# How Do Neutrino Oscillations Reveal Their Masses?

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• snorkack
snorkack
What´s really experimentally known about neutrino masses for now?
- There are two neutrino oscillation periods, each with a mass square difference, so at least two neutrino mass eigenstates of the three have nonzero masses. (Note that "imaginary" is also "nonzero"!)
- There is an upper bound on mass (below 100 meV) but there does not seem to be an observed lower bound on mass.

Is there any theoretical reason forbidding oscillations between an eigenstate of nonzero rest mass and an eigenstate of exactly zero rest mass?
The observed mass square differences are quoted as 76 and 2440 meV2. If the light eigenstate is, say, 2 meV, does it mean that the heavy eigenstates are 9 meV and 50 meV, or that they are 49 meV and 50 meV? Or that we do not know which?
How do oscillations work between mass eigenstates if the total energy is smaller than the mass of an eigenstate partaking in oscillation? Relic neutrinos should now be 0,5 meV if massless, less if massive. What did oscillations do on cooling?

snorkack said:
- There is an upper bound on mass (below 100 meV) but there does not seem to be an observed lower bound on mass.
Lower Bounds On Neutrino Mass Eigenstates From Neutrino Oscillation

The lower bound comes from the minimum sum of neutrino masses from the oscillation numbers (about 66 meV for a normal ordering of neutrino masses and about 106 meV for an inverse hierarchy of neutrino masses). See, e.g., here and here.

The 95% confidence interval minimum value of the mass difference between the second and third neutrino mass eigenstate is 48.69 meV, and the corresponding value of the mass difference between the first and second neutrino mass eigenstate is 8.46 meV. This implies that with a first neutrino mass eigenstate of 0.1 meV, a sum of the three neutrino masses is 0.01 + 8.47 + 57.16 = 65.64 meV in a normal hierarchy and 0.01 + 48.70 + 57.16 = 105.87 meV in an inverted hierarchy. The often quoted figure of 0.06 eV for the minimum sum of the neutrino masses in a normal ordering and 0.1 eV for the minimum sum of the neutrino masses in an inverted ordering are just order of magnitude approximations (or may reflect outdated measurements).

The sum of the three neutrino masses could be greater than these minimums. If the sum of the three masses is greater than these minimums, the smallest neutrino mass is equal to a third of the amount by which the relevant minimum is exceeded to the extent that it is not due to uncertainty in measurements of the two mass differences.

So, for example, if the lightest of the three neutrino masses is 10 meV, then the sum of the three neutrino masses is about 96 meV in a normal mass ordering and about 136 meV in an inverted mass ordering.

The latest measurement of neutrino properties from T2K from March of this year favors a normal ordering of neutrino masses strongly but not decisively. We should be able to know the neutrino mass ordering more definitively in less than a decade according to a Snowmass 2021 paper released in December of 2022:

We have made significant progress since neutrino mass was first confirmed experimentally (also from the Snowmass 2021 paper):

Upper Bounds On Neutrino Mass From Direct Measurement

Direct measurement bounds the lightest neutrino mass at not more than about 800 meV, which isn't very constraining. This is potentially reducible to 200 meV within a few years according to physics conference presentations, which also isn't competitive with cosmology based bounds set forth below.

The tightest proposed constraints from cosmology (see below) are that this absolute mass value is actually 7 meV or less (with 95% confidence), although many cosmology based estimates are more conservative and would allow for a value of this as high as 18 meV or more (with 95% confidence). The one sigma (68% confidence) values are approximately 3.5 meV or less, and 9 meV or less, respectively.

Direct measurements of the neutrino masses are not anticipated to be meaningfully competitive with other means of determining the neutrino masses for the foreseeable future.

Upper Bounds On Neutrino Mass From Cosmology

The upper bound on the mass of the sum of the three neutrino masses is a cosmology based. As the Snowmass 2021 paper explains:
Cosmological measurements of the cosmic microwave background temperature and polarization information, baryon acoustic oscillations, and local distance ladder measurements lead to
an estimate that the sum of for all i of m(i) < 90 meV at 90% CL which mildly disfavors the inverted ordering over the normal ordering since the sum of for all i of m(i) greater than or equal to 60 meV in the NO and greater than or equal to 110 meV in the IO; although these results depend on one’s choice of prior of the absolute neutrino mass scale.
Significant improvements are expected to reach the σ(the sum of for all m(ν)) ∼ 0.04 eV level with upcoming data from DESI and VRO, see the CF7 report, which should be sufficient to test the results of local oscillation data in the early universe at high significance, depending on the true values.

According to Eleonora di Valentino, Stefano Gariazzo, Olga Mena, "Model marginalized constraints on neutrino properties from cosmology" arXiv:2207.05167 (July 11, 2022), cosmology data favors a sum of three neutrino masses of not more than 87 meV (nominally ruling out an inverted mass hierarchy at the 95% confidence interval level, which oscillation data alone favor at a 2-2.7σ level), implying a lightest neutrino mass eigenstate of about 7 meV or less. Other estimates have put the cosmological upper bound on the sum of the three neutrino masses at 120 meV.

The upper bound from cosmology is model dependent, but it is also quite robust to a wide variety of assumptions in those models. Of course, if future cosmology data implies that the sum of the three neutrino masses is lower than the lower bound from neutrino oscillation data (since all cosmology bounds to date are upper bounds), then there is a contradiction which would tend to cast doubt on the cosmology model used to estimate the sum of the three neutrino masses.

Upper Bounds On Majorana Neutrino Mass

There is also an upper bound on the Majorana mass of neutrinos, if they have Majorana mass, from the non-observation of neutrinoless double beta decay. As of July of 2022 (from here (arXiv 2207.07638)), we could determine with 90% confidence, based upon the non-detection of neutrinoless beta decay in a state of the art experiment establish a minimum half-life for the process of 8.3 * 1025 years. As explained by this source, an inverted mass hierarchy for neutrinos (with purely Majorana mass) is ruled out at a half life of about 1029 years (an improvement by a factor of 1200 in the excluded neutrinoless double beta decay half life over the current state of the art measurement). Exclusively Majorana mass becomes problematic even in a normal mass hierarchy in about 1032 or 1033 years (an improvement by a factor of 1.2 million to 12 million over the current state of the art). These limitations, however, are quite model dependent, in addition to being not very constraining.

On the other hand, if one is a supporter of the Majorana neutrino mass hypothesis, it is somewhat reassuring to know that we shouldn't have been able to see neutrinoless double beta decay yet if the neutrino masses are as small as neutrino oscillation data and cosmology data suggests.

snorkack said:
Is there any theoretical reason forbidding oscillations between an eigenstate of nonzero rest mass and an eigenstate of exactly zero rest mass?
Not a strong one, although there are suggestive reasons why it would make more sense if it had a tiny, but non-zero rest mass.

All neutrinos do interact directly via the weak force, and every single other Standard Model particle with a non-zero rest mass also interacts directly via the weak force (while all Standard Model particles that do not interact directly via the weak force, i.e. photons and gluons, and the hypothetical graviton which doesn't have weak force "charge") have zero rest mass. Similarly, all other Standard Model fermions have rest mass.

Possibly, the weak force self-interaction of the neutrinos ought to give rise to some rest mass. If the electron and lightest neutrino mass eigenstate both reflected predominantly the self-interactions of these particles via Standard Model forces (as some papers have suggested), a lightest neutrino mass eigenstate of the right order of magnitude given the combination of neutrino oscillation and cosmology bounds would flow from the relative values of the electromagnetic force and weak force coupling constants.

A massless neutrino would always travel at precisely the speed of light and would not experience the passage of time internally, while a massive neutrino would travel at a speed slightly less than the speed of light depending upon its kinetic energy due to special relativity, and would experience the passage of time internally, which makes more sense for a particle whose oscillations are not direction of time symmetric (because the PMNS matrix appears to have a non-zero CP violating term).

But none of this is ironclad theoretical proof that the lightest neutrino mass eigenstate can't be zero.
snorkack said:
How do oscillations work between mass eigenstates if the total energy is smaller than the mass of an eigenstate partaking in oscillation?
There is no reason that virtual particles in a series of neutrino oscillations shouldn't be possible, but the end states of any interaction need to conserve mass-energy.

In practice, we generally don't observe neutrinos with exceedingly low kinetic energy, from either reactors or nuclear decays or cosmic sources. We don't have the tools to do so, and don't know of processes that should give rise to them that we can observe.

All observed neutrinos have relativistic kinetic energy (i.e. kinetic energy comparable to or in excess of their rest mass), even though very low energy neutrinos are theoretically possible. Observations of relic neutrinos with very low kinetic energy are a scientific goal rather than a scientific achievement.

Last edited:
vanhees71
Consider oscillations like this:
https://en.wikipedia.org/wiki/Neutrino_oscillation#/media/File:Oscillations_electron_long.svg
left edge enlarged to
https://en.wikipedia.org/wiki/Neutrino_oscillation#/media/File:Oscillations_electron_short.svg
These make certain assumptions:
• all three mass eigenstates are highly relativistic;
• all three mass eigenstates are present in equal numbers and energies
Correct?
What would happen to the oscillations if we watched them in a reference frame where the heaviest mass eigenstate is stationary?
What would happen to the oscillations if we watched them in a reference frame where the mass eigenstates propagate in different directions?

> all three mass eigenstates are highly relativistic;

Yes. That's true for all neutrinos we measure today.

> all three mass eigenstates are present in equal numbers and energies

No. It says "initial electron neutrino", which is a superposition with different contributions of the mass eigenstates (mostly m1 and m2, not much m3).

If the neutrinos are so slow that their propagation differs measurably then you get sensitive to mass eigenstates instead of flavor eigenstates.

vanhees71 and ohwilleke
mfb said:
> all three mass eigenstates are highly relativistic;

Yes. That's true for all neutrinos we measure today.

> all three mass eigenstates are present in equal numbers and energies

No. It says "initial electron neutrino", which is a superposition with different contributions of the mass eigenstates (mostly m1 and m2, not much m3).
Ah, thanks! The link actually provides the breakdown for the flavour eigenstates per mass eigenstate, not vice versa.
In a mixture where all flavour eigenstates are equally represented, are mass eigenstates also equally represented?
mfb said:
If the neutrinos are so slow that their propagation differs measurably then you get sensitive to mass eigenstates instead of flavor eigenstates.
Yes.
So if a superposition of neutrino mass eigenstates is Lorentz transitioned/Doppler shifted, does the count of each mass eigenstate remain unchanged at its previous value?

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