Of Particular Significance

The Standard Model More Deeply: The Electron and its Cousins (Part 2)

POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON 05/15/2024

From the structure of the Standard Model of particle physics, one might wonder if the electron, muon and tau, so similar except for their masses, might really be the same object seen in three different guises. Last week, starting with a post for general readers, and then looking at the situation in more detail (Part 1 of this two-part series), I showed one way in which this idea fails to agree with experiment. Today I’ll give you another point of failure, focused on a property of particles known as “spin”.

Spin: The Idea In Brief

What is “spin” in physics? It’s related to the word “spin” in English, but with some adjustment. I’ll write a longer article about spin elsewhere, but here’s a brief introduction.

The Earth “spins”, both in the English sense and in the physics sense: it rotates. The same is true of a tennis ball. The rotation can be changed when the ball interacts with a tennis racket or with the ground. It can also be changed when it interacts with another tennis ball. In fact, if two balls strike each other, some of the spin of one ball can be transferred to the other, as in Fig. 1.

Figure 1: (Left) Two balls, the upper one spinning, approach and collide; (Right) The balls recoil from the collision, with some of the spin of the upper ball now transferred to the lower ball.

Elementary “particles” can have spin too, which can be (in part) changed by or transferred to other objects that they interact with. But this type of spin is somewhat different from ordinary rotation.

For example, an electron “spins”. Always. An interaction can change the direction of an electron’s spin, but it cannot change its overall amount. I’ll tell you what I mean by “amount” in a moment, but the quantity of the amount is a famous number:

  • h / 4π = ℏ / 2

where h, Planck’s constant, shows up whenever quantum physics matters. This amount of spin is tiny; if a tennis ball had this amount of spin, it wouldn’t have even rotated once since the universe was born. But for an electron, it’s quite a lot.

Usually, as shorthand, one takes the ℏ to be implicit, and just says that “the electron has spin 1/2”. I’ll do that in what follows.

How can we visualize this spinning? It’s not easy. It’s best not to visualize an electron as a small rotating ball; it’s the wrong picture. Quantum field theory, the modern theory of “particles”, gives a different picture. Just as we should think of light as a wave made of wave-like photons, we should think of the electron as a wave — specifically, a wave in the electron field. [This wave is not to be confused with a wave function, which is something else]. A wave in the electron field can rotate in ways that a little ball cannot. This, however, is hard to draw, and in any case is a story for another day.

The important point is that the spin of an elementary “particle” is not like the simple rotational spinning of a tennis ball, even though it is in the same category. An electron has spin intrinsically, by its very nature; there is no electron without its spin, just as there is no electron without its internal energy and rest mass (as emphasized in my recent book, Chapter 17). That’s certainly not true for a tennis ball, which can have any spin, including none.

These details won’t matter much in what follows, though, so let’s step away from these subtleties and move on.

Ways to Spin

So far, I’ve suggested two types of spinning:

  • the intrinsic spinning (or “intrinsic angular momentum”) of elementary particles
  • the ordinary spinning (or “ordinary angular momentum”) of objects that are rotating, or are in orbit around other objects.

Both contribute to the surprising way that physicists use the word “spin”.

Suppose we have an object that is made of multiple elementary particles. Its total angular momentum involves combining the intrinsic angular momentum of its elementary particles with the ordinary angular momentum that the particles may have as they move around each other. Physicists now do something unexpected: they refer to the object’s total angular momentum as its spin. They do this even though the object is not elementary, meaning that its spin may potentially combine both intrinsic spinning and ordinary spinning of the objects inside it.

So the real meaning of spin, as particle physicists use the term, is this: it is the total angular momentum of an isolated object — an object that may be elementary, or that may itself be formed from multiple elementary objects. That means that the spin of an object like an atom, made of electrons, protons and neutrons, or of a proton, made of quarks, antiquarks and gluons, may potentially arise from multiple sources.

Atoms, Protons and Strings

For example, let’s take a hydrogen atom, made from an electron and a proton. (For starters we’ll treat both of these subatomic particles as though they were elementary. We’ll return to their possible internal structure later.)

I’ll refer in the following to four atomic states, illustrated below in Fig. 2.

  • The ground state of a hydrogen atom (known as the “1s state”) has spin 0. Nevertheless, it is made of an electron with spin 1/2 surrounding a proton of spin 1/2. The two spin in opposite directions so that their angular momentum cancels.
  • There is a very slightly excited state, often neglected in first-year physics courses or in quick summaries of atomic physics, where the electron and proton spin in the same direction, and their spins add instead of cancelling. This state of the atom has spin 1; I’ll call it the “spin-flipped 1s state”. (The transition from this excited state to the ground state involves the emission of a radio wave photon with a wavelength of 21 cm, leading to the so-called “21 cm line” widely observed in astronomy. )
  • There are more dramatically excited states known as the “2s and 2p states”. The 2s state has spin 0, while the 2p state has spin 1. But even though the 2p state and the spin-flipped state both have spin 1, their spins have different origins. The total angular momentum of the 2p state does not come from the intrinsic angular momenta of the electron and proton; those cancel out, just as they do in the 1s and 2s states. Instead, the spin of the 2p state comes from a sort of rotational motion of the electron around the proton.

These four states are sketched in Figure 2. (Spin-flipped versions of the 2s and 2p states exist but are not shown.)

Figure 2: (Bottom left) In the ground state of hydrogen, the spin of the proton (central red dot) is opposite to that of the electron (surrounding blue cloud), so that the atom has spin 0. (Bottom right) If the electron spin is flipped, both the proton and electron spin in the same sense, giving the atom spin 1. (Top) While the 2s state is similar to the 1s state, the 2p state has the electron and proton spinning in opposite directions but has the electron moving around the proton (dashed black line), giving the atom spin 1.

The fact that the ground state has spin 0, and yet the 2p state has larger spin specifically due to the electron’s motion around the proton, illustrates the main point of this post. If an object is made from multiple constituent objects, nothing can prevent those constituents from moving around one another. That means they can have ordinary (or “orbital”) angular momentum, which then contributes, along with the constituents’ spin, to the combined object’s spin — i.e., to its total angular momentum.

Therefore, an object that is not elementary, and so contains multiple objects inside it, will inevitably have excited states with different amounts of spin. Indeed, the hydrogen atom has excited states of total angular momentum 0, 1, 2, 3, and so on. All atoms exhibit similar behavior.

The same applies for protons, which aren’t elementary either. The proton has spin 1/2, but its excited states have spin 1/2, 3/2, 5/2, 7/2, and so on. The first excited state of the proton, the Delta, has spin 3/2. This is most easily understood as a rearrangement of the spins of the quarks, gluons and anti-quarks that it contains (though the exact rearrangement is not obvious, due to the complexity of a proton; see also chapter 6 of the book.) The next excited state, the p(1440), has spin 1/2 like the proton. But many other excited states have been observed, with spin 3/2, 5/2, 7/2, 9/2, and perhaps even 11/2.

A string, such as one finds in string theory, is another object with internal constituents, which one might call bits-of-string. A string can always be spun faster. That’s why, in the superstring theory that is sometimes touted as a potential “theory-of-everything” (or in less grandiose language, a complete theory of space, fields and particles), there are states of all possible spin — any integer times 1/2. Unfortunately, were this really the theory of our universe, the higher-spin states of the string would probably have far too much mass for us to make them in near-term experiments, putting this prediction of the theory out of reach for now.

But string theory isn’t just useful in this rarified context. It can also be used to describe the physics of “hadrons” — objects made from quarks and gluons, including protons. All indications from experiments and numerical calculations do indeed suggest that hadrons come in all possible spins; this includes the excited states of the proton already mentioned above. (That said, the higher the spin, the harder it is to make the states, making it more and more challenging to observe them.)

Spin and the Electron, Muon and Tau

None of this is true for electrons, muons or taus, all of which have spin 1/2. No electron-like particle with spin 3/2 has ever been observed.

This argues strongly against the electron, muon and tau all being made from the same object. Atoms, protons and strings all have excited states with the same spin as the ground state, but at roughly the same mass, they also have excited states with more spin. If the muon were an excited state of the electron, we would expect to see an object with spin 3/2 that has a mass comparable to the muon, and certainly below the mass of the tau. Such a state would easily have been observed decades ago, so it doesn’t exist.

Are there loopholes to this logic? Yes. It is possible, in special circumstances, for the excited states with higher spin to have much larger masses than the excited states which share the same spin as the ground state. This is a long story which I won’t try to tell here, but examples arises in the context of extra dimensions, and others in the related context of exotic theories of quark-like and gluon-like objects (with buzzwords such as “AdS/CFT” or “gauge/string duality”).

However, it’s hard to apply the loophole to the muon and tau. In such a scenario, the electron should have many more cousins than just two, and some of the others should have observed by now.

Furthermore, data now confirms that both the tau and muon get their rest mass from the Higgs field; see Fig. 3. For such particles, the loopholes I just mentioned don’t apply.

Figure 3: The interaction strengths of various types of elementary particles with the Higgs field, plotted versus those particles’ rest masses. Any particle whose rest mass comes entirely or largely from the Higgs field should lie on or near the dashed line. The data shows this is true of both the tau τ and the muon μ, as well as of the bottom and top quarks b, t, and of the W and Z bosons.

We must also recall the arguments given in the first part of this series. If muons and taus are excited states of electrons, it should be possible for any sufficiently energetic collision to turn an electron into a muon or tau, and for decays via photons to do the reverse. But these processes are not observed.

In short, the properties of the electron, muon and tau disfavor the idea that they are somehow secretly the same object in three different quantum states. The explanation of their similarities must lie elsewhere.

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17 Responses

  1. Dr.Strassler:
    I’m not sure where I remember reading this, but i believe I read that neutrinos can “morph” into one another? Does this mean heavier neutrinos can morph into lighter neutrinos, and give off energy, for energy conservation? However, how would lighter neutrinos morph into heavier neutrinos, without violating energy conservation?

      1. Dr.Strassler:
        Ok, that helped immensely. I see the flaw in my original question. I have another question, if I may. The muon is the heavier cousin of the electron, more massive, but with the same charge. If I was to slam an electron, with enough kinetic energy, into a massive “stationary” target, could I get a muon to form? Charge would be conserved, the energy for the greater mass of the muon coming from the collision. I realize muons are unstable, and it would rapidly decay into other stuff, but is the above event possible?

        1. The only way to do it is for the electron, via a weak nuclear interaction, to be transformed into a disturbance in the neutrino field of electron type (i.e. to be come a “virtual neutrino” of electron type), which then via mixing becomes a disturbance in the neutrino field of muon type, and finally via another weak interaction becomes a muon. The rate for this is exceedingly small, because the weak nuclear interaction in this context is indeed weak, and because neutrino mixing is a very small effect. Off the top of my head I think the rate for this is suppressed by at best (m_neutrino/m_W)^4 ~ 10^(-44). But it’s not zero.

          1. Dr.Strassler:
            Thanks, my thought experiment was really just to see if I understood the concept. I understand, the probability is very small….but not zero
            BTW: do you know Dr. Ethan Siegel? He has a blog, similar to yours, with some good information on it also.

            1. Yes, we are in touch occasionally. He’s good, and generally very reliable. We rarely disagree. I’ve just missed meeting him in person a couple of times.

  2. Would you say that the electron and positron are the same object but seen in different guises.

    Or the left-handed and right-handed neutrino ( if it exists ), which would have different masses.

    1. No; the electron and positron are two different types of ripples in the same field — which we may call the “electron field”, [or the “positron field” — but the convention is the former.]

      The electron field is a complex function. For the electron, the waves in the real and imaginary parts of that function are +90 degrees out of phase; for the positron, they are -90 degrees out of phase. In other words, one is cosine + i sine, the other is cosine – i sine .

      Left and right-handed neutrinos (like left- and right-handed electrons) are yet another story; that’s too long for a comment. To answer this one has to go carefully through how the electron is built in the Standard Model from left- and right-handed electrons [and similarly for the positron] and explain that this happens because the electron field, as I’ve referred to it above, is itself built by using the Higgs field to tie together two fields, one for the left-handed electron and one for the right-handed electron. I’ve given some hint of this here: https://profmattstrassler.com/2022/07/18/celebrating-the-standard-model-why-are-neutrino-masses-so-small/

      1. So an object is a particular type of ripple in a field, and experiment doesn’t rule out the possibility that the muon and electron could be two different types of ripples in the same field

        1. They are definitely ripples in two fields, in the sense that these two fields, despite having many of the same properties, have different interactions with the Higgs field, and as a result, two completely different resonant frequencies.

          To be sure, there is a matter of definition here. We could decide to call them “two components of one field” simply by redefining what we mean by “field”. But my point is, no matter what we call them, they satisfy very different equations. Electrons and positrons, by contrast, are two different solutions to essentially the same equation, which is why their masses are identical.

  3. Leonard Susskind was the first person I know who correctly described the bonding action of quark color as like a “rubber band,” an elastic string. In his 1969 “Harmonic-Oscillator Analogy for the Veneziano Model” paper, he applied the then-new Feynman QED model to show that the extensive data on Regge trajectories compressed down to a strikingly simple statement: There are tiny rubber bands inside of hadrons!

    I am baffled beyond words why he did not share a Nobel Prize for that paper. Others realized that Susskind’s strings were composed of physical objects (quarks) with finite rest masses swinging at the ends of a new force that behaved radically differently from 1/r^2 forces like those between electric charges. But given that Susskind figured all of this out _before_ the existence of these new particles and forces was firmly established and that those who quantified those forces followed Susskind’s lead on how the new color charges worked, doesn’t that make Susskind’s accomplishment more remarkably rather than less? In effect, Susskind _predicted_ the color force before its full elucidation by others. If that’s not Nobel Prize-worthy, I’m not what is.

    As for Scherk and Schwarz’s clever hypothesis five years later (Dual Models for Non-Hadrons, 1974) that this string-like pattern might repeat itself at vastly higher energies to explain seemingly point-like electrons as especially tiny “super” strings, they neglected to define or even hypothesize the nature of the new particles and forces that would make these “super” strings into a physics hypothesis. Without that, it’s not physics, just exceptionally incomplete math with insufficient bounds.

    I know you are not a devoted “super” string advocate, Professor Strassler, but you know this community better than I do. I’m just a poor, bewildered computer scientist. Did anyone ever bother to complete Scherk and Schwartz’s interesting but incomplete hypothesis by defining the new particles and forces needed to complete their “super” string hypothesis?

    1. Well, I wasn’t there (too young, of course) so I can’t be 100% sure I understand the sociology and the history. But I can tell you my understanding. (Incidentally, this is related to a talk I gave almost 20 years ago, but still relevant; unfortunately an amateur recorded it, so the sound quality is poor. https://www.youtube.com/watch?v=yEkqcQH4zgA)

      What you have to understand is this: in the late ’60s and early ’70s, people were trying to understand the proton and neutron, and other hadrons, in detail. A rough model wasn’t the goal. A detailed set of equations which could be used to make precise predictions was the goal.

      There were various models for hadrons around. One was the “dual resonance model”, which Susskind showed was actually a theory of relativistic strings. It did indeed seem to capture some properties of hadrons. But it didn’t precisely agree with data. And it didn’t explain baryons — protons and neutrons — very well. Another model was the quark model, which had its own problems, and there were various versions of it.

      Then in 1973 it became clear that a theory of quarks and gluons based on Yang-Mills theory was a good candidate for the strong interactions. Within months, precise calculations had been done in certain contexts, and had been shown to agree, in detail, with data. No such precise, detailed agreement with data was ever shown for the dual resonance model. The conclusion, which was backed by more and more data over time, was this: the quark/gluon theory (known as QCD) is the correct theory of hadrons, and the dual resonance model, though it captured some of the phenomena, is not. Nobel prize for the former, and no Nobel prize for the latter, even if it still deserved (and received) some smaller prizes and some pats on the back.

      Now, this still left open the question: if the quark/gluon theory is the right one for explaining hadrons and their behavior, why does the dual resonance model, i.e. string theory, work for hadrons at all? It’s imperfect, but it still seems to capture something. Hints at answers to this question appeared first in work of ‘t Hooft and of Polyakov, and analogies with type II superconductors were also discussed, for instance by S. Weinberg. But these did not explain any direct link between these theories and the details of the dual resonance model — the string theory in flat space that Susskind and others were studying. Moreover, they only potentially explained why a theory with many colors and no quarks would behave sort of like a closed-string theory, and why a theory with many colors and just a few quark flavors would behave sort of like an open-and-closed-string theory, but not why a theory with roughly equal numbers of colors and quark flavors — which is true of real-world QCD — would do so. In such a theory, the strings break very rapidly, and it is not clear that the strings ever really exist except in extreme circumstances. Furthermore, this approach did not explain what protons and neutrons are; they simply don’t appear in ‘t Hooft’s approach. In addition, it’s clear that whatever string theory this is, it is NOT the dual resonance model — i.e. string theory in flat space. For instance, if two hadrons scatter at a fixed angle, the scattering rate falls exponentially with energy in the dual resonance model, while it falls as a power of energy in QCD and in data. Worst of all, later studies of string theory showed that it is only obviously consistent if it has a massless spin 2 particle, while there is no such particle seen in data, in agreement with QCD calculations.. If you like, this is a crucial failed prediction of the model that Susskind was studying. So the links between the real world’s quarks and gluons and the string theory that Susskind and others were studying remained tentative and tantalizing… [even as string theory was meanwhile repurposed as a possible theory-of-the-universe in which the massless spin-2 particle would be the particle of gravity.]

      It was only in the very late 90s that these puzzles were finally resolved — qualitatively. Strong indications that all theories of quarks and gluons, whether they have hadrons or not, are string theories on curved spacesnot on flat space — was hinted at by Polyakov and discovered independently and definitively by Maldacena. In theories that have hadrons, there’s something that happens in the curved space, such as a floor that prevents things from falling indefinitely, that generates the hadrons. The strings moving around on that floor of the space act like the strings of Susskind’s flat space string theory, and explain qualitatively why it sometimes roughly matches data. The strings moving in the space well above the floor provide the power laws that are found in high energy scattering at fixed angle (this was shown by myself and Joe Polchinski). So it turns out that only string theory in a special curved space can even qualitatively match the physics of a QCD-like theory. This was not known to Susskind, or to anyone prior to the 1990s.

      Now, should Susskind, Polyakov and Maldacena collectively get a Nobel for their profound insights? It wouldn’t trouble me if they did, but here’s the problem. These links between string theory in curved space and theories of quarks and gluons are only precise for theories that do not appear in the real world. We do not know the details of the string theory or the curved space that are needed to describe real-world hadrons. One can use the equations of quarks and gluons to make calculations of many processes at the Large Hadron Collider that are accurate to 10% or better. By contrast, there is no string theory that can do anything better than give an insightful but qualitative view of any LHC processses, or other, lower-energy hadron phenomena. (Some interesting parallels between string-based calculations and real-world data have been shown, but nobody knows why they work; other calculations clearly do not apply.)

      Worse, in most cases the string theory is unable to reproduce what we actually observe. For example, a crucial feature of Large Hadron Collider experiments is the presence of jets; a high-energy quark or gluon is transformed by the physics of QCD into a spray of hadrons. (A crude introduction to this phenomenon is here.) As I and several others argued (and as Hoffman and Maldacena proved), within these curved-space string theories in the regimes where calculations are currently possible, jets do not occur. Instead, jets can only appear in the string theory when the space is so curved that our current calculational methods do not work, and so no one can yet use the curved-space string theories to make direct contact even with this crucial fact about the real world.

      Generally, Nobel prizes are not awarded for qualitative understanding. They are awarded for experimental discoveries and for clear, unambiguous interpretation of those discoveries. The quark/gluon theory is clear and unambiguous. The corresponding string theory to which it is equivalent is still not fully known and would be far from useful, with current mathematics, even if it were precisely known. So I believe this explains the situation.

      Oh, and also, about protons and neutrons: I mentioned that they do not appear in the theory that Susskind was studying. It wasn’t until Polchinski’s 1995 work on D-branes was combined with Maldacena’s work (by Witten and by Gross and Ooguri) that it became clear what the hadrons we’re made from actually are. https://profmattstrassler.com/2018/02/05/a-brilliant-light-disappears-over-the-horizon-in-memory-of-joe-polchinski/ Again, this was unknown to physicists of the 1970s, and was another reason that a string theory of hadrons remained in question.

  4. Now consider the three types of neutrinos. Neutrino mixing shows that neutrinos have a complex inter-relationship. But there are only three distinct neutrinos. How would one show that the neutrinos are distinct and independent particles?

    1. That is indeed trickier, but not much more so. There are no spin-3/2 neutrino observed, either. Z boson decays to neutrinos are consistent with them being three independent spin-1/2 particles. And most important, W bosons convert electrons, muons and taus to various combinations of the three neutrinos, and vice versa. It’s essentially impossible for the neutrinos to be composites of one another while the electron, muon and tau are independent particles.

  5. You mention, “there is no electron without its internal energy”. Can you please explain as best as possible where this internal energy comes from? Thanking you for the updates, much appreciated.
    As a Theologian, I delight in the transference and application of Quantum Physics as you describe into Theological applications Kind regards Joseph Atwater

    1. A central goal of my recent book was to answer precisely this question. Almost nobody ever really answers it in non-technical language, because you have to deeply understand what an electron is before the answer can make sense. Even undergraduate university quantum physics doesn’t cover this.

      It takes 200 pages to provide a clear and complete answer — passing through the concepts of motion, mass, waves, fields, and finally the quantum, culminating in Chapter 17. (There’s also a relevant discusion of how the proton gets its internal energy, which is initiated in Chapter 6 but only really completed in Chapter 24.)

    2. I can recommend Waves In An Impossible Sea for the detailed answer.

      From what I understand, if we imagine running the clock backwards, the Universe gets more dense and energetic. The microwave background radiation blueshifts into visible light. At the point where it gets too hot for interstellar hydrogen, it gets stripped into a plasma, and the universe becomes opaque to light. As it continues to get hotter and denser, atomic nuclei can longer maintain a distinct state, and the universe becomes filled with a quark, gluon, electron plasma. As it gets hotter still it no longer makes sense to take about stable wavicles but just so much energy crashing about in the quantum fields.

      So the internal energy that makes up fundamental particles today is condensation from the hot dense state known as the big bang.

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