# Paper Explainer: Cornering Natural SUSY at LHC Run II and Beyond

This is a blog post on my most recent paper, written with my fellow Rutgers professor David Shih, a Rutgers NHETC postdoc Angelo Monteux, and two Rutgers theory grad students: David Feld (my student) and Sebastian Macaluso (David’s student). It was a pretty big project, as the large (for a theory paper) author list indicates, and in fact the end result was split into two papers for publication, with the 2nd paper coming along shortly.

This summer, the two multipurpose LHC experiments (ATLAS and CMS) released a number of their early results from the newest data collection run (“Run II”). The biggest news was of course that they did not see the famed diphoton bump that we theorists had been collectively losing our minds over since the previous December

However, the 2nd biggest news was that they didn’t find anything else either. As I’ll explain, we have long expected to discover something at the LHC, and not finding anything yet is obviously not good news. This lack of discovery only heightened the sense in certain quarters that there was nothing to find: that the LHC was going to find the Higgs boson and nothing else. There’s been a not insignificant amount of doom and gloom in certain quarters of the theory world.

We wanted to dig into this concern a bit, and see how much the LHC was really pushing against our theory expectations that we should have discovered something already. The framework we are laboring under is supersymmetry (the “SUSY” of the title). Supersymmetry is the leading idea for new physics at the LHC, and it predicts a host of new particles: a copy of each known particle in fact. For example, the quarks have their squark partners, with the up, down, charm, strange, bottom, and top quarks each having their partners: the sup, sdown, scharm, sstrange, sbottom, and stop squarks. The electron has the selectron, the gluon the gluino, the photon, $W$, $Z$, and higgs having their photino, wino, zino, and higgsino partners. (the names are, if not the greatest part of SUSY, one of the greatest parts).

We like SUSY as theorists for a lot of reasons. One of these reasons is that SUSY can solve the “Naturalness” (or “Hierarchy”) problem. I’ve written about it before , but the general idea is that we cannot explain why the Higgs boson is as “light” as it is (only 125 GeV). The Higgs, being (as far as we can tell) an elementary spin-0 particle, should get corrections to its mass as large as the heaviest particles to which it couples. This is a problem, because we believe that there is new physics at incredibly high energy scales ($10^{16-19}$ GeV). Thus, the Higgs mass we measure should be a combination of the “bare” mass of the Higgs (some random parameter, plus these quantum corrections. “Naturally,” the sum of two numbers, one of which is extremely large, should also be extremely large. So the “Natural” mass of the Higgs is far beyond the mass we measure it at.

This is a generic problem for any version of the Higgs which is an elementary spin-0 particle. You might posit that there is no new physics at all above the energies of the Standard Model, which would certainly fix this problem, but then you can’t solve any other problem in particle physics (for example, neutrino masses or dark matter).

What SUSY does to solve this problem is introduce all these new partners of the known fields. The partners are chosen to have spins that differ by exactly 1/2 of a unit of spin from the known particles: so a spin-1/2 quark gets a spin-0 squark partner, and so on. This difference in spin means that the contribution any superpartner gives to the Higgs mass will be exactly opposite the contribution from the particle itself. Thus: the bare mass of the Higgs will be protected, and can be close to the small number we measure it as. Ta-da, if Nature is supersymmetric, Naturalness is solved.

However, that can’t be the full answer. We don’t see a spin-0 selectron floating around with the same mass as the electron. And the LHC now doesn’t see gluinos with masses below roughly at TeV. So SUSY, if it exists, is a broken symmetry. All the superpartner masses must be shifted up, heavier than the light particles we know and love.

And here’s where the problem comes in. The LHC is very effective at looking for certain superpartners: the ones charged under the strong nuclear force (i.e. the colored superpartners, the squarks and the gluinos). As the LHC continues to look and not see these particles, it sets heavier and heavier lower limits on their masses. These lower limits mean that supersymmetry starts reintroducing the Naturalness problem: the cancellation between the particles and their superpartners is no longer perfect, and so it is harder and harder to have a light Higgs but not have light superpartners.

We quantify this “difficulty” through a tuning measure, which is basically asking “how carefully chosen would the superparticle and bare Higgs masses have to be in order to get a Higgs mass of 125 GeV?” The heavier the superpartners are, the more delicate this balance is: you have to caaaarefully subtract off two big numbers to get just a small remainder (the Higgs mass) at the end of the day. (Note: really the tuning is done with masses squared, rather than masses. If you know what this means, you already knew this, if you don’t, don’t worry about it.)

The next paper out will address how we calculated this tuning measure. David Shih has spent a lot of time thinking about these issues, and he noted a number of simplifications that have been made over the years that - it turns out - overestimate the amount of tuning needed. I’ll leave off discussion of that for the next paper, but the punch line is this: we had reason to expect that SUSY with a given set of superpartner masses could be less tuned than typically assumed.

For this paper, the primary facts you need to know are as follows. First, the tuning of the Higgs mass cares most about the mass of three particles: the Higgs partner the Higgsino, the top partner the stop, and the gluon partner the gluino. That the mass of the Higgsino and the Higgs are related should come as no surprise, supersymmetry wants them to be the same after all, before it was broken at any rate. Small tuning requires a pretty light Higgsino, but unfortunately Higgsinos are not strongly interacting particles, so we can’t produce them directly. The top quark is the heaviest quark, which means it has the largest interaction with the Higgs. So it and its partner, the stop, directly affect the Higgs parameters the most, thus the need for a light stop. The gluino doesn’t affect the Higgs, but it does affect the stop mass, so for a light stop, we also want a light gluino. So, at minimum, Natural SUSY should have a light gluino, stop, and higgsino.

The 2nd fact you need is that that the tuning measure depends on the scale at which supersymmetry breaking is communicated to the superpartners, the so-called messenger scale $\Lambda$. Larger $\Lambda$ tends to require more tuning for a given set of masses, as you’ll see.

So the name of the game is to pick a level of tuning (we picked 10%, meaning that you need to balance the bare masses and the superpartner masses only to 1 part in 10. This is an arbitrary choice, but it serves as a useful benchmark), and figure out what superparticle masses are required in order to have this level of tuning. Then, we need to see if the LHC experiments could have seen such superparticles with masses at or below the values we just calculated. If the LHC could have, and didn’t, then “Natural SUSY” at 10% tuning is ruled out, which would be seen as bad news. If the LHC couldn’t see such particles, then that’s interesting, it means that the Grand Old Model of supersymmetry is still not completely covered by experiments, even in the most attractive “Natural” region.

So first we had to cook up models. The first one was just supersymmetry in all its usual glory. Or, at least the most simple version thereof. We called this “vanilla” supersymmetry, and it just had the gluinos at one mass and all the squarks at another mass. The Higgsino is the lightest SUSY particle, and it is invisible in our detector, and stable. So what we are looking for is the production of all these strongly interacting particles which then decay down to Higgsinos and lots of quarks and gluons, resulting in events with lots of visible “jets” of energy and two invisible particles which show up as an imbalance in the summed momentum of the visible particles (this is called Missing Transverse Momentum — or Missing Transverse Energy — MET).

The LHC didn’t set limits on this model exactly, they actually set limits on various Simplified Models where there was at most two or three superparticles capable of being produced in the LHC collisions. Thus, to set limits on our vanilla SUSY model, we had to “recast” (or reinterpret) the LHC results: we generated fake events that simulate the LHC collisions, used fake detector simulators, and coded up fake versions of the analysis the experimentalists themselves developed to find new physics.

We did this with 8 different LHC searches, and this is where most of the work actually occurred: generating millions of fake LHC events, writing the code, validating it, and matching to the public results. Notably, we didn’t actually recast any of the searches for the partner of the top quark: the stop squark. We are tackling these now, and basically we omitted them because they are more finicky, specialized searches. It also turns out that, for our purposes, we didn’t need them, we can get very strong bounds regardless.

Our results for vanilla SUSY are here: we’re showing the regions of stop and gluino masses we can exclude, assuming the Higgsino is as heavy as possible (lighter Higgsinos turn out to be easier to find). The grey regions are the regions already ruled out by “Run I” data. The various curved colored lines are the lower limits on viable masses set by the different searches we reanalyzed. The colored green regions are the regions of less than 10% tuning, assuming a messenger scale of $\Lambda = 20$ TeV or 100 TeV. These are very low messenger scales, by the way, and even so, we see that vanilla SUSY is totally excluded anyway.

The problem with vanilla SUSY is that there are too many strongly interacting superpartners being produced: all the partners of all the quarks. We don’t really need them for Naturalness, we just need the gluino and the stop. So we next consider “Effective SUSY,” which is vanilla SUSY with every superparticle other than the gluino, stop, and Higgsino made heavier than anything the LHC could produce (we picked 5 TeV). This reduces the rate at which new particles would be produced by up to a factor of 10, which makes “hiding” Natural SUSY a bit easier.

Here’s the same type of plot I showed previously, but for Effective SUSY. You can see that there are regions of 10% tuning which are not yet ruled out by the LHC. Not big regions, but they’re there: Natural SUSY isn’t dead yet.

Now, those two models we picked have something in common: lots of invisible momentum carried away by the Higgsinos. That’s exactly the signature the LHC people have been working hard to find, and they are very good at it. But what if we throw them for a loop, and dilute that missing energy away by letting the Higgsino decay into some other particles. We picked two models that can do this: RPV SUSY and Stealth/Hidden Valley SUSY. The details are not important right now, just know that both pick different ways to convert the energy carried by the Higgsino into visible particles. You might think this is worse, in terms of hiding a signal, but in fact, events with lots and lots of jets of energy at the LHC are very common. Events with lots of MET are rarer. So trading MET for jets is a viable strategy, and here are the limits.

Again, you see we have a region of untuned parameter space that the LHC hasn’t gotten to yet.

Finally, we take two great tastes and make them go great together: combine RPV SUSY and Effective SUSY, both of which were successful at allowing Natural SUSY to evading existing LHC bounds. Combined, there is lots of parameter space left.

So now what? Well, we can ask what messenger scale would be allowed for a given value of tuning, which I show here for the models we considered. The parameter $\Delta$ is one over the tuning amount, so 10% tuning is $\Delta = 10$. Higher $\Delta$ is more tuning, and thus “less Natural” SUSY. We see that any model requires very low messenger scales if you want to be Natural. That’s a model-building hurdle that must be overcome, and might push us in particular directions about what is a good idea for SUSY at this stage in the game.

We can also ask about what advances we might see at the LHC over the next years and decades. The results this summer used about 12-15 fb$^{-1}$ of data (see here for a discussion of the unit involved). Right now, ATLAS and CMS have 30-40 fb$^{-1}$ on tape, but not analyzed. In the next few years, we expect 300 fb$^{-1}$ to be collected, and long term, the “High Luminosity” LHC will collect 3000 fb$^{-1}$. What does such data gain us?

Using a very simple extrapolation, we find that, for particles which currently have a lower limit from the LHC of 1-3 TeV, we might expect to see a 1.2 TeV increase in experimental reach over the life of the LHC. This is not an advance we get for free: the assumptions we made do require continual effort on the part of the experimentalists to improve their searches in a steady way (this also doesn’t account for surprising new techniques that beat expectations). With these advances, we can completely exclude 10% Natural SUSY (or find it), at the LHC.

In the meantime, we should be thinking about models which can be Natural with these low messenger scales, and thinking about gaps in our coverage which we didn’t think of in this paper. We should also not lose hope: the LHC is doing great work, but we are not yet in the era when we can say with confidence that Natural SUSY is dead. That day may come, and we certainly need new ideas, but the LHC is here, the results it is producing are very interesting, and there is lots of work to be done.