Renormalization Group Flows for Brane Couplings

TL;DR

The paper addresses how brane-localized couplings renormalize in theories with branes, distinguishing quantum-induced brane divergences from classical short-distance singularities due to zero brane thickness. Using a six-dimensional scalar model with a 3-brane in a conical background, it shows that classical divergences on codimension-two branes are logarithmic and generate running of brane couplings, deriving tree-level RG equations such as $\mu d\lambda_2/d\mu = \lambda_2^2/(2\pi\alpha)$ and related higher-order beta functions. It further computes one-loop bulk-to-brane corrections that feed into the brane RG equations, e.g. $\mu d\lambda_0/d\mu$, $\mu d\lambda_2/d\mu$, and $\mu d\lambda_4/d\mu$, illustrating how bulk interactions influence brane renormalization. The findings highlight a nontrivial classical running of brane-localized operators with potential implications for brane-world scenarios and large extra dimensions, and suggest that finite-thickness physics may resolve singularities while preserving long-wavelength RG behavior.

Abstract

Field theories in the presence of branes encounter localized divergences that renormalize brane couplings. The sources of these brane-localized divergences are understood as arising either from broken translation invariance, or from short distance singularities as the brane thickness vanishes. While the former are generated only by quantum corrections, the latter can appear even at the classical level. Using as an example six-dimensional scalar field theory in the background of a 3-brane, we show how to interpret such classical divergences by the usual regularization and renormalization procedure of quantum field theory. In our example, the zero thickness divergences are logarithmic, and lead classically to non-trivial renormalization group flows for the brane couplings. We construct the tree level renormalization group equations for these couplings as well as the one-loop corrections to these flows from bulk-to-brane renormalization effects.

Figures (1)

• Figure 1: Brane mass corrections to the scalar propagator. A $\bullet$ denotes an insertion of the coupling $\lambda_2$.