Let me also suggest a very interesting paper related to the topic, hope it would be useful: http://prd.aps.org/abstract/PRD/v85/i11/e113003

Dear Thomas,

Attempts to derive the neutrino masses (are any masses actually) from more fundamental principles are always welcome, so let me first of all congratulate you for your efforts.

As I am one of the "neutrino guys" as my colleagues call me, let me help you with some concepts and data that is already established by the community and what is still open to discussions (what is the major part).

1) Neutrinos live in a quantum superposition of states. They can be defined from a flavor base, from which they are recognized by their interactions with the charged leptons, what give these states their names: electron-neutrino, muon-neutrino and tau-neutrino. On this base, the observable is "Flavor" and the eigen-values are the "leptonic family number" (http://en.wikipedia.org/wiki/Lepton_number). On the other hand, neutrinos can be defined by their kinematics, or a mass base. In this base, neutrinos are recognized by their mass, which is the fundamental observable. The eigen-values are m1, m2 and m3. So, and this is very important, the electron neutrino does not have a single mass... it actually has a superposition of the three masses m1, m2 and m3. The "amount" of each mass is given by the leptonic mixing matrix. Another important remark is that we don't know yet which mass is the largest. We may have a regular hierarchy: m1 < m2 < m3, but we could also have a inverted hierarchy m3 < m1 < m2. So, for now, we only have upper limits for the largest mass (whichever it is).

2) Neutrinos cannot have the same mass. There is no lower limit for the neutrino mass (to my knowledge) and your value is well within the upper limit for the largest mass. But there is a very precise measure (less than 5% uncertainty) saying that the three masses MUST be different. There was a time when the available data would permit something like m1 > m2 = m3... but not anymore. Now we know that any degeneracy between masses is strongly disfavored (more than 5sigma) by all data.

With these two points clear, I have to ask: Does your value for mass comes from interaction properties? Because if it does, then we have a problem. You see, when neutrinos interact, their masses are mixed up by the quantum mechanical character of the superposition. So there is no way an electron neutrino could have a definite mass. This is a great problem for neutrino physics today because any proposed mass generation mechanism must be on the kinematic base, and not on the flavor base. That is why, for instance, we know for sure that IF the Higgs mechanism really is the responsible for giving mass to all particles, it cannot give mass to the neutrinos. That is because the Higgs mechanism happens as a flavor interaction, and not as a kinematic effect. An preferable alternative (perhaps because of the lack of other alternatives) is the Seesaw mechanism, which occurs only on the mass base.

Well, that is not all we know, but it is pretty much what everything related to mass. There are tons of data th...

Hi Gustavo,

Thanks for the nice discussion. Could you please explain a bit more why Higgs mechanism can not give mass to neutrino ? Just another related question what do you mean while referring kinematic base and flavor base ? I was thinking that Higgs is an outcome of kinematic effect also.

Hi Tapasi,

I am not a theoricist, so what I am going to say in the following lines may need some review by a serious "neutrino theory guy". So if anyone finds some inconsistency reading this, please let me know.

> Could you please explain a bit more why Higgs mechanism can not give mass to neutrino ?

The Higgs mechanism gives mass to the known fermions by making use of a Yukawa coupling:

http://en.wikipedia.org/wiki/Yukawa_interaction

This kind of Lagrangian employs both left and right-hand part of the field. As we know, neutrinos only have the left-hand mode (while antineutrinos are right handed). So there comes the first challenge: even if the mass of neutrinos would come from the Higgs, they would have to have a different kind of interaction with the Higgs field. Another difficulty comes from the fact that the Yukawa Lagrangian is well defined on the flavor base while the neutrino mass must come from a different base than this, or the oscillation experiments could not be explained. And so, there comes the second answer:

> what do you mean while referring kinematic base and flavor base ?

Well, I admit that this one might have come from a poor choice of words. By "kinematics base" I am referring to the quantum base here the Hamiltonian is diagonal. This is the base where the masses are defined and it is the one that describes the neutrino propagation through space (and that is why I used "kinematic base"). On the other hand, the weak interactions involving neutrinos are defined on the "flavor base", where we know which one is the eigenstate related to the charged fermions (electron-neutrino, muon-neutrino and tau-neutrino). This is also the base where usually the Higgs mechanism acts. But the neutrinos are a really pathological case, since the base where the flavor is define is the one where the mass is mixed, and vice-versa. So we need two basis in order to fully describe a neutrino: one where mass is defined (so kinematics happens on this one) and another where flavor is defined (interactions happens on this one). Is it clear now? As I understood, you mention that the Higgs mechanism is also a kinematic effect. Well, it is. But it works because for all other particles, kinematics and interactions happens on the same base and this is not true for neutrinos.

If you have any further questions, I would love to extend the discussion, so feel free to ask.

Gustavo

Thomas, I am reading your linked document with quite some interest. I would suggest that until you have a formula that also encompasses the muon as well as the electron and Tau in a consistent formula then it will be difficult to draw any conclusions quite yet. The other advantage of including the muon is that its properties are more accurately known, and so the chances of numerical coincidences become much less. I would also take on board Gustav's comments about the chance of neutrino masses being identical as "not likely" and look at your theory to see how that might impact upon it. Apart from the occasional glitch like faster than light neutrinos, these empirical folk are quite trustworthy ;-) Keep up the great effort though, that link of yours has some pretty interesting stuff in it.

You might also consider my own little "discovery" in light of your theory:

m_electron / (m_Pr * alpha^10 ) =(7/10)^2 to well within CODATA 2010 error accuracy. m_Pr = reduced Planck mass (not the Planck mass), alpha fine structure constant.

Coincidence? Well the power of 10 in alpha (fine structure constant) has a viable theoretical explanation. The (7/10) was supposed to be an integer if my theoretical derivation made sense, so the fact it is 7/10ths rather than a small integer remains at present a numerical mystery. I have not published (well I guess now I have!) it because I cannot (quite) explain the 7/10'ths - the CODATA 2010 error analysis gives 0.699999 (21)

I am however more than satisfied now that alpha will likely play a vital role in our understanding of mass. (As Feynman said, yes indeed we should lie at wake at night wondering why alpha is what it is)

@Denys Poda - that linked paper is behind a paid firewall - no use to me at all. I would love to read it, so http://arxiv.org/pdf/1205.4068v2.pdf is the place to go.

@Gustavo,

So, if I have interpreted you correctly, any formula that predicted neutrino mass from lepton mass would also have to have within it an explanation and values for the lepton mixing matrix you refer to?

My thanks to people who participated regarding this question. You have helped me update my thinking. Likely, based on newer data (and some of your comments) the Quasi-Degenerate concept (noted in the J. Beringer article I cited in the original question) does not apply.

I can still obtain candidate masses for neutrinos from an extension of a periodic-table-like array I developed for masses of quarks and charged leptons. The approximate formula that generates such masses also generates (for integer parameters seemingly appropriate for neutrinos) the following (and some lighter) masses (in units of eV/(c^2)). Presumably, only 3 apply. Likely, but not certainly, these are the first, second, and fourth of … 9.161×10^(−2) , ~1.5×10^(−4) , ~2.5×10^(−7) , and 4.100×10^(−10).

I attach the first half of a new book. This book updates my research and includes the above results. Again, thanks.

@Andrew

Yes, you interpreted me correctly =) So far, any attempt to describe a mechanism for neutrino mass generation has also to deal with the mixing. But that is not an absolute requirement.

As I said before, the neutrino masses are not defined in the flavor base. However, if you devise a mechanism that is not related to the Electroweak interation (not using alpha, e, theta_w...) then you don't need to bother with the mixing. The thing is that we are "attracted" to think about the neutrino as being "electron neutrino" but this definition is linked with the Electroweak interaction (thus the "flavor base"). If one abandon any labels that come with flavodynamics and think about the neutrino while it is not interacting (OR it is interaction, but with a new type of interaction) then you can think of mass separated of mixing. In short: mixing is related to weak interactions not to mass it self.

An example would be if one comes up with a theory that says that neutrino masses comes from gravitational interactions and these interactions do not take place in the same base as the weak ones. Another example would be to talk about a completely new interaction, perhaps with dark matter, that generates mass for the neutrinos. Last, but not least, you can always talk about spontaneous mass generation, as an fundamental property, that would not be related to any electroweak property. It could be related to the structure of space, for example.

So, to conclude: Whenever you talk about "mass of the electron, muon and tau neutrino" you have to say something about the mixing. But if you abandon these "weak labels" and say "mass 1, 2 and 3", then you don't have to bother about the mixing.

Gustavo:

Thanks.

- Tom

@Gustavo,

Do you happen to know a good empirical reference work that fills in the known values and accuracies of all the empirical mixing angles and eigenvalues in the Standard Model..i.e. the detail behind : http://en.wikipedia.org/wiki/File:Standard_Model.svg

I have a single Lie Algebra I am investigating that seems to have all the right structures and nodes and in the absence of any physicist being at all interested in what I am doing, I need to check the detail myself. Unfortunately I am weak in group theory so have had to spend several months just learning how to do all this.

I presume with an equation that predicts all these angles and eigenvalues, the mass predictions should then be a done deal direct from the quantum numbers/flavors if I understand your talk about neutrinos?

@Andrew

Yes, I know quite a few. I work with both experiment and theory of neutrinos, always trying to be in touch with the most up to date data. But it is more straightforward to check the needed values on Wikipedia, as I do my best to keep them up to date as well ;)

Go to http://en.wikipedia.org/wiki/Neutrino_oscillation

And then to the section "Observed values of oscillation parameters"

There you will find the latest measurements of the mixing angles and the squared mass differences. If you are curious about why and how these properties are measured, check the PDG (Particle Data Group) review of neutrino mixing:

http://pdg.lbl.gov/2013/reviews/rpp2012-rev-neutrino-mixing.pdf

As about your theory: if you feel like showing it, don't be shy ;) If I can be of any help, please let me know.

@Gustavo,

I have attempted to show it. I have spent three years attempting to show it, after ten years of reviewing the background of the unification problem and proposing (yet another) hypothesis to finally make sense of it. (yes I do have a sense of humor about the endeavor, and yes, maybe I can present it in a more engaging way)

I quote below a typical response from a distinguished professor of QM to an inquiry from me to "come to his physics department xyz and discuss a possible topic of study for a PhD thesis in unified theory"

Here is the reply:

"I guess I should start out by being direct and say that I share the commonly encountered scepticism regarding proposed theories that aim to solve all issues of quantum and gravitational physics in one fell swoop. Beyond that scepticism---which might be criticised as no more than evidence of a closed mind---there are the endless demands on time…we are all very busy. "

And this from the alma mater where I did my engineering degrees (with first class honors - not to be too proud about it)! I suspect all physicists wear those peril sensitive sunglasses form "Hitchhikers Guide to the Galaxy" that turn instantly black whenever confronted by any possibly dangerous knowledge from outside their physics department.

@Andrew,

I feel sorry and ashamed at the same time. I understand a little of the so called "skepticism" about new theories and models, since it appears we have an abundance of inspired people who produces little sweat with theirs studies, when it should be the opposite. What I really see is a kind of phobia, disguised by skepticism: there is this general fear that you will loose your time evaluating and collaborating with people that are too much inspired to sit on their chairs for a few years and apply scientific method on their ideas, like everybody should. I don't deny that these people are around and they are many. I think it is Gerard t'Hooft who call this kind of person "a quack", since like a duck they tend to be loud about accomplishing little. Well, personally, I see them as misguided. I can not blame people for being more inspired than me, can I? All I can do is advise when needed to show how to be critical and to keep in touch with reality when developing a theory. Of course most theories don't survive their youth but that is just part of the process. When the overflowing inspiration is mature enough to becomes sweat, I guarantee these people will be incredibly more capable than those who today deny their chance. I don't say this because I think of it as a poetical justice of some sort, but actually because these people, the "mature quacks" will have a different history and this will reflect on their accomplishments.

I am "a quack" myself, since behind my works with simulation and data analysis (let us call them my official research), I also work in a particular unification scheme. What makes me a quack is that I have a lot of inspiration but no time to study the necessary elements to develop my model to the point where it can be confronted by real data. I hope one day I will find that time in my life and that, being it right or wrong, I will be able to write down on paper what my inspiration tells me to (I have to sweat on it). So being a quack is not a bad thing, if you know what "lacks to the quack" and pursue it. That is my case.

@Gustavo,

Yes, but I can understand the reluctance of "engaging" with outsiders having worked in engineering research in a university department for 12 years myself - the last thing you wanted is an obsessive pest who just won't go away. Also, for the more rational inquiries, there is not much up side in it for you to listen - they either have a valid point , (which can cause embarrassment for your department), or they are wrong, in which case you have wasted your time! But where has the passion for their subject gone - that is what I found most disturbing and surprising.

As far as quacks are concerned, they can usually be found out because they will forever keep sidestepping simple direct questions and will never admit they "don't know", or made a mistake, and will never ask questions to clarify a point of view. I call it gratuitous pontification.

I for one do not want to keep working on an hypothesis that has turned out to be a mathematical curiosity. I have a day job, and lots of other interests besides. What irks me is the lack of opportunity to get a professional opinion, which at this stage of the work would provide very valuable guidance, and would only take a couple of hours over coffee and a white board.

@Andrew

I could not have said it better =)

I don't think I am the most qualified person to look at your work, but I would like to spend some coffee time over it (just curiosity). Feel free to send me whatever you want to show (sometimes we don't because it is too early). You can reach me at [email protected]

Permit me to post a follow-up note. And, again, thanks to people who participated regarding this question.

I think I have sharpened my attempted research, in part to include possibly of being able to anticipate the extent to which neutrinos are Direct or Majorana fermions. The Dirac-and/or-Majorana result comes from math that seems to correlate with numbers of instances of elementary particles.

Here is a summary of some findings.

We predict masses {or mass-math eigenvalues} for neutrinos, of which 2 would be Majorana fermions (~0.125 eV/(c^2) and ~0.058 eV/(c^2)) and 1 would be a Dirac fermion (~0.058 eV/(c^2)).

I am unable to determine uncertainty ranges for the masses. So, for example, a previous estimate of ~0.092 eV/(c^2) for 1 mass could be as good as the current ~0.125 eV/(c^2) for the same mass. And, the ~0.092 eV/(c^2) value might correlate with a sum of masses that is less than the astrophysics-based observations suggesting an upper limit of ~0.23 on the sum of the masses.

I attach information related to this work.

Again, thanks.

@Martin

Thanks for calling my attention on the fact that I didn't mention the quarks mixing in my previous post. You are right on remembering that quarks also experience mixing and we know that for far longer than neutrinos. In fact, it was the kaon mixing that inspired Pontecorvo to propose the same as an possible explanation for the neutrino disappearance. 

On the other hand, your statement about how the Higgs could give mass to the neutrinos is not quite correct. It would be 100% correct if right handed neutrinos were known to exist. It is not by a meter of theoretical choice that we don't put right hand neutrinos on our lagrangians,on the contrary :it is purely experience. In order to skip our detection for so long, right handed neutrinos would have to be sterile or too massive. On the first case,how would the Higgs interact with them?  On the second,  how could the Higgs be responsible for giving different masses via a coupling that is left -right symmetric? 

Thomas

I agree with the astrophysics-based observations suggesting an upper limit of ~0.23 eV on the sum of the neutrino masses.

The 2014 issue of the Review of Particle Physics (Chinese Physics C, vol 38, nr 9, September 2014) recommends the upper limit 2 eV for the mass of the e-neutrino, the other species having mass limits in the MeV range.