content_image/312841.jpg content_image/312842.jpg content_image/312843.jpg content_image/312844.jpg content_image/312845.jpg
reaches $^{g}$∗ , the coupling constant stays constant. This is a potential, but unproven resolution of the Landau pole § in QED: the Standard Model has such a pole at 10 $^{34}$MeV [23]. However this analysis assumes a monotonic beta function. If the beta function of QED has a non-trivial zero, then the Landau pole would not exist. 

Explicitly, near the fixed point of figure 1.3(a), the β function can be approximated by 

or 

which has the solution 

<img src='content_image/312841.jpg'>

<img src='content_image/312842.jpg'>

<img src='content_image/312843.jpg'>

This has important implications for the exact solution of the renormalization group equation (1.8). For sufficiently large p , the integral in the exponential of (1.8) will be dominated by those values of p where g ( p ) is close to $^{g}$∗ . Then 

<img src='content_image/312844.jpg'>

<img src='content_image/312845.jpg'>

So the two-point function returns to a simple scaling law, as might be ex- pected in a naive dimensional analysis argument. But there is an important difference: we would expect the power law to be p − $^{2}$, but instead it is 

