Decaying Sterile Neutrinos and the Short Baseline Oscillation Anomalies

PHYSICAL REVIEW D 101, 115013 (2020)

Decaying sterile neutrinos and the short baseline oscillation anomalies

† ‡ Mona Dentler ,1,* Ivan Esteban ,2, Joachim Kopp ,3,4, and Pedro Machado 5,§ 1Institut für Astrophysik, Georg August-Universität Göttingen, 37077 Göttingen, Germany 2Departament de Fisíca Qu`antica i Astrofísica and Institut de Ciencies del Cosmos, Universitat de Barcelona, 08028 Barcelona, Spain 3Theoretical Physics Department, European Organization for Nuclear Research, 1211 Geneva, Switzerland 4PRISMA Cluster of Excellence and Mainz Institute for Theoretical Physics, Johannes Gutenberg-Universität Mainz, 55128 Mainz, Germany 5Fermi National Accelerator Laboratory, Batavia, 60510 Illinois, USA

(Received 18 November 2019; accepted 6 May 2020; published 11 June 2020)

The MiniBooNE experiment has observed a significant excess of electron neutrinos in a muon neutrino beam, in conflict with standard neutrino oscillations. We discuss the possibility that this excess is explained by a sterile neutrino with a mass ∼1 keV that decays quickly back into active neutrinos plus a new light boson. This scenario satisfies terrestrial and cosmological constraints because it has neutrino self- interactions built-in. Accommodating also the LSND, reactor, and gallium anomalies is possible, but requires an extension of the model to avoid cosmological limits.

DOI: 10.1103/PhysRevD.101.115013

I. INTRODUCTION earlier LSND experiment [4], and with several reported hints for anomalous disappearance of electron neutrinos in reactor Many major discoveries in neutrino physics have started experiments [5,6] and in experiments using intense radio- out as oddball anomalies that gradually evolved into incon- active sources [7,8].2 However, the sterile neutrino parameter trovertible evidence. In this work, we entertain the possibility space consistent with MiniBooNE and these other anomalies that history is repeating itself in the context of the MiniBooNE is in severe tension with the nonobservation of anomalous νμ anomaly. From 2002 to 2019, the MiniBooNE experiment disappearance [9–19], unless several additional new physics has been searching for electron neutrinos (νe) appearing in a 1 effects are invoked concomitantly [20,21]. muon neutrino (νμ)beam[1–3], and has found a corre- 4 8σ In this work, we propose a different explanation for the sponding signal at . statistical significance. For some MiniBooNE anomaly, and possibly also for the LSND, time, the simplest explanation for this signal appeared to be reactor, and gallium anomalies. In particular, we consider a the existence of a fourth neutrino species νs, called “sterile ” sterile neutrino that rapidly decays back into Standard neutrino because it would not couple to any of the Standard Model (“active”) neutrinos ν [22–24]. The MiniBooNE Model interactions, but would communicate with the a ν excess is then interpreted as coming from these decay Standard Model only via neutrino mixing. If s has small products. We will see that this scenario requires only very but nonzero mixing with both ν and νμ and if the corre- e small mixing between ν and νμ, thus avoiding the strong νμ ν s sponding mostly sterile neutrino mass eigenstate 4 is some- disappearance constraints. It also requires somewhat larger what heavier (∼1 eV)thantheStandardModelneutrinos,the mixing between νs and νe, in line with the hints from MiniBooNE signal could be explained. This explanation reactor and radioactive source experiments. Finally, we will 3 8σ would also be consistent with a similar . anomaly from the argue that decaying sterile neutrinos may avoid cosmological constraints because the model automatically * endows sterile neutrinos with self-interactions (“secret [email protected] † ” [email protected] interactions [25,26]). ‡ [email protected] §[email protected] II. DECAYING STERILE NEUTRINO FORMALISM 1Here and in the following, when we say neutrino we mean also the corresponding antineutrinos. We extend the Standard Model by a sterile neutrino νs (a Dirac fermion) and a singlet scalar ϕ. The relevant Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to 2The latter class of experiments is usually referred to as the author(s) and the published article’s title, journal citation, “gallium experiments,” based on the active component of their and DOI. Funded by SCOAP3. target material.

2470-0010=2020=101(11)=115013(13) 115013-1 Published by the American Physical Society DENTLER, ESTEBAN, KOPP, and MACHADO PHYS. REV. D 101, 115013 (2020)

ˆ ˆ interaction and mass terms in the Lagrangian of the basis, and Γ ¼ Γ4Π4 is the decay term, which contains the ˆ model are projection operator Π4 ¼jν4ihν4j onto the fourth, mostly X sterile, mass eigenstate as well as the decay width Γ4 of ν4 L ⊃− ν¯ ν ϕ − ν¯ ν ð Þ g s s mαβ α β: 1 in its rest frame. Similarly, Γϕ is the rest frame decay width a¼e;μ;τ;s of ϕ. The functional Rν½ρˆν; ρϕ;E;t describes the appearance of the daughter neutrinos from ν4 and ϕ decay. The neutrino flavor eigenstates να are linear combinations Neglecting the masses of ν1, ν2, and ν3,itisgivenby of the mass eigenstates νj (j ¼ 1, 2, 3, 4) according to the relation να ¼ Uα ν , where U is the unitary 4 × 4 leptonic j j Rν½ρˆν; ρϕ;E;t mixing matrix. The first term in Eq. (1) can thus be Z ∞ X Γlabðν → ν ϕÞ rewritten as ˆ d 4 k ¼ ΠF dE4 ρˆν;44ðE4;tÞ E 2 dEk 2 1−x k − ν¯ ν ϕ − j j ν¯ ν ϕ − ð ν¯ ν ϕ þ Þ ð Þ ϕ4 g F F g Us4 4 4 gUs4 4 F H:c: ; 2 Z X ∞ dΓlabðϕ → ν ν¯ Þ þ Πˆ ρ ð Þ k j ð Þ with F dEϕ ϕ Eϕ;t ; 6 k;j E dEk X3 where dΓlabðX → YÞ=dE are the differential decay νF ≡ Usiνi: ð3Þ k i¼1 widths for the various decays X → Y in the lab frame, and xϕ4 ≡ mϕ=m4. The projection operator We assume initially that the fourth, mostly sterile, mass ν ≃ ν Oð Þ 3 eigenstate 4 s has a mass m4 between eV and jν ihν j X U U Oð100 Þ ϕ Πˆ ¼ F F ¼ P si sj jν ihν jðÞ keV , and that the mass of is of the same order, F jhν jν ij2 j j2 i j 7 F F ¼1 k Usk but smaller. The last term in Eq. (2) will then induce ν4 → i;j νF þ ϕ decays, while the first term is responsible for ϕ → ν þ ν¯ decays. When these decays occur in a neutrino isolates the specific combination of mass eigenstates that F F ν ϕ beam, they will produce lower-energy neutrinos at the appears in 4 and decays, and the integrals run over all expense of higher-energy ones, and they may also alter the parent energies E4, Eϕ that lead to daughter neutrinos of R ½ρˆ flavor structure of the beam. In particular, they can produce energy E. Analogously, ϕ ν;E;t describes the appearance of scalars from ν4 decay: excess low-energy νe in a νμ beam, as suggested by the MiniBooNE anomaly. Z E=x2 X Γlabðν → ν ϕÞ The phenomenology of the model depends mainly on ϕ4 d 4 k Rϕ½ρˆν;E;t¼ dE4 ρˆν;44ðE4;tÞ five new parameters. Besides m4 and mϕ, these are the E k dEϕ j j2 j j2 ν coupling g and the mixings Ue4 , Uμ4 between 4 and Γlabðν¯ → ν¯ ϕÞ þ ρˆ¯ ð Þ d 4 k ð Þ νe, νμ. We will assume the mixing with ντ to be zero and ν;44 E4;t : 8 dEϕ neglect the complex phases, as these parameters do not play an important role in explaining the MiniBooNE excess. For With the appearance terms Rν½ρˆν; ρϕ;E;t and Rϕ½ρˆν;E;t Γ practical purposes, it is convenient to quote m4 4 instead of defined, the equations of motion (4) and (5) can be solved g,asm4Γ4 appears directly in the laboratory frame decay ð Γ Þ analytically if we neglect matter effects. Neglecting fur- length E= m4 4 . Also, it is convenient to use the ratio thermore the small mass splittings between the three light mϕ=m4 instead of just mϕ because the ratio measures more neutrino mass eigenstates, the electron neutrino flux directly the kinematic suppression in ν4 decays. ϕeðL; EÞ appearing in a muon neutrino beam of energy The evolution in energy E and time t of a neutrino beam E after a distance L due to oscillations and decay is in our model can be described by a neutrino density matrix given by ρˆ ð Þ 4 4 ν E; x (a × matrix in flavor space), the corresponding ρˆ¯ ð Þ antineutrino density matrix ν E; x , and the scalar density 2 2 −m4Γ4L ϕ ðL;EÞ¼ϕμð0;EÞjU 4j jUμ4j 1 þ e E function ρϕðE; tÞ. The evolution equations are [27,28], e e 2 2 dρˆνðE;tÞ 1 m4 −m4Γ4L Δm41L 2 jhνejνFij ¼ − ½ ˆ ρˆ − Γˆ ρ þ R ½ρˆ ρ ðÞ − 2 2E þj j I i H; ν ; ν ν; ϕ;E;t 4 e cos 2 Uμ4 2 : dt 2 E E jhνFjνFij dρϕðE; tÞ mϕ ð9Þ ¼ − Γϕρϕ þ Rϕ½ρˆν;E;tð5Þ dt E Here, jνFi is the superposition of mass eigenstates into ˆ ¼ 1 ð0 Δ 2 Δ 2 Δ 2 Þ ν where H 2E diag ; m21; m31; m41 is the standard which the 4 decay [defined in Eq. (3)], and the decay neutrino oscillation Hamiltonian, written here in the mass integral I is given by

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Z ∞ Γ X lab −m4 4L 1 dΓ ðν4 → νjϕÞ I ¼ ð1 − E4 Þϕ ð0 Þ dE4 e μ ;E4 m4 ð1− 2 Þ Γ4 dE E= xϕ4 j E4 Z Z ∞ 2 mϕΓϕL Γ Eϕ=xϕ4 1 − Γ m4 4L mϕΓϕL E m4 4L − þ dEϕ dE4 ð1 − e ϕ Þ − ð1 − e E4 Þ m4Γ4L mϕΓϕL E Eϕ − E4 Eϕ E4 Eϕ X lab lab X lab 1 dΓ ðν4 → ν ϕÞ dΓ ðν¯4 → ν¯ ϕÞ 1 dΓ ðϕ → ν ν¯ Þ ϕ ð0 Þ j þ ϕ¯ ð0 Þ j i j ð Þ × m4 μ ;E4 μ ;E4 mϕ : 10 Γ4 dE dE Γϕ dE E4 j i;j Eϕ

¯ In the above equations, ϕμð0;EÞ and ϕμð0;EÞ are the unoscillated beam following the formulas given above. initial νμ and ν¯μ fluxes, respectively. A completely analo- We then follow the fitting procedure recommended by the gous equation describes ν¯e appearance. MiniBooNE collaboration (see the data releases accompa- The physical interpretation of Eq. (9) is straightforward: nying Refs. [1,3]), but go beyond it by accounting for the ν ν the first term on the right-hand side describes νμ → νe impact of μ and e disappearance on the signal and oscillations, altered by the removal of neutrinos at energy background normalization (see Appendix for details). E due to ν4 decay. In fact, this contribution matches the result Illustrative results are shown in Fig. 1, where we have of Ref. [29], on invisible ν4 decay. The second term gives the chosen parameter values that give an optimal fit to contribution from neutrinos generated in ν4 and ϕ decays. MiniBooNE data while being consistent with null results 2 The factor jUμ4j arises because ν4 is the only mass eigenstate from other oscillation experiments, as well as nonoscilla- Γ ¼ 2 1 2 ν that decays. It describes the amount of ν4 in the νμ beam. The tion constraints. At m4 4 . eV , most 4 will have ¼ factor jhν jν ij2=jhν jν ij2 is the probability of the decay decayed before reaching the detector. The value mϕ=m4 e F F F 0 82 ν product to be detected as an electron neutrino, and the integral . implies mild phase space suppression in 4 decays, ν I controls the energy distribution of the decay products. which tends to shift the e spectrum to lower energies, in Analytic expressions for the decay widths appearing in excellent agreement with the data. Compared to models ϕ Eqs. (4)–(12) are given in Appendix B. with massless [22,23], our scenario also has the advan- tage that it allows ϕ → νFν¯F decays, further boosting the νe flux at low energies. It is therefore favored compared to the III. FIT TO MINIBOONE DATA mϕ ¼ 0 case at more than 99% confidence level. The fit in To compare the predictions of the decaying sterile our model is better than in oscillation-only scenarios (blue neutrino scenario to MiniBooNE data, we evolve the dotted histogram in Fig. 1) [19], which by themselves

FIG. 1. Comparison of MiniBooNE neutrino-mode (left) and antineutrino-mode (right) data [3] to the predictions of the neutrino oscillation þ decay scenario discussed in this work. We show the expected spectrum at the point which optimally fits MiniBooNE data, while being consistent with all null results (orange histogram with systematic error band; parameters given in the plot). We also show the 2 2 MiniBooNE-only best point for 3 þ 1 oscillations without decay (blue dotted histogram, parameter values Δm41 ¼ 0.13 eV , 2 2 jUe4j ¼ 0.024, jUμ4j ¼ 0.63).

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detector, which is a ∼6 m sphere located ∼500 m from the primary target, under essentially the same angle as the parent neutrino would have done.

IV. CONSTRAINTS We now discuss the various constraints that an explanation of the MiniBooNE anomaly in terms of decaying sterile neutrinos has to respect. The most relevant constraints are also summarized in Figs. 2 and 3. FIG. 2. Nonoscillation constraints on decaying sterile neutrinos (1) Oscillation null results. Putting MiniBooNE into ν for parameters favored by the global fit without LSND (shaded), context with other e appearance searches, we show in and by the global fit without the free-streaming constraint Fig. 3 two slices through the 5-dimensional parameter (hatched). space of the decaying sterile neutrino model along the plane 2 2 spanned by jUe4j and jUμ4j . To produce this figure, we already offer an excellent fit as long as only MiniBooNE have used fitting codes from Refs. [9,12,19] (based partly data are considered (MiniBooNE quotes a χ2 per degree of on Refs. [37–39]). We see that most of the parameter region freedom of 9.9=6.7 [3]). Our model, however, is also preferred by MiniBooNE is well compatible with the consistent with all constraints. Notably, it reproduces the KARMEN short-baseline oscillation search [40] and with angular distribution of the neutrino interaction products in the OPERA long-baseline experiment [41]. We have MiniBooNE because it predicts an actual flux of electron checked that the limits from ICARUS [42–44] and E776 neutrinos instead of attempting to mimic the signal with [45] are significantly weaker. – 2 2 other particles [30 35]. In particular, the angle between the All constraints on jUe4j (jUμ4j ) from νe (νμ) disap- parent νs and the daughter νe is suppressed by a large pearance experiments are avoided [17,19,46]. This is Lorentz boost γ ∼ Oð1000Þ [36]. This boost is sufficient to mostly because in pure oscillation scenarios the number ensure that the daughter neutrinos enter the MiniBooNE of excess events in MiniBooNE and LSND is proportional

2 2 FIG. 3. Allowed values of the squared mixing matrix elements jUe4j and jUμ4j (measuring the mixing of νs with νe and νμ, respectively) in the decaying sterile neutrino scenario. We show two representative slices through the 5-dimensional 99% confidence regions. Our fits include MiniBooNE, OPERA, ICARUS, E776, and KARMEN data, as well as constraints from nuclear beta decay spectra and from the requirement of neutrino free-streaming in the early Universe. For the null results from oscillation experiments, the region to the right of the curves is excluded. For the free-streaming constraint, the region to the left of the gray dotted contour is 2 excluded. We also show, as a black rule at the bottom of the plot, the jUe4j range preferred by the reactor neutrino anomaly. Constraints on νμ disappearance are significantly weaker here than in the 3 þ 1 scenario without decay, and are hence not shown. We also do not show a fit including both LSND and cosmology as the goodness of fit would be very poor. Note that the global combinations are 2 2 sensitive to five degrees of freedom, namely m4, jUe4j , Uμ4, m4Γ4, and mϕ=m4; oscillation experiments are sensitive only to the last four 2 2 of these; beta decay spectra depend on two degrees of freedom (m4 and jUe4j ); reactor experiments depend only on jUe4j ; and the free- streaming constraint depends only on the parameter combination m4=jUs1j.

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2 2 to jUe4j jUμ4j , while in our scenario it is proportional only is easily satisfied because in the interesting parameter 2 2 2 ≫ 1 Γ ≳ 1 2 ν to jUμ4j as long as jUe4j ≫ jUμ4j . Therefore, it agrees range with m4 eV and m4 4 eV ,any 4 that well even with the tightest constraints [47,48]. are produced in the early Universe will have decayed via ν → ν þðϕ → ν ν¯ Þ We can already see from Fig. 3 that MiniBooNE is also 4 1;2;3 1;2;3 1;2;3 long before recombination 2 compatible with LSND and with the jUe4j range preferred and the onset of structure formation. by the reactor anomaly, but only in a parameter region that (6) Neutrino free-streaming (blue region in Fig. 2 and ν would unacceptably reduce free-streaming of active neu- gray dotted lines in Fig. 3). Via the mixing with s, also ν ϕ trinos in the early Universe. We will see below that this the light neutrino mass eigenstates 1;2;3 feel -mediated tension can be avoided in extensions of the model. interactions and are therefore not fully free-streaming. This (2) Beta decay spectra (purple regions in Fig. 2 and may put the model in tension with CMB observations, black dashed lines in Fig. 3). Direct searches for sterile which require that neutrinos should free-stream from about 5 3 neutrinos looking for anomalous features in beta decay redshift 10 onwards [56–60]. This requirement bounds spectra [49–52] suggest that Oð0.001 − 0.01Þ mixings the squared coupling among the lightest neutrino mass 2 2 between active and sterile neutrinos—as required by eigenstate and the scalar ϕ, i.e., ðgjUs1j Þ . (Heavier mass MiniBooNE—are allowed for m4 ≲ few keV. eigenstates are not relevant as they decay quickly.) Here we (3) Neutrinoless double beta decay. If neutrinos are are taking ν1 to be the lightest mass eigenstate, as favored Majorana particles, the nonobservation so far of neutrino- by current data. Quantitatively, 2 less double beta decay requires m4jU 4j ≲ 0.2 eV [53]. e 4 0 1 4 This is the reason we always focus on Dirac neutrinos in eff −10 2 m4 . 2 ðm4Γ4Þ ≲ 4 × 10 eV xϕ4: ð13Þ this work. eV jUs1j (4) Neff, a measure for the energy density of relativistic 2 −6 particles in the early Universe (green region in Fig. 2). The with m4 ≲ 200 eV required for g ≳ 10 [60]. Note that in ∼3 measured value of Neff is very close to the SM value of Fig. 3, this constraint is present even for very small both at the BBN and recombination epochs [54,55]. mixings. This is because, at fixed m4Γ4, small mixings Naively, one might expect that this observation precludes need to be compensated by a large coupling g, strengthen- 2 the existence of a fourth neutrino species with m4 ≲ MeV. ing the free streaming constraint. The value of jUs1j is 2 2 In our model, however, the Neff constraint is avoided by fixed in terms of jUe1j and jUμ1j by unitarity, assuming the “secret interactions” mechanism [25,26]: any small the active neutrino mixing angles to be fixed at their values abundance of νs generates a temperature-dependent from Ref. [66]. ∝ 2 ν –ν potentialpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVeff g T, reducing the s a mixing by a factor However, the constraint could be substantially weakened Δ 2 ð Þ ν in extensions of our model, see for instance Refs. [67–70]. m = EVeff . Hence, the production of s is suppressed until the temperature drops low enough. For the parameter A minimalist example is the production of extra species of range that the short-baseline anomalies are pointing to, this light particles at the expense of the neutrino sector after can easily be postponed to late times (T ≪ MeV), after neutrino decoupling. These would compensate for the lack of free-streaming in active neutrinos. neutrino-electron decoupling. Consequently, when νs are eventually produced, they are produced at the expense of (7) SN 1987A. The fact that neutrinos from supernova 1987A could be observed at Earth without being absorbed active neutrinos, so Neff does not change any more and constraints are automatically satisfied. More quantitatively, through scattering on the cosmic neutrino background constrains neutrino self-interactions [71]. We have checked Neff constraints are avoided when that, due to mixing suppression, these constraints are 4 avoided in our scenario. Note that supernova cooling, eff −14 2 m4 ðm4Γ4Þ ≳ 2 × 10 eV ; ð11Þ eV which is sensitive to noninteracting sterile neutrinos, does not constrain our model as ν4 and ϕ quickly decay to lighter where we have defined neutrinos that remain trapped in the supernova core. (8) Decays of SM neutrinos. We have checked that ν → ν¯ þ 2ν m4Γ4 decays of the form 2;3 1 1, mediated by an off- ð Γ Þeff ≡ ð Þ m4 4 2 2 : 12 ϕ 2 2 2 mϕ shell , are always sufficiently rare to be consistent with jUs4j ðjUe4j þjUμ4j Þ 1 − 2 m4 solar neutrino constraints [29,72]. Note, however, that we predict the cosmic neutrino background today to consist This constraint can be easily satisfied in the mass range exclusively of ν1 or ν3, for normal and inverted neutrino allowedP by beta decay limits. mass ordering, respectively. ð5Þ mν, the sum of neutrino masses. Massive neutrinos affect the CMB as well as structure formation, and 3It is noteworthy, though, that some cosmological fits this hasP for instance allowed the Planck collaboration to set have actually found a preference for neutrino self-interactions a limit mν ≲ 0.12 eV [55]. In our model, this constraint [56,61–65] that could be accommodated in our model.

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(9) Perturbativity (red region in Fig. 2). Requiringp thatffiffiffiffiffiffi the νs–ϕ coupling constant g in Eqs. (1) and (2) is < 4π imposes the bound 2 eff 2 m4 ðm4Γ4Þ ≲ 0.25 eV : ð14Þ eV

Similarly to the free-streaming bound, this constraint applies even for very small mixing when m4Γ4 is fixed. This bound restricts m4 in our model to be ≳100 eV for m4Γ4 values large enough to explain the MiniBooNE anomaly. In summary, the sterile neutrino mass range to explain FIG. 4. Comparison of the reactor antineutrino spectrum the MiniBooNE anomaly is between 100 eV and 2.5 keV. predicted in the decaying sterile neutrino scenario discussed in this work (blue) to the standard Huber–Mueller prediction V. THE LSND AND REACTOR ANOMALIES (orange-dashed) [76,77] and to Daya Bay data (black data points with error bars) [75]. For model parameters motivated by the AsshowninFig.3, decaying sterile neutrinos can MiniBooNE and LSND anomalies, a flux deficit consistent with simultaneously fit the MiniBooNE and LSND anomalies, the reactor anomaly can be accommodated. (See text for details, but only if cosmological neutrino free-streaming con- and for a discussion of how possible cosmological constraints can straints can be avoided (see discussion under point (6) be avoided.) above for possible scenarios). Quantitatively, a parameter goodness-of-fit test [73] reveals that LSND is beta decay spectra, and cosmological free-streaming con- 4 7σ incompatible with the rest of the data at the . level if straints; bright red regions show instead a global fit to free-streaming constraints hold. If the free-streaming pro- MiniBooNE, LSND, OPERA, ICARUS, E776, KARMEN, 2 1σ blem is solved by other means, this reduces to . , and nuclear beta spectra, but excluding the free-streaming implying consistency. The best fit to all data including constraint. Solid lines indicate constraints from OPERA ¼ LSND, but excluding free-streaming is found at m4 (blue), ICARUS (purple), KARMEN (cyan), E776 (green), 97 j j2 ¼ 0 018 j j2 ¼ 0 0015 Γ ¼ 0 87 2 eV, Ue4 . , Uμ4 . , m4 4 . eV , nuclear beta decay spectra (black dashed), and free- mϕ=m4 ¼ 0.89. streaming in the early Universe (black dotted). The region 2 Interestingly, at this value of jUe4j , the model can also to the right of the lines is excluded. explain the flux deficit observed in reactor and gallium We observe that, at smaller values of m4Γ4, the experiments [5–8,19,74]. We test our model against reactor allowed parameter regions from short-baseline oscillations data by comparing to Daya Bay’s generic flux-weighted (MiniBooNE, LSND, KARMEN) shift towards larger 2 2 cross section [75]. To estimate the viable parameter space values of jUe4j and jUμ4j . In this case, only a small we perform a chi-square-test using the covariance matrix fraction of neutrinos decays before reaching the detector, given in the same reference. In addition we introduce a making the phenomenology more similar to that of 3 þ 1 2.4% systematic flux normalization error corresponding to models without decay. Strong constraints from beta decay the theoretical uncertainty, in accordance with Fig. 28 of spectra and from cosmology imply that a good global fit j j2 2 Ref. [75]. The Ue4 region preferred by reactor experi- cannot be achieved at m4Γ4 ≪ 1 eV . ments is included in Fig. 3, and a comparison of the reactor Regarding the dependence of the fit on m4, we note that neutrino spectrum to our model prediction is shown smaller values of m4 are favored by beta decay spectra, but in Fig. 4. disfavored by cosmology, in agreement with Fig. 2. Exclusion limits from oscillation experiments do not VI. DETAILED INVESTIGATION OF THE depend on m4 for m4 ≫ eV. PARAMETER SPACE Comparing Fig. 5 with mϕ=m4 ¼ 0.5 and Fig. 5 with ¼ 0 9 To supplement Fig. 3 and give the reader a broader mϕ=m4 . , we see that it becomes in general more overview of the preferred parameter regions of decaying difficult to fit all experiments at smaller mϕ=m4. The reason sterile neutrinos, we show in Figs. 5 and 6 additional slices is that, at small mϕ=m4, the active neutrinos produced in ν4 through the 5-dimensional parameter space. and ϕ decays have a harder spectrum. This in particular The color coding in the figure is the same as in makes it more difficult to explain the MiniBooNE low- Fig. 3: the yellow, banana-shaped regions are preferred energy excess. In fact, for even smaller values of mϕ=m4, by MiniBooNE, the large dark red ones by LSND; the and in particular for nearly massless ϕ (as considered in 2 orange regions at low jUe4j correspond to a global fit to Refs. [22,23]), the MiniBooNE-preferred region would MiniBooNE, OPERA, ICARUS, E776, KARMEN, nuclear disappear completely from the plots.

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FIG. 5. Slices through the 5-dimensional parameter space of decaying sterile neutrinos at mϕ=m4 ¼ 0.5 fixed. The color code is the same as in Fig. 3.

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FIG. 6. Slices through the 5-dimensional parameter space of decaying sterile neutrinos at mϕ=m4 ¼ 0.9 fixed. The color code is the same as in Fig. 3.

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VII. CONCLUSIONS νμ events. Note, however, that in a 3 þ 1 model, the ν In summary, we have shown that scenarios in which the measured μ flux will be reduced by an amount ∼j j2 ν → ν SM is extended by a sterile neutrino that has a decay mode Uμ4 due to μ s oscillations. (This effect is to active neutrinos can well explain the MiniBooNE unimportant in a 2-flavor model, where the deficit is 2 2θ θ anomaly without violating any constraints. An explanation only of order sin μe, where μe is the effective of the LSND and reactor/gallium anomalies is possible if 2-flavor mixing angle.) We account for this effect by the model is extended to avoid constraints on neutrino free- first computing the expected νe signal based on the streaming in the early Universe. The preferred mass of the unoscillated MiniBooNE flux, and then diving it by sterile neutrino is of order few hundred eV. the νμ survival probability in each bin. The impact of this change in normalization is ACKNOWLEDGMENTS illustrated in the top panels of Fig. 7.Thecolored region in panel (a) of this figure shows our repro- We would like to thank Andr´e de Gouvea, Frank duction of the official MiniBooNE fit, which is shown Deppisch, Matheus Hostert, Bill Louis, and Michele as black contours. In panel (b), we have included the Maltoni for very helpful discussions. We are particularly change in normalization for the signal. grateful to Thomas Schwetz for sharing his fitting codes, (2) Oscillations of the νe backgrounds. Part of the from which some of our own codes took inspiration. We also MiniBooNE background is constituted by the in- thank Andr´e de Gouvea, O. L. G. Peres, Suprabh Prakash trinsic νe contamination in the beam. In a 2-flavor fit, and G. V.Stenico for sharing their draft on a similar neutrino this contribution to the total event rate is only decay solution to the SBL anomalies [24]. We thank Miguel 2 modified by a factor of order sin 2θμe, but in the Escudero and Samuel J. Witte for sharing their cosmological full 3 þ 1 framework, it is reduced by a factor of constraint. I. E. would like to thank CERN and Johannes 2 order jUe4j instead. The impact of this modification Gutenberg University Mainz for their kind hospitality. This to the background sample is shown in Fig. 7(c). work has been funded by the German Research Foundation ν – (3) Oscillations of the μ sample. The fit described in the (DFG) under Grant Nos. EXC-1098, KO 4820/1 1, FOR supplemental material to Ref. [1] which we are 2239, GRK 1581, and by the European Research Council following includes also MiniBooNE’s sample of νμ (ERC) under the European Union’s Horizon 2020 research events. This is necessary to properly account for and innovation programme (grant agreement No. 637506, systematic uncertainties which are correlated between “νDirections”). Fermilab is operated by Fermi Research the two samples. But of course, in a 3 þ 1 scenario, Alliance, LLC under contract No. DE-AC02-07CH11359 the νμ sample suffers from νμ disappearance into νs, with the United States Department of Energy. I. E. acknowl- j j2 edges support from the FPU program fellowship FPU15/ proportional to Uμ4 . (Once again, in a 2-flavor ∝ 2 2θ 03697, the EU Networks FP10ITN ELUSIVES (H2020- model, only a much smaller fraction sin μe will MSCA-ITN-2015-674896) and INVISIBLES-PLUS disappear, which is usually negligible.) The impact of (H2020-MSCA-RISE-2015-690575), and the MINECO including νμ disappearance is shown in panel (d) Grant No. FPA2016-76005-C2-1-P. M. D. is supported by of Fig. 7. the Alexander von Humboldt Foundation and the German We see that including the effect of 3 þ 1 oscillations on the Federal Ministry of Education and Research. normalization in the control regions and on the background prediction reduces the significance of the MiniBooNE anomaly, though it remains above 3σ. These effects are APPENDIX A: IMPACT OF OSCILLATIONS thus unable to fully explain the MiniBooNE anomaly, but ON THE BACKGROUND PREDICTION “ ” IN MINIBOONE they could well be part of an Altarelli cocktail of several effects conspiring to lead to the large observed excess [78]. In this appendix, we briefly discuss our fit to Let us finally mention one caveat with the above MiniBooNE data, and in what ways it differs from the corrections to the MiniBooNE fit. Namely, we can only collaborations’ fit as described in the supplemental material apply the corrections at the level of reconstructed events as to Ref. [1], and using the data released with Ref. [3]. the mapping between true and reconstructed neutrino In particular, we consider the following three effects, energies is not publicly available for muon neutrinos. which are relevant in a fit to a 3 þ 1 scenario, but are This means we have to assume that the reconstructed not encountered in a 2-flavor fit. neutrino energy is a faithful representation of the true (1) Normalization of the νμ → νe oscillation signal. neutrino energy. While this is true for quasielastic scatter- ν To predict the number of expected e events from ing events which constitute the majority of events, it is not ν → ν μ e oscillations for a given set of oscillation the case for other event categories. For instance, a neutrino– parameters, the initial νμ flux must be known. It is nucleon interaction may create an extra pion, and if this obtained in situ using MiniBooNE’s own sample of pion is reabsorbed as it propagates out of the nucleus, the

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(a) (b)

FIG. 7. Impact of oscillations in the background and control regions on the MiniBooNE fit in a simple 3 þ 1 model (oscillations only, 2 2 2 2 no decay). All panels show Δm41 vs the effective 2-flavor mixing angle sin 2θμe, which in a 3 þ 1 scenario is given by 4jUe4j jUμ4j . Panel (a) shows our reproduction (colored regions) of the official MiniBooNE fit (black contours), based on the instructions given in the supplemental material to Ref. [1] and using the data released with Ref. [3]. In panel (b), we include in addition the impact of νμ → νs disappearance on the normalization of the signal in each bin. The colored contours in panel (c) include on top of this the effect of νs disappearance on the intrinsic νe contamination in the beam. Panel (d) finally shows the additional impact of νμ disappearance on the sample of νμ events that is included in the fit along with the νe sample. In all panels, we show projections of the three-dimensional 2 2 2 2 2 2 parameter space spanned by Δm41, jUe4j , and Uμ4j onto the Δm41– sin 2θμe plane, imposing the constraint jUe4j < 0.2 due to bounds from reactor neutrino experiments. event will be misinterpreted as a quasi-elastic interaction, APPENDIX B: DECAY WIDTHS AND and the kinematic reconstruction of the neutrino energy TRANSITION PROBABILITY based on the observed charged lepton energy and direction Based on the interaction terms from Eq. (2), we can will fail. compute the differential decay rates of the heavy neutrino

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ν4 and of the scalar ϕ. In the massless light neutrino limit, the daughter neutrino and scalar energies, respectively. ν we obtain for the 4 decay width in the lab In the ν4 rest frame, Ej is restricted to the interval 2 ½0;m4ð1 − xϕ4Þ. lab 2 1 dΓ ðν4 → ν ϕÞ jU j E j ¼ P sj j ð Þ The lab frame decay rate of the scalar ϕ is m4 3 2 2 2 2 ; B1 Γ4 dE jU j ð1 −x Þ E E4 j k¼1 sk ϕ4 4 X 1 dΓlabðϕ → ν ν¯ Þ 1 i j ¼ ð Þ X Γlabðν → ν ϕÞ mϕ ; B4 1 d 4 j 1 1 Γ ¼ ð Þ i;j ϕ dEi Eϕ m4 2 : B2 Eϕ Γ4 dEϕ 1 − x E4 j E4 ϕ4 with the total rest frame decay width of ϕ In these expressions,

2 X3 2 X3 Γ ¼ g j j2 ð Þ g 2 2 2 ϕ mϕ UsiUsj : B5 Γ4 ¼ m4ð1 − xϕ4Þ jU 4Usjj ðB3Þ 8π 16π s i;j¼1 j¼1 is the total rest frame decay width of ν4, xϕ4 ≡ mϕ=m4 is The kinematic constraint on the daughter neutrino energies the ratio of scalar and neutrino masses, and Ej, Eϕ are is Ei;Ej ∈ ½0;mϕ.

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