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arxiv: 2604.18678 · v1 · submitted 2026-04-20 · ✦ hep-th · gr-qc· hep-lat

Recognition: unknown

The perturbative Ricci flow in gravity

Henry Werthenbach, Jonas T. Kohnen, Robert V. Harlander, Yannick Kluth

Authors on Pith no claims yet

Pith reviewed 2026-05-10 04:09 UTC · model grok-4.3

classification ✦ hep-th gr-qchep-lat
keywords Ricci flowperturbative gravityrenormalization schemeNewton's constantnon-Gaussian fixed pointgradient flowquantum gravityasymptotic safety
0
0 comments X

The pith

A perturbative Ricci flow in gravity defines a renormalization scheme for Newton's constant that exhibits a non-Gaussian fixed point.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a perturbative formulation of the Ricci flow for gravity by adapting the gradient flow technique from QCD. Standard Feynman rules are supplemented with flowed propagators, vertices, and graviton flow lines that track the evolution of the metric under the flow. Vacuum expectation values of independent operators are computed at two-loop order to extract the necessary counterterms for the flowed action. These counterterms enable a new renormalization scheme for Newton's constant G_N. The renormalization group trajectory in this scheme produces a non-Gaussian fixed point that matches expectations from non-perturbative analyses.

Core claim

Following steps analogous to the gradient flow in QCD, we supplement the usual Feynman rules for perturbative gravity by flowed propagators and vertices as well as graviton flow lines which describe the evolution of gravity along the Ricci flow. By calculating vacuum expectation values of a number of independent operators at the two-loop level, we derive the required counterterms of the flowed action. Our results allow us to define a Ricci-flow based renormalization scheme for Newton's constant G_N. Studying its renormalization group behavior, we recover a non-Gaussian fixed point in accordance with well-known non-perturbative considerations.

What carries the argument

The flowed action supplemented with graviton flow lines and two-loop counterterms derived from operator vacuum expectation values, which implements the Ricci-flow renormalization scheme for Newton's constant.

If this is right

  • The scheme supplies a perturbative definition of the renormalization-group trajectory for Newton's constant.
  • It reproduces the non-Gaussian fixed point at two-loop order in agreement with non-perturbative results.
  • The same construction can be applied to additional gravitational operators and couplings.
  • It provides a controlled perturbative window onto aspects of quantum gravity previously accessible only non-perturbatively.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Higher-loop extensions of the same flowed diagrams could test whether the fixed point remains stable under further corrections.
  • Inclusion of matter fields within the flowed framework would allow study of their influence on the fixed-point structure.
  • The method may be compared with other perturbative approaches to the renormalization group in gravity to identify shared features.

Load-bearing premise

The direct analogy between QCD gradient flow and Ricci flow in gravity preserves the necessary perturbative structure so that two-loop counterterms suffice to define a consistent renormalization-group trajectory.

What would settle it

A three-loop computation of the flowed counterterms that produces inconsistent renormalization-group equations or eliminates the non-Gaussian fixed point.

Figures

Figures reproduced from arXiv: 2604.18678 by Henry Werthenbach, Jonas T. Kohnen, Robert V. Harlander, Yannick Kluth.

Figure 1
Figure 1. Figure 1: FIG. 1. Example diagrams contributing at [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
read the original abstract

We develop a perturbative formulation of the Ricci flow in gravity. Following steps analogous to the gradient flow in QCD, we supplement the usual Feynman rules for perturbative gravity by flowed propagators and vertices as well as graviton flow lines which describe the evolution of gravity along the Ricci flow. By calculating vacuum expectation values of a number of independent operators at the two-loop level, we derive the required counterterms of the flowed action. Our results allow us to define a Ricci-flow based renormalization scheme for Newton's constant $G_N$. Studying its renormalization group behavior, we recover a non-Gau{\ss}ian fixed point in accordance with well-known non-perturbative considerations

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a perturbative formulation of the Ricci flow in gravity by supplementing standard Feynman rules with flowed propagators, vertices, and graviton flow lines, following an analogy to QCD gradient flow. Two-loop vacuum expectation values of independent operators are computed to obtain counterterms of the flowed action. These results are used to define a Ricci-flow based renormalization scheme for Newton's constant G_N, whose renormalization-group flow is shown to recover a non-Gaussian fixed point consistent with non-perturbative expectations.

Significance. If the construction is free of higher-order obstructions, the work supplies a concrete perturbative handle on geometric flows in gravity and an explicit two-loop calculation that grounds a new renormalization scheme for G_N. Credit is due for performing the two-loop VEV computations and for making the flowed Feynman rules explicit.

major comments (2)
  1. [two-loop VEV calculations] § on two-loop counterterm extraction: the central claim that two-loop VEVs of independent operators suffice to define a consistent renormalization scheme for G_N lacks any discussion of possible obstructions at three loops or higher that could arise from the diffeomorphism invariance of the Einstein-Hilbert action or from the geometric constraints of the Ricci flow itself; without such an argument the extracted beta function for G_N cannot be guaranteed to define a reliable RG trajectory.
  2. [RG behavior of G_N] § on RG behavior of G_N: the non-Gaussian fixed point is recovered 'in accordance with well-known non-perturbative considerations'; this phrasing leaves open whether the scheme is independently predictive or whether the two-loop truncation and normalization have been chosen to reproduce an already-known result, undermining the claim that the fixed point emerges from the flowed theory.
minor comments (2)
  1. [Abstract] Abstract: 'Gaußian' should read 'Gaussian'.
  2. [Feynman rules] Notation: the definitions of the flowed graviton propagators and vertices should be collected in a single subsection with explicit Feynman rules for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and indicate the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [two-loop VEV calculations] § on two-loop counterterm extraction: the central claim that two-loop VEVs of independent operators suffice to define a consistent renormalization scheme for G_N lacks any discussion of possible obstructions at three loops or higher that could arise from the diffeomorphism invariance of the Einstein-Hilbert action or from the geometric constraints of the Ricci flow itself; without such an argument the extracted beta function for G_N cannot be guaranteed to define a reliable RG trajectory.

    Authors: We agree that an explicit discussion of the perturbative order would improve clarity. Our construction is defined at two-loop order, where the vacuum expectation values determine the counterterms and yield a well-defined beta function for G_N within the Ricci-flow scheme. Diffeomorphism invariance is maintained order-by-order in the expansion, and no inconsistencies appear in the two-loop computation. Potential obstructions at three loops or higher would require separate higher-order calculations that lie beyond the present scope; we will add a short paragraph in the conclusions noting the two-loop limitation and the absence of obstructions at the computed order, while stating that higher-loop consistency remains an open question for future work. revision: partial

  2. Referee: [RG behavior of G_N] § on RG behavior of G_N: the non-Gaussian fixed point is recovered 'in accordance with well-known non-perturbative considerations'; this phrasing leaves open whether the scheme is independently predictive or whether the two-loop truncation and normalization have been chosen to reproduce an already-known result, undermining the claim that the fixed point emerges from the flowed theory.

    Authors: The beta function is obtained directly from the two-loop counterterms computed with the flowed propagators, vertices, and flow lines; the non-Gaussian fixed point is then located by solving the resulting renormalization-group equation. No free parameters are adjusted to enforce the fixed point, and the normalization is fixed by the perturbative matching conditions. The agreement with non-perturbative expectations is therefore a consistency check rather than an input. We will revise the abstract and the relevant discussion section to make this explicit, replacing the current phrasing with language that stresses the fixed point as an output of the two-loop calculation. revision: yes

Circularity Check

0 steps flagged

No significant circularity: perturbative counterterm calculation and RG analysis are independent of the recovered fixed point

full rationale

The derivation proceeds by supplementing Feynman rules with flowed propagators/vertices and graviton flow lines, computing explicit two-loop VEVs of independent operators to obtain counterterms, defining a renormalization scheme for G_N from those counterterms, and extracting the beta function whose flow exhibits a non-Gaussian fixed point. This fixed point is reported as matching external non-perturbative results, but the paper presents it as an outcome of the two-loop computation rather than an input or fitted parameter. No quoted step equates the final result to its own inputs by construction, no self-citation chain bears the central claim, and the perturbative structure is built from first-principles Feynman rules without smuggling ansatze or renaming known patterns. The scheme and beta function therefore retain independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on standard perturbative quantum-field-theory axioms for gravity and introduces technical flow objects whose independent justification is limited to the QCD analogy.

axioms (2)
  • standard math Standard perturbative expansion and Feynman rules for Einstein gravity
    Used as the base that is supplemented by flowed objects.
  • domain assumption Analogy between QCD gradient flow and Ricci flow holds at the level of propagator and vertex modifications
    Invoked to justify the addition of flow lines and flowed rules.
invented entities (1)
  • Flowed graviton propagators and vertices no independent evidence
    purpose: To encode the evolution of the metric under Ricci flow inside perturbative diagrams
    New technical objects introduced by direct analogy to QCD; no independent falsifiable prediction supplied in the abstract.

pith-pipeline@v0.9.0 · 5410 in / 1454 out tokens · 53418 ms · 2026-05-10T04:09:24.318186+00:00 · methodology

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