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

  1. Jun 2, 2015 #1
    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. jcsd
  3. Jun 2, 2015 #2

    fzero

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    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).
     
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