I recently ran into a description of the Mu3e experiment, and got curious about it and the physics it studies. So after giving it a look, I am able to explain that shortly here - I think it is a great example of how deep our studies of particle physics are getting; or, on the negative side, how deep our frustration has gotten with the unassailable agreement of our experiments with Standard Model predictions.
Matter stable and unstable in the Standard Model
In the Standard Model matter is constituted by two different kinds of particles: leptons and quarks. There are three families of each of these, and only the first family is normally found inside and around us; members of the other families can be produced in energetic processes, but are otherwise of little practical relevance (except that some of us make a living out of studying them, and for a few good reasons other than the low salaries we get!)
The quarks in the first generation are called up, u and down, d; in uud and udd triplets they form protons and neutrons, which are the ingredients of the atomic nuclei. First generation leptons are instead the electron e and the neutron v. Electrons contribute to forming neutral atoms by equilibrating the positive electric charge of nuclei; neutrinos participate in weak interactions inside stars, or in radioactive decays, but they form no bound states.
If all we had were the particles I mentioned above, particle physics would be quite simple and boring. But as I mentioned above, there are two more generations of particles. These have generally a higher mass than that of first generation ones, which gives them a chance to exploit Einstein's famous equivalence of mass and energy to disintegrate, turning their mass into energy plus the mass of lighter particles: this is how unstable particles decay. They do so because in physics whatever process is not forbidden by an explicit law, does happen - with a rate that depends on specific details, such as in this case how much energy is there to distribute to the final state products.
In this post I will discuss the muon, the second generation replica of the electron. Being 200 times heavier than the electron the muon does decay, and it does so invariably by generating an electron, an electron antineutrino, and a muon neutrino. The three bodies it produces are almost weightless in comparison with their parent, so almost all of the muon energy is spent in kicking them in different directions.
Why doesn't a muon decay into an electron without neutrinos? We could imagine, e.g., the muon to "shrug off" the extra mass by emitting an energetic photon, with an electron being the result. This "muon to electron-photon" decay is a process that the Standard Model strictly forbids, in force of an apparently unbreakable rule called "conservation of the leptonic number". In a nutshell, there is a thing called "muonness" which you cannot destroy. You can transfer it from a muon to a muon neutrino (they both are endowed with one unit of muon-ness), but you cannot create a process whereby the sum of the number of muons plus muon neutrinos, minus the number of anti-muons and muon antineutrinos, changes by any amount. The same, by the way, holds for electrons.
But we know that the conservation of leptonic number is somehow violated in the phenomenology of neutrinos, because we have observed that neutrinos do change their leptonic number! This fact, discovered in 1998, has made the plot considerably thicker. We have never observed a violation of leptonic number conservation in the physics of electrons, muons, and taus - the charged members of the lepton family. So we would like to study the matter in more detail!
Let us look into muon decay
The decay of the muon can be described pictorially in what is called a Feynman diagram: in it, full lines describe the propagation of matter particles in space and time (the two coordinates in the plane), and dashed lines describe particles that mediate the interaction between matter. In the picture below you can e.g. see how a muon can turn into a muon neutrino as it emits a W boson. The W boson is the electrically charged carrier of the weak interaction; it lives a fantastically short life and then decays into an electron and an electron antineutrino.
In Feynman diagrams as the one above time flows from left to right, and the vertical axis indicates one spatial direction. Hence you can read it as "A muon is resting at some point of space for some time, then it decays, emitting a neutrino and a W boson in opposite directions; the W at some point materializes an electron and an antineutrino". In the diagram, you can also see as the antineutrino is seen as an arrow pointing backwards - this allows us to identify it as a particle carrying a negative leptonic number of the electron kind.
If we imagine lepton number to be violated by neutrinos only, we could conceive a process whereby the muon decays to an electron and a photon - the process I mentioned above - or even one where the muon decays to two electrons and a positron! The relevant Feynman diagrams are shown below. In them you can see that the muon may duly yield its muon-ness to a muon neutrino, but then if the latter forgets about it and turns into an electron neutrino, things make sense again.
In this diagram you can see that the foul-play particle is the neutrino in-between the muon and electron lines: it talks to the muon and says "I'll make good use of your muon-ness!", and then it finds another W boson in its way, and says "ok, here is some electron-ness, please create an electron!".
And above, the same trick is played by the neutrino at the top of the diagram, while the rest of the particles are busy doing their duties. At the bottom, a W creates a neutrino-antielectron pair, and the neutrino later creates an electron. This one can has the right leptonic number to do so!
Diagrams like the two above are the motivation for two very focused experiments: the Mu2e one, which looks for muon-to-electron transitions, and the Mu3e one, which looks for muon-to-two-electrons-and-a-positron ones. The experiment takes a very intense beam of muons generated at the PSI laboratories, and looks for transitions generating the wanted three particles in a carefully designed tracker.
You might be unimpressed by the whole story, but consider this: the experiment aims to be sensitive to decays that have a relative probability of one part in ten million billions. That is a 10^-16 probability, which we could exemplify as the odds that tomorrow you win 10M$ at the lottery, and then the plane you are flying on crashes on tall mountains, killing everybody onboard except you. Not your everyday schedule, by any means.
Below is a picture of the Mu3e experimental layout - the experiment is the endpoint of the muons path, which as you can see goes through a number of elements that can focus them and direct them with the right momentum into the mu3e apparatus, where they hit a target and stop there, ready to decay in utterly implausible ways.
In the drawing, the protons enter on the right (red arrow), hit a target, and the produced hadrons then are directed to a set of magnets that allow them to decay into muons; the latter are directed to the Mu3e detector on the left, where they may decay. The decays will normally just produce a single electron, plus two invisible neutrinos; but if three tracks are observed by the detector, that's something to scrutinize with extreme care! It will usually be a background event where multiple muons have chosen to decay in the same point of space, but who knows?
I must say I do believe neutrinos can shrug off their leptonic number somehow, but I have the feeling that this experiment will only manage to put a limit - a very stringent one at that - to the branching ratio of muons to three electrons. Still, I think it is quite a cool endeavour, isn't it?