How does neutrino oscillation imply that neutrinos have mass?

[umagongdi]

Sorry if this qs been asked before, i couldn't find one similar.

 

[genneth]

Massless things travel at the speed of light, and so in their rest frames, no proper time ever passes. If proper time does not proceed, then no oscillations are possible!

 

[Meir Achuz]

Oscillation must go like sin[(E_1-E_2)t].
The only way that E_2 can differ from E_1 for the same momentum is if there is a mass.

 

[Bill_K]

Write down the Hamiltonian as a 2x2 matrix spanning the two states: electron neutrino and muon neutrino. The neutrino masses will be the eigenvalues of this matrix. The existence of neutrino oscillations implies that the two states mix, i.e. the off-diagonal terms of the matrix are nonzero. Consequently when you diagonalize it, the two eigenvalues will be unequal, implying that at least one of the eigenstates has a nonzero mass.

 

[JustinLevy]

The "proper time" argument and the other arguments seem incompatible with the following situation:

If one of the neutrino flavors is massless, and the other are massive... can there be oscillation between all the flavor pairs?

So which line of argument is correct? I assume the 'proper time' argument is the flawed one, but it is the most intuitive one. So is there something wrong with the proper time argument, and where is the implicit flaw? Is the issue that there is no inertial rest frame for massless objects and therefore proper-time (while well defined), is not an appropriate way to discuss their evolution?

 

[genneth]

The "proper time" argument and the other arguments seem incompatible with the following situation:

If one of the neutrino flavors is massless, and the other are massive... can there be oscillation between all the flavor pairs?

So which line of argument is correct? I assume the 'proper time' argument is the flawed one, but it is the most intuitive one. So is there something wrong with the proper time argument, and where is the implicit flaw? Is the issue that there is no inertial rest frame for massless objects and therefore proper-time (while well defined), is not an appropriate way to discuss their evolution?

I suspect the actual problem is the existence of different proper times for different masses, and what it means to oscillate between them. The "intuitive" argument, as you put it, technically shows that not all the neutrinos can be massless.

 

[Subluminal]

Massless things travel at the speed of light, and so in their rest frames, no proper time ever passes. If proper time does not proceed, then no oscillations are possible!

Does this model exclude a non-zero mass particle from achieving and maintaining the speed of light?

 

[PAllen]

Oscillation must go like sin[(E_1-E_2)t].
The only way that E_2 can differ from E_1 for the same momentum is if there is a mass.

Just curious, why must the two neutrino states differ in either energy, momentum, or mass? Suppose they are both massless, momentum and energy same, just some quantum number flips?

 

[Parlyne]

Just curious, why must the two neutrino states differ in either energy, momentum, or mass? Suppose they are both massless, momentum and energy same, just some quantum number flips?

Neutrino oscillation necessarily is a process that occurs during the free propagation between source and detector. As long as you're working in a basis where every state has a well defined mass (that is, the mass matrix is diagonal), there is nothing that can change any quantum numbers.

A little more technically, neutrinos are created in flavor states, which are generically superpositions of the mass eigenstates. This superposition, then, evolves under time evolution defined by the free Hamiltonian and is eventually detected. The detection method, of course, since it will involve charged leptons, is sensitive to flavor. So, the evolved state should be looked at in the flavor basis.

As an example, we can consider a toy model where there are only two states. Let's call the flavor states $|a>$ and $|b>$ and the mass states $|1>$ and $|2>$. And, just to be really silly about it, let's assume that the flavor states are given by
$$|a> = \frac{1}{\sqrt{2}}\left(|1>+|2>\right)$$
and
$$|b> = \frac{1}{\sqrt{2}}\left(|1>-|2>\right)$$

This is, of course, horribly unphysical; but, it will illustrate my point nicely.

If we assume the particle is created in the a state and allowed to evolve freely for time T, we will find that, after the evolution, the state looks like
$$\begin{array}{rcl} |a(T)> & = & e^{i\hat{H}T}\frac{1}{\sqrt{2}}\left(|1>+|2>\right)\\ & = & \frac{1}{\sqrt{2}}\left(e^{iE_1T}|1>+e^{iE_2T}|2>\right)\\ & = & \frac{1}{2}\left(\left(e^{iE_1T}+e^{iE_2T}\right)|a>+\left(e^{iE_1T}-e^{iE_2T}\right)|b>\right) \end{array}$$

If the two states have the same energy, the term proportional to $|b>$ vanishes, leaving the state changed only by an overall phase. And, of course, if they don't have the same energy, for most values of T, there's a non-zero probability of detecting the particle as a b, even though it was produced as an a.

The reason this is generally stated to require different masses is that the kinematics of production certainly won't allow two states in a superposition that have the same mass to have different energies.

 

[PAllen]

Neutrino oscillation necessarily is a process that occurs during the free propagation between source and detector. As long as you're working in a basis where every state has a well defined mass (that is, the mass matrix is diagonal), there is nothing that can change any quantum numbers.

A little more technically, neutrinos are created in flavor states, which are generically superpositions of the mass eigenstates. This superposition, then, evolves under time evolution defined by the free Hamiltonian and is eventually detected. The detection method, of course, since it will involve charged leptons, is sensitive to flavor. So, the evolved state should be looked at in the flavor basis.

As an example, we can consider a toy model where there are only two states. Let's call the flavor states $|a>$ and $|b>$ and the mass states $|1>$ and $|2>$. And, just to be really silly about it, let's assume that the flavor states are given by
$$|a> = \frac{1}{\sqrt{2}}\left(|1>+|2>\right)$$
and
$$|b> = \frac{1}{\sqrt{2}}\left(|1>-|2>\right)$$

This is, of course, horribly unphysical; but, it will illustrate my point nicely.

If we assume the particle is created in the a state and allowed to evolve freely for time T, we will find that, after the evolution, the state looks like
$$\begin{array}{rcl} |a(T)> & = & e^{i\hat{H}T}\frac{1}{\sqrt{2}}\left(|1>+|2>\right)\\ & = & \frac{1}{\sqrt{2}}\left(e^{iE_1T}|1>+e^{iE_2T}|2>\right)\\ & = & \frac{1}{2}\left(\left(e^{iE_1T}+e^{iE_2T}\right)|a>+\left(e^{iE_1T}-e^{iE_2T}\right)|b>\right) \end{array}$$

If the two states have the same energy, the term proportional to $|b>$ vanishes, leaving the state changed only by an overall phase. And, of course, if they don't have the same energy, for most values of T, there's a non-zero probability of detecting the particle as a b, even though it was produced as an a.

The reason this is generally stated to require different masses is that the kinematics of production certainly won't allow two states in a superposition that have the same mass to have different energies.

Thanks, that's actually quite helpful. A university website I found had a nice calculation showing different neutrino masses lead to osciallation, but that only validates (mass different) -> oscillation; I was looking for (oscillation)->(mass <>0), for some very broad framework. I couldn't find that. What you've shown convinces me that within any reasonable quantum model, oscillation itself implies mass difference.

 

[Parlyne]

Thanks, that's actually quite helpful. A university website I found had a nice calculation showing different neutrino masses lead to osciallation, but that only validates (mass different) -> oscillation; I was looking for (oscillation)->(mass <>0), for some very broad framework. I couldn't find that. What you've shown convinces me that within any reasonable quantum model, oscillation itself implies mass difference.

Glad to be of service.

I will also point out that, in fact, mass difference is not, by itself, sufficient for oscillation. If, for instance, the flavor states were, themselves, states of definite mass, there would be no oscillation, even if the masses were different.