Majorana Neutrinos: Different Physics than Oscillation

[Vanadium 50]

You talked about actual leptons coming out of actual processes that are detected in the lab. Don't those processes normally involve a specific flavor of lepton? If so, it would seem like they should also involve a specific flavor of neutrino.

I tried to avoid getting into the question of mass vs. flavor eigenstates. Neutrinos are leptons, anti-neutrinos are anti-leptons. Yes, we can go into mroe detail and talk about individual flavors, but I deliberately avoided doing this.

 

[Vanadium 50]

I think you'd better let @Vanadium 50 speak for themself,

And yet you then jumped in and put some words in my mouth., Please don;t do that. hat's disrepsectful, unhelpful, unbcollegial and beneath you.

 

[Ken G]

And yet you then jumped in and put some words in my mouth., Please don;t do that. hat's disrepsectful, unhelpful, unbcollegial and beneath you.

I put no words in your mouth, it was exactly what you said. You suggested defining antineutrinos by any neutrino that makes an antilepton, is that not just what you said? And does it not immediately follow that by this definition, a neutrino is not its own antiparticle, even if it is a pure Majorano fermion? What is clear from the thread is terms like "is its own antiparticle", and "the particle and antiparticle are the same thing" do not have a consensus interpretation. This is precisely the reason I reframed the question some time ago, yet without any answer from anyone so far:

If neutrinos are established as being Majorana fermions, will there still be value in the term "antineutrino", and if so, what? (And the related question, will the term antineutrino be retired?)

 

[Vanadium 50]

that is why we call it "its own antiparticle".

I am trying to get us away from that woolly language.

I don't want people thinking a beam ofgneutrinos and a beam of antineutrinos are the same thing. That is experimentally not the case: I can run the current in the neutrino horn and get positive leptons in the detector, and then switch the current direction and now get negative leptons in the detector. This is not theory - this is an observed fact.

That's why I went through all the rigamarole of Weyl spinors.

 

[PeterDonis]

I can run the current in the neutrino horn and get positive leptons in the detector, and then switch the current direction and now get negative leptons in the detector. This is not theory - this is an observed fact.

Can you be more specific about which experiment this refers to? The only specific experiment you've referred to so far is the Cowan Reines experiment, which used a nuclear reactor as the neutrino source and so could not choose what kind of neutrinos were being detected--it was whatever kind were being emitted by the nuclear reactions in the reactor. What you're describing here seems like a neutrino source where we can choose what kind of neutrinos it emits at will.

 

[malawi_glenn]

Careful now, who is "we"? I think you'd better let @Vanadium 50 speak for themself

I meant "that is why particle physicsists in general write that pure majorana fermions are their own antiparticle". You should be able to read this in any quantum field theory textbook.

Hang on, we do say the "photon is its own antiparticle." Maybe not "all day long", but we do say it. Are you saying the situation is different with the Z boson? I mean, can it annihilate with another one or not?

What I meant is that we do not make a "fuzz" about the photon, the higgs, or the Z boson being their own antiparticles (whatever that means). Is there any value to you, the fact that those particles are their own antiparticle? To me, its not at all important. What is important to me is that if neutrinos are not pure Dirac, we have a strong case for beyond the standard model physics - there are more things to be discovered after this :)

what are we using to communicate? Words!

I can write down equations for you if you prefer that, after all - the language of physics is math. It would be better I think if we just said "the quantum field is transformed back to itself after successive CPT-transformations" ;) Or just, "pure Dirac fermion", "pure Majorana fermion" or "pseudo-Dirac fermion". Or, just stick with the Weyl spinors...

The device here is "if and when it is discovered that neutrinos are Majorana fermions, what will we do."

Ok now I understand better what you meant by that.

 

[malawi_glenn]

What you're describing here seems like a neutrino source where we can choose what kind of neutrinos it emits at will.

Perhaps this https://lbnf-dune.fnal.gov/how-it-works/neutrino-beam/

You basically create (charged) pions which decays to muon and neutrino. The pions can be deflected using an electromagnet. In this way you can get a beam of neutrinos from $\pi^+$ and from $\pi^-$ separetely, and thus one can measure if there is any difference between those beam compositions and interactions.

 

[PeterDonis]

Perhaps this https://lbnf-dune.fnal.gov/how-it-works/neutrino-beam/

You basically create (charged) pions which decays to muon and neutrino. The pions can be deflected using an electromagnet. In this way you can get a beam of neutrinos from $\pi^+$ and from $\pi^-$ separetely, and thus one can measure if there is any difference between those beam compositions and interactions.

Yes, this is very helpful, thanks!

 

[PeterDonis]

You basically create (charged) pions which decays to muon and neutrino.

This would be a weak interaction decay conserving lepton number, correct? So if we're getting positive pions, those would decay into positive muons (antimuons, lepton number -1) and muon neutrinos (lepton number +1), whereas if we're getting negative pions, those would decay into negative muons (lepton number +1) and muon antineutrinos (lepton number -1), correct?

 

[Ken G]

What I meant is that we do not make a "fuzz" about the photon, the higgs, or the Z boson being their own antiparticles (whatever that means). Is there any value to you, the fact that those particles are their own antiparticle? To me, its not at all important.

I never asked "is it a fuss if neutrinos are their own antiparticle," I asked, "if neutrinos are pure Majorano fermions, will there be any value in the term antineutrino, or will the term be retired." No one has yet answered that, even the person who said that there is no possible way that neutrinos are their own antiparticle.

What is important to me is that if neutrinos are not pure Dirac, we have a strong case for beyond the standard model physics - there are more things to be discovered after this :)

Yes of course, that is the exciting physics here. But to understand the significance to the lexicon, we need the answer to the question I have posed so many times now.

I can write down equations for you if you prefer that, after all - the language of physics is math.

Actually, the language of physics is the language we find in physics books, and that is not all math. Words matter, that's why we are literally using them right now. Why do I have to justify the importance of words just to get a fairly straightforward question answered?

 

[PeterDonis]

if neutrinos are pure Majorano fermions, will there be any value in the term antineutrino, or will the term be retired.

Does the Majorana Collaboration have anything to say about this? At any rate they would seem to be the ones in the best position to give an argument for the "just retire the term" position.

 

[PeterDonis]

if neutrinos are pure Majorano fermions

I'm not sure if this is even a possibility. It seems to be established that the Standard Model could be expanded to accommodate a Majorana mass for neutrinos, but I'm not sure whether it could accommodate neutrinos only having a Majorana mass. (If nothing else, one would have to explain why the standard Higgs mechanism doesn't give neutrinos Dirac masses, even though left-handed neutrinos, at least, are weak isospin doublets with the charged leptons.)

 

[PeterDonis]

This paper looks like a more detailed treatment of the Majorana neutrino question by a Fermilab physicist:

https://arxiv.org/pdf/0903.0899.pdf

There's an item in this paper that I'm not sure I understand. Fig. 1 in the paper shows the Feynman diagram that is expected to dominate neutrinoless beta decay. The accompanying text notes that each vertex has to conserve lepton number; that means the neutrino coming out of the left vertex has to be an antineutrino (since it's created along with an electron), but the neutrino going into the right vertex has to be a neutrino (since it gets converted to an electron). Hence, according to the text, the diagram cannot happen unless the neutrino is its own antiparticle, so the same particle can participate in both vertices.

But if the neutrino is its own antiparticle, wouldn't its lepton number have to be zero? And if that is the case, wouldn't both vertices then violate lepton number conservation? And wouldn't that then undermine the argument that led to the inference that the left vertex has to emit an antineutrino whereas the right vertex has to absorb a neutrino?

 

[Ken G]

Does the Majorana Collaboration have anything to say about this? At any rate they would seem to be the ones in the best position to give an argument for the "just retire the term" position.

Yes I agree. Their paper (https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.130.062501 ) does not speak directly to the issue of neutrinos being their own antiparticle, instead they focus on violation of lepton number conservation, but the violation can be a rare thing. Their interest is in the creation of a matter universe, because they say: "Neutrinoless double-beta decay (0νββ) is a hypothetical nuclear process involving the unbalanced creation of two new matter particles but no antimatter [1–5]. This lepton number-violating process is predicted generically by many grand-unification theories, as well as by leptogenesis [6], a leading explanation of why the Universe is matter dominated."

So from that statement alone, it seems like they would only need lepton numbers to be rarely unconserved, and not necessarily enough to notice in the experiments @Vanadium 50 is talking about. Reasoning from that, it would seem that what the Majorana Collaboration is really saying is that neutrinos could behave like Majorana fermions a small fraction of the time, and that might be enough to explain how our universe got to be matter. Yet much of their other language seems to focus on neutrinos acting like Majorana fermions essentially all the time, i.e., being Majorana fermions. So it's difficult to understand language like: (https://enapphysics.web.unc.edu/research/majorana/ ) "If observed, 0νββ implies that the neutrino is its own antiparticle, and therefore a Majorana fermion.” So that's why I'm asking the question of whether the term "antineutrino" still has value even neutrinos are found to Majorana fermions. I don't at present understand why the argument @Vanadium 50 is presenting has not already completely ruled out that possibility, because if a neutrino made by a negative lepton "remembers that" well enough that it can only then make a negative lepton the vast majority of the time, it's quite clearly an antineutrino.

Perhaps the term "Majorana fermion" is being generalized to "occasionally shows Majorana characteristics", and "is its own antiparticle" is being generalized to "can occasionally annihilate with another like itself". But if so, then they are between a rock and a hard place-- they need the neutrinos to usually act like Dirac fermions to agree with what @Vanadium 50 is saying, but they need them to be Majorana just often enough to give their experiment a chance of detecting neutrinoless double-beta decay. That's a lot harder to justify than testing if they are Majorana fermions.

It is starting to sound like, neutrinos are not really particles because they are not mass eigenstates, but they can be thought of as superpositions of particles; and they are not really their own antiparticles but they might be superpositions of particles that are their own antiparticles, combined with those that (mostly) are not.

 

[Ken G]

We can delve deeper into the thinking of the Majorana Collaboration. At https://enapphysics.web.unc.edu/research/majorana/ , they reveal their main motivations:

"First, the neutrino mass is over a million times smaller than the mass of other particles that make up matter. If the neutrino is a Dirac particle, there is no natural explanation for this. However, if neutrinos are Majorana particles, a simple theory called the “see-saw mechanism” can account for their tiny mass.

Second, we live in a universe composed almost entirely of matter, but the big bang produced equal amounts of matter and antimatter. Where did all the antimatter go? The neutrino acting as its own antiparticle could help explain how matter came to dominate over antimatter – and therefore, how we are able to exist at all."

Unfortunately, that first one only muddies the waters. I can see how the second one might apply if neutrinos are mostly a superposition of Dirac fermions, and only a little Majorana, but the first one seems to require they be mostly Majorana, since how else could it explain their small mass? But if they are mostly Majorana, then how does a beam of them made from negative leptons know to make negative leptons? It sounds like having one's cake and eating it too.

 

[Ken G]

I'm not sure if this is even a possibility. It seems to be established that the Standard Model could be expanded to accommodate a Majorana mass for neutrinos, but I'm not sure whether it could accommodate neutrinos only having a Majorana mass. (If nothing else, one would have to explain why the standard Higgs mechanism doesn't give neutrinos Dirac masses, even though left-handed neutrinos, at least, are weak isospin doublets with the charged leptons.)

Yes I see your point, reflected in @Vanadium 50 's argument as well, but you see the issue with the first quote I gave above from the Majorana Collaboration. If they want to explain the tiny neutrino mass, and they say that Dirac fermions have no natural explanation there, they must be relying pretty heavily on the Majorana characteristics. How does a tiny Majorana mass term account for a tiny total mass?

The quote from the Majorana Collaboration goes on:
"However, if and only if the neutrino is its own antiparticle, it is possible for the two neutrinos to “cancel,” leaving behind only the electrons. Thus, observing 0νββ would provide conclusive evidence that the neutrino is a Majorana particle. Unfortunately, this process is incredibly rare – the half-life of the decay is over a billion times longer than the age of the universe."

So there is the issue I mentioned above-- they are looking for a rare rate that they think they can estimate, but how could they estimate it if they don't know how strong the Majorana elements are? They don't imply the rate is rare because the particles are mostly Dirac in nature, they imply it's just a rare rate. So it sounds like they actually looking for a doubly rare rate, and they don't even know how to estimate it, but they are hoping it lives just below the level where we would have seen lepton number violation in regular neutrino-type beams.

 

[PeterDonis]

How does a tiny Majorana mass term account for a tiny total mass?

In at least one version of the seesaw mechanism, as I understand it, the Majorana mass is tiny and the Dirac mass is very large, so there are two sets of neutrinos, a set of very light ones (which are the ones we have detected) and a set of very heavy ones (which we haven't detected yet). I don't know how much of the potential parameter space of such theories the LHC results have ruled out.

 

[Ken G]

Here is a recent paper on the Dirac vs. Majorana issue as it relates to dipole moments:
https://lss.fnal.gov/archive/2022/pub/fermilab-pub-22-663-t.pdf
What is interesting in this paper is that once again, they do not seem to be talking about mostly Dirac with some tiny Majorana aspect, they are literally talking about particles that have antiparticles, vs. particles that are their own antiparticles. For example, they say:
"We find that a next-generation experiment two orders of magnitude more sensitive to the neutrino electromagnetic moments via νµ elastic scattering may discover that the neutrino electromagnetic moments are nonzero if the neutrinos are Dirac fermions. Instead, if the neutrinos are Majorana fermions, such a discovery is ruled out by existing solar neutrino data, unless there are more than three light neutrinos."

They add to this either/or perspective:
"It is well known that diagonal dipole moments for Majorana fermions are forbidden and hence these only have transition dipole moments. Dirac fermions, instead, are allowed to have both diagonal and transition dipole moments."

But then there is the cryptic:
"We also discuss how Majorana neutrinos can “mimic” Dirac neutrinos in the presence of new light neutral fermions."
Could this "mimicking" include acting to preserve lepton number most of the time? I'm speculating, but they do say:
"If the neutrinos are Majorana fermions, one can mimic the Dirac case by adding more neutrino mass eigenstates."
So although what they mean by "the Dirac case" is strictly about the dipole moment, one cannot help but wonder if one can connect negative leptons to negative leptons using particles that are their own antiparticles if one simply embeds the lepton number information into additional degrees of freedom such as additional mass eigenstates. What's to stop a Majorana neutrino from being able to annihilate with an identical particle, yet still "know" it came from a negative lepton? Then it would be its own antiparticle, so it would not strictly be an "antineutrino", yet it would mimic an antineutrino based on its memory of the lepton it is associated with.

I presume the issue is clarified in the sources referred to when they say:
"In recent years, there have been many efforts connecting these constraints to the parameters of the Lagrangian (see, for example, [6, 7, 15–25]). In most of these studies, special attention was dedicated to the Majorana-neutrino hypothesis." Surely, no one would dedicate so much effort to an idea that was obviously contradicted by simple neutrino beams, so I think they must have ways to conceptualize neutrinos as pure Majorana fermions, yet still equip them with the ability to "remember" which type of lepton they came from.

 

[Ken G]

In at least one version of the seesaw mechanism, as I understand it, the Majorana mass is tiny and the Dirac mass is very large, so there are two sets of neutrinos, a set of very light ones (which are the ones we have detected) and a set of very heavy ones (which we haven't detected yet). I don't know how much of the potential parameter space of such theories the LHC results have ruled out.

Interesting, but that would still imply that the neutrinos we are detecting are pure Majorana neutrinos, so would need to have some way of knowing what type of lepton they came from, while still showing the CPT symmetry of a particle that is its own antiparticle. I'm wondering if the pro-Majorana folks have a trick up their sleeve that we are overlooking.

 

[PeterDonis]

that would still imply that the neutrinos we are detecting are pure Majorana neutrinos

Not necessarily. In the seesaw mechanism, as I understand it, the actual physical neutrinos (the ones we detect in detectors) are neither the pure Majorana ones nor the pure Dirac ones; they are combinations of them.

 

[Ken G]

Not necessarily. In the seesaw mechanism, as I understand it, the actual physical neutrinos (the ones we detect in detectors) are neither the pure Majorana ones nor the pure Dirac ones; they are combinations of them.

Well that brings us back to the original issue, if the neutrino beam experiments force neutrinos to be mostly Dirac-type, or if they could still be essentially pure Majorana. I think the quotes I just gave make it pretty clear that there are still a lot of neutrino experts that think neutrinos could be essentially pure Majorana (after all, that paper was talking about dipole moment matrices that have zero diagonal elements, not diagonal elements that are a tiny bit smaller than you'd expect, etc.). Yet of course they know about the beam experiments, so it seems to me there must be a way that a neutrino that is flat out "its own antiparticle" (based on ability to annihilate, CPT symmetry, etc.) can still create mostly only negative leptons if it came from negative leptons. It must therefore come from some property other than lepton number, that is the possibility we have not taken into account.

 

[Vanadium 50]

Can you be more specific about which experiment this refers to?

Lots and lots. MINOS did it, and before that CCDR did it, and it goes all the way back to the CERN Wide-Band-Beam in the 70's and early 80's, and experiments like CDHS. Van der Meer invented the horn in the 1960's.

Here is one table from the latest Review of Particle Properties:

You can get a slightly more comprehensive view by looking at the experiments who have results on the top and bottom lines. (Also note that the nu cross-section is larger than nubar, in contrast to some claims made upthread)

 

[PeterDonis]

there must be a way that a neutrino that is flat out "its own antiparticle" (based on ability to annihilate, CPT symmetry, etc.) can still create mostly only negative leptons if it came from negative leptons

I looked through the "Publications" section of the Majorana Collaboration website in the hope of finding a paper that dealt with any proposed theoretical basis for such a possibility, but couldn't find one.

 

[PeterDonis]

Lots and lots.

Are all of these experimental results possible if neutrinos are pure Majorana fermions (i.e., no Dirac masses, just Majorana masses)? It doesn't seem like they would be, since pure Majorana fermions can't have a meaningful lepton number, but these results certainly seem to indicate that neutrinos do. Unless there is some other mechanism involved, but I'm not sure what it could be, and so far I've found no discussion of any such thing in the literature I've looked at.

 

[Ken G]

Exactly my question as well. I think the answer must be "yes", given all the work being done on Majorana neutrinos, but I don't know the mechanism proposed to essentially "hide" a lepton number like that.

 

[Ken G]

You can get a slightly more comprehensive view by looking at the experiments who have results on the top and bottom lines. (Also note that the nu cross-section is larger than nubar, in contrast to some claims made upthread)

Not sure what "upthread" statement is being referenced, but is it not apparent that the cross-sections could be different (here by a factor remarkably close to 2) if it is true that the target of what is being called a neutrino is a neutron, and the target of what is being called an antineutrino is a proton? So if you are implying that a different cross section requires a different neutrino, I would point to the differing target. What seemed more telling is your point that one does not see both of those reactions happening for a given beam of neutrinos, which would seem to require that if they are Majorana neutrinos, they need internal degrees of freedom that are not lepton number. But that would seem to be all they would need.

 

[malawi_glenn]

This would be a weak interaction decay conserving lepton number, correct? So if we're getting positive pions, those would decay into positive muons (antimuons, lepton number -1) and muon neutrinos (lepton number +1), whereas if we're getting negative pions, those would decay into negative muons (lepton number +1) and muon antineutrinos (lepton number -1), correct?

In the context of the standard model (pure Dirac-neutrinos) yes.

"if neutrinos are pure Majorano fermions, will there be any value in the term antineutrino, or will the term be retired." No one has yet answered that

I thought you could read between the lines in my posts. "No" (if you ask me or basically every theoretician, what experimentalists would call things - I do not know).

 

[Ken G]

Thanks for your answer, at least we have one theorist who would say that, and a claim that most any theorist would. I'm not sure if the implied separation between theorists and experimentalists is really valid though, it would be pretty strange for experimentalists to see value in a term that theorists don't. But if we take it as likely that there is not value in the term "antineutrino" if neutrinos are pure Majorana particles, we now have the issue of whether or not neutron beam experiments have already ruled that out. It looks like the answer to that has to be "no, otherwise not so many people would still be doing research on neutrinos as pure Majorana particles." Combining those two conclusions takes us all the way back to the beginning of the thread: we do not know that neutrinos are not their own antiparticles, and it would not invalidate a long list of everything else we know if they were. But we still have to understand how all that can be consistent with @Vanadium 50 's point that neutrino beams built from antileptons tend to only produce antileptons in the detectors, there has to be a way for Majorana neutrinos to pull off that trick.

 

[malawi_glenn]

at least we have one theorist who would say that, and a claim that most any theorist would.

I base this on the analogy that we do not speak of "antiphotons", "anti-Z" nor "anti-higgs".

 

[vanhees71]

I tried to avoid getting into the question of mass vs. flavor eigenstates. Neutrinos are leptons, anti-neutrinos are anti-leptons. Yes, we can go into mroe detail and talk about individual flavors, but I deliberately avoided doing this.

The point is that this is not yet unanimously decided for neutrinos. Whether they are Majorana or Dirac particles is subject of active research, e.g., by the search for neutrinoless double $\beta$-decay. For a pretty comprehensive review on Majorana vs. Dirac neutrinos, see

https://arxiv.org/abs/1412.3320v1

 

[Vanadium 50]

Are all of these experimental results possible if neutrinos are pure Majorana fermions

Yes,

If that were not the case, Dirac or Majorana would not be an open issue.,

It doesn't seem like they would be, since pure Majorana fermions can't have a meaningful lepton number

You're falling back on the popularization phrase "Majorana neutrinos are their own antiparticles". The question is not "is there lepton number violation" but "how much" (and maybe "do we care?") The scale will be of order m/.E, or ~10^-12 or so in most experiments, No experiment can reach that level. because no experiment will have enough data to see even one event.,

You should consider "Majorana neutrinos are their own antiparticles" in the same category as "relativistic mass" - a story we tell so that non-experts get a glimmer of understanding, but one that leads to incorrect conclusions if taken too far. And 'too far" is not very far..

 

[Ken G]

I base this on the analogy that we do not speak of "antiphotons", "anti-Z" nor "anti-higgs".

Exactly, I agree.

 

[Ken G]

The point is that this is not yet unanimously decided for neutrinos. Whether they are Majorana or Dirac particles is subject of active research, e.g., by the search for neutrinoless double $\beta$-decay. For a pretty comprehensive review on Majorana vs. Dirac neutrinos, see

https://arxiv.org/abs/1412.3320v1

That's a very helpful article. It does make the interesting point: "In the limit of vanishingly small mass the difference between Dirac and Majorana fermions disappears. Therefore the observed smallness of the neutrino mass makes it very difficult to discriminate between different types of massive neutrinos, and it is not currently known if neutrinos are Majorana or Dirac particles." So we have that point of contact between Dirac and Majorana neutrinos, the question that seems outstanding is how do Majorana neutrinos, once they do have mass, still manage to "remember" what kind of lepton made them, to explain why neutrino beams from negative leptons act differently from those from positive leptons, even though they have zero lepton number?

 

[Ken G]

You're falling back on the popularization phrase "Majorana neutrinos are their own antiparticles".

How is it "falling back on a popularization phrase" to quote exactly from language in expert articles on the topic? From the paper @vanhees71 just cited:
"What are Majorana particles? These are massive fermions that are their own antiparticles."
It's a chapter from a book, but certainly it is written for physicists, and is not "popularized." (When is the last time you saw a popularized article contain 118 equations?)

The question is not "is there lepton number violation" but "how much" (and maybe "do we care?")

Actually, the question of whether there is complete lepton violation in neutrino reactions remains open. From the article:
"Obviously, Majorana particles must be genuinely neutral, i.e. they cannot possess any conserved charge-like quantum number that would allow one to discriminate between the particle and its antiparticle."
Lepton number is not necessarily "charge-like", but it is certainly implied by the logic of the statement that they are taking ability to discriminate a particle and its antiparticle as the given impossibility for Majorana neutrinos. Nonzero lepton number would invalidate the logic of this statement.

The scale will be of order m/.E, or ~10[suo]-12[/sup] or so in most experiments, No experiment can reach that level. because no experiment will have enough data to see even one event.,

The neutrinoless double-beta decay is indeed a very unlikely event, so they need to look at a huge sample. This article describes what is needed: https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.130.062501
It is true that this experiment could get a result even if neutrinos are some kind of mix between Dirac and Majorana type, but that does not deny the truth that the experimenters think they might be of pure Majorana type, and I think it's pretty clear they are hoping that is the case (to make the reaction happen more often).

You should consider "Majorana neutrinos are their own antiparticles" in the same category as "relativistic mass" - a story we tell so that non-experts get a glimmer of understanding, but one that leads to incorrect conclusions if taken too far. And 'too far" is not very far..

At this point, this is sounding like a personal opinion masquerading as authoritative, and worse, being used as a kind of club to dismiss discussion of this very interesting topic, among "non-experts" who can hope for no more than a "glimmer of understanding."

 

[PeterDonis]

Yes,

Is there a mathematical model that explains how? I've been unable to find one in any of the papers I've read (including the ones that have been linked in this thread, in post #22 and another linked by @vanhees71 in post #100).