# Paper Explainer: Two is not always better than one: Single Top Quarks and Dark Matter

A few months ago, I was lucky enough to be contacted by an experimental student in the CMS collaboration, Deborah Pinna. Deborah had a question for me: in a certain set of dark matter models that I had written one of the early papers on, we only considered one particular class of final states, namely production of dark matter at the LHC along with a pair of top quarks. Why, she asked, did we not also consider the production of a single top quark, along with dark matter?

The answer was that everyone, including myself, just assumed that this channel didn’t matter. I’ll explain why in a bit, but I had just assumed that the rate at which this sort of event could occur would be so low that I never actually bothered to check. It turned out that my intuition was wrong. Deborah did check, and upon finding out that this single-top channel mattered, contacted me, assuming perhaps there was a good reason for ignoring it. There wasn’t.

I was really happy to contribute to Deborah’s project, and I want to emphasize that she and a postdoc, Alberto Zucchetta, did all of the heavy lifting on this paper.

So what was the idea? What is single versus pair production of tops, and why does it matter?

We know dark matter exists due to its gravitational interactions, but we have no idea what it is. We suspect that dark might be a particle that interacts with the known Standard Model particles with a very weak non-gravitational force. This “weak force” might be The weak nuclear force that we already know about, or it might be due to some other new particle and new force we have not yet measured. The possibilities are not endless, but they are immense.

One attractive possibility is that the dark matter is a relic of the time when the Universe was hot and dense. If that is the case, the strength of the interaction dark matter must have with some other particle must be at least as strong as the weak nuclear force. If that other particle is in the Standard Model, that means we can find it at the LHC. The point is that if dark matter is a relic of the time when the Universe was very hot, there must be a minimum interaction strength, which is good news if you want to find dark matter in an experiment.

This logic takes us to the point of thinking it is reasonable to look for dark matter at the LHC, but since we don’t know what dark matter is, there are a lot of options left. You can try to build a full theory of how dark matter interacts with the Standard Model, which might fit into a model like supersymmetry. That approach has its positives, but since the model will have many parameters, it is also possible that you’ll miss something because it is hard to fully explore the space of possibilities.

So you can try to build a Simplified Model, one with only a minimum number of particles with which to communicate between the Standard Model and dark matter. One of the simplest Simplified Models is one where the new particle is a spin-0 particle — I wrote one of the early papers presenting this in the context of Simplified Models, but of course others had worked on versions of it before.

The nice thing about this idea from a theorist point of view is that is easy to embed this Simplified Model into a more complicated full theory. For example, the Higgs boson is a spin-0 particle, so you can ask if the “new” mediating particle is just the Higgs, or a Higgs in some extended Higgs model. Since we don’t understand the full physics behind the Higgs boson, and we don’t understand dark matter, maybe this is two great tastes that go great together.

In fact, if you introduce a new scalar that connects dark matter to the Standard Model quarks, you run into a particular problem. As I’ve written about, the left- and right-handed fermions in the Standard Model are different. The entire reason that the Higgs boson “gives mass” to the matter particles in the Standard Model is due to this difference: the Higgs is said to “break chiral symmetry.”

Therefore, if we want a new particle that connects dark matter to the known fermions, it too must break chiral symmetry. If this is some new contribution to chiral symmetry breaking, then we would naively expect to see evidence of that in precision tests elsewhere (or, if we don’t, it’s now a very interesting question of why we don’t see that). This gives credence to the idea that, if the dark matter connection is via a spin-0 particle, it connects to the Standard Model in the same way the Higgs boson does. Most critically, it couples more to the heavier particles.

The heaviest fermion is the top quark. So now we have a reason to explore the hypothesis that the dark matter can interact with pairs of top quarks. Through interactions that look like those shown here.

Here, the dashed line is the new mediating particle, and the $\chi$ particles are the dark matter. Now, this is the “theory” part of particle physics. We have a model which we like for various reasons, even if one of those reasons is just “well, we don’t know it’s wrong yet.” Next, the job is to find out if it is wrong, which means looking for signatures of this model in various experiments, including the LHC.

I, and other people, had considered the following types of production at the LHC:

The top quark is a strongly interacting particle. So op and antitop pairs will be produced in huge quantities at the LHC. Sometimes, one of those tops or antitops could radiate off one of these new mediating particles, which in turn decays into a pair of dark matter. The result: a top-antitop pair and invisible particles, which we “see” as an imbalance in the momentum of the visible particles — what is called “missing transverse momentum” or “missing transverse energy” (MET). So, one of the places that I and others suggested experimentalists could look for dark matter was in “top pairs” at the LHC. This is great, because tops can decay to spectacular signatures, so looking for this is not necessarily easy, but it is doable.

However, there is another way you can produce a top at the LHC. The top (and any of the heavy fermions), decays through the weak nuclear force. In the top quark’s case, turning into a bottom quark and a $W$ boson. That means, if you have $b$ quarks or $W$ bosons floating around in the proton (and there are, flickering in and out of existence briefly), you can produce a single top quark. That single top can then radiate off a mediator, and produce dark matter. The processes look like those shown here.

The “weakness” of the weak force means that this process would be suppressed compared to the strong interactions producing pairs of tops. So I just assumed this type of process was completely negligible compared to the pair production. So I ignored it. Deborah and Alberto showed it wasn’t negligible.

Figures showing distributions of kinematic variables. Compared to the single top production (colored regions), the top pair (black line) tends to drop faster at high energies.

A couple of quirky things make it not nearly as unimportant as you’d naively expect. First, you only have to make one top quark, not two. Since tops are heavy, you need less energy, and the less energy you need, the more often something will happen at the LHC, all else being equal. Second, the types of particles in the initial state that can produce single tops are more common at high energies than the ones making top pairs. So when you do make a single top plus dark matter, the particles produced tend to be more energetic. This makes it easier to pass various criteria at the LHC that are necessary for the event to be recorded for later analysis. Deborah and Alberto point this out in the paper when they investigate the “hardness” of the events

So, this process matters. Now, there are two options. We can either design a way to look for single top plus dark matter from scratch, or take an existing search and see how sensitive it would be to this new way of making dark matter. In the paper, we do the latter. We adapt the existing CMS search for pairs of tops plus dark matter, and see how the limits improve by including the extra events where dark matter is produced along with a single top. Obviously this isn’t as good as you can do, but designing realistic searches from scratch is difficult to do with any certainty. For theorists like myself, it is very difficult since we can’t fully simulate the detector response, and for experimentalists like all my coauthors on this paper, there is no point doing it in a way that doesn’t use internal CMS information about the detector. In which case, just do the search for real and publish the results. So think of this paper as the proof of concept, that will be followed up (eventually) by real data by the experimentalists.

The "propaganda plot" of our conclusions of the paper is shown here, where we show the strength of the interaction that can be probed using the existing analysis assuming only top pair production, compared to what you can do (again with the existing analysis) but including this neglected production mode. It's an ${\cal O}(1)$ improvement, which isn't necessarily earthshattering, but that is still a significant improvement. It is even more significant when you consider that this is what you can do when you throw these new dark matter events into analysis totally unsuited to find them. Without a more detailed study, one can't know for certain, but there is every reason to suspect that an analysis tuned to look for single top events would, when combined with the top-pair search, result in pretty significant improvements on the existing state of the art. And when you're looking for something you know very little about, any improvement is welcome.

Existing search limits (black) compared to projections when the new single top modes are included (red). Lower limits on the couplings imply a more powerful search.