# Neutrinos and Conservation Laws

1. Jun 21, 2015

### Buzz Bloom

I am confused about the current physics regarding neutrinos and implications about the conservation laws of mass-energy and linear momentum. I have read the threads listed for similar discussions, including, "How does conservation of energy/mass apply to neutrinos?", and none of them seem to address the issue I am raising here.

The following are the "facts" about neutrinos according to current theory as I understand them:
Neutrinos have a non-zero rest mass.
Neutrinos travel at less than the speed of light.
There are three species of neutrinos, each with a rest mass different than the other two.
When a neutrino is created it travels at a velocity near the speed of light. That is, it has a relativistic speed.
As a neutrino of a given species moves, it randomly changes into another species.
When a neutrino interacts, it will become a specific one of the three possible species.
The probability that an interacting neutrino becomes any particular species is related to that species rest mass. (What is this relationship?)

The following is an interpretation regarding these facts. I am not sure of the status of this interpretation within the physics community.
Before a neutrino interacts, it has no individual species. Rather, it is a probabilistic superimposed state of all three species, and it's interaction changes its state into a particular species.

If this interpretation is correct, then the neutrino's rest mass and velocity are also not specific until an interaction. However, to satisfy conservation of mass-energy and momentum, these quantities must be the same for all three species. The following are the relativistic equations for these conservation laws. m01, m02, and m03 are the rest masses for the three species, and v1, v2, and v3 are their corresponding linear velocities.

It can be shown that if the three rest masses are not all equal, then there are no values for the three velocities that will satisfy these equations. If the masses are equal, then the velocities must also be equal.

Does the current theory regarding neutrinos allow violations of the conservation laws? If not, can anyone help me resolve my confusion.

Last edited: Jun 21, 2015
2. Jun 21, 2015

### Staff: Mentor

They don't have to be. The neutrino does not have to be emitted with an exact well-defined energy and momentum. The same applies to the detection.

3. Jun 21, 2015

### Buzz Bloom

Hi mfb:

I think you are referring to the fact that when an interaction takes place that emits a neutrino, there is some variability in the three rest masses of the three specias in their superimposed state. I am thinking about a repeated sequence of experiments, for example the beta decay emting of a neutrino from a tritium neucleus . This is actually being done this year (2015) with a facility built by a European consortium to measure the rest mass of neutrinos. See https://en.wikipedia.org/wiki/KATRIN .

If an experiment like this is repeated many times, I would expect the variance of the values about the mean to be relatively small. As usual, equations that represent physics must be expected to predict only approximate experimental results. I understand that the three neutrino species rest mass values are expected to be significantly different. The conservation equations require that the experimentally determned mean rest mass values have corresponding mean momentum values that approximately satisfy these equations within some predictable error range. The theorectically expected size of the mass differences will make satisfying the equations within a predicted error range impossible.

Buzz

Last edited: Jun 21, 2015
4. Jun 22, 2015

### Staff: Mentor

The rest masses of the mass eigenstates are fixed. But there is no such thing as "the exact energy of the neutrino" or "the exact momentum". If those two would exist, the neutrino would be in one of the mass eigenstates - which are not the states that interact with other particles (flavor eigenstates).

Which values, which mean? KATRIN cannot measure the mass of neutrinos directly. It will perform a fit to the energy spectrum of electrons, a nonzero neutrino mass changes the shape of this spectrum slightly. If KATRIN sees a deviation from non-zero neutrino masses at all, the masses will be so close together that the experiment cannot do more than giving a mass scale for all neutrinos. But we can imagine a hypothetical better experiment, which would be sensitive to something like an effective electron neutrino mass which is a combination of the three mass eigenstates.

5. Jun 22, 2015

### Buzz Bloom

Hi mfb:

I confess that I do not have the education to understand the techincal details of the KATRIN experiment. I understand that their experiment is based on the ananlysis of electrons. Here are some quotes from the KATRIN site https://www.katrin.kit.edu/ .

KATRIN measures the neutrino mass in a model-independant way via ultrahigh precision measurements of the kinematics of electrons from beta-decay. To detect the subtle effects of a massive neutrino on the kinematics of the beta electrons requires on one hand the provision of a strong gaseous windowless Tritium source with well-known properties and precision control. On the other hand it requires a high resolution spectrometer (MAC-E filter) with large diameter (10 m) to analyze precisely the electron energies from the source. All major components are under construction or already in place and in the commissioning phase.

The high sensitivity of the KATRIN experiments to analyze precisely the electrons kinetic energy will be reached by a special type of spectrometers, so-called MAC-E-Filters (Magnetic Adiabatic Collimation combined with an Electrostatic Filter).

Sensitivity:
(90% upper limit if neutrino mass is zero)

0.2 eV
with about equal contributions of statistical and systematical errors.

Discovery potential:
A neutrino mass of 0.35 eV would be discovered with 5 sigma significance.
A neutrino mass of 0.30 eV would be discovered with 3 sigma significance.

Systematic uncertainties are expected to amount to an equal size as the statistical errors after a measuring time of 3 full years, using an analyzing interval of 30 eV below the endpoint. These are especially:
Time variation of parameters of the Windowless Gaseous Tritium Source (WGTS),
description of space charging within the WGTS,
determination of scattering probabilities of β-electrons within the WGTS,
description of the final state distribution of (3HeT)+ ions after tritium decay,
variations of the retarding potential,
and the limited uniformity of the magnetic and electrostatic fields in the spectrometer analyzing plane.​

My possibly incorrect understanding from these quotes is that the KATRIN experimental team expect to obtain a much better precision than you seem to think is possible.

Having read the above quotes more carefully than I did before, I now (pehaps mistakenly) understand that the KATRIN measurement will not provide rest masses for all three species, but only for the species that is associated with the beta decay of Tritium, that is, an electron neutrino. I am not sure I correctly understand the process for calculating a neutrino rest mass from the electron data. I seems to me that the raw electron data is an energy meaurement for the electrons. From this and the theoretical knowledge about the total energy emitted during the decay, the neurtino total dynamic mass-energy is calculated. To get the rest mass, the kinectic energy needs to be subtracted. I have not read all the difficult to understand technical material at the KATRIN site, so I have no understanding how the KATRIN team will do this. I gather form other material at the site that a neutrino from a Tritium beta decay is not strongly relativistic, so Newtonian dynamics would provide a resonable approxination. Do you have any ideas about this?

I do not understand what "mass scale for all neutrinos" means. I tried to find a definition on the internet, but all I could find were articles I could not understand that used the term without definition,

Buzz

Last edited: Jun 22, 2015
6. Jun 22, 2015

### Staff: Mentor

I know those KATRIN numbers.

From mixing experiments, it is known that the largest mass squared difference is about 2*10-3 eV2. To get detected by KATRIN; the heaviest neutrino type needs a mass of at least ~300 meV, this makes the mass of the lightest one at least ~297 meV (edit: fixed a typo), for heavier neutrinos the difference gets even smaller. As you can see, if the neutrinos are heavy enough to get detected by KATRIN, their masses have to be very similar.
If the lightest neutrino has a mass of just a few meV, the heaviest one could have a mass of ~45 meV, but then KATRIN cannot see the mass.

Right, and even that is very optimistic. It would require a weird fine-tuning for the masses, where three particles have masses very close to each other. Not impossible, but not very likely and competely different from the patterns we see for other particles.

KATRIN measures the rate of electrons above a specific energy, and repeats this measurement for different energies. This allows to calculate back to the original energy spectrum of the electrons. And that spectrum depends on neutrino masses. A non-zero neutrino mass makes the upper edge "sharper" (it goes to zero faster). They don't need the total energy released, it is part of the measurement result (using the theoretical value would introduce calibration problems, in addition to the problem how to measure the mass of tritium better than 30 parts in a trillion).

I moved the thread to the particle physics subforum.

Last edited: Jun 23, 2015
7. Jun 22, 2015

### Orodruin

Staff Emeritus
Careful here. If by "three species" you mean e, mu, and tau, they are not states with definite masses.
Not random as in "without rules". The rules of neutrino oscillations and flavour conversion are very well defined.
This is a misconception. If the neutrino interacts with a charged current interaction, the interaction will produce a charged lepton of a flavour whih may be measured. The neutrino before that was in a linear combination of different mass eigenstates.

Until measured, the neutrino is in a linear combination of the mass eigenstates. These are coherent due to the small mas squared difference. Simplified, the mass squared difference is small enough for wave packets of different mass eigenstates not to separate and thus be able to interfere in the detection process.

You can only make those statements at levels where the different mass eigenstates are so well separated that coherence between them is lost. Essentially, this requires you to know the momentum of the neutrino with sufficient precision. If you could design an experiment where neutrino momenta were determined to a sufficient level, you would not observe oscillations but only the exchange of three different massive particles with different coupling constants to the different charged leptons (as occurs with weak interactions in the quark sector).

What you would actually see in a sufficiently precise tritium beta decay with mixed neutrinos would be several bumps in the energy spectrum, each proportional to the mixing of the corresponding mass eigenstate. The effective approach is only applicable when the experiment is not sensitive enough to resolve these.

As mfb said before, if KATRIN discovers a non zero neutrino mass, the mass of all neutrino mass eigenstates need to be of that size. Also, there is no such thing as the "mass of an electron neutrino". The electron neutrino is a linear combination of several mass eigenstates.

8. Jun 23, 2015

### Buzz Bloom

Hi mfb:

Thank you for moving the thread to a more appropriate subforum. When I started the thread I wasn't sure which subforum was best.

I assume you made a typo and intended "297 meV" rather than "297 eV". The quote from KATRIN indicates that a mass of ~300 mev would be expected to have 3σ significance. In a KATRIN paper preseted in November 2014 ( http://www.ba.infn.it/~now/now2014/web-content/TALKS/aMon/Plen/Robertson_NOW2014.pptx ) the following is on one of the slides:

If I understand this correctly, this means KATRIN expects to be able to measure the μe mass with reasonabve accuracy, although less than 3σ. if it is greater than 200 meV. I understand your statement to mean that the range of rest masses of t he e, μ, and τ neutrinos are all within about a 1% range of each other. Canyou give a refeence that explains this?

9. Jun 23, 2015

### Orodruin

Staff Emeritus
The e, mu, and tau neutrinos do not have definite masses. They are all different linear combintations of three mass eigenstates. Understanding this is crucial in order to understand neutrino oscillations.

If KATRIN finds evidence of non-zero neutrino mass, the mass eigenstates will all be very close in mass (so close that they cannot be separated).

10. Jun 23, 2015

### Buzz Bloom

Hi Oroduin:

The more I try to understand neutrinos, the more confused I get. I think I now understand this concept: there are three mass eigen states (all within about 1% of each other?) for each of the tree flavors of neutrinos. The phenomenon of neutrino oscillation (which I understand, perhaps incorrectly) refers to oscillation among the three flavors while a neutrino is in motion. I now get that this implies that while a neutrino is in motion, from its wave equation nine probabilities (which add to unity) for nine distinct possible masses may be calculated. If I get this correctly, then the average of the three mass eigen state for an e neutrino may be substantially different from the average of the three mass eigen state for an μ neutrino, and both of these averages may be substantially different from the average of the three mass eigen state for a τ neutrino.

If all this is correct, then the original conservation law question I asked should be rephrased. First, for this purpose, I will suppose that there are also nine (eigen?) velocity values, each corresponding to one of the nine eigen masses. The rephrasing would then be:

If it can be shown that the three average rest eigen masses for the three flavors are not all equal, then there are no values for the three corresponding average eigen velocities that will satisfy these equations within an acceptable error range prediction. If the three average masses are (almost) equal, then the corresponding three average velocities must also be (almost) equal.

Does the current theory regarding neutrinos allow violations of the conservation laws?​

It seems to me that all of the discussion about eigen masses does not address the conservation law question, except to refer to averages of three values rather than distinct values.

I can imagine the possibility that the answer to the conservation law question is that some aspect of the uncertainty principle does not permit sufficiently accurate measurements of both neutrino mass and velocity that would demonstrate this violation. However, I am not aware of any such aspect of uncertainty regarding mass and velocity having such a limitation, in the way that position and momentum (or time and energy) does.

Buzz

Last edited: Jun 23, 2015
11. Jun 23, 2015

### Staff: Mentor

Should have been 297 meV, not eV, sure.
200 meV for the heaviest neutrino and 192 meV for the lightest neutrino are also possible - again, very close together.
See the mixing measurements, they measure the differences of squared masses, my numbers are just different solutions for the measured differences.

200 meV is the expected upper limit if the neutrinos are significantly lighter than that. Upper limit means "if the neutrinos have a mass of 200 meV (see above: type is not very relevant if they are that heavy), the probability to see a larger signal than we saw is 90%" (or 95%, not sure which one they use here). As they do not see a larger signal than they see (by definition...), if they set an exclusion limit of 200 meV it is quite likely that the neutrinos are lighter. If the neutrino masses are 200 meV, the KATRIN result will probably be inconclusive - not clear enough to rule out much smaller masses, just with some hint of a signal.

Actually, the Planck telescope that measured the CMB is sensitive to the sum of all three neutrino eigenstates - a bit more model-dependent, but more sensitive than KATRIN. Together with other cosmology measurements it showed that the sum of all three neutrino mass eigenstates should be below 180 meV (Source). A solid measurement of non-zero neutrino masses at KATRIN would be a big surprise.

Edit: Wrote my message while you were posting, I'll update this post.
Only if they are very heavy.

There are three mass eigenstates in total, not three per flavor.

Do you know vector spaces? Imagine a 2-dimensional vector space: you have two coordinates defined by some axes. You can express every vector based on those two coordinates. Now you draw two vectors in this space that are not aligned with the axes. You can also use those vectors as axes and express vectors in this system. That's how the different eigenstates work (just with three dimensions instead of two). Different ways to express a state.
There are three states that have a well-defined mass, those are the mass eigenstates.
There are three states that have a well-defined flavor, those are the flavor eigenstates.
The mass eigenstates are a linear combination of flavor eigenstates.
The flavor eigenstates are a linear combination of mass eigenstates.

Last edited: Jun 23, 2015
12. Jun 23, 2015

### Buzz Bloom

Hi mfb:

I do know about vector spaces, but I expect it will take me some time to digest the physical implications of your description of the math,

Thanks very much for your elaboration,
Buzz

13. Jun 23, 2015

### Buzz Bloom

Hi mfb:
I get that these vectors are mathematical, and that their physical interpretation requires some additional assumptions or defintions. Here are some guesses about possible alternative interpretations:
Regarding mass:
A vector in this space corresponding to 3 eigen masses repesents the values of the 3 eigen masses (which I assume must be distinct). This requires the units of a coordinate to be interpreted as a mass unit, like 1meV. I also guess the coodinates may represent the eigen values in some standard order, perhaps from smallest to largest.
Alternatively, the coordinates may be pure numbers representing probabilities. If that is so, I have no idea about how the three probaility values of a vector should be interpreted.
Regarding flavor:
This seems more straight forward. Each coordinate for a vector corresponding to flavors represents a particular flavor. Now the value of a vector's coordinates can be interpreted as the probability that a particulate neutrino might manifest itself when measured to be of that paticular flavor. In this case, the coordinate values, say x, y, and z, must add to unity. Therefore if one endpoint of all such vectors are at the origin of the vector space, then the other endpoint for all such vectors will lie in the plane x+y+z+1.
Or alternatively, the coordinates may be complex numbers, like the values of a probability quantum wave function where the square of the absoute value of the complex number represents the probability (or probability density). Then the vector enpoints would be on the surface of the unit sphere.

I have no confidence at all that any of these guesses are even close to the theory of neutrinos. I would very much appreciate some further education here.

I am also curious about the theory that predicts the relative closeness (variance) of the three eigen masses in terms of the actual mass max, min, or average. Can you point me at some not too difficult-to-undestand source? I can deal with difficult math, but in many articles I try to read, there is a lot of undefined physics jargon to cope with.

One more particular question: As I have read various articles I came across a disagreement about whether neutrinos have anti-neutrino anti-particles, or neutrinos are their own anti-particles. Has this point been resolved, or is it still a controversy?

Thanks again for the discussion,
Buzz

Last edited: Jun 23, 2015
14. Jun 24, 2015

### ChrisVer

If you are interested in talking about probabilities, then you are not taking the components of the vectors to be the probabilities...
In the neutrino oscillations you write the flavor eigenstates $|\nu_\alpha>~,~a=e,\mu,\tau$ as a combination of the mass eigenstates $|\nu_i> ~,~i=1,2,3$ via a unitary matrix (PMNS matrix):
$| \nu_\alpha > = \sum_{i=1}^3 U_{i \alpha} |\nu_i>$
Or in matrix form:
$\begin{bmatrix} |\nu_e> \\ |\nu_\mu> \\ |\nu_\tau > \end{bmatrix} = \begin{pmatrix} U_{1e} & U_{2e} & U_{3e} \\ U_{1\mu} & U_{2\mu} & U_{3\mu} \\ U_{1 \tau} & U_{2 \tau} & U_{3 \tau} \end{pmatrix} \begin{bmatrix} |\nu_1> \\ |\nu_2> \\ |\nu_3 > \end{bmatrix}$
Now the amplitude of getting an electron neutrino from this combination would be the product:
$<\nu_e | \nu_\alpha>$
If on the other hand you put in the case of $\nu_e$ any kind of neutrino, you would have the amplitude of $\alpha \rightarrow \beta$: $<\nu_\beta | \nu_\alpha>$
The probability will then be the absolute value of it squared:
$P(\alpha \rightarrow \beta) = | <\nu_\beta | \nu_\alpha> |^2$
Now if you sum the probability to get any result $\beta = e, \mu , \tau$ you will get unity (because that's the total probability of the possible outcomes- either $\alpha$ will remain $\alpha$ or it'll oscillate/change to something else):
$\sum_{\beta = e, \mu ,\tau} P(\alpha \rightarrow \beta) =1$
$\sum_{\beta = e, \mu ,\tau} | <\nu_\beta | \nu_\alpha> |^2=1$
Now you could also write the above inner product in terms of the mass eigenstates.

Last edited: Jun 24, 2015
15. Jun 24, 2015

### Buzz Bloom

Hi ChrisVer:

I am confused about the physics represented by the matrix equation in your post, I am also confused by the term "eigenstate". I understand the mathematics of vectors and matrices and also about the eigenvectors and eigenvalues of matrices. Here is a definition of "eigenstate" I found on the internet:
In the matrix equation in your post, "|νe>" is referred to as an eigenstate, but in the equation it is a scaler component of a vector rather than an "eigenvector".
My mathematics understanding of "eigenvector" is that it is the solution of a matrix-vector equation of the following form:
M × V = a × V. ​
M is a given square matrix, V is an unknown vector, and a is an unknown scaler.

The vector V that is a solution of this equation is an eigenvector, and the correponding solution value for a is its corresponding eigenvalue.
(See https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors .)

I interpret the definition of "eigenstate" to be that of a physics term rather than a math term. Can you help me understand the mathematical concept of "eigenstate"? That is, what mathematical equation has a corresponding eigenvector solution which is physically interpreted to be an eigenstate? If this is an equation of the form I give above, then what do the component values in the matrix M represent physically, and what would the corresponding eigenvalue a represent physically?

Buzz

Last edited: Jun 24, 2015
16. Jun 25, 2015

### vanhees71

I haven't read this thread completely. I lack of time these days to read and post to PF :-(((. If I find the time, I'll write another Insights article in my "Didactic Sins" series (of which so far only one appeared ;-)) about neutrino oscillations. It's one of the very interesting topics which is confused maximally in the literature. The handwaving treatment is very misleading although leading to an approximately correct mixing formula.

First of all within QFT only mass eigenstates are sensible asymptotic states that can be interpreted as particles. Neutrinos are, however, never produced and observed in their mass eigenstates but in their flavor eigenstates (according to the Standard Model extended by neutrino masses and mixing). Consequently, strictly speaking, you don't observe the neutrinos but the well-defined particle-asymptotic states created by the interaction of the neutrino quantum field with the detector material. So, if you consider the entire production process (e.g., a positive pion decay to anti-muon and muon neutrino) and detection process in the detector (e.g., by elastic scattering with an electron in the detector and looking for the electrons to detect the appearance of electron neutrinos).

Quantum-field theoretically you have to work with wave-packet asymptotic states in the production and the detection channel, because you need the space-time information for these processes. Of course there's no violation of energy-momentum conservation for each energy-momentum eigenmode in the Fourier decomposition of the wave packet, as usual in QFT. Taking into account the full system, there's no such violation due to the neutrino oscillations. Of course, often the time information is not observed. Then you have to integrate the neutrino detection rate (neutrino current) at the place of the detector over time. There are many good and, unfortunately, also many bad paper and (even worse) treatments in textbooks, confusing the students and researchers alike. It's an example to take Einstein's advice seriously to make things as simple as possible but not simpler!

17. Jun 25, 2015

### ChrisVer

A mass eigenstate is the eigenstates of the mass matrix...
In particular the basis in which the mass matrix becomes diagonal...
If you have the mass-term $v^T M v$ you can try to look for a transformation $v \rightarrow S v$ such that $S^T M S \equiv M_{diag}$ will be a diagonal matrix... In that case the state $Sv$ is a mass eigenstate...
The same is true for what you've written for eigenstate/eigenvalue with $Mv=av$...
In a ompletely analogous way a flavor eigenstate is given by doing the same for the interaction term in the Lagrangian... However for the neutrinos it is impossible to diagonalize both the mass matrix $M$ and the interaction matrix/hamiltonian $H$ since they are not commuting..and that's also how you end up with the superposition of the one eigenstate with the other's and the PMNS matrix. That thing is known from QM.
That is also why I wrote $| \nu_e >$ and not a scalar... the PMNS matrix is giving the mixing of the mass eigenstates of the flavor ones... It's like a change of basis.

Last edited: Jun 25, 2015
18. Jun 25, 2015

### snorkack

Suppose that you can detect the rest mass of a neutrino from a specific interaction.
Say, a beta decay of a tritium atom. A beta particle is a stable particle, so whatever the practical difficulties, in principle you can measure its energy and momentum with arbitrary precision. Suppose that you do so, and also measure the momentum of the recoiling He-3 atom with arbitrary precision, as well as the initial momentum of the tritium atom (here the finite lifetime of triton does pose a lower bound, but it is very low).
Do you then get the energy and momentum of that individual electron neutrino with precision to identify it as also belonging to one specific mass eigenstate?

19. Jun 25, 2015

### vanhees71

The electron neutrino is not a mass eigenstate, but a flavor eigenstate. So the tritium decays to a anti-electron-neutrino + electron+He3. Of course, you have always energy-momentum conservation in each event (note that we measure asymptotic states, for which you have full energy-momentum conservation). However there's a uncertainty due to the fact that the flavor eigenstate is not a mass eigenstate. So with some probability you decay to one of (at least) three neutrino mass-eigenstates, and correspondingly the total invariant mass of the electron+He3 has a finite width! Of course, we are very far from measuring this. The upper bounds for the neutrino masses from direct measurement of the endpoint of the tritium beta-decay spectrum is pretty much larger than the bounds from neutrino-mass measurements, if I remember right. Maybe there's an improvement by KATRIN soon?

20. Jun 25, 2015

### Orodruin

Staff Emeritus
Even KATRIN will not have sufficient resolution to tell the mass eigenstates apart.

I have already said this in the thread, but it is worth repeating: If you have enough energy resolution to tell the mass eigenstates apart, you will not see any oscillations and the tritium electron spectrum will be a superposition of the spectra of each mass eigenstate with coefficients proportional to their mixing with the electron neutrino state. This would be completely analogous to what can be observed in the charged current interactions of quarks.