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This is a great answer. Thank you.

So in these situations how do you tell apart electrons from one source compared to another? In the article they mention how the LHC collides particles at a rate of "40 million times each second". I can imagine there are a lot of electrons and other particles flying around from other collisions. What makes an electron discernible between one type of particle and another?

Statistics like the op said. If you are expecting your decay products from an interaction to be 20% X and 80% Y but after 100 billion attempts which should have averaged out to the expected outcome you instead get 21% X and 79% Y something in your calculation is wrong.

Truly, you never really know which pair of particles came from a specific decay, and which come from some other processes and just happen to line up with the mass/energy you're looking at. Fortunately, for most particles, the combinatorial background signal follows a smooth curve around the energies you're looking at, so you can fit a curve to that background signal and then attribute the rest of the signal to the particle production. For an example, see the main plot on the Upsilon page in Wikipedia (https://en.wikipedia.org/wiki/Upsilon_meson). You can see there is a linear decline (in log space) of the background signal but then there's another peak around 9.5 GeV which is the additional signal from the Upsilon decay.

The point is, we cannot tell which pair of electrons/muons come from the decay of a specific particle, but we can tell how many extra occurred beyond what we would expect from all other known processes.

Thanks so much for the explanation and the example. I at least know a little more about these complicated endeavours now.

Happy to do it! Thanks for asking insightful questions. :)

I did an undergrad in physics, and at one point I wanted to be a particle physicist. But I didn't want to go straight to grad school, so I spent a couple years teaching middle school math and science. I loved that, and spent 25 years teaching instead of going back to physics.

If you don't mind my asking, what are you doing now after spending years studying such a specific area of particle physics?

I've been a software engineer for the better part of a decade now. My areas of focus during that time have been NLP, messaging, teaching, politics, and now data analysis and processing around criminal justice.

Are the decay modes (3% one way, 3% another, 94%?) experimentally determined, or does theory predict this distribution?

As in most things with particle physics, it's a combination of both. Theory predicts a wide band and then experimentalists come in with the best estimate they can make. Some theories are excluded and other are refined. There are another round of predictions and when the experiments get powerful enough, they can challenge or support some predictions.

Some of the best data for branching ratios comes from e+e- (electon-positron) colliders such as LEP (literally, the Large Electron-Positron collider). In these colliders, we can fine-tune the energy to produce massive amounts of particles we care about. From that, we can see how they decay. Mostly, Upsilons decay into massive sprays of hadrons and leptons (called jets in particle physics). These can come from decaying Tau particles (the much much heavier cousins of muons and electons) or from quarks/hadrons decaying over and over and over again into things like Kaons, pions, muons, electrons, photons, and other lightish particles. In the relatively clean environment of a e+e- collider, we can reconstruct these jets and determine which may have come from Upsilons. Combining this with a whole bunch of other measurements (and some theory) lets us determine the branching ratio (how often a particle decays into certain things).

Something I never quite understood: When you shoot particle A at particle B in some accelerator, how does the formation-and-almost-immediate-decay of particle C affect the end result? Like how can we know that those leptons didn't just come from the initial collision?

The scattering matrix predicts the distribution of output particle momenta given input particle moments and output particles.

The scattering matrix is calculated by including all the possible interactions you expect. So a matrix including some intermediate C will be different from one that does not.

Then you can line up what you actually observe and select the matrix that most accurately describes it.

So if I understand it correctly, it's a statistical phenomenon. Looking at one particular (heh) collision, you cannot be sure. But if you run it a lot of times, the frequency of the occurence will be the tell-tale signal. And what the existence of the particle does is provide an additional path. It allows the pair to be formed in two different ways, which raises the overall probability. That right?

Yes it’s statistical. In fact all physical measurements are statistical. When you measure the length of a table, there is some error term due to imperfections in your measuring stick as well as possible errors when you aligned/looked at the stick and table.

The extra path does not necessarily raise the probability. The interactions are more complex than that. The simplest thing to say is that it affects the distribution.

s/mass/rest mass/g

It's 511 keV at rest, and you are sensing a 4.73 GeV electron coming out of the decaying Upsilon. E=m and c=1 in the units you are using.