By and large, particle physicists confronted with the need to awe and enthuse an audience of laypersons will have no hesitation in choosing to speak about the Higgs boson and its mysteries - undoubtedly a fascinating story that requires one to start with the 1960ies and the intuition of a handful of theoretical physicists, and then grows epic in a crescendo of colliders that sought and missed the Higgs boson, and then the LHC which finally found the elusive signal of production and decay of that particle.
However, there are heaps of other small beautiful pearls waiting to be dug out in the proton-proton collisions that the Large Hadron Collider has been delivering to the ATLAS and CMS experiments since 2010. The world of subnuclear particles is simple if you grind it down to the smallest building blocks of matter: a handful of quarks, a handful of leptons, and a few force carriers that keep them together; the Higgs boson belongs to that basic level. But if you go just one step above, you find an enormous wealth of structure and beauty in the physics of hadrons.  

Mesons, the next simplest thing

Hadrons are particles that withstand the action of the strong force: Adrós in ancient Greek means dense, strong. Both adjectives are good qualifiers to connotate hadrons, which are bound states of a quark and an antiquark or three-quark states, but also, as we have recently learned, more complex four- or five-quark states. These particles are held together by the strong force, which explicitates through the exchange of a boson of zero mass, called gluon. So if you could zoom inside a hadron (as we effectively do when we collide these particles at high energy) you would not only find those quarks that make it up, but you would also meet a sea of other quark-antiquark pairs that are constantly popping up and being reabsorbed, and a sea of gluons in the process of carrying strong forces from one quark to the next. 

The simplest hadrons are bound states called mesons, of which the lightest and oldest member is the charged pion - a particle discovered already in the late 1940ies in cosmic rays. Mesons like the pion are made up by a quark and an antiquark pair, but as I mentioned above, their internal structure is the result of such a mess of virtual particle interactions that we have not fully understood their internal dynamics yet. That is, we do understand the forces that take place between meson constituents, but we cannot compute them exactly because of a special feature of quantum chromodynamics, the theory of strong interactions, which prevents our main mathematical weapon - series expansion - to converge to finite results. 

The above is in contrast to quantum electrodynamics, a theory developed in the 1950ies with dominant contributions by Richard Feynman and Julian Schwinger (Nobel prizes in 1965). They showed how we can compute everything that happens to particles endowed with electric charge by just considering their interaction as the sum of an infinite series of smaller and smaller nudges caused by the contribution of more and more photon exchanges. The contribution of more photons is less probable and smaller, and thus it counts less than that of fewer photons. By computing the interaction as the sum of smaller and smaller terms in infinite series that converged to finite values, you can obtain exact predictions for anything from the electron magnetic moment to orbits in the hydrogen atom. 

The "solution" of quantum electrodynamics was a giant success of physics in the XXth century, but alas - the next force that physicists tried to compute exactly did not lend itself to the same treatment: the strong force, due to the exchange of gluons instead of photons, has the annoying property that for low-energy processes the contribution of more gluons is not small enough to allow the infinite sum to converge. We have a beautiful theory in our hands - quantum chromodynamics - but we can only use it to compute very high-energy processes, where the sums converge; while the most basic properties of matter, like those of the light mesons, is impervious to our calculation methods. [The difference of low and high energy behaviour of the series is that the coupling constant, which is the term at the basis of the series expansion, is too high at low energy for higher-order terms to grow sufficiently smaller at low energy. The couling decreases with energy, eventually making series convergent for high-energy phenomena.]

So all the above is to say that there is still a lot to learn in the physics of light particles. And there is a lot of physics to dig out, in fact: if you put together a meson, it will decay very soon into lighter particles, in all conceivable ways that are not forbidden by physical laws.

How to decay

Now, the pion has very little choice in how to decay: its mass (139 MeV) is too light for it to choose among many different options. Basically, a negatively charged pion can only decay to a muon and a neutrino, or to an electron and a neutrino - there is nothing else you can buy with those 139 MeV. You can figure that out if I give you the mass of the lightest elementary particles:

- the neutrino is practically massless, but it is also electrically neutral;
- the electron weighs a mere 0.511 MeV, and has negative charge like our negative pion;
- the muon weighs 105 MeV, and also has one unit of negative charge.

Since the negative pion must pass its electric charge to somebody, it will either produce an electron or a muon; those particles, however, cannot be summoned from the vacuum by themselves, as they carry a quantum number called "lepton number" which you can't produce out of the blue. Rather, if you summon an electron AND an electron anti-neutrino, the total lepton number of the pair is zero, and you are allowed to do so. The same goes with a muon - muon antineutrino pair. And that is about it for the negative pion: no more fancy things to do when it decays.

So is that it? Well, yes, and yet already in such a limited choice there is a lot going on. For you can measure the lifetime of the pion, and also the relative probability of its decay into either of the two final states we discussed. And here, already there is a nice detail - the pion likes way more to decay to a muon rather than to an electron. That is because of angular momentum conservation the electron that is generated in the decay is forced to get a spin aligned with its momentum, which the electron simply hates, as it is very light and the alignment almost breaks a fundamental rule of the standard model of particle physics. The muon is instead okay with it, so we get many more muons from pion decay than electrons!

Well, the above is just an example to show how complex and rich can be the physics of even the simplest meson. But now, let me tell you about a slightly higher-mass meson, the eta particle. The eta is a particle made by a strange-antistrange quark pair, and this already makes things more interesting - suffices to say that strange quarks are not part of ordinary matter. But at a mass of 548 MeV, the eta is sufficiently massive to have a host of possible ways to decay. So studying all these possibilities has kept physicists busy for a while, soon after its discovery. 

If we look at the particle decay database in the PDG live web site, we find a rather rich table of possible decays, with their measurable quantities tidily listed:

So we learn that the eta likes to decay to a pair of photons (39.4% of the times) or to three neutral pions (32.6% of the times). But it can also decay to a positive, a negative, and a neutral pion (23.0% of the times). All other decay modes are rather rare. Are they un-interesting because of their rather infrequent occurrence? On the contrary! Because of their rarity, they are extremely important to study. Because these decays are mediated by weak or electromagnetic interactions, and not by strong ones, they can be calculated to high precision. 

Typically in an experiment you have uncertainties in the decay probability you measure for a particle, which scale with the probability itself - so, e.g., that large 39.4% decay to two photons has a 0.2% uncertainty associated with it, which is a relative 0.5% error. If we instead study a much rarer decay mode, such as the decay to a pair of muons, we get a larger relative error - 0.8 millionths out of 5.8 millionths, which is a 14% relative error - but in absolute terms the uncertainty is much _smaller_!

Because of this, by measuring extremely small probabilities of decay of these mesons, physicists hope to become sensitive to very tiny deviations from theory predictions. This would be the case if, e.g., a super-weak force existed in nature, modifying ever so slightly those decay probabilities. Where would be most likely to spot such an effect? Not on the decay of the eta to two photons, because the 0.2% error overall would hide those small effects; instead, the dimuon decay, with its precision of 0.8 millionths, would be a much better bet.

A nice recent CMS result

If you look further down in the table above, you can see that the 2023 PDG listing shows an upper limit for the decay of the eta to two muon pairs (the 18th in the list): previous experiments could only place that limit by not seeing any such decay in their data. Indeed, we are talking about something exceedingly rare. But soon after the update of that table, CMS was able to spot the decay and measure the rate! It is of 5 billionths, give or take some 20% total relative uncertainty. A wonderful result!

But how exactly does an eta meson pull that off? Well, the strange and antistrange quarks inside the eta orbit around one another until they annihilate, leaving behind energy - those 548 MeV that made up its mass - and nothing else. The energy is emitted by the annihilation point in the form of two photons, as in the dominant decay mode, but then, very infrequently, each of the photons can turn into a pair of muons. The photons can do so because they have each an energy of 274 MeV, which is larger than twice the mass of the muon (210 MeV). But it is a very rare occurrence!

The conversion into two muons of each of the photon is called "internal", because these photons are virtual particles, not asymptotic states in this case. CMS searches for these four final-state muons in its data by first isolating a sample of events with at least two muons, where the more frequent decay of the eta into only one muon pair can be spotted without problem in the large data sample collected by the experiment in the 2017-2018 run of the LHC. The dimuon decay mode provides for an excellent normalization point to check muon detection efficiency.

There are very few events with four muons in the data, so the search for the signal is not very hard: you just select the signal of interest (two positive and two negative muons), clean it up a bit, and compute the total invariant mass of the four particle. Voila: a neat peak in the distribution appears!

Physicists are very skeptical guys and gals, but if there is something that makes them stop arguing is a clean mass distribution, and the one above is about as clean as any - there is absolutely nothing that can explain that bump at the correct mass (0.55 GeV) other than the fact that those are really eta particles that chose that very odd decay mode to disintegrate.

The fit, which includes a model of non-resonant background (the rising data toward the right) and the sharp peak on the left, allows to estimate that about 50 eta -> 4 muon events are present in the data. From that number, and consideration of the total integrated luminosity that the data correspond to, plus an account of detection efficiency and selection probabilities, allows CMS to estimate the decay probability of the eta into four muons. 

Regrettably, the number found by CMS is in excellent agreement with the predictions of theoretical calculations. That means that if there is some new physics contribution to these rare decay processes, it must be exceedingly tiny - much smaller than a one-in-a-billion effect.