# 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!

## 2 thoughts on “Renormalization and Super Yang Mills Theory”

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