I Majorana Neutrinos: Different Physics than Oscillation

[Vanadium 50]

"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.

 

[PeterDonis]

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.

 

[Vanadium 50]

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.

 

[Ken G]

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."

 

[PeterDonis]

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.

 

[PeterDonis]

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

 

[Ken G]

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.

 

[PeterDonis]

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.

 

[PeterDonis]

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.

 

[Ken G]

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.

 

[Ken G]

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.)

 

[PeterDonis]

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.

 

[PeterDonis]

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.

 

[Ken G]

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.

 

[Ken G]

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.

 

[vanhees71]

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!

 

[vanhees71]

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

 

[vanhees71]

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]

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 ;).

 

[Ken G]

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.

 

[Ken G]

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.

 

[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.

 

[Ken G]

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?

 

[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.

 

[Ken G]

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?

 

[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.

 

[Ken G]

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.

 

[Ken G]

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?

 

[Ken G]

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.

 

[PeterDonis]

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.

 

[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.

 

[Ken G]

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).

 

[PeterDonis]

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.

 

[Ken G]

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?

 

[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 $\Psi$.

 

[PeterDonis]

Are we disagreeing about anything?

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.

 

[PeterDonis]

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.

 

[Ken G]

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 $\pi^-$?

 

[Vanadium 50]

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)

 

[PeterDonis]

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?

 

[Ken G]

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).

 

[vanhees71]

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

 

[Ken G]

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

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

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

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

 

[Vanadium 50]

And to clarify that last question, this entire thread has been about not mistaking arbitrary theoretical choices for "experimental fact"

Let's be clear then..

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

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

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

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

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

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

 

[PeterDonis]

those quotes make it seem like the difference between the two particle models is not just that some tiny mass has the ability to very rarely cause the helicity to flip in the Majorana case, it is that even if that could happen to a Dirac neutrino, they still could not annihilate

This can't be right, because electrons and positrons are known to be Dirac fermions, but they can annihilate each other. If neutrinos turn out to be Dirac fermions, that just means they work like electrons and positrons. It doesn't mean neutrinos and antineutrinos can't annihilate each other.

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

 

[Ken G]

Let's be clear then..

In my message above, the implication is that a neutrino detector looking at a beam from positive (negative) pion decays will see negative (positive muons) and not a 50-50 mix. Do you dispute this?

No one on this thread has ever for one instant disputed any experimental evidence. Why would you even ask?

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

More "reasoning by analogy"? No, the issue is whether or not it is a "beam of antineutrinos", not "The Odyssey."

You can object to the L and R subscripts.

Why would you put words in my mouth? What is the actual one thing I have actually objected to? ("A beam of antineutrinos.")

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

Again, irrelevant.

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

I understand that there are historical prejudices involved here, largely based on the early expectation that neutrinos will turn out to be Dirac fermions. I'm trying to anticipate the future if neutrinos are discovered to actually be Majorana fermions. Further, I'm pointing to the usual application of "Occam's Razor": (no additional features to the language, like "aether," except those that are actually required to predict the observations.)

If you want to argue that the bar on the nubar is unnecessary if neutrinos are Majorana, you'tr tilting at windmills. We should redefine the electron charge as positive.

The question is whether the term "beam of antineutrinos" actually means anything at all for a beam of Majorana neutrinos. This is the question you have ducked in those straw-man analogies-- does the phrase have any meaning.

We should drop stellar magnitudes in favor of something physical like Janskys. We should call the first population of stars Population I and not Population III. We shouldn't drive on a parkway and park on a driveway. None of these are going to happen, so I am least have decided to move on with my life,

It's not just a question of awkward language, it's a question of using language that suggests something exists that does not exist at all. It's not like deciding if the universe is all matter or all antimatter, it is deciding if a universe of nothing but Majorana neutrinos would have any concept of that distinction.

 

[Ken G]

This can't be right, because electrons and positrons are known to be Dirac fermions, but they can annihilate each other.

I didn't mean that two Dirac neutrinos can never annihilate under any circumstances, the statement was in the same context as the quote that preceded it: two Dirac neutrinos that come from the same process, say beta decays that also produced electrons, could never annihilate each other even if one of them managed to flip helicity. Flipping helicity is all that is required for Majorana, but annihilation is still impossible for Fermions. The helicity flip is a very rare process due to the small mass, but the point being made is that it relates to a fundamental difference that goes beyond those rare helicity flips.

If neutrinos turn out to be Dirac fermions, that just means they work like electrons and positrons. It doesn't mean neutrinos and antineutrinos can't annihilate each other.

The question is still this: does a mass eigenstate of a Majorana neutrino (which @vanhees71 has defined as the free particle here) ever have any sense to which it is more of an antineutrino than a neutrino? What is the answer to that question, and how does it relate to the common phrase "a beam of antineutrinos"?