## 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.