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arxiv: 2605.29159 · v2 · pith:NKEWLXEGnew · submitted 2026-05-27 · ✦ hep-th

Asymptotically Safe Gravitational Form Factors from the Proper-Time Flow Equation

Emiliano Maria Glaviano This is my paper

Pith reviewed 2026-06-29 10:19 UTC · model grok-4.3

classification ✦ hep-th
keywords asymptotically safe gravitygravitational form factorsproper-time flow equationnon-Gaussian fixed pointrenormalization group flowquantum gravitymomentum dependence
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The pith

Asymptotically safe gravitational form factors are finite and scale-independent only if the ultraviolet boundary condition selects the 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 integrates the proper-time flow equations for non-local form factors in four-dimensional quantum gravity down to zero cutoff scale at quadratic order in curvature. This allows full reconstruction of their momentum dependence. Asymptotic safety does not automatically produce a finite limit as the cutoff Lambda goes to infinity, since solutions typically develop logarithmic divergences ln(q squared over Lambda squared). A finite limit is achieved only by selecting the non-Gaussian fixed point in the ultraviolet boundary condition, which removes the divergences and makes the form factors independent of the renormalization scale mu.

Core claim

Within the proper-time formalism at quadratic order in the curvature expansion, the flow equations for gravitational form factors integrate to k equals zero. Asymptotic safety alone leaves a logarithmic divergence in the Lambda to infinity limit, requiring an additional renormalization condition. Only the non-Gaussian fixed point as ultraviolet boundary condition removes these divergences, yielding finite dimensionful form factors that are independent of the renormalization scale mu, decay as one over q squared in the ultraviolet, and exhibit the expected logarithmic structure in the infrared with the Planck scale in place of mu.

What carries the argument

The proper-time flow equation for non-local gravitational form factors, integrated from the non-Gaussian fixed point at quadratic order in curvature, which enforces finiteness and scale independence

If this is right

  • The renormalized form factors become independent of the renormalization scale mu
  • The ultraviolet momentum dependence follows a power-law decay proportional to 1/q squared
  • The infrared structure matches expected logs but with the Planck scale setting the scale
  • Finite dimensionful form factors are obtained without ultraviolet logarithmic contributions

Where Pith is reading between the lines

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

  • If higher curvature orders preserve the fixed point selection, the result would hold more generally
  • This method could be compared to other flow equation schemes for consistency in form factor predictions
  • The Planck scale dominance in IR might influence effective field theory matching at low energies

Load-bearing premise

Integrating the flow at only quadratic order in the curvature expansion captures the momentum dependence without higher terms affecting the UV finiteness or boundary condition

What would settle it

Computing the form factors including cubic curvature terms and checking if the Lambda to infinity limit remains finite when starting from the non-Gaussian fixed point would test the claim

Figures

Figures reproduced from arXiv: 2605.29159 by Emiliano Maria Glaviano.

Figure 1
Figure 1. Figure 1: FIG. 1. Fixed-point form factors for [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Plots of the perturbations [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Plots of the perturbations [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Flow of the asymptotically safe solution in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Numerical form factors at [PITH_FULL_IMAGE:figures/full_fig_p033_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Reference asymptotically safe form factors at [PITH_FULL_IMAGE:figures/full_fig_p033_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Absolute difference [PITH_FULL_IMAGE:figures/full_fig_p033_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Numerical form factors at [PITH_FULL_IMAGE:figures/full_fig_p035_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Asymptotically safe form factors at [PITH_FULL_IMAGE:figures/full_fig_p035_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Absolute difference [PITH_FULL_IMAGE:figures/full_fig_p035_10.png] view at source ↗
read the original abstract

We study the renormalization group flow of non-local form factors in four-dimensional quantum gravity within the proper-time formalism at quadratic order in the curvature expansion. We show that the flow equations can be integrated down to $k=0$, allowing the reconstruction of the full momentum dependence of the form factors. Within this framework, we construct asymptotically safe solutions at this order. We find that asymptotic safety of the flow does not automatically ensure a finite cutoff-independent $\Lambda\to\infty$ limit for the integrated solutions, which in general develop a logarithmic divergence $\ln(q^2/\Lambda^2)$, so that a renormalization condition is still required. A finite $\Lambda\to\infty$ limit compatible with asymptotic safety is obtained only when the ultraviolet boundary condition selects the non-Gaussian fixed point. This yields finite dimensionful form factors, removes UV logarithmic contributions, and ensures independence from the renormalization scale $\mu$. The resulting renormalized asymptotically safe form factors display a power-law decay $\sim1/q^2$ in the ultraviolet and reproduce the expected logarithmic structure in the infrared, with the Planck scale replacing the renormalization scale.

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 studies the renormalization group flow of non-local gravitational form factors in four-dimensional quantum gravity within the proper-time formalism, truncated at quadratic order in the curvature expansion. The flow equations are integrated explicitly down to k=0 to reconstruct the full momentum dependence. Asymptotically safe solutions are constructed, but a finite, cutoff-independent limit as Λ→∞ is not automatic and requires selecting the non-Gaussian fixed point as the ultraviolet boundary condition; this removes logarithmic divergences ln(q²/Λ²), yields finite dimensionful form factors independent of the renormalization scale μ, produces a power-law decay ∼1/q² in the ultraviolet, and reproduces the expected logarithmic structure in the infrared with the Planck scale replacing μ.

Significance. If the central result holds within the stated truncation, the work supplies an explicit construction of renormalized asymptotically safe form factors with cutoff-independent physical properties. Credit is due for the explicit integration of the flow equations to k=0, which is a concrete strength of the proper-time approach and allows direct reconstruction of momentum dependence. This contributes a calculational example to the asymptotic-safety program for momentum-dependent quantities.

major comments (1)
  1. [Discussion of UV boundary conditions and integrated solutions (following the statement of the flow equations)] The demonstration that only the non-Gaussian fixed point as UV boundary condition produces a finite Λ→∞ limit (removing the logarithmic divergence) is performed exclusively at quadratic order in the curvature expansion. Higher-order curvature invariants enter the proper-time beta functions for the form factors and can alter the fixed-point structure or the integrated flow, potentially reintroducing divergences; the manuscript provides no estimate or argument for the stability of the finiteness property under truncation extension. This is load-bearing for the claim that the NGFP selection ensures a finite cutoff-independent result compatible with asymptotic safety.
minor comments (2)
  1. [Abstract and introduction] The abstract and early sections introduce the form factors without an explicit definition or reference to their precise operator structure; adding this would improve readability.
  2. [Section on the proper-time flow equation] The manuscript could clarify the precise form of the proper-time regulator and cutoff function used in the flow equation, including any assumptions about its shape.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the constructive major comment. We respond to it below.

read point-by-point responses

  1. Referee: The demonstration that only the non-Gaussian fixed point as UV boundary condition produces a finite Λ→∞ limit (removing the logarithmic divergence) is performed exclusively at quadratic order in the curvature expansion. Higher-order curvature invariants enter the proper-time beta functions for the form factors and can alter the fixed-point structure or the integrated flow, potentially reintroducing divergences; the manuscript provides no estimate or argument for the stability of the finiteness property under truncation extension. This is load-bearing for the claim that the NGFP selection ensures a finite cutoff-independent result compatible with asymptotic safety.

    Authors: We agree that the demonstration is performed exclusively within the quadratic curvature truncation, as stated repeatedly in the manuscript (including the abstract and the section following the flow equations). At this order the selection of the non-Gaussian fixed point removes the logarithmic divergence and yields a finite, cutoff-independent result. Higher-order curvature invariants would indeed enter the beta functions and could modify the fixed-point structure or the integrated flow. The manuscript contains no estimate of stability under truncation extension because performing such an estimate requires a systematic extension of the truncation, which lies outside the scope of the present work. We have revised the conclusions to include an explicit statement that the finiteness property is established at quadratic order and that its robustness against higher-order corrections is an open question for future investigations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows from explicit integration of flow equations

full rationale

The paper integrates the proper-time flow equations at quadratic curvature order to reconstruct form-factor momentum dependence, then demonstrates by direct solution that a generic UV boundary produces ln(q²/Λ²) divergences while the non-Gaussian fixed point boundary removes them and yields μ-independent finite results. This outcome is obtained from the differential equations themselves rather than by definitional fiat or self-citation. No load-bearing self-citations, ansatze imported from prior author work, or fitted inputs renamed as predictions appear in the abstract or described chain. The truncation limitation is acknowledged as an assumption but does not render the reported computation circular within the stated order.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the proper-time flow equation at quadratic curvature order, the existence of a non-Gaussian fixed point that can be used as a UV boundary condition, and the assumption that the truncation does not introduce spurious divergences. No new particles or forces are postulated.

free parameters (1)
  • UV boundary condition at non-Gaussian fixed point
    The choice of which fixed point supplies the initial condition at high k is selected by hand to obtain finiteness; its value is not derived from external data.
axioms (2)
  • domain assumption The proper-time flow equation remains reliable when truncated to quadratic order in curvature.
    Invoked throughout the abstract as the framework within which the flow is integrated.
  • domain assumption A non-Gaussian fixed point exists and can be used as a UV boundary condition without higher-order corrections altering its location.
    Required for the finite Λ→∞ limit to be compatible with asymptotic safety.

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Forward citations

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Reference graph

Works this paper leans on

112 extracted references · 100 canonical work pages · cited by 4 Pith papers · 65 internal anchors

  1. [1]

    Status of the asymptotic safety paradigm for quantum gravity and matter

    A. Eichhorn, Status of the asymptotic safety paradigm for quantum gravity and matter, Found. Phys.48, 1407 (2018), arXiv:1709.03696 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  2. [2]

    Bonanno, A

    A. Bonanno, A. Eichhorn, H. Gies, J. M. Pawlowski, R. Percacci, M. Reuter, F. Saueressig, and G. P. Vacca, Critical reflections on asymptotically safe gravity, Front. in Phys.8, 269 (2020), arXiv:2004.06810 [gr-qc]. 52

    work page arXiv 2020

  3. [3]

    The Asymptotic Safety Scenario in Quantum Gravity -- An Introduction

    M. Niedermaier, The Asymptotic safety scenario in quantum gravity: An Introduction, Class. Quant. Grav.24, R171 (2007), arXiv:gr-qc/0610018

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  4. [4]

    Quantum Einstein Gravity

    M. Reuter and F. Saueressig, Quantum Einstein Gravity, New J. Phys.14, 055022 (2012), arXiv:1202.2274 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  5. [5]

    Dupuis, L

    N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, and N. Wschebor, The nonperturbative functional renormalization group and its applications, Phys. Rept.910, 1 (2021), arXiv:2006.04853 [cond-mat.stat-mech]

    work page arXiv 2021

  6. [6]

    Pawlowski and M

    J. Pawlowski and M. Reichert, Quantum gravity: A fluctuating point of view, Frontiers in Physics 8, 551848 (2021)

    2021

  7. [7]

    Percacci,An Introduction to Covariant Quantum Gravity and Asymptotic Safety(World Scien- tific, 2017) pp

    R. Percacci,An Introduction to Covariant Quantum Gravity and Asymptotic Safety(World Scien- tific, 2017) pp. 1–300

    2017

  8. [8]

    Saueressig, The Functional Renormalization Group in Quantum Gravity (2023) arXiv:2302.14152 [hep-th]

    F. Saueressig, The Functional Renormalization Group in Quantum Gravity (2023) arXiv:2302.14152 [hep-th]

    work page arXiv 2023

  9. [9]

    Reuter and F

    M. Reuter and F. Saueressig,Quantum Gravity and the Functional Renormalization Group: The Road towards Asymptotic Safety, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2019)

    2019

  10. [10]

    T. R. Morris, The Exact renormalization group and approximate solutions, Int. J. Mod. Phys. A 9, 2411 (1994), arXiv:hep-ph/9308265

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  11. [11]

    Wetterich, Exact evolution equation for the effective potential, Physics Letters B301, 90 (1993)

    C. Wetterich, Exact evolution equation for the effective potential, Physics Letters B301, 90 (1993)

    1993

  12. [12]

    Flow Equations for N Point Functions and Bound States

    U. Ellwanger, FLow equations for N point functions and bound states, Z. Phys. C62, 503 (1994), arXiv:hep-ph/9308260

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  13. [13]

    P. F. Machado and F. Saueressig, On the renormalization group flow of f(R)-gravity, Phys. Rev. D 77, 124045 (2008), arXiv:0712.0445 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  14. [14]

    Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation

    A. Codello, R. Percacci, and C. Rahmede, Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation, Annals Phys.324, 414 (2009), arXiv:0805.2909 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  15. [15]

    A bootstrap strategy for asymptotic safety

    K. Falls, D. F. Litim, K. Nikolakopoulos, and C. Rahmede, A bootstrap towards asymptotic safety (2013), arXiv:1301.4191 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  16. [16]

    Further evidence for asymptotic safety of quantum gravity

    K. Falls, D. F. Litim, K. Nikolakopoulos, and C. Rahmede, Further evidence for asymptotic safety of quantum gravity, Phys. Rev. D93, 104022 (2016), arXiv:1410.4815 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  17. [17]

    K. G. Falls, D. F. Litim, and J. Schr¨ oder, Aspects of asymptotic safety for quantum gravity, Phys. Rev. D99, 126015 (2019), arXiv:1810.08550 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  18. [18]

    Ultraviolet properties of f(R)-Gravity

    A. Codello, R. Percacci, and C. Rahmede, Ultraviolet properties of f(R)-gravity, Int. J. Mod. Phys. A23, 143 (2008), arXiv:0705.1769 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  19. [19]

    Asymptotically safe f(R)-gravity coupled to matter I: the polynomial case

    N. Alkofer and F. Saueressig, Asymptotically safef(R)-gravity coupled to matter I: the polynomial case, Annals Phys.396, 173 (2018), arXiv:1802.00498 [hep-th]. 53

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  20. [20]

    N. Ohta, R. Percacci, and G. P. Vacca, Renormalization Group Equation and scaling solutions for f(R) gravity in exponential parametrization, Eur. Phys. J. C76, 46 (2016), arXiv:1511.09393 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  21. [21]

    N. Ohta, R. Percacci, and G. P. Vacca, Flow equation forf(R) gravity and some of its exact solutions, Phys. Rev. D92, 061501 (2015), arXiv:1507.00968 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  22. [22]

    Renormalization Group Equation for $f(R)$ gravity on hyperbolic spaces

    K. Falls and N. Ohta, Renormalization Group Equation forf(R) gravity on hyperbolic spaces, Phys. Rev. D94, 084005 (2016), arXiv:1607.08460 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  23. [23]

    N. Ohta, R. Percacci, and A. D. Pereira, Gauges and functional measures in quantum gravity I: Einstein theory, JHEP06, 115, arXiv:1605.00454 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  24. [24]

    J. D. Gon¸ calves, T. de Paula Netto, and I. L. Shapiro, Gauge and parametrization ambiguity in quantum gravity, Phys. Rev. D97, 026015 (2018), arXiv:1712.03338 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  25. [25]

    Field Parametrization Dependence in Asymptotically Safe Quantum Gravity

    A. Nink, Field Parametrization Dependence in Asymptotically Safe Quantum Gravity, Phys. Rev. D91, 044030 (2015), arXiv:1410.7816 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  26. [26]

    Renormalisation of Newton's constant

    K. Falls, Renormalization of Newton’s constant, Phys. Rev. D92, 124057 (2015), arXiv:1501.05331 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  27. [27]

    H. Gies, B. Knorr, and S. Lippoldt, Generalized Parametrization Dependence in Quantum Gravity, Phys. Rev. D92, 084020 (2015), arXiv:1507.08859 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  28. [28]

    Gauge Dependence of the Effective Average Action in Einstein Gravity

    S. Falkenberg and S. D. Odintsov, Gauge dependence of the effective average action in Einstein gravity, Int. J. Mod. Phys. A13, 607 (1998), arXiv:hep-th/9612019

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  29. [29]

    Gauge and Cutoff Function Dependence of the Ultraviolet Fixed Point in Quantum Gravity

    W. Souma, Gauge and cutoff function dependence of the ultraviolet fixed point in quantum gravity (2000), arXiv:gr-qc/0006008

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  30. [30]

    V. F. Barra, P. M. Lavrov, E. A. Dos Reis, T. de Paula Netto, and I. L. Shapiro, Functional renormalization group approach and gauge dependence in gravity theories, Phys. Rev. D101, 065001 (2020), arXiv:1910.06068 [hep-th]

    work page arXiv 2020

  31. [31]

    Bonanno, G

    A. Bonanno, G. Oglialoro, and D. Zappal` a, Gauge and parametrization dependence of quantum Einstein gravity within the proper time flow, Phys. Rev. D112, 026002 (2025), arXiv:2504.07877 [hep-th]

    work page arXiv 2025

  32. [32]

    N. Ohta, R. Percacci, and A. D. Pereira, Gauges and functional measures in quantum gravity II: Higher derivative gravity, Eur. Phys. J. C77, 611 (2017), arXiv:1610.07991 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  33. [33]

    Asymptotic safety of quantum gravity beyond Ricci scalars

    K. Falls, C. R. King, D. F. Litim, K. Nikolakopoulos, and C. Rahmede, Asymptotic safety of quantum gravity beyond Ricci scalars, Phys. Rev. D97, 086006 (2018), arXiv:1801.00162 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  34. [34]

    M. H. Goroff and A. Sagnotti, The ultraviolet behavior of einstein gravity, Nuclear Physics B266, 709 (1986)

    1986

  35. [35]

    Reconstructing the Universe

    J. Ambjorn, J. Jurkiewicz, and R. Loll, Reconstructing the universe, Phys. Rev. D72, 064014 (2005), arXiv:hep-th/0505154. 54

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  36. [36]

    Evidence for Asymptotic Safety from Lattice Quantum Gravity

    J. Laiho and D. Coumbe, Evidence for Asymptotic Safety from Lattice Quantum Gravity, Phys. Rev. Lett.107, 161301 (2011), arXiv:1104.5505 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  37. [37]

    Nonperturbative Quantum Gravity

    J. Ambjorn, A. Goerlich, J. Jurkiewicz, and R. Loll, Nonperturbative Quantum Gravity, Phys. Rept.519, 127 (2012), arXiv:1203.3591 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  38. [38]

    Exploring Euclidean Dynamical Triangulations with a Non-trivial Measure Term

    D. Coumbe and J. Laiho, Exploring Euclidean Dynamical Triangulations with a Non-trivial Mea- sure Term, JHEP04, 028, arXiv:1401.3299 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  39. [39]

    Lattice Quantum Gravity and Asymptotic Safety

    J. Laiho, S. Bassler, D. Coumbe, D. Du, and J. T. Neelakanta, Lattice Quantum Gravity and Asymptotic Safety, Phys. Rev. D96, 064015 (2017), arXiv:1604.02745 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  40. [40]

    The phase structure of Causal Dynamical Triangulations with toroidal spatial topology

    J. Ambjørn, J. Gizbert-Studnicki, A. G¨ orlich, J. Jurkiewicz, and D. N´ emeth, The phase structure of Causal Dynamical Triangulations with toroidal spatial topology, JHEP06, 111, arXiv:1802.10434 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  41. [41]

    Loll, Quantum Gravity from Causal Dynamical Triangulations: A Review, Class

    R. Loll, Quantum Gravity from Causal Dynamical Triangulations: A Review, Class. Quant. Grav. 37, 013002 (2020), arXiv:1905.08669 [hep-th]

    work page arXiv 2020

  42. [42]

    Ambjorn, Z

    J. Ambjorn, Z. Drogosz, J. Gizbert-Studnicki, A. G¨ orlich, J. Jurkiewicz, and D. N` emeth, CDT Quantum Toroidal Spacetimes: An Overview, Universe7, 79 (2021), arXiv:2103.15610 [gr-qc]

    work page arXiv 2021

  43. [43]

    Bassler, J

    S. Bassler, J. Laiho, M. Schiffer, and J. Unmuth-Yockey, The de Sitter Instanton from Euclidean Dynamical Triangulations, Phys. Rev. D103, 114504 (2021), arXiv:2103.06973 [hep-lat]

    work page arXiv 2021

  44. [44]

    J. F. Donoghue, Leading quantum correction to the Newtonian potential, Phys. Rev. Lett.72, 2996 (1994), arXiv:gr-qc/9310024

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  45. [45]

    J. F. Donoghue, General relativity as an effective field theory: The leading quantum corrections, Phys. Rev. D50, 3874 (1994), arXiv:gr-qc/9405057

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  46. [46]

    B. R. Holstein and J. F. Donoghue, Classical physics and quantum loops, Phys. Rev. Lett.93, 201602 (2004), arXiv:hep-th/0405239

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  47. [47]

    A. A. Akhundov, S. Bellucci, and A. Shiekh, Gravitational interaction to one loop in effective quantum gravity, Phys. Lett. B395, 16 (1997), arXiv:gr-qc/9611018

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  48. [48]

    On the covariant formalism of the effective field theory of gravity and leading order corrections

    A. Codello and R. K. Jain, On the covariant formalism of the effective field theory of gravity and leading order corrections, Class. Quant. Grav.33, 225006 (2016), arXiv:1507.06308 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  49. [49]

    ’t Hooft and M

    G. ’t Hooft and M. J. G. Veltman, One-loop divergencies in the theory of gravitation, Ann. Inst. H. Poincare Phys. Theor. A20, 69 (1974)

    1974

  50. [50]

    A. O. Barvinsky and G. A. Vilkovisky, Covariant perturbation theory. 2: Second order in the curvature. General algorithms, Nucl. Phys. B333, 471 (1990)

    1990

  51. [51]

    A. O. Barvinsky and G. A. Vilkovisky, Beyond the schwinger-dewitt technique: Converting loops into trees and in-in currents, Nuclear Physics282, 163 (1987)

    1987

  52. [52]

    A. O. Barvinsky and G. A. Vilkovisky, Covariant perturbation theory. 3: Spectral representations of the third order form-factors, Nucl. Phys. B333, 512 (1990). 55

    1990

  53. [53]

    A. O. Barvinsky, Y. V. Gusev, V. V. Zhytnikov, and G. A. Vilkovisky, Covariant perturbation theory. 4. Third order in the curvature (1993), arXiv:0911.1168 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  54. [54]

    I. G. Avramidi,Heat kernel and quantum gravity, Vol. 64 (Springer, New York, 2000)

    2000

  55. [55]

    Knorr, C

    B. Knorr, C. Ripken, and F. Saueressig, Form Factors in Asymptotic Safety: conceptual ideas and computational toolbox, Class. Quant. Grav.36, 234001 (2019), arXiv:1907.02903 [hep-th]

    work page arXiv 2019

  56. [56]

    Bosma, B

    L. Bosma, B. Knorr, and F. Saueressig, Resolving Spacetime Singularities within Asymptotic Safety, Phys. Rev. Lett.123, 101301 (2019), arXiv:1904.04845 [hep-th]

    work page arXiv 2019

  57. [57]

    Knorr, C

    B. Knorr, C. Ripken, and F. Saueressig, Form Factors in Asymptotically Safe Quantum Gravity (2024) arXiv:2210.16072 [hep-th]

    work page arXiv 2024

  58. [58]

    Knorr and M

    B. Knorr and M. Schiffer, Non-Perturbative Propagators in Quantum Gravity, Universe7, 216 (2021), arXiv:2105.04566 [hep-th]

    work page arXiv 2021

  59. [59]

    Polyakov Effective Action from Functional Renormalization Group Equation

    A. Codello, Polyakov Effective Action from Functional Renormalization Group Equation, Annals Phys.325, 1727 (2010), arXiv:1004.2171 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  60. [60]

    A. Satz, A. Codello, and F. D. Mazzitelli, Low energy Quantum Gravity from the Effective Average Action, Phys. Rev. D82, 084011 (2010), arXiv:1006.3808 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  61. [61]

    Fixed points and infrared completion of quantum gravity

    N. Christiansen, D. F. Litim, J. M. Pawlowski, and A. Rodigast, Fixed points and infrared com- pletion of quantum gravity, Phys. Lett. B728, 114 (2014), arXiv:1209.4038 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  62. [62]

    Global Flows in Quantum Gravity

    N. Christiansen, B. Knorr, J. M. Pawlowski, and A. Rodigast, Global Flows in Quantum Gravity, Phys. Rev. D93, 044036 (2016), arXiv:1403.1232 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  63. [63]

    Local Quantum Gravity

    N. Christiansen, B. Knorr, J. Meibohm, J. M. Pawlowski, and M. Reichert, Local Quantum Gravity, Phys. Rev. D92, 121501 (2015), arXiv:1506.07016 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  64. [64]

    T. Denz, J. M. Pawlowski, and M. Reichert, Towards apparent convergence in asymptotically safe quantum gravity, Eur. Phys. J. C78, 336 (2018), arXiv:1612.07315 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  65. [65]

    Bonanno, T

    A. Bonanno, T. Denz, J. M. Pawlowski, and M. Reichert, Reconstructing the graviton, SciPost Phys.12, 001 (2022), arXiv:2102.02217 [hep-th]

    work page arXiv 2022

  66. [66]

    Bonanno and M

    A. Bonanno and M. Reuter, Proper time flow equation for gravity, JHEP02, 035, arXiv:hep- th/0410191

    work page arXiv

  67. [67]

    E. M. Glaviano and A. Bonanno, Proper-time flow equation and non-local truncations in quantum gravity, in17th Marcel Grossmann Meeting: On Recent Developments in Theoretical and Exper- imental General Relativity, Gravitation, and Relativistic Field Theories(2024) arXiv:2410.23696 [gr-qc]

    work page arXiv 2024

  68. [68]

    Towards an accurate determination of the critical exponents with the Renormalization Group flow equations

    A. Bonanno and D. Zappala, Towards an accurate determination of the critical exponents with the renormalization group flow equations, Phys. Lett. B504, 181 (2001), arXiv:hep-th/0010095

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  69. [69]

    Proper time regulator and Renormalization Group flow

    M. Mazza and D. Zappala, Proper time regulator and renormalization group flow, Phys. Rev. D 64, 105013 (2001), arXiv:hep-th/0106230. 56

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  70. [70]

    A. M. Bonanno, D. Buccio, E. M. Glaviano, and F. Saueressig, Scaling Solutions of Matter Form Factors in Asymptotically Safe Quantum Gravity (2026), arXiv:2605.11805 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  71. [71]

    Giacometti, D

    G. Giacometti, D. Rizzo, and D. Zappala, On the universal content of the proper time flow in scalar and Yang-Mills theories (2025), arXiv:2510.04896 [hep-th]

    work page arXiv 2025

  72. [72]

    Quantum gravity contributions to the gauge and Yukawa couplings in proper time flow

    G. Giacometti, K. Kowalska, D. Rizzo, E. M. Sessolo, and D. Zappala, Quantum gravity contribu- tions to the gauge and Yukawa couplings in proper time flow (2026), arXiv:2604.03033 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  73. [73]

    A. M. Bonanno, R. A. Konoplya, G. Oglialoro, and A. Spina, Regular black holes from proper-time flow in quantum gravity and their quasinormal modes, shadow and Hawking radiation, JCAP12, 042, arXiv:2509.12469 [gr-qc]

    work page arXiv

  74. [74]

    Giacometti, A

    G. Giacometti, A. Bonanno, S. Plumari, and D. Zappal` a, Spontaneous breaking of diffeomorphism invariance in conformally reduced quantum gravity (2024), arXiv:2410.08916 [gr-qc]

    work page arXiv 2024

  75. [75]

    Bonanno, M

    A. Bonanno, M. Conti, and D. Zappal` a, The conformal sector of Quantum Einstein Gravity beyond the local potential approximation, Phys. Lett. B847, 138311 (2023), arXiv:2309.15514 [hep-th]

    work page arXiv 2023

  76. [76]

    A. M. Bonanno, M. Conti, and S. L. Cacciatori, Ultraviolet behavior of conformally reduced quadratic gravity, Phys. Rev. D108, 026008 (2023), arXiv:2304.12011 [hep-th]

    work page arXiv 2023

  77. [77]

    Universality and symmetry breaking in conformally reduced quantum gravity

    A. Bonanno and F. Guarnieri, Universality and Symmetry Breaking in Conformally Reduced Quan- tum Gravity, Phys. Rev. D86, 105027 (2012), arXiv:1206.6531 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  78. [78]

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

    A. Bonanno, E. Glaviano, and G. P. Vacca, Proper-time functional renormalization inO(N) scalar models coupled to gravity (2025), arXiv:2508.00807 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  79. [79]

    A. M. Bonanno and E. M. Glaviano, Gravitationally Induced UV Completion of anO(N) Scalar Theory (2026), arXiv:2601.20820 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  80. [80]

    Spina, Black Holes in Asymptotic Safety: A Review of Solutions and Phenomenology (2025), arXiv:2510.14552 [gr-qc]

    A. Spina, Black Holes in Asymptotic Safety: A Review of Solutions and Phenomenology (2025), arXiv:2510.14552 [gr-qc]

    work page arXiv 2025

Showing first 80 references.