Probability of collision

A very long time ago, in internet time, I wrote about the Colour Glass Condensate (or CGC for short) and promised to talk about what I did with it in my MSc work. The CGC describes a particle like a proton or the nucleus of an atom which is moving very very fast — such as in a particle accelerator experiment (and not anywhere else, in this epoch of the universe). As one might guess from the name, CGC particles form thin, glass-like sheets that interact through the colour force of quantum chromodynamics. One of the results of being thin and glass-like is that if, as it’s travelling down the beam pipe in a particle accelerator, it collides with another particle, the other particle is likely to smash right through it. That leaves us with one smashed up proton or nucleus (we have deeply inelastic scattering, in technical terms) and one colliding particle that whizzes on. The collision is unlikely to significantly affect the path of the particle that smashes through, but it may have a big effect on the colour. If we want to calculate the probability that the particles collide and do something interesting, a good place to start is by looking at whether or not the colours change. This is the calculation my MSc work focused on; more specifically, the calculation shows how the probability of a colour change depends on the energy of the particles in the collider.


In the details of the interaction we looked at, the electron smashing through the nucleus produces a quark and an antiquark. These are the only way that the electron can change the colour of the nucleus, so if we track what they’re doing, we’ll know whether or not there’s been a colour exchange with the nucleus. Has a gluon been transmitted between them?

The easiest way to track that turns out to be by checking whether or not the quark and the antiquark are still correlated. When they’re created, they have exactly opposite properties, but if one of them interacts with the nucleus, only that one will change. Our job becomes to track how closely the quark and antiquark are correlated, depending on the energy involved and the difference between them. The equation that does this is called the Balitsky-Kovchegov equation* (after Ian Balitsky of Old Dominion University and JLab and Yuri Kovchegov of Ohio State University). The BK equation looks like this (<Sxy>Y means the energy-dependent average value of the correlation between the particles at positions x and y):

It’s fairly complicated. Analytically, it can’t be solved — that is to say, we can’t simplify the symbols any further without putting numbers in.. Therefore, we put numbers in. But solving differential equations numerically is an art in itself: the derivative is full of statements about “tending to zero” which can’t be applied to actual numbers the way they can to symbols. However, it’s certainly possible, if time consuming. Sufficiently time consuming that even when we get a computer to do all the number crunching, it takes days or weeks to get to an answer with reasonable accuracy. To do better than that, we have to call in more advanced computing techniques. In our case we made use of a GPU — which is designed to run high-intensity graphics on the computer — to do our number crunching.

It turns out that we need a substantial detour through numerical methods and computer science to calculate the correlation function for the quark-antiquark pair. In fact the only other physics we’ll talk about in this series is when we finally get results from the BK equation. Before that, we’ll think about thinking like a computer.

*Technically, the BK equation is a truncation of the full result, which can be achieved either through the JIMWLK equation or the Balitsky hierarchy of equations. The fine print of the mathematical details of the truncation doesn’t affect the broad sweep of the results, although it has some surprisingly visible consequences.

The Colour Glass Condensate

So far, this series has been a crash course in the basics of the standard model of particle physics, Feynman diagrams and quantum chromodynamics. We’ve spoken a little about quantum mechanics on the way. Now, so that we can talk about this state of matter known as the colour glass condensate (or CGC for short), we’ll have to think about another really weird part of physics: special relativity. Relativity is really about asking what happens when we change velocity.

Changing velocity doesn’t change the laws of physics: this is why tray tables in aeroplanes make sense. Despite the plane’s tremendous velocity, a cup placed on the table stays on the table, just like it would on the ground. Of course, aeroplane tray tables don’t work during takeoff and landing, but the issue there is the acceleration — the change in the velocity — not the velocity itself. It’s also why many physics experiments can ignore the fact that the Earth is a big rock flying through space at a terrifying speed. (Technically the Earth’s velocity is not constant, particularly because it keeps turning, and this gives rise to effects like the Coriolis force.) This idea — that you can pick an constant velocity and physics will work the same way as at any other constant velocity — is one that every idea of relativity keeps front and centre. In fact, velocity-independence (or “frame-independence”) has become a requirement for anything proposed as a law of physics.

So what does change if  you change velocities? Most obviously, the relative velocity of everything else. I may think I’m walking at half a metre per second down the plane aisle, but from the ground you’d say I’m rushing overhead at several hundred kilometres per hour, with a slight modification based on whether I’m walking towards the front or the back of the plane. For centuries, the slight modification was assumed to mean that my velocity relative to you was just the sum of my velocity relative to the plane and the plane’s velocity relative to you. However, when one gets more accurate — historically this came from various attempts to measure and describe light, notably by Maxwell and Michelson and Morley — just adding velocities doesn’t work out. In order to get the best theories of light to square up with the best measurements, velocity had to combine in some way other than simple addition. (One of the results of the new method of combination is that a light wave travels at the same speed — whether you run towards it or away from it, it has exactly the same speed relative to you. This is very weird — but we just said we were going to redefine addition, so we should expect things to be weird.) Simple addition is a very good approximation to the new rule for slow-moving objects, but it’s not so good for very very fast ones. This is why ordinary addition of velocities seems to make perfect sense. It’s only when we start measuring fairly esoteric things (like the speed of light) that we come across the new velocity combination rule.

In all of this messing around with velocity, we’re actually messing around with distance and time too. That shouldn’t be too surprising, since velocity is just distance divided by time. In general, if an object is moving very fast compared to you, then when you measure its length, you’ll get an answer slightly smaller than you would if it wasn’t moving. This is called “length contraction.” It’s very weird, but all the evidence points to it being true. There’s a similar, but opposite effect for time. If two event happen on an object moving very fast compared to you, you’ll measure the time between the events as longer than you would if the object wasn’t moving (compared to you). This is called “time dilation.” Again, it’s very strange, but it’s the least strange thing we can do to make sense of the measurements.

So how does all of this relate back to particle physics? In the simplest sense1, it’s because the particles in a collider like the LHC are hurtling towards each other at very nearly the speed of light. To describe a particle about to enter a collision, we will need to take into account the length contraction and time dilation that it will experience. This is where the colour glass condensate comes in.

A particle travelling very fast will be contracted in the direction it travels. So while we might have said that a round ball is a reasonable approximation of a proton or an atomic nucleus, that round ball now becomes more like pancake. If you like (and we do) you could even say it’s like a sheet of glass. Now our particle pancake is not just length contracted, it’s time dilated. That means that the time between events within the particle is larger than usual — in other words, it changes very very slowly. A similar very slow change is a property that is sometimes attributed to glass, which encourages us to name this slowly-changing particle pancake after a sheet of glass.

The glass is colour glass because, as we said last week, there are going to be lots and lots of gluons around — all of which have colour charge. “Condensate” refers to the huge number of gluons and to the fact that if there are enough gluons (and in the CGC there are), the system becomes saturated, so that adding more gluons doesn’t really change anything.

Now, after a brief detour via special relativity, we have our setup. A target particle (a proton or atomic nucleus) is moving so incredibly fast that it becomes a colour glass condensate particle-pancake. A probe particle travels towards the target and interacts with it — probably punching right through it and probably shattering it. (The “inelastic” in “deeply inelastic scattering” means that the target is probably shattered. And the “deeply” means that it’s really, really probably shattered.) Boom! Now all we need are some equations to calculate the probability of this collision actually happening, and perhaps to tell us what we might get out the other side. And maybe some tools that will allow us to calculate with those equations . . .

1 In a more complicated sense, the new velocity combination rules put time and distance on the same footing, which can’t be done in ordinary quantum mechanics and require the development of ideas like quantum field theory.