Thinking about angular momentum

For many years, I have felt daunted by the quantum structure of nature. Don’t get me wrong – I studied it in lab class and I read a lot about it in my textbooks. It’s one thing to repeat an old experiment, or read a book; it’s quite another to be involved in a physical system where it’s do or die – get it, or don’t get it.

The bottomonium system seemed very mysterious at first. The upsilons, the etas, the h’s, the chis. So many Greek symbols, so many states, such a seeming mystery. I know I’m supposed to “just get” this stuff, and many of my colleagues do. But I also know who I am, and I am not afraid to accept that sometimes it takes a lot of time for a concept to puncture my cranium and sink comfortably into my brain.

Since I’ve been going around and giving all these seminars, I’ve had a lot of time to think about angular momentum. In nature, there are several kinds. The first is the most familiar, and it’s called “orbital” angular momentum. Find a friend, lock arms, and rotate yourselves around a common center until you puke. That’s orbital angular momentum – the rotating, I mean, not the puking. I guess the latter is linear momentum.

The other kind is internal angular momentum, commonly known as “spin”. It takes its name from the analogy that it’s a quantity that seems inherent to a particle, as if that particle were storing energy in its own rotation around an arbitrary axis. But the particle is not, in fact, spinning. That’s where the classical analogy breaks down. Do the calculation of how fast the electron, for instance, would need to be spinning and you quickly find that it violates a lot of what we know to be true about nature.

I’ve come to think of spin as a kind of vertigo. If grasping arms and physically rotating in space is orbital angular momentum, that sense of rotation you get in your head when no physical rotation is actually present – vertigo – is much like spin. It’s as real as actually spinning when you find you can’t walk straight as a result, and it plays a real role in defining how you interact with your environment.

Spin and orbital momentum, together, create a rich structure in even the simplest multi-particle quantum systems. It is the richness, evolving out of just a few particles, that defines the spectrum of states in a physical system. Let’s go back to bottomonium. Like the system from which bottomonium takes its name – positronium, an orbiting pair of an electron and a positron – this system has a ground state, and a multitude of states above that with unique combinations of orbital and spin angular momentum.

For example, positronium’s ground state is also called “para-positronium”, where the spins are oriented opposite one another in an antisymmetric configuration and the electron and positron have no orbital angular momentum. para-positronium is the state which, when formed, annihilates into a pair of gamma rays – this is the basis of a PET scan. If the spins align with one another, or are anti-parallel in a symmetric state, then we find ourselves in the ortho-positronium state. This lives much longer, because it has to annihilate into at least three gamma rays to conserve total momentum.

In the bottomonium system, the ground state is called the “eta” (instead of “para-bottomonium”, I guess) and borrows its nomenclature from its cousin quark, the charm quark, and her quantum state names. Because quarks are ruled mainly by the strong force, when the eta decays it does so into a pair of gluons instead of a pair of photons. The ortho-bottomonium state, called the “upsilon”, lives longer because it must decay into at least three gluons.

Nature is full of symmetry. The symmetry between positronium, its structure the result of the richness of electromagnetism, and bottomonium, its structure due to the strong force, strikingly unifies ones understanding of the two systems. There are key differences – coupling in electromagnetism is an order of magnitude weaker than coupling in the strong force – and this changes a lot of realities. But, fundamentally, if you can think about positronium, or the hydrogen atom, you are equipped to think about bottomonium.

The more I dig into the data of quantum mechanics, the less I think I do not understand it.

Charm Jets are Tricky Bastards

The continuum. It’s not an “organization of intergalactic intelligent designers whose purpose seems to be constant irritation of Jean-Luc Picard”: The continuum is the ultimate expression of quantum mechanics. It is what nature does with energy when it converts to mass: generate a continuous distribution of random particles whose production doesn’t violate the conservation of energy or quantum numbers, like “lepton number”:

The continuum is the white noise of nature. Smash together matter and antimatter, create pure energy, and what will most often come out are simply random combinations of quarks and leptons with no particular structure to their energy. That is, their mass and kinetic energy form a continuous distribution, a continuum. If I could convert the appearance of these random particles to a singular image, I would invoke the snow on a cathode-ray tube TV screen, hissing and popping.

But the continuum isn’t harmless. Because nature produces anything allowed by conservation of energy, all kinds of stuff that isn’t necessarily of particular interest to your research comes popping out of a particle collider. It may not be of interest, but it can cause real trouble. Particle and nuclear physicists speak of the *probability of production*, a geometry-independent quantity called the *production cross-section*, or just the *cross-section*. The cross-section of various non-B-meson processes at BaBar, for instance, far outranks that of B-meson processes. Overwhelmed by the TV snow of the continuum, we have to be clever and reject as many of these events as possible. But their overwhelming production rate, coupled with their somewhat random kinematics, means that sometimes lots of useless events will happen to match the signature of signal, and must be measured and treated to account for them. Think of trying to watch your favorite TV show on a station that barely comes into your TV, the sitcom visible but buried in a sea of flickering dots.

I am in the process of struggling with the continuum. In my current reseach, I am working with several of the students in my group to reconstruct B mesons decaying into a strange meson (one containing at least one strange quark) and a photon. The photon is very high energy, a gamma ray, with one to several billion electron volts of energy. You’d think that would be a fairly unique signature – a whopping gamma ray slamming into our Cesium-Iodide calorimeter – but you’d be wrong.

For instance, one continuum process is the generation of a charmed meson, whose mass is large and thus can be mistaken to a bottom meson. What’s worse, the decays of charmed mesons can produce jet-like structures of *very* high-energy neutral pions, and the preferred decay channel of these pions is photons. Getting a multi-billion electron volt photon is cheap, and then said photon ends up contaminating our analysis.

We’re in the process of minimizing the time needed to run over all of BaBar’s real and simulated data. We’ve learned that events from charm continuum production not only take the longest to process (they contain lots of high energy photons), but are also the biggest events (because they often have a large multiplicity of low-mass hadrons). Consequently, to make economical usage of disk space and CPU time, we must make an early effort to remove these continuum reactions.

The continuum at first sounds like innocent white noise, a hissing TV, but in fact it is a rich and complex structure of the quantum mechanical world that challenges and vexes.