# 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##.

vanhees71
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