Two-loop Renormalization in Quantum Gravity near Two Dimensions

TL;DR

This paper investigates two-loop renormalization in (2+ε)-dimensional quantum gravity, demonstrating the potential for a consistent quantum gravity theory via ε-expansion near two dimensions, with focus on matter field divergences.

Contribution

It provides the first calculation of two-loop divergences in (2+ε)-dimensional quantum gravity, showing the existence of a covariant fixed point and supporting the ε-expansion approach.

Findings

Nonlocal and infrared divergences cancel among diagrams.

A fixed point with general covariance exists in the renormalization group.

Results support the construction of a consistent quantum gravity theory via ε-expansion.

Abstract

We study two--loop renormalization in $(2+\epsilon)$--dimensional quantum gravity. As a first step towards the full calculation, we concentrate on the divergences which are proportional to the number of matter fields. We calculate the $\beta$ functions and show how the nonlocal divergences as well as the infrared divergences cancel among the diagrams. Although the formalism includes a subtlety concerning the general covariance due to the dynamics of the conformal mode, we find that the renormalization group allows the existence of a fixed point which possesses the general covariance. Our results strongly suggest that we can construct a consistent theory of quantum gravity by the $\epsilon$ expansion around two dimensions.