Invisible Neutrino Decay Could Resolve IceCube’s Track and Cascade Tension

The IceCube Neutrino Observatory detects high energy astrophysical neutrinos in two event topologies: tracks and cascades. Since the flavor composition of each event topology differs, tracks and cascades can be used to test the neutrino properties and the mechanisms behind the neutrino production in astrophysical sources. Assuming a conventional model for the neutrino production, the IceCube data sets related to the two channels are in > 3 σ tension with each other. Invisible neutrino decay with lifetime τ =m ¼ 10 2 s = eV solves this tension. Noticeably, it leads to an improvement over the standard nondecay scenario of more than 3 σ while remaining consistent with all other multimessenger observations. In addition, our invisible neutrino decay model predicts a reduction of 59% in the number of observed ν τ events which is consistent with the current observational deficit.

According to the conventional framework, high energy neutrinos in astrophysical sources are expected to be produced primarily as a result of charged pion decay.Charged pions decay to a muon and a muon neutrino, and the muon in turn decays on to a positron, electron neutrino, and a muon antineutrino resulting in a neutrino flavor ratio at the source of ν e : ν µ : ν τ = 1 : 2 : 0, each with approximately the same energy.After neutrino oscillations, the flavor ratio at the Earth is roughly 1 : 1 : 1 leading to the expectation that the spectral distributions of neutrinos will be the same for any flavor.We stress that this last statement is independent of the source class since any mechanism that produces high energy neutrinos will do so dominantly as a result of charged pion decays.Hence, within this picture, the only possible result is equal fluxes for each flavor.
IceCube is partially sensitive to the flavor state of the neutrino through two distinct event topologies: track events resulting dominantly from ν µ interactions [3], and nearly spherical cascade events resulting dominantly from ν e and ν τ interactions [16].The IceCube Collaboration has interpreted each of these data sets in terms of the true per-flavor neutrino flux at the Earth under the assumption that the flavor ratio remains constant at 1 : 1 : 1 for all energies and that the flux follows a SPL [8].The IceCube Collaboration finds that those two different channels produce results in tension with each other [3], as shown in Fig. 1.  [3] and cascade [16] data samples.The tension between the two data samples is driven on the high energy end by the observation of six tracks with energies E > 1 PeV and the lack of Glashow Resonance events at E = 6.3 PeV in the cascade channel [3].On the low energy side there is also an apparent excess of events in the cascade channel [15].
In this Letter, for the first time, we combine spectral and flavor information simultaneously to investigate the tension between the data sets associated to the two event topologies.We investigate several modifications to the standard picture of the high energy astrophysical neutrino flux beyond what is foreseen within the Standard Model [17].

arXiv:1805.05950v1 [hep-ph] 15 May 2018
We determine the diffuse intensity at the Earth after oscillations, convert this into the per-flavor intensity from each of the track and cascade channels, and fit a power law to each assuming a 1 : 1 : 1 flavor ratio in order to compare a model to IceCube's observations.We then compare the normalizations and spectral indices to the measured ones by combining both tracks and cascades under the assumption that the correlation between the normalizations and spectral indexes are small.It turns out that invisible neutrino decay provides a good fit to the data and is preferred over the Standard Model at more than 3σ removing the tension.Our proposed solution is not in contradiction with existing multi-messenger constraints and also explains the current deficit in the observation of ν τ events.
Standard Neutrino Source Model.-For the sake of generality, we model the neutrino spectral distribution in such a way to be agnostic about the mechanism of the neutrino production, i.e. pγ or pp interactions.We consider a general BPL model parameterized by the break energy in the source frame Ẽν,b and the change in the spectral index ∆, such that the spectral index below the break energy is γ and it is γ + ∆ above it.The SPL case is then recovered for ∆ = 0.
This model is further generalized to the case where the break energy for neutrinos coming from muon decay (ν e and ν µ ) is different than that from pion decay (ν µ ).In fact, pions and muons lose energy in pγ sources, e.g. in the presence of magnetic fields due to synchrotron losses and they may have separate break energies, Ẽν,b,µ and Ẽν,b,π .For example, for synchrotron losses, the neutrino break energy scales like m 5/2 i τ −1/2 i for i ∈ {π, µ} where m (τ ) is the mass (lifetime) of the particle, so the ratio of the neutrino break energies is R π,µ ≡ Ẽν,b,π / Ẽν,b,µ 18.4 when synchrotron cooling dominates.The simpler BPL model introduced above is recovered when R π,µ = 1.Thus there are at most five free parameters in the BPL model: γ, ∆, Ẽν,b , R π,µ , and the neutrino flux normalization Φ ν .
The IceCube neutrino flux is considered to be dominantly extragalactic and compatible with a diffuse origin [8,[18][19][20].Hence, the expected diffuse neutrino intensity at the Earth for the flavor ν β (β = e, µ, τ ) is [21].For the redshift evolution ρ(z), we assume as benchmark case that the source luminosity density evolves as (1 + z) θ for θ = 3 up to a certain z c 1.5 and it is constant for z > z c [22].Different redshift scalings for θ ∈ [0, 4] and z c ∈ [0.5, 2] do not significantly affect our conclusions.The averaged oscillation probability is where U is the standard mixing matrix [23,24].For the mixing angles we take the latest global fit results [25,26].
We then compute the corresponding per-flavor intensity expected in the two event topologies; the track intensity roughly corresponds to the ν µ one, while the cascade one corresponds to the ν e + ν µ one (see the Appendix for technical details).A scan over all possible values of each model parameter is done to compare with the IceCube neutrino data through a χ 2 test: where the sum runs on both neutrino event topologies.
For the SPL case with two free parameters (γ and Φ ν ) we find χ 2 = 13.4 which corresponds to 3.23σ of tension.
When we expand the source model to the BPL case with four free parameters (γ, ∆, Ẽν,b , Φ ν ) and R π,µ = 1, we find that the χ 2 does not improve which results in > 3.66σ tension.That is the BPL case is not preferred by the data with respect to the SPL.In addition, letting R π,µ float freely only improves the fit to χ 2 = 10.7 which is disfavored at > 3.27σ and provides only marginal improvement (1.64σ) over the BPL case.In this case, the best fit point has R π,µ > 100 and ∆ large, similar to a fully damped muon source.
Our findings confirm that adding a break to the source spectra provides only marginal improvement to the data fit and that a SPL fit is justified.Most importantly, the standard neutrino source scenario is disfavored at > 3.2σ by the IceCube track and cascade data (see the left columns of Table I for a summary).While muon cooling does provide both an energy and flavor dependent effect, it is not enough to resolve the tension due to the large mixing angles.Thus we expect that any mechanism which increases the relative number of ν µ 's at the source (such as muon damping from synchrotron cooling) at high energy will equally increase the relative number of ν τ 's after oscillations since θ 23 ∼ 45 • is minimizing the effect.
Invisible Neutrino Decay.-Inorder to resolve the tension between the fits provided by the two event topologies, an interesting model that modifies the flavor ratio in an energy dependent fashion during propagation is neutrino decay [27].Neutrino decay is typically described by a new interaction term of the form L ⊃ g ij ν i ν j φ where φ is a new light (m φ < ∼ m ν ) or massless scalar known as the Majoron, which could provide neutrinos with their masses [28][29][30].Specifically, we here focus on the invisible decay scenario where the decay products are a Majoron and a right handed neutrino (left handed antineutrino) [29,30]; another interesting model of invisible neutrino decay is to unparticles [31,32].Noticeably, depending on the neutrino mass ordering and absolute mass scale, the decay products of visible neutrino decay may have significantly less energy.For a steeply falling
We assume that ν 1 is stable since it has the least ν µ fraction since this can suppress the ν µ fraction at low energies.This may be the case if the mass ordering is normal, as is currently favored at 2−3.4σ [25,26,33,34], and the Majoron has a mass between ν 1 and ν 2 , or if ν 1 is massless (or very light) and has no (significant) coupling to the Majoron.
The oscillation averaged probability for invisible neutrino decay is where is the corrected cosmological distance scaling for neutrino decay [35].Thus in our model Λ 1 = 0 and Λ 2 = Λ 3 and τ /m for ν 2 and ν 3 is the one new free parameter.
Figure 2 shows the modification of the track vs. cascade ratio due to invisible neutrino decay within the model introduced above.One can check that in order to have an effect within the region of interest of IceCube, we should have τ /m ∼ 10 2 s/eV.
Minimizing the χ 2 in the SPL only case with neutrino decay, we find χ 2 = 1.57with log 10 [(τ /m)/(s/eV)] = 1.93 +0.26  −0.40 .At 1 d.o.f.this represents a good fit, consistent with the data at 1.25σ.It is an improvement over the stable neutrino case of ∆χ 2 = 11.8 showing that the neutrino decay scenario is preferred by the data over the standard stable neutrino case by 3.4σ.The 2D χ 2 pro- jection of the source spectral index γ and the neutrino lifetime τ /m is shown in Fig. 3.We note that τ /m is fairly well determined since it must give observable consequences within IceCube's region of interest.Varying the redshift evolution power θ produces a fairly small effect with the best fit value of τ /m and the χ 2 changes only slightly with τ /m increasing with θ.If we extend our fit to the BPL source model, the best fit point does not change at all and ∆ = 0 is preferred.The results are summarized in Table I [36].
Our findings should be compared with existing bounds on invisible neutrino decay.The best terrestrial constraints on invisible ν 3 decay come from atmospheric and long-baseline data: log 10 [(τ 3 /m 3 )/(s/eV)] > −9.52 [37], while the best terrestrial constraints on invisible ν 2 decay are from solar neutrinos and are log 10 [(τ 2 /m 2 )/(s/eV)] > −3.15 [38,39].Strong constraints, in apparent contra-TABLE I.The χ 2 and significance for the single power law (SPL) and broken power law (BPL) models, along with the best fit source spectral index and neutrino lifetime.Here we fix Rπ,µ = 1 for the BPL model, see text.The BPL models have as many or more parameters than data points, thus only a lower limit on the significance can be placed by taking  diction with our findings, have been derived from SN 1987A: log 10 [(τ /m)/(s/eV)] > ∼ 5 [40]; however these constraints only apply to νe measurements under the assumption that all neutrino mass eigenstates are decaying and should be considered with caution.Even in the case of full ν 2 and ν 3 decay, the νe → νe oscillation averaged probability would be suppressed by 16% which is still smaller than the SN 1987A statistical uncertainties (∼ 20%) and current theoretical uncertainties.IceCube data has been used to place a constraint on the neutrino lifetime at log 10 [(τ /m)/(s/eV)] > ∼ 1 by assuming that neutrinos do not fully decay within the IceCube energy range [41,42] which is not the case considered here.The most stringent constraints on the lifetime of neutrinos have been derived from cosmic microwave background data at the level of log 10 [(τ /m)/(s/eV)] > ∼ 11 [43].Noticeably, these bounds can be alleviated in the event that only one or two neutrinos decay and the remaining ones are free streaming [44,45] and are therefore not in contradiction with our findings.
Other Possible Interpretations.-Anotherpossible explanation of the tension between the track and cascade data sets is the decay of dark matter (DM) to electron neutrinos (χ → ν e νe ).We focus on DM decay instead of annihilation as the galactic anisotropy constraints [18,19,46] are weaker for DM decay since the DM annihilation more peaked towards the galactic center.In order to estimate the expected track and cascade distribution, the galactic and extragalactic diffuse intensity of neutrinos is computed, including electroweak corrections, by using Pythia 8.2 [47] and a Navarro-Frenk-White galactic DM profile [48].
While a good quality of fit (χ 2 < 1) is found in a SPL+DM model with 4 parameters (τ χ s, m χ TeV, Φ ν , γ), this model has a number of undesirable properties.The galactic contribution to the flux peaks at energies below the cascade flux sensitivity and contributes due to the typical energy uncertainty of cascades are > ∼ 15% [49]; this results in a contribution to the cascade flux at low energies due to the energy uncertainty, but a minimal contribution to the track flux (after oscillations).The resultant peak flux is much larger than the measured flux at energies just below the region of interest for IceCube's cascade analysis.From SU(2) symmetry there will also be an e + +e − channel, further leading to γ-rays from electroweak corrections constrained by Fermi-LAT [50].Finally, this fit requires a relatively short DM lifetime which is strongly constrained by cosmic microwave background data and bounds from the reionization epoch since the best fit values are τ χ ∼ 10 23 s, m χ ∼ 10 TeV [51,52].All considered, we find that DM decay does not resolve this tension.
Several additional effects could provide an energy and flavor dependent modification of the standard neutrino flux from an astrophysical source.For example, the Glashow resonance occurs when a νe with E ν = 6.3 PeV scatters off an electron in the ice creating an on-shell W − [53] increasing IceCube's sensitivity in that energy range considerably.IceCube performs their fits assuming that I ν = I ν .While this is generally the case if neutrinos are mainly produced through pp interactions, it won't be the case if the main neutrino production channel is pγ interactions.For the SPL case I νe /I νe 3.5 which would somewhat harden the cascade spectrum, but would not be enough to reduce the tension of the fit.
Another option that could alleviate the track vs. cascade fit tension is neutron decay sources.Neutrons decay to νe 's and are produced alongside charged pions in pγ interactions (as well as in pp interactions) and are thus expected to provide an additional contribution of ν e 's to the high energy astrophysical neutrino flux.The energy of neutrinos from neutrons is suppressed by about two orders of magnitude compared with those from pion decay.However, for a spectral index > ∼ 2 as in our case, this contribution is subleading.Neutrons also result from photodisintegration of heavy ions in dense sources, although this flux is also suppressed compared to the standard contribution by at least an order of magnitude [54,55].
In addition, non-standard neutrino interactions with ultralight mediators (m Z 1 eV) as well as pseudo-Dirac neutrino models [56,57] may also affect the track vs. cascade ratio.However, in both cases, we expect an impact on the neutrino data set that is smaller than the one induced by the invisible neutrino decay scenario.
A Solution to the ν τ Observational Deficit.-TheIce-Cube detector is expected to observe 2 or 3 ν τ events in the energy range of interest [58].However, currently no ν τ events are observed.The assumption of invisible neutrino decay for the ν 2 and ν 3 eigenstates would induce a reduction of I ντ of 80% below 1 PeV which convolved with the detection efficiency leads to a ∼ 59% reduction in the number of ν τ events for our best fit value m/τ = 10 2 eV/s.The invisible neutrino decay could then also explain the current deficit of ν τ events.
Conclusions.-TheIceCube Neutrino Observatory detects high energy astrophysical neutrinos through two event topologies: tracks and cascades.By simultaneously taking advantage of the energy and flavor information present in the two data sets, for the first time we have placed strong constraints on the consistency of the data with the standard source picture.In fact a conventional model for the neutrino production in astrophysical sources is unable to simultaneously explain the track and cascade data at > 3σ.
We tested several New Physics models and found that the invisible neutrino decay of ν 2 and ν 3 with τ /m = 10 2 s/eV is preferred by the IceCube data by 3.4σ and is consistent with all other existing constraints.While this model is more natural in the normal mass ordering, it is consistent with either ordering.Interestingly, our model also predicts a 59% reduction in the number of expected ν τ events reconciling the current observational deficit.
As more high energy neutrino data arrives with the advent of IceCube-Gen2 [59] and KM3NeT [60] and the spectral distributions will be defined more precisely for both event topologies, it will be possible to further test our result.
FIG.2.The track to cascade ratio as a function of the neutrino energy.The invisible neutrino decay of ν2 and ν3 reduces the track and cascade ratio below 1 PeV up to 75% with respect to the case where all neutrinos are stable.The deviation from the expected value of 0.5 for the standard case is mostly due to track misidentification.