Light Sterile Neutrinos: Status and Perspectives

The indications in favor of the existence of light sterile neutrinos at the eV scale found in short-baseline neutrino oscillation experiments is reviewed. The future perspectives of short-baseline neutrino oscillation experiments and the connections with beta-decay measurements of the neutrino masses and with neutrinoless double-beta decay experiments are discussed.


Introduction
The 2015 Nobel Prizes in Physics is a great acknowledgment of the fundamental importance of the model-independent discoveries of neutrino oscillations in the Super-Kamiokande atmospheric neutrino experiment [1] and in the SNO solar neutrino experiment [2]. These discoveries, which proved that neutrinos are massive and mixed particles, led to the standard three-neutrino mixing paradigm (3ν), in which the three active neutrinos ν e , ν µ , ν τ are superpositions of three massive neutrinos ν 1 , ν 2 , ν 3 with respective masses m 1 , m 2 , m 3 (see Ref. [3]). There are two independent squared-mass differences, the small solar ∆m 2 SOL 7.5 × 10 −5 eV 2 and the larger atmospheric ∆m 2 ATM 2.3 × 10 −3 eV 2 , which can be interpreted as ∆m 2 SOL = ∆m 2 21 and ∆m 2 ATM = |∆m 2 31 | |∆m 2 32 |, with ∆m 2 k j = m 2 k − m 2 j (see Refs. [4][5][6]). The completeness of the 3ν mixing paradigm has been challenged by the following indications in favor of short-baseline neutrino oscillations, which require the existence of at least one additional squaredmass difference, ∆m 2 SBL ∆m 2 ATM (see the review in Ref. [7]): 1. The reactor antineutrino anomaly [8], which is an about 2.8σ deficit of the rate ofν e observed in several short-baseline reactor neutrino experiments in comparison with that expected from the calculation of the reactor neutrino fluxes [9,10]. 2. The Gallium neutrino anomaly [11][12][13][14][15], consisting in a short-baseline disappearance of ν e measured in the Gallium radioactive source experiments GALLEX [16] and SAGE [17] with a statistical significance of about 2.9σ.
$ Invited contribution to the Nuclear Physics B Special Issue on Neutrino Oscillations celebrating the Nobel Prize in Physics 2015. 3. The LSND experiment, in which a signal of short-baselineν µ →ν e oscillations has been observed with a statistical significance of about 3.8σ [18,19].
The additional squared-mass difference ∆m 2 SBL requires the existence of at least one massive neutrino ν 4 in addition to the three standard massive neutrinos ν 1 , ν 2 , ν 3 . Since from the LEP measurement of the invisible width of the Z boson we know that there are only three active neutrinos (see Ref. [3]), in the flavor basis the additional massive neutrinos correspond to sterile neutrinos [20], which do not have standard weak interactions.
The possible existence of sterile neutrinos is very interesting, because they are new particles which could give us precious information on the physics beyond the Standard Model (see Refs. [64,65]). The existence of light sterile neutrinos is also very important for astrophysics (see Ref. [66]) and cosmology (see Refs. [7,[67][68][69][70]).
In this review, we consider 3+1 [71][72][73][74] and 3+2 [75][76][77][78][79], neutrino mixing schemes in which there are one or two additional massive neutrinos at the eV scale 1 and the masses of the three standard massive neutrinos are much smaller. We do not consider schemes in which ∆m 2 SBL is obtained with one or more very light (or massless) non-standard massive neutrinos and the three standard massive neutrinos have almost degenerate masses at the eV scale (e.g., the 1+3, 1+3+1 and 2+3 schemes), because this possibility is strongly disfavored by cosmological measurements [88] and by the experimental bound on neutrinoless double-β decay (assuming that massive neutrinos are Majorana particles; see Ref. [89]).
complex phase η appears with different signs in the effective 3+2 probabilities of short-baseline ν µ → ν e andν µ →ν e transitions, it can generate measurable CP violations.
The MiniBooNE data require a special treatment, because they show an anomalous excess in the lowenergy bins [121,159] which, as explained later, induces a tension in the global analysis of the data of short-baseline neutrino oscillation experiments [115,116]. Hence, we will discuss two types of global fits: "total" (TotGLO) and "pragmatic" (PrGLO). In the total fits all the data listed above of short-baseline neutrino oscillation experiments are taken into account. In the pragmatic fits [85] the anomalous low-energy bins of the MiniBooNE experiment [121,159] are omitted. Table 1 summarizes the statistical results obtained from global fits of the data above in the 3+1 and 3+2 schemes. Besides the total and pragmatic fits there is also a 3+1-noMB fit without MiniBooNE data and a 3+1-noLSND fit without LSND data which are explained below.
From Tab. 1, one can see that in all fits which include the LSND data the absence of short-baseline oscillations is nominally disfavored by about 6σ, because the improvement of the χ 2 with short-baseline oscillations is much larger than the number of oscillation parameters.
In both the 3+1 and 3+2 schemes the goodnessof-fit in the total analysis is significantly worse than that in the pragmatic analysis and the appearancedisappearance parameter goodness-of-fit is much worse. This result confirms the fact that the Mini-BooNE low-energy anomaly is incompatible with neutrino oscillations, because it would require a small value of ∆m 2 41 and a large value of sin 2 2ϑ eµ [115,116], which are excluded by the data of other experiments (see Ref. [85] for further details) 4 . Note 4 One could fit the three anomalous MiniBooNE low-energy that the appearance-disappearance tension in the 3+2-TotGLO fit is even worse than that in the 3+1-TotGLO fit, since the ∆χ 2 PG is so much larger that it cannot be compensated by the additional degrees of freedom 5 . Therefore, we think that it is very likely that the MiniBooNE low-energy anomaly has an explanation which is different from neutrino oscillations 6 . The cause of the MiniBooNE low-energy excess of ν e -like events is going to be investigated in the Micro-BooNE experiment at Fermilab [163], which is a large Liquid Argon Time Projection Chamber (LArTPC) in which electrons and photons can be distinguished 7 .
In the following we adopt the "pragmatic approach" advocated in Ref. [85] which considers the PrGLO fits, without the anomalous MiniBooNE low-energy bins, as more reliable than the TotGLO fits, which include the anomalous MiniBooNE low-energy bins.
The 3+2 mixing scheme was considered to be interesting in 2010 when the MiniBooNE neutrino [159] and antineutrino [164] data showed a CP-violating tension, but this tension almost disappeared in the final MiniBooNE data [121]. In fact, from Tab. 1 one can see that there is little improvement of the 3+2-PrGLO fit with respect to the 3+1-PrGLO fit, in spite bins in a 3+2 scheme [90] by considering the appearance data without the ICARUS [125] and OPERA [126] constraints, but the required large transition probability is excluded by the disappearance data.   of the four additional parameters and the additional possibility of CP violation. Moreover, the p-value obtained by restricting the 3+2 scheme to 3+1 disfavors the 3+1 scheme only at 1.1σ. Therefore, we think that considering the larger complexity of the 3+2 scheme is not justified by the data and in the following we consider only the 3+1 mixing scheme. Figure 1 shows the allowed regions in the sin 2 2ϑ eµ -∆m 2 41 , sin 2 2ϑ ee -∆m 2 41 and sin 2 2ϑ µµ -∆m 2 41 planes obtained in the 3+1-PrGLO fit. These regions are relevant, respectively, for ν µ disappearance data. One can see that the combined disappearance constraint in the sin 2 2ϑ eµ -∆m 2 41 plane excludes a large part of the region allowed by ν e appearance data, leading to the well-known appearance-disappearance tension [7, 85-87, 90, 114-116, 118, 160, 165] quantified by the parameter goodness-of-fit in Tab. 1. The best-fit values of the oscillation parameters are (∆m 2 41 ) bf = 1.6 eV 2 , (|U e4 | 2 ) bf = 0.028, (|U µ4 | 2 ) bf = 0.013, which imply (sin 2 2ϑ eµ ) bf = 0.0014, (sin 2 2ϑ ee ) bf = 0.11 and (sin 2 2ϑ µµ ) bf = 0.050.
It is interesting to investigate what are the impacts of the MiniBooNE and LSND experiments on the global analysis of short-baseline neutrino oscillation data. With this aim, we consider two additional 3+1 fits: a 3+1-noMB fit without MiniBooNE data and a 3+1-noLSND fit without LSND data. From Tab. 1 one can see that the results of the 3+1-noMB fit are similar to those of the 3+1-PrGLO fit and the nominal exclusion of the case of no-oscillations remains at the level of 6σ. On the other hand, in the 3+1-noLSND fit, without LSND data, the nominal exclusion of the case of no-oscillations drops dramatically to 2.6σ. In fact, in this case the main indication in favor of shortbaseline oscillations is given by the reactor and Gallium anomalies which have a similar statistical significance [15]. Therefore, it is clear that the LSND experiment is still crucial for the indication in favor of short-baselineν µ →ν e transitions.

Experimental perspectives
There is an impressive program of many experimental projects which will explore the existence of light sterile neutrinos at the eV scale in the next years (see also the reviews in Refs. [185][186][187][188][189][190][191]). It is convenient to divide them in the following categories.

3.1.
The aim of these experiments is to reveal shortbaseline oscillations in a robust way by measuring distortions of the neutrino spectrum or variations of the flavor neutrino detection probability as a function of distance. They can be divided in the following subcategories.
Source experiments. These experiments use radioactive sources of ν e orν e placed near or inside a large detector [192].   the projects which have been proposed (see also Ref. [188]).
In source experiments with monochromatic ν e 's generated by nuclear electron capture (for example SAGE [166] and CrSOX [167]), ν e disappearance can be measured as a function of distance.
In source experiments with a continuousν e spectrum generated by nuclear β decay (for example CeSOX [167,168]) also the distortions of the neutrino spectrum can be measured.
Reactor experiments. These experiments use a reactorν e source with a detector placed at a distance of the order of 10 m. There are several experiments in preparation, as shown by the list in Tab. 3 (see also Ref. [189]). They are planned to have a sufficient energy resolution in order to be sensitive to the distortions in the neutrino spectrum due to the oscillations. Some experiments (for example Stereo [176]) will have a length which may allow to observe the variations of theν e survival probability as a function of distance. Others use will use two detectors at different distances (for example PROSPECT [193] and CARR [184]) or a movable detector (for example DANSS [177]).

3.2.
For accelerator experiments a crucial ingredient for reaching a robust result is the presence of "near" and "far" detectors (as, for example, in the SBN [202] experiment, where there will be even the "middle" Mi-croBooNE detector, albeit smaller than the near detector). The near detector provides a normalization of the  [206] 0.6 1000 ∼ 6.5 10 − 22 proposal IsoDAR-JUNO (CHN) [171] 0.6 20000 ∼ 6.5 20 − 100 proposal OscSNS (USA) [207] 1.4 450 ∼ 40 50 − 70 proposal  Figure 2: Comparison of the allowed region in the sin 2 2ϑ ee -∆m 2 41 plane obtained in the pragmatic 3+1 global fit PrGLO of short-baseline neutrino oscillation data with the sensitivities of the CeSOX [167,168] source experiment, of the Stereo [176], SoLid [178], DANSS [177] and NEOS [180] reactor experiments and of the KATRIN [201] β-decay experiment. neutrino flux and cross section which allows to measure the oscillations between the two detectors with small systematic uncertainty. Figure 3 shows the sensitivities in the sin 2 2ϑ eµ -∆m 2 41 plane of the SBN [202] and nuPRISM [205] accelerator experiments, which are expected to observe with a convincing statistical significance   The accelerator experiments in Tab. 4 (see also the NESSiE proposal in Ref. [209] and the low-energy neutrino factory studies in Refs. [198,200,208]) can measure also the short-baseline ν µ andν µ disappearance which is necessarily associated with ν e oscillations is given by the Gallium and reactor anomalies. The consistency of the short-baseline neutrino oscillation scenario with any number of sterile neutrinos requires that also ν µ andν µ disappearance must be observed [165]. Figure 4 shows the sensitivities in the sin 2 2ϑ µµ -∆m 2 41 plane of the SBN [202] and KPipe [204] accelerator experiments. One can see that also (−) ν µ disappearance should be observed if the short-baseline neutrino oscillations indicated by the LSND, reactor and Gallium anomalies really exist.

Neutral-current measurements
In principle, measuring the neutral-current scattering of active neutrinos is the best way to probe their disappearance into sterile states. However, neutralcurrent measurements are extremely difficult, because plane obtained in the pragmatic 3+1 global fit PrGLO of shortbaseline neutrino oscillation data with the sensitivities of the SBN [202] and KPipe [204] accelerator experiments.
the only observable signal is the recoil of the target particle.
The signal can be enhanced at low neutrino energies by the coherent scattering on nuclei [210,211] for which the cross section is approximately proportional to the square of the number of neutrons in the nucleus (the proton contribution is suppressed by 1 − 4 sin 2 ϑ W 1, where ϑ W is the weak mixing angle). This process has not been observed so far, but it is actively searched for [212][213][214][215][216]. In the future it may lead to the direct measurement of active-sterile transitions [217][218][219].

β-decay mass measurements
The most sensitive experiments on the search of the effects of neutrino masses in β decay use the Tritium decay process 8 (3.1) Non-zero neutrino masses distort the measurable spectrum of the emitted electron. It is convenient to consider the Kurie function (see Ref. [3]) where T e is the electron kinetic energy, Q = M3 H − M3 He − m e 18.574 keV is the Q-value of the process, and Θ is the Heaviside step function. Considering an experiment in which the energy resolution is 8 Other methods are described in the reviews in Refs. [220][221][222][223][224] such that m k Q − T e for the three standard light neutrino masses (k = 1, 2, 3), the Kurie function can be approximated by with the effective light neutrino mass m β given by Hence, m β causes a distortion of the end-point of the electron kinetic energy spectrum and a heavy nonstandard neutrino mass m k with k ≥ 4 can be measured by observing a kink of the kinetic energy spectrum of the emitted electron at Q − m k below the end point [116,[225][226][227][228][229][230][231][232]. Recently, the Mainz [233] and Troitsk [234,235] collaborations obtained upper bounds for the mixing factor |U e4 | 2 for m 2 4 10 eV 2 . In the 3+1 scheme these bounds imply an exclusion curve in the sin 2 2ϑ ee -∆m 2 41 plane for ∆m 2 41 10 eV 2 [119], which is well above the allowed region obtained in the pragmatic 3+1 global fit PrGLO shown in Fig. 1.
The experiment KATRIN [236], which is under construction and is scheduled to start data taking in 2016, will aim to reach a sensitivity of 0.2 eV at 90% C.L. for m β in five years of running. Some studies have been performed to analyze the sensitivity of the KATRIN experiment to the effects of heavy sterile neutrinos with keV-scale masses [201,[237][238][239] and light eV-scale sterile neutrinos [201,232,[240][241][242]. Figure 2 shows the KATRIN sensitivity presented in Ref. [201]. One can see that it covers a significant portion of the PrGLO allowed region. Hence, there is a concrete possibility that KATRIN can observe the effect of m 4 if ν 4 exists and both m 4 and |U e4 | 2 are not too small.
If massive neutrinos are Majorana particles (see the recent reviews in Refs. [89,252]), in the case of 3+1 mixing the rate of neutrinoless double-β decay is proportional to the square of the effective Majorana mass |m ββ | = |U e1 | 2 m 1 + |U e2 | 2 e iα 2 m 2 + |U e3 | 2 e iα 3 m 3 +|U e4 | 2 e iα 4 m 4 .   Figure 5: Value of the effective Majorana mass |m ββ | as a function of the lightest neutrino mass in the cases of 3ν and 3+1 mixing with normal and inverted ordering of the three lightest neutrinos [251]. The horizontal band is an estimate of the current experimental 90% C.L. upper limit for |m ββ | taking into account the uncertainties of the nuclear matrix element calculations [89].
In this expression there are three completely unknown complex phases α 2 , α 3 , α 4 which depend on the Majorana phases in the neutrino mixing matrix. These unknown complex phases can generate cancellations between the different mass contributions. Figure 5 shows the range of allowed values of |m ββ | as a function of the lightest neutrino mass in the cases of 3ν and 3+1 mixing with normal and inverted ordering of the three lightest neutrinos [251]. The 3ν mixing parameters are those obtained in Ref. [253] and the sterile neutrino mixing is that obtained in the global pragmatic 3+1 PrGLO fit of short-baseline neutrino oscillation data discussed in Section 2.
From Fig. 5 one can see that the presence of an additional massive neutrinos at the eV scale can change dramatically the predictions for the possible range of values of |m ββ | [15,102,[243][244][245][246][247][248][249][250][251][252]. In the case of a normal 3ν mass hierarchy (m 1 m 2 m 3 ) the value of |m ββ | is dominated by the contribution of ν 4 , which implies that 1×10 −2 |m ββ | 7×10 −2 eV. This range of values of |m ββ | is larger than that predicted by the standard 3ν mixing in the case of a normal hierarchy and similar to that predicted in the case of an inverted hierarchy in the standard 3ν mixing scheme. On the other hand, in the case of an inverted 3ν mass ordering there can be a complete cancellation between the contribution of ν 4 and those of the three standard light neutrinos, leading to the disappearance of the lower limit for |m ββ | predicted by the standard 3ν mixing scheme.
The next generation of neutrinoless double-beta decay experiments (see Refs. [254][255][256][257][258][259]) is planned to explore the range of |m ββ | between about 1 × 10 −2 and 5 × 10 −2 eV predicted by the standard 3ν mixing in the case of an inverted hierarchy. They are not expected to reach the range of |m ββ | between about 8 × 10 −4 and 5 × 10 −3 eV predicted by the standard 3ν mixing in the case of a normal hierarchy. From Fig. 5 it is clear that the predictions are dramatically changed in the 3+1 neutrino mixing scheme and a positive result in these experiments is guaranteed in the case of a normal mass hierarchy, whereas in the case of an inverted mass hierarchy the allowed range of |m ββ | goes from zero to about 0.1 eV.

Conclusions
The reactor, Gallium and LSND anomalies can be explained by neutrino oscillations if the standard three-neutrino mixing paradigm is extended with the addition of light sterile neutrinos which can give us important information on the new physics beyond the Standard Model.
The global fits of short-baseline neutrino oscillation data in the framework of mixing schemes with one or more sterile neutrinos suffer from a tension between the results of appearance and disappearance short-baseline neutrino oscillation experiments. This tension can be alleviated adopting the "pragmatic approach" advocated in Ref. [85], in which the anomalous MiniBooNE low-energy excess of ν e -like events is neglected from the global analysis of short-baseline neutrino oscillation data. The cause of the Mini-BooNE low-energy excess is going to be investigated in the MicroBooNE experiment at Fermilab [163].
Moreover, the cosmological data indicate a tension between the necessity to have a sterile neutrino mass at the eV scale and the expected full thermalization of the sterile neutrinos through active-sterile oscillations in the early Universe [118,160,[260][261][262]. Hence, the possible existence of light sterile neutrinos at the eV scale is controversial and needs new reliable experimental checks.
The impressive program of new experiments reviewed in Section 3 gives us confidence that the question of the existence of the light sterile neutrinos indicated by the reactor, Gallium and LSND anomalies will be answered in a definitive way in the next years.
Let us finally emphasize that the discovery of the existence of sterile neutrinos would be a major discovery which would have a profound impact not only on neutrino physics, but on our whole view of fundamental physics, because sterile neutrinos are elementary particles beyond the Standard Model. The existence of light sterile neutrinos would prove that there is new physics beyond the Standard Model at low-energies and their properties can give important information on this new physics. Without any doubt, such a discovery would deserve a new Nobel Prize in Physics.