# Why We Need the Higgs, or Something Like It

In the comments to the previous post, Monty asks a perfectly good question, which can be shortened to: “Is the Higgs boson really necessary?” The answer is a qualified “yes” — we need the Higgs boson, or something like it. That is, we can’t simply take the Standard Model as we know it and extend it to arbitrarily high energies without new physics kicking in.

The role of the Higgs field is to break the symmetry of the electroweak interactions, as discussed here. We think that there is a lot of symmetry underlying particle interactions, but that much of it is hidden from our low-energy view. In technical terms, the electroweak theory of Glashow, Weinberg and Salam posits an “SU(2)xU(1)” symmetry, that somehow gets broken down to “U(1).” That unbroken symmetry gives us electromagnetism, a force carried by a massless particle, the photon. The broken symmetries are still there, but their force-carrying particles become massive when the symmetry breaks — those are the W^{+}, W^{–}, and Z^{0} bosons.

There’s no question that something breaks the symmetry. The question that is worth asking is: “Can we imagine breaking the symmetry without introducing any new particles?”

Let’s first think about how the Higgs mechanism actually works. What we call the “Higgs field” is actually a collection of four fields that rotate into each other under the symmetry. But it’s hard to draw a picture of a four-dimensional field, so consider instead this picture of the potential for a two-dimensional field (φ_{1} and φ_{2}). Notice that there are two kinds of ways the field can oscillate: the flat direction around the circle, and the radial direction in which the potential is highly curved. For the realistic four-dimensional case, there would be three flat directions and one radial one.

All of these directions are important. At high temperatures in the early universe, the Higgs bounces around in its potential, and its average value is zero (near the origin). But at lower temperatures things settle down, and the Higgs can oscillate around some point in that circle of minimum energy. Here is a crucial point: vibrations of the field in each direction are associated with particles, and the curvature of the potential corresponds to the *mass* of the associated particle. So the flat directions are massless particles, and the curved radial direction is a massive one.

But massless particles are easy to produce, so why don’t we see all of these massless Higgs bosons? The answer is: we have! Due to their interactions with the force-carrying particles, the three kinds of particles that would be massless Higgs bosons get “eaten” by the three W and Z bosons, which in turn gives them mass. That’s the miracle of the Higgs mechanism: where you might expect three massless Higgs particles and three massless force particles, instead you just get three massive force particles. So that’s the most simple reason why we need a Higgs field or something like it: we have already observed it, in the form of the massive W and Z bosons.

So what about the radial vibrations of the Higgs field, the ones that have a large mass? Those are what we call the actual “Higgs boson,” and that’s what we’re looking for at particle accelerators.

You are welcome to imagine a theory that has the three massless Higgs particles that get eaten by the W’s and the Z, but nevertheless lacks the radial component that we’re still searching for. Indeed, you can easily construct such a theory by cranking up the mass of the Higgs — as the mass approaches infinity, the field doesn’t oscillate at all, and there’s no corresponding particle. This is known in the trade as a non-linear sigma model. So could we have a model like that explain electroweak symmetry breaking, and do without the visible Higgs boson?

No. The argument here is more subtle, but nonetheless airtight. It comes from the fact that a Higgsless version of the Standard Model would “violate unitarity,” which is a fancy way of saying it would give nonsensical predictions.

Think about the scattering of two W bosons. It’s easy to use Feynman diagrams to calculate the probability that two W’s that pass by each other will actually scatter. The problem is, the result is a quantity that gets bigger and bigger as the energy increases — without limit. In other words, the probability that two W’s will interact becomes larger than one! That can’t really happen.

The Higgs comes to the rescue. That increasing probability is what you would get if you only considered the force-carrying particles, not the Higgs. But if we allow for the Higgs, it will also contribute to W scattering. It’s contribution also grows with energy, but with the opposite sign of the troublesome contributions from the W’s and Z’s themselves! That’s the miracle of quantum mechanics — different contributions to the same final state can actually interfere with each other. So the Higgs can save the W bosons from the catastrophic result of getting probabilities that add up to more than one.

At least, it can do that if its mass is low enough that it kicks in in time. Running the numbers, we find that the Higgs mass has to be lower than 800 GeV or so in order for W scattering to make sense. That’s why the Large Hadron Collider is built to look at energies of up to 1000 GeV; it’s important to make sure we should be able to find the Higgs. (Although there are still no guarantees.)

The above argument is airtight, but its conclusion is that *something* needs to happen before 800 GeV. That something might be the Higgs, or it might be something even more exotic. It’s the closest we have to a “no-lose” theorem in physics — either the Higgs boson will be there, or something more exciting. The experiments will have the final say, as they tend to do.

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