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Flip Tanedo | USLHC | USA

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Mysteries of Mesons: The Eightfold Way

Hi everyone! I’m going to make a brief digression from my Feynman diagram posts because there are a few important ideas that I wanted to explain before I get to the Higgs and more speculative scenarios. I’ve been meaning to explore some of these ideas in the context of meson physics for some time, but my draft post ended up getting longer and longer until I decided to cut it up into shorter bite-sized pieces; this is the first piece.

Recall that mesons are bound states of a quark and an antiquark (a kind of quark ‘atom’).  They are interesting because they capture a lot of “known unknowns.” Quantum chromodynamics can, in principle, tell us everything we would want to know about the meson system, but it’s very difficult (in many cases practically impossible) to calculate anything from these first principles. We already know why: non-perturbativity.

But here’s the funny thing: we’ve known about mesons for a very long time, much longer than we’ve known about the fundamental quarks and gluons that make up a meson. Instead of discovering the “fundamental” objects first and then observing the complicated dynamics that the “fundamental” theory (QCD) generates, physicists at early colliders found a plethora of these funny particles and had no idea where they came from and why there were so many of them. They knew that these particles could interact with one another, for example by looking at bubble chamber tracks (image from BNL):

In these posts we’ll explore a little about how past generations of physicists developed some theories of mesons. Even though this type of physics is more than half a century old, it represents a fantastic time when new particles being discovered every month. There are lessons from that time that will carry over to the interpretation of new results from the LHC. Further, we’ll see how some of the theoretical ideas developed at the time have continued to develop in surprising new ways.

The Eightfold Way

One of the first things that physicists wanted when they found all of these new particles was to find a way to classify them.

Jim’s inaugural post gave a nice example of the “Eightfold Way,” a sort of periodic table of hadrons originally developed by Murray Gell-Mann (and independently by Yuval Ne’eman) in the 60s. Jim showed the baryon table showing the proton, neutron, and some of their more exotic cousins. Here is the analogous meson table:

Before explaining what’s going on here, we can learn a few things just by staring at this picture.

  1. Each dot represents a meson. There are three types of particle names: the pions (?), the kaons (K), and the etas (?).
  2. Evidently there’s some meaning to the placement of each particle relative to the others.
  3. The mesons each have an electric charge: +, -, or neutral (0).
  4. It looks like opposite points of the hexagon are antiparticles of one another since we expect antiparticles to have the opposite charge. (This is indeed the case.)

So we’ve met our first nine mesons. These turn out to be the lightest mesons, and in fact the pions are the very lightest mesons. There are actually many, many, many mesons out there, but for now let’s focus on the lightest ones. The pions are all made up of up and down quarks, the kaons contain a strange quark, and the etas are quantum superpositions up–anti-up / down–anti-down / strange–anti- strange quarks.

Just like the periodic table of chemistry, however, the peculiar arrangement of this diagram is also trying to teach us something. You might think that it would be useful to arrange these mesons according to the quark content. There are two problems with this:

  1. The eightfold way was developed before quarks were experimentally discovered. (Actually, the eightfold way provided and important part of the theoretical structure that led people to suspect that quarks might be real!)
  2. As we saw for the etas, some mesons are not well defined in terms of individual quark/anti-quark pairs but rather as quantum superpositions of several types of quark/anti-quark pair. In fact, this is true for the neutal pions and kaons as well.

So the Eightfold Way is not quite organized according to quark content, at least not directly. The structure of the diagram is actually based on the symmetries of the mesons. The branch of mathematics that describes symmetries is called group theory (in particular, representation theory) and is now a staple in the education of every particle physicist. Back in the 1960s, however, the field was not so well known to physicists and Murray Gell-Mann essentially re-invented the relevant mathematics for himself. (Historically this has happened fairly often between mathematicians and physicists.)

On the horizontal axis of the diagram is something called isospin, I. On the vertical axis is something else called hypercharge, Y.  For now all that matters is that the usual electric charge is given by Q = I + Y/2 (Edit 31 Jan: thanks to reader Stan for pointing out the factor of 1/2 that I originally missed!). This is indeed the pattern that we see: mesons that are higher and further right tend to be positively charged, while mesons that are lower and to the left tend to be negatively charged. By the way, at this point we don’t need to know very “deeply” what these things mean, but they are properties which particles have. Just as we can describe a circle simply by specifying its radius, we can describe particles by listing some set of properties that include isospin and hypercharge.

I should say that the diagram above shows what is called the pseudoscalar nonet (or octet + singlet) because they describe nine particles. (“Pseudoscalar” tells us about the angular momentum of the particle.) These are mesons which do not have any intrinsic spin. There are also heavier versions of each of those particles, for example the vector nonet of spin-1 particles. This is analogous to the component quark and anti-quark having some angular momentum, just like the excited states of electrons in the hydrogen atom.

You can see that the spin-1 pions are called rhos (?), the spin-1 kaons are called K-stars (K*), and the spin-1 versions of the etas are called the phi (?) and omega (?). In fact, there are even higher spin copies of these guys, not to mention analogous mesons formed out of the heavier quarks. Indeed, now you can see why the 1960s were “boom” years in experimental physics where new particles were being discovered almost weekly.

Relation to modern ideas

This has an interesting relation to very modern ideas for of physics beyond the Standard Model. Models of extra dimensions predict an analogous “tower” of copies of known particles, the so-called Kaluza-Klein tower. Because this KK tower looks just like the tower of mesons. We understand that the meson tower comes from the fact that they are composite particles, so it looks like theories of extra dimensions mimic theories of composite particles like mesons. This is one of the key observations underlying the so called holographic principle or gauge/gravity correspondence in which theories of extra dimensions are “dual” to strongly coupled theories.
In a broader sense, the discussion above represents a deep theme in particle physics where symmetry became the central principle for how we understand nature (I’ve mentioned this before!). These days one of the fundamental tools of a theoretical physicist is group theory (the mathematical description of symmetries) and models of new physics aren’t described so much by the individual particles but by the symmetry content of the theory.
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