# Supersymmetry, to the Rescue!

Part Three of a Series on N=4 Super Yang-Mills Theory

This is the third in a series of articles that will explain N=4 super Yang-Mills theory. In this series I take that phrase apart bit by bit, explaining as I go. Because I’m perverse and out to confuse you, I started with the last bit here, and now I’m working my way up.

N=4 Super Yang-Mills Theory

Ah, supersymmetry…trendy, sexy, mysterious…an excuse to put “super” in front of words…it’s a grand subject.

If I’m going to manage to explain supersymmetry at all, then I need to explain spin. Luckily, you don’t need to know much about spin for this to work. While I could start telling you about how particles literally spin around like tops despite having a radius of zero, and how quantum mechanics restricts how fast they spin to a few particular values measured by Planck’s constant…all you really need to know is the following:

Spin is a way to categorize particles.

In particular, there are:
Spin 1: Yang-Mills fields are spin 1, carrying forces with a direction and strength.
Spin ½: This spin covers pretty much all of the particles you encounter in everyday matter: electrons, neutrons, and protons, as well as more exotic stuff like neutrinos. If you want to make large-scale, interesting structures like rocks or lifeforms you pretty much need spin ½ particles.
Spin 0: A spin zero field (also called a scalar) is a number, like a temperature, that can vary from place to place. The Higgs field is an example of a spin zero field, where the number is part of the mass of other particles, and the Higgs boson is a ripple in that field, like a cold snap would be for temperature.

While they aren’t important for this post, you can also have higher numbers for spin: gravity has spin 2, for example.

With this definition in hand, we can start talking about supersymmetry, which is also pretty straightforward if you ignore all of the actual details.

Supersymmetry is a relationship (or symmetry) between particles with spin X, and particles with spin X-½

For example, you could have a relationship between a spin 1 Yang-Mills field and a spin ½ matter particle, or between a spin ½ matter particle and a spin 0 scalar.

“Relationship” is a vague term here, much like it is in romance, and just like in romance you’d do well to clarify precisely what you mean by it. Here, it means something like the following: if you switch a particle for its “superpartner” (the other particle in the relationship) then the physics should remain the same. This has two important consequences: superpartners have the same mass as each-other and superpartners have the same interactions as each-other.

The second consequence means that if a particle has electric charge -1, its superpartner also has electric charge -1. If you’ve got gluons, each with a color and an anti-color, then their superpartners will also have both a color and an anti-color. Astute readers will have remembered that quarks just have a color or an anti-color, and realized the implication: quarks cannot be the superpartners of gluons.

Other, even more well-informed readers will be wondering about the first consequence. Such readers might have heard that the LHC is looking for superpartners, or that superpartners could explain dark matter, and that in either case superpartners have very high mass. How can this be if superpartners have to have the same mass as their partners among the regular particles?

The important point to make here is that our real world is not supersymmetric, even if superpartners are discovered at the LHC, because supersymmetry is broken. In physics, when a symmetry of any sort is broken it’s like a broken mirror: it no longer is the same on each side, but the two sides are still related in a systematic way. Broken supersymmetry means that particles that would be superpartners can have different masses, but they will still have the same interactions.

When people look for supersymmetry at the LHC, they’re looking for new particles with the same interactions as the old particles, but generally much higher mass. When I talk about supersymmetry, though, I’m talking about unbroken supersymmetry: pairs of particles with the same interactions and the same mass. And N=4 super Yang-Mills is full of them.

How full? N=4 full. And that’s next week’s topic.

# Why I Study a Theory That Isn’t “True”

I study a theory called N=4 super Yang-Mills. (There’s a half-decent explanation of the theory here. For now, just know that it involves a concept called supersymmetry, where forces and matter are very closely related.) When I mention this to people, sometimes they ask me if I’m expecting to see evidence for N=4 super Yang-Mills at the Large Hadron Collider. And if not there, when can we expect a test of the theory?

Never.

Never? Yep. N=4 super Yang-Mills will never be tested, because N=4 super Yang-Mills (sYM for short) is not “true”.

We know it’s not “true”, because it contains particles that don’t exist. Not just particles we might not have found yet, but particles that would make the universe a completely different and possibly unknowable place.

So if it isn’t true, why do I study it?

Let me give you an analogy. Remember back in 2008 when Sarah Palin made fun of funding “fruit fly research in France”?

Most people I talked to found that pretty ridiculous. After all, fruit flies are one of the most stereotypical research animals, second only to mice. And besides, hadn’t we all grown up knowing about how they were used to research HOX genes?

Wait, you didn’t know about that? Evidently, you weren’t raised by a biologist.

HOX genes are how your body knows what limbs go where. When HOX genes activate in an embryo, they send signals, telling cells where to grow arms and legs.

Much of HOX genes’ power was first discovered in fruit flies. With their relatively simple genetics, scientists were able to manipulate the HOX genes, creating crazy frankenflies like Antennapedia (literally: antenna-feet) here.

A fruity fly’s HOX genes, and the body parts they correspond to.

Old antenna-feet. Ain’t he a beauty?

It was only later, as the science got more sophisticated, that biologists began to track what HOX genes do in humans, making substantial progress in understanding debilitating mutations.

How is this related to N=4 super Yang-Mills? Well, just as fruit flies are simpler to study than humans, sYM is simpler to study than the whole mess of unconnected particles that exist in the real world. We can do calculations with sYM that would be out of reach in normal particle physics. As we do these calculations, we discover new patterns and new techniques. The hope is that, just like HOX genes, we will discover traits that still hold in the more complicated situation of the real world. We’re not quite there yet, but it’s getting close.

By the way, make sure to watch Big Bang Theory on Thursday (11/29, 8/7c on CBS). Turns out, Sheldon is working on this stuff too, and for those who have read arXiv:1210.7709, his diagrams should look quite familiar…