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### Why do we expect a Higgs boson? Part I: Electroweak Symmetry Breaking

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In recent posts we’ve seen how the Higgs gives a mass to matter particles and force particles. While this is nice, it is hardly a requirement there must be a Higgs boson—maybe particles just happen to have mass and there’s no “deeper” origin of that mass. In fact, there’s a different reason why particle physicists are obsessed about finding the Higgs (or something like it)—that’s called electroweak symmetry breaking.

The statement that we’d like to understand is the following:

The Higgs boson breaks electroweak symmetry spontaneously.

That’s pretty heady stuff, but we’ll take it one piece at a time. Write it down and use it to impress your friends. Just be sure that you read the rest of this post so you can explain it to them afterward. (There’s a second part to the statement that we’ll examine in a follow up post.)

Electroweak symmetry

You might be familiar with the idea that electricity and magnetism are two manifestations of the same fundamental force. This is manifested in Maxwell’s equations and is often seen written on t-shirts worn by physics undergraduates. (If you happen to own such a t-shirt, I refer you to this article.) Electroweak symmetry is, in a sense, the next step in this progression, by which the electromagnetic force is unified with the weak force. This unification into an ‘electroweak’ theory and the theory’s subsequent ‘breaking’ into separate electromagnetic and weak forces led to the 1979 Nobel Prize in Physics.

So what’s going on here? We know that the force particle for electromagnetism is the photon, and we know that the force particles for the weak force are the W+, W-, and Z bosons. Permit me to make the a priori bold claim that the “unified” set of particles are actually the following: three W bosons and something we’ll call a B boson.

What? Now there are three W particles? And what’s this funny B boson; we never drew any diagrams with that weirdo in our guide to Feynman diagrams! Don’t worry, we’ll see shortly that because of the Higgs, these particles all mix up into the usual gauge bosons that we know and love. This should at least be plausible, since there are four particles above which we know must give us the four electroweak particles that we know: the W+, W-, photon, and Z.

Note that this new “unified” batch of gauge bosons don’t really look very unified: The Ws look completely different from the B. This illustration reflects an actual physical difference: the Ws mediate one type of force while the B mediates a different force. In this sense, the “unified” electroweak symmetry isn’t actually so unified!

Remark: The next natural step in unifying the forces would be  to actually unify the W and B particles with one another. In fact, mathematically one can find ways to combine the B, all three Ws, and all eight gluons in what is referred to as a grand unified theory (GUT). The next step beyond this would be to unify those forces with gravity, which is referred to in popular literature as a `theory of everything.’ Unlike electroweak unification, however, there’s no reason to suspect that either of these phenomena should be accessible at the TeV scale.
Technical remark: mathematically the unification of forces falls under the representation theory of continuous groups (or rather, their algebras). The electroweak group is the product SU(2) × U(1). Note that SU(2) has three generators—this is precisely why there are three W bosons.

## Electroweak symmetry  is broken

In everyday phenomena, we observe electricity and magnetism as distinct phenomena. The same thing happens for electromagnetism and the weak force: instead of seeing three massless Ws and a massless B, we see two massive charged weak bosons (W+ and W-), a massive neutral weak boson (Z) and a massless photon. We say that electroweak symmetry is broken down to electromagnetism.

Now that masses have come up you should suspect that the Higgs has something to do with this. Now is a good time to remember that there are, in fact, four Higgs bosons: three of which are “eaten” by the weak gauge bosons to allow them to become massive. It turns out that this “eating” does more that that: it combines the ‘unified’ electroweak bosons into their ‘not-unified’ combinations!

The first two are easy; the W1 and W2 combine into the W+ and W- by “eating” the charged Higgs bosons. (Technically we should now call them “Goldstone” bosons.)

We’ll say a bit more about why eating a Higgs/Goldstone can cause the W1 and W2 particles to combine into, say, a W+. For now, note that the number of “degrees of freedom” match. Recall that ‘degree of freedom’ roughly translates in to the number of distinct particle states. In the electroweak theory we have two massless gauge bosons (2 × 2 polarizations = 4 degrees of freedom) and two charged Higgses (2 degrees of freedom) for a total of six degrees of freedom. In the broken theory, we have two massive gauge bosons (2 × 3 polarizations) which again total to six degrees of freedom.

A similar story goes through for the W3, B, and H0 (recall that this is not the same as the Higgs boson, which we write with a lowercase h). The W3 and B combine and eat the neutral Higgs/Goldstone to form the massive Z boson. Meanwhile, the photon is the leftover combination of the W3 and B. There are no more Higgses to eat, so the photon remains massless.

It’s worth noting that the Ws didn’t combine into charged Ws until electroweak symmetry breaking. This is because [electric] charge isn’t even well-defined until the electroweak theory has broken to electromagnetic theory. It’s only after this breaking that we have a photon that mediates the force that defines electric charge.

## Electroweak symmetry is broken spontaneously

Alright, we have some sense of what it means that “electroweak symmetry” is broken. What does it mean that it’s broken spontaneously, and what does this whole story have to do with the Higgs? Now we start getting into the thick of things.

The punchline is this: the Higgs vacuum expectation value (“vev” for short) is what breaks electroweak symmetry. You might want t quickly review this post where we first introduced the Higgs vev in the context of particle mass. For those who like hearing fancy physics-jargon, you can use the following line:

The Higgs vev is the order parameter for electroweak symmetry breaking.

First, let’s see why the Higgs obtains a vacuum expectation value at all. We can draw nice pictures since the vev is a classical quantity. The potential is a function that tells you the energy of a particular configuration. You might recall problems in high school physics where you had to find the minimum of an electric potential, or determine the gravitation potential energy of a rock being held at some height. This is pretty much the same thing: we would like to draw the potential of the Higgs field. (To be technically clear: this is the potential for the combined bunch of four Higgses.)

Let’s start with what a “normal” potential looks like. Here on the x and y axes we’ve plotted the real and imaginary parts of a field ϕ; all that’s important is that a point on the x-y plane corresponds to a particular field configuration. If the particle is sitting at the origin (in the middle) then it has no vacuum expectation value, otherwise, it does obtain a vacuum expectation value.

On the z axis we draw the potential V(ϕ). The particle wants to roll to the minimum of the potential, so in the cartoon above—the “normal” case—the particle obtains no vacuum expectation value. I’ll mention in passing that concave of the potential is related to the particle’s mass.

Now let’s examine what the Higgs potential looks like. Physicists refer to this as the “Mexican hat” potential (These images are based on an illustration that is often used in physics talks. Unfortunately I am unable to find the original source of this graphic and ended up re-drawing it.):

What we observe is that the origin is no longer a minimum of the potential. In other words, the Higgs wants to roll down the hill where it can have lower potential energy. I’m not telling you why the potential is shaped this way (there are a few plausible guesses), and within the Standard Model this is an assumption about the Higgs.

So the Higgs must roll off of its hill into the ravine of minimum potential energy. This happens at every point in spacetime, meaning that the Higgs vev is “on” everywhere and matter particles can bounce off it to obtain mass. There’s something even more important though: this vev breaks electroweak symmetry.

In the cartoons above, there’s something special about the origin. If the particle sits at the origin, you can do a rotation about the x-y plane and the configuration doesn’t change. On the other hand, if the particle is off of the origin, then doing a rotation will send the particle around along a circular trajectory (shown as a solid green line). In other words, the rotational symmetry is broken because the physical configuration changes.

The case of electroweak symmetry is the same, though it requires more dimensions than we can comfortably draw. The point is that there are originally four Higgses which are all parts of a single “Higgs.” In the unified theory where electroweak symmetry is unbroken, these four Higgses can be rotated into one another and the physics doesn’t change. However, when we include the Mexican hat potential, the system rolls into the bottom of the Mexican hat: one of the Higgses obtains a vev while the others do not. Performing a “rotation” then moves the vev from one Higgs to the others and the symmetry is broken—the four Higgses are no longer being treated equally.

Now to whet your appetite for my next post: you can see that once electroweak symmetry is broken, there is a “flat direction” in the potential (the green circle). Remember when I said that the concave of the potential has to do with the particle’s mass? The fact that there is a flat direction means that there are massless particles. In fact, for the Higgs, there are three flat directions that correspond to—you guessed it—the three massless Higgs/Goldstone particles which are eaten by the weak gauge bosons: the H+, H-, and H0. The fourth Higgs—the particle that we usually call the Higgs—corresponds to an excitation in the radial direction where there is a concave, so the Higgs boson has mass.

## Do we really need a Higgs?

Okay, so if you’ve followed so far, you have an idea of how electroweak symmetry breaking explains how the massless W and B bosons combine with the Higgses to form the usual W+W-Z, and photon. We’ve also reviewed how matter particles get mass (by bumping into the resulting vev) and how some of those gauge bosons got mass (by eating some of the Higgses). But was all of this necessary, or did we just cook it all up because we liked the idea of electroweak unification?

We will see in one of my follow up posts that in fact, electroweak symmetry breaking is almost necessary for our theory to make sense. (I’ll quantify the “almost” when we get there, but the technical phrase will be “perturbative unitarity.”) Note that I said that electroweak symmetry breaking is the important thing. Throughout this entire post you could have replaced the Higgs boson with “something like it.” There are plenty of theories out there with multiple Higgs bosons, no Higgs bosons, or generically Higgsy-things-but-not-quite-the-Higgs. That’s fine—in all of these theories, the “Higgsy-thing” always breaks electroweak symmetry. In doing so, you always end up with Goldstone bosons that are eaten by the W+W-, and Z. And you always end up with some kind of particle like the Higgs that we expect to find at the LHC.

## One last request: vote to support this blog

Hi everyone, if you liked this post (or any of my other posts, e.g. the Feynman diagram series) I’d like to ask you to vote for me (Philip Tanedo) for the 2011 Blogging Scholarship. The voting goes on for about another week and you can vote once per day. If you re-blog any of my posts, it would mean a lot if you could encourage your readers/friends/Facebook friends, etc. to also vote for me. For the past two years I’ve been able to blog due to support from the National Science Foundation and the Paul and Daisy Soros foundation, but without additional support like the Blogging Scholarship for next year I would be unable to continue with US LHC / Quantum Diaries.