# Majorana neutrinos

Dr.AbeNikIanEdL
both fall equally under the "standard model."
Or equally under “beyond the standard model”, depending on who you ask ;).

There is still one field that is produced when particles decay, and its charge conjugate is produced when antiparticles decay. Its natural to call these ”neutrino” and “anti-neutrino”.

Preference for lepton number conservation is not arbitrary, it is based on lepton number being conserved either exactly or to an extremely good approximation.

Gold Member
There is still one field that is produced when particles decay, and its charge conjugate is produced when antiparticles decay. Its natural to call these ”neutrino” and “anti-neutrino”.
Was it not established above that what is produced when leptons decay preserves chirality? So neutrino beams preserve the chirality of the particles that created them, that much is clear. But on what basis does one claim that the fields that are produced are charge conjugations of each other, rather than chiral opposites? Does that not already assume something about the neutrino fields that has not been established experimentally?
Preference for lepton number conservation is not arbitrary, it is based on lepton number being conserved either exactly or to an extremely good approximation.
Again that refers to the particles that have lepton number, such as the leptons that created the neutrino beam, and the leptons produced by the beam. But if antileptons produce a beam of neutrinos that turn out to be Majorana neutrinos, is that not a situation where the beam itself drastically violates lepton number conservation, even if the lepton number is recovered when the beam interacts in the detector and creates antileptons again? I thought that is what had been established above.

Dr.AbeNikIanEdL
But on what basis does one claim that the fields that are produced are charge conjugations of each othe
Well, they have opposite (weak hyper-) charges…
the beam itself drastically violates lepton number conservation
I don’t understand what this phrase is supposed to mean.

vanhees71
Gold Member
Well, they have opposite (weak hyper-) charges…
It is often said that a Majorana neutrino cannot be distinguished as a particle or an antiparticle, are you saying that is not true?
I don’t understand what this phrase is supposed to mean.
To my understanding, leptons have lepton number +1, while antileptons have lepton number -1. It is also my understanding that Majorana neutrinos would not have a lepton number (or would have lepton number zero, if there is any distinction there). Is that not correct? Because if it is correct, then when antileptons interact to create "antineutrinos" in a beam, we have particles with lepton number -1 creating no particles with lepton number -1. So that would be nothing close to lepton number conservation. However, when the "antineutrinos" are detected, they will create particles with lepton number -1, so the lepton number is in some sense recovered, even though it was not present in the beam itself. Is that not what would be happening if neutrinos are Majorana particles?

Gold Member
By the way, this talk http://www.maria-laach.tp.nt.uni-siegen.de/downloads/files/2016/Mertens-2016-2.pdf was entered into a discussion on Majorana neutrinos by @vanhees71 some years ago. To my read, it's quite clear on the meaninglessness of the "antineurino" concept, if neutrinos are Majorana particles. It also makes a key distinction between chirality and helicity, which @Vanadium 50 also alluded to above, saying that to order m/E, a particle with a given chirality can mix in some opposite helicity. It seemed to say that low-mass neutrinos can be clear about their own chirality but a little ambiguous about their helicity, which relates to the issues of both creating leptons or antileptons, and also how neutrinoless double beta decay can occur (though the points it was making about helicity being a property of the particle and chirality a property of the interaction were not clear to me, helps to have the spoken part!).

But none of that seems to have much to do with the antimatter issue-- there simply appears to be no point in calling any Majorana neutrino an "antineutrino" regardless of chirality or helicity issues, the term simply doesn't seem to have any meaning.

Mentor
they have opposite (weak hyper-) charges…
Below the electroweak symmetry breaking scale, we no longer have weak hypercharge, we only have electric charge, and neutrinos are electrically neutral.

Above the electroweak symmetry breaking scale (i.e., at energies where electroweak symmetry is not broken), all of the fermions in the Standard Model are massless, so it doesn't even make sense to ask whether neutrinos (or any other fermions) have Majorana or Dirac masses. So the entire discussion in this thread already assumes that the electroweak symmetry is broken, so that we can meaningfully discuss fermion masses.

malawi_glenn and Ken G
Dr.AbeNikIanEdL
It is often said that a Majorana neutrino cannot be distinguished as a particle or an antiparticle, are you saying that is not true?
Majorana fermions can be written in terms of a Weyl field ##\chi## as (see post #167)

## \Psi = \chi + \chi^c## .

Under charge conjugation it changes form ##\chi\to\chi^c##, so ##\Psi## stays the same. But ##\chi## and ##\chi^c## have opposite charges, and they are the things that appear in interaction terms. It does not matter if we are in the broken phase, they have a charge with which they couple to leptons and Ws and that is reversed between the two upon charge conjugation.
It is also my understanding that Majorana neutrinos would not have a lepton number (or would have lepton number zero, if there is any distinction there).
Yes, ##\Psi## would have lepton number of 0, by virtue of ##\chi## and ##\chi^c## having opposite lepton numbers. The mass term is the only one that mixes ##\chi## and ##\chi^c##, and hence violates lepon number. In the interactions it is still conserved.

malawi_glenn and vanhees71
Gold Member
Majorana fermions can be written in terms of a Weyl field ##\chi## as (see post #167)

## \Psi = \chi + \chi^c## .

Under charge conjugation it changes form ##\chi\to\chi^c##, so ##\Psi## stays the same. But ##\chi## and ##\chi^c## have opposite charges, and they are the things that appear in interaction terms. It does not matter if we are in the broken phase, they have a charge with which they couple to leptons and Ws and that is reversed between the two upon charge conjugation.
I see, so that explains what you meant above, but what we are talking about is the concept of a "beam of antineutrinos." You are saying we can see, in that beam, neutrino fields and antineutrino fields that are superimposed, but we see nothing we could call a "beam of antineutrinos" if they are Majorana.

Then the question is, what controls the chirality of that beam? We still have to have a difference between a neutrino beam created by leptons vs. antileptons. If this was analogous to linearly polarized light seen as a superposition of circular polarizations, it would be the difference between ##\chi + \chi^c## and ##\chi - \chi^c##, what is the difference in the case of Majorana neutrinos?
Yes, ##\Psi## would have lepton number of 0, by virtue of ##\chi## and ##\chi^c## having opposite lepton numbers. The mass term is the only one that mixes ##\chi## and ##\chi^c##, and hence violates lepon number. In the interactions it is still conserved.
That language confuses me, because when I see a field being described as ##\chi + \chi^c##, I don't see something I am forced to say is mixed by a mass term, I see an equal superposition of a matterlike field and an antimatterlike field. In the viewgraphs I linked to above, the mass effect was that it made a definite chirality have a nondefinite helicity, saying the opposite helicity was present to order m/E since the neutrino isn't moving at the speed of light any more. But we've heard that Majorana neutrinos have zero lepton number, so that doesn't sound like the same thing. Also, when I see a supernova that neutronizes some ##10^{57}## electrons and protons, generating neutrinos in the process which we say might have zero lepton number, I do not see some tiny lepton number violation, I see ##10^{57}## leptons that might not be there any more had those neutrinos been able to escape into space (generally they don't, but that's a detail of supernovae).

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Mentor
It does not matter if we are in the broken phase, they have a charge with which they couple to leptons and Ws
Below the electroweak symmetry breaking energy, the only such coupling is the weak isospin coupling, and it only applies to the left-handed neutrino/right-handed antineutrino two-component spinor (which is a weak isospin doublet with the left-handed lepton/right-handed antilepton two-component spinor in each generation). This coupling has already been discussed in quite some detail in previous posts.

and that is reversed between the two upon charge conjugation.
As I have already described in a previous post, whether a given instance of this coupling involves a "neutrino" or "antineutrino" depends on the specific event and how it is oriented in spacetime. There is only one coupling; there aren't two, one for "neutrinos" and one for "antineutrinos". And all that is true whether or not neutrinos have Majorana masses or Dirac masses.

vanhees71
Gold Member
Perhaps it will focus the discussion if this question is answered: if we assume neutrinos are pure Majorana particles, then what is the difference between what by current convention gets called a "beam of neutrinos" and a "beam of antineutrinos"? I thought the answer was going to be, the chirality of the particles in the beam, and that this was going to closely map to the helicity of those particles, which was going to generate mostly either leptons or antileptons in the detector, with some order m/E discrepancy that is very difficult to detect. I'm basing the claims about chirality and helicity on those viewgraphs (http://www.maria-laach.tp.nt.uni-siegen.de/downloads/files/2016/Mertens-2016-2.pdf), though I still do not understand why they say that chirality is a property of the interactions and helicity is a property of the particles (I thought they were going to say the opposite!).

Dr.AbeNikIanEdL
Below the electroweak symmetry breaking energy, the only such coupling is the weak isospin coupling, and it only applies to the left-handed neutrino/right-handed antineutrino two-component spinor (which is a weak isospin doublet with the left-handed lepton/right-handed antilepton two-component spinor in each generation). This coupling has already been discussed in quite some detail in previous posts.

As I have already described in a previous post, whether a given instance of this coupling involves a "neutrino" or "antineutrino" depends on the specific event and how it is oriented in spacetime. There is only one coupling; there aren't two, one for "neutrinos" and one for "antineutrinos". And all that is true whether or not neutrinos have Majorana masses or Dirac masses.
Ehm, sure. Are we disagreeing about anything?

vanhees71
Dr.AbeNikIanEdL
Then the question is, what controls the chirality of that beam? We still have to have a difference between a neutrino beam created by leptons vs. antileptons. If this was analogous to linearly polarized light seen as a superposition of circular polarizations, it would be the difference between χ+χc and χ−χc, what is the difference in the case of Majorana neutrinos?
What the beam consists of will depend on what process procuded the neutrinos. For example if ##\pi^+## decays you will get positively charged leptons and neutrinos described by lets say ##\chi##, and if ##\pi^-## decays you get negatively charged ones and ##\chi^c##. There is no interaction that produce ##\Psi##.

Mentor
I'm not sure. You appear to be saying that interactions producing ##\chi## and interactions producing ##\chi^c## are distinct interactions. They're not. They're the same term in the Lagrangian and the same vertex in Feynman diagrams; that term/vertex involves the two-component spinor whose two components are ##\chi## and ##\chi^c##. Whether we describe the neutrinos corresponding to external legs in the Feynman diagram as ##\chi## or ##\chi^c## depends on the specific details of the particular instance of the interaction.

For example, in a previous post I described how beta decay, which produces an outgoing leg that you would describe with ##\chi^c##, is the same interaction (same term in the Lagrangian, same Feynman diagram vertex) as the interaction that produces electrons when neutrino beams hit a target containing neutrons; but in that interaction, the neutrino leg is ingoing and you would describe it with ##\chi##.

Dr.AbeNikIanEdL
that term/vertex involves the two-component spinor whose two components are χ and χc.
That does indeed not sound right to me. ##\chi## is supposed to be a Weyl spinor, i.e. it has two components, its not a component of a spinor.
You appear to be saying that interactions producing χ and interactions producing χc are distinct interactions. They're not.
That depends somewhat on your counting. Are a term and its conjugate “the same term”?

For example in the decays I gave above in #187, one decay involves a vertex where a ##W^+## decays into a positively charged lepton and ##\chi##, and the other a vertex where a ##W^-## decays into a negative lepton and a ##\chi^c##. These are of course in some sense “the same vertex”, I agree.

For example, in a previous post I described how beta decay, which produces an outgoing leg that you would describe with χc, is the same interaction (same term in the Lagrangian, same Feynman diagram vertex) as the interaction that produces electrons when neutrino beams hit a target containing neutrons; but in that interaction, the neutrino leg is ingoing and you would describe it with χ.
Well yes, and the beta decay produces anti-neutrinos while the beam that produces electrons should contain neutrinos, so that checks out.

vanhees71
Mentor
These are of course in some sense “the same vertex”, I agree.
Ok, then I think we're just using different ordinary language to describe the same physics.

Gold Member
What the beam consists of will depend on what process procuded the neutrinos. For example if ##\pi^+## decays you will get positively charged leptons and neutrinos described by lets say ##\chi##, and if ##\pi^-## decays you get negatively charged ones and ##\chi^c##. There is no interaction that produce ##\Psi##.
If neutrinos are Majorana particles, then you cannot get neutrinos described by only ##\chi## when ##\pi^+## decays, because Majorana particles are not described only by ##\chi##. There must be some other reason that a Majorana neutrino that comes from a ##\pi^+## decay is vastly more likely to make ##\pi^+## rather than ##\pi^-##, but it cannot be that it is described by only ##\chi## and not ##\chi^c## since that would not be a Majorana particle. So how does a Majorana particle favor making ##\pi^+## over ##\pi^-##?

Staff Emeritus
None of that is right,

Pion decay (how they make beams of neutrinos) proceeds via:
$$\pi^+ \rightarrow \mu^+ + \nu_L$$$$\pi^- \rightarrow \mu^- + \overline{\nu}_R$$

One can detect neutrinos by the inverse reaction
$$\nu_L + N \rightarrow \mu^- + X$$$$\overline{\nu}_R+ N \rightarrow \mu^+ + X$$

These are experimental facts, supported by the dozens of references in the PDG article I linked to. If you disagree, we need to address this going further.

Note that the fields $\nu_R$ and $\overline{\nu}_L$ do not participate. The whole question of Dirac vs. Majorana is whether I link together $\nu_L$ and $\nu_R$ to make a physical neutrino (Dirac) or $\nu_L$ and $\overline{\nu}_R$ (Majorana),

Note that this has nothing to do with the first half of the message. (Ignoring some higher-order ppb-level effects)

vanhees71 and Dr.AbeNikIanEdL
Mentor
None of that is right,

Pion decay (how they make beams of neutrinos) proceeds via:
$$\pi^+ \rightarrow \mu^+ + \nu_L$$$$\pi^- \rightarrow \mu^- + \overline{\nu}_R$$

One can detect neutrinos by the inverse reaction
$$\nu_L + N \rightarrow \mu^- + X$$$$\overline{\nu}_R+ N \rightarrow \mu^+ + X$$

These are experimental facts, supported by the dozens of references in the PDG article I linked to. If you disagree, we need to address this going further.

Note that the fields $\nu_R$ and $\overline{\nu}_L$ do not participate. The whole question of Dirac vs. Majorana is whether I link together $\nu_L$ and $\nu_R$ to make a physical neutrino (Dirac) or $\nu_L$ and $\overline{\nu}_R$ (Majorana),
Just to be clear, in the notation you are using, ##\bar{\nu}_R## is the CPT conjugate of ##\nu_L##, i.e., it is the same as what in another common notation would be called ##\nu_L^c##, correct?

Gold Member
And to clarify that last question, this entire thread has been about not mistaking arbitrary theoretical choices for "experimental fact", so we certainly don't want to leave it on exactly that same mistake. Experimental facts are the outcomes of experiments, not notational choices.

It sounds like you are saying that the minimum theoretical requirement in the standard model that agrees with experiment is that leptons connect with left-handed neutrinos and antileptons connect with right-handed neutrinos. I didn't see "antineutrino" in there anywhere. It certainly seems to me at this point that the entire term "antineutrino" has no real reason to exist, because we don't currently need it, until the experiments that show neutrinos are Dirac neutrinos are carried out. Otherwise, the term is pure common convention, carrying no requirement that the antiparticle be meaningfully different from the particle beyond its handedness (which is not the same thing).

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Gold Member
2022 Award
I'm not sure. You appear to be saying that interactions producing ##\chi## and interactions producing ##\chi^c## are distinct interactions. They're not. They're the same term in the Lagrangian and the same vertex in Feynman diagrams; that term/vertex involves the two-component spinor whose two components are ##\chi## and ##\chi^c##. Whether we describe the neutrinos corresponding to external legs in the Feynman diagram as ##\chi## or ##\chi^c## depends on the specific details of the particular instance of the interaction.
Of course. That's called "crossing symmetry" and "particle" and "antiparticle" (positive-frequencey/negative-frequency parts in the mode decomposition of the (asymptotic) free fields) always come together for all fields in a specific way such as to fulfill the microcausality constraint. Then the possible interaction terms are determined by Poincare invariance of the (variation of) the action.

For neutrinos (no matter whether they are Dirac or Majorana particles) you have, according to the phenomenology of the weak interactions, left-handed neutrinos and right-handed anti-neutrinos coupled to the charged leptons and the quarks.

Neutrinos don't make much sense as external legs due to mixing, since they represent mass eigenstates, which you however never detect due to the said nature of the weak interaction. You have to consider the complete creation (established by "near-side detectors" in long-base-line experiments) and detection process ("far-side detector"), i.e., what you can observe are cross sections for creating neutrinos of a given flavor and detecting neutrinos of a given flavor, but you never can measure neutrinos as asymptotic free particles, which would be mass eigenstates.
For example, in a previous post I described how beta decay, which produces an outgoing leg that you would describe with ##\chi^c##, is the same interaction (same term in the Lagrangian, same Feynman diagram vertex) as the interaction that produces electrons when neutrino beams hit a target containing neutrons; but in that interaction, the neutrino leg is ingoing and you would describe it with ##\chi##.
Due to the Majorana mass term the ##\chi^c## at the production vertex can interact with your neutron as a ##\chi##, i.e., you have lepton-number violation, and this is what enables neutrinoless double-##\beta## decay for Majorana neutrinos, which is forbidden for Dirac neutrinos. That's what all the hype is about!

Once more, here's the link to the nice paper describing all kinds of spin-1/2 fields (Weyl, Dirac, and Majorana):

https://arxiv.org/abs/1006.1718
https://doi.org/10.1119/1.3549729

malawi_glenn
Gold Member
The thing that is still not answered for me is this: you have a bunch of antileptons interacting to make a beam of neutrinos that are said to oscillate between flavors but do not oscillate between particle and antiparticle. Let's say for argument's sake it is determined that neutrinos are totally of Majorana type, not at all Dirac type. Now people keep using the phrase "Majorana mass term", but I'm saying, the particle is a Majorana fermion, which we have seen in quite a few places means the particle is of a very different nature than a Dirac neutrino.

For example, in the viewgraphs both you and I linked to, we have the statements that Dirac would say: “The neutrino is not identical to the known antineutrino” whereas Majorana would say: “The neutrino is identical to the known antineutrino”. And, Dirac would say: "Even if the neutrino flips its helicity it is still a fundamentally different particle. The reaction is not possible” whereas Majorana would say: “Neutrino only has the wrong helicity, if it can flip the helicity this reaction should be possible. (note: lepton number would be violated)”. I don't see how those oppositely framed statements is addressed in the above.

In other words, those quotes make it seem like the difference between the two particle models is not just that some tiny mass has the ability to very rarely cause the helicity to flip in the Majorana case, it is that even if that could happen to a Dirac neutrino, they still could not annihilate. The quotes also clearly imply that one of the two cases resoundingly invalidates the entire concept of an "antineutrino" (and a theorist on here has already stated the term "antineutrino" would likely be retired from the lexicon, while an experimentalist said that surely would not happen), which is what initiated this spinoff thread. So this is about way more than whether or not neutrinoless double beta decay happens in one reaction out of a ghastly number.

Indeed, since we have @vanhees71 's point that the flavored neutrinos we detect are not the things we would call free particles in the beam (because the latter should be mass eigenstates), then how much more is it true that the free particles in a Majorana neutrino beam cannot be antineutrinos either? Is it not correct that any Majorana neutrino mass eigenstate has no more antimatter identity than matter identity?

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Staff Emeritus
And to clarify that last question, this entire thread has been about not mistaking arbitrary theoretical choices for "experimental fact"
Let's be clear then..

In my message above, the implication is that a neutrino detector looking at a beam from positive (negative) pion decays will see negative (positive muons) and not a 50-50 mix. Do you dispute this? If you do, I refer you to the PDG summary upthread and references therein.

If you agree with that, but doin't like the notation,

The particles designed by ν act exactly as neutrinos are expected to. Calling them somthing is logically equivalent to "Homer didn't write The Odyssey. It was written by another blind poet of the same name."

You can object to the L and R subscripts. The fact that these decay in Lines 1 and 2 go through a chiral neutrino is undisputed. The fact that the two processes in 1 ans 2 have opposite chirality is undisputed. The assignment of L and R has extremely strong evidence behind it.

If you want to argue that most papers drop the L and R as redundant, you got me., Many do.

If you want to argue that the bar on the nubar is unnecessary if neutrinos are Majorana, you'tr tilting at windmills. We should redefine the electron charge as positive. We should drop stellar magnitudes in favor of something physical like Janskys. We should call the first population of stars Population I and not Population III. We shouldn't drive on a parkway and park on a driveway. None of these are going to happen, so I am least have decided to move on with my life,

Mentor
those quotes make it seem like the difference between the two particle models is not just that some tiny mass has the ability to very rarely cause the helicity to flip in the Majorana case, it is that even if that could happen to a Dirac neutrino, they still could not annihilate
This can't be right, because electrons and positrons are known to be Dirac fermions, but they can annihilate each other. If neutrinos turn out to be Dirac fermions, that just means they work like electrons and positrons. It doesn't mean neutrinos and antineutrinos can't annihilate each other.

(And the above does not even take into account the complications I have already pointed out that are involved in "annihilation".)

Gold Member
Let's be clear then..

In my message above, the implication is that a neutrino detector looking at a beam from positive (negative) pion decays will see negative (positive muons) and not a 50-50 mix. Do you dispute this?
No one on this thread has ever for one instant disputed any experimental evidence. Why would you even ask?
The particles designed by ν act exactly as neutrinos are expected to. Calling them somthing is logically equivalent to "Homer didn't write The Odyssey. It was written by another blind poet of the same name."
More "reasoning by analogy"? No, the issue is whether or not it is a "beam of antineutrinos", not "The Odyssey."
You can object to the L and R subscripts.
Why would you put words in my mouth? What is the actual one thing I have actually objected to? ("A beam of antineutrinos.")
The fact that these decay in Lines 1 and 2 go through a chiral neutrino is undisputed. The fact that the two processes in 1 ans 2 have opposite chirality is undisputed. The assignment of L and R has extremely strong evidence behind it.
Again, irrelevant.
If you want to argue that most papers drop the L and R as redundant, you got me., Many do.
I understand that there are historical prejudices involved here, largely based on the early expectation that neutrinos will turn out to be Dirac fermions. I'm trying to anticipate the future if neutrinos are discovered to actually be Majorana fermions. Further, I'm pointing to the usual application of "Occam's Razor": (no additional features to the language, like "aether," except those that are actually required to predict the observations.)
If you want to argue that the bar on the nubar is unnecessary if neutrinos are Majorana, you'tr tilting at windmills. We should redefine the electron charge as positive.
The question is whether the term "beam of antineutrinos" actually means anything at all for a beam of Majorana neutrinos. This is the question you have ducked in those straw-man analogies-- does the phrase have any meaning.
We should drop stellar magnitudes in favor of something physical like Janskys. We should call the first population of stars Population I and not Population III. We shouldn't drive on a parkway and park on a driveway. None of these are going to happen, so I am least have decided to move on with my life,
It's not just a question of awkward language, it's a question of using language that suggests something exists that does not exist at all. It's not like deciding if the universe is all matter or all antimatter, it is deciding if a universe of nothing but Majorana neutrinos would have any concept of that distinction.

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Gold Member
This can't be right, because electrons and positrons are known to be Dirac fermions, but they can annihilate each other.
I didn't mean that two Dirac neutrinos can never annihilate under any circumstances, the statement was in the same context as the quote that preceded it: two Dirac neutrinos that come from the same process, say beta decays that also produced electrons, could never annihilate each other even if one of them managed to flip helicity. Flipping helicity is all that is required for Majorana, but annihilation is still impossible for Fermions. The helicity flip is a very rare process due to the small mass, but the point being made is that it relates to a fundamental difference that goes beyond those rare helicity flips.
If neutrinos turn out to be Dirac fermions, that just means they work like electrons and positrons. It doesn't mean neutrinos and antineutrinos can't annihilate each other.
The question is still this: does a mass eigenstate of a Majorana neutrino (which @vanhees71 has defined as the free particle here) ever have any sense to which it is more of an antineutrino than a neutrino? What is the answer to that question, and how does it relate to the common phrase "a beam of antineutrinos"?

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Gold Member
Here's another question that might shed light on that one. The first paper that @vanhees71 cited above noted that Dirac neutrinos are the general class that includes both Majorana and Weyl type, but Weyl type only applies for massless neutrinos, though they can be regarded as building blocks for Majorana types and even for the full Dirac type. It also sounds to me like the Majorana type is the one that is built not to distinguish neutrinos and antineutrinos, so that requires some special construction. The Dirac type is more general, and does allow a distinction between neutrinos and antineutrinos, expressly because it has a more general construction. So if experiments show that neutrinos are in fact of Majorana type, why doesn't that quite clearly indicate that they are built specially to provide no meaning to the phrase "beam of antineutrinos"?

Given this, I could see an argument for retaining the idea that neutrinos can have an antimatter character, because we allow the more general theory until the more specifically built one is demonstrated. But what I don't see is why it is controversial that if neutrinos are found to be purely of Majorana type, then there is no such thing as a "beam of antineutrinos."

Mentor
I believe the implication of the quote is that two neutrinos that come from the same process, say beta decays that also produced electrons, could never annihilate each other even if one of them managed to flip helicity, if they are Dirac fermions. Flipping helicity is all that is required for Majorana, impossible in all cases for Fermions.
I don't think this is correct. (Whether it's what the viewgraphs are saying is hard to say, since, as I note below, the viewgraphs don't give any math, and ordinary language is a bad tool for describing the physics involved.)

In two-component spinor language, a Dirac mass term couples two two-component spinors together, while a Majorana mass term couples the two components of one two-component spinor. But in either case the "flipping" that the coupling induces ends up producing the CPT conjugate of the original. The Dirac mass term just does it by "flipping" two two-component spinors instead of only one.

You appear to be thinking of a "flipping" that only flips one two-component spinor even if the fermion is a Dirac fermion with two two-component spinors, but I'm not aware of any such operation.

I'm certain the person who presented those viewgraphs knows that
The statements you quoted from the viewgraphs talk about helicity, not chirality. Helicity is frame dependent. Chirality is not. I have been talking about chirality, and the ##L## and ##R## subscripts in the post by @Vanadium 50 refer to chirality.

The viewgraphs talk about helicity vs. chirality, but appears to focus on helicity because "physical particles occur with a definite helicity in nature". But that statement appears to contradict the statement just above it on the same slide: "helicity of massive particle depends on reference frame".

The viewgraphs don't give any math, so I don't think they are a good reference for this discussion, since many of the communication issues we are having are due to ordinary language being a bad tool for describing the physics involved.

vanhees71
Gold Member
In two-component spinor language, a Dirac mass term couples two two-component spinors together, while a Majorana mass term couples the two components of one two-component spinor. But in either case the "flipping" that the coupling induces ends up producing the CPT conjugate of the original. The Dirac mass term just does it by "flipping" two two-component spinors instead of only one.
Yes, but as I understand what those quotes mean, the flipping is only happening to one of the two neutrinos in the double beta decay. For Majorana, that would be enough for annihilation, for Dirac, not enough.
The statements you quoted from the viewgraphs talk about helicity, not chirality. Helicity is frame dependent.
Yes, this is why the effect only happens at order m/E, a measure of the accessibility of a frame that flips the helicity.
Chirality is not. I have been talking about chirality, and the ##L## and ##R## subscripts in the post by @Vanadium 50 refer to chirality.
Yes, I know, I am merely quoting from the viewgraphs, which are talking about flipping helicity not chirality.
The viewgraphs talk about helicity vs. chirality, but appears to focus on helicity because "physical particles occur with a definite helicity in nature". But that statement appears to contradict the statement just above it on the same slide: "helicity of massive particle depends on reference frame".
I mentioned that above parenthetically, I expected them to say that chirality was the particle-dependent thing and helicity was more malleable. They seemed to say the opposite, so that is indeed confusing. It's not clear the distinction matters in the rest of their argument, maybe it was even a typo.
The viewgraphs don't give any math, so I don't think they are a good reference for this discussion, since many of the communication issues we are having are due to ordinary language being a bad tool for describing the physics involved.
Still, the math must be interpreted. The issue of the thread is what is the meaning of the phrase "a beam of antineutrinos." That's language, not math. If it was true that all we need is math, we'd never even need the word "antineutrino." But math is not good enough, because physics does more than just predict quantitative outcomes, it creates a way to talk about nature. That gets sticky, I know, but that's why it requires a discourse like this one. The question is, if neutrinos are pure Majorana particles, is there any such thing as a "beam of antineutrinos", and if not, what is the better way to say a beam of neutrinos that is going to produce a whole lot of positrons and not many electrons when you bash it into neutrons and protons?

Staff Emeritus
Why would you put words in my mouth?
I did no such thing,.

Four times, I have asked you to be civil.

vanhees71
Mentor
as I understand what those quotes mean, the flipping is only happening to one of the two neutrinos in the double beta decay
There is only one neutrino line in the lowest order diagram for double beta decay. That diagram is made with two ordinary beta decay vertices (I described this vertex in an earlier post), and one neutrino line linking them (so the neutrino line is internal and there is no outgoing neutrino external leg). That means the neutrino line must have opposite chirality at the two vertices, which means that, somewhere on that internal neutrino line, there must be a "flipping" operation that flips the chirality. But of course, since the neutrino line has the same weak interaction vertex at both ends, the "flipping" must also preserve the weak interaction coupling.

A Majorana mass term couples the two components of the ##\nu_L## two-component spinor that has a weak interaction coupling, so it could serve as the required "flipping" operation above. It would "flip" ##\nu_L## to its CPT conjugate ##\bar{\nu}_R##.

A Dirac mass term, however, would couple ##\nu_L## with the separate "sterile" neutrino ##\nu_R##, which has no weak interaction coupling, so it can't serve as the required "flipping" operation above.

The above seems to me to be what the viewgraphs with the contrasting "Dirac" and "Majorana" statements were trying to get at. But I'm not sure they did a very good job.

vanhees71
Mentor
The question is whether the term "beam of antineutrinos" actually means anything at all for a beam of Majorana neutrinos.
@Vanadium 50 already gave an operational definition of "beam of antineutrinos"--a beam that produces negative leptons when it hits a target containing baryons. By his definition, of course the term has meaning, since such beams have been produced many, many times in experiments.

We had a question under discussion for a while in this thread about how a model of neutrinos as pure Majorana fermions could produce such a beam, but that has been resolved (the chirality of the neutrinos emitted at the source is sufficient for that given the smallness of neutrino masses).

There might be physicists who would want to adopt a different term than "beam of antineutrinos" to describe the kind of beam that the above operational definition refers to, if neutrinos turn out to be Majorana fermions. But it does not appear that @Vanadium 50 is one of them, and if that is the case, it is pointless to continue to ask him the question you are asking him. He's already given all the answer he's going to give.

vanhees71
Gold Member
There is only one neutrino line in the lowest order diagram for double beta decay. That diagram is made with two ordinary beta decay vertices (I described this vertex in an earlier post), and one neutrino line linking them (so the neutrino line is internal and there is no outgoing neutrino external leg). That means the neutrino line must have opposite chirality at the two vertices, which means that, somewhere on that internal neutrino line, there must be a "flipping" operation that flips the chirality. But of course, since the neutrino line has the same weak interaction vertex at both ends, the "flipping" must also preserve the weak interaction coupling.

A Majorana mass term couples the two components of the ##\nu_L## two-component spinor that has a weak interaction coupling, so it could serve as the required "flipping" operation above. It would "flip" ##\nu_L## to its CPT conjugate ##\bar{\nu}_R##.

A Dirac mass term, however, would couple ##\nu_L## with the separate "sterile" neutrino ##\nu_R##, which has no weak interaction coupling, so it can't serve as the required "flipping" operation above.

The above seems to me to be what the viewgraphs with the contrasting "Dirac" and "Majorana" statements were trying to get at. But I'm not sure they did a very good job.
OK thanks, that makes sense-- the "flip" is not going to produce annihilation for the Dirac neutrino because there are those sterile states to flip into, not available for the Majorana neutrinos. So that's what the quote means.

Now for the question this is all leading to: if neutrinos are Majorana, their mass eigenstates do, or do not, distinguish matter from antimatter in any way?

Mentor
this is why the effect only happens at order m/E, a measure of the accessibility of a frame that flips the helicity
This doesn't make sense as it is stated (and not just by you, the viewgraphs appear to be stating it the same way). I can't make a piece of actual physics happen or not happen just by changing which frame I use to describe it.

This is one of the items that I think really suffers from the lack of math in the viewgraphs.

Gold Member
@Vanadium 50 already gave an operational definition of "beam of antineutrinos"--a beam that produces negative leptons when it hits a target containing baryons. By his definition, of course the term has meaning, since such beams have been produced many, many times in experiments.
Yes I know, that is ducking the question completely. One could give the same operational definition for Bobneutrinos, that isn't the issue. The issue is, is there any justification for calling that beam a beam of antineutrinos, or is there not? The term "antineutrino" is supposed to mean more than some operational definition that could also apply to "Bobneutrinos". In particular, it is supposed to mean that if you take the C conjugate, you are supposed to get a beam of neutrinos that makes leptons instead of antileptons. Would that indeed happen? It all gets to the core issue-- what is the difference between a beam of neutrinos that makes leptons, and one that makes antileptons? Is it purely the chirality? If so, that's not matter/antimatter.
We had a question under discussion for a while in this thread about how a model of neutrinos as pure Majorana fermions could produce such a beam, but that has been resolved (the chirality of the neutrinos emitted at the source is sufficient for that given the smallness of neutrino masses).
Yes, this is what I thought the resolution was. So imagine my confusion when chirality is getting given "operational definitions" that don't mean chirality, but rather matter/antimatter. If Majorana neutrinos do not distinguish matter from antimatter, but do distinguish chirality, then clearly those are not the same things if neutrinos turn out to be of Majorana type, even if they are shoehorned in with unjustified operational definitions that seem to stem from nowhere except an accident of history that Dirac fermions were favored over Majorana. That's precisely the way we got rid of the geocentric model of the solar system, even though it is a perfectly acceptable "operational definition" of how the solar system works (in, for example, Tycho's model, a model which stressed the idea that the Earth was not allowed to move, rather than that it was the arbitrary origin of the coordinates.) It's not about tautologies, it is about using words that connect with their implied meanings to avoid promoting misconceptions, like that neutrinos are different from antineutrinos if they are Majorana.
There might be physicists who would want to adopt a different term than "beam of antineutrinos" to describe the kind of beam that the above operational definition refers to, if neutrinos turn out to be Majorana fermions. But it does not appear that @Vanadium 50 is one of them, and if that is the case, it is pointless to continue to ask him the question you are asking him. He's already given all the answer he's going to give.
The issue is what is the justification. It is always pointless to ask someone who feels allowed to define any term how they wish without justification, but the term "antimatter" is supposed to mean something, so does require justification to use, even if "Bobneutrino" does not.

Mentor
if neutrinos are Majorana, their mass eigenstates do, or do not, distinguish matter from antimatter in any way?
It depends on what you mean by "distinguish matter from antimatter".

Very roughly speaking, if neutrinos are Majorana, a neutrino mass eigenstate would look something like ##\nu_L + \nu_L^c = \nu_L + \bar{\nu}_R##, which obviously goes into itself under CPT conjugation. But if neutrinos are Dirac, we would have one mass eigenstate that looks like ##\nu_L + nu_R##, and its CPT conjugate would be ##\nu_L^c + \nu_R^c = \bar{\nu}_R + \bar{\nu}_L##, which is a mass eigenstate, but not the same state as before. (Of course we already have an example of this in the Standard Model since this is how electrons and positrons work.)

So in the above sense, yes, a Majorana mass eigenstate would be "its own antiparticle", whereas a Dirac mass eigenstate would not.

However, as I believe has already been noted in this thread, the actual interactions that neutrinos undergo do not involve mass eigenstates. They involve two-component spinors where the "thing" component is of left-handed chirality, and the "antithing" component, the CPT conjugate of the "thing" component, is of right-handed chirality. The difference in chirality can be used to distinguish "things" from "antithings" regardless of what kind of mass eigenstates are present in the theory. This is the basis for the distinction @Vanadium 50 made between "beams of neutrinos" and "beams of antineutrinos".

Of course if you let such a beam propagate long enough, it will lose its definite nature as a "beam of neutrinos" or a "beam of antineutrinos", because of the mixing induced by the mass terms in the propagator. But how long "long enough" is depends on the masses--the smaller the masses, the longer you have to wait for a detectable amount of mixing to occur. In practical terms, it seems like "long enough" is way longer than any length of time we have so far probed in experiments.

vanhees71