# Majorana neutrinos

Gold Member
Let me summarize it again. I hope I get it right.

For Majorana neutrinos the only difference between "neutrinos" and "antineutrinos" is the chirality. Neutrinos are left-handed and antineutrinos are right-handed. There are no other "charge-like" quantities like lepton number. In other words: For Majorana neutrinos the charge-conjugate of the left-handed neutrino is the right-handed neutrino and vice versa. Since there's no "lepton number" there's also no lepton-number conservation.
Perhaps one additional point to stress: transformations that normally turn particles into their antiparticles turn the Majorana neutrino into the identical state, just as for photons. So one could not say that neutrinos have one chirality and antineutrinos have another, because there would be no such thing as antineutrinos, there are just two different chiral eigenstates.
The lepton-number violations in the case of Majorana neutrinos are small due to the smallness of the neutrino masses.
Not sure what this means. Since pure Majorana neutrinos have zero lepton number, or no lepton number which seems like more or less the same thing, when an electron strikes a proton and makes a neutron and a Majorana neutrino, which is involved in neutronization in supernovae, lepton number violation is dramatic to say the least.
A very concise treatment is in the Book

S. Bilenky, Introduction to the Physics of Massive and Mixed Neutrinos, 2nd Ed., Springer (2018)
So it sounds like the answer to how they know to make leptons or antileptons is chirality, but not lepton number, and not particle vs. antiparticle.

Gold Member
But if neutrinos are Majorana, a neutrino beam is one of pure left handed chirality (not helicity - not polarization) and an anti-neutrino beam is one of pure right handed chirality
So, if neutrinos are Majorana, then you can have a beam of pure left-handed chirality, or a beam of pure right-handed chirality, and neither is a beam of antineutrinos. This is what I have been saying all along, and you said would require that quantum mechanics and special relativity had to be wrong.

Last edited:
Motore
Gold Member
I think the point that @Vanadium 50 has been trying to get across in this connection is that this business of "particle" vs. "antiparticle" is not that simple.
Is there not a transformation that turns particles into antiparticles? And can that transformation not be applied to a theoretical neutrino state, and see if the identical state results?
Consider the basic weak interaction lepton doublet in the Standard Model (looking just at the first generation for simplicity, having multiple flavors doesn't change anything in what I'm about to say). It's a doublet of the left-handed electron and the left-handed electron neutrino. But what do these terms actually refer to? They refer to two-component Weyl spinors. The "left-handed electron" Weyl spinor is actually a left-handed electron/right-handed antielectron (positron), and the "left-handed electron neutrino" Weyl spinor is actually a left-handed electron neutrino/right-handed electron antineutrino.
Is it necessary to call that last pair, left-handed electron neutrino and right-handed electron antineutrino, or when is it better to just call them left-handed electron neutrino and right-handed electron neutrino? That's the issue here-- just as with the aether, the question was never "can you imagine antineutrinos exist", the question was always "do you have some way to establish the existence of an antineutrino." A word that has no meaning is not kept in the lexicon, like the aether.
In both cases, which "particle" you describe the spinor as depends on which interaction you are looking at and how it is oriented in spacetime. For example, the "beta decay" interaction has both the "electron" and the "electron neutrino" lines as outgoing lines, so we describe the outgoing electron as a "left-handed electron" and the outgoing neutrino as a "right-handed electron antineutrino".
Same question again.
In other words, at the level of single Weyl spinors, it doesn't even make sense to differentiate "particles" and "antiparticles"--they're just different ways of looking at the same 2-component Weyl spinor.
That is precisely what I have been saying (though not at the level of rigor you are providing). The issue is not if one can refute the existence of antineutrinos, or the aether, the issue is, is it an angel on the head of a pin? If it is, you jettison it, that's what always happens.
So why do we say that the electron is not its own antiparticle? Because the "electron" we actually observe in experiments is not just one 2-component Weyl spinor. It's two of them put together, i.e., a Dirac spinor. In the Standard Model there is, in addition to the "left-handed electron/right-handed positron" 2-component Weyl spinor (part of the weak doublet I described above), a "right-handed electron/left-handed positron" 2-component Weyl spinor, which is a weak singlet--it has no weak interaction couplings. So the "electron" we actually observe is a mixture of the "left-handed electron" component of the weak doublet "electron" Weyl spinor, and the "right-handed electron" component of the weak singlet "electron" Weyl spinor. And the "positron" we actually observe is a mixture of the "right-handed antielectron" component of the weak doublet and the "left-handed antielectron" component of the weak singlet. These are distinct "particles"; we can't invoke what we said above about Weyl spinors and "particles" vs. "antiparticles" because we aren't dealing with a single Weyl spinor; we are dealing with a pair of them coupled by a Dirac mass term.
I appreciate this quite detailed explanation, this is not something I know already and you always explain things well.
The reason why the target in the experiments we've been discussing can still "tell" what kind of neutrinos it is seeing is that the source produces (with a small inaccuracy that we can ignore here) either pure left-handed neutrinos or pure right-handed neutrinos, and because neutrino masses are so small, the amount of "change of handedness" due to the mass term is too small to produce any detectable results at the target. So at the target we can still treat the beam as containing either all left-handed or all right-handed neutrinos, and the two possible interactions each require opposite handedness: the "produces electrons" interaction requires left-handed neutrinos, and the "produces positrons" interaction requires right-handed neutrinos.
It does appear that this resolves the question we have been grappling with about how neutrinos, be they Dirac or Majorana, can behave as they do in experiments. But my point, is that does not establish the claim that we can have beams of neutrinos or beams of antineutrinos. Until Majorana neutrinos are ruled out, the entire idea of a "beam of antineutrinos" is no different from "light propagating through the aether." And note that physicists of the late 1800s used the latter idea just as often as accelerator physicists use the former idea today. I'm not saying the idea will suffer the fate of the aether (which who knows, could reappear some day), I'm saying that what we do and do not know presently is quite similar to that situation.
And even if we find that neutrinos only have Majorana masses so that they "are their own antiparticles", they are still 2-component Weyl spinors with a left-handed and a right-handed component, and interactions that can distinguish between handedness can distinguish between the components. (Or, to put it in terms of a previous question you posed, the additional degree of freedom that lets these interactions discriminate between the neutrinos is chirality, not particle vs. antiparticle.)
Yes, my bold, that's the point.

Mentor
Is there not a transformation that turns particles into antiparticles?
It is often said that particles and antiparticles are "CPT conjugates" of each other, where "CPT" refers to the CPT transformation (the composition of charge conjugation, parity inversion, and time reversal). However, one has to be careful interpreting that. See below.

And can that transformation not be applied to a theoretical neutrino state, and see if the identical state results?
It depends on what you mean by "the identical state".

If I have a 2-component Weyl spinor ##(1 \ 0)##, applying CPT to it gives the 2-component Weyl spinor ##(0 \ 1)##. In post #140 I described a 2-component neutrino Weyl spinor as "left-handed electron neutrino/right-handed electron antineutrino". The term "left-handed neutrino" referred to ##(1 \ 0)##, and the term "right-handed antineutrino" referred to ##(0 \ 1)##.

Are those "the identical state"? As I said in post #140, that question really isn't even well-defined. They're the two "basis" components of the same 2-component Weyl spinor; which one you use to describe a particular interaction depends, as I said, on how you look at the interaction. So you could say they are "the identical state", because they're both components of the same 2-component Weyl spinor; or you could say they're not "the identical state", because they're different components that are CPT conjugates of each other. People who say that Majorana neutrinos are "their own antiparticles" would appear to lean towards the first interpretation.

For the case of electrons, as I said in post #140, electrons are not described by a single 2-component Weyl spinor, but by two of them, the "left-handed electron/right-handed antielectron" one and the "right-handed electron/left-handed antielectron" one. So an "electron" is the 4-component Dirac spinor ##(1 \ 0)_L \otimes (1 \ 0)_R##, and applying CPT to this gives the CPT conjugate 4-component Dirac spinor ##(0 \ 1)_L \otimes (0 \ 1)_R##, where now note that the ##L## and ##R## subscripts are opposite to the chirality of the Weyl spinor they are subscripts of. All this, plus the fact that electrons are charged, makes it easier to say that electrons and positrons are not "identical states".

Mentor
Is it necessary to call that last pair, left-handed electron neutrino and right-handed electron antineutrino, or when is it better to just call them left-handed electron neutrino and right-handed electron neutrino?
That would probably depend on which physicist you ask.

From an experimentalist's viewpoint, such as the viewpoint @Vanadium 50 took in his earlier post where he defined "neutrino" and "antineutrino" as "produces electrons" and "produces positrons", keeping the differentiation might still be useful even for Majorana fermions, as long as their masses are small enough that mixing of chiralities is negligible during the experiments they are doing. Since the two are CPT conjugates of each other, this usage is still consistent with the general rule that applying CPT "turns a particle into its antiparticle".

From a theorist's viewpoint, the terminology "is its own antiparticle", as it's used in some of the papers referenced in this thread when talking about Majorana fermions, basically means "only requires one field to describe, not two", where "field" here means one scalar, 2-component Weyl spinor, or vector. Majorana fermions would be the general form of the Weyl spinor case, and the photon would be an example of the vector case. (If we include composite particles at the "effective field theory" level, I believe the ##\pi^0## meson would be a scalar example, or more precisely a pseudoscalar.) The reason only one field is required in these cases is that CPT takes the field into itself (though it might mix up the components for the fields, spinor and vector, that have more than one component).

Gold Member
It is often said that particles and antiparticles are "CPT conjugates" of each other, where "CPT" refers to the CPT transformation (the composition of charge conjugation, parity inversion, and time reversal). However, one has to be careful interpreting that. See below.

It depends on what you mean by "the identical state".

If I have a 2-component Weyl spinor ##(1 \ 0)##, applying CPT to it gives the 2-component Weyl spinor ##(0 \ 1)##. In post #140 I described a 2-component neutrino Weyl spinor as "left-handed electron neutrino/right-handed electron antineutrino". The term "left-handed neutrino" referred to ##(1 \ 0)##, and the term "right-handed antineutrino" referred to ##(0 \ 1)##.

Are those "the identical state"? As I said in post #140, that question really isn't even well-defined. They're the two "basis" components of the same 2-component Weyl spinor; which one you use to describe a particular interaction depends, as I said, on how you look at the interaction. So you could say they are "the identical state", because they're both components of the same 2-component Weyl spinor; or you could say they're not "the identical state", because they're different components that are CPT conjugates of each other. People who say that Majorana neutrinos are "their own antiparticles" would appear to lean towards the first interpretation.
Ok, well there is certainly a significant degree of technical issues to consider here. But of course, the tried-and-true litmus test in science is always this: is there a good reason to preserve the term "antineutrino" to distinguish the chirality states of the Majorana neutrinos? The analogy that comes to my mind here is, imagine we have some particle that is always left-hand circularly polarized if it is matter, and right-hand circularly polarized if it is antimatter, and physicists got very used to dealing with this particle. Then let's say they discovered the photon, created in situations where it is either left-hand polarized or right-hand polarized. Would this make a reasonable analogy to what you are saying, because a CPT transformation turns left-hand polarization into right-hand polarization? And does it seem natural they would assume they have matter and antimatter there, if they had never seen two left-hand circular polarized photons annihilate each other. But after awhile, they notice that there is no real need to attribute matter or antimatter to photons, it serves no purpose at all, but they cling to that interpretation anyway because they are used to it. Then, they discover two left-hand circular polarized photons that annihilate each other (in analogy to neutrinoless double beta decay). Do they not at this point retire the concept of an "antiphoton"? I don't know how close the analogy works, but exploring it might help establish some difference here.
For the case of electrons, as I said in post #140, electrons are not described by a single 2-component Weyl spinor, but by two of them, the "left-handed electron/right-handed antielectron" one and the "right-handed electron/left-handed antielectron" one. So an "electron" is the 4-component Dirac spinor ##(1 \ 0)_L \otimes (1 \ 0)_R##, and applying CPT to this gives the CPT conjugate 4-component Dirac spinor ##(0 \ 1)_L \otimes (0 \ 1)_R##, where now note that the ##L## and ##R## subscripts are opposite to the chirality of the Weyl spinor they are subscripts of. All this, plus the fact that electrons are charged, makes it easier to say that electrons and positrons are not "identical states".
Perhaps the issue is not how easy it is to say that, but how useful it is to say it. For example, it is often noted that Michelson-Morley did not prove no aether exists, they proved that the aether covers its tracks so perfectly in SR that there is just no point to it any more.

Last edited:
Gold Member
That would probably depend on which physicist you ask.

From an experimentalist's viewpoint, such as the viewpoint @Vanadium 50 took in his earlier post where he defined "neutrino" and "antineutrino" as "produces electrons" and "produces positrons", keeping the differentiation might still be useful even for Majorana fermions, as long as their masses are small enough that mixing of chiralities is negligible during the experiments they are doing. Since the two are CPT conjugates of each other, this usage is still consistent with the general rule that applying CPT "turns a particle into its antiparticle".
I would stress the word "might" you just used in that statement. So that's the whole issue-- would it be useful, or wouldn't it? Some people still like to imagine there is an aether, but SR covers its tracks so completely that it is normally regarded as superfluous. Is the "neutrino" vs. "antineutrino" distinction superfluous when all that is different there is the chirality and nothing else? I know that @Vanadium 50 's definition was tautological, I pointed that out at the time, but the issue is if it is a good definition. The problem is, if it turns out that either type of neutrino is able to annihilate with either type, then how does it serve us to claim one is matter and the other antimatter, even if we are confident the annihilation will not actually occur in practical situations?
From a theorist's viewpoint, the terminology "is its own antiparticle", as it's used in some of the papers referenced in this thread when talking about Majorana fermions, basically means "only requires one field to describe, not two", where "field" here means one scalar, 2-component Weyl spinor, or vector. Majorana fermions would be the general form of the Weyl spinor case, and the photon would be an example of the vector case. (If we include composite particles at the "effective field theory" level, I believe the ##\pi^0## meson would be a scalar example, or more precisely a pseudoscalar.) The reason only one field is required in these cases is that CPT takes the field into itself (though it might mix up the components for the fields, spinor and vector, that have more than one component).
And we did see a theorist on this thread who said they would favor retiring the term "antineutrino" if neutrinos are found to be of Majorana type. The question is, would observers still continue to use the term, creating a "Tower of Babel" with the theorists? That rarely happens in science, the lexicon works its way out. There certainly seems to be a division there between theorists and experimentalists, but how much of that is just clinging to a certain way of thinking about it? I wager that observers of the solar system typically clung to the geocentric model long after the theorists had converted to heliocentric (it's almost the relationship between Tycho and Kepler right there).

Mentor
if it turns out that either type of neutrino is able to annihilate with either type
This is misstating things. Consider the simplest Feynman diagram for, say, electron-positron annihilation. How many electron/positron lines does it have? The answer is not two. It's one. The ingoing electron and ingoing positron lines are the same line. (Inside the diagram, if we are considering annihilation in isolation with no other particles present, this single line passes through two vertices that have photon lines; those are the outgoing photon lines. But it's the same, single electron/positron line all the way through.)

You could, in principle, have a "neutrino annihilation" diagram that worked the same way (although the lines coming out could not be photon lines, since neutrinos are uncharged; I think you could do it with Z bosons), and that would be true regardless of whether neutrinos turn out to be Majorana or Dirac fermions; it would be the same diagram in either case. In short, this type of diagram doesn't care about the Majorana/Dirac distinction, and can't tell you anything useful about it.

Staff Emeritus
You could, in principle, have a "neutrino annihilation" diagram that worked the same way (
I don't think so. What does the neutrino annihilate into?

Not photons, as you say.

The only particle that is light enough would be other neutrinos, and we wouldn't call that annihilation; we would call it scattering.

dextercioby
Mentor
What does the neutrino annihilate into?
Obviously the energy would have to be high enough to produce something like a pair of Z bosons. So actually observing this diagram would be very rare. Perhaps at those energies "annihilation" isn't the term that would be used to describe what happens. But at the Feynman diagram level, it would still be the same kind of diagram as the electron annihilation one.

malawi_glenn
Staff Emeritus
"Tiny" as in "none has yet been measured", correct? I.e., this is an experimental constraint.
I think "hopeless"m, as @vanhees71 called it is more appropriet.

Because an actual physical beam is not pure, you find yourself looking for a part per zillion effect on top of a percent-level effect. If a paper published, "We expected the bean to be 98% neutrinos but it's actually only 97.99999999%. Discovery!" would that be credible? I think not.

Mentor
If a paper published, "We expected the bean to be 98% neutrinos but it's actually only 97.99999999%. Discovery!" would that be credible? I think not.
Eventually it might be if detection technology advanced far enough. We already know of some theoretical predictions that have been confirmed to 13 or 14 decimal places.

Staff Emeritus
It's not a technology issue. It's physics.

Neutrino beamlines work by takint a beam of pions, letting them decay to muons (which live ~100x longer) and neutrinos, and then stopping the muons. The problem is that they only live 100x longer, so ~1% of the time they decay in the same volume as the pion. Muons give neutrinos of the opposite sign than pions., If I make the decay pipe shorter by a factor or 2, I reducxe the numbers of muons that decay by a factor of 2, but I also reduce the number of pions that decay by a factor of 2, so the contamination is pretty much constant.

Furthermore, the highhest rate neutrino detector proposed is the DUNE near detector, which will say a few million events per day. That's with a beam power of over a megawatt. If you want to see a 10-9 effect, you need 1018 events to do it. Run for a year means terawatt scale beams, Not only does nobody know how to do this, not only does the target magically need to survive this much power without evaporating, not only would it take up all the US electrical production capacity and then some, but this is probably optimistic by several orders of magnitude.

This is what drives people to giant underground tanks of xenon.

Gold Member
This is misstating things. Consider the simplest Feynman diagram for, say, electron-positron annihilation. How many electron/positron lines does it have? The answer is not two. It's one. The ingoing electron and ingoing positron lines are the same line. (Inside the diagram, if we are considering annihilation in isolation with no other particles present, this single line passes through two vertices that have photon lines; those are the outgoing photon lines. But it's the same, single electron/positron line all the way through.)

You could, in principle, have a "neutrino annihilation" diagram that worked the same way (although the lines coming out could not be photon lines, since neutrinos are uncharged; I think you could do it with Z bosons), and that would be true regardless of whether neutrinos turn out to be Majorana or Dirac fermions; it would be the same diagram in either case. In short, this type of diagram doesn't care about the Majorana/Dirac distinction, and can't tell you anything useful about it.
I take your point, but I think the key issue is what the leftward and rightward parts of that single neutrino line connect to. That's what distinguishes the electron/positron side of the diagram you describe, from the photon/photon side. In other words, the reason we take your diagram and decide we have a positron and electron on one side is the need to change charge, which changes the behaviors at the two ends of that line, whereas the photon side doesn't need that. But neutrinos also don't need to change charge, so we would need some other reason to decide we need to call them a neutrino and an antineutrino. That seems to be the crux of the matter-- if the processes that those ends of the neutrino line connect to can be the same process, as can be true for the photons on the other side of the diagram, then that's what I mean by "annihilating with another particle like itself." That's what happens in neutrinoless double beta decay, the two beta decays that "anchor" the ends of the neutrino line you are describing are exactly the same process, so there's no need to say the neutrino turns into an antineutrino when it emerges from those vertices.

So even though that process is very rare, it is a process that requires the neutrinos not be interpreted as neutrino/antineutrino, whereas there is no process that requires they do be interpreted that way. In fact, maybe we've been going about this all wrong-- we've been asking for the Majorana neutrino camp to show an experiment that proves neutrinos are Majorana particles. Why have we not asked the Dirac neutrino camp to show an experiment that proves they are Dirac particles? Again the aether analogy comes to mind-- SR did not come from an experiment that tried to show there wasn't an aether, it came from an experiment that tried to show that the aether that was widely being assumed was there was actually there-- and failed to do so. If someone would claim there is such a thing as an antineutrino, let them design an experiment that proves such a particle exists, and if they fail to do so enough times, eventually we say "I guess we don't need the idea after all."

Mentor
I think the key issue is what the leftward and rightward parts of that single neutrino line connect to.
They don't connect to anything. They are the two incoming external legs.

the reason we take your diagram and decide we have a positron and electron on one side is the need to change charge
That automatically gets taken care of because, since they are the two incoming external legs, the two ends of the same line must be CPT conjugates (since they're the same line). It doesn't matter what quantum numbers (if any) the CPT conjugation changes.

whereas the photon side doesn't need that
It doesn't need "that" because the two outgoing photon external lines are not the same line. They are photon lines coming from two different vertices inside the diagram and they never connect to each other. So they aren't the same as the external electron lines.

if the processes that those ends of the neutrino line connect to can be the same process
This can't possibly be the "crux" of the matter because it is true for the electron annihilation diagram: both vertices that the single electron line goes through are the same process--the same kind of vertex. So you can't distinguish between electrons and neutrinos this way.

Mentor
Just to be clear, here is the electron-positron annihilation Feynman diagram I have been describing:

Gold Member
They don't connect to anything. They are the two incoming external legs.
I know, I mean the way you define what those particles are is all the processes they could connect to. You just label the lines, but that's what the label implies-- all those processes left off the end.
That automatically gets taken care of because, since they are the two incoming external legs, the two ends of the same line must be CPT conjugates (since they're the same line). It doesn't matter what quantum numbers (if any) the CPT conjugation changes.
It matters to the labels of the lines, that are the implied particles-- they have those quantum numbers. What we are deciding is whether it is necessary (given current observations), or even makes sense (in the case of future neutrinoless double beta decay), to label one of the lines as a particle and the other an antiparticle in the case where a diagram like that would apply to neutrinos.
It doesn't need "that" because the two outgoing photon external lines are not the same line. They are photon lines coming from two different vertices inside the diagram and they never connect to each other. So they aren't the same as the external electron lines.
Ah, I had not appreciated the subtlety that the central line is attributed to the electron/positron and not to the photon. I presume because the electron/positron has rest mass so if we go into the COM frame, we see something there that is kind of instantaneously stationary, though I'm not quite sure what that "thing" is that corresponds to that central line! But all I mean is that the photon lines are not labeled photon and antiphoton because there are no processes that connect to the ends of the photon lines, and define those labels, that require or can even define that distinction.
This can't possibly be the "crux" of the matter because it is true for the electron annihilation diagram: both vertices that the single electron line goes through are the same process--the same kind of vertex. So you can't distinguish between electrons and neutrinos this way.
It's not the vertices in the diagram I'm talking about, it's the hypothetical vertices left off the ends of the lines that define the meaning of those particles, the meaning of the labels that go with the lines. That's what is different for the electrons and positrons, they spiral in different directions about magnetic fields and so on. But that's what would not be different for Majorana neutrinos, if they both come from precisely the same process like beta decays.

Mentor
I had not appreciated the subtlety that the central line is attributed to the electron/positron and not to the photon. I presume because the electron/positron has rest mass
It's because QED (and a fortiori the Standard Model) has a vertex with two electron legs (one with the arrow pointing into the vertex and one with the arrow pointing out) and one photon leg, but does not have a vertex with two photon legs and one electron leg.

This would be equally true for a massless fermion interacting with a massless boson. The constraint is not the fermion having nonzero rest mass but the kinds of vertices that can occur in a Lorentz invariant Lagrangian.

Mentor
it's the hypothetical vertices left off the ends of the lines that define the meaning of those particles
The "meaning" of the particles isn't defined by hypothetical vertices, it's defined by what asymptotic free particle states exist in the theory. For both electrons and neutrinos, these will be mass eigenstates. For electrons we know these are Dirac mass eigenstates. For neutrinos, what kind of states they are will depend on how the question of whether neutrino masses are Dirac or Majorana is finally resolved.

malawi_glenn
Gold Member
It's because QED (and a fortiori the Standard Model) has a vertex with two electron legs (one with the arrow pointing into the vertex and one with the arrow pointing out) and one photon leg, but does not have a vertex with two photon legs and one electron leg.

This would be equally true for a massless fermion interacting with a massless boson. The constraint is not the fermion having nonzero rest mass but the kinds of vertices that can occur in a Lorentz invariant Lagrangian.
Ok thanks, fair enough.

Gold Member
The "meaning" of the particles isn't defined by hypothetical vertices, it's defined by what asymptotic free particle states exist in the theory.
But what can that possibly mean other than the types of vertices those states can participate in? What else could possibly define a theoretical construct that can only make its presence felt by what it interacts with, in any empirical science? The reason I'm bringing this up is it is essential to the empirical meaning of any concept of an "antineutrino." I can call it "Bob", but if it does not have specific interactions that define its meaning, it's just a word. Same for the theory that describes the free states-- more words and equations that are a mathematical game unless they have interactions that can be tracked and observed. That's what I mean by "hypothetical vertices at the ends of the lines", the things that make this a testable science-- crucial for the issue of what it means to claim "antineutrinos exist."
For both electrons and neutrinos, these will be mass eigenstates. For electrons we know these are Dirac mass eigenstates. For neutrinos, what kind of states they are will depend on how the question of whether neutrino masses are Dirac or Majorana is finally resolved.
Precisely, which is why I pointed out that it is odd the Majorana camp is being asked to serve up neutrinoless double beta decay, but the Dirac camp has been presented with no equivalent challenge. It seems there is no way at all to test that neutrinos are of Dirac nature, which is already a significant problem in an empirical science. In other words, it's worse than saying the Majorana interpretation requires observing a rare event, we have the Dirac interpretation with no observation to point to at all!

Normally we adjudicate such issues using Occam's Razor, so we attach no unnecessary parts to a theory. That's why there is no aether any more, not that observations showed it doesn't exist, but observations did not require it. It's hard for me to see the concept of an antineutrino, instead of merely a chiral state of a neutrino, in that exact same light. Hence my analogy with left- and right-circularly polarized photons having no reason to be considered as photons and antiphotons. (I realize the neutrino physics is more complicated, but it's an analogy.)

Mentor
what can that possibly mean other than the types of vertices those states can participate in?
I believe @vanhees71 explained that in a post a while back.

vanhees71
Mentor
it is odd the Majorana camp is being asked to serve up neutrinoless double beta decay, but the Dirac camp has been presented with no equivalent challenge
One of the papers linked to in the thread, IIRC, described experimental tests that could only be passed by Dirac neutrinos.

vanhees71
Gold Member
One of the papers linked to in the thread, IIRC, described experimental tests that could only be passed by Dirac neutrinos.
OK, good to know such tests exist. If such tests are conducted, and the neutrinos don't pass, I presume there will be a tendency to become skeptical of the "antineutrino" concept-- with no damage to QM or SR or supernova theory.

vanhees71
Gold Member
I believe @vanhees71 explained that in a post a while back.
Any explanation based on theory is of course just going to trace right back to what I'm talking about-- the vertices are the experiments that tested the theory. This is just science. If I'm not being clear, let me put it another way: where is the experimental apparatus in the Feynman diagram? Of course it would be unwieldy to include them, but they must certainly be implied, or there is no test of the theory included in the theory.

Last edited:
vanhees71
Gold Member
2022 Award
As in the early days of neutrino observations, in the early-mid 1950ies, the way to proceed is to write down the most general effective theory and then figure out, how to test the parameters with dedicated experiments.

The history of the neutrino is pretty interesting and paradigmatic for how research in physics and the interplay between theory and experiment works. In 1930 Pauli stated the neutrino hypothesis in an informal letter to some people attending a conference on ##\beta## decay, which was quite a riddle in these days. One must remember that the neutron wasn't discovered yet, and there where ideas around that a nucleus consists of protons and electrons, and the electrons come out somehow, leading to ##\beta## decay. That didn't work out, because the electrons had a continuous spectrum and not a line spectrum you'd expect from electrons somehow bound within the nuclei.

Then in 1934 Fermi came up with his QFT for neutrinos, and in these days it was clear that parity had to be conserved. He could explain the ##\beta## spectrum by assuming a three-body decay.

Then more and more details about neutrino interactions were discovered, and the picture of the original Fermi theory didn't fit anymore. There also also the famous "##\theta \tau## puzzle". There were apparently two particles called ##\theta## and ##\tau## which looked pretty much the same except that one decayed to 2 pions, the other to 3 pions. The resolution was the hypothesis that in fact there's only one particle, which we call ##\text{K}^+## today, which decays in both channels, ##2 \pi## (parity even) and ##3 \pi## (parity odd), and that implies the violation of parity conservation, i.e., the invariance under space reflection must be broken by the weak interaction.

This opened the way to adapt the theory. Feynman and Gell-Mann wrote down the most general Lagrangian with all kinds of couplings of neutrinos and antineutrinos. Experimentalists, as always with neutrinos, had a hard time to figure out, which coupling is right. A while the situation was pretty ambigous, but after some time the famous "vector-minus-axial-vector coupling", i.e., maximal violation of chiral symmetry with purely left-handed neutrinos (right-handed anti-neutrinos) turned out to be right, and the parity violation was established, e.g., by the Wu experiment, leading to one of the quickest Nobel prizes in history for Lee and Yang just one year after the experimental discovery.

Then in the mid 60ies the Standard Model started to take its so far final form, with the neutrinos conjectured to be massless and the assumption of lepton-number conservation with only left-handed neutrinos and the charge-conjugated neutrino being distinguished from the neutrino by lepton number (+1 for neutrinos, -1 for antineutrinos).

This, however lead to the famous "solar-neutrino puzzle", which lasted for decades, until in the 1990ies neutrino-flavor oscillations have been discovered, leading to the conclusion that neutrinos must have a non-zero mass and that there must be flavor-mixing matrix.

The question today is, whether the neutrinos are Dirac fermions with the corresponding mass terms and there's a conserved lepton number. Then with the weak coupling being strictly V-A, still only left-handed neutrinos (and right-handed anti-neutrinos) are interacting, and the right-handed parts which are coupled to the game solely by the Dirac masses are in this sense "sterile neutrinos".

The other possibility is that lepton number is not conserved and that the charge conjugated neutrinos are identical with the neutrinos, i.e., that the only difference in the neutrino and the charge-conjugated "anti-neutrinos" is their chirality. The masses are then given by the Majorana mass terms, which violates lepton-number conservation and with the possibility of "neutrino-less double ##\beta## decay", which seems to be considered the most promising way to discover that neutrinos are Majorana particles. This picture has the appealing effect that there are no "sterile neutrinos". Just recently the hints at sterile neutrinos in various neutrino experiments seem to get weakter again, and maybe that's a hint that maybe after all the neutrinos might be Majorana fermions.

Finally there's also the possibility to have both, Dirac and Majorana mass terms. Then lepton number is not conserved but there are sterile right-handed neutrinos.

It's clear that the question, which is the right neutrino model is still open!

dextercioby and Dr.AbeNikIanEdL
Gold Member
2022 Award
You did, but...

I don't think this is right. Go back to the Weyl fields:

$$C | \psi_L> = \overline{\psi_L} \neq \psi_R$$

I would say it is nothing uncer C or at least sterile.
Let's follow Coleman's lectures, which are most lucid on this topic.

The most simple way to treat Majorana Fermions is to use a Majorana representation of the ##\gamma## matrices, which are then purely imaginary, i.e., ##\gamma^{\mu *}=-\gamma^{\mu}## (where the star of a matrix means just take the conjugate complex values of its matrix elements) Then charge conjugation is given simply by ##\psi^c=\psi^*##, where the star on the field operator means to just take the conjugate complex of the entries in the operator-valued column spinor, i.e., ##\psi^*=\psi^{\dagger \text{T}}##.

Then obviously the free Dirac equation is invariant under charge conjugation, and the corresponding unitary charge-conjugation operator maps particle annihilation operators to antiparticle-annihilation operators and vice versa, and the same holds for the creation operators. By definition the vacuum state is invariant under ##C##.

Now since the ##\gamma^{\mu}## are purely imaginary matrices, ##\gamma_5=\mathrm{i} \gamma^0 \gamma^1 \gamma^2 \gamma^3## is also purely imaginary. Now
$$\psi_L=\frac{1}{2}(1-\gamma_5) \psi$$
and
$$\psi_L^c=\psi_L^* = \frac{1}{2} (1-\gamma^5)^* \psi^*=\frac{1}{2} (1+\gamma^5) \psi^*,$$
i.e., it is right-handed.

A Majorana fermion is now a fermion with mass, i.e., it must have both left- and right-handed components, i.e., it's not a Weyl field. A Majorana fermion is its own charge-conjugated state, i.e., ##\psi^c=\psi^*=\psi##. So you can represent it by a left-handed Weyl spinor, ##\chi_L## and write
$$\psi=\chi_L+\chi_L^c=\chi_L+\chi_L^*,$$
and ##\chi_L^*## is a right-handed Weyl spinor (note that I work in a Majorana representation of the Dirac matrices).

A very nice paper, working out all the features of Weyl, Dirac, and Majorana fermions, is

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

Gold Member
2022 Award
Perhaps one additional point to stress: transformations that normally turn particles into their antiparticles turn the Majorana neutrino into the identical state, just as for photons. So one could not say that neutrinos have one chirality and antineutrinos have another, because there would be no such thing as antineutrinos, there are just two different chiral eigenstates.
Note that for photons ##A^{\mu c}=-A^{\mu}## in order to make QED C invariant.
Not sure what this means. Since pure Majorana neutrinos have zero lepton number, or no lepton number which seems like more or less the same thing, when an electron strikes a proton and makes a neutron and a Majorana neutrino, which is involved in neutronization in supernovae, lepton number violation is dramatic to say the least.

So it sounds like the answer to how they know to make leptons or antileptons is chirality, but not lepton number, and not particle vs. antiparticle.
Neglecting the neutrino masses means that there are only left-handed neutrinos and lepton number is conserved with neutrinos having lepton number 1 and anti-neutrinos lepton number -1. The violation of lepton number conservation by small Majorana mass terms is small, and thus lepton-number violation is NOT dramatic. If it were, we'd have discovered it, and (pure) dirac neutrinos were ruled out already!

Dr.AbeNikIanEdL
Dr.AbeNikIanEdL
This picture has the appealing effect that there are no "sterile neutrinos". Just recently the hints at sterile neutrinos in various neutrino experiments seem to get weakter again, and maybe that's a hint that maybe after all the neutrinos might be Majorana fermions.
However, within the SM you can’t write down plain majorana mass terms, it would have to be via the (dimension 5) Weinberg operator. So this would only be an effective theory that still needs some UV completion. If this is via a seesaw-like mechanism, you get your ‘sterile’ neutrinos right back ;).

vanhees71 and malawi_glenn
Gold Member
Neglecting the neutrino masses means that there are only left-handed neutrinos and lepton number is conserved with neutrinos having lepton number 1 and anti-neutrinos lepton number -1. The violation of lepton number conservation by small Majorana mass terms is small, and thus lepton-number violation is NOT dramatic. If it were, we'd have discovered it, and (pure) dirac neutrinos were ruled out already!
Thanks for the wonderful above summary of the situation. I'm still trying to understand what it means to say that the violation of lepton number conservation is weak. I get that in accelerator experiments, it wouldn't show up often, and this is related to the fact that neutrinoless double beta decay is an incredibly rare phenomenon. But I think what you must mean is that the detection of the violation is hard to do. If neutrinos are Majorana particles, then every time we make a neutrino beam from a bunch of leptons, or what we thought was an antineutrino beam from a bunch of antileptons, isn't the neutrino beam itself an example of drastic lepton number violation? Then when the neutrino beam strikes a detector, the lepton number is in a sense recovered when new leptons and antileptons are created. So the violation "covers its tracks," but it would be there in the Majorana theory, just hard to see if we tend to regard neutrinos as being Dirac particles so able to have a lepton number as the "middlemen" in that experiment.

Gold Member
However, within the SM you can’t write down plain majorana mass terms, it would have to be via the (dimension 5) Weinberg operator. So this would only be an effective theory that still needs some UV completion. If this is via a seesaw-like mechanism, you get your ‘sterile’ neutrinos right back ;).
But don't you "hide" those sterile neutrinos at very high mass, whereas the Dirac version doesn't have that attractive feature? I thought this was argued as an advantageous element of the Majorana picture, in the sense that it helps explain why neutrino masses are so low.

vanhees71
Dr.AbeNikIanEdL
Sure, that’s where the popularity of the seesaw mechanism comes from. My point was that while you can write down a Majorana mass term with only the experimentally observed fields, you need something additional to make it work in the whole standard model, as you need in the Dirac case.
Note that in seesaw the neutrinos fundamentally have both Dirac and Majorana mass terms (the latter one only for the sterile ones), so at that point you have not decided between those two cases. Only the effective degrees of freedom you see at low energies are purely Majorana.

vanhees71
Gold Member
Sure, that’s where the popularity of the seesaw mechanism comes from. My point was that while you can write down a Majorana mass term with only the experimentally observed fields, you need something additional to make it work in the whole standard model, as you need in the Dirac case.
Note that in seesaw the neutrinos fundamentally have both Dirac and Majorana mass terms (the latter one only for the sterile ones), so at that point you have not decided between those two cases. Only the effective degrees of freedom you see at low energies are purely Majorana.
Ah, good point, so one cannot even say that Dirac neutrinos are part of the standard model and Majorana neutrinos are not, and what's more, what is actually being debated is which type is exhibited at low energies (because the standard model always involves some combination of both). So would you say that within the standard model, two contrasting types of neutrino behavior are distinguished at low energy: the Dirac type, for which the nonsterile neutrinos are most naturally massless, and the Majorana type, which by the seesaw mechanism attributes low mass to the nonsterile versions and high mass to the sterile ones? Then when neutrinos are found to have mass, you must kind of scramble to shoehorn the Dirac model into that new picture, whereas Majorana was there all along?

vanhees71
Dr.AbeNikIanEdL
If you want massive Dirac neutrinos you can just add the right handed neutrinos and write down the mass term/a corresponding higgs coupling and be done. That’s literally copy-and-paste of what you do with any other fermion in the SM. You likewise might say this is what was there all along, there is no scrambling or any ‘real’ problem.

Its a matter of taste, and preconceptions about how a fundamental model should look beyond describing observations, which of the versions appears more ‘natural’ to you.

vanhees71
Gold Member
OK, so there is no theoretical reason to favor either type without getting into personal preferences, and both fall equally under the "standard model." So given that, would it be fair to say that characterizing a beam of neutrinos that were generated by antileptons as "a beam of antineutrinos," an extremely common phrase in the current lexicon, is purely a kind of historical accident without any strong theoretical foundation other than an arbitrary preference for lepton number conservation over the other possible theoretical preferences?