Review of sterile neutrino searches at very short-baseline reactor experiments

Mikhail Danilov

doi:10.1088/1402-4896/ac81fd

1. Introduction

The Standard Model (SM) is probably the most elaborate theory of Matter ever developed (for a recent review see e.g. [1]). It describes practically everything that we observe very often with very high precision. The final confirmation of SM was the discovery of the famous Higgs Boson at the Large Hadron Collider (LHC) in 2012 [2, 3]. Still, it is commonly believed that SM is not the ultimate theory. First of all, it does not describe what we do not see, namely, the Dark Matter that is about 6 times more abundant in the Universe than the ordinary matter that is described by SM [4]. The dominance of matter over antimatter in the Universe is also not explained. Fine tuning of the models parameters is required to explain a relatively small mass of the Higgs Boson. And the number of these free model parameters is very large. SM does not predict the masses of fundamental particles, the couplings between quarks of different generations, and the strength of the weak, electromagnetic and strong interactions. It does not explain why there are just 3 generations of quark and leptons. Finally, SM does not include gravity. Therefore, search for phenomena beyond SM is now the main direction in particle physics.

There are several indications of Physics beyond SM. The long-standing discrepancy between the very precise experimental measurements (relative accuracy is 0.35 ppm) and theoretical predictions (relative accuracy is 0.37 ppm) for the anomalous muon magnetic moment has about 4.2 standard deviation (σ) statistical significance [5]. A considerable improvement in the experimental accuracy is expected in the near future.

There are several discrepancies at the level of 3σ between experimental data and SM predictions in beauty hadron decays. Probably the most remarkable is the difference in the probabilities of decays with μ⁺ μ⁻ and e⁺ e⁻ in the final state which should be equal in SM (see [6] and references therein). Again, a better accuracy is required in order to make definite conclusions. A considerable improvement in experimental accuracy is expected in this field in experiments at LHC as well as at the SuperB Factory at KEK.

Finally, several indications of New Physics beyond SM exist in the neutrino sector. Discovery of neutrino oscillations [7, 8] demonstrated that neutrinos have masses. This fact can be considered as a sign of New Physics since neutrinos are massless in SM. However, it is possible to extend SM and include neutrino masses to it. There is another hint of New Physics in the neutrino sectorsterile neutrinos. There are several experimental indications of existence of sterile neutrinos with masses of the order of 1 eV. A detailed discussion of these indications and future prospects for sterile neutrino searches with the emphasis on reactor experiments are the main topics of the present paper.

A possible discovery of the sterile neutrinos would require a modification of SM and also a modification of the Standard Model of Cosmology. Searches for New Physics beyond SM is the main direction in particle physics nowadays. Experimental evidences of the existence of sterile neutrinos with a mass of about 1 eV are the most statistically significant indications of New Physics. They are still controversial and new experiments are planned to clarify the situation. Very important results in this field were obtained in 2021 and new crucial results are expected in the near future.

2. Hints for sterile neutrinos

Oscillations of atmospheric neutrinos were discovered by the Super Kamiokande experiment [7]. The deficit in the detected solar neutrino flux observed by a variety of experiments has been interpreted as due to neutrino flavor oscillations. This process implies that neutrinos emitted in the core of the Sun as electron neutrinos can be detected as neutrinos of another flavor in detectors on the Earth. The existence of solar electron neutrino deficit has been discovered by several solar neutrino experiments [9–11]. The SNO experiment then demonstrated [8] that the deficit in the electron neutrino flux was compensated by the appearance of muon and tau neutrinos. Thus, the neutrino oscillations explain the reduction of the solar electron neutrino flux. The observation of neutrino oscillations demonstrates that neutrino have masses and that the lepton mixing matrix (Pontecorvo, Maki, Nakagava, Sakata matrix) [12–14] is non trivial. This matrix connects the neutrino flavor eigenstates ν_e, ν_μ, and ν_τ with the neutrino mass eigenstates ν₁, ν₂, and ν₃. Since the measured neutrino mass-squared differences demonstrate a hierarchical structure ( $| {\rm{\Delta }}{m}_{{atm}}^{2}| \approx 2.5\times {10}^{-3}$ eV², ${\rm{\Delta }}{m}_{{solar}}^{2}=7.53\pm 0.18\times {10}^{-5}$ eV² [1]) the probabilities of neutrino flavor changes from α to β are often well approximated by a two-neutrino formula(see e.g. [15])

$\begin{eqnarray}&&{Prob}({\nu }_{\alpha }\to {\nu }_{\beta })\approx {\sin }^{2}2{\theta }_{\alpha \beta }{\sin }^{2}\left(\displaystyle \frac{1.27{\rm{\Delta }}{m}_{{ij}}^{2}[\,{\mathrm{eV}}^{2}]L[{\rm{m}}]}{{E}_{\nu }[\mathrm{MeV}]}\right),\end{eqnarray} \tag{ 1 }$

with only one mixing angle θ_{α
β} and one difference of neutrino masses squared ${\rm{\Delta }}{m}_{{ij}}^{2}={m}_{i}^{2}-{m}_{j}^{2}$ of the relevant neutrino mass states. In this expression E_ν and L are the neutrino energy and the path length. The survival probability is then given by

$\begin{eqnarray}&&{Prob}({\nu }_{\alpha }\to {\nu }_{\alpha })\approx 1-{\sin }^{2}2{\theta }_{\alpha \alpha }{\sin }^{2}\left(\displaystyle \frac{1.27{\rm{\Delta }}{m}_{{ij}}^{2}[\,{\mathrm{eV}}^{2}]L[{\rm{m}}]}{{E}_{\nu }[\mathrm{MeV}]}\right).\end{eqnarray} \tag{ 2 }$

At large distances the oscillation pattern is smeared out because of finite accuracy of energy and/or distance measurements and only the initial neutrino flux is reduced:

$\begin{eqnarray}&&{Prob}({\nu }_{\alpha }\to {\nu }_{\alpha })\approx 1-0.5{\sin }^{2}2{\theta }_{\alpha \alpha }.\end{eqnarray} \tag{ 3 }$

Oscillations of the three known neutrino flavors have been measured with a good precision. The two mass-squared differences and three angles describing such oscillations are well known by now [1].

Z boson can decay into a neutrino-antineutrino pair. Therefore, measurements of the Z boson decay width can be used to determine the number of active neutrinos with masses smaller than the half of the Z boson mass. This number was found to be equal to 3 [16]. However, additional sterile neutrinos are allowed. They are singlets of the SM gauge group and do not interact directly with gauge bosons. Usually only one sterile neutrino ν_s is considered which consists mainly of the heavy ν₄ neutrino mass state while the 3 active neutrinos ν_e, ν_μ, and ν_τ are mainly composed of light neutrino mass states ν₁, ν₂, and ν₃. This is the so called 3 + 1 sterile-active neutrino mixing model which we will call the 4ν model for brevity. In this model θ_ee = θ₁₄.

Sterile neutrinos appear naturally in many extensions of SM. Moreover, there are several experimental hints of their existence. The GALLEX and SAGE solar neutrino gallium experiments performed calibrations with very powerful radioactive sources. The radioactive sources were placed in the centers of the gallium containing volumes with the average neutrino path length of 72.6 cm (SAGE) and 190 cm (GALLEX). SAGE and GALLEX observed only (88 ± 5)% ν_e events of the expected number [10]. The deficit of ν_e events in the calibration runs of the SAGE and GALLEX experiments [10, 11] ('Gallium Anomaly'(GA) [17]) can be explained by electron neutrino to sterile neutrino oscillations at very short distances with quite a large ${\sin }^{2}2{\theta }_{{ee}}\approx 0.24$ according to equation (3). Since the baselines were quite short in the both experiments the Δm² for the additional mass eigenstate should be quite large at least in the eV² range.

Analogously the 6% deficit of ${\tilde{\nu }}_{e}$ in comparison with the re-evaluated reactor fluxes [18, 19] ('Reactor Antineutrino Anomaly'(RAA) [20]) can be explained by the active-sterile neutrino oscillations again at very short distances. The required short distances of the oscillations imply that the mass-squared difference Δm² should be about 1eV², much larger than the two known mass-squared differences. Therefore an additional neutrino is needed and, as explained above, it should be sterile. Since the mass-squared difference between a new and known neutrinos is much larger than the mass-squared differences between known neutrinos the oscillations between known and sterile neutrinos can be well described by a two-neutrino oscillation formula (1) with only one ${\rm{\Delta }}{m}_{41}^{2}$ . RAA leads to the best-fit values for sterile neutrino parameters of ${\rm{\Delta }}{m}_{41}^{2}=2.3$ eV² and ${\sin }^{2}2{\theta }_{{ee}}=0.14$ .

More recent estimates of the ${\tilde{\nu }}_{e}$ fluxes from reactors [21, 22] have not solved the problem with RAA since one of them [22] predicts a smaller ${\tilde{\nu }}_{e}$ flux while the other one [21] predicts a larger ${\tilde{\nu }}_{e}$ flux in comparison with the Huber-Mueller (H-M) model [18, 19]. The Daya Bay [23] and RENO [24] collaborations have measured the ²³⁵U contribution to the reactor ${\tilde{\nu }}_{e}$ flux multiplied by the Inverse Beta Decay (IBD) reaction cross section. The IBD process is used to detect ${\tilde{\nu }}_{e}$ at reactors. The results are about 10% smaller than the H-M model predictions.

The H-M model converts the measurements of the beta spectra from thermal neutron fission products of ²³⁵U, ²³⁹Pu, and ²⁴¹Pu at ILL [25–28] to the ${\tilde{\nu }}_{e}$ spectra. Recent measurements at the Kurchatov Institute [29, 30] of the ratio of beta spectra of fission products of ²³⁵U and ²³⁹Pu give (5.4 ± 0.2)% smaller ratio than the ILL results.

Smaller values for ²³⁵U contribution to the IBD rate obtained by Daya Bay and RENO, and the Kurchatov Institute experiment result reduce considerably the significance of RAA.

There is also a difference between measured and predicted ${\tilde{\nu }}_{e}$ energy spectra. Their ratio has a bump at about 6 MeV [31–33]. A recent discussion of different approaches to the reactor antineutrino spectrum calculations as well as the comparison with the experimental data can be found in [34]. However, all modern searches for sterile neutrinos measure the ratio of the ${\tilde{\nu }}_{e}$ energy spectra at different distances from reactors and therefore they do not rely on the absolute ${\tilde{\nu }}_{e}$ flux predictions as well as on the shape of the predicted ${\tilde{\nu }}_{e}$ energy spectrum.

Very recently the BEST experiment has confirmed GA [35, 36]. The observed deficit of ν_e events is even larger ((79 ± 5)% and (77 ± 5)% in the inner and outer volumes of the detector) and has a significance of above 5σ [37]. For ${\rm{\Delta }}{m}_{41}^{2}\lt 5$ eV² the very large values of ${\sin }^{2}2{\theta }_{{ee}}\approx 0.4$ preferred by BEST (see equation (3)) have been already excluded by DANSS [38, 39] and NEOS [40]. This will be discussed in the next section. For large mass square differences ${\rm{\Delta }}{m}_{41}^{2}\gt 5$ eV² and ${\sin }^{2}2{\theta }_{{ee}}$ the BEST results are in tension with the limits obtained by Daya Bay, Bugey-3 and RENO (see e.g. [41]) using predictions for the absolute ${\tilde{\nu }}_{e}$ flux from reactors including their large uncertainties. A part of the BEST preferred region at large Δm² was already excluded by the PROSPECT [42] and STEREO [43] experiments.

The LSND collaboration obtained evidence for electron antineutrino appearance in the muon antineutrino beams at distances where known neutrino oscillations can not contribute [44]. The corresponding mass-squared difference should be larger than ∼ 0.2 eV² [44] that is much bigger than the mass-squared differences of known neutrino oscillations. The first MiniBooNE checks of the LSND signal were not conclusive [45–47]. Later the MiniBooNE collaboration obtained the 4.8σ evidence for electron (anti)neutrino appearance in the muon (anti)neutrino beams [48, 49]. The significance of the signal grows to 6.1σ if the LSND and MiniBooNE results are combined [49]. The best-fit point in the sterile neutrino parameter space is close to the maximal mixing ( ${\sin }^{2}2{\theta }_{e\mu }=0.807$ ) and a small mass-square difference of ${\rm{\Delta }}{m}_{14}^{2}=0.043\,{\mathrm{eV}}^{2}$ (see figure 1). However, this area is excluded by OPERA [50] and only a region with larger mass-squared differences and smaller mixing is still allowed.

Figure 1. Refer to the following caption and surrounding text. — **Figure 1.** MiniBooNE allowed regions for combined neutrino and antineutrino mode data sets for events with 200 < E_ν < 3000 MeV within a two-neutrino oscillation model. The shaded areas show the 90% and 99% Confidence Level (C.L.) LSND $\tilde{{\nu }_{\mu }}\to \tilde{{\nu }_{e}}$ allowed regions. The black point shows the MiniBooNE best-fit point. Also shown are 90% C.L. limits from the KARMEN [83] and OPERA [50] experiments. Figure is adopted from [49].
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**Figure 1.** MiniBooNE allowed regions for combined neutrino and antineutrino mode data sets for events with 200 < E_ν < 3000 MeV within a two-neutrino oscillation model. The shaded areas show the 90% and 99% Confidence Level (C.L.) LSND $\tilde{{\nu }_{\mu }}\to \tilde{{\nu }_{e}}$ allowed regions. The black point shows the MiniBooNE best-fit point. Also shown are 90% C.L. limits from the KARMEN [83] and OPERA [50] experiments. Figure is adopted from [49].
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The sterile neutrino explanation of the LSND and MiniBooNE results requires mixing of sterile neutrinos both with electron and muon neutrinos. The probability of the electron neutrino appearance in the muon neutrino beam is proportional to the probabilities of the ν_e disappearance and ν_μ disappearance ( ${\sin }^{2}2{\theta }_{e\mu }\approx 0.25{\sin }^{2}2{\theta }_{{ee}}{\sin }^{2}2{\theta }_{\mu \mu }$ for small angles). Hence the strong upper limits on ${\tilde{\nu }}_{e}$ and ν_μ disappearance lead to strong limits on the ν_e appearance in ν_μ beams that contradict the LSND and MiniBooNE results [51, 52]. However one should mention that a weak indication of ν_μ disappearance was obtained by the Ice Cube experiment ( ${\sin }^{2}2{\theta }_{\mu \mu }=0.10,{\rm{\Delta }}{m}_{14}^{2}=4.5\,{\mathrm{eV}}^{2}$ ) [53]. With such a large mixing parameter appearance and disappearance results are marginally compatible at large ${\rm{\Delta }}{m}_{14}^{2}$ . On the other hand such large mixing is in contradiction with strong limits on ν_μ disappearance mainly form the MINOS/MINOS+ experiments [54].

The overall situation with the ν_e appearance is not yet clear and more results are needed to clarify it. The MiniBooNE results are scrutinized by the MicroBooNE experiment at the same neutrino beam. MicroBooNE observed even smaller number of ν_e events than expected and established upper limits on the possible excess [55]. However, the sterile neutrino explanation of the MiniBooNE excess is not completely ruled out [56] (see figure 2).

Figure 2. Refer to the following caption and surrounding text. — **Figure 2.** MicroBooNE constraints on the sterile neutrino parameter space at 3σ C.L. (blue, inclusive and orange, Charged-Current Quasielastic). Also are shown the MiniBooNE 1−, 2−, and 3−σ preferred regions in shades of grey [49], the future sensitivity of the three SBN detectors (pink [84]), and existing constraints from KARMEN (green [83]) and OPERA (gold [50]). Figure is adopted from [56].
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The Neutrino-4 collaboration claimed in 2018 an observation of ${\tilde{\nu }}_{e}$ oscillations to sterile neutrinos with very large values of ${\rm{\Delta }}{m}_{41}^{2}\simeq 7$ eV² and ${\sin }^{2}2{\theta }_{{ee}}\simeq 0.4$ although the significance of the result was only 2.8σ [57, 58] (see figure 3).

Figure 3. Refer to the following caption and surrounding text. — **Figure 3.** Normalized L/E dependence of the IBD rate in the Neutrino-4 experiment (triangles) and theoretical predictions (dots). Figure is adopted from [57].
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The Neutrino-4 analysis was criticized in [59–62]. Neutrino-4 initially refused these comments [63, 64] but later took into account two of them [65]. This reduced the significance of the result by ≈ 0.5σ. The best-fit point for the improved analysis and increased data sample is ${\rm{\Delta }}{m}_{41}^{2}=7.3\pm 1.17$ eV², ${\sin }^{2}2{\theta }_{{ee}}=0.36\pm {0.12}_{\mathrm{stat}}$ [65]. The statistical significance of the result is 2.7σ. The Neutrino-4 claim is in tension with the PROSPECT results [42], with the measurements of the absolute ${\tilde{\nu }}_{e}$ flux from reactors [41], and Solar neutrino measurements [62]. On the other hand the Neutrino-4 result is in a perfect agreement with the recent BEST result [35, 36].

3. Searches for sterile neutrinos at very short base-line reactor experiments

This section is a cardinal update of the presentation at ICCPA-2018 [59].

3.1. The DANSS experiment

The DANSS detector [66] consists of 2500 scintillator strips (1 × 4 × 100 cm³) with a thin (∼0.2mm) Gd-containing reflective surface coating. The strips are arranged in 100 layers. The strips in the adjacent layers are orthogonal. Light from the scintillator strip is collected with three wavelength-shifting fibers glued into grooves along the strip. The central fiber is read out with a Silicon PhotoMultiplier (SiPM). The side fibers from 50 parallel strips are bundled together and read out with a compact PhotoMultiplier tube (PMT). The scintillator counters are surrounded with a composite shielding to suppress backgrounds. DANSS is placed on a movable platform under the core of the 3.1 GW_th industrial power reactor at the Kalinin Nuclear Power Plant (KNPP) in Russia. The detector distance to the reactor core center is changed from 10.9 m to 11.9 m, and 12.9 m 2–3 times a week. Reactor materials provide ∼50 m of water equivalent (m.w.e.) shielding that removes the hadron component of the cosmic background and reduces the cosmic muon flux by a factor of 6. The very good suppression of the cosmic background and the high granularity of the detector allow DANSS to achieve a very high signal/background (S/B) ratio of more than 50 (at 10.9 m from the reactor, after model independent subtraction of the accidental background). The size of the reactor core is quite big (3.7 m in height and 3.2 m in diameter) which leads to smearing of the oscillation pattern. This drawback is compensated by a high $\tilde{{\nu }_{e}}$ flux which allows DANSS to detect more than 5 thousand ${\tilde{\nu }}_{e}$ per day at a distance of 10.9 m. The energy resolution of the DANSS detector is very modest (σ_E/E ∼ 34% at E = 1 MeV). This leads to additional smearing of the oscillation pattern, comparable with the smearing due to the large reactor core size.

The IBD reaction ${\tilde{\nu }}_{e}+p\to {e}^{+}+n$ is used to detect ${\tilde{\nu }}_{e}$ . Positrons immediately deposit their energy in the detector and annihilate with production of 2 or 3 gammas with the total energy of 1.02 MeV. Neutrons are slowed down to thermal energies where the capture cross-section is high and then captured by Gd which in turn emits gammas with the total energy of about 8 MeV. Thus the IBD reaction produces two signals, prompt and delayed. The delayed coincidence of these two signals allows to suppress the background drastically. The positron and neutron signals are close to each other not only in time but also in space. The neutron scatters many times before the capture but still a typical distance between the positron and neutron capture point is about 10 cm. This spacial coincidence can be used to further reduce the accidental background in detectors with high granularity capable to reconstruct the positron and the neutron capture positions. The ${\tilde{\nu }}_{e}$ energy can be inferred from the positron energy ${E}_{\tilde{\nu }}\approx {E}_{{e}^{+}}+1.8\,\mathrm{MeV}$ where 1.8 MeV is the IBD reaction threshold energy or from the prompt signal energy that includes 1.02 MeV from the annihilation gammas.

The DANSS experiment compares the positron energy spectra measured with the same detector at the three distances from the reactor core using only relative IBD counting rates. This is the most conservative approach which does not depend on the predicted ${\tilde{\nu }}_{e}$ flux and spectrum as well as on the detector efficiency (only short term variations of the efficiency can influence the results). Figure 4 shows the ratio of positron energy spectra at the bottom and top detector positions. It is consistent with the 3ν hypothesis (black curve). There is no statistically significant evidence for sterile neutrinos. The best 4ν hypothesis fit (red curve) is only slightly better (1.3σ). Hence there is no statistically significant evidence for sterile neutrinos. The best-fit point of RAA (cyan curve) is clearly excluded at more than 5σ. Figure 5 (left) shows the 90% Confidence Level (C.L.) area excluded by DANSS in the ${\rm{\Delta }}{m}_{14}^{2},\,{\sin }^{2}2{\theta }_{{ee}}$ plane [67]. The excluded area covers a large fraction of regions indicated by the RAA. In particular, the best fit point ${\rm{\Delta }}{m}_{14}^{2}=2.3\,{\mathrm{eV}}^{2},\,{\sin }^{2}2{\theta }_{{ee}}=0.14$ [20] is excluded at more than 5σ C.L. A large sterile neutrino parameter region preferred by the recent BEST results [35, 36] including the best fit point was already excluded by DANSS and NEOS (see figure 5(right)). After modernization in 2022 DANSS will be able to scrutinize much larger area including the Neutrino-4 preferred region (see figure 5(right)). Upgraded DANSS will have almost 3 times better energy resolution and 1.7 times larger volume [68].

Figure 4. Refer to the following caption and surrounding text. — **Figure 4.** Ratio of positron energy spectra measured at the bottom and top DANSS detector positions (statistical errors only). The black curve is the prediction for the 3ν case, the red curve corresponds to the best fit in the 4ν mixing scenario ( ${\sin }^{2}2{\theta }_{{ee}}=0.014$ , ${\rm{\Delta }}{m}_{41}^{2}=1.3\,{\mathrm{eV}}^{2}$ ) for the full analysis , the cyan curve is the expectation for the best-fit point for RAA [20] ( ${\sin }^{2}2{\theta }_{{ee}}=0.14$ , ${\rm{\Delta }}{m}_{41}^{2}=2.3\,{\mathrm{eV}}^{2}$ ).
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**Figure 4.** Ratio of positron energy spectra measured at the bottom and top DANSS detector positions (statistical errors only). The black curve is the prediction for the 3ν case, the red curve corresponds to the best fit in the 4ν mixing scenario ( ${\sin }^{2}2{\theta }_{{ee}}=0.014$ , ${\rm{\Delta }}{m}_{41}^{2}=1.3\,{\mathrm{eV}}^{2}$ ) for the full analysis , the cyan curve is the expectation for the best-fit point for RAA [20] ( ${\sin }^{2}2{\theta }_{{ee}}=0.14$ , ${\rm{\Delta }}{m}_{41}^{2}=2.3\,{\mathrm{eV}}^{2}$ ).
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Figure 5. Refer to the following caption and surrounding text. — **Figure 5.** Left panel: DANSS 90% C.L. exclusion area (cyan) and the boundary of the 90% C.L. sensitivity area (red dashed). Grey areas show predictions from RAA. Left panel is adopted from [67]. Right panel: 90% C.L. exclusion area for the current analysis (cyan), 90% C.L. expected sensitivity area after the DANSS upgrade (red). Grey areas show results of the BEST experiment, star marker indicates the best-fit point in the Neutrino-4 experiment.
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3.2. The NEOS experiment

The NEOS detector [40] had one unsegmented volume filled with 0.87 ton of 0.5% Gd-doped liquid scintillator. It was installed at a 2.8 GW_th reactor unit 5 of the Hanbit Nuclear Power Complex (Korea) at 23.7 ± 0.3 m from the reactor core center. As in the DANSS case a quite large active core size (3.1 m in diameter, 3.8 m in height) leads to the oscillation pattern smearing. On the other hand the IBD counting rate is quite high (∼2000 events/day). The background caused by the neutron scattering which imitates the positron signal and subsequent capture of neutrons is rejected with a high efficiency of 73% using pulse shape discrimination (PSD) that suppresses signals from the recoil protons. Together with the sizable overburden of ∼20 m.w.e. this allows to achieve a very good S/B ratio of 22.

NEOS took data only at one distance from the reactor. Initially NEOS normalized its data on the ${\tilde{\nu }}_{e}$ energy spectrum measured at a different reactor by the Daya Bay collaboration and obtained strong limits on the sterile neutrino parameters [40] shown in figure 6 (left). Later the RENO and NEOS collaborations established even stronger limits using a joint analysis of the data from the same reactor complex [69] (see figure 6 (left)). It is not clear why the Feldman-Cousine statistical analysis method [70] leads to much stronger limits in comparison with the raster scan or CL_s [71] methods. Monte Carlo simulations of generic experiments searching for sterile neutrinos predicted comparable sensitivities for these methods [71]. Figure 6 (right) shows that the DANSS and NEOS limits obtained with the same raster scan method complement each other. The RENO-NEOS best-fit point ${\rm{\Delta }}{m}_{14}^{2}=2.41\pm 0.03,\,{\sin }^{2}2{\theta }_{{ee}}=0.08\pm 0.03$ agrees nicely with the RAA best-fit point. However these parameters are already excluded by DANSS.

Figure 6. Refer to the following caption and surrounding text. — **Figure 6.** Left: Comparison of the exclusion limits on sterile neutrino oscillations and an allowed region. The right side of each contour indicates an excluded region. The black curve (cyan filled region) represents a 95% (68%) C.L. exclusion contour (allowed region) obtained from the RENO and NEOS combined search using the Feldman and Cousins method [70]. The orange (blue) curve represents a 90% C.L. exclusion contour obtained from the RENO and NEOS combined search using the raster scan method where the spectral comparison is made at NEOS (RENO) detector. The best fit parameter (black point) is found at $| {\rm{\Delta }}{m}_{41}^{2}| =2.41\,{\mathrm{eV}}^{2}$ and ${\sin }^{2}2{\theta }_{{ee}}=0.08$ . For the comparison, shown are the NEOS+Daya Bay [40] 90% C.L. (gray shaded) and RENO far/near [85] 95% C.L. (blue dotted) limits on the disappearance. Also shown is a 95% C.L. allowed region of RAA [20] (black dotted) with the best fit [86] (star). Figure is adopted from [69]. Right: 90% C.L. exclusions obtained with the raster scan method. Cyan area—DANSS exclusions based on 5 millions of IBD events, black line NEOS 2017 results [40], orange line—combined NEOS + RENO results [69].
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**Figure 6.** Left: Comparison of the exclusion limits on sterile neutrino oscillations and an allowed region. The right side of each contour indicates an excluded region. The black curve (cyan filled region) represents a 95% (68%) C.L. exclusion contour (allowed region) obtained from the RENO and NEOS combined search using the Feldman and Cousins method [70]. The orange (blue) curve represents a 90% C.L. exclusion contour obtained from the RENO and NEOS combined search using the raster scan method where the spectral comparison is made at NEOS (RENO) detector. The best fit parameter (black point) is found at $| {\rm{\Delta }}{m}_{41}^{2}| =2.41\,{\mathrm{eV}}^{2}$ and ${\sin }^{2}2{\theta }_{{ee}}=0.08$ . For the comparison, shown are the NEOS+Daya Bay [40] 90% C.L. (gray shaded) and RENO far/near [85] 95% C.L. (blue dotted) limits on the disappearance. Also shown is a 95% C.L. allowed region of RAA [20] (black dotted) with the best fit [86] (star). Figure is adopted from [69]. Right: 90% C.L. exclusions obtained with the raster scan method. Cyan area—DANSS exclusions based on 5 millions of IBD events, black line NEOS 2017 results [40], orange line—combined NEOS + RENO results [69].
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In the phase-II NEOS collected data during the whole reactor cycle of 500 days and 2 reactor-off periods. There was some instability in the liquid scintillator response. The data analysis is ongoing.

3.3. The Neutrino-4 experiment

The Neutrino-4 detector is made of 50 liquid scintillator sections with a total (fiducial) volume of 1.8(1.42)m³ [65]. Each section is read out with a single PMT at the top of the section via an 8cm air gap. The air gap reduces the light collection efficiency from the scintillator volume close to the PMT and makes the light collection efficiency reasonably uniform for the whole section. However, the reduction of the light yield makes PSD not efficient. The detector is installed on a movable platform near a very compact (42 × 42 × 35 cm³) and powerful (100 MW) SM-3 research reactor at Dmitrovgrad (Russia). The distance to the reactor core is changed every 10-15 days which allows to perform measurements in the range from 6 m to 12 m. This is an enormous asset in the control of systematic uncertainties. A small overburden of 3.5 m.w.e. and absence of PSD lead to a very modest S/B ratio of 0.54. The energy resolution is also modest (∼16% at 1 MeV). In the analysis of the data the energy resolution is assumed to be equal to 250 keV for all energies. Such unusual energy independence of the resolution is inferred from the background shape that is dominated by several γ lines. However the resolution for γ sources that is dominated by measurements of several Compton electrons can be different from the resolution for IBD positrons.

There were several critical remarks on the Neutrino-4 analysis [59–62]. Two of them were taken into account in the recent analysis (see figure 7) [65]. The results were obtained assuming that the test statistics has the χ² distribution with 2 degrees of freedom. However, this assumption is not valid. Therefore, in order to determine the signal significance Neutrino-4 performed Monte Carlo studies and obtained the 2.7σ significance of the sterile neutrino signal including systematic uncertainties. The best fit point is ${\rm{\Delta }}{m}_{41}^{2}=7.3\pm 1.17$ eV², ${\sin }^{2}2{\theta }_{{ee}}=0.36\pm {0.12}_{\mathrm{stat}}$ [65]. The obtained sterile neutrino parameters are in tension with the limits obtained by the reactor ${\tilde{\nu }}_{e}$ flux measurements at larger distances even when the large uncertainties in the predictions of the flux are taken into account (see, for example [41]). The PROSPECT and STEREO results [42, 43] are also in tension with the Neutrino-4 claim as will be discussed below. The analysis of solar neutrino data excludes ${\sin }^{2}2{\theta }_{{ee}}\gt 0.22$ at 95% C.L. [62] that is a large fraction of the Neutrino-4 preferred area including the best-fit point.

Figure 7. Refer to the following caption and surrounding text. — **Figure 7.** Accepted (1σ -blue, 2σ -green, 3σ - yellow), and excluded (3σ -pink) areas for 3 + 1 neutrino oscillations obtained by Neutrino-4. Figure is adopted from [65].
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The Neutrino-4 collaboration is constructing a new detector that will have 3 times better sensitivity [65]. The new liquid scintillator with larger amount of Gd will decrease the neutron capture time and hence the background. PSD will further decrease the background. The energy resolution will be considerably improved by using a two-sided readout of horizontal sections (now they are vertical with one PMT). The new detector should be completed already in 2022.

3.4. The PROSPECT experiment

The PROSPECT detector [42, 72] consists of 154 optically isolated rectangular segments (14.5 cm × 14.5 cm × 117.6 cm) filled with liquid scintillator loaded with ⁶Li in order to capture and detect neutrons and read out by two PMT each. The PROSPECT detector is installed at the High Flux Isotope Reactor (HFIR) at Oak Ridge National Laboratory with less than 1 m.w.e. overburden. Nevertheless, a very good PSD and 3D reconstruction of events allowed PROSPECT to achieve a decent S/B ratio of 1.36. A high reactor power (85 MW), the large detector (∼4 ton), and a small distance from the reactor (6.7 m) allow PROSPECT to collect about 800 IBD events per day with excellent energy resolution of 4.5% at 1 MeV. Unfortunately 42% of segments were not operational because of problems with PMTs. Nevertheless PROSPECT excluded a sizable part of the sterile neutrino parameters (see figure 8) [42]. The obtained limits are in tension with the Neutrino-4 result but can not exclude it.

Figure 8. Refer to the following caption and surrounding text. — **Figure 8.** Left: Oscillation exclusion contours derived using the Gaussian CL_s and Feldman-Cousins (FC) methods. Also pictured are the 1σ and 2σ (green and yellow) exclusion ranges produced by PROSPECT toy MC data sets, as well as the RAA preferred parameter space and best-fit point from [20]. Figure is adopted from [42]. Right: Comparison of sterile oscillation sensitivities for different current and projected PROSPECT [42] and PROSPECT-II data sets. Figure is adopted from [73].
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PROSPECT plans to upgrade the detector and increase considerably the sensitivity (see figure 8 right) [73].

3.5. The SoLid experiment

The extremely highly segmented SoLid detector consists of 12 800 polyvinyltoluene cubes ((5 × 5 × 5) cm³) with thin sheets of ⁶LiF:ZnS(Ag) to capture and detect neutrons [74]. Light from each cube is collected with three orthogonal wavelenght-shifting fibers read out with SiPMs. The SoLid detector is installed at a distance of ∼ 6 m of the 60 MW SCK·CEN BR2 research reactor in Belgium. It has a modest energy resolution of ∼14% at 1 MeV and good PSD for background rejection. Unfortunately the internal radioactivity from the contamination of ⁶LiF:ZnS layers leads to a very high background. The IBD signals are imitated by electrons from ²¹⁴ Bi decays followed by α particles from ²¹⁴ Po decays. With complicated machine learning techniques SoLid managed to extract the IBD signal but so far has no results on sterile neutrino searches. SoLid upgrades the detector with new SiPMs to mitigate the problems [75]. Unfortunately, it is too expensive to change the contaminated ⁶ LiF: ZnS(Ag) sheets.

3.6. The STEREO experiment

The STEREO detector [43] is made of six optically separated cells filled with a gadolinium loaded liquid scintillator. STEREO was installed at the High Flux Reactor of the Institute Laue-Langevin. The cell distances from the core range from 9.4 m to 11.1 m. Elaborated PSD technique was used to suppress the background from fast neutrons. Still, the achieved S/B ratio of 0.9 was quite modest. STEREO excluded a sizable fraction of the sterile neutrino parameter space [43] (see figure 9). As in the case of NEOS, the Feldman-Cousines analysis method gives much stronger limits in comparison with other methods. Moreover, the limits are much stronger than the experiment sensitivity.

Figure 9. Refer to the following caption and surrounding text. — **Figure 9.** Comparison of the exclusion contours (solid) and exclusion sensitivity contours (dashed) at 95% C.L. of phase I + II, for the two-dimensional method (red), the raster-scan method (dark blue), and the CL_s method (light blue). Overlaid are the allowed regions of the RAA (grey) and its best-fit point (star) [86]. Figure is adopted from [43].
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STEREO finished the data taking but the data analysis is still ongoing.

3.7. The JUNO-TAO experiment

The Juno-TAO is a ton-level liquid scintillator detector at ∼30 meters from a reactor core of the Taishan Nuclear Power Plant in Guangdong, China [76]. The detector will have extremely good energy resolution of 2% at 1 MeV and will detect 2000 ${\tilde{\nu }}_{e}$ per day. The expected Juno-TAO 90% C.L. sensitivity on ${\sin }^{2}2{\theta }_{{ee}}$ with 3 years of data taking is about 0.01-0.02 in the wide range of ${\rm{\Delta }}{m}_{14}^{2}$ between 0.06 eV² and 3 eV². JUNO-TAO plans to start data taking in 2022.

3.8. Summary on very short base-line (VLBS) experiments

The advantages (red color) and disadvantages (blue color) of recent VSBL experiments are summarized in table 1. All modern experiments dont rely on the predicted reactor antineutrino spectrum since it has large uncertainties. Instead, they compare measured antineutrino spectra at different distances from the reactor core. The experiments at industrial reactors (DANSS and NEOS) benefit from the high counting rates up to 5000 ${\tilde{\nu }}_{e}$ /day. They have the highest sensitivity of ${\sin }^{2}2{\theta }_{{ee}}\lt 0.01$ at ${\rm{\Delta }}{m}_{14}^{2}\,\approx$ (1-2) eV². At larger ${\rm{\Delta }}{m}_{14}^{2}$ oscillations are averaged out already inside the large reactor core and the sensitivity deteriorates. For large ${\rm{\Delta }}{m}_{14}^{2}$ the experiments at the research reactors in particular upgraded Neutrino-4 and PROSPECT have higher sensitivity. The upgraded DANSS, Neutrino-4, and PROSPECT experiments will soon test the Neutrino-4 claim of the sterile neutrino observation and scrutinize even larger part of the sterile neutrino parameters preferred by the recent BEST results.

Table 1. Parameters of the VSBL experiments. Advantages and disadvantages are indicated with the red and blue colors correspondingly.

4. Global fits

It is not trivial to combine the results of different experiments on sterile neutrino searches. Initially this was done (see e.g. [51, 52]) assuming that the test statistics Δχ² = χ²(3ν) − χ²(4ν) follows the χ² distribution with 2 degrees of freedom which is not corrects in the sterile neutrino searches (see e.g. [77]). The Monte Carlo simulation of the Δχ² distribution is needed for correct estimates of the significance that requires deep knowledge of experiments. To circumvent this very serious complication several authors performed a simplified simulation of experiments used in their analysis [38, 77–79]. With the Monte Carlo determination of the Δχ² distribution the significance of the evidence for sterile neutrinos in ${\nu }_{e}/{\tilde{\nu }}_{e}$ disappearance experiments (without Neutrino-4 and BEST) was reduced from 2.4σ to 1.8σ (see figure 10) [77]. In the 3 + 1 sterile neutrino model the probability of the electron (anti)neutrino appearance in the muon (anti)neutrino beam is proportional to the probabilities of the ${\tilde{\nu }}_{e}$ disappearance and ν_μ disappearance. Therefore, the strong limits on ${\tilde{\nu }}_{e}$ and ν_μ disappearance do not allow to explain by sterile neutrinos simultaneously the LSND/MiniBooNE excess in the ( ${\tilde{\nu }}_{e}$ )ν_e appearance and RAA/GA deficit (see figure 11) [40]. However, the Neutrino-4 result and the highly significant BEST result were not included into the comparison.

Figure 10. Refer to the following caption and surrounding text. — **Figure 10.** Contours of the 1σ (blue), 2σ (red), and 3σ (green) allowed regions in the ( ${\rm{\Delta }}{m}_{41}^{2}$ , ${\sin }^{2}2{\theta }_{{ee}}$ ) plane obtained with the combined analysis [77] of the data of the four reactor spectral ratio experiments NEOS [40], DANSS [87], Bugey-3 [88], and PROSPECT [72]. The solid lines represent the contours obtained with Monte Carlo evaluation of the distribution of Δχ², and the dashed lines depict the contours obtained assuming the χ² distribution. Also shown are the marginal Δχ² s (black) for ${\rm{\Delta }}{m}_{41}^{2}$ , ${\sin }^{2}2{\theta }_{{ee}}$ together with the Δχ² values corresponding to 1σ (blue), 2σ (red), and 3σ (green) obtained with the χ² (dashed) and Monte-Carlo (solid) distributions. The blue cross indicates the best-fit point. Figure is adopted from [77].
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**Figure 10.** Contours of the 1σ (blue), 2σ (red), and 3σ (green) allowed regions in the ( ${\rm{\Delta }}{m}_{41}^{2}$ , ${\sin }^{2}2{\theta }_{{ee}}$ ) plane obtained with the combined analysis [77] of the data of the four reactor spectral ratio experiments NEOS [40], DANSS [87], Bugey-3 [88], and PROSPECT [72]. The solid lines represent the contours obtained with Monte Carlo evaluation of the distribution of Δχ², and the dashed lines depict the contours obtained assuming the χ² distribution. Also shown are the marginal Δχ² s (black) for ${\rm{\Delta }}{m}_{41}^{2}$ , ${\sin }^{2}2{\theta }_{{ee}}$ together with the Δχ² values corresponding to 1σ (blue), 2σ (red), and 3σ (green) obtained with the χ² (dashed) and Monte-Carlo (solid) distributions. The blue cross indicates the best-fit point. Figure is adopted from [77].
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Figure 11. Refer to the following caption and surrounding text. — **Figure 11.** Comparison of the MINOS, MINOS+, Daya Bay, and Bugey-3 combined 90% CLs limit on ${\sin }^{2}2{\theta }_{e\mu }$ to the LSND and MiniBooNE 90% C.L. allowed regions. Regions of parameter space to the right of the red contour are excluded. The regions excluded at 90% C.L. by the KARMEN2 Collaboration [83] and the NOMAD Collaboration [89] are also shown. The combined limit also excludes the 90% C.L. region allowed by a fit to global data by Gariazzo *et al* where MINOS, MINOS+, Daya Bay, and Bugey-3are not included [51, 90], and the 90% C.L. region allowed by a fit to all available appearance data by Dentler *et al* [52] updated with the 2018 MiniBooNE appearance results [48]. Figure is adopted from [41].
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**Figure 11.** Comparison of the MINOS, MINOS+, Daya Bay, and Bugey-3 combined 90% CLs limit on ${\sin }^{2}2{\theta }_{e\mu }$ to the LSND and MiniBooNE 90% C.L. allowed regions. Regions of parameter space to the right of the red contour are excluded. The regions excluded at 90% C.L. by the KARMEN2 Collaboration [83] and the NOMAD Collaboration [89] are also shown. The combined limit also excludes the 90% C.L. region allowed by a fit to global data by Gariazzo *et al* where MINOS, MINOS+, Daya Bay, and Bugey-3are not included [51, 90], and the 90% C.L. region allowed by a fit to all available appearance data by Dentler *et al* [52] updated with the 2018 MiniBooNE appearance results [48]. Figure is adopted from [41].
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The validity of existing limits on sterile neutrino parameters was questioned recently [80]. For very small neutrino wave packet sizes evolution of the neutrino beams with time becomes more complicated than equation (2) and the oscillation behavior is smeared out (see figure 12). Existing experimental limits on the size of reactor antineutrino wave packets are quite weak. If one uses the present limits as the actual values of the reactor antineutrino wave packet size, the limits on the sterile neutrino mixing parameter ${\sin }^{2}2{\theta }_{{ee}}$ become considerably weaker and the tension between the results on neutrino disappearance in radioactive sources experiments and reactor experiments vanishes at least for some values of ${\rm{\Delta }}{m}_{41}^{2}$ . The contradiction between appearance and disappearance experiments is also reduced. However there is no reason for the wave packet size to be equal to its current limit. Most probably the size is much larger than the current limit and it does not change the present limits on the sterile neutrino parameters.

Figure 12. Refer to the following caption and surrounding text. — **Figure 12.** Illustration of the decoherence effect, for σ_x = 2.1 × 10⁻⁴ nm(blue curve), with the reactor antineutrino anomaly (RAA) best-fit parameters: ${\rm{\Delta }}{m}_{14}^{2}=2.32\,{\mathrm{eV}}^{2},\,{\sin }^{2}2{\theta }_{{ee}}=0.14$ (red curve). In the y-axis, the ratio between expected number of events in 4ν and 3ν models as well as the experimental results from NEOS [40]. Figure is adopted from [80].
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**Figure 12.** Illustration of the decoherence effect, for σ_x = 2.1 × 10⁻⁴ nm(blue curve), with the reactor antineutrino anomaly (RAA) best-fit parameters: ${\rm{\Delta }}{m}_{14}^{2}=2.32\,{\mathrm{eV}}^{2},\,{\sin }^{2}2{\theta }_{{ee}}=0.14$ (red curve). In the y-axis, the ratio between expected number of events in 4ν and 3ν models as well as the experimental results from NEOS [40]. Figure is adopted from [80].
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The standard cosmological model strongly disfavors the eV-scale sterile neutrinos (see e.g. [15]). However there are models which include eV-scale sterile neutrinos and reduce the tension between different determinations of the Hubble constant [81].

5. Conclusions

The situation with the experimental indications of eV-scale sterile neutrino existence is quite controversial. Indications from all but one reactor experiments have statistical significance below 2σ. This should be compared with the initial estimates of the RAA significance of about 3σ. On the other hand Neutrino-4 claims the observation of sterile neutrinos with 2.7σ significance. Moreover the large ν_e deficit observed recently by BEST with more than 5σ significance perfectly agrees with the Neutrino-4 result. However a large fraction of sterile neutrino parameters preferred by Neutrino-4 and BEST are either excluded or are in tension with results from several experiments. This is nicely illustrated by figure 13 [36].

Figure 13. Refer to the following caption and surrounding text. — **Figure 13.** Exclusion contours of all gallium anomaly experiments: two GALLEX, two SAGE and two BEST results. The blue solid line and the blue dotted line show the 2σ and 3σ confidence level, respectively. The figure also presents the exclusion contours from Prospect [42], DANSS [91], Stereo [43], KATRIN [92], the combined analysis of RENO and NEOS data [69], reactor anti-neutrino anomalies (RAA) [20], interpretations of the MicroBooNE result for the oscillation hypothesis with fixed mixing angle ( ${\sin }^{2}2\theta$ ) and profiled over the angle [56], and the model-independent 95% upper bound on ${\sin }^{2}2\theta$ from all solar neutrino experiments [62]. The 2σ allowed region of Neutrino-4 [58] is also presented and the grey shading represents the merged exclusion of the very short baseline (VSBL) null results. Figure is adopted from [36].
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${\sin }^{2}2\theta $ — **Figure 13.** Exclusion contours of all gallium anomaly experiments: two GALLEX, two SAGE and two BEST results. The blue solid line and the blue dotted line show the 2σ and 3σ confidence level, respectively. The figure also presents the exclusion contours from Prospect [42], DANSS [91], Stereo [43], KATRIN [92], the combined analysis of RENO and NEOS data [69], reactor anti-neutrino anomalies (RAA) [20], interpretations of the MicroBooNE result for the oscillation hypothesis with fixed mixing angle ( ${\sin }^{2}2\theta$ ) and profiled over the angle [56], and the model-independent 95% upper bound on ${\sin }^{2}2\theta$ from all solar neutrino experiments [62]. The 2σ allowed region of Neutrino-4 [58] is also presented and the grey shading represents the merged exclusion of the very short baseline (VSBL) null results. Figure is adopted from [36].
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The upgraded DANSS, Neutrino-4, and PROSPECT experiments as well as JUNO-TAO will clarify the situation in a few years.

More than 6σ evidence for ν_e( ${\tilde{\nu }}_{e}$ ) appearance from LSND/MiniBooNE is not confirmed by MicroBooNE, but not excluded completely. The Fermilab Short Baseline Neutrino program and the JSNS² experiment at J-PARK [82] will clarify the situation in a few years.

The next several years will be crucial for the searches for eV-scale sterile neutrinos.

Acknowledgments

Author would like to thank N.Skrobova for the invaluable help in the preparation of the manuscript. This work is supported by the Ministry of Science and Higher Education of the Russian Federation under the Contract No. 075-15-2020-778.

Data availability statement

All data that support the findings of this study are included within the article (and any supplementary files).

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