Earlier this week I explained how neutrinos can get their mass within the Standard Model of particle physics, either by engaging with the Higgs field once, the way the other particles do, or by engaging with it twice. In the first case, the neutrinos would be “Dirac fermions”, just like electrons and quarks. In the second, they’d be “Majorana fermions”. Decades ago, in the original Standard Model, neutrinos were thought not to have any mass at all, and were “Weyl fermions.” Although I explained in my last post what these three types of fermions are, today I want go a little deeper, and provide you with a diagrammatic way of understanding the differences among them, as well as a more complete view of the workings of the “see-saw mechanism”, which may well be the cause of the neutrinos’ exceptionally small masses.
[N.B. On this website, mass means “rest mass” except when otherwise indicated.]
The Three Types of Fermions
What’s a fermion? All particles in our world are either fermions or bosons. Bosons are highly social and are happy to all do the same thing, as when huge numbers of photons are all locked in synch to make a laser. Fermions are loners; they refuse to do the same thing, and the “Pauli exclusion principle” that plays a huge role in atomic physics, creating the famous shell structure of atoms, arises from the fact that electrons are fermions. The Standard Model fermions and their masses are shown below.
The simplest and most basic type of fermion is a Weyl fermion, though we are not aware of any such particles in nature. A particle which is a Weyl fermion must have zero mass; this is what neutrinos were once thought to be, before evidence of their masses was uncovered. I’ll draw a Weyl fermion as a single line, indicating a particle traveling from, say, left to right.
The second type of fermion, a Dirac fermion, is common in the Standard Model; the electron is a Dirac fermion, as are its its cousins the tau and the muon, and all the quarks. A Dirac fermion is made from the marriage of two Weyl fermions
A Dirac fermion is made from the marriage of two Weyl fermions. It is sometimes convenient mathematically to view the mass as the mechanism that makes the particle switch from one Weyl fermion to the other. (But do not take this too seriously, because it is only strictly true in processes where the fermion is “virtual”. [For mathy folks: this is a generalization of the statement that 1/(e-m)=1/e+m/e2+m2/e3+… to a matrix calculation known as a Neumann series; this works for m<<e, but a real particle would have the equivalent of m=e, where the expansion is ill-defined.])
This viewpoint is illustrated in Figure 3 below, where a Dirac fermion moves from left to right; one of its Weyl fermions is shown in blue and the other in green, flipping back and forth at a rate set by the mass m. If the mass were zero, the flip would never happen, and the particle would be two separate Weyl fermions, one blue, one green.
For this mass to be possible, the two Weyl fermions must have the same properties; they must interact with all the forces of nature in the same way. For instance, they must have the same electric charge.
The third type, a Majorana fermion, marries a Weyl fermion to itself; even more precisely, it marries a Weyl fermion to its anti-particle. This is shown below, where the antiparticle is shown as a dashed line of the same color. [Physicists often use arrows to show the difference between the particle and its anti-particle, but this can get confusing, because the anti-particle traveling to the right is often depicted with an arrow pointing to the left.] Just as for the Dirac fermion, this can only work if the mass connects objects with the same properties. That means that a Weyl fermion can only be a Majorana fermion if it and its anti-particle have the same properties. For instance, since an anti-particle has the opposite electric charge as the corresponding particle, both particle and anti-particle must have electric charge zero. This isn’t true for the electron, muon, tau or the quarks, but it is true for the neutrinos, which is why they will eventually have the option of being Majorana fermions. But not quite yet.
The Standard Model, the Higgs Field, and Dirac Fermions
One thing that makes the Standard Model truly remarkable is that in the absence of the Higgs field, all of its fermions would be massless Weyl fermions. What we call “an electron” would actually be two different massless particles with different properties; they wouldn’t even deserve the same name. Here, looking ahead to the effect of the Higgs field, we will call them both “electron”, but I’ll color them differently to remind you that they are really different. One half of the electron is affected by the weak nuclear force (in particular, the W field and its corresponding particle, the W boson), while the other half is not. Because of this, they cannot marry… not until the Higgs field comes along and, by switching on and taking on a non-zero value, hides the effect of the weak nuclear force. (In fact this hiding, which involves giving the W and Z bosons a big mass, is what makes this force weak, in the sense discussed here and here.)
By hiding the weak nuclear force, and making it an effect that is small for processes at low energy (hence the enhanced lifetimes of particles that decay via the weak nuclear force), the Higgs field makes the two halves of the electron compatible. The two Weyl fermion halves were unable to marry because of the weak nuclear force, but with that issue hidden away, they are now free to marry, and a Dirac fermion results. This is indicated in Figure 4.
You see that we begin with an interaction at left between the two halves of the electron (blue and green) and the Higgs field. Once the Higgs field has a constant non-zero value across the universe, the Higgs field acts just as a Dirac mass term would, flipping the two halves of the electron from one to the other. The mass of the electron is then the product of two quantities:
- the strength of the Higgs field’s interaction with the electron, called ye (the “electron Yukawa coupling”), which was present in the original diagram at left, and
- the magnitude of the value of the Higgs field, v, of about 250 GeV .
(There is also a technical square root of 2, not worth discussing here.) As promised, if v were zero the electron would be massless. Also, since the electron’s mass is only 0.0005 GeV, the quantity ye is only 0.000003. The reason for its small size is unknown.
If neutrinos are Dirac fermions, the same mechanism applies as for electrons. Figure 6 reminds us that one of the two half-neutrinos is “sterile”, meaning it responds neither to the electromagnetic, strong nuclear, or weak nuclear force; but other than that, Figure 6 looks the same as Figure 5. In fact, in this case, all the Standard Model’s fermions get their masses in the same way.
However, to explain the fact that the neutrinos have mass no more than a millionth of the electron’s, their Yukawa couplings must also be at least a million times smaller, i.e. yν < 10-12. There’s no obvious reason why this should be the case. (Perhaps I’ll tell you at a later time how new forces impacting the sterile neutrinos might be able to cause this, but that’s too advanced a subject for today.) With Dirac neutrinos, there’s no obvious principle to make their masses small.
The Original See-Saw
But with Majorana neutrinos, there is a possible principle, and this involves the “see-saw” mechanism that I sketched last time. This is part of what makes it appealing (but remember, that’s far from saying it’s true, even though some physicists occasionally seem to get mixed up about that.)
In the mid-to-late 1970s, a number of people pointed out the basic idea, to different degrees of detail. They observed that
- although, without the Higgs field being switched on, an ordinary neutrino cannot have a Majorana mass, as it is not the same as its anti-neutrino [technically, they have different “hypercharges”, the name give for the U(1) in SU(3)xSU(2)xU(1)],
- a sterile neutrino is the same as its anti-particle, so it can have a Majorana mass, as in Figure 4, even without the Higgs field, and
- a sterile neutrino with a very large mass and an ordinary interaction with the Higgs field (no tiny Yukawa coupling needed) would lead to an ordinary neutrino with a very small mass.
This is illustrated in Figure 7. There are now several steps to the logic.
- First, we take the same Higgs/neutrino/sterile-neutrino interaction shown in Figure 6 above; but now we use it twice, and in between, we add a huge Majorana mass M for the sterile neutrino.
- Second, with such a huge mass, this sterile neutrino is irrelevant for any current experiments; all it does is create an interaction between a neutrino, its anti-particle, and two instances of the Higgs field; and this interaction (which would disappear altogether if M were infinite) is suppressed by 1/M.
- Third, the Higgs field switches on, and gives the neutrino a mass that contains two factors of yν, two factors of v, and one factor of 1/M .
A key question you should ask: if you look at the discussion of how the weak nuclear force works in this post, (see its Figure 6), you’ll see there is a similar suppression that comes from a W boson because of its large mass mW — except that there the suppression is 1/mW2. Well, this is a key difference between bosons and fermions: where a boson suppresses effects by the square of its mass, a fermion does so by only one factor of its mass.
What we see in Figure 7 tells us that both yν and M enter into the neutrinos’ masses. This means we could get neutrino masses a million times smaller than the electron’s in different ways. For instance, we could have yν = 1 and have M be up around 1015 GeV. Or we could have yν = ye = 0.000003, and M somewhere around or just above 1000 GeV, even possibly within range of the Large Hadron Collider [LHC]. Or anything in between.
So you see, even if we measure the neutrino masses, we won’t actually know M. We’ll only know a maximum for it: Yukawa couplings much, much bigger than 1 don’t really make sense (this is a long story) so if the largest neutrino mass is about a trillion times smaller than v, as experiment currently suggests, we have an estimate that M can’t ever be much larger than a trillion times bigger than v. It could be much smaller.
But remember this estimate only holds if the see-saw mechanism is in operation!
The Generalized See-Saw
A sterile neutrino with a very large mass is a particularly simple idea, but reality might be a lot more complex than this. Because that reality may be hidden up at very high energy, we may not learn anything about it anytime soon. Other high-energy phenomena, of an unknown source, could potentially mimic the effect of a sterile neutrino, and we wouldn’t easily know. All we can potentially do in current experiments is probe the neutrino/anti-neutrino interaction with two Higgs fields, as shown in Figure 8, without knowing how it was generated. Any such interaction must be suppressed by a high-energy scale, which we can call Emax.
In my previous post I wrote that Emax becomes M in the original see-saw. But now you can see that this was a bit of cheat. Comparing Figure 7 to Figure 8, you can see that Emax is really M/yν2 (ignoring factors of 2), and could, in the original see-saw, be much, much larger than the actual sterile neutrino mass M.
Or, said differently: if we measure the largest neutrino mass mν and try to infer Emax by using the neutrino masses, namely by using the estimate Emax < v2/mν, we have to remember that the scale at which new phenomena appear might be much, much lower than this estimate.
This is why physicists at the LHC actively look for sterile Majorana neutrinos under the rationale of the see-saw. They can do this because sterile neutrinos, despite being impervious to the electromagnetic, strong nuclear and weak nuclear forces, still interact with ordinary matter a little bit. This can occur via the interaction with the Higgs field seen in Figure 6 and 7, and moreover, after the dust settles, it’s always the case that ordinary and sterile neutrinos mix; sterile neutrinos will be a little bit ordinary, and the ordinary neutrinos a little bit sterile. These effects potentially allow a sterile neutrino to be produced at the LHC and to decay to observable particles, as long as M is small enough. (See for example https://arxiv.org/abs/1506.06020 .) I think most theoretical physicists view this effort as a long shot, but theorists might well be wrong here, since we are really quite confused about the fermions’ masses… and if you don’t check via an experiment, you don’t know for sure.
But it could easily be the case that sterile neutrinos or whatever mimics them is far out of reach of current experiments. In that case, we are limited to detailed study of the neutrinos we know — and there is still a lot more to learn there. That, however, is for another day.
Technical Epilogue: Three Neutrinos are Better than One
Let me just finish with a comment about the fact that there are, of course, three neutrinos, and everything I’ve said so far has focused on one at a time. Doing three completely is more complicated, of course, but if you learned linear algebra in school then you can follow it. For Dirac neutrinos, the Yukawa couplings yν become a 3×3 matrix Yν, and the 3×3 mass matrix for the neutrinos is just
- Mν = Yν v .
The actual masses of the neutrinos are the three eigenvalues of the matrix Mν .
In the Majorana case, the Yukawa couplings are still a 3×3 matrix Yν, but now we also have the sterile neutrino masses, which also form a 3×3 matrix M . Define the inverse of M, which we call M-1, as the matrix such that M M-1 = 1, where 1 is the 3×3 identity matrix. Then the ordinary neutrino masses are
- Mν = v2 Yν M-1 (Yν)T
where (Yν)T is the conjugate of the Yukawa matrix. Again, the actual masses of the neutrinos are the three eigenvalues of the matrix Mν ; but this time they involve the combinations of the two matrices Yν and M, so unfortunately neither matrix could be uniquely determined even if the matrix Mν were fully known. This does represent a significant limitation on what we can hope to learn about neutrinos in the near future, if they are indeed of Majorana type.
26 Responses
Dimensions and motions, could these be analogous to space and forces? Is space and energy all you need to create an universe?
Space is easy to visualize, energy (the thing that create the unstable space), not so much. But it’s there and maybe one day someone will figure out the 5W’s for energy.
But, back to my concept that the tendency for all the motions to return back to the zero dimension is all that is required to decipher the present day physics.
I added a remarkable illustration how particles (particular motions of points in space) can get trapped to form hadrons and maybe even electrons and neutrinos. Please notice it only takes three identical parts, particles ( motions) to create a very stable 3-dimensional cube, confidence?
What is energy?! Anyone?
[Editor’s note: link removed, this is not the place for advertising one’s pet theories of the universe.]
I pressed the post comment button too fast, I wanted to include this article which I think is relevant to my concept of trapped dimensions.
Again, I ask, what is energy? Visualizing the universe dissipating down to zero dimension is easy but it doesn’t make it all the way down because of all the energy, why? Yes, a cyclic universe makes sense, but that would mean we need to stop searching for a beginning and an end. I guess we’re not wired up to think like that because we are born and die. Maybe if we start thinking collectively then we see that life, after all, is about existing.
https://www.quantamagazine.org/colliders-and-supercomputers-force-fresh-hints-in-quark-mystery-20220214/
Dr. Strassler, Thank you so much for taking the time to response, I greatly appreciate it. I have a lot more questions; however it is not fair to take your time to give me a class. For class instruction I should have continued on with my post grad work. But, I will ask one more question that is not related to this paper, but to your answer above, top quark decay, I thought that quarks were not found in nature alone, unless they were part of a quark gluon plasma as in the very early stages of the big bang. So is the top quark decay part of a quark gluon plasma losing energy?
It’s a subtle point. Quarks, if created from scratch, will cease to be isolated after about a trillionth of a trillionth of a second, which is about how long it takes for the strong nuclear force to surround them with gluons and other quarks and anti-quarks. But the top quark decays about ten times faster than that, so its decay occurs while it is still largely isolated.
So, no, it is not an issue of quark/gluon plasma, which is a high-temperature effect; this is instead a short-time effect.
Is there any evidence for non-zero neutrino masses besides the flavor oscillations?
I get why the observation of neutrinos changing flavor is considered strong evidence for them having non-zero rest mass. But as far as I’m aware it is all indirect evidence; nobody has actually directly measured a nonzero neutrino mass, and the experiments (like KATRIN) trying to do that have only established upper bounds. And unlike all the other SM particles, those “rest masses” would be highly abnormal. When we observe other fundamental particles with rest mass, we always observe them with a specific fixed one- electrons have X, W bosons have Y- but the neutrinos would ALWAYS be in a superposition of 3 different neutrino rest masses. Because the only way to explain neutrino flavor oscillation with massive neutrinos is to have each flavor eigenstate correspond to a superposition of the rest mass states (and vice versa).
So, are there any viable “massless neutrino” mechanisms for neutrino flavor oscillations that are still allowed by the data? Or mechanisms where the flavor oscillation is more fundamental, so the neutrinos do have a tiny mass but it is because of the oscillation (or both have a common underlying cause)? Because it sounds like tiny neutrino masses are a good initial hypothesis, but would introduce at least as many mysteries/problems as they’d solve.
The answer to your first question is “so far, no.”
It is not true, however, that “those “rest masses” would be highly abnormal”. When a top quark decays, it decays to a bottom quark/strange quark/down quark superposition; this quark has an ill-defined mass. The point is, however, that this superposition very quickly decoheres and one always observes one of the three mass eigenstates. This would happen with neutrinos also, in some circumstances that are currently impractical; your statement that “the neutrinos would ALWAYS be in a superposition of 3 different neutrino rest masses” is not correct. It is a subtle point, but roughly, decoherence in the neutrino case takes a long time in all current practical experiments; but still, the word “ALWAYS” in your statement is wrong. You may get something out of this discussion: https://indico.cern.ch/event/645835/contributions/2777732/attachments/1592421/2520651/cern1a.pdf , where the conditions for coherence or decoherence are discussed.
I don’t agree that “tiny neutrino masses are a good initial hypothesis, but would introduce at least as many mysteries/problems as they’d solve.” To explain mixing without the masses introduces all sorts of problems. For example, this one (involving violations of Einstein’s relativity) was tried and didn’t work: https://arxiv.org/abs/1109.6860 . I vaguely remember there were some proposals for getting mixing without masses about 20-25 years ago, involving large extra dimensions of space, but at the moment I can’t find them.
Thanks. I wasn’t aware that the issue was coherence rather than superposition. I’d always seen the top quark described as having a rest mass of about 172 GeV, with its short life meaning that this is really the central peak of a range of possible rest masses for any given top. Same goes for the other SM particles, like bottom quarks or Ws, with the stable ones having much narrow rest mass distributions because they don’t have the lifetime uncertainty. It sounded like neutrinos’ rest mass distribution would instead have 3 isolated, narrow peaks (for the 3 mass eigenstates) rather than a single one. So that unlike all the other particles, stable or not, there isn’t a single central or average value that can be called “the” electron neutrino’s mass.
But it sounds like nobody’s come up with a good alternative to tiny neutrino masses for the neutrino oscillation explanations in the ~30 years since they were discovered. So it makes sense that theorists would assume their existence for the time being (like they did for many years with the Higgs).
The issue of a “range of rest masses” is yet a third, unrelated issue: neither coherence nor superposition, but uncertainty. You’re mixing up three very different quantum mechanics issues. The neutrinos do indeed have three narrow mass peaks, if the standard theory of their masses and mixing is correct. Whether a particular experiment has to treat the neutrinos as coherent waves is an issue of how far apart those mass peaks are (relative to the energies and times of the experiment), not whether the peaks are narrow.
Dr. Strassler; Regarding my questions about epochs. I may have been a bit to linear in my thinking when reading this paper. I was looking at the fact that Weyl fermions have to have mass zero until they interact with the Higgs field/boson. I was thinking along a timeline here, first exist as a Weyl fermion until higgs interaction then becoming a massive particle in order to have the boson interaction.
I was using the timeline also in thinking about the early big bang particle soup. Until there is enough cooling for the higgs to occur the particle soup would be Weyl fermions. Once higgs cools out then in would be in a Weyl particle soup, causing lots of making of mass because the universe would not be very big This would lead to in a lot of neutrino – W boson interactions which could create a plus or minus charge disparity.
Rather than saying a timeline, maybe I should say I was thinking of a step process as the big bang progressed.
Okay, I think I understand your question. You’re a little mixed up but not entirely off the mark.
During the earliest parts of the Hot part of the Big Bang (it may have been cold earlier, during inflation), the universe was hot enough that the Higgs field was switched off. In that period all the known fermions were Weyl fermions and that period, you might say, was a Weyl-fermion soup. A tiny fraction of a second later, as the expanding universe cooled, the Higgs field switched on; all the fermions then immediately developed their masses, becoming Dirac fermions or Majorana fermions. So there is no Weyl soup after that; it is a Dirac and Majorana fermion soup. However, both before and after there are also photons and gluons, and before the Higgs switches on there are W bosons too; so it’s a fermion *and* boson soup at all times.
However, your statements “..causing lots of making of mass because the universe would not be very big This would lead to in a lot of neutrino – W boson interactions which could create a plus or minus charge disparity” don’t make any sense to me; they connect unrelated issues.
Dimitris, First off, My personal opinion I think that it is important to see different ways of thinking about problems physics is faced with today. We have to move out of comfort zones and have discussions. So thank you for posting another idea. As for me, I look at zero, one and two dimensions as mathematical constructions, just like adding additional dimensions down below the Planck Length. I would “discuss” that existence doesn’t start until three dimensions. I am also not a unified theory fan because we still have so much more to learn before we should ever start thinking about a GUT, or a ToE for that matter.
And I too am old with thick skin which is why I offered my thoughts. And, I too am always looking for comments, good or bad, because it is the only way to learn and move forward
I guess I should have written it better, my point is both the photon (bosons) and the electron (fermions) are both point particles (zero dimension). What makes them different is their motions through space.
The “forces” only come into play when one particle interacts with another. The motions each particle, every point in space, was acquired at the Big Bang. Since then the universe has been in a transient phase with one rule, return to it’s fundamental state, zero dimension.
Postulate:
1. All particles are at zero dimension, a point.
2. The difference in particles trapped in the same space, sphere (atoms), is their 3D motion, specific to each, and no two particles can have the same motion, i.e. Pauli exclusion principle is applicable to particles already trapped in 3D motions.
3. Photons, bosons, can occupy the same space because they remain zero dimension, (points) particles, hence can have the same motion.
So, since bosons, zero dimension particles, are the “force” carries, then a unified theory can be postulated as the requirement of all particles returning down the fundamental dimension, zero. Hence, the strong force is the interaction of a set of quarks, trapped in the same space, trying to return down to the same point. The energy that keeps them in motion comes from the Big Bang.
This postulate implies a balance between the strong force and initial energy, so this implies that, indeed, there could be an inifinite number of hadrons, composed of different combinations of quarks that have achieved the mentioned balance.
Conclusion: All you need to create a universe is space and energy (motion). The forces we have calculated are those that we can measure as of today. I predict there should be different magnitudes of strong force w.r.t. the different set of quark combinations.
Comments? Please good and bad, I got thick skin, old, but thick.
Foe electron,as per the text discussed here-
the strength of interaction is combined to higgs field can give rise to mass of electron,but in unique arbitrary path giving rise to coordination with strong force and other usual components of research at LHC for quark-gluon.
Great article! Thanks for taking the time to write it.
Dr. Strassler, thank you very much, this was very informative and it really makes me think harder on things. I have a lot of questions, but the biggest one that comes to mind for me is that if we go back to the big bang, does there now have to be a Weyl Epoch that happens before the formation of quarks and neutrinos? This would have to be followed by a boson epoch and we could get the formation of quarks and neutrinos. I am using the term “epoch” very loosely here. Also, I am a little lost on charge conservation. The W boson come in plus and minus charge versions. Neutrinos have no charge so in the weak interaction how is charge conservation maintained.
Thank you Dr. Strassler.
The W- boson can convert a neutrino to an electron, or a positron to an anti-neutrino; the W+ boson can convert an anti-neutrino to a positron or an electron to a neutrino. That should answer the last question.
About the first question, I don’t unfortunately understand what you are referring to; there is no “boson epoch” and quarks and neutrinos are, as far as as we know, elementary and therefore present at the very beginning. Is this something you read about somewhere?
It’s like 1=-1 or 1= other than 1 then
A neutrino is always a Mayorana fermion,full stop.
One question. What turns the Higgs Field on? I kinda got the impression that it was always constant and on.
This is kind of a linguistic problem: “switched on” is the best verbiage I can come up with in English. What turns it on, historically, is that the universe was too hot and the Higgs field was highly variable but on average was zero, and then as it cooled the Higgs field was able to settle down. What assures that it is switched on today are its interactions with itself, but these in turn are generated by a whole host of effects whose details we do not know anything about at this stage. So let us simply say: it is not known why the Higgs field is switched on or why its value is 246 GeV.
I hope the science will soon find the way how to trigger the decay of Thorium and U238 without using fast neutrons…
Great post. I think this one and the last one are the best explanations of the standard model that I have come across. Thanks for explaining it beautifully.
If it isn’t too much of a digression, I was wondering about the difference between an electron and a muon, in the context of figure 5. Does the muon have a larger mass simply because it has a larger Yukawa coupling? Or is it more complicated than that?
Simply because it has a larger Yukawa coupling, yes.
Thank you, Prof. Strassler, superbly done. I understand and appreciate your most laudable didactic effort, but a bit of algebra and maybe even an occasional matrix or a d/dt doesn’t subtract from a layperson’s understanding, whereas it is of great value for those who thirst for the added insight it provides. Perhaps, like above, keep adding a “technical epilogue” to your pages, please?
WOW! … I wish I had taken the math scholarship Prof. Kenneth MacKenzie offered me instead of staying in engineering.
WOW! Bloody fascinating. So, how much math does one need to get into this wonderful infinitesimal world of particle physics?
BTW: Is it valid to generalize v*ye ~ A sin(wt)?