The perturbative Ricci flow in gravity

1. Problem Statement

The paper addresses the long-standing challenge of formulating a consistent quantum theory of gravity. Specifically, it tackles the issue that Einstein-Hilbert gravity in four dimensions is non-renormalizable in the Dyson sense due to Newton's constant ($G_N$) having a negative mass dimension. This necessitates an infinite tower of counterterms to absorb ultraviolet (UV) divergences.

While the Asymptotic Safety scenario suggests gravity might possess a non-trivial UV fixed point (making it predictive at high energies), finding this fixed point perturbatively is difficult:

• Standard mass-independent renormalization schemes (like $\overline{\text{MS}}$) often miss power divergences, potentially hiding non-Gaussian fixed points.
• Non-perturbative Functional Renormalization Group (FRG) approaches face challenges with truncations and the breaking of diffeomorphism invariance by regulators.
• Previous perturbative attempts often relied on 2D gravity, which introduces complications distinguishing UV poles from kinematic poles.

The authors aim to develop a perturbative formulation of gravity that maintains BRST invariance, is sensitive to power divergences, and does not rely on lower-dimensional gravity, using the Ricci flow as a novel tool.

2. Methodology

The authors adapt the concept of the gradient flow (widely used in QCD lattice calculations) to the context of gravity.

• Flowed Metric and Action:

  • They define a "flowed" metric $\hat{g}_{\mu\nu}(t, x)$ evolving with an auxiliary "flow time" $t$ according to the (gauged) Ricci flow equation:
    $$\partial_t \hat{g}_{\mu\nu} = -2 \hat{R}_{\mu\nu} + 2\alpha_0 \hat{\nabla}_{(\mu} \hat{F}_{\nu)}$$
  • The initial condition at $t=0$ is the physical metric plus counterterms.
  • They construct a flowed action $S_{\text{flow}}$ in $(d+1)$ dimensions (where $t$ is the extra dimension), introducing a Lagrange multiplier field $\hat{L}_{\mu\nu}$ to enforce the flow equation.

• Perturbative Expansion:

  • The metric is expanded around a flat background: $g_{\mu\nu} = \delta_{\mu\nu} + h_{\mu\nu}$.
  • The flowed metric is similarly expanded: $\hat{g}_{\mu\nu} = \delta_{\mu\nu} + \hat{h}_{\mu\nu}$.
  • The flow equation is solved iteratively, generating a perturbative series for $\hat{h}_{\mu\nu}$ in terms of convolutions of the original graviton field $h_{\mu\nu}$.

• Feynman Rules:

  • The standard Einstein-Hilbert Feynman rules are supplemented by flowed propagators, flowed vertices, and graviton flow lines.
  • Flow lines: Directional lines connecting the Lagrange multiplier $\hat{L}$ to the flowed field $\hat{h}$, representing evolution in flow time. They contain a Heaviside step function $\theta(t-s)$, ensuring time ordering.
  • Propagators: The flowed graviton propagator connects fields at different flow times $t$ and $s$, decaying as $e^{-(t+s)p^2}$.
  • UV Divergences: Since flow time acts as a regulator, UV divergences are strictly associated with fields at vanishing flow time ($t=0$).

• Calculation:

  • The authors calculate vacuum expectation values (VEVs) of diffeomorphism-invariant operators (specifically the volume $I_1$ and integrated curvature $I_{\hat{R}}$) up to the two-loop level.
  • They utilize a software framework (similar to that used for QCD gradient flow) to reduce integrals to master integrals.

3. Key Contributions

1. First Perturbative Ricci Flow in Gravity: The paper establishes the first complete perturbative framework for the Ricci flow in 4D gravity, analogous to the gradient flow in QCD.
2. Derivation of Counterterms: They explicitly derive the required counterterms ($\hat{c}_1, \hat{c}_2$) for the flowed action to render Green's functions finite at the two-loop level.
3. Gauge Independence: They demonstrate that the final physical results (VEVs and counterterms) are independent of the gauge-fixing parameters ($\alpha, \beta$), confirming the consistency of the approach.
4. New Renormalization Scheme: They propose a Ricci-flow based renormalization scheme (Fixed-Volume Scheme, FVS) that naturally captures power divergences due to the dimensionful flow time parameter $t$.

4. Results

• Counterterms: The authors determined the coefficients for the counterterms in the $\overline{\text{MS}}$ scheme:
  $$\hat{c}_1^{\overline{\text{MS}}} = -\frac{107}{30} \frac{1}{\epsilon}, \quad \hat{c}_2^{\overline{\text{MS}}} = \frac{407}{30} \frac{1}{\epsilon}$$
  These are independent of gauge parameters.
• Renormalized Observables: They provided the finite, renormalized expressions for the VEVs of the volume and integrated curvature operators up to two loops.
• Beta Function and Fixed Point:
  • By defining a renormalized Newton coupling $g_{RF}$ based on the integrated curvature in the FVS, they derived the $\beta$-function:
    $$\beta_{RF} = 2g_{RF}(1 - 5\nu g_{RF})$$
  • Non-Gaussian Fixed Point: The $\beta$-function exhibits a zero at a non-vanishing coupling $g^*_{RF} = 1/(5\nu)$.
  • Perturbative Nature: The critical exponent at this fixed point is $\theta = 2$, indicating the coupling is relevant in the UV. The authors argue that with reasonable choices of the scheme parameter $\nu$, this fixed point lies within the perturbative regime.

5. Significance

• Bridging Perturbative and Non-Perturbative: The work provides a perturbative method to access non-trivial UV fixed points, a feat usually reserved for non-perturbative FRG or lattice methods. It validates the existence of an asymptotically safe fixed point using standard diagrammatic techniques.
• Sensitivity to Power Divergences: Unlike $\overline{\text{MS}}$, the Ricci flow scheme is sensitive to power divergences (via the flow time $t$), which is crucial for uncovering the fixed point in gravity.
• BRST Invariance: The approach maintains BRST invariance throughout the calculation, avoiding the symmetry-breaking issues often encountered in FRG truncations.
• Future Applications: The framework is extendable to include matter fields, gauge fields, and non-zero cosmological constants. It also opens the door for phenomenological applications, such as studying black hole physics and gravitational waves via the short-flow-time expansion of composite operators.
• Lattice Compatibility: The method is designed to be compatible with lattice formulations of quantum gravity, offering a bridge between continuum perturbation theory and non-perturbative lattice simulations.

In conclusion, Harlander et al. successfully demonstrate that the Ricci flow can serve as a robust renormalization scheme in perturbative gravity, yielding a non-Gaussian fixed point consistent with asymptotic safety scenarios.