A broad experimental programme has shown that the three quantum-mechanical eigenstates of neutrino flavour, ν e , ν μ and ν τ , are related to the three eigenstates of neutrino mass, ν 1 , ν 2 and ν 3 , 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 \(L/{E}_{
u }\propto {(\Delta {m}_{ji}^{2})}^{-1}\), where L is the distance travelled by the neutrino, E ν 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 ν i and ν j . 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, ν 4 , 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 ν 4 and a single sterile flavour state ν s , the PMNS matrix becomes a 4 × 4 unitary matrix described by six real mixing angles θ ij (1 ≤ i < j ≤ 4). Oscillations driven by the two measured mass-squared splittings have not had time to evolve for small values of L/E ν . The ν μ to ν e flavour-change probability, \({P}_{{
u }_{{\rm{\mu }}}\to {
u }_{{\rm{e}}}}\), and the ν e and ν μ survival probabilities, \({P}_{{
u }_{{\rm{e}}}\to {
u }_{{\rm{e}}}}\) and \({P}_{{
u }_{{\rm{\mu }}}\to {
u }_{{\rm{mu}}}}\), can then, to a very good approximation, be described by
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