# Neutrino masses and oscillation

• I

This depends on the absolute neutrino mass, which we do not know.
We do, however, have a lower limit, because of the solar neutrino experiment: too light and the neutrinos wouldn't have enough proper time to oscillate among the 3 flavors before reaching the Earth. (And, of course, an upper limit: WP lists ~ .1 eV as the upper limit. But that only gives us a lower limit on clustering.)

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Orodruin
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We do, however, have a lower limit, because of the solar neutrino experiment: too light and the neutrinos wouldn't have enough proper time to oscillate among the 3 flavors before reaching the Earth.
No, this is incorrect. You could perfectly well have neutrino oscillations with one of the mass states having identically zero mass. You have a lower bound on the second lightest state.

No, this is incorrect. You could perfectly well have neutrino oscillations with one of the mass states having identically zero mass. You have a lower bound on the second lightest state.
My understanding is that the oscillation requires all three to have masses that differ by only a small fraction of their mean value (or else the solar neutrino measurements disagree). Not true?

Orodruin
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My understanding is that the oscillation requires all three to have masses that differ by only a small fraction of their mean value (or else the solar neutrino measurements disagree). Not true?
Indeed, this is incorrect. Where did you get that from?

kimbyd
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My understanding is that the oscillation requires all three to have masses that differ by only a small fraction of their mean value (or else the solar neutrino measurements disagree). Not true?
Right, the solar neutrino oscillation observations only put a lower bound on the difference between masses. I don't think there's any dependence at all of the oscillation on the absolute values of the masses, only their differences.

However, that does implicitly place a lower-bound on the average neutrino mass, because the mass can't be negative. I would tend to think that would be relevant to cosmic neutrino clustering.

Probably WP recently (possibly Quartz or Phys.org, though I don't recall anything recent there), plus long-ago familiarity with the solar-neutrino "paradox". But if it's not true, then how has the paradox been resolved?

That is, if the emitted electron (anti-?)neutrinos don't oscillate into all three alternatives, then isn't the bookkeeping wrong? Or is one of the alternatives (e.n. -> massless) effectively a one-way transition?

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Probably WP recently (possibly Quartz or Phys.org, though I don't recall anything recent there), plus long-ago familiarity with the solar-neutrino "paradox". But if it's not true, then how has the paradox been resolved?

That is, if the emitted electron (anti-?)neutrinos don't oscillate into all three alternatives, then isn't the bookkeeping wrong? Or is one of the alternatives (e.n. -> massless) effectively a one-way transition?
I am sorry, I don't understand the problem here. Even if one neutrino state is massless, all of the mass squared differences (which is what neutrino oscillations depend on) can be non-zero.

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It should also be pointed out that solar neutrino flavour transitions are not really "oscillations" in the most common sense, but rather resonant flavour conversions due to the MSW matter effect. Alexei Smirnov wrote a short note on that part.

I am sorry, I don't understand the problem here. Even if one neutrino state is massless, all of the mass squared differences (which is what neutrino oscillations depend on) can be non-zero.
My reasoning: We have an emitted electron neutrino ("e.n.") -> {no transition, tau n., mu n.}. The e.n. is observed (inferred) to transition on its path to the Earth, so it's got non-zero mass. If one of the others had 0 mass, then you could still observe just 1/3 of the emitted flux as (still) being e.n's, but only if the Earth were at a very special distance, right?

That is, the e.n. would oscillate into the 3rd flavor (assumed massive), and perhaps back again. But even if the 3rd flavor only oscillated back to the e.n. flavor, each time the oscillation produced an e.n., there would be a finite (~ 1/3) probability of transitioning into the massless flavor, and never back again. So the fraction that were observable as e.n's. would drop as a function of distance from the Sun. By luck, it might be 1/3 at the Earth, but it could as well be any other fraction. Conversely, if all three masses are sufficiently non-zero, then the oscillations can occur frequently enough that, at any distance greater than some threshold (which I do not know, but obviously much smaller than 1 AU), the fraction that are e.n's. will always be 1/3.

kimbyd
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My reasoning: We have an emitted electron neutrino ("e.n.") -> {no transition, tau n., mu n.}. The e.n. is observed (inferred) to transition on its path to the Earth, so it's got non-zero mass. If one of the others had 0 mass, then you could still observe just 1/3 of the emitted flux as (still) being e.n's, but only if the Earth were at a very special distance, right?

That is, the e.n. would oscillate into the 3rd flavor (assumed massive), and perhaps back again. But even if the 3rd flavor only oscillated back to the e.n. flavor, each time the oscillation produced an e.n., there would be a finite (~ 1/3) probability of transitioning into the massless flavor, and never back again. So the fraction that were observable as e.n's. would drop as a function of distance from the Sun. By luck, it might be 1/3 at the Earth, but it could as well be any other fraction. Conversely, if all three masses are sufficiently non-zero, then the oscillations can occur frequently enough that, at any distance greater than some threshold (which I do not know, but obviously much smaller than 1 AU), the fraction that are e.n's. will always be 1/3.
A zero-mass flavor wouldn't prevent oscillations. The oscillations come from two features:
1) The masses are different.
2) The mass eigenstates and flavor eigenstates are different.

That second point is key, though can be tricky to understand. It means that there are two different ways of representing the neutrino wave functions: as functions of flavor states, or mass states. Each flavor state is actually a mixture of the three mass states. For example:
$$|\nu_e\rangle = a_1 |m_1\rangle + a_2 |m_2\rangle + a_3 |m_3\rangle$$

Here ##|\nu_e\rangle## is the electron-neutrino flavor eigenstate, and ##|m_n\rangle## are the three mass eigenstates. The factors in front determine how much of the electron state is made up of each of the mass eigenstates.

The mass of each of the three mass states determines their frequency of oscillation. Over time, the three mass states will evolve at different rates, related to their masses. Even if one of their masses is zero, the other two will oscillate. So even if you start out with a neutrino in a pure flavor state, over time it will become a mixture of flavor states.

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• Orodruin
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My reasoning: We have an emitted electron neutrino ("e.n.") -> {no transition, tau n., mu n.}. The e.n. is observed (inferred) to transition on its path to the Earth, so it's got non-zero mass. If one of the others had 0 mass, then you could still observe just 1/3 of the emitted flux as (still) being e.n's, but only if the Earth were at a very special distance, right?
The flavour neutrino states do not have definite masses and therefore it makes no sense to talk about the "mass of an electron neutrino". Neutrino oscillations are an interference phenomena between the neutrino mass eigenstates that accumulate different phases depending on their mass.

That is, the e.n. would oscillate into the 3rd flavor (assumed massive), and perhaps back again. But even if the 3rd flavor only oscillated back to the e.n. flavor, each time the oscillation produced an e.n., there would be a finite (~ 1/3) probability of transitioning into the massless flavor, and never back again.
I think you need to reexamine your understanding of neutrino oscillations. This is not how neutrino oscillations work (see kimbyd's post). The amplitude of oscillations depends on what admixture of the neutrino flavour states constitute the mass eigenstates and the masses of the mass eigenstates. Also, again, solar neutrinos do not oscillate in the true sense of the word. The flavour transitions are related to adiabatic flavour conversion.

A zero-mass flavor wouldn't prevent oscillations. The oscillations come from two features:
1) The masses are different.
2) The mass eigenstates and flavor eigenstates are different.

That second point is key, though can be tricky to understand. It means that there are two different ways of representing the neutrino wave functions: as functions of flavor states, or mass states. Each flavor state is actually a mixture of the three mass states. For example, in an extreme case you might have:
$$|\nu_e\rangle = a_1 |m_1\rangle + a_2 |m_2\rangle + a_3 |m_3\rangle$$

Here ##|\nu_e\rangle## is the electron-neutrino flavor eigenstate, and ##|m_n\rangle## are the three mass eigenstates. The factors in front determine how much of the electron state is made up of each of the mass eigenstates.

The mass of each of the three mass states determines their frequency of oscillation. Over time, the three mass states will evolve at different rates, related to their masses. Even if one of their masses is zero, the other two will oscillate. So even if you start out with a neutrino in a pure flavor state, over time it will become a mixture of flavor states.
Understood. But if one of the masses is 0, but the other 2 oscillate, doesn't that produce the exponential loss I described, in which, over time, everything transitions into the 0-mass eigenstate (and therefore stops oscillating)?

Orodruin
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Understood. But if one of the masses is 0, but the other 2 oscillate, doesn't that produce the exponential loss I described, in which, over time, everything transitions into the 0-mass eigenstate (and therefore stops oscillating)?
No. The mass eigenstates do not oscillate, they acquire phases. The only thing that matters in the end is the acquired phase differences. Even if one phase is zero, it can still have a phase difference with the others. It is unclear to me why you think this would lead to an exponential loss.

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The flavour neutrino states do not have definite masses and therefore it makes no sense to talk about the "mass of an electron neutrino". Neutrino oscillations are an interference phenomena between the neutrino mass eigenstates that accumulate different phases depending on their mass.

I think you need to reexamine your understanding of neutrino oscillations. This is not how neutrino oscillations work (see kimbyd's post). The amplitude of oscillations depends on what admixture of the neutrino flavour states constitute the mass eigenstates and the masses of the mass eigenstates. Also, again, solar neutrinos do not oscillate in the true sense of the word. The flavour transitions are related to adiabatic flavour conversion.
Aha. This begins to be clearer. (Tricky business, this neutrino physics!)

Orodruin
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