I Majorana Neutrinos: Different Physics than Oscillation

[Vanadium 50]

Majorana neutrinos violate lepton number. But the degree of violation in accelerator experiments is tiny - part in a billion, part in a trillion, whatever. In 0νββ decay, the violation is significant (changes lepton number by 2) but the rates are low if Majorana, zero if Dirac.

 

[vanhees71]

Maybe this helps to clarify precisely the issues with phenomenology referred to above:

https://link.springer.com/article/10.1007/JHEP05(2017)024 (open access)

 

[vanhees71]

Majorana neutrinos violate lepton number. But the degree of violation in accelerator experiments is tiny - part in a billion, part in a trillion, whatever. In 0νββ decay, the violation is significant (changes lepton number by 2) but the rates are low if Majorana, zero if Dirac.

Exactly, and because of this indeed today the question whether neutrinos are Majorana or Dirac particles is undecided. To decide it by simply looking at production of charged leptons by scattering of neutrinos seems hopeless and the neutrino-less-double-$\beta$ decay is ultra challenging but presumably the only chance to decide the question.

 

[PeterDonis]

Maybe this helps to clarify precisely the issues with phenomenology referred to above

I don't see anything in that paper that addresses the issue I described in post #117 (and in earlier posts as well).

 

[vanhees71]

I thought we were still discussing the issue discussed in #107.

 

[PeterDonis]

the degree of violation in accelerator experiments is tiny

"Tiny" as in "none has yet been measured", correct? I.e., this is an experimental constraint.

If you are saying that theoretically, the violation in accelerator experiments done to date is expected to be tiny for Majorana neutrinos, that is what I don't understand; I described the issue in post #117 (and in earlier posts). In terms of lepton number violation, with Majorana neutrinos, it seems like the violation of lepton number conservation in the experiments you described earlier in the thread should be huge: even with a source that generates neutrinos with all leptons (for example, from decays of $\pi^-$ mesons to muons, $\mu^-$) or all antileptons (for example, from decays of $\pi^+$ mesons to antimuons, $\mu^+$), with a target that contains roughly equal numbers of protons and neutrons, the detector should detect roughly equal numbers of leptons and antileptons. But that doesn't seem to be what is observed.

 

[Dr.AbeNikIanEdL]

The problem is that our ability to tune the neutrino source so that leptons of only one charge get produced in the detector (electrons or positrons, but not both) does not seem to be consistent with a Majorana neutrino model--on such a model, electrons and positrons should be produced in the detector in roughly equal numbers no matter what we do to the neutrino source.

Whether neutrinos are Dirac or Majorana (or massless in a different world), the coupling structure in the weak sector is always such that you have a left-chiral Weyl field coupling to leptons, and the charge conjugate of that Weyl field coupling to anti-leptons. This determines what interactions happen, and at this stage there is no lepton number violation.
Now the two Weyl fields are either combined into a Majorana field, or you add an additional right-handed field and build a Dirac field out of those, and the two fields you have combined end up in the mass term together.
So if the mass term is Majorana, you have terms in the propagator that mix between these two* (in terms of the Weyl fields the term is more a "vertex with two legs", see for example here https://arxiv.org/abs/2012.09882). There you have the lepton number violation, but the terms are proportional to the neutrino masses, so any violation of lepton number is suppressed by those.

Edit: * what I mean here by "the two" is "the field interacting with leptons" and "the field interacting with anti-leptons"

 

[PeterDonis]

the coupling structure in the weak sector is always such that you have a left-chiral Weyl field coupling to leptons, and the charge conjugate of that Weyl field coupling to anti-leptons

I agree the weak couplings involve only left-chiral Weyl fields and their charge conjugates, but I don't think it's quite as simple as you describe. In particular, I don't think it helps with the issue I described, because Weyl fields don't propagate by themselves. What propagates from the source to the target in the experiments that have been described are mass eigenstates.

Suppose, for example, that we have set the source to produce neutrinos--for example, suppose the source is NuMi at Fermilab and we've selected the mode where the neutrinos come from decay of positive pions and kaons, which produce antimuons $\mu^+$ and muon neutrinos, i.e., neutrinos described by left-handed Weyl fields. (Note that this is not the same as the coupling you describe--the antimuons and muon neutrinos are both outgoing from the reaction, and no leptons are coming in, so they must have opposite lepton numbers since this is a Standard Model, lepton number conserving process.)

But what propagates from the source to the target is not just the left-handed Weyl field, but a mass eigenstate. The question is, which mass eigenstate?

If it's a Dirac mass eigenstate that propagates from the source to the target, then it's easy to understand why, at the target, only electrons are produced, even though the target contains roughly equal numbers of protons and neutrons. A Dirac mass eigenstate couples the left-handed neutrino to a right-handed neutrino, which has no weak couplings, so the only possible interaction at the target is the coupling of the left-handed neutrino to an electron (which happens when the neutrino hits a neutron and turns it into a proton).

But if it's a Majorana mass eigenstate that propagates from the source to the target, then we have a left-handed neutrino coupled to a right-handed antineutrino, and both of those have weak couplings. So at the target, both interactions are possible: left-handed neutrino to electron (along with neutron to proton) and right-handed antineutrino to positron (along with proton to neutron), and with a target having roughly equal numbers of protons and neutrons, we should expect to observe roughly equal numbers of electrons and positrons. But that's not what we observe, so it seems like observations in these experiments should rule out a Majorana mass.

 

[Dr.AbeNikIanEdL]

But if it's a Majorana mass eigenstate that propagates from the source to the target, then we have a left-handed neutrino coupled to a right-handed antineutrino, and both of those have weak couplings. So at the target, both interactions are possible: left-handed neutrino to electron (along with neutron to proton) and right-handed antineutrino to positron (along with proton to neutron).

Yes, as I said the propagator mixes the two fields in the Majorana case. But only proportional to the neutrino mass. When you say

we should expect to observe roughly equal numbers of electrons and positrons

you did not just assume

a target having roughly equal numbers of protons and neutrons

but also that the mixing in the propagator produced an equal mixing of the two neutrino fields, which is not true.
In principle lepton number violation is possible in this kind of scenario, for example by observing $\mu^- \to e^+$ conversion (+ nuclear conversions to make charges work out), see e.g. https://arxiv.org/abs/2110.07093. But all these effect are suppressed by the small neutrino masses.

(Note that this is not the same as the coupling you describe--the antimuons and muon neutrinos are both outgoing from the reaction, and no leptons are coming in, so they must have opposite lepton numbers since this is a Standard Model, lepton number conserving process.)

Probably, what I meant was that you have left- and right-handed doublets and that determines the coupling structure and hence possible interactions/decays.

 

[PeterDonis]

the propagator mixes the two fields in the Majorana case. But only proportional to the neutrino mass

Ah, I see. So if the source is producing muon neutrinos, we would only expect a fraction of antineutrinos at the target proportional to the mixing rate in the propagator and the time of propagation. Which for the experiments we have been discussing will be too small to measure. But in principle, if we could measure it, finding it would be a way of showing the existence of Majorana masses for neutrinos (since Dirac masses would not produce such effects no matter how long the propagation time).

 

[Dr.AbeNikIanEdL]

Yes, my understanding is that neutrino-less double beta-decay is your better bet in most scenarios.

 

[vanhees71]

The flavor eigenstate is a superposition of the mass eigenstates, and what propagates is this superposition. You can only detect the reaction of this propagating state with your detector material, and this interaction is again governed by the weak-interaction vertices, and that's why there's some probability to detect the neutrino which started as an electron neutrino as a neutrino of another flavor. That's neutrino mixing, and occurs no matter whether the neutrinos are Majorana or Dirac fermions.

Your second paragraph is also right of course, but that's just lepton-number violation, which necessarily occurs for Majorana neutrinos.

Ah, I see. So if the source is producing muon neutrinos, we would only expect a fraction of antineutrinos at the target proportional to the mixing rate in the propagator and the time of propagation. Which for the experiments we have been discussing will be too small to measure. But in principle, if we could measure it, finding it would be a way of showing the existence of Majorana masses for neutrinos (since Dirac masses would not produce such effects no matter how long the propagation time).

The flavor-mixing matrix mixes the flavors (left-handed with left-handed and right-handed with right-handed), mass terms mix the left-handed and right-handed parts of the field.

 

[Ken G]

The point I was making is that I don't see anyone doing that. @Vanadium 50 doesn't seem to be; he explicitly said the experimental results he describes are consistent with a Majorana neutrino model (though he hasn't explained how that can be the case).

I quoted just above the text that inspired that phrase. Again:
"So we're left with two possibilities:
1. We see antineutrinos, more or less as expected.
2. Our understanding of stellar collapse is grossly wrong (no neutronization), and QM is wrong, and SR is wrong, and all three are wrong in just the right way to conspire to give a signal that looks exactly like expected."
That seems literally the definition of demanding there be a difference between neutrinos and antineutrinos, because nothing else makes any sense. If that statement is no longer @Vanadium 50 's position, I'm glad that the thread has produced some progress there. But above all, I agree that "who said what when" is of no great consequence, what we need to understand is how Majorana neutrinos could be completely consistent with all current observations, while not exhibiting any distinction between neutrinos and antineutrinos, historically analagous to how the absence of an aether was completely consistent with all observations circa 1887.

 

[vanhees71]

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.

For Dirac neutrinos there are independent left- and right-handed components and they carry lepton number as a "charge-like quantity", and neutrinos and antineutrinos are distinguished by opposite lepton numbers. The right-handed neutrinos (left-handed anti-neutrinos) are "sterile", i.e., non-interacting.

Whether neutrinos are Majorana or Dirac particles is not yet empirically decided, i.e., all data are consistent with both Majorana or Dirac neutrinos.

The lepton-number violations in the case of Majorana neutrinos are small due to the smallness of the neutrino masses.

A very concise treatment is in the Book

S. Bilenky, Introduction to the Physics of Massive and Mixed Neutrinos, 2nd Ed., Springer (2018)

 

[Vanadium 50]

and as he describes it, the source can be made to produce a beam that only produces electrons when fired into the target (indicating that only the neutron reaction above is happening) or a beam that only produces positrons when fired into the same target (indicating that only the proton reaction above is happening).

The question is, if neutrinos are pure Majorana fermions, how is it that possible?

First, it's not just as "He describes it". I showed the data, and provided a pointer to a collectgioon of references..

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

 

[Vanadium 50]

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

You did, but...

For Majorana neutrinos the charge-conjugate of the left-handed neutrino is the right-handed neutrino and vice versa. Si

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.

 

[Vanadium 50]

But in principle, if we could measure it, finding it would be a way of showing the existence of Majorana masses for neutrinos (since Dirac masses would not produce such effects no matter how long the propagation time).

Yes, but this is hard.

A "pure neutrino beam" is limited to about 99% purity, and there is an anti-corrleation between intensity and purity so most real beams don't do even this well.

 

[PeterDonis]

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

But if neutrinos have non-zero masses, there is no such thing--at least not once the beam has propagated for any appreciable time from the source compared to the mixing rate given by its mass. The mass term mixes left-handed and right-handed fields, whether it's Dirac or Majorana.

 

[PeterDonis]

this is hard.

Agreed. I'm just trying to make sure I'm clear about all of the possibilities allowed by the model, including ones that are extremely hard to detect in actual experiments.

 

[PeterDonis]

That seems literally the definition of demanding there be a difference between neutrinos and antineutrinos

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.

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.

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". But the interaction that produces the electrons detected in the experiments we've been discussing has the neutrino line as an incoming line, not an outgoing line, so we describe it as a left-handed electron neutrino. But in both cases, it's the same interaction (same Feynman diagram vertex) involving the same Weyl spinors.

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.

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.

If neutrino masses are pure Majorana masses, OTOH, then there is no weak singlet "right-handed electron neutrino/left-handed electron antineutrino" 2-component Weyl spinor. (Or at least, there is no reason to include one in the model.) The Majorana mass term couples the two components of the same Weyl spinor. So the "neutrino" we actually observe in experiments would just be one 2-component Weyl spinor, and what we said above about Weyl spinors and "particles" vs. "antiparticles" would apply to it.

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.

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.)

 

[Ken G]

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.

 

[Ken G]

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.

 

[Ken G]

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.

 

[PeterDonis]

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".

 

[PeterDonis]

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).

 

[Ken G]

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.

 

[Ken G]

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).

 

[PeterDonis]

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.

 

[Vanadium 50]

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.

 

[PeterDonis]

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.