New high-sensitivity searches for neutrons converting into antineutrons and/or sterile neutrons at the HIBEAM/NNBAR experiment at the European Spallation Source

According to Sakharov [7], there must be baryon number violating processes to explain the Universe's baryon asymmetry as observed today. Early grand unified theories (GUTs) such as SU(5) [58–61] that contained BNV do not provide a good source of baryogenesis. The original baryon asymmetry generated by such models conserves $\mathcal{B}-\mathcal{L}$ and violates $\mathcal{B}+\mathcal{L}$, just as in the SM, and any leftover asymmetry below the unification scale would be erased by electroweak sphaleron interactions. A more promising class of models attempting to explain the origin of matter are those focused on electroweak baryogenesis, which do not succeed in the SM but could work in some SM extensions (see, e.g. reference [62] and references therein). Alternatively, baryogenesis can be generated via leptogenesis [63], which utilizes the seesaw mechanism [64, 65] of neutrino masses and allows for an initial lepton asymmetry to be converted into a baryon asymmetry via the sphaleron processes [66]. The simplest examples of such models require the baryogenesis scale to be very high and are very hard to test experimentally. More specific lepto–baryogenesis models include νMSM [67, 68] and co-leptogenesis models via the neutrino interactions with sterile neutrinos from a dark sector [69–71].

A subset of weak-scale baryogenesis models have the attractive feature of being experimentally testable. Explicit UV-complete models featuring post-sphaleron baryogenesis (PSB) [22, 72, 73] use interactions that violate baryon number by two units and predict magnitudes of observable phenomena such as $n\to \bar{n}$ oscillation [21] periods. These models also connect the neutrino's Majorana mass to $n\to \bar{n}$ transformations and present an upper limit for the $n\to \bar{n}$ oscillation time which can be accessible at next-generation facilities like NNBAR at the ESS. Other simplified models that could realize PSB with a connection to neutrino masses and dark matter, while simultaneously giving rise to an observable $n\to \bar{n}$ rate, have been studied in references [23, 24]. Scenarios of co-baryogenesis with a dark sector have been discussed in references [17, 20, 74].

These searches represent dedicated probes of selection rules (${\Delta}\mathcal{B}=1,\mathbf{2};{\Delta}\mathcal{L}=0$), which fulfill a Sakharov condition but have been comparatively overlooked in the program of experiments probing fundamental symmetries and lepto/baryogenesis.

The existence of a dark sector, interacting primarily gravitationally with our familiar visible sector, has long been postulated as a means of explaining astronomical data [75]. When such a dark (sterile) sector is assumed to have particles having gauge interactions similar to our own SM interactions, one easily implies the existence of, e.g. sterile neutrinos and sterile baryons which could represent asymmetric dark matter induced by a baryon asymmetry in the dark sector. Self-interacting and dissipative characteristics of such dark matter would have interesting astrophysical implications [76–80]. In principle, observable portals onto such a sector can occur, i.e. mixing phenomena between any stable or meta-stable electrically neutral particles, allowing for conversion into a dark partner particle. For example, photons may become 'dark photons' via kinetic mixing [81], while neutrinos can oscillate into sterile neutrinos of the dark sector [82, 83]. The neutron represents another possible generic portal.

One of the simplest examples of a hidden sector is the theory of mirror matter, a dark sector represented by a replica of the SM (for reviews see [79, 84, 85], for a historical overview see [86]). The assumption of this minimal model forms the basis of the phenomenological framework for the sterile neutron transition searches considered in this work, though these searches have some sensitivity to a more generic dark sector. Forms of n → n' transitions have been proposed [87, 88], to which HIBEAM is sensitive, that can also shed light onto the apparent anomaly present between experimental free cold and ultracold neutron 'beam' and 'bottle' measurements of the neutron lifetime [89].

BNV is a generic feature of GUTs [90] and many other proposed extensions of the SM [23–27, 49, 51].

Classic BNV signatures include proton and bound neutron decay, mediated by four-fermion operators, and $n\to \bar{n}$ oscillations [8], mediated by six-quark operators [21, 91]. In SM effective field theory, the lowest orders of these operators have mass dimensions +6 and +9, respectively; also, in supersymmetric models these can take dimensions +4 and +5. Thus, if there were only one mass scale M_BNV characterizing BNV processes, $n\to \bar{n}$ oscillations would be more highly suppressed (like $1/{M}_{\text{BNV}}^{5}$) compared with proton and bound neutron decay, for which the effective Lagrangian would only involve a suppression by $1/{M}_{\text{BNV}}^{2}$. However, there is no good reason to assume that BNV processes correspond to a single scale nor is it known which processes nature has chosen should there be one BNV scale. There are a number of approaches where $n\to \bar{n}$ oscillations are the dominant manifestation of BNV, while proton decay is either absent or suppressed well below experimental limits [9, 21, 27, 92, 93]. Some early studies of $n\to \bar{n}$ oscillations include [9, 11–15, 21]; a recent review is [10].

Basic dimensional analysis based on the above considerations implies sensitivity to mass scales of $\mathcal{O}\left(10\right)-\mathcal{O}\left(1000\right)$ TeV, accessible via a precision $n\to \bar{n}$ oscillation search such as the experimental program proposed in this article. Such a scale is substantially in excess of the reach of current or planned colliders. This is complementary to the large-volume single nucleon decay experiments that are sensitive to a different set of BNV processes with scales near the GUT energy.

Examples of models predicting observable $n\to \bar{n}$ arising from BNV at TeV and PeV scales include R-parity violating supersymmetry scenarios [25, 26] and extra-dimensional models. Extra-dimensional scenarios arise from the leading candidate for quantum gravity, i.e. superstring theory [94], which predicts extra spatial dimensions beyond the three which are observed. Compactification radii characterizing these extra dimensions might be much larger than the Planck length. A model of this type [27, 28] provides an explicit example of how proton decay can be strongly suppressed, while $n\to \bar{n}$ transformations can occur at levels within reach of the new, high-sensitivity experiment proposed here. This is also true of an extra-dimensional model with a left–right gauge symmetry broken at the scale of $\mathcal{O}\left(1{0}^{3}\right)$ TeV [29].

Another topical issue in which the $n\to \bar{n}$ process may play a role concerns the origin of neutrino mass. There is a symbiosis between $n\to \bar{n}$ transitions and neutrinoless double β decays: they both violate $\mathcal{B}-\mathcal{L}$ (the anomaly-free SM symmetry) by two units and imply Majorana masses, and both processes are connected in unification models, such as the left–right-symmetric model [95–97] based on the gauge group G_LRS = SU(3)_c × SU(2)_L × SU(2)_R × U(1)_B−L . In these models, the spontaneous symmetry breaking of G_LRS to the SM is produced by the vacuum expectation value (VEV) of a Higgs field transforming as (1, 1, 3)₂ under G_LRS, so that its VEV breaks $\mathcal{B}-\mathcal{L}$ by two units. This gives rise to both an operator with ${\Delta}\mathcal{L}=0$ and ${\Delta}\mathcal{B}=2$, such as the six-quark operator mediating $n\to \bar{n}$ oscillations and to an operator such as the bilinear Majorana product of right-handed neutrinos, with ${\Delta}\mathcal{B}=0$ and ${\Delta}\mathcal{L}=2$ that is responsible for a seesaw mechanism leading to Majorana neutrino masses. Thus in theories with Majorana neutrino masses and quark–lepton unification, it is natural to expect both Majorana neutrinos as well as $n\to \bar{n}$ transitions [21]. In fact, there exist both left–right-symmetric and SO(10) models with observable $n\to \bar{n}$ oscillation where this connection is explicit; there has also been recent work on these connections within SU(5) effective field theory [98].

Setting aside specific theories of physics beyond the SM, the sphaleron interaction itself being a nine-quark–three-lepton interaction can be written as QQQQQQ QQQL LL. This implies that if any two of the following processes are seen, then the other should exist: $n\to \bar{n}$ transition (the first six-quark operator), proton decay (the second four-fermion operator) and ${\Delta}\mathcal{L}$ = 2, lepton number violation. The last term implies low energy processes such as neutrinoless double-beta decay and direct lepton number violation in the form of same sign charged lepton pairs [99] at hadron colliders, possibly even at the Large Hadron Collider (LHC). The neutrinoless double-beta decay can result from the neutrino Majorana mass or the new physics [97] that leads to same sign dileptons at colliders, and in the left–right symmetric model of neutrino mass there is a deep connection between the two processes [100].

In summary, together with the discovery of proton decay, an observation of $n\to \bar{n}$ oscillations, could, therefore, establish evidence for the Majorana nature of neutrinos and/or probe the theory behind neutrino Majorana mass. Equivalently, discoveries of $n\to \bar{n}$ oscillations and ${\Delta}\mathcal{L}$ = 2 lepton number violation would imply proton decay. Searches for $n\to \bar{n}$ oscillations thus play a key and complementary role in a wider experimental program of $\mathcal{B}$ and $\mathcal{L}$ violation searches [101].

In the SM frames, the neutron has only the Dirac mass term $m\bar{n}n$ which conserves $\mathcal{B}$. However, as mentioned in section 2.2, $n\to \bar{n}$ can proceed by effective six-quark (dimension 9) operators. These involve light quarks u and d and violate $\mathcal{B}$ by two units:

$$\begin{equation}{\mathcal{O}}_{{\Delta}\mathcal{B}=2}=\frac{1}{{\mathcal{M}}^{5}}{\left(udd\right)}^{2}\enspace +\enspace \mathrm{h}.\mathrm{c}.,\end{equation} \tag{ 1 }$$

with $\mathcal{M}$ being a large cutoff scale originated from new physics beyond the SM. This can induce a Majorana mass term:

$$\begin{equation}\frac{{{\epsilon}}_{n\bar{n}}}{2}\left({n}^{\mathrm{T}}Cn+\bar{n}C{\bar{n}}^{\mathrm{T}}\right)=\frac{{{\epsilon}}_{n\bar{n}}}{2}\left(\bar{{n}_{\mathrm{c}}}n+\bar{n}{n}_{\mathrm{c}}\right),\end{equation} \tag{ 2 }$$

where C is the charge conjugation matrix and ${n}_{\mathrm{c}}=C{\bar{n}}^{\mathrm{T}}$ stands for the antineutron field. ⁶⁹ Thus, the $n\to \bar{n}$ matrix element/mixing mass term ${{\epsilon}}_{n\bar{n}}$ depends on the scale of new physics:

$$\begin{equation}{{\epsilon}}_{n\bar{n}}=\frac{C{{\Lambda}}_{\text{QCD}}^{6}}{{\mathcal{M}}^{5}}=C{\left(\frac{500\enspace \mathrm{T}\mathrm{e}\mathrm{V}}{\mathcal{M}}\right)}^{5}{\times}7.7\cdot 1{0}^{-24}\enspace \mathrm{e}\mathrm{V},\end{equation} \tag{ 3 }$$

with C = O(1) being the model-dependent factor in the determination of the matrix element $\langle \bar{n}\vert {\mathcal{O}}_{{\Delta}\mathcal{B}=2}\vert n\rangle$. This mixing between the neutron and antineutron fields gives rise to the phenomenon of $n\to \bar{n}$ oscillation [8, 9]. The direct bound on $n\to \bar{n}$ oscillation time ${{\epsilon}}_{n\bar{n}}^{-1}={\tau }_{n\bar{n}}{ >}0.86{\times}1{0}^{8}$ s [37], i.e. ${{\epsilon}}_{n\bar{n}}{< }7.7{\times}1{0}^{-24}$ eV, corresponds to $\mathcal{M}{ >}500$ TeV or so. By improving the experimental sensitivity by two orders of magnitude, one could test the new physics above the PeV scale.

Conversion of $n\to \bar{n}$ can be understood as the evolution of a beam of initially pure neutrons

$$\begin{equation}\vert {\Psi}\left(t\right)\rangle ={\psi }_{n}\left(t\right)\hspace{2pt}{\psi }_{\bar{n}}\left(t\right)={\text{e}}^{-\text{i}\hat{\mathcal{H}}t}\vert {\Psi}\left(t=0\right)\rangle ,\hspace{25.0pt}\vert {\Psi}\left(t=0\right)\rangle =\vert n\rangle =1\hspace{2pt}0,\end{equation} \tag{ 4 }$$

described by the 2 × 2 Hamiltonian

$$\begin{equation}\hat{\mathcal{H}}=\left(\begin{matrix}\hfill {E}_{n}\hfill & \hfill {{\epsilon}}_{n\bar{n}}\hfill \\ \hfill {{\epsilon}}_{n\bar{n}}\hfill & \hfill {E}_{\bar{n}}\hfill \end{matrix}\right)\quad \end{equation} \tag{ 5 }$$

where E_n and ${E}_{\bar{n}}$ are the neutron and antineutron energies, respectively. While the neutron and antineutron masses are equal by CPT invariance, E_n and ${E}_{\bar{n}}$ are not generically equal due to the environmental effects which act differently on the neutron and antineutron states, as a presence of matter medium or magnetic fields, or perhaps some hypothetical fifth forces [104, 105].

The probability of finding an antineutron at a time t is given by ${P}_{n\bar{n}}\left(t\right)=\vert {\psi }_{\bar{n}}\left(t\right){\vert }^{2}$ or explicitly:

$$\begin{equation}{P}_{n\bar{n}}\left(t\right)=\frac{{{\epsilon}}_{n\bar{n}}^{2}}{{\left({\Delta}E/2\right)}^{2}+{{\epsilon}}_{n\bar{n}}^{2}}\enspace {\mathrm{sin}}^{2}\left[t\enspace \sqrt{{\left({\Delta}E/2\right)}^{2}+{{\epsilon}}_{n\bar{n}}^{2}}\right]\enspace {\mathrm{e}}^{-t/{\tau }_{n}},\end{equation} \tag{ 6 }$$

where ${\Delta}E={E}_{n}-{E}_{\bar{n}}$ and τ_n denotes the mean life of the free neutron. It thus becomes immediately clear that the probability of a conversion is suppressed when the energy degeneracy between neutron and antineutron is broken. In particular, for free neutrons, suppression occurs due to the interaction of the magnetic field (B ≃ 0.5 G on Earth) with the neutron and antineutron's magnetic dipole moments (${ \overrightarrow {\mu }}_{n}=-{ \overrightarrow {\mu }}_{\bar{n}}$), equivalent to ${\Delta}E/2=\vert { \overrightarrow {\mu }}_{n} \overrightarrow {B}\vert \approx \left(B/1\enspace \mathrm{G}\right){\times}1{0}^{-11}$ eV in equation (6). To prevent significant suppression of $n\to \bar{n}$ conversion, one must maintain the so called quasi-free regime |ΔE|t ≪ 1, which can be realized in vacuum in nearly zero magnetic field [106–108]. In this case, equation (6) reduces to:

$$\begin{equation}{P}_{n\bar{n}}\left(t\right)={{\epsilon}}_{n\bar{n}}^{2}{t}^{2}=\frac{{t}^{2}}{{\tau }_{n\bar{n}}^{2}}={\left(\frac{t}{0.1\enspace \mathrm{s}}\right)}^{2}{\left(\frac{1{0}^{8}\enspace \mathrm{s}}{{\tau }_{n\bar{n}}}\right)}^{2}{\times}1{0}^{-18},\end{equation} \tag{ 7 }$$

where ${\tau }_{n\bar{n}}=1/{{\epsilon}}_{n\bar{n}}$ is the characteristic oscillation time. Since in real experimental situations the neutron flight time is small, t ∼ 0.1 s or so, the exponential factor related to the neutron decay can be neglected in equation (6).

This necessitates magnetic shielding for searches utilizing a neutron beam [37, 109–111]. HIBEAM and NNBAR must employ such shielding, as will be discussed in section 8.1.

In the experiment performed at the ILL [37], the magnetic field was suppressed below 10 mG or so and the lower limit ${\tau }_{n\bar{n}}{ >}0.86{\times}1{0}^{8}$ s (90% CL) was obtained. In turn, this translates into upper limit ${{\epsilon}}_{n\bar{n}}{< }7.7{\times}1{0}^{-24}$ eV, which still stands as the strongest limit on the $n\to \bar{n}$ mass mixing obtained with free neutrons. The effects of imperfect vacuum (residual gas pressure) on the observation of neutron to antineutron transformation were discussed in references [37, 112–114].

As for bound neutrons in a nucleus, the potential energy difference experienced between a neutron and antineutron in the strong nuclear field (ΔE ∼ 10–100 MeV, depending on nuclei) introduces a suppression of ∼10⁻³¹ with respect to the conversion of a free neutron. This of course inhibits the conversion of neutrons bound in nuclei, with sensitive searches only possible with large-volume detectors [53, 115–121], such as Super-Kamiokande [53, 57], SNO [122], DUNE [123–125] or Hyper-Kamiokande [126]. The comparatively large number of neutrons permits searches with currently complementary limits. However, event identification is obscured by atmospheric backgrounds, intranuclear scattering of the decay products and other nuclear physics effects.

A limit on $n\to \bar{n}$ conversion time in a specific nucleus (T) can be related to that of a free neutron (${\tau }_{n\bar{n}}$) via a nuclear suppression factor, R ∼ 10²² s⁻¹, which can be calculated with phenomenological nuclear models [124, 127–133] and predict quadratic scaling such that:

$$\begin{equation}T=R\cdot {\tau }_{n\bar{n}}^{2}\sim {\left(\frac{1{0}^{8}\enspace \mathrm{s}}{{\tau }_{n\bar{n}}}\right)}^{2}{\times}1{0}^{31}\enspace \mathrm{y}\mathrm{r}.\end{equation} \tag{ 8 }$$

At present, the strongest limit obtained by Super-Kamiokande [57] for oxygen is ${\tau }_{n\bar{n}}{ >}4.7{\times}1{0}^{8}$ s, or equivalently ${{\epsilon}}_{n\bar{n}}{< }2.5{\times}1{0}^{-24}$ eV. Super-Kamiokande has also carried out searches for ΔB = −2 dinucleon decays to specific multi-meson and leptonic final states [54–56, 134]. More details on current limits and future sensitivities are in section 4.1.

Caution is required when comparing limits and sensitivities for free and bound neutron searches. Calculations relating T and ${\tau }_{n\bar{n}}$ rely on underlying model assumptions, such as a point-like conversion process, while the physics behind $n\to \bar{n}$ conversion is a priori unknown ⁷⁰ . The visibility of a signal in a bound neutron search could therefore be arbitrarily suppressed compared to a free search, or vice versa. For example, a recently proposed model of low-scale BNV contains the possibility of a suppressed (or even enhanced) bound neutron conversion probability [33]. There can be also some environmental effects that can affect free $n\to \bar{n}$ oscillations even if the magnetic field is properly shielded. These effects can be related, e.g. with long-range fifth-forces induced by very light $\mathcal{B}-\mathcal{L}$ baryophotons. Present high-sensitivity limits on such forces [137] with Yukawa radius comparable to the Earth's radius or to Sun–Earth distance still allow significant contribution to the neutron–antineutron energy level splitting, which in fact can be as large as ΔE ∼ 10⁻¹¹ eV or so [104, 105].

Consideration of free and bound neutron searches is thus complementary: neither makes the other redundant, and indeed they require one another to help constrain the underlying physical process.

Within the framework of an assumed ultraviolet extension of the SM that features $n-\bar{n}$ transitions, one has a prediction for the coefficients of the various types of six-quark operators in the resultant low-energy effective Lagrangian, and the next step in obtaining a prediction for the $n-\bar{n}$ transition rate of free neutrons is to estimate the matrix elements of these six-quark operators between |n⟩ and $\vert \bar{n}\rangle$ states. Since the six-quark operators have dimension 6, their matrix elements are of the form ${{\Lambda}}_{\text{eff}}^{6}$. The relevant scale is set by the quantum chromodynamics (QCD) confinement scale, Λ_QCD ∼ 0.25 GeV, so one expects, roughly speaking, that the matrix elements are of order $\sim {{\Lambda}}_{\text{QCD}}^{6}\simeq 2.4{\times}1{0}^{-4}\enspace {\mathrm{G}\mathrm{e}\mathrm{V}}^{6}$, and this expectation is borne out by both early estimates using the MIT bag model [14, 15] and recent calculations using lattice QCD (LQCD) [138–142], including approximate assessments of modeling uncertainties. The LQCD results in [139, 140] indicate that, for most operators, the corresponding Λ_eff is larger, by ∼10%–40%, than the Λ_eff characterizing the MIT bag model results, i.e. a factor ∼2–8 for ${{\Lambda}}_{\text{eff}}^{6}$, and thus for the matrix elements themselves. This suggests that overall experimental sensitivities may reach higher than previously expected [138]. This being said, direct constraint of PSB and its predicted upper bound on ${\tau }_{n\to \bar{n}}$ [73] is slightly different, as this limit is derived not from tree-level amplitudes but instead from loop diagrams involving W-boson exchange. In [73], larger MIT bag-model estimates are used, and so the LQCD matrix element for this particular amplitude appears smaller by some ∼15% [140]; this leads to an expectation that the upper limit for ${\tau }_{n\to \bar{n}}$ will be shifted slightly up by roughly the same proportion. The community's integration of this new knowledge is continuing, and still more accurate predictions are being actively discussed and developed [143]. Similar computational methods may eventually advance peripheral modeling of secondary processes, such as the annihilation itself and background interactions.

A key attribute of a traditional free $n\to \bar{n}$ conversion search is a neutron beam focused onto an annihilation target to minimize interactions with, e.g. a guide wall. The difference in neutron and antineutron interactions with wall material have been assumed to act as a large potential difference, suppressing the oscillation. The interaction can be seen as destroying any wave function component, which effectively 'resets the clock' for the oscillation time measurement. With this assumption, only the neutron's free-flight time since last wall interaction contributes to the probability to find an antineutron, necessitating a large-area experimental apparatus in practice.

An almost free $n\to \bar{n}$ oscillation search has recently been proposed [144, 145] in which one allows slow, cold neutrons (and antineutrons) (with energies of <10⁻² eV) to reflect from effective n/$\bar{n}$ optical mirrors. Although the reflection of n/$\bar{n}$ had been considered in the 1980s for ultra cold neutrons (UCNs) [146–149] and recently in [145] for proposed experiments to constrain ${\tau }_{n\to \bar{n}}$, the authors now extended this approach to higher energies, namely where nominally cold, initially collimated neutrons can be reflected from neutron guides when their transverse velocities with respect to the wall are similarly very or ultracold. Conditions for suppressing the phase difference for n and $\bar{n}$ were studied, and the required low transverse momenta of the n/$\bar{n}$ system was quantified, leading to new suggestions for the nuclei composing the reflective guide material. It was shown that, over a broad fraction of phase space, the relative phase shift of the n and $\bar{n}$ wave function components upon reflection can be small, while the probability of coherent reflection of the n/$\bar{n}$ system from the guide walls can remain high. The theoretical uncertainties associated with a calculation of the experimental sensitivity, even in the absence of direct measurements of low energy $\bar{n}$ scattering amplitudes, can be small. A new calculation of the low energy antineutron–nucleus scattering amplitudes needed to evaluate these uncertainties has recently appeared [150].

An important consequence would be that the conversion probability depends on the neutron's total flight time, as wall interactions no longer reset the clock. Such an experimental mode relaxes some of the constraints on free n oscillation searches, and in principle allows a much higher sensitivity to be achieved at reduced complexity and costs. The above represents a new idea from within the community which requires a program of simulation and experimental verification. While this does not form part of the current core plan for the HIBEAM/NNBAR experiment, it is considered as a promising future research direction, and experimental verification of this concept is under investigation.

Though the HIBEAM searches are generic in nature, the mixing of n and its sterile twin ${n}^{\prime }$ is considered here within the paradigm of a parallel gauge sector in a mirror matter model [17, 30]. There are a range of possible conversion processes which can be explored experimentally, motivating a suite of searches outlined in section 7.

In addition to external fields and interactions in the standard sector, the possibility of equivalent fields in the sterile sector must be taken into consideration. There can exist also some hypothetical forces between ordinary and sterile sector particles which can be induced e.g. by the photon kinetic mixing with dark photon [81, 151–153], or by new gauge bosons interacting with particles of both sectors as e.g. common $\mathcal{B}-\mathcal{L}$ gauge bosons [104] or common flavor gauge bosons of family symmetry [154–156]. The respective forces can provide portals for direct detection of dark matter components from a parallel sterile sector and give a possibility for identification of their nature [157–159]. In addition, flavor gauge bosons can induce mixing between neutral ordinary particles and their sterile partners and induce oscillations, e.g. between kaons and sterile kaons, conversion of muonium into hidden muonium, etc [155, 156].

The possibility of neutron–mirror neutron mass mixing ${\alpha }_{n{n}^{\prime }}\bar{n}{n}^{\prime }+\mathrm{h}.\mathrm{c}.$ was proposed in [17]. It can be induced by six-fermion effective operators $\frac{1}{{M}^{5}}\left(\bar{u}\bar{d}\bar{d}\right)\left({u}^{\prime }{d}^{\prime }{d}^{\prime }\right)$ similar to operator (1) but involving three ordinary quarks and three quarks of the sterile (mirror) sector. This mixing violates conservations of both baryon number and mirror baryon number (${\Delta}\mathcal{B}=1$, ${\Delta}{\mathcal{B}}^{\prime }=-1$) but it conserves the combination $\mathcal{B}+{\mathcal{B}}^{\prime }$. The mixing mass α_nn' can be estimated as:

$$\begin{equation}{\alpha }_{n{n}^{\prime }}=\frac{C{{\Lambda}}_{\text{QCD}}^{6}}{{M}^{5}}={C}^{2}{\left(\frac{10\enspace \mathrm{T}\mathrm{e}\mathrm{V}}{M}\right)}^{5}{\times}2.5{\times}1{0}^{-15}\enspace \mathrm{e}\mathrm{V}.\end{equation} \tag{ 9 }$$

It was shown that no direct, astrophysical or cosmological effects forbid that n → n' oscillation time τ_nn' = 1/α_nn' can be smaller than the neutron decay time, and in fact it can be as small as a second. So rapid n → n' oscillations could have interesting implications for the propagation of ultra-high CRs [30, 31] or for neutrons from solar flares [160]. Thus, the effective scale M of underlying new physics can be of a few TeV, with direct implications for the search at the LHC and future accelerators. Effects of n → n' oscillation can be directly observed in experiments searching for anomalous neutron disappearance (n → n') and/or regeneration (n → n' → n) processes [17]. Experimental sensitivities of such searches with ultracold and cold neutrons were discussed in references [161, 162].

The Hamiltonian for n → n' is given in equation (10). The presence of a static magnetic moment shifts the n and n's total energies. The Hamiltonian is expressed for the general case of neutrons propagating in magnetic fields B (of the standard sector) and B' (of the sterile sector); the former of these is generated by the magnetic poles of the Earth, the latter by hypothetical ionization and flow of gravitationally captured dark material in and around the Earth [18]. Such an accumulation could occur due to ionized gas clouds of sterile atoms captured by the Earth, e.g. due to photon–sterile photon kinetic mixing; present experimental and cosmological limits on such mixing [163, 164] and geophysical limits [165] still allow the presence of a relevant amount of sterile material on the Earth [166]. Then a sterile magnetic field can be induced by the drag of dark electrons due to the Earth's rotation via the mechanism described in [167], which can be enhanced through the dynamo effect [18]. In addition to the static magnetic moments, ${ \overrightarrow {\mu }}_{n}$ and ${ \overrightarrow {\mu }}_{{n}^{\prime }}$, and unlike for the $n\to \bar{n}$ transition ⁷¹ , transition magnetic moments (TMMs) [74] may also be present in the off-diagonal TMMs ${ \overrightarrow {\mu }}_{n{n}^{\prime }}=\kappa \overrightarrow {{\mu }_{n}}$ between the neutron and sterile neutron term ⁷² .

As analogous contributions to the Hamiltonian, TMMs can be seen as quantities which are as fundamental as the more familiar static magnetic moments. TMMs contribute to the mixing via the interaction of new physics processes with the external magnetic fields, leading to terms $\kappa { \overrightarrow {\mu }}_{n} \overrightarrow {B}$ and ${\kappa }^{\prime } \overrightarrow {{\mu }_{n}}{ \overrightarrow {B}}^{\prime }$, where the dimensionless parameter κ ≪ 1 measures the magnitudes of the TMM in units of neutron magnetic moment μ_n .

$$\begin{equation}\hat{\mathcal{H}}=\left(\begin{matrix}\hfill {m}_{n}+{ \overrightarrow {\mu }}_{n} \overrightarrow {B}\hfill & \hfill {\alpha }_{n{n}^{\prime }}+\kappa { \overrightarrow {\mu }}_{n} \overrightarrow {B}+{\kappa }^{\prime }{ \overrightarrow {\mu }}_{n}{ \overrightarrow {B}}^{\prime }\hfill \\ \hfill {\alpha }_{n{n}^{\prime }}+\kappa { \overrightarrow {\mu }}_{n} \overrightarrow {B}+{\kappa }^{\prime }{ \overrightarrow {\mu }}_{n}{ \overrightarrow {B}}^{\prime }\hfill & \hfill {m}_{{n}^{\prime }}+{ \overrightarrow {\mu }}_{{n}^{\prime }} \overrightarrow {{B}^{\prime }}\hfill \\ \hfill \hfill \end{matrix}\right).\end{equation} \tag{ 10 }$$

Beyond removing exponential decay, a number of simplifications can be made to the picture of n → n' mixing described above. In the simplest model, it is assumed that n and n' share degeneracies such as m_n = m_n', $\vert \overrightarrow {{\mu }_{n}}\vert =\vert \overrightarrow {{\mu }_{n}^{\prime }}\vert$ and κ = κ'. The magnitude, direction and time dependence of the mirror magnetic field ${ \overrightarrow {B}}^{\prime }$ is a priori unknown.

If only n → n' mass mixing α_nn' is present, i.e. assuming for the moment that κ = 0, then the probability of n → n' oscillation at time t is given by [18, 45]:

$$\begin{equation}{P}_{n{n}^{\prime }}\left(t\right)=\frac{{\alpha }_{n{n}^{\prime }}^{2}\enspace {\mathrm{cos}}^{2}\enspace \frac{\beta }{2}}{{\left(\omega -{\omega }^{\prime }\right)}^{2}}\enspace {\mathrm{sin}}^{2}\left[\left(\omega -{\omega }^{\prime }\right)t\right]+\frac{{\alpha }_{n{n}^{\prime }}^{2}\enspace {\mathrm{sin}}^{2}\enspace \frac{\beta }{2}}{{\left(\omega +{\omega }^{\prime }\right)}^{2}}\enspace {\mathrm{sin}}^{2}\left[\left(\omega +{\omega }^{\prime }\right)t\right],\end{equation} \tag{ 11 }$$

where 2ω = |μ_n B| and 2ω' = |μ_n B'|, β is the angle between the directions of magnetic fields $\overrightarrow {B}$ and ${ \overrightarrow {B}}^{\prime }$, and the contribution of α_nn' in the oscillation frequencies is neglected assuming that α_nn' < |ω − ω'|. If |ω − ω'|t ≫ 1, the oscillations can be averaged in time and one obtains:

$$\begin{equation}{\bar{P}}_{n{n}^{\prime }}=\frac{{\alpha }_{n{n}^{\prime }}^{2}\enspace {\mathrm{cos}}^{2}\enspace \frac{\beta }{2}}{2{\left(\omega -{\omega }^{\prime }\right)}^{2}}\enspace +\enspace \frac{{\alpha }_{n{n}^{\prime }}^{2}\enspace {\mathrm{sin}}^{2}\enspace \frac{\beta }{2}}{2{\left(\omega +{\omega }^{\prime }\right)}^{2}}.\end{equation} \tag{ 12 }$$

In particular, if B' = 0 (i.e. ω' = 0), from (11) the expression ${P}_{n{n}^{\prime }}\left(t\right){< }{\left({\alpha }_{n{n}^{\prime }}/\omega \right)}^{2}$ is obtained if ωt > 1, and ${P}_{n{n}^{\prime }}\left(t\right)\approx {\left({\alpha }_{n{n}^{\prime }}t\right)}^{2}$ if ωt ≪ 1.

As B approaches B', |ω − ω'| decreases and the probability P_nn'(t) resonantly increases. In a quasi-free regime, when |ω − ω'|t ≫ 1, the probability reaches the value:

$$\begin{equation}{P}_{n{n}^{\prime }}\left(t\right)\approx \frac{1}{2}{\left({\alpha }_{n{n}^{\prime }}t\right)}^{2}\enspace {\mathrm{cos}}^{2}\enspace \frac{\beta }{2}={\mathrm{cos}}^{2}\enspace \frac{\beta }{2}{\left(\frac{t}{0.1\enspace \mathrm{s}}\right)}^{2}{\left(\frac{1\enspace \mathrm{s}}{{\tau }_{n{n}^{\prime }}}\right)}^{2}{\times}5{\times}1{0}^{-3},\end{equation} \tag{ 13 }$$

where τ = 1/α_nn' is the characteristic n → n' oscillation time for free neutrons in a field-free vacuum. Therefore, this leads to a situation where the n → n' oscillation probability non-trivially depends on the value (and direction) of the magnetic field. This effect can be observed in experiments searching for anomalous neutron disappearance (n → n') and regeneration (n → n' → n) [17]. Experimental sensitivities of such searches with cold and ultracold neutrons were discussed in reference [161].

Several dedicated experiments searching for n → n' oscillation with UCNs were performed in the last decade [40–46]. Under the hypothesis that there is no mirror magnetic field on the Earth, i.e. B' = 0, the strongest lower limit τ_nn' > 414 s (90% CL) was obtained by comparing the UCN losses in zero (B < 10⁻³ G) and non-zero (B = 0.02 G) magnetic fields [41]. However, this limit becomes invalid in the presence of B'. Lower limits on τ_nn' and ${\tau }_{n{n}^{\prime }}/\sqrt{\mathrm{cos}\enspace \beta }$ in the presence of non-vanishing B' following from experiments [40–46] are summarized in reference [46]. In fact, some experiments show deviations from the null hypothesis, which may point toward τ_nn' ∼ 10 s and B' ∼ 0.1 G. For B' > 0.5 G or so, an oscillation time as small as 1 s is not excluded [46].

In the case of TMM-induced n → n' transition, the average oscillation probability becomes [88]:

$$\begin{equation}{\bar{P}}_{n{n}^{\prime }}=\frac{2{\kappa }^{2}{\left(\omega +{\omega }^{\prime }\right)}^{2}{\mathrm{cos}}^{2}\enspace \frac{\beta }{2}}{{\left(\omega -{\omega }^{\prime }\right)}^{2}}\enspace +\enspace \frac{2{\kappa }^{2}{\left(\omega -{\omega }^{\prime }\right)}^{2}{\mathrm{cos}}^{2}\enspace \frac{\beta }{2}}{{\left(\omega +{\omega }^{\prime }\right)}^{2}}.\end{equation} \tag{ 14 }$$

The limits on parameter κ and the TMM κμ_n that can be obtained by data analysis of these experiments [40–46] are given in reference [88]. In the case when α_nn' and κ are both present, the average probability of n → n' transition is given just by a sum of terms (12) and (14).

In the following, we choose ${ \overrightarrow {B}}^{\prime }$ to be zero, leaving only three parameters determining the probability of the n → n' process: α_nn', κ and $\overrightarrow {B}$. Figure 1 illustrates the interplay between these parameters and their impact on the oscillation, taking α_nn' = 5.68 × 10⁻¹⁸ eV (τ = 500 s). For TMM-only transitions (when α_nn' = 0), one has:

$$\begin{equation}{P}_{n{n}^{\prime }}=2{\kappa }^{2}.\end{equation} \tag{ 15 }$$

Figure 1 (right) compares the fraction of converted neutrons after having traveled 25 m in a vacuum for the case of a TMM term and non-zero mass mixing versus zero mass mixing.

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The TMM can also lead to an enhanced n → ${n}^{\prime }$ transformation in a gas atmosphere due to the creation of a positive Fermi potential along the neutron path [88]. A constant magnetic field $\overrightarrow {B}$ in the flight volume can be chosen such that for one polarization of neutron it will provide a negative magnetic potential compensating the positive Fermi potential of the gas: ${V}_{\mathrm{F}}= \overrightarrow {\mu } \overrightarrow {B}$. Thus, for example, the Fermi potential of air at normal temperature and pressure corresponds to a constant magnetic field of ∼10 G. Continuing to assume that |B'| = 0, the oscillation Hamiltonian becomes:

$$\begin{equation}\mathcal{H}=\left(\begin{matrix}\hfill {V}_{\mathrm{F}}-\mu B\hfill & \hfill {\alpha }_{n{n}^{\prime }}+\kappa \mu B\hfill \\ \hfill {\alpha }_{n{n}^{\prime }}+\kappa \mu B\hfill & \hfill 0\hfill \end{matrix}\right).\quad \end{equation} \tag{ 16 }$$

With a zero diagonal term (in the resonance), this corresponds to a pure oscillation with probability:

$$\begin{equation}{P}_{n{n}^{\prime }}={\left({\alpha }_{n{n}^{\prime }}+\kappa \mu B\right)}^{2}{t}^{2}.\end{equation} \tag{ 17 }$$

The probability due to the mass mixing term here is enhanced by the term due to the TMM that is proportional to field B.

Sections 3.1 and 3.3 address the transformations of $n\to \bar{n}$ and n → n', respectively. 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 references [74, 170]. It is essential to test the full range of conversions between the sectors: $n\to \left\{{n}^{\prime },{\bar{n}}^{\prime }\right\},\bar{n}\to \left\{{n}^{\prime },{\bar{n}}^{\prime }\right\}$, ${n}^{\prime }\to \left\{n,\bar{n}\right\}$ and ${\bar{n}}^{\prime }\to \left\{n,\bar{n}\right\}$.

In principle, a transformation to four states mixed in the n, $\left(n,\bar{n},{n}^{\prime },{\bar{n}}^{\prime }\right)$, in free space without any fields can be described by the symmetric Hamiltonian:

$$\begin{equation}\hat{\mathcal{H}}=\left(\begin{matrix}\hfill {m}_{n}+{ \overrightarrow {\mu }}_{n} \overrightarrow {B}\hfill & \hfill {\varepsilon }_{n\bar{n}}\hfill & \hfill {\alpha }_{n{n}^{\prime }}\hfill & \hfill {\alpha }_{n{\bar{n}}^{\prime }}\hfill \\ \hfill {\varepsilon }_{n\bar{n}}\hfill & \hfill {m}_{n}-{ \overrightarrow {\mu }}_{n} \overrightarrow {B}\hfill & \hfill {\alpha }_{n{\bar{n}}^{\prime }}\hfill & \hfill {\alpha }_{n{n}^{\prime }}\hfill \\ \hfill {\alpha }_{n{n}^{\prime }}\hfill & \hfill {\alpha }_{n{\bar{n}}^{\prime }}\hfill & \hfill {m}_{{n}^{\prime }}+{ \overrightarrow {\mu }}_{{n}^{\prime }}{ \overrightarrow {B}}^{\prime }\hfill & \hfill {\varepsilon }_{n\bar{n}}\hfill \\ \hfill {\alpha }_{n{\bar{n}}^{\prime }}\hfill & \hfill {\alpha }_{n{n}^{\prime }}\hfill & \hfill {\varepsilon }_{n\bar{n}}\hfill & \hfill {m}_{{n}^{\prime }}-{ \overrightarrow {\mu }}_{{n}^{\prime }}{ \overrightarrow {B}}^{\prime }\hfill \end{matrix}\right).\end{equation} \tag{ 18 }$$

Here, ${\varepsilon }_{n\bar{n}}$ is the $n\bar{n}$ Majorana mass mixing parameterand α_nn' and ${\alpha }_{n{\bar{n}}^{\prime }}$ are mass mixing parameters for nn' and for $n{\bar{n}}^{\prime }$ correspondingly. In the following we neglect possible TMM terms between $n,\bar{n}$ and ${n}^{\prime },{\bar{n}}^{\prime }$ states and assume m_n' = m_n , μ_n' = μ_n .

Thus, in this case, the final state antineutron can be a result of the classical $n\to \bar{n}$ with mixing mass amplitude ɛ, and with baryon number change ${\Delta}\mathcal{B}=-2$ and probability ${P}_{n\to \bar{n}}={\varepsilon }_{n\bar{n}}^{2}{t}^{2}$. It can also arise due to the second-order oscillation process: $n\to {n}^{\prime }\to \bar{n}$ with amplitude (${\alpha }_{n{n}^{\prime }}{\alpha }_{n{\bar{n}}^{\prime }}$) or $n\to {\bar{n}}^{\prime }\to \bar{n}$ with amplitude $\left({\alpha }_{n{\bar{n}}^{\prime }}{\alpha }_{n{n}^{\prime }}\right)$ plus interference of all three channels. If ${\varepsilon }_{n\bar{n}}$ is very small and α_nn' and ${\alpha }_{n{\bar{n}}^{\prime }}$ are relatively large, then $n\to \bar{n}$ could be observed for a non-zero sterile magnetic field. However, neither previous limits on free $n\to \bar{n}$ oscillation from experiments in which the magnetic field was suppressed [37], nor nuclear stability limits from $n\to \bar{n}$ conversion in nuclei [53], would be valid for this scenario, since a fixed field $\overrightarrow {B}$ compensating for the magnetic field in the sterile sector would be needed to allow the full process $n\to {n}^{\prime }\to \bar{n}$ to proceed. In fact, $n\to \bar{n}$ conversion in free neutron experiments can emerge as second-order process induced by n → n' and n → n' conversions, with the probability ${P}_{n\bar{n}}\left(t\right)\simeq {P}_{n{n}^{\prime }}\left(t\right){P}_{n{\bar{n}}^{\prime }}\left(t\right)$. Existing limits allow the oscillation times τ_nn' and ${\tau }_{n{\bar{n}}^{\prime }}$ to be as small as 1 ÷ 10 s (for a summary of present experimental situation see [46]). Therefore, by properly tuning the value of magnetic field B resonantly close to B' (with precision of mG or so) and thus achieving the quasi-free regime, the probability of induced $n\to \bar{n}$ oscillation can be rendered as large as:

$$\begin{equation}{P}_{n\bar{n}}\left(t\right)=\frac{1}{4}{\alpha }_{n{\bar{n}}^{\prime }}^{2}{\alpha }_{n{\bar{n}}^{\prime }}^{2}{t}^{4}\enspace {\mathrm{sin}}^{2}\enspace \beta =\frac{{\mathrm{sin}}^{2}\enspace \beta }{4}{\left(\frac{t}{0.1\enspace \mathrm{s}}\right)}^{4}{\left(\frac{1{0}^{2}\enspace {\mathrm{s}}^{2}}{{\tau }_{n{n}^{\prime }}{\tau }_{n{\bar{n}}^{\prime }}}\right)}^{2}{\times}1{0}^{-8},\end{equation} \tag{ 19 }$$

where β is an (unknown) angle between the directions of $\overrightarrow {B}$ and ${ \overrightarrow {B}}^{\prime }$ [170]. Hence, the probability of induced $n\to \bar{n}$ transition can be several orders of magnitude larger than the present sensitivity in direct $n\to \bar{n}$ conversion (7). Once again, for achieving such enhancement, 10 orders of magnitude or perhaps more, the magnetic field should not be suppressed but one must scan over its values and directions for finding the resonance when magnitudes B ≈ B' and angle ϕ is non-zero. In addition, different from direct $n\bar{n}$ mixing, $n\to \bar{n}$ transitions induced via n → n' and n → n' mixings has a tiny effect on the stability of nuclei [170].

The distribution of final states following $\bar{n}$ annihilation in target nuclei is of critical importance for understanding the needs of the annihilation detector system for the $n\to \bar{n}$ (and $n\to {n}^{\prime }\to \bar{n}$) searches. To date, the target material has been ¹²C, with four to five pions in the final state, but their intranuclear origin and their cascade dynamics thereafter can be significant. Studies of these effects have been made [124, 171, 172]. In this approach, a model of elementary $\bar{p}N$ annihilation is used, described in detail in [124, 171], taking into account ∼100 annihilation channels for $\bar{p}p$ and ∼80 channels for $\bar{p}n$, including heavy resonances. The simulations of the elementary annihilation agree well with $\bar{p}$–p interaction data sets (for instance, see table 2 of [124]). For $\bar{n}N$ annihilation, it is assumed that annihilation channels for $\bar{n}n$ are identical to $\bar{p}p$, and annihilation channels for $\bar{n}p$ are charge-conjugated to $\bar{p}n$. Thus, annihilation processes can indeed be considered for $\bar{n}p$ and $\bar{n}n$. Further computations model the particle transport through the nuclear medium (final state interactions). The proposed model for $\bar{n}{\;}^{12}\mathrm{C}$ was tested on available experimental data sets from $\bar{p}{\;}^{12}\mathrm{C}$ annihilations at rest, showing good agreement.

Table 1 shows the simulated and measured particle multiplicities following $\bar{p}{\;}^{12}\mathrm{C}$ interactions based on 10 000 Monte Carlo events. Pionic states dominate after the decay of heavy resonances and are in good agreement with experimental data. The total energy of the final state particles is also shown, for which the measurement is well reproduced by the calculations.

Table 1. A list of average multiplicities M from experimental data and simulations taking into account $\bar{p}$ annihilation branching ratios [124, 172] while also considering intranuclear (anti)nucleon potentials with associated nucleon mass defects and nuclear medium response. Based on simulations of 10 000 events. Measurements of proton and neutron multiplicities were not made.

	M(π)	M(π+)	M(π−)	M(π0)	Etot (MeV)	M(p)	M(n)
$\bar{p}\mathrm{C}$ experiment	4.57 ± 0.15	1.25 ± 0.06	1.59 ± 0.09	1.73 ± 0.10	1758 ± 59	—	—
$\bar{p}\mathrm{C}$ simulation	4.60	1.22	1.65	1.73	1762	0.96	1.03

The momentum distribution of positively charged pions is shown in figure 2. The momentum peaks around ∼250 MeV, albeit within a broad distribution which extends up to around 1000 MeV. The data are reasonably well described by the simulation. The figure also shows the distribution of kinetic energies of protons. As before, the data are well described. The contribution from evaporative protons is also shown and is seen to correspond to the data quite well even toward lower values of kinetic energy (∼20 MeV). Given the agreement with the data observed in figure 2, there is some measure of confidence in the simulations of extranuclear $\bar{n}{\;}^{12}\mathrm{C}$ final states.

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We see from figure 3 the expected final state π^0,± counts and momentum spectra arising from a slow $\bar{n}$ annihilating on ¹²C nucleus. One expects approximately four to five pions of various charges to enter the detector, where it is critically important to identify π⁰ species due to their proportionally high population. Because these are so short lived, it is incumbent on the detector to capably reconstruct the π⁰, as well as track π^±, over a rather large range of momenta.

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Figure 4 shows the total momentum of the final state system of emitted mesons and photons versus the system's total invariant mass. Owing to nuclear effects (final state interactions, rescattering, absorption), the final state invariant mass distribution for mesons and photons can fall to less than 1 GeV, lower than would be expected for a naive $\bar{n}N$ annihilation at around 1.9 GeV. The figure also shows the distribution of invariant mass arising only from original annihilation mesons before and after transport. The kinematic distributions shown in figures 2–4 have implications for the detection strategy of annihilation events, particularly affecting the expected signal efficiency.

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As illustrated in figure 5, free (or extranuclear) searches consist of a beam of focused free neutrons propagating in field-free (or quasi-free) regions to an annihilation detector at which any antineutrons would annihilate with a thin target, giving rise to a final state of charged pions and photons. Searches for free $n\to \bar{n}$ oscillation have taken place at the Pavia Triga Mark II reactor [109, 110] and at the ILL [37, 111]. The latter ILL search [37] provides the most competitive limit for the free neutron oscillation time: ∼8.6 × 10⁷ s.

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The figure of merit (FOM) of sensitivity for a free $n\to \bar{n}$ search is best estimated not by the oscillation time sensitivity but by the quantity below:

$$\begin{equation}\mathrm{F}\mathrm{O}\mathrm{M}=\sum\limits _{i}{N}_{{n}_{i}}\cdot {t}_{{n}_{i}}^{2}\sim \langle {N}_{n}\cdot {t}_{n}^{2}\rangle ,\end{equation} \tag{ 20 }$$

where, for velocity spectrum bins i, ${N}_{{n}_{i}}$ is the number of neutrons per unit time reaching the annihilation detector after ${t}_{{n}_{i}}$ seconds of flight through a magnetically protected, quasi-free conditioned vacuum region. As equation (7) shows, the probability of a conversion is proportional to the (transit time)². Thus $\mathrm{F}\mathrm{O}\mathrm{M}=\langle {N}_{n}\cdot {t}_{n}^{2}\rangle$ is proportional to the approximate number of the conversions per unit time in a neutron beam which impinge on a target ⁷³ .

A high-precision search therefore requires a large flux of slow neutrons produced at a low emission temperature which are allowed to propagate over a long time prior to allow conversions to antineutrons. As shown in the subsequent sections, these conditions are satisfied in searches at the ESS.

Searches for $n\to \bar{n}$ in bound neutrons in large volume detectors look for a signature of pions and photons consistent with a $\bar{n}N$ annihilation event inside a nucleus, as illustrated in figure 5. Searches have taken place at Homestake [115], KGF [116], NUSEX [117], IMB [118], Kamiokande [119], Frejus [120], Soudan-2 [121], the Sudbury Neutrino Observatory [122] and Super-Kamiokande [53, 54]. A signature of pions and photons consistent with an $\bar{n}N$ annihilation event was sought, with Super-Kamiokande providing the most competitive search, for which an inferred free neutron oscillation time lower limit of ∼2.7 × 10⁸ s was obtained. Super-Kamiokande has also searched for dinucleon decays to specific hadronic final states, such as nn → 2π⁰ and np → π⁺ π⁰, as well as dinucleon decays into purely leptonic and lepton + photon final states [55, 176, 177]. Further limits on BNV decays have been obtained by relating these types of decays [29, 56, 134, 178].

Two main experimental approaches are used to search for sterile neutrons: measurements of neutrons trapped in a UCN bottle and measurements of beam neutrons ⁷⁴ . The principles behind these approaches are illustrated in figure 6. There would be anomalous loss of neutrons from the UCN trap via their conversion to sterile neutrons (figure 6(a)). With beam neutrons, experiments can look for the regeneration of neutrons following a beam stop (figure 6(b)), an unexplained disappearance of neutron flux (figure 6(c)) and $n\to \bar{n}$ via a sterile neutron state. For a comprehensive set of searches with both UCN and beam neutrons, the experiments should scan as wide a range of magnetic fields as possible to induce a neutron–sterile neutron transitions.

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Early searches for sterile neutrons were performed using UCN gravitational storage traps to correlate the possible disappearance of neutrons with the variation of the laboratory magnetic fields [40, 41, 43, 44, 179], assuming that Earth's magnetic field should be compensated to near zero (to satisfy the quasi-free condition) to permit the n → n' process to occur. With this assumption, the best limit for a free oscillation time τ_n→n' was obtained by [44], where τ_n→n' ⩾ 448 s (90% CL). More recent measurements and analyses [42, 45, 47] have accounted for the possibility of a modest sterile sector magnetic field by including a wider variation of the laboratory magnetic field B in the UCN traps. From the analysis of all existing UCN experimental data, the lower limits on τ_n→n' as a function of |B'| were obtained [46] in the range of tens of seconds for the sterile sector magnetic fields less than ∼0.3 G. However, one UCN experiment [41, 44] reanalyzed in [45] has reported a non-zero asymmetry with a significance of 5σ in the storage time of unpolarized neutrons in a Be-coated trap when a laboratory magnetic field was regularly changed from +0.2 G to −0.2 G. This anomalous result was interpreted [45] as an n → n' oscillation signal with the asymmetry caused by the variation of the angle β between vectors of magnetic fields of sterile and laboratory fields, B' and ±B. Thus, n → n' transitions with, e.g. τ_n→n' ∼ 30 s are not excluded for a region of |B'| ∼ 0.25 G.

In 2018, a re-optimization of the ESS schedule took place, due mainly to the fact that the original ESS timeline was established before construction began in 2014, and so was impacted by building delays resulting from the ESS's implementation of updated seismic and security standards adopted in recent years within Sweden (and globally). This has primarily impacted the design and construction of the ESS target building, as the strengthened standards were formulated in the aftermath of the Fukushima accident and amid a general increase in concern over global security threats. Following this new baseline, the new plan has the goal to start instrument commissioning in 2022, with early scientific experiments to be carried out on the first three instruments as soon as possible thereafter. The start of the user program at the ESS is expected to begin in 2023. In the ESS initial stages (July 2022–December 2023), the first neutrons will be produced from the target at very low beam power, where commissioning will include ramp up and testing from accelerator, target and integrated control systems (ICSs). The planned ESS ramp-up time frame is shown in table 2.

The configuration of the ESS neutron source and moderator systems presents excellent opportunities for the fundamental physics research of HIBEAM and NNBAR. This is achieved thanks to the high brightness of the ESS source, the configuration of the beam extraction system and the upgradeability options available for the source, which are attractive for proposed fundamental physics applications. The upper moderator has been designed for the initial suite of 16 instruments (15 neutron scattering instruments, plus the test beamline, located at W11, the same position as NNBAR). The design of the moderator is fully described in [183, 184]. The features of interest for HIBEAM and NNBAR include the following:

• (a)  
  The retaining of a monolith configuration for shielding openings and beam extraction ports such that moderators can be placed above and below the tungsten target. Since a design optimization for the initial instrument suite led to the choice of a single (upper) moderator system, this left open an option for future upgrades below the target useful for NNBAR.
• (b)  
  The upper cold parahydrogen moderator is 3 cm thick, with a shape optimized for beam extraction in the 42 beamports arranged in two 120° sectors, as seen in figure 9. For HIBEAM/NNBAR, the total available width of the moderator for beam extraction is about 17 cm.
• (c)  
  Moderators are placed in plugs that are replaced frequently (the average lifetime of a moderator system at full power is presently assessed to be ∼1 year)
• (d)  
  In consideration of possible future upgrades and fundamental physics experiments like NNBAR (which may need a larger moderator system), the inner shielding openings have been designed to be taller at the bottom than at the top of the tungsten target. A cross-sectional view of the region of the target showing the moderators, inner shielding and beam extraction openings is shown in figure 10.

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The HIBEAM experiment will use the upper moderator. By the time the experiment starts, it is expected that the Mark II moderator ('butterfly-1'), optimized for maximum cold brightness to all the instruments, will be in place. This moderator's main characteristic is its reduced height (3 cm) compared to its lateral dimensions (∼24 cm), which was found to deliver maximum brightness to the sample areas across the instrument suite [183, 184].

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A design study, termed HighNESS, has just began for a new cold neutron source and associated instruments at the ESS, including NNBAR. This is funded as a Research and Innovation Action within the EU Horizon 2020 program under Grant Agreement 951782 [185, 186]. A liquid deuterium moderator is envisaged, which provides the high-flux, slow-neutron source required by NNBAR. Such a high-intensity source, with a large emission surface and a wavelength spectrum shifted toward the longer wavelengths, compared to the upper moderator, would be highly beneficial not only to NNBAR, but also to several neutron scattering applications, such as SANS, imaging and spin-echo [183]. The lower moderator to be used by the NNBAR experiment should be tailored specifically for high-intensity neutron extraction, unlike the ESS upper moderator which is generally developed for brightness. This high intensity can be achieved by increasing the dimensions of the moderator. However, it has been shown [183] that a 3 cm parahydrogen moderator already delivers about 80% of the maximum intensity achievable by increasing the moderator height. The only way to have a worthwhile increase in intensity is by using a different type of moderator. The choice under study is a liquid ²H moderator, similar to what is used at reactor sources like the ILL (also in dimensions) or at the SINQ facility. Preliminary studies of these options were performed in [187], indicating an increase in intensity of a factor of ∼3 compared to a 3 cm flat moderator. This increase in intensity is related to the overall larger dimensions, as well as the negligible neutron absorption in deuterium. The possible geometry of the lower large ²H moderator is shown in figure 11, which is updated from the originally proposed geometry of reference [184].

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Some expected parameters of the neutron source for HIBEAM and NNBAR are listed in table 3.

5.2.1. The large beam port for NNBAR

In the current baseline design of the ESS monolith, a critical provision has been made for the NNBAR experiment. A normal ESS beamport would be too small for NNBAR to reach its ambitious sensitivity goals. Therefore, part of the beam extraction system in the ESS monolith has been engineered so that a large frame covering the size of three beamports will be constructed. Initially, the frame will be filled by three regular-size beamports plus additional shielding for other experiments, including the ESS test beamline. The three beamports can be removed to provide a LBP to NNBAR for the duration of the experiment and eventually replaced at the end of the experiment. Two views of the beam extraction region at the NNBAR beamport are shown in figures 10 and 12, both from the moderator to the monolith exit. At the time of this writing, no other existing or planned neutron facility will have an LBP of similar dimensions, making the ESS the ideal site for a full-scale NNBAR experiment.

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Ample discussion on the properties and usage of the ANNI beamline for fundamental physics searches has been considered in [39]. Figure 13 gives a schematic outline of the ESS/ANNI fundamental physics beamport. Neutrons emerging from the moderator pass through a guide system (figure 14) and then into an experimental area (figure 13) with a length of around 54 m and a width of around 5 m (visualized here without all other experimental apparatuses in the hall). Due to the beam hall size constraints, there would be very limited room to use focusing reflectors to increase sensitivity for n oscillation searches. HIBEAM considerations are thus minimally based on the full-length beamline with possible aperture collimation.

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Simulations using the ESS butterfly moderator design in McStas 2.4 [188] have been performed by the ANNI collaboration, fully modeling the S-curved n guide in a background-reduced, cold-spectrum selected, beam-shape-optimized way [39]. A McStas output of an n source file with coordinates, momenta and weights of the neutrons normalized to one initial proton with energy 2 GeV on the ESS tungsten target transported through the ANNI beam optics to the collimator exit at z = 22 m was produced. Figure 15 shows the spectrum of velocities of neutrons from this simulation flying through the ANNI beamline.

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As seen in figure 16, the initial beam characterization shows a large swathe of slow neutrons over the beam length of 50 m having larger divergence at smaller velocities. Most neutrons do not fall outside a 1.5 m radius (angle ⩽ 1.72 degree), though even that would constitute an enormous and financially untenable detector. Thus there is a sacrifice of a fraction of the valuable slowest neutrons by choosing a detector with a practical radius of 0.25 m–0.50 m. Assuming 1 MW of operating power, simulations show an absolute beam normalization of 1.5 × 10¹¹ n s⁻¹ (at the beamport exit). For a conservatively designed 1 m diameter detector downstream of the 50 m propagation length, this flux becomes 6.4 × 10¹⁰ n s⁻¹. This is without the installation of any further beam optimization or neutron reflectors, no lowering of the detector due to gravitational drop or optimization of beampipe shape, and assuming an inherently perfect detector efficiency.

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Figure 17 shows a top view of the neutron tracks estimated by Phits due to the interaction of the ANNI neutron beam with the ¹²C target. Most of the neutrons pass directly through the target—a small fraction are scattered off of it given low n¹²C cross section and some are absorbed, inducing the emission of ∼MeV photons. We take the origin of the coordinate system to be the experimental area, after ANNI's curved guide extraction, i.e. the origin is located in the so-called 'available envelope' as shown in figure 13.

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Figure 18 shows a cross-sectional view of the ANNI neutrons at the annihilation target. The observed neutron interference pattern is caused by the different bounce distances the ANNI neutrons take when being transported through the S-curved guide.

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The capabilities of the HIBEAM beamline can be further contextualized when the full final flux is considered as a function of a detector radius, as seen in figure 19. This hints at the need for greater beam control via n reflectors. However, space constraints will limit this prospect.

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The disappearance of $n\to \left\{{n}^{\prime },{\bar{n}}^{\prime }\right\}$ is sensitive to a scenario in which at least one of the mass mixing parameters α_nn' or ${\beta }_{n{\bar{n}}^{\prime }}$ (see equation (18)) is non-zero. The search assumes the presence of an unknown sterile magnetic field B', which would be matched by a magnetic field in the visible sector.

A schematic overview of the experiment is shown in figure 20. A more detailed diagram showing all relevant apparatus is shown, together with a simpler schematic picture illustrating the basic principles of the search. Neutrons propagate along an aluminum vacuum tube of around 50 m in length, with a varying diameter separated by a window; neutron rates at the start and the end of the propagation zone are precisely measured. The symbol M (left side) represents a current-integrating beam monitor with efficiency 20%–30%. The symbol C (right side) represents a current-integrating beam absorption counter with an efficiency ∼100%. The assumed beam intensity used here and for subsequent HIBEAM projections is 6.4 × 10¹⁰ n s⁻¹ per 1 MW of operating power [39], scaling linearly thereafter. The sterile magnetic field is assumed to be constant, uniform and not exceeding the magnitude of Earth's magnetic field [18]. The measurements of the change of neutron flux will be made for a range of axial laboratory magnetic field values in different directions in a tetrahedral configuration for a range of −0.5 G to +0.5 G with a step of only a few mG, a few times less than the resonance width to achieve adequate resolution of the resonance trough. Thus, the counting rate (determined by charge integration) of the counter C in figure 20 will be controlled by the magnitude of magnetic field. The charge integrating counter M will monitor variations of the beam intensity independent of variations of magnetic field.

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The detection of a resonance would appear as the reduction of the total counting rate in the C/M ratio vs |B − B'|. This experimental signal is sensitive to multiple parameters of the sterile sector. From this measurement, a limit on the mass mixing parameter , or the n → n' oscillation time ${\tau }_{n\to {n}^{\prime }}^{\text{dis}}$, can be extracted. A positive signal would indicate not only the existence of the sterile state n', but also the existence of sterile photons γ', required for the transformation to occur at non-zero B'. With more detailed scans, the three-dimensional direction of the sterile magnetic field B' can be established. As also shown in references [162, 189], due to its first-order nature, the disappearance method is the most statistically sensitive approach for setting a limit on or ${\tau }_{n\to {n}^{\prime }}^{\text{dis}}$.

Measurements of n → n' disappearance with a cold n beam will require full magnetic control in the flight volume of the vacuum tube shown in figure 20. The 3D magnetic field should be uniform and be preset to the desired 3D value in any direction with accuracy better than 2 mG in the range from 0 mG to ∼500 mG, presenting a technical challenge for this experiment. Another challenge will be the construction of the charge-integrating counters which can achieve a measured charge proportional to the n flux with high accuracy, typically 10⁻⁷. It has been recently shown [190] that such stability and accuracy can be achieved with a ³He detector in charge-integration mode. Measurements for positive and negative B-field magnitudes would allow the determination of the oscillation time ${\tau }_{n\to {n}^{\prime }}^{\text{dis}}$ independently of the value of the unknown angle β between the vectors B and B', as well as an estimate of the angle β itself.

The dependence of the sensitivity of n → n' searches on the properties of the beam and apparatus can be rather complex given the regenerative nature of the oscillation under certain magnetic field conditions, n monitor efficiencies and environmental background rates. The sensitivity for low magnetic field disappearance in the absence of visible resonance signal was best (if briefly) discussed in [162], but is reiterated here. For disappearance, the main dependencies concern the bare and square-normalized integrals of the neutron velocity spectrum, S(v):

$$\begin{equation}{J}_{0}=\int S\left(v\right)\mathrm{d}v,\hspace{25.0pt}{J}_{2}=\int \frac{S\left(v\right)}{{v}^{2}}\enspace \mathrm{d}v,\end{equation} \tag{ 21 }$$

which are then used to calculate the lower limit for the n → n' oscillation time, ${\tau }_{n\to {n}^{\prime }}^{\text{dis}}$

$$\begin{equation}{\tau }_{n\to {n}^{\prime }}^{\text{dis}}{ >}{\left(\frac{{J}_{2}\sqrt{{J}_{0}T{\epsilon}}}{{J}_{0}}\cdot \frac{{L}^{2}}{2}\cdot \frac{1-1.7{\epsilon}+0.76{{\epsilon}}^{2}}{g\sqrt{1-{\epsilon}}}\cdot \frac{\sqrt{2K}}{\sqrt{2+K}}\right)}^{\frac{1}{2}},\end{equation} \tag{ 22 }$$

where T (s) is the accumulated time for each individual magnetic field measurement point, is the n-monitor efficiency (taken to be, e.g. 30%), L (m) the length of magnetically controlled flight, the g-factor parameterizes the confidence level (for instance, g_95% = 3.283) obtained from statistical Monte Carlo simulations and K is the number of 'zero-effect' measurements (e.g. at magnetic field B = 0) exceeding the time T for the effect measurement by factor K. This calculation is based on the single maximum deviation of one of the +B and −B folded together measured points from the mean value of 200 individual measurements of the ratio C/M. With appropriate calculation of J₀ and J₂ (see equation (21)), equation (22) can be used for scaling different measurements and configurations at the same beamline. However, better limits can be obtained with more detailed analysis based on the line shape fit to experimental magnetic field scan data, as well as beam pulse timing.

The structure of equation (22) is mainly analytical in origin, though their dependence upon factors of g was ascertained by thousands of independent Monte Carlo experiments. To obtain signal sensitivity to a 95% CL with 200 separate magnetic field point measurements and an additional 25 background runs with equidistributed run folding over a finite magnetic field range (e.g. [−200, 200] mG), a (conservative) background of 1 n s⁻¹, a 30% n monitor efficiency and two 25 m magnetically controlled sections of beamline, the sensitivity in oscillation time τ_n→n' can be calculated for various detector radii over different periods of running without any exploitation of the pulsed beam time structure. One ESS operating year is considered to be approximately 200 days (see table 2) when discounting for routine maintenance and seasonal shutdowns. The sensitivity of the disappearance method as a function of detector radius for ${\tau }_{n\to {n}^{\prime }}^{\text{dis}}$ is shown in figure 21 together with sensitivities for regeneration modes (discussed in sections 7.2 and 7.3).

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Figure 22 shows the current limits from trapped UCN experiments together with the expected sensitivity of the HIBEAM experiment (in the disappearance mode) after 1 year's ESS running for a power usage of 1 MW. Increases in disappearance sensitivity of greater 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 (up to and beyond an order of magnitude), many of which are unexplored by UCN-based experiments, and remain free of the model assumptions of those searches. In the limit of a vanishing TMM, the sensitivity increases quadratically with the observation time. However, the possible contribution of a TMM complicates this picture (section 3.3). For simplicity, the FOM of sensitivity when comparing experiments is therefore taken here to be the oscillation time.

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It should be noted that the interpretations and limits rely on the experimental assumptions, which may be poorly understood for neutron collisions on UCN material trap walls [161]. This source of systematic uncertainty can be removed by performing dedicated searches with propagating cold neutrons in a magnetic field, as planned for the HIBEAM experiment.

High-precision searches for n → n' are also being pursued using UCN at the PSI by the nEDM collaboration [191], albeit for the magnetic field range |B| < 0.2 G so far considered. A series of searches for n → n' conversions, due to various processes along a beamline (e.g. figure 6), are planned at the High Flux Isotope Reactor (HFIR) at Oak Ridge National Laboratory [189, 192]. The higher beam intensity of the ESS and the longer available flight paths will allow exploring these mechanisms with higher sensitivity at the ESS than at the HFIR reactor.

The regeneration search derives from a similar theoretical basis as the disappearance search [162, 189], though corresponding to a two-stage (second-order) process with a consequently quadratically smaller probability. In the first stage, the n → n' transformation takes place in an intense cold n beam at the quasi-free environment limit corresponding to |B − B'| ∼ 0. The largely untransitioned n beam will be blocked by a high suppression beam trap, while the remaining sterile n' will continue unabated through the absorber; in a second volume behind the absorber (stage two), maintaining the same conditions of |B − B'| ∼ 0 as the section before it, the n' → n transformation produces detectable ns with momentum conserved, as though the totally absorbing wall were not present at all. The resonance behaviour depends primarily on the magnitude of the laboratory B; if the vectors of B and B' are not well aligned, i.e. the angle β Z 0, the oscillation can still occur with somewhat reduced amplitude [18]. This feature provides a robust systematic check for the experiment: oscillations can be turned off simply by changing the magnitude of B out-of-resonance in the volume before and/or after absorber. Taking measurements at the positive and negative magnitudes of the field B in both volumes (four combinations) allows for a determination of the oscillation time independent of the angle β.

A schematic overview of the principle of the regeneration experiment is shown in figure 24. The lowest possible n-background rate in the counter R will be important for a high sensitivity of regeneration search in B-scan, and sufficiently shielding R represents an important challenge for this measurement.

The observation of the resonance in the B-scan would be defined by a sudden appearance of regenerated ns when the B − B' ≈ 0 condition in both volumes is met. Like with disappearance, a positive signal would be a demonstration of the n' → n transformation as well as the existence of the sterile n' and γ'. The requirement of matching conditions in both volumes ensures this type of measurement is significantly more robust to systematic uncertainties that could cause a false signal to be observed and provides an unambiguous test of that hypothesis.

For the regeneration experiments, in the absence of an observed signal above background level, an upper limit on the oscillation time ${\tau }_{n\to {n}^{\prime }}^{\text{reg}}$ can be established from a statistical analysis. This limit was parameterized in [162] through the quartic-normalized integral of the velocity spectrum

$$\begin{equation}{J}_{4}=\int \frac{S\left(v\right)}{{v}^{4}}\enspace \mathrm{d}v\end{equation} \tag{ 23 }$$

providing the following estimate for the oscillation n → n' time

$$\begin{equation}{\tau }_{n\to {n}^{\prime }}^{\text{reg}}{ >}{\left(\sqrt{4T}\cdot \frac{{L}^{4}}{4g\sqrt{{\bar{n}}_{b}}}\cdot {J}_{4}\right)}^{\frac{1}{4}},\end{equation} \tag{ 24 }$$

where ${\bar{n}}_{b}$ is the average background rate in the detector R (see figure 24) in n s⁻¹. This calculation is again based on the single maximum deviation of one of the folded magnetic scans of 200 measurements and T is time for one individual measurement. Running over possible values of this background rate, the behavior of the upper oscillation limit can be constructed and is shown in figure 23. It should be noted that the regeneration mode searches are susceptible to environmental background rates only due to full beam absorption at the halfway point of the beamline. From equation (24), it can be seen that the length L of each of two vacuum tubes in the regeneration scheme is the only parameter that can essentially increase the limit for ${\tau }_{n\to {n}^{\prime }}^{\text{reg}}$. The regeneration oscillation time sensitivity is lower for the same running time than for the disappearance mode as the former (latter) is a two-transition (single-transition) process. However, both processes are complementary with different experimental configurations and neither sharing the same sets of experimental uncertainties. Furthermore, any observation in the disappearance mode could be verified by a regeneration experiment running for a longer time.

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As discussed in section 3.3, the symmetry between ordinary matter and mirror matter in general allows a range of transformations between the visible and sterile neutron sectors, beyond the simple n → n' process tackled in sections 7.1 and 7.2. A neutron can be transformed into a mirror (sterile) antineutron which then regenerates back to a detectable n state: $n\to {\bar{n}}^{\prime }\to n$. Since the angular momentum of the neutron is conserved, the magnetic moment of the mirror antineutron will be oppositely aligned to the magnetic moment of the sterile neutron due to the mirror CPT theorem. Therefore, a resonance should be observed when magnetic fields in the first and second flight tubes are opposite in direction. This field configuration is included in the anticipated set of measurements shown in figure 24. This search will therefore be made as a complement to n → n' → n discussed in section 7.2 and with a similar sensitivity.

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Searches for $n\to \bar{n}$ assume that the transformation occurs via mixing with a non-zero mass amplitude ${\varepsilon }_{n\bar{n}}$ term; this necessitates magnetic shielding in a search. However, $n\to \bar{n}$ can also arise due to the second-order oscillation processes: $n\to {n}^{\prime }\to \bar{n}$ and $n\to {\bar{n}}^{\prime }\to \bar{n}$, with an amplitude comprising ${\beta }_{n{\bar{n}}^{\prime }}{\alpha }_{n{n}^{\prime }}$ and associated interference terms. The earlier body of searches [37, 109–111] for free $n\to \bar{n}$ would be insensitive to this scenario.

A schematic layout of the search for $n\to \bar{n}$ through regeneration is shown in figure 25. Construction of the $\bar{n}$ annihilation detector at the end of second vacuum volume will be required for this experiment. A search for $n\to \bar{n}$ through regeneration is complementary to the classic $n\to \bar{n}$ search assuming no sterile neutron mixing. Discussion of the details of the annihilation detector is therefore deferred to section 8.2, where it is described in the context of the classic $n\to \bar{n}$ search.

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The program of searches has the potential to both make a fundamental discovery of a dark sector and to quantify the processes underpinning the observations. For example, if the search discussed in section 7.1 could detect a signal in the disappearance mode, this would imply a disappearance for all possible final states of oscillating neutrons. Since the magnetic moment of a neutron is oppositely aligned to that of an antineutron due to the CPT theorem (and as for sterile neutron and antineutrons), it will be advantageous to use non-polarized beams. Here, the compensation B ≈ B' would imply transformations to $\bar{n},{n}^{\prime },{\bar{n}}^{\prime }$ for different initial polarization states of neutrons. In regeneration searches, all four magnetic field combinations in two flight volumes would be possible, with ±B₁ versus ±B₂ used to detect all possible channels of n → regeneration and $n\to \bar{n}$ due to mass mixing amplitudes α_nn' and ${\beta }_{n{\bar{n}}^{\prime }}$.

As discussed in section 3.3 and shown in equation (15), the probability of the process n ↔ n' due to a TMM with magnitude κ for sufficiently large magnetic fields is approximately constant, P_n' = κ². Due to the independence of P_n' of the magnitude of magnetic fields, the oscillating (n, n') system can travel through strong magnetic fields while retaining the same probability of transformation. In a classical sense, the gradients of the magnetic field potential μ ⋅ B(r) cause a force acting only upon the neutron part of the (n, n') system, and not on the sterile neutron part. Thus, the components of the (n, n') system are separated in space as with two spin components in a Stern–Gerlach experiment. When passing through a difference of magnetic potential, the difference in kinetic energies of the (n, n') components can become larger than the energy width of the wave packet of the system: this 'measures' the system by collapsing it into either a pure n or pure n' state. The required magnetic field gradient corresponding to the 'measurement' event can be found according to [88] from the following equation:

$$\begin{equation}\frac{{\Delta}B}{{\Delta}x}{ >}\frac{1}{\mu v{\left({\Delta}t\right)}^{2}}=\frac{v}{\mu {\left({\Delta}x\right)}^{2}},\end{equation} \tag{ 25 }$$

where Δx is the distance traveled in the magnetic field for time Δt, μ is neutron magnetic moment and v is the neutron velocity.

The presence of strong magnetic field gradients destroys the 'entanglement' of the oscillating (n, n') system. A surprising consequence is that as the system continues through the gradient and has its initial state repeatedly reset, this creates additional opportunities for the transformations n → n' or n' → n, effectively increasing the transformation rate. In reference [88] this mechanism was suggested as an explanation for the neutron lifetime anomaly [89]. The disappearance of neutrons due to the magnetic gradients present in UCN trap experiments could explain the ∼1% lower value of the measured n lifetime in bottle experiments than is seen in measurements using the beam method. This explanation of the n lifetime anomaly together with existing limits from the direct experimental n → n' searches implies [88] that κ is in the range of ∼10⁻⁴–10⁻⁵.

To test this hypothesis, solenoidal coils with alternating currents in each coil can be implemented around the two vacuum tubes to create a magnetic field along the beam axis with a 'zig-zag' shape, providing an almost constant gradient along the tube length. These coils can be applied in a regeneration experiment scheme as shown in figure 24 to search for the nTMM-induced n → ${n}^{\prime }$ → n regeneration effect with κ < 10⁻⁵. Another opportunity still under development is the use of a cylindrical Halbach array.

As described in [88] and section 3.3, an enhanced n → ${n}^{\prime }$ transformation rate can be produced in a gas atmosphere due to the nTMM. A constant magnetic field B will be applied in the flight volume to give rise to a negative magnetic potential which will compensate the positive Fermi potential of the gas. The gas density should be sufficiently low to avoid incoherent scattering or absorption of the neutrons, resulting in a pure oscillation with probability ${P}_{n{n}^{\prime }}={\left({{\epsilon}}_{n{n}^{\prime }}+\kappa \mu B\right)}^{2}{t}^{2}$. The magnetic field is then scanned in order to search for a resonance condition resulting in a regeneration signal. The magnitude of the laboratory magnetic field at which the resonance might occur depends on the magnitude of the neglected hypothetical sterile magnetic field. The magnitude of B' can be determined by setting the laboratory magnetic field to zero and instead scanning the pressure in the flight tube. In this scenario, the probability is described by the corresponding equation:

$$\begin{equation}{P}_{n{n}^{\prime }}={\left({{\epsilon}}_{n{n}^{\prime }}{\pm}\kappa \mu {B}^{\prime \prime }\right)}^{2}{t}^{2},\end{equation} \tag{ 26 }$$

where ± is due to the different possible parities of the sterile magnetic field. Thus, the magnitude of B' can be also independently determined.

All forms of the sterile neutron searches 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 utilize gas detectors based on ³He in a single-wire proportional chamber. The detectors can be operated at low gain since the n + ³He → t + p reaction produces a very large ionization signal. While this is the baseline technology assumed for HIBEAM, it is also possible that modern readout solutions from high energy physics can augment the performance of such a neutron detection scheme further. As an example, the most challenging readout scenario is considered here, in which each neutron in the flux would be individually detected. Assuming a neutron flux of 10¹¹ n s⁻¹ evenly spread out over a circular surface of diameter 2 m, and assuming each detector element to be about 1 square cm in transverse area, one would have 30 000 readout channels with a singles counting rate in each channel of about 3 MHz. This rate is indeed not trivial to deal with but it can be accommodated by the ATLAS TRT detector, where each detector element is a single-wire proportional chamber (operated at high avalanche gain) read out on the wire. Typical singles counting rates in the ATLAS TRT are in the range of 6–20 MHz, so 3 MHz seems quite feasible in comparison. The ATLAS TRT electronics will be taken out from the ATLAS setup in 2024, presenting a timely opportunity for HIBEAM, and are therefore an interesting possibility to investigate for use in HIBEAM. The energy of the reaction products do not provide any useful information about the originating neutron, therefore only the number of neutrons detected would be recorded in the data stream. The use of coincidence criteria with neighboring detector cells will also be investigated, to avoid double counting when nuclear fragments leak into neighboring cells.

If the individual counting of neutrons is not necessary, an integration of the released charge in the detector material gives a measurement proportional to the number of incoming neutrons. After proper calibration, such current integration is less sensitive to the actual rate of neutrons. It should also be mentioned that, in view of the shortage of ³He, much R&D is done for scattering experiments on other methods for cold neutron detection. Synergies with readout systems developed for high energy experiments are expected, and neutron detection in HIBEAM is a good example of an application with a specific scientific use as motivation for exploring such synergies.

As explained in section 3.1, the ns must be transported in a magnetically shielded vacuum. For quasi-free ns, this corresponds to a vacuum of 10⁻⁵ mbar and a magnetic field of less than around 10 nT along the n flight path [108].

The target vacuum can be achieved with a vacuum chamber comprising highly non-magnetic materials, e.g. Al, with turbo molecular pumps mounted outside of the magnetically shielded area. Magnetic fields of less than 10 nT have been achieved over large volumes (see, for example, reference [193]). For the planned experiment, a shielding concept will be used based on an aluminum vacuum chamber, a two-layer passive shield made from magnetizable alloy for transverse shielding, and end sections made from passive and active components for longitudinal shielding, as shown in figure 26.

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A key experimental task of any $n\to \bar{n}$ search is to isolate and detect the annihilation of $\bar{n}$s from a beam of free ns. The transformation has an extremely low probability, and although an ESS experiment could have by far the longest experimental observation time of a free n beam, it may be probable that any experiment would measure only $\mathcal{O}\left(1\right)$ candidates.

The overarching goal for the detector system is to provide the highest possible sensitivity for detecting an $\bar{n}$ annihilation. These ambitions go beyond a statistical significance analysis, allowing for a claim of discovery from even only a few observed annihilation events; in the case of non-observation, a robust upper limit can be imposed, and multiple compelling theories of baryogenesis eliminated or severely constrained.

As much as possible, the detector system must provide a reliable and complete reconstruction of each annihilation event. Statistical correction of experimental shortcomings cannot be performed on the individual event level; thus, the design goal must be to record as many observable parameters as possible, taking in all available information about the subsequent annihilation products and, if possible, compensate directly for detector effects that are statistical in nature via over-sampling. From this, one understands immediately that special attention must also be paid to ∼4π detector coverage and similarly must avoid permanently and temporarily dead detection areas, due either to support structures or failing detector components, respectively. Serviceability, too, must then be a key design feature.

An important detector constraint is that a magnetic field cannot be used. Thus, momentum cannot be directly measured, only the kinetic energy deposited in the detector and the direction of the particles. A sensitive $n\to \bar{n}$ experiment should therefore:

• (a)  
  identify all charged and neutral pions, properly reconstructing their energy and direction; and
• (b)  
  reconstruct the energy and direction of most higher energy, charged nuclear fragments (mostly protons; fast ns will likely escape undetected).

The positive identification of an annihilation event would ideally comprise the identity of all pions, a total invariant mass which amounts to two nucleons (∼1.9 GeV) and a reconstructed common point of origin in space and time of all emitted particles including nuclear fragments from the plane of the annihilation foil. The directionality of an event should be verified by checking that particles move outwards, acting as an important discriminant against backgrounds from CRs and atmospheric neutrinos. A generic detector must thus include tracking, energy loss, calorimetry, timing and CR veto systems, as illustrated in figure 27.

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The optimization of the diameter of the annihilation foil is a balance between a large diameter that maximizes the visible n flux and overall manufacturing cost, which grows dramatically with diameter. The baseline design assumes a 200 cm diameter for NNBAR and substantially lower for HIBEAM. The discussion below is guided by NNBAR while HIBEAM features are presented in section 6. During the long time of flight, the n beam will be affected by gravitation such that the flux incident upon the annihilation foil will not be regular and uniform but rather elongated vertically and containing a vertical gradient in observation time such that the slowest ns in the velocity spectrum will pass through the bottom of the foil. Since the oscillation probability (and so too the annihilation probability) grows as t², the largest number of annihilations will hypothetically take place at the bottom of the foil.

The geometry of the detection system will also be optimized balancing practical considerations for event reconstruction. A consequence of the broad spread of annihilations in the foil disk is that there is no strong argument for having a cylindrical detector layout. A rectangular layout is likely to be cheaper and will serve the purpose equally well, if not even better than a circular cross-section setup. On the one hand, a large area for annihilation presents a difficulty since particle detection normally assumes some average angle of incidence and a central point of emission. On the other hand, a common vertex for a number of secondaries in a large area becomes a strong constraint for an annihilation event: they contain at least two tracks in the majority of cases.

The momentum for each particle can be reconstructed if particles are identified and their direction and kinetic energies measured. If the missing momentum (higher than the Fermi momentum) points into any uninstrumented areas of the detector, one can still have a good understanding of the event, though the topology would not be as well characterized as in the case of all secondaries being observed. While full efficiency and 4π coverage for tracking and reconstruction of annihilation products is impossible due to the n beam path, very high coverage should be achievable. A ∼10 m long detector with a 1 m inner radius with full coverage in azimuth would result in a geometrical acceptance of 90%, such that about half of all annihilation events will be fully reconstructed. Even with excellent geometric efficiency, loss of information about the total energy is unavoidable. The energy carried away by nuclear fragments may in most cases go unrecorded, though this should be a small fraction of the kinetic energy imparted to other heavy charged fragments like protons. On the other hand, up to 10% of the annihilation energy is lost to fast neutrons emitted from the nucleus. For momentum balance, the loss of nuclear fragments can be more significant due to their higher mass.

8.4.1. The ¹²C annihilation foil

8.4.2. The vacuum vessel

8.4.3. Background considerations for higher sensitivity searches

8.5.1. Tracking outside the vacuum

8.5.2. Tracking inside the vacuum chamber

The energy measurement by the calorimeters will be a crucial part of the evidence that an annihilation event has been observed in two ways. It will identify the neutral pions that occur in 90% of the annihilations and ensure that the sum of absorbed energy (kinetic and tied up in pion rest masses) shall sum to two nucleon masses. In addition, the calorimetry shall provide the kinetic energy of the charged pions and charged nuclear fragments. Energy measurements at these energies are notoriously difficult. Several processes are involved.

8.6.1. Charged hadronic particles

8.6.2. Photons from neutral pion decay

8.7.1. Cosmic veto

8.7.2. Timing

8.7.3. Triggering

For $n\to \bar{n}$ via mass mixing, the quasi-free condition is needed, implying magnetic field-free transmission of neutrons. Any antineutrons which are produced would then annihilate with a target surrounded by a detector. The detector would reconstruct the characteristic multi-pion signal to infer the existence of $n\to \bar{n}$.

As shown in section 4.1, the FOM for a free $n\to \bar{n}$ search is given by $\langle {N}_{n}\enspace {t}_{n}^{2}\rangle$. The FOM is proportional to the rate of converted neutrons impinging on a target to achieve a high FOM, the following criteria must be met:

• (a)  
  The n source must deliver a beam of slow, cold ns (energy <5 meV) at high intensity, maximising both t_n and N_n , respectively, for a given beamline length.
• (b)  
  The beamport must correspond to a large opening angle for n emission.
• (c)  
  A long beamline is needed to maximize t_n .
• (d)  
  A long overall running time is needed due to the rareness of the process.

Note that neutron beams with lower average energies have higher transport efficiencies when supermirror reflectors are utilized, as in the second-stage NNBAR experiment described in section 9.

Considering figure 25, if the central n absorber were to be removed and two vacuum tubes were be combined to one with the common magnetic compensating/shielding system, one would recover the essential elements of a $n\to \bar{n}$ experiment at the HIBEAM/ANNI beamline, although with a shorter neutron flight path. Figure 28 shows the sensitivity in ILL units per year normalized to the ESS running year, i.e.

$$\begin{align}\text{ILL}\;\text{units}\;\text{per}\;\text{year}& =\frac{{\langle {N}_{n}\enspace {t}_{n}^{2}\rangle }_{\text{ESS}}}{{\langle {N}_{n}\enspace {t}_{n}^{2}\rangle }_{\text{ILL}}\cdot \enspace \left(\text{Operational}\;\text{Factor}\right)}\\ & =\frac{{\langle {N}_{n}\enspace {t}_{n}^{2}\rangle }_{\text{ESS}}}{\left(1.5{\times}1{0}^{9}\right)\cdot \left(1.2\right)}\end{align} \tag{ 27 }$$

of a $n\to \bar{n}$ search as a function of the radius of the detector, assuming a 1 MW operating power. One ILL unit is defined using the FOM and the flux and running time used in reference [37] as an observable for the number of converting neutrons for a given mass mixing term. The operational factor is a correction factor for the different annual running times expected at the ESS compared to the ILL for the latter's total running period. The sensitivity estimate given by this approach conservatively assumes that the detection efficiency at HIBEAM would be the same as at the ILL experiment (∼50%).

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The sensitivity reaches a plateau for a detector radius of ∼2 m. It can be seen that an ILL-level sensitivity can be achieved after running for several (∼3) years with an appropriately sized detector, but this should be considered a generous possibility, as cost considerations may lead to a smaller detector. These can be only linearly offset by a longer running period and higher operating power. $\bar{n}{\;}^{12}\mathrm{C}$ annihilation and outgoing product tracking efficiencies, along with their associated cosmic, atmospheric and fast-n backgrounds, have not yet been considered entirely, though state-of-the-art simulations of the underlying microscopic processes have been completed [124, 172].

Nonlinear, though fractional, increases in sensitivity can be achieved with the design and construction of focusing (pseudo-)ellipsoidal superupmirrors starting near the beamport to increase (anti)neutron flux on the ¹²C annihilation target. Highly preliminary computations using 0.25 m minor-axes and major-axis lengths of 27–50 m for half-ellipsoidal reflector geometries assuming perfect n reflectivity have shown some ∼40% increase in overall sensitivity. More realistic configurations and reflectivity modeling must be completed and geometrically optimized.

A simple baseline NNBAR experiment is shown in figure 30. A longitudinal distance of 200 m separates the source and the target foil. Neutrons passing through the foil are absorbed in a beam trap.

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In order for NNBAR to make the most of the impressive neutron intensity at the ESS, there must a be a 'gathering' reflector to ensure that a large neutron flux is directed and focused via a magnetically shielded region on the annihilation target. To first order, the most obvious reflector geometry is that of an ellipsoid [196], which would facilitate efficient transport of neutrons that otherwise would miss the annihilation target completely. The start and end points of the reflector on the longitudinal axis are 10 m and 50 m, respectively. The semi-minor axis of the ellipsoid is 2 m. The radius of the annihilation foil is 1 m.

An effective reflector must be fully illuminated by the source. This means that a substantial amount of the overall intensity will have trajectories that deviate significantly from the nominal beam trajectory axis. To achieve this, reflecting angles for even fairly cold neutrons (∼1000 m s⁻¹) will exceed that of the limit of the best traditional reflectors. NNBAR will thus use neutron supermirror technology [197] which has been used with success for many years as a means to guide thermal and cold neutrons to many scattering instruments at both pulsed and continuous neutron sources. NNBAR will utilize the same multi-layered surface treatment on its reflector to gather and focus the wide range of neutron trajectories at the source. To do so, surfaces with surface reflectivity up to m = 6, i.e. a reflection capability as high as six times better than the reflectivity limit for polished nickel. Neutrons from a liquid deuterium source are collimated in the structure housing the moderator. Neutrons emerging are reflected via an ellipsoid supermirror along a magnetically shielded region toward a target foil, surrounded by an annihilation detector. A beam trap absorbs the beam. The detection efficiency of an annihilation event in the foil is 50%.

At the LBP, the liquid deuterium source would be offset from symmetry axis of the LBP which is aligned with the target. To deal with this, a segmented differential reflector was designed. This has a distorted ellpsoid-like shape, albeit one which can be optimized via the solution of coupled differential equations for neutron reflections to allow specific reflection angles at certain distances from the foci, providing maximum intensity to the target foil [194]. Segmentation also allows optimization of the m value for different panels in the supermirror complex, reducing costs and allowing for easier large scale manufacturing.

Figure 31 (top) shows a simulation of a neutron reflection and focusing toward a target at 200 m distance along the longitudinal axis (referred to as beam trajectory position) from the position of the reflector in a differential reflector. It is shown how the neutron emerges from a source which is offset from the central-axis of the ellipsoid-like reflector. Figure 31 (bottom) shows a sampling of traced rays representing neutron trajectories, also including gravity. The neutrons can be restricted to a range in the vertical direction transverse to the longitudinal axis of around 2 m.

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Figure 32 shows a sketch of a differential reflector configuration with parameters that can be optimized to maximize the sensitivity of the NNBAR experiment. Shown are the source focal point position, $\overrightarrow {{x}_{\mathrm{s}}}$, the target focal point position, $\overrightarrow {{x}_{\mathrm{f}}}$, the reflector start position, z_i, and the reflector end position, z_f.

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An optimization of the shape parameters (keeping the others constant) was made. Figure 33 show how the sensitivity for NNBAR varies as a function of the source focal point vertical position, the source focal point horizontal position, the target foil focal point horizontal position, the reflector start position, the reflector end position and the surface reflectivity m. The sensitivity is expressed as ILL units per year, as defined in section 8.9.

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As can be seen, a sensitivity of around 350 ILL units per year is obtained. The sensitivity is strongly dependent on the surface reflectivity m. Indeed, developments in the supermirror technology are one of the driving factors in allowing NNBAR a large sensitivity increase since the ILL experiment. The peak sensitivity of ∼350 ILL units per year is achieved for m ∼ 6, 7 and falls to ∼10 ILL units per year for m ∼ 1. A doubling of start position from 10 m to 20 m leads to an approximate halving of the sensitivity. Extending the end position from 40 m to 60 m increases the sensitivity by around 20%. The foil focal point horizontal position is less sensitive, changing by about 10% for shifts of up to 4 m. Optimising the source focal points (horizontal and vertical) changes the sensitivity by up to around 30% in the considered range.

As was shown in section 9.3, simulations predict a sensitivity of around 350 ILL unit per year. For an experiment running for three years this provides an improvement of three orders of magnitude. The increase in sensitivity can be broadly decomposed into the gain factors given in table 4. Sensitivity increases are due to the greater source intensity, propagation length and run time. The largest gain is from the now widely available high m reflectors.

To further investigate differences between the ILL and the ESS for a search for $n\to \bar{n}$, the performance of only the moderators was studied. In this case, the calculated performance of a liquid deuterium lower moderator of ESS was compared to the ILL horizontal cold source (HCS), used for the original ILL search. For this comparison, updated brightness data was considered, indicating that the cold brightness of the ILL moderators is about three times larger than the official data [198]. Even with this correction, it was found that an optimized ESS lower moderator together with the upper 'butterfly' moderator would deliver a higher intensity, larger than the HCS used for the ILL search. The expected gain is due to several factors, including an optimal beam extraction (aiming at viewing the full surface of the moderator) and an optimization of the moderator specifically for the NNBAR experiment.

The results shown above exploit a simulation of a liquid deuterium lower moderator. One of the goals of the HighNESS project [185, 186] will be to deliver an engineering design of such a moderator. The performance will be different to that previously simulated and a new NNBAR sensitivity will be quantitatively determined. Based on experience and results from previous moderator designs, it is possible to list and give quantitative estimates of factors influencing the performance of such a moderator. This includes several contributions, which either increase or decrease the performance of the moderator for NNBAR's ultimate capability.

The first group of contributions enhance NNBAR's sensitivity and include optimizations of the design, including moderator size, positioning with respect of the target and the use of reentrant holes, a proven technology to increase neutron intensity by a factor of 1.5× for a specific direction [199]. Furthermore, if pure orthodeuterium is assumed instead of the mixture of ortho- and paradeuterium used in reference [187], the expected spectrum should be colder than previously calculated in the simulation [187] used for the NNBAR sensitivity estimates above. Another option might be to employ a single-crystal reflector filter in front of the moderator to enhance the thermal and cold flux while reducing the epithermal and fast flux [200].

The second group comprises contributions which can potentially degrade the sensitivity of NNBAR. This can occur with the refinement of engineering details: the fact that other beamlines, other than NNBAR, might view the moderator and may consequently require the removal of some reflector material surrounding the moderator, resulting in decreased performance [183]. Also, a possible shadowing effect of the inner shielding in the monolith could reduce the effective (viewed) surface area of the moderator. Naive estimations suggest a cancellation of these competing effects.

A full quantification of the NNBAR sensitivity is part of the HighNESS program. However, it should be noted that, in principle, running times can be extended to mitigate against any loss of sensitivity. Furthermore, estimates provided in this paper are rather conservative with an assumed selection efficiency for an annihilation event of around 50%, as obtained at the ILL; indeed, detector technology and data analysis methods in experimental particle physics are substantially more advanced compared to the early 1990s and so a far higher efficiency would be expected for a modern-day experiment. Finally, only a lower liquid deuterium moderator was considered for the sensitivity calculations given in this section. The upper butterfly moderator would also provide an additional flux of cold neutrons. Mitigation by longer running and the conservative nature of the current estimates could thus also protect against an unexpected lowering of the planned full power of the ESS from 5 MW to 2–3 MW.

Limits on the free $n\to \bar{n}$ oscillation time, together with the potential sensitivities of HIBEAM (section 8) (assuming 3 years running at 1 MW) and NNBAR (assuming 3 years running at 5 MW, and a three orders of magnitude improvement in ILL units) are shown in figure 34.

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Also shown in figure 34 is a projected converted free oscillation time lower limit for bound neutron conversions within ⁴⁰Ar nuclei within the future DUNE experiment, where ${\tau }_{n\to \bar{n}}{\geqslant}5.53{\times}1{0}^{8}$ s [123] for an assumed exposure of 400 kt years. Note that this limit does not take account of new ⁴⁰Ar intranuclear suppression factor calculations completed in [124]; systematic simulation studies continue within the DUNE collaboration and further automated analysis improvements are underway [123] in hopes of eliminating atmospheric neutrino backgrounds. There is as yet no estimate for the expected $n\to \bar{n}$ sensitivity for Hyper-Kamiokande.

As discussed in section 3.1, consideration of limits or sensitivities on the free neutron oscillation time for free and bound neutron searches is, to a certain extent, an apples and pears comparison. There is overlap in physics potential, but neither renders the other redundant. Indeed, both would play essential and complementary roles in both the definitive establishment of and cross reference for any future discovery.

As discussed in section 3.5 the experimental signature for $n\to \bar{n}$ is striking: the annihilation of the antineutron, releasing roughly 1.9 GeV of total energy, typically in the form of pions (four to five on average).

It is instructive to consider the background mitigation strategy of the most sensitive cold neutron experiment performed to date at the ILL [175]. Using a steady reactor neutron source, this experiment confirmed an absolutely zero background while expecting a signal efficiency of ∼50% by using multiple track and kinematic cuts. Such a unique, state-of-the-art feature is impressive, especially given the large integrated flux of slow neutrons which passed through the annihilation target every second (10¹¹ n s⁻¹). Developing a detector scheme with zero background is extremely important for the potential detection of an antineutron appearance where even a single event can represent a fundamental discovery.

Both HIBEAM and NNBAR must contend with cold neutron beam-generated and CR backgrounds. Beam-generated backgrounds comprise fast neutrons and high-energy particles from the spallation source, as well as charged and neutral CRs generated in the upper atmosphere. NNBAR will be especially likely to be affected by high-energy products from the spallation process given the large opening angle of the beamport. These all are now discussed.

10.1.1. Cold neutron beam backgrounds

10.1.2. Cosmic rays

10.1.3. High energy products from the spallation source—NNBAR only

Both disappearance and regeneration of sterile neutrons would rely on the observation of a magnetic field dependent resonant oscillation signal. These two methods have different systematic difficulties: (i) for disappearance, one must detect a small reduction in the counting rate of total incident neutrons to the order of ∼10⁻⁶–10⁻⁸ of the total flux, a challenge due to the need for precisely characterized, ultrahigh efficiency neutron monitors able to handle the large beam intensity; (ii) for regeneration, one must have a very low background count rate in the final neutron detector following total incident beam absorption, requiring an essential detector shielding effort.

From knowledge of other currently operating cold neutron sources such as the HFIR and the Spallation Neutron Source at Oak Ridge National Laboratory, it is known that the ambient background rates in ³He tube-style cold neutron detectors are rather low (few n s⁻¹), even when large in overall area and closer to their respective sources than HIBEAM plans to be; this is also the case despite higher duty cycles compared to the ESS. This generalized rate usually includes no vetoes or purpose-built particle-tracking equipment around the detectors and only modest shielding and so can be still improved. Assuming an effective background rate of ∼1 n s⁻¹ should thus be achievable for a regeneration experiment at HIBEAM.

The main methods to suppress the expected backgrounds for HIBEAM and NNBAR in neutron–antineutron searches are:

• Prepare a control background sample by 'switching off' all active magnetic shielding elements of the experiment, allowing the Earth's field to suppress the neutron transformation effect, but leaving the entire experiment unperturbed in the process. This could be done during beam-off and beam-on periods, as well as partially characterized with a similar target-detector configuration during an HIBEAM $n\to {n}^{\prime }\to \bar{n}$ experiment
• Add one or more targets downstream of the annihilation target but within the sensitive volume of the detector to produce additional 'sources' for background events without an annihilation signal. Any antineutron produced in the original cold neutron beam would be removed by the primary annihilation target [110].
• Suppress the generation of gamma backgrounds produced by neutron capture on target via demanding multiple 'track-like' cuts on the detector, since these events do not create tracks.
  • Tracking these particles back to a common vertex (± several mm³) to resolution smaller than the total beam spot on target will be important.
• Design a CR veto or, with a modern fast-timing calorimeter, use the entire calorimeter as a CR veto, coupled with vertex reconstruction capability, in order to reduce muon events.
  • Rejection of neutral CR events is to be accomplished by background subtraction via directly measured rates within the detector setup, along with energy deposition and multiple track cuts.

10.3.1. Gamma backgrounds from target

A preliminary estimation of gamma emission from the annihilation target due to neutron capture was done for HIBEAM. McStas simulated events [39] for the ANNI beam were used for the neutron source assuming ESS operating at 1 MW and the ANNI neutron current 6.4 × 10¹⁰ neutrons s⁻¹. The simulation was implemented in Phits [201] using a carbon-12 target of 1 m diameter. The distance from the neutron source to the target was assumed to be of 53 m, with a target thickness of 100 μm. Gravitational effects acting on the neutrons were not taken into account in the simulation.

Figure 35 shows a top view of the photon tracks obtained by Phits due to the interaction of the ANNI neutron beam with the carbon-12 target. As the HIBEAM detector will completely surround the target, these photon tracks represent an important background source that will be further studied in dedicated detector simulations in the near future.

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The photon current from the target is obtained by Phits. The current is calculated such that any photon crossing the target surface adds 1 to the current. By multiplying the photon current with the incident neutron current and target area, the photon rate emission from the target over the full energy spectrum can be estimated. The photon rate calculated by Phits is 3.15 × 10⁵ photons s⁻¹. Furthermore, since the collaboration is also exploring the use of beryllium-9 as the material for the target, the photon rate from neutron capture for it was also estimated for it in Phits and found to be 6.68 × 10⁵ photons s⁻¹.

As described in section 8 an annihilation detector must provide as reliable and selective information as possible in each individual event. The crucial information is event topology and conservation of energy and momentum. For free $\bar{n}n$ annihilation events, identifying and measuring total energy and momentum of all outgoing particles gives unique annihilation signatures and one should measure these parameters with the best possible resolution since the discriminating cuts will be limited by the resolution by which the kinetic energy has been measured. However, $\bar{n}n$ annihilation in nuclei (carbon in this case) introduces additional means to dissipate energy and momentum by nuclear fragments. Some of the energy cannot be captured as it is carried by neutrons. Energy carried with protons (and to some minor extent composite fragments) can be measured, but particle identification is important since the proton rest mass shall not be included in the total invariant mass. Taken together, the total invariant mass of pions and photons will only be equal to two neutron masses in about 30% of the cases when the annihilation takes place in a carbon nucleus (see figure 4).

If the kinetic energy of charged nuclear fragments is also measured, this will narrow the distribution, but it is clear that a selection on invariant mass has to be quite generous in order to avoid cutting away good annihilation events. As a consequence, it makes no sense to strive at the highest possible energy resolution. This is actually quite satisfying, since, in particular the charged hadrons are in a very difficult energy regime where energies are often too high to be absorbed by electromagnetic processes only, and the statistical significance on energy deposit by strong interactions is extremely poor. Thus, optimization of the calorimetry is important. Figure 2 illustrates the simulated kinetic energy/momentum distributions of the charged hadrons expected in an annihilation event. Given the distributions shown, an energy resolution for charged hadrons of ±5 MeV appears to be more than sufficient.

The principles

The topological aspects of annihilation events to be handled by the tracking must be considered together with the need that the tracking will provide a $\frac{\mathrm{d}E}{\mathrm{d}x}$ measurement. Since tracks are resolved in three-dimensional space, the track direction for charged particles into the calorimeter is accurately determined. Thus, safe particle identification by combining $\frac{\mathrm{d}E}{\mathrm{d}x}$ and energy is achieved. Background of gamma radiation in the MeV range will be very high. To cope with this background, a high granularity, in space and time, using a large number of detector elements and electronic readout channels is needed. This will be one guideline for a calorimeter design. As pointed out, the kinetic energy resolution can be rather relaxed. For charged hadrons this may pay off in a simple calorimeter design. For photons, however, one should still strive at good energy resolution in order to obtain a narrow invariant mass peak for the identification of the neutral pions.

A.1.1. The tentative calorimeter design

A hybrid approach is adopted for the calorimeter with ten, 3 cm thick layers of plastic scintillators reaching a total thickness of 30 cm followed by a 25 cm (ca 20 radiation lengths) of lead glass for the electromagnetic calorimetry. A Geant-4 [204] visualization of the response of the set-up to a charged pion with 240 MeV kinetic energy is shown in figure A1. The pion punches through the scintillators leaving a cone a Cherenkov light in the lead glass.

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The hadron calorimeter with 30 cm thickness is sufficient to stop protons with more than 200 MeV kinetic energy. About 80% of protons in the high energy end of the spectrum will come to rest without a nuclear interaction. This will cover effectively all protons that happen to be emitted from the annihilation point (figure 2, bottom). However, 30 cm of plastic will only stop pions up to about 80 MeV kinetic energy (260 MeV in momentum, figure 2, top) which is only about 30% of the expected pion spectrum. Charged pions passing through additional 25 cm of lead glass will have about 250 MeV kinetic energy i.e. momentum about 400 MeV which would account for another 30% of the pion spectrum. Even if one can calibrate the energy response by Cherenkov radiation, one cannot expect to make a proper energy measurement since most charged hadrons passing this amount of matter will suffer one or a few hadron nucleus collisions with a rather unpredictable energy signal as result. The very low number of collisions is the reason that hadron calorimetry by showering is not a viable technique at these energies. Actually, the nuclear fragments from these hadronic collisions will not produce Cherenkov light. Thus, calibrated Cherenkov light for charged pions will mostly reflect the range of the pion until it makes a collision. Figure A2 shows the clear, almost linear relation between range and the amount of Cherenkov light, as predicted using Geant-4. Note that only pions with 240 MeV have been injected. Ideally one should have only a sharp peak at about 7000 photons, but anything lower than that is due to collisions on the way.

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Considering all of the above, the strategy is to use the plastic scintillator stack for a range measurement, where each scintillator defines a kinetic energy bin in the proton and pion spectra and the Cherenkov light in the lead glass as a measure of the charged pion range in the lead glass. For cases where a particle comes to rest by electromagnetic energy loss only, the range will give the kinetic energy (since the particle mass is known by the particle identification) with an adequate resolution. In case of a hadronic interaction in the detector materials, one must differ between protons and pions. For protons one should be able to know if a nuclear reaction has taken place since there will be a clear correlation between $\frac{\mathrm{d}E}{\mathrm{d}x}$ and range which is violated if a reaction takes place. The $\frac{\mathrm{d}E}{\mathrm{d}x}$ value itself can, if good enough energy loss resolution is achieved, provide complementary energy information, unaffected by the hadronic collision. For pions however, which almost all will be near minimum ionizing, there is no such correlation between $\frac{\mathrm{d}E}{\mathrm{d}x}$ and range, and one will only be able to state the range of the pion for as long as it was a pion. The charged pions produce Cherenkov light down to quite low kinetic energies in lead glass.

By limiting the information from the plastic layers to hit/no hit information, the light readout and the readout electronics can be made simple (a discriminator threshold only) and cheap, thus allowing a very high segmentation to the benefit of handling the gamma background. The scintillator thicknesses will be chosen such that minimum ionizing particles will give substantially larger signals than Compton electrons from gammas. With 3 cm thickness of scintillating plastic, an MIP will deposit about 6 MeV, much above gammas with nuclear physics origin.

The plastic scintillator layers will be segmented in a way such that coarse tracking in 3D can be done on the hit/no-hit information that is available on a nanosecond time scale. Thus a powerful and fast track trigger can be constructed. Different ways of extracting and sensing the light from the scintillator segments will be investigated in order to optimize the solution for cost, performance and segmentation, taking advantage of the moderate resolution required. The lead glass will have a good energy resolution, being sensitive over the whole volume. Cherenkov light has the advantage of being direction-sensitive. This gives the possibility to measure the direction of particles or even make it blind to particles in the wrong direction. This would be important to discriminate against fake events of CR origin. This is an important R&D topic. The resolution by which the point of impact of gammas from π⁰ decay shall be good in order to obtain good resolution in the reconstruction of π⁰.

This will drive the lateral segmentation of the lead glass either as lead glass blocks or a segmentation of the light readout. Whether or not the readout is direct by photomultipliers or via wavelength shifting fibers is also an R&D item. One could think of segmenting the 25 cm of lead glass in the depth dimension to better measure the range of charged pions. It is questionable if it is worth the increased cost of light sensors and electronics. Possibly, the cost of the lead glass itself is balancing the cost of more readout.

To conclude, it is considered that this unusual detector application aimed for too high energy for nuclear physics and too low energy for particle physics methods offers many interesting detector R&D challenges. It is anticipated that it would be deployed in the ESS test beam described below. The prototype would be supplemented with a scintillator cosmic shield and an inner TPC.

A dedicated test beam line (TBL) will be build at the ESS to be used initially to verify that the accelerator has successfully delivered beam on target, and to characterize the pulsed neutron beam emitted from the upper moderator (e.g. time structure, spatial distribution, energy dependence etc), while it can be adjusted to also view a future lower moderator. In the longer term, it will provide supporting measurements for the user program and also serve for the development of key neutron technologies, such as optical components, choppers, and detector systems. For the last mentioned purpose, the ESS TBL is an ideal place to perform tests of the prototypes of HIBEAM/NNBAR detectors. Figure A3 shows a schematic overview of the TBL. In its basic configuration, it is a pin-hole (camera obscura) imaging station for viewing the entire width and height of the moderator assembly, with a double-disc chopper at the pin-hole position to provide tuneable wavelength resolution and wavelength band selection. For the moderator characterization, a position sensitive detector will be placed at 17 m from the moderator with an adjustable pin-hole at the half-way position, enabling the spatial imaging of neutrons of different wavelengths emerging from the moderator. The instrument will be in direct line-of-sight and not be equipped with optics, but in addition to the double disk chopper will have a series of collimators (one stationary and one adjustable in size), a range of optional beam attenuators and filters, and a heavy shutter. The test for the HIBEAM/NNBAR detectors will be done in combination with the annihilation target. After hitting the annihilation target, the beam will be absorbed in a beam stop that is already part of the TBL. With a direct view of the source, the TBL represents also an optimal place to make background measurements since it will not only provide cold but also fast and high energy neutrons as can be seen in figure A4.

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Planned measurements include:

• Test of the full prototype detector
• Benchmark against experimental data of Monte Carlo background simulations
• Gamma background measurements from neutron interaction and activation with the annihilation target and surrounding materials
• Fast neutron background measurements
• CRs and skyshine background characterization

All these measurements will allow a deep understanding of the background and provide a better mitigation strategy both for HIBEAM and the NNBAR experiment. Finally, it is worth mentioning that most of the backgrounds, along with the signal Monte Carlo events provided by the authors reference [172], have already been preliminarily simulated and interfaced with a simulation of the NNBAR–HIBEAM detector in Geant4 via a coherent software framework. The results will be described in a forthcoming publication.