# Paper Explainer: Vector Boson Fusion Searches for Dark Matter at the LHC

Here, I describe a recent paper I wrote with a group of experimentalists (Jim Brooke, Patrick Dunne, Bjoern Penning, and Miha Zgubič) and a Rutgers undergrad, John Tamanas. We investigated the ability of the Large Hadron Collider (LHC) to find dark matter using a particular type of event, one called “vector boson fusion,” or VBF. I’ll describe what VBF is in a bit.

Bjoern and the other experimentalists had gotten in touch with me a while back, letting me know they were interested in looking at these VBF-type events in the context of dark matter. Now, there are certainly theory papers out there which have considered VBF-searches for dark matter before us. However, the entire point of writing theoretical papers on experimental phenomenology is to get the experimentalists interested in actually doing the search; so here were some experimentalists who were interested, and they wanted to know if I would help. Of course you say yes at that point.

Now, the experimentalists were not looking to get me involved in a real search at the LHC: to do that requires being part of the actual experiments (ATLAS or CMS), otherwise you can’t look at the data. Being a member of the experimental collaborations is a possibility for theorists, and there are valuable contributions you can make in them. However, there are trade-offs: being a member of an experiment means you have experimental service requirements, and you have tasks you need to accomplish. Plus, if you’re a member of CMS, you can’t spend as much time investigating what ATLAS is doing (and vice versa). So far I’ve remained unaffiliated, as have most theorists.

But the experimentalists weren’t looking to do a real search with real data. Instead, they wanted to determine the possibilities for VBF-type searches, using publicly available information, and then use this study to justify the “real” search using the internal experimental code and resources. In a sense, we determine that something is a workable idea, then the same people I’m collaborating with take the idea “behind the firewall” and do it again, for real. A little circuitous, but part of science is telling people what you’re doing and why you’re doing, and part of that communication is through papers like this, directed as much at the rest of the experiment(s) as at the rest of the community.

The experimentalists were all set to perform a mock analysis on simulated LHC data (what I call pseudodata), using public codes that they had validated did a very good job of giving similar results as the real ATLAS and CMS analyses. What they wanted me to do was to give them a set of models to work with — theoretical constructions for ways dark matter might be produced at the LHC that VBF techniques would be good for, to help with the implementation of these models to generate the necessary pseudodata, and to put the projected LHC constraints in context with other ways to look for dark matter.

So, what models to work with?

This is where what “VBF” is becomes important. But first, let me set some things up. To look for dark matter at a proton-proton collider is tricky. After all, a necessary condition for a particle to be dark matter is for it to be “dark.” That is, it needs to be invisible.

Now, we hope (and have decent theoretical reasons to expect) that dark matter is actually a particle that has weak interactions with particles like protons. Weak here might be “colloquially” weak, as in “not very significant, but not zero,” but it also might be “theorist-weak,” as in “an interaction that is mediated by the particles of the weak nuclear force.” This dual meaning of the word “weak” is a source of endless confusion, even amongst physicists. I would say that there are good theoretical arguments that dark matter has “theorist-weak” interactions, but those arguments could also suggest that dark matter might just have “colloquially-weak” interactions. Either way, the arguments imply that producing dark matter by smacking two protons together might not be a fool’s errand, as either way the level of “weakness” is just big enough for the LHC to produce non-negligible numbers of dark matter particles.

However, producing dark matter at the LHC isn’t enough. Because, you know, we can’t see it, what with the darkness and all. Even if the energy of the protons in the collider itself is sufficient to overcome the weakness of the interactions and we make dark matter, now all you have is some extra particles of dark matter flying through your detector. There are a lot of particles of dark matter flying through the detector already, and we can’t see those. Adding more doesn’t help.

Instead, what you need to do is create dark matter in a collision that also pushes visible matter aside with a lot of energy. If I’m sitting on ice, and I throw a ball sideways, due to conservation of momentum, I’ll recoil against it, sliding across the ice. If you couldn’t see the ball being thrown, you could still infer that I had thrown something by just watching my motion.

So, we look for dark matter in events were there is a lot of “missing traverse momentum” (sometimes misleadingly called missing transverse energy or “MET” for “missing energy, transverse). Transverse means that the particles are moving perpendicular to the beam of incoming protons. We care about the transverse motion because we know that nothing before the collision was moving in this direction, so by conservation of momentum anything after the collision moving perpendicular to the beam is balanced by something moving in the opposite transverse direction. However, there was already momentum parallel to the beam (the protons themselves), so we can’t use the motion along the beam of particles after the collision to infer the existence of invisible dark matter. If we see a particle shooting down the direction of the beam, with nothing obvious balancing it in the other direction, maybe that’s a sign of something like dark matter, but we can’t tell, because we don’t actually know how much momentum in the beam direction was involved in the particular collision we’re looking at.

The simplest type of such recoiling search is to look for a single “jet” of particles moving with lots of transverse momentum which aren’t balanced by any other visible particle. This type of search is called a “mono-jet” search, for obvious reasons. Here’s a picture of what a mono-jet event looks like at ATLAS. The rings in the picture are various layers of the detector, which are really cylinders extending around the proton collision point. Each layer is designed to pick up particles with particular qualities. The inner grey region is a silicon tracker that sees most things that have either charge or “strong nuclear” interactions (the interaction force that keeps protons and neutrons together), but isn’t great at measuring their energy. The green ring is the electromagnetic calorimeter (E-Cal), designed to measure the particles with that have electromagnetic interactions (photons and charged particles). The red ring is the hadronic calorimeter (H-Cal), designed to get lots of energy deposition from particles which have strong-nuclear interactions. Way out in the edge is the muon-chamber, designed to see muons, the heavy partners of the electron. The H- and E-Cals are what give you measurements of energy and momentum for particles that reach them.

In this event, you see a bunch of stuff in the inner tracker, but only one big spike of energy in the E- and H-Cals. This means that the energy was deposited by a jet of strongly interacting particles, some of which were charged. Unfolding the cylinder that makes up the H-cal and indicating where the energy deposition was along the cylinder gives you the plot on the right: the “LEGO” plot, where height indicates the amount of energy deposited in each region of the detector. Clearly, one big jet of energy and nothing else anywhere. Balancing that jet requires the existence of some invisible particle(s), moving in the opposite direction as the visible jet.

ATLAS event display showing one monojet event recorded during the 7 TeV collisions.

Radiating off an extra jet with lots of transverse momentum is rarer than just producing dark matter by itself. But we need that jet or else we can’t see the event. However, many confounding issues make this a very technically difficult search to engage in. First, Standard Model processes can give invisible particles: neutrinos. They aren’t dark matter, but they’ll look dark matter at the LHC: that is, they look like nothing. Or, if you mismeasure that one big jet or fail to measure a bunch of little jets, you’ll think there was some imbalance of momentum that is indicative of an invisible particle. Each of these possibilities is rare, but unfortunately, so is producing dark matter, so you need to do better than “eh, probably doesn’t happen that often.”

Joe Lykken, theorist and deputy director of Fermilab, once called monojet searches the “trash collection” of particle colliders: anything that goes wrong ends up in this search. Fortunately, the experimentalists are good at their jobs, so they can deal with these problems, but it’s a lot of work.

One problem with monojet searches is that you have to demand a lot of energy in the visible jet or in the MET. If you try to lower this threshold, too many “boring” events would fall into the category of events that you’d be looking at. The LHC actually can’t store the data for each collision of protons that occurs. The proton collisions occur 20 million times a second; the LHC can only write a thousand or so events per second out to permanent storage. Fortunately, most proton-proton collisions are boring: they don’t produce Higgses, they don’t produce dark matter, they don’t do anything but throw a crapton of jets out in the detector. You might want to look at some of these events, because even “boring” physics at LHC energies can be interesting, but you don’t need to look at all of them.

What you want is a way to sift through the many pedestrian events to get the interesting ones, the ones that might contain something new and unusual. So you have to build “triggers” into ATLAS and CMS that decide which events are interesting. Unfortunately, due to the finite speed of light, the physical detectors of ATLAS and CMS can’t go ask a computer somewhere outside of the room to tell them whether an event is interesting: by time the electrical impulses crawled their way out of the room, and the answer returning, hundreds of millions more collision events would have occurred. So the triggers are hardwired into the machines themselves; the snap decision to keep an event is made by the ATLAS and CMS detectors, then passed along to higher decision-making levels until the event is finally “written to tape.”

Those hardwired triggers can’t really be changed, so we really hope that we covered all the bases for interesting physics. Part of the trade-off to trigger on dark matter type events is to demand lots of MET or a high momentum jet; otherwise ATLAS and CMS would do nothing but write “boring” events that almost look like dark matter, but aren’t.

Given this, there is motivation to find new ways to look for dark matter at the LHC. These other techniques won’t be as generic as the monojet search, since “accidentally sending off a jet of energy during a collision” is something that happens to protons regardless of the new physics that may or may not exist to create dark matter. But the other ways might be more powerful in their domain of validity.

The next thing you might try after one jet is two (two jets! ah-ah-ah). Clearly, we are thinking hard over here. But, let’s try to get clever. The problem with the monojet search is that the required transverse momentum has to be very high, which makes the events rarer than they would otherwise be. If we can use some shape of the event to pick out interesting new physics, then we might do better.

So now, finally, I can introduce vector boson fusion. The idea of VBF is that, as the protons come into the collision, two parts of the proton (cleverly called “partons,” these can be quarks, antiquarks, or gluons) have a glancing collision. Usually, when a collision happens at the LHC, the partons sort of hit “head-on” and transfer all their energy into whatever is about to be created. In VBF, the two partons come in, then radiate off force-carrying particles called “vector bosons.” Photons are vector bosons, but so are gluons (the force carriers of the strong nuclear force), and the $W$ and $Z$ bosons which carry the weak nuclear force. These two vector bosons then collide with each other (that’s the fusion part of vector boson fusion), and may create some new interesting physics. The incoming partons get slightly deflected by this exchange of the force-carrying bosons, and end up hitting the detector elements at glancing angles.

Example Feynman diagram for vector-boson fusion producing dark matter. Time increases as you read the diagram from left-to-right.

In this figure, I’m showing a toy Feynman diagram that is a sketch of what VBF looks like. Time increases as you read the picture left-to-right: two incoming quarks radiate off vector bosons (those are the wavy lines), which fuse in some interaction that I’ve drawn as a big circle, and hopefully dark matter gets spit out. If I drew a toy diagram of what this would look like in the ATLAS or CMS detector, it might look like this sketch (ignore the angles and such). The cylinder is the detector, the proton beams travel along the center line, and the radiated-off dark matter are the two red lines, while the deflected jets are the blue.

Why is this advantageous? Well, by asking for two jets, we can start looking at ways that interesting physics that is produced by VBF might look different from just generic “two jets and some missing energy.” Demanding two glancing jets with MET oriented opposite to the combination allows us to ask for much less missing momentum in the event than we could get away with in a monojet search: our trigger threshold has been lowered.

Cartoon sketch of a vector-boson fusion event, as seen in a detector like ATLAS or CMS (the cylinder). Two jets (blue) appear in the "forward" regions of the detector, while invisible dark matter (red) is produced in the "central" part of the detector. Sketch taken from my paper 1008.5151, written with Michael Ramsey-Musolf.

So maybe something that wouldn’t have gotten the attention of the ATLAS and CMS triggers in monojets would get past the VBF trigger. My experimentalist colleagues want to look, and I want to help.

So, now I can re-ask the question: what models to consider?

We looked at three general types of dark matter related physics, but I’m only going to discuss one here, since introducing the ideas behind the other two types really require separate write-ups to get all the ideas clear. (for completeness, the other two I don't talk about are a set of effective operators and a set of simplified models with connections to extended Higgs sectors)

One of the best cases to look at is producing dark matter through the Higgs boson. We have been measuring the properties of the Higgs boson carefully since it was discovered in 2012; and it appears to look like the Higgs we were expecting, no signs of dark matter or anything. However, there is still room for improvement.

The Higgs boson interacts with $W$ and $Z$ bosons, so VBF production is relatively common. If the Higgs decays invisibly (into dark matter, say), we could pick that up in this VBF search. Right now, we know that the Higgs bosons produced at the LHC can’t decay invisibly more than 25% of the time (Standard Model prediction is much less than this).

This is an indirect measurement. It wasn’t made by looking at events where the Higgs was made and then decayed into something we can’t see, it was made by looking at times where the Higgs was made and then decayed into visible stuff, and combining the results of all those searches. It is a pretty robust conclusion, but a direct measurement would be nice.

Here I’m showing our expected results: how well we could directly measure the invisible decay of the Higgs boson using VBF techniques. Right now, the inferred measurement is 25%, and lower is better: we could see rarer decays. The horizontal axis is the amount of data collected, in units of fb$^{-1}$. The 8 TeV run of the LHC collected 20 fb$^{-1}$.

Expected limits on percentage of Higgs boson decays which are invisible, using VBF searches, as a function of collected LHC data. From 1603.07739.

We had two assumptions: in the first, we assumed that we would never understand the potential errors in making the VBF measurement better than we do now (red line). If you make this assumption, eventually you run into a limit in how well you can make this measurement, even if you collect more data — you are “systematically limited.” And by sheer coincidence, this limit would allow us to measure the Higgs decaying invisibly 25% of the time, the same as our current inferred measurement. The second assumption is that we’d improve our understanding of the potential error sources as we collected more data. In that case, we’d eventually be systematically limited, but only much later. Here, we find that we could measure 5% invisible decays once the LHC had collected the “full” data set (there’s an assumption that the LHC will eventually collect 3000 fb$^{-1}$ of data, but only after many years). That’s pretty good, and it’s a direct measurement.

So, what to take from this?

VBF-type searches aren’t as generic as monojets. There are dark matter models where this would be a bad idea. However, when it works, it works really well. We can use this to make a direct measurement of Higgs properties, without having to make any theorist model-dependent assumptions, which would be much better than what is possible otherwise. If the dark matter happens to interact with the visible particles via the Higgs, than this VBF search is probably the best single measurement to find it.

So, I’m glad to know that experimentalists are going to be continuing to push this forward, and I’m glad I was helpful. Because right now, as the LHC takes data, being helpful for the experimentalists is one of the most useful things a theorist can be doing.