On The Decaying-Sterile Neutrino Solution to the Electron (Anti)Neutrino Appearance Anomalies

We explore the hypothesis that the unexplained data from LSND and MiniBooNE are evidence for a new, heavy neutrino mass-eigenstate that mixes with the muon-type neutrino and decays into an electron-type neutrino and a new, very light scalar particle. We consider two different decay scenarios, one with Majorana neutrinos, one with Dirac neutrinos; both fit the data equally well. We find a reasonable, albeit not excellent, fit to the data of MiniBooNE and LSND. The decaying-sterile-neutrino hypothesis, however, cleanly evades constraints from disappearance searches and precision measurements of leptonic meson decays, as long as $1~{\rm MeV}\gtrsim m_4\gtrsim 10$~keV. The SBN program at Fermilab should be able to definitively test the decaying-sterile-neutrino hypothesis.


Introduction
Over the last several decades, a variety of revolutionary neutrino puzzles evolved into our current understanding of the neutrino sector of fundamental particle physics. A few of these puzzles, however, remain unresolved. Among them are data from the LSND and MinBooNE experiments.
The LSND collaboration looked forν e -candidate events at a detector located dozens of meters away from a stopped-pion target. Stopped π + decay into µ + ν µ and the muon subsequently decays, µ + → e + ν eνµ , yielding a well-characterized flux of ν e , ν µ ,ν µ and, most relevant, noν e . LSND observes a very significant excess -more than 4 sigma -ofν e -candidate events [1].
The MiniBooNE experiment was designed to test the oscillation-interpretation of the LSND data, discussed in more detail in the next paragraph. The detector was located downstream of a pion-decay-in-flight neutrino or antineutrino beam (mostly π + → µ + ν µ or π − → µ −ν µ ). The experimental baseline L was chosen such that, for typical neutrino energies E, the value of L/E matched that of the LSND experiment. The MiniBooNE collaboration reported a combined 4.7 sigma excess of ν e - [2,3] andν e -candidate events [4] the detector has very limited charge-discrimination capabilities while running in both the neutrino-beam and antineutrino-beam modes. If both the LSND and MiniBooNE data are a consequence of the same unexplained phenomenon, the combined evidence is at the 6 sigma level [3].
Under the assumption that there are no unaccounted for "mundane" explanations to these two excesses -unidentified background processes, problems with modeling the neutrino scattering process, detector-related effects, etc -these so-called short-baseline anomalies 1 translate into new more physics -on top of nonzero active neutrino masses -in the neutrino sector. The simplest new-physics interpretation to the data from LSND and MiniBooNE is to postulate that a ν µ (ν µ ) has nonzero probability of being detected as a ν e (ν e ). Neutrino oscillations can lead to this phenomenon. In light of all other evidence for neutrino oscillations, the neutrino oscillation interpretation to the the short-baseline anomalies requires the introduction of a fourth neutrino mass eigenstate ν 4 associated to a mass-squared difference ∆m 2 ∼ 1 eV 2 . The data point to new mixing parameters such that |U e4 | 2 |U µ4 | 2 ∼ 10 −3 [3]. In this scenario, the new flavor eigenstate is postulated to have no gauge quantum numbers and is hence dubbed a sterile neutrino. While this eV-scale sterile-neutrino hypothesis fits all data associated with searches for ν µ → ν e appearance, it is in conflict with other data, including neutrino disappearance data at short-baselines. Very roughly, the reason for this is that there is no incontrovertible evidence for neutrino disappearance at short-baselines. These failed searches constrain |U e4 | 2 and |U µ4 | 2 to be less than several percent and hence fail to satisfy |U e4 | 2 |U µ4 | 2 ∼ 10 −3 . More quantitatively, global fits to the world's neutrino data indicate that the eV-scale sterile-neutrino hypothesis is not a satisfactory explanation for the short-baseline anomalies. See, for example, Refs. [5][6][7][8] for recent analyses and discussions.
Here, we revisit a different solution to the LSND and MiniBooNE puzzle. Instead of assuming that a fourth eV-scale neutrino is produced coherently during pion or muon decay, we postulate that a heavier fourth neutrino mass eigenstate is produced in the neutrino source and that this new neutrino state decays into an electron-type neutrino and a new, effectively massless scalar particle [9]. The decay is prompt enough such that, a significant portion of the time the daughter neutrino can interact in the detector and lead to an excess of ν e -andν e -candidate events. This hypothesis was first raised to explain the LSND results [9]. Radiative sterile-neutrino decays were also explored as a potential explanation to the observations reported by MiniBooNE and LSND [10][11][12][13]. We do not consider these here.
We extend the analysis in Ref. [9] to include the most recent data from the MiniBooNE experiment, and ask whether the decaying-sterile-neutrino hypothesis is a good fit to the data. We explore different decay scenarios with Majorana neutrinos and Dirac neutrinos. These are spelled out in Section 2. We also explore how well the decaying-sterile-neutrino hypothesis will be tested by the Short-Baseline-Neutrino Program (SBN Program) at Fermilab. Details and results, along with a description of how we treat the data from LSND and MiniBooNE, are discussed in Section 3. A short summary of our findings is presented in Section 4. 1 The short-baseline anomalies also include the reactor and gallium anomalies. For recent summaries of these data, see, for example, Refs. [5][6][7][8]. We will have nothing to say about these here other than the fact that the hypothesis we will be investigating cannot account for either of them.

Formalism
We postulate the existence of a fourth neutrino mass eigenstate. Since we want to explain the data from LSND and MiniBooNE, the fourth neutrino must have a nonzero ν µ component. We don't need a nonzero ν τ or ν e component so we set these to zero. A very small ν e or ν τ component would not modify our results in a significant way. In other words, U e4 = U τ 4 = 0 and ν µ = U µi ν i , i = 1, 2, 3, 4, U µ4 = 0.
We further introduce a new interaction that allows ν 4 , with mass m 4 , to decay into a new, very light scalar field φ and a ν e . There are different effective Lagrangians capable of mediating this phenomenon [14][15][16][17][18][19]. Here, we concentrate on two possibilities.
If the neutrinos are Dirac fermions, there are several distinct ways of coupling a heavy neutrino to a light neutrino and a scalar field φ. These are associated with the transformation properties of φ under lepton-number (or lepton-number-minus-baryon-number) and the parity-violating properties of the new interaction. We will assume that the new scalar field φ is a standard model gauge singlet and that it carries zero lepton number. We will also assume that the new interaction violates parity maximally and, like the weak-interactions, only couples to left-chiral light neutrinos. Since we are interested in the decay to ν e , at low-energies, the interaction that mediates the heavy neutrino decay is where, to facilitate comparisons to the Majorana case, we express the neutrino fields as twocomponent Weyl fermions. 2 ν e is the field associated with the left-chiral ν e and the ν c 4 is the field associated with the left-chiralν 4 . Note that ν e is a linear superposition of the light neutrino mass eigenstates, ν e = U ei ν i , i = 1, 2, 3. We choose this specific decay in order to maximize the effect of the sterile-neutrino decay at MiniBooNE and LSND and in order to minimize the effect at experiments sensitive to ν µ or ν τ in the final state, as we discuss in more detail later. We will be interested in L/E values such that ordinary neutrino oscillations, driven by m 2 2 − m 2 1 and m 2 3 − m 2 1 , do not have time to modify neutrino flavor evolution. We will also be interested in m 4 values that are much larger than m 1,2,3 and will treat the ν e as a massless particle. This means that all daughters of the ν 4 decay mediated by Eq. (2.1) are left-handed ν e while all the daughters of theν 4 decay mediated by Eq. (2.1) are right-handed ν e . Other choices lead to different final states. For example, one can choose φ to carry lepton number two in such a way that the decay process is ν 4 →ν e φ, or one can choose an effective Lagrangian proportional to ν 4 ν c e so that all ν e produced in the decay of ν 4 are right-handed. The choice above -Eq. (2.1) -maximizes the "visibility" of the daughter ν e . In the limit where the light neutrino masses are negligible -an excellent approximation here -the lefthanded ν e is perfectly aligned with the left-chiral interaction field, while the right-handed ν e are perfectly sterile as far as the weak interactions are concerned. For a more detailed, recent discussion of these issues, see, for example, Ref. [20]. Note that it is easy to express Eq. (2.1) in a way that explicitly preserves the SU (2) × U (1) gauge symmetry of the standard model: ν c 4 ν e φ → ν c s (L e H)φ/Λ, where L e is the electron-flavor lepton-doublet, H is the Higgs boson field, and Λ is the effective scale of the physics that leads to the decay Lagrangian.
If the neutrinos are Majorana fermions and one only adds one new Weyl fermion to the low-energy particle content of the standard model -ν 4 -along with the gauge-singlet scalar field φ, the Lagrangian that mediates tree-level ν 4 decay at low energies is Here it is not meaningful to assign lepton-number charge to the φ-field. Since we are interested in the limit where the light neutrino masses are negligible, it is meaningful and convenient to talk about ν e -always left-handed -andν e -always right-handed. In this case, Eq. (2.2) mediates both ν 4 → ν e φ and ν 4 →ν e φ, both with the same branching ratio at the tree-level.
Here it is also easy to express Eq. (2.2) in a way that explicitly preserves the SU (2) × U (1) gauge symmetry of the standard model: in the limit U µ4 1, ν 4 ν e φ → ν s (L e H)φ/Λ, where ν s is the left-handed sterile neutrino field and Λ is the effective scale of the physics that leads to the decay Lagrangian. 3 Again, for a more detailed, recent discussion of these issues, see, for example, [20].
Henceforth, we will treat ν e and φ as massless particles. For Dirac neutrinos, in the ultra-relativistic approximation (β 4 → 1), the differential decay rate of a ν 4 with helicity r and energy E 4 into a ν e with helicity s and energy E e is [9] dΓ ν r and the matrix element is The same expression, of course, holds for the decay ofν 4 . In the scenario of interest (Eq. 2.1), all daughter ν e are left-handed and all ν 4 are produced via the weak interactions and are relativistic in the laboratory reference frame. We are also interested in ν 4 masses that are much smaller than the mass of the muon. Therefore, in the lab frame, virtually all ν 4 are left-handed and the energy spectrum of the daughter neutrinos is proportional to E e /E 4 . The total decay width for The situation is very similar for Majorana neutrinos. Eq. (2.3) holds along with Eq. (2.4) as long as one replaces g D → g M and allows for both ν 4 → ν e φ and ν 4 →ν e φ, keeping in mind that the ν e are all left-handed and theν e are all right-handed. Here, the ν 4 are produced via the weak interactions, are relativistic in the laboratory reference frame, and are much lighter than the muon. 4 Hence, in the lab frame, we expect the energy spectrum of the daughter neutrinos to be proportional to E e /E 4 (harder) while that of the daughter antineutrinos to be proportional to ( The total decay rate of ν 4 is given by Eq. (2.5) with g D → g M and an overall factor of 2 [21], accounting for the fact that there are two different allowed decay modes: It is straight-forward to compute, for a neutrino produced in a charged-current process involving muons, the energy and flavor of the neutrinos that reach the detector. In our computations, we make use of the results in [9], to which we refer to in more details, adapting the relevant expressions for the decay-scenarios of interest. For a recent, more complete treatment, that combines oscillation and decay effects, see, for example, Ref. [22]. Here, instead, we summarize the qualitative impact of ν 4 production and decay. This discussion will help inform the results we present in the following sections.
We are interested in m 4 m 1,2,3 and, in a charged-current process involving muons, the ν 4 is produced incoherently relative to ν 1,2,3 . Hence, when, for example, a pion decays into a muon and a neutrino, the neutrino is either a ν 4 , with probability |U µ4 | 2 , or the orthogonal state, 5 with probability 1 − |U µ4 | 2 . If the initial state is a ν 4 , it will reach the detector with probability e −Γ 4e L , where L is the baseline and Γ 4e is the ν 4 decay width, see Eq. (2.5) or (2.6). Hence, the probability that the neutrino will behave like a ν µ in the detector is (2.7) In the limit where ν 4 is very long-lived, Γ 4e L 1, . This agrees with the ν µ survival probability assuming there is a stable ν 4 and it is produced incoherently or, equivalently for the purposes of this setup, the new mass-squared difference is very large, ∆m 2 14 L/E 1 and the oscillations average out. Instead, in the limit where the decay is fast Γ 4e L 1, P µµ = (1 − |U µ4 | 2 ) 2 . The parent particle will yield a ν e in the final state only if ν 4 decays because U e4 ≡ 0 and both ν 4 and the state proportional to U µi ν i , i = 1, 2, 3, are orthogonal to ν e . If the ν 4 decays before reaching the detector -this happens with probability (1 − e −Γ 4e L ) -a ν e or aν e with some energy less than the original parent energy will arrive at the detector 6 with probability B e or Bē. In the Dirac case of interest here, B e = 1, Bē = 0, while in the Majorana case B e = Bē = 0.5. The probability that the ν e orν e emerges with energy E e is proportional to Eq. (2.4). The same happens forν 4 decays. In summary, (2.8) and the same-helicity (opposite-helicity) final state has a harder (softer) spectrum. Note that, strictly speaking, P µe and P µē are not probabilities. Qualitatively, it is easy to see why this hypothesis can outperform the standard (3+1)oscillation hypothesis [5][6][7][8]. In the (3+1)-oscillation scenario, P µe ∝ |U µ4 | 2 |U e4 | 2 while the survival probabilities of ν µ and ν e are, respectively, P µµ ∝ |U µ4 | 2 (1 − |U µ4 | 2 ) and P ee ∝ |U e4 | 2 (1 − |U e4 | 2 ). A sizable P µe requires both a non-negligible |U µ4 | 2 and |U e4 | 2 which, in turn, are constrained by disappearance searches. In the sterile-decay scenario, P ee = 0 and, P µµ ∝ |U µ4 | 2 (1 − |U µ4 | 2 ), similar to the oscillation scenario, especially in the limit of small Instead, it is constrained by non-oscillation experiments, as we quickly summarize in the next subsection, and we find that reasonably large values of Γ 4e L are allowed for the L/E values of interest. In the case of Majorana neutrinos, one half of the neutrinos will decay into antineutrinos, and vice-versa. This means that, in the case of the LSND experiment, some of theν e -excess events arises from parent ν µ created in the decay of the stopped π + , while half of the decaying-component associated with theν µ from the Michel decay will behave like a ν e and will not contribute to theν e -excess. In the case of MiniBooNE, the excess of ν e andν e events will be associated to both ν µ andν µ parents. Since the wrong-sign contamination is different between neutrino-mode running and antineutrinomode running, we expect the excesses observed in the case of the neutrino and antineutrino beams to be slightly different. We return to these issues in the discussion of our results, in Sec. 3.

Constraints on New Neutrinos and Neutrino-Scalar Interactions
There are several bounds on the new-physics parameters we are introducing here: m 4 , g D,M ≡ g, and |U µ4 |. We will discuss oscillation-related bounds in the next sections and here we summarize non-oscillation results.
Searches for neutral heavy leptons constrain |U µ4 | 2 as a function of m 4 . Keeping in mind that we are interested in constrains assuming ν 4 decays, as far as non-neutrino-oscillation experiments are concerned, invisibly, |U µ4 | 2 10 −2 for m 4 1 MeV (see Refs. [23,24] for recent quantitative analyses). The bounds are significantly weaker for smaller values of the m 4 . For m 4 1 MeV, the strongest bounds come from precision measurements of π → µν. Bounds from ν µ disappearance, as we will discuss later, are around |U µ4 | 2 10 −2 for m 4 10 eV and hence will dominate for m 4 1 MeV.
The couplings g of neutrinos to other neutrinos and a scalar particle, in the region of parameter space of interest here, are also best constrained by leptonic meson decays, especially the decays of pions and Kaons (e.g. K → µνφ). The bound on g depends on both the nature of the decay and on |U µ4 | 2 . Here, conservatively, we use the results from Ref. [25], which translate into (2.9) As far as short-baseline experiments, we are sensitive to |U µ4 | 2 and Γ 4e ∝ (gm 4 ) 2 , see Eqs. (2.5,2.6). As will be discussed in great detail in the next couple of sections, we will be interested in (gm 4 ) 2 |U µ4 | 2 ∼ 1eV 2 or so the constrain Eq. (2.9) can be easily satisfied for m 4 10 keV.
In summary, for 1 MeV m 4 10 keV, we expect to avoid all non-oscillation bounds with relative ease. We return to these in Sec. 3.

Simulations and Results
Here we provide details of the data we analyze and discuss how well they fit the decayingsterile-neutrino hypothesis. We also discuss the details of our simulation of data from the SBN program and how sensitive it is to the decaying-sterile-neutrino hypothesis.

LSND
The Liquid Scintillator Neutrino Detector (LSND) experiment [26] ran at the Los Alamos Neutron Science Center (LASCE) from 1993 to 1998. The experiment was designed to look forν e from a pion-decay-at-rest neutrino source [1]. LSND consisted of a cylindrical tank filled with 167 tons of mineral oil doped with a low concentration of liquid scintillator. This combination allows the detection of both Cherenkov and scintillation light, which are collected by 1220 photo-multiplier tubes (PMT) that surround the detector inner wall. Neutrinos are produced by the interaction of a 798 MeV proton beam with a production target, where positive pions stop at the beam dump and decay at rest into positive muons (π + → µ + + ν µ ). The distance between the beam dump and the longitudinal center of LSND is 30 meters. The positive muons also decay at rest (µ + → e + + ν e +ν µ ). The Michelν µ would lead to aν e signal in the presence of neutrino oscillations or other flavor-changing mechanism. Theν e are detected via inverse beta decay (IBD),ν e +p → n+e + , where the positron leads to Cherenkov and scintillation light inside mineral oil. The outgoing neutron manifests itself as subsequent scintillation light as it is captured on hydrogen and a 2.2 MeV photon is emitted [27]. LSND makes use of this two-component signal to select aν e -candidate event sample.
In order to generate expected event rates for the different decay scenarios and fit them to the available data, we make use of the GLoBES [28,29] c-library. Decay-at-rest fluxes were obtained from Ref. [1], and we use the IBD cross-section from Ref. [30]. In the case of Majorana neutrinos, we expectν e appearance from not only theν µ but also from the ν µ parents from π + decay, as discussed in the previous section. We considered events associated to neutrino energies between 20 and 60 MeV. Finally, a Gaussian energy smearing with σ(E) = 17%/E[MeV] was implemented to take into account the energy resolution of the experiment.
We perform a χ 2 -analysis, including an overall normalisation error of 25% for signal and background. Uncertainties in the neutrino flux, cross-section and efficiency lead to systematic errors between 10% and 50%, as discused in Ref. [1]. The LSND background sources come mainly from intrinsic beamν e andν µ events and are summarized in Table VIII of Ref. [1]. In order to validate our analysis procedure, we first fit the two-flavor oscillation hypothesis and compare our results with those presented by the LSND Collaboration [1]. When generating events, we introduce a normalization factor that allows us to mimic the total rates of the best-fit spectrum obtained by the LSND collaboration (Figure 24 of Ref. [1]). Our best-fit oscillation spectrum (green histogram), in bins of L/E, is depicted in Fig. 1, along with the data and backgrounds published by the collaboration; the best-fit point for the oscillation analysis is sin 2 2θ, ∆m 2 = 0.0063, 7.2 eV 2 and the minimum value of χ 2 is χ 2 min = 10.19. Given the eleven bins we included in our analysis (and hence nine degrees of freedom), we conclude that two-flavor-oscillations are a good fit to the LSND data, as expected. The allowed regions of the (sin 2 2θ, ∆m 2 ) parameter space match well with those published by the LSND collaboration. With this agreement, we are confident we are capable of faithfully reproducing the data-analysis of LSND well enough to repeat the procedure for the decayingsterile-neutrino hypothesis. We generate neutrino event spectra for each set of decay parameters |U µ4 | 2 , gm 4 and attempt to fit them to the LSND data, using a χ 2 -fit. The best-fit spectra in the case of Dirac and Majorana neutrinos are depicted, respectively, in black and blue in Fig. 1. The results for the two hypotheses are very similar. Ignoring the effect of the ν 4 →ν e φ decays, the Majorana and Dirac cases are identical, except for the fact that B e = 1 in the Dirac case and B e = 0.5 in the Majorana case. Since the effect of the decay is proportional to |U µ4 | 2 B e , one can compensate for the change in B e by changing |U µ4 | 2 by a factor of two. The ν 4 produced in DAR are monochromatic, with energy around 30 MeV. Hence, theν e produced in ν 4 →ν e φ have very low energies and only populate the highest L/E-bins. The situation is made worse by the fact that the energy spectrum of the daughterν e , in the antineutrino decay, are soft, peaking (linearly) at zero energy. The overall result is that mostν e from ν 4 →ν e φ have too low energy to significantly contribute to the LSND excess.
The best fit point falls in the region where the decay is fast so that, to zeroth order, all ν 4 decay between production and detection. We estimate the goodness-of-fit by comparing χ 2 min =19.53 (20.17) in the Dirac (Majorana) cases with nine degrees of freedom and conclude the fit is acceptable (p-value around two percent). The quality of this fit is worse than that of the oscillation fit. This is due to fact that the energy spectrum of the daughterν e is distorted towards lower energies compared with the energy spectrum of the parentν 4 . The allowed regions of the parameter space, along with the best-fit points, are depicted in Fig. 2. As advertised, the results of the two decay scenarios are similar once one rescales the value of |U µ4 | 2 by a factor of 2. The dots indicate the best-fit-point.

MiniBooNE
The MiniBooNE experiment was designed to test the oscillation interpretation of the LSND data [31]. It consisted of a spherical tank filled with 800 tons of mineral oil and internally covered with 1280 PMTs to collect, mostly, Cherenkov light. The MiniBooNE detector is located 540 meters downstream from the neutrino source. In order to generate a neutrino flux, the booster neutrino beam (BNB), located at Fermilab, delivers 8.89 GeV protons that interact with a beryllium target. Charged mesons, like pions and kaons, are then produced and decay predominantly into muon neutrinos and antineutrinos. A magnetic focusing horn was used to sign-select the charged mesons, allowing, depending on the polarity of the horn, two neutrino-beam configurations: 1) neutrino mode: positively-charged mesons are focused to create a high-intensity flux of neutrinos; 2) antineutrino mode: negatively-charged mesons are focused to create a high-intensity flux of antineutrinos. MiniBooNE measures both ν e and ν µ , plus their antiparticles, and is sensitive to ν e andν e appearance and ν µ andν µ disappearance. ν µ,e andν µ,e are identified as they scatter through the charged-current quasielastic (CCQE) process, yielding µ ± , e ± , respectively. These particles emit Cherenkov and scintillation light inside the detector, and muon-candidates are distinguished well from electron-candidates.
We analyse MiniBooNE appearance data collected when the neutrino-beam was running in both neutrino and antineutrino modes [3,32]. The MiniBooNE data set corresponds to 12.84 × 10 20 protons on target (POT) in neutrino mode and 11.27 × 10 20 POT in the antineutrino mode. We analyze the different data sets separately and combined.
We simulate MiniBooNE events in GloBES, where the CCQE cross-section information is available. Flux information was obtained from Ref. [33] and we include a Gaussian energy smearing function with σ(E) = 30%/ E[GeV] to mimic the detector energy resolution. For the electron-like events, the analysis is done in the neutrino energy range E QE ν ∈ [0.2, 3.0] GeV and the signal detection efficiencies for electron-like events are taken from [34]. Background events are summarized in Table 1 and Figure 1 of Ref. [3]. Note that, while neutral current events are impacted by the ν 4 decay, we estimate that the effect is negligible in the region of the parameter space in which we are interested. Hence, we do not include decay effects in the background events.
In our χ 2 analysis, we take statistical and systematic errors into account by using the official MiniBooNE covariance matrices, available in Ref. [32]. These include correlations among ν e (ν e ) signal and background events and ν µ (ν µ ) events for the neutrino (antineutrino) mode. In the combined analysis, the correlations among all neutrino and antineutrino samples are considered. Here, we are ultimately interested in the region of the parameter space where the impact of the new physics on ν µ -disappearance is very small, thanks to strong bounds from other experiments. Hence, the only impact of the ν µ part of the data is to provide information concerning the neutrino flux and the neutrino scattering parameters. In other words, we are interested in gauging the impact of fitting the ν e andν e appearance data assuming the same new physics does not impact the ν µ andν µ data. In order to achieve this, we followed the prescription, discussed in Appendix E.4 of Ref. [35], of considering only the contribution of electron neutrino and antineutrino events (signal and background) in the fit, along with an extra component related to the uncertainty in the overall normalization of the spectrum. Our data, therefore, consists of 11 ν e energy bins in both the neutrino and antineutrino modes. We will use the minimum value of the χ 2 in order to gauge the goodness-of-fit, using the 11 bins to compute the number of degrees of freedom.
As in the LSND case, we first fit the MiniBooNE neutrino-mode and antineutrino-mode data with the two-flavor oscillation hypothesis. For the neutrino-mode data, our best-fit oscillation spectrum (green histogram), in bins of L/E, is depicted in Fig. 3, along with the excess data published by the collaboration; the best-fit point for the oscillation analysis is sin 2 2θ, ∆m 2 = 0.83, 0.036 eV 2 and the minimum value of χ 2 is χ 2 min = 9.46. Given the eleven bins we included in our analysis (and hence nine degrees of freedom), we conclude that two-flavor-oscillations are a good fit to the MiniBooNE neutrino data, as expected. The allowed regions of the (sin 2 2θ, ∆m 2 ) parameter space match very well those published by the MiniBooNE collaboration. We obtain similarly satisfactory results with the MiniBooNE antineutrino-mode data. With this agreement, we are confident we are capable of faithfully reproducing the data-analysis of MiniBooNE well enough to repeat the procedure for the deacying-sterile-neutrino hypothesis. We generate neutrino event spectra for each set of decay parameters |U µ4 | 2 , gm 4 and attempt to fit them to the MiniBooNE data, using a χ 2 -fit. The best-fit spectra to neutrino-mode data, in the case of Dirac and Majorana neutrinos are depicted, respectively, in black and blue in Fig. 3. As in the LSND case, the results for the two hypotheses are rather similar. The reason here, however, is different. In the Majorana case, both the ν e and thē ν e daughters of the decay contribute to the excess of ν e -candidate events. The scattering cross-sections are a little different and, as discussed in Sec. 2, the wrong-helicity daughters have a softer spectrum. Ultimately, we expect that similar values of the parameters will yield similar-quality fits for the Majorana and Dirac hypothesis, especially in the case of antineutrino-mode data since the ν e detection cross-section is larger than that ofν e .
For both neutrino-mode and antineutrino-mode data, the best fit point falls in the region where the decay is relatively slow. Hence, to zeroth order, a lower-energy ν 4 decay more often than a higher-energy ν 4 . For the neutrino mode, we estimate the goodness-of-fit by comparing χ 2 min =11.08 (11.56) in the Dirac (Majorana) cases with nine degrees of freedom and conclude the fit is acceptable. For the antineutrino-mode, we estimate the goodnessof-fit by comparing χ 2 min =7.71 (6.66) in the Dirac (Majorana) cases with nine degrees of freedom and conclude the fit is also acceptable. The quality of these fits is similar to that of the oscillation fit. The allowed regions of the parameter space are depicted in Figs. 4 (neutrino mode), 5 (antineutrino mode), and 6 (neutrino and antineutrino modes combined). Unlike the LSND case, as advertised, the results of the two decay scenarios are similar for roughly similar values of |U µ4 | 2 . There is no obvious factor of two map between the Dirac and Majorana hypotheses, especially in the case of the antineutrino mode.

LSND and MiniBooNE Combined
Next, we evaluate how well the decaying-sterile-neutrino hypothesis fits both LSND and MiniBooNE data by adding the χ 2 obtained in the two independent analyses. The LSND-only and MiniBooNE-only allowed regions of the parameter space are depicted in Fig. 7 to facilitate comparisons, along with the allowed regions of the parameter space. The best-fit point, for the Dirac-neutrino scenario, is at |U µ4 | 2 , g D m 4 = (0.063, 1.17 eV) and χ 2 min = 45.33. For 31 degrees of freedom (11+11+11-2), we estimate a p-value of several percent, which we deem to be reasonable. The best-fit spectra, for the Majorana-neutrino case, for LSND (gold color) and MiniBooNE (neutrino-mode) (magenta) are depicted in Figs. 1 and 3. Note that the best-fit slightly undershoots the LSND data, and slightly overshoots those from MiniBooNE. The situation of the Majorana-neutrino scenario is similar; the quality of the fit is a little worse: χ 2 min = 48.34.

MINOS and MINOS+
MINOS [37] is a long-baseline superbeam experiment based at Fermilab. The source of neutrinos is the NuMI beam facility at Fermilab [38]. The experimental setup consists of a 1 kton near detector situated 1.04 km downstream and a 5.4 kton far detector situated 735 km away, on-axis in the Soudan underground laboratory. The primary goal of the MINOS experiment was to confirm, with an accelerator-based ν µ -beam, the evidence for ν µ -disappearance first seen in atmospheric experiments, measure the oscillation parameters sin 2 2θ 23 and |∆m 2 31 |, and look for the subleading long-baseline ν e -appearance signal. For these purposes, MINOS looked at charged-current ν µ -disappearance and ν e -appearance events in both neutrino and antineutrino modes [39]. It also measures neutral current events that are helpful in sterile-neutrino searches. Initially, MINOS operated with the low-energy tune of the NuMI beam that peaks at neutrino energies around 3 GeV. This was followed by running, referred to as MINOS+, with the medium-energy tune of the NuMI beam, where the flux peaks at neutrino energies around 7 GeV. The most recent sterile neutrino searches were presented in [36]. These results correspond to an exposure of 10.56 × 10 20 POT for MINOS and 5.80 × 10 20 POT for the MINOS+ experiment. Assuming the neutrino mass-eigenstates are stable, for m 4 10 eV, the collaboration claims that the data constrain |U µ4 | 2 < 2.3 × 10 −2 at the 90% confidence level. Here, we take this result at face value and apply it to the decaying-sterile-neutrino scenarios of interest.
Strictly speaking, the analysis presented in [36] does not apply if the ν 4 is unstable, for two reasons. One was already discussed. If one ignores the daughters of the neutrino decay, the ν µ survival probability depends on the ν 4 lifetime, see Eq. (2.7). However, the difference between a stable and unstable ν 4 , as far as this contribution is concerned, is proportional to |U µ4 | 4 , a factor |U µ4 | 2 smaller than the leading contribution. Since MINOS(+) is sensitive to |U µ4 | 2 values of order 10 −2 , the fact that ν 4 can decay is irrelevant for this contribution to the disappearance analysis. The other potential impact of the decay is that the daughter ν e of the ν 4 decay can oscillate into a ν µ by the time it reaches the far detector. This extra contribution to the ν µ survival probability is, relative to the leading |U µ4 | 2 -effect, suppressed by |U e3 | 2 ∼ 0.02 and hence very small.
For the reasons discussed above, we take the constraint from the ν µ disappearance data to be |U µ4 | 2 < 2.3 × 10 −2 at the 90% confidence level for all values of gm 4 of interest. This is represented by a horizontal line in Fig. 7. This constraint rules out the region of parameter corresponding to small gm 4 but leaves behind a healthy portion of the parameter space, including values of gm 4 small enough that the decay of ν 4 is not necessarily prompt for the energies of interest. Since the Dirac hypothesis points to relatively smaller values of |U µ4 | 2 , the allowed region of parameter space is "larger" in this case.
One final note before proceeding. Given that, for large gm 4 , we require |U µ4 | 2 10 −2 (and independent of gm 4 ), the bounds from meson leptonic decays on g and |U µ4 | 2 , discussed in Sec. 2.1, translate into gm 4 10 3 eV, saturated as m 4 approaches 1 MeV.

SBN
The Short-Baseline Neutrino (SBN) Program is a set of three liquid argon detectors that will be aligned with the central axis of the BNB at Fermilab. According to the proposal [40], the SBN Program is designed to address several anomalies in neutrino physics and will test, with the most sensitivity, the oscillation-interpretation to LSND and MiniBooNE data. In order to explore the potential of the SBN Program to test the decaying-sterile neutrino model scenarios discussed here, we performed a sensitivity analysis considering only the neutrino-mode running for the BNB (see Section 3.2). The generation of events as well as the χ 2 function were implemented in GLoBES. The relevant details regarding the three detectors, the flux, the scattering cross-sections and efficiencies are described in Ref. [41] and the assumptions we make here are the same. We are considering only the ν e -appearance channel in order to estimate the sensitivity of the SBN Program. In the analysis, we correlate the spectrumnormalization error among the three detectors and set the uncertainty at 15%. Detailed descriptions of the signal and background for the appearance channel can also be found in Ref. [41].
The sensitivity of the SBN Program, assuming 6.6 × 10 20 POT for SBND and ICARUS (three nominal years of running) and 1.32 × 10 21 POT for MicroBooNE (six nominal years of running), is depicted in Fig. 8. The regions of the parameter space preferred by combined LSND and MiniBooNE are also depicted in order to facilitate comparisons. The SBN program can definitively test the decaying-sterile neutrino solution to the LSND and MiniBooNE data.

Summary and conclusions
The excess of ν e -andν e -candidate events at MiniBooNE and LNSD remains unexplained. The, arguably, simplest solution -3+1 neutrino-oscillation with a new mass-squared difference around 1 eV 2 -is, however, severely constrained. If these data are indeed pointing to more new physics in the neutrino sector, it is likely that the new physics contains more ingredients than new neutrino mass-eigenstate that mix slightly with the active neutrinos. Here, we explored the hypothesis that there is a new neutrino mass-eigenstate ν 4 and a new very light scalar particle φ. ν 4 and φ interact in such a way that ν 4 → ν e φ. Here, the excess of ν eandν e -candidate events at MiniBooNE and LNSD are the daughter ν e andν e from ν 4 andν 4 decay. This hypothesis was first proposed in Ref. [9] in order to address the LSND anomaly.
We find a reasonable fit to the data of MiniBooNE and LSND, albeit the quality of the fit to only MiniBooNE and LSND data is not as good as the one obtained with the 3+1 neutrino-oscillations hypothesis. The decaying-sterile-neutrino hypothesis, however, can cleanly evade data from ν µ -disappearance searches, which constrain |U µ4 | 2 10 −2 , and is immune to searches involving ν e -disappearance. We find that precision measurements of meson leptonic decays can also be satisfied as long as 1 MeV m 4 10 keV. The SBN program at Fermilab should be able to definitively test the decaying-sterile-neutrino hypothesis. We considered two different decay scenarios, one with Majorana neutrinos, one with Dirac neutrinos. The MiniBooNE and LSND data are such that both models fit the data with very similar efficacy.
There are a few other new-physics solutions to the LSND and MiniBooNE data. Several, however, address only one data set or the other, including some recent, very interesting ideas [42][43][44] that also postulate the existence of new light particles and new interactions. While the decaying-sterile-neutrino hypothesis explored here is not an excellent fit to both data sets -especially the LSND data -it seems to provide an interesting possibility. We hope the results presented here will inspire the collaborations -they are the only ones capable of performing a proper fit to their data -to investigate this possibility.
We did not consider bounds from early-universe cosmology. The relatively large mixing between ν s and ν µ indicates that it should be in thermal equilibrium in the early universe [45]. The fact that they decay quickly, however, should loosen bounds from, for example, big-bang nucleosynthesis. The new interaction between active and sterile neutrinos will also impact the dynamics of the early universe, and so will the new light degree of freedom φ. More dynamics, including, for example, other couplings of φ to the active neutrinos, may help alleviate some of the potential tension. The exploration of these types of constraints is beyond the ambitions of this manuscript.
Other manifestations of the sterile-neutrino decay hypothesis have been, very recently, discussed in the literature, including [46]. The work presented here share several similarities with these efforts but we explore, for the most part, a different region of the -very largespace of decaying-sterile-neutrino models.