Neutrino theory regarding rest masses

[snorkack]

I am confused by your two questions here, especially the second.

Why solid lattice?​

For each the nucleon to have a zero momentum, the temerpature T must be 0. I think this requires that the collection must be a solid, since a liquid of gas will have a non-zero T.

Um. Both isotopes of He are liquid at absolute zero.
Say you have a pool of liquid He-3 at absolute zero, so no vapour pressure and vacuum above the surface.
And then you are operating it as electron antineutrino detector. By reaction
He-3+nuebar->t+e+
Can you measure the energy of the positron emitted?

A solid made up of identical nucleons I think must form a crystal lattice.

Why attempt to localize?​

I do not understand "localize" in this context. I did not specify that the solid lattice is limited in size, but I did suggest that a thickness be determined to optimize the tradeoff between a (1) large frequency of interation events and (2) a low frequency of disturbing the energy and momentum of the produced electron e and/or nucleon N_out. I did suggest the lattice might be a thin sheet, but it could also be a thin spherical shell with the neutrino generator at it's center.

So how are electron energies measured with a great precision, like in these tritium decay experiments?

 

[Buzz Bloom]

Hi Orodruin:

I suggest you familiarise yourself with the CKM matrix and quark mixing.

Before tracking down a textbook, I decided to look at the Wikipedia article
https://en.wikipedia.org/wiki/Cabibbo–Kobayashi–Maskawa_matrix .
Here is a quote I would like to ask about:

The constraints of unitarity of the CKM-matrix​

This seems to be saying that the CKM-matrix is unitary. To refresh my memory from a linear algebra course I took as an undergraduate in the 1950s, I found the definition at
https://en.wikipedia.org/wiki/Unitary_matrix :

In mathematics, a complex https://www.physicsforums.com/javascript:void(0) https://www.physicsforums.com/javascript:void(0) U is unitary if its conjugate transpose U* is also its inverse.​

Since you suggested I might learn about the PMNS matrix by studying the CKM-matrix,
does the first Wkipedia quote imply that the PMNS matrix is also unitary?

BTW, I confess I tend to stay away from trying to use a textbook as a reference source. If it is about a topic I know little about, I find it very difficult to learn anything specific I want to understand. Most textbooks I've looked at recently require reading thoroughly from the beginning, and remembering what is read, since any later disccusion does not refer back to definitions of technical terms or notation, and usually there is no glossary or suitable index. In other words, they are terrible reference sources unless you have previously taken a course using the particular textbook, and you still retain a reasonably good memory.

Thanks for your suggestion,
Buzz

 

[Orodruin]

Since you suggested I might learn about the PMNS matrix by studying the CKM-matrix,
does the first Wkipedia quote imply that the PMNS matrix is also unitary?

The unitarity of the CKM matrix is a prediction from the Standard Model and has to be tested experimentally (and it has been). The PMNS matrix is also generally assumed to be unitary under some conditions, but there are some theoretical ideas which would make it almost unitary, but with small corrections.

 

[Buzz Bloom]

Hi Orodruin:

The PMNS matrix is also generally assumed to be unitary under some conditions, but there are some theoretical ideas which would make it almost unitary, but with small corrections.

Can you post citations of articles that explain
"assumed to be unitary under some conditions",
and
"some theoretical ideas which would make it almost unitary, but with small corrections"?

Also re

The unitarity of the CKM matrix is a prediction from the Standard Model and has to be tested experimentally (and it has been).

Can you post citations of articles about the experiments that confirmed the unitarity of the CKM matrix?

Thanks for your discussion,
Buzz

 

[Orodruin]

Can you post citations of articles about the experiments that confirmed the unitarity of the CKM matrix?

See http://ckmfitter.in2p3.fr and references from there.

Can you post citations of articles that explain
"assumed to be unitary under some conditions",
and
"some theoretical ideas which would make it almost unitary, but with small corrections"?

See hep-ph/0607020 and references therein.

 

[Buzz Bloom]

Hi Orodruin:

Thanks for the citations. I am sure it will take me quite a while to digest them.

Regards,
Buzz

 

[Buzz Bloom]

Hi Orodruin:

I scanned the article hep-ph/0607020 you cited regarding my question:

Can you post citations of articles that explain
"assumed to be unitary under some conditions",
and
"some theoretical ideas which would make it almost unitary, but with small corrections"?​

As whole, it is clearly way over my head, but it what I asked for. Here is a quote that I think I almost understand.

Without attaching ourselves to any particular model, we have studied a minimal
scheme of unitarity violation -MUV-, considering only three light neutrino species and
with the usual unitary matrix U_PMNS replaced by the most general non-unitary one.​

I underlined the phrase I would particularly like to undestand. I found several articles on the internet that discussed "light neutrino species", but none defined it. Could you do that for me please.

Thanks for your help,
Buzz

 

[ChrisVer]

Light neutrino species is that you have 3 light (rest mass less than 45 GeV) neutrinos , such as \nu_e, \nu_\mu, \nu_\tau.

 

[Buzz Bloom]

Hi ChrisVer:

Light neutrino species is that you have 3 light (rest mass less than 45 GeV) neutrinos , such as νe,νμ,ντ\nu_e, \nu_\mu, \nu_\tau.

The article was from 2007. I understand that it is now generally accepted, as very likely to be so, that the sum of these 3 masses is about 430 meV. This is about 11 orders of magnitude less than this 45 GeV threshold for being a "light neutrino". Can you summarize the likely implications regarding the conclusions of this paper about the possible non-unitarity of the neutrino mixing matrix in the light of this enormous difference between the concept then of a light neutrino and the reality of today's understanding about these masses?

Thanks for you post,
Buzz

 

[ChrisVer]

The experiment in LEP showed that there are three active (=subject to weak interactions) light (of mass less than 45GeV) neutrino species...that's what fitted the experimental data best... this doesn't seem such an enormous difference, at least not to me... even 400meV (their sum) is less than 45GeV... it's just that there are no other light neutrinos in the inbetween spectrum.
Plus I don't know about PMNS non-unitariness.

 

[Buzz Bloom]

Hi ChrisVer:

For some reason it appears weirdly in your PC...

I think this must be a font problem. Thanks for interpreting it for me.

Can I ask you what your background is

I was trained in mathematics, mostly applied, and I also had some introductory physics cources. Befor retiring, I had a career in software development, especially concerning databases. As a lifetime hobby, I have tried to educate myself about a variety of scientic topics: mostly in (1) molecular biology (relating to the origin of life), and (2) in physics, especially GR and cosmology. Earlier this year I began to study some atmospheric physical-chemistry regarding global warming.

Very recently cosmology had lead me to issues about the neutrino, and also QM. I have found the theoretical and experimental physics about the neutrino to be a continuous fascinating mystery. I have formed the opinion that one cannot know just a small amount about the neutrino. Every time I thought I had learned something new about the neutrino, it soon became clear that what I had learned was not quite completely correct. It was just the tip of an iceberg which needed deeper study to clarify what I had thought I had just learned.

BTW, my first job after graduating college involved using a Marchant electro-mechanical calculator to find several eigen values and vectors of a 41x41 matrix. The method involved iteratively re-multiplying an arbitrary initial unit vector by the matrix until the process converged within the precision achievable with the calculator. After an eigen vector was found, a new initial unit vector was selected which was then made normal to all the previously found eigen vectors.

The physics forum has been extremely helpful.

Thank you for your patience and for all your help,
Buzz

 

[Buzz Bloom]

Hi ChrisVer:

even 400meV (their sum) is less than 45GeV

Another small but curious mystery. What makes the threshold value of 45GeV particulary special?

Thanks again,
Buzz

 

[Orodruin]

Hi ChrisVer:
Another small but curious mystery. What makes the threshold value of 45GeV particulary special?

Thanks again,
Buzz

It is half the mass of the Z boson and the constraints come from the Z boson hidden decays.

 

[Buzz Bloom]

Hi Orodruin:

It is half the mass of the Z boson and the constraints come from the Z boson hidden decays.

Thanks for your post,
Buzz

 

[Buzz Bloom]

Hi fzero:

Then this [(M)] is a real matrix with entries involving the eigenvalues and products of sines and cosines of the PMNS angles.

I have been pondering this for a while, trying to remember what I think I learned while an undergraduate. I am pretty sure that (M) can not have real components, since the components of it's eigenvectors are complex. I think that the eigenvectors of any real matrix must have real components. If you are not sure whether this is correct or not, I will start a thread in the math sub-forum.

Thanks for the discussion,
Buzz

 

[ChrisVer]

I am not really sure, but think about this... if u is an eigenvector, isn't iu an eigenvector too? Eigenvectors are defined from:
A \cdot \textbf{v} = \lambda \textbf{v}
So both \textbf{v} or i\textbf{v} satisfy the above.
But you may be right in figuring out that M doesn't have to be real - it should only be hermitian since the mass eigenvalues cannot be imaginary... I am saying "may be right" because I haven't performed the calculations and I am not going to do that either (since wolfram is really bad in helping me perform them computationally). Maybe someone with a better program at hand can input the PMNS matrix with the cos,sin and exp[i*delta] and find whether the result is purely real or not.

 

[fzero]

$M$ must be Hermitian (and you can prove that from the expression $UDU^\dagger$), but my comment was based on looking at the actual entries of $U$ and how the phases entered that product. I suspect that trig identities will make the terms proportional to $e^{\pm i \delta}$ vanish. The actual calculation is tedious, but not too difficult, so you're invited to check.

 

[ChrisVer]

Let's try my luck... Instead of taking all the products etc, I will only check a suspicious term (from the form of the PMNS matrix):]

Now I will take the M_{12} that is the:
\begin{align}
M_{12} &= U_{1m} (DU^\dagger)_{m2} \notag \\
&=-c_{12} c_{13} s_{12}c_{23} d_{11}- c_{12}^2 s_{23} c_{13} s_{13} e^{-i\delta} d_{11}+ s_{12} c_{13} c_{12}c_{23} d_{22} - s_{12}^2 s_{13}c_{13} c_{23} d_{22} e^{-i\delta} + s_{13}c_{13}s_{23}e^{-i\delta} d_{33} \notag
\end{align}

I don't see how you could get rid of the e^{i\delta}'s...

 

[Orodruin]

I don't see how you could get rid of the e^{i\delta}'s...

You cannot. If you could, there would be no possibility for neutrino oscillations to violate CP.

Note that the M you are talking about here is actually $mm^\dagger$ if neutrinos are Dirac, where $m$ is the neutrino mass matrix (which is proportional to its Yukawa couplings). The matrix $m$ would be a completely general complex matrix and it would take a biunitary transformation to diagonalise it. Having the product between it and its conjugate results in a Hermitian matrix.

If neutrinos would be Majorana, then $m$ is a complex symmetric matrix, diagonalisable as $d = UmU^T$, where the phases of the entries in the diagonal matrix $d$ depend on $U$ (which is not unique). Again, the expression $mm^\dagger$ is Hermitian and has absolute values squared of the entries of $d$ as eigenvalues.

 

[Buzz Bloom]

Hi ChrisVer:

I think that the eigenvectors of any real matrix must have real components.

if uu is an eigenvector, isn't iuiu an eigenvector too? Eigenvectors are defined from:
A⋅v=λv A \cdot \textbf{v} = \lambda \textbf{v}
So both v\textbf{v} or ivi\textbf{v} satisfy the above.

You are correct. I should have said, "may have all real components." I was confused by my experience when I was only working with real normalized eigenvectors.

Thanks for your post,
Buzz

 

[Buzz Bloom]

Hi fzero, ChrisVer, and Orodruin:

MM must be Hermitian

I don't see how you could get rid of the eiδe^{i\delta}'s...

You cannot.

To summarize from the discussion: it is certain that:

(1) M must be Hermitian, that is, it's conjugate transpose is it's own inverse
(2) at least one of M's nine components is not real.​

Thanks for the discussion,
Buzz

 

[Orodruin]

(1) M must be Hermitian, that is, it's conjugate transpose is it's own inverse
(2) at least one of M's nine componets is not real.

1) No. M is hermitian, but you have given the description of a unitary matrix. A hermitian matrix is equal to its own hermitian conjugate.
2) No. We do not know this. This is still to be determined experimentally.

 

[ChrisVer]

This is still to be determined experimentally.

You mean like determining δ=0?

 

[Orodruin]

Yes, if $\delta = 0$ or $\pi$, neutrino oscillations are not violating CP.

 

[Buzz Bloom]

Hi Orodruin:

M must be Hermitian (and you can prove that from the expression UDU†UDU^\dagger)

Sorry about the appearance of the special characters in the quote, I think there is somthing flaky in my computer.

M is hermitian, but you have given the description of a unitary matrix.

Underlining in above quotes is mine.

Sorry about my confusion. The vocabulary for the variety of complex matrix types is not yet well re-embeded in my mind.

I also said:

at least one of M's nine componets is not real.​

You commented:

No. We do not know this. This is still to be determined experimentally.​

If my mind is now working OK, if M is Hermetian, then M's three diagonal components must all be real.
Also, Wikperida https://en.wikipedia.org/wiki/Hermitian_matrix defines a Hermetian matrix:

a square matrix with [at least some] complex entries that is equal to its own conjugate transpose (bracketed text my addition)​

Therefore M is Hermetian also implies that two or four or all six of it's non-diagonal element are complex.

Thanks for your post,
Buzz

 

[Orodruin]

Therefore M is Hermetian also implies that two or four or all six of it's non-diagonal element are complex.

No, your logic is failing here. The off diagonal terms can also be real, they do not need to be, but they may be. There may be zero non-real elements in the matrix.

 

[ChrisVer]

A real symmetric matrix is Hermitian... Hermitianity is the relation that A^\dagger = A... A real symmetric matrix is satisfying the hermitianity condition.
A = \begin{pmatrix} a & b \\ b &c \end{pmatrix} with a,b,c \in \mathbb{R} has A^\dagger = A^T =\begin{pmatrix} a& b \\ b & c\end{pmatrix}= A.

In this case again, it's as I asked Orodruin too, if \delta =0 (or \pi) then the e^{\pm i \delta} doesn't stand for a complex number...it's \pm 1...and there are no possible complex elements in M (of course the real numbers are just a subset of the complex numbers)...except for a Majorana case(?).

 

[Buzz Bloom]

Hi ChrisVer:

Hermitianity is the relation that A^†=A. ... A real symmetric matrix is satisfying the hermitianity condition.

(I edited the garbled quote that my computer put above to try to make it look like the original. How did you enter the dagger? is it a TeX command? I think my computer flakiness is related to TeX.)

I will try to change the definition at Wikipedia to make it clear that a Hermitian matrix may have complex components.

Thanks for your post,
Buzz

 

[ChrisVer]

The dagger I use in latex is " ^\dagger ".. ^ is for the powering.
Well again I'm saying that the real numbers case is just a special case of the complex numbers [where the imaginary part vanishes] so there is nothing wrong in saying it is complex and it happening to be real...it's just that the extra operation of complex conjugation * is trivial.

 

[Buzz Bloom]

Hi ChrisVer:

I made a correction at https://en.wikipedia.org/wiki/Hermitian_matrix . I hope it sticks.

Regards,
Buzz