Open Questions about Neutrinos Today

Merry Christmas!

Neutrinos are mysterious. I just blogged about some of the big open questions involving neutrinos:

• Neutrino puzzles.

In brief, nontechnical terms they are these:

• What is the correct theory of neutrinos?
• Why are they almost but not quite massless?
• Do all three known neutrinos—electron, muon, and tau—all have a mass?
• Are the two neutrinos with very close masses the light ones, or the heavy ones?
• Is any kind of neutrino its own antiparticle?
• Are there right-handed neutrinos: that is, neutrinos that spin counterclockwise along their direction of motion when moving at high speeds?
• Are there sterile neutrinos: that is, neutrinos that don't interact with other particles via the strong, weak or electromagnetic force?

In my blog post I go into some more details.

I'm looking for the best review articles on neutrino puzzles, to help update this physics FAQ:

• Open questions in physics.

Do you know some good review articles?

john baez said:

What is the correct theory of neutrinos?

I don't like this. Suppose you had the correct theory. How do you know that you do? is QED the "correct theory" if photons and electrons? How do you know that the next digit of g-2 won't falsify it?

john baez said:

Why are they almost but not quite massless?

Why stop with neutrinos? Only three particles have sensible masses - i.e. on the same scale as the Higgs vev - the W, the Z and the top quark.

john baez said:

Do all three known neutrinos—electron, muon, and tau—all have a mass?

I would say none of those are particles - i.e. the flavor eigenstates don't have plane wave solutions. Only the mass eigenstates are particles. But it is known that all three flavor eigenstates have non-zero mass expectation values. That's as good as you can hope for.

john baez said:

Are the two neutrinos with very close masses the light ones, or the heavy ones?

I think you already need to start thinking about what it would take to convince you. The ensemble of measurements favors normal ordering by about 2σ, or 95% CL. Is that enough? If not, what do you need? 3σ? 5σ? 4σ in one experiment?

john baez said:

Are there right-handed neutrinos: that is, neutrinos that spin counterclockwise along their direction of motion when moving at high speeds?

This is known, and the answer is "yes". But while you described helicity, I suspect you meant chirality, which is an open question.

john baez said:

Are there sterile neutrinos: that is, neutrinos that don't interact with other particles via the strong, weak or electromagnetic force?

I would argue that a fermion that feels the strong force is more properly called a quark than a neutrino. But anyway...

If you had a truly sterile neutrino, you not only couldn't detect it, you couldn't produce it either. We could discuss this, or we could discuss angels...pins...dancing...

What most people think of as sterile neutrinos are actually only semi-sterile. You introduce a sterile neutrino into the theory, it mixes with one or more active neutrinos, and now one neutrino has almost the same coupling strength as an active neutrino and the other has only an itty-bitty one.

Proponents would say nothing precludes us from adding a few sterile neutrinos into the theory. I might respond by saying I agree - I agree so much, I don't want to stop at two. Why not a million? Then I am off their Christmas card list.

Vanadium 50 said:

But it is known that all three flavor eigenstates have non-zero mass expectation values. That's as good as you can hope for.

"Is there a massless mass eigenstate?" is an open question that could be answered experimentally (at least in principle).

I'm not 100% sure it's possible to put together a one-massless and two-massive theory of neutrinos consistent with observation.

Someone proposed a model a while ago where this was an explicit prediction. Is there a deeper problem with just setting one mass (or mass-producing coupling) to zero? As far as I understand it wouldn't show up in neutrino mixing, and current sum of mass or direct measurements are not sensitive enough to rule out zero for the lightest state.

mfb said:

Is there a deeper problem with just setting one mass (or mass-producing coupling) to zero?

Every mixing derivation I have worked through uses six Weyl fields (or their equivalent). Such a model would have five. I would not want to say it is possible with 100% certainty without re-working it through under those assumptions.

One other thing: the oscillation goes as L/E. Or does it? Is it L/E or L/(gamma m)? Or maybe (L/gamma)/m? It's not obvious to me one can just set m = 0 in the final expression without more thought.

Vanadium 50 said:

Do all three known neutrinos—electron, muon, and tau—all have a mass?

I would say none of those are particles - i.e. the flavor eigenstates don't have plane wave solutions. Only the mass eigenstates are particles.

I meant the three mass eigenstates that most closely line up with the electron, muon, and tau flavor eigenstates. I need some nontechnical way to say what I mean. This is for ordinary folks, so words like "eigenstate" are not allowed. I guess the best way is something like this: "Is there a massless neutrino, or do they all have nonzero mass?"

I'm interested that you think maybe it's no longer possible to get a theory with one massless neutrino and two massive ones to fit the data. Back when I was paying attention it was theoretically possible.

Are the two neutrinos with very close masses the light ones, or the heavy ones?

I think you already need to start thinking about what it would take to convince you. The ensemble of measurements favors normal ordering by about 2σ, or 95% CL. Is that enough? If not, what do you need? 3σ? 5σ? 4σ in one experiment?

Okay. I haven't been keeping up with this stuff for the last decade or so. If experts feel pretty convinced that normal ordering is the right scenario, this question shouldn't be on the list!

Are there right-handed neutrinos: that is, neutrinos that spin counterclockwise along their direction of motion when moving at high speeds?

This is known, and the answer is "yes". But while you described helicity, I suspect you meant chirality, which is an open question.

I meant chirality, and I meant "in the limit as the speed approaches c". I don't know a great way to state this question to nonexperts without using words like "helicity" and "chirality", which I'd rather avoid.

Are there sterile neutrinos: that is, neutrinos that don't interact with other particles via the strong, weak or electromagnetic force?

If you had a truly sterile neutrino, you not only couldn't detect it, you couldn't produce it either. We could discuss this, or we could discuss angels...pins...dancing...

What most people think of as sterile neutrinos are actually only semi-sterile. You introduce a sterile neutrino into the theory, it mixes with one or more active neutrinos, and now one neutrino has almost the same coupling strength as an active neutrino and the other has only an itty-bitty one.

How does the sterile one mix with the others? Does it only couple to them via the Higgs or some other mass-generating mechanism? If so I think my description was not too bad.

Proponents would say nothing precludes us from adding a few sterile neutrinos into the theory. I might respond by saying I agree - I agree so much, I don't want to stop at two. Why not a million? Then I am off their Christmas card list.

As long as it's a question lots of physicists are asking, it can be on this list. But actually it could be better to ask a more general question, like "Are there extra neutrinos beyond the three known ones and their three possible right-handed counterparts?"

john baez said:

I'm interested that you think maybe it's no longer possible to get a theory with one massless neutrino and two massive ones to fit the data. Back when I was paying attention it was theoretically possible.

I'm not saying that I think it's possible. Just that I'd like to look fairly carefully at the assumptions made i the derivations before I felt one could put a zero in for a mass.

john baez said:

I meant the three mass eigenstates that most closely line up with the electron, muon, and tau flavor eigenstates.

Turns out there's no such thing!

Things are very different in the neutrino and quark sectors.

• ν₁ is about 2/3 ν_e and the remainder is about an even split of ν_μ and ν_τ.
• ν₂ is a roughly even mix of the three flavors.
• ν₃ is about an even split of ν_μ and ν_τ with a little ν_e sprinkled in.

You are free to write whatever you want on your blog of course, but keeping the idea alive that flavor eigenstates are "good enough" to discuss neutrinos overlooks one of the biggest and most surprising differences between neutrinos and quarks. It also obscures the fact that neutrinos don't oscillate. They interfere.

Why do neutrinos look so different than quarks? I'd say that is exceptionally poorly understood. There may not even be an answer beyond "it's got to be something". It's interesting that the CKM matrix gets less diagonal as you go lighter and the PMNS matrix is very non-diagonal. Is there some symmetry enforcing this? The "textures" gang would argue yes.

john baez said:

If experts feel pretty convinced that normal ordering is the right scenario, this question shouldn't be on the list!

I hardly consider myself an expert. To me, 2σ for an either-or proposition is fairly strong, but even so, it's one-in-twenty that it is wrong. But it's getting close. Certainly other people are becoming convinced. I would say at this point I personally wouldn't support any future experiment that could only measure the mass hierarchy. By the time it runs, we'll know.

john baez said:

I meant chirality

Then you probably shouldn't use helicity.

Again, it's your blog, but it would be good if you could use your considerable talents to find a way to explain chirality to your audience. "Wrong but easy to explain" is how we got into the "relativistic mass" mess. Tell me you aren't tired of explaining why really really fast objects don't turn into black holes.

john baez said:

How does the sterile one mix with the others? Does it only couple to them via the Higgs or some other mass-generating mechanism?

You know, I hadn't really thought about this. Certainly a Higgs Yukawa would do this, but I always kind of imagined that this was through some process with larger coupling at a higher mass scale: sneutrino or other SUSY loops or a see-saw.

Thanks! Especially thanks for emphasizing how "heavily blended" the flavor eigenstates are, as linear combinations of mass eigenstates.

By the way, I'm not talking about a blog, where it's easy to spend lots of time explaining stuff properly. I'm talking about the Physics FAQ, and I'm trying to update an ancient article on "open questions in physics", which is supposed to succinctly list a lot of open questions. I can't explain eigenstates, chirality vs. helicity, or any of that. If I did that for all the main open questions in physics, I'd need to write a book.

I can say "chirality" and let people scratch their heads, or I can eliminate the terminology altogether and replace it by something ordinary folks can understand. I believe

"Are there right-handed neutrinos: that is, in the limit as their speed approaches that of light, are there neutrinos that spin counterclockwise along their direction of motion?"

is okay. Even this "limit" stuff is probably going to confuse people. If anyone can think of something clearer to ordinary folks that's just as short, that would be great.

Dumb question, instead of "in the limit" etc can't you just go "near the speed of light" or similar?

Something like "that is, are there neutrinos that spin counter-clockwise along the direction of motion when traveling near the speed of light?"

Or is that too imprecise?

Vanadium 50 said:

It also obscures the fact that neutrinos don't oscillate. They interfere.

Interesting... Does that appear by squaring the sum with the PMNS coefficients?

john baez said:

I can say "chirality" and let people scratch their heads, or I can eliminate the terminology altogether and replace it by something ordinary folks can understand. I believe

I would like to see a way to describe chirality to a layman without making them scratch their heads, given that it's a very mathematical construct (in contrast to helicity).
Wouldn't it be enough to ask questions about them in terms of participating in interactions (for example with the [itex]W^\pm[/itex]-bosons)? E.g. we know that (anti)neutrinos don't interact with [itex]W^{-(+)}[/itex] (results from not having a right-handed singlet representation, as for example quarks do).

Vanadium 50 said:

One other thing: the oscillation goes as L/E. Or does it? Is it L/E or L/(gamma m)? Or maybe (L/gamma)/m? It's not obvious to me one can just set m = 0 in the final expression without more thought.

The neutrino oscillation formula often presented looks simple but is quite oversimplified. There's a long record of confusion in the literature about how neutrino oscillations are to be understood and reconciled with energy and momentum conservation.

The correct formal answer is, imho, to treat the entire experiment from the production (emission) of the neutrinos at the "near-side detector" to the measurement (absorption) of ht neutrinos at the "far-side detector", using wave packets. In terms of Feynman diagrams the neutrino line is then always and internal line, i.e., a neutrino propagator, and with this all the quibbles are resolved cleanly. See, e.g.,

https://arxiv.org/abs/0905.1903

Qualitatively the quibbles are due to the fact that you can interpret only asymptotic free mass eigenstates as "particles". To say it in an abstract way in QFT an elementary particle is defined as being described by an irreducible representation of the proper orthochronous Poincare group. The trouble with the neutrinos is that you cannot detect mass eigenstates but only flavor eigenstates which are superpositions of mass eigenstates. That's why we never observer neutrinos as particles (mass eigenstates) directly but only scattering events with other particles or by the other decay products of a ##\beta## decay, where it occurs as "missing energy and/or momentum" (which was the original starting point of the neutrino story by Pauli's hypothesis in 1930).

john baez said:

I can say "chirality" and let people scratch their heads, or I can eliminate the terminology altogether and replace it by something ordinary folks can understand. I believe

"Are there right-handed neutrinos: that is, in the limit as their speed approaches that of light, are there neutrinos that spin counterclockwise along their direction of motion?"

is okay. Even this "limit" stuff is probably going to confuse people. If anyone can think of something clearer to ordinary folks that's just as short, that would be great.

Well, this is a question I've also asked myself very often when trying to explain "chirality" in introductory lectures. For a theorist it's simple to formally say a state of chirality is described by a wave mode of the Dirac field which is an eigenstate of ##\gamma_5## with eigenvalues ##-1## ("left-handed") or ##+1## ("right-handed").

The trouble however is to have some intuition behind this, and I so far failed to come up with a completely right one. I've also no better idea than what you suggest, just referring to the massless (or ultrarelativistic) limit, using the fact that for massless Dirac particles chirality is the same as helicity, but strictly speaking that indeed only holds for massless particles exactly.

Much of the confusion about oscillation vs. interference comes about from thinking of the flavor states as the "real particles" that somehow don't have a well-defined mass.

If I say "I produce an electron neutrino", I really mean "I produce a particular admixture of mass eigenstates". If the three mass states were far apart - far enough apart that we could tell them apart - in mass, one would see three entries in the PDG:

• [itex]\pi \rightarrow \mu \nu_1[/itex]
• [itex]\pi \rightarrow \mu \nu_2[/itex]
• [itex]\pi \rightarrow \mu \nu_3[/itex]

with appropriate branching fractions. There would be no oscillations: you'd produce a ν₂ or whatever and that would be the end of it. Of course you could detect an electron in a beam of pion-produced neutrinos: ν₂ has a (fixed) probability of producing an electron.

However, for neutrinos in our universe, we cannot distinguish which mass eigenstate was in flight, so QM says we have to add amplitudes. And thus the neutrinos interfere.

This is exactly the same as in the double-slit experiment. Slits far apart and you have no interference. Slits close together so you can't tell which slit the light wave went through, and you do.

Now, you might say "This is all well and good with muon decay, where the Michel electron has a range of energies. But if I make my neutrinos via pion decay, it's a two-body decay. I make my muon beam monochromatic, and I measure the energy of the decay muon amd now I know the 4-momentum of the neutrino: contract it and I know the mass. It doesn't matter that this is impractical. It's possible, and that's all I need to add intensities and not amplitudes."

There are several reasons this line of reasoning doesn't work, all of them subtle. I think the easiest to understand is the relation E²-p²=m² is a statement about plane waves. But the pion and muon are not exactly plane waves. The neutrinos are produced in a decay pipe, so the muon and pion don't have plane-wave wavefunctions. They have particle-in-a-box wavefunctions. (Because they are particles in a box.) And while the deviation from E²-p²=m² is small because they are in a n equals a zillion state, it's still large enough to make the mass of the neutrino produced uncertain. And so you need to add amplitudes, not intensities.

So [itex]\pi \rightarrow \mu \nu \implies \nu + N \rightarrow N + e [/itex] is actually the interference of three processes: [itex]\pi \rightarrow \mu + \nu_1 \implies \nu_1 + N \rightarrow N + e [/itex], [itex]\pi \rightarrow \mu + \nu_2 \implies \nu_2 + N \rightarrow N + e [/itex] and [itex]\pi \rightarrow \mu + \nu_3 \implies \nu_3 + N \rightarrow N + e [/itex]. The process appears to evolve with time just as a two-slit interference pattern appears to evolve in space.

PS In a decay pipe, the particle isn't even in an energy eigenstate. It's in a mixture. But all have quantum numbers in the zillions.

Indeed you cannot measure mass eigenstates, because what's detected is due to the weak interactions and thus you detect flavor eigenstates. The same holds for the creation of the neutrinos, which is also in terms of flavor eigenstates. Thus you have from the beginning a superposition of mass eigenstates. When the neutrino is then traveling in free space, through the quantum mechanical time evolution you get relative phases between the mass eigenstates and thus also a superposition in terms of flavor eigenstates at the far-side detector and thus you measure with the corresponding probability depending on the path length one of the flavor eigenstates. That's what's called "neutrino oscillation".

That's of course the somewhat oversimplified explanation. If you want to get it right you need QFT and wave packets as described in #13 and the quoted paper by Akhmedov et al.

vanhees71 said:

Well, this is a question I've also asked myself very often when trying to explain "chirality" in introductory lectures. For a theorist it's simple to formally say a state of chirality is described by a wave mode of the Dirac field which is an eigenstate of ##\gamma_5## with eigenvalues ##-1## ("left-handed") or ##+1## ("right-handed").

The trouble however is to have some intuition behind this, and I so far failed to come up with a completely right one. I've also no better idea than what you suggest, just referring to the massless (or ultrarelativistic) limit, using the fact that for massless Dirac particles chirality is the same as helicity, but strictly speaking that indeed only holds for massless particles exactly.

Okay, so I'm not missing anything obvious! I actually think it's pretty good to take advantage of the fact that chirality reduces to helicity in the ultrarelativistic limit, since everyone can imagine a neutrino zipping along near the speed of light spinning clockwise along its axis of motion. Also, it's easy to convince people that the concept of "spinning clockwise along its axis of motion" isn't Lorentz-invarant except for particles moving at the speed of light: if it's moving slower than light, just outrun the particle and now look at it.

All this "gut-level intuition" is no substitute for talking about eigenvectors of ##\gamma_5## - but also, all that stuff about ##\gamma_5## is no substitute for these gut-level intuitions. Ideally we eventually get both, and then we can play them off against each other to figure stuff out efficiently.

Vanadium 50 said:

Much of the confusion about oscillation vs. interference comes about from thinking of the flavor states as the "real particles" that somehow don't have a well-defined mass.

If I say "I produce an electron neutrino", I really mean "I produce a particular admixture of mass eigenstates". If the three mass states were far apart - far apart that we could tell them apart - in mass, one would see three entries in the PDG:

• π→μν1
• π→μν2
• π→μν3

with appropriate branching fractions. There would be no oscillations: you'd produce a ν₂ or whatever and that would be the end of it. Of course you could detect an electron in a beam of pion-produced neutrinos: ν₂ has a (fixed) probability of producing an electron.

However, for neutrinos in our universe, we cannot distinguish which mass eigenstate was in flight, so QM says we have to add amplitudes. And thus the neutrinos interfere.

This is exactly the same as in the double-slit experiment. Slits far apart and you have no interference. Slits close together so you can't tell which slit the light wave went through, and you do.

It is something unclear to me.
Neutrinos ν1,2,3 are basic states. Electron, muon, and tau are also basic states and they are flavour states. Here is some inconsistency (or not a perfect symmetry.), ν1,2,3 are not flavour states, but e, μ and τ are. Is it true what I wrote? Maybe this inconsistency is not a problem or ..?
Maybe this is a naive question.

(There are some problems with editor, now I cannot edit inline formulae.)

Does a newly created neutrino have a specified mass?
Consider a beta decay event.
The mother nucleus may have a long half-life. Therefore its momentum can, in principle, be measured with a great precision.
The daughter nucleus may be stable, in which case its momentum can be measured with unlimited precision.
And electron is stable (ditto).
There are certain practical issues with reaching the above Heisenberg precision. But if you followed a number of beta decay events with such precision, what would be your prediction for neutrino rest mass? Same in all beta decay events, or different in different beta decay events of same nucleus, with a continuum distribution?

Lord Crc said:

Dumb question, instead of "in the limit" etc can't you just go "near the speed of light" or similar?

Something like "that is, are there neutrinos that spin counter-clockwise along the direction of motion when traveling near the speed of light?"

Or is that too imprecise?

It all depends on how nitpicky people are. A grumpy pedant will point out that however close to the speed of light it moves, it's possible for a left-handed particle to be observed spinning counter-clockwise along its direction of motion. But I should probably not aim to please the grumpy pedants too much, since the more I do that, the less the intended audience of the Physics FAQ will understand what I'm saying!

snorkack said:

Does a newly created neutrino have a specified mass?

No.

Consider a beta decay event.
The mother nucleus may have a long half-life. Therefore its momentum can, in principle, be measured with a great precision.
The daughter nucleus may be stable, in which case its momentum can be measured with unlimited precision.
And electron is stable (ditto).
There are certain practical issues with reaching the above Heisenberg precision. But if you followed a number of beta decay events with such precision, what would be your prediction for neutrino rest mass? Same in all beta decay events, or different in different beta decay events of same nucleus, with a continuum distribution?

That's a fun puzzle! Here are some facts that may help you solve it:

• We shouldn't mix up energy and mass. ##E = mc^2## is only true for particles at rest, which is why another name for mass is "rest mass". As far as we can tell, neutrinos have only 3 possible values for their mass. Given this, it's impossible to have a neutrino with a continuum distribution of mass. But it can have a continuum distribution for energy.
• A related point: energy is conserved, but the total mass of all particles in a system is not conserved. So when nucleus X decays into nucleus Y, an electron and an antineutrino, you can't measure the mass of the antineutrino by measuring the mass of nucleus X and subtracting the mass of nucleus Y and the electron.
• Here's an issue vaguely analogous to yours. When a spin-zero particle decays into two photons, the angular momentum of each individual photon is uncertain - each one separately is not in an angular momentum eigenstate. But because angular momentum is conserved, when you measure the angular momentum of one of these photons and get a definite value, you also know the angular momentum of the other (as long as neither have interacted with anything else in the meantime). This works because angular momentum is conserved. So it also works with energy - but not mass.

john baez said:

But I should probably not aim to please the grumpy pedants too much, since the more I do that, the less the intended audience of the Physics FAQ will understand what I'm saying!

Matt Strassler IIRC used colors to indicate when he hand-waved over some important math/physics details in some of his blog posts about the standard model. I thought that worked well, as it gave some idea where there might be dragons while keeping it accessible. Perhaps something similar could work.

In any case thanks for the work, I've found the Physics FAQ useful many times.

john baez said:

That's a fun puzzle! Here are some facts that may help you solve it:

• We shouldn't mix up energy and mass. ##E = mc^2## is only true for particles at rest, which is why another name for mass is "rest mass". As far as we can tell, neutrinos have only 3 possible values for their mass. Given this, it's impossible to have a neutrino with a continuum distribution of mass. But it can have a continuum distribution for energy.
• A related point: energy is conserved, but the total mass of all particles in a system is not conserved. So when nucleus X decays into nucleus Y, an electron and an antineutrino, you can't measure the mass of the antineutrino by measuring the mass of nucleus X and subtracting the mass of nucleus Y and the electron.

Not by simply substracting mass.
The approach would be:

1. Measure the rest masses of, say, triton, He-3 and electron with great precision
2. Measure precise momentum and therefore also energy of triton before, and He-3 and electron after, a number of beta decay events
3. Use the relationship m²c⁴=E²-p²c², to measure the rest mass of neutrino in eac event.

Supposing you manage to diminish experimental errors to the extent that the individual neutrino rest masses are measured with greater precision than the difference between different neutrino rest mass eigenstates, what is your prediction for the measurement outcome?

1. All neutrinos turn out to have been emitted in same eigenstate of flavour (electron!) and same eigenstate of rest mass?
2. All neutrinos are emitted in same eigenstate of flavour, but they have 3 discrete options for rest mass, being emitted each in a single but different eigenstate?
3. Each neutrino, being emitted in the electron eigenstate of flavour, is emitted in a mixture of mass eigenstates, and mass measurement at that point will produce a continuum of results, mixture of mass eigenstates but not bound to match any single value of them?

john baez said:

Okay, so I'm not missing anything obvious! I actually think it's pretty good to take advantage of the fact that chirality reduces to helicity in the ultrarelativistic limit, since everyone can imagine a neutrino zipping along near the speed of light spinning clockwise along its axis of motion. Also, it's easy to convince people that the concept of "spinning clockwise along its axis of motion" isn't Lorentz-invarant except for particles moving at the speed of light: if it's moving slower than light, just outrun the particle and now look at it.

All this "gut-level intuition" is no substitute for talking about eigenvectors of ##\gamma_5## - but also, all that stuff about ##\gamma_5## is no substitute for these gut-level intuitions. Ideally we eventually get both, and then we can play them off against each other to figure stuff out efficiently.

That's indeed also my way out of this dilemma. I've no better idea yet.

exponent137 said:

It is something unclear to me.
Neutrinos ν1,2,3 are basic states. Electron, muon, and tau are also basic states and they are flavour states. Here is some inconsistency (or not a perfect symmetry.), ν1,2,3 are not flavour states, but e, μ and τ are. Is it true what I wrote? Maybe this inconsistency is not a problem or ..?
Maybe this is a naive question.

(There are some problems with editor, now I cannot edit inline formulae.)

For formulae use LaTeX. Just click the "LaTeX Guide" on the left below the input field:

https://www.physicsforums.com/help/latexhelp/

It's pretty much analogous to the case of the quarks in the electroweak sector of the standard model. Also there by convention when introducing the mixing matrix (in this case the CKM=Cabibbo-Kobayashi-Maskawa matrix) you keep the up-like quarks (u,c,t) as mass-eigenstates but "mix" the down-like quarks (d,s,b). That's possible by using all the freedom to define the various physically unimportant phases of the fields.

For the neutrinos the mixing matrix is named PMNS=Pontekorvo-Maki-Nakagawa-Sakati matrix.

https://en.wikipedia.org/wiki/Cabibbo–Kobayashi–Maskawa_matrix
https://en.wikipedia.org/wiki/Pontecorvo–Maki–Nakagawa–Sakata_matrix

snorkack said:

Not by simply substracting mass.
The approach would be:

1. Measure the rest masses of, say, triton, He-3 and electron with great precision
2. Measure precise momentum and therefore also energy of triton before, and He-3 and electron after, a number of beta decay events
3. Use the relationship m²c⁴=E²-p²c², to measure the rest mass of neutrino in eac event.

Supposing you manage to diminish experimental errors to the extent that the individual neutrino rest masses are measured with greater precision than the difference between different neutrino rest mass eigenstates, what is your prediction for the measurement outcome?

1. All neutrinos turn out to have been emitted in same eigenstate of flavour (electron!) and same eigenstate of rest mass?
2. All neutrinos are emitted in same eigenstate of flavour, but they have 3 discrete options for rest mass, being emitted each in a single but different eigenstate?
3. Each neutrino, being emitted in the electron eigenstate of flavour, is emitted in a mixture of mass eigenstates, and mass measurement at that point will produce a continuum of results, mixture of mass eigenstates but not bound to match any single value of them?

Am I misunderstanding something? This is a typical approach taken to measure the neutrino [effective in case of degeneracy/low resolution] mass in beta-decay experiments (e.g. by KATRIN) through the Kurie's plot. The idea is still taken from the conservation of energy momentum, and some of the energy of the rested tritium must be attributed to become the neutrino mass (thus you'd observe an earlier cut-off to the electron's momentum). In principle, if you had super-resolution, you'd be able to see the effect of the different neutrino masses kicking in (see e.g. sl12 here Giuliani_PIC_2005.pdf (infn.it) ).
Thing is the electron gets produced with an electron neutrino, and the electron neutrino is decomposed into the 3 mass eigenstates, in a similar way the electrons that are produced from a decay of a particle will have spin up or down.

snorkack said:

Does a newly created neutrino have a specified mass?
Consider a beta decay event.
The mother nucleus may have a long half-life. Therefore its momentum can, in principle, be measured with a great precision.
The daughter nucleus may be stable, in which case its momentum can be measured with unlimited precision.
And electron is stable (ditto).
There are certain practical issues with reaching the above Heisenberg precision. But if you followed a number of beta decay events with such precision, what would be your prediction for neutrino rest mass? Same in all beta decay events, or different in different beta decay events of same nucleus, with a continuum distribution?

A newly created "neutrino" has not a specified mass but a specified flavor, which is a superposition of the mass eigenstates. For ##\beta## decay you have ##\text{n} \rightarrow p+\text{e}^-+\overline{\nu}_{\text{e}}##, i.e., the emitted anti-neutrino is an anti-electron neutrino.

Concerning energy-momentum conservation you have to use the mass eigenstates and you see that you get a superposition of entangled proton-electron-antineurtrino-mass-eigenstates. To really understand the long-baseline experiments for neutrino oscillations within quantum theory you need to use wave packets for the initial and final states and take the production process of the neutrinos at the near-side detector and the annihilation process of the neutrinos in the far-side detector into account. Then the neutrino line is an internal line, and there is no more any problem with the fact that neutrinos are produced in flavor eigenstates (which cannot be interpreted as particles) and are thus a superposition of mass eigenstates (which can in principle be interpreted as particles but which are not detectable, because only flavor eigenstates can be absorbed by the detector). There's also no problem with energy-momentum conservation.

ChrisVer said:

Am I misunderstanding something? This is a typical approach taken to measure the neutrino [effective in case of degeneracy/low resolution] mass in beta-decay experiments (e.g. by KATRIN) through the Kurie's plot. The idea is still taken from the conservation of energy momentum, and some of the energy of the rested tritium must be attributed to become the neutrino mass (thus you'd observe an earlier cut-off to the electron's momentum). In principle, if you had super-resolution, you'd be able to see the effect of the different neutrino masses kicking in (see e.g. sl12 here Giuliani_PIC_2005.pdf (infn.it) ).
Thing is the electron gets produced with an electron neutrino, and the electron neutrino is decomposed into the 3 mass eigenstates, in a similar way the electrons that are produced from a decay of a particle will have spin up or down.

So to answer the question in #23,

snorkack said:

1. All neutrinos turn out to have been emitted in same eigenstate of flavour (electron!) and same eigenstate of rest mass?
2. All neutrinos are emitted in same eigenstate of flavour, but they have 3 discrete options for rest mass, being emitted each in a single but different eigenstate?
3. Each neutrino, being emitted in the electron eigenstate of flavour, is emitted in a mixture of mass eigenstates, and mass measurement at that point will produce a continuum of results, mixture of mass eigenstates but not bound to match any single value of them?

it's item 3, as explained on Slide 6 of the above quoted talk.

vanhees71 said:

Then the neutrino line is an internal line, and there is no more any problem with the fact that neutrinos are produced in flavor eigenstates (which cannot be interpreted as particles) and are thus a superposition of mass eigenstates (which can in principle be interpreted as particles but which are not detectable, because only flavor eigenstates can be absorbed by the detector). There's also no problem with energy-momentum conservation.

In principle, you could have neutrino detection with such precision that you can simultaneously measure flavour eigenstate and rest mass eigenstate.

snorkack said:

In principle, you could have neutrino detection with such precision that you can simultaneously measure flavour eigenstate and rest mass eigenstate.

That exact same argument suggests you can measuree Lx and Lz simultaneously. Youyr argument is with quantum mechanics, not particle physics.

The answer to this "dilemma" (surprised nobody called it a "paradox" yet) is in Message #9.

A. You need to use an apparatus large enough. The scale is usually measured in kilometers.
B. Even if you could do this, you would see three decays:

• [itex]X \rightarrow Y + e^- + \overline{\nu}_1[/itex]
• [itex]X \rightarrow Y + e^- + \overline{\nu}_2[/itex]
• [itex]X \rightarrow Y + e^- + \overline{\nu}_3[/itex]

You would not somehow "find the mass of the flavor eigenstate".

Vanadium 50 said:

That exact same argument suggests you can measuree Lx and Lz simultaneously. Youyr argument is with quantum mechanics, not particle physics.

The answer to this "dilemma" (surprised nobody called it a "paradox" yet) is in Message #9.

A. You need to use an apparatus large enough. The scale is usually measured in kilometers.
B. Even if you could do this, you would see three decays:

• [itex]X \rightarrow Y + e^- + \overline{\nu}_1[/itex]
• [itex]X \rightarrow Y + e^- + \overline{\nu}_2[/itex]
• [itex]X \rightarrow Y + e^- + \overline{\nu}_3[/itex]

You would not somehow "find the mass of the flavor eigenstate".

I think what's measured as particles are indeed the "other" particles rather than neutrinos as "particles". You cannot measure neutrinos as a "particle", because there's no interaction that projects to the mass eigenstates but only such that project to flavor eigenstates which are not mass eigenstate

Take the "neutrino-mass" determination with, e.g., the strategy to measure the endpoint of the electron-energy spectrum very accurately (a la KATRIN) by measuring the electron energy spectrum in tritium decay, i.e., ##\text{t} \rightarrow ^3\text{He}+\text{e}^-+\bar{\nu}_{\text{e}}##. Let's make it idealized considering fully accurate measurements of arbitrary precision. So what's measured? You take an ensemble of tritium nuclei at rest and measure precisely the energy of the electron from the ##\beta## decay of each triton. Of course you don't measure the (anti-)neutrino but only the energy of the electron. Since the decay is not into a mass eigenstate of the anti-neutrino but in the electron-flavor eigenstate (which is established by measuring definitely an electron in the ##\beta## decay). Now the energy of the electron depends on the mass of the neutrino. So when measuring the endpoint of the electrons' energy spectrum you don't get a definite value but a distribution of end-point energies due to the superposition of neutrino-mass eigenstates. So what you'll measure is an average of the neutrino masses weighted with the corresponding PMNS matrix elements (modulus squared).

vanhees71 said:

I think what's measured as particles are indeed the "other" particles rather than neutrinos as "particles". You cannot measure neutrinos as a "particle", because there's no interaction that projects to the mass eigenstates but only such that project to flavor eigenstates which are not mass eigenstate

Take the "neutrino-mass" determination with, e.g., the strategy to measure the endpoint of the electron-energy spectrum very accurately (a la KATRIN) by measuring the electron energy spectrum in tritium decay, i.e., ##\text{t} \rightarrow ^3\text{He}+\text{e}^-+\bar{\nu}_{\text{e}}##. Let's make it idealized considering fully accurate measurements of arbitrary precision. So what's measured? You take an ensemble of tritium nuclei at rest and measure precisely the energy of the electron from the ##\beta## decay of each triton. Of course you don't measure the (anti-)neutrino but only the energy of the electron.

Does not follow.
If you don´ t measure the antineutrino, but are otherwise free to make measurements to arbitrary precision, you can also measure the recoil of He-3, and the angle between electron and He-3 recoil. Which combined would give you the specific momentum and energy of antineutrino.

vanhees71 said:

Since the decay is not into a mass eigenstate of the anti-neutrino but in the electron-flavor eigenstate (which is established by measuring definitely an electron in the ##\beta## decay). Now the energy of the electron depends on the mass of the neutrino. So when measuring the endpoint of the electrons' energy spectrum you don't get a definite value but a distribution of end-point energies due to the superposition of neutrino-mass eigenstates.

But if you measure the momenta of both electron and the nucleus recoil, you get the energy, momentum and rest mass of antineutrino for each decay event - near endpoint (antineutrino energy is small) or far from endpoint.

vanhees71 said:

So what you'll measure is an average of the neutrino masses weighted with the corresponding PMNS matrix elements (modulus squared).

But if you measure momenta of both electron and nucleus, what would you get for neutrino rest mass?
A spectrum where all neutrinos fit, within measurement precision, to one of the three mass eigenvalues, and the three decay paths have a specific branching factor?

snorkack said:

Does not follow.
If you don´ t measure the antineutrino, but are otherwise free to make measurements to arbitrary precision, you can also measure the recoil of He-3, and the angle between electron and He-3 recoil. Which combined would give you the specific momentum and energy of antineutrino.

But if you measure the momenta of both electron and the nucleus recoil, you get the energy, momentum and rest mass of antineutrino for each decay event - near endpoint (antineutrino energy is small) or far from endpoint.

But if you measure momenta of both electron and nucleus, what would you get for neutrino rest mass?
A spectrum where all neutrinos fit, within measurement precision, to one of the three mass eigenvalues, and the three decay paths have a specific branching factor?

Yes, and with this measurement you'd project to a corresponding neutrino-mass eigenstate, but the state you are measuring is not a neutrino-mass eigenstate. That's why you won't find the same neutrino mass in each measurement but with some probability, given by the corresponding modulus squared PMNS matrix elements, one of the three neutrino masses.

First, it is absolutely not true that neutrinos only interact with matter in flavor eigenstates. There are neutral current events: ν + X → ν + X.

Second, the term "flavor eigenstate" is confusing people. It may be helpful to think "flavor projection of the mass eigenstate" instead.

If you had mass resolution that was good enough, the decay [itex]X \rightarrow Y + e^+ + \nu_1[/itex] would occur with twice the rate as [itex]X \rightarrow Y + e^+ + \nu_2[/itex] (I am going to use positron emission as an example so I don't have a zillion overlines to include.)

If I had a beam of pure [itex]\nu_1[/itex], it would have twice the reaction cross-section on a target of Y as [itex]\nu_2[/itex]. (The inverse process)

If I get a beam of neutrinos from X decay and use them to induce the inverse process on a target of Y, the rate is 5/9 in appropriate units.

Now, if I cannot tell whether I have a [itex]\nu_1[/itex] or [itex]\nu_2[/itex] in flight, the two states interfere (which some people call "oscillate") and the strength varies (with L/E) between 1 and 1/9.

The Particle Data Group has five review articles on neutrinos.