Tuesday, December 18, 2007

Neutrino Masses and Angles

Neutrinos come in three known flavors. These flavors correspond to the three charged leptons, the electron, the muon and the tau. The neutrino flavors can change during the neutrino's travel, and one flavor can be converted into another. This happens periodically. The neutrino flavor oscillations have a certain wavelength, and an amplitude which sets the probability of the change to happen. The amplitude is usually quantified in a mixing angle θ. sin2(2 θ) = 1, or θ = π/4 corresponds to maximal mixing, which means one flavor changes completely into another, and then back. For a brief introduction, see also our earlier post Neutrinos for Beginners.

This neutrino mixing happens when the mass-eigenstates of the Hamiltonian are not the same as the flavor eigenstates. The wavelength λ of the oscillation turns out to depend (in the relativistic limit) on the difference in the squared masses Δm2 (not the square of the difference!) and the neutrino's energy E as λ = 4Em2. The larger the energy of the neutrinos the larger the wavelength. For a source with a spectrum of different energies around some mean value, one has a superposition of various wavelengths. On distances larger than the typical oscillation length corresponding to the mean energy, this will average out the oscillation.

The plot below from the KamLAND Collaboration shows an example of an experiment to test neutrino flavor conversion. The KamLAND neutrino sources are several Japanese nuclear reactors that emit electron anti-neutrinos with a very well known energy and power spectrum, that has a mean value around some MeV. The average distance to the reactors is ~180 km. The plot shows the ratio of the observed electron anti-neutrinos to the expected number without oscillations. The KamLAND result is the red dot. The other data points were earlier experiments in other locations that did not find a drop. The dotted line is the best fit to this data.

[Figure: KamLAND Collaboration]

One sees however that there is some kind of redundancy in this fit, since one can shift around the wavelength and stay within the errorbars. These reactor data however are only one of the measurements of neutrino oscillations that have been made during the last decades. There are a lot of other experiments that have measured deficites in the expected solar and atmospheric neutrino flux. Especially important in this regard was the SNO data that confirmed that indeed not only there were less solar electron neutrinos than expected, but that they actually showed up in the detector with a different flavor, and the KamLAND analysis of the energy spectrum that clearly favors oscillation over decay.

The plot below depicts all the currently available data for electron neutrino oscillations, which places the mass-square around 8×10-5 eV2, and θ at about 33.9° (i.e. the mixing is with high confidence not maximal).

[Figure: Hitoshi Murayama, see here for references on the used data]

The lines on the top indicate excluded regions from earlier experiments, the filled regions are allowed values. You see the KamLAND 95%CL area in red, and SNO in brown. The remaining island in the overlap is pretty much constrained by now. Given that neutrinos are so elusive particles, and this mass scale is incredibly tiny, I am always impressed by the precision of these experiments!

To fit the oscillations between all the known three neutrino flavors, one needs three mixing angles, and two mass differences (the overall mass scale factors out and does not enter, neutrino oscillations thus are not sensitive to the total neutrino masses). All the presently available data has allowed us to tightly constrain the mixing angles and mass squares. The only outsider (that was thus excluded from the global fits) is famously LSND (see also the above plot), so MiniBooNE was designed to check on their results. For more info on MiniBooNE, see Heather Ray's excellent post at CV.

This post is part of our 2007 advent calendar A Plottl A Day.


  1. When did MiniBooNE become 'Boomerang'?

    And when can we get a post on the naming of experiments and the increasingly bizarre-looking but easily pronounceable combinations of capital and lowercase letters that are now in fashion?

  2. *gurk* Thanks, I've corrected that. I guess this was a broken link in my brain. Yeah, I find these acronymes increasingly weird. The weirdest one I've seen so far is CATFISH, which is supposed to mean "Collider grAviTational FIeld Simulator for black Holes", which not only doesn't make much sense given what it does, it also sounds awful. However, it seems they actually picked the name before they figured out its 'meaning'. To understand that please note that the inventors sit in Mississippi ;-) Best,


  3. Yea, the naming convention that is now in fashion confuses the hell out of me sometimes.

    Especially when one of the letters of the name is a part of another acronym. I sometimes feel I need a flowchart to understand the name - then I stop caring and forget what the name means.

  4. I know a fellow at nearby Jefferson Lab who now works for "Muons Inc.", which is a private company (often with government contracts of course) developing Muon production and control resources of all things. Muons will decay into electrons and electron and muon neutrinos, and that is one of the reasons for doing this. I helped him do some calcs of absorption and scattering of muons using g4beamline etc., which is available on their site.

    Their website is http://www.muonsinc.com/, and you can find some mention of neutrino factories.

    BTW, how can neutrinos change flavor in empty space in light of conservation issues? I gather it depends on the subtle superposition issues, etc, but if they have different masses they can't just "turn into a particle of a different mass", true?

  5. Hi Neil'

    That's interesting. I like Muons Inc, much better than MiniBones or things like that. I don't quite understand your question though, conservation of what? Best,


  6. Ok Bee, I thought that a particle can't just change "rest mass" without violating conservation of energy and/or momentum. IOW, if it slowed down in your frame to keep its total energy the same, then it can't conserve energy/momentum in other frames. That's why I can't imagine how neutrinos can just change into other kinds, if the masses are different (are they?) That's why I thought they needed to interact weakly with a nucleus etc, but your and other discussions say they can do it in empty space. If the masses of all flavors are the same, well I still think some "quantum number" must be preserved, no?

  7. Hi Neil',

    Thanks for the clarification. The flavor states are just not the mass eigenstates. Thus, they do not have specific masses, but are superpositions of different mass eigenstates. If you'd prepare the correct superposition of flavor states that corresponds to a mass eigenstate it wouldn't change and be time-independent. Best,


  8. If they'd known all along that neutrinos have mass, they'd have called what we now call "\nu_1" the "electron neutrino", etc., and the whole thing would be less confusing.

    In the mass eigenstate language, instead of having neutrino oscillation of a single flavor, you have the same old everyday interference. (I.e. neutrino oscillation can also be described as mass eigenstate interference.)

  9. Thanks for the great "Advent Calendar" -- I've really enjoyed the postings.

    A question about neutrino oscillation, though: the explanations I've seen about the probability of neutrino oscillation seem frame dependent -- they include a term for the "distance travelled". Is there a way to describe neutrino oscillation in the rest frame of the neutrino itself? Or is the probability of observing different flavors indeed frame dependent? The latter would seem very bizarre to me, but then quantum mechanics as a whole seems very bizarre to me (a layman).


  10. Hi Eric:

    The neutrinos are ultrarelativistic, traveling with the speed of light, and therefore the distance traveled is essentially the time they traveled. Given that all the masses were nonezero, one could in principle do that in the restframe, but then you couldn't use the ultrarelativistic limit, and everything becomes very messy. The above nice equation for the wavelength being proportional to 1/Δm^2 uses the limit of E>>m, to get rid of the square roots. Best,


  11. Hi Bee and Stefan,

    This speculation may link your posts on phase diagrams to differences in neutrino mass?

    Molecules are known to move at different speeds [or velocities if direction is known] in the different phases of gas, liquid or solid [sub-plasmas?] with related enthalpy.

    Perhaps this is also why neutrinos have different masses?
    This may also correspond to the Einstein theory that relativistic mass increases as the speed of light is approached?

    Perhaps the generation 1 Electron is slowest, perhaps as sub-plasma matter?
    Perhaps the generation 2 Muon is intermediate, perhaps as plasma?
    Perhaps the generation 3 Tau is fastest, perhaps as super-plasma?

  12. Hi Doug:

    Thanks for your comment. I am afraid though I don't quite know what you mean with 'slow'. If they have the same velocity, then they have the same velocity? An electron can be fast, or not so fast, but if an electron has the same velocity as a muon, then it has the same velocity, so what do you mean with the electron is the slowest? Best,


  13. Hi Bee,

    I was attempting to discuss the different generations of neutrinos.

    By speed variance, I mean that the 3 neutrinos may travel at different relative speeds such as these arbitrary examples:

    Electron neutrino ~0.90c
    Muon neutrino ~0.95c
    Tau neutrino ~ 0.99c

    Granted that this is speculative, but different relative speeds may be equivalent to different relative masses as per Einstein.

    The electron neutrino, which interacts with "ordinary" matter by Chlorine-37 + electron_neutrino = Argon-37 + electron, may be the slowest since it has the smallest mass.

    See Solar Neutrino Telescope section of this HyperPhysics page;
    How the Sun Shines by John N Bahcall 29 June 2000.

  14. Hi Doug:

    If the different flavors were the same as different mass eigenstates, then two different flavor neutrinos with the same energy would travel with different speed. Is that what you are trying to say? I don't see however how that would explain that one flavor changes into another flavor? Best,


  15. Hi, I'm not sure what Doug is getting it, but in case he was/still is confused about relativistic versus rest mass: Normally, when we speak of particle "mass" it is the mass the particle would have at rest. The different neutrino flavors are supposed to have different rest masses IIUC, but the whole thing is confused (for a middle-brow like me at least) if different types are in quantum superposition.

    Some think we should not speak of "relativistic mass" but use "mass" as an invariant. Even then the relativistic "mass-energy" (how much mass and/or energy can be extracted in our lab frame out of converting the moving particle, such as into gamma rays etc.) is still given as gamma times the rest value (gamma = (1 - v^2/c^2)^-1/2).)

    What I don't get is, if neutrinos can be superpositions of different mass states, how does that effect the relationship between energy and velocity? Given an energy range of the total neutrino wave function, it seems the velocity must be in a superposition too (i.e., the higher mass going slower to represent the same average energy etc.). That seems to me, it would be strange because the superposed "heavy" wave would get out of step with the lighter one. Then, as the neutrino travels, it becomes a spread out sort of wave?

  16. Hi Neil',

    Thanks, it didn't occur to me this could be misunderstood, but yes, with mass I (always) mean the relativistic invariant rest-mass of a particle.

    Your question is a good one, the above sketched procedure is of course only approximate. In practice we always have to deal with wave-packets rather than plane waves. As I've said above, these calculations are done in the ultrarelativistic limit, were the speed of (all) the neutrinos is the speed of light. We know that the total neutrino masses are extremely small (< eV) and in the energy ranges considered (MeV - hundreds of GeV, depending on the experiment), this is an excellent approximation. If one has much heavier neutrinos, you'll have to take into account that E is not just p, and translations in space can not be as conveniently be expressed through translations in time. In principle however the treatment remains the same, i.e. diagonalize the mass matrix, apply translation operators to get from (x,t) to (x',t'), eigenstates remain eigenstates, everything else mixes into each other. There are a couple of very well written introcution into the subject by Akhmedov, that I can really recommend e.g. hep-ph/0001264, and then there is also this nice paper

    "Quantum Theory of Neutrino Oscillations for Pedestrians - Simple Answers to Confusing Questions"
    Harry J. Lipkin hep-ph/0505141



  17. Thanks for the further clarification, Bee (and others on this thread). I think I see my error now -- I was naively assuming that there was "a" rest frame for the neutrino, but it appears each eigenstate of mass must have a different inertial rest frame. No matter which one you pick, there will be interference with the others.

  18. Hmmm, further musing about superpositions of mass eigenstates. The gravitational field of a propagating neutrino must be itself in a superposition, right? Over short ranges this wouldn't be noticeable, presumably, since the neutrinos are travelling so close to c. But over astronomical distances they might be. It looks like some people are trying to detect gravitational lensing of neutrinos; I wonder if they will also see any quantum gravitational effects?

  19. Hi Bee,

    Thank you.
    I see that I failed to read footnote #2 RE Mass in the neutrino table as expectation values.

    However, what I am [perhaps was] trying to do is use a physlink equation relating mass to velocity m=m0/((1-(v/c)^2)^0.5

    I suspect that this formula can somehow be useful but probably not with expectation values.


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