neutron beam to search for a deflection due to a transversely applied electric
field. In such an experiment, a neutron with a hypothetical non-zero electric
charge $Q$ would follow a parabolic path resulting in a deflection

$y=\frac{QEL^{2}}{2m_{n}v^{2}}$ (9)

with $E$ the electric field applied over the length $L$, $m_{n}$ the neutron
mass, and $v$ the neutron velocity. The shift is determined by measuring the
neutron count rate as a function of the lateral position of a narrow aperture
in front of a detector for the two cases $E=0$ and $E\neq 0$. In 1967 a
similar experiment was performed, however, using a double-crystal spectrometer
sensitive to changes of the angular orientation of a neutron beam [362]. In
the 1980’s Baumann, Gähler, Kalus and Mampe performed two experiments using a
single- and a multi-slit aperture approach, respectively [364, 348]. They were
able to obtain a several tens of micrometer wide neutron beam using neutron
optical lenses and eventually achieved the current best result for the neutron
electric charge of $(-0.4\pm 1.1)\times 10^{-21}~{}e$. Improvements of this
limit by 1-2 orders of magnitude have been proposed using spin interferometry
with cold neutrons [365], Ramsey spectroscopy of gravitationally bound quantum
states of UCNs [366] (see also Section 7.3), and a deflection method with UCNs
[367, 368].

A novel method using a Talbot-Lau neutron grating interferometer has been
proposed recently [369]. The approach plans to employ neutron absorption
gratings on silicon or quartz wafers with grating periods on the order of a
few micrometers, which are commonly used in neutron phase contrast radiography
and neutron dark-field imaging in various physics and industry applications
[370, 371, 372, 373, 374]. A symmetric Talbot-Lau setup consists of three
identical absorption gratings placed at equal distances to each other and
allows to detect tiny deflections of the neutron beam [375]. The first grating
forms several coherent line sources. These beams are then diffracted by the
second grating which produces an overlapping interference pattern in the plane
of the third grating. This last grating is placed directly in front of the
neutron detector and serves to analyse the microscopic pattern without the
need of a high spatial-resolution detector. Since the method relies on
diffraction of the neutrons on the gratings, there exists a strong wavelength
dependency which can wash out the interference pattern if a continuous white
beam is employed. However, by combining the high-intensity pulsed neutron beam
of the ANNI beamline at the ESS with a time-of-flight Talbot-Lau setup it is
possible to obtain and measure interference patterns for several distinct
wavelengths. Overall, this substantially increases the sensitivity to
deflection. The electric charge of the neutron is then measured by fixing the
position of the gratings such that the incident intensity on the detector is
most sensitive to beam deflections, i.e. at the steepest slope of the
interference pattern. By inverting the polarity of the applied electric field,
one then searches for changes in the measured neutron count rate which would
indicate a non-zero electric charge.

Another important key feature of the proposed setup is to use two separate
neutron beams which are exposed to electric fields of opposite polarity –
similar to the Beam EDM experiment. This allows for a better systematic
control by compensating for common noise and drifts in the individual beams.
It was estimated that in a full-size experiment with a length $L=5$ m and an
electric field $E=100$ kV/cm a sensitivity on the order $10^{-23}~{}e$ can be
achieved within 100 days of measurement time [369].

###  5.5 Hadronic weak interaction

Due to the complex QCD structure of baryons, the Hadronic Weak Interaction
(HWI) is one of the final frontiers of electroweak phenomenology. It is
customarily characterised by either 6 parity-odd meson exchange coupling
constants in the Desplanques-Donoghue-Holstein (DDH) model [376, 377] or the
more modern approach of five low-energy effective field theory (EFT) couplings
[378, 209, 379, 380]. First attempts to compute these couplings on the lattice
are made in [211]. In the limit of large number of colours $N_{c}$, there is a
hierarchy of the expected relative magnitudes which can be tested
experimentally [381, 382, 217].

The challenging experimental program of mapping out these coefficients began
shortly after postulation of parity violation (PV) in 1957 [28] with first
observation of PV in the circular polarisation of a gamma transition in
[181]Ta [383]. PV effects in HWI are usually small but may be amplified by a
factor of up to $10^{6}$ (in [139]La [384]) by the interference of closely
spaced S and P resonances. These data have confirmed statistical models [385,
386], but the results are not directly interpretable in terms of the
underlying couplings due to the complicated nuclear wave function structure.
Thus measurements in the last four decades have focused on few-body
observables with exactly calculable wave functions [212]. Former results
include the longitudinal single-spin asymmetry from polarised proton
scattering from hydrogen $A^{pp}_{p}$ [387, 388], deuterium $A^{pd}_{p}$
[388], and helium [389]. Measurements of $n+p\to d+\gamma$ (both gamma
asymmetry with respect to the neutron spin $A^{np}_{\gamma}$ [390] and gamma
circular polarisation [391]), $n+d\to t+\gamma$ (gamma asymmetry
$A^{nd}_{\gamma}$) [392], and neutron spin rotation in [4]He
($\phi^{\text{He-4}}_{n}$) [393] had uncertainties that were still too large
to identify the HWI contributions to the observables.

Two recent high-precision hadronic PV inaugural experiments on the SNS FnPB of
$n+p\to d+\gamma$ ($A^{np}_{\gamma}$) and $n+^{3}\mathrm{He}\to p+t$
($A^{n\text{He-3}}_{p}$) have ushered a new era of precision HWI
characterisation. They offer results of 2.1 $\sigma$ [283] and 1.6 $\sigma$
[394], respectively, from zero, and complete the set of experiments needed to
give a first experimental determination of the couplings introduced above. A
three times improvement in each of these experiments at ANNI would have
significant impact in constraining parameters of the HWI when taken with
$A_{p}^{pp}$ and $A_{p}^{pd}$. The next-generation measurement of
$\phi_{n}^{\text{He-4}}$ is being planned at the NIST NG-C beamline [393,
395]. The addition of a new counting-mode measurement of $A_{\gamma}^{nd}$ at
ANNI would over-constrain the theory, allowing for an extraction of all five
pionless couplings and test the EFT formalism. In addition, first results in a
new experiment to measure spin rotation in hydrogen ($\phi_{n}^{\text{p}}$)
[395] would help to isolate the $I=2$ coupling, which may be calculated on the
lattice in the near future. The estimated effect $\phi_{n}^{\text{p}}\approx
9\times 10^{-7}$ rad/m, is large enough to be observed at an intense slow
neutron beam such as ANNI. The scientific motivation to pursue this experiment
is therefore very high.

####  5.5.1 Improvement of the SNS experiments NPDGamma and n3He

The NPDGamma and n3He measurements at the SNS were both statistics limited,
with a figure of merit $P^{2}N$, where $P$ is the polarisation and $N$ the
total number of neutrons captured on target. They both have similar
requirements of a high flux neutron beamline [267], with low background gamma
rates from captures outside the target. High polarisation helps the figure of
merit, but $P\approx 100\%$ is not required to minimise the related systematic
correction at the limited relative uncertainty of the $\mathcal{O}(10^{-8})$
asymmetries that can be achieved. The beamline requires a thin neutron monitor
to correct for fluctuations in the neutron flux. A polarised 3He spin filter
was used for the preliminary measurement of NPDGamma at LANL, but it was
determined that a higher figure of merit and more stable neutron polarisation
could be achieved with a supermirror bender polariser. A $\sim$1 mT holding
field is needed to maintain the neutron polarisation and define the spin
direction. A relative gradient of less than $1\times 10^{-4}$ cm-1 is required
to prevent Stern-Gerlach steering [283, 394] and allow for efficient spin
flipping. The direction of the magnetic field must be known to $0.1^{\circ}$
with respect to the detector to prevent the mixing of the parity-allowed
nuclear spin asymmetry associated with
$\bm{k}_{n}\cdot\left(\bm{\sigma}_{n}\times\bm{k}_{\gamma,p}\right)$. A
resonant neutron spin rotator [396] has been used to alternate the spin
direction on a pulse-by-pulse basis to reduce systematics associated with
instrumental drift. These experiments use a novel double cos-theta RF coil
with fringeless transverse fields that operates for both longitudinally and
transversely polarised neutrons [397].

The NPDGamma experiment employs a liquid hydrogen target operated at 16 K with
catalysts to thermalise into the lower energy parahydrogen state [398], with a
greatly reduced scattering cross section, and in which the spin-flip
transitions are energetically forbidden, reducing beam depolarisation in the
target. A key element in the design of such a target is the balance between
thin neutron windows for gamma background, and safety of the cryogenic target
against explosion [393]. A new target vessel would need to be designed and
fabricated for the higher flux ANNI beamline – the old vessel was cut up to be
used as a target for background asymmetry measurements [283]. The 48-segment
CsI-Tl scintillator array [394] with $3\pi$ angular coverage detector used at
LANL and SNS could also be used in a future measurement at the ESS. Vacuum
photodiodes were used to reduce systematic uncertainty from dependence of
detection efficiency on the magnetic field from the spin rotator.

The n3He experiment uses a combined 3He target and ion chamber to measure the
direction of the protons. A grid of vertical wires can measure the proton
asymmetry in one transverse direction as a function of depth into the target.
It was designed to measure both the longitudinal (PV) and transverse (both PV
and parity conserving) asymmetries, but was run only in transverse mode, which
has a figure of merit two times larger than for longitudinal mode. This
detector was run in current mode, with separate preamplifiers and 24-bit
digitisers for each of the 144 sense wires. This same apparatus would be
suitable for a higher-statistics measurement at the ESS.

The sensitivities of both NPDGamma and n3He were entirely determined by
statistics. Thus the higher polarised neutron flux at ANNI (the simulated gain
in event rate for NPDGamma is included in Table 2) would allow to raise the
significance of the observed asymmetries above the $5\sigma$ level with about
6 months of data per target (compare Table 3 of Section 5.6).

####  5.5.2 New experiments: NDTGamma, spin rotation

The NDTGamma experiment presents a special challenge with the neutron capture
cross section 642 times smaller than hydrogen, so that a similar design would
be completely dominated by background radiation. This was a major concern of a
previous measurement of $A^{nd}_{\gamma}$ at the ILL [392]. In addition, the
spin-1 ground state of deuterium allows for spin-flip scattering. However, the
parity odd amplitude is expected to be similar to NPDGamma, giving a 25 times
larger asymmetry, which would result in similar experimental sensitivity to
the PV amplitudes. Given the lower capture rate, and the advent of new high
precision digital spectroscopy [399], this presents the opportunity to perform
the first hadronic PV measurement of a few-body observable in counting mode
with waveform digitisers and digital pulse processing. With spectroscopic
separation of backgrounds, D2O could be used as the target material, which is
nonflammable, and would not require a pressure vessel. The 6.2 MeV capture
gamma from deuterium is well above the gamma energy from capture of other
background materials and oxygen in the target, which would only contribute 20%
of the event rate. Preliminary measurements at LANL have shown that the
average neutron polarisation at capture in D2O is about 50%.

The beamline design is similar to that of NPDGamma, but with a [3]He spin
filter if the gamma background from ANNI’s polarising bender should be too
high in spite of its location far upstream in the guide (see Figure 9). The
D2O target would be cooled to reduce spin-flip scattering. The entire neutron
path through the detector needs to be shielded with 6Li to reduce the
background capture gamma rate to a level comparable to that from deuterons,
not to limit the total count rate. In a highly segmented scintillator array
such as the WASA detector [400], the count rate would be $\sim$500 kHz in each
channel. In 6 months of beamtime on ANNI, the experiment would collect
$7\times 10^{15}$ gammas, resulting in $\delta A^{\prime}=1/P\sqrt{N}G=4\times
10^{-8}$ given the total neutron polarisation $P=0.6$ and geometry factor
($\cos\theta$ coverage and detection efficiency) $G=0.5$. This is 0.15 times
the range of asymmetries from the extremities of the DDH reasonable range of
$(-0.62\mbox{~{}to~{}+}2.1)\times 10^{-7}$ [212] .

Neutron spin rotation experiments are also well-suited for the ANNI beamline.
The apparatus being developed to measure $\phi_{n}^{\text{He-4}}$ at the NIST
NG-C beamline [393, 395, 401] would also be well-suited to measure
$\phi_{n}^{\text{p}}$ at ANNI where the pulse structure allows separating the
velocity-independent physics effect from velocity-dependent systematics such
as Stern-Gerlach steering. The apparatus includes two room temperature
cylindrical magnetic shields, and a superconducting, magnetic shield around
the nuclear target to suppress the spin rotation due to magnetic fields. The
target is divided into two segments separated by a spin flipper, with the
liquid pumped back and forth between the upstream and downstream chambers to
isolate nuclear PV from magnetic effects. In addition, the beam is segmented
into two channels with the target upstream in one half and downstream in the
other to suppress noise due to fluctuations in the beam intensity. Special
spin transport coils guide the spins of the polarised neutrons up to the
target and then abruptly terminate the magnetic field so that neutron spins
drift freely in the target chamber. A second spin transport coil rotates the
neutron spins 90∘ CW or CCW to extract the small rotation angle with a
supermirror analyser followed by a 4-quadrant [3]He ionisation chamber.

The sensitivity of this apparatus to $\phi_{n}^{\text{p}}$ is about one third
as large as that of $\phi_{n}^{\text{He-4}}$ because the optimised mean free
path in hydrogen is about one third that of helium. With the flux at ANNI it
is about $1.5\times 10^{-7}$ rad/m per month, to be compared with the “best-
value” expectation $9.1\times 10^{-7}$ rad/m [217]. The most technically
demanding aspect of this experiment would be to engineer a cryogenic chamber
which could non-magnetically transport the hydrogen between chambers and
satisfy the hydrogen safety demands of the facility. The nonmagnetic bellows
pump under development for the liquid helium experiment is a positive
displacement pump which would work as well for liquid hydrogen in principle as
it does not exploit any special properties of liquid helium. Cryogenic
chambers have been constructed and used in the past at Indiana University for
the NPDGamma experiment [230] and at Thomas Jefferson National Accelerator
Facility, for example, for the Qweak PV electron scattering experiment [402],
and for the PRAD experiment [403] to measure elastic electron scattering at
small angles. These demonstrate the feasibility of running a spin rotation
experiment in hydrogen at ANNI.

####  5.5.3 Hadronic TRIV measurements

The same amplification of PV effects of up to $10^{6}$ in [139]La applies also
to time reversal invariance violating (TRIV) effects, in the total absorption
cross section, which is equal to the imaginary component of the forward
scattering amplitude by the optical theorem:

$f(0)=A+B\bm{\sigma}_{n}\cdot\bm{I}+C\bm{\sigma}_{n}\cdot\bm{k}_{n}+D\bm{\sigma}_{n}\cdot\left(\bm{k}_{n}\times\bm{I}\right).$
(10)

The last term, which is both parity and time-reversal odd, can be extracted by
passing an unpolarised beam through a [3]He target and a nuclear target with
transverse polarisation at 90∘ with respect to each other separated by a
neutron spin rotator. Explicit time reversal symmetry without altering any of
the misalignment angles can be performed by rotating the platform holding the
two targets and spin flipper, but not the upstream and downstream collimators
[404]. A modular apparatus has been constructed at Los Alamos National
Laboratory by the NOPTREX collaboration to perform research and development
for this experiment [405]. The collaboration is carrying out an experimental
search for nuclear targets with large nuclear amplification. Promising nuclear
resonances in [139]La ($\Delta\sigma/\sigma=0.097$ at 0.734 eV), [131]Xe
(0.042 at 2.2 eV), [81]Br (0.019 at 0.88 eV), [115]In (0.014 at 7 eV), and
[117]Sn (0.008 at 1.3 eV) were measured at LANL by the TRIPLE collaboration
[406, 385]. These resonances are at epithermal neutron energies and would not
be suitable for the ANNI beamline, but would require a thermal or hot
beamline, ideally at a short-pulse station (Section 9.1.3) where these
resonances may be resolved by neutron time-of-flight.

###  5.6 ANNI summary tables

Tables 3 and 4 summarise status, sensitivity, addressed physics and
requirements with respect to the ESS for the experiments discussed in the
previous sections.

Proposed experiment Measurement Quantity Last measured Current value / limit
Statistical uncertainty ($\bm{1}\sigma$) @ANNI $[\text{100 days}]$ $n\to
p+e+\bar{\nu}_{e}$ ep/n $A$ Beta asymmetry Perkeo III@PF1B [276] $-0.11985\pm
0.00017\pm 0.00012$ $1\times 10^{-5}$ ep/n $C$ Proton asymmetry Perkeo II@PF1B
[314] $-0.2377\pm 0.0010\pm 0.0024$ $1\times 10^{-4}$ ep/n $a$
$e$-$\bar{\nu}_{e}$ correlation from $p$ recoil spectrum $a$SPECT@PF1B [278]
$-0.10430\pm 0.00084$ $1\times 10^{-4}$ ep/n $b$ Fierz interference from beta
asymmetry Perkeo III@PF1B [277] $0.017\pm 0.020\pm 0.003$ $6\times 10^{-4}$
CRES $b$ Fierz interference from beta spectrum UCNA@UCN-LANL [407] $0.067\pm
0.005{}^{+0.090}_{-0.061}$ $1\times 10^{-4}$ BRAND $a$ $e$-$\bar{\nu}_{e}$
correlation from $e$-$p$ correlation $a$CORN@NG-C [320] $-0.10758\pm
0.00136\pm 0.00148$ $5\times 10^{-5}$ BRAND $B$ Neutrino asymmetry Perkeo
II@PF1B [313] $0.9802\pm 0.0034\pm 0.0036$ $5\times 10^{-5}$ BRAND $D$ Triple
correlation $D$ emiT@NG-6 [316] $(-0.94\pm 1.89\pm 0.97)\times 10^{-4}$
$5\times 10^{-5}$ BRAND $R$ Triple correlation $R$ nTRV@FUNSPIN [299] $(4\pm
12\pm 5)\times 10^{-3}$ $1\times 10^{-3}$ BRAND $N$
$\sigma_{n}$-$\sigma_{e,\perp}$ Correlation nTRV@FUNSPIN [299] $0.067\pm
0.011\pm 0.004$ $1\times 10^{-3}$ BRAND $H$, $L$, $S$, $U$, $V$ Other
correlations with $\sigma_{e,\perp}$ unmeasured unmeasured $1\times 10^{-3}$
Beam EDM $d_{n}$ Neutron electric dipole moment @UCN-PSI [57] $(0.0\pm 1.1\pm
0.2)\times 10^{-26}e\,\text{cm}$ $5\times 10^{-27}e\,\text{cm}$ EDMn $d_{n}$ "
" " $1.5\times 10^{-29}e\,\text{cm}$ QNeutron $q_{n}$ Neutron electric charge
@ILL [348] $(-0.4\pm 1.1)\times 10^{-21}e$ $10^{-23}e$ NPDGamma
$A_{\gamma}\left(n(p,d)\gamma\right)$ Gamma asymmetry NPDGamma@FnPB [283]
$(-3.0\pm 1.4\pm 0.2)\times 10^{-8}$ $7\times 10^{-9}$ n3He
$A_{p}\left(n(\isotope[3]{He},t)p\right)$ Proton asymmetry n3He@FnPB [394]
$(1.55\pm 0.97\pm 0.24)\times 10^{-8}$ $4\times 10^{-9}$ NDTGamma
$A_{\gamma}\left(n(d,t)\gamma\right)$ Gamma asymmetry unmeasured unmeasured
$5\times 10^{-8}$ $\phi^{\text{p}}_{n}$ $\phi^{\text{p}}_{n}$ Spin rotation in
para-hydrogen unmeasured unmeasured $8\times 10^{-8}$ rad/m

Table 3: Summary table of observables at ANNI: Currently most precise
measurement (measured by experiment@facility) and expected sensitivity of the
proposed experiment at ANNI in 100 days at 5 MW ESS power. NG-6 is a cold beam
line at NIST and UCN-LANL and UCN-PSI are the UCN sources at LANL and PSI,
respectively. If available, current values are given with statistical and
systematic error.

Experiments New physics Relevant BSM Addresses ESS edge Requirements ep/n,
BRAND, CRES CKM non-unitarity, Scalar, tensor couplings, CP violation
4${}^{\text{th}}$ quark generation, New interactions Fundamental symmetries,
Baryogenesis, Dark matter Neutron flux, Pulse structure Neutron polarimetry,
Beam stability, Background & Environment Beam EDM, EDMn CP violation coupled
to strong sector CP-violating Higgs couplings, High-scale SUSY, Baryogenesis
models, Axion-like dark matter Baryon asymmetry, Baryogenesis, Strong CP
problem Neutron flux, Pulse structure & Neutron flight length (Beam EDM)
Background & Environment QNeutron $(\mathcal{B}-\mathcal{L})$ violation Lepto-
quarks in GUTs, $(\mathcal{B}-\mathcal{L})$-violating models Charge
quantisation, Neutrality of matter Neutron flux, Pulse structure Environment
NPDGamma, n3He, NDTGamma, $\phi^{\text{p}}_{n}$ nonperturbative QCD quark-
quark correlations EFTs for hadronic (weak) interaction Neutron flux, Pulse
structure Beam stability, Environment

Table 4: Summary table of observables at ANNI: addressed physics, ESS edge and
Requirements. “Neutron polarimetry” includes in particular the availability of
polarised 3He at the ESS. “Background” means low background at the site of the
experiment. “‘Environment” comprises, for example, stable temperature and low
and stable magnetic fields.

##  6 The HIBEAM/NNBAR program

###  6.1 Introduction

Of all the empirically observed conservation laws, the conservation of baryon
number is arguably the most fragile. Should baryon number violation (BNV) not
be a feature of Nature, the Standard Model would be falsified [408] and the
dynamic generation of the observed matter-antimatter symmetry would not be
possible [61]. It is perhaps most appropriate to discuss not whether baryon
number is violated but in which channels baryon number is violated. The
HIBEAM/NNBAR program414141The acronym HIBEAM stands for High Intensity Baryon
Extraction And Measurement. [6, 409] takes advantage of the unique opportunity
provided by the world’s brightest neutron source, the ESS, for a two-stage
(HIBEAM and then NNBAR) and long-term series of experiments of increasing
sensitivity to search for BNV via the conversion of neutrons to antineutrons
and/or sterile neutrons. The program encompasses $\Delta B=1,2$ processes and
will achieve an improvement in the sensitivity for discovery of neutron
conversions by three orders of magnitude compared with previous searches [410,
411, 412, 8].

Like the photon and neutrino, which may potentially mix with axion-like
particles [128, 413] and sterile neutrinos [414], respectively, neutron mixing
is an ideal portal to physics beyond the SM. It is a quasi-stable and
electrically neutral particle that can be copiously produced. The observation
of neutron mixing would address open questions in modern physics. As described
further in Section 4.1.3, BNV-violating neutron conversions into antineutrons
and/or sterile neutrons are generic probes of new physics, address the matter-
antimatter asymmetry and could point to the existence of a hidden sector of
particles, thus shedding light on dark matter[110, 109, 415, 99, 105, 416,
103, 106, 417, 121, 122]. It should also be added that unlike the photon and
neutrino, there have been comparatively few high sensitivity searches with
neutrons, owing to infrastructure requirements, as would be satisfied at the
ESS.

The HIBEAM stage of the program emphasises searches for sterile neutrino
processes, induced by non-zero magnetic field values, and includes a search
for neutrons to antineutrons via sterile neutron states. The neutron-
antineutron conversion search at HIBEAM is based on an assumption that sterile
neutrons exist in a hidden sector, which would be affected by a sterile
magnetic field. The NNBAR stage is a dedicated search for free neutrons to
antineutrons. This is a more generic search for neutron-antineutron
conversions and is not based on any assumptions of a hidden sector. HIBEAM
also includes preparations for the NNBAR stage via a low-sensitivity prototype
experiment for free neutron conversion to antineutrons.

The full program probes a range of possible mixing possibilities,
characterised by the off-diagonal elements of the mixing matrix shown in
Equation 11.

$\hat{\mathcal{H}}=\left(\begin{array}[]{cccc}m_{n}+\vec{\mu}_{n}\vec{B}&\varepsilon_{n\bar{n}}&\alpha_{nn^{\prime}}&\alpha_{n\bar{n}^{\prime}}\\\
\varepsilon_{n\bar{n}}&m_{n}-\vec{\mu}_{n}\vec{B}&\alpha_{n\bar{n}^{\prime}}&\alpha_{nn^{\prime}}\\\
\alpha_{nn^{\prime}}&\alpha_{n\bar{n}^{\prime}}&m_{n^{\prime}}+\vec{\mu}_{n^{\prime}}\vec{B}^{\prime}&\varepsilon_{n\bar{n}}\\\
\alpha_{n\bar{n}^{\prime}}&\alpha_{nn^{\prime}}&\varepsilon_{n\bar{n}}&m_{n^{\prime}}-\vec{\mu}_{n^{\prime}}\vec{B}^{\prime}\end{array}\right)$
(11)

Here, $\varepsilon_{n\bar{n}}$ is the $n\bar{n}$ Majorana mass mixing
parameter, and $\alpha_{nn^{\prime}}$ and $\alpha_{n\bar{n}^{\prime}}$ are
mass mixing parameters for $nn{{}^{\prime}}$ and for $n\bar{n}{{}^{\prime}}$,
respectively.

When discussing the mixing matrix, it is convenient to first consider the
physics of NNBAR which looks for the classical $n\rightarrow\bar{n}$
($\Delta\mathcal{B}=-2$) signature, following the quasi-free propagation of
neutrons in low magnetic fields ($\lessapprox 10$ nT). This can arise due to a
mixing mass amplitude $\varepsilon$ and a probability
$P_{n\rightarrow\bar{n}}=\varepsilon_{n\bar{n}}^{2}t^{2}$. More details on
NNBAR are given in Section 6.3.

The conversion of neutrons to antineutrons can also arise via the second order
oscillation process: $n\rightarrow n{{}^{\prime}}\rightarrow\bar{n}$ with
amplitude ($\alpha_{nn^{\prime}}\alpha_{n\bar{n}^{\prime}}$) or
$n\rightarrow\bar{n}{{}^{\prime}}\rightarrow\bar{n}$ with amplitude
$(\alpha_{n\bar{n}^{\prime}}\alpha_{nn^{\prime}})$ and the interference of all
three channels. If $\varepsilon_{n\bar{n}}$ is very small and
$\alpha_{nn^{\prime}}$ and $\alpha_{n\bar{n}^{\prime}}$ are relatively large,
then $n\rightarrow\bar{n}$ could be observed for a non-zero sterile magnetic
field. Previous limits on free $n\rightarrow\bar{n}$ oscillation from
experiments with a suppressed magnetic field [8] and nuclear stability limits
from $n\rightarrow\bar{n}$ conversion in nuclei [418] would not strongly
constrain this scenario, since a field $\vec{B}$ is necessary to compensate
the magnetic field in the sterile sector in order to permit the full process
$n\rightarrow n^{\prime}\rightarrow\bar{n}$. The probability of such a process
can be written as $P_{n\bar{n}}(t)\simeq
P_{nn^{\prime}}(t)P_{n\bar{n}^{\prime}}(t)$. Existing constraints[157] permit
$\tau_{nn^{\prime}}$ and $\tau_{n\bar{n}^{\prime}}$ to be as low as $1\div 10$
s

By optimising the magnetic field $\mathbf{B{}}$ to be resonantly close to
$\mathbf{B{{}^{\prime}}}$ (with a precision of several mG), the quasi-free
regime is thus reached and the probability of induced $n\rightarrow\bar{n}$
oscillation can become as large as

$P_{n\bar{n}}(t)=\frac{1}{4}\alpha_{n\bar{n}^{\prime}}^{2}\alpha_{n\bar{n}^{\prime}}^{2}t^{4}\sin^{2}\beta=\frac{\sin^{2}\beta}{4}\left(\frac{t}{0.1~{}{\rm
s}}\right)^{4}\left(\frac{10^{2}\,{\rm
s}^{2}}{\tau_{nn^{\prime}}\tau_{n\bar{n}^{\prime}}}\right)^{2}\times 10^{-8}.$
(12)

Here, $\beta$ is the (unknown) angle between the directions of $\mathbf{B{}}$
and $\mathbf{B{{}^{\prime}}}$ [163]. The probability of induced
$n\rightarrow\bar{n}$ transition can therefore be several orders of magnitude
larger than the present sensitivity in direct $n\rightarrow\bar{n}$ conversion
[8]. As explained in Section 6.2 the HIBEAM part of the program looks for
neutrons ”disappearing", regenerating and converting to antineutrons.

###  6.2 HIBEAM

At the most basic level, HIBEAM will address transformations of
$n\rightarrow\bar{n}$ and $n\rightarrow n^{\prime}$. However, should a sterile
neutron sector exist, processes connecting the visible and sterile sectors
need not be restricted to the above processes, as proposed in Refs. [138,
163]. As a neutron beam experiment, HIBEAM can look for the regeneration of
neutrons following a beam stop (Figure 16 (a)), an unexplained disappearance
of neutron flux (Figure 16 (b)), as well as $n\rightarrow\bar{n}$ via a
sterile neutron state (Figure 16 (c)). These searches are described in further
detail in the following sections.

Figure 16: Illustration of the principles of searches for sterile neutron
oscillation with neutron beams. Both regeneration (a) ($n\rightarrow
n^{\prime}$) and disappearance (b)
($n\rightarrow\\{\bar{n},n^{\prime}\\}\rightarrow n$) search modes are
possible. Another possibility (c) is regenerative in style, though instead
leads to further mixing to an antineutron
($n\rightarrow\\{\bar{n},n^{\prime}\\}\rightarrow\bar{n}$), requiring an
annihilation detector. Each regeneration mode requires a neutron absorber to
be placed at the halfway point of the beamline, preventing all ordinary
neutrons from proceeding downbeam while permitting sterile species to pass
unencumbered.

####  6.2.1 Search for $n\rightarrow n{{}^{\prime}}$ via disappearance

In HIBEAM’s $n\rightarrow\\{n{{}^{\prime}},\bar{n}^{\prime}\\}$ disappearance
search, neutrons would move through a 50 m long aluminum vacuum tube, with
neutron rates at the start and the end of the propagation measured precisely.
The presence of an unknown sterile magnetic field $\mathbf{B{{}^{\prime}}}$ is
taken into account by performing a magnetic field scan such that the sterile
and visible magnetic fields are matched and a $n\rightarrow n{{}^{\prime}}$
transition is less suppressed. The detection of a resonance would manifest
itself as a reduction in the total counting rate.

Charge-integrating counters are needed which are able to measure a charge
proportional to the neutron $n$ flux with an relative accuracy up to the order
$10^{-7}$. One possibility is the use of a [3]He detector [419] in charge-
integration mode.

####  6.2.2 Search for the regenerative $n\rightarrow
n{{}^{\prime}},\bar{n}^{\prime}\rightarrow n$ process

The regeneration search derives from a similar theoretical basis as the
disappearance search [150, 420], though corresponding to a two-stage (second
order) process with a consequently quadratically smaller probability. In the
first stage, the $n\rightarrow n{{}^{\prime}},\bar{n}^{\prime}$ transformation
takes place in an intense cold $n$ beam at the quasi-free environment limit
corresponding to $|\mathbf{B}-\mathbf{B{{}^{\prime}}}|\sim 0$. The largely
untransitioned $n$ beam will be blocked by a high suppression beam trap, while
the remaining sterile neutron will continue unabated through the absorber. In
a second volume behind the absorber (stage two), maintaining the the same
conditions of $|\mathbf{B}-\mathbf{B{{}^{\prime}}}|\sim 0$ as the section
before it, the $n{{}^{\prime}},\bar{n}^{\prime}\rightarrow n$ transformation
produces detectable neutrons with momentum conserved, as though the totally-
absorbing wall were not present.

The observation of the resonance in the $\mathbf{B}$-scan would be defined by
a sudden appearance of regenerated $n$’s when the
$\mathbf{B}-\mathbf{B{{}^{\prime}}}\approx 0$ condition in both volumes is
met. Like with disappearance, a positive signal would be a demonstration of
the $n{{}^{\prime}},\bar{n}^{\prime}\rightarrow n$ transformation as well as
the existence of the sterile neutrons and photons. The requirement of matching
conditions in both volumes ensures this type of measurement is significantly
more robust to systematic uncertainties that could otherwise cause a false
signal to be observed.

####  6.2.3 Search for $n\rightarrow\bar{n}$ by regeneration through mirror
states

The $n\rightarrow\bar{n}$ process can also arise due to the second order
oscillation processes: $n\rightarrow
n{{}^{\prime}},\bar{n}^{\prime}\rightarrow\bar{n}$. The earlier body of
searches for free neutrons converting to antineutrons [410, 411, 412, 8] were
insensitive to this scenario.

####  6.2.4 Detectors for sterile neutron searches

Discussion of antineutron detection is deferred to Section 6.4.4 which
describes the detector for the search for free neutrons converting to
antineutrons. A smaller version of the detector described there would be
deployed at HIBEAM.

The other sterile neutron searches at HIBEAM rely on measurement of the
visible state of the neutrons. Detection of cold and thermal neutrons requires
major technical competence for the scattering experiments of the ESS, though a
standard solution for neutron detection may utilise gas detectors based on
[3]He in a single wire proportional chamber. The detectors can be operated at
low gain since the $n+\isotope[3]{He}\rightarrow t+p$ reaction produces a very
large ionisation signal.

####  6.2.5 HIBEAM sensitivity

Figure 17 shows the current limits from trapped ultracold neutron experiments
together with the expected sensitivity of the HIBEAM experiment at a beamline
of the ANNI design [262, 7] (see Section 5) after one year of ESS running at
1MW power (taken from Ref. [6]). Increases in disappearance sensitivity of
more than an order of magnitude are possible depending on the value of the
magnetic field used. It can be seen that HIBEAM covers a wide range of
oscillation times for a given magnetic field value, many of which are
unexplored by UCN-based experiments, and remain free of the model assumptions
of those searches. As a second order process, the regeneration search
sensitivity is lower though can approach that of the disappearance mode for
appropriate running time and a design power of 5 MW. [6]. As an expected zero-
background experiment, the search for neutrons converting to antineutrons via
sterile neutrons, the sensitivity of this mode on
$(\tau_{nn^{\prime}}\tau_{n\bar{n^{\prime}}})^{\frac{1}{2}}$ can exceed $1000$
s. Finally, the aforementioned searches will also probe the multi-Gauss region
in which constraints are extremely weak. These data are also summarised at the
end of this section in Table 6.8.

Figure 17: Excluded neutron oscillation times in blue for $n\rightarrow
n^{\prime}$ disappearance from UCN experiments [152, 156, 154, 161, 157, 158]
as a function of the magnetic field $\mathbf{B{{}^{\prime}}}$. The projected
sensitivity for HIBEAM (disappearance mode) is also shown in magenta for one
year’s running at the ESS assuming a power of 1MW.

###  6.3 NNBAR at the ESS - measurement methodology

The NNBAR experiment seeks to measure the signal of a spontaneous conversion
of a neutron to an antineutron by focusing the cold neutron beam onto a thin
carbon target foil. Should one of the neutrons oscillate to an antineutron on
the way to the target, a striking multipion event signature, with energy
typically well above existing backgrounds, could be observed. In principle,
one should be able to reconstruct the 1.88 GeV two-neutron mass, although,
since the annihilation is not to happen in free space but inside a nucleus,
final state interaction effects and meson absorption can significantly distort
the expected invariant mass resolution, even before any detector effects are
taken into account [421].

The neutrons used in the NNBAR experiment are produced by the spallation
process following moderation, after which they reach typical velocities of
about 1200 m/s. The beam of cold neutrons travels through a vacuum in an
ultra-low magnetic field such that the quasi-free condition holds. The
motivation for choosing an annihilation target made of carbon resides in its
low cross section for neutron absorption and high cross section for
annihilation. More specifically, the annihilation cross section is
proportional to $1/v$, where $v$ is the neutron velocity. For the average
spectrum of neutron velocities of HIBEAM/NNBAR it is $\sim$ 4 kb [96] for
annihilation and 4 mb for absorption i.e., a six orders of magnitude
difference. The moderator-target longitudinal distance is $\sim 200$ m for
NNBAR and $\sim 50$ m for HIBEAM, with sensitivity depending quadratically on
the transit time. A schematic of the experiment is shown in Figure 18.
Neutrons passing through the foil are absorbed in a beam dump, which will
possibly be made of 6Li. A second target foil can also be installed for
background control purposes allowing the background estimations for NNBAR to
be data$-$driven. Figure 19 shows the future location of NNBAR within the ESS
instrument suite.

Figure 18: Baseline NNBAR experiment. Neutrons from the moderator are focused
on a distant target foil surrounded by an annihilation detector. Figure 19:
Overview of the ESS instruments and location of the NNBAR experiment.

###  6.4 NNBAR design

The key components of the NNBAR experiment are: 1) a cold moderator source; 2)
a high precision neutron focusing system; 3) field-free propagation along a
dedicated beamline and 4) detection of the annihilation signal. Each of these
elements is described in the following sections.

####  6.4.1 The NNBAR Large Beam Port

NNBAR will strongly profit from the large liquid deuterium moderator LD2
described in Section 3.3. This moderator is currently under design and it will
be optimized for delivering a high integrated flux to the NNBAR experiment.
The neutrons produced by the LD2 moderator will travel through the ESS Large
Beam Port, a special beam port installed in the ESS target monolith for NNBAR
to reach its goal. The Large Beam Port covers the size of three ESS standard
beam ports and in the early days of ESS will be filled by three regular size
beam ports. The central beam port will be used for the test beamline whose
purpose is to characterize the ESS source at the beginning of the user
program. The engineering drawing of the Large Beam Port is shown in Figure 20
and the picture of the Large Beam Port as installed in the ESS target monolith
can be seen in Figure 21. Once the Large Beam port will be opened for the
operation of the NNBAR experiment it will permit the use of a large fraction
of the solid angle as can be seen in Figure 20.

Figure 20: Top view of the NNBAR beamport. Reproduced with permission from
Mats Segerup and Rickard Holmberg. Figure 21: The ESS Large Beam Port.

With the extractions of an extremely high intensity cold beam the Large Beam
Port will however transport also a very high fraction of fast neutrons and
spallation background. This will require that the NNBAR beamline in the first
meters close to the monolith be heavily shielded in order to be operated
safely. Beamline simulations are being carried out in order to design the
shielding with the required thickness [422] and also to assess if any
components of the ESS facility like the ESS bunker would need additional
shielding due to the opening of the large beam port.

####  6.4.2 NNBAR optics

In order to make use of the Large Beam Port it is necessary to design special
optics that allow the exploitation of such a large view of the source. The
reflector needs to ensure that the large neutron flux coming from the source
is directed and focused to the annihilation target $200$ m away, maximizing
the NNBAR figure of merit $FOM=N\cdot<t^{2}>$, where $N$ is the number of free
neutrons with $t$ is the transit time of a neutron in the magnetically
shielded region prior to reaching the annihilation target. The flight time is
measured from the last reflection that the neutron had in the optics. The
baseline design of the NNBAR optics is a an elliptic guide made of reflecting
material. This is an ellipsoid with a focal distance $200\text{\,}\mathrm{m}$
and a short-axis $b$ of $2\text{\,}\mathrm{m}$. The center of the moderator is
located at focal point, while the center of the detector is located at the
other focal point. The reflector covers the part of the ellipse that starts
$10\text{\,}\mathrm{m}$ from the source and ends at a distance of
$50\text{\,}\mathrm{m}$.

Other designs are currently under study using McStas, a neutron ray tracing
code, including a differential reflector that gives a gain of 23% compared to
the baseline reflection and a "nested reflector". These reflectors are easy to
assemble. Different designs are currently being optimized within the HighNESS
project and are shown in Figure 22. So far the obtained value of the nested
mirror options seems to deliver the highest $FOM$. Further studies are to be
performed to allow an optimal choice of reflector for NNBAR. It is important
to point it out that the reflector is being designed together with the
moderator in an iterative process to enhance the sensitivity of the final
experiment.

Figure 22: NNBAR nested optics options a) mono planar b)double planar c)
torodail (cylindrical)

####  6.4.3 Magnetic shielding and vacuum

To match the quasi-free condition $\Delta E\ll t$ [6] (where $E$ and $t$ are
the energy and the propagation time of the neutron) the magnetic field should
be small enough to avoid suppressing the probability of oscillations. This can
be achieved by shielding it at a level below 10 nT along the 200 m neutron
propagation length. This requirement can be satisfied using a two-layer
mumetal shield combined with a 316L stainless steel vacuum pipe that is also
part of the magnetic shielding system [409]. This shielding layout with the
mumetal sheets arranged in octagonal shape and clamped together is based on
the magnetic shield of the atomic fountain used for the Hannover Very Long
Baseline Atom Interferometry facility [423]. A set of coils is needed around
the mu-metal in order to equilibrate them.

Simulations performed with COMSOL [424] a finite element calculation tool has
shown that this design can satisfy the 10 nT requirement. The layout of the
shielding is shown in Figure 23.

Figure 23: Overview of the NNBAR magnetics shield: left the 200m magnetic
shielded vacuum pipe. Right: cut through the shield from COMSOL simulations.
(1) Outer mumetal layer (2) vacuum pipe (3) inner mumetal layer

####  6.4.4 Preliminary detector design

The NNBAR detector model [425, 164, 409] has been implemented in GEANT4, and a
range of different technologies are being considered including lead-glass,
Crystal (CsI(Na)) and sampling calorimeters (liquid argon/steel). A Time
projection chamber (TPC) (outer tracking) and a layer of silicon (inner
tracking) are also included. The following main elements are part of the
current baseline detector, starting from the center, radially outwards: a) the
aforementioned annihilation target, a $100\,\mu$m thick carbon disk, with
$1\,$m diameter for the initial HIBEAM stage, and $2\,$m diameter for the full
NNBAR second stage; b) a charged particle tracker, necessary for the tracking
and identification of pions and the annihilation vertex. The inner tracking is
accomplished with a silicon layer placed within a $2\,$cm thick aluminium
beampipe. The TPC surrounds the beampipe and provides particle identification
through measurements of the specific continuous energy loss, $\frac{dE}{dx}$;
c) a hadronic range detector for charged pion measurement composed of 10
inter-orthogonal scintillator slats and, for photon measurement (neutral pion
reconstruction), an electromagnetic calorimeter comprising lead-glass modules.
Surrouding the detector d), a scintillator-based cosmic ray background veto
would be deployed. A selective trigger system to be able to gather signal and
signal-like background candidate events. Construction of a prototype [426]
detector system comprising a TPC, scintillator range and lead-glass
calorimeter is underway.

The detector is optimised so as to allow high efficiency ($>50$%) for signal
tagging and reconstruction while offering sufficient discrimination power so
as to reject backgrounds. A range of background processes are being studied
including cosmic rays, gamma emission due to activated detector and other
material, and interactions of fast and slow neutrons from the spallation
source. The aim is for a zero-background search, as achieved at Ref.[8]. This
requires precision capabilities in timing (provided by the scintillator
system) and particle identification (eg via continuous ionisation energy loss
in the TPC $\frac{dE}{dx}$ and $\pi^{0}$-tagging in the calorimeter).

Figure 24 shows an overview of the NNBAR detector. The detector is 600 cm long
in the longitudinal ($z$) direction. In the transverse ($x-y$) plane, the
detector has a width and height of 515 cm. The detector components and their
dimensions are labelled in the figure.

Figure 24: Schematic overview of the NNBAR detector design in the x-y (a) and
y-z (b) views.

Figure 25 (a) shows a signal event with five final-state pions in the NNBAR
detector with a nuclear fragment from the carbon target. Figure 25 (b) depicts
a cosmic ray muon traversing the NNBAR annihilation detector from top to
bottom.

(a)

Figure 25: Event displays with the NNBAR detector in the plane transverse to
the beamline showing (a) a signal event with five pions; (b) a cosmic muon.
Red (blue) color in the figure represents a negatively (positively) charged
particle.

###  6.5 Previous free neutron oscillation searches and current limits

Previous free neutron $n\rightarrow\bar{n}$ experiments were performed at the
Pavia Triga Mark II reactor [410, 411] and at the Institute Lauen
Langevin(ILL) [412, 8], the latter which [8] provides the most stringent and
comprehensive limit for the free neutron oscillation time: $\sim 8.6\times
10^{7}\,$s.

###  6.6 NNBAR sensitivity at the ESS

The sensitivity of NNBAR is proportional to the number of
$n\rightarrow\bar{n}$ transitions which could take place. In other words, the
sensitivity of the experiment is considered in regards to the discovery reach
of oscillation events. It is best quantified by the figure of merit
$N_{n}\cdot t^{2}$ where $N_{n}$ is the free neutron flux reaching the target.

As both neutron flux on target as well as free neutron flight time are the
crucial variables influencing the figure of merit, all improvements on
experimental parameters influencing either one or the other will lead to
increased sensitivity. The neutron flight time is uniquely influenced by the
propagation length, whereas several neutronic parameters impact $N_{n}$.
Neutron reflector technology, in particular, is expected on its own to
increase the experimental sensitivity for neutron-antineutron oscillation by a
factor of around 40 when compared to the previous ILL experiment.

###  6.7 The gain from using the ESS

Table 5 depicts the gain factors affecting the experiment’s sensitivity. The
largest gain comes from the high reflectivity ($m$) neutron mirrors. While
such mirrors are now extensively available, the combination of factors makes
the ESS unique in terms of pushing the limits of neutron oscillation
observation. In particular, the long neutron flight path as well as a large
intensity due to ESS’ large beam port are unequalled at any other existing or
planned neutron research facility.

Factor | Gain wrt ILL
---|---
Source Intensity | $\geq 2$
Neutron Reflector | $40$
Length | $5$
Run time | $3$
Total gain | $\geq 1000$
Table 5: Breakdown of gain factors for NNBAR with respect to the last search
for free neutron-antineutron conversions at the ILL.

###  6.8 Comparison with other future experiments

It is instructive to compare the sensitivity of other planned
$n\rightarrow\bar{n}$ experimental initiatives to NNBAR at the ESS. All other
internationally competitive programs that are planned focus on searches using
bound neutrons, which are complementary to searches using free neutrons. As
such, NNBAR is unique, representing the most sensitive $n\rightarrow\bar{n}$
proposal using free neutrons. A comparison between sensitivities on neutron
oscillations for free and bound neutron searches is difficult and model
dependent [6]. In particular, such comparisons need to focus on limits that
can be set on oscillation time using intra-nuclear suppression factors and not
on the discovery reach itself. With that in mind, there are essentially three
future experiments that include $n\rightarrow\bar{n}$ as part of their
programs a) Hyper-Kamiokande - there are no existing official estimates on the
expected $n\rightarrow\bar{n}$ sensitivity from the collaboration; b) Deep
Underground Neutrino Experiment (DUNE) - the projected converted free
oscillation time lower limit for bound neutron conversions within [40]Ar
nuclei within the future DUNE experiment is $\tau_{n\rightarrow\bar{n}}\geq
5.53\times 10^{8}$ s [427] for an assumed exposure of 400 kt$\cdot$years ; c)
NOvA - there are no estimates on future neutron oscillations time limits for
NOvA. It is worth mentioning, however, that an analysis using data from 4
months of Far Detector exposure [428] found the NOvA sensitivity to be below
the limits set by the Super-Kamiokande experiment[429, 430]. A summary, taken
from Ref. [6] for projected future sensitivities, as well as past limits
obtained by other experiments can be see in Figure 26.

Figure 26: Lower limits on the free neutron oscillation time from past
experiments on free and bound neutrons. Projected future sensitivities from
HIBEAM and NNBAR are also shown, together with the expected sensitivity for
DUNE. Searches with free neutrons include those done at the Pavia Triga Mark
II reactor[410, 411] and the ILL [412, 8]. The most recent and competitive
result from the ILL is shown and is denoted ILL-2 [8]. Limits from bound
neutron searches are given from Homestake [431], KGF [432], NUSEX [433], IMB
[434], Kamiokande [435], Frejus [436], Soudan-2 [437], SNO [438], and Super-
Kamiokande [439]. For the bound neutron experiments, various model-dependent
intranuclear suppression factors are used to estimate a free neutron
oscillation time lower limit.

Sensitivities for the HIBEAM/NNBAR program on oscillation times are summarised
in Table 6.8. Results from earlier experiments and the gain from making
searches at the ESS are also shown. Note that the increase of a factor $\sim
10^{3}$ for NNBAR corresponds to the rise in the discovery potential due to
the rise in the figure of merit at the ESS. The limit on the oscillation time
that can be reached with respect to the ILL experiment sensitivity increases
by $\sqrt{10^{3}}\sim 30$.

Table 6: Summary table of NNBAR and HIBEAM sensitivities. Limits from previous
searches are also given.

Experiment Measurement Quantity Recent measurements Current limit Gain (ESS)
Experimental conditions NNBAR $n\rightarrow\bar{n}$ Free neutron oscillation
time ($\tau$) [8] $\tau$ $>$ $8.7\times 10^{7}$ s $\sim$ $10^{3}$ (discovery)
$\sim$ 30 ($\tau$) beam energy = 800 MeV; power = 2 MW HIBEAM
$n\rightarrow[n^{\prime},\bar{n}^{\prime}]$
$n\rightarrow[n^{\prime},\bar{n}^{\prime}]\rightarrow n$
$n\rightarrow[n^{\prime},\bar{n}^{\prime}]\rightarrow\bar{n}$ neutron
disappearance, regeneration and antineutron conversion [153, 157, 158] [157,
158] $\tau_{nn^{\prime},n\bar{n^{\prime}}}>~{}50$ s
$(\tau_{nn^{\prime}}\tau_{n\bar{n}}^{\prime})^{\frac{1}{2}}>~{}1000$ s
$B$-field dpendent $\sim 10$ ($B$-field dependent) beam energy = 800 MeV;
power = 2 MW

###  6.9 An option to use the coherent reflection of antineutrons and neutrons

The interesting option to simultaneously increase the sensitivity of NNBAR and
decrease its cost is to implement the new idea of a guide both for neutrons
and antineutrons [440, 441, 442]. A typical effective critical velocity of the
guide material for antineutrons is $\sim 4$ m/s. A typical lifetime of
antineutrons in the guide is $\sim 2~{}$s; it is defined both by antineutron
annihilation and dephasing between neutrons and antineutrons. With these
values and a specially designed guide, one could probably deliver a large
fraction of the total neutron/antineutron intensity to the annihilation
detector while the systematic uncertainties associated with the interaction of
antineutrons with the guide walls are quite small.

This approach has been applied in the analysis of a possible (quite short)
experiment at the PF1B facility at ILL [263]. It was found that an increase of
$3-10$ times in sensitivity over the best existing constraint is feasible
[443]. An analogous study for the NNBAR experiment is planned in the near
future. Potential advantages of the new experimental design, to be verified by
a detailed analysis, include a larger sensitivity in a longer setup, a lower
radiation around the exit from the ESS radiation protection, a smaller
detector background, a significantly lower cost due to a more compact design.
An additional order of magnitude gain in sensitivity might be possible if a
dedicated VCN source is built at ESS.

##  7 UCN/VCN

The energy of ultracold neutrons (UCNs) is so low ($\sim 10^{-7}~{}\text{eV}$)
that they can be reflected from many materials at all angles of incidence
(different energy ranges correspond to different materials). UCNs can be
stored in material traps [444], and such low energies also make it possible to
store them in magnetic traps [445]. The possibility of long holding times
(several hundred seconds) for UCNs in experimental setups makes these neutrons
extremely sensitive to small effects. In some cases, UCNs turn out to be a
more sensitive probe in comparison with thermal and cold neutrons, despite
significantly lower fluxes. In fact they are the only means by which some
topics, such as gravitationally bound states of the neutron, can be
experimentally addressed. The work of F.L. Shapiro [446] first drew attention
to this possibility for precision measurements using UCNs, and the first
experimental observations of UCNs [447, 448] led to a rapid development in
this area of research beginning in the 1970s [449]. This in turn led to
significant improvements in accuracy for measurements of the neutron lifetime
[450], constraints on the size of a neutron electric dipole moment (EDM) [57],
and the discovery and study of UCN quantum states above a flat mirror in a
gravitational field [190]. Since the first detection of UCNs, a great deal of
methodological experience has been accumulated in the production, detection,
spectrometry, and storage of UCNs. UCN losses on the walls of the storage
volumes can be lower than a few percent of the losses associated with neutron
decay. Processes involving small energy transfers were also discovered in the
interaction of UCNs with material surfaces [451], which had not been taken
into account in early experiments. These processes can be a source of
systematic errors in experiments measuring the neutron lifetime, on the one
hand, and on the other hand, they open up prospects for studying dynamics on
the surfaces of condensed matter. It remains a challenge for experimentalists
to obtain the theoretically predicted loss factors on weakly-absorbing
materials such as solid oxygen and beryllium [452].

Throughout this time many advances in experimental accuracy were obtained by
using more intense UCN sources, while others – especially for neutron lifetime
measurements – were driven by improvements of experimental apparatus. The
program for the use of UCN to study the surface of condensed matter was
proposed some time ago [453], but even today the intensity of existing sources
is not enough to implement this program. The active development of other,
established laboratory methods for studying surfaces, and the limited coverage
of $(|\bm{q}|,\omega)$ for UCN, seems to indicate that UCN science is not yet
competitive in this area. Today, the potential of very cold neutron (VCNs,
energies $<10^{-4}~{}$eV) experiments is not fully exploited. Despite the
periodic discussion of the advantages of their use [454, 455, 456], the
development of this field is hampered both by the lack of intense sources of
these neutrons (the only available source of VCN is PF2 at the ILL [457]) and
by the need to adapt experimental setups and existing techniques. The
development of much more intense UCN and VCN sources could possibly enlarge
the field of UCN applications, and create whole new fields of application for
VCNs in particle physics – as well as in neutron scattering.

The concrete implementation of UCN and VCN sources for the ESS is presently
under study. Because experiments exploiting such sources are strongly affected
by specific source parameters – especially the energy spectrum and phase-space
density – the experimental concepts at the present time can only be discussed
at a schematic level. The remainder of this section surveys a range of science
experiments in these fields that could potentially be pursued at the ESS,
provided a competitive UCN density is produced, and comments on certain
aspects of the source design that could impact their practical implementation
or sensitivity.

###  7.1 Implementing a UCN source at the ESS

Delivering higher numbers of UCN into experiments remains an important
possibility for improving experimental accuracy. For storage experiments in
which the free neutron lifetime is not the limiting loss mechanism, such as
measurements of neutron EDM, significant gains may also be made by increasing
the holding time. One should also note that some experiments rely on a high
transmitted flux of UCNs, rather than a high stored density. In all cases,
however, present-day experiments with UCNs are strongly limited by counting
statistics.

The longest-running and most reliable user-mode UCN source is PF2 at the
Institut Laue-Langevin (ILL), based on the concept of a Doppler-shifting
neutron turbine [457]. This source has a total UCN current density of about
$2.6\times 10^{4}$ cm-2 s-1 up to $v_{z}=6.2$ m/s, and $3.3\times 10^{4}$ cm-2
s-1 up to $v_{z}=$ 7 m/s, and corresponding stored UCN densities up to about
40 neutrons per cm3 (or $20~{}\text{cm}^{-3}$ with an aluminum safety foil
installed, in the usual configuration). All UCN sources implemented after PF2
use solid deuterium or superfluid helium as superthermal converters for cold
neutrons [458]. In contrast to moderation, superthermal conversion allows to
accumulate phase-space density of UCN over the equilibrium value, via
inelastic scattering where a Boltzmann factor suppresses those interactions
which lead to an increase of the neutron energy. Both of these converter
materials are well studied [459, 460, 461, 462, 463].

Solid deuterium provides a higher conversion rate, where the optimal spectrum
for incident neutrons has an effective temperature of $\sim 30\,$K, and the
optimal converter temperature is $\sim 5\,$K. The high UCN production rate
makes this an attractive option for pulsed neutron sources, but absorption
losses limit the lifetime of UCNs inside the source. While it would be
desirable for a UCN source at the ESS to deliver UCN densities limited only by
the pulsed neutron flux density, no experimentally-validated implementation of
such a concept is presently available. The theoretical approaches to such a
concept [464] would require placing the UCN source very close to the
spallation target, since velocity dispersion leads to pulse spreading after
neutrons have left the source. The prospective gain factor relative to the
average UCN density is ultimately limited by the ratio of the time interval
between pulses to the duration of a single pulse. For the ESS (with a pulse
duration of 2.86 ms and a pulse frequency of 14 Hz) it would be $\sim 25$
times, neglecting any additional losses. A further theoretical proposal
building on this concept is the “time-focusing” of neutrons, to re-compress
pulses that have undergone some spreading outside the source [465]. Non-
adiabatic spin-flips in a variable magnetic field were later considered as a
means to implement time-focusing; this idea has been tested at some level
[466, 467], but further developments are required for practical use. Recently,
two other mechanisms have also been proposed for time-focusing of neutron
pulses: non-stationary diffraction by a moving grating [468], and a
homogeneous time-varying magnetic field accompanying deceleration of very cold
neutrons to UCN energies [469]. The practical implementation of these ideas
requires further investigation of the proposed techniques.

Superfluid helium has a specific rate of UCN production almost 10 times lower
than solid deuterium (assuming a neutron spectrum with effective temperature
of around $30$ K), and the optimal spectrum for incident neutrons has an
effective temperature of $\sim 6$ K. At converter temperatures below $\sim
0.8$ K, superfluid helium permits very long UCN lifetimes in the converter
(finally limited by beta decay below $\sim 0.6$ K), making it possible to
accumulate UCNs over long times and obtain high “ _in-situ_ ” densities within
the converter volume. The _in-situ_ UCN density $\rho_{\text{source}}$ will be
reduced by dilution and transfer losses, for any extraction-based experiment
in which the UCNs are transferred to an _ex-situ_ storage volume for
measurement. For a perfect guide system in which dilution losses dominate, the
density $\rho_{\text{expt}}$ of UCNs delivered into such an experiment is
determined by the volumes of the source ($V_{\text{source}}$), guide system
($V_{\text{guide}}$), and experiment ($V_{\text{expt}}$):

$\rho_{\text{expt}}=\frac{V_{\text{source}}\rho_{\text{source}}}{V_{\text{source}}+V_{\text{guide}}+V_{\text{expt}}}.$
(13)

The long times required for obtaining the saturated UCN density in a helium
converter make it unrealistic to exploit the time-structure of a pulsed
neutron source for further gains in UCN density. The time-structure associated
with neutron delivery can, however, be experimentally useful for disentangling
beam-related backgrounds. In addition, since UCN production in liquid helium
is dominated by neutrons in a narrow wavelength range around 8.9 Å, the time
structure allows for significant suppression of the thermal load, activation,
and requirements for biological shielding. This can be achieved by using
choppers to remove the other wavelengths without attenuating the time-averaged
flux at 8.9 Å (see also the discussion in Section 5.3.2). A similar effect can
be achieved using monochromators at a steady state neutron source, but only
with significant loss of 8.9 Å neutrons. There are also possibilities to
achieve significant gains in UCN statistics at the ESS via delivery of large-
area neutron beams, or through use of novel imaging concepts [470] which are
not practical to implement at completed, running facilities.

The ideas for intense UCN sources proposed in relation to the WWR reactor
[471] or the PIK reactor [472] cannot be implemented directly in the ESS, due
to design features of the spallation source: a very compact and bright zone
for neutron production, and a fast decrease of neutron density away from this
zone accompanied by intense gamma-ray loads.

Various ideas and possibilities for implementing UCN sources at the ESS are
presently under discussion [456] and require further evaluation. Many are
linked to the availability of a high-intensity lower moderator, described in
Section 3.3. The option of _in-situ_ UCN production at the ANNI facility is
discussed more thoroughly in Section 5.3.2. Possible source locations close to
the target are outlined in Figure 27 [473]. UCN sources can be placed in
several locations inside the target monolith structure (in red in Figure 27,
right) which has a radius of 5.5 m. Location “1” is inside the “twister”, a
structure that contains the upper and lower moderators. In this position the
UCN source is closest to the spallation target, and would either be placed
below the liquid deuterium moderator or replace it. Location “2” is inside a
shielding plug (called Moderator Cooling Block) which is originally intended
merely for shielding, but can be adapted to accommodate a secondary source.
This position is at a further distance from the target, facilitating the
cooling of the UCN converter, but still receives a large flux from the high-
intensity moderator. The use of the Large Beam Port (LBP) is also very
attractive due to a large solid angle viewed from the moderator and a high
calculated brightness for 8.9 Å neutrons of $3.4\times 10^{11}~{}\text{n
cm}^{-2}\text{s}^{-1}\text{sr}^{-1}\text{\AA{}}^{-1}$, averaged from a large
emission surface of $40\times 24~{}\text{cm}^{2}$. This brightness is
comparable to that of the ILL’s cold source at 8.9 Å, $5\times
10^{11}~{}\text{n cm}^{-2}\text{s}^{-1}\text{sr}^{-1}\text{\AA{}}^{-1}$ [474].
A large He-II vessel placed in location “3” is also a possibility [475], while
due to constraints of space the use of a standard beamport at location “4"
presently appears less favorable. Another option could be to use innovative
reflector optics [476] to maximise the neutron flux at position “5”. Based on
the same idea, a neutron-optical device is being developed to feed a He-II
vessel located a greater distance away [477]. Position “5” represents also the
in-beam option of a UCN source, placed in a beamline pointing at the cold
moderator.

Figure 27: Horizontal cut through the target and bunker region (cf. Figure 5),
at the height of the liquid-deuterium moderator (shown in green) situated
below the spallation target (not visible). The drawings are obtained from a
detailed MCNP [478] geometry. The cylindrical region of radius of 5.5 m around
the center represents the shielding monolith (shown in red). About half of its
42 standard beamports are visible in the cut plane. Possible locations of UCN
sources, as studied within the HighNESS project are: (1) inside the “twister”,
a structure which contains the upper (liquid-parahydrogen, high-brightness)
and lower (liquid-deuterium, high-intensity) moderators; in this position the
UCN source is closest to the spallation target and would either be placed
below the lower moderator or replace it, (2) inside the moderator cooling
block, (3) in the large beamport (shown as a white segment in the monolith)
that is initially foreseen to be used for the NNBAR experiment, (4) in a
standard beamport, (5) outside the “bunker”, a heavy concrete shielding
structure (shown in orange) placed around the monolith; the minimum distance
of this location from the moderator is 15 m, the large beamport being used to
image the neutron emission surface of the lower moderator onto a volume of
superfluid helium for down-conversion of cold neutrons in a UCN source or _in-
situ_ in an experiment.

All potential source positions should be considered in more detail in the
future. Both helium and deuterium converters present a problem for cooling
when placed in intense radiation fields. Recall that the optimum temperature
for helium is below 0.8 K, and for deuterium is about 5 K. A liquid deuterium
source of cold neutrons maintained at 20 K is much easier to cool, and is a
good spectrum shaper for both converters. The thermal load is mainly provided
by gamma radiation, and to a first approximation depends only on the total
converter mass. Because the required temperature is lower, and the required
mass higher, in the case of superfluid helium for a similar complexity of
cooling apparatus the distance to the target should be increased relative to
solid deuterium. Thermal conductivity of the converter medium is an additional
concern, and cooling technologies in the ranges above and below 1.2 K are
fundamentally different. The optimal placement for “in-pile” UCN production
(i.e., very close to the primary neutron source) thus depends strongly on the
source configuration.

The reasonable thickness of the converter is limited by the depth of UCN
escape from its material. Thus, it is practically unlimited for helium and
does not exceed a few centimeters for deuterium due to elastic scattering by
crystal inhomogeneities at the indicated temperature. Technologies for
creating a perfect deuterium crystal are a subject of ongoing research, and
there are promising recent results in the field [479]. When comparing sources
based on superfluid helium and solid deuterium for the ESS, with horizontal
beam lines, it should be kept in mind that these converters have quite
different neutron-optical potentials (18 neV for helium versus 105 neV for
deuterium). Accordingly, the extraction system must be optimised for the
respective UCN spectrum. In the case of a deuterium source, for certain
specific experimental configurations the experimental apparatus can be raised
to shift the spectrum into the energy region that provides long storage times.
This spectrum transformation could be a source of additional losses and should
be taken into account in a detailed comparison.

Solid deuterium also offers a possibility to produce Very Cold Neutrons (VCN)
from the same converter (see Section 7.6).

An alternative approach for some applications is to set up experiments within
the converter of a superfluid helium source. This approach makes it possible
to use the full accumulated _in-situ_ UCN density, which at
$220~{}\text{cm}^{-3}$ [480] for existing prototype sources already exceeds
the values anticipated for next-generation storage experiments. (Only much
lower densities can presently be delivered to _ex-situ_ experiments, due to
extraction and transport losses.) A further advantage is that by eliminating
transport losses, one automatically uses the low-energy part of the UCN
spectrum which can be most efficiently stored. Extraction-based measurements
at any UCN source suffer from the practical compromise that faster UCNs are
more easily transported, whereas slower UCNs are more easily stored.

The clear advantages of the _in-situ_ approach have been recognized in the
past, leading to at least three distinct experimental efforts in this
direction [345, 481, 482, 483, 484]. All have confronted significant
challenges in the technical implementation of the _in-situ_ measurement
concept, due to challenges of scale and the challenge of jointly satisfying
the stringent requirements for UCN production and for precision measurements
in the same apparatus. _In-situ_ experiments may provide the most promising
avenue to increase the storable density of UCNs with velocities up to 4 m/s,
since the required technology for UCN production has already been
demonstrated.

The phase-space density in the experimental apparatus is most important: this
results from not only the available UCN density at storable velocities, but
also extraction efficiency and all loss processes that must be taken into
account for UCN delivery to the experiment. New UCN sources delivering higher
phase-space densities in experiments (e.g., storage experiments with number
density exceeding $10^{2}~{}\text{cm}^{-3}$ for UCN velocities $<4$ m/s) would
enable significantly improved experimental sensitivities. In particular, the
following cases would profit from increases of flux or stored density:

  * –

improving the accuracy of measuring the neutron lifetime;

  * –

precision gravitational neutron spectroscopy;

  * –

improving the sensitivity to the electric dipole moment of the neutron;

  * –

experiments to study non-stationary quantum effects [485, 486, 487, 488].

In the following sections, these experiments are briefly discussed.

###  7.2 Measurements of the neutron lifetime

As mentioned in Section 4.1.1 precision measurements of neutron beta decay,
including the neutron lifetime, are of key theoretical importance. The neutron
lifetime is used to determine coupling constants of the electroweak
interaction, and to check the unitarity of the CKM quark mixing matrix. The
neutron lifetime is also an important parameter for astrophysics and
cosmology.

One may conceptually distinguish two different experimental approaches for
measuring the neutron lifetime: (1) those which directly measure the total
decay rate, and (2) those which are sensitive only to a specific decay branch.
The first approach describes all UCN-based approaches of which we are aware,
including those detecting decay products: time-dependence in the detected rate
of decay products is driven by the surviving number of UCN. The proposal to
measure the neutron lifetime in a beam-based experiment, by observing
transformation of the neutron time-of-flight spectrum [489], also belongs to
the first approach. Experiments following the second approach include the
remaining beam-based lifetime measurements, which in addition to measuring the
absolute rate of decay products for a specific branch, rely on absolute
measurements of the total neutron flux in order to determine the specific
activity for producing the observed decays. This second approach thus enables
checking if the detected branch fully explains the decay rate measured via the
first approach.

In particular the second approach is insensitive to any decay branches that
are not directly detected, and thus comparing determinations of the neutron
lifetime between these two approaches provides a conceptual method to
constrain invisible decay channels, such as the decay into a hydrogen atom and
a neutrino [490]. One or more “dark” decay channels could hypothetically
explain the observed discrepancy between beam-based and storage-based
measurements of the neutron lifetime [36]. Subsequent evaluations of this
hypothesis have indicated that it is not well-supported by available data
[491, 492].

Experiments based on UCN storage in traps measure the neutron lifetime by
observing the time-evolution of the number of UCNs in the trap. In the case of
magnetic storage, it is assumed that when the UCN spectrum is formed below the
magnetic potential, in the absence of depolarisation or inelastic
interactions, the storage time is equal to the neutron lifetime. In
experiments, researchers try to control the process of depolarisation or to
detect depolarised UCN and try to reduce the influence of marginally-trapped
neutrons. Marginally-trapped neutrons have energies above the value of the
trap potential, but could be stored for a long time in trajectories that
provide a small velocity component parallel to the magnetic field gradient (or
normal to the trap walls in the case of material traps). In the case of
storage in a material trap, one must take into account UCN losses due to the
interaction with the walls by measuring the dependence of the storage time on
the frequency of collisions of neutrons with the walls and extrapolating this
dependence to zero frequency [493]. Changing the frequency is achieved either
by changing the spectrum of stored neutrons (energy calibration), or by adding
additional surfaces to the storage volume, or by changing the volume of the
trap (geometric calibration) [494]. The losses can be controlled by detecting
neutrons escaping the trap via the inelastic scattering channel [495]. It is
postulated that additional surfaces or surfaces of different traps provide the
same loss coefficients. It is possible to calibrate the loss at a fixed
collision frequency and change the loss factor, allowing one to get away from
this postulate [496].

Possibilities for improving the statistical uncertainty in this approach, by
increasing the phase-space volume of the experimental setup, are significantly
limited. A significant increase in the intensity of permanent magnetic fields
is technically unrealistic to implement in such experiments. Increasing the
geometric volume is possible as long as the characteristic filling times
remain shorter than the neutron lifetime, but this can lead to an increase in
systematic effects: in particular, it would be necessary to increase the
spectral “cleaning time” for removing marginally-trapped neutrons that can
escape the trap before decaying, and thus shorten the apparent neutron
lifetime. The best measured statistical uncertainty to date for the neutron
lifetime is 0.28 s [497], measured with magnetic UCN storage.

Similarly, in experiments with material traps, the best statistical
uncertainty to date is 0.7 s [498], obtained in a large setup. Further
increases in the geometric dimensions of the trap are not expected to lead to
improvements, since the characteristic time for filling the trap will exceed
its storage time.

The broadening of the spectrum of stored neutrons will automatically lead to
an increase in systematic errors (due to the energy dependence of all
characteristic parameters describing the storage and detection of UCNs). It
should be noted that in experiments with material traps, it is still tempting
to use materials with theoretically low loss factors (such as beryllium and
solid oxygen), which have not yet been observed in practice [452]. It follows
from the discussion above that improving the accuracy in any of these
approaches requires an increase in the UCN density delivered to an
experimental setup. Provided high densities can be reached, it is interesting
to consider storage of UCNs on the surface of superfluid helium [499] and
experiments on transformations of the UCN phase-space volume [500]. Such
transformations could make it possible to efficiently fill the traps with the
soft part of the UCN spectrum.

To date, UCN measurements detecting charged decay products [501, 502] have not
reached a competitive level of accuracy in comparison to those experiments
discussed above that count stored UCN directly.

For measuring the neutron lifetime at the ESS, one can draw attention to the
proposal to determine it from the transformation of the neutron time-of-flight
spectrum [489]. This proposal has nothing to do with UCN directly, but the
observed effect in the proposed experimental design becomes larger for slower
neutrons. It is interesting to consider the possibility of implementing such
an approach with UCN, although a special implementation involving vertical
extraction may be required to account for effects due to gravity.

One may also consider implementing experiments at the ESS that follow the
second approach of detecting a specific decay branch, either UCN-based or
otherwise. Such implementations require very good knowledge of the detection
efficiency for neutron decay products in order to determine the absolute rate
for the observed decay branch, as well as absolute measurement of the neutron
flux or stored neutron number. Beam-based neutron lifetime measurements
already use this approach [270, 271, 272, 35], which is further discussed in
Section 5.1.1. We are not aware of any measurement approaches based on UCN
storage that follow this concept; in principle it could be implemented via
substantially-improved knowledge of both the absolute UCN number and the
detection efficiency for decay products.

###  7.3 Gravitational spectroscopy with neutrons

This section is devoted to the control and understanding of the behavior of an
elementary quantum system, such as a neutron with extremely low vertical
velocity component, in a gravitational field above a mirror [503, 190, 504]
using resonance spectroscopy methods [505]. It offers a way, based on quantum
interference and spectroscopy, to search for new hypothetical interactions
beyond the standard model and modifications of gravity at short distances. The
neutron gives access to parameters that characterize gravity in the standard
framework of general relativity, as well as more speculative extensions such
as torsion [506, 507]. This is possible because the neutron is bound between a
flat lying mirror and the rising potential of gravity above, $V=mgz$, where
$m$ denotes the neutron mass, $g$ the local gravitational acceleration, and
$z$ the distance above the mirror. Every bound system in quantum mechanics has
discrete energy levels. In our case the lowest energy eigenvalues $E_{n}$
($n=1,2,3,4,5$) are 1.41 peV, 2.46 peV, 3.32 peV, 4.08 peV, and 4.78 peV. In
Figure 28 they are shown together with the corresponding neutron
wavefunctions, given by the well-known mathematical Airy functions that solve
Airy’s differential equation.

Figure 28: Energy eigenvalues and eigenfunctions of a neutron bound in the
gravity potential of the Earth, and corresponding equivalent frequencies. By
oscillating the mirror at a frequency that corresponds to the energy
difference between two quantum states, transitions are introduced. Shown is a
resonant transition between $|1\rangle\leftrightarrow|6\rangle$ at a frequency
of 972.35 Hz. The oscillation frequencies are in the acoustic frequency range.

In 2002 the lowest stationary quantum state of neutrons in the gravitational
field was clearly identified [190, 508, 509, 510]. The proof of principle
triggered new experiments and activity in this direction, namely the setting
up of the GRANIT [511], the qBOUNCE [512] collaborations, and teams from Tokyo
[513] and Los Alamos National Laboratory. Spectroscopy is a technique used to
assess – in most cases – an unknown quantity of energy by means of a frequency
measurement. Basic quantities like atomic masses, particle energies, momenta,
and magnetic moments have been determined in this way. Transitions can be
driven between the gravitational energy eigenstates, by a time-dependent
perturbation provided by an external oscillator. These transitions occur with
a characteristic energy transfer, and the resonance condition is observed when
the oscillator frequency matches the difference between two eigenstates. The
transition is observed via the change in population of the various
eigenstates, after the time-dependent perturbation has been applied.

In this Gravity Resonance Spectroscopy (GRS) technique [514], the energy
difference between these states has a one-to-one correspondence to the
frequency of the modulator, in analogy to the Nuclear Magnetic Resonance
technique, where the energy splitting of a magnetic moment in an outer
magnetic field is related to the frequency of a radio-frequency field. The
linear gravity potential leads to discrete non-equidistant energy eigenstates
$|n\rangle$. A combination of any two states can therefore be used, as each
transition can be addressed by its unique energy splitting, or, by vibrating
the mirror mechanically with the appropriate frequency. In that way GRS is
becoming a tool, which combines the virtues of UCNs – namely long observation
time together with insensitivity to electrostatic and van der Waals
interactions – with spectroscopy. Highest sensitivity can be reached by
applying Ramsey’s method of separated oscillating fields [504] to GRS. This
method will allow a precise measurement of energy differences with a precision
similar to the magnetic resonance technique. In a beam experiment, this
precision is determined by the level’s width according to the ratio: $\Delta
E/E\sim\hbar v/LE$, where $\Delta E$ is the level’s energy width, $E$ is an
energy difference between gravitational levels, $v$ is the neutron velocity,
and $L$ is the mirror length defining the duration of the interaction; for
qBOUNCE, $\Delta E/E\sim 3\%$. As a consequence, the coupling of residual
fluctuations of the magnetic field to the magnetic moment of the neutron must
be highly suppressed. The search for generalized gravitational theories is
highly topical now. Observational cosmology led to the introduction of the
concepts of dark matter (DM) and dark energy (DE). At the level of precision,
GRS provides constraints on some possible gravity-like interactions. If some
as yet undiscovered particles interact with neutrons, this should result in a
measurable energy shift of the observed quantum states shown in Figure 28
[514, 515, 516].

The accuracy of measuring the gravitational quantum states of neutrons depends
on two factors: a) the UCN density in phase space, and b) the observation time
for UCN in quantum states. Therefore an _in-situ_ experiment at the ESS would
be of interest, using the highest UCN densities inside a helium converter
(without extracting the UCNs). The observation time for quantum states could
ultimately be comparable to the lifetime of the neutron, as proposed in [517].
The practical implementation of such an experiment, however, is challenging. A
convenient method of analysis of gravitational quantum states in such an
experiment is time-resolved spectroscopy as proposed in [518].

###  7.4 Measurements of the neutron’s permanent electric dipole moment (EDM)

Ramsey’s method of separated oscillatory fields is the foundational technique
used for measuring the neutron EDM, whether using cold neutrons or UCNs – see
also Section 5.3. (Other approaches have been proposed, e.g. continuous
measurements exploiting spin-dependent absorption of neutrons on 3He nuclei
[345], or beam experiments exploiting neutron spin-rotation in crystals with
very high internal electric fields [519]. To date these approaches have not
delivered results at a competitive sensitivity level.) The Ramsey method was
initially applied in beam experiments [339], until that approach reached
systematic limitations due to beam divergence and motional magnetic fields
[344]; see Section 5.3.1 for a discussion of how these effects might be
mitigated in modern experiments at the ESS. At that juncture, UCNs offered a
viable way forward both by mitigating these particular systematic effects and
by opening the door to much longer observation times [520]. Since then, all
subsequent advances in the precision and accuracy of neutron EDM measurements
have been based on storage experiments with UCNs in a measurement volume
separate from the UCN source.

A key subsequent development was the concept of “comagnetometry” [521], in
which an atomic species whose intrinsic sensitivity to magnetic fields is
similar – but whose EDM is expected, or has been measured, to be small –
occupies the measurement volume together with stored UCNs. This enables a
simultaneous differential measurement, in which all effects seen by the
comagnetometer are subtracted in common-mode. This approach greatly improves
the sensitivity of such measurements, at the expense of experimental
complexity and certain limitations on the experimental conditions (e.g.,
electric field strength) that are required in order for the comagnetometer to
function. We note, however, that the leading systematic errors are quite
different for atomic comagnetometers and UCNs. In fact, the largest source of
error in the most recent neutron EDM measurement was associated with the
comagnetometer correction, arising from differences in how spatially
inhomogeneous fields within a measurement cell are probed by comagnetometer
atoms and UCNs. This leads to the possibility that high-order gradients are
sampled differently by the two ensembles [57]. Recent proposals for new
measurements of the neutron EDM (e.g., reference [480] as well as the
proposals discussed in Section 5.3) in some cases favor other approaches to
achieve stringent control of experimental conditions, rather than contending
with the increasing challenges of comagnetometry at higher levels of
sensitivity.

A further conceptual advance, now universally adopted in the present
generation of neutron EDM experiments, was to perform two measurements
simultaneously with a common magnetic field and oppositely-directed electric
fields [522]. This additional common-mode correction effectively removes
offset field drifts within the pairs of measurements that must be combined in
order to determine the EDM. When in addition a comagnetometer is present for
both measurements, one can also remove in this way the leading effects
connected with magnetic field gradients. A generalization of this concept to
many cells has been proposed [523], which in suitably stable conditions would
permit controlling this type of systematic effect at arbitrary order. A
conceptually similar approach employing four cells (and no comagnetometer) is
used in measurements of the 199Hg atomic EDM [524]; the multicell concept is
also central to the implementation of the EDMn project that has been proposed
for the ESS, in which cells with zero electric field are intended to be used
for averaging and correction of systematic errors (see Section 5.3.2).

As for other precision physics experiments, the major effort in EDM
measurements is spent in connection with constraining systematic errors.
Important systematic effects include (1) those arising from spatial and
temporal variation of magnetic fields (and in particular, of magnetic field
_gradients_), (2) nondynamical phases arising from a combination of motional
magnetic fields and gradients, and (3) direct false-EDM effects that correlate
with the applied electric field, e.g. those arising from leakage currents or
induced magnetization of inner components. Achieving higher densities of
slower UCNs in smaller storage chambers would directly reduce the impact of
(1) and (2), while also increasing the rapidity with which systematic studies
can be conducted by intentionally varying or amplifying particular systematic
errors: the analysing power available from UCN statistics is frequently the
rate-limiting consideration. While these classes of effects have been
extensively studied at present-day sensitivities, constraining them in future
experiments will rely on dedicated research-and-development efforts. It is
also to be expected that new systematic effects will become relevant as
experimental precision increases.

While increasing the number of stored UCNs may be the main route to long-term
improvements in measurements of the neutron EDM, one should also note that
gains can still be had from increasing the observation time and electric field
strength – parameters for which the scaling of sensitivity for a single
measurement is actually more favorable, see Equation 6. Unlike experiments
measuring the neutron lifetime, the longest holding times achieved so far in
neutron EDM measurements are on the order of $200~{}\text{s}$ – significantly
shorter than the ultimate limit imposed by the neutron lifetime. This holding
time must be chosen as a compromise among the cycle time for re-filling the
measurement cells, the storage time constant of the measurement cells, and
numerous constraints including the performance of other key components such as
a comagnetometer (if any). Recent demonstrations of long UCN storage lifetimes
in small bottles [525, 526], with material coatings intended for EDM
applications and using soft UCN spectra extracted from a superfluid-helium
source, have given rise to hopes that the storage time in particular may
provide a route toward improvements in the next generation of experiments. The
electric field strength is typically limited to the order of $2~{}\text{MV/m}$
(or less, in the presence of a comagnetometer) by the surface conductivity of
the insulating ring which forms one wall of a UCN storage cell. However,
recent advances in cryogenic technology developed for an _in-situ_ neutron EDM
measurement by the American collaboration nEDM@SNS have indicated that much
higher fields, on the order of $10~{}\text{MV/m}$, can be sustained when the
cell is filled with superfluid helium [343]. This brings the electric field
strength into the same range as for the Beam EDM experiment (see Section
5.3.1).

In contrast to the scenario described in Section 5.3.2, one may also envision
implementing a multicell _in-situ_ EDM measurement with a smaller number of
cells at a dedicated UCN facility. Through use of multimirror imaging optics
[470, 527] it may be possible to deliver very high neutron flux in the 8.9 Å
wavelength range, from a large area of the lower moderator surface at the ESS,
albeit only over a short axial range due to the beam divergence. This might
open the possibility of achieving UCN densities in the range of
$10^{4}~{}\text{cm}^{-3}$ – if such cold-neutron-extraction concepts can be
efficiently realized, without excessive backgrounds or unanticipated losses.
While the 8.9 Å component of the feeding beam cannot be fully exhausted within
a few-meter installation, such an approach has the advantage that any
requirements for guiding cold neutrons within the multicell stack could be
substantially relaxed. This concept requires further study.

###  7.5 Study of non-stationary quantum effects

The development of ideas about the optical phenomenon called the accelerating
matter effect [528, 529, 530], which consists in a shift in the frequency of a
wave when passing through an accelerating refracting sample, led to the
hypothesis of the existence of a very general acceleration effect [531]. As
applied to the physics of the microscopic world, its formulation is that the
result of a particle interacting with any accelerating object should be a
change of its energy and frequency determined by the ratio
$\Delta\omega=ka\tau$, where $k$ is the wave number, $a$ is the acceleration
of the object, and $\tau$ is the interaction time.

It should be expected that any elementary scatterer moving with acceleration
should change the frequency of the wave, and this means an effect that
complements the usual Doppler shift but is proportional not to speed, but to
acceleration. The validity of the acceleration effect hypothesis in quantum
mechanics has recently been confirmed by numerically solving a number of
problems related to the interaction of a wave packet with potential structures
moving with acceleration [532]. If the hypothesis on universality of the
acceleration effect is true, it applies to the case of neutron scattering on
the atomic nuclei of accelerating matter. Assuming that this elementary
process can change the frequency of the neutron wave function, we will be
forced to take a new look at the problem of neutron optics of an accelerating
medium.

Different aspects of this hypothesis must be proved experimentally. The use of
UCN for this aim is highly preferable, because the interaction time of
neutrons with nuclei is inversely proportional to the neutron velocity. In
addition, the study of non-stationary quantum effects requires either
monochromatisation or interruption of the neutron beam. Thus for these
experiments, the high phase-space density that could potentially be achieved
at the ESS is advantageous.

###  7.6 Implementation of a VCN Source at the ESS

Intense fluxes of neutrons in the range between 10 Å to about 120 Å are of
interest for a variety of applications, some of which are listed in Section
7.7; however, so far the development of applications for neutrons in this
wavelength range has been hindered by the challenging task of realizing VCN
sources. At present the only existing VCN beamline is PF2 at the ILL, where
slow neutrons are extracted vertically from one of the cold moderators, but
this beamline has low intensity for realization of most potential
applications. Its neutron flux at
$v=40~{}\textrm{m}/\textrm{s}~{}(100~{}\textrm{\AA})$ is about
$10^{5}~{}\textrm{cm}^{-2}\textrm{s}^{-1}(\textrm{m}/\textrm{s})^{-1}~{}(=0.4\times
10^{5}~{}\textrm{cm}^{-2}\textrm{s}^{-1}\textrm{\AA}^{-1})$. In recent years
however there have been promising developments that hint at the possibility of
realizing new sources. On one hand research has progressed in identifying
materials for production of intense VCN fluxes: at least an order of magnitude
higher, relative to PF2. The innovative use of cascaded cooling in
paramagnetic systems such as deuterated clathrate hydrates has been suggested
[533], and experimental investigations of these materials are ongoing within
the HighNESS project. Solid deuterium is another candidate material for VCN
production.

One of the most promising developments related to VCN sources concerns the use
of effective reflectors which allow to direct neutrons to experimental set-
ups. In recent years it was shown that diamond nanoparticle powders can be
used as such reflectors [534], and VCN storage in a cavity made of the powder
was realised [535]. The influence of hydrogen impurities [536], powder
structure [537] and particle size [538] on reflection was determined and a
method was found to remove hydrogen from the powder [539, 540]. The
interaction of neutrons with powders of nanostructures, possible applications,
and neutron transport simulation inside, are widely studied in various
scientific centers both theoretically and experimentally. The possibility to
use these reflectors for cold neutrons was also shown [541, 542].

The neutronic study of VCN sources within the HighNESS project has started,
considering either the use of dedicated VCN converter materials, or the design
of an optimized geometry based on reflectors such as nanodiamonds [473], which
can increase the reflection of the longer wavelength neutrons from a cold
moderator [543].

It is possible to combine the use of a dedicated VCN converter material with
advanced reflectors for a more efficient VCN source. A possible configuration
of the VCN source at ESS (or another intense neutron facility) could look like
this: a large liquid deuterium moderator is located under the ESS target. A
capsule of a nanoparticle reflector, open on one side, is buried with its
closed end into this source. At the bottom of the capsule is a relatively
small solid deuterium converter (this principal is illustrated at Figure 29).
Solid deuterium as VCN moderator is attractive because it will produce a large
amount of VCN and it is possible to combine a source of UCN/VCN. The use of
reflectors made of nanoparticles will significantly increase the direction
flux of extracted VCN [544] and allow reducing the thickness of the solid
deuterium by a factor of two at fixed VCN flux. Calculations for realistic
geometries, and optimization, are planned to be carried out in the near
future.

Figure 29: A conceptual diagram of the VCN source lay-out.

###  7.7 Scientific Prospects for Use of VCN

We are aware of two ideas where large potential gains from the use of VCN have
been investigated in detail:

  * 1.

Search for neutron-antineutron oscillations [545]. The idea of an experiment
with cold neutrons was discussed in Section 6. The sensitivity of the
experiment is proportional to the product of the observation time of
oscillations and the square root of the neutron flux. The switch to slower
neutrons with the same experimental setup does not give a gain at first
glance. The observation time is inversely proportional to the neutron
longitudinal velocity, and the neutron flux in the neutron guide is
proportional to the square of the velocity (under the natural assumption that
the neutron phase-space density in the source is constant). However, the use
of the above-mentioned reflectors can in principle make it possible to
significantly increase the phase-space density of the VCN [546].

  * 2.

Search for additional types of interactions [547, 548] by studying the angular
distribution of neutrons scattered by atoms of noble gas. Full details of the
method are presented in reference [547].

In contrast to the above ideas which are relatively well-developed in terms of
estimated gain factors, the following points can also be considered but remain
at an earlier stage of development where the balance of potential gains (if
any) against various disadvantages are less clear at this point:

  * 1.

Search for the electric dipole moment (EDM) of the neutron [549]. In
comparison to the cold-neutron beam experiment discussed in Section 5.3.1, the
use of VCN for an apparatus with the same length could deliver a gain in
statistical sensitivity due to longer observation times. However at the
present level of analysis it is unclear if this approach could be competitive
with experiments using cold neutrons or UCN. A detailed analysis is also
needed for systematic effects, which are different for different neutron
velocities.

  * 2.

The sensitivity of certain interferometers (for instance based on diffraction
gratings) depends on both the intensity and the parameters of the optical
system. The deviation of the refractive index from unity is greater, the
slower the neutrons. As result the the optical effects are more pronounced and
requirements for the optical system become not so strong. Interferometers can
be used for precision measurements of the coherence length of neutron
scattering by nuclei, or searches for new types of interactions.

  * 3.

The use of VCN in a beam experiment to measure the neutron lifetime, similarly
to [550], where cold neutrons were used, will lead to an increase in the
probability per neutron to observe a decay, that will affect the statistical
accuracy of the result. It should be noted that the systematic error in this
experiment, associated with the determination of the absolute neutron flux,
still exceeds the statistical one, although some progress has been made
towards its reduction [551].

  * 4.

Increasing the sensitivity of commonly used neutron scattering techniques such
as spin echo [552, 553, 554] (by increasing the observation/interaction time)
and reflectometry (by expanding the dynamic range of the setup) [555].

  * 5.

Use of tomography with VCN for solving specific problems [556].

A detailed analysis of the proposed UCN/VCN sources should be performed in
order to reliably evaluate their performance. The types of chosen sources
would determine the list of scientific problems that could be solved using
them. An assessment of the accuracy and sensitivity of prospective research
could only be made after evaluating the actual source parameters.

##  8 CE$\nu$NS at the ESS

###  8.1 Introduction

Coherent elastic neutrino-nucleus scattering (CE$\nu$NS) is the most probable
mechanism for low-energy neutrino interaction. Nucleons participate coherently
in this neutral-current scattering process, resulting in a coupling
effectively proportional to the square of the number of neutrons in the target
nucleus [232, 557]. This large enhancement to the scattering cross section
facilitates the detection of these elusive particles while considerably
reducing the target mass involved. Nevertheless, this process remained
undetected until recently [233, 558], more than four decades following its
theoretical description. This somewhat puzzling delay was due to a combination
of factors: the modest energy of the nuclear recoils induced -the single
observable from this process-, and the limited intensity of available neutrino
sources in the favorable energy range -coherence is lost above few tens of
MeV-.

Coherent nuclear scattering mediates the interactions of popular dark matter
candidates (Weakly Interacting Massive Particles), and dominates neutrino
transport within supernovae and neutron stars [559, 560]. These aspects added
to a broad interest in obtaining an experimental verification for CE$\nu$NS.
Most importantly, the observation of this process unlocked four decades of
phenomenological proposals aiming to exploit this new mechanism of neutrino
interaction. CE$\nu$NS measurements promise to improve our knowledge of both
neutrino properties and nuclear structure, each providing new opportunities
for revealing physics beyond the Standard Model. The fact that evidence for
the incompleteness of the Standard Model comes from the neutrino sector, an
area of still incipient knowledge, adds to the appeal of these investigations.
For instance, a neutral-current detector responds identically to all known
neutrino types [561]. Therefore, observation of neutrino oscillations in such
a device would provide unequivocal evidence for sterile neutrinos [557, 562,
563, 564]. In addition to this, the differential cross section for this
process is strongly dependent on values of the neutrino magnetic moment
otherwise beyond reach [565, 566, 567, 568]. Numerous recent studies have
described the sensitivity of CE$\nu$NS to non-standard neutrino interactions
with quarks [569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581],
to the effective neutrino charge radius [582, 583], to neutron density
distributions [584, 585, 586, 587, 588, 589], and to accelerator-produced dark
matter [590, 591, 592, 593]. A precise measurement of the CE$\nu$NS cross
section would also provide a sensitive appraisal of the weak nuclear charge
[594, 595, 596, 597]. There is a crucial interplay between nuclear structure
and CE$\nu$NS. The low energy scattering probes the neutron distribution of
ground state and low–energy excitations. Therefore, nuclear structure provides
key information to correctly interpret the measured cross section, and the
cross section probe the weak sector of the nucleus [598, 599]. In turn, this
input will contribute to calibrate other standard model tests in nuclei based
on parity violation [600]. A complete review of the vibrant field of CE$\nu$NS
phenomenology is beyond the scope of this paper: a more extensive compilation
can be found in [9].

In a seminal paper [557], Drukier and Stodolsky described the prospects for
CE$\nu$NS detection from a variety of low-energy neutrino sources (solar,
terrestrial, supernova, reactor, and spallation source). Of these, spallation
sources were highlighted as the most convenient, due to higher recoil energies
-easier to detect- and pulsed operation -leading to a favorable steady-state
environmental background reduction-. While the main use for these facilities
is as intense neutron sources, their protons-on-target (POT) produce positive
pions, which decay at rest generating a monochromatic flash of 30 MeV prompt
$\nu_{\mu}$. This is followed by delayed $\nu_{e}$ and $\bar{\nu}_{\mu}$
emissions with a broad energy (Michel spectrum, up to few tens of MeV), over
the $\sim$2.2 $\mu$s timescale characteristic of $\mu$ decay. Specifically,
the neutrino flux 20 m away from target at a 5 MW ESS is expected to be $\sim
1.6\times 10^{8}$ cm-2 s-1 [9].

The physics potential for CE$\nu$NS at the ESS has been recently described in
[9]. Additional studies have further examined it [564, 601, 602, 603]. A 5 MW,
2 GeV ESS is expected to provide an order of magnitude increase in neutrino
flux with respect to the Spallation Neutron Source (SNS) at ORNL, the site of
the first CE$\nu$NS observation. Compact, low-cost detectors (few kg to few
tens of kg) operated at this new European facility, will produce CE$\nu$NS
measurements not limited by statistics of the signal, but instead by
systematics of order few percent, such as the specific response of different
detector materials to low-energy nuclear recoils [604]. As a result, the ESS
will provide a definitive sensitivity to the broad body of new physics
reachable via CE$\nu$NS, unsurpassed for the foreseeable future [9, 601, 564]
(Figure 30).

Figure 30: Left: Balanced comparison of pulsed spallation facilities as
CE$\nu$NS sources. Future (red diamond) and past (black square) sources are
shown relative to the SNS at the time of the first CE$\nu$NS measurement (blue
circle). The separate impact of proton current and proton energy on neutrino
flux are correctly included as in [9]. Similarly, the effect of beam pulse
time-profile on signal-to-background ratio is properly computed assuming 10
$\mu$s windows for full detection of the delayed $\nu_{e}$,$\bar{\nu}_{\mu}$
component, and accounting for the ability to subtract steady-state backgrounds
when characterized in anti-coincidence [233, 9]. This plot is in contrast to
more simplistic comparisons [605, 606]. Top right: (adapted from [9])
Improvement in sensitivity to a parametrization of non-standard neutrino
interactions, expected from use of high-pressure Xe and Ar TPCs at a 5 MW, 2
GeV ESS (see text). Parameter space outside of the tori is excluded
(“COHERENT" indicates present SNS limits). The advantage of combining
information from different targets (solid line contour) is evident. Bottom
right: Further improvements expected from the squeezed beam bunching in a
ESSnuSB configuration, leading to a steady-state background reduction [607].
Tori are for ESS sensitivity, colored regions for ESSnuSB.

###  8.2 CE$\nu$NS at the ESS

A point emphasized in [9] is that the best foreseeable neutrino source
deserves the most-advanced next-generation nuclear recoil detectors, so as to
obtain the best possible results from the extraordinary opportunity the ESS
provides. Several promising new technologies were described in that
publication. In the interest of brevity, only two examples are highlighted
here. The first is the proposed use of undoped cryogenic (80 K) CsI as a
CE$\nu$NS detecting medium. This material holds vast potential for this
application [608] when operated in combination with state-of-the-art light
sensors and novel waveshifters [9]. To be specific, for comparable neutrino
fluxes (i.e., before accounting for the $\times$10 increase at the ESS), the
CE$\nu$NS rate per kg of scintillator can be increased by a factor of eight
with respect to the room-temperature CsI[Na] employed for the first CE$\nu$NS
detection at the SNS, reaching in the process a sensitivity to nuclear recoils
down to a $\sim$1 keVnr threshold [9, 608] (the suffix “nr" makes reference to
nuclear recoil energy, as opposed to “ee" for electron-equivalent, i.e.,
observable energy). As a reference, the CsI[Na] threshold in [233] was 5
keVnr, with a significant signal acceptance achieved only above 10 keVnr.
Besides increasing the signal rate, a reduced energy threshold maximizes the
sensitivity to new neutrino physics, which arises preferentially at the lowest
recoil energies [9]. For perspective, a compact 31.5 kg cryogenic CsI detector
will register $\sim$12,000 CE$\nu$NS events per year at the ESS, with a
superior energy resolution derived from the production of $\sim$47 primary
electron-hole pairs per keVee at a silicon light sensor [9]. As a measure of
potential for CE$\nu$NS at the ESS, these figures can be contrasted with the
$\sim$3,000 events/yr, $\sim$ 4.2 photoelectrons per keVee, and $\sim$20 keVnr
threshold that a massive 750 kg liquid argon detector would achieve at the SNS
[590, 609].

Innovative high-pressure gaseous xenon TPC detectors have been recently
developed for neutrinoless double-beta decay searches [610]. These devices are
sensitive to both primary scintillation (S1) and electroluminescent
amplification of charge ionization (S2) signals, a property that can be
exploited to separate nuclear recoil from electron recoil signals [611]. They
are also insensitive to charge-trapping effects at low-energy that can limit
the application of two-phase noble liquid detectors to CE$\nu$NS
experimentation [9]. Additionally, they feature an outstanding energy
resolution [612], and the ability to easily swap the active gas, which can
help enhance the sensitivity to some aspects of CE$\nu$NS phenomenology
(Figure 30). For all these reasons, they are prime candidates to be adopted as
CE$\nu$NS detectors at the ESS. Their response to low-energy nuclear recoils
is an incipient area of knowledge [9], to be studied in the interim until ESS
operation.

A curious coincidence increases the interest in simultaneously using these two
technologies for this application: due to their similar number of neutrons per
nucleus, with cesium and iodine immediately surrounding xenon in the table of
elements, CsI and Xe detectors are essentially indistinguishable in their
expected response to CE$\nu$NS [9]. In contrast to their identical CE$\nu$NS
response, these two detector technologies, and in turn their expected
systematics, are fundamentally different. Their side-to-side use at the ESS
would bring about a unique situation in experimental particle physics: two
highly-complementary and yet distinct technologies, able to provide robust
independent confirmation for any subtle signatures of new physics that might
be found in one of them.

###  8.3 Timeline and expected sensitivity

The presently envisioned slow ramp-up leading to a 5 MW, 2 GeV ESS performance
(Section 3.4) is an excellent match to the R&D necessary to develop next-
generation nuclear recoil detectors able to fully profit from this unique
neutrino source. The construction and characterization of advanced high-
pressure noble gas, cryogenic CsI, and low-noise germanium detectors for this
application [9] has been recently supported by two ERC Horizon 2021 actions.
This timeline provides room for the development of the novel detector
technologies envisioned, as well as for the required studies of detector
response to low-energy nuclear recoils. The latter are of crucial importance
in order to correctly identify indications of new physics via CE$\nu$NS [604,
608, 613].

It is worth emphasizing at this point that CE$\nu$NS detectors are devoid of a
significant burden on the ESS, being nonintrusive to its established neutron
science goals. Due to their compact design they require only the allocation of
a modest floor area for their installation. As a result, in case of
unanticipated ESS delays in achieving design power, these technologies can
continue to be improved in situ for a period of time (e.g., by characterizing
and abating backgrounds) up until maximum ESS power is achieved. New physics
results can be generated in the interim by virtue of the mentioned improved
signal rate per unit detector mass that is expected from advanced detectors.
Table III in [9] provides detailed information on the increase in sensitivity
to several areas of neutrino phenomenology expected from next-generation
CE$\nu$NS detectors at the ESS. Further significant improvements can be
generated by an eventual upgrade of the facility to neutrino Super Beam
specifications (Figure 30, Section 9).

Suitable potential locations for CE$\nu$NS experimentation within the ESS
facility have already been identified. Besides a proximity to the ESS target
able to maximize neutrino flux, such sites must guarantee that backgrounds
from prompt neutrons able to escape the target monolith do not compete with
CE$\nu$NS signals. Profiting from previous experience in site selection at the
SNS, two promising locations are being investigated: a utility room 15 m from
target, with 5.5 meters of steel monolith and a minimum of 6 meters of
magnetite-loaded heavy concrete (3.8 g/cm3) separating them, and additional
underground galleries as close as 23 m to target, with soil and concrete
pylons as intervening shielding. Full Monte Carlo simulations of neutron
generation and transport in the ESS building are under way: first results
confirm the suitability of these locations. These use MCNPX and GEANT4
geometries of the target monolith, and detailed NAVISWORKS 3-D building
layouts. Once neutron production has commenced at the ESS, neutron background
measurements will be performed to confirm these predictions, a process similar
to that followed at the SNS previous to the first CE$\nu$NS measurement [233,
558].

The advent of CE$\nu$NS has opened new exciting avenues in the search for
physics beyond the Standard Model, spanning both particle and nuclear realms.
CE$\nu$NS studies will expand our incipient knowledge of neutrino properties
through multiple new paths, each providing an opportunity for discovery. The
ESS will soon become the most intense pulsed neutrino source suitable for
CE$\nu$NS experimentation, remaining the leader for the foreseeable future.
Advanced next-generation CE$\nu$NS detector technologies are set to profit
from the unique opportunity that the ESS provides.

##  9 The ESS neutrino Super Beam (ESS$\nu$SB)

In the standard three flavour scenario, the phenomenon of neutrino oscillation
can be described by three mixing angles: $\theta_{12}$, $\theta_{13}$ and
$\theta_{23}$, two mass squared differences $\Delta$m${}_{21}^{2}$ (=
m${}_{2}^{2}$ \- m${}_{1}^{2}$) and $\Delta$m${}_{31}^{2}$ (= m${}_{3}^{2}$ \-
m${}_{1}^{2}$) and one Dirac type phase $\delta_{CP}$. During the past few
decades, some of these parameters were measured with good precision. At the
moment, the unknown parameters are: (i) the mass hierarchy of the neutrinos,
which can be either normal, i.e. $\Delta$m${}_{31}^{2}$ $>$ 0, or inverted,
i.e. $\Delta$m${}_{31}^{2}$ $<$ 0, (ii) the octant of the mixing angle
$\theta_{23}$, which can be either at the lower octant, i.e. $\theta_{23}$ $<$
45$\degree$, or at the higher octant, i.e. $\theta_{23}$ $>$ 45$\degree$ and
(iii) the Dirac violating phase, $\delta_{CP}$. In the search for the CP-
violation in the leptonic sector, crucial information has been obtained from
reactor and accelerator experiments [614, 615]. The discovery and measurement
of the third neutrino mixing angle, $\theta_{13}$, with a value $\sim$
$9\text{\,}\mathrm{\SIUnitSymbolDegree}$, corresponding to
$\sin^{2}$2$\theta_{13}$ $\sim$ 0.095, confirmed the possibility of
discovering and measuring a non-zero value of the Dirac leptonic CP violating
angle, $\delta_{CP}$. Before this measurement, a significantly smaller value
of $\theta_{13}$ was assumed, with a range of values of
$\sin^{2}$2$\theta_{13}$ $\sim$ 0.01 and 0.09, with 0.04 as a standard value.
In the light of this new finding, the sensitivity to CP violation observation
and measurement, with precision, of $\delta_{CP}$ has shown a strong
enhancement at the second oscillation maximum compared to that at the first
oscillation maximum [616, 239, 617]. This can be seen in Figure 31 where the
change in the probability, upon changing the values of $\delta_{CP}$, is much
more significant at the second peak maximum, as counted from the right side in
the figure, as compared to the first maximum. Moreover, by placing the far
detector at the second oscillation maximum, the experiment is significantly
less affected by, and hence more robust against, systematic uncertainties.
This is particularly important since the improvement of the present systematic
errors is known to be very hard. However, placing the far detector at the 2nd
oscillation maximum implies the need to use very high intensity "super"
neutrino beams to compensate for the longer baseline, hence lower statistics.

The long-baseline experiments which are currently running to measure these
unknowns are T2K [618] in Japan and NO$\nu$A [619] in the USA. Regarding the
true hierarchy of the neutrino mass, the results of both T2K and NO$\nu$A
favour normal hierarchy over inverted hierarchy. Regarding the true nature of
the octant of $\theta_{23}$, both these experiments support a higher octant,
however the maximal value, i.e. $\theta_{23}$ = 45$\degree$, is also allowed
within 1$\sigma$. Regarding the value of $\delta_{CP}$, there is a mismatch
between T2K and NO$\nu$A. Considering the branch for $\delta_{CP}$ as
-180$\degree$ $\leq$ $\delta_{CP}$ $\leq$ 180$\degree$, T2K supports the best-
fit value of $\delta_{CP}$ around -90$\degree$, i.e. maximal CP violation, and
the best-fit value measured by NO$\nu$A is around 0$\degree$, i.e. CP
conservation. However, it is important to note that both the values of
$\delta_{CP}$ = 0$\degree$ and -90$\degree$ are allowed at 3$\sigma$ and it
requires more data to establish the true value of $\delta_{CP}$. It is
believed that these two experiments will give a hint towards the true nature
of these unknown oscillation parameters, while the future generation of the
long-baseline experiments will establish these facts with a significant
confidence level.

Figure 31: Oscillation probabilities for the
$360\text{\,}\mathrm{k}\mathrm{m}$ (red curves) and
$540\text{\,}\mathrm{k}\mathrm{m}$ (purple curves) baselines as a function of
the energy for neutrinos (left panel) and antineutrinos (right panel). The
solid (dotted) lines are for $\delta=-90^{\circ}$ ($\delta=0$). The grey
dotted lines show the convolution of the signal component of the neutrino flux
with the detection cross section. Thus, they serve as a guide of what energies
of the oscillation probability would be well-sampled by the ESS$\nu$SB setup.

The European Spallation Source neutrino Super-Beam (ESS$\nu$SB) is a proposed
accelerator-based long-baseline neutrino experiment in Sweden [4]. In this
project, a high intensity neutrino beam will be produced at the upgraded ESS
facility in Lund. The neutrinos will be detected in a large water Cherenkov
detector located at a distance of $360\text{\,}\mathrm{km}$ from the ESS site,
which is the baseline distance, or at a distance of $540\text{\,}\mathrm{km}$,
which is a possible back-up option. The primary goal of this experiment is to
measure the leptonic CP phase, $\delta_{CP}$, by probing the second
oscillation maximum. As the variation of neutrino oscillation probability with
respect to $\delta_{CP}$ is much higher in the second oscillation maximum, as
compared to the first oscillation maximum [616, 239, 617], ESS$\nu$SB, as a
second generation super-beam experiment has the potential of measuring
$\delta_{CP}$ with unprecedented precision compared to the first generation
long-baseline experiments.

An EU supported design study of ESS$\nu$SB has been conducted by 17 laboratory
and university groups in 11 European countries during the period January 2018
till April 2022. The detailed results of this study are described in the
ESS$\nu$SB Conceptual Design Report to appear in The European Physical Journal
Special Topics [620]. In the following sections 9.1 to 9.3 the most important
results of this study are outlined. In July 2022 EU decided to support the
continuation during the period 2023-2026 of this design study under the
project name ESSnuSBplus, which shall result in a Technical Design Report to
be published at the end of this period. During this continued design study the
details of the civil engineering required for the realization of the project
shall be studied as well as the further plans outlined in the following
section 9.4.

###  9.1 Instrumentation

An initially pure $\nu_{\mu}$ beam is obtained from the decays of charged
pions produced in the colliosins of high energy protons with a target. The
determination of the leptonic CP violating phase angle $\delta_{CP}$ from the
oscillations of such a $\nu_{\mu}$ beam is made by measuring at the baseline
distance the absolute number of the appearing $\nu_{e}$ and the shape of the
spectrum of their relative momentum as well as from the comparison of these
quantities for a neutrino and an antineutrino beam, respectively. To be able
to produce a neutrino beam sufficiently intense for such measurements,
concurrently with the intense spallation neutron beam of the ESS, it is
necessary to apply a number of upgrades to the ESS facility. The proposed
upgrades are schematically presented in Figure 32. The pulse frequency of the
ESS LINAC (the proton driver) must be increased from $14\text{\,}\mathrm{Hz}$
to $28\text{\,}\mathrm{Hz}$ to obtain additional acceleration cycles that will
be used for neutrino production, without affecting the neutron programme.
Moreover, during neutrino cycles, H- ions instead of protons need to be
accelerated in order to ease the filling of the accumulator ring. An
accumulator ring will be built to compress the ESS pulse to about
$1.2\text{\,}\mathrm{\SIUnitSymbolMicro s}$. A neutrino production target
station, composed of four identical targets enveloped by four magnetic
focusing devices (horns), will be built. The horns will be used to sign select
and focus the pions, and thus also the neutrinos resulting from their decay,
toward the near and far detectors. A near detector will be used to monitor the
neutrino beam and to measure neutrino interaction cross-sections, especially
electron neutrino cross-sections, at a short baseline from the neutrino
source, while the far detector will be used to detect the oscillated neutrino
beam at the long baseline distance. In the following sections, these parts are
discussed together with the physics potential of the experiment.

Figure 32: Schematic layout of the required ESS upgrades. The ESS linear
accelerator is shown in black, the transfer line from the LINAC to the
accumulator ring is shown in blue, the accumulator ring and the switchyard are
shown in black, the target station is shown in green, and the near detector
site is shown in magenta. (Units in mm)

####  9.1.1 Proton Driver

The ESS LINAC, currently being constructed, will accelerate 14 proton pulses
of $2.86\text{\,}\mathrm{ms}$ length per second and $62\text{\,}\mathrm{mA}$
current to $2\text{\,}\mathrm{GeV}$ energy, implying the production of a
$5\text{\,}\mathrm{MW}$ proton beam. Figure 33 shows the layout of the ESS
proton driver. The proposed power increase of the LINAC from
$5\text{\,}\mathrm{MW}$ to $10\text{\,}\mathrm{MW}$ will be realised by
increasing the pulse frequency from $14\text{\,}\mathrm{Hz}$ to
$28\text{\,}\mathrm{Hz}$, adding 14 more H- pulses of $50\text{\,}\mathrm{mA}$
to be accelerated to $2.5\text{\,}\mathrm{GeV}$, interleaved with the proton
pulses. The reason for accelerating H- ions is that they can be stripped of
their electrons to inject protons in a manner that increases the phase space
density for an existing proton beam, otherwise limited by Liouville’s theorem.
Each of the H- pulses will be chopped into four sub-pulses separated by gaps
of $\sim$$100\text{\,}\mathrm{\SIUnitSymbolMicro s}$ length. In addition,
there are $133\text{\,}\mathrm{ns}$ gaps every
$1.33\text{\,}\mathrm{\SIUnitSymbolMicro s}$, corresponding to the revolution
period in the ring. The need for these gaps is dictated by the rise-time
requirement of the extraction kicker magnets in the accumulator ring and the
time needed to reconfigure the ring for the injection of the next sub-pulse.

The total number of particles delivered to the accumulator ring will be
$8.9\times 10^{14}$ per pulse cycle (macro-pulse), divided into four batches
of $2.2\times 10^{14}$. Each batch is stacked in the accumulator ring over
about 600 turns, compressing the $3.5\text{\,}\mathrm{ms}$, including gaps, to
four $1.2\text{\,}\mathrm{\SIUnitSymbolMicro s}$ pulses, which are
subsequently extracted to the target. The acceleration of H- ions requires the
addition of a H- ion source to the side of the LINAC proton source and a
doubling of the front-end accelerator elements up to the point where the
proton and H- beam-lines are merged. For the H- production, the Penning source
[621] at ISIS-RAL [622] and the RF source at SNS [623, 624] were identified as
the most promising ion sources to meet the ESS$\nu$SB requirements.

Figure 33: Proton driver layout.

A LINAC-to-ring (L2R) transfer line has been designed to transport the
$2.5\text{\,}\mathrm{GeV}$ H- beam output from the upgraded high-$\beta$ line
(HBL) at the end of the LINAC to the Accumulator Ring [625]. Several
mechanisms of unwanted H- stripping mechanisms, that would lead to large beam
losses, have been considered in the design [626]. At beam energies greater
than ${\sim}$$100\text{\,}\mathrm{MeV}$, activation of machine components
could become a concern if the loss values would exceed acceptable limits. The
beam-loss will be kept below $1\text{\,}\mathrm{W}$/m, which ensures a maximum
value of 1 mSv/h ambient dose rate due to activation at
$30\text{\,}\mathrm{cm}$ from a surface of any given accelerator component,
after 100 days of irradiation and $4\text{\,}\mathrm{h}$ of cool-down [627].
Moreover, as mentioned before, several hardware modifications will need to be
applied on the ESS LINAC in order to make it able to produce the intense
neutrino beam. The modification programme includes, but is not limited to: the
upgrade of the low energy beam transport (LEBT), the medium energy beam
transport (MEBT), the radiofrequency quadrupole (RFQ), the drift-tube linac
(DTL) tank, the Modulator Capacitor and the Cooling System.

####  9.1.2 Accumulator Ring

There are several reasons for having an Accumulator ring to compress the
$2.86\text{\,}\mathrm{ms}$ ($3.5\text{\,}\mathrm{ms}$ including beam-free
gaps) long ESS linac pulse to $1.2\text{\,}\mathrm{\SIUnitSymbolMicro s}$. One
is that the background in the large Far Detectors caused by neutrinos from