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### Gell-Mann’s Eight Fold Way

I thought I might touch on the topic of a recent post by one of my fellow bloggers here at Quantum Diaries.  As was mentioned, researchers at Fermilab have recently discovered the Ξ0b (read Xi-0-b) a “heavy relative of the neutron.”  The significance of this discovery was found to 6.8σ [1]!  The CDF Collaboration has prepared a brief article regarding this discovery, which is is being submitted to Physical Review Letters (a peer-review journal).  A pre-print has been made available on arXiv.

But rather than talk about what’s already been written, let’s discuss something new.  Namely, how on earth did physicists know to look for such a particle?

The answer to this question takes us back to the year 1961.  An American physicist (and now Nobel Laureate, 1969)  by the name of Murray Gell-Mann proposed a way to arrange baryons and mesons based on their properties.  In a sense, Gell-Mann did for particle physics what Dmitri Mendeleev did for chemistry.

Gell-Mann decided to use geometric shapes for arranging particles.  He placed a baryon/meson onto these geometric shapes.  The location a particle was given went according to that particle’s properties.  All his diagrams however were incomplete.  Meaning, there were spaces on the shapes that a particle should have went, but the location was empty.  This was because Gell-Mann had a much smaller number of particles to work with, today more have been discovered; but we still have holes in the diagrams.

But to illustrate how Gell-Mann originally made these diagrams, I’ve shown an example using a triangle, which is part of a larger diagram that appeared in the previous post on this  subject.  I’ve also added three sets of colored lines to this diagram.

Let’s talk about the black set of lines first.  If you go along the direction indicated by each of these lines you’ll notice something interesting.  On the far right line (labeled Qe=+1, Up =2), there is only one particle along this direction, the Σ+b.  This baryon is composed of two up quarks, a beauty quark, and has an electric charge of +1.

Let’s go to the second black line (labeled Qe = 0, Up =1).  Here there are four particles (the Σ0b has yet to be discovered).  But all of these four particles have one up quark, and zero electric charge.

See the pattern?

But just to drive the point home, look at the orange lines.  Each line represents the number of strange quarks found in the particles along the line’s direction (0, 1 or 2 strange quarks!).  The blue lines do the same thing, only for the number of down quarks present in each particle. Also, for all the particles shown on this red triangle, each particle has one beauty quark present!

In fact, if you go back to the original post on the Ξ0b discovery, you’ll notice the diagram has three “levels.”  All the particles on the top level have two beauty quarks present.  Then the red triangle appears (that I’ve shown in detail above).  Then finally in the bottom level, all the particles have zero beauty quarks.

Also, if you spend some time, you can see the black, orange and blue lines I’ve drawn at right actually form planes in this 3D diagram.  And all the particles on one of these planes will have the properties of the plane (electric charge, quark content)!

Well, when Gell-Mann first created the Eight-Fold-Way in the early 1960s, none of the shapes were “filled.”  But just like Dmitri Mendeleev, Gell-Mann  took this to mean that there were undiscovered particles that would go into the empty spots!!!!!

So this seemingly abstract ordering of particles onto geometric shapes (called the Eight-Fold-Way) gave Gell-Mann a way to theoretically predict the existence of new particles.  And just like Mendeleev’s periodic table, the Eight-Fold-Way went one step further, by immediately giving us knowledge on the properties these undiscovered particles would have!

If you’re not convinced, let’s come back to the experimental discovery of the Ξ0b, which is conveniently encompassed by the yellow star in the diagram above.  This particle was experimentally discovered just a few weeks ago.  But Murray Gell-Mann himself could have made the prediction that the Ξ0b existed decades earlier.  Gell-Mann would have even been able to tell us that it would have zero electric charge and be made of a u,s and b quark!!!

In fact, Gell-Mann’s Eight-Fold-Way tells high energy physicists that there is still one particle left to be discovered before this red triangle may be completed.  So, to all my colleagues in HEP, happy Σ0b hunting!

But in summary, it was the Eight-Fold-Way that gave physicists the clue that the Ξ0b was lurking out there in the void, just waiting to be discovered.

Until Next Time,

-Brian

References

[1] T. Aaltonen (CDF Collaboration), “Observation of the Xi_b^0 Baryon,” arXiv:1107.4015v1[hep-ex], http://arxiv.org/abs/1107.4015

### 8 Responses to “Gell-Mann’s Eight Fold Way”

1. FrankH says:

I don’t understand why there are two particles on each of the sides of your orange triangle. Is there a simple explanation?

2. Brian Dorney says:

FrankH,

This is a very good question.

Let’s take the simple example of the Hydrogen atom. When hydrogen exists in the ground state, its electron is in the 1s orbital. But I could easily excite a hydrogen atom by shining some light (of the proper wavelength) onto the atom. Then the electron would jump to a higher orbital, say the 2p orbital. In each case, the atom in question is made of a proton and an electron, but in one case the electron is excited.

Coming back to the diagram, see the Particles labeled as Lambda_B and the Xi^0_b on the top side of the yellow triangle?

These particles are made of a u,d and a b quark. But in one case, these three quarks are in an excited state (this would be the case of the Xi^0_b). But since the Xi^0_b hasn’t been discovered yet let’s go to a case where I can give you some numbers to illustrate this. Look at the bottom level here:

There is the Lambda and the Sigma^0 at the same point in the middle. Both particles are made of a u, d and s quark. But the Lambda has a mass of ~1115.7 MeV/c^2 while the Sigma^0 has a mass of ~1192.6 MeV/c^2.

This difference in mass between the two particles means that in one case (the Sigma^0) the quarks are excited, and form a unique and different state (a new particle).

Hope this helps,
-Brian

3. Carlo Marchiori says:

I think that showing how these extravagant patterns fit into group theory would be rewarding for readers. Is there any theoretical physicist out there?!?

4. Gavin Flower says:

If the pair of particles shown at the mid points indicates an excited state – why are there not double particles at the corners?

Also, surely there are higher exited states corresponding to yet more particles to fit into the diagram?

5. Brian Dorney says:

@Carlo Marchiori,

I would absolutely agree with you; however, that’s a bit beyond my technical expertise. The quark flavors constitute a symmetry group (in a Lie Algebra). The Proton and the neutron (and any other hadrons made of only up and down quarks) are part of the SU(2) group.

If you allow the strange quark as well, you get SU(3); with the charm it becomes SU(4), and then with the beauty quark it becomes SU(5).

You can represent particles in group theory with matrices. These matrices have the property of their parent group. For example, particles (like the proton) belonging to the SU(2) group are Special (S), and Unitary (U) and of dimension 2 X 2. Special means they have determinant 1, and unitary means the matrix’s inverse is equal to its conjugate transpose.

So the symmetry group SU(n) consists of all n x n matrices with determinant 1 and whose inverse are equal to their conjugate transpose.

Performing calculations in this sense is rather complicated, and a bit outside my ability as an experimentalist currently. However, if you’d like more information, please see:

http://pdg.lbl.gov/2011/reviews/rpp2011-rev-su3-isoscalar-factors.pdf

and

http://pdg.lbl.gov/2011/reviews/rpp2011-rev-young-diagrams.pdf

For more details (both links are by the Particle Data Group).

Best,
-Brian

6. Uri Karshon says:

It is a pity that the author of this comment forgot to mention that the 8-fold way was discovered independently at the same time by Murray Gell-Mann and Yuval Ne’eman.

7. alexandra says:

si deseo encontrar simetría en un conjunto de partículas tanto bariones como en mesones utilizando los diagramas de gell-mann como puedo organizarlos para encontrar su simetria

8. Phillipp says:

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