# Neutrino mass and velocity
**URL:** https://boards.straightdope.com/t/neutrino-mass-and-velocity/197536
**Category:** Factual Questions
**Created:** [August 25, 2003, 4:02pm UTC](https://boards.straightdope.com/t/neutrino-mass-and-velocity/197536 "2003-08-25T16:02:10Z")
**Posts on this page:** 8
**Page:** 1
### Author:  [@Grey](https://boards.straightdope.com/u/Grey)
#### Post date: [August 25, 2003, 4:02pm UTC](https://boards.straightdope.com/t/neutrino-mass-and-velocity/197536/1 "2003-08-25T16:02:10Z")
If a neutrino’s mass oscillates as it travels, does its velocity change? It would seem that to conserve energy the velocity much decrease as the mass increase. Anyone know?
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### Author:  [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)
#### Post date: [August 25, 2003, 4:13pm UTC](https://boards.straightdope.com/t/neutrino-mass-and-velocity/197536/2 "2003-08-25T16:13:15Z")
Are you talking about the oscillations of a muon-neutrino changing back and forth between (probably) a tauon-neutrino?
These oscillations suggest that the two types have very slightly diffreent masses. This is one of two major experiments (the other perfomed in 1985 gave a mass of 10[sup]-4[/sup] m[sub]e[/sub] but could not be reproduced) that suggest that the neutrinos have non-zero masses, tho’ it’s still not known for sure if neutrinos have mass or not.
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### Author:  [@Grey](https://boards.straightdope.com/u/Grey)
#### Post date: [August 25, 2003, 4:24pm UTC](https://boards.straightdope.com/t/neutrino-mass-and-velocity/197536/3 "2003-08-25T16:24:52Z")
I was basing the question off the SNO results that indicate neutrino mass oscillation
> [@](#):
>
> The new results, obtained entirely from the SNO, are so accurate that it is 99.999% probable that solar neutrinos change type before reaching Earth. The results, which have been submitted to Physical Review Letters, are of great importance because the way in which the neutrinos - long thought to be massless particles - change types is thought to be linked to neutrino mass and mass differences between various neutrino types.
[link](http://www.sno.phy.queensu.ca/sno/results_04_02/)
So, does the momentum remains constant when the mass changes? I’m guessing so but I’m no expert.
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### Author:  [@Chronos](https://boards.straightdope.com/u/Chronos)
#### Post date: [August 25, 2003, 8:28pm UTC](https://boards.straightdope.com/t/neutrino-mass-and-velocity/197536/4 "2003-08-25T20:28:55Z")
This is a very interesting question. I’ll just keep on explaining it, and you can stop me when I get to one that makes sense, ok?
The key problem here is that an electron neutrino, say, doesn’t have a mass. It has mass, and it has an expected mass, but it doesn’t have one single mass. If you specify an exact mass for a neutrino, then you’re not talking about a nu-e, a nu-mu, or a nu-tau; you’re talking about a nu-1, a nu-2, or a nu-3. A “flavor state” neutrino (e, mu, or tau) can be considered as a combination of “mass neutrinos” (1, 2, and 3), or vice-versa.
To put it another way: The neutrino state lives in a three-dimensional space, but that space can be spanned either by the states e, mu, and tau, or by the states 1, 2, and 3. The three flavor states have definite flavor, but not definite mass, and the three mass states have definite mass, but not definite flavor.
Indeed, it’s the mass discrepancy which causes oscillation in the first place. Usually, when a neutrino is formed, it’s formed in a flavor state. That flavor state is a linear combination of the three mass states. But since those mass states have different masses, they propogate differently. After some distance, the mass states are “out of synch” with each other, and the combination of those out of synch mass states is some other flavor state.
As for what remains fixed: A neutrino with a definite flavor can be produced in a state of definite momentum, or a state of definite energy, but it can’t be in a state of definite mass. It also can’t be in a state of both definite energy and definite momentum, since those two together would give you a definite mass. For most purposes, it doesn’t matter whether it’s definite energy or momentum, since for low-mass particles, the energy is almost exactly equal to the momentum. And a neutrino which starts off with one definite flavor will eventually become another definite flavor, but the energy or momentum (whichever one was definite) will stay fixed. A neutrino can also have a definite mass, energy, and momentum, which will all stay fixed, but in this case, it can’t ever have a definite flavor.
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### Author:  [@Grey](https://boards.straightdope.com/u/Grey)
#### Post date: [August 26, 2003, 5:51am UTC](https://boards.straightdope.com/t/neutrino-mass-and-velocity/197536/5 "2003-08-26T05:51:31Z")
> [@](#):
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> \*Originally posted by Chronos \*
> \*\*This is a very interesting question. I’ll just keep on explaining it, and you can stop me when I get to one that makes sense, ok? \*\*
You know, it’s times like this when I really wish we had a whiteboard for these type of things.
> [@](#):
>
> \*Originally posted by Chronos \*
> \*\*The key problem here is that an electron neutrino, say, doesn’t have a mass. It has mass, and it has an expected mass, but it doesn’t have one single mass. If you specify an exact mass for a neutrino, then you’re not talking about a nu-e, a nu-mu, or a nu-tau; you’re talking about a nu-1, a nu-2, or a nu-3. A “flavor state” neutrino (e, mu, or tau) can be considered as a combination of “mass neutrinos” (1, 2, and 3), or vice-versa.
>
> To put it another way: The neutrino state lives in a three-dimensional space, but that space can be spanned either by the states e, mu, and tau, or by the states 1, 2, and 3. The three flavor states have definite flavor, but not definite mass, and the three mass states have definite mass, but not definite flavor. \*\*
So a nu-1 neutrino would be a linear combination of “flavour states” like [A nu-e, B nu-mu, C nu-tau] with each component (A, B, C) lying in the range 0 to 1? So an electron neutrino would be [1,0,0] and a muon neutrino would be [0,1,0] etc.? Is it possible to have a fractional A, B and C? Does this mean if A, B and C = 0 the neutrino could possibly have zero mass?
For some reason I’m picturing a unit sphere with nu-e, nu-mu and nu-tau on the axis and the mass neutrino being the radius.
> [@](#):
>
> \*Originally posted by Chronos \*
> \*\*Indeed, it’s the mass discrepancy which causes oscillation in the first place. Usually, when a neutrino is formed, it’s formed in a flavor state. That flavor state is a linear combination of the three mass states. But since those mass states have different masses, they propagate differently. After some distance, the mass states are “out of synch” with each other, and the combination of those out of synch mass states is some other flavor state. \*\*
Would that be similar to propagation delay in optical wavelengths? The peaks separating as the differing velocities create differing arrival rates? Optically though, you can recover the separate peaks/wavelengths, but with a neutrino you recover a different flavour. What do you recover when the trailing states arrive; a new neutrino?
> [@](#):
>
> \*Originally posted by Chronos \*
> \*\*As for what remains fixed: A neutrino with a definite flavor can be produced in a state of definite momentum, or a state of definite energy, but it can’t be in a state of definite mass. It also can’t be in a state of both definite energy and definite momentum, since those two together would give you a definite mass. For most purposes, it doesn’t matter whether it’s definite energy or momentum, since for low-mass particles, the energy is almost exactly equal to the momentum. And a neutrino which starts off with one definite flavor will eventually become another definite flavor, but the energy or momentum (whichever one was definite) will stay fixed. A neutrino can also have a definite mass, energy, and momentum, which will all stay fixed, but in this case, it can’t ever have a definite flavor. \*\*
So, basically, it depends. 🙂
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### Author:  [@Chronos](https://boards.straightdope.com/u/Chronos)
#### Post date: [August 26, 2003, 9:13pm UTC](https://boards.straightdope.com/t/neutrino-mass-and-velocity/197536/6 "2003-08-26T21:13:05Z")
> [@](#):
>
> So a nu-1 neutrino would be a linear combination of “flavour states” like [A nu-e, B nu-mu, C nu-tau] with each component (A, B, C) lying in the range 0 to 1? So an electron neutrino would be [1,0,0] and a muon neutrino would be [0,1,0] etc.? Is it possible to have a fractional A, B and C? Does this mean if A, B and C = 0 the neutrino could possibly have zero mass?
More or less. A, B, and C can be fractional (so long as they’re all in the range [0,1]), but you have to normalize it (A[sup]2[/sup] + B[sup]2[/sup] + C[sup]2[/sup] = 1) so you can’t have all three be zero.
> [@](#):
>
> For some reason I’m picturing a unit sphere with nu-e, nu-mu and nu-tau on the axis and the mass neutrino being the radius.
Not quite. All neutrino states are on that unit sphere, and you’re describing the points on the unit sphere using its coordinates in a Cartesian coordinate system (x, y, and z, or whatever). The catch is that there are two different Cartesian coordinate systems you can use: You can use e, mu, and tau as your axes, or you can use 1, 2, and 3 as your axes. These two sets of axes are tilted at some angle (or angles) from each other, and the values of those angles is one of the things which determines the oscillation.
I suppose that propogation delay in light is similar, except that with neutrinos, you have names for certain combinations of phases. And if you know how far the neutrino travelled (and you know all the oscillation parameters, which we don’t yet), then you could determine what it was when it started, but it’s not that anymore.
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### Author:  [@Pasta](https://boards.straightdope.com/u/Pasta)
#### Post date: [August 26, 2003, 10:59pm UTC](https://boards.straightdope.com/t/neutrino-mass-and-velocity/197536/7 "2003-08-26T22:59:18Z")
> [@](#):
>
> \*Originally posted by Chronos \*
> **More or less. A, B, and C can be fractional (so long as they’re all in the range [0,1]), but you have to normalize it (A[sup]2[/sup] + B[sup]2[/sup] + C[sup]2[/sup] = 1) so you can’t have all three be zero.**
Couldn’t resist the minor nitpick… 🙂
A, B, and C can be any complex numbers satisfying |A|[sup]2[/sup]+|B|[sup]2[/sup]+|C|[sup]2[/sup]=1.
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### Author:  [@Grey](https://boards.straightdope.com/u/Grey)
#### Post date: [August 27, 2003, 1:37am UTC](https://boards.straightdope.com/t/neutrino-mass-and-velocity/197536/8 "2003-08-27T01:37:54Z")
Thanks very much **Chronos** , it is a little clearer.