# Can there by a theory that is both UV and IR stable?

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1. Jun 2, 2015

### metroplex021

The question is in the title: is it possible for a theory to be both UV and IR stable? And concrete models would be much appreciated!

2. Jun 2, 2015

### fzero

If you mean: can a theory have a stable RG fixed point in both the UV and IR, the answer is yes. A concrete example is $g\phi^4$ theory in 3 dimensions. The 1-loop beta function for $g$ is

$$\beta(g) = \frac{1}{16\pi^2} ( - g + 3 g^2).$$

There are 2 fixed points: the free theory at $g=0$ and an interacting theory at $g\sim 1/3$ (higher loops would be expected to change the position slightly). We can integrate the beta function to get

$$g(\mu) = \frac{ 16\pi^2 g(\Lambda) }{ 48\pi^2 g(\Lambda) - \frac{\mu}{\Lambda} ( g(\Lambda) -1)}.$$

For fixed $\Lambda$, we see that taking the scale to $\mu \rightarrow \infty$ sends $g(\mu) \rightarrow 0$, so this fixed point corresponds to the UV. Similarly, sending $\mu\rightarrow 0$ takes us to $g(\mu)\rightarrow 1/3$, so this is an IR fixed point (generally known as the Wilson-Fisher fixed point).