Neutrino oscillation with very low energy neutrinos

[PAllen]

This question just occurred to me recently. Assuming the different neutrinos (electron, muon, tau) do not have the same mass, then if their energy is very low, such that they are moving non-relativistically, an oscillation must substantially change the velocity (to conserve momentum). It seems this must be the case, because any neutrino ascillation could be viewed in a frame where neutrinos are slow moving. Yet this is certainly an odd mental image - spontaneous speed change.

Just checking that this is expected?

 

[mfb]

If they are moving non-relativistically, the three mass eigenstates quickly get decoherent, and you don't see oscillation any more. You'll either detect the lightest mass eigenstate arriving first, or the middle mass eigenstate arriving a bit later, or the heaviest eigenstate arriving even later. No speed change.
Note: every plausible detection mechanism sees flavor eigenstates - but you can calculate the probability to see them for each mass eigenstate separately.

 

[PAllen]

So, hypothetically, if an atom undergoing beta decay is moving near c away from me, and also is very far away, and its neutrino happens to be heading toward me at extremely low energy, I might detect a muon neutrino at a time consistent with its always having that mass, rather than a time corresponding to part of the flight at electron neutrino speed?

 

[mfb]

Flight times exist for the mass eigenstates only. There is no "speed of an electron neutrino", just a "speeds of the three mass eigenstates". Neutrino mixing is strong, you cannot even make an approximate association of mass and flavor eigenstates like you can do for quarks.

 

[Vanadium 50]

might detect a muon neutrino at a time consistent with its always having that mass

That object doesn't exist.

The way to think of it that minimizes conceptual misunderstandings is that there are three neutrinos, nu1, nu2 and nu3. They have definite masses. Nu3 has about half the coupling strength to muons and taus that it would if it only coupled to one, and a tiny coupling to electrons. Nu2 has equal couplings, again, one-third the maximum, and nu1 has about three-quarters of the maximum coupling to electrons, and 10% of the maximum to the other two. Those are the particles in the theory, not nu_e, nu_mu and nu_tau.

If I set up an experiment where I produce neutrinos of a given flavor and detect neutrinos of a given flavor, I do not know if the neutrino in flight is a nu1, nu2 or nu3, so I need to add amplitudes, not intensities, so I get interference - which we call oscillations. (This is not the greatest terminology.) Asking what mass eigenstate was in flight is exactly equivalent to asking "which slit did the electron go through"?

If I have some other measurement, like timing, it tells me which mass eigenstate it was, and that breaks the interference, just as identifying which slit breaks the interference. You will know you had a nu2 and that you always had a nu2. A nu2 was emitted and a nu2 was detected.

 

[PAllen]

Thank to both of you. That is very clear now.

 

[MRBlizzard]

mfb please

If the three neutrino mass eigenstates have different velocities, then, as they propagate over the lightyears, won’t the mass eigenstates draw away from each other. Wouldn’t that make the neutrino mixing impossible. Obviously, I’m missing something. I only got as far as Merzbacher.

 

[Orodruin]

If the three neutrino mass eigenstates have different velocities, then, as they propagate over the lightyears, won’t the mass eigenstates draw away from each other

Yes. Heuristically, the wave packets of the different mass eigenstates will separate. You do not need to go to lightyear distances (depending on neutrino energy). This occurs for solar neutrinos that reach the Earth.

Wouldn’t that make the neutrino mixing impossible.

No. Mixing is not something that happens because the state wave packets overlap, it is a fundamental property - the mismatch in weak interactions between the mass eigenstates of neutrinos and those of charged leptons.

However, as the states separate you lose coherence between them and therefore the oscillatory behaviour disappears. The flavor transition probabilities then become constant in distance - where the constants are given by the mixing matrix.

 

[MRBlizzard]

Thank you.
Could you point me towards a reading list towards Electroweak Theory.