# Renormalization and Super Yang Mills Theory

It’s well known that $\mathcal{N}=4$ super Yang-Mills theory is perturbatively finite. This means that there’s no need to introduce a regulating cutoff to get sensible answers for scattering amplitude computations. In particular the $\beta$ and $\gamma$ functions for the theory vanish.

Recall that the $\gamma$ function tells us about the anomalous dimensions of elementary fields. More specifically, if $\phi$ is some field appearing in the Lagrangian, it must be rescaled to $Z \phi$ during renormalization. The $\gamma$ function then satisfies

$\displaystyle \gamma(g)=\frac{1}{2}\mu\frac{d}{d\mu}\log Z(g,\mu)$

where $g$ is the coupling and $\mu$ the renormalization scale. At a fixed point $g_*$ of the renormalization group flow, it can be shown that $\gamma(g_*)$ exactly encodes the difference between the classical dimension of $\phi$ and it’s quantum scaling dimension.

Thankfully we can replace all that dense technical detail with the simple picture of a river above. This represents the space of all possible theories, and the mass scale $\mu$ takes the place of usual time evolution. An elementary field operator travelling downstream will experience a change in scaling dimension. If it happens to get drawn into the fixed point in the middle of the whirlpool(!) the anomaly will exactly be encoded by the $\gamma$ function.

For our beloved $\mathcal{N}=4$ though the river doesn’t flow at all. The theory just lives in one spot all the time, so the elementary field operators just keep their simple, classical dimensions forever!

But there’s a subtle twist in the tale, when you start considering composite operators. These are built up as products of known objects. Naively you might expect that these don’t get renormalized either, but there you would be wrong!

So what’s the problem? Well, we know that propagators have short distance singularities when their separation becomes small. To get sensible answers for the expectation value of composite operators we must regulate these. And that brings back the pesky problem of renormalization with a vengeance.

The punchline is the although $\mathcal{N}=4$ is finite, the full spectrum of primary operators does contain some with non-trivial scaling dimensions. And that’s just as well really, because otherwise the AdS/CFT correspondence wouldn’t be quite as interesting!

# Scattering Without Scale, Or The S-Matrix In N=4

My research focuses on an unrealistic theory called massless $\mathcal{N}=4$ super Yang-Mills (SYM). This sounds pretty pointless, at least at first. But actually this model shares many features with more complete accounts of reality.  So it’s not all pie in the sky.

The reason I look at  SYM is because it contains lots of symmetry. This simplifies matters a lot. Studying SYM is like going to an adventure playground – you can still have great fun climbing and jumping, but it’s a lot safer than roaming out into a nearby forest.

Famously SYM has a conformal symmetry. Roughly speaking, this means that the theory looks the same at every length scale. (Whether conformal symmetry is equivalent to scale invariance is a hot topic, in fact)! Put another way, SYM has no real notion of length. I told you it was unrealistic.

This is a bit unfortunate for me, because I’d like to use SYM to think about particle scattering. To understand the problem, you need to know what I want to calculate. The official name for this quantity is the S-matrix.

The jargon is quite straightforward. “S” just stands for scattering. The “matrix” part tells you that this quantity encodes many possible scattering outcomes. To get an S-matrix, you have to assume you scatter particles from far away. That’s certainly the case in big particle accelerators – the LHC is huge compared to a proton!

But remember I said that SYM doesn’t have a length scale. So really you can’t get an S-matrix. And without an S-matrix, you can’t say anything about particle scattering. Things aren’t looking good.

Fortunately all is not lost. You can try to define an S-matrix using the usual techniques that worked in normal theories. All the calculations go through fine, unless there are any low energy particles around. Any of these so-called soft particles will cause your S-matrix to blow up to infinity!

But hey, we should expect our S-matrix to be badly behaved. After all, we’ve chosen a theory without a sense of scale! These irritating infinities go by the name of infrared divergences. Thankfully there’s a systematic way of eliminating them.

Remember that I said our SYM theory is massless. All the particles are like photons, constantly whizzing about that the speed of light. If you were a photon, life would be very dull. That’s because you’d move so fast through space you couldn’t move through time. This means that essentially our massless particles have no way of knowing about distances.

Viewed from this perspective it’s intuitive that this lack of mass yields the conformal symmetry. We can remove the troublesome divergences by destroying the conformal symmetry. We do this in a controlled way by giving some particles a small mass.

Technically our theory is now called Coulomb branch SYM. Who’s Coulomb, I hear you cry? He’s the bloke who developed electrostatics 250 years ago. And why’s he cropped up now? Because when we dispense with conformal symmetry, we’re left with some symmetries that match those of electromagnetism.

In Coulomb branch SYM it’s perfectly fine to define an S-matrix! You get sensible answers from all your calculations. Now imagine we try to recover our original theory by decreasing all masses to zero. Looking closely at the S-matrix, we see it split into two pieces – finite and infinite. Just ignore the infinite bit, and you’ve managed to extract useful scattering data for the original conformal theory!

You might think I’m a bit blasé in throwing away these divergences. But this is actually well-motivated physically. The reason is that such infinities cancel in any measurable quantity. You could say that they only appear in the first place because you’re doing the wrong sum!

This perspective has been formalized for the realistic theories as the KLN theorem. It may even be possible to get a rigorous version for our beloved massless $\mathcal{N}=4$ SYM.

So next time somebody tells you that you can’t do scattering in a conformal theory, you can explain why they’re wrong! Okay, I grant you, that’s an unlikely pub conversation. But stranger things have happened.

And if you’re planning to grab a pint soon, make it a scientific one!