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.)

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”).