# Superposition of neutrinos with different masses

• Larry Pendarvis

#### Larry Pendarvis

<<Mentor note: This post and its answers were moved from this thread.>>

How do you reconcile a superposition of flavors, with different masses, with conservation of energy and of momentum? Say the rest masses of three flavors are known, and you want to measure the velocity of the neutrinos, and sometimes you get flavor #1, and sometimes flavor #2, and sometimes you get flavor #3. Conservation of energy will say one velocity, but conservation of momentum will require a different velocity. Regardless of what you are measuring, it would seem that you cannot reconcile conservation of energy with conservation of momentum.
I reckon we don't see such mixing in electrons and muons and taus because of the greater difference in rest mass, but that is only a matter of degree, not fundamental.

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How do you reconcile a superposition of flavors, with different masses, with conservation of energy and of momentum?

First of all, the flavours are not mass eigenstates and therefore do not have definite masses. Thus, you cannot talk about things such as "electron neutrino mass" without specifying exactly what you mean by this. Second, the mass differences between the mass eigenstates are so small that for many practical purposes, you can regard them as a superposition. If you really want to treat things in a strict fashion, you need to do QFT and describe the creation and destruction processes. However, it can be shown that for the applications that are available, the treatment of flavour states as a linear combination of mass eigenstates is fine.

Note that there is a priori nothing conceptually different from the quark sector here. Just as neutrino mass eigenstates are not perfectly aligned with the charged lepton mass eigenstates, the up quark mass eigenstates are not aligned with the down quark mass eigenstates. In the latter case, the mixing is given by the CKM matrix, which dictates how strongly different flavour combinations couple to the W. The big difference is that the quark masses have very distinct masses and the mass eigenstates very quickly are subjected to decoherence. Therefore, we see the quark mass eigenstates and these are what we usually call up, down, strange, charm, bottom, and top. If we did the corresponding thing in the neutrino sector, we would call the leptons electron, ##\nu_1##, muon, ##\nu_2##, tau, and ##\nu_3##.
Say the rest masses of three flavors are known, and you want to measure the velocity of the neutrinos, and sometimes you get flavor #1, and sometimes flavor #2, and sometimes you get flavor #3.

Again, neutrino flavours do not have definite masses. A priori, you could measure the different velocities of the neutrino mass eigenstates. However, the typical uncertainty in energy is larger than the typical difference between the energies of the different mass eigenstates. The error in the measured velocity would typically be large in comparison to the velocity differences.

I reckon we don't see such mixing in electrons and muons and taus because of the greater difference in rest mass, but that is only a matter of degree, not fundamental.

Mixing is not something that happens to neutrinos, it is something that happens to neutrinos in relation to the charged leptons. The word "neutrino mixing" is a bit misleading as it really should be "lepton mixing". In the same way, quark mixing is the mismatch between the mass eigenstates of the up and down type quarks. The reason we chose to call the mass eigenstates of the charged leptons as "flavour basis" is that it is very convenient to do so (charged lepton masses are very different and therefore will essentially never be causing interference - for possible exceptions to this essentially: doi:10.1088/1126-6708/2007/09/116).

• Larry Pendarvis
First of all, the flavours are not mass eigenstates and therefore do not have definite masses. Thus, you cannot talk about things such as "electron neutrino mass" without specifying exactly what you mean by this. Second, the mass differences between the mass eigenstates are so small that for many practical purposes, you can regard them as a superposition. If you really want to treat things in a strict fashion, you need to do QFT and describe the creation and destruction processes. However, it can be shown that for the applications that are available, the treatment of flavour states as a linear combination of mass eigenstates is fine.

Note that there is a priori nothing conceptually different from the quark sector here. Just as neutrino mass eigenstates are not perfectly aligned with the charged lepton mass eigenstates, the up quark mass eigenstates are not aligned with the down quark mass eigenstates. In the latter case, the mixing is given by the CKM matrix, which dictates how strongly different flavour combinations couple to the W. The big difference is that the quark masses have very distinct masses and the mass eigenstates very quickly are subjected to decoherence. Therefore, we see the quark mass eigenstates and these are what we usually call up, down, strange, charm, bottom, and top. If we did the corresponding thing in the neutrino sector, we would call the leptons electron, ##\nu_1##, muon, ##\nu_2##, tau, and ##\nu_3##.

Again, neutrino flavours do not have definite masses. A priori, you could measure the different velocities of the neutrino mass eigenstates. However, the typical uncertainty in energy is larger than the typical difference between the energies of the different mass eigenstates. The error in the measured velocity would typically be large in comparison to the velocity differences.

Mixing is not something that happens to neutrinos, it is something that happens to neutrinos in relation to the charged leptons. The word "neutrino mixing" is a bit misleading as it really should be "lepton mixing". In the same way, quark mixing is the mismatch between the mass eigenstates of the up and down type quarks. The reason we chose to call the mass eigenstates of the charged leptons as "flavour basis" is that it is very convenient to do so (charged lepton masses are very different and therefore will essentially never be causing interference - for possible exceptions to this essentially: doi:10.1088/1126-6708/2007/09/116).
Are attempts to measure the mass of the electron neutrino, the mass of the muon neutrino, and the mass of the tau neutrino, then, doomed to failure?

Are attempts to measure the mass of the electron neutrino, the mass of the muon neutrino, and the mass of the tau neutrino, then, doomed to failure?
No, people are generally very aware of what they are trying to measure. Generally, when people say "electron neutrino mass" in different experiments, this refers to an effective mass which is a combination of the mass eigenstate masses and the mixing parameters. The combination measured depends on the experiment, for example tritium beta decay experiments and neutrino less double beta decay experiments measure different combinations (and 0nubb only if neutrinos are Majorana). You simply have to be aware of what you are measuring and report accordingly.

No, people are generally very aware of what they are trying to measure. Generally, when people say "electron neutrino mass" in different experiments, this refers to an effective mass which is a combination of the mass eigenstate masses and the mixing parameters. The combination measured depends on the experiment, for example tritium beta decay experiments and neutrino less double beta decay experiments measure different combinations (and 0nubb only if neutrinos are Majorana). You simply have to be aware of what you are measuring and report accordingly.
Say you are aware of what you are measuring, and you measure the effective mass of whatever combination we call an electron neutrino, and you measure the effective mass of whatever combination we call the tau neutrino, and you want to measure the velocity of those neutrinos, and you are at some distance from where the neutrinos are generated so they change flavor or type or whatever the right word is that they change, then sometimes your measurement detects an electron neutrino, and sometimes your instruments detect a tau neutrino. Conservation of energy will say one velocity, but conservation of momentum will require a different velocity. Regardless of what you are measuring, it would seem that you cannot reconcile conservation of energy with conservation of momentum.
Granted the conservation discrepancy will be small, but isn't a small discrepancy what confirmed the existence of the neutrino in the first place?

Say you are aware of what you are measuring, and you measure the effective mass of whatever combination we call an electron neutrino, and you measure the effective mass of whatever combination we call the tau neutrino,

Fine so far.

and you want to measure the velocity of those neutrinos,

Wrong. The flavour eigenstates are not the propagation eigenstates. A flavour eigenstate can never have a well defined velocity, momentum, or energy.

Regardless of what you are measuring, it would seem that you cannot reconcile conservation of energy with conservation of momentum.

As I said in my first post, this view is too naïve. If you want to do things very properly you need to use QFT and your velocity measurement is not really a velocity measurement, the velocity is definable up to uncertainties in the creation and destruction processes and the QFT process is a combination of creation, propagation, and destruction. The masses of the neutrino mass eigenstates are so close together that interference between their different contributions to the full process occurs. Or you can just make the usual approximation (which is a very very good one) that they are plane waves and compute the oscillation probabilities, creation process, and destruction process separately based on this.

Granted the conservation discrepancy will be small, but isn't a small discrepancy what confirmed the existence of the neutrino in the first place?
The point is the discrepancy is so vanishingly small that inherent uncertainties in energy will wash out any chance of measuring the velocity. Technically, over astronomical distances, there could be a possibility of separating the wave packets of different mass eigenstates, but then you run into problems of having production processes that last during short enough periods so that you can make a distinction.

Fine so far.

Wrong. The flavour eigenstates are not the propagation eigenstates. A flavour eigenstate can never have a well defined velocity, momentum, or energy.

As I said in my first post, this view is too naïve. If you want to do things very properly you need to use QFT and your velocity measurement is not really a velocity measurement, the velocity is definable up to uncertainties in the creation and destruction processes and the QFT process is a combination of creation, propagation, and destruction. The masses of the neutrino mass eigenstates are so close together that interference between their different contributions to the full process occurs. Or you can just make the usual approximation (which is a very very good one) that they are plane waves and compute the oscillation probabilities, creation process, and destruction process separately based on this.

The point is the discrepancy is so vanishingly small that inherent uncertainties in energy will wash out any chance of measuring the velocity. Technically, over astronomical distances, there could be a possibility of separating the wave packets of different mass eigenstates, but then you run into problems of having production processes that last during short enough periods so that you can make a distinction.
When the first "neutrinos" from a supernova arrive here 3 hours before the first photons, a velocity could be measured, assuming the distance to the explosion is known and the magnitude of the photon delay from inside the star is known.

assuming the distance to the explosion is known and the magnitude of the photon delay from inside the star is known.

This only put an upper bound on the masses. And the crux is that such considerations inherently are dependent on the supernova process and in order to measure the velocities of the different mass eigenstates you would have to create the neutrinos in a very short time period. This is further complicated by the fact that the neutrinos are not monochromatic, but come in several different energies, which also creates a spread.

The problem is exactly the fact that the flavor and mass matrices don't commute. So if you measure an electron neutrino, you can't know at the same time its mass. This is pure QM...

This only put an upper bound on the masses. And the crux is that such considerations inherently are dependent on the supernova process and in order to measure the velocities of the different mass eigenstates you would have to create the neutrinos in a very short time period. This is further complicated by the fact that the neutrinos are not monochromatic, but come in several different energies, which also creates a spread.
Assuming there was a short-time process that generated one mass eigenstate, would it then be possible to measure and calculate the mass and the velocity of the first to arrive many light-years away? They would be the fastest (highest-energy) ones. Or would there still be some kind of Mixing that would spread it out?

You would get the mass of the lightest mass eigenstate. However, this includes knowing the total time of flight or a correlation with a light signal from the same event. Things then start to depend on (and introduce uncertainties based on) astrophysics. A priori it is a possible measurement, but it is questionable if it is ever going to be practical.

Say you are aware of what you are measuring, and you measure the effective mass of whatever combination we call an electron neutrino, and you measure the effective mass of whatever combination we call the tau neutrino, and you want to measure the velocity of those neutrinos, and you are at some distance from where the neutrinos are generated so they change flavor or type or whatever the right word is that they change, then sometimes your measurement detects an electron neutrino, and sometimes your instruments detect a tau neutrino. Conservation of energy will say one velocity, but conservation of momentum will require a different velocity. Regardless of what you are measuring, it would seem that you cannot reconcile conservation of energy with conservation of momentum.

To take a concrete example, consider the decay ##\pi^+ \rightarrow \mu^+ + \nu_\mu##. I think you have to consider the final state as a superposition of three possible outcomes, one for each neutrino mass state: (a) ##\mu^+ + \nu_1##, (b) ##\mu^+ + \nu_2##, and (c) ##\mu^+ + \nu_3##. These three outcomes have slightly different momenta and energies for the ##\mu^+## and the ##\nu##, corresponding to the different masses of the ##\nu_1##, ##\nu_2##, ##\nu_3##. In each outcome, both energy and momentum are conserved. If we could make sufficiently precise measurements of the momenta and energies of the ##\mu^+## and the ##\nu##, we could identify which neutrino mass state is involved, and thereby "collapse" the superposition; but that "if" is a huge hurdle.

• Orodruin
To take a concrete example, consider the decay ##\pi^+ \rightarrow \mu^+ + \nu_\mu##. I think you have to consider the final state as a superposition of three possible outcomes, one for each neutrino mass state: (a) ##\mu^+ + \nu_1##, (b) ##\mu^+ + \nu_2##, and (c) ##\mu^+ + \nu_3##. These three outcomes have slightly different momenta and energies for the ##\mu^+## and the ##\nu##, corresponding to the different masses of the ##\nu_1##, ##\nu_2##, ##\nu_3##. In each outcome, both energy and momentum are conserved. If we could make sufficiently precise measurements of the momenta and energies of the ##\mu^+## and the ##\nu##, we could identify which neutrino mass state is involved, and thereby "collapse" the superposition; but that "if" is a huge hurdle.

So when you look into a Feynman graph and want to determine the cross sections, is that achieved by plugging in the vertices the PMNS matrix elements?
In fact I find it a little complicated to determine which representation to choose for a Feynman graph...the flavor or the mass?

So when you look into a Feynman graph and want to determine the cross sections, is that what you achieve by plugging in the vertices the PMNS matrix elements?
In fact I find it a little complicated to determine which representation to choose for a Feynman graph...the flavor or the mass?

The point is that you are not going to be able to separate them unless you measure the neutrino state (which you typically will not). Technically the correct thing would be to do the rates of ##\pi \to \mu + \nu_i## for ##i = 1, 2, 3## and then add them up (there is no interference between them). In each of these there will appear a mixing matrix element, but when adding them to find the decay rate you have the sum of the squared absolute values of a full row of a unitary matrix, which is equal to one. Therefore, it is fine to just use one neutrino with coupling one (the correction is proportional to the neutrino masses).

To take a concrete example, consider the decay ##\pi^+ \rightarrow \mu^+ + \nu_\mu##. I think you have to consider the final state as a superposition of three possible outcomes, one for each neutrino mass state: (a) ##\mu^+ + \nu_1##, (b) ##\mu^+ + \nu_2##, and (c) ##\mu^+ + \nu_3##. These three outcomes have slightly different momenta and energies for the ##\mu^+## and the ##\nu##, corresponding to the different masses of the ##\nu_1##, ##\nu_2##, ##\nu_3##. In each outcome, both energy and momentum are conserved. If we could make sufficiently precise measurements of the momenta and energies of the ##\mu^+## and the ##\nu##, we could identify which neutrino mass state is involved, and thereby "collapse" the superposition; but that "if" is a huge hurdle.
Suppose we know that a power plant is producing only electron antineutrinos. When we detect them close to the point of generation, we do in fact detect only electron antineutrinos. But if we pull back some distance, is it not true that we can (theoretically) detect all three types? And is it not considered to be the case that some of the electron antineutrinos changed their type during the time of flight? In order for that theory to be correct, would it not involve a change from one rest mass to another? Our measurements do not show a superposition; the measurement itself "collapses" the superposition if it has not already "collapsed".

And is it not considered to be the case that some of the electron antineutrinos changed their type during the time of flight?

No. In quantum mechanics, if it's not measured it doesn't have a definite value (except arguably in the specific case of a system prepared in an eigenstate of the property we're talking about, and then only if that property commutes with the Hamiltonian and no other interaction happens - and this dubious and arguable exception doesn't apply here). If something doesn't have a definite value it's meaningless to talk about that value changing. Likewise, it's meaningless to talk about energy or momentum conservation being violated by quantities that have no definite value.

No. In quantum mechanics, if it's not measured it doesn't have a definite value (except arguably in the specific case of a system prepared in an eigenstate of the property we're talking about, and then only if that property commutes with the Hamiltonian and no other interaction happens - and this dubious and arguable exception doesn't apply here). If something doesn't have a definite value it's meaningless to talk about that value changing. Likewise, it's meaningless to talk about energy or momentum conservation being violated by quantities that have no definite value.
But if we detect only electron antineutrinos in a million experiments close to the reactor, can we not say that we do know what type they are at that location and thus at that amount of time since their generation? And then if we detect all three types at a greater distance, in a million experiments, would that not indicate that something has been going on in between?

But if we pull back some distance, is it not true that we can (theoretically) detect all three types?

This depends what you mean by "detect". The definition of a muon neutrino is that it produces a muon in charged current interactions. Reactor neutrinos have very low energies which are not large enough to produce muons (much less taus). As such, you cannot detect a (anti)neutrino from reactors as a mu or tau flavour. What you can detect is a defecit of electron neutrinos (this is known as a disappearance channel).

In beam produced neutrinos, energies are higher and you can produce muons or even taus (e.g., at the OPERA detector).
And is it not considered to be the case that some of the electron antineutrinos changed their type during the time of flight? In order for that theory to be correct, would it not involve a change from one rest mass to another?
No. As stated earlier, flavour eigenstates do not have definite masses. Neutrino oscillations are due to the interference of the contributions from the different mass eigenstates. At short distances, the interference phase is small and this means the linear combination of the mass eigenstates is the same as that created. Therefore it will interact in the same way in a charged current interaction. As the mass eigenstates propagate, they acquire different phases and the linear combination therefore changes

And then if we detect all three types at a greater distance, in a million experiments, would that not indicate that something has been going on in between?

Something is going on in between. The mass eigenstates are accumulating different phases.

This depends what you mean by "detect". The definition of a muon neutrino is that it produces a muon in charged current interactions. Reactor neutrinos have very low energies which are not large enough to produce muons (much less taus). As such, you cannot detect a (anti)neutrino from reactors as a mu or tau flavour. What you can detect is a defecit of electron neutrinos (this is known as a disappearance channel).

In beam produced neutrinos, energies are higher and you can produce muons or even taus (e.g., at the OPERA detector).

No. As stated earlier, flavour eigenstates do not have definite masses. Neutrino oscillations are due to the interference of the contributions from the different mass eigenstates. At short distances, the interference phase is small and this means the linear combination of the mass eigenstates is the same as that created. Therefore it will interact in the same way in a charged current interaction. As the mass eigenstates propagate, they acquire different phases and the linear combination therefore changes
So the answer to the original question, "What is the current upper limit on the mass of the most massive neutrino?" is that it is an invalid question, since there is no most massive neutrino; all three have indefinite masses, which are exactly and equally indefinite.
And we can discount all those popular-science "missing-neutrino" reports because no neutrinos were ever missing, there were only overspecialized detectors.
In fact, there is only ONE type of neutrino, not three. It is a superposition, and anyone who thinks he has found a different type has only found the same type a little later.

So the answer to the original question, "What is the current upper limit on the mass of the most massive neutrino?" is that it is an invalid question, since there is no most massive neutrino; all three have indefinite masses, which are exactly and equally indefinite.
The question is perfectly fine. What you have to realize is that the states with definite masses are the mass eigenstates and that these are different from the flavour eigenstates. The heaviest mass eigenstate is the the one with the largest mass and it is perfectly fine to ask what this mass is.

In fact, there is only ONE type of neutrino, not three. It is a superposition, and anyone who thinks he has found a different type has only found the same type a little later.
No, this is wrong. There most certainly are three types of neutrinos. Neutrino oscillations are an effect of how these interfere during weak interactions.

The definition of a muon neutrino is that it produces a muon in charged current interactions. Reactor neutrinos have very low energies which are not large enough to produce muons (much less taus)...
In beam produced neutrinos, energies are higher and you can produce muons or even taus (e.g., at the OPERA detector).
So are you saying that a muon neutrino is exactly the same as an electron neutrino, the only difference being that the muon neutrino has had its velocity increased? And at that velocity, there is no difference at all between a high-speed electron neutrino and a high-speed muon neutrino? But in a different inertial frame, those same neutrinos might have a too-low velocity; they can produce a muon only if the muon's frame is fast enough. So the definition of "muon neutrino" depends on the inertial frame, not on the neutrino itself.

The question is perfectly fine. What you have to realize is that the states with definite masses are the mass eigenstates and that these are different from the flavour eigenstates. The heaviest mass eigenstate is the the one with the largest mass and it is perfectly fine to ask what this mass is.
Yes. That is what I was talking about, mass eigenstates. What you said. So...

The question is perfectly fine. What you have to realize is that the states with definite masses are the mass eigenstates and that these are different from the flavour eigenstates. The heaviest mass eigenstate is the the one with the largest mass and it is perfectly fine to ask what this mass is.

No, this is wrong. There most certainly are three types of neutrinos. Neutrino oscillations are an effect of how these interfere during weak interactions.
And is it not considered to be the case that some of the electron antineutrino mass eigenstates changed their type during the time of flight? In order for that theory to be correct, would it not involve a change from one rest mass to another?

But if we detect only electron antineutrinos in a million experiments close to the reactor, can we not say that we do know what type they are at that location and thus at that amount of time since their generation? And then if we detect all three types at a greater distance, in a million experiments, would that not indicate that something has been going on in between?

If you look at the measurements and make the additional assumption that a property has a definite value even when it's not measured, then yes, you could draw that conclusion. But without that additional assumption, no, you cannot.

This thread has moved well beyond the original topic ("what is the upper bound on the mass?"). It's time to close it.

So are you saying that a muon neutrino is exactly the same as an electron neutrino,

No. They are different linear combinations of the mass eigenstates. The rest of that post is misconceptions based on this interpretation.

And is it not considered to be the case that some of the electron antineutrino mass eigenstates changed their type during the time of flight?

There are no electron antineutrino mass eigenstates. Flavour eigenstates are not mass eigenstates or vice versa, therefore you cannot talk about something being an electron (i.e., flavour) eigenstate at the same time as it is a mass eigenstate.

Currently we are getting nowhere and you are repeating the same mistakes over and over. I suggest you go back to read through our comments several times and think about the implications. Perhaps even go to a textbook on neutrino oscillations.