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Flip Tanedo | USLHC | USA

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Symmetry in Physics, Pt. 3: Internal Symmetries

Now time for another installment of “symmetry in physics.” For those of you tuning in late (or who have forgotten what we’ve been discussing), we started out in part 1 with a very general discussion of the symmetries of spacetime and how this constrains the form of our theories. Next, in part 2 we looked at discrete symmetries and how they relate the notion of antimatter to charge and parity conjugation. We’ll be using some of the jargon of part 2, so make sure you brush up and remember what “CP” means. Now we’d like to address another mystery of the Standard Model: why is there so much repetition?

Family Symmetry

Let’s review the matter content of the Standard Model:


The top two rows are quarks, the bottom two are leptons (charged leptons and neutrinos). Each row has a different electric charge. The top row has charge +2/3, the 2nd row has charge -1/3, the third row has charge -1, and the last has charge 0. As discussed in part 2, there are also the corresponding anti-particles with opposite charges [note 1]. Just about all of the matter that you’re used to is made up of only the first column. All atoms and everything they’re made of are more-or-less completely composed of up and down quarks and electrons (the neutrinos haven’t done much since early in the universe).

The replication of the structure of the first column is known as family symmetry. For each particle in the first column, there are two other particles with nearly the same properties. In fact, they would have exactly the same properties, except that they are sensitive to the Higgs field in different ways so that the copies end up having heavier masses. Technically the Higgs discriminates between different generations and breaks this symmetry, but we are still left with the question: why are there two other families of matter?

(A lexicographic note: familes are also known as generations. The way I hear it most is that we say ‘family symmetry’ but refer to ‘generations of particles’ which exhibit it.)

“Who ordered that?”

This was theoretical physicist I.I. Rabi’s famous quote upon learning about the muon, the first discovery of matter beyond the first generation. Certainly it makes our matter content a bit simpler: instead of having to memorize the 12 particles above, we only really need to learn the first four and know that each has two heavier copies. But why should there be three generations?

In short, nobody really knows. But one of the consequences of the existence of three generations is that we are able to construct theories which naturally have CP-asymmetry, i.e. theories which treat matter and antimatter differently. Were it not for such a CP asymmetry, matter and antimatter would evolve analogously over the history of the universe and we’d never be able to end up with the excess of matter over antimatter that we observe today. It is this excess, of course, which led to the existence of things like galaxies and, ultimately, humans.

The detailed reason why having three generations allows for CP asymmetry is too technical for our purposes here, but it boils down again to the how these generations interact with the Higgs boson. The Higgs interacts with particles and their parity conjugates, e.g. coupling a left-handed electron with a right-handed electron. Remembering that these are two totally different species of particles, the family symmetry means that we can write out the Higgs interaction as a 3×3 matrix. Each element describes how the Higgs connects left-handed electron-like particles to right-handed electron-like particles. For example, there is an element that tells us how the Higgs connects a left-handed muon to a right-handed tau. This is a matrix of complex numbers [note 2] whose “complexity” (“imaginary-ness,” or more precisely its complex phase) is physically manifested as a CP-asymmetry.

This is rather important, since a 2×2 complex matrix wouldn’t have the same complex phase. Two of the winners of the 2008 Nobel Prize (Kobayashi and Maskawa) were recognized for making the critical leap from two generations to three generations and discovering this CP asymmetry [note 3]. Again, this is not a technical explanation, but for those with more physics background, I can recommend this summary as a starting point.

“Internal symmetry”

Family symmetry is what is known as an internal symmetry because we’re not actually doing any transformation on space(time). One can compare this to the spacetime symmetries that we discussed in Part 1; we can’t “reorient ourselves” in space to view an electron as a muon the way that we can perform a change of coordinates to turn a left-moving particle into a right-moving particle.

Charge (which we mentioned as a “discrete [spacetime] symmetry” previously) can also be thought of an internal symmetry. Indeed, this is probably a more natural way to think about charge. The Standard Model exhibits a few different kinds of charge. We are all familiar with electric charge, and most readers of this blog will know that quarks also carry a kind of ‘color charge.’ Electric charge is actually a subset of a larger class of charges called electroweak charge, but we’ll get to this next time.

What kind of internal symmetry do all of these charges correspond to? This ends up being a rather deep question which we’ll mention in the next post,  but the main idea is:

  • Electric charge comes from saying that a particle is identical to a particle multiplied by a phase (i.e. multiplied by a complex number of length 1.)
  • Electroweak charge comes from saying that up-type and down-type quarks should be identical (and similarly with charged leptons and neutrinos).
  • Color charge comes from saying that quarks of different color should be identical.

The key word is that these come from special internal symmetries called gauge symmetries and are responsible for the existence of forces. This is actually a straightforward statement: the existence of electric charge tells us that there is a force that leads two electric charges to attract or repel based on the charge assignments. In quantum theory these forces are mediated by particles, in this case the photon. Similarly, the existence of electroweak charge leads to the W and Z gauge bosons and the existence of color charge leads to gluons that… well… “glue” quarks together into hardrons.

Charges are rather powerful because they are conserved. Thus these gauge symmetries play a very tangible role in constraining the form of our theory by imposing that at an interaction point, the charges coming in must equal those coming out. In this way electromagnetic symmetry prohibits me from saying that I can have three electrons interact without any additional particles coming in or out. It also means that the electron is a stable particle, even though there are lighter particles in the theory. Usually heavy things decay into lighter things (e.g. an electron would ‘want’ to decay into neutrinos), but if the heavy thing (in this case the electron) has a charge that the lighter things do not, then there is no way for the heavy thing to decay without violating the charge symmetry

Broken and approximate symmetries

As we saw above with the different flavors/generations in the Standard Model particle content, internal symmetries can be broken. In particular, the Higgs boson discriminates between different flavors in how it interacts with fermions. It prefers to interact with higher-generation quarks and leptons, causing these generations to be heavier (since the Higgs ‘gives mass’ to these particles). We note also, however, that this explicit ‘breaking’ of the flavor symmetry is completely isolated in the Higgs interactions. If we were to ignore the Higgs boson, the symmetry is robust. Thus properties which have nothing to do with the Higgs, such as a particle’s charge and spin, must be ‘symmetric’ between different generations. The up, charm, and top quarks all must have the same charge. (Alternately, the observation that they do have the same charge tells us of the existence of an internal family symmetry that is broken by the Higgs.)

Next, you should have been upset that in the previous section I told you that “up-type and down-type quarks should be identical.” This statement is obviously not true: up quarks have charge +2/3 while down quarks have charge -1/3. They’re clearly very different things. Thus we say that electroweak symmetry is broken. By now you will not be surprised if I told you that this is deeply connected to the Higgs boson. It turns out that the Higgs is the particle which is responsible for breaking electroweak symmetry. If you’re keeping score, this means that the Higgs breaks family symmetry by giving different generations different masses, but also breaks “electroweak symmetry” that makes up/down quarks behave differently and charged leptons/neutrinos to behave differently. That Higgs sure is pesky!

Just as some symmetries are broken, there are other kinds of internal symmetries which are called approximate symmetries or accidental.  These are symmetries which are not “fundamental” principles, but rather that are imposed by virtue of the structure of the Standard Model. For example, it appears that baryon number (quark number) and lepton number are conserved in interactions. This wasn’t something that we imposed on the theory, it just so happens that such interactions are not allowed by the pre-existing structure of the Standard Model.  This symmetry is an “accident” of the theory. (It’s a good accident, too, since otherwise protons would decay too quickly for interesting things to form in our universe.)

Summary and Outlook, Epilogue 3

This was our foray into more abstract symmetries. What I hope to have conveyed is that a lot of the structure of the Standard Model can be summarized in terms of a symmetry principle. When looking at our Standard Model particle content, we can now say that the particles really could have been summarized by saying that there are quarks and leptons and a set of symmetries. For example, starting with just a single quark and a single lepton,

  • Electroweak symmetry imposes that the quark comes in up and down varieties while the lepton comes in charged and neutral varieties.
  • Color symmetry imposes that each quark comes in 3 colors.
  • Family symmetry imposes that there are three generations of each of the above particle.

Thus we end up with a much bigger set of particles, but they all came from symmetry principles. We made an analogous statement in Part 1 that, at the time, probably sounded very trivial: a particle is a particle is a particle! I.e. a particle that is moving to the left is identical (by translation symmetry) to the particle when it’s moving to the right. In the same way, an up quark is “identical” to a charm quark since it’ related by an internal family symmetry.

… Except that it’s not. And this is where the Standard Model gets really interesting. Symmetries can be broken, and that can lead to interesting behavior. If you’re wondering why we’re searching so hard for the Higgs at the LHC, one reason is that it broke so many of our symmetries (and so we’ll probably have to throw it in particle jail or something).

Next time we’ll go into more detail into how gauge symmetries are special, the short answer is that they tell us about the interactions of our theory.




[1] In part 2 we went into detail about the relation between antimatter and CP. To be completely accurate, I should list the particles and their chiralities. Here’s a more honest list of the Standard Model matter content:

  • Left handed quarks with charge +2/3: up, charm, top.
  • Left handed quarks with charge -1/3: down, strange, bottom.
  • Left-handed anti-quarks with charge -2/3: anti-up, anti-charm, anti-top
  • Left-handed anti-quarks with charge +1/3: anti-down, anti-strange, anti-bottom.
  • Left-handed leptons with charge -1: electron, muon, tau.
  • Left-handed anti-lepton with charge +2: positron, anti-muon, anti-tau.
  • Left-handed leptons with charge 0: electron-neutrino, muon-neutrino, tau-neutrion.

In addition to these there are right-handed CP conjugate particles (i.e. antiparticles). The CP partner of a left-handed anti-up quark is a right-handed up quark. After “electroweak symmetry breaking” (i.e. the Higgs mechanism) This right-handed up quark pairs with the left-handed up quark to form a massive up-quark which can be left- or right-handed. Note that there are no anti-neutrinos in the vanilla version of the Standard Model. We now know that neutrinos have mass, but the nature of neutrino mass so there is more to this story, but this is another topic that would require its own dedicated discussion. This is all kind of technical and not central to the point of this post, but I figured someone out there might want to hear it.

[2] The fact that this matrix is made of complex numbers is a manifestation of a rather deep and surprising notion that nature knows about complex numbers at all. (The standard joke is that the biggest discovery in theoretical physics was the complex plane.)

[3] The quantity in the Standard Model that represents the quark ‘flavor’ (or ‘generational’) structure is called the CKM matrix, named after Cabbibo, Kobayashi, and Maskawa. Cabbibo wrote down the 2×2 matrix for two generations of quarks. Kobayashi and Maskawa extended this to a 3×3 matrix and noted the existence of a complex phase leading to CP violation. The latter two were awarded the Nobel prize, though many thought that the exclusion of Cabbibo was rather controversial.


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