What should I think of a diverging beta function (in Renormalisation Group flow)?

Question

I have written a set of RG flow equations using Functional Renormalization Group methods. I am looking at the flow of a well known problem with an additional original coupling. I did not do anything systematic. The first step of my work is just to see if my new coupling grows under iteration of the RG flow.

I find something strange and have no idea how (and if) I should interpret it: For a large set of initial conditions, the RG flow goes towards a point where the beta functions diverge. The RG flow stops at a finite cut-off value (that depends on the initial conditions) and is not defined below that.

I am looking into the RG flow of the $O(N)$ model. I have included a 4-point coupling that contains two spatial derivatives. When the field expectation value $\rho_0$ is very small, one of the new couplings decouples from all the other couplings. I get the following beta functions $$\beta_{\text{new}} = -\frac{4 \lambda_{\text{new}}}{\left(8\pi + \lambda_{\text{new}}\right) \rho_0} \, ,$$ $$\beta_{\rho_0} = 0 \, .$$ If this is initiated with $-8\pi<\lambda_{\text{new}}<0$, the flow equation $$\partial_t\lambda_{\text{new}} = \beta_{\text{new}}\, ,$$ decreases $\lambda_{\text{new}}$ until it becomes equal to $-8\pi$ and the RG flow stops.

I find this very strange and do no know what do think about it. The only thing that I can think of is that my approximation scheme really bad and that I should not trust this part of my calculation. What do you think?

Answer 1

This is essentially a Landau pole. A Landau pole describes a renormalization group (RG) flow where the coupling diverges at a finite value of the renormalization scale (see, e.g., Landau pole in Wikipedia). According to the classification of Bogoliubov and Shirkov (N. N. Bogoliubov; D. V. Shirkov (1980). Introduction to the Theory of Quantized Fields (3rd ed.). John Wiley & Sons.) if the beta function for a coupling $g$, $\beta_{g}$, scales like $g^{\alpha}$ with $\alpha>1$ at large values of $g$ then $g$ will have a Landau pole.

To see that $\lambda$ in the OP possesses, essentially, a Landau pole, it is convenient to switch variables to \begin{equation} h=\frac{1}{8\pi+\lambda} \end{equation} so that \begin{equation} \beta_h = \frac{4}{\rho_0}(1-8\pi h)h^2. \end{equation} Notice that, $\beta_h \sim h^3$ as $h$ becomes large (or as $\lambda$ approaches $-8\pi$ from above). Then, following the classification of Bogoliubov and Shirkov, $h$ has a Landau pole.