Neutrino Interferometry for High-Precision Tests of Lorentz

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Neutrino Interferometry for High-Precision Tests of Lorentz

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Marquette University
[email protected]
Physics Faculty Research and Publications

Physics, Department of

Neutrino Interferometry for High-Precision Tests of Lorentz Symmetry with IceCube
Karen Andeen
Marquette University, [email protected]
IceCube Collaboration

Accepted version. Nature Physics, Vol. 14 (2018):961-699. DOI. © 2018 Springer Nature Limited. Used with permission. Shareable link provided by the Springer Nature SharedIt content-sharing initiative. A complete list of authors available in article text.

Marquette University
[email protected]
Physics Faculty Research and Publications/College of Arts and Sciences
This paper is NOT THE PUBLISHED VERSION; but the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation below.
Nature Physics, Vol. 14, No. 9 (September 2018): 961-966. DOI. This article is © Springer Nature Publishing Group and permission has been granted for this version to appear in [email protected] Springer Nature Publishing Group does not grant permission for this article to be further copied/distributed or hosted elsewhere without the express permission from Springer Nature Publishing Group.
Neutrino interferometry for high-precision tests of Lorentz symmetry with IceCube
Karen Andeen
Department of Physics, Marquette University, Milwaukee, WI
The IceCube Collaboration
(See complete list of IceCube contributors at end of article.)
Lorentz symmetry is a fundamental spacetime symmetry underlying both the standard model of particle physics and general relativity. This symmetry guarantees that physical phenomena are observed to be the same by all inertial observers. However, unified theories, such as string theory, allow for violation of this symmetry by inducing new spacetime structure at the quantum gravity scale. Thus, the discovery of Lorentz symmetry violation could be the first hint of these theories in nature. Here we report the results of the most precise test of spacetime symmetry in the neutrino sector to date. We use high-energy atmospheric neutrinos observed at the IceCube Neutrino Observatory to search for anomalous neutrino oscillations as signals of Lorentz violation. We find no evidence for such phenomena. This allows us to constrain the size of the dimension-four operator in the standard-model extension for Lorentz violation to the 10−28 level and to set limits on higher-dimensional operators in this framework. These are among the most stringent limits on Lorentz violation set by any physical experiment.

Very small violations of Lorentz symmetry, or Lorentz violation (LV), are allowed in many ultrahigh-energy theories, including string theory1, non-commutative field theory2 and supersymmetry3. The discovery of LV could be the first indication of such new physics. Worldwide efforts are therefore underway to search for evidence of LV. The standard-model extension (SME) is an effective-field-theory framework to systematically study LV4. The SME includes all possible types of LV that respect other symmetries of the standard model such as energy– momentum conservation and coordinate independence. Thus, the SME can provide a framework to compare results of LV searches from many different fields such as photons5,6,7,8, nucleons9,10,11, charged leptons12,13,14 and gravity15. Recently, neutrino experiments have performed searches for LV16,17,18. So far, all searches have obtained null results. The full list of existing limits from all sectors and a brief overview of the field are available elsewhere19,20. Our focus here is to present the most precise test of LV in the neutrino sector.
The fact that neutrinos have mass has been established by a series of experiments21,22,23,24,25,26. The field has incorporated these results into the neutrino standard model (νSM)—the standard model with three massive neutrinos. Although the νSM parameters are not yet fully determined27, the model is rigorous enough to be brought to bear on the question of LV. In the Methods, we briefly review the history of neutrino oscillation physics and tests of LV with neutrinos.
To date, neutrino masses have proved to be too small to be measured kinematically, but the mass differences are known via neutrino oscillations. This phenomenon arises from the fact that production and detection of neutrinos involves the flavour states, while the propagation is given by the Hamiltonian eigenstates. Thus, a neutrino with flavour |𝜈𝜈𝛼𝛼⟩ can be written as a superposition of Hamiltonian eigenstates |𝜈𝜈𝑖𝑖⟩; that is, |𝜈𝜈𝛼𝛼⟩ = �3𝑖𝑖=1 𝑉𝑉𝛼𝛼𝛼𝛼(𝐸𝐸)|𝜈𝜈𝑖𝑖⟩,where V is the unitary matrix that diagonalizes the Hamiltonian and, in general, is a function of neutrino energy E. When the neutrino travels in vacuum without new physics, the Hamiltonian depends only on the neutrino masses, and the Hamiltonian eigenstates coincide with the mass eigenstates. That is, H = 21𝐸𝐸 𝑈𝑈†diag(𝑚𝑚12, 𝑚𝑚22, 𝑚𝑚32)𝑈𝑈, where 𝑚𝑚𝑖𝑖 are the neutrino masses and U is the Pontecorvo–Maki–Nakagawa– Sakata matrix that diagonalizes the mass matrix m (ref. 27).
A consequence of the flavour misalignment is that a neutrino beam that is produced purely of one flavour will evolve to produce other flavours. Experiments measure the number of neutrinos of different flavours, observed as a function of the reconstructed energy of the neutrino, E, and the distance the beam has travelled, L. The microscopic neutrino masses are directly tied to the macroscopic neutrino oscillation length. In this sense, neutrino oscillations are similar to photon interference experiments in their ability to probe very small scales in nature.
Lorentz-violating neutrino oscillations
Here, we use neutrino oscillations as a natural interferometer with a size equal to the diameter of Earth. We look for anomalous flavour-changing effects caused by LV that would modify the observed energy and zenith angle distributions of atmospheric muon neutrinos observed in the IceCube Neutrino Observatory28 (see Fig. 1). Beyond flavour change due to small neutrino masses, any hypothetical LV fields could contribute to muon neutrino flavour conversion. We therefore look for distortion of the expected muon neutrino distribution. As this analysis does not distinguish between a muon neutrino (𝜈𝜈𝜇𝜇) and its antineutrino (𝜈𝜈𝜇𝜇), when the word ‘neutrino’ is used, we are referring to both.

Fig. 1: Test of LV with atmospheric neutrinos.

Muon neutrinos are produced in the upper atmosphere by the collisions of cosmic rays with air molecules. These atmospheric muon neutrinos pass through the entire Earth and are then detected by IceCube in Antarctica. The LV, indicated by arrows, permeates space and could induce an anomalous neutrino oscillation to tau neutrinos. Therefore, a potential signal of LV is the anomalous disappearance of muon neutrinos. Note, here we test only the isotropic component.
Past searches for LV have mainly focused on the directional effect in the Sun-centred celestial-equatorial frame19 by looking only at the time dependence of physics observables as direction-dependent physics appears as a function of Earth’s rotation. However, in our case, we assume no time dependence, and instead look at the energy distribution distortions caused by direction- and time-independent isotropic LV. Isotropic LV may be a factor ~103 larger than direction-dependent LV in the Sun-centred celestial-equatorial frame if we assume that the new physics is isotropic in the cosmic microwave background frame20. It would be most optimal to simultaneously look for both effects, but our limited statistics do not allow for this.

To calculate the effect, we start from an effective Hamiltonian derived from the SME4, which can be written as

𝐻𝐻 ≈ 𝑚𝑚2 + 𝑎∘𝑎(3) − 𝐸𝐸𝑐∘𝑐(4) + 𝐸𝐸2𝑎∘𝑎(5) − 𝐸𝐸3𝑐∘𝑐(6) ⋯ (1)

The first term of equation (1) is from the νSM; however, its impact decreases at high energy. The remaining




















symbol on the top indicates isotropic coefficients, and the number in the bracket is the dimension of the

operator. These terms are typically classified by charge, parity and time reversal (CPT) symmetry; CPT-odd (å(d))




















equation (1) can be expressed as 2 × 2 matrices, such as


𝑐𝑐(6) = ( ∘ (6)∗


𝑐∘𝑐𝜇(𝜇6𝜇𝜇) ∘ (6)) (2) −𝑐𝑐𝜇𝜇𝜇𝜇

Without loss of generality, we can define the matrices so that they are traceless, leaving three independent parameters, in this case: 𝑐∘𝑐𝜇(𝜇6𝜇𝜇), Re(𝑐∘𝑐𝜇(𝜇6𝜇𝜇)) and Im(𝑐∘𝑐𝜇(𝜇6𝜇𝜇)). The off-diagonal Lorentz-violating term 𝑐∘𝑐𝜇(𝜇6𝜇𝜇) dominates

neutrino oscillations at high energy, which is the main interest of this paper. In this formalism, LV can be

described by an infinite series, but higher-order terms are expected to be suppressed. Therefore, most

terrestrial experiments focus on searching for effects of dimension-three and -four















E2  å(5),



(6), E4  å(7)and E5  𝑐∘𝑐∘(8). Such higher orders are accessible by IceCube, which observes high-energy neutrinos where

we expect an enhancement from the terms with dimension greater than four. In fact, some theories, such as

non-commutative field theory2 and supersymmetry3, allow for LV to appear in higher-order operators. As an

example, we expect dimension-six new physics operators of order









mass, which is the natural energy scale of the unification of all matter and forces including gravity. We assume

that only one dimension is important at any given energy scale, because the strength of LV is expected to be

different at different orders.

We use the νμ → ντ two-flavour oscillation scheme following ref. 29. This is appropriate because we assume there is no significant interference with νe. Details of the model used in this analysis are given in the Methods. The oscillation probability is given by

𝑃𝑃(𝜈𝜈𝜇𝜇 → 𝜈𝜈𝜏𝜏) = −4𝑉𝑉𝜇𝜇1𝑉𝑉𝜇𝜇2𝑉𝑉𝜏𝜏1𝑉𝑉𝜏𝜏2sin2(𝜆𝜆2−2𝜆𝜆1 𝐿𝐿) (3)
where Vαi are the mixing matrix elements of the effective Hamiltonian (equation (1)), and λi are its eigenvalues. Both mixing matrix elements and eigenvalues are a function of energy, νSM oscillation parameters and SME coefficients.

The IceCube neutrino observatory
The IceCube Neutrino Observatory is located at the geographic South Pole30,31. The detector volume is one cubic kilometre of clear Antarctic ice. Atmospheric muon neutrinos interacting on surrounding ice or bedrock may produce high-energy muons, which emit photons that are subsequently detected by digital optical modules (DOMs) embedded in the ice. The DOMs consist of a 25-cm-diameter Hamamatsu photomultiplier tube, with readout electronics, contained within a 36.5 cm glass pressure housing. These are installed in holes in the ice with roughly 125 m separation. There are 86 holes in the ice with a total of 5,160 DOMs, which are distributed at depths of 1,450 m to 2,450 m below the surface, instrumenting 1 Gt of ice. The full detector description can be found in an earlier study31.

This detector observes Cherenkov light from muons produced in charged-current νμ interactions. Photons detected by the DOMs allow for the reconstruction of the muon energy and direction, which is related to the energy of the primary νμ. As the muons are above critical energy, their energy can be determined by measuring the stochastic losses that produce Cherenkov light. See earlier work28 for details on the muon energy proxy used in this analysis. In the teraelectronvolt (TeV) energy range, these muons traverse distances of the order of kilometres, and have a small scattering angle due to the large Lorentz boost, resulting in 0.75° resolution on the reconstructed direction at 1 TeV (ref. 32). We use up-going muon data of TeV-scale energy from two years of detector operation28 representing 34,975 events with a 0.1% atmospheric muon contamination.

Analysis set-up
To obtain the prediction for LV effects, we multiply the oscillation probability, given in equation (3), with the predicted atmospheric neutrino flux calculated using the matrix cascade equation (MCEq)33. These ‘atmospheric neutrinos’ originate from the decay of muons and various mesons produced by collisions of primary cosmic rays and air molecules, and consist of both neutrinos and antineutrinos. The atmospheric neutrinos have two main components: ‘conventional’, from pion and kaon decay, and ‘prompt’, from charmed meson decay. The conventional flux dominates at energies less than 18 TeV because of the larger production cross-section, whereas the harder prompt spectrum becomes relevant at higher energy. In the energy range of interest, the astrophysical neutrino contribution is small. We include it modelled as a power law with normalization and spectral index, ~ ΦE−γ. The absorption of each flux component propagating through Earth to IceCube is properly modelled34,35. Muon production from νμ charged-current events at IceCube proceeds through deep inelastic neutrino interactions as calculated in ref. 36.
The short distance of travel for horizontal neutrinos leads to negligible spectral distortion due to LV, whereas the long path length for vertical neutrinos leads to modifications. Therefore, if we compare the zenith angle distribution (θ) of the expectation from simulations and νμ data from cosθ = −1.0 (vertical) to cosθ = 0.0 (horizontal) (see Fig. 1), then one can determine the allowed LV parameters. Figure 2 shows the ratio of transition probabilities of vertical events to horizontal events. The data transition probability is defined by the ratio of observed events to expected events, and the simulation transition probability is defined by the expected events in the presence of LV to the number of events in the absence of LV. In the absence of LV, this ratio equals 1. Here, as an example, we show several predictions from simulations with different dimension-six LV parameters |𝑐∘𝑐𝜇(𝜇6𝜇𝜇)|. In general, higher-order terms are more important at higher energies. To assess the existence of LV, we perform a binned Poisson likelihood analysis by binning the data in zenith angle and energy. We use 10 linearly spaced bins in cosine of zenith angle from −1.0 to 0.0 and 17 logarithmically spaced bins in reconstructed muon energy ranging from 400 GeV to 18 TeV. Systematic uncertainties are incorporated as nuisance parameters in our likelihood. We introduce six systematic parameters related to the neutrino flux prediction: normalizations of conventional (40% error), prompt (no constraint) and astrophysical (no constraint) neutrino flux components; ratio of pion and kaon contributions for conventional flux (10% error); spectral index of primary cosmic rays (2% error); and astrophysical neutrino spectral index (25% error). The absolute photon detection efficiency has been shown to have negligible impact on the exclusion contours in a search for sterile neutrinos that uses an equivalent analysis technique for a subset of the IceCube data considered here34,37. The impact of light propagation model uncertainties on the horizontal to vertical ratio is less than 5% at a few TeV, where this analysis is most sensitive35. Thus, the impact of these uncertainties on the exclusion contours is negligible.

Fig. 2: The ratio of vertical to horizontal neutrino transition probabilities at IceCube.
Here, vertical events are defined by cosθ ≤ −0.6 and the horizontal events are defined by cosθ > −0.6. The transition probability ratio with 1 s.d. statistical errors (error bars), extracted from the data, is compared to the prediction for various dimension-six operator values. The range of uncorrelated systematic uncertainties is shown as a light grey band. This is constructed from ensembles of many simulations where the nuisance parameters are varied within their uncertainties.
To constrain the LV parameters, we use two statistical techniques. First, we performed a likelihood analysis by profiling the likelihood over the nuisance parameters per set of LV parameters. From the profiled likelihood, we find the best-fit LV parameters and derive the 90% and 99% confidence levels (CLs) assuming Wilks’s theorem with three degrees of freedom37. Second, we set the priors to the nuisance parameter uncertainties and scan the posterior space of the likelihood by means of a Markov chain Monte Carlo (MCMC) method38. These two procedures are found to be complementary, and the extracted LV parameters agree with the null hypothesis. For simplicity, we present the likelihood results in this paper and show the MCMC results in the Methods.
Figure 3 shows the excluded region of dimension-six SME coefficients. The results for all operators are available in the Supplementary Information. The fit was performed in a three-dimensional phase space; however, the complex phase of the off-diagonal terms is not important at high energy, and we choose the following representation methods. The horizontal axis shows the strength of LV, ρ6 ≡  �(𝑐∘𝑐𝜇(𝜇6𝜇𝜇))2 + Re(𝑐∘𝑐𝜇(𝜇6𝜇𝜇))2 + Im(𝑐∘𝑐𝜇(𝜇6𝜇𝜇))2, and the vertical axis represents a fraction of the diagonal element, 𝑐∘𝑐𝜇(𝜇6𝜇𝜇)/𝜌𝜌6. The best-fit point shown by the marker is compatible with the absence of LV; therefore, we present 90% CL (red) and 99% CL (blue) exclusion regions. The contour extends to small values, beyond the phase space explored by previous analyses16,17,18. The

leftmost edge of our exclusion region is limited by the small statistics of high-energy atmospheric neutrinos. The rightmost edge of the exclusion region is limited by fast LV-induced oscillations that suppress the flux but lead to no shape distortion. This can be constrained only by the absolute normalization of the flux. In the case of the dimension-three operator, the right edge can be excluded by other atmospheric neutrino oscillation measurements18,39. We have studied the applicability of Wilks’s theorem via simulations. Near-degenerate real and imaginary parameters reduce the expected degrees of freedom from three and the results here are interpreted as conservative confidence intervals. Fig. 3: The excluded parameter space region for the dimension-six SME coefficients.
The figure shows the exclusion region of the dimension-six SME coefficients in the space of two parameters: ρ6 and 𝑐∘𝑐𝜇(𝜇6𝜇𝜇)/𝜌𝜌6. Here, ρ6 ≡  �(𝑐∘𝑐𝜇(𝜇6𝜇𝜇))2 + Re(𝑐∘𝑐𝜇(𝜇6𝜇𝜇))2 + Im(𝑐∘𝑐𝜇(𝜇6𝜇𝜇))2 (horizontal axis) and 𝑐∘𝑐𝜇(𝜇6𝜇𝜇)/𝜌𝜌6 (vertical axis) are a combination of the three SME coefficients: ρ represents LV strength, and 𝑐∘𝑐𝜇(𝜇6𝜇𝜇)/𝜌𝜌6 represents a fraction of the diagonal element, while the subscript 6 indicates the dimension. In the white region, LV is allowed. The best-fit point of this sample for this operator is shown by the black cross. The blue and red regions are excluded at 99% CL and 90% CL, respectively.
Unlike previous results16,17,18, this analysis includes all parameter correlations, allowing for certain combinations of parameters to be unconstrained. This can be seen near 𝑐∘𝑐𝜇(𝜇6𝜇𝜇)/𝜌𝜌6 = −1 and 1, where LV is dominated by the large diagonal component. This induces the quantum Zeno effect40, where a neutrino flavour state is ‘arrested’ in one state by a continuous interaction with a LV field suppressing flavour transitions. Thus, the unshaded regions below and above our exclusion zone are very difficult to constrain with terrestrial experiments. Table 1 summarizes the results of this work along with representative best limits. A comprehensive list of LV tests is available in ref. 19. To date, there is no experimental indication of LV, and all of these experiments have maximized their limits by assuming that all but one of the SME parameters are zero19. Therefore, to make our results comparable with previous limits, we adopt the same convention. For this, we set the diagonal SME

parameters to zero and focus on setting limits on the off-diagonal elements. The details of the procedure used to set limits are given in the Supplementary Information.

Table 1 Comparison of attainable best limits of SME coefficients in various fields

Dimension Three

Method Cosmic microwave background polarization He-Xe co-magnetometer Torsion pendulum Muon g-2 Neutino oscillation

Type Astrophysical
Tabletop Tabletop Accelerator Atmospheric


Gamma-ray-burst (GRB) Astrophysical

vacuum birefringence

Laser interferometer



Sapphire cavity oscillator Tabletop

Ne-Rb-K co-



Trapped Ca+ ion


Neutrino oscillation



GRB vacuum birefringence Astrophysical

Ultrahigh-energy cosmic Astrophysical


Neutrino oscillation



GRB vacuum birefringence Astrophysical

Ultrahigh-energy cosmic Astrophysical


Gravitational Cherenkov Astrophysical


Neutrino oscillation



GRB vacuum birefringence Astrophysical

Neutrino oscillation



Gravitational Cherenkov radiation Neutrino oscillation

Astrophysical Atmospheric

Sector Photon

Limits ~10−43 GeV

Reference 5

Netron Electron Muon Neutrino

~10−34 GeV ~10−31 GeV ~10−24 GeV
�Re�𝑎∘𝑎𝜇(𝜇3𝜇𝜇)�� , �Im�𝑎∘𝑎𝜇(𝜇3𝜇𝜇)�� < 2.9 × 10−24GeV(99%CL)
< 2.0 × 10−24GeV(90%CL)

10 12 13 This work

Photon ~10−22


Photon ~10−18


Neutron ~10−29


Electron Neutrino
Photon Proton

~10−19 Re(𝑐∘𝑐𝜇(𝜇4𝜇𝜇)), |Im(𝑐∘𝑐𝜇(𝜇4𝜇𝜇))|
< 3.9 × 10−28(99%CL)
< 2.7 × 10−28(90%CL)
~10−34 GeV−1
~10−22 to 10−18 GeV−1

14 This work
6 9

Photon Proton

�Re�𝑎∘𝑎𝜇(𝜇5𝜇𝜇)�� , �Im�𝑎∘𝑎𝜇(𝜇5𝜇𝜇)�� < 2.3 × 10−32GeV−1(99%CL) < 1.5 × 10−32GeV−1(90%CL) ~10−31 GeV−2 ~10−42 to 10−35 GeV−2

This work
6 9

Gravity ~10−31 GeV−2


Photon Neutrino

�Re�𝑐∘𝑐𝜇(𝜇6𝜇𝜇)�� , �Im�𝑐∘𝑐𝜇(𝜇6𝜇𝜇)�� < 1.5 × 10−36GeV−2(99%CL)
< 9.1 × 10−37GeV−2(90%CL) ~10−28 GeV−3
�Re�𝑎∘𝑎𝜇(𝜇7𝜇𝜇)�� , �Im�𝑎∘𝑎𝜇(𝜇7𝜇𝜇)�� < 8.3 × 10−41GeV−3(99%CL)
< 3.6 × 10−41GeV−3(90%CL) ~10−46 GeV−4

This work
6 This work


�Re�𝑐∘𝑐𝜇(𝜇8𝜇𝜇)�� , �Im�𝑐∘𝑐𝜇(𝜇8𝜇𝜇)�� < 5.2 × 10−45GeV−4(99%CL) < 1.4 × 10−45GeV−4(90%CL)

This work

Let us consider the limits from the lowest to highest order. Dimension-three and -four operators are included in the renormalizable sector of SME. These are the main focus of experiments using photons7,8, nucleons10,11 and charged leptons12,13,14. Going beyond terrestrial experiments, limits arising from astrophysical observations provide strong constraints5,6. Among the variety of limits coming from the neutrino sector, the attainable best limits are dominated by atmospheric neutrino oscillation analyses16,17,18, where the longest propagation length and the highest energies enable us to use neutrino oscillations as the biggest interferometer on Earth. The results from our analysis surpass past ones due to the higher statistics of high-energy atmospheric neutrinos and our improved control of systematic uncertainties. Using a traditional metric, which assumes neutrinos to be massless, we can recast our result as an upper limit on any deviation of the speed of massless neutrinos from the speed of light due to LV. That is less than 10−28 at 99% CL. This is about an order-of-magnitude improvement over past analyses16,17,18, and is of the same order as the deviation in speed that is expected due to the known neutrino mass at the energies relevant for this analysis.
Searches of dimension-five and higher LV operators are dominated by astrophysical observations6,9,15. Among them, ultrahigh-energy cosmic rays have the highest measured energy41 and are used to set the strongest limits on dimension-six and higher operators9. However, these limits are sensitive to the composition of ultrahighenergy cosmic rays, which is currently uncertain20,42. These limits assume that the cosmic rays at the highest energies are protons, but if they are in fact iron nuclei, then the ultrahigh-energy cosmic ray limits are significantly reduced. Our analysis sets the most stringent limits in an unambiguous way across all fields for the dimension-six operator. Such high-dimension operators are generic signatures of new physics43. For example, the dimension-five operator is an attractive possibility to produce neutrino masses, and dimension-six operators represent new physics interactions that can, for example, mediate proton decay. Although LV dimension-six operators, such as 𝑐∘𝑐𝜇(𝜇6𝜇𝜇), are well motivated by certain theories including non-commutative field theory2 and supersymmetry3, they have so far not been probed with elementary particles due to the lack of available highenergy sources. Thus, our work pushes boundaries on new physics beyond the standard model and general relativity.
We have presented a test of LV with high-energy atmospheric muon neutrinos from IceCube. Correlations of the SME coefficients are fully taken into account, and systematic errors are controlled by the fit. Although we did not find evidence for LV, this analysis provides the best attainable limits on SME coefficients in the neutrino sector along with limits on the higher-order operators. Comparison with limits from other sectors reveals that this work provides among the best attainable limits on dimension-six coefficients across all fields: from tabletop experiments to cosmology. This is a remarkable point that demonstrates how powerful neutrino interferometry can be in the study of fundamental spacetime properties.
Further improvements on the search for LV in the neutrino sector using IceCube will be possible when the astrophysical neutrino sample is included44. Such analyses45,46 will require a substantial improvement of detector and flux systematic uncertainty evaluations47,48. In the near future, water-based neutrino telescopes such as KM3NeT49 and the ten-times-larger IceCube-Gen250 will be in a position to observe more astrophysical neutrinos. With the higher statistics and improved sensitivity, these experiments will have an enhanced potential for discovery of LV.
Neutrino oscillations and tests of LV
The field of neutrino oscillations has been developed through a series of measurements of solar51,52,53,54,55, atmospheric56,57,58, reactor59,60,61,62 and accelerator neutrinos57,63,64. In the early days, the cause of neutrino