Ultraviolet Closure in Asymptotically Weyl-Invariant Gravity

Daniel Coumbe

REVIEW

Ultraviolet Closure in Asymptotically Weyl-Invariant Gravity

Daniel Coumbe

General Relativity and Quantum Cosmology · Submitted 1 Sep 2026 · Published version · 34 references

Pith's verdict: The on-shell counterterm-collapse argument is new and coherent inside its stated frame, but the 'arbitrary loop order' claim rests on an unverified assumption of anomaly absence that the paper itself concedes.

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arXiv ID2609.01047 (version 1)

Pith ID3FGSIT3F, Pith's permanent record of this paper and its review

CitationDaniel Coumbe. "Ultraviolet Closure in Asymptotically Weyl-Invariant Gravity." arXiv:2609.01047 (2026). https://pith.science/paper/3FGSIT3F

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Editor's note

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Two things to know. The genuinely new piece is Proposition 4.1: inside a restricted sector of local, parity-even, dimension-four, Weyl- and projectively-invariant counterterms, every admissible divergence in Palatini R^2 gravity collapses on shell, on the regular branch R≠0, to the original R^2 density plus an Euler term. The paper is also unusually honest about what this is not: Sec. 4.2 states plainly that anomaly absence is unestablished, that the sector excludes torsion, nonmetricity, curvature derivatives, and parity-odd terms, and that the result is on shell. The stress-test concern lands, but it lands on a paper that already concedes the point. The real weakness is the "arbitrary loop order" framing, which outruns what the algebraic classification alone supports.

I checked the proof steps. The mechanics are coherent: dimension four plus algebraicity forces quadratic curvature; the metric equation forces the traceless Ricci to vanish; the connection equation makes Γ the Levi-Civita connection of h=|R|g; h is then Einstein with unit scalar curvature, which collapses the Ricci-squared and R-squared contributions so that Riemann-squared equals the Euler density on shell. That cancellation is real and specific to the R^2 endpoint. The Section 2 uniqueness result is standard and the paper cites the right prior. Section 3 is labeled phenomenological, and Propositions 3.1 and 3.2 are fine on those terms.

Soft spots, in proportion. The load-bearing one: the loop-order claim needs Weyl and projective symmetries to survive quantization, and the paper gives no reason to expect that. Weyl anomalies are generic in curved-space QFT, so a fair summary is: a one-loop-consistent counterterm check in the anomaly-free sector, extended to all orders only by assumption. A referee should ask the author to either state the claim at one loop or offer an argument for anomaly absence. Secondary: excluding torsion and nonmetricity by fiat limits the significance for metric-affine readers, though the torsionless-representative logic is handled carefully, and the parameters R0, c1, c3 are free.

Who this is for: Palatini-gravity and asymptotic-safety people tracking Weyl-invariant gravity as a UV endpoint. It is a conditional consistency check, not a resolution of quantum gravity, and the paper says so. It deserves a serious referee: the central argument is new, self-contained, and correct inside its stated frame.

Recommendation: accept for peer review, expect revision on the loop-order claim.

Referee report

Major objections (3)

[Sec. 4.2, Proposition 4.1] The central proof of on-shell closure assumes that the classical Weyl and projective symmetries survive quantization without anomalies. As the paper explicitly states in Sec. 4.2, this assumption is not addressed and is an open problem. Weyl anomalies are generic in curved-spacetime quantum field theories (Duff 1994, Deser & Schwimmer 1993) and could introduce additional dimension-four counterterms that are not Weyl invariant, potentially invalidating the symmetry classification used in Steps 6–8 of the proof. The claim of closure 'at arbitrary loop order' therefore rests on an unverified premise. The paper should discuss the status of Weyl anomalies in the Palatini R^2 theory more thoroughly, including whether any known features (e.g., the absence of a spin-2 ghost, the topological character of the Euler term, or special properties of the regular branch) might suppress anomalies. At minimum, the conditions under which anomaly cancellation could occur should be outlined. Without such discussion, the result remains a formal algebraic classification conditional on an undefended assumption.
[Sec. 4.1, assumptions 2(a)–(f)] The restricted theory space excludes explicit torsion, nonmetricity, derivatives of curvature, and parity-odd terms. While the paper notes that this is a deliberate choice to test the 'minimal sector', the physical motivation for believing that such terms cannot appear in the counterterm spectrum is not fully developed. In a generic effective field theory, all invariants allowed by symmetries and power counting can arise. The paper should justify why these omissions are natural or argue that their inclusion would not affect the closure result (e.g., by showing they vanish on shell or are not generated). Without such justification, the generality of the result is limited.
[Sec. 4.1, Step 3 (connection equation)] The connection field equation (59) is stated to hold 'up to projective indeterminacy'. The paper then chooses a torsionless representative of the projective class. However, the equation ∇_λ(√{-g}R g^{μν})=0 is derived under the assumption that the connection is symmetric in the lower indices? In Palatini f(R), the connection is generally not symmetric; torsion can appear. The paper assumes from the start that the projective class admits a torsionless representative (Assumption 1). This is a strong geometric condition; the discussion in Sec. 4.1 about projective invariance and torsionlessness should be expanded to clarify when such a representative exists and whether it is always possible on the regular branch. The proof of Proposition 4.1 relies on this existence, but the condition is not checked for the explicit field configuration (e.g., solutions of the Palatini R^2 theory). A brief remark or reference would help.

Minor objections (5)

[Abstract and Sec. 1] The phrase 'nontrivial evidence for improved ultraviolet behaviour' should be qualified more strongly given the conditional nature of the result, e.g., 'provides conditional evidence under the stated assumptions'.
[Sec. 3.4, Eq. (39)] The representative n(R*) = 2 - 1/(1+c1 R* + c3 R*^3) is presented with c1>0, c3>0. The derivation in Propositions 3.1 and 3.2 yields constraints on the flow function β_eff, not directly on the functional form. It would be helpful to explicitly show that the given n(R*) satisfies those constraints (a=1, b=1, p(2)<-2, p(1)=-1). A short verification would strengthen the phenomenological motivation.
[Sec. 4.1, Step 5] The sentence 'No prior classification of projectively invariant affine-curvature scalars is needed' is accurate but could be preceded by a reminder that projective invariance is still being used to argue that dependence on the antisymmetric part of Ricci is absent. This logical step is somewhat compressed; a brief expansion would improve clarity.
[Sec. 4.2, Eq. (83)] The coefficient c_2 multiplies ∫ d^4x √{-h} E_4(h). In the conclusion, it is stated that √{-h}E_4(h) is equivalent to √{-g}E_4(g) up to a boundary term. Since the Euler term is topological, its coefficient is not fixed by the closure argument; it would be helpful to note that such a term does not affect the equations of motion and can be absorbed into the renormalization of the topological sector.
[References] The reference list is adequate, but the paper could cite more recent works on anomalies in Palatini gravity (e.g., Alkofer et al. on anomaly cancellation in metric-affine theories) to strengthen the discussion in Sec. 4.2.

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The paper

The paper claims

Weyl invariance picks R^2 as the UV gravity endpoint; on-shell counterterms close.

The reading

This paper argues that within the minimal scalar Palatini f(R) sector, exact Weyl invariance uniquely selects the $R^{2}$ action as the ultraviolet endpoint of gravity. It further shows that in the strict ultraviolet limit, within a restricted parity-even, torsionless, curvature-built sector, all admissible on-shell counterterms at any loop order reduce to the original √-g $R^{2}$ density plus a topological Euler term. This conditional on-shell closure provides evidence for improved ultraviolet behaviour compared to Einstein gravity, where non-renormalizable divergences proliferate. The result is limited by the restriction to a specific theory space and the assumption of no quantum anomalies among the classical symmetries.

What carries the argument

The central object is the Palatini curvature R(g,Γ) and the Weyl transformation that rescales only the metric while leaving the connection fixed. Under this condition √-g $R^{2}$ is exactly invariant. The key mechanism is the on-shell reduction: the metric equation forces the traceless Ricci tensor to vanish, and the connection equation forces Γ to become the Levi-Civita connection of the conformally related metric hμν = |R| gμν. In the h-geometry, the Riemann-squared term simplifies to the Euler density due to the Einstein condition on h, leading to the counterterm collapse.

Extended reading notes

Core claim

The paper discovers that asymptotically Weyl-invariant gravity (AWIG) in the Palatini formalism has a tightly controlled ultraviolet structure: the endpoint theory is uniquely fixed to √-g $R^{2}$ by Weyl invariance, and within a controlled set of assumptions (no torsion, no nonmetricity, parity-even, no curvature derivatives, regular branch R ≠ 0), every admissible on-shell divergent counterterm at arbitrary loop order collapses to a linear combination of the original $R^{2}$ density and the Euler density. This is Proposition 4.1, which implies that the restricted on-shell counterterm sector is closed modulo a topological term, a property not shared by Einstein gravity.

Load-bearing premise

The argument assumes that no quantum anomaly breaks the classical Weyl and projective symmetries, a condition explicitly left unexamined; also, the counterterm classification is restricted to a parity-even, torsionless, curvature-built sector that excludes many potentially relevant operators.

Editorial extensions

If this is right

The Palatini R^2 theory is singled out by Weyl invariance alone, providing a first-principles selection of the ultraviolet endpoint action.
Within the restricted sector, all loop divergences are multiplicatively renormalizable on shell, modulo a topological term, indicating that the theory's UV behavior is stronger than power-counting renormalizability suggests.
The combination of Weyl and projective symmetries, if anomaly-free, forces the counterterm structure to be extremely constrained, offering a concrete target for a future functional renormalization-group calculation.
The result implies that AWIG can be consistently truncated to the Einstein-like infrared and the Weyl-invariant ultraviolet without introducing new gravitational degrees of freedom or higher-derivative ghosts in the UV.
It raises the possibility that the on-shell closure survives off-shell with additional symmetry constraints, motivating further investigation of the quantum effective action in the full theory space.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the Weyl anomaly is absent or cancelled, AWIG would be one of the few quantum gravity candidates with fully controlled all-loop divergences in a symmetry-selected sector.
The restriction to torsionless, parity-even curvature invariants suggests that the broader metric-affine theory space may introduce additional operators, but the on-shell closure in the minimal sector provides a baseline for classifying such extensions.
The curvature-flow analysis of the exponent n(R*) could be reinterpreted as the beta function of a quantum fixed point, potentially linking AWIG to asymptotic safety without requiring a dimensionful Newton constant.

Original abstract

We investigate asymptotically Weyl-invariant gravity (AWIG) in the Palatini formulation, defined by an exponent n that interpolates between an Einstein-like infrared regime, where n goes to one, and a Weyl-invariant ultraviolet regime, where n goes to two. In four dimensions, we show that exact Weyl invariance within the minimal scalar Palatini f(R) sector uniquely selects the R^2 action as the ultraviolet endpoint. Interpreting the exponent as an effective curvature-dependent quantity, we constrain a phenomenological class of smooth autonomous flows compatible with the required infrared and ultraviolet endpoints. We then analyse the strict ultraviolet theory. On the regular branch R not equal to zero, and within a restricted torsionless, parity-even sector constructed purely from curvature, all admissible divergent on-shell counterterms reduce at arbitrary loop order to the original \sqrt{-g}R^2 term plus a topological Euler density. Although this conditional result does not establish full off-shell perturbative renormalizability, it provides nontrivial evidence for improved ultraviolet behaviour relative to Einstein gravity and identifies a concrete target for a future first-principles renormalization-group calculation.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,178 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.97 · human participants 0.01 · controlled comparison 0.16 · randomized assignment 0.01 · states quantitative result 0.23 · compares with prior results 0.72 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We investigate asymptotically Weyl-invariant gravity (AWIG) in the Palatini formulation, defined by an exponent n that interpolates between an Einstein-like infrared regime, where n goes to one, and a Weyl-invariant ultraviolet regime, where n goes to two. In four dimensions, we show that exact Weyl invariance within the minimal scalar Palatini f(R) sector uniquely selects the R^2 action as the ultraviolet endpoint. Interpreting the exponent as an effective curvature-dependent quantity, we constrain a phenomenological class of smooth autonomous flows compatible with the required infrared and ultraviolet endpoints. We then analyse the strict ultraviolet theory. On the regular branch R not equal to zero, and within a restricted torsionless, parity-even sector constructed purely from curvature, all admissible divergent on-shell counterterms reduce at arbitrary loop order to the original \sqrt{-g}R^2 term plus a topological Euler density. Although this conditional result does not establish full off-shell perturbative renormalizability, it provides nontrivial evidence for improved ultraviolet behaviour relative to Einstein gravity and identifies a concrete target for a future first-principles renormalization-group calculation.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We investigate asymptotically Weyl-invariant gravity (AWIG) in the Palatini formulation, defined by an exponent n that interpolates between an Einstein-like infrared regime, where n goes to one, and a Weyl-invariant ultraviolet regime, where n goes to two.	contribution 0.71	0.46	0.07	asserted 0.98
s1	In four dimensions, we show that exact Weyl invariance within the minimal scalar Palatini f(R) sector uniquely selects the R^2 action as the ultraviolet endpoint.	result 0.98	0.93	0.28	asserted 1.00
s2	Interpreting the exponent as an effective curvature-dependent quantity, we constrain a phenomenological class of smooth autonomous flows compatible with the required infrared and ultraviolet endpoints.	contribution 0.52	0.89	0.07	asserted 0.87
s3	We then analyse the strict ultraviolet theory.	contribution 0.85	0.81	0.03	unclear 0.84
s4	On the regular branch R not equal to zero, and within a restricted torsionless, parity-even sector constructed purely from curvature, all admissible divergent on-shell counterterms reduce at arbitrary loop order to the original \sqrt{-g}R^2 term plus a topological Euler density.	result 1.00	0.88	0.61	asserted 1.00
s5	Although this conditional result does not establish full off-shell perturbative renormalizability, it provides nontrivial evidence for improved ultraviolet behaviour relative to Einstein gravity and identifies a concrete target for a future first-principles renormalization-group calculation.	interpretation 0.79	0.86	0.57	hedged possibility 0.76

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (34)

[1]

José M. M. Senovilla and David Garfinkle. The 1965 Penrose singularity theorem.Class. Quant. Grav., 32(12):124008, 2015

View on Pith 1965
[2]

Gerard ’t Hooft and M.J.G. Veltman. One loop divergencies in the theory of gravitation. Annales Poincare Phys. Theor., A20:69–94, 1974. 15

View on Pith 1974
[3]

Marc H. Goroff and Augusto Sagnotti. The Ultraviolet Behavior of Einstein Gravity.Nucl. Phys. B, 266:709, 1986

View on Pith 1986
[4]

David B. Malament. The class of continuous timelike curves determines the topology of spacetime.J. Math. Phys., 18(7):1399–1404, 1977

View on Pith 1977
[5]

S. W. Hawking, A. R. King, and P. J. McCarthy. A New Topology for Curved Space-Time Which Incorporates the Causal, Differential, and Conformal Structures.J. Math. Phys., 17:174–181, 1976

View on Pith 1976
[6]

H. Weyl. Gravitation and electricity.Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), 1918:465, 1918

View on Pith 1918
[7]

Daniel F. Litim. Fixed points of quantum gravity.Phys. Rev. Lett., 92:201301, 2004

View on Pith 2004
[8]

Assaf Shomer. A Pedagogical explanation for the non-renormalizability of gravity. 2007, arXiv:0709.3555

View on Pith Pith review arXiv 2007
[9]

Steven Weinberg. Critical Phenomena for Field Theorists. In14th International School of Subnuclear Physics: Understanding the Fundamental Constituents of Matter, 1976

View on Pith 1976
[10]

Thomas P. Sotiriou and Valerio Faraoni. f(R) Theories Of Gravity.Rev. Mod. Phys., 82:451–497, 2010

View on Pith 2010
[11]

Gonzalo J. Olmo. Palatini Approach to Modified Gravity: f(R) Theories and Beyond.Int. J. Mod. Phys. D, 20:413–462, 2011

View on Pith 2011
[12]

Ariel Edery and Yu Nakayama. Palatini formulation of pureR2 gravity yields Einstein gravity with no massless scalar.Phys. Rev. D, 99(12):124018, 2019

View on Pith 2019
[13]

K. S. Stelle. Classical Gravity with Higher Derivatives.Gen. Rel. Grav., 9:353–371, 1978

View on Pith 1978
[14]

Daniel Coumbe. Asymptotically Weyl-Invariant Gravity.Int. J. Mod. Phys. A, 34(31):1950205, 2019

View on Pith 2019
[15]

Daniel Coumbe. Is Asymptotically Weyl-Invariant Gravity Viable?Phys. Rev. D, 103(8):084050, 2021

View on Pith 2021
[16]

Daniel Coumbe and Aria Rahmaty. Falsifiable alternative to general relativity.Phys. Rev. D, 112(8):084005, 2025

View on Pith 2025
[17]

Daniel Coumbe. What is dimensional reduction really telling us? InProceedings, 14th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories (MG14) (In 4 Volumes): Rome, Italy, July 12-18, 2015, volume 4, pages 3960–3965, 2017

View on Pith 2015
[18]

D. N. Coumbe. Renormalizing Spacetime.Int. J. Mod. Phys. D, 27(16):1950008, 2018

View on Pith 2018
[19]

Daniel Coumbe. Quantum gravity without vacuum dispersion.Int. J. Mod. Phys. D, 26(10):1750119, 2017

View on Pith 2017
[20]

D. N. Coumbe. Hypothesis on the Nature of Time.Phys. Rev. D, 91(12):124040, 2015

View on Pith 2015
[21]

Mohamed M. Anber and John F. Donoghue. On the running of the gravitational constant. Phys. Rev. D, 85:104016, 2012

View on Pith 2012
[22]

Christian Brouder and Michael Dutsch. Relating on-shell and off-shell formalism in pertur- bative quantum field theory.J. Math. Phys., 49:052303, 2008. 16

View on Pith 2008
[23]

Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexan- der Postnikov, and Jaroslav Trnka.Grassmannian Geometry of Scattering Amplitudes. Cambridge University Press, 2016

View on Pith 2016
[24]

John F. Donoghue. A Critique of the Asymptotic Safety Program.Front. in Phys., 8:56, 2020

View on Pith 2020
[25]

M. Reuter. Nonperturbative evolution equation for quantum gravity.Phys. Rev. D, 57:971–985, 1998

View on Pith 1998
[26]

Holger Gies and Abdol Sabor Salek. Asymptotically safe Hilbert–Palatini gravity in an on-shell reduction scheme.Eur. Phys. J. C, 83(2):173, 2023

View on Pith 2023
[27]

Oleg Melichev. On the Renormalization of Metric-Affine Gravity Theories, 2024. arXiv:2406.14146 [hep-th]

View on Pith Pith review arXiv 2024
[28]

Thomas P. Sotiriou and Stefano Liberati. Metric-affine f(R) theories of gravity.Annals Phys., 322:935–966, 2007

View on Pith 2007
[29]

Cecilia Bejarano, Adria Delhom, Alejandro Jiménez-Cano, Gonzalo J. Olmo, and Diego Rubiera-Garcia. Geometric inequivalence of metric and Palatini formulations of General Relativity.Phys. Lett. B, 802:135275, 2020

View on Pith 2020
[30]

Andrei O. Barvinsky, Diego Blas, Mario Herrero-Valea, Sergey M. Sibiryakov, and Chris- tian F. Steinwachs. Renormalization of gauge theories in the background-field approach. JHEP, 07:035, 2018

View on Pith 2018
[31]

S. A. Fulling, Ronald C. King, B. G. Wybourne, and C. J. Cummins. Normal forms for tensor polynomials. 1: The Riemann tensor.Class. Quant. Grav., 9:1151–1197, 1992

View on Pith 1992
[32]

D. Lovelock. The Einstein tensor and its generalizations.J. Math. Phys., 12:498–501, 1971

View on Pith 1971
[33]

M. J. Duff. Twenty years of the Weyl anomaly.Class. Quant. Grav., 11:1387–1404, 1994

View on Pith 1994
[34]

Stanley Deser and A. Schwimmer. Geometric classification of conformal anomalies in arbitrary dimensions.Phys. Lett. B, 309:279–284, 1993. 17

View on Pith 1993

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

José M. M. Senovilla and David Garfinkle. The 1965 Penrose singularity theorem.Class. Quant. Grav., 32(12):124008, 2015

View on Pith 1965

[2] [2]

Gerard ’t Hooft and M.J.G. Veltman. One loop divergencies in the theory of gravitation. Annales Poincare Phys. Theor., A20:69–94, 1974. 15

View on Pith 1974

[3] [3]

Marc H. Goroff and Augusto Sagnotti. The Ultraviolet Behavior of Einstein Gravity.Nucl. Phys. B, 266:709, 1986

View on Pith 1986

[4] [4]

David B. Malament. The class of continuous timelike curves determines the topology of spacetime.J. Math. Phys., 18(7):1399–1404, 1977

View on Pith 1977

[5] [5]

S. W. Hawking, A. R. King, and P. J. McCarthy. A New Topology for Curved Space-Time Which Incorporates the Causal, Differential, and Conformal Structures.J. Math. Phys., 17:174–181, 1976

View on Pith 1976

[6] [6]

H. Weyl. Gravitation and electricity.Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), 1918:465, 1918

View on Pith 1918

[7] [7]

Daniel F. Litim. Fixed points of quantum gravity.Phys. Rev. Lett., 92:201301, 2004

View on Pith 2004

[8] [8]

Assaf Shomer. A Pedagogical explanation for the non-renormalizability of gravity. 2007, arXiv:0709.3555

View on Pith Pith review arXiv 2007

[9] [9]

Steven Weinberg. Critical Phenomena for Field Theorists. In14th International School of Subnuclear Physics: Understanding the Fundamental Constituents of Matter, 1976

View on Pith 1976

[10] [10]

Thomas P. Sotiriou and Valerio Faraoni. f(R) Theories Of Gravity.Rev. Mod. Phys., 82:451–497, 2010

View on Pith 2010

[11] [11]

Gonzalo J. Olmo. Palatini Approach to Modified Gravity: f(R) Theories and Beyond.Int. J. Mod. Phys. D, 20:413–462, 2011

View on Pith 2011

[12] [12]

Ariel Edery and Yu Nakayama. Palatini formulation of pureR2 gravity yields Einstein gravity with no massless scalar.Phys. Rev. D, 99(12):124018, 2019

View on Pith 2019

[13] [13]

K. S. Stelle. Classical Gravity with Higher Derivatives.Gen. Rel. Grav., 9:353–371, 1978

View on Pith 1978

[14] [14]

Daniel Coumbe. Asymptotically Weyl-Invariant Gravity.Int. J. Mod. Phys. A, 34(31):1950205, 2019

View on Pith 2019

[15] [15]

Daniel Coumbe. Is Asymptotically Weyl-Invariant Gravity Viable?Phys. Rev. D, 103(8):084050, 2021

View on Pith 2021

[16] [16]

Daniel Coumbe and Aria Rahmaty. Falsifiable alternative to general relativity.Phys. Rev. D, 112(8):084005, 2025

View on Pith 2025

[17] [17]

Daniel Coumbe. What is dimensional reduction really telling us? InProceedings, 14th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories (MG14) (In 4 Volumes): Rome, Italy, July 12-18, 2015, volume 4, pages 3960–3965, 2017

View on Pith 2015

[18] [18]

D. N. Coumbe. Renormalizing Spacetime.Int. J. Mod. Phys. D, 27(16):1950008, 2018

View on Pith 2018

[19] [19]

Daniel Coumbe. Quantum gravity without vacuum dispersion.Int. J. Mod. Phys. D, 26(10):1750119, 2017

View on Pith 2017

[20] [20]

D. N. Coumbe. Hypothesis on the Nature of Time.Phys. Rev. D, 91(12):124040, 2015

View on Pith 2015

[21] [21]

Mohamed M. Anber and John F. Donoghue. On the running of the gravitational constant. Phys. Rev. D, 85:104016, 2012

View on Pith 2012

[22] [22]

Christian Brouder and Michael Dutsch. Relating on-shell and off-shell formalism in pertur- bative quantum field theory.J. Math. Phys., 49:052303, 2008. 16

View on Pith 2008

[23] [23]

Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexan- der Postnikov, and Jaroslav Trnka.Grassmannian Geometry of Scattering Amplitudes. Cambridge University Press, 2016

View on Pith 2016

[24] [24]

John F. Donoghue. A Critique of the Asymptotic Safety Program.Front. in Phys., 8:56, 2020

View on Pith 2020

[25] [25]

M. Reuter. Nonperturbative evolution equation for quantum gravity.Phys. Rev. D, 57:971–985, 1998

View on Pith 1998

[26] [26]

Holger Gies and Abdol Sabor Salek. Asymptotically safe Hilbert–Palatini gravity in an on-shell reduction scheme.Eur. Phys. J. C, 83(2):173, 2023

View on Pith 2023

[27] [27]

Oleg Melichev. On the Renormalization of Metric-Affine Gravity Theories, 2024. arXiv:2406.14146 [hep-th]

View on Pith Pith review arXiv 2024

[28] [28]

Thomas P. Sotiriou and Stefano Liberati. Metric-affine f(R) theories of gravity.Annals Phys., 322:935–966, 2007

View on Pith 2007

[29] [29]

Cecilia Bejarano, Adria Delhom, Alejandro Jiménez-Cano, Gonzalo J. Olmo, and Diego Rubiera-Garcia. Geometric inequivalence of metric and Palatini formulations of General Relativity.Phys. Lett. B, 802:135275, 2020

View on Pith 2020

[30] [30]

Andrei O. Barvinsky, Diego Blas, Mario Herrero-Valea, Sergey M. Sibiryakov, and Chris- tian F. Steinwachs. Renormalization of gauge theories in the background-field approach. JHEP, 07:035, 2018

View on Pith 2018

[31] [31]

S. A. Fulling, Ronald C. King, B. G. Wybourne, and C. J. Cummins. Normal forms for tensor polynomials. 1: The Riemann tensor.Class. Quant. Grav., 9:1151–1197, 1992

View on Pith 1992

[32] [32]

D. Lovelock. The Einstein tensor and its generalizations.J. Math. Phys., 12:498–501, 1971

View on Pith 1971

[33] [33]

M. J. Duff. Twenty years of the Weyl anomaly.Class. Quant. Grav., 11:1387–1404, 1994

View on Pith 1994

[34] [34]

Stanley Deser and A. Schwimmer. Geometric classification of conformal anomalies in arbitrary dimensions.Phys. Lett. B, 309:279–284, 1993. 17

View on Pith 1993