interactions that will be recorded on argon in its high precision detectors. SBND's anticipated high statistics, in particular, provide a unique opportunity for first-ever measurements of rare SM-predicted neutrino interaction processes at 0.1-1 GeV, including rare photon production processes such as coherent NC single-photon production, NC $\Delta \rightarrow N\gamma$ radiative decay, or production and radiative decay of heavier resonances.  Additionally, the high statistics coupled with the unprecedented event reconstruction, excellent particle identification, and fine-sampling calorimetry of the SBN detectors' LArTPC technology opens up invaluable opportunities for new physics searches.

In particular, the capabilities of LAr detectors will allow for greater discrimination between $e^-, \, e^+e^-, \,  \gamma,$ and $\gamma\gamma$ final states, as well as to identify final state hadron multiplicities.
For models in which new particles are produced in neutrino-nucleus scattering, it will be possible to search for a hadronic vertex associated with the displaced decay position. The EM showers in this case may not point back to the original vertex due to missing energy.  Furthermore, for models that explain MiniBooNE with new heavy particles produced in meson decays, the SBN detectors can also leverage the decays-at-rest of kaons produced in the NuMI absorber <cit.>.

Reference <cit.> provides a broad overview of such new physics, including their signatures in SBN. Here, we limit the discussion to a summary of ones suggested as interpretations to short-baseline anomalies, extending beyond eV-scale sterile neutrinos:

* SBN can probe eV-scale sterile neutrinos decaying to active neutrinos and a Majoron or gauge boson, which would lead to new features in the active neutrino energy spectrum with respect to 3+$N$ scenarios.

* Large extra-dimension models, such as ones proposed as an explanation of the reactor anomaly, would affect both appearance and disappearance channels at SBN.

* Resonant $\nu_\mu\rightarrow\nu_e$ oscillations that arise in the presence of a light scalar boson that couples only to neutrinos and could induce a MSW effect sourced by the cosmic neutrino background could also be probed with SBN, through the search of $\nu_\mu\rightarrow\nu_e$ transitions and lack of $\nu_\mu$ disappearance (as the latter would be suppressed compared to a vanilla 3+1 scenario).

* Violation of Lorentz and CPT symmetry
would lead to modifications in the oscillation probability measurable at SBN, such as direction-dependent effects, neutrino-antineutrino mixing, annual modulations, and energy dependent effects on observable mass splittings and mixing angles.

* Sterile neutrinos and altered dispersion relations (ADR), also proposed as an explanation of the short-baseline anomalies,
would have a similar phenomenology in SBN to that of the usual 3+3 sterile scenario (while evading the constraints from long-baseline and atmospheric neutrino experiments).

* Charged current non-standard interactions (CCNSI) in the lepton sector
could lead to a number of observable effects, such as (1) deviations of the SM CC quasi-elastic cross section, (2) modification of angular and energy distributions due to the presence of new Lorentz structures, and (3) flavor violation such as $\nu_\mu n \rightarrow e^- p$;
At SBN, CCNSIs can lead to an apparent baseline-independent $\nu_\mu\rightarrow\nu_e$ conversion.

* Dark neutrino sectors connected to the standard neutrino sector, allowing for neutrino
upscattering into a heavy state which could then decay to a light neutrino and a gauge boson within a detector, followed by the gauge boson decay to visible particles such as $e^+e^-$
could also be measurable at SBN. As shown in  Fig. <ref>, an $e^+e^-$ pair can give rise to four distinct topologies in LAr, depending on the lifetime of the parent particle and on the angle between the charged leptons. A variety of signatures could be probed, including pair production of $e^+e^-$, $\mu^+\mu^-$ or $\pi^+\pi^-$ induced by neutrino interactions, with little to no hadronic activity and with the same signal strength at all three detectors, since there is no $L/E$ dependence.

* Heavy neutrinos and transition magnetic moment
proposed as an explanation of the LSND/MiniBooNE anomalies
would be observable at SBN as anomalously large single-photon production with
small hadronic activity. LArTPC $e-\gamma$ discrimination capability places SBN in a special
position to probe these scenarios.
Again, the signal strength would be the same at all three detectors, since there is no L/E dependence in these models either.

The four main topologies of $e^+e^-$ LEE models at MicroBooNE: coherent and incoherent scattering with well-separated or overlapping $e^+e^-$ pairs.

The 2020 Update of the European Strategy for Particle Physics
(ESPP) <cit.> recommended that muon beam
R&D should be considered a high-priority future initiative and that a
programme of experimentation be developed to determine the neutrino
cross-sections required to extract the most physics from the DUNE and
Hyper-K long-baseline experiments.
The ENUBET <cit.> and
nuSTORM <cit.>
collaborations have begun to work within and alongside the CERN
Physics Beyond Colliders study group <cit.> and the
international Muon Collider Collaboration <cit.> to carry out
a joint, five-year design study and R&D programme to deliver a
concrete proposal for the implementation of an infrastructure in

* ENUBET and nuSTORM deliver the neutrino cross-section
measurement programme identified in the ESPP and allow sensitive
searches for physics beyond the Standard Model to be carried out;
and in which

* A 6D muon ionisation cooling experiment is delivered as part
of the technology development programme defined by the
international Muon Collider Collaboration.

With their existing proton-beam infrastructure, CERN and Fermilab are
both uniquely well-placed to implement ENUBET, nuSTORM, and the
6D-cooling experiment as part of the required muon collider
The design of ENUBET, carried out within the framework of a European
Research Council funded design study, includes the precise layout of
the kaon/pion focusing beamline, photon veto and timing system as well
as the development and test of a positron tagger together with the
required electronics and readout.
The feasibility of implementing nuSTORM at CERN has been studied by
the CERN Physics Beyond Colliders study group while a proposal to site
nuSTORM at FNAL was developed for the last Snowmass study in 2013.
The FNAL study focused on the optimisation of the muon storage ring to
provide exquisite sensitivity in the search for sterile neutrinos.
In the Physics Beyond Colliders study, the muon storage ring was
optimised to carry out a definitive neutrino-nucleus scattering
programme using stored muon beams with momentum in the range 1 GeV to
6 GeV while maintaining its sensitivity to physics beyond the
Standard Model.

The study of nuSTORM is now being taken forward in the context of the
demonstrator facility required by the international Muon Collider
Collaboration that includes the 6D muon ionization cooling experiment.
The muon-beam development activity is being carried out in close
partnership with the ENUBET collaboration and the Physics Beyond
Colliders Study Group.
In consequence we now have the outstanding opportunity to forge an
internationally collaborative activity by which to deliver a concrete
proposal for the implementation of the nuSTORM infrastructure.

On top of the program outlined above, nuSTORM can still provide unprecedented sensitivity to light sterile neutrinos. In particular, it allows to search for short-baseline oscillations in $\nu_e \to \nu_\mu$ appearance, the CPT-conjugate channel of the appearance hypothesis at LSND. This would be possible due to charge selection of muons in a magnetic field, which can discriminate the $\overline{\nu}_\mu$ produced in $\mu^+$ decays from the $\nu_e\to\nu_\mu$ oscillations. A detailed study of nuSTORM's sensitivity to sterile neutrinos was performed in Ref. <cit.>, which focused on a $3.8$ GeV muon ring design for siting at FNAL. The study considered a $1.3$ kt magnetized iron-scintillator detector at $2$ km from the ring with an exposure of $10^{21}$ POT, corresponding to $\approx 2\times 10^{18}$ useful muon decays. The sensitivity curves covered the entire LSND and MiniBooNE regions of preference at more than $5\sigma$. This impressive sensitivity was achieved thanks to the muon signature, which is subject to low levels of backgrounds, and the low systematic uncertainties on the neutrino flux.

Thanks to its unique neutrino beam, nuSTORM is also sensitive to other explanations to short-baseline anomalies. It would stand out as a unique test of lepton-flavor-violation in muon decays. Because the beam is derived from $\mu^+$ decays, any exotic branching ratio of the muon, such as $\mu^+\to e^+ \nu_\alpha \overline{\nu}_e$, would be a striking signature in a near detector with electron-positron discrimination capabilities, such as in a magnetized, low-density detector. This type of near detector would also benefit the sensitivity to models with neutrino upscattering to new dark particles with decays to $e^+e^-$.

§.§.§ Long-Baseline Experiments

The DUNE experiment is a next-generation, long-baseline neutrino oscillation experiment, designed to be sensitive to $\numu$ to $\nue$ oscillations.
The experiment consists of a high-power, broadband neutrino beam, a powerful precision multi-instrument Near detector complex located at Fermi National Accelerator Laboratory in Batavia, Illinois, 574 m away from the neutrino production target, and a massive Liquid Argon Time Projection Chamber Far detector located at the 4850 ft level of the Sanford Underground Research Facility (SURF), 1300  km away in Lead, South Dakota, USA.
The anticipated total fiducial mass of the Far detector is 40 kton. The long baseline of 1300 km provides sensitivity, in a single experiment, to all parameters governing neutrino oscillations.

Due to the high-power proton beam facility, the Near detector consisting of precision detectors capable of off-axis data taking for improved constraining of systematics, and the massive Far detector, DUNE provides enormous opportunities to probe BSM phenomena in both new particle production and interactions, and in neutrino propagation effects.

LBNF neutrino beam fluxes at the DUNE Far detector for neutrino-enhanced Forward Horn Current (FHC) beam running (left), and antineutrino-enhanced Reverse Horn Current (RHC) beam running (right).

DUNE expects to begin data taking operations in 2029 with half of the full Far detector, and start beam data taking operations in 2031 with the 1.2 MW Long-Baseline Neutrino Facility (LBNF) beam, upgradable to 2.4 MW.
The LBNF neutrino beam flux sampled on-axis by the Far detector is shown in Fig. <ref>.
The wide-band range of energies provided by the LBNF beam afford DUNE significant sensitivity to probe sterile mixing, which would typically cause distortions of standard oscillations in energy regions away from the three-flavor $\numu\rightarrow\numu$ disappearance maximum.
Therefore, DUNE sterile mixing probes reach a broad range of potential sterile neutrino mass splittings by looking for disappearance of CC and NC interactions over the long distance separating the ND and FD, as well as over the short baseline of the ND.
The DUNE sterile neutrino mixing studies shown below assume a minimal 3+1 oscillation scenario with three active neutrinos and one sterile neutrino, with a new independent neutrino mass-squared difference, $\Delta m^2_{41}$, and for which the mixing matrix is extended with three new mixing angles, $\theta_{14}$, $\theta_{24}$, $\theta_{34}$, and two additional phases $\delta_{14}$ and $\delta_{24}$.

Figure <ref> shows how the standard three-flavor oscillation probability is distorted at neutrino energies above the standard oscillation peak when oscillations into sterile neutrinos are included and the energy ranges DUNE ND and FD are sensitive to those distortions.

[Regions of $L/E$ probed by DUNE for
3-flavor and 3+1-flavor $\nu$ oscillations]
Regions of $L/E$ probed by the DUNE Near and Far detectors compared to 3-flavor and 3+1-flavor neutrino disappearance and appearance probabilities. The gray-shaded areas show the range of true neutrino energies probed by the ND and FD. The top axis shows true neutrino energy, increasing from right to left. The top plot shows the probabilities assuming mixing with one sterile neutrino with $\Delta m^2_{\rm{41}}=0.05$ eV$^2$, corresponding to the slow oscillations regime. The middle plot assumes mixing with one sterile neutrino with $\Delta m^2_{\rm{41}}=0.5$ eV$^2$, corresponding to the intermediate oscillations regime. The bottom plot includes mixing with one sterile neutrino with $\Delta m^2_{\rm{41}}=50$ eV$^2$, corresponding to the rapid oscillations regime. As an example, the slow sterile oscillations cause visible distortions in the three-flavor  survival probability (blue curve) for neutrino energies. Reproduced from Ref. <cit.>.

The sterile neutrino effects have been implemented in GLoBES via the existing plug-in for sterile neutrinos and NSI <cit.>. The DUNE ND plays a very important role in the sensitivity to sterile neutrinos both directly, for rapid oscillations with $\Delta m_{41}^2 > 1$ eV$^2$ where the sterile oscillation matches the ND baseline, and indirectly, at smaller values of $\Delta m_{41}^2$ where the ND is crucial to reduce the systematic uncertainties affecting the FD to increase its sensitivity. For these studies, the DUNE ND is assumed to be an identical scaled-down version of the FD, with identical efficiencies, backgrounds and energy reconstruction. The full set of systematic uncertainties employed in the sterile neutrino studies, as well as the methodology accounting for non-negligible beam-induced baseline spreads between production target and ND, are described in Ref. <cit.>.

By default, GLoBES treats all systematic uncertainties included in the fit as normalization shifts. However, depending on the value of $\Delta m^2_{41}$, sterile mixing will induce shape distortions in the measured energy spectrum beyond simple normalization shifts. As a consequence, shape uncertainties are very relevant for sterile neutrino searches, particularly in regions of parameter space where the ND, with virtually infinite statistics, has a dominant contribution. The correct inclusion of systematic uncertainties affecting the shape of the energy spectrum in the two-detector fit GLoBES framework used for this analysis posed technical and computational challenges beyond the scope of the study.
Therefore, for each limit plot, we present two limits bracketing the expected DUNE sensitivity limit, namely: the black limit line, a best-case scenario, where only normalization shifts are considered in a ND+FD fit, where the ND statistics and shape have the strongest impact; and the grey limit line, corresponding to a worst-case scenario where only the FD is considered in the fit, together with a rate constraint from the ND.

For sensitivity to $\theta_{14}$, the dominant channels are those regarding $\nu_e$ disappearance.
For simplicity, only the $\nu_e$ CC sample is analyzed and the NC and $\nu_{\mu}$ CC disappearance channels are not taken into account. This is expected to be improved by using more complex multi-channel fits in future studies, as highlighted and recommended by Ref. <cit.>.
The sensitivity at the 90% C.L., taking into account the systematic uncertainties mentioned above, is shown in Fig. <ref>, along with a comparison to current constraints.
For the $\theta_{24}$ mixing angle, the $\nu_{\mu}$ CC and NC disappearance samples are analyzed jointly.
Results are shown in Fig. <ref>, along with comparisons with present constraints.

The left plot shows the DUNE sensitivities to $\theta_{14}$ from the $\nu_e$ CC samples at the ND and FD, assuming $\theta_{14}$=0, along with a comparison with the combined reactor result from Daya Bay and Bugey-3. The right plot is adapted from Ref. <cit.> and displays sensitivities to $\theta_{24}$ using the $\nu_\mu$ CC and NC samples at both detectors, along with a comparison with previous and existing experiments. In both cases, regions to the right of the contours are excluded.  This figure is from Ref. <cit.>.

In the case of the $\theta_{34}$ mixing angle, disappearance in the NC sample, the only contributor to this sensitivity, is probed.  The results are shown in Fig. <ref>. Further, a comparison with previous experiments sensitive to ,  mixing with large mass-squared splitting is possible by considering an effective mixing angle $\theta_{\mu\tau}$, such that $\sin^2{2\theta_{\mu\tau}}\equiv 4|U_{\tau4}|^2|U_{\mu 4}|^2=\cos^4\theta_{14}\sin^22\theta_{24}\sin^2\theta_{34}$, and assuming conservatively that $\cos^4\theta_{14}=1$, and $\sin^22\theta_{24}=1$. This comparison with previous experiments is also shown in Fig. <ref>.
The sensitivity to $\theta_{34}$ is largely independent of
$\Delta m^2_{41}$, since the term with $\sin^2\theta_{34}$ in
the expression describing ${P(\nu_{\mu}\rightarrow\nu_s)}$, depends solely on the $\Delta m^2_{31}$ mass splitting.

Left: Comparison of the DUNE sensitivity to $\theta_{34}$ using the NC samples at the ND and FD with previous and existing experiments. Regions to the right of the contour are excluded.  Right: DUNE sensitivities to $\theta_{\mu e}$ from the appearance and disappearance samples at the ND and FD are shown on the top plot, along with a comparison with previous existing experiments and the sensitivity from the future SBN program.  Regions to the right of the DUNE contours are excluded.  Both plots from Ref. <cit.>.

Finally, sensitivity to the $\theta_{\mu e}$ effective mixing angle, defined as $\sin^2{2\theta_{\mu e}}\equiv 4|U_{e4}|^2|U_{\mu 4}|^2=\sin^22\theta_{14}\sin^2\theta_{24}$, is shown in Fig. <ref>, which also displays a comparison with the allowed regions from LSND and MiniBooNE, as well as with present constraints and projected constraints from Fermilab's SBN program.

DUNE will also have the ability to conduct short-baseline sterile probes, for instance, by searching for anomalous sterile-driven $\nu_\tau$ appearance in the Near detector. The $\tau$ lepton is not directly observable in the DUNE detectors due to its short $2.9\times 10^{-13}$ s lifetime, and it is only produced for interactions where the incoming $\nu_\tau$ has an energy of $\sim3.5$ GeV due to the relatively large $\tau$ mass of 1776.82 MeV. However, the final states of $\tau$ decays, $\sim65\%$ into hadrons, $\sim18\%$ into $\nu_\tau+e^-+\bar{\nu}_e$, and $\sim17\%$ into $\nu_\tau+\mu^-+\bar{\nu}_\mu$, are readily identifiable in the DUNE ND, given the excellent spatial and energy resolution of the ND instruments, namely, ND-LAr, ND-GAr, and SAND, which are complementary in providing sensitivity to different decays channels. While within a three-flavor scenario, the DUNE ND baseline is far too short for $\nu_\mu\rightarrow\nu_\tau$ oscillations to occur, $\nu_\tau$ originating in sterile-neutrino driven fast oscillations could be detected. In particular, probing the $\tau\rightarrow\mu$ detection channel with high-energy muons in the final state, which is challenging due to muon containment and backgrounds, becomes very accessible through the use of the ND-GAr magnetic field and the SAND detector further downstream, with studies indicating that ND-GAr's reconstructible muon momentum via curvature extends beyond 15 GeV/$c$. This sensitivity will be strongly enhanced when operating LBNF in the high-energy tune, aimed at enriching the available sample of $\nu_{\tau}$ at the Far detector, while extending sensitivity to anomalous $\nu_{\tau}$ appearance at the Near detector <cit.>. Preliminary studies using LBNF's nominal flux, and including ND-LAr and ND-GAr, estimate that DUNE's sensitivities to anomalous $\nu_\tau$ appearance may extend beyond those of previous searches carried out by NOMAD and CHORUS.

Hyper-Kamiokande (HK) is a large-scale water Cherenkov detector with a fiducial volume of about 188 kton which is approximately 8.4 times larger than Super-Kamiokande.

HK is currently under construction in Japan and operations are scheduled to begin in 2027 together with the upgraded J-PARC neutrino beam.

The physics capabilities of HK cover a broad range of topics including a search for sterile neutrino mixing <cit.>.

There are various major approaches currently being considered.

While we focus in the following on the sensitivity to sterile neutrino searches, HK can also investigate other exotic scenarios like the breaking of Lorenz and CPT invariance, as demonstrated in T2K in Ref. <cit.>, and non-standard neutrino interactions, as studied with Super-K's atmospheric neutrino observations <cit.>.

Mixing of light sterile neutrino will be investigated with the Hyper-Kamiokande data at a baseline of 295 km baseline.
T2K reported a limit on $\sin^2 \theta_{24}$ for $10^{-4}\,{\rm eV^2} < \Delta m^{2}_{41} < 3 \times 10^{-4}\,{\rm eV^2}$ using both CC and NC samples at Super-Kamiokande <cit.>.
More stringent limit will be set by Hyper-Kamiokade with more than 20 times higher statistics by the combination of a larger fiducial volume and an upgraded J-PARC neutrino beam.

Hyper-Kamiokande's near detectors will measure the neutrino beam flux and cross-section at different baselines.
Each detector has the capability to test the existence of sterile mixing at certain values of $L/E$ but it should be noted that the sensitivity will be further enhanced by combined measurements among the detectors where the ND280 works as a near detector and give constraint to the IWCD measurement.

The IWCD (Intermediate Water Cherenkov Detector) instrument has sensitivity to sterile neutrino mixing in the $\nu_{\mu} \rightarrow \nu_{e}$ channel with a baseline of $\sim$1 km and energy of 0.5 - 1 GeV, which matches the $L/E$ at LSND and MinoBooNE.
As a remarkable advantage, IWCD can measure the neutrino flux at different off-axis angles by moving the detector along its vertical pit.
As the energy spectrum changes with the off-axis angle, IWCD can rule out some potential interpretations by the combination of the measurement, such as feed-down from high energy due to nuclear effects or unexpected background.
The design of the IWCD detector is still under investigation, but it has potential to test the allowed region given by LSND, as indicated by the studies from the NuPRISM collaboration <cit.>.

The off-axis ND280 has sensitivity to few eV$^{2}$ sterile neutrinos.
A first search was published in Ref. <cit.>.
It is anticipated that the upgrades currently being done for T2K, consisting in one fully active target (Super-FGD), two High-Angle TPCs, and a Time Of Flight system, and possible further upgrades under study for Hyper-K, will boost the sensitivity of ND280.
Searches for sterile neutrinos with the upgraded ND280 will have several advantages with respect to Ref. <cit.>, including a larger target mass, a lower threshold to reconstruct leptons, better performances in distinguishing electrons from gammas, and the larger exposure that will be collected in HK.
ND280 has also sensitivity to search for relatively heavy sterile neutrinos produced by the decay of Kaons produced by the beamline, as demonstrated in Ref. <cit.>.

The ESSnuSB (European Spallation Source Neutrino Super Beam) experiment <cit.> is a proposed long-baseline neutrino oscillation experiment to use a neutrino superbeam produced using 2.0 GeV protons from the ESS Linac in Lund, Sweden, resulting in a 5 MW beam peaked at $E_{nu} = 0.4$ GeV. By sampling this beam at a distance of 540 km from Lund, using a large underground Water Cherenkov detector with a 500 kton fiducial mass, ESSnuSB will make measurements of the three-flavor oscillation second maximum, which would enable discovery of leptonic CP violation for 56% of $\delta_{CP}$ values for 10 years of data taking, and 65% of $\delta_{CP}$ values if an upgrade of the beam power to 10 MW and of the proton energy to 2.5 GeV is realized.

With this experimental setup, ESSnuSB is also able to place bounds on the sterile mixing parameters, but the CP sensitivity of ESSnuSB may also be affected by the existence of light sterile neutrinos. The studies below assume two possible detector configurations for ESSnuSB and compare them: 1) combined far (FD) and near (ND) detectors with correlated systematics; and 2) an FD only with an overall systematic uncertainty.

The analysis uses the GLoBES <cit.> software to simulate ESSnuSB. An ND and an FD are explicitly simulated to reduce systematic uncertainties <cit.>. The FD is a 1 Mt MEMPHYS-like water-Cherenkov detector <cit.> located at a distance of 540 km from the source, while the ND is assumed to have the same efficiency and background rejection capabilities as the FD <cit.> with a fiducial volume of 0.1 kt and placed at a distance of 0.5 km from the source. A beam power of 5 MW with 2.5 GeV protons capable of producing $2.7\times 10^{23}$ POT/year is assumed. The results are presented for a total data exposure 10 years, with 5 (5) years running in neutrino (antineutrino) mode. Throughout the simulations of the combined near and far detectors, the same treatment of systematics as in Ref. <cit.> was employed. In all cases, the best-fit values of the parameters from Ref. <cit.> assuming Normal Ordering were used.

The upper panels of Fig. <ref> display the sensitivity of ESSnuSB to the sterile mixing angles $\theta_{14}$ and $\theta_{24}$ for particular values of $\Delta m_{41}^2$ at 95 % CL.

Bounds (95 % CL) on sterile mixing parameters in the $\sin^2\theta_{14}$ – $\sin^2\theta_{24}$ plane for ESSnuSB (left and middle panels). The panel on the right shows a comparison of ESSnuSB with other long-baseline neutrino oscillation experiments.

The left panel shows the results for combined FD+ND, while the middle panel presents the results for the FD using the ND as a counting experiment. From the left panel, it is apparent that for $\Delta m^2_{41} = 0.01$ eV$^2$ the bounds are weak, as for this value of $\Delta m^2_{41}$ the oscillations have not yet developed for the ND, and therefore, the existing sensitivity comes from the FD, where the oscillations are averaged out. Increasing the value of $\Delta m^2_{41}$, the oscillations become more developed in the ND resulting in the stronger bound for $\Delta m^2_{41} = 1$ eV$^2$. Increasing the value of $\Delta m^2_{41}$ further, the oscillations tend to become averaged out and again the sensitivity decreases. From the middle panel, where there is no spectral information on the ND, the sensitivity for  $\Delta m^2_{41} \sim 1$ eV$^2$ is lost and similar sensitivities  are obtained for all values of $\Delta m^2_{41}$.
The right panel of Fig. <ref> compares the sensitivity of ESSnuSB to the sterile mixing parameters with other future long-baseline neutrino oscillation experiments, specifically T2HK <cit.>, T2HKK <cit.>, and DUNE <cit.>. The comparison shows that the sensitivity of ESSnuSB is slightly worse than the other experiments in most of the region of the parameter space. However, as can be seen from the left panel in Fig. <ref>, it is expected that the ESSnuSB sensitivity would be considerably improved in the presence of an ND.

Figure <ref> presents the CP violation sensitivity of ESSnuSB due to $\delta_{13}$ for four different values of the sterile mixing phase $\delta_{24}$, illustrating how the presence of sterile neutrino oscillations would modify the interpretation of leptonic CP violation measurements.

Leptonic CP violation sensitivity of ESSnuSB for four different values of the sterile mixing phase $\delta_{24}$ as a function of true $\delta_{13}$.

The sterile mixing parameters assumed are $\sin^2\theta_{14} = \sin^2\theta_{24} = 0.025$ <cit.>, $\theta_{34} = \delta_{34} = 0^\circ$, and $\Delta m^2_{41} = 1$ eV$^2$. The left panel shows the sensitivity for FD+ND, and in the right one, the sensitivities considering only the FD with 8 % (10 %) overall systematics in signal (background). Again,when including the ND, the sterile mixing parameters are better constrained and the sensitivity to CP violation improves.

The ESS Linac proton beam is expected to turn on in 2025. Construction of the neutrino beam facility is under study and may begin with a low-energy nuSTORM-like ND complex starting in 2024 with operations starting in 2027.

§.§ Reactor Neutrino Experiments

In the period following Snowmass 2021, emphasis in the reactor sector will be placed on 50 km-scale baseline experiments probing the SM neutrino mass hierarchy and the solar mixing angle, such as JUNO, and on and very short-baseline experiments aiming to probe sterile oscillations and perform high-statistics measurements of reactor fluxes and spectra.
Short-baseline reactor experiments will be particularly important in furthering understanding of the source of enduring reactor-sector anomalies, in addressing existing claims of non-standard oscillation observation in recent reactor experiments, and in providing orthogonal experimental datasets valuable in global BSM physics fits.

Data from multiple channels, energies, and sources will be crucial for disentangling possible BSM effects manifested in the anomalies, since these effects may manifest differently in different experimental regimes.
In the stable of global measurements, short-baseline reactor measurements are unique in their capability to very purely probe sterile oscillation effects.
This is due to the lower energies involved in interactions and decays in the reactor, which prohibits production and decay of heavier hidden-sector particles, and their very short baselines, which minimize the impact of non-standard interactions.
Even in the case of purely oscillation-driven BSM explanations, reactor experiments offer unique benefits due to their unambiguously pure flavor content: this is in contrast to interpretations of short-baseline decay-in-flight and decay-at-rest experiments, which will be complicated by competition between appearance and disappearance effects in individual flavor channels <cit.>.
These points serve to emphasize the substantial value added by acquiring datasets from all of experiment types: short-baseline reactor datasets will find greater utility and application in the coming decade if they are accompanied by short-baseline accelerator datasets, and vice versa.

Experiment     Baseline (m)     Reactor     Reactor     Detector     Target     Sterile $\nu$

Type     Power (MWth)     Size         Search Strategy

DANSS <cit.>      11–13     LEU     3000     1 m$^3$     Segmented PS with Gd coating     Multi-Site
JUNO-TAO <cit.>      30     HEU     4600     2.8 ton     Single GdLS     Single-Site
NEOS-II       24     LEU     2800      1 m$^3$     Single-volume GdLS + PSD     Single-Site
Neutrino-4 Upgrade     6–12     HEU     90      2 m$^3$     Segmented GdLS     Multi-Site/Zone
PROSPECT-II <cit.>      7–9     HEU     85     4 ton      Segmented $^6$LiLS + PSD     Multi-Zone

List of the future short-baseline reactor neutrino experiments that will be able to search for sterile neutrino oscillations.

A summary of new short-baseline reactor oscillation measurements currently underway or planned for the next five years is provided in Table <ref>.
For future experiments, some specific reactor experiment parameters are particularly valuable in achieving the future short-baseline reactor physics goals mentioned above.
While reactor experiments in the past decade have provided excellent limits in the oscillation phase space region below a few eV$^2$, limits above this mass splitting are substantially weaker; to address this, very short ($<$10 m) baseline experiments using compact cores and segmented detectors are particularly valuable.
To provide more robust probes of isotopic IBD yields and spectra and thus better understanding of the fidelity of existing flux predictions, high-statistics dataset are needed from reactors of widely varying fuel content.
In this context, experiments running at HEU reactors and over a full cycle at a single LEU reactor core would be particularly valuable.
As demonstrated in Table <ref>, most of these expected future experiments fulfill one or more of these key experimental requirements.

Left: Claimed sensitivities (90% CL) to sterile neutrinos of the upcoming reactor neutrino experiments listed in Tab. <ref>. Also shown is the unexplored parameter space obtained by comparing reactor rates + evolution datasets to the reevaluated HM model based on the KI suggested normalization correction as discussed in Sec. <ref>. NEOS and JUNO-TAO are shown in dashed lines to indicate that the experiments make use of the absolute spectral shapes. Also shown (solid) is a version of JUNO-TAO's curve that is produced by making use of the experiment's position reconstruction to generate virtual segments that provides the ability to perform a completely absolute shape independent analysis. The plot was generated using the same approach as described in Fig. <cit.>. Right: Sensitivities of the future reactor neutrino experiments in comparison with the Gallium Anomaly and Neutrino-4 suggested parameter space. Also included is the line at $\sin^2 2 \theta=0.3$ (dashed black) corresponding to the limit which has to be excluded to avoid ambiguities in the future LBL CP-violation measurements.

The sterile neutrino oscillation probing powers of these experiments, expressed in terms of sensitivity to sin$^2$2$\theta_{14}$ in a 3+1 oscillation framework, are shown on the right plot in Figure <ref>, with the left plot also indicating some phase space parameter ranges of interest.
In addition to showing individual experiments, the right hand plot indicates the style of analysis assumed in each curve: one using only relative measured spectral shapes, one using knowledge of the absolute measured spectrum, and one using only rate information.
Rate-based analyses provided by reactor flux measurements clearly lack the  sensitivity of shape-based ones; combined with limitations in knowledge of absolute reactor fluxes (Section <ref>), rate-based analyses appear, at present, unlikely to yield much fruit in the pursuit of sterile neutrinos.
Of the shape-based measurements, DANSS and TAO are likely to dominate sensitivities in the lower-$\Delta $m$^2$ regime, while PROSPECT-II is likely to dominate future sensitivity in the less-explored region above $\sim$3 eV$^2$.
DANSS and TAO may achieve some power in probing the oscillation phase space region suggested by Neutrino-4, while PROSPECT-II should address this region at extremely high confidence level while also covering phase space suggested by the Gallium anomaly that is currently unaddressed by spectrum-based reactor analyses.
Finally, the combination of DANSS, TAO, and PROSPECT-II datasets should generate percent-level sensitivity to active-sterile couplings in the electron disappearance channel for all $\Delta m^2$ space below roughly 10 eV$^2$, enabling greater clarity in interpretation of CP-violation results from DUNE <cit.>.

§.§.§ DANSS upgrade
One of the limitations of DANSS experiment was that the oscillation spectrum was smoothed out by the finite energy resolution. Resolution is limited mainly by the light collection system which makes further progress in DANSS physics program very challenging. Because of which, it has become necessary to upgrade the detector design in order to improve energy resolution.

The upgrade plan is to replace current strips with new strips with larger cross section <cit.>. In the current design, the strips are coated with titanium oxide for light reflection and gadolinium oxide for neutron capture which produces relatively thick dead layer with titanium and gadolinium. The non-uniformities in the light collection system is the another limiting factor for resolution. The light collection system consists of three wavelength shifting fibers which are readout by SiPMs and PMTs are on the same side of the strip which introduces non-uniformities. To tackle this issue, the upgrade plan is to have SiPM only readout from both sides of the strips. Removing the PMTs from light collection system will allow to increase the size of sensitive volume by 70%. The sensitivity to sterile neutrinos with the upgraded detector after 1.5 years of running is shown in Fig. <ref>.

Expected sensitivity of upgraded DANSS detector to sterile neutrino oscillations after 1.5 years of running in comparison with the sensitivity of the DANSS experiment.

§.§.§ JUNO-TAO
The Taishan Antineutrino Observatory (TAO) will be a satellite detector of the JUNO experiment <cit.>. Located at a baseline of approximately 30 m from one of the Taishan 4.6 GW$_\mathrm{th}$ cores in a basement 9.6 m below ground level, TAO's neutrino target will consist of 2.8 tons of gadolinium-doped liquid scintillator contained in a 1.8 m diameter acrylic sphere. This sphere will be surrounded by a copper spherical shell supporting an array of Silicon Photomultipliers (SiPMs) with 94% surface coverage, allowing to reach a light level of 4500 photoelectrons per MeV and an energy resolution of $\lesssim$2% at 1 MeV. This volume will be placed inside a cylindrical stainless steel tank with 2.1 m of outer diameter and 2.2 m of height filled with a liquid scintillator buffer. The tank will be cooled to -50$^{\circ}$C to mitigate the dark noise of the SiPMs, and will be surrounded by an active 1.2 m thick water Cherenkov veto tank, as well as high-density polyethylene in the top and lead in the bottom. Approximately 4,000 IBD interactions are expected per day in a central 1 ton fiducial volume with a detection efficiency of about 50%. TAO is scheduled to begin operations around the same time as the rest of the JUNO experiment in 2023.

90.0% and 99.7% C.L. exclusion contours expected at JUNO-TAO with 3 years of data taking. The exclusion contours from the Daya Bay <cit.>, PROSPECT phase-I <cit.>, and NEOS <cit.> experiments are also shown for comparison purposes. The parameter space to the right of all curves is excluded. Plot obtained from <cit.>.

TAO's primary purpose is to make a precise measurement of the reactor antineutrino spectrum that will serve as a benchmark for JUNO, other experiments, and nuclear databases. However, TAO will also be in an excellent position to search for a distortion in the shape of the reactor antineutrino spectrum caused by sterile neutrino mixing. The exclusion contours expected in JUNO-TAO at 90% and 99.7% C.L. in the absence of a sterile neutrino signal are shown on the right panel of Fig. <ref> <cit.>. The analysis behind these contours takes into account the physical dimensions of both the detector and the reactor core, and assumes 3 years of exposure with 80% reactor time on. A conservative bin-to-bin uncorrelated uncertainty of 5% (with 50 keV bin width) on the predicted reactor spectrum is assumed. The exclusion contours from the Daya Bay <cit.>, PROSPECT phase-I <cit.>, and NEOS <cit.> experiments are also shown for comparison purposes. TAO's baseline, abundant statistics, and exquisite energy resolution, afford it a leading sensitivity to $\sin^2 2\theta_{14}$ in the $10^{-1}$ eV$^2$ $\lesssim |\Delta m^2_{41}| \lesssim 3$ eV$^2$ region.

§.§.§ NEOS II
NEOS-II has taken about 400 live days of reactor-ON data to cover a full fuel cycle and this is to observe time evolution of reactor neutrino spectrum and flux and to investigate whether there is any relation between the "5 MeV excess" and $^{235}$U (or $^{239}$Pu). For background subtraction two periods (total $\sim$125 days) of reactor-OFF data were taken before and after the reactor-ON period. NEOS-II has observed light-yield decrease due to precipitation in the GdLS target from the early stage of the data-taking. The light-yield decrease resulted in worse energy resolution, from $\sim$5.6%/$\sqrt{1~\mathrm{MeV}}$ at the beginning to $\sim$7.7%/$\sqrt{1~\mathrm{MeV}}$ at the end of the data-taking, and increased time between prompt and delayed signals from $\sim$28 $\mu$s to $\sim$40 $\mu$s. According to our study, however, these are found to be minor effects in the NEOS-II data analysis.

Currently two main analyses are underway in NEOS-II.
One is on the decomposition of $^{235}$U and $^{239}$Pu spectra to better understand the origin of the “5 MeV excess” and the other is a light sterile neutrino search using rate+shape analysis.

Figure <ref> shows the 90% C.L. sensitivity (exclusion) curve of NEOS-II (NEOS-I) on a light sterile neutrino search where the Daya Bay spectrum is used as a reference spectrum. However, we plan to use the RENO spectrum as a reference for the final result of NEOS-II. The exclusion curve and sensitivity are drawn using a raster scan. A slight improvement in the sensitivity is expected in the NEOS-II using rate+shape analysis and updated reference from RENO, while a shape-only analysis was used in NEOS-I.

NEOS-II 90% C.L. sensitivity (orange) curve to light sterile neutrino oscillations using 400 live days of data. Also overlayed is the NEOS-I final exclusion (black) curve.

§.§.§ Neutrino-4 Upgrade
The Neutrino-4 experiment is planning for an upgrade of their experiment to increase statistical precision.
The upgraded detector will contain 4 identical modules, each consisting of a square grid of 25 segments–amounting to 3$\times$ larger volume.
The horizontally positioned segments will have read-out at both the ends, as opposed to vertically placed single read-out segments in Neutrino-4.
The target volume will consist of Gd-doped PSD-capable liquid scintillator to reduce background.
The combination of the increased volume and reduced backgrounds from higher Gd-loading and addition of the PSD capability is expected to have a three-fold increase in statistical power.

§.§.§ PROSPECT-II

The PROSPECT experiment has placed some of the most stringent limits on the eV-scale sterile neutrino oscillations <cit.>.
The detector was decommissioned in 2020 after an unexpected HFIR down time. The results are currently statistically limited and could be significantly improved by a longer dataset.
PROSPECT-II <cit.> is an upgrade of the PROSPECT experiment with evolutionary modifications intended to take advantage of the beneficial design aspects of the PROSPECT detector and mitigate the drawbacks in the PROSPECT detector to extend the experiment's statistical reach.
The suggested design changes were based on the lessons learned after the decommissioning of the PROSPECT detector.
To eliminate the chance of liquid scintillator contacting the PMT electronics, a separate subsystem is being designed that holds the PMTs completely outside the liquid scintillator volume.
Additionally, in order to reduce the contact of liquid scintillator with other detector components, a simplified calibration system that runs around the periphery of the detector is being developed.
These modifications are expected to mitigate the risk of PMT failure, highly simplify inner detector assembly, significantly improve the choice of detector materials, and allow for increase in the target volume while still maintaining the footprint approved for operations at HFIR.

PROSPECT-II design choices are based on the data collected by PROSPECT and data-validated Geant4 simulations.
Considering all the design modifications, PROSPECT-II is expected to have $\sim$30% increase in exposure and a three-fold increase in S:B from 1.4 to 4.3.
Figure <ref> shows the projected sensitivity of the experiment to the sterile neutrino-induced oscillations.
To mitigate the reliance on the reactor neutrino models the sensitivities are estimated
by comparing Asimov datasets of relative spectra estimated at multiple baselines within the detector.
As illustrated in Fig. <ref>, within two calendar years, PROSPECT-II will be able to fully cover the RAA and Gallium Anomaly suggested regions below 15 eV$^2$

While longer baseline and LEU experiments cover the lower oscillation frequencies and the $\beta$-decay endpoint experiments like KATRIN and Project 8 cover the higher oscillation frequencies, PROSPECT-II is uniquely situated to complement these results by providing the most stringent limits on disappearance between 1–20 eV$^2$.

Sensitivity of the PROSPECT-II experiment to sterile neutrinos for 2 years at HFIR (in black), 2 years at HFIR and 2 years at a power reactor (in purple), and 4 years at HFIR (in black) and 2 years at a power reactor (in green). PROSPECT will cover major portion of the suggested sterile neutrino parameter space. A combination of PROSPECT, KATRIN, and Daya Bay experiments will be able to cover significant portion of the parameter space down to the long baseline CP-violation disambuguity limit <cit.>.

§.§.§ Other experiments

Plastic Scintillator Experiments
Several plastic scintillator experiments including  CHANDLER <cit.>, and ISMRAN <cit.>, NuLat <cit.>, and SoLID <cit.> that have the capability to provide modest sensitivity to sterile neutrinos have either been built or planned to be built in the near future.
Similar to the experiments described above, these experiments all use segmented detectors for background reduction which would also be useful to perform oscillation search with a single detector.
Plastic scintillators provide a convenient and robust way to segment the detectors either in two or three dimensions.
A ability to achieve the high signal-to-background ratios necessary to perform precision oscillation searches has yet to be demonstrated for these experiment types.

CHANDLER CHANDLER is a near-field reactor neutrino detector technology consisting of an array of wavelength shifting plastic scintillator cubes, alternating in layers with thin sheets of $^6$Li loaded zinc sulfide (ZnS) scintillator.  Positrons from reactor $\bar{\nu}_e$ interactions deposit their energy in the plastic scintillator creating a prompt pulse of light, while the neutrons thermalize and capture on $^6$Li in the ZnS to create a delayed pulse.  The ZnS releases its scintillation light 20 times slower than the plastic scintillator, which makes an unmistakable neutron capture tag.   Similarly, the high segmentation of the plastic cubes is used to tag the positron annihilation gamma, forming a powerful positron tag.  The combination, in conjunction with tight spatial and temporal coincidence requirements, form a strong discriminator against all backgrounds, since correlated, or even random, coincidences of a positron and a neutron are vey rare in nature.  Small air gaps between the cubes allow the light to transmitted along the rows and columns of cubes, by total internal reflection (TIR), to the surface of the detector.  Light from neutron capture in the ZnS is absorbed by the wavelength shifter in the cube layers on either side of the sheet and re-emitted, so that it too can be transmitted by TIR. The position of the cube where the light originated sits at the intersection of the hit row and column.

The CHANDLER technology was demonstrated in a 2017 deployment to the North Anna Nuclear Generating Station, in Mineral, Virginia <cit.>.  There, the 80 channel, 80 kg MiniCHANDLER prototype was housed in a 14-foot trailer parked 25 meters from Reactor 2, and ran for four and a half months, including one month of reactor off.  These data were used to isolate a sample of 2881 events.  In this analysis, topological selections for the positron annihilation gammas played a critical role in the observation of reactor inverse beta decay events, shifting the signal significance from less than 3$\sigma$ to 5.5$\sigma$.  This demonstration used old PMTs, which were only marginally sensitive the Compton edge of the 511 keV gammas.  The CHANDLER collaboration is undertaking an upgrade of the MiniCHANDLER prototype with new PMT combined with compound parabolic light guides which will improve the light collection efficiency by a factor of four.  This will greatly enhance the efficiency of the positron tag, and achieve an average energy resolution of around 6% at 1 MeV.

CE$\nu$NS Experiments
The discovery of coherent elastic neutrino-nucleus scattering (CE$\nu$NS) by the COHERENT experiment <cit.> using a decay-at-rest source, opened up the possibility of using this mechanism to perform several BSM searches.
Multiple global efforts <cit.> are underway to leverage the intense source of neutrinos from reactors to perform CE$\nu$NS measurements.
CE$\nu$NS has not yet been observed in these experiments owing to a combination of high backgrounds at reactor facilities and a need for extremely low threshold detectors.
However, enabling detector technologies are progressing at a fast pace.
The observation reactor CE$\nu$NS opens doors to a vibrant BSM physics program, including searches for sterile neutrino oscillations.

§.§.§ Joint Analyses

As discussed in Sec. <ref>, a single experiment can't have the capability to cover all the sterile neutrino parameter space of interest.
Thus it is beneficial to perform joint fits of experimental datasets each of which provide sensitivity to different regions of parameter space.
When performed using proper statistical techniques and by including relevant systematics and correlations properly, such joint fits can be extremely powerful in covering wider parameter space as well as in alleviating any unknown
systematic effects specific to a single experiment that could mimic oscillations.
As highlighted in Ref. <cit.>, there are attractive opportunities in combining data from multiple current and upcoming reactor experiments such as Daya Bay, PROSPECT, STEREO, NEOS, and JUNO-TAO.
Reactor experiments could further be combined with accelerator experiments such as MINOS and MINOS+.
It is worth noting that the Daya Bay collaboration plans to publicly release its full data set once all final results have been released <cit.>, allowing such combinations to occur even well after the collaboration has dissolved.
Similarly, following the end of data taking in 2016, the MINOS/MINOS+ $\mathrm{CL}_s$ surfaces remain available for use in future combinations.
Other experiments such as PROSPECT and STEREO are also moving towards releasing data as part of their publications in a format that could be used to perform such joint analyses.

§.§ Radioactive Source Experiments

The combination of the results obtained by GALLEX, SAGE and BEST leads to a gallium anomaly (Sec. <ref>) with a large significance, allowing the possibility that electron neutrinos may disappear through oscillations with the participation of a sterile neutrino at the $\mathcal{O}(\rm{eV})$ mass scale. The following are experimental proposals aiming to perform additional studies to confirm or refute the gallium anomaly.

§.§.§ BEST-2

The BEST Collaboration has proposed to use their experimental configuration (see Sec. <ref>) to run the same type of experiment by measuring the electron neutrinos produced by an artificial source made of $^{65}\rm{Zn}$ <cit.>.

When the isotope $^{65}\rm{Zn}$ decays (Fig. <ref>, left), a neutrino of energy 1.35 MeV is emitted in nearly half of the decay events. The remaining events involve the emission of a neutrino with an energy of
235 keV close to the threshold for capture by gallium (233 keV), and the corresponding cross section is small. On the other hand, the cross section for the capture of 1.35 MeV neutrinos by $^{71}\rm{Ga}$ nuclei is approximately three times as large as that for 0.75 MeV neutrinos from a $^{51}\rm{Cr}$ source <cit.>. Then, the expected rate of capture of neutrinos from a $^{65}\rm{Zn}$ source of activity 3 MCi in a single zone at identical dimensions of the sources and target zones is $n_0 = 108$ d$^{-1}$. Since the $^{65}\rm{Zn}$ lifetime
is longer than the $^{51}\rm{Cr}$ lifetime ($T_{1/2} = 244.1$ and $27.7$ d, respectively), BEST-2 could perform measurements with $^{65}\rm{Zn}$ for a longer time, so that an accumulation of a commensurate data sample may require a substantially lower source activity <cit.>.

Left. Nuclear levels for the $^{65}\rm{Zn}$ radioactive source decay. Right. Number of events, $N$ as a function of the number of exposures, $m$, for experiments with $^{51}\rm{Cr}$ and $^{65}\rm{Zn}$ sources <cit.>.

To compare against the previously published results, with the data sample accumulated in the BEST experiment, it was estimated that the number of extracted $^{71}\rm{Ge}$ atoms expected in the absence of oscillations was 1657. For the proposed source based on $^{65}\rm{Zn}$, the total number of extracted $^{71}\rm{Ge}$ atoms as a function of the number of target exposures, $N(m)$, is given in the right panel of Fig. <ref> for various values of the source activity. The total event number $N = 1657$ may be attained even with a $^{65}\rm{Zn}$ source of activity nearly ten times lower than the activity of the $^{51}\rm{Cr}$ source in the BEST experiment <cit.>.

The differences (especially the higher energy of the emitted neutrinos) to be implemented in BEST-2 are expected to reach an extended range of $\Delta m^2$ values. In fact, the areas of sensitivity to this parameter in the two experiments (BEST and BEST-2) prove to be shifted in such a way that the highest sensitivity of one experiment corresponds to the lowest sensitivity of the other experiment, and vice versa. According to <cit.>, a global result of the two
experiments would provide a better statistical significance of measured oscillations (if there are any in the region of searches), but it will also permit precisely measuring the parameters of these oscillations.

§.§.§ Neutrino oscillometry with Jinping

Another neutrino artificial source proposed to study the possible oscillation of active to sterile neutrinos is the isotope $^{144}\rm{Ce} - ^{144}\rm{Pr}$. In this case, the idea is to use such a source for the future solar neutrino experiment Jinping <cit.>.

s a source of $\bar{\nu}_e$, the decay chain of isotopes $^{144}\rm{Ce} - ^{144}\rm{Pr}$ is a suitable option for oscillometry experiment.

The antineutrino energy spectrum of $^{144}\rm{Pr}$ is continuous with the end point around 3 MeV and with an overall half-life of 285 days. About 48.5% of the emitted antineutrinos are at energies above the detection threshold of the inverse beta decay (IBD) reaction (the value of the threshold is 1.8 MeV) and thus can be used for the measurements. Based on  previous calculations, the maximal source activity can reach 100 kCi <cit.>.

Following <cit.>, the Jinping neutrino detector will be located in the Jinping Mountain, Sichuan Province, China with a maximum overburden around 2400 meters.
The Jinping collaboration plans to build a 2 kton detector using slow liquid scintillator (LSc). This delays the scintillation process and thus separates from the Cherenkov light, significantly increasing the background rejection capability using the particle identification method. The inner detector volume will have a spherical shape with a radius around 8.2 meters. The expected energy resolution will be 5%$\sqrt{E\rm{[MeV]}}$ and the position resolution  10 cm/$\sqrt{E\rm{[MeV]}}$.

The schematic layout of the proposed experiment at Jinping. Two positions for the radioactive source are assumed: at the center (position I) and outside of the detector (position II). Two fiducial volume cuts are applied. Inner cut with radius 100 cm and outer cut with width 70 cm. The yellow area is the active volume for position I, the yellow area plus the inner cut region is active volume for position II <cit.>.

The proposed experimental setup is shown in Fig. <ref>. Two possible locations for the point source are being considered: one at the center of the sphere (position I) and another at a distance 3 m from the edge of the detector (position II). The source at the detector center case gives a higher statistics but a shorter range of baseline. Technically, the more realistic case is when the source is outside of the detector. At both source positions the exposure time is assumed to be 450 days. Initial source activity will be 50 and 100 kCi for position I and position II respectively. The expected non-oscillation event rate is 28.5K and 73.8K for position I and position II respectively.

The exclusion contours on a two-dimensional parameter space. 90% and 95% C.L. are shown for two possible setups. Combined reactor anomaly is also shown <cit.>. The stars indicate the current best-fit value for the sterile neutrino hypothesis <cit.>, the bes-fit value of the reactor antineutrino anomaly (RAA)  <cit.>, and the best-fit result of the Neutrino-4 experiment <cit.> from left to right respectively <cit.>.

The expected sensitivity to sterile neutrinos, represented as a two-dimensional exclusion plot, compared with another proposed experiments is drawn on Fig. <ref>. All currently interesting regions for sterile neutrino searching are covered if position I is implemented. For position II, some part of the RAA and best fit are still not fully covered.

§.§.§ THEIA

The most recently updated sterile neutrino landscape based on the results from BEST and MicroBooNE combined with short-baseline reactor experiments point to the possibility of active to sterile neutrino oscillations in the region with $\Delta m^2  > 1$ eV$^2$ which implies that oscillation pattern  is only measurable at distances of less than 10 m.

The three direct independent methods to test sterile neutrino hypothesis include a very short-baseline reactor experiment, accelerator experiment and use of neutrino and antineutrino generators in the vicinity of the large liquid scintillator (LS)  or water based liquid scintillator detector (WbLS). The 100 kton (25 kton)  WbLS THEIA detector  <cit.> represents a particularly promising venue for a decisive test using neutrino and antineutrino generators,  to observe a distance-dependent neutrino flux from the source at the distances of the order of oscillation length.

The neutrino oscillation length is given by the following formula:

\begin{equation}
L_{osc}[\rm{m}] = 2.48 \frac{E_{\bar \nu_e} [\rm{MeV}]}{\Delta m^2_{new} [\rm{eV}^2]}.
\end{equation}

In the case of sterile neutrino $\Delta m^2 > 1$ eV$^2$, the oscillation distance is of the order of a couple of meters. THEIA spans over tens of meters of distance and the source can be placed within meters of the target, creating a baseline comparable to the sterile neutrino oscillation length.  Observation of the oscillation pattern in the distant-dependent measurement in WbLS/LS  would represent convincing proof of the existence of sterile neutrinos and their oscillation with the other three flavors.

Sterile neutrino search with antineutrino generators in THEIA
THEIA  can detect $\bar \nu_e$ via inverse beta decay reaction (IBD) on hydrogen that makes it easily distinguishable from the backgrounds thanks to the double coincidence signature. The most promising antineutrino generator for this search is a pair of beta decaying nuclei $^{144}$Ce-$^{144}$Pr thanks to high yield of  $^{144}$Ce in the spent nuclear fuel combined with high  endpoint at 3 MeV with respect to 1.8 MeV IBD  interaction threshold, and relatively long half-life of 285 days.  There are two possible configurations: placing the source in the center of the detector for maximum interaction rate or on the side of the detector which is less disruptive and preferred but leads to reduced antineutrino flux. Nevertheless, due to its huge volume of 100 kton THEIA  will observe over 100 million antineutrino events in case of no oscillation (a couple of million less in case of sterile neutrino oscillations) with 3.7 PBq source running for 18 months. This measurement is practically background free and statistical error is negligible leading to a very sensitive measurement of large mass sterile neutrino oscillations even if the mixing angle is very small using distance-dependent  rate and spectrum measurement. It is worth noting that even with an order of magnitude weaker $^{144}$Ce-$^{144}$Pr source, THEIA would still make a robust measurement of sterile neutrino oscillations. A 0.37 PBq source has been successfully made in the past and is logistically easier for shielding, handling and transport.

In addition, the IsoDAR<cit.> collaboration is working on producing a feasible decay at rest source that is based on irradiation of $^7$Li with a strong neutron flux to produce $^8$Li that beta decays within  838 milliseconds. Emitted antineutrinos have a Gaussian energy spectrum in the range from 3 MeV to 13 MeV. This work is progressing steadily, but is not available for production. The advantage of this source is that it produces antineutrinos at higher energy and at a steady rate.  IsoDAR would also provide practically background-free high-statistics measurements of sterile neutrino oscillations.

Sterile neutrino search with neutrino generators in THEIA
The most exciting opportunity for THEIA to test the sterile neutrino hypothesis would be to directly cross-check the GA and BEST result by deploying $^{51}$Cr electron neutrino source. As opposed to radiochemical measurements THEIA would measure  a Compton-like shoulder due electron neutrino backscatters produced by the $^{51}$Cr source.  As opposed to bulk rate measurement of the event rate, THEIA would allow spectrum measurement with electron neutrino detection in real time. $^{51}$Cr emits a monoenergetic  753 keV gamma 90% of the time. This energy translates at the Compton-like shoulder at 0.5 MeV, which is slightly below 0.6 MeV energy threshold of the THEIA pure WbLS phase. However, as part of the Neutrinoless Double Beta Decay search, a 8 m radius balloon filled with pure liquid scintillator (LS) will be deployed. The LS will have a high scintillation yield, resulting in better energy resolution and lower energy threshold required to detect  electron neutrinos from $^{51}$Cr. This would present an excellent opportunity for sterile neutrino search with $^{51}$Cr electron neutrino source, directly testing GA and BEST results with electron neutrino detection in real-time.

In conclusion, a large WbLS detector such as THEIA  will provide an exceptional playground for the detailed study of sterile neutrino oscillations, requiring modest energy and vertex resolution. Additional important ingredients include knowledge of the absolute incoming antineutrino (neutrino) flux that can be measured in sources and a few meter distance between the source and the detector.
THEIA WbLS detector has a high potential to carry out a powerful search for sterile neutrinos free of statistical limitations that plague current generation of experiments, in the high sterile neutrino mass regime of $\Delta m^2 > 1$ eV$^2$. With its large target mass of up to 100 kton,  a tremendous number of IBD interactions from $^{144}$Ce-$^{144}$Pr source  will be collected allowing detailed, high statistics study of the position dependent neutrino flux. THEIA can also directly cross-check GA and BEST high confidence  electron neutrino measurement with an independent detection method.

§.§ Atmospheric Neutrino Experiments

§.§.§ IceCube Upgrade

The IceCube Upgrade <cit.> will begin operation in 2026 and provide both a lower energy threshold and more precise calibration for the full IceCube Neutrino Observatory.
Presently dominant sources of systematic uncertainty for IceCube sterile neutrino analyses include absorption and scattering in South Pole glacial ice <cit.>, both expected to be significantly reduced by improved precision of detector calibrations from the Upgrade with its smaller optical module spacing and enhanced calibration systems.
Reduced thresholds will enable unprecedented precision for $\nu_\tau$ appearance searches from standard oscillations, and will similarly contribute to improvements of low-energy sterile neutrino searches at IceCube.

§.§.§ DUNE

As noted in Section 6.2.2.1 (DUNE LBL), DUNE <cit.> is primarily designed as an accelerator experiment to accurately measure the value of $\delta_{\rm CP}$ using a neutrino beam produced from the LBNF accelerator complex at Fermilab.
It will employ a group of Near Detector(ND) and a Far Detector (FD) to measure neutrino oscillation parameters.
The DUNE FD will be located at the Sanford Underground Research Facility, South Dakota.
Benefiting from low cosmic-muon background, it will also act as an atmospheric neutrino detector, and will make measurements of several oscillation parameters for SM and BSM physics models, which will be complementary to the measurementss made with the accelerator neutrino data.

The FD will be constructed as four separate LArTPC modules, with a total fiducial mass of 40 kt.
While DUNE will collect significantly lower event statistics compared to very large volume neutrino telescopes such as IceCube <cit.>, the excellent energy and direction reconstruction will enable competitive atmospheric physics measurements.
With LArTPC detectors, very high-resolution spatial tracking (up to millimeter accuracy) of particles produced in neutrino interactions is possible and it can resolve final state morphologies to an extent not possible with other detector technologies.
DUNE will also have a finer control of detector and neutrino interaction systematics, enabled by its Near Detectors in the beamline.
Between IceCube and DUNE atmospheric neutrino event measurements, there will be a complementarity in terms of large sample size and high resolution.
Notably, using atmospheric neutrino events in a few GeV range, DUNE FD will be able to discover the Neutrino Mass Ordering (NMO) with a statistical significance of 3$\sigma$ or higher with 10 years of data <cit.>.
DUNE also has potential to measure the value of $\delta_{\rm CP}$ using sub-GeV atmospheric neutrino event samples <cit.>.
It will constrain many BSM physics models such as sterile neutrinos, NSI, neutrino decays and Lorentz Invariance Violation (LIV).

Atmospheric neutrinos <cit.> are produced in the interaction of the cosmic rays with atmospheric nuclei and contain both $\overset{(-)}{\nu_e}$ and $\overset{(-)}{\nu_\mu}$ components.
They span a broad energy range from MeV to PeV, with a peak around 1 GeV. Propagation of atmospheric neutrinos through the Earth gives accessible L/E from 10 to $10^4$ kM/GeV and their measurements can provide sensitivities to $\Delta m^2 \; \sim$  $(10^{-4} - 1) \; \rm{eV}^2$.
The atmospheric data at DUNE FD will make measurement of $\nu_e$ and $\nu_\mu$ disappearance as well as $\nu_\tau$ appearance channels.
Hence, the DUNE FD can constrain all three new mixing angles $\theta_{14}, \; \theta_{24}$ and $\theta_{34}$ over the aforementioned broad range of $\Delta m^2_{41}$ with the atmospheric data.
Sensitivities to these parameters are expected to be enhanced around $10^{-2} - 10^{-3} \; \rm{eV}^2$ thanks to matter effects, as neutrinos cross the Earth's mantle and core during their propagation.
Preliminary sensitivity study suggests that with conservative assumptions on detector properties such as resolutions and PID classification, the DUNE FD can probe sterile parameter space with sensitivities comparable to the current limits from other experiments.
Dedicated sensitivity studies to these parameters using DUNE atmospheric neutrino MC simulations is planned in the near future.
A joint fit of atmospheric and beam neutrino data is expected to improve parameter constraints over both individual fits. Using DUNE atmospheric data, constraining more complicated models such as decaying sterile neutrinos is also foreseen.
Additionally, due to DUNE's capacity to inspect in detail the energy losses of muons, they can infer the energy of uncontained high-energy muons.
This provides sensitivity to DUNE to high-energy atmospheric neutrino analysis as shown in Ref. <cit.>, which can provide complementary constraints in light sterile neutrino to the ones provided by the contained events.

§.§.§ Hyper-Kamiokande

Hyper-Kamiokande (Hyper-K) is the next-generation large water Cherenkov detector to be deployed in Japan.
Beyond their accelerator program discuss previously in this white paper, they also are able to measure atmospheric neutrinos much like its predecessor Super-K.

The expected sensitivity to light sterile neutrinos from Hyper-K has been reported in Ref. <cit.>.
As in the case of Super-K they are predominantly sensitive to $|U_{\mu 4}|^2$ and $|U_{\tau 4}|^2$ mixing elements by looking for distortions in the angular and energy distributions of their events.
The expected sensitivity can be seen in Fig. <ref> compared to the previous constraints from Super-K.
As can be seen from this figure the improvement on $|U_{\mu 4}|^2$ is marginal; however, significant improved sensitivity can is expected in $|U_{\tau 4}|^2$.
This is interesting since currently ANTARES has reported a small preference for non-zero $|U_{\tau 4}|^2$, which would be tested by Hyper-K.

Hyper-K expected 90% C.L. upper limits on $|U_{\mu 4}^2$ appear as red lines in the left figure.
The right figure shows 90% (solid) and 99% (dashed) C.L. limits on $|U_{\mu 4}^2$ vs $|U_{\tau 4}^2$ for a 5.6 Mton·year exposure (red) in comparison with recent limits from Super-K (black) Figure from Ref. <cit.>.

§.§.§ Atmospherics in THEIA

THEIA is a proposed multi-kiloton hybrid optical neutrino detector designed with multiple physics goals in mind <cit.>. The experiment plans to leverage novel scintillating materials to fully exploit information from scintillation and Cherenkov light production mechanisms using advancements in fast photodetectors and spectral sorting technologies, on a large scale. This “hybrid” technology will simultaneously enable high light yields for superb position and energy reconstruction, direction reconstruction from Cherenkov light and particle identification (PID) through knowledge of both mechanisms. There is also the potential to employ isotope loading techniques for various physics needs (Sec. <ref>).

While physics potentials have been evaluated for long baseline physics, solar neutrinos, supernova neutrinos, DSNB, geoneutrinos, reactor neutrinos, neutrinoless double beta decay and nucleon decay in <cit.>, the potential for leveraging atmospheric neutrinos, especially in regards to the picture of the short-baseline anomalies and possible sterile neutrinos and non-standard neutrino interactions (NSIs) has not been previously examined. Given the ring-imaging capability of a Cherenkov detector, THEIA would be capable of an analysis similar to that performed by Super-Kamiokande <cit.>. THEIA would have similar sensitivity to Super-Kamiokande, with enhanced samples based on PID techniques –for example, by tagging neutrons more effectively.  Yet, it would be many years before THEIA would equal the exposure used in the analysis of <cit.>. However, as discussed in <cit.>, Hyper-Kamiokande may be able to enhance this picture by simultaneously employing beam and atmospheric neutrinos for both sterile neutrino and NSI searches, which also has complementarity with DUNE. A placement of THEIA in the LBNF beamline could further supplement these experiments, due to utilization of similar detection technology to Hyper-Kamiokande with a location in the same beam as DUNE. This would provide similar detectors in two different locations and beams (Hyper-Kamiokande and THEIA) as well as two different detection technologies in the same location and beam (DUNE and THEIA), which could open significant opportunities in further exploring parameter space and understanding systematic uncertainties across the three experiments. Further studies are planned to examine this possibility.

§.§.§ KM3NeT and ORCA

Neutrino telescopes offer a complementary approach to investigate the anomalies found in short-baseline experiments.
In this section we will focus on the capabilities of the KM3NeT detector <cit.>, which is now under construction in the Mediterranean Sea.
It should be also mentioned that KM3NeT's predecessor, ANTARES, also in the Mediterranean Sea, has also looked for sterile neutrinos with the data gathered in ten years <cit.> as discussed in prior sections of this work.

The detection principle is as follows.
Neutrinos (originated in cosmic sources or in the Earth's atmosphere) can interact in the surroundings of the detector and produce particles which will induce Cherenkov light that can be detected by an array of photomultipliers encapsulated in the so-called Optical Modules (OMs).
The KM3NeT neutrino telescope is made of two subdetectors, ORCA and ARCA.
ORCA is installed at a depth of 2500 m, off the French coast in Toulon.
ARCA is located 3000 m deep, close to Porto Palo, in Sicily.
Both use the same technology, being the main difference the distance between the OMs.
ORCA is denser and will have 115 lines with 18 OMs each distributed in a volume equivalent to 8 MTon.
The energy threshold estimated for this configuration is about 3 GeV.
ARCA, with 230 lines, will have a sparser distribution of OMs, which will instrument about one cubic kilometer of water.
The difference in size and density drives the physics focus of each configuration.
ORCA is more suited for studies on atmospheric neutrinos and low energy astrophysical events, while ARCA's main goals are high energy neutrinos from astrophysical sources (with particular interest in Galactic sources given its geographical location).
At the time of this writing, there are 10 lines of ORCA and 8 lines of ARCA already installed and taking data, with plans to install more lines in the following months.

ORCA will precisely measure various parameters of the three-neutrino standard oscillation framework, namely the neutrino mass ordering, $\sin^{2}\theta_{23}$, $\Delta m_{13}^{2}$, and PMNS unitarity <cit.>.
As a matter of fact, the first oscillation study with a partial configuration and less than one year of exposure finds that
oscillations are preferred with a confidence level of $5.9\sigma$ over “no oscillations” <cit.>.
The main analysis selects events induced by atmospheric neutrinos coming below the horizon and compares the observed energy, direction and topology distributions to predictions under different oscillation hypotheses.
In the energy range 3-100 GeV, in which the ORCA detector is most sensitive to neutrinos, events are categorised in two event topologies: tracks, characterised by a long muon track (mostly from muon-neutrino CC interactions), and showers, characterised by events with no distinguishable tracks.
ORCA is an excellent instrument to constrain oscillation parameters, primarily due to better energy and angular resolutions than other neutrino telescopes in this energy regime as shown in Fig. <ref>.

Left: Median direction resolution as a function of true neutrino energy for showers and tracks. Right: Probability distribution of the reconstructed energy as a function of true neutrino energy for upgoing showers and tracks (black lines indicate the quantiles). Credit: KM3NeT Collaboration.

The presence of light-sterile neutrinos, enhanced by matter effects, would alter the observed rate of events in the GeV regime.
For scenarios with large mass splitting and non-zero $\theta_{24}$ and $\theta_{34}$, a deficit of high-energy track-like events is expected.
This region of the phase space has been probed by neutrino telescopes, such as ANTARES <cit.> and IceCube/DeepCore <cit.>.
In addition, ORCA is well suited to test non-zero $\theta_{14}$ models, as the rate of showers around 20 GeV depends on this mixing angle.

The 99% C.L. ORCA sensitivity after three years of data taking for different combination of sterile parameters. Top left: $\theta_{24}-\theta_{34}$, with $\Delta m_{41}^{2}=1$ eV${^2}$ (ANTARES <cit.>, IceCube <cit.>, and SK <cit.>). Top right: $|U_{\mu e}|^{2}$ (Daya Bay+Bugey-3+MINOS+ <cit.>, LSND <cit.>, and MiniBooNE <cit.>). Bottom: $\sin^2\theta_{24}$ (SK <cit.>, IceCube <cit.>, MINOS+ <cit.>, and cosmology <cit.>), $\sin^2\theta_{14}$ (Daya Bay+Bugey-3+MINOS+ <cit.>, STEREO <cit.>, Neutrino-4 <cit.>, global fit <cit.>, reactor fit <cit.>, and cosmology <cit.>), and $\sin^2\theta_{34}$ (SK <cit.>, IceCube <cit.>, and cosmology <cit.>). Credit: KM3NeT Collaboration.

First sensitivity studies have been performed using a joint binned likelihood for three samples (track, shower, and others), in which systematic uncertainties on the neutrino flux, detector response and standard oscillations were included <cit.>.
Fig. <ref> shows the ORCA sensitivity to sterile mixing angles in different scenarios after three years of data taking.
In the large $\Delta m_{41}^{2}$ limit, ORCA can set competitive constraints in $\theta_{24}$ and $\theta_{34}$[In this analysis $\delta_{24}$ is kept free in the fit and $\theta_{14}$ and $\delta_{14}$ are fixed to zero].
It will be able to improve current limits on $\sin^2\theta_{24}$, $\sin^2\theta_{34}$, $\sin^2\theta_{14}$ and $\sin^2 2\theta_{\mu e}$ for low values of the mass splitting[These analyses kept free all relevant sterile neutrino parameters], not yet constrained by cosmology or appearance/disappearance measurements.
Furthermore, ORCA will be able to test the majority of the LSND and MiniBoone anomaly region as well as part of the Neutrino-4 allowed region.

Other BSM scenarios can be also tested with ORCA detector. In particular, competitive sensitivities to NSI and invisible neutrino decay can already be achieved with partial configurations of the detector <cit.>. When completed, the ORCA detector will potentially become a leading tool for probing these models.

ARCA will be able to observe neutrinos in the high-energy regime <cit.>, exceeding IceCube's capabilities for up-going muon neutrinos as shown in Fig. <ref>.
Several studies have shown that the presence of sterile neutrinos or NSI result in the disappearance of TeV-scale muon neutrinos passing through the Earth <cit.>.
Recently, IceCube has set world-leading constraints in some of these models characterising atmospheric neutrinos in the few TeV range <cit.>. Preliminary studies are currently underway to assess ARCA sensitivities.

Left: ARCA angular resolutions as function of the neutrino energy for $\nu_\mu$ CC events. IceCube resolution was extracted from the point source analyses <cit.>. Right: ARCA effective area as function of the neutrino energy for the selected event sample. IceCube effective area was extracted from the diffuse analyses <cit.>. Credit: KM3NeT Collaboration.

§.§ Beta and Electron Capture Decay Experiments

Beta decay and electron capture experiments are complementary to oscillation experiments in searches for sterile neutrinos.
Exploiting the kinematics of the weak process, these experiments have a clean probe of the coupling of the electron flavor neutrino to the mass states.
As described in Sec. <ref>, the observed spectrum is a superposition of spectra corresponding to each mass state with endpoints shifted by the different neutrino masses.
These experiments have the capability to search for sterile neutrinos ranging in mass between sub-eV/$c^2$ and MeV/$c^2$.

§.§.§ KATRIN/TRISTAN

As described in Sec. <ref> KATRIN is sensitive to light sterile neutrinos, providing complementary information to oscillation-based searches.
As the analysis interval of KATRIN is currently constrained to 40 below $E_0$, only sterile neutrinos up to a corresponding mass of about $m_4 < \SI{40}{\electronvolt}$ can be searched for.

To search for heavier sterile neutrinos, a larger range of the tritium $\beta$-spectrum must be measured. In particular, keV-scale sterile neutrinos would be of interest, as they are potential dark matter candidates <cit.>. However, indirect observations and cosmological considerations limit their mixing with active neutrinos to $\sin^2\Theta<10^{-6}$.

As the count rate increases further away from the endpoint, the statistical uncertainty decreases. Assuming the source strength of KATRIN and a measurement time of about one year, a statistical sensitivity better than $\sin^2\Theta<10^{-6}$ could be reached <cit.>. However, a high-statistics measurement of the entire tritium $\beta$-decay spectrum poses a new technical challenge: electron rates exceeding $10^8$ cps and several new systematic uncertainties become relevant when describing the experimental tritium spectrum far away from the endpoint.

The TRISTAN project is exploring the sensitivity of such a search and is developing a new silicon drift detector focal-plane array for KATRIN with more than 1000 pixels <cit.>. This technology allows a measurement of a high $\beta$-electron flux with an energy resolution of 300 eV (FWHM) for 20 keV electrons, and large (3 mm) pixel footprints. Such a high-resolution detector would enable a differential measurement of the full tritium $\beta$-decay spectrum, in order to search for signatures of physics beyond the SM. The new detector will be installed after completion of the neutrino mass measurements.

§.§.§ Project 8

Project 8 is a $\beta$-decay endpoint experiment with an aim to precisely measure the neutrino mass using cyclotron radiation emission spectroscopy (CRES) and atomic tritium as the source <cit.>.
CRES is a frequency-based spectroscopic method that leverages the energy dependence of cyclotron radiation to perform a high resolution differential spectrum measurement over a wide energy range <cit.>.
The use of atomic tritium provides a tight control on systematics by eliminating the uncertainty associated with the final states in the decay of molecular tritium <cit.>.

Project 8 has undertaken a phased approach with each phase demonstrating a  critical technological milestone.
Phase-I of the experiment demonstrated CRES using internal conversion electrons from $^{83m}$Kr <cit.>.
Phase-II performed the first CRES-based molecular tritium endpoint measurement and placed an upper limit on neutrino mass <cit.>.
Critical R&D for Phase-III of the experiment is ongoing with two parallel efforts to demonstrate: i) CRES in larger volumes and ii) the use of atomic tritium.
Phase-III of the experiment plans to initially use molecular tritium followed by atomic tritium and has a projected sensitivity to the neutrino mass $m_{\beta}$ of 0.4 eV/c$^2$ and 0.3 eV/c$^2$ using molecular and atomic tritium, respectively.
Phase-IV is planned to be the ultimate Project 8 mass measurement experiment, with a goal to reach an unprecedented sensitivity to $m_{\beta}$ of $<$ 0.04 eV(90% C.L.).

As a differential spectroscopy method, CRES allows Project 8 to make simultaneous searches for both active and sterile masses.
Furthermore, the benefits of the CRES technique for neutrino mass measurement—namely low backgrounds, good resolution, and high event rates—also apply to a search for sterile neutrinos.
Hence, a superior sensitivity to the direct neutrino mass also provides superior sensitivity to a sterile neutrino.

Sensitivity of the upcoming phases of Project 8 experiment to light sterile neutrinos in the 3+1 framework. All curves, including reactor and BEST suggested parameters, are shown at 95 % C.L. The sensitivity is statistically limited and includes the current best knowledge of the systematics from energy resolution arising from thermal doppler broadening, frequency to energy conversion, and variations in magnetic field. Control of systematic uncertainties leading to sensitivities in $\sin^2(2\theta) > 0.1 $ (shown as a solid line) is expected to be straightforward. Further careful systematic control could enable experimental sensitivities down to $\sin^2(2\theta) \sim 0.01$ (shown as dashed lines). In the near future, with Phase-III, Project 8 aims to reach down to $\Delta m^2_{14}\sim$ eV$^2$ (C.L. 95%) and cover major portions of the reactor <cit.> and BEST <cit.> gallium anomaly suggested parameter spaces including BEST best-fit point. Phase-IV of Project 8 aims to completely cover the reactor and BEST gallium anomalies at high significance.

The sensitivity of Project 8 to sterile neutrinos is determined using the analytical neutrino mass sensitivity method as suggested in Ref. <cit.>.
Figure <ref> shows the upcoming phases of Project 8 to be capable of a competitive sterile neutrino search over several orders of magnitude in $\Delta m^2_{14}$.
The sensitivity estimates include statistical and systematic uncertainties as noted in the caption.
The experiment's sensitivity to higher $\Delta m^2_{14}$ is primarily limited by the efficiency of the cyclotron frequency detection method for lower energy electrons since a higher value of $m_4$ would manifest as a kink at a lower energy in the $\beta$-decay spectrum.
In generating sensitivity curves in Fig. <ref>, the efficiency was assumed to be well-understood for energies tens of eV below the endpoint.
This is a fair assumption based on Project 8's ability to quantify the efficiency of the complex Phase-II detector over 2.5 keV below the endpoint.
The two different detection methods being investigated for upcoming phases are yet to demonstrate the control of efficiency over the wide energy range, but are expected to have lower complexity in efficiency than in Phase-II.
Project 8 through the differential $\beta$-decay spectrum measurement using CRES thus provides a promising avenue to search for light sterile neutrinos in the near future.

§.§.§ HUNTER

The HUNTER spectrometer.

The HUNTER Collaboration[Temple University, UCLA, University of Houston, Racah Institute of Physics, and Princeton University] brings together techniques from high energy, nuclear, and AMO physics to perform a laboratory sterile neutrino search aimed at the mass range 10-300 keV/$c^2$ <cit.>.
The neutrinos are emitted in the electron capture (EC) decay of $^{131}$Cs  atoms captured in a magneto-optical trap and cooled to milliKelvin temperatures where they are essentially at rest.
In electron capture, the initial final state is an excited atom recoiling against the emitted neutrino – for a given neutrino mass, the neutrino is mono-energetic.  All the observable particles in the EC decay and subsequent de-excitation of the recoiling atom are reconstructed.  These particles are a $^{131}$Xe ion, an x-ray, and one or more Auger electrons, each of which requires different measurement techniques.
Figure <ref> shows a CAD drawing of the HUNTER  apparatus.
The meter-long horizontal arms of the vessel contain spectrometers for the ion (left) and electrons (right).  In the ion spectrometer, the electrostatic field focuses the ions onto an MCP which allows determination of the ion's vector momentum using the impact location and time of flight.  The electron spectrometer is similar, but requires an axial magnetic field to contain the electrons.  X-rays are detected via scintillator panels, a signal that triggers the readout and provides the start time for the time of flight measurements.

Left: simulated spectrum (blue histogram) of one year of livetime including a hypothetical 60 keV/$c^2$ sterile neutrino with $\sin^2\theta=3\times 10^{-4}$.  Right: HUNTER's sensitivity for three phases of the experiment, compared to other laboratory limits.

The left panel of Fig. <ref> is a simulated spectrum including the apparatus resolution and the major backgrounds.
A simulated neutrino signal within the reach of the first phase of HUNTER with a year of live-time is shown.  The right panel shows the reach of the initial HUNTER configuration and two potential upgrades.

§.§.§ DUNE

Large liquid argon detectors making use of atmospheric argon, associated with $\mathrm{{}^{39}Ar}$ beta decay activity of roughly 1Bq/kg <cit.>, will enable the probing of $m_{4}$ values below the $\mathrm{{}^{39}Ar}$ beta decay end point of $Q = \SI{565}{keV}$.  By utilizing very large detectors with large volumes of liquid argon, a substantial amount of beta decays can be detected as to enable sensitive measurements of $|U_{e4}|^2$ at larger $m_{4}$ values than for $\mathrm{{}^{3}H}$.  Given that liquid argon detectors (such as liquid argon time projection chambers, or LArTPCs) function as total absorption calorimeters, the liquid argon provides both the source and the detector for such measurements, in contrast to measurements using $\mathrm{{}^{3}H}$.

The Deep Underground Neutrino Experiment (DUNE) <cit.> at the Long-Baseline Neutrino Facility will use massive LArTPCs to study accelerator neutrinos (∼1GeV) undergoing flavor oscillations over a long baseline (∼1300km) in order to probe leptonic CP violation.
A search for heavy neutral leptons can also be carried out using ionization charge measurements of $\mathrm{{}^{39}Ar}$ beta decays in the DUNE far detector, enabling sensitivity to $|U_{e4}|^2$ in the 20450keV mass range.
Figure <ref> shows projected statistical upper limits (95% C.L.) for $|U_{e4}|^2$ at DUNE as a function of $m_{4}$.
Recording the full DUNE $\mathrm{{}^{39}Ar}$ beta decay dataset toward maximum sensitivity to $|U_{e4}|^2$ (better than $10^{-6}$) requires substantial development of the trigger system at DUNE, including selectively recording low-energy activity within “regions of interest” inside of the detector; this capability is currently under development.
Reconstruction of $\mathrm{{}^{39}Ar}$ beta decays has been previously carried out at MicroBooNE <cit.>, a LArTPC neutrino experiment at Fermilab, demonstrating that low thresholds (roughly 100keV) and good energy resolution from low TPC noise levels (roughly 50keV) are achievable in large LArTPC detectors <cit.>; more comprehensive studies at ProtoDUNE-SP <cit.> are currently in progress.
Further studies on the impact of $\mathrm{{}^{39}Ar}$ beta decay spectrum theoretical uncertainties <cit.>, experimental systematic uncertainties, and radiological backgrounds in the DUNE far detector are also in progress.
While these additional considerations may lower the sensitivity to $|U_{e4}|^2$ considerably, given the currrent global limits on $|U_{e4}|^2$ of $10^{-2}$ to $10^{-4}$ in the relevant range of $m_{4}$ values <cit.>, significant improvement is expected.

Projected upper limits (95% C.L.) for $\sin^{2}\theta = |U_{e4}|^2$ as a function of heavy neutral lepton mass $m_{\mathrm{heavy}} = m_{4}$ for DUNE. “DUNE-ALL” refers to a limiting dataset containing every single $^{39}$Ar beta decay occurring during detector operations ($2 \times 10^{16}$ electrons), while “DUNE-EXT” refers to a baseline dataset obtained using external triggers and full detector readout without special localized triggering for low-energy activity in the detector ($1 \times 10^{13}$ electrons).  All upper limits shown are purely statistical (no theoretical/experimental systematic uncertainties included) and assume zero background.

§.§.§ BeEST

The BeEST (Beryllium Electron-capture with Superconducting Tunnel junctions) experiment is a model-independent search for sub-MeV sterile neutrinos using implanted radioactive $^7$Be atoms into high-rate superconducting quantum sensors <cit.>.
The BeEST concept uses momentum reconstruction of the low-energy $^7$Li daughter atom following EC decay of $^7$Be to infer direct mass information of the neutrino.
$^7$Be ($T_{1/2}=53.22(6)$ days <cit.>) is a pure EC decaying isotope, and the ideal case for neutrino studies via momentum reconstruction due to its large decay energy $Q_{EC}=861.89(7)$ keV <cit.>, relatively high maximum value for the recoil kinetic energy $T_{D(max)}=56.826(9)$ eV, and simple atomic and nuclear structure <cit.>.
Due to the explicit neutrino mass dependence on the recoil kinetic energy, the existence of a heavy mass state, $m_i$, would cause the nuclear recoils to have a lower kinetic energy, and the relative fraction of these shifted events to the total determined by the mixing with the $\nu_e$ flavor, $|U_{ei}|^2$.

The BeEST experiment uses Superconducting Tunnel Junction (STJ) sensors to detect the low-energy radiation following EC decay.
STJs are high-speed quantum sensors that were originally developed for high-resolution X-ray spectroscopy in astronomy and material science <cit.>.
STJs are a type of Josephson junction that consists of two superconducting electrodes separated by a thin insulating tunneling barrier.
The absorption of radiation in one of the electrodes breaks the Cooper pairs of the superconducting ground state and excites free excess charge carriers above the superconducting energy gap $\Delta$ in proportion to the absorbed energy.
This results in exceptionally high energy resolution ($\sim1$ eV) for low-energy radiation relevant to nuclear recoils.
Each STJ detector pixel is able to operate at rates up to $10^4$ counts/s with exceptionally high predictability on the detector response <cit.>, making them ideal for searches of this type.

The $^7$Be$^+$ rare isotopes are implanted into the STJ detectors through Si collimators at TRIUMF's Isotope Separator and ACcelerator (ISAC) facility in Vancouver, Canada at an energy of 25 keV.
The $^7$Be$^+$ beam is produced using the isotope separation on-line technique via spallation reactions from a 10 $\mu$A, 480-MeV proton beam incident on a stack of thin uranium carbide targets.
Following implantation of the rare isotope, The $^7$Li recoil spectrum from the decay of $^7$Be is measured at Lawrence Livermore National Laboratory (LLNL) with the STJ detector at a temperature of $\sim$0.1 K in a two-stage adiabatic demagnetization refrigerator (ADR) and the signals are read out with a specialized current-sensitive preamplifier <cit.>.
For a precision energy calibration, the STJs are simultaneously exposed to 3.49965(15) eV photons from a pulsed Nd:YVO$_4$ laser triggered at a rate of 100 Hz <cit.>.
The measured $^7$Li recoil spectrum contains four peaks that result from the two nuclear and two atomic processes following EC decay.
These include one for $K$-capture decay to the nuclear ground state (K-GS), one to the excited state of $^7$Li (K-ES), and two for the corresponding $L$-capture decays (L-GS and L-ES, respectively) (Fig. <ref>).

Due to the relative simplicity of the $^7$Be$\rightarrow^7$Li EC decay system, the spectral response is able to be precisely evaluated using modern theoretical tools.
The atomic de-excitation and auto-ionization (electron shake-up and -off) effects that follow EC decay generate higher-order features in the spectrum and are accessible to ab-initio atomic theory.
Additionally, the sequestration of beryllium in the sensors material generates small (eV-scale) chemical shifts in the EC decay energies.
To this end, density functional theory is used to model the electronic structure of beryllium and lithium atoms in different atomic environments of a polycrystalline absorber film of the sensor to provide a further improvement of sensitivity to new physics.
This work is already in an advanced stage, and will be employed to provide improved limits to the high-statistics data from the BeEST experiment planned for upcoming phases, and may allow for the extraction of significantly improved limits in the $10-100$ keV mass range.

The BeEST experiment has completed its proof-of-concept (Phase-I) in 2020 with the first demonstration of high-resolution nuclear recoil detection with STJs <cit.>.
Following this, the first low-activity (10 Bq) data set using a single tantalum-based STJ detector was taken for 28 days - including the use of precision in-situ laser calibration of the sensor response.
The resulting Phase-II data allowed for the first extraction of an upper limit for $|U_{ei}|^2$ from the BeEST experiment in the 100–850 keV mass range <cit.> , and improved upon previous decay measurements by up to an order of magnitude (Fig. <ref>).
Within the next 5 years, the BeEST will complete its four phases of the experiment and achieve limits approaching $|U_{ei}|^2\approx10^{-7}$ in the few-hundred keV mass range, as described further in Ref. <cit.>.
Following Phase-IV of the BeEST, a dramatic improvement in sensitivity is planned using large arrays ($10^4$ pixels) of STJs with new materials.
This development is ongoing, with the goal to probe relative couplings to the electron neutrino flavor of $|U_{ei}|^2\leq10^{-9}$.
The current and projected limits for the BeEST experiment are presented in Fig. <ref> and incorporate known and simulated detector responses under conservative assumptions on the atomic and materials theory work described above.

(left) Image of a 32-pixel Al-based STJ array in preparation for Phase-IV of the BeEST <cit.>.  (middle) The low-energy $^7$Li nuclear recoil spectrum for 28 days of acquisition from a single pixel counting at low-rate <cit.>.  The spectrum generated by a hypothetical 300 keV sterile neutrino signal with 1% mixing is shown in red (dashed). (right) Current and projected limits for the BeEST and SuperBeEST experiments compared to existing laboratory decay limits (shaded areas) and the TRISTAN projection.

§.§.§ PTOLEMY

The PTOLEMY experiment is designed to look for relic neutrinos by means of neutrino capture on a tritium target.
Searching for spectral distortions near the endpoint with its very intense source, the experiment also projects sensitivity to eV-scale sterile neutrinos <cit.>.

§.§.§ Double Beta Decay

While nuclear beta-decay has become a valuable tool for searching for MeV-scale sterile neutrinos, the relatively rare double beta-decay process is also sensitive to the presence of sterile neutrinos.
Often thought of solely as a background to searches for neutrinoless double-beta decay ($0\nu\beta\beta$), double beta decay occurs when an atom undergoes a simultaneous emission of two electrons and two antineutrinos.
This process leads to a continuous electron energy spectrum that cuts off at the characteristic Q-value for that isotope. The region of this spectrum that is typically studied in detail is the end-point because this is where the signal of neutrinoless double-beta decay will populate.
As discussed throughout this section, the presence of heavy sterile neutrinos will lead to deformation of the beta-decay spectra, and this is also true for double beta-decay <cit.>.

The summed energy distribution from the standard $2\nu\beta\beta$ spectrum (solid) and the impact from the presence of a 1 MeV sterile neutrino (dashed) for $^{100}$Mo (purple) and $^{136}$Xe (blue). Figure from Ref. <cit.>.

In the presence of a heavy sterile neutrino, the kinematics of standard two-neutrino double beta decay is modified in both the energy and angular distribution.
The modifications to the summed electron energy is spread across a relatively large range of energies but lead to an overall modulation of the spectrum at low energies, as shown in Fig. <ref>.
In addition to the impact on the energy of the electrons, the angular distribution is also strongly impacted by the presence of heavy sterile neutrinos.

The constraint offered by current $0\nu\beta\beta$ experiments sensitive to only the energy (solid blue) and those also sensitive to the angular distributions (solid red). Future (next-generation) experimental sensitivity are shown in dashed lines. Figure from Ref. <cit.>.

Constraints from current and planned neutrinoless double beta decay experiments have been explored in Ref <cit.>. These can be separated into experiments sensitive to just energy that has taken data (GERDA II, CUPID-0, NEMO-3, CUPID-Mo, and KamLAND-Zen) and planned to take data (LEGEND, CUPID, and DARWIN) and those sensitive to the angular distributions that have taken data (NEMO-3) and those planned (SuperNEMO). These can be seen in Fig. <ref>, where limits are placed on the sterile neutrino mass and the mixing strength. In Fig. <ref>, limits coming from current neutrinoless double beta decay experiments use exposures of O(100) kg-years, which corresponds to $10^4$ $2\nu\beta\beta$ events, while future next-generation experiments (those targeting the inverted hierarchy region of the neutrinoless double-beta decay phase space) use exposures of O($10^3 - 10^4$) kg-years, with $10^6$ $2\nu\beta\beta$ events. In the far future, experiments that wish to target the normal hierarchy region of the neutrinoless double beta decay will require exposures of O($10^6$) kg-years <cit.>, this increase in the number of events will enable further explorations of this sterile neutrino phase-space.

Leveraging data from $0\nu\beta\beta$ experiments to search for the presence of sterile neutrinos and other BSM processes <cit.> demonstrate the power diverse experimental data can offer for exploring and searching for anomalies.

§.§ Meson factories

§.§.§ NA62

Rare meson decays are well-known for providing some of the strongest constraints on light dark sectors. Two experimental efforts to measure the flavor-changing NC decay of kaons to neutrinos, $K^+ \to \pi^+ \nu\overline{\nu}$, are underway: NA62, with a $K^+$ beam <cit.>, and KOTO, with a $K_L$ beam <cit.>. In this section we discuss some prospects of NA62 to

The NA62 experiment is a kaon-decay-in-flight experiment located at CERN <cit.>. To achieve its primary goal, NA62 aims to have a total exposure of $\mathcal{O}(10^{13})$ kaon decays over a few years of run time. The detectors are designed for sub-percent momentum resolution on the kaon and final charged-track momenta. Kaons have an average of $75$ GeV momentum and their decays in flight throughout a distance of $\mathcal{O}(65)$ m are detected by a combination of spectrometers arranged in a cylindrical fashion around the beam.

NA62 can also perform several searches for new physics by searching for exotic kaon decays. Light particles have been searched for in $K^+ \to \ell^+ N_{\rm inv}$ <cit.>, as well as in $\pi^0 \to (\gamma) +\text{inv}$ <cit.>. In the context of short-baseline anomalies, NA62 has a world-leading sensitivity to HNLs invoked in some explanations of the MiniBooNE and LSND anomalies. The current limits on long-lived or invisible HNLs produced in $K_{2\ell}$ decays span the mass region between $100 \lesssim m_N < m_K - m_{\ell}$.

Two channels are of particular interest for HNL searches:

* $K^+ \to \ell^+ N$, where $N$ is long-lived or invisible,

* $K^+ \to \ell^+ (N \to \nu \ell^+ \ell^-)$, where $N$ decays inside the detector.

The first channel has already been searched for in both flavors, $\ell^+ = e^+$ and $\mu^+$ <cit.>. The limits reach mixing angles as low as $\mathcal{O}(10^{-9})$, in the region of interest for the Type-I seesaw mass mechanism. The strategy in this case consists in searching for the presence of an invisible resonance in $M_{\text{inv}}^2 = (p_K - p_\ell)^2$, therefore it is mostly sensitive to long-lived or invisibly-decaying HNLs. This constraint applies to the models discussed in Sec. <ref>.

Limits on $|U_{\mu}|^2$ as a function of the mass of the HNL $N$ (left) and the dark photon $A^\prime$ mass (right) in the dark sector model of Ref. <cit.>. The NA62 sensitivity to the prompt decays of HNLs in $K^+\to \mu^+ \nu e^+e^-$ and $K^+\to \mu^+ \nu \mu^+\mu^-$ searches is shown as a solid black line.

The second channel with three charged tracks is instead sensitive to short-lived HNLs. This possibility was first discussed in the context of a transition magnetic moment model <cit.>, discussed in Sec. <ref>. In that model, the electromagnetic decays of the HNL with an off-shell photon, $N \to \nu (\gamma^* \to e^+e^-)$, could dominate the SM rate. The measurement by BNL-E865 <cit.>, $\mathcal{B}(K^+ \to \mu^+\nu e^+e^-) = ( 7.06 \pm 0.3)\times 10^{-8}$ allowed to set a constraint on the mixing and branching ratio of the HNL of $|U_{\mu N}|^2 \times \mathcal{B}(N\to \nu \gamma) < 0.5\times 10^{-6}$ <cit.>. This constraint, however, only applies to HNLs with masses $m_N > 145$ MeV due to experimental cuts on  $m_{ee} > 145$ MeV, designed to suppress $\pi^0_D$ decays. A more precise measurement was performed at NA48, finding
$\mathcal{B}(K^+ \to \mu^+\nu e^+e^-) = (7.81\pm 0.23)\times 10^{-8}$, applicable to the region $m_{ee} > 140$ MeV.

Three charged track decays are also sensitive to HNLs that decay electromagnetically through a dark photon, $Z^\prime$, such as in the models discussed in Sec. <ref>. In this case, the HNLs can decay to either $\nu e^+e^-$ or $\nu \mu^+\mu^-$. Ref. <cit.> derived constraints in a specific dark neutrino model with pseudo-Dirac HNLs. The reach of NA48 to the $|U_{\mu 4}|^2$ mixing element was around $\mathcal{O}(10^{-8})$ or better. A dedicated search at NA62 can improve upon that reach by one to two orders of magnitude. Backgrounds from $\pi^0_D$ decay present the biggest challenge for this search, but can be suppressed by the requirement that the measured $M_{\nu ee}^2 \equiv (p_K - p_\ell)^2 = m_N^2$. In addition, if the mediator is produced on-shell, then a resonance search in the $\ell^+\ell^-$ invariant mass can be performed. In the case of a signal, NA62 could measure both the dark photon and the HNL mass. The NA62 sensitivity to the dark sector model in Ref. <cit.> is shown in Fig. <ref> under the assumption of no backgrounds. Based on Ref. <cit.>, for $K^+\to \mu^+\nu e^+ e^-$, a total of $N_K^{\rm Fid} = 2.14\times10^{11}$ fiducial kaon decays were assumed with an acceptance of ${\rm A}_\beta = 4\%$., while for $K^+\to \mu^+\nu \mu^+ \mu^-$, a total of $N_K^{\rm Fid} = 7.94\times 10^{11}$ and ${\rm A}_\beta = 10\%$ were assumed.

§.§ Collider Experiments

§.§.§ FASER$\nu$ and FLArE

The flux of broadband neutrinos from the LHC with energies around $\sim100$ GeV to $\sim1$ TeV in the forward direction provides a new opportunity to look for neutrino oscillations.
Existing experiments such as FASER$\nu$ <cit.> and SND@LHC <cit.> as well as proposed experiments which go under the umbrella term of Forward Physics Facility (FPF) will be sensitive to sterile neutrino oscillations.
Given a typical baseline of 600 m and typical energies in the 100 GeV to 1 TeV range, this corresponds to sensitivities to $\Delta m^2_{41}\sim1000$ eV$^2$, or $m_4\sim30$ eV.
While some existing constraints apply at this mass range in the oscillation averaged limit, there are no oscillation probes at this $L/E$ providing a new direct test of oscillations for larger $\Delta m^2$'s than are usually considered.
The forward physics program at the LHC also benefits from the production of all three flavors of neutrinos with hierarchical production rates that differ each by $\gtrsim1$ order of magnitude.
In addition to having all three flavors available at the source, FASER$\nu$ and SND@LHC both have flavor discrimination capabilities among the three flavors allowing for, in principle, probes of all 9 oscillation channels, subject to backgrounds and flux uncertainties.

The expected sensitivity to sterile neutrino oscillations in the $\nu_\mu$ disappearance channel at the LHC using Asimov and Feldman-Cousins for FASER$\nu$ at the upcoming LHC run and a proposed 10-ton LAr detector in a future HL-LHC run.
For comparison, the existing oscillation averaged constraints coming mostly from MINOS+ and MiniBooNE are also shown.
Figure is from <cit.>.

One major challenge is the presence of significant flux uncertainties which affect the normalization and, more importantly, the shape.
The relative contribution to the neutrino flux from different particles is rather poorly understood <cit.> and these shape effects could conceivably mimic a neutrino oscillation signature <cit.>.
Thus significantly more theoretical work to understand these fluxes is essential to use this unique neutrino environment to probe neutrino oscillations.

Nonetheless, it is possible to estimate the sensitivity to sterile neutrino oscillations at FASER$\nu$ and FLArE, a proposed LAr detector in the forward direction at the LHC <cit.>.
The most sensitive channel relative to existing constraints is the $\nu_\mu$ disappearance channel which shows sensitivity at the $\Delta m^2_{41}\sim100-1000$ eV$^2$ range down to mixings $|U_{\mu4}|^2<10^{-2}$, better than existing constraints from <cit.> which are dominated by MINOS/MINOS+ <cit.> and MiniBooNE <cit.>.
The sensitivity for FASER$\nu$ in the upcoming LHC run at 150 fb$^{-1}$ and FLArE-10 (10 fiducial tons of LAr) with 3 ab$^{-1}$ in the HL-LHC is shown in Fig. <ref>.
The primary uncertainty is the flux uncertainty.
This is accounted for in a fairly conservative fashion by including an estimate of the impact of shape effects by varying the flux across the range of predictions from different models <cit.> with an associated $1\sigma$ pull term.
This sensitivity is then calculated using the Feldman-Cousins procedure <cit.> at 95% CL including the flux systematic uncertainty.
The median sensitivity is estimated with the Asimov method.

It is anticipated that these LHC experiments could have sensitivity to sterile oscillations for the other channels as well, although those will depend on the details of the flux uncertainties which are still quite large and difficult to systematically quantify.

§ THE PATH FORWARD TO RESOLVING THE ANOMALIES

§.§ Reflecting on the Path of the Past Decade

Sections <ref> and <ref> serve to demonstrate that a significant portion of the neutrino community's efforts over the past decade has been driven by the goal of establishing or refuting the existence of light sterile neutrinos motivated by these anomalies.
To this end, following the requirements identified in <cit.> has led to a broader understanding of viable interpretations of the anomalies and strengthened experimental efforts – and experimental capabilities – in that direction.
Notably, the requirement to probe the anomalies with multiple and orthogonal approaches (accelerator-based short/long-baseline, reactor-based short-baseline, atmospheric neutrinos, and radioactive source) in the same spirit as employed for neutrino oscillations has now been realized through recent, ongoing, or impending experimental programs:

* The development of new radioactive sources and detectors for improved tests of the Gallium Anomaly has been pursued and realized in the form of the BEST experiment.

* The Reactor Antineutrino Anomaly and subsequent reactor-based activities and new results have placed a required emphasis on oscillation-testing short-baseline reactor experiments and on improved understanding of reactor neutrino fluxes.

* The community has just begun a comprehensive multi-channel/multi-baseline accelerator-based short-baseline program to search for 3+$N$ oscillations while directly addressing the MiniBooNE anomaly both in regards to oscillatory and non-oscillatory solutions.

* A direct test of the LSND Anomaly using an improved decay-at-rest beam facility and experimental arrangement has just begun in the form of the JSNS$^2$ experiment.

* Beyond direct anomaly tests, many alternate techniques/facilities, including direct neutrino mass measurements, long-baseline oscillation experiments, and atmospheric and astrophysical neutrino experiments, have been applied to the sterile neutrino explanation of the anomalies.

§.§ Primary Focuses of the Next Decade

As the question of light sterile neutrino oscillations is further explored over the next several years, the community's efforts should be directed toward disentangling the plethora of possibilities that have been identified over the past ten years as viable interpretations of the experimental anomalies in the neutrino sector.
The goal of these collective efforts will be to validate and solidify our understanding of the neutrino sector.
Regardless of what the ongoing and upcoming experiments observe — be it a deviation from the three-neutrino picture or otherwise — the community should be prepared to address how to put these anomalies to rest or adequately distinguish between different interpretations.
We summarize the main experimental, analysis, and theory-driven thrusts that will be essential to achieving this goal as follows: