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 forelectroweak
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 th*e *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 an*effective* 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.