Of Particular Significance

Celebrating the Standard Model: Seeing in Triplicate

POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON 05/06/2024

Within the Standard Model of particle physics, one finds three almost identical types of particles: the electron, the muon and the tau. Their interactions with the electromagnetic force, the weak nuclear force and the strong nuclear force are exactly the same. In particular, all three have electric charge -1 (which in first-year physics classes we would usually write as “-e”).

They’re not entirely identical, however. For instance, they have different masses.

  • Electron: 0.000511 GeV/c2
  • Muon: 0.105658 GeV/c2
  • Tau: 1.777 GeV/c2

Compare these to a hydrogen atom, which has mass 0.938783 GeV/c2 . (For the definition of “GeV”, which is an amount of energy, click here.) [Specifically, these are their “rest masses”. Rest mass is the type of mass that is intrinsic to objects and does not change with speed; see Chapter 5 of Waves in an Impossible Sea.]

Why does nature have these three similar particles, collectively called the “charged leptons“? We don’t know. But they’re not alone.

There are also

  • three quarks with electric charge 2/3 (the up-type quarks)
  • three quarks with electric charge -1/3 (the down-type quarks)
  • three types of neutrinos with no electric charge (the neutral leptons)

and within each trio of particles, the only differences are found in their masses (and in other things that directly relate to their masses, such as their lifetimes and their interactions with the Higgs field.)

These four classes of particles make up the twelve types of “fermions” of the Standard Model [also sometimes called the “matter” particles, though I dislike that terminology due to its inconsistency.] Why are there four classes, and why does each class have three members?

Figure 1: The three particles in each of the four classes of fermions share idental interactions with the electromagnetic, weak nuclear and strong nuclear forces.

A set of four particles containing one member of each class is called a “generation”. For instance, the up quark, the down quark, the electron and the lowest-mass neutrino form what is usually called the “first generation”. Why does the Standard Model have three complete generations of fermions?

No one knows. The Standard Model itself doesn’t address these questions, so their answers lie beyond it. I’ll briefly describe three possible explanations, one of which we can discard, but the other two of which remain open.

Three Version of the Same Thing?

One natural guess is that perhaps the three particles in each class are actually the same object, just behaving in three different ways. For instance, perhaps the electron, muon and tau are really a single entity, acting in three different ways that have different amounts of energy, and therefore, by E=mc2, different amounts of rest mass.

We know examples of this kind of multiplicity. It shows up in atoms, and also in their protons and neutrons. For example, a hydrogen atom (an electron bound to a proton) takes many forms, with slightly different energies; these are known as the hydrogen atom’s “quantum states”. The quantum state of lowest energy is called the “ground state” of hydrogen. The others are called “excited states”, because to make them requires energy be added to the ground state; the electron then becomes more active (“excited”) than it is in the ground state.

As a result, nature presents us with the hydrogen atom in many guises. All are built from the same ingredients, but those ingredients are arranged in slightly different ways, leading each state of hydrogen to have a slightly different mass.

A proton, too, is the ground state of an object that has many excited states in which its quarks and gluons, having absorbed some energy, are more active than they are in a proton. There are many such states observed in experiments, each with its own mass.

So then, by analogy, might the electron be the ground state of an object, with the muon and tau excited states of the same object? It’s a natural idea, but the answer is no.

I’ll explain the reasoning in more detail tomorrow. But here are two basic arguments:

  • An atom in its ground state can readily be kicked into an excited state by the effects of a collision. The same is true of a proton. But a collision of an electron with another object — perhaps a proton, a photon (a particle of light), or another electron — never has been seen to turn an electron into a muon or a tau.
  • The excited states of atoms and protons vary not only in mass but also in “spin” (their internal angular momentum.) By contrast, although the electron, muon and tau differ in their masses, they all have exactly the same spin.

Similar but more subtle arguments apply to the three neutrinos, the three up-type quarks, and the three down-type quarks.

A Real Almost-Symmetry?

Another possibility is that the electron, muon and tau might be three objects related by a real, almost-symmetry. [Physicists would describe this as an “approximate, explicitly-broken symmetry”.] What is meant by this?

The three sides of a equilateral triangle, shown in Fig. 2, are objects related by a symmetry; they are equivalent to one another, as one can see by rotating the triangle by 120 degrees.

Figure 2: The three sides of an equilateral triangle are related by a symmetry; a rotation by 120 degrees leaves the triangle unchanged.

But in Fig. 3 is a shape whose three sides have equal length, and yet they have different thickness. In one way they are identical; in another way, they are not. A rotation of the triangle by 120 degrees looks different.

Figure 3: The three sides of this triangle are related by an approximate symmetry; a rotation of the triangle by 120 degrees leaves it similar but not identical. The symmetry is “explicitly broken”; the sides are similar, yet different.

Perhaps, by a loose analogy to this last figure, the three generations of fermions are related by a symmetry, but one that is only partial. There might even be a hidden form of geometry lying behind this near-symmetry. That geometry could be literal: it could lie within the shape of “extra dimensions” of space, ones far too small for us to detect with today’s technology. Or perhaps the geometry involved is just mathematical. But the idea that there could be a real but partially broken symmetry is one to take seriously.

An “Accidental” Almost-Symmetry

Or perhaps the similarity between the electron, muon and tau is only skin deep — only an apparent (or “accidental”) symmetry. It’s a symmetry that seems to be there only because of the limitations of our current technology.

As an analogue, consider the four large moons of the planet Jupiter. At first glance, through binoculars or a small amateur telescope, they seem almost identical; one can barely see any differences among them

Figure 4: Through a small telescope, the four large moons of Jupiter look almost identical. Copyright Jan Sandberg.

But that’s only because of the limitations of one’s optics. Close-up photographs by NASA spacecraft show the moons to be remarkably different:

Figure 5: Close-up photographs from NASA spacecraft reveal that the four moons are quite different; the apparent similarity seen in Fig. 4 was due to a lack of knowledge, caused by limited technology.

Although the differences between the electron, muon and tau appear today to be limited to their masses and related issues, perhaps experiments done by our (possibly distant) descendants will reveal additional dramatic differences between them. In this case, the symmetry that seems to be there will be understood someday to have been an accident.

Unsolved Problems

But all this is speculation. The question of why the Standard Model has three generations is one of the great unsolved puzzles of particle physics.

Also unsolved is the question of why the masses of the three particles in each class are so different. In a sense, we know what creates that difference. The Higgs field is responsible for the masses of all these particles; without the Higgs field, their masses would all be zero. But the Higgs field doesn’t determine what the ratios of the masses are; that honor goes to the various strengths with which the various types of particles interact with the Higgs field. Unfortunately, no one knows what determines those strengths; their origin is a complete mystery. [See Chapter 21 and 22 of Waves in an Impossible Sea for further discussion.]

So the Standard Model, wonderfully successful as it is, leaves enormous questions unanswered. Attempts to address these questions have led to a wide variety of theoretical speculations. But until future experiments give us new insights, we won’t know which (if any) of our many speculations is actually correct.

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25 Responses

  1. Children are now taught the Standard model: https://kids.kiddle.co/Standard_Model 😲

    Way back in 2011 when you set up your blog, I thought it was well meaning but unrealistic to teach particle physics to us the general public because it was too sophisticated and difficult. It’s a humbling reminder to us older folk that cultures move on and with it; the abilities, ambitions and expectations of the current younger generation.

      1. Indeed you have; so thanks for the time, energy and competence you’ve put into building this site over the past thirteen years, as well as patiently answering the many questions, including mine, that has encouraged me to abandon an archaic, incorrect way of interpreting the world. Interpreting the world now as vibrations in cosmic fields via the Standard Model has been a huge, disorientating change in how I once viewed the universe.

  2. My understanding was that each generation “flavor” is a (nearly) conserved symmetry based on electroweak and baryon/lepton number conservations, so the butterfly on closer examination falls apart into three (or four) pieces. A “composite” symmetry, if you will.

    Is the higher energy electroweak symmetry breaking forcing the lower energy QCD phase change flavors – “effective” symmetries? Or is it assumed to be one internal symmetry associated with all these fields at all energies?

    1. That’s an older perspective, and it may well not be true. Some papers assume it, many don’t. (In my best paper on flavor, actually, there’s no such nearly-conserved symmetry.)

  3. I think the decay products provide evidence, Matt. The muon typically decays into an electron, an electron neutrino, and a muon antineutrino (μˉ → eˉ + veˉ + vμ⁺). So you can gain a concept of the muon if you have a concept of the electron and the neutrino. I am reminded of George Zweig’s Origins of the Quark Model. His 1963 thesis was about particles being composed of “aces”, rather like tinker-toy parts that snap together. I think the same principle applies to leptons. However this sort of thing just doesn’t seem to feature in the Standard Model.

    1. The statement “you can gain a concept of the muon if you have a concept of the electron and the neutrino” is not correct. The only thing you learn is the muon’s charges under the various forces. Tau leptons can decay in the same way. But they can also decay to a rho meson and a neutrino. The rho meson and neutrino together tell you absolutely nothing about a tau except for its charges.

  4. I think the clues are in the decays, Matt. For example μˉ → eˉ + veˉ + vμ⁺

  5. “That geometry could be literal: it could lie within the shape of “extra dimensions” of space, ones far too small for us to detect with today’s technology. ” – Does M theory (or string theory in general) offer explicit explanations for the 3 versions of particles and all observed experiements that have defined the standard model?

    1. Any extra-dimensional theory can explain the three generations in broad outline. Getting the details of the masses and other subtleties right is not so easy, but that may be more of a technical problem than anything else.

      The problem with string/M theory is that when it comes to the particles of nature, it can do almost anything, and thus (so far) it teaches us almost nothing.

      The biggest problem of all with any extra-dimensional theory — or even with a four-dimensional theory of gravity — is that we don’t understand how nature, faced with a choice of many possible universes, selects the one we’re in. Maybe it’s just chance, in which case there would be no underlying explanation for the details of what we observe. In that case, there would just be correlation among details. [The same is true of the orbits of the planets; there is no underlying explanation for why there are 8 planets and what their orbits are, but there is an understanding of planetary history that explains why the planets’ orbits should lie in a plane, and why it makes sense that between Jupiter and Mars there might be an asteroid belt.]

      1. Thanks Matt. “The problem with string/M theory is that when it comes to the particles of nature, it can do almost anything, and thus (so far) it teaches us almost nothing.” – That made a little light bulb go off in my head. This blog and your book are a treasure to me!

      2. Can cells exist in four macroscopic dimensions, wouldn’t their contents leak?

        1. Even in three dimensions, atoms are mostly empty, and yet cells don’t leak. Small extra dimensions will make absolutely no difference — and even rather large ones make no difference if matter is stuck on walls or in corners of the extra dimensions.

    2. Hi, AnchorVW (I have this image of a battleship using a VW Bug as an anchor),

      First, I must warn you that I am nothing more than a poor, bewildered information specialist, not a physicist. Still, even with only that paltry background, I’ve seen enough of the curiously random potpourri of ideas called “M-theory” (formerly “super” string theory) to get a big grin when anyone asks M-theory folks to make experimentally verifiable phenomena.

      My mental analogy is always the same: Given that E-theory enables precise capture of the entirety of the Standard Model, what does E-theory predict about this or that unresolved issue in the Standard Model?

      The answer is “nothing.”

      To which folks should rightly respond, “What is E-theory?” The English language, of course. It is not a theory if your theory is so expressive that one can say anything imaginable and then some (how many vacua?). It is a mathematical language in which you can say anything but predict nothing.

      But why did I say “superstring” instead of “string” theory?

      Professor Leonard Susskind proposed string theory in 1969 [1] through an astonishingly insightful feat of applying new mathematical methods, particularly the then-new QED theory, to a mystifying but solid collection of data from particle accelerators worldwide. He correctly predicted what we now call the color force that binds what we now call quarks together in particles such as protons, neutrons, and many others. Unlike electric attraction, these bindings had an unprecedentedly strange behavior: They acted like a rubber band or a bungee cord, storing energy until the cord either broke and formed new ones or pulled the quarks back together. Most remarkably, Professor Susskind figured all of this out _before_ the existence of quarks and the color force was recognized. Now that’s good science! He collapsed all that particle accelerator data into a small set of precise equations whose behavior could be summed up accurately and intuitively as vibrating rubber bands.

      Professor Susskind, of course, got his share of a Nobel Prize for making this remarkable prediction of how an unknown force would behave when found experimentally… hold on… wait a second… What? Really? Are you _kidding_ me!? Sorry folks, correction: Professor Susskind did _not_ share in any Nobel Prize, then or later, for uncovering the deepest and most unexpected mathematical structures known then or now, the hadronic strings or flux tubes of the color force. Despite his smartly applied data and advanced theory to achieve this prediction of color force behavior, which is now a major part of the Standard Model, he lost out. Wow! He lost out on the Nobel Prize in Physics for being _too quick_ to figure out vibrating string behavior from the data. Now that’s ironic!

      The “super” string stuff came years later. It began as an especially extreme hypothesis by other folks: firecrackers versus 100,000 Barringer Meteor Craters. “Super” strings are non-predictive precisely because they dropped the energy constraints of real charges, forces, and particles of Susskind’s extremely predictive “rubber bands” (color force) hadronic string theory. The “super” theory became a powerful, playful mathematical language — an E-theory — capable of expressing anything while predicting nothing.

      The originator of the original, highly predictive, and deeply data-based string theory, Professor Susskind, is very much alive. It would be delightful to see him get that Nobel Prize in Physics that he so inexplicably did not win back in the early 1970s. The physics prize process is all about reducing data complexity to succinct math theory, even if belatedly.

      ———-
      [1] L. Susskind, “Harmonic-Oscillator Analogy for the Veneziano Model,” PRL 23 (10), 545 (1969). https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.23.545

      1. Thanks Terry! Appreciate the insight and history. 1969, wow I didnt know that was the origin. Very interesting! A vw bug as an anchor on a battleship I love it. 🙂

        1. Uhh, some of this history is correct, but by no means all of it. Susskind was one of my two main mentors when I was a graduate student, and large fractions of Terry’s history simply don’t accord with what Susskind has said to me and to those around him.

      2. On colors, the superior and hence modern theory is QCD, which is a quantum field theory.

        “QCD is a type of quantum field theory called a non-abelian gauge theory, with symmetry group SU(3).” [“Quantum chromodynamics”, Wikipedia]

        I guess it’s up to debate but string theory seems to naturally propose superstring symmetry and its kicking in on energy scales where LHC (and electron sphericity) experiments should see them – they do not. Isn’t that sufficient testing/disfavoring?

        1. Approximate supersymmetry is probably needed for string theory’s stability, but you can have supersymmetry without string theory and you can have string theory without supersymmetry. If the LHC had discovered supersymmetric particles, that would not have been evidence for string theory (despite some claims otherwise.) Conversely, LHC’s non-discovery of supersymmetric particles proves very little; supersymmetry may or may not be present at higher energy scales, and string theory may or may not have to do with nature.

  6. Is there a possibility of a fourth undiscovered generation?

    Or it doesn’t exist in practice because of issues such as instability / mass approaching the planc limit etc. ?

    1. A fourth generation like the previous three, in that its particles get their mass from the Higgs field, is excluded by data. I briefly addressed this in https://profmattstrassler.com/articles-and-posts/the-higgs-particle/implications-of-higgs-searches-as-of-92011/

      One example of this type that is relatively easy to exclude is known as the “Standard Model with a Fourth Generation” — meaning that in each of the four classes of known matter particles (charged leptons, neutrinos, up-type quarks and down-type quarks) of which there are three flavors (for instance, the up, charm and top quarks are the three flavors of up-type quarks) we should add a fourth: a fourth charged lepton, a fourth [heavy!] neutrino, a fourth up-type quark and a fourth down-type quark. In such a world, 1) the rate for the main Higgs production process goes way up, 2) the rate for the Higgs to decay to two photons goes way down, relative to a Standard Model Higgs.

      Since data agrees well with the expectations for a Standard Model Higgs, a fourth generation is completely ruled out.

      A similar set of particles could exist if they get their masses in a completely different way. But this would really not be a fourth generation of the Standard model, and instead would be quite different. In particular, their weak nuclear interactions could not be the same as for the other three generations.

      1. Matt Strassler, thank you for pointing this out so clearly. I recall the little thrill I felt the first time I saw the analysis that limited the number of generations to exactly three. That is science at its best: Good, solid collection of data followed by analyses that produce insights not possible by any other route. Why three? That is as delicious of a mystery as the existence of generations!

        1. There is also a confirming ruling out of specifically a 4th generation of neutrinos in cosmology (c.f. DESI “DESI 2024 VI: Cosmological Constraints from the Measurements
          of Baryon Acoustic Oscillations”, arXiv 2404.03002)

  7. what lies behind a quantum field and how far back can one go empirically smaller than tau and then further back as there is no beginning or end

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