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Flip Tanedo | USLHC | USA

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Why do we expect a Higgs boson? Part II: Unitarization of Vector Boson Scattering

Hi everyone—it’s time that I wrap up some old posts about the Higgs boson. Last December’s tantalizing results may end up being the first signals of the real deal and the physics community is eagerly awaiting the combined results to be announce at the Rencontres de Moriond conference next month. So now would be a great time to remind ourselves of why we’re making such a big deal out of the Higgs.

Review of the story so far

Since it’s been a while since I’ve posted (sorry about that!), let’s review the main points that we’ve developed so far. See the linked posts for a reminder of the ideas behind the words and pictures.

There’s not only one, but four particles associated with the Higgs. Three of these particles “eaten” by the W and Z bosons to become massive; they form the “longitudinal polarization” of those massive particles. The fourth particle—the one we really mean when we refer to The Higgs boson—is responsible for electroweak symmetry breaking. A cartoon picture would look something like this:

The solid line is a one-dimensional version of the Higgs potential. The x-axis represents the Higgs “vacuum expectation value,” or vev. For any value other than zero, this means that the Higgs field is “on” at every point in spacetime, allowing fermions to bounce off of it and hence become massive. The y-axis is the potential energy cost of the Higgs taking a particular vacuum value—we see that to minimize this energy, the Higgs wants to roll down to a non-zero vev.

Actually, because the Higgs vev can be any complex number, a more realistic picture is to plot the Higgs potential over the complex plane:


Now the minimum of the potential is a circle and the Higgs can pick any value. Higgs particles are quantum excitations—or ripples—of the Higgs field. Quantum excitations which push along this circle are called Goldstone bosons, and these represent the parts of the Higgs which are eaten by the gauge bosons. Here’s an example:

Of course, in the Standard Model we know there are three Goldstone bosons (one each for the W+, W-, and Z), so there must be three “flat directions” in the Higgs potential. Unfortunately, I cannot fit this many dimensions into a 2D picture. 🙂 The remaining Higgs particle is the excitation in the not-flat direction:

Usually all of this is said rather glibly:

The Higgs boson is the particle which is responsible for giving mass.

A better reason for why we need the Higgs

The above story is nice, but you would be perfectly justified if you thought it sounded like a bit of overkill. Why do we need all of this fancy machinery with Goldstone bosons and these funny “Mexican hat” potentials? Couldn’t we have just had a theory that started out with massive gauge bosons without needing any of this fancy “electroweak symmetry breaking” footwork?

It turns out that this is the main reason why we need the Higgs-or-something-like it. It turns out that if we tried to build the Standard Model without it, then something very nefarious happens. To see what happens, we’ll appeal to some Feynman diagrams, which you may want to review if you’re rusty.

Suppose you wanted to study the scattering of two W bosons off of one another. In the Standard Model you would draw the following diagrams:

There are other diagrams, but these two will be sufficient for our purposes. You can draw the rest of the diagrams for homework, there should be three more that have at most one virtual particle. In the first diagram, the two W bosons annihilate into a virtual Z boson or a photon (γ) which subsequently decay back into two W bosons. In the second diagram it’s the same story, only now the W bosons annihilate into a virtual Higgs particle.

Recall that these diagrams are shorthand for mathematical expressions for the probability that the W bosons to scatter off of one another. If you always include the sum of the virtual Z/photon diagrams with the virtual Higgs diagram, then everything is well behaved. On the other hand, if you ignored the Higgs and only included the Z/photon diagram, then the mathematical expressions do not behave.

By this I mean that the probability keeps growing and growing with energy like the monsters that fight the Power Rangers. If you smash the two W bosons together at higher and higher energies, the number associated with this diagram gets bigger and bigger. If  these numbers get too big, then it would seem that probability isn’t conserved—we’d get probabilities larger than 100%, a mathematical inconsistency. That’s a problem that not even the Power Rangers could handle.

Mathematics doesn’t actually break down in this scenario—what really happens in our “no Higgs” theory is something more subtle but also disturbing: the theory becomes non-perturbative (or “strongly coupled”). In other words, the theory enters a regime where Feynman diagrams fail. The simple diagram above no longer accurately represents the W scattering process because of large corrections from additional diagrams which are more “quantum,” i.e. they have more unobserved internal virtual particles. For example:

In addition to this diagram we would also have even more involved diagrams with even more virtual particles which also give big corrections:

And so forth until you have more diagrams than you can calculate in a lifetime (even with a computer!). Usually these “very quantum” diagrams are negligible compared to the simpler diagrams, but in the non-perturbative regime each successive diagram is almost as important as the previous. Our usual tools fail us. Our “no Higgs theory” avoids mathematical inconsistency, but at the steep cost of losing predictivity.

Now let me be totally clear: there’s nothing “wrong” with this scenario… nature may very well have chosen this path. In fact, we know at least one example where it has: the theory of quarks and gluons (QCD) at low energies is non-perturbative. But this is just telling us that the “particles” that we see at those energies aren’t quarks and gluons since they’re too tightly bound together: the relevant particles at those energies are mesons and baryons (e.g.pions and protons). Even though QCD—a theory of quarks and gluons—breaks down as a calculational tool, nature allowed us to describe physics in terms of perfectly well behaved (perturbative) “bound state” objects like mesons in aneffective theory of QCD. The old adage is true: when nature closes a door, it opens a window.

So if we took our “no Higgs” theory seriously, we’d be in an uncomfortable situation. The theory at high energies would become “strongly coupled” and non-perturbative just like QCD at low energies. It turns out that for W boson scattering, this happens at around the TeV scale, which means that we should be seeing hints of the substructure of the Standard Model electroweak gauge bosons—which we do not. (Incidentally, the signatures of such a scenario would likely involve something that behaves somewhat like the Standard Model Higgs.)

On the other hand, if we had the Higgs and we proposed the “electroweak symmetry breaking” story above, then this is never a problem. The probability for W boson scattering doesn’t grow uncontrollably and the theory remains well behaved and perturbative.

Goldstone Liberation at High Energies

The way that the Higgs mechanism saves us is somewhat technical and falls under the name of the Goldstone Boson Equivalence Theorem. The main point is that our massive gauge bosons—the ones which misbehave if there were no Higgs—are actually a pair of particles: a massless gauge boson and a massless Higgs/Goldstone particle which was “eaten” so that the combined particle is massive. One cute way of showing this is to show the W boson eating Gold[stone]fish:

Indeed, at low energies the combined “massless W plus Goldstone” particle behaves just like a massive W. A good question right now is “low compared to what?” The answer is the Higgs vacuum expectation value (vev), i.e. the energy scale at which electroweak symmetry is broken.

However, at very high energies compared to the Higgs vev, we should expect these two particles to behave independently again. This is a very intuitive statement: it would be very disruptive if your cell phone rang at a “low energy” classical music concert and people would be very affected by this; they would shake their heads at you disapprovingly. However, at a “high energy” heavy metal concert, nobody would even hear your cell phone ring.

Thus at high energies, the “massless W plus Goldstone” system really behaves like two different particles. In a sense, the Goldstone is being liberated from the massive gauge boson:

Now it turns out that the massless W is perfectly well behaved so that at high energies. Further, the set of all four Higgses together (the three Goldstones that were eaten and the Higgs) are also perfectly well behaved. However, if you separate the four Higgses, then each individual piece behaves poorly. This is fine, since the the four Higgses come as a package deal when we write our theory.

What electroweak symmetry breaking really does is that it mixes up these Higgses with the massless gauge bosons. Since this is just a reshuffling of the same particles into different combinations, the entire combined theory is still well behaved. This good behavior, though, hinges on the fact that even though we’ve separated the four Higgses, all four of them are still in the theory.

This is why the Higgs (the one we’re looking for) is so important: the good behavior of the Standard Model depends on it. In fact, it turns out that any well behaved theory with massive gauge bosons must have come from some kind of Higgs-like mechanism. In jargon, we say that the Higgs unitarizes longitudinal gauge boson scattering.

For advanced readers: What’s happening here is that the theory of a complex scalar Higgs doublet is perfectly well behaved. However, when we write the theory nonlinearly (e.g. chiral perturbation theory, nonlinear sigma model) to incorporate electroweak symmetry breaking, we say something like: H(x) = (v+h(x)) exp (i π(x)/v). The π’s are the Goldstone bosons. If we ignore the Higgs, h, we’re doing gross violence to the well behaved complex scalar doublet. Further, we’re left with a non-renormalizable theory with dimensionful couplings that have powers of 1/v all over the place. Just by dimensional analysis, you can see that scattering cross sections for these Goldstones (i.e. the longitudinal modes of the gauge bosons) must scale like a positive power of the energy. In this sense, the problem of “unitarizing W boson scattering” is really the same as UV completing a non-renormalizable effective theory. [I thank Javi S. for filling in this gap in my education.]

Caveat: Higgs versus Higgs-like

I want to make one important caveat: all that I’ve argued here is that we need something to play the role of the Higgs in order to “restore” the “four well behaved Higgses.” While the Standard Model gives a simple candidate for this, there are other theories beyond the Standard Model that give alternate candidates. For example, the Higgs itself might be a “meson” formed out of some strongly coupled new physics. There are even “Higgsless” theories in which this “unitarization” occurs due to the exchange of new gauge bosons. But the point is that there needs to be something that plays the role of the Higgs in the above story.