Two-loop beta-function from the exact renormalization group

TL;DR

This paper derives the two-loop beta-function for a massless scalar field theory using the exact renormalization group, emphasizing the importance of momentum dependence and potential applications to non-renormalizable systems.

Contribution

It presents a method to compute the two-loop beta-function from the exact RG flow equation, including full momentum dependence, applicable to non-renormalizable theories.

Findings

Correct two-loop beta-function obtained with full momentum dependence

Flow equation method applicable to non-renormalizable systems

Analysis of field rescaling and anomalous dimension effects

Abstract

We calculate the two-loop renormalization group (RG) beta-function of a massless scalar field theory from the irreducible version of Polchinski's exact RG flow equation. To obtain the correct two-loop result within this method, it is necessary to take the full momentum-dependence of the irreducible four-point vertex and the six-point vertex into account. Although the same calculation within the orthodox field theory method is less tedious, the flow equation method makes no assumptions about the renormalizability of the theory, and promises to be useful for performing two-loop calculations for non-renormalizable condensed-matter systems. We pay particular attention to the problem of the field rescaling and the effect of the associated exponent eta on the RG flow.