Last time I posted, we looked at the “Eightfold way” classification of mesons. We argued that this is based only on **symmetry** and allowed physicists in the 60s to make meaningful predictions about mesons even though mesons are ultimately complicated “non-perturbative” objects where quarks and anti-quarks perform an intricate subatomic ‘dance’ (more on this below!).

# Historical models of mesons

In fact, physicists even developed theories of mesons as *fundamental* particles—rather than bound states of quarks—which accurately described the observed light meson masses and interactions. These theories were known as “**phenomenological**” models, **c****hiral perturbation theory**, or **nonlinear sigma models**. These are all fancy names for the same idea.

**little Higgs models**. In these models the Higgs boson is relatively light due to a mechanism called

**collective symmetry breaking**in which multiple symmetries must be broken to generate a Higgs mass. (For technical introductions for physicists, see here and here.) This idea that light particles come from broken symmetries has its origin in “phenomenological” models of mesons via the Goldstone mechanism.

From a formal point of view these models suffered a theoretical sickness: while they agreed well with experiment at *low* energies, they didn’t seem to make much sense if you used them to calculate predictions for *high* energies. It’s not that the predictions didn’t match with experiments, it’s that theory seemed to make no predictions! (Alternately, its predictions were nonsense.) The technical name for this illness is **non-renormalizability**, and it was American Nobel Laureate Ken Wilson who really clarified the correct way to understand these theories.

**Ken Wilson**(b. 1936) may not have the public fame as Richard Feynman or Robert Oppenheimer, but he is without a doubt one of the great American theoretical physicists of the century. His research focused on the theoretical framework of quantum field theory and its applications to both particle physics and condensed matter physics. He was one of the great thinkers of our field who really understood the “big idea,” and I think he is nothing short of a hero of modern physics.

Rather than going into the precise sense in which a non-renormalizable theory is a ‘sick’ theory, let’s emphasize Wilson’s key insight: these sick theories are fine as long as we are careful to ask the right questions. Wilson made this statement in a much more mathematically rigorous and elegant way—but in this post we’ll focus on getting the *intuition* correct.

# Effective theories

The point is that these “non-renormalizable” theories are just approximations for the behavior of a more fundamental theory, which we call an **effective theory **(here’s a very old post on the big idea). These approximations get the “rough behavior” correct, but doesn’t sweat the details. If you then try to ask the approximate theory something about the details that it neglects, then it gives you a gibberish response. Wilson taught us how to understand the gibberish as the theory saying, “*I’m not sophisticated enough to answer that*!”

Here’s a concrete example. One of my previous posts presented a pixelated image of the Mona Lisa to demonstrate “lattice QCD.” (This is actually exactly the effective theory that Ken Wilson was working on.)

The pixelated Mona Lisa is an “effective” image with details blurred out compared to the “fundamental” image. Even with these details removed, from far away the images look the same. In fact, the effective image is sufficient to answer questions like

- What is the overall color of the image or of different patches of the image? (Beige/brown)
- How many figures are in the image? (One… but keep this in mind for later)

On the other hand, effective Mona Lisa is completely unequipped to answer more subtle questions like

- Where is the Mona Lisa looking?
- Is the Mona Lisa happy or sad?

Okay, arguably even art historians can’t come up with answers to those questions. But the point is that the pixelated image can’t even begin to try to answer them—the questions ask about details that were left out of the “effective” image. Such questions are outside of the **domain of validity** of the effective image.

Now here’s a very important lesson in particle physics:

Modelsof particle physics also have a domain of validity, beyond which they are ill equipped to make sensible predictions.

For some models, like the effective theories of mesons, asking questions outside of the model’s domain of validity leads to nonsense answers. On the other hand, within the domain of validity the models are perfectly predictive. In fact, different “effective models” have to agree when their domains of validity overlap. Here’s an example from an old post where classical electromagnetism is an effective theory for quantum electrodynamics, as manifested by the formula for the electric field.

# Dancing with the quarks

Now let’s get back to mesons, albeit though an analogy. We know that a pion is really a quark–anti-quark caught up in a subatomic dance. They spin about one another, exchange gluons, and can even interact with other particles as a joint entity. Here’s a rough picture:

But here’s the thing: that’s the picture that we see only if we can really look *very* closely and observe the quarks directly. This requires having front row seats at “Dancing with the Quarks” (or at least an HDTV). For someone who can only watch the broadcast at low resolution, the dance looks very different: everything is blurred out:

In fact, this is now just like the case of the pixelated Mona Lisa. Note that because the quarks are *so* meticulously coordinated, the blurry picture looks like there’s only *one* object dancing! We call that object a pion and we can make careful measurements of how it spins and interacts… all without knowing that if we only had better resolution we would actually see *two *quarks dancing in unison rather than one pion.

Things brings us back to the state of particle physics in the 1960s. We can create an entire **effective theory** to describe the pion, but we have to accept that we’ve put on our fuzzy glasses and can’t make out any details. We can’t ask our effective theory something like “how many hands are in the picture above?” Well, it looks like two… but it’s hard to be sure. I could ask an even more difficult question: what is the gender of the dancers in the picture above? Now the effective theory completely falls apart. Any answer that it can give must be manifestly wrong because it doesn’t even know that there are two dancers, much less the particular gender of either. In the same way, the effective theories of mesons seemed to fall apart when you asked questions about energies higher than their regime of validity.

# Modern Effective Theories

Let me end by remarking that even though the underlying goal of high energy physics is to probe nature at a *fundamental* level, effective theories are *still* incredibly useful tools.

**Matching theories to low-energy experiments**. It is often the case that theories of exotic new particles at high energies are constrained by experiments that are conducted at much lower energies. For example, many models of new physics are limited by how they would affect the physics of ordinary*W*and*Z*bosons. By writing an effective theory of*W*and*Z*bosons that parameterizes the effect of new physics, we can provide robust bounds on the properties of whatever new particles appear at high energies. (For experts: these are the electroweak precision constraints, see hep-ph/0405040, hep-ph/0412166, hep-ph/0604111) The analogy to the dancing quarks is to use the blurry picture to tell us that, “*I don’t know how many hands there are, but if there are more than two, then they have to be pretty close to one another*.” (For experts: this approach has recently been applied to direct detection of dark matter.)- “
**Phenomenological models**.” In the previous case we simplify a calculation of a fundamental theory by working with an effective theory; this is a top-down approach. We can also consider the bottom-up approach where we write down a model that describes known low-energy physics and figure out at what energy it breaks down. We can then predict that there should be some new physics not encapsulated in our model appearing at those energies. This is where we are with particle physics: we have observed a bunch of neat particles and measured their properties—but the entire framework breaks down somewhere around the TeV scale unless we have something like the Higgs boson appearing. **Strong coupling**and**duality**. This brings us back to mesons. Recall that our effective meson theory was a way for 1960s physicists to describe the particles coming out of early colliders*without*ever having to worry about the horrible non-perturbative QCD substructure that we now know is actually there. In some cases, there is a much stronger relation between the fundamental and effective theories and the two theories are said to be**dual**to one another. The 1990s were revolutionary for the development of formal dualities between seemingly unrelated theories: Witten’s web of dualities in*M*-theory, Seiberg duality in supersymmetric gauge theories, and gauge/gravity dualities like the AdS/CFT correspondence proposed by Maldacena. (For theoretical physics fans: those are some really big names in the field; each one of them is a MacArthur”Genuis” fellow!)

Anyway, there’s a surprising amount of “deep” physics that one can glean from thinking about mesons… even if they are somewhat “boring” particles that aren’t even fundamental. The notion of effective field theory is one of the central pillars of particle physics (as well as statistical physics), and in fact perhaps provides the most solid intuition about the entire field of high energy physics.