Of Particular Significance

Celebrating the Standard Model: Why Are Neutrino Masses So Small?

POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON 07/18/2022

For the general reader interested in particle physics or astronomy:

Most of the Standard Model’s particles have a mass [a rest mass, to be precise], excepting only the photon (the particle of light) and the gluon (found in protons and neutrons.) For reasons not understood at all, these masses stretch out over a range of a trillion or more.

If it weren’t for the three types of neutrinos, the range would be a mere 400,000, from the top quark’s mass (172 GeV/c2) to the electron’s (0.000511 GeV/c2), still puzzling large. But neutrinos make the puzzle extreme! The universe’s properties strongly suggest that the largest mass among the neutrinos can’t be more than 0.0000000001 GeV/c2 , while other experiments tell us it can’t be too much less. The masses of the other two may be similar, or possibly much smaller.

Figure 1: The masses of the known elementary particles, showing how neutrino masses are much smaller and much more uncertain than those of all the other particles with mass. The horizontal grey bar shows the maximum masses from cosmic measurements; the vertical grey bars give an idea of where the masses might lie based on current knowledge, indicating the still very substantial uncertainty.

This striking situation is illustrated in Figure 1, in which

  • I’ve used a “logarithmic plot”, which compresses the vertical scale; if I used a regular “linear” plot, you’d see only the heaviest few masses, with the rest crushed to the bottom;
  • For later use, I’ve divided the particles into two classes: “fermions” and “bosons”.
  • Also, though some of these particles have separate anti-particles, I haven’t shown them; it wouldn’t add anything, since the anti-particle of any particle type has exactly the same mass.

As you can see, the neutrinos are way down at the bottom, far from everyone else? What’s up with that? The answer isn’t known; it’s part of ongoing research. But today I’ll tell you why

  • once upon a time it was thought that the Standard Model solved this puzzle;
  • today we know of two simple solutions to it, but don’t know which one is right;
  • each of these requires a minor modification of the Standard Model: in one case a new type of particle, in another case a new phenomenon.

How the Majority of Standard Model Particles Get Mass

The story begins with Steven Weinberg in 1967 and Abdus Salam in 1968, who first introduced the basic concept of how the Standard Model’s “fermions” (often referred to, to my dismay, as “matter particles”) get their masses. This is illustrated in Figure 2.

It’s a weird, awkward idea, certainly at first glance, so much so that if experiment didn’t confirm it, it would be hard to believe. The idea is that particles like the electron are really put together from two particles, not one. These half-electrons are not half-particles, though; they are particles in and of their own right, except that a particle like this (called a “Weyl fermion“) must have zero mass.

Figure 2: An electron is a Dirac fermion, formed through the marriage of two Weyl fermions, made possible by the Higgs field. The two parts of the electron differ only in their interaction with the weak nuclear force, especially though the W field and its boson. Without the marriage arranged by the Higgs field, neither “half” of the electron could have any mass.

Without the Higgs field, there is a fundamental obstruction to the electron having any mass at all. Although both halves of the electron have “electric charge” (meaning they are affected by electric and magnetic forces), only one half of the electron interacts with the particle known as the W boson, a crucial component of the weak nuclear force. You can’t “marry” two half-particles into one if they behave in fundamentally different ways. They have to behave the same way with respect to all the elementary forces of nature. So on the face of it, these two half-electrons must remain unmarried, and with zero mass, forever.

But when the Higgs field switches on, it changes the rules, giving the W boson its mass. Along the way, as Weinberg and Salam pointed out, it allows these two discrepant electron-halves to be married into one. The resulting electron is a “Dirac fermion“, with a mass.

If the Higgs field’s interaction with the electron’s halves were very strong, then this would be a strong marriage and the newly formed electron would have a very large mass, like the top quark. But instead this interaction is very weak, and the marriage is a loose one, resulting in an electron whose mass is much smaller than the top quark’s.

The same logic applies for the heavier cousins of the electron (the muon and the tau), as well as for the six types of quarks. Each one is really made from two half-particles — two Weyl fermions — of which only one half interacts with W bosons, and each of which would have zero mass were it not for the marriage engineered by the Higgs field.

Massless Neutrinos? Standard Model 1.0

But the logic for neutrinos has a twist. As far as experiment was able to tell, for several decades after the discovery of the first neutrino in 1956, each of the three neutrinos is a sort of half-neutrino… a Weyl fermion that interacts with the W boson and thus experiences the weak nuclear force (but is unaffected by the electromagnetic and strong nuclear forces.) By the logic I just gave you, this Weyl fermion can only have zero mass; it has nothing to marry.

Well, for decades that seemed fine; there wasn’t any experimental sign that neutrinos have any mass. It seemed that the Standard Model gave a simple explanation as to why neutrinos were (apparently) massless: they alone among the fermions of the Standard Model (version 1.0) lack their other half, which is needed for a mass-making marriage.

Figure 3: In the original version of the Standard Model, one imagined neutrinos had zero mass because they had no other half to marry with.

But gradually, evidence accumulated that neutrinos can change their type while in flight. This “neutrino mixing”, as it is called, is a long story (here’s my article about it), and today it is a major area of particle physics research. The mixings are most easily explained by at least two of the three types of neutrinos having mass. Well, if any of them do, then the logic that predicted all neutrinos have zero mass must be wrong; and if two of them do, it may well be all three.

Meanwhile, because neutrinos are easily made during the early era of the universe, they are abundant in the universe even today. Were their masses large, this would have had an impact on how galaxies and clusters of galaxies form, and more generally on why the universe isn’t uniform and instead has lots of structures in it. Careful study of these structures indicates that neutrino masses must all be more than several million times smaller than the electron’s mass, and more than a trillion times smaller than the top quark’s.

So that poses a puzzle. The neutrinos aren’t “Weyl fermions” with zero mass; they have mass just like all the other fermions do. But if that’s so, why are all their masses so much smaller than those of all the other fermions?

For that matter, can the Standard Model even accommodate these experimental discoveries? This risks getting us into a semantic argument about what is and isn’t “the Standard Model” when it comes to neutrinos, an argument I will take pains to avoid. Let’s just ask: What can we do to the Standard Model 1.0 to make it consistent with experiment? Theoretical physics offers two very different possible stories, and either one (or a combination of the two) is consistent with all experimental data.

Dirac-type Neutrinos?

Let me start with the easy one: let’s just make the neutrinos like the electron. We will imagine that there is an as-yet-undiscovered other half to each type of neutrino, a Weyl fermion with initially zero mass, until the Higgs field switches on. But there’s something unique about neutrinos: the second half of each neutrino is “sterile,” in the sense that its interactions with ordinary matter are staggeringly weak. This is because:

  • neither half of the neutrino is affected by the electromagnetic force (it is electrically neutral, hence its name)
  • neither half of the neutrino is affected by the strong nuclear force
  • unlike the known half of the neutrino, the added half does not interact with the W boson — and more generally it is unaffected by the weak nuclear force too.

No electromagnetic force, no strong nuclear force, no weak nuclear force. That means the only known forces that affect it are gravity and the Higgs force. Such a particle would leave no direct traces in any current experiments.

Figure 4: One can allow neutrinos to have mass by adding their other halves to the Standard Model; they then become Dirac fermions just like the electron and the other fermions. Their masses can only be small because their interactions with the Higgs field is tiny, a fact for which there’s no definite explanation, though the “sterility” of the added half-neutrinos offers some opportunities.

These sterile half-neutrinos would then allow the neutrinos to be married by the Higgs field to form “Dirac fermions” with a mass, as shown in Figure 4; this is the same as Figure 2 for the electron. But I haven’t addressed why these masses are much smaller than an electron’s. In fact, the small masses for these Dirac neutrinos would have to arise from very tiny interactions with the Higgs fields. Why should those interactions be so tiny?

No one knows. Yet the very fact that these half-neutrinos are sterile suggests a direction for speculation. Their lack of interaction with the known particles and forces offers these particles possibilities to interact easily with as-yet-unknown strong forces and as-yet-unknown particles. (Such possibilities are highly restricted for non-sterile particles, like electrons, quarks and W bosons.) These additional interactions, in turn, could potentially suppress the sterile half-neutrinos’ interactions with the Higgs field. So if neutrinos are Dirac fermions, there may be ways to explain their small masses, though it will likely require a big addition to the Standard Model.

Majorana-type Neutrinos?

The second possible origin of the neutrino masses is quite different. It uses a unique feature of the known half-neutrino; by marrying itself, it can become a “Majorana neutrino”, with a mass. It does this by visiting the Higgs field twice, as shown in Figure 5. [This all may seem socially awkward to some readers, but that is completely appropriate given its namesake.] As this self-marriage is only possible for an electrically neutral fermion, it’s not an option for the electron or the quarks.

There’s really nothing wrong with this idea, so why wasn’t it introduced very early on in the Standard Model’s history? The problem is that visiting the Higgs field twice comes at a price: it inevitably makes the Standard Model incomplete. Though the Standard Model may still work fine at current experiments, giving neutrinos a mass by two visits to the Higgs field assures that there must be a maximum energy Emax beyond which the Standard Model’s equations cannot do their entire job. If you do experiments above that energy, there will be measurements for which the Standard Model makes no predictions at all. That means that someday, something else will have to be added to the Standard Model to fix this problem.

Figure 5: A single Weyl neutrino, unaffected by the electromagnetic or strong nuclear force, can “marry itself” by interacting with the Higgs field twice, making it a Majorana neutrino with a mass.

Well, so what? There was a time when this bothered people; they felt that incomplete theories weren’t really consistent. That time is past; with better understanding of quantum field theory (the math that underlies the Standard Model,) we no longer view this with concern. After all, we already know the Standard Model is incomplete: gravity is not a part of it, because we aren’t sure how best to combine Einstein’s theory of gravity with the other forces. The incompleteness for gravity may only show up at an Emax a million trillion times higher than we currently access at our particle accelerators. For Majorana neutrinos it won’t be so extreme; Emax may be no more than one trillion times higher than we currently can reach. But the conceptual issue of incompleteness isn’t necessarily so different for the two examples.

So yes, neutrinos can be Majorana fermions. It comes at the price of an incomplete theory, but conversely it avoids the need to add three new half-neutrinos to the theory. More importantly, it provides the seeds of an explanation here as to why neutrino masses are so small! The larger the energy Emax at which the Standard Model ceases to work, the smaller the neutrino masses mneutrino have to be.

This is known as the “generalized see-saw mechanism”, and the fulcrum of the see-saw is the Higgs field’s value: about 250 GeV, and usually called “v“. However small v is relative to Emax, the neutrino masses mneutrino must be even smaller compared to v.

For instance, if v is a trillion times smaller than Emax, then the largest mneutrino must be at least a trillion times smaller than v. If you like equations, you can say this more briefly:

  • mneutrino < v2/Emax .

So if the Standard Model (or at least its neutrino portions) were to continue to be valid to energies a trillion times higher than the Large Hadron Collider, and the neutrinos are Majorana fermions, then we would be guaranteed that the neutrinos’ masses must be comparable to or smaller than what is shown in Figure 1.

Figure 6: The “generalized seesaw” effect, where a Majorana neutrino is formed by a single Weyl neutrino marrying itself as in Figure 5; it involves two interactions with the Higgs field but requires phenomena not included within the Standard Model. This naturally produces small neutrino masses at the expense of making the Standard Model incomplete (and thus non-predictive) above a limiting energy Emax.

But don’t misread the logic. If neutrino masses are someday measured to be a trillion times smaller than the Higgs field’s value, then although Emax cannot be larger than a trillion times the Higgs field’s value, it could be much smaller. It could, for example, be just ten times larger than v, and within reach of the Large Hadron Collider or its successor. If that were the case, we might see experimental evidence of the Standard Model’s breakdown in the relatively near future.

In short, what the see-saw mechanism does is promise us that the Standard Model will break down at or before the energy scale v2/mneutrino, the square of the Higgs field’s value divided by the largest mass among the neutrinos. [Remember, though, that this assumes that neutrinos are Majorana. If they’re Dirac, then this isn’t true!]

The Original See-Saw

If you’re curious (otherwise you can skip the section), let me describe to you the original see-saw mechanism. Introduced in the mid-1970s, it involves adding the same sterile neutrinos needed for Dirac fermion neutrinos in Figure 4, and repurposing them for the see-saw mechanism. What happens is that our familiar but hapless neutrino marries a sterile neutrino as in Figure 4, unaware that this sterile neutrino has an enormous mass Msterile. (Wait; wasn’t the sterile neutrino a massless Weyl neutrino? Ah, in Figure 4 it was; but because it is sterile it doesn’t have to be massless! Being sterile, it can have a Majorana mass all on its own, without any Higgs field!)

Figure 7: The original see-saw mechanism, in which a sterile neutrino with a large mass plays the role of the unknown phenomenon in Figure 6. The more mass the sterile neutrino has, the less mass the known neutrinos will have.

The presence of this sterile neutrino with a large mass causes the Standard Model’s equations to break down; what we generically called Emax above is simply Msterile. And since this sterile neutrino’s mass is so big, we cannot see it in any ongoing experiments; as far as our experiments are concerned, our familiar neutrino is lonely, essentially married to itself after all — a Majorana neutrino, despite its efforts to be Dirac. The larger is Msterile, the smaller is the familiar Majorana neutrino’s mass — hence the name “see-saw”.

The Many Open Questions

There’s a huge question looming over us that I haven’t even addressed yet: what are the actual neutrino masses? We know some relations between them; we know a maximum for them; and we know a minimum mass for at least one of them; but beyond that, the issue remains open. Experiments are ongoing. But that’s a story for another day.

Once we know their masses, will we be able to tell whether the neutrinos are of Dirac-type or of Majorana-type? Not unless we’re lucky. It could be that particularly prominent clues are accessible to current technology; but if not, we may not know the answer for decades. Why is it so difficult to figure out? It’s not hard in principle; if you’re an impractical sort of theoretical physicist, you can imagine all sorts of of methods. Here’s a fun one: if you could increase the Higgs field’s value v inside a box by one percent, the mass of an electron in the box would increase by one percent; similarly a Dirac neutrino’s mass would increase by one percent; but because the Majorana neutrino visits the Higgs field twice, its mass would increase by two percent. Very easy, conceptually. But unfortunately, neither this nor any related method will work; principle is not practice, especially when it comes to neutrinos, whose interactions with ordinary matter are too few and far between for anything about them to be easy.

Clearly, when it comes to neutrinos, there’s a lot more research to be done. But meanwhile, I hope this post gives you some insight into how the very structure of the Standard Model plays into these issues. The fact that neutrinos are affected only by the weak nuclear force, and that their other half, if it exists, would be a sterile particle, gives them unique character, allowing them potentially to be either Dirac fermions or Majorana fermions. This distinctiveness makes them a breed apart within the Standard Model, and many physicists suspect that this is in some way responsible for their remarkably small masses.

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35 Responses

  1. Could two sterile half neutrinos marry to become a candidate for dark matter? If it could only be detected via a Higgs interaction (or gravity which we will ignore for being too weak), would it be possible to design an experiment to detect it?

    1. Even a single sterile neutrino can have a Majorana mass and be a candidate for dark matter, yes. Two sterile neutrinos cannot be married by the Higgs field, however, because the Higgs field itself is not sterile… so a sterile neutrino with a Majorana mass, or two neutrinos married by something other than the Higgs field, will not have a Higgs interaction. But if we allow mixing between these objects and the known neutrinos, then the story becomes more complex. The short answer to your second question, therefore, is: it depends on a bunch of details. Indeed, existing experiments are already sensitive in some cases, but in other cases it would be difficult to imagine… and in fact, sterile objects sometimes have *only* gravitational interactions, in which case the answer to the second question is “probably no”.

  2. I am a novice, but I like your article, it is a shame I don’t know more about the experiments (especially the details), I noticed a comment about left/right handedness chirality and helicity, and I believe without understanding these, you won’t be able to solve the problem. I came up with a theoretical framework back in 1998, that explained a mechanism for quantum gravity, energy and matter that is created by ‘one’ dynamic multidimensional quantum of space, (a ‘complex volume’) I call ‘APE’ s, and it is essential that you understand both chirality and helicity so that the interactions of the ‘APE’ s create chirality of both energy and matter dependant on their structures and interactions with the flow(field) of free ‘APE’ s. Everything emerges from the bottom up, so that our space(dimensions) can expand and contract without adding or deleting any ‘APE’ s. Ps. in case anyone is interested the ‘APE’ has fixed dimensions(therefore finite) that has wavelike characteristics(therefore has fixed limits) so that it is its own opposite half of its cycle(2 revolutions). In each revolution it expands and contracts(4 phases) (creating the curvature of space in time). The curvature changes with the interactions of the different structures of energy and matter, so all the forces are a characteristic of the curvature of space in that structure, be it a black hole … or the vacuum of space. I hope that’s not too much in one go, but some may find it interesting.

  3. Are neutrinos their own antiparticles? If we were to find that they are, can this be used to distinguish between the Dirac and Majorana alternatives?

    1. That’s precisely the question! If the answer is yes, then they are Majorana; otherwise they are Dirac. But that’s precisely what’s so hard to measure experimentally… there are only a few methods and they will only work if certain quantities in nature turn out to be large.

  4. Lets try to figure this out … I can’t find it out in your post, and I failed in a 15 minute look throudh all “the usual” books, including Duncan.

    “All neutrinos are left handed”. OK, is this chirality or helicity?

    We do have one piece of information that’s obvious: all neutrinos were left handed back when they were all massless. Now some are massive, and
    they are still, and always, left handed, no matter whether coming or going.
    This is true even if one flavor is massless and the other two massive.

    So it must be chirality? Or am I mixed up in trying this argument?

    If so … does that generate a preferred frame, or, of course, one can always tell if it is coming or going in your own frame.

  5. error 3*10^52 kg.
    Matt addresses a huge number of particels. It used to be 42 , the secret of the universe. We used this number to model oil and gas formation on the from a complex modge of former fossilised organic matter called kerogen. Eventually, a simpler quantum field approach reduced it to 3 .
    Let’s get down to initial conditions or hidden assumptions. Well it all happened in the Big Bang. I think Matt will agree. All those particles, fields were there at the Planck limit to creation ~10^-42 sec. Flying around at light speed. U1 (sU2) (SU3) and.. Steven Weinberg hoped (SU5) or SO10 was there, but sadly not , particularly for specialists whose whole careers were so devoted.
    With the conventional big Bang after ca. under 4 minutes everything we now observe in the heavens lay within a size of 1 light year in dimension. It is now in our epoch out at 46*!0^10 light year . mind you we only see out to 14*10^9 ly. That’s because one of the assumptions is that the expansion of empty space ( and I mean everything, forget vev and big lambda).There are all told 12 flaky initial conditions.
    Take an expression that incorporates h*c in the expression. Firstly , gravimetric equations dont’ have h planck’s constant in them.
    In h*c , now as the world becomes more classical in its physics description as h goes to the limit zero, more quantum as it increases. However, as c goes to infinity, the world becomes less relativistic and more classical. A balance appears to have been struck. Or as they say we wouldn’t be here anytime, not just now. Life would be be simpler if someone could claim h* c is a constant for all time.

  6. Greater than c exclamation? obviously not c prime lol.
    Er.. the Higgs field(s) were bolted onto the SM prior to 2012. In 2011 they weren’t sure that the particle existed. The Cern director was concerned about its existence. The search was about SUSY too. Over 50% of the scientists involved asked ” Does the SM predict mass less neutrinos.” the answer then was that all the particles of the SM were mass less”.It’s the Higg s fields(4 of them) as Matt mentions , the massive guage bosons of the +- w (s) and the z massive guage bosons..So the Higgs field has been swiched on for the last 14 billion years. Of course if it had switched off formerly, then everything would fly apart at the speed of light and we wouldn’t be here to question it. They had found 3 of them already so it appeared a certain bet for me.
    We are asked Why is the neutrino so puny with regard to its tiny mass. I state there is a simple answer; it’s because the universe is so big and massive.A 1 millieV electron neutrino ca. 2*10^-39 kg is a tiny mass no doubt but that is significant compared with an axion. Were they to exist i would simply add, it’s because the Universal mss is a million fold greater than a current mass (mass energ equiv.) of ~3*10^54kg.

  7. ” the Standard Model will break down at or before the energy scale v2/mneutrino, the square of the Higgs field’s value divided by the largest mass among the neutrinos.

    It won’t break down if the neutrinos are moving at speeds, Vn > c! 🤔

    Dirac neutrinos are the ones with V <= c

      1. So, you agree that neutrinos, some anyway, may indeed be traveling faster than c and could be the remnants of the very early universe where particles started to form BELOW Planck’s scale. That must of been an extremely “hot” space that could have velocities faster than the now constant c for the particles that we can measure, at least.

        I guess my point is are the constants we are using, were they the same at or below Planck time of the universe? Are neutrinos fossils of the early universe?

  8. Planck Units for Universe.
    Mass 3*10^52 kg. For a neutrino degeneracy pressure relevant to a 0.00117eV electron neutrino mass the calculated mass
    that a photon neutrino bath at 2.737K that would d collapse is 2*10^54kg.
    The density is very low, cf 9.9*10^-27 kg/ m^-3 for the critical density with a Hubble constant at this epoch of ca. 70/km/sec/mega parsec; equivalent to a neutrino cold bath of ca. 1 electron mass per m^3 02 ca. 1/12000 of the published critical density.
    Note that the !0^80 proton number for the mass of the universe was based on old ideas of the numerosity of galaxies. For this cold neutrino/photon partical pairing, a Frank Wilczek quantum fermion/boson condensate or “aether” would gravitationally collapse , since Hubble and now Web this old galaxy numerosity is in serious error . It was also postulated that the early universe would reveal a thinning of galaxies , in contradiction to steady state models.
    Such a quantum cold condensate of this low densty ca. equiv to 1 electron mass per metre cube of space would give rise to a REVERSE Hubble flow of ca. 0.6 km/sec/megaparsec, cf. 70km/sec….
    Hence rather than an initial big bang of unknown max temperature, higher temperatures arise as a consquence of lepto genesis. and then on to barygenesis. The photons of the cmb are then extant and not fossil photons their energies reduced by expansion of a supposable empty space

  9. The rest mass energy of a particle is given by , in electron volt nomenclature, 4.964 k *T where k = 0.00008616 ev per Kelvin K , T
    For example, when applied to a collapsing massive star , 200-300 solar mass , now referred to as a particle pair instability process ,a consequence of electron antielectron (positron) degeneracy pressure , there is no feedback mechanism to prevent further collapse. Furthermore, the process cannot be halted by further neutron degeneracy pressure etc. The result leads to either a neutron star or for masses greater than 1.4 solar mass, a blackhole..
    Jim Peebles and Steven Weinberg alluded to this.
    Thus for a a 511,000 eV electron., I’ll ignore the species number, here 2,
    Flip Toneda many years ago gave an excellent detailed description of chirality ,helicity and parity for the electron. Organic chemists are familiar with epimerisation and racemisation.. It took a week of corrections to account for the socalled non physical mathematical soutions, 4 in total.

    For electron positron pair creation from high energy gamma ray photons the rest mass pair production temperature for 2 particles (charged electron species) the temperature is close to 1.2 *10^9 K. At higher temperatures there’s a kinetic energy contribution of ca. kT per species.
    As an aside neutrino production in nuclear reactors is insignificant, although ca. 40% 0f the energy is lost to neutrinos, these are high kinetic energy particles that pass through us, several trillions per second for the workers there, to no health affects. The cold cmb neutrinos amount 400 million per metre cubed of space and an equivalent number of similar low energy photons.
    The core temperature of the Sun is only 15* 10^6 K. low in comparison to all other particle pair production temperature.

  10. Can there be three Weyl half-neutrinos that marry in differing ways to create the three Dirac neutrinos?

    1. To make all three neutrinos into Dirac neutrinos, we need three sterile neutrinos. But are you asking whether some neutrinos could be Dirac and the others Majorana? Not sure what you meant by “differing ways.” If you asked why their masses are different, it’s the same idea as for the electron, muon and tau; their interactions with the Higgs field differ in strength. Maybe I have misunderstood your question?

  11. Fascinating stuff! As a layman, I congratulate you on making a complex and counterintuitive concept so comprehensible. I envy your students …

  12. As a non scientist a couple of questions occur to me now.

    1. Does the First model double the number of particles in the SM?
    2. Clearly particle marriage is a metaphor; what is actually going on?

    1. 1. No, it just doubles the number of half-neutrinos; everything else remains the same. The total number of *names* of particles remains the same… because of the marriages… but the total number of half-fermions in the Standard Model 1.0 goes from 45 to 48. The counting is not obvious; that’s a story for another day.

      2. If it were easy to explain in a few words what is really going on as the Higgs field gives mass to particles, I would have done it. I’ve never succeeded in doing it in less than a 90 minute lecture; even an hour isn’t enough. I am writing a book on it, though, so hopefully in a couple of years (everything is so, so slow in publishing) you’ll have that as a resource.

    2. For 2., I think you can get a rough non-rigorous picture by interpolating between two of my posts: [this one](https://4gravitons.com/2021/05/21/alice-through-the-parity-glass/) on violation of parity by the weak nuclear force and [this one](https://4gravitons.com/2021/07/30/lessons-from-neutrinos-part-ii/) on why neutrino oscillation implies neutrino masses.

      The story is that you can very loosely think of any massive particle you can detect as the two “married” partners constantly interchanging. They interchange because they interact with the Higgs field, and because that interaction involves energy the particles you detect appear to have mass.

      This is definitely still mixing intuitions in a sloppy way, so I get why Matt hasn’t said something along these lines. But I think it’s a decent place to start if you’re clear it’s just a starting-point.

      1. I have a slightly more technical post coming in which that approach will be taken. But I don’t think it really answers the question in a deep way, it’s really just touches on how we physicists see it, not how it really is.

  13. While discussing mass of electron in Fig.1 do you want to add the statement that one half of Weyl electron is left handed, the other is right handed, in case it is correct?

    1. I don’t want to add it! 🙂 Because then I have to explain what “handed” means, and that (as my students can attest) becomes very confusing; you have to understand *both* helicity *and* chirality, and that is not an easy business even for grad students. If I decide to explain it, I need to have a better way of doing so, involving a discussion of parity and its violation, with experimental examples.

      1. Ok! Thanks. That is your choice. But really my ulterior motive in the comment was to check if what I remembered was right!!

        1. Well here’s a challenge: take a left-handed electron (which interacts with the weak nuclear force), meaning that its spin axis is opposite to its motion. Now I’ll stay put, but you get into your rocket ship and go zipping past the electron. From your perspective, it is now right-handed, because its motion seems to be in the other direction although the spinning orientation is unchanged. So. Does this (from-your-perspective)-right-handed electron interact with the weak nuclear force, or not? From my perspective, it’s still left-handed…

          1. Thanks. This is a difficult challenge. I thought Higgs Lagrangian to particles was Lorentz covariant. So if under your transformation, the two Weyl electrons change into each other, I do not know what happens. But on the other hand you say Higgs couples them anyway. So may be in the end it sorts out anyway!!

            1. Ah, you see now… the point is that there is a difference between left-handed *chirality* and left-handed *helicity*; they’re the same for massless particles, but not for ones with mass. And so this gets quite confusing… even for grad students. Now you know why I don’t go into this on the blog! 🙂

      2. Hello Matt
        Since the neutrinos now have rest mass one might presume that with their antineutrino they may undergo a pair production process as do all other SM fermion partners.
        electron positron pair instability is known now to occur in huge mass stars called hypernovae.
        Consider neutrino degeneracy pressure allied to a pair production temperature. A 1 millielectron volt electron neutrino anti pair would exhibit a temperature very similar to that of the CMB radiation temperature.. Also, the mass required to create such a process is ca. 2*10^54 g. Such a mass equates that of the universe. The Hubble ultra deep and Webb extreme new data have increased the number of galaxies by at least one hundred fold . This universal mass in natural units is given by the ratio of the cube of the Planck mass to the square of the electron neutrino masses. Simply stated, the neutrino masses are so small because the universe miss is so large.

        1. “A 1 millielectron volt electron neutrino anti pair would exhibit a temperature very similar to that of the CMB radiation temperature.. Also, the mass required to create such a process is ca. 2*10^54 g. Such a mass equates that of the universe. ” I’m not following the calculation you did to obtain the mass you claim; what exactly is the logic?

          Note that pair production of neutrino/anti-neutrino happens in Z boson decays, of which we humans have created an enormous number.

          1. I think the observations are

            1) mc² for largest neutrino mass is comparable to kT for CMB temperature

            2) total energy in the CMB is comparable to mass-energy of the universe

            I think (2) is just a consequence of LambdaCDM / big bang cosmology and the (log scale) recency of matter-radiation equality, but (1) is in fact interesting (in a dangerous numerology sort of way).

            I’ve given it a little bit of thought in the form of “neutrino mass² ~= cosmological constant * Planck scale”, or “neutrino mass² ~= cosmological constant” depending on how you define CC, and I’ve seen at least one paper that uses this “coincidence” to design a dynamical solution to the CC problem. I forget the details though.

            (These are only equivalent given the observation that radiation density in the present day is close to *dark energy* density, which relies on the more famous cosmological coincidence that we live find ourselves near both matter-rad equality and matter-DE equality.)

            Personally I think these scale coincidences have a good track record, with lambdaCDM vindicating many of Dirac’s numerological coincidences and the seesaw mechanism also arguably being an example!

            (In fact, you can try to bring the seesaw mechanism into the mess of conjectures above, but this is probably enough numerology for one comment.)

            1. Numerology, and more precisely dimensional analysis, have an excellent track record. The only problem is that for every ten or a hundred dimensional analysis arguments you can invent, only one of them (or less) will turn out to reflect something in actual physics. Still, they do often guide theoretical thinking and have led to many interesting and creative ideas, which have merit even if nature turns out not to use most of them.

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