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arxiv: 2508.00807 · v2 · submitted 2025-08-01 · ✦ hep-th · gr-qc

Proper-time functional renormalization in O(N) scalar models coupled to gravity

Alfio M. Bonanno , Emiliano M. Glaviano , Gian Paolo Vacca This is my paper

Pith reviewed 2026-05-19 01:15 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords functional renormalization groupproper-time regulatorO(N) scalar modelquantum gravityscaling solutionscritical exponentsasymptotic safety
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0 comments X

The pith

Switching to a proper-time regulator in the functional renormalization group largely reproduces the scaling solutions and critical exponents previously found for O(N) scalars coupled to gravity.

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

The paper tests a proper-time regulator version of the functional Wilsonian renormalization group on an O(N) scalar multiplet coupled to gravity in three and four dimensions. It adopts the identical background-fluctuation splitting and gauge-fixing procedure used in an earlier effective-average-action study and employs a comparable truncation. The explicit aim is to check whether the scaling solutions and their critical properties survive the change of regulator. Most qualitative and quantitative features are recovered, although small differences appear at finite N and in the large-N limit, and these differences depend on the choice of improved scheme in each framework.

Core claim

In the proper-time regulator framework the scaling solutions for the O(N) model coupled to gravity remain essentially the same as those obtained with the effective average action, both in d=4 and d=3, with only minor quantitative shifts that depend on the value of N and on which 'improved' scheme is selected in each formulation.

What carries the argument

The proper-time regulator applied to the Wilsonian functional renormalization group flow, using the same background-fluctuation splitting and gauge fixing as the effective-average-action calculation.

If this is right

  • The existence and stability of the scaling solutions are robust against the choice between proper-time and effective-average-action regulators.
  • Critical exponents extracted at finite N show only small scheme-dependent corrections.
  • In the large-N limit the differences between the two regulator schemes remain visible but do not alter the overall qualitative picture.
  • The use of improved schemes in either framework can produce modest quantitative shifts without destroying the fixed-point structure.

Where Pith is reading between the lines

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

  • Results obtained with one common regulator choice may be trusted more when they reappear under a second, technically distinct regulator.
  • The same truncation can now be tested in other gravity-matter systems by switching regulators without re-deriving the entire gauge-fixing machinery.
  • Small scheme differences at large N suggest that large-N expansions in these models should be cross-checked with at least two regulator families.

Load-bearing premise

The background-fluctuation splitting and gauge-fixing procedure chosen in the earlier effective-average-action study remains valid when the regulator is changed to proper time.

What would settle it

A direct computation of the same critical exponents in the proper-time scheme that yields values differing by more than a few percent from the effective-average-action results at identical truncation order would undermine the reported agreement.

Figures

Figures reproduced from arXiv: 2508.00807 by Alfio M. Bonanno, Emiliano M. Glaviano, Gian Paolo Vacca.

Figure 1
Figure 1. Figure 1: FIG. 1: Log plot of the relative difference [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Log plot of the relative difference [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Scaling solutions for [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Scaling solutions for [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Plot of the difference [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Scaling solutions for [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Scaling solutions for [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Log plot of the relative difference [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Log plot of the relative difference [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
read the original abstract

We focus on the use of the functional Wilsonian renormalization group framework characterized by a proper time regulator and test its use in the search of the scaling solutions and the critical properties of an O(N)-invariant scalar field multiplet coupled to gravity in d=4 and d=3 dimensions. We employ the same background-fluctuation splitting and gauge fixing procedure, already adopted in a previous study based, instead, on the effective average action framework and a similar truncation of the effective action. Our main goal is to compare the results for the scaling solutions and some of the associated critical exponents. In this analysis, performed in a different framework, most of the picture previously uncovered is confirmed both at qualitative and quantitative level. There are, neverthelss, few differences both at finite N and in its large value limit, depending also on the schemes which in both frameworks are called 'improved'

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

1 major / 2 minor

Summary. The manuscript investigates scaling solutions and critical properties of O(N)-invariant scalar fields coupled to gravity in d=4 and d=3 using the proper-time functional renormalization group framework. It adopts the background-fluctuation splitting and gauge fixing from a prior effective average action study with a similar truncation, and compares the results for fixed points and critical exponents, claiming qualitative and quantitative confirmation with some differences at finite and large N depending on 'improved' schemes.

Significance. This work offers a valuable cross-check between the proper-time regulator and the effective average action approaches in the context of asymptotically safe gravity-matter systems. Confirming most results across frameworks would bolster confidence in the truncation's reliability and the existence of non-trivial fixed points. The explicit comparison at both finite N and in the large-N limit is a strength, as is the focus on scheme dependence.

major comments (1)
  1. [Setup section (background-fluctuation splitting and gauge fixing)] The paper reuses the identical background-fluctuation splitting (metric = background + fluctuation) and gauge-fixing procedure from the previous effective average action study without providing a re-derivation or explicit verification that diffeomorphism invariance and background independence are preserved under the proper-time flow equation. Since the proper-time regulator involves an integral over the proper-time parameter s rather than the Wetterich kernel, this transfer is not automatically guaranteed and could impact the treatment of gauge modes and the Hessian. This assumption is central to the comparability of results and thus to the claim of confirmation.
minor comments (2)
  1. [Abstract] There is a typo: 'neverthelss' should be 'nevertheless'.
  2. [Abstract] The abstract reports qualitative and quantitative agreement but supplies no explicit pointers to equations, tables, or error estimates in the main text; adding such references would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of our work and for the detailed comment, which helps improve the clarity of the manuscript. We address the point below.

read point-by-point responses

  1. Referee: [Setup section (background-fluctuation splitting and gauge fixing)] The paper reuses the identical background-fluctuation splitting (metric = background + fluctuation) and gauge-fixing procedure from the previous effective average action study without providing a re-derivation or explicit verification that diffeomorphism invariance and background independence are preserved under the proper-time flow equation. Since the proper-time regulator involves an integral over the proper-time parameter s rather than the Wetterich kernel, this transfer is not automatically guaranteed and could impact the treatment of gauge modes and the Hessian. This assumption is central to the comparability of results and thus to the claim of confirmation.

    Authors: We appreciate the referee drawing attention to this aspect of the setup. The background-fluctuation splitting and gauge-fixing were taken over from the earlier effective average action analysis precisely to enable a controlled comparison between the two renormalization-group frameworks. The proper-time flow equation is obtained by inserting a regulator that depends on the background metric into the functional integral; the resulting integral over the proper-time parameter s is a specific representation of the cutoff, but the underlying construction preserves the same background covariance and gauge-fixing structure as in the Wetterich case. Consequently, the treatment of the Hessian and the projection onto gauge modes follows identically within the truncation employed. Nevertheless, we acknowledge that the manuscript does not contain an explicit re-derivation of these properties for the proper-time regulator. In the revised version we will add a concise paragraph in the setup section, together with references to existing literature on proper-time FRG in curved backgrounds, to make this reasoning explicit and thereby strengthen the justification for the direct comparison. revision: yes

Circularity Check

0 steps flagged

No significant circularity; independent proper-time regulator computation

full rationale

The paper derives scaling solutions and critical exponents for the O(N) model coupled to gravity by applying the proper-time functional renormalization group flow equation in d=3 and d=4, using a truncation of the effective action. Although the background-fluctuation splitting and gauge-fixing procedure are adopted from a prior effective-average-action study, the flow equation itself is replaced by an integral over the proper-time parameter s, producing new numerical or approximate results for the fixed-point values and exponents. The comparison to the earlier framework is presented as an external check, with explicit differences noted at finite N and in the large-N limit; no equation or result in the present work reduces by construction to the inputs or outputs of the cited prior study. The derivation chain remains self-contained within the proper-time regulator and the chosen truncation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The comparison rests on the validity of the shared truncation ansatz and the background-fluctuation splitting procedure transferred from the earlier effective average action study; no explicit free parameters or new entities are named in the abstract.

axioms (1)
  • domain assumption Background-fluctuation splitting and gauge fixing procedure from prior study remains suitable for proper-time regulator
    Explicitly adopted from previous effective average action analysis

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Forward citations

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