# A different presentation of the Higgs

There have been several very clever attempts to explain the Higgs to a general audience using analogies; one of my favorites is a CERN comic based on an explanation by David Miller. Science-by-analogy, however, is a notoriously delicate tightrope to traverse. Instead, we’ll take a different approach and jump straight into the physics. We can do this because we’ve already laid down the ground work to use Feynman diagrams to describe particle interactions.

In the next few posts we’ll proceed as we did with the other particles of the Standard Model and learn how to draw diagrams involving the Higgs. We’ll see what makes the Higgs special from the diagrammatic point of view, and then gradually unpack the deeper ideas associated with it. The approach will be idiosyncratic, but I think it is closer to the way particle physicists really think about some of the big ideas in our field.

This first post we’ll start very innocently. We’ll present simplified Feynman rules for the Higgs and then use them to discuss how we expect to produce the Higgs at the LHC. In follow-up posts we’ll refine our Feynman rules to learn more about the nature of mass and the phenomenon called electroweak symmetry breaking.

# Feynman Rules (simplified)

First off, a dashed line represents the propagation of a Higgs boson:

You can already guess that there’s something different going on since we haven’t seen this kind of line before. Previously, we drew matter particles (fermions) as solid lines with arrows and force particles (gauge bosons) as wiggly lines. The Higgs is indeed a boson, but it’s different from the gauge bosons that we’ve already met: the photon, W, Z, and gluon. To understand this difference, let’s go into a little more depth on this:

• Gauge bosons, things which mediate “fundamental” forces, carry angular momentum, or spin. Gauge bosons carry one unit of spin; roughly this means if you rotate a photon by 360 degrees, it returns to the same quantum mechanical state.
• Fermions, matter particles, also carry angular momentum. However, unlike gauge bosons, they carry only half a unit of spin: you have to rotate an electron by 720 degrees to get the same quantum state. (Weird!)
• The Higgs boson is a scalar boson, which means it has no spin. You can rotate it by any amount and it will be the same state. All scalar particles are bosons, but they don’t mediate “fundamental” forces in the way that gauge bosons do.

This notion of spin is completely quantum mechanical, and it is a theorem that any particle with whole number spin is a boson (“force particle”) and any particle with fractional spin is a fermion (“matter particle”). It’s not worth dwelling too much about what kind of ‘force’ the Higgs mediates—it turns out that there are much more interesting things afoot.

Now let’s ask how the Higgs interacts with other particles. There are two Feynman rules that we can write down right away:

Here we see that the Higgs can interact with either a pair of fermions or a pair of gauge bosons. This means, for example, that a Higgs can decay into an electron/positron pair (or, more likely, a quark/anti-quark pair). For reasons that will become clear later, let’s say that the Higgs can interact with any Standard Model particle with mass. Thus it does not interact with the photon or gluon, and for argument’s sake we can ignore the interaction with the neutrino.

The interaction with fermions is something that we’re used to: it looks just like every other fermion vertex we’ve written down: one fermion coming in, one fermion coming out, and some kind of boson. This reflects the conservation of fermion number. We’ll see later that because the Higgs is a scalar, there’s actually something sneaky happening here.

Finally, the Higgs also interacts with itself via a four-point interaction: (This is similar to the four-gluon vertex of QCD.)

There are actually lots of subtleties that we’ve not mentioned and a few more Feynman rules to throw in, but we’ll get to these in the next post when we will see what happens with the Higgs gets a “vacuum expectation value”. Please, no comments yet about how I’m totally missing the point… we’ll get to it all gradually, I promise.

# Higgs Production

Thus far all we’ve been doing is laying the groundwork in preparation for a discussion of the neat things that make the Higgs special. Even before we get into that stuff, though, we can use what we’ve already learned to talk about how we hope to produce the Higgs at the LHC. This is an exercise in drawing Feynman diagrams. (Review the old Feynman diagram posts if necessary!)

The general problem is this: at the LHC, we’re smashing protons into one another. The protons are each made up of a goop of quarks, antiquarks, and gluons. This is important: the protons are more than just three quarks! As we mentioned before, protons are terribly non-perturbative objects. Virtual (anti-)quarks and gluons are being produced and reabsorbed all over the place. It turns out that the main processes that produce Higgs bosons from proton collisions comes from the interaction of these virtual particles!

One of the main “production channels” at the LHC is the following gluon fusion diagram:

This is kind of a funny diagram because there’s a closed loop in the middle. (This makes it a very quantum effect… and somewhat more tricky to actually calculate.) What’s happening is that a gluon from one proton and a gluon from the other proton interact to form a Higgs. However, because the gluons don’t directly interact with a Higgs, they have to do so through quarks. It turns out that the top quark—which is heaviest—has the strongest interaction with the Higgs, so the virtual quarks here are tops.

Another way to get a Higgs is associated production with a top pair. The diagram looks like this:

Here gluons again produce a Higgs through top quarks. This time, however, a top quark and an anti-top quark are also produced along with the Higgs. We can draw a similar diagram without the gluons:

This is called vector fusion, because virtual W or Z bosons produce a Higgs. Note that we have two quarks being produced as well.

Finally, there is associated production with a W or Z. As homework you can fill in the particle labels assuming the final gauge boson is either W or Z:

There are other ways of producing a Higgs out of a proton-proton collision, but these are the dominant processes. While we know a lot about the properties of a Standard Model Higgs, we still don’t know its mass. It turns out that the relative rates of these processes depends on the Higgs mass, as can be seen in the plot below (from the “Tevatron-for-LHC” report):

The horizontal axis is the hypothetical HIggs mass, while the vertical axis measure the cross section for Higgs production by the various labeled processes. For our purposes, the cross section is basically the rate at which these processes occur. (Experimentally, we know that a Standard Model Higgs should have a mass between about 115 GeV and 200 Gev.) We can see that the gg → h is the dominant production mechanism throughout the range of possible Higgs masses—but this is only half of the story.

We don’t actually directly measure the Higgs in our detectors because it decays into lighter Standard Model particles. The particular rate at which it decays to different final states (“branching ratios”) are plotted above, image from CDF. This means we have to tell our detectors to look for the decay products of the Higgs in addition to the extra stuff that comes out of producing the Higgs in the first place. For example, in associated production with a top pair, we have gg → tth. Each of the tops decay into a b quark, a lepton, and a neutrino (can you draw the diagram showing this?), while the Higgs also decays—say, into a pair of b quarks. (For now I’m not distinguishing quarks and anti-quarks.) This means that one channel we have to look for is the rather cumbersome decay,

gg → tth →blν blν bb

Not only is this a lot of junk to look for in the final state (each of the b quarks hadronizes into a jet), but there are all sorts of other Standard Model processes which give the same final state! Thus if we simply counted the number of “four jets, two leptons, and missing energy (neutrinos)” events, we wouldn’t only be counting Higgs production events, but also a bunch of other background events which have nothing to do with the Higgs. One has to predict the rate of these background events and subtract them from the experimental count. (Not to mention the task of dealing with experimental uncertainties and possible mis-measurements!)

The punchline is that it can be very tricky to search for the Higgs and that this search is very dependent on the Higgs mass. This is why we may have to wait a few years before the LHC has enough data to say something definitive about the Higgs boson. (I’ve been somewhat terse here, but my main point is to give a flavor of the Higgs search at the LHC rather than explain it in any detail.)

As a single concrete example, consider the gluon fusion production channel, gg → h. This seems nice since there’s no extra particles in the production process. However, from the plot above, we can see that for relatively light masses (less than 140 GeV) the Higgs will want to decay into b quarks. This is no good experimentally since the signal for this has hopelessly large background from non-Higgs events.

In fact, rather counter intuitively, that one of the best ways to use gluon-fusion to search for a light-mass Higgs is to look for instances where it decays into a pair of photons! This is really weird since the Higgs doesn’t interact directly with photons, so this process must occur through virtual quarks, just like the Higgs-gluon coupling above. As the branching ratio chart above shows, this is a very rare process: the Higgs doesn’t want to decay into photons very often. However, the upshot is that there aren’t many things in the Standard Model which can mimic this “two photon” signal so that there is very little background. You can see that this stops working if the Higgs is too heavy since the decay rate into photons shrinks very quickly.

# Next time

In our next post we’ll introduce a completely new type of Feynman rule representing the Higgs “vacuum expectation value.” In doing so we’ll sort out what we really mean when we say that a particle has mass and continue our march towards the fascinating topic of electroweak symmetry breaking (“the Higgs mechanism”).

• It is difficult to find a black cat in a dark room if it there isn’t present.

• TimG

Thanks for another great post, Flip! I’m a little confused about why there’s a large background for b anti-b, but a small background for the two photon decay. Couldn’t b anti-b decay into two photons, or for that matter any particle/antiparticle pair could decay into two photons? My semi-educated guess is that things would rather decay into quarks than photons since the strong force is stronger, but even if that’s right I’m not sure why the Higgs going to two photons wouldn’t be just as suppressed as everything else.

• gunn — it makes me very happy that you say this, even though I suspect you meant it half-jokingly. One of the really important things that I want to get into is why we believe that the Higgs *or something like it* should exist and should be observable at the LHC. Indeed, there are many speculative alternatives to the Standard Model Higgs boson—but there are very good reasons to expect that *something* needs to do what the Standard Model Higgs does (“electroweak symmetry breaking”). This is actually the very exciting part of the story.

TimG — I’m not an expert on this, so I invite others to chime in and correct me. However, the reason is more or less what you have said: decaying into “hard” (high energy, so that they can be see in our detectors) photons tends to be suppressed because of the smallness of the electromagnetic coupling.

Let me flesh this idea out a bit more. Typically in a proton-proton collision, only two of the colliding things are actually very high energy. The other stuff (nonperturbative junk) just sprays off along in the beam direction and is ignored.

The great thing about the two photon signal is that the two high energy quarks produce two high energy photons. (Since photons are massless, all of the Higgs mass must be converted into photon ‘kinetic’ energy.) The point is that there are *no* other high energy particles in this process. Thus this signal, even though it happens very rarely (due to the weakness of the electromagnetic force), is very clean.

In contrast, if we have something where we produce b quarks, these things decay into jets. (See the end of this post: http://blogs.uslhc.us/qcd-and-confinement ) Jets are kind of annoying signals because all sorts of processes produce jets. For example, we could have quark-quark -> gluon -> quark quark. This ‘background’ process happens all the time and gives a very similar signal to the higgs-decay-into-b quarks. Thus one has to subtract away all of these non-Higgs processes. (Which means one has to accurately *calculate* the rate for these processes… which is non-trivial.)

A good question, then, is to ask what the background is for the high-energy two photon signal from non-Higgs process. For example, you could imagine a gluon decaying into two photons via a quark loop (the same trick we used to get the Higgs to couple to photons). In this case, however, the intermediate state is massless so that the photons wouldn’t be high energy.

Does this help? (I apologize for the somewhat discombobulated response, I’m currently traveling.) If not, let me know and I’ll try to write something a bit more clearly after conferring with some of my experimentalist colleagues!

Best,
F

• TimG

Thanks, Flip, that’s actually quite helpful. I do have a follow up question, though. You point out that we can tell it’s a Higgs and not a gluon decaying to two photons, because the gluon wouldn’t produce such high energy photons. (I’m guessing this means in particular with the gluon decay you wouldn’t see a sharp peak in the two photon signal around some mass in the expected range for the Higgs mass.) But how can we tell it’s the Higgs we’ve discovered, and not some other new particle that happens to have a mass in the same range? E.g. a heavy gluon, if such a thing is possible?

• Hi Tim—in retrospect, I suspect my original response might have been a little misleading. In the next week or so I’ll try to write up a follow up post about the diphoton signal for the Higgs. (It turns out that this signal is one of the main goals of the CMS detector!)

Cheers,
F

• Mario.

“…and for argument’s sake we can ignore the interaction with the neutrino.”