aainstitutetext: Donostia International Physics Center
Manuel Lardizabal Ibilbidea, 4, 20018 Donostia, Spainbbinstitutetext:
Instituto de Física Teórica (IFT-CFTMAT), CSIC-UAM, Calle de Nicolás Cabrera
13–15, Campus de Cantoblanco, E-28049 Madrid, Spain

# Search for Hidden Neutrinos at the European Spallation Source: the SHiNESS
experiment

Stefano Roberto Soleti,111Corresponding author. b Pilar Coloma a Juan Jose
Gómez Cadenas<EMAIL_ADDRESS>
###### Abstract

The upcoming European Spallation Source (ESS) will soon provide the most
intense neutrino source in the world. We propose the Search for Hidden
Neutrinos at the ESS (SHiNESS) experiment, highlighting its unique
opportunities to search for the existence of sterile neutrinos across a wide
range of scales: anomalous oscillations at short baselines; non-unitarity
mixing in the active neutrino sector; or an excess of events with multiple
leptons in the final state, produced in the decay of heavy neutrinos. The
baseline design of the detector comprises a tank filled with 42 ton of liquid
scintillator, located 25 m far from the ESS beam target. We show that SHiNESS
will be able to considerably improve current global limits for the three cases
outlined above. Although in this work we focus on new physics in the neutrino
sector, the proposed setup may also be used to search for signals from weakly
interacting particles in a broader context.

††preprint: IFT-UAM/CSIC-23-150

##  1 Introduction

A plethora of neutrino experiments carried out in the last few decades have
confirmed existence of neutrino oscillations, which implies nonzero neutrino
masses Super-Kamiokande:1998kpq ; SNO:2002tuh ; KamLAND:2004mhv . Neutrino
mixing can be described by a unitary matrix that parametrizes the change of
basis between the flavor states and mass eigenstates. After the measurement of
the last mixing angle $\theta_{13}$ DayaBay:2012fng , our knowledge of the
parameters that govern neutrino mixing in three families is almost complete,
with the notable exception of the CP-violating phase and the neutrino mass
ordering, which are still unknown. In the meantime, the neutrino research
program has entered a _precision_ era Esteban:2020cvm ; T2K:2021xwb ;
NOvA:2021nfi  in the realm of neutrino oscillations, which will be further
improved with the upcoming experiments DUNE DUNE:2015lol , JUNO JUNO:2020bcl
and Hyper-Kamiokande Abe:2011ts .

However, our understanding of the neutrino sector is still limited. Although
the observation of neutrino oscillations implies that at least two neutrinos
are massive, the values of their masses remain unknown. Moreover, the Standard
Model (SM) of particle physics does not provide a mechanism to give neutrinos
a mass, which necessarily requires to enlarge the SM particle content. Even in
the simplest scenario where right-handed neutrinos are added to the SM in
order to generate neutrino masses, new questions arise regarding the specific
mechanism at play, the Majorana/Dirac nature of neutrinos, and the scale of
new physics associated to the new particles introduced. In fact, neutrinos
offer a window to explore multiple effects stemming from new physics beyond
the SM (BSM) across a wide range of scales.

The existence of additional neutrino mass eigenstates would have different
phenomenological consequences, depending on the scale associated to the new
particles. In fact, leaving theoretical arguments aside, the existence of new
neutrino mass eigenstates (and the scale of new physics associated to their
mass) can only be probed experimentally. While precision measurements at
collider experiments have confirmed that the number of light neutrinos that
participate in the weak interactions is three ALEPH:2005ab , a number of
neutrino experiments have produced results not in full agreement with this
landscape. These somehow controversial _short-baseline_ anomalies include a
significant excess of electron (anti)neutrinos in a mainly muon (anti)neutrino
beam, as observed by the LSND LSND:2001aii  and MiniBooNE MiniBooNE:2020pnu
experiments (the _LSND/MiniBooNE_ anomalies, each of them with a significance
above $4\sigma$); and an evidence of electron neutrino disappearance observed
by several experiments employing a gallium detector exposed to a radioactive
source Abdurashitov:2005tb ; GALLEX:1994rym ; Barinov:2021asz  (the _gallium
anomaly_ Laveder:2007zz ; Acero:2007su ; Giunti:2010zu , with a significance
above $5\sigma$ Barinov:2022wfh ; Barinov:2021mjj ; Berryman:2021yan ;
Elliott:2023cvh ; Brdar:2023cms ). The most naive explanation for these
anomalies is the existence of a new neutrino with a mass squared splitting of
around 1 eV2, called _light sterile neutrinos_. However, the experimental
landscape is not fully consistent Boser:2019rta , given the absence of an
excess in the KARMEN KARMEN:2002zcm  and MicroBooNE MicroBooNE:2021tya
experiments, the significant tension between appearance and disappearance
measurements MiniBooNE:2012meu ; IceCube:2016rnb ; Dentler:2018sju ;
Diaz:2019fwt , and that the so-called _reactor antineutrino anomaly_ has
recently vanished in light of new reactor flux reevaluations Berryman:2019hme
; Estienne:2019ujo ; Giunti:2021kab . Thus, the community still awaits new
experimental data that could shed some light on the origin of the reported
anomalies.

Sterile neutrinos at higher scales may not lead to a new oscillation frequency
but could yield interesting phenomenological signals elsewhere. To begin with,
additional neutrinos could induce an apparent violation of the unitarity of
the neutrino mixing matrix in the $3\times 3$ active block (which governs
oscillations among active neutrinos), potentially even involving additional
sources of CP violation Langacker:1988up ; Bilenky:1992wv ; Bergmann:1998rg ;
Czakon:2001em ; Bekman:2002zk ; Antusch:2006vwa ; Fernandez-Martinez:2007iaa ;
Escrihuela:2016ube . While non-unitarity constraints from heavy neutrinos
(above the EW scale) are severely constrained from global fits to electroweak
precision data Blennow:2023mqx ; Antusch:2016brq , neutrinos at lower scales
could still lead to observable effects Blennow:2016jkn . For neutrino masses
below the pion mass, the effect would be that of an averaged-out oscillation,
leading to an overall constant effect on the transition probabilities. On the
other hand, a sterile neutrino with a mass in the MeV-GeV range, usually
called _heavy neutral lepton_ (HNL), may be produced in meson decays through
its mixing with the active neutrinos (for recent reviews on HNL phenomenology,
see e.g. refs. Abdullahi:2022jlv ; Antel:2023hkf ). After propagating over
some distance, it could decay back into SM particles, leaving a visible signal
in neutrino detectors. Typical decay signals for a HNL below the GeV scale
that only interacts via mixing involve a combination of light mesons,
electrons/muons and neutrinos (which lead to missing energy).

In view of the general features outlined above, an intense, pulsed, source of
well-characterized neutrinos would be the ideal tool to probe for any of the
new physics signals outlined above. Spallation sources are excellent in this
regard: in the proton collisions in the target, an intense flux of pions is
produced that in turn yields a high-intensity neutrino flux from their decay
at rest (DAR). The upcoming European Spallation Source (ESS) currently under
construction in Lund, Sweden, will provide the most intense neutron beam in
the world Garoby:2017vew  and therefore offers an exceptional opportunity to
search for new physics in the neutrino sector. To this end, we propose the
Search for Hidden Neutrinos at the European Spallation Source (SHiNESS)
experiment, which will be able to: (1) probe the unitarity of the active
neutrino mixing matrix with unprecedented sensitivity; (2) set world-leading
limits on heavy neutral leptons, and (3) definitely clarify whether the origin
of the short-baseline neutrino anomalies are due to an eV-scale sterile
neutrino. We note that, while studies exist to use the future ESS Neutrino
Super Beam Alekou:2022emd  (ESS$\nu$SB) for light sterile neutrino searches
Blennow:2014fqa ; KumarAgarwalla:2019blx ; Ghosh:2019zvl , SHiNESS does not
require any upgrade to the beamline and can start acquiring data as soon as
the ESS turns on. Moreover, we highlight its complementarity to the current
neutrino physics program of the ESS Baxter:2019mcx , which focuses on coherent
elastic neutrino-nucleus scattering (CE$\nu$NS) detection to pursue new
physics signals: both experiments may take place simultaneously without
interfering with any of the main physics goals of the ESS (or with each
other).

This document is organized as follows. Section 2 describes in detail the
experimental setup at the ESS and the detector technology, while section 3
discusses the phenomenological signals expected for the three cases under
consideration in this work: zero-distance effects induced by a non-unitarity
leptonic mixing matrix, oscillations at short baselines induced by light
sterile neutrinos, and decay signals from heavy neutral leptons. Section 4
discusses the expected number of events for each case, while section 5
describes the main backgrounds expected. Section 6 contains the calculated
sensitivity of the experiment to the different scenarios considered. A summary
of the proposal and a comparison with the current experimental landscape are
provided in section 7.

##  2 The experimental setup

###  2.1 The beam

The European Spallation Source is an upcoming facility which will combine a 2
GeV proton linac with an advanced hydrogen moderator, generating the most
intense neutron beam in the world GUTHRIE2017637 . The proton beam impinges on
a rotating tungsten target, producing fast, high-energy neutrons that are
slowed down by the moderator. The 5 MW design power will deliver a total of
$2.8\times 10^{23}$ protons-on-target per calendar year, assuming 5000 hours
of beam delivery. The proton beam pulse has a rate of 14 Hz with 2.8 ms long
spills, giving a duty factor of $4\times 10^{-2}$. A comparison of the ESS
beam pulse with other spallation sources is shown in figure 1.

Figure 1:  ESS pulse structure in terms of neutron brightness compared to
other facilities. The total number of neutrons is proportional to the area
under the curve. Adapted from ref. GUTHRIE2017637 .

Compared to the Spallation Neutron Source (SNS) in Oak Ridge, United States
and the J-PARC facility in Tokai, Japan, the ESS has a much larger duty
factor. This feature, in principle, could increase the background from steady-
state processes such as cosmic rays. However, steady-state backgrounds can be
accurately characterized during the long anti-coincident periods between beam
spills Baxter:2019mcx , and the signal-to-background figure of merit (FOM), as
defined in ref. Abele:2022iml  and reproduced in figure 2, is comparable for
the three facilities.

Figure 2:  Comparison of past (black squares) and future (red diamonds)
spallation sources relative to the SNS (blue circle), in terms of neutrino
flux and signal-to-background FOM. Adapted from ref. Abele:2022iml .

Spallation sources produce both $\pi^{-}$ and $\pi^{+}$ in the proton-nucleus
collisions in the target. Negative pions are efficiently absorbed by nuclei
before they can decay, while positive ones lose energy as they propagate and
finally decay at rest through:

$\displaystyle\pi^{+}\rightarrow$ $\displaystyle\mu^{+}+\nu_{\mu}$ (1)
$\displaystyle\mu^{+}\rightarrow e^{+}+\nu_{e}+\bar{\nu}_{\mu}.$

The energy spectra produced by these processes are well known Burman:2003ek ;
Baxter:2019mcx . The $\nu_{\mu}$ flux is monochromatic with
$E_{\nu_{\mu}}\approx 29.7$ MeV, since the $\pi^{+}$ decay is a two-body
process, while the $\nu_{e}$ and the $\bar{\nu}_{\mu}$ fluxes exhibit a Michel
distribution at energies $E_{\nu_{e},\bar{\nu}_{\mu}}<m_{\mu}/2\approx 52.8$
MeV. The goal of SHiNESS is to compare the expected amounts of $\pi^{+}$ DAR
neutrino interactions with the measured ones.

A simplified simulation of the beam was performed for exploratory studies
regarding the detection of coherent elastic neutrino-nucleus scattering
(CE$\nu$NS) at the ESS. The neutrino yield obtained by this simulation is 0.3
per proton Baxter:2019mcx , resulting in an expected $8.5\times 10^{22}$
$\pi^{+}$ DAR neutrinos per flavor per year. Figure 3 shows the neutrino flux
as a function of the energy, obtained with a Geant4 GEANT4:2002zbu  simulation
using the BERT-HP physics list Heikkinen:2003sc .

Figure 3:  Flux of each neutrino flavor for an ESS beam spill as a function of
the neutrino energy. The simulation has been performed with Geant4 using the
BERT-HP physics list.

Apart from the monoenergetic peak at 29.7 MeV and the two Michel distributions
for the $\bar{\nu}_{\mu}$ and the $\nu_{e}$ components, the spectrum presents
several interesting features:

  * •

muons captured by an atom can decay in orbit, producing $\nu_{\mu}$ with an
endpoint energy of around 105 MeV;

  * •

kaons produced by the beam impinging on the target can also decay and produce
monoenergetic 236 MeV $\nu_{\mu}$ through $K^{+}\rightarrow\mu^{+}\nu_{\mu}$;

  * •

the small amount of $\mu^{-}$ that decay before being absorbed can produce
$\bar{\nu}_{e}$ through $\mu^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\mu}$. This
component represents an irreducible background for the $\bar{\nu}_{e}$
appearance search (see section 5.1).

The COHERENT collaboration has shown that the systematic uncertainty
associated to the neutrino flux of the SNS can be constrained at the 10% level
COHERENT:2021yvp . This is also the uncertainty assumed in the proposal for
CE$\nu$NS detection at the ESS Baxter:2019mcx . Thus, in this document, we
adopt a 10% systematic uncertainty for the $\nu_{e}$, $\nu_{\mu}$, and
$\bar{\nu}_{\mu}$ components. However, the $\bar{\nu}_{e}$ component from
$\mu^{-}$ decay has a very poor normalization constraint, since the $\mu^{-}$
decay and pion production points are not well known. This component can depend
on the detail of the simulation and on the physics model adopted, so we assign
a systematic uncertainty of 25%.

Total neutrino flux measurement could be performed through the neutral current
(NC) channel of eq. (19) (see section 4.1.3). The detector could then be used
also as a neutrino flux monitor for the proposed CE$\nu$NS experiments at the
ESS Baxter:2019mcx . The CE$\nu$NS detection proposal also mentions the
possibility to replicate at the ESS a small heavy water detector (1 m3 D2O)
proposed to independently measure the neutrino yield of the SNS
COHERENT:2021xhx .

###  2.2 The detector

####  2.2.1 Location

The D03 experimental hall of the ESS offers unallocated space for a medium-
scale experiment. The detector can be placed around 25 m far from the proton
beam target, with an angle of approximately $35^{\circ}$ in the backward
direction of the beam, in order to suppress forward-boosted particles (see
figure 4).

In the short baseline limit of $\Delta m_{21}^{2}L/E_{\nu}\ll 1$ and $\Delta
m_{31}^{2}L/E_{\nu}\ll 1$, ordinary, three-flavor neutrino oscillations have
yet to develop Kopp:2013vaa . Thus, the eventual departure from unitarity will
manifest itself through zero-distance effects, as detailed in section 3.1 and
heavy neutral leptons, described in section 3.3, can be long-lived enough to
reach the detector. In addition, at this distance from the beam target, the
$L/E$ of the neutrino oscillation is also well matched to the LSND allowed
region in the (3+1) light sterile neutrino model, as shown in section 3.2.

Figure 4:  Layout of the ESS experimental areas. The proton beam direction is
from right to left. The red circle represents the proposed SHiNESS tank, drawn
to scale.

####  2.2.2 Technology

The detection of $\pi^{+}$DAR neutrino interactions can be efficiently
performed with a detector filled with liquid scintillator, which is the
solution adopted by the KARMEN Zeitnitz:1994kz , LSND LSND:1996jxj , and JSNS2
JSNS2:2013jdh  experiments. In this proposal, as a baseline approach, we
consider a 4 m-high tank with a radius of 2 m, filled with liquid scintillator
doped with gadolinium222Incidentally, this is a similar size of the tank used
for the Water Cherenkov Test Experiment at CERN, which is currently under
preparation and whose goal is to validate the technology of the proposed
Intermediate Water Cherenkov Detector of the Hyper-Kamiokande experiment wcte
.. We adopt 2,5-diphenyhloxazole (PPO) as the fluor and linear alkylbenzene
C6H5C10-13H2(10-13)+1 (LAB) as the solvent. This cocktail is largely
available, has a high light yield and a high flash point, with a successful
track record in neutrino physics Beriguete:2014gua ; JUNO:2020bcl ;
SNO:2020fhu ; Park:2013nsa . A possible alternative is represented by a
mixture of phenyl-xylylethane (PXE) and dodecane (C12H26), which was
successfully used by the Double Chooz experiment DoubleChooz:2006vya ;
Aberle:2011ar . The presence of a small concentration of gadolinium (around
0.1% in mass) greatly increases the neutron absorption probability, given its
very high neutron cross section (48800 barn) leinweber2006neutron . The Gd
atom releases a sum of 8 MeV energy in a $\gamma$ cascade after neutron
capture Tanaka:2019hti , which has the advantage of being sensibly higher than
the energy released upon capture on hydrogen (2.2 MeV).

The design of the SHiNESS detector is driven by the following considerations:

  * •

in order to reach competitive sensitivities on the non-unitarity of the
neutrino mixing matrix (section 3.1) and on the light sterile neutrino
searches (section 3.2), accidental backgrounds need to be rejected at a level
of at least $10^{2}$, as shown in section 5.1.

  * •

the detection of HNLs producing a $e^{+}e^{-}$ pair (section 3.3) requires the
ability to reconstruct the direction of the two leptons.

While signal events for inverse beta decay (IBD) and carbon charged-current
interactions produce prompt and delayed signals in close vicinity (see section
4), interactions caused by accidental coincidences will be usually separated
by tens of centimeters. Thus, a good vertex resolution is essential to reject
this background. Figure 5 shows the background rejection power as a function
of the position resolution. If we require a signal efficiency of 50%, a vertex
resolution of 25 cm corresponds to an accidental background rejection factor
of approximately $10^{2}$.

Figure 5:  Background rejection for IBD events as a function of the vertex
resolution, assuming a 90% (dark blue) and a 50% (light blue) signal selection
efficiency.

In order to satisfy these requirements, the SHiNESS baseline design includes
an array of 38 Hamamatsu photomultipliers R12860 Hamamatsu  with a diameter of
20”.

The detection of Cherenkov photons, in addition to the scintillation light,
can improve the vertex resolution and allow to reconstruct the directionality
of the particles produced in the interaction, thus enabling HNL searches.
Separation of Cherenkov and scintillation light can be achieved through
timing, directionality, or chromatic separation (or a combination of the
three). This is because Cherenkov light is emitted promptly and along the
direction of the particle, while scintillation light has a typical emission
time constant and is emitted isotropically. Also, the scintillation emission
spectrum is relatively narrow, while the Cherenkov radiation follows a power-
law distribution given by the Frank-Tamm formula Frank:1937fk . Separation has
been demonstrated in LAB and LAB/PPO exploiting both the different time
profile Li:2015phc ; Kaptanoglu:2018sus  and the wavelength spectrum
Kaptanoglu:2018sus  of the two types of light emission.

In the baseline SHiNESS design, shown in figure 6, separation between
Cherenkov and scintillation time is enabled by 32 Incom Large Area Picosecond
Photodetectors (LAPPDsTM) Incom  of $20\times 20$ cm2. LAPPDs offer single
photoelectron time resolutions below 100 ps and sub-cm spatial resolution
Lyashenko:2019tdj , allowing to discriminate in time between scintillation and
Cherenkov light emission. As a reference, the ANNIE experiment is deploying
one LAPPD for every 25 PMTs (whose diameters vary between 8” and 11”), with an
expected vertex resolution improvement of a factor of three (from 38 cm to 12
cm) and an expected angular resolution improvement of a factor of two (from
11∘ to 5∘) ANNIE:2017nng . In the case of SHiNESS, 32 LAPPDs allow to detect
at least 10 Cherenkov photons in the first 5 ns for 50% of the inverse beta
decay (IBD) events.

Figure 6:  Three-dimensional drawing of the SHiNESS detector, which comprises
of a cylindrical tank (4 m high and 4 m wide) filled with 42 t of liquid
scintillator. The light is collected by 38 Hamamatsu PMTs R12860 (in white)
and 32 Incom LAPPDs (in blue).

The photosensors, placed uniformly on all sides, provide a photocoverage of
10.1%. Assuming a light yield of $11,590$ photons/MeV GUO201933 , this
photocoverage corresponds to an energy resolution of $\sim
15\%/\sqrt{E/\mathrm{MeV}}$. The energy resolution is not a critical parameter
for the physics searches and the main driver of the photocoverage is the
ability to accurately reconstruct the interaction vertex.

Figure 7:  Simulated event display of an inverse beta decay in the liquid
scintillator tank, showing the charge collected by the photosensors. The
circles correspond to Hamamatsu R12860 PMTs and the squares to Incom LAPPDs.
The true trajectories correspond to the colored segments (in red the particle
tracks generated by the positron, in green the ones generated by the neutron).

A simulated event display of an IBD event in the detector (process described
in detail in section 4.1.1) is shown in figure 7. The positron emission and
the neutron absorption happen in close vicinity.

Since the detector will have only a limited overburden, a cosmic-muon veto
must be implemented. Two main options are being considered. The first one is
represented by an outer layer of liquid scintillator surrounding the main
detector and equipped with PMTs. This is the solution adopted by the LSND
experiment, which achieved a cosmic-ray veto inefficiency $<10^{-5}$
Conrad:2013mka , and by the JSNS2 experiment, which employs also a second
layer as _gamma-catcher_ JSNS2:2021hyk . A second option is represented by
panels made of plastic scintillator which surround the detector, as
implemented by e.g. the KARMEN KARMEN:1994xse  and MicroBooNE
MicroBooNE:2019lta  experiments.

####  2.2.3 Performances

The long attenuation length of the liquid scintillator cocktail (14 m at 430
nm for LAB/PPO Beriguete:2014gua ), combined with a uniform photosensor
coverage, allows achieving an almost uniform energy resolution as a function
of the radius, as shown in figure 8(a) for a Geant4 simulation of 30 MeV
positrons and 8 MeV Gd-capture $\gamma$.

(a) Energy resolution as a function of the distance $r$ from the detector
center for a 8 MeV $\gamma$, corresponding to a neutron absorption, and a 30
MeV $e^{+}$.

(b) Position resolution in the $x$ axis for a 20 MeV $e^{+}$ and a 1 MeV
neutron. The $x$ coordinate has been reconstructed with a centroid algorithm
using the scintillation light in the PMTs.

Figure 8:  Energy (left) and position (right) resolution in the SHiNESS
detector for a positron and a neutron.

A naive centroid algorithm using only the light detected by the PMTs gives a
vertex resolution of approximately 20 cm (see figure 8(b)). This value can be
significantly improved by reconstructing the Cherenkov cone using the first
hundreds of picoseconds of the LAPPDs signals, as demonstrated by the ANNIE
experiment ANNIE:2017nng .

Figure 9 shows the spatial distribution for Cherenkov and scintillation
photons and the relative charge collected by the photosensors in the SHiNESS
detector for a $e^{+}e^{-}$ pair produced by a HNL decay. Isolating the
Cherenkov component allows to reconstruct the direction of the two leptons,
which would not be possible using only the scintillation light.

Figure 9:  Spatial distribution of scintillation and Cherenkov photons (top)
and the corresponding charge collected by the photosensors placed on the side
of the SHiNESS detector (bottom) for a $e^{+}e^{-}$ pair produced by a HNL
decay. The red dot corresponds to the interaction vertex.

The time discrimination between Cherenkov and scintillation light could
eventually be increased by loading the scintillator with _slow fluors_ , as
detailed in ref. Biller:2020uoi . Figure 10(a) shows the arrival time
distributions for the photons generated by a 30 MeV electron in the SHiNESS
detector, using PPO or acenaphthene, one of the slow fluor candidates. In the
latter case, the separation in time between Cherenkov and scintillation
photons increases up to few ns. After requiring a Cherenkov light detection
efficiency of 50%, the purity of the Cherenkov selection goes from 23% in the
case of PPO to 91% in the case of acenaphthene.

(a) Light emission time profile for Cherenkov and scintillation photons.

(b) Wavelength spectrum for Cherenkov and scintillation photons (left axis)
and QE for the two kinds of photosensor (right axis).

Figure 10:  Scintillation with PPO (light red), scintillation with
acenaphthene (red), and Cherenkov (blue) photon distributions as a function of
time (left) and wavelength (right) for a 30 MeV electron in the SHiNESS
detector. Quantum efficiency (QE) for the Hamamatsu PMT and the Incom LAPPD
are also provided.

The R12860 and LAPPD quantum efficiencies are well matched to the
scintillation and Cherenkov emission spectrum (see figure 10(b)). However,
absorption length of LAB/PPO increases rapidly in the UV, so only a small
portion of the UV Cherenkov photons will reach the photosensors. A rich R&D
program is devoted at increasing the discrimination between scintillation and
Cherenkov photons in liquid scintillator Wonsak:2018uby ; Aberle:2013jba . A
notable example is represented by quantum dots (QDs): these are semiconducting
nanocrystals that can be loaded in the liquid scintillator and can shift the
UV Cherenkov photons towards the visible with a high efficiency
Gruszko:2018gzr . In the SHiNESS detector, a simplified simulation of
perovskite quantum dots loading plasmachem  gives an improvement of
approximately a factor of 1.7 in the number of detected Cherenkov photons.
Table 1 shows a summary of the possible cocktails for the liquid scintillator
and the relative Cherenkov detection performances.

Cocktail |  | Cherenkov sample purity
---
at 50% efficiency
| Detected Cherenkov photons
---
with $\Delta t<5$ ns (median)
LAB/PPO | 23% | 10.4
LAB/Acenaphthene | 91% | 10.4
LAB/PPO + QDs | 23% | 18.2
Table 1: Purity of the sample of detected Cherenkov photons, requiring a 50%
efficiency, and median number of detected Cherenkov photons in the first 5 ns
for various liquid scintillator cocktails. The numbers were obtain by
simulating a sample of positrons produced by IBD events in the SHiNESS
detector.

Alternatively, several collaborations are developing water-based liquid
scintillators (WbLS) ANNIE:2017nng ; Theia:2019non ; WATCHMAN:2015lcq , which
uniquely combine the high light yield and low threshold of scintillator with
the directionality of a Cherenkov detector Alonso:2014fwf . As an example, the
THEIA collaboration Theia:2019non  is exploring the separation of Cherenkov
and scintillation photons for direction reconstruction of few-MeV electrons
using a WbLS cocktail.

##  3  Phenomenological implications of an extended neutrino sector

As outlined in the introduction, an extended neutrino sector is well-motivated
as the SM particle content needs to be enlarged in order to generate neutrino
masses. Depending on the nature of the new particles and the values of their
masses, different phenomenological consequences arise. We start discussing the
implications of a non-unitary leptonic mixing matrix in section 3.1, as it
provides a model-independent way to search for indirect evidence of new states
in the neutrino sector, that is valid across a wide range of masses and for a
plethora of models.

However, SHiNESS will also be able to search for direct signals due to
additional neutrinos, either through the observation of oscillation signals
(if there are new neutrinos present at the eV scale) or from their decays (for
heavy neutrinos, with masses around the MeV scale or above). These two
scenarios are discussed separately, in sections 3.2 and 3.3, respectively.

###  3.1  Probing the unitarity of the lepton matrix

Neutrino oscillations generally assume a unitary $3\times 3$ lepton mixing
matrix $U$. In the Pontecorvo–Maki–Nakagawa–Sakata parametrization this matrix
is written by means of three mixing angles and one complex (Dirac) CP phase
ParticleDataGroup:2022pth , plus two additional CP phases if neutrinos are
Majorana. The unitarity of the matrix, however, only holds in a limited amount
of neutrino mass models. In general, in the presence of $n$ additional
neutrinos, the mass Lagrangian in the extended neutrino sector is diagonalized
by a $(n+3)\times(n+3)$ mixing matrix:

$\mathcal{U}=\begin{pmatrix}[l]U&W\\\ Z&V\\\ \end{pmatrix},$ (2)

where $V$ is a $n\times n$ matrix and $W$ and $Z$ are $3\times n$ and $n\times
3$ rectangular matrices, respectively. Here, $U$ is the $3\times 3$ active
neutrino block of the matrix, which does not have to be unitary anymore. In
the simple case of a type-I Seesaw Minkowski:1977sc ; Mohapatra:1979ia ;
Yanagida:1979as ; Gell-Mann:1979vob , the presence of right-handed heavy
neutrinos causes unitarity deviations for $U$ which are proportional to the
ratio between the masses of the light and heavy states. While a naive type-I
seesaw would yield an exceedingly small effect ($\mathcal{O}(10^{-13})$ for
$\mathcal{O}(\mathrm{TeV})$ heavy states), there are several examples of
neutrino mass models predicting larger deviations Mohapatra:1979ia ;
Antusch:2014woa ; Nelson:2007yq ; Antusch:2006vwa . Thus, searching for
unitarity violation represents a model-independent way to probe for non-
standard neutrino states and interactions Blennow:2016jkn , with direct
sensitivity to BSM physics. Moreover, if the mixing matrix in the active
block, $U$, is not unitary, then additional CP phases would enter its
definition. In fact, deviations from unitarity in the active block are often
described using a lower triangular parametrization Xing:2007zj ; Xing:2011ur ;
Escrihuela:2015wra , which includes three off-diagonal complex parameters,
each of them with an associated CP phase. Therefore, non-unitarity effects
could a priori be relevant for the upcoming neutrino oscillation experiments
DUNE and Hyper-Kamiokande, which will aim to determine whether there is CP
violation in the lepton sector.

A $3\times 3$ unitary matrix should meet a total of 9 conditions, which are
easily derived from the requirement that
$U^{{\dagger}}U=UU^{{\dagger}}=\mathbb{I}$. They boil down to requiring that
all its rows are normalized to one, and that the so-called _unitarity
triangles_ close:

$\displaystyle r_{\alpha}$ $\displaystyle\equiv$ $\displaystyle|U_{\alpha
1}|^{2}+|U_{\alpha 2}|^{2}+|U_{\alpha 3}|^{2}=1\,,$ (3) $\displaystyle
t_{\alpha\beta}$ $\displaystyle\equiv$ $\displaystyle U_{\alpha 1}^{*}U_{\beta
1}+U_{\alpha 2}^{*}U_{\beta 2}+U_{\alpha 3}^{*}U_{\beta
3}=0\quad(\alpha\neq\beta)\,,$ (4)

where $\alpha,\beta$ indicate active neutrino flavor indices $(e,\mu,\tau)$.
If $U$ is not unitary then the conditions in eqs. (3), (4) do not have to be
satisfied. Thus, a way to constrain the unitarity of the leptonic mixing
matrix consists in probing the quantities $r_{\alpha}$ and $t_{\alpha\beta}$
defined above. In this case it is worth noting though, that if the whole
matrix $\mathcal{U}$ is unitary, then the application of Cauchy-Schwarz
inequalities imposes restrictions on the values of $r_{\alpha}$ and
$t_{\alpha\beta}$. This allows to place indirect limits on unitarity
deviations which can in some cases be stronger than direct bounds
Antusch:2006vwa ; Parke:2015goa . However, in order to be as general as
possible, here we adopt instead an _agnostic_ assumption (as in ref.
Ellis:2020hus ) and do not assume that $\mathcal{U}$ is unitary.

Experimentally, a non-unitary matrix leads to a non-vanishing probability to
have flavor transitions at negligible distances, also known as _zero distance_
effects Antusch:2006vwa . The oscillation probabilities in this case read

$\displaystyle P(\nu_{\alpha}\rightarrow\nu_{\alpha})$
$\displaystyle=r_{\alpha}^{2}\,,$ (5) $\displaystyle
P(\nu_{\alpha}\rightarrow\nu_{\beta})$
$\displaystyle=|t_{\alpha\beta}|^{2}\,.$ (6)

A short-baseline neutrino experiment such as SHiNESS may therefore be
sensitive to such effects since standard, three-flavor oscillations have yet
to develop. From the expressions above, however, it is evident that the
sensitivity to $r_{\alpha}$ will be directly limited by the systematic
uncertainties associated to the normalization of the event sample and
therefore will not be discussed here. A search for anomalous appearance is
much more promising, provided that backgrounds can be sufficiently reduced. In
particular, SHiNESS is expected to be sensitive to the closure of the
electron-muon triangle parameter $t_{e\mu}$ through the appearance probability
$P(\bar{\nu}_{e}\rightarrow\bar{\nu}_{\mu})$, looking for an excess of IBD
events in the detector.

###  3.2  Light sterile neutrinos

If the extra neutrinos are light enough, they could participate in
oscillations, leading to a new oscillation frequency and a smoking signal at
neutrino detectors. As outlined in the introduction, throughout the past two
decades a plethora of experiments have reported contradicting signals in the
context of sterile neutrino searches in oscillation experiments Diaz:2019fwt ;
Dentler:2018sju . If the reported anomalies are caused by the existence of a
fourth mass neutrino eigenstate, the best value for the mass splitting $\Delta
m_{41}^{2}=m_{4}^{2}-m_{1}^{2}$ is around the $\mathcal{O}(\mathrm{eV}^{2})$
scale Dentler:2018sju  and much larger than the other two mass splittings
$\Delta m_{31}^{2}$, $\Delta m_{21}^{2}$. For this reason this is usually
referred to as a _(3+1) scenario_. For sufficiently short baselines, $\Delta
m_{21}^{2}L/E_{\nu}\ll 1$ and $\Delta m_{31}^{2}L/E_{\nu}\ll 1$, so that
standard three-flavor neutrino oscillations have yet to develop. In this
limit, appearance and disappearance oscillations can be approximated as:

$\displaystyle P(\nu_{\alpha}\to\nu_{\alpha})$ $\displaystyle\simeq
1-\sin^{2}\left(2\theta_{\alpha\alpha}\right)\sin^{2}\left(\frac{\Delta
m_{41}^{2}L}{4E_{\nu}}\right)\,,$ (7) $\displaystyle
P(\nu_{\alpha}\to\nu_{\beta})$
$\displaystyle\simeq\sin^{2}\left(2\theta_{\alpha\beta}\right)\sin^{2}\left(\frac{\Delta
m_{41}^{2}L}{4E_{\nu}}\right)\,,\quad$ (8)

where $E_{\nu}$ is the neutrino energy, $L$ is the distance traveled by the
neutrino Dasgupta:2021ies , and the last expression only holds for
$\alpha\neq\beta$. Here we have adopted the usual definitions for the
effective mixing angles $\theta_{\alpha\alpha},\theta_{\alpha\beta}$ that
control the amplitude of the oscillation (which eventually can be mapped onto
the elements of the full mixing matrix, $\mathcal{U}_{\alpha 4}$ and
$\mathcal{U}_{\beta 4}$).

The SHiNESS experiment has the ability of definitely clarifying the complex
(and, at times, contradictory) experimental scenario revolving the existence
of a sterile neutrino, thanks to its key features:

  * •

will detect $\pi^{+}$ DAR neutrinos, whose spectrum is well understood and
shape systematic uncertainties are small;

  * •

will sit at a distance from the neutrino source similar to the one of the LSND
experiment (25 m vs. 30 m). This will allow to cover the entire parameter
space in the (3+1) model allowed by the LSND result, complementary to the
space covered by the JSNS2 experiment JSNS2:2013jdh ;

  * •

is sensitive to both the $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$ appearance
channel and the $\nu_{e}\rightarrow\nu_{e}$ disappearance channel, thus
probing the light sterile neutrino model for both the LSND/MiniBooNE and the
gallium anomaly.

We finalize this discussion by stressing the similarities between the
phenomenology associated to a non-unitary mixing matrix and that of an
averaged-out sterile neutrino oscillation. Generally speaking, oscillations
are said to be averaged-out when the corresponding oscillation frequency leads
to oscillations that are too fast to be resolved by the detector, considering
its energy and spatial resolution as well as the distance to the detector and
the energy of the neutrinos. In the case of SHiNESS this translates into the
condition $\Delta m_{41}^{2}\gtrsim 20~{}\mathrm{eV}^{2}$. As discussed in
refs. Blennow:2016jkn ; Coloma:2021uhq , at leading order the phenomenological
consequences of a non-unitarity mixing matrix are mostly equivalent to those
of an averaged-out sterile neutrino, up to quartic terms in the mixing between
the heavy and active states (see refs. Fong:2016yyh ; Fong:2017gke  for a
discussion on such effects). Thus, above this value the limits on the
unitarity of the lepton mixing matrix (obtained in a model-independent way)
can be mapped onto the limits on the mixing angle for a sterile neutrino (and
_viceversa_). Conversely, for sterile neutrino masses below this scale the
corresponding oscillations could be resolved and measured at the detector. As
the two limits yield very different phenomenological implications, we will
study them separately.

###  3.3  Heavy Neutral Leptons

If the new neutrino states have their masses at the MeV or GeV scale, they are
usually refereed to as Heavy Neutral Leptons (HNL). HNLs can couple to SM
particles through their mixing with the SM neutrinos and therefore may be
produced copiously in meson decays as well as muon or tau decays. Once
produced, a HNL may travel for hundreds or thousands of meters before decaying
back to a combination of SM leptons and mesons. This makes fixed-target
experiments an ideal setup to search for their decay signals (for a recent
review of the strongest constraints on HNLs see e.g. ref. Fernandez-
Martinez:2023phj ). Specifically, the large number of stopped pions and muons
at spallation sources makes them outstanding facilities to search for HNLs
below the pion mass Ema:2023buz .

Once produced near the ESS beam target, HNLs can be long-lived enough to reach
the SHiNESS detector and decay visibly inside as $N\to\nu_{\alpha}e^{+}e^{-}$,
where the flavor of the neutrino in the final state will depend on the mixing
of the HNL with the active neutrinos. Hereafter we follow the analysis of ref.
Ema:2023buz  and consider two cases separately, depending on whether the HNL
mixes predominantly with muon or electron neutrinos.

In the muon mixing case, the HNL is mainly produced from the decay of muons,
although for sufficiently light masses ($m_{N}<m_{\pi}-m_{\mu}$) it may also
be produced subdominantly from pion decays. The HNL then decays via the
neutral current, with the following decay width:

$\displaystyle\Gamma(N\to
e^{+}e^{-}\nu_{\mu})=\frac{G_{F}^{2}m_{N}^{5}}{768\pi^{3}}\left|{U_{\mu
N}}\right|^{2}\left(1-4\sin^{2}\theta_{W}+8\sin^{4}\theta_{W}\right),$ (9)

where $\theta_{W}$ is the Weinberg angle.

In the electron mixing case, the HNL is abundantly produced both from the
decay of muons and pions. Contrary to the muon mixing case, in this case the
HNL can decay both via the neutral current and the charged current. The
corresponding decay rate reads Gorbunov:2007ak ; Atre:2009rg :

$\displaystyle\Gamma(N\to e^{+}e^{-}\nu_{e})$
$\displaystyle=\frac{G_{F}^{2}m_{N}^{5}}{768\pi^{3}}\left|{U_{eN}}\right|^{2}\left(1+4\sin^{2}\theta_{W}+8\sin^{4}\theta_{W}\right).$
(10)

In the limit of a very long-lived HNL, the total number of HNL decays in the
detector can be obtained approximately as:

$\displaystyle N^{(i)}_{ee}$ $\displaystyle\simeq
N_{i}\times\frac{1}{\Gamma_{i}}\int dE_{N}\frac{d\Gamma(i\to
fN)}{dE_{N}}\times\epsilon_{\mathrm{det}}^{(i)}(m_{N})\times\frac{L_{\mathrm{det}}}{\gamma\beta
c\tau_{N}}\times\mathrm{BR}(N\to\nu ee),$ (11)

where $N_{i}$ stands for the total number of parent particles ($i=\mu,\pi$),
$f$ represents the other particles produced together with the HNL (which
depend on the decay channel), $d\Gamma(i\to fN)/dE_{N}$ is the differential
production rate as a function of the HNL energy $E_{N}$ (which can be found
e.g. in ref. Ema:2023buz ), and $\Gamma_{i}$ is the total decay width of
particle $i$. The selection efficiency is given by
$\epsilon_{\mathrm{det}}^{(i)}(m_{N})$, see figure 13. The length of the
detector is $L_{\mathrm{det}}$, and $c\beta\tau_{N}$ is the decay length of
the HNL in the lab frame, while $\mathrm{BR}(N\to\nu ee)$ is the branching
ratio into the decay channel $N\to\nu ee$. It follows from eq. (11) that, in
this limit, the result only depends on $\Gamma(N\to\nu ee)$, while the
dependence on the total width of the HNL effectively drops out.

##  4 Expected number of signal events

###  4.1 Neutrino detection

####  4.1.1 Inverse beta decay

The interaction of $\bar{\nu}_{e}$ in the liquid scintillator can be detected
through to observation of the IBD process both on protons:

$\bar{\nu}_{e}+p\rightarrow e^{+}+n.$ (12)

and on carbon atoms:

$\bar{\nu}_{e}+^{12}\mathrm{C}\rightarrow e^{+}+n+^{11}\mathrm{B}.$ (13)

The experimental signature of this process is the emission of a positron
followed by the delayed neutron capture and its $\gamma$ emission. Figure 11
shows the time and charge distribution for $10^{4}$ IBD events in the SHiNESS
detector, obtained with a Geant4 simulation. The positron generates a _prompt_
signal, while the neutron absorption generates a _delayed_ signal with a
characteristic exponential time constant of around 28 . Neutrons generate
signal in the lower end of the charge spectrum, while the positrons follow a
broad Michel distribution.

(a) Stacked histogram of the sum of the charge collected by the PMTs for each
event.

(b) Stacked histogram of the time difference between the IBD interaction and
the signal generated by each particle.

Figure 11: Charge and time distribution for $10^{4}$ IBD events in the SHiNESS
detector. The positrons produce a prompt signal with a broad charge
distribution, while the neutrons are absorbed with a characteristic time
constant of around 28 and are concentrated at lower energies, corresponding to
the 8 MeV capture by Gd and the 2.2 MeV capture by hydrogen.

Assuming an active mass of 42 ton of liquid scintillator and knowing the flux-
averaged cross sections we can calculate the number of inverse beta decay
events we expect in case of $\bar{\nu}_{e}$ appearance. The yearly flux of
$\bar{\nu}_{\mu}$ from $\pi^{+}$ DAR that can oscillate into $\bar{\nu}_{e}$
at a distance of $L=$25 m from the beam target is given by:

$\displaystyle\phi_{\bar{\nu}_{\mu}}=$ $\displaystyle\frac{8.5\times
10^{22}}{4\pi L^{2}}~{}\mathrm{events/year}$ (14) $\displaystyle=$
$\displaystyle 1.08\times 10^{15}~{}\mathrm{cm}^{-2}~{}\mathrm{events/year}.$
(15)

The total number of oscillated $\bar{\nu}_{e}$ events, assuming a 100%
oscillation probability and a 10% systematic uncertainty, is given by the sum
of the interactions on the proton and carbon targets:

$\displaystyle N_{\bar{\nu}_{e}}^{\mathrm{IBD}}=($ $\displaystyle
N_{\mathrm{H}}\cdot\sigma_{\bar{\nu}_{e}p\rightarrow
e^{+}n}^{\bar{\nu}_{\mu}}+$ (16) $\displaystyle
N_{\mathrm{C}}\cdot\sigma_{\bar{\nu}_{e}^{12}\mathrm{C}\rightarrow
e^{+}n^{11}\mathrm{B}}^{\bar{\nu}_{\mu}})\cdot\phi_{\bar{\nu}_{\mu}}\cdot\epsilon_{\mathrm{det}}$
$\displaystyle=$ $\displaystyle(1.48\pm 0.15)\times
10^{5}~{}\mathrm{events/year},$

where:

  * •

$N_{\mathrm{H}}=2.76\times 10^{30}$ is the number of hydrogen targets;

  * •

$N_{\mathrm{C}}=1.70\times 10^{30}$ is the number of carbon targets;

  * •

$\sigma_{\bar{\nu}_{e}p\rightarrow e^{+}n}^{\bar{\nu}_{\mu}}=93.5\times
10^{-42}$ cm2 is the $\bar{\nu}_{\mu}$ flux-averaged cross section for the IBD
process on proton KARMEN:2002zcm ; Vogel:1999zy ;

  * •

$\sigma_{\bar{\nu}_{e}^{12}\mathrm{C}\rightarrow
e^{+}n^{11}\mathrm{B}}^{\bar{\nu}_{\mu}}=8.5\times 10^{-42}$ cm2 is the
$\bar{\nu}_{\mu}$ flux-averaged cross section for the IBD process on carbon
KARMEN:2002zcm ; Kolbe:2000np ;

  * •

$\epsilon_{\mathrm{det}}=0.5$ is the assumed SHiNESS detector efficiency.

####  4.1.2 Charged-current interaction

The $\nu_{e}$ can interact on carbon via the charged-current channel:

$\displaystyle\nu_{e}+^{12}\mathrm{C}\rightarrow$
${}^{12}\mathrm{N_{gs}}+e^{-}$ (17)
${}^{12}\mathrm{N_{gs}}\rightarrow^{12}\mathrm{C}+e^{+}+\nu_{e}.$

The experimental signature of this process is the emission of the electron and
the subsequent ${}^{12}\mathrm{N_{gs}}$ beta decay in delayed coincidence
Conrad:2011ce . The scattered electron has a maximum kinetic energy
$E_{e^{-}}$ of 35.5 MeV due to the $Q$ value of 17.3 MeV. The recoil energy of
the ${}^{12}\mathrm{N_{gs}}$ nucleus is negligible, so
$E_{\nu_{e}}=E_{e^{-}}+17.3$ MeV LSND:2001fbw . The positron has a maximum
kinetic energy of 16.3 MeV and the decay has a lifetime of 15.9 ms
Krakauer:1991rf , which is significantly longer than the beam spill. Thus, in
order to reduce the amount of steady-state backgrounds, we apply a timing cut
of 2 beam spills (7.6 ms), which has an efficiency
$\epsilon_{\mathrm{time}}=30\%$. Figure 12 shows the time distribution of the
${}^{12}\mathrm{N_{gs}}$ $\beta^{+}$-decay for two consecutive beam spills.

Figure 12:  Time distribution of $\nu_{e}+^{12}\mathrm{C}\rightarrow
e^{-}+^{12}\mathrm{N_{g.s.}}$ events (solid blue line) for two consecutive
beam spills. The beam-spill time window is shown in light red and the
$\nu_{e}$ disappearance signal time window is defined by the dashed black
lines. The fraction of events inside the signal time window is 30%.

The yearly $\phi_{\nu_{e}}$ flux from $\pi^{+}$ DAR is the same as the
$\phi_{\bar{\nu}_{\mu}}$ one of eq. (14), given the production process of eq.
(1). The number of $\nu_{e}$ yearly charged-current interactions in the
detector, assuming a 10% systematic uncertainty, is then given by:

$\displaystyle N_{\nu_{e}}^{\mathrm{CC}}=$ $\displaystyle
N_{\mathrm{C}}\cdot\sigma_{\nu_{e}^{12}\mathrm{C}\rightarrow
e^{-}{{}^{12}}\mathrm{N_{gs}}}\cdot\phi_{\nu_{e}}\cdot\epsilon_{\mathrm{det}}\cdot\epsilon_{\mathrm{time}}$
(18) $\displaystyle=$ $\displaystyle(2.45\pm 0.25)\times
10^{3}~{}\mathrm{events/year},$

where $\sigma_{\nu_{e}^{12}\mathrm{C}\rightarrow
e^{-}{{}^{12}}\mathrm{N_{gs}}}=8.9\times 10^{-42}$ cm2 is the flux-averaged
cross section as calculated in ref. Kolbe:1999au . Both the KARMEN
KARMEN:1994xse  and the LSND experiment LSND:2001fbw  found this calculation
in good agreement with the data.

####  4.1.3 Neutral-current interaction

All the three neutrino types $\bar{\nu}_{\mu}$, $\nu_{\mu}$ and $\nu_{e}$ can
interact with the carbon atom through the neutral current (NC) channel and the
subsequent emission of 15.11 MeV $\gamma$:

$\displaystyle^{12}\mathrm{C}+\nu\rightarrow$ ${}^{12}\mathrm{C}^{*}+\nu$ (19)
${}^{12}\mathrm{C}^{*}\rightarrow^{12}\mathrm{C}+\gamma.$

The flux-averaged cross sections for $\nu_{e}$ and $\bar{\nu}_{\mu}$ and for
the mononenergetic $\nu_{\mu}$ were calculated in ref. Kolbe:1994xb  as
$\sigma^{\nu_{e},\bar{\nu}_{\mu}}_{{}^{12}\mathrm{C}\nu\rightarrow^{12}\mathrm{C}^{*}\nu}=10.5\times
10^{-42}$ cm2 and
$\sigma^{\nu_{\mu}}_{{}^{12}\mathrm{C}\nu\rightarrow^{12}\mathrm{C}^{*}\nu}=2.8\times
10^{-42}$ cm2, respectively. Both cross sections were measured by the KARMEN
experiment and found in good agreement with the calculation KARMEN:1995dwo ;
KARMEN:1998xmo . Given the large beam spill of the ESS, it is not possible to
distinguish between $\nu_{\mu}$, produced by prompt $\pi^{+}$ decay ($\tau=26$
ns), and $\nu_{e}$ and $\bar{\nu}_{\mu}$, produced by slower $\mu^{+}$ decay
($\tau=2.2$ $\mathrm{\SIUnitSymbolMicro s}$). Thus, we consider a total cross
section for the entire flux
$\sigma_{{}^{12}\mathrm{C}\nu\rightarrow^{12}\mathrm{C}^{*}\nu}=13.3\times
10^{-42}$ cm2.

The total number of neutral current $\pi^{+}$ DAR neutrino interactions on
carbon, assuming a 10% systematic uncertainty, is then given by:

$\displaystyle N_{\nu}^{\mathrm{NC}}=$ $\displaystyle
N_{\mathrm{C}}\cdot\sigma_{{}^{12}\mathrm{C}\nu\rightarrow^{12}\mathrm{C}^{*}\nu}\cdot\phi_{\nu}\cdot\epsilon_{\mathrm{det}}$
(20) $\displaystyle=$ $\displaystyle(7.33\pm 0.73)\times
10^{4}~{}\mathrm{events/year},,$

where $\phi_{\nu}=3.24\times 10^{15}$ cm-2 events/year is the sum of the
$\nu_{e}$, $\nu_{\mu}$, and $\bar{\nu}_{\mu}$ fluxes.

###  4.2 Decays of Heavy Neutral Leptons

As outlined in section 3.3, the main signal expected at SHiNESS from a HNL
with mass below the pion mass is an excess of events with an electron positron
pair in the final state (plus a neutrino, which of course exists the detector
unobserved). The expected number of signal events can be estimated inserting
the values for SHiNESS into eq. (11), for sufficiently low values of the
mixing (so that the approximation of a very long-lived particle holds). For
the electron-mixing case we obtain:

$\displaystyle N_{ee}^{(\mu)}$ $\displaystyle\simeq\displaystyle 3.8\times
10^{6}\left(\frac{\left|{U_{eN}}\right|^{2}}{10^{-6}}\right)^{2}\left(\frac{m_{N}}{m_{\mu}}\right)^{6}\left(1-\frac{m_{N}}{m_{\mu}}\right)^{4}\left(1+\frac{4m_{N}}{m_{\mu}}+\frac{m_{N}^{2}}{m_{\mu}^{2}}\right)\,,$
(21) $\displaystyle N_{ee}^{(\pi)}$ $\displaystyle\simeq 1.9\times
10^{6}\left(\frac{\left|{U_{eN}}\right|^{2}}{10^{-6}}\right)^{2}\left(\frac{m_{N}}{m_{\mu}}\right)^{7}\frac{m_{N}\,m_{\pi}\,(m_{\pi}^{2}-m_{N}^{2})}{(m_{\pi}^{2}-m_{\mu}^{2})^{2}}\,,$
(22)

where in the last expression we have neglected the mass of the electron for
simplicity. In the muon-mixing scenario, on the other hand, we get:

$\displaystyle N_{ee}^{(\mu)}$ $\displaystyle\simeq 6.9\times
10^{5}\left(\frac{\left|{U_{\mu
N}}\right|^{2}}{10^{-6}}\right)^{2}\left(\frac{m_{N}}{m_{\mu}}\right)^{6}\left(1-\frac{m_{N}}{m_{\mu}}\right)^{4}\left(1+\frac{4m_{N}}{m_{\mu}}+\frac{m_{N}^{2}}{m_{\mu}^{2}}\right)\,,$
(23) $\displaystyle N_{ee}^{(\pi)}$ $\displaystyle\simeq 4.0\times
10^{5}\left(\frac{\left|{U_{\mu
N}}\right|^{2}}{10^{-6}}\right)^{2}\left(\frac{m_{N}}{m_{\mu}}\right)^{6}\frac{m_{\pi}\left[(m_{\mu}^{2}+m_{N}^{2})m_{\pi}^{2}-(m_{\mu}^{2}+m_{N}^{2})^{2}+4m_{\mu}^{2}m_{N}^{2}\right]}{m_{\mu}(m_{\pi}^{2}-m_{\mu}^{2})^{2}}\,.$
(24)

For each case, and for a given value of the mass and the mixing, the total
number of events is obtained adding both contributions from pion and muon
decays.

Experimentally, the $e^{+}e^{-}$ pair will be emitted in the laboratory frame
with a certain angle, which is essentially controlled by $m_{N}$. The
selection efficiency in the SHiNESS detector for HNL events produced by muons
and pions decay is shown in figure 13, assuming an angular resolution of
$10^{\circ}$. This value represents an educated guess on the capability of the
detector and is comparable with the $\sim 12^{\circ}$ of LSND Conrad:2013mka ,
which only employed PMTs. A cut on the reconstructed energy of 17 MeV and a
cut on the angle between the two leptons of $15^{\circ}$ are applied. The
dependence on the parent particle comes from the different energy spectra,
which leads to a slightly different boost and therefore a different opening
angle for the electron pair.

Figure 13: Selection efficiency in the SHiNESS detector for HNLs in the case
of pion (blue) and muon (light blue) production. The angle of the $e^{+}e^{-}$
pair is required to be larger than $15^{\circ}$ and the reconstructed energy
is required to be larger than 10 MeV.

##  5 Expected backgrounds

###  5.1 Backgrounds to neutrino detection

####  $\bar{\nu}_{e}$ beam component

Although the vast majority of $\mu^{-}$ is absorbed in the target before it
can decay, a small fraction can decay in flight and emit $\bar{\nu}_{e}$,
which are indistinguishable from the ones caused by the eventual
$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$ oscillation. A simulation of the ESS
beam with the BERT-HP physics list gives a ratio
$\phi_{\bar{\nu}_{e}}/\phi_{\bar{\nu}_{\mu}}=2.2\times 10^{-3}$, after
requiring the interaction to happen during the beam spill time window.The
number of background events due to intrinsic $\bar{\nu}_{e}$ beam component
assuming a 25% systematic uncertainty is then given by:

$\displaystyle N_{\bar{\nu}_{e}}^{\mathrm{beam}}=($ $\displaystyle
N_{\mathrm{H}}\cdot\sigma_{\bar{\nu}_{e}p\rightarrow e^{+}n}^{\bar{\nu}_{e}}+$
(25) $\displaystyle N_{\mathrm{C}}\cdot\sigma_{\bar{\nu}_{e}^{12}C\rightarrow
e^{+}n^{11}\mathrm{B}}^{\bar{\nu}_{e}})\cdot\phi_{\bar{\nu}_{e}}\cdot\epsilon_{\mathrm{det}}$
$\displaystyle=$ $\displaystyle(254\pm 65)~{}\mathrm{events/year},$

where:

  * •

$\sigma_{\bar{\nu}_{e}p\rightarrow e^{+}n}^{\bar{\nu}_{e}}=72\times 10^{-42}$
cm2 is the $\bar{\nu}_{e}$ flux-averaged cross section for the IBD process on
proton KARMEN:2002zcm ;

  * •

$\sigma_{\bar{\nu}_{e}^{12}\mathrm{C}\rightarrow
e^{+}n^{11}\mathrm{B}}^{\bar{\nu}_{e}}=7.4\times 10^{-42}$ cm2 is the
$\bar{\nu}_{e}$ flux-averaged cross section for the IBD process on carbon
KARMEN:2002zcm .

This is an irreducible background which needs to be precisely assessed with a
detailed simulation of the ESS beam.

#### Cosmic-ray backgrounds

Although steady-state backgrounds can be precisely quantified during beam-off
periods, it is important to suppress them as much as possible to avoid losing
signal sensitivity. Cosmic rays can be efficiently vetoed by scintillator
panels surrounding the detectors, as in the KARMEN experiment Eitel:2000by ,
or by an outer layer of liquid scintillator equipped with PMTs, which is the
solution chosen by the JSNS2 experiment JSNS2:2021hyk . The main background
caused by cosmic rays is represented by cosmic neutrons entering the detector,
which can’t be vetoed, and by fast neutrons produced by cosmic muons impinging
on the materials surrounding the detector. These neutrons can enter the
detector, recoil the protons being thermalized, and finally be captured,
emitting 8 MeV gammas. They can mimic both the prompt signal (with the recoil
proton) and the delayed coincidence (with the neutron absorption). They
represent the dominant background of the JSNS2 experiment Maruyama:2022juu .

A CORSIKA simulation Heck:1998vt  of the cosmic-ray rate at the ESS site gives
a cosmic neutron rate in the detector of 5.8 Hz, assuming a 30 cm concrete
roof. This rate can be reduced by a factor of $10^{3}$ after applying
conservative selection cuts on the energy, time, and position of the event in
the detector, which translates into $1.6\times 10^{-5}$ events/spill.

A further reduction in the number of selected events can be obtained by
applying particle identification (PID) techniques. The LSND collaboration
pioneered a method which combines scintillation and Cherenkov light emissions
and achieves a rejection factor of around 100 LSND:2001aii .

The JSNS2 collaboration has developed a pulse-shape discrimination algorithm
by dissolving 10% Di-Isopropyl-Naphthalene (DIN) by weight into the liquid
scintillator, achieving a 97.4% fast neutron rejection power Hino:2021uwz .

This is also a typical task that can be performed by machine learning
algorithms. As an example, a convolutional neural network has been used to
discriminate between neutrinoless double-beta decay events and cosmic
spallation background in a liquid scintillator detector. The algorithm
achieved 61.6% background rejection at 90% signal efficiency, without using
vertex or energy reconstruction Li:2018rzw .

A Geant4 simulation of the SHiNESS detector shows that a further reduction of
a factor of 2 can be achieved by surrounding the steel tank with a layer of 20
cm of paraffin. Thus, assuming a paraffin shield and a conservative PID
rejection factor of 20, the rate of the cosmic-induced neutron background
during beam-on periods is $98\pm 10$ events/year and in any case can be
precisely quantified during beam-off periods. This background must be
multiplied by a factor of 2 for the $\nu_{e}$ CC search, given the extended
time window.

#### Accidental backgrounds

The detection of both the IBD and CC events relies on the detection of a
_prompt_ signal and a _delayed_ coincidence. Several background processes,
described below, can mimic at least one of the two. The accidental rate will
be given by the sum of the prompt-like background rates multiplied by the sum
of the delayed-like background rates, with the same procedure described in
ref. JSNS2:2013jdh .

#####  $\bar{\nu}_{e}$ beam interactions

The intrinsic $\bar{\nu}_{e}$ beam component, described in section 5.1, can
contribute to the accidental background when the prompt or the delayed signal
is not detected. This translates into irreducible prompt and delayed
accidental rates of $1\times 10^{-6}$ for the IBD channel. The IBD positron
can also mimic the electron of the CC channel.

##### Cosmic rays

The cosmic-ray background, described in detail in section 5.1, can contribute
to the accidental background when the prompt recoil proton or the captured
neutron is not detected. This corresponds to a small prompt and delayed
accidental rate of $3.8\times 10^{-7}$ ($7.5\times 10^{-7}$) events/spill for
IBD (CC) events.

##### Charged-current $\nu_{e}$ interactions

The charged-current channel
${}^{12}\mathrm{C}(\nu_{e},e^{-})^{12}\mathrm{N_{gs}}$ of eq. (17) can
represent a background for the IBD channel search. The $e^{-}$ emitted
promptly can mimic the IBD $e^{+}$ and the delayed beta decay can fake the
neutron absorption. Given the long lifetime of ${}^{12}\mathrm{N_{gs}}$ (15.8
ms), the probability of detecting both the $e^{-}$ and the $e^{+}$ during the
beam spill and within the IBD time window (around $100~{}$) can be considered
negligible. However, they both contribute to the a prompt accidental rate of
$6.4\times 10^{-4}$ events/spill and a delayed accidental rate of $0.8\times
10^{-4}$ events/spill. The $\nu_{e}$ can also interact with the carbon atom
via ${}^{12}\mathrm{C}(\nu_{e},e^{-})^{12}\mathrm{N}^{*}$, which does not beta
decay and contributes only to a prompt accidental rate of $3\times 10^{-4}$
events/spill. This background could be eventually suppressed by the
topological capabilities of the detector, which permit to distinguish point-
like events (such as electrons) from multi-site events (such as photons) in
the material Wonsak:2019lqq ; Dunger:2019dfo .

##### Neutron beam component

Low-energy neutrons (tens of MeV) produced by the beam can be absorbed by the
Gd atoms, emitting 8 MeV $\gamma$, mimicking neutron absorption from IBD
events. The proposal of detecting CE$\nu$NS at the ESS Baxter:2019mcx
requires a POT-coincident flux of less than $2\times 10^{-3}$ neutrons/cm2 s.
With this same requirement, and assuming 0.1% Gd mass concentration, this flux
produces an accidental rate in the detector of $4.5\times 10^{-3}$
events/spill after applying preliminary selection cuts. However, detailed
simulations will be useful to confirm the feasibility of this requirement in
the detector location. A dedicated measurement might also be necessary to
quantify this component: the JSNS2 collaboration used a 1-ton scintillator
detector to precisely assess the neutron rate at J-PARC JSNS2:2013jdh .

##### Gamma beam component

Neutrons produced by the beam can be captured in the materials surrounding the
detector, producing gamma particles. A fraction of these gammas can eventually
enter the detector and mimic a delayed signal in both the IBD and the CC
channels. In order to precisely quantify this background, a detailed
simulation of the ESS beam and of the experimental hall is necessary. A
preliminary result gives a rate of beam-induced gamma of $6.1\times 10^{-3}$
events/spill after event selection.

Accidental background rates for $\bar{\nu}_{e}$ IBD and $\nu_{e}$ CC channels
are summarized in table 2. A conservative cut on the spatial distance between
the prompt and the delayed signal can achieve an accidental background
rejection power of 90%, as shown in figure 5. With this assumption, the
accidental background rate is $130\pm 17$ events/year for the $\bar{\nu}_{e}$
IBD channel and $31\pm 6$ events/year for the $\nu_{e}$ CC channel, assuming a
10% systematic uncertainty.

Channel | Background |  | Prompt rate
---
(events/spill)
| Delayed rate
---
(events/spill)
$\bar{\nu}_{e}$ IBD | Cosmic rays | $3.8\times 10^{-7}$ | $3.8\times 10^{-7}$
|  $\bar{\nu}_{e}$ beam | $1.0\times 10^{-6}$ | $1.0\times 10^{-6}$
| ${}^{12}\mathrm{C}(\nu_{e},e^{-})^{12}\mathrm{N_{g.s.}}$ | $6.4\times 10^{-4}$ | $0.8\times 10^{-4}$
| ${}^{12}\mathrm{C}(\nu_{e},e^{-})^{12}\mathrm{N}^{*}$ | $3.0\times 10^{-4}$ | -
| Beam neutrons | - | $4.5\times 10^{-3}$
| Beam gammas | - | $6.1\times 10^{-3}$
| Total | $9.4\times 10^{-4}$ | $1.1\times 10^{-2}$
$\nu_{e}$ CC | Cosmic rays | $7.5\times 10^{-7}$ | $7.5\times 10^{-7}$
|  $\bar{\nu}_{e}$ beam | $1.0\times 10^{-6}$ | -
| ${}^{12}\mathrm{C}(\nu_{e},e^{-})^{12}\mathrm{N}^{*}$ | $3.0\times 10^{-4}$ | -
| Beam gammas | - | $8.1\times 10^{-3}$
| Total | $3.0\times 10^{-4}$ | $8.1\times 10^{-3}$
Table 2: Rates of prompt and delayed accidental backgrounds for the
$\bar{\nu}_{e}$ IBD and $\nu_{e}$ CC channels in the SHiNESS detector.

###  5.2 Backgrounds to HNL decay

HNL candidate events are selected by requiring the presence of two leptons
originating from the same vertex. In order to obtain an estimate of the
background for these events in the experiment, we require the detection of two
separate signals in the photosensors within a time window of 100 ns, which
makes the accidental background negligible. A primitive lepton-finding
reconstruction algorithm has been implemented. The energy of the two signals
is required to be above 17 MeV and the angle between the two reconstructed
direction must be larger than 15∘. Assuming these requirements, the main
backgrounds for HNL events are detailed below.

##### Cosmic-ray background

Cosmic muons can produce radioactive isotopes by spallation on the matter
surroundings the detector. These muons do not necessarily pass through the
veto, so they can’t be tagged. The 17 MeV cut removes the vast majority of
these products. The remaining events, misreconstructed as two separate
leptons, give a background rate of $18$ events/year for the HNL search.

##### Charged-current $\nu_{e}$ interactions

The interaction of a $\nu_{e}$ with the carbon atom can produce nitrogen atoms
in excited states, through the
${}^{12}\mathrm{C}(\nu_{e},e^{-})^{12}\mathrm{N}^{*}$ interaction. Most of
these states decay by prompt proton emission to 11C, which in turn emit gammas
between 2 MeV and 7 MeV LSND:2001fbw . The 17 MeV cut, coupled with the good
energy resolution of the detector, is able to reject essentially all these
events and this component is considered negligible. However, the outgoing
electron can be misreconstructed as two separate leptons, when the gammas or
the delayed beta decay are not detected, giving a background contribution of
$14$ events/year.

##### Neutral-current $\nu$ interactions

The neutral current interaction between all three neutrino types and carbon
atoms, detailed in section 4.1.3, produce a 15.11 MeV gamma, which could be
misreconstructed as two distinct leptons. The vast majority of these events is
rejected by the 18 MeV cut and a preliminary simulation gives a background
rate for this component of $8$ events/year.

#####  $\bar{\nu}_{e}$ beam component

The intrinsic $\bar{\nu}_{e}$ beam component can represent a background also
for the HNL search when the delayed neutron is not detected and the positron
is misreconstructed as two distinct leptons. This component gives a background
of $21$ events/year.

Given the large uncertainties regarding the reconstruction algorithm and the
need for a detailed simulation of the beam gammas and cosmic-ray component, we
assign a systematic uncertainty of 50% to the total HNL background. Thus, the
HNL sensitivity calculation of section 6.3 assumes a background of $(61\pm
31)$ events/year.

##  6  Results

This section presents our results for the expected sensitivity of SHiNESS for
the three scenarios outlined in section 3.

###  6.1  Non-unitarity

As outlined in section 3.1, SHiNESS is expected to be sensitive to the closure
of the unitarity triangle in the $e\mu$ sector, parametrized by $t_{e\mu}$ in
eq. (4). In order to determine the attainable sensitivity, we perform a binned
analysis of the expected number of IBD interactions. Specifically, we define a
Gaussian $\chi^{2}$ as:

$\chi^{2}_{\mathrm{IBD}}\left(|t_{e\mu}|^{2}\right)=\sum_{i}{\left[\frac{N_{\mathrm{obs,i}}^{\mathrm{IBD}}-N_{\mathrm{exp,i}}^{\mathrm{IBD}}(\xi_{\mathrm{sig}},\xi_{\bar{\nu}_{e}},t_{e\mu})}{{\sigma_{\mathrm{stat,i}}}}\right]^{2}+\left(\frac{\xi_{\mathrm{sig}}}{\sigma_{\mathrm{sig}}}\right)^{2}+\left(\frac{\xi_{\bar{\nu}_{e}}}{\sigma_{\bar{\nu}_{e}}}\right)^{2}},$
(26)

where the sum extends over all energy bins. Here, we assume the experiment
will observe a number of events
$N_{\mathrm{obs,i}}^{\mathrm{IBD}}=N_{\mathrm{exp,i}}^{\mathrm{IBD}}+N_{\mathrm{bkg,i}}$
consistent with the expected IBD interactions from the intrinsic
$\bar{\nu}_{e}$ beam component, calculated in eq. (25), plus the steady-state
background contribution. In presence of non-unitarity effects, the expected
number of events in each bin is obtained as

$N_{\mathrm{exp,i}}^{\mathrm{IBD}}=(1+\xi_{\mathrm{sig}})|t_{e\mu}|^{2}N^{\bar{\nu}_{\mu}}_{\mathrm{exp,i}}+(1+\xi_{\bar{\nu}_{e}})N_{\mathrm{exp,i}}^{\mathrm{IBD}}+N_{\mathrm{bkg,i}}\,,$
(27)

where $\xi_{sig},\xi_{\bar{\nu}_{e}}$ are the nuisance parameters associated
to the signal and the $\bar{\nu}_{e}$ beam component-normalization, while we
assume that uncertainties on the steady-state background are negligible as it
can be characterized using beam-off data. This implies that the steady-state
background contributions cancel in the numerator of eq. (26) and only
contribute to the statistical error in the denominator:
$\sigma_{\mathrm{stat,i}}=\sqrt{N_{\mathrm{exp,i}}^{\mathrm{IBD}}+N_{\mathrm{bkg,i}}}$.
Finally, $\sigma_{\mathrm{sys}}=10\%$ is the systematic uncertainty on the
normalization, while $\sigma_{\bar{\nu}_{e}}=25\%$ is the systematic
uncertainty on the $n_{\bar{\nu}_{e}}$ beam component. The minimum of the
$\chi^{2}$ is found after minimizing eq. 26 with respect to the nuisance
parameters $\xi_{sig},\xi_{\bar{\nu}_{e}}$.

Our results are shown in figure 14 for 2 and 4 years of data taking by the
blue and orange lines, respectively. As can be seen from the comparison
between the blue and orange lines, the improvement obtained by doubling the
data taking period is limited by the systematic uncertainties. For comparison,
we also show limits on this parameter obtained from the averaged-out region of
light sterile neutrino searches at short-baseline neutrino experiments (see
the discussion in refs. Blennow:2016jkn ; Coloma:2021uhq ), using the relation

$\left|t_{e\mu}\right|^{2}=\frac{1}{2}\sin^{2}\theta_{\mu e}\,.$ (28)

Our results indicate that SHiNESS can significantly improve the limits
provided by KARMEN KARMEN:2002zcm  and NOMAD NOMAD:2003mqg , indicated by the
shaded regions.

Figure 14: $\chi^{2}$ analysis with one degree of freedom for the
$\bar{\nu}_{e}$ appearance channel as a function of the absolute value of the
electron-muon triangle closure parameter $t_{e\mu}$ for 2 years (blue) and 4
years (orange) of data taking. Existing limits at 90% C.L. from the KARMEN
KARMEN:2002zcm  and NOMAD NOMAD:2003mqg  experiments are shown in blue and
red, respectively.

###  6.2  Light sterile neutrinos

The search for $\bar{\nu}_{e}$ appearance at spallation sources to look for
light sterile neutrinos is not a novel concept: it has been already performed
by the KARMEN experiment at the ISIS facility KARMEN:2002zcm , was proposed by
the OscSNS collaboration at the SNS OscSNS:2013vwh , and is currently being
carried out by the JSNS2 collaboration at J-PARC JSNS2:2013jdh . However, the
ESS beam will provide a neutrino flux around 1 order of magnitude larger than
J-PARC and SNS, allowing a competitive sensitivity in this channel with a
relatively small detector (42 ton for SHiNESS, compared to the 450 ton of
OscSNS). Using eq. (8) and assuming the MiniBooNE best-fit point at
$(\sin^{2}2\theta_{\mu e},\Delta m_{41}^{2})=(0.807,0.043~{}\mathrm{eV}^{2})$
MiniBooNE:2020pnu  gives a flux-averaged appearance probability of 0.21% at
SHiNESS location. We can multiply this probability by the number of
$\bar{\nu}_{e}$ interactions expected with 100%
$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$ transmutation of eq. (16), obtaining
an appearance signal of $N_{\bar{\nu}_{e}}^{app}=304\pm 23$ events/year.
Signal and background events for this channel and 2 calendar years of data
taking are summarized in the first row of table 3. Figure 15 shows the
reconstructed energy distribution for $\bar{\nu}_{e}$ candidate events (left
panel), assuming a $15\%/\sqrt{E/\mathrm{MeV}}$ energy resolution.

|
---|---
Figure 15:  Left: Reconstructed energy spectrum for IBD candidate events. The filled light blue, blue, and green histograms correspond to the cosmic-ray background, the accidental background, and the $\bar{\nu}_{e}$ beam component, respectively. The solid green line corresponds to the full spectrum in the presence of $\bar{\nu}_{e}$ appearance, assuming the MiniBooNE best-fit point $(\sin^{2}2\theta_{\mu e},\Delta m_{41}^{2})=(0.807,0.043~{}\mathrm{eV}^{2})$ MiniBooNE:2020pnu . Right: Reconstructed energy spectrum for $\nu_{e}$ CC candidate events. The filled light blue, blue, and red histograms correspond to the cosmic-ray background, the accidental background, and the $\nu_{e}$ beam component in the absence of oscillation, respectively. The solid red line corresponds to the full spectrum in the presence of $\nu_{e}$ disappearance, assuming the GALLEX+SAGE+BEST best-fit point $(\sin^{2}2\theta_{ee},\Delta m_{41}^{2})=(0.38,7.3~{}\mathrm{eV}^{2})$ MiniBooNE:2020pnu .  | Channel
---
$(\sin^{2}2\theta,\Delta m_{41}^{2})$
Signal | Background |
|  $\bar{\nu}_{e}$ app.
---
$(0.807,0.043~{}\mathrm{eV}^{2})$
$608\pm 66$ |  $\bar{\nu}_{e}$ beam | $508\pm 129$
|  | Cosmic rays | $196\pm 14$
|  | Accidental | $259\pm 31$
|  | Total | $963\pm 133$
|  $\nu_{e}$ disapp.
---
$(0.38,7.3~{}\mathrm{eV}^{2})$
$938\pm 402$ | Unoscillated $\nu_{e}$ | $3974\pm 402$
|  | Cosmic rays | $372\pm 19$
|  | Accidental | $62\pm 10$
|  | Total | $4408\pm 403$
Table 3: Signal and background events for the $\bar{\nu}_{e}$ appearance and
$\nu_{e}$ disappearance channels for 2 calendar year of exposure (10000 hours)
at full beam power. In the case of $\nu_{e}$ disappearance, the signal
corresponds to the deficit of events. The $(\sin^{2}2\theta,\Delta
m_{41}^{2})$ values correspond to the MiniBooNE best fit point
MiniBooNE:2020pnu  for the $\bar{\nu}_{e}$ appearance and to the
GALLEX+SAGE+BEST best fit point Barinov:2021mjj  for the $\nu_{e}$
disappearance. The uncertainties include a 10% systematic on the normalization
and a 25% on the $\bar{\nu}_{e}$ beam component.

The SHiNESS experiment also has the ability to probe $\nu_{e}$ disappearance
at a short baselines, which is the channel of both the _gallium_ and the
_reactor antineutrino_ anomaly. The best-fit point given by a combined
analysis of the BEST, SAGE, and GALLEX results occurs at
$(\sin^{2}2\theta_{ee},\Delta m_{41}^{2})=(0.38,7.3~{}\mathrm{eV}^{2})$
Barinov:2021mjj . Inserting this value into eq. (7) gives a flux-averaged
disappearance probability of 19.1% at SHiNESS baseline. We can multiply this
probability by the number of $\nu_{e}$ interactions of eq. (17), obtaining a
disappearance signal (thus a deficit of events) of
$N_{\nu_{e}}^{disapp}=469\pm 108$ events/year. Signal and background events
for this channel and 2 calendar years of data taking are summarized in the
second row of table 3. The right panel in figure 15 shows the reconstructed
energy distribution for $\nu_{e}$ candidate events, assuming a
$15\%/\sqrt{E/\mathrm{MeV}}$ energy resolution.

The sensitivity of SHiNESS to light neutrino oscillations is shown in figure
16, computed using a $\chi^{2}$ analysis analogous to the one of eq. (26).
Results are shown separately for oscillation searches in the appearance (left)
and disappearance (right) channels. As shown in the left panel, the SHiNESS
experiment is able to fully cover the LNSD 99% C.L allowed region at a
$5\sigma$ level, in a region complementary to the one covered by the JSNS2
experiment JSNS2:2013jdh . As for the search in the disappearance channel, our
results indicate that SHiNESS can cover almost entirely the region allowed by
a GALLEX+SAGE+BEST combined analysis with a sensitivity better than $5\sigma$,
allowing to definitely rule out the (3+1) light sterile neutrino explanation
for the gallium anomaly with an independent measurement. Also, as can be seen
in both panels, in the limit of large mass-squared splittings the averaged-out
oscillations translate into a flat sensitivity. It is easy to check that, once
the corresponding value of the mixing angle is mapped onto a unitarity
constraint as outlined in refs. Blennow:2016jkn ; Coloma:2021uhq , the results
in section 3.1 are correctly recovered, as expected.

We finish this section with a remark regarding the validity of the sensitivity
limits shown in figure 16. In our calculations, the test statistics is assumed
to be distributed according to a $\chi^{2}$ with 2 degrees of freedom,
assuming Wilks’ theorem holds. Although this is a common approach in the
literature, it has been shown Agostini:2019jup ; Giunti:2020uhv ;
PROSPECT:2020raz ; Coloma:2020ajw ; Berryman:2021yan  that the actual
distribution typically differs from a $\chi^{2}$ distribution, since the
conditions needed to apply Wilks’ theorem are violated for sterile neutrino
oscillation searches. Therefore our results should only be regarded as
indicative of an _approximate_ value for the attainable sensitivity. An
accurate result requires proper evaluation through Monte Carlo simulation:
this falls beyond the scope of this article and is left for future work.

|
---|---
Figure 16:  Left: The SHiNESS sensitivity to neutrino oscillations in the
appearance channel. The LSND allowed region is shown in light blue
LSND:2001aii . The thin solid line corresponds to the expected sensitivity of
the JSNS2 experiment JSNS2:2013jdh . The dashed lines correspond to exclusion
contours of the KARMEN2 KARMEN:2002zcm  and NOMAD NOMAD:2003mqg  experiments.
Right: The SHiNESS sensitivity to neutrino oscillations in the $\nu_{e}$
disappearance channel. The region allowed by the GALLEX+SAGE+BEST combined
analysis Barinov:2021asz ; Berryman:2021yan  is shown in light blue. The
dashed lines correspond to the exclusion contours of the RENO+NEOS
RENO:2020hva , PROSPECT PROSPECT:2020sxr , and KATRIN KATRIN:2022ith
experiments. Contours labeled as ‘Wilks’ have been obtained assuming the test
statistics follows a $\chi^{2}$ distribution with two degrees of freedom, see
text for details.

###  6.3  Heavy neutral leptons

Figure 17 shows the sensitivity at 90% C.L. for electron and muon mixing in
the left and right panels, respectively. In this case, the sensitivity has
been computed using just the total rates (unbinned $\chi^{2}$) for the signal
(see section 4.2) and backgrounds (see section 5.2), assuming a 50% systematic
uncertainty on the background. Within the considered mass range, our results
indicate that SHiNESS can set world-leading constraints in both cases,
improving over current limits by over an order of magnitude in a sizable
region of the parameter space.

(a) Electron mixing.

(b) Muon mixing.

Figure 17:  Constraints at 90% C.L. on the mixing of a HNL $|U_{lN}|^{2}$ with
electron (left) and muon (right) neutrinos, for 2 years of data taking. The
gray areas correspond to existing limits from PSI Daum:1987bg , PIENU
PIENU:2017wbj ; PIENU:2019usb ; Bryman:2019bjg , TRIUMF Britton:1992xv , T2K
Arguelles:2021dqn , and MicroBooNE MicroBooNE:2021usw , obtained from ref.
Fernandez-Martinez:2023phj .

##  7 Summary and conclusions

The existence of neutrino masses is irrefutable, and requires the extension of
the particle content of the Standard Model (SM). Such extension may lead to
additional consequences, including an apparent non-unitary mixing matrix in
the active neutrino sector, anomalous oscillations at short-baselines, or even
the decay signals of heavy neutral leptons that interact with the visible
sector through their mixing with the SM neutrinos.

The ESS provides an exceptional opportunity to carry out neutrino experiments
at the edge of the intensity frontier, and to pursue new physics signals. The
facility will provide the most intense source of $\pi^{+}$DAR neutrinos in the
world, with one order of magnitude improvement with respect to current
facilities. Here we propose to carry out a new experiment, SHiNESS, and
highlight the novel opportunities it offers to explore the existence of new
physics in the neutrino sector.

The experiment will be able to collect a high-statistics sample of neutrino
interactions with only two calendar years of full-power beam. The baseline
detector design includes a tank filled with 42 ton of liquid scintillator
whose light is detected by large-area PMTs and LAPPDs. Liquid scintillators
are a low-risk, proven technology with a solid track record in particle
physics. The introduction of LAPPDs allows to separate the Cherenkov light
from the scintillation light and enables the reconstruction of topological
features. A possible alternative is represented by water-based liquid
scintillators, which have the potential to improve the background-rejection
capabilities of the experiment. The proposed detector could also operate as a
neutrino flux monitor for the proposed CE$\nu$NS detection experiments
Baxter:2019mcx  and represents a perfect fit for the rich ESS particle physics
program Abele:2022iml .

We have identified a potential location for the detector at about 25 m from
the beam target (see figure 4). The detector could eventually be replicated at
a different baseline in order to further reduce flux systematic uncertainties,
as done by the JSNS2 collaboration with the JSNS2-II detector Maruyama:2022juu
.

To illustrate its physics potential, we have computed the expected
sensitivities for three well-motivated examples of new physics in the neutrino
sector, as outlined above: searches for a non-unitary leptonic mixing matrix
(summarized in figure 14), for anomalous oscillations induced by a sterile
neutrino at the eV scale (see figure 16), and decay signals of heavy neutral
leptons with masses above the MeV scale (shown in figure 17). In all three
cases considered, we find that the SHiNESS experiment would be able to improve
significantly over current constraints, targeting unexplored regions of the
parameter space. A notebook has also been made available online, which allows
to reproduce the sensitivities and energy spectra of the channels used for our
oscillation analysis binder .

The timing of this proposal is timely and appropriate, as the ESS user program
is expected to start in 2025. The search for light sterile neutrinos has
spurred a very active physics program in several laboratories worldwide. At
the same time, the renewed interest in heavy neutral leptons at the MeV-GeV
scale has triggered an intense activity in the field. In this sense we
underscore the short timescale of the proposed experiment, which could deliver
results in a very short period of time. To begin with, the SHiNESS detector
could start acquiring data as soon as the beam turns on, and does not require
any beamline upgrade or modification. Moreover, thanks to the intensity of the
ESS beam, SHiNESS has the capability to reach competitive sensitivities for
several new physics scenarios in a couple of years of data taking.

###### Acknowledgements.

The authors are thankful to R. Castillo Fernandez, P. Ferrario, P. Huber, J.
Martín-Albo, F. Monrabal, P. Novella, M. Sorel, and D. Zerzion for the
precious suggestions. SRS acknowledges the support of a fellowship from “la
Caixa Foundation” (ID 100010434) with code LCF/BQ/PI22/11910019 and of the
Severo Ochoa Program grant CEX2018-000867-S. This work has received partial
support from the European Union’s Horizon 2020 Research and Innovation
Programme under the Marie Sklodowska-Curie grant agreement no. 860881-HIDDeN
and the Marie Sklodowska-Curie Staff Exchange grant agreement no.
101086085–ASYMMETRY. PC acknowledges financial support by grant
RYC2018-024240-I, funded by MCIN/AEI/10.13039/501100011033 and by “ESF
Investing in your future”. She is also supported by the Spanish Research
Agency (Agencia Estatal de Investigación) through the grant IFT Centro de
Excelencia Severo Ochoa no. CEX2020-001007-S funded by
MCIN/AEI/10.13039/501100011033/, and by Grant PID2022-142545NB-C21, funded by
MCIN/AEI/10.13039/501100011033/ FEDER, UE.

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