Paper Explainer: Hiding Thermal Dark Matter with Leptons

This is a description of my recent paper with my student David Feld. 

Dark matter is a problem. We know that there is a gravitational anomaly in galaxies: the stuff we can see is moving far too fast to be held together by its own gravity. Add to this the precision measurements of the echoes of the Big Bang (the Cosmic Microwave Background), which tells us that the way the Universe was expanding and matter was clumping cannot be explained without some new stuff that didn’t interact with light, and you have very solid evidence for the existence of dark matter. Then of course, there is the Bullet Cluster, where we can see the gravitational imprint of dark matter directly.

So we know it exists. We just don’t know what it is.


There are, of course, many ideas. Ideas are, after all, theoretical physicists’ stock in trade. One very good idea is that dark matter is a thermal relic. When the Universe was very young, it was very dense and very hot, so every particle zipping around had enough energy that, when they banged into other particles, they could create other very massive particles. The heavy particles themselves would smash into each other and annihilate, creating other particles.

As the Universe expanded, this creation of new heavy particles would stop, as the other particles gradually cooled to low energies. However, at the same time, the heavy particles would find a partner to annihilate with less and less often. At some point, both creation and annihilation of heavy particles would cease, and the Universe would be left with some relic abundance of these heavy particles. This would happen regardless of the existence of dark matter, or what it is.

Most heavy particles are unstable, but you can construct scenarios where they aren’t, in which case these relics of the early Universe would still be around today. The interesting thing is that if dark matter happened to be a very massive particle that interacted only slightly with the Standard Model particles we know about, then we’d had a lot of them around today. Enough to be the dark matter. This is the idea of a Weakly Interacting Massive Particle.

Notice that this idea requires dark matter to interact with the Standard Model. And while that interaction must be small, it must be there. In fact, the larger the interaction, the longer in the history of the Universe a dark matter particle can “find” a partner to annihilate with, and the less we would have today. So, this means that if “thermal dark matter” is the right answer, it has a minimum interaction strength with the Standard Model. This is good news, since it means dark matter can’t be totally invisible.


 The various ways to look for dark matter, assuming that dark matter is a thermal relic.

The various ways to look for dark matter, assuming that dark matter is a thermal relic.

We’ve been looking for just this kind of dark matter for a long time. The basic idea is that the idea of thermal dark matter requires a way for dark matter pairs to turn into pairs of Standard Model particles, and vice versa. So you can start turning this diagram on its side, and get various ways to look for dark matter today. Start with a dark matter particle and a Standard Model particle and you can ricochet the dark matter off the visible particle: what we call direct detection. If two dark matter particles find each other in the depths of space (which will continue to happen, albeit slowly) and you get indirect detection of whatever Standard Model particles are produced. Or you can just smash two protons together at the LHC and look for dark matter in the rubble: collider production

Each of these types of searches have been going on for a while. And we’re getting to the point where we expect to have seen dark matter with the minimum interaction strength needed for thermal dark matter. We haven’t seen it though. So, is thermal dark matter dead?

This is an important question. If thermal dark matter isn’t the right idea, we need to look elsewhere: dark matter remains a problem that needs a solution, but this idea of dark matter as a relic of the Early Universe would be the wrong solution. So: have we ruled out thermal dark matter?

There are two key observations. First, just because we know that the interaction strength in the early Universe was sufficient to produce the “right” amount of dark matter doesn’t allow us to predict exactly how much dark matter would interact in any experiment today, there are just many ways to change, for example, how the rate of indirect detection. So there’s always a bit of wiggle room. Second, we just know that thermal dark matter must interact with some particle in the early Universe, not necessarily which one. Collider production is through protons hitting other protons. Direct detection is dark matter bouncing off of nuclei. What if dark matter doesn’t interact with the constituents of protons and neutrons?

So we have killed many models of thermal dark matter. But we have not gotten all of them. It is timely therefore to ask “what’s left?” Which models of thermal dark matter survive, and how sensible are they?


In this paper, my graduate student David Feld and I decided to look at a model we viewed as “maximally difficult” to find. There may be models that are harder to experimentally search for, but this is certainly up there. The idea is to have dark matter interact only with leptons. Even worse, we make the interaction with leptons through a “spin-0” particle. This makes the interactions today even harder to see, for various technical reasons. We can then ask: how visible is this very invisible dark matter?

It turns out that it is surprisingly visible. We can’t eliminate all possible versions of “lepto-philic” thermal dark matter, but we knew that wasn’t going to be possible going in (you can just make the dark matter heavier and that evades most bounds pretty easily). But if this type of dark matter only interacted with leptons, we’d be pretty screwed: it would be almost impossible to see this sort of particle, even if it was light enough that the LHC had enough energy to produce it.

But it turns out that you can’t easily make leptophilic dark matter not talk to quarks at all. The problem is as follows: leptons are really two different particles (see my explanation here). A left-handed lepton (an electron, muon, or tau) is a “doublet” of the weak nuclear force, while the right-handed lepton is a “singlet” (i.e. uncharged) under this force. To interact with both pieces, the connecting particle must itself be a doublet. This is the realization behind the Higgs mechanism, and in fact this doublet mediator is going to take part in this mechanism. Like it or not, this mediator is a type of Higgs boson (though, as has been known for a while and as we recap in the paper, experimentally it can’t be the Higgs boson we found at 125 GeV at the LHC).

Dark matter must be electrically neutral. If I don’t want dark matter to have charged partners (which would be something I can look for in experiments, and remember I want “maximally invisible dark matter” here), then this means dark matter must be a singlet of the weak nuclear force. So the mediator that connects to it must also be a singlet.

 The 1 million parameter points in the leptophilic dark matter model, applying successive experimental tests: a thermal cross section (top), indirect detection (middle), direct detection (bottom). Grey points are excluded, the colored points survive.

The 1 million parameter points in the leptophilic dark matter model, applying successive experimental tests: a thermal cross section (top), indirect detection (middle), direct detection (bottom). Grey points are excluded, the colored points survive.

So here’s the problem: the doublet mediator that talks to leptons must “mix” with the singlet mediator that talks to dark matter. At the same time, the Higgs bosons (including the doublet) are undergoing “electroweak symmetry breaking,” which gives mass to all the matter fields in the Universe. As we show in the paper, this mixing combined with electroweak symmetry breaking will always mix the leptophilic mediators with the mediators that interact with quarks. There is no “natural” way to forbid this mixing, the only way to stop it would be to tune particular parameters to be small, without any deeper reason for them to be small. 

What this means is that, when you try to build a realistic model of leptophilic dark matter (with a spin-0 mediator), you almost always end up with dark matter that interacts with quarks. To quantify this, we had to numerically scan over relevant parameters, since even after simplifying as much as possible there were 14 of them, which encode the masses and interactions of the mediator with the Higgs bosons and the dark matter. So we generated a million points, which populated this parameter space, and asked which point could a) give us viable thermal dark matter, b) had a 125 GeV Higgs with parameters like the ones we measure at the LHC, and c) have dark matter that would not have been seen by the experimental searches.

The points that survive each of these tests are shown here. All in all, only a few percent of the initial million parameter points pass all the constraints, and those are ones that are some combination of somewhat heavy dark matter (above 100 GeV or so), and accidentally leptophilic. That is, the only versions of this model that survive are ones that happen to accidentally have a small coupling that allows the interactions to be leptophilic after electroweak symmetry is broken.

So we can’t rule out leptophilic thermal dark matter, and it will take some time before we are able to. But it also not easy. We picked this scenario because it was supposed to be a slam dunk example of dark matter that would be impossible to find. However, it is typically just as visible as versions of dark matter that don’t only interact with leptons, a point that only becomes apparent when you look at how the dark matter part of the theory would have to fit in with how the Higgs mechanism works. So, when I think about nightmare scenarios of dark matter — dark matter that is impossible to find, no matter how hard we try — I’m now going to worry less about this sort of lepton-specific interaction. Because, yes, I could get it to work, but only by twisting some values of parameters specifically to make it invisible, and not for any deeper reason.