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 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 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 $^{§}$A Landau pole means that the coupling constant becomes infinite at finite energy. Technically QCD has a Landau pole at ∼ Λ$_{QCD}$, but the phrase is usually used to refer to non-asymptotically free theories only 16 β ∼ − B ( g − $^{g}$∗ ) (1.22) d d log p g ∼ − B ( g − $^{g}$∗ ) (1.23) ¯ g ( p ) = $^{g}$∗ + C ( M p ) B (1.24) G ( p ) ≈ G ( $^{g}$∗ ) exp [ − (log p M 2(1 − γ ( $^{g}$∗ )) ] (1.25) 1 γ ( g ∗ ) ≈ C ( 1 p 2 ) 1 − γ ( g ∗ ) (1.26)