The continuum limit of the conformal sector at second order in perturbation theory

TL;DR

This paper extends a novel perturbative continuum limit approach for quantum gravity to second order, demonstrating the existence of a well-defined renormalized trajectory under certain conditions and discussing the recovery of diffeomorphism invariance.

Contribution

It shows that the continuum limit at second order is well-defined with a proper choice of cutoff scale and clarifies the role of irrelevant and relevant couplings in restoring diffeomorphism invariance.

Findings

Existence of a well-defined renormalized trajectory at second order

Conditions on cutoff scale for avoiding singularities

Recovery of diffeomorphism invariance in the limit

Abstract

Recently a novel perturbative continuum limit for quantum gravity has been proposed and demonstrated to work at first order. Every interaction monomial $\sigma$ is dressed with a coefficient function $f^\sigma_\Lambda(\varphi)$ of the conformal factor field, $\varphi$. Each coefficient function is parametrised by an infinite number of underlying couplings, and decays at large $\varphi$ with a characteristic amplitude suppression scale which can be chosen to be at a common value, $\Lambda_\text{p}$. Although the theory is perturbative in couplings it is non-perturbative in $\hbar$. At second order in perturbation theory, one must sum over all melonic Feynman diagrams to obtain the particular integral. We show that it leads to a well defined renormalized trajectory and thus continuum limit, provided it is solved by starting at an arbitrary cutoff scale $\Lambda=\mu$ which lies in the range $0<\mu<a\Lambda_\text{p}$ ($a$ some non-universal number). If $\mu$ lies above this range the resulting coefficient functions become singular, and the flow ceases to exist, before the physical limit is reached. To this one must add a well-behaved complementary solution, containing irrelevant couplings determined uniquely by the first-order interactions, and renormalized relevant couplings. Even though some irrelevant couplings diverge in the limit $\Lambda_\text{p}\to\infty$, domains for the underlying relevant couplings can be chosen such that diffeomorphism invariance will be recovered in this limit, and where the underlying couplings disappear to be replaced by effective diffeomorphism invariant couplings.