# How does conservation of energy/mass apply to neutrinos?

## Main Question or Discussion Point

I’m thinking as I go, here, so I am numbering points for ease of reference, correction etc.

1. Neutrinos (excluding the anti- and sterile varieties) come in three flavours.
2. It is known that neutrinos have mass.
3. The masses of individual neutrinos are not known precisely.
4. Cosmological studies indicate that the combined mass of all three flavours is not less than 0.5 eV.
5. Also, the heaviest neutrino flavour cannot be less massive than about 0.05 eV.
6. It seems there is a considerable difference between the masses, with a very slight possibility (mathematically?) that the lightest could be massless.
7. The flavours in order of ascending mass are: electron-, muon- and tau-neutrinos.
8. It appears that as they travel through space, neutrinos mutate between flavours.

Now for the question! When the mutation sequence is tau > muon > electron, where does the mass/energy go? Similarly, where does the energy come from when the sequence is reversed?

My guess is that vacuum energy might come in here somewhere, but that would raise questions about the extent to which vacuum energy plays a part in the conservation of energy generally.

## Answers and Replies

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jtbell
Mentor
7. The flavours in order of ascending mass are: electron-, muon- and tau-neutrinos.
The flavors don't have definite masses. The neutrino "flavor states" are each superpositions (linear combinations) of the three "mass states." If you could measure the mass of an electron neutrino, you would get one of three values at random, according to their "weights" in the superposition.

mfb
Mentor
The composition of mass eigenstates do not change, so you have a conserved superposition of them.

By the way: Most of the energy of neutrinos in the universe is kinetic energy, and mass is not a conserved quantity.

mass is not a conserved quantity.
Is that because there can be an exchange between mass and energy?

jtbell said:
If you could measure the mass of an electron neutrino, you would get one of three values at random, according to their "weights" in the superposition.
Does this mean that the assigning of masses to flavours of neutrinos is an oversimplification for the benefit of non-scientists like myself?

This question is probably too “classical” even to ask. If, for example, a neutrino left its source as a tau- and was later detected as an electron-, would its mass be that of a tau-?

If that is the case, in what sense could it be said to be an electron-?

mfb
Mentor
Is that because there can be an exchange between mass and energy?
No, but you can create and annihilate particles, and change the sum of masses of all involved particles. A simple example is the annihilation of electron+positron to two photons: Two particles with 511keV each annihilate and produce two particles without mass. The total energy is conserved, mass is not.

Does this mean that the assigning of masses to flavours of neutrinos is an oversimplification for the benefit of non-scientists like myself?
If the flavour eigenstates would be mass eigenstates, we would not have any neutrino mixing. Mixing indicates that the flavour eigenstates are superpositions of the mass eigenstates (and the mass eigenstates are superpositions of the flavour eigenstates).
Flavour measurements are different from mass measurements.

Bill_K
Does this mean that the assigning of masses to flavours of neutrinos is an oversimplification for the benefit of non-scientists like myself?
The Wikipedia article on neutrino masses is careful to call them mass eigenstates 1, 2 and 3.

A recent review paper goes further, writing the three flavor eigenstates as a unitary matrix U times the mass eigenstates, and saying, "ν1 contains mostly νe, while ν3 contains mainly νμ and ντ with very little νe. The state ν2 contains roughly equal amounts of νe, νμ and ντ."

Later on, "Tritium beta decay end point experiments measure the “electron neutrino mass” defined by m(νe) ≡ √(∑ |Uei|2mi2)."

If, for example, a neutrino left its source as a tau- and was later detected as an electron-, would its mass be that of a tau-?
Not necessarily!

Please could someone give an explanation of eigenstates that a non-mathematician might be able to understand? :)

jtbell
Mentor
An "x eigenstate" or "eigenstate of (property) x" is a state in which x has a definite value, that is, there is no uncertainty. An energy eigenstate has a definite precise energy, a mass eigenstate has a definite precise mass, etc.

Thanks, jtbell. It's really that straightforward??

All I need now is a translation of Bill K's "m(νe) ≡ √(∑ |Uei|2mi2)" and I'll feel as though I'm getting somewhere. I can cope with the left hand side of the equation. :)

Based on responses I have received in this thread, and bits of information from elsewhere, I have come up with some thoughts about neutrinos. Before trying to go any further I would like to share these thoughts with a view to further comments and corrections.

First possibility:

1. A neutrino is a neutrino… is a neutrino… is a neutrino… The flavours do not represent different types of neutrino.

2. The neutrino flavours indicate different relationships between the particulate (massive) facet of the neutrino and its associated wave package.

3. Most of the energy associated with a neutrino is kinetic energy; it can, therefore, be ignored when considering the actual mass/energy of the neutrino.

4. If one considers the particulate facet of the neutrino as containing the mass, and the associated wave as containing the energy; changes of flavour simply signify changes in mass/energy proportions.

5. A possible drawback to this model is that a less massive flavour will necessarily have greater energy than will a more massive flavour. Is this really a problem?
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Alternative possibility:

1. As above.

2. In order to mutate between flavours neutrinos must have mass.

3. Neither the mass of a (generic) neutrino, nor the specific masses of the neutrino flavours has been determined precisely.

4. There is only one neutrino mass.

5. Flavours are virtual particles which are not really particles at all. They are disturbances in the associated fields (e.g. the electromagnetic field) which are described in physics in terms of an exchange of virtual particles.

6. A possible drawback is that if the masses of neutrinos could be determined precisely, they would be invariable. This would seem to make nonsense of the whole idea of flavours.
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The inclusion of “possible drawbacks” indicates that I am not kidding myself that I have reached any conclusions.

Bill_K
ALT 2 is True. All the rest are False!
Well, each of the 3's is half True.

Thanks Bill K.

On the principle that we learn from our mistakes, that's not a bad outcome. :)
Of course, I would need to know why they are false in order to learn much.

mfb
Mentor
Instead of working from your wrong concepts, try to learn about the standard model first.
There is no single point where you can say "that is wrong" - even assumptions which are necessary to make your statements might be wrong (but it is hard to tell).

Bill_K
Endervhar, I second mfb's comment. The neutrino system has some subtleties that can be difficult to grasp at first. But your comments indicate that it is not really these which you are puzzled about, rather some of the more basic underlying ideas. And so for us to try to explain neutrinos will not help you that much, and may even lead to further confusion. I suggest you back up and get a better footing first on some simpler examples.

Thanks mfb & Bill K. Of course your comments make perfect sense. If I were 50 or so years younger I would go back to basics, give myself a good grounding in maths and physics and, hopefully, be able better to cope with things like the Standard Model. However, at 72, with probably little more than a half-hour or so a day available, I need to take every available short cut. There are so many things I want to know and understand. Wishful thinking?

It was not my intention to delve into the complexities of neutrinos; it was conservation of energy I was really looking at, but as so often happens, one thing leads to another.

If I’m going to have any chance of following up on the neutrino questions, would you have any suggestions as to the best place for a non-scientist to start?

mfb
Mentor
I need to take every available short cut.
There is no free lunch, and a reason why a master degree (or similar) in physics takes about 5 years.
A shorter version has to use some simplified models - they can be interesting, but they all have their limits.

I'm not looking for a free lunch, unless seeking help in my efforts to understand falls into that category.