Fixed Points and Asymptotic Freedom

Abstract

In our derivation of the RG-improved coupling constant (6.9), we noted that $\lambda _r(\mu )$ grows with $\mu$; in fact it blows up when $\mu$ reaches exponentially large values:

$$\begin{aligned} \lambda _r(\mu ) \rightarrow \infty \quad \hbox {as}\quad \mu \rightarrow \mu _0 e^{16\pi ^2\over 3\lambda _r(\mu _0)} \end{aligned}$$

which is called the Landau singularity—Landau observed that the same thing happens to the electron charge in QED. However, we cannot be sure that this behavior really happens unless we investigate the theory nonperturbatively. Since the coupling constant becomes large, perturbation theory is no longer reliable, and it is conceivable that $\lambda (\mu )$ could turn around at some scale and start decreasing again. Lattice studies of $\phi ^4$ theory have confirmed that the Landau singular behavior is in fact what happens. One finds that it is not possible to remove the cutoff in this kind of theory while keeping the effective coupling nonzero at low energies.