A tale of two fermions

Last year, when the rise of the bottomonium began, I found myself returning to basic quantum mechanics in order to make sure I understood the coming landscape. A primary goal of our effort to secure the Upsilon(3S) and Upsilon(2S) data was to discover the ground state of bottomonium. But, what does it mean to be bottomonium, and how do you know how to look for the ground state?

Quantum theory, without which we cannot understand the subatomic, is needed to begin answering these questions. Let’s keep things simple. Let us imagine a universe in which there are just two fermions, attracted to one another by some force. For now, let us not concern ourselves with the nature of that force – only that it is attractive. Fermions are particles with half-integer spin – 1/2, 3/2, etc. For the sake of this matter, let us assume that, like the bottom quark, these fermions have 1/2 spin.

Spin is intrinsic angular momentum; that is, even if the particles are not in motion, they carry an irreducible internal quantum number which behaves like an angular momentum. The particle is not spinning; in fact, you can do some math and show that in order to get this behavior we call “spin” an electron would have to be spinning at a speed in excess of light. The word “spin” is a guide; it helps human to relate to intrinsic momentum, but it’s a misnomer.

There is also the angular momentum related to real motion – orbital angular momentum. Together, spin and orbital angular momentum tell us the total angular momentum of a pair of particles. The rule is pretty simple (although deriving it is, well, a long lesson). The total angular momentum, denoted by “j”, allowed for a given spin (“s”) and orbital (“l”) angular momentum state, is given by:

| l – s |  <= j <= | l + s |

What are the allowed spin states of a pair of spin 1/2 fermions? The spin is either 0 (spins in an anti-symmetric state) and 1 (spins in a symmetric state). So, s = 0 or 1.

What are the allowed orbital angular momentum states? Well, the pair could have l = 0, 1, 2, . . . With this knowledge, we can begin to map out the structure of our two-particle universe.

The bound states of the pair can have s = 0,1 and l = 0, 1, 2, . . . So, what is the total angular momentum of all of these possibilities?

The first is s = 0 and l = 0, giving us j = 0. This is the least angular momentum the pair can have, and this defines the ground state of the system. There is no lower-energy state, because this combination of total spin and total orbital angular momentum is as low as it goes. Can’t have less than zero!

What about s = 1, l = 0? Here, j = 1, and the two particles are now above the ground state – but only just. You can get here by flipping the spin of one of the particles in the pair, for instance, from the ground state.

This is fun! Let’s keep going. OK. Let’s try s = 0, l = 1. Now, again, j = 1, but here we have a state like the ground state in spin, but with orbital momentum in the pair.

Let’s do one more, our first big one: s = 1, l = 1. Now, j gets complicated. The total angular momentum can be either of j = 0, 1, 2. Refer to our formula above. | l – s | = | 1 – 1 | = 0. | l + s | = | 1 + 1 |  = 2. That mean that 1, in between 0 and 2, is also allowed, giving us three angular momentum configurations for s = 1, l = 1. A rich structure begins to emerge.

Going from our generic example of a pair of spin 1/2 particles, we can get back to our universe for a moment. Let’s pretend that the two particles are a bottom quark and a bottom anti-quark. (s,l) = (0,0) is the ground state – the eta_b [1].(s,l) = (1,0) is the Upsilon, the lowest energy vector state of the system. (0,1) is the undiscovered h_b, an analog to the eta_b but with orbital momentum. The (1,1) system are the three chi_bJ states – chi_b0, chi_b1, and chi_b2.

You can play this game with an electron and a positron, and many other particles. What you learn is that it doesn’t matter what the particles are – you know a lot just by knowing their spin, and that they bind.

Now if you want to know the MASSES of the states, things get more complicated. Mass is a function not just of the masses of the two particles, but also the force between them. Now is when things get complicated, and you have to know more about how the force binds them – how strong it is, for instance.

A lot can be learned about the world from so humble a beginning. It makes you realize just how simple knowledge can teach you deep truths about the cosmos.

[1] http://steve.cooleysekula.net/goingupalleys/2008/07/07/behold-the-elusive-ground-state-of-bottomonium/

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