One of the fundamental concepts in particle physics is the measure of the probability that a specific process will happen when a particle scatters with another particle or a nucleus. Such a process is inherently stochastic since it is governed by the laws of quantum mechanics.

In particle physics we call the cross section (σ) of a certain process a quantity that is proportional to the rate that a specific outcome happens in a collision. It has the units of an area and it is measured in barns. 1 barn (b) represents an area of 10^-28m.

You can think about the “cross section” as the effective area between two interacting particles. It’s an easy concept to understand if you have some physical objects, like when you play pool and the balls on the table can scatter against each other, but the same concept can be extended also to the case when particles can interact through long-range interactions, such as via electromagnetic force.

[Further information can be found at: https://en.wikipedia.org/wiki/Cross_section_(physics)]

Figure 1: A schematic depiction of the interaction between two particles. The quantity b is usually called the impact parameter. From https://link.springer.com/article/10.1140/epjc/s10052-016-4585-8/figures/5

Cross sections at the LHC

The fundamental interaction between known particles are described by the Standard Model of Particle Physics, or SM in short. SM is a Quantum Field Theory, a very advanced theory that combines together the principles of quantum mechanics and special relativity. The SM is able to describe, with amazing precision, the dynamics between fundamental particles mediated by the electromagnetic, weak and strong force. Gravity is still out of the picture and won’t matter at the LHC energy scale (unless extra-dimensions exist but it’s out of the scope of this post).

The question now is: what does the Standard Model predict?

Among various things, it predicts the rate that certain processes might happen. Basically it provides a framework to determine the cross sections of scattering between particles. In particular, it can predict the cross section of proton-proton scattering, both elastic, when protons just scatter on each other without producing other particles, and inelastic, when new particles are produced in the interaction and the protons are fragmented.

The cross sections are the observables of our theory as they can be measured experimentally in a high energy physics experiment.

We can compute how many times a proton-proton collision will result in the production of, let’s say, a W or a Higgs boson from our theory and we can measure it in our experimental apparatus by careful data analysis.

Finally, let's circle back to our original question: How often do we produce something interesting in the LHC?

First let’s define “interesting” a process of our choice, let’s say something rare like the production of two W-bosons (usually denoted as pp->WW). Let’s take our cross section plot predicted by the Standard Model (see Figure 2) and we find out that at a centre of mass energy of 13 TeV the WW production rate is about 150pb (8th column from the left, purple point). If we compare this with the TOTAL pp cross section of ~10^11pb (from the first column), we have that we have a WW production every 670 million proton-proton collisions.

Figure 2: The proton-proton cross sections at the LHC for various energies in the centre of mass and compared to theory computations. From: https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PUBNOTES/ATL-PHYS-PUB-2023-039/fig_01a.png

Figure 2 also shows the comparison between measured cross sections and computed cross sections using the SM. This is an amazing example about how the theory is able to predict the cross sections of a large variety of processes across many orders of magnitude.

You might be wondering now: since interesting processes are the rare ones, how do I find them in an experimental apparatus at the LHC?
If there is enough interest about this post, I will be happy to follow up with another one.

100%

Discussion