[Paper Review] Curvature dependence of quantum gravity with scalars

Benjamin Bürger, Jan M. Pawlowski+2 more|arXiv (Cornell University)|Dec 3, 2019

Black Holes and Theoretical Physics2 references32 citations

TL;DR

The paper computes curvature-dependent graviton correlators, couplings, and the full f(R) potential in asymptotically safe gravity coupled to minimally coupled scalars, extends to negative curvature, and analyzes quantum and background EoMs. It finds two EoM solutions (one at negative curvature, one at positive), with stability analyses suggesting large positive curvature can stabilize the system for many scalar flavors.

ABSTRACT

We compute curvature-dependent graviton correlation functions and couplings as well as the full curvature potential $f(R)$ in asymptotically safe quantum gravity coupled to scalars. The setup is based on a systematic vertex expansion about metric backgrounds with constant curvatures initiated in arXiv:1711.09259 for positive curvatures. We extend these results to negative curvature and investigate the influence of minimally coupled scalars. The quantum equation of motion has two solutions for all accessible numbers of scalar fields. We observe that the solution at negative curvature is a minimum, while the solution at positive curvature is a maximum. We find indications that the solution to the equation of motions for scalar-gravity systems is at large positive curvature, for which the system might be stable for all scalar flavours.

Motivation & Objective

• Study curvature dependence of graviton n-point functions in asymptotically safe gravity.
• Compute curvature-dependent f(R) background potential and its quantum equation of motion.
• Assess the impact of minimally coupled scalar fields on fixed points and EoMs.
• Investigate stability and asymptotic behavior of scalar-gravity systems across curvature regimes.

Proposed method

• Use functional renormalisation group with a vertex expansion around backgrounds of constant curvature.
• Truncate to leading Einstein–Hilbert tensor structures and define g(r), mu(r), lambda3(r) as dimensionless couplings.
• Evaluate traces on curved backgrounds using spectral sums for positive curvature and spectral integrals for negative curvature.
• Employ a heat-kernel expansion to set initial conditions and derive differential equations for f(r) and f1(r).
• Study background and quantum EoMs by solving flow equations at fixed points and analyzing asymptotics with Nielsen identities.

Experimental results

Research questions

• RQ1 How do curvature and scalar field content affect curvature-dependent graviton couplings and the f(R) potential in asymptotically safe gravity?
• RQ2 What are the fixed-point structures g*(r), mu*(r), lambda3*(r) across curvature, and how do scalars modify them?
• RQ3 Do background and quantum equations of motion admit solutions, and how do these solutions depend on N_s and curvature sign?
• RQ4 What is the asymptotic behavior of the fixed-point functions and are the EoMs stable at large curvature or scalar number?

Key findings

• Two solutions to the quantum EoM exist for all accessible N_s, one at negative curvature (minimum) and one at positive curvature (maximum).
• The fixed-point effective couplings mu_eff*(r) and lambda3_eff*(r) are nearly curvature-independent, while g*(r) retains curvature dependence.
• Increasing N_s shifts fixed-point values; for large N_s the positive-curvature solution remains stable and the Newton coupling stays small.
• There is no background EoM solution (rf′(r)−2f(r)=0) for any curvature or N_s within the studied regime, extending prior results.
• The large-curvature asymptotics imply g*(r) ~ c_+/r with c_+ depending on N_s, and compatibility with Nielsen identities constrains the asymptotic behavior of f and f1.
• The divergence of g*(r) at negative curvature for some N_s is attributed to truncation and regulator artifacts rather than a physical bound.