The article "Invisible Neutrino Decay Could Resolve IceCube's Track and Cascade Tension" by Peter Denton and Irene Tamborra tries to explain this problem by assuming that τ and μ neutrinos can decay to a superparticle called majoron (see this).

The standard model for the production of neutrinos is based on the decays of pions producing e^{+}ν_{e} and

μ^{+} ν_{μ}. Also μ^{+} can travel to the direction of Earth and decay to e^{+} ν_{e} ν_{μ} and double the electron neutrino fraction. The flavor ratio would be 2:1:0.

** Remark**: The article at (see this) claims that the flavor ratio is 1:2:0 in pion decays, which is wrong: the reason for the lapsus is left as an exercise for the reader.

Calculations taking into account also neutrino oscillations during the travel to Earth to be discussed below leads in good approximation to a predicted flavor ratio 1:1:1. The measurement teams suggest that measurements are consistent with this flavor ratio.

There are however big uncertainties involved. For instance, the energy range is rather wide [13 TeV, 7.9 PeV] and if neutrinos are produce in decay of third generation weak boson with mass about 1.5 PeV as TGD predicts, the averaging can destroy the information about branching fractions.

In TGD based model (see this) third generation weak bosons - something new predicted by TGD - at mass around 1.5 TeV corresponding to mass scale assignable to Mersenne prime M_{61} (they can have also energies above this energy) would produce neutrinos in the decays to antilepton neutrino pairs.

- The mass scale predicted by TGD for the third generation weak bosons is correct: it would differ by factor 2
^{(89-61)/2}= 2^{14}from weak boson mass scale. LHC gives evidence also for the second generation corresponding to Mersenne prime M_{79}: also now mass scale comes out correctly. Note that ordinary weak bosons would correspond to M_{89}.

- The charge matrices of 3 generations must be orthogonal and this breaks the universality of weak interactions. The lowest generation has generation charge matrix proportional to (1,1,1) - this generation charge matrix describes couplings to different generations. Unit matrix codes for universality of ordinary electroweak and also color interactions. For higher generations of electro-weak bosons and also gluons universality is lost and the flavor ratio for the produced neutrinos in decays of higher generation weak bosons differs from 1:1:1.

One example of charge matrices would be 3/2

^{1/2}×(0,1,-1) for second generation and (2,-1,-1)/2^{1/2}for the third generation. In this case electron neutrinos would be produced 2 times more than muon and tau neutrinos altogether. The flavor ratio would be 0:1:1 for the second generation and 4:1:1 for the third generation in this particular case.

- This changes the predictions of the pion decay mechanism. The neutrino energies are above the energy about 1.5 PeV in the range defined by the spectrum of energies for the decaying weak boson. If they are nearly at rest the energie are a peak around the rest mass of third generation weak boson. The experiments detect neutrinos at energy range [13 TeV, 7.9 PeV] having the energy of the neutrinos produced in the decay of third generation weak bosons in a range starting from 1.5 PeV and probably ending below 7.9 PeV. Therefore their experimental signature tends to be washed out if pion decays are responsible for the background.

- Suppose that L+ν
_{L}pair is produced. It can also happen that L^{+}, say μ^{+}travels to the direction of Earth. It can decay to e^{+}ν_{μ}ν_{e}. Therefore one obtains both ν_{μ}and ν_{e}. From the decy to τ^{+}ν_{τ}one obtains all three neutrinos. If the fractions of the neutrinos from the generation charge matrix are (X^{e},X^{μ},X^{τ}), the fractions travelling to each are proportional to

x

^{α}↔ X^{α}=(X^{e},X^{μ},X^{τ}) =(x^{e}+x^{μ}+x^{τ},x^{μ}+ x^{τ},x^{τ}) .

and the flavor ratio in the decays would be

X

^{e}:X^{μ}:X^{τ}=x^{e}+x^{μ}+x^{τ}: x^{μ}+x^{τ}:x^{τ}.

The decays to lower neutrino generations tend to increase the fraction of electronic and muonic neutrinos

in the beam.

- Also neutrino oscillations due to different masses of neutrinos (see this) affect the situation. The analog of CKM matrix describing the mixing of neutrinos, the mass squared differences, and the distance to Earth determines the oscillation dynamics.

One can deduce the mixing probabilities from the analog of Schrödinger equation by using approximation E= p+m

^{2}/2p which is true for energies much larger than the rest mass of neutrinos. The masses of mass eigenstates, which are superpositions of flavour eigenstates, are different.

The leptonic analog of CKM matrix U

_{α i}(having in TGD interpretation in terms of different mixings of topologies of partonic 2-surfaces associated with different charge states of various lepton families allows to express the flavor eigenstates ν_{α}as superpositions of mass eigenstates ν_{i}. As a consequence, one obtains the probabilities that flavor eigenstate ν_{α}transforms to flavour eigenstate ν_{β}during the travel. In the recent case the distance is very large and the dependence on the mass squared differences and distance disappears in the averaging over the source region.

The matrix P

_{αβ}telling the transformation probabilities α→β is given in Wikipedia article (see this) in the general case. It is easy to deduce the matrix at the limit of very long distances by taking average over source region to get exressions having no dependence

P

_{αβ}= δ_{αβ}- 2 ∑_{i>j}Re[U_{β i}U^{†}_{i α}U_{α j}U^{†}_{jβ}] .

Note that ∑

_{β }P_{αβ}=1 holds true since in the summation second term vanishes due to unitary condition U^{†U=1 and i>j condition in the formula. } - The observed flavor fraction is Y
_{e}:Y_{μ}:Y_{τ}, where one has

Y

_{α}= P_{αβ}X^{β . }It is clear that if the generation charge matrix is of the above form, the fraction of electron neutrinos increases both the decays of τ and μ and by this mechanism. Of course, the third generation could have different charge matrix, say (3/2

^{1/2}(0,1,-1). In this case the effects would tend to cancel.

For a summary of earlier postings see Latest progress in TGD.

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