Main

A broad experimental programme has shown that the three quantum-mechanical eigenstates of neutrino flavour, νe, νμ and ντ, are related to the three eigenstates of neutrino mass, ν1, ν2 and ν3, by the unitary Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix11,12. This mixing between flavour and mass states gives rise to the phenomenon of neutrino oscillation, in which neutrinos transition between flavour eigenstates with a characteristic wavelength in \(L/{E}_{\nu }\propto {(\Delta {m}_{ji}^{2})}^{-1}\), where L is the distance travelled by the neutrino, Eν is the neutrino energy and \(\Delta {m}_{ji}^{2}={m}_{j}^{2}-{m}_{i}^{2}\) is the difference between the squared masses of the mass eigenstates νi and νj. The three known neutrino mass states give rise to two independent mass-squared differences and thus to two characteristic oscillation frequencies that have been well measured with neutrinos from nuclear reactors13,14, the Sun15, the atmosphere of Earth16,17 and particle accelerators18,19,20.

In apparent conflict with the three-neutrino model, several experiments during the past three decades have made observations that can be interpreted as neutrino flavour change with a wavelength much shorter than is possible given only the two measured mass-squared differences3,4,5,6,7,8,9. These observations are often explained as neutrino oscillations caused by at least one additional mass state, ν4, corresponding to a mass-squared splitting of \(\Delta {m}_{41}^{2}\gtrsim 1{0}^{-2}\,{{\rm{eV}}}^{2}\), which is much greater than the measured \(\Delta {m}_{21}^{2}\) and \(\Delta {m}_{32}^{2}\). New mass states would require the addition of an equivalent number of new flavour states, in conflict with measurements of the Z-boson decay width21, which have definitively shown that only three light neutrino flavour states couple to the Z boson of the weak interaction. Therefore, these additional neutrino flavour states must be unable to interact through the weak interaction and are thus referred to as ‘sterile’ neutrinos. In this analysis, we focus specifically on light sterile neutrinos—those with masses below at least half the mass of the Z boson. It should be noted that the term ‘sterile neutrino’ has also been used to describe new particles, such as heavy right-handed lepton partners, that are potentially more massive than the Z boson. However, our study does not directly test these scenarios. The discovery of additional neutrino states would have profound implications across particle physics and cosmology, for example, on our understanding of the origin of neutrino mass, the nature of dark matter and the number of relativistic degrees of freedom in the early universe.

With the addition of a single new mass state ν4 and a single sterile flavour state νs, the PMNS matrix becomes a 4 × 4 unitary matrix described by six real mixing angles θij (1 ≤ i < j ≤ 4). Oscillations driven by the two measured mass-squared splittings have not had time to evolve for small values of L/Eν. The νμ to νe flavour-change probability, \({P}_{{\nu }_{{\rm{\mu }}}\to {\nu }_{{\rm{e}}}}\), and the νe and νμ survival probabilities, \({P}_{{\nu }_{{\rm{e}}}\to {\nu }_{{\rm{e}}}}\) and \({P}_{{\nu }_{{\rm{\mu }}}\to {\nu }_{{\rm{mu}}}}\), can then, to a very good approximation, be described by

$${P}_{{\nu }_{{\rm{\mu }}}\to {\nu }_{{\rm{e}}}}={\sin }^{2}(2{\theta }_{{\rm{\mu e}}}){\sin }^{2}\left(\frac{\Delta {m}_{41}^{2}L}{4{E}_{\nu }}\right),$$
(1)
$${P}_{{\nu }_{{\rm{e}}}\to {\nu }_{{\rm{e}}}}=1-{\sin }^{2}(2{\theta }_{{\rm{ee}}}){\sin }^{2}\left(\frac{\Delta {m}_{41}^{2}L}{4{E}_{\nu }}\right),$$
(2)
$${P}_{{\nu }_{{\rm{\mu }}}\to {\nu }_{{\rm{\mu }}}}=1-{\sin }^{2}(2{\theta }_{{\rm{\mu \mu }}}){\sin }^{2}\left(\frac{\Delta {m}_{41}^{2}L}{4{E}_{\nu }}\right),$$
(3)

where θee ≡ θ14, \({\sin }^{2}(2{\theta }_{{\rm{\mu \mu }}})\equiv 4{\cos }^{2}{\theta }_{14}{\sin }^{2}{\theta }_{24}(1-{\cos }^{2}{\theta }_{14}{\sin }^{2}{\theta }_{24})\) and \({\sin }^{2}(2{\theta }_{{\rm{\mu e}}})\equiv {\sin }^{2}(2{\theta }_{14}){\sin }^{2}{\theta }_{24}\), following the common parameterization22. Flavour transitions due to these new oscillation parameters are experimentally probed by observing unexpected deficits or excesses in charged current (CC) νe and νμ interactions in a flavour-sensitive neutrino detector from a source of well-defined neutrino flavour content.

Observations compatible with a fourth neutrino mass state have been made in measurements of intense electron-capture decay sources6,7,8, in which a deficit in detected νe rates implies non-unity \({P}_{{\nu }_{{\rm{e}}}\to {\nu }_{{\rm{e}}}}\) from a \(\Delta {m}_{41}^{2} > {\mathcal{O}}(1\,{{\rm{eV}}}^{2})\). Although a hint of non-unity \({P}_{{\overline{\nu }}_{{\rm{e}}}\to {\overline{\nu }}_{{\rm{e}}}}\) is provided by the nuclear-reactor-based Neutrino-4 experiment9, this result is in conflict with other reactor-based observations from DANSS, NEOS, PROSPECT and STEREO, which see no evidence for L/Eν-dependent \({\overline{\nu }}_{{\rm{e}}}\) disappearance23,24,25,26. Two accelerator-based experiments, LSND and MiniBooNE, have observed potential evidence of non-zero \({P}_{{\nu }_{{\rm{\mu }}}\to {\nu }_{{\rm{e}}}}\) associated with large mass splittings of \(\Delta {m}_{41}^{2} > {\mathcal{O}}(1{0}^{-2}\,{{\rm{e}}{\rm{V}}}^{2})\). The LSND experiment observed an anomalous excess of \({\overline{\nu }}_{{\rm{e}}}\) interactions in a π+ decay-at-rest beam3. The MiniBooNE experiment, situated downstream from the Booster Neutrino Beam (BNB) proton target facility generating a beam of GeV-scale νμ and \({\overline{\nu }}_{{\rm{\mu }}}\) from decays of boosted π+ and π−, observed an excess of electromagnetic showers indicative of νe interactions that would imply a non-zero \({P}_{{\nu }_{{\rm{\mu }}}\to {\nu }_{{\rm{e}}}}\) (refs. 4,5). Observations of νe disappearance and νe appearance should be accompanied by νμ disappearance (non-unity \({P}_{{\nu }_{{\rm{\mu }}}\to {\nu }_{{\rm{\mu }}}}\)) if the PMNS matrix is unitary. No conclusive observation of this νμ disappearance has been reported27,28,29. The overall picture of the existence and phenomenology of sterile neutrino states thus remains inconclusive.

In this article, we present new results on sterile neutrino oscillations from the MicroBooNE liquid-argon time projection chamber (LArTPC) experiment at Fermilab10. Situated along the same BNB beamline hosting the MiniBooNE experiment, MicroBooNE was conceived to directly test the non-zero \({P}_{{\nu }_{{\rm{\mu }}}\to {\nu }_{{\rm{e}}}}\) observation of MiniBooNE. By supplanting the Cherenkov detection technology of MiniBooNE with the precise imaging and calorimetric capabilities of a LArTPC, MicroBooNE can reduce backgrounds and select a high-purity sample of true νe-generated final-state electrons. The first νe measurement results of MicroBooNE using differing final-state topologies showed no evidence for an excess of νe-generated electrons from the BNB30,31,32,33. These results were used to set limits on νμ → νe flavour transitions, excluding sections of the region in \((\Delta {m}_{41}^{2},{\sin }^{2}(2{\theta }_{{\rm{\mu e}}}))\) space favoured by LSND and MiniBooNE data34. As the BNB has an intrinsic contamination of electron neutrinos, the disappearance of electron neutrinos can cancel the appearance of electron neutrinos from νμ → νe oscillations35. This effect leads to a degeneracy between the impact of the mixing angles θμe and θee of equations (1) and (2) that weakens the sensitivity to the parameters of the expanded 4 × 4 PMNS matrix.

We overcome the limitations of the degeneracy between νe appearance and νe disappearance by performing one of the first oscillation searches using two accelerator neutrino beams: the BNB and the Neutrinos at the Main Injector (NuMI) beam. The MicroBooNE detector is aligned with the direction of BNB and is at an angle of about 8° relative to the NuMI beam. Beam timing information is used to distinguish and record events from each beam separately. This configuration results in two neutrino datasets differing in the intrinsic electron-flavour fraction. The electron-flavour content of BNB is 0.57% and that of the NuMI beam is 4.6%. These two independent sets of data, with substantially different electron-flavour contents, break the degeneracy between νe appearance and disappearance. We show the impact of using two beams in Fig. 1a,b, in which we compare simulated νe energy spectra from the BNB and the NuMI beam for the three-flavour (3ν) hypothesis and for two sets of parameters of the expanded four-flavour (4ν) PMNS model with \(\Delta {m}_{41}^{2}=1.2\,{{\rm{e}}{\rm{V}}}^{2}\) and \({\sin }^{2}(2{\theta }_{{\rm{\mu e}}})=0.003\). For \({\sin }^{2}{\theta }_{24}=0.0045\) the νe appearance and disappearance cancel in the BNB, leaving a νe spectrum that is almost identical to the 3ν case, whereas the NuMI beam shows an indication of νe disappearance. The appearance and disappearance effects almost fully cancel in the NuMI beam for \({\sin }^{2}{\theta }_{24}=0.018\), whereas the BNB shows a clear indication of νe appearance. In the Methods, we provide further discussion of this degeneracy over a broader range of mass-squared splittings and mixing angles.

Fig. 1: Illustration of the breaking of the degeneracy between νe appearance and disappearance enabled by the independent BNB and NuMI datasets in MicroBooNE.
Fig. 1: Illustration of the breaking of the degeneracy between νe appearance and disappearance enabled by the independent BNB and NuMI datasets in MicroBooNE.
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a,b, Simulated reconstructed energy spectra of FC CC νe interactions in MicroBooNE from the BNB (a) and the NuMI beam (b). The dark blue histograms show the 3ν expectation for \({\sin }^{2}{\theta }_{24}={\sin }^{2}(2{\theta }_{{\rm{\mu e}}})=0\). The light blue and red histograms show expectations for two sets of parameters of the 4ν model, both with \(\Delta {m}_{41}^{2}=1.2\,{{\rm{e}}{\rm{V}}}^{2}\) and \({\sin }^{2}(2{\theta }_{{\rm{\mu e}}})=0.003\). The light blue histograms show the expectation for \({\sin }^{2}{\theta }_{24}=0.018\) and the red histograms show the expectation for \({\sin }^{2}{\theta }_{24}=0.0045\). Note that these parameters were chosen specifically to highlight differences in the oscillated spectra between BNB and NuMI and do not imply that parameter spaces associated with these values are newly excluded by this result.

Using the two-beam technique, this new MicroBooNE analysis achieves marked improvements in sensitivity to the parameters \({\sin }^{2}(2{\theta }_{{\rm{ee}}})\) and \({\sin }^{2}(2{\theta }_{{\rm{\mu e}}})\) relative to MicroBooNE’s prior sterile neutrino analysis over a broad range of \(\Delta {m}_{41}^{2}\) values. These improvements are shown by the sensitivities presented in Extended Data Fig. 2. The results presented here using two neutrino beams place robust new constraints on the validity of the sterile neutrino hypothesis in explaining existing short-baseline anomalies in neutrino physics. This analysis strengthens the direct test of the sterile neutrino interpretation of the MiniBooNE anomaly and allows MicroBooNE to probe the \({\sin }^{2}(2{{\theta }}_{{\rm{\mu }}{\rm{e}}})\) parameter space favoured by LSND. We also constrain \({\sin }^{2}(2{\theta }_{{\rm{ee}}})\), complementing existing exclusions from reactor, solar36,37 and β-decay38 experiments, thereby further restricting the sterile neutrino parameter space relevant to the gallium anomaly.

We use data corresponding to 6.369 × 1020 protons on target (POT) in the BNB, with magnetic van der Meer horns configured to focus positively charged hadrons, leading to a νμ-dominated beam with a 5.9% \({\overline{\nu }}_{{\rm{\mu }}}\) component and a 0.57% \({\nu }_{{\rm{e}}}+{\overline{\nu }}_{{\rm{e}}}\) component. From the NuMI beam, a total of 10.54 × 1020 POT are used, in which 30.8% were taken with horns configured to focus positively charged hadrons and the remainder with horns focusing negatively charged hadrons. The NuMI flux observed in the MicroBooNE detector, with both horn configurations combined, is νμ dominated with a 42.1% \({\overline{\nu }}_{{\rm{\mu }}}\) component and a 4.6% \({\nu }_{{\rm{e}}}+{\overline{\nu }}_{{\rm{e}}}\) component. In the rest of this paper, we do not discriminate between neutrinos and antineutrinos and refer to the \({\nu }_{{\rm{\mu }}}+{\overline{\nu }}_{{\rm{\mu }}}\) and \({\nu }_{{\rm{e}}}+{\overline{\nu }}_{{\rm{e}}}\) samples as νμ and νe samples for brevity. For both BNB and NuMI, the POT used in this analysis represent roughly half of the total data collected by the MicroBooNE detector; additional data remain available for future studies.

The LArTPC detector of MicroBooNE has an active volume of 10.4 × 2.6 × 2.3 m3 containing 85 tonnes of liquid argon. Charged particles passing through the argon create ionization trails. A 273 V cm−1 electric field drifts the ionization electrons towards an anode plane consisting of three layers of wires separated by 3 mm and each with a 3-mm wire pitch that collects the electrons and enables three-dimensional imaging of the neutrino interactions. The passage of charged particles through the argon also produces scintillation light that is collected by a system of photomultiplier tubes to provide timing information. Signal processing and calibrations of MicroBooNE data are described in refs. 39,40,41,42,43,44.

Neutrino interactions in the LArTPC are reconstructed with the Wire-Cell analysis framework45. The techniques for identifying and reconstructing neutrino interactions and their energies have been described elsewhere33. We select a sample of CC νe interactions from the BNB (NuMI beam) with 82% (91%) purity and 46% (42%) efficiency, and a sample of CC νμ interactions with 92% (78%) purity and 68% (62%) efficiency. The CC νe and CC νμ samples are divided into fully contained (FC) and partially contained (PC) samples, depending on whether all charge depositions are contained in a fiducial volume 3 cm within the TPC boundary. The CC νμ events that contain a reconstructed π0 are separated into two additional FC and PC samples per beam. Neutral current (NC) interactions that produce a π0 are distinguished by the absence of a long muon-like track and the presence of detached reconstructed electromagnetic showers. These form an additional sample. In total, we define 14 distinct event categories, seven for each beam.

We produce a Monte Carlo prediction of our 14 samples, to which we compare the data. There is substantial systematic uncertainty creating this Monte Carlo simulation. The uncertainty on the predicted rates of the 14 samples is given in Table 1 and is referred to as the unconstrained systematic uncertainty. The largest uncertainties come from neutrino interaction modelling for the BNB samples and from a combination of neutrino flux and interaction uncertainties for the NuMI samples. Many of these uncertainties are highly correlated. Thus, a combined fit of all samples effectively constrains the uncertainties on the CC νe prediction and at the same time allows the CC νe prediction to be modified, as can be seen from Table 1. The pionless samples constrain uncertainties on CC νe signal events, whereas the π0 samples constrain uncertainties on the dominant background.

Table 1 Event counts and systematic uncertainties
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Uncertainties on the neutrino flux prediction arise from uncertainties in the production of charged pions and kaons in the BNB and NuMI targets and the material around the target halls and hadron-decay volumes. These uncertainties are evaluated through comparison with external hadron production data46,47,48, following a procedure similar to that described in ref. 49. The νe flux from three-body K and μ decays is highly correlated with the νμ flux from two-body π and K decays, allowing our νμ samples to effectively constrain the uncertainties on the νe flux predictions. The neutrino interaction model is tuned using datasets of pionless CC interactions from the T2K experiment50. Uncertainties on this neutrino interaction model are evaluated by varying the input parameters within their allowed uncertainties. These uncertainties are correlated between the BNB and NuMI datasets and between the CC νμ and νe samples because of the lepton universality of the weak interaction. Uncertainties on the simulation of the detector include uncertainties on the response of the detector to ionization, uncertainties on the amount of ionization charge freed by passing charged particles through the detector, uncertainties on the electric field map of the TPC, uncertainties on the production and propagation of scintillation light, uncertainties on backgrounds from interactions occurring outside the cryostat, and uncertainties on finite statistics of the simulation samples used for predictions.

The simultaneous fit to the 14 samples from the BNB and the NuMI beam incorporates all sources of systematic uncertainty through a covariance matrix. We allow \({\sin }^{2}(2{\theta }_{{\rm{\mu e}}})\), \({\sin }^{2}(2{\theta }_{{\rm{ee}}})\) and \(\Delta {m}_{41}^{2}\) complete freedom within unitarity bounds as parameters of the fit. The covariance-matrix formalism χ2 test of the fit can be found in the Methods. The constrained predictions shown in Fig. 2 assume the 3ν hypothesis of \({\sin }^{2}(2{\theta }_{{\rm{\mu e}}})={\sin }^{2}(2{\theta }_{{\rm{ee}}})=0\). They agree well with the data, with a P-value of 0.92. The best-fit values for the oscillation parameters in the 4ν hypothesis are \(\Delta {m}_{41}^{2}=1.30\times 1{0}^{-2}\,{{\rm{e}}{\rm{V}}}^{2}\), \({\sin }^{2}(2{\theta }_{{\rm{\mu e}}})=0.999\), and \({\sin }^{2}(2{\theta }_{{\rm{ee}}})=0.999\), with a χ2 difference with respect to the 3ν hypothesis of

$$\Delta {\chi }^{2}={\chi }_{{\rm{null}},3\nu }^{2}-{\chi }_{\min ,4\nu }^{2}=0.228.$$
(4)

We observe no marked preference for the existence of a sterile neutrino with a P-value of 0.96 evaluated using the Feldman–Cousins procedure.

Fig. 2: Observed CC νe candidate events.
Fig. 2: Observed CC νe candidate events.
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a–d, Reconstructed energy spectra of events selected as FC CC νe candidates in the BNB (a), PC CC νe candidates in the BNB (b), FC νe candidates in the NuMI beam (c) and PC νe candidates in the NuMI beam (d). The data points are shown with statistical error bars. The constrained predictions for each sample are shown for the 3ν hypothesis as the solid histograms, with the blue showing the true CC νe events and the green showing the background events. The background category contains CC νμ interactions, NC neutrino interactions, cosmic rays and interactions occurring outside the fiducial volume of the detector. The yellow band shows the total constrained systematic uncertainty on the prediction.

Exclusion contours are calculated using the frequentist CLs (confidence level as a function of s) method51. The exclusion contour in any two-dimensional parameter space is obtained by profiling the third free parameter. At any point in the two-dimensional space, the value of the profiled parameter that minimizes the χ2 with respect to the data is chosen. Figure 3a shows the 95% CLs exclusion contour in the \((\Delta {m}_{41}^{2},{\sin }^{2}(2{\theta }_{{\rm{\mu e}}}))\) parameter space. The region allowed at 99% CL by the LSND measurement and the vast majority of the region allowed at the 95% CL by the MiniBooNE experiment are excluded. Figure 3b shows the 95% CLs exclusion contour in the \((\Delta {m}_{41}^{2},{\sin }^{2}(2{\theta }_{{\rm{ee}}}))\) parameter space. A notable portion of the region allowed by gallium measurements and part of the region derived from the Neutrino-4 measurement are excluded. In the Methods and Extended Data Fig. 2, we compare our exclusions with the expected median sensitivities.

Fig. 3: Constraints on parameters of the 4ν oscillation model.
Fig. 3: Constraints on parameters of the 4ν oscillation model.
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a,b, The red lines show exclusion limits at the 95% CLs level in the plane of \(\Delta {m}_{41}^{2}\) and \({\sin }^{2}(2{\theta }_{{\rm{\mu }}{\rm{e}}})\) (a) or \({\sin }^{2}(2{\theta }_{{\rm{e}}{\rm{e}}})\) (b). All the regions to the right of these lines are excluded by the MicroBooNE data. In a, the yellow shaded area is the LSND 99% CL allowed regions3, which neglects the degeneracy between νe disappearance and appearance. The light blue area is the MiniBooNE 95% CL allowed region58, considering both νe disappearance and appearance. In b, the purple shaded area is the 2σ allowed region of the gallium anomaly59. The dark blue shaded area is the 2σ allowed region from the Neutrino-4 experiment9. For context, note that the stronger-than-expected constraint on \({\sin }^{2}(2{\theta }_{{\rm{\mu e}}})\), driven by the deficit observed in the BNB νe CC FC sample and the excess in the NuMI νμ CC sample, is discussed in detail in the Methods and Extended Data Fig. 2.

In summary, using data from the MicroBooNE detector, we report one of the first searches for a sterile neutrino using two accelerator neutrino beams. The oscillation fit to the 4ν model using a total of 14 CC νe, CC νμ and NC π0 samples from the BNB and the NuMI beam in a single detector achieves a marked reduction of systematic uncertainties and a powerful mitigation of degeneracies between νe appearance and disappearance. The result shows no evidence of oscillations induced by a single sterile neutrino and is consistent with the 3ν hypothesis with a P-value of 0.96. We comprehensively exclude at a 95% CL the 4ν parameter space that would explain the LSND and MiniBooNE anomalies through the existence of a light sterile neutrino in a model with an extended 4 × 4 PMNS matrix. Our result expands the diverse range of experimental approaches, excluding regions that would explain the gallium anomaly and the Neutrino-4 observation with a light sterile neutrino. This work, therefore, provides a robust exclusion of a single light sterile neutrino as an explanation for the array of short-baseline neutrino anomalies observed over the past three decades, representing the strongest constraint from a short-baseline experiment using accelerator-produced neutrinos. Expanded models, including several light sterile neutrinos52, neutrino decay effects53,54 or production and decay of new particles connected with the dark sector55,56 might explain the anomalies. The Short Baseline Neutrino (SBN) Programme57 at Fermilab adds two new LArTPC detectors in the BNB, at different distances from the proton target. Future measurements by MicroBooNE and the broader SBN Programme can shed light on this expanded model space, with future comprehensive insights provided by near-term short-baseline measurements from diverse flavour channels and energy regimes.

Methods

Neutrino beams at MicroBooNE

The BNB and the NuMI are conventional neutrino beamlines that use intense proton beam pulses to generate on-axis neutrino fluxes peaking in the neutrino-energy range 0.5–5 GeV. Boosted charged mesons escaping the target are focused into a decay pipe using magnetic horns, allowing decaying mesons to impart much of their kinetic energy directly to their neutrino product. The MicroBooNE detector is on the axis of the BNB, 468.5 m from the proton target. The detector is about 8° off-axis and 679 m from the proton target of the NuMI beam. The muon flavour components of these beams are mostly generated via the primary decay channel of the dominant π mesons, \({\pi }^{\pm }\to {\mu }^{\pm }+{\nu }_{{\rm{\mu }}}({\overline{\nu }}_{{\rm{\mu }}})\), whereas the electron-flavour component is generated by decay of the pion’s boosted μ± daughter, by \({\mu }^{\pm }\to {e}^{\pm }+{\overline{\nu }}_{{\rm{\mu }}}({\nu }_{{\rm{\mu }}})+{\nu }_{{\rm{e}}}({\overline{\nu }}_{{\rm{e}}})\), and by semileptonic decays of sub-dominant K mesons, specifically \({K}^{\pm }\to {\pi }^{0}+{e}^{\pm }+{\nu }_{{\rm{e}}}({\overline{\nu }}_{{\rm{e}}})\) and \({K}_{{\rm{L}}}^{{\rm{0}}}\to {\pi }^{\mp }+{e}^{\pm }+{\nu }_{{\rm{e}}}({\overline{\nu }}_{{\rm{e}}})\).

Neutrinos in the BNB are created by colliding protons with a kinetic energy of 8 GeV on a beryllium target, whereas in the NuMI beamline, 120 GeV protons collide with a carbon target. These differences serve to generate a higher on-axis beam energy in the NuMI beam, as well as a greater proportion of K meson production, leading to a higher νe content at the highly off-axis position of MicroBooNE. The NuMI beam also incorporates a longer charged-meson decay pipe (675 m) than the BNB (50 m), which increases its electron-flavour content by facilitating a higher proportion of decays of secondary μ+ from upstream π+ decay. Although the latter effect drives higher electron-flavour content on-axis for NuMI relative to the BNB, it is the larger proportion of unfocused or poorly focused K mesons of NuMI that drives its elevated electron-flavour content at the off-axis angle of MicroBooNE relative to the on-axis flux of BNB.

Neutrino flux simulation

The simulation of the neutrino flux at MicroBooNE accounts for the production of hadrons from the initial interaction of the proton beam on the target and the propagation of these hadrons through a detailed beamline geometry description, achieved using the Geant4 toolkit60. Hadron production cross-sections are constrained by dedicated external measurements where available, tailored to the specific beam parameters, including target differences and initial proton beam energy. The BNB simulation, identical to that used at MiniBooNE61, uses Geant v.4.10.4 with a custom physics list and constrains π± yields with data from the HARP experiment62, along with an updated K+ production constraint from SciBooNE63,64. The NuMI simulation has been updated to Geant v.4.10.4 with the FTFP-BERT physics list65. The constraints on π± and K± yields from the NA49 experiment at CERN46,47 are implemented using the PPFX toolkit49, which has been updated to use the new Geant version. Uncertainties are estimated for each process and various components of the beamline geometry, resulting in a combined systematic uncertainty of approximately 13% for the BNB and 26% for the NuMI beam on the integrated flux. These uncertainties on the fluxes are different from the uncertainties quoted in Table 1, which are on the overall event rates. These uncertainties are dominated by hadron production rates and are larger for the NuMI beam because of the lack of constraints at large off-axis angles from dedicated hadron production experiments. As the nature of neutrino production in the two beams is different, the uncertainties are considered uncorrelated across the respective beams.

Degeneracy between ν e appearance and disappearance

The interplay between oscillation of intrinsic νμ and νe components in the BNB and the NuMI beam is shown in Extended Data Fig. 1. For various combinations of the expanded 4ν PMNS mixing angles, and assuming \(\Delta {m}_{41}^{2}=1.4\,{{\rm{e}}{\rm{V}}}^{2}\), the ratio of predicted νe signal events with 0 < Eν < 2.5 GeV in MicroBooNE relative to the 3ν prediction is shown for the BNB on the x-axis and for the NuMI beam on the y-axis. By tracing vertically along x = 1, we observe that a MicroBooNE BNB νe measurement could be consistent with the 3ν hypothesis of θee = θμe = 0 as well as with non-zero mixing angles in the alternate 4ν case. The addition of a NuMI νe measurement enables a much clearer interpretation of the allowed oscillation behaviour, while also strengthening the constraining power of the analysis. Specifically, perfect agreement between data and the 3ν prediction for both BNB and NuMI would favour the null oscillation case, whereas a large deficit in the high νe-content NuMI beam would indicate competing appearance and disappearance effects in the BNB νe sample. Moreover, we can see that the range of allowed 4ν predictions in NuMI and BNB when taken together is quite restricted, allowing us to set tighter limits on this oscillation model based on the observed CC νe event rates after constraints in Table 1 (x ≈ 0.9, y ≈ 1).

Extended Data Fig. 2 shows the impact of the degeneracy breaking on the sensitivity to the 4ν parameter space. The dashed lines show the exclusion regions of the previous BNB-only analysis of MicroBooNE34, compared with the solid red lines, which show the exclusions obtained by this analysis when including the NuMI beam data.

Statistical methods for oscillation analysis

The combined statistical and systematic uncertainties on the 14 event samples are described by the covariance matrix

$$\Sigma =\left(\begin{array}{ccc}{C}_{1,1} & \cdots \, & {C}_{1,14}\\ \vdots & \ddots & \vdots \\ {C}_{14,1} & \cdots \, & {C}_{14,14}\end{array}\right),$$
(5)

where the Ci,j are the bin-by-bin covariance matrices between the ith and jth event samples. These covariance matrices are the sums of the covariance matrices arising from statistical uncertainties and from each source of systematic uncertainty,

$${C}_{i,j}={C}_{i,j}^{{\rm{stat.}}}+\sum _{k}{C}_{i,j}^{{{\rm{syst.}}}_{k}},$$
(6)

where the sum runs over the k sources of systematic uncertainty. The covariance matrix for the statistical uncertainty follows the Pearson format.

Figure 2 and Table 1 demonstrate the power of the 14 event samples to effectively constrain the systematic uncertainties due to the correlations present in the covariance matrix. To produce the constrained predictions shown in Fig. 2 and the constrained systematic uncertainties in Table 1, a conditional constraint formalism66 is used, which uses the 14 event samples simultaneously to constrain the systematic uncertainties and to provide updated predictions for each event sample. To understand how the constraint is applied to the ith event sample, the full unconstrained covariance matrix can be written as

$$\Sigma =\left(\begin{array}{cc}{C}_{i,i} & {C}_{i,x}\\ {C}_{x,i} & {C}_{x,x}\end{array}\right),$$
(7)

where elements with a subscript x represent the remaining 13 blocks of the full matrix. An updated, constrained covariance matrix for the ith event sample is obtained as

$${C}_{i,i}^{{\rm{constr}}.}={C}_{i,i}-{C}_{i,x}\cdot {({C}_{x,x})}^{-1}\cdot {C}_{x,i}.$$
(8)

Given the unconstrained binned prediction for the ith event sample, μi, and the remaining binned prediction and data samples, μx and nx, a constrained binned prediction can also be formed as

$${\mu }_{i}^{{\rm{constr}}.}={\mu }_{i}+{C}_{i,x}\cdot {({C}_{x,x})}^{-1}\cdot ({n}_{x}-{\mu }_{x}).$$
(9)

To perform the oscillation fit, a χ2 test statistic,

$${\chi }^{2}={(N-M)}^{{\rm{T}}}\cdot {\Sigma }^{-1}\cdot (N-M),$$
(10)

is formed using the 14 binned event samples from the data, N = (n1, …, n14), where ni is the ith event sample from the data, and the corresponding unconstrained binned predictions for the 14 event samples, M = (μ1, …, μ14). The oscillation parameters used to produce the prediction are varied until χ2 is minimized. As the bin-to-bin correlations between the 14 event samples are contained in the matrix Σ, minimizing this χ2 intrinsically incorporates the constraint procedure into the measurement of the oscillation parameters. As χ2 is minimized, the absolute systematic uncertainty varies as the number of oscillated neutrino interactions within the fiducial volume changes, whereas the fractional systematic uncertainty remains constant. By contrast, the absolute systematic uncertainty related to non-neutrino backgrounds and out-of-fiducial-volume neutrino interactions remains unchanged. Consequently, the total covariance matrix is updated in accordance with the oscillation parameters of interest.

Exclusion limits on the oscillation parameters are calculated using the frequentist-motivated CLs method51, which is commonly used for determining exclusion limits in high-energy physics. The CLs test statistic is defined as CLs = P4ν/P3ν, where P4ν and P3ν are the one-sided P-values of \(\Delta {\chi }_{{\rm{C}}{{\rm{L}}}_{{\rm{s}}},{\rm{d}}{\rm{a}}{\rm{t}}{\rm{a}}}^{2}\) under the 4ν and the null 3ν hypotheses, respectively, where

$$\Delta {\chi }_{{{\rm{CL}}}_{{\rm{s}}}}^{2}={\chi }_{4\nu }^{2}-{\chi }_{3\nu }^{2}$$

at a given point in the 4ν parameter space. These P-values are one-sided because the test statistic measures the deviation from the null in the specific direction of the alternative hypothesis. The P-values are determined using a frequentist approach by generating pseudo-experiments with the full covariance matrix, assuming the respective hypothesis is true. The region in which CLs ≤ 1 − α is excluded at the confidence level α. By generating pseudo-experiments under the null 3ν hypothesis, expected exclusion limits are calculated across the 2D parameter spaces of \(\Delta {m}_{41}^{2}\) and \({\sin }^{2}(2{\theta }_{{\rm{\mu e}}})\) or \({\sin }^{2}(2{\theta }_{{\rm{ee}}})\).

To quantify the expected sensitivity of the analysis, the median \({\sin }^{2}(2{\theta }_{{\rm{\mu e(ee)}}})\) value from all expected exclusion limits is determined for each \(\Delta {m}_{41}^{2}\) value. These median sensitivities are shown in Extended Data Fig. 2 and are compared with the exclusions set using the data. To show the expected level of fluctuations of the measured limit from the median sensitivity, 1σ and 2σ bands are also shown in Extended Data Fig. 2, which encompass the central 68.3% and 95.5% of exclusions from the pseudo-experiments. In \({\sin }^{2}(2{\theta }_{{\rm{\mu e}}})\) space, our exclusion is stronger than our median expected sensitivity. Two factors contribute to this stronger exclusion. First, a deficit in the BNB CC νe sample more strongly disfavours νe appearance. Second, the excess in the NuMI CC νμ sample leads to a reduction in the constrained fractional uncertainty on the NuMI νe prediction through the joint fit procedure, which in turn further strengthens the exclusion limit. In \({\sin }^{2}(2{\theta }_{{\rm{ee}}})\) space, the deficit in the BNB CC νe FC sample plays the opposite role, slightly favouring νe disappearance and making the exclusion contour weaker than the median sensitivity.

Impact of the NuMI CC ν μ sideband

As shown in Table 1, a combined fit of the 14 reconstructed samples constrains the signal CC νe prediction and its uncertainties due to the correlations between the sideband and the signal channels. For the NuMI CC νe signal sample in particular, a crucial driver of the constraint is the corresponding CC νμ sideband, shown in Extended Data Fig. 3. Before the fit, the normalization difference between data and the prediction is 24.5% with an overall uncertainty of 21.1% on the prediction.

To evaluate the impact of this difference on the combined fit, we can extend the covariance matrix used in the analysis (364 × 364, corresponding to the energy spectra of the 14 channels) by adding a bin representing the overall NuMI CC νμ normalization (combining FC and PC) and computing the respective covariances with the other 364 analysis bins, resulting in a 365 × 365 matrix. We can then obtain a post-fit mean and error on this normalization parameter by constraining the 364 × 364 block to the data using equations (8) and (9). This gives us an estimate of how much this parameter is effectively being pulled in the combined fit. The post-fit value of this parameter is 1.28 ± 0.058, indicating consistency with the corresponding observed value as well as a modest pull of  about 1.3σ.