pith. machine review for the scientific record. sign in

arxiv: 2510.17320 · v3 · submitted 2025-10-20 · ✦ hep-th · gr-qc

FeynGrav 4.0

Boris Latosh This is my paper

Pith reviewed 2026-05-18 06:31 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords FeynGravFeynman rulesBRST formalismquadratic gravityCheung-Remmen variablesghost interactionsgeneral relativityquantum gravity
0
0 comments X

The pith

FeynGrav 4.0 implements a refined BRST formalism and Cheung-Remmen variables to produce finite Feynman rules for ghosts and gravitons in general relativity and quadratic gravity.

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

The paper updates the FeynGrav package to handle Feynman rules in gravity theories more cleanly. It introduces a more sophisticated BRST implementation that limits ghost-graviton interactions to a finite set instead of an infinite collection. It also adds Feynman rules based on Cheung-Remmen variables, which rewrite the general relativity action into polynomial form and thereby generate only a finite number of interaction vertices. A higher-derivative gauge-fixing term is included for quadratic gravity models. Minor usability improvements round out the release.

Core claim

The new version of the package supplies a finite collection of interaction rules between ghosts and gravitons for both general relativity and quadratic gravity through an improved BRST formalism, together with a complete set of Feynman rules derived from the polynomial Cheung-Remmen formulation of the Einstein-Hilbert action.

What carries the argument

The BRST formalism implementation combined with Cheung-Remmen variable rewriting, which together convert the gravitational action into a form that yields only finitely many interaction vertices.

If this is right

  • Calculations involving ghosts in quadratic gravity can now be performed with a closed, finite set of diagrams.
  • The polynomial form supplied by Cheung-Remmen variables allows systematic generation of all interaction terms without truncation.
  • Higher-derivative gauge fixing becomes available as a built-in option for quadratic gravity models.
  • Users obtain a practical tool for automated Feynman-rule extraction in theories that previously required manual handling of infinite vertex towers.

Where Pith is reading between the lines

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

  • The finite-rule structure may reduce the computational cost of one-loop and higher calculations in effective field theories of gravity.
  • Similar rewriting techniques could be tested on other higher-curvature or modified-gravity actions to check whether they also admit finite ghost expansions.
  • Automated consistency checks between the BRST-derived rules and the Cheung-Remmen rules could be added in future package versions to cross-validate the implementation.

Load-bearing premise

The new BRST code and the Cheung-Remmen rewriting have been implemented without algebraic omissions or errors that would restore infinite or incorrect interaction vertices.

What would settle it

Generate the full set of ghost-graviton vertices for a simple four-graviton process using the updated package and verify by direct comparison that no additional vertices appear beyond those explicitly listed in the finite rule set.

read the original abstract

We present the new version of FeynGrav, a package that provides a set of tools to work with Feynman rules for gravity models. The new version addresses two principal issues and includes changes that improve user experience. Firstly, we present a more sophisticated implementation of the BRST formalism for general relativity and quadratic gravity, which results in a finite set of interaction rules between ghosts and gravitons. We also implement a realisation of a higher derivative gauge fixing term for quadratic gravity. Secondly, we implement Feynman rules for Cheung-Remmen variables. These variables present the general relativity action in a polynomial form and produce a finite set of Feynman rules. Lastly, we introduce some minor quality-of-life changes to the package to improve the user experience.

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

3 major / 1 minor

Summary. The paper presents FeynGrav 4.0, an update to a Mathematica package for computing Feynman rules in gravity models. It describes a refined BRST implementation for general relativity and quadratic gravity that yields a finite set of ghost-graviton interaction rules, introduces a higher-derivative gauge-fixing term for quadratic gravity, implements Feynman rules using Cheung-Remmen variables that also produce a finite set of rules, and includes minor quality-of-life improvements.

Significance. If the implementations correctly achieve the claimed finiteness without algebraic omissions, the package would provide a practical tool for perturbative gravity calculations, reducing the burden of infinite vertex series in BRST-quantized theories and polynomial formulations of GR.

major comments (3)
  1. Abstract and BRST implementation section: the claim that the new BRST formalism 'results in a finite set of interaction rules between ghosts and gravitons' is presented without an explicit list of the retained vertices, a termination proof, or sample output showing absence of higher-order terms, which is required to substantiate the central finiteness assertion.
  2. Cheung-Remmen variables section: the statement that these variables 'produce a finite set of Feynman rules' lacks accompanying explicit vertex expressions or cross-checks against known amplitudes, leaving the correctness of the rewriting unverified in the manuscript.
  3. Overall package description: no test suite, reproducible code snippets, or verification examples are supplied to allow independent confirmation that the BRST and Cheung-Remmen modules do not reintroduce infinite or incorrect interactions, directly bearing on the soundness of the reported updates.
minor comments (1)
  1. The manuscript would benefit from clearer notation distinguishing the new BRST module from prior versions and from specifying the exact Mathematica version and dependencies required for the package.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript describing FeynGrav 4.0. We address each major point below, clarifying the basis for the finiteness claims and indicating where we will strengthen the presentation with additional evidence in the revised version.

read point-by-point responses

  1. Referee: Abstract and BRST implementation section: the claim that the new BRST formalism 'results in a finite set of interaction rules between ghosts and gravitons' is presented without an explicit list of the retained vertices, a termination proof, or sample output showing absence of higher-order terms, which is required to substantiate the central finiteness assertion.

    Authors: The finiteness follows from the structure of the refined BRST transformations combined with the higher-derivative gauge-fixing term, which truncates ghost-graviton interactions at a finite order by construction. The manuscript emphasizes the implementation rather than exhaustive enumeration. In the revision we will add an appendix containing the complete list of retained vertices, a concise explanation of the truncation mechanism, and sample package output confirming the absence of higher-order terms. revision: yes

  2. Referee: Cheung-Remmen variables section: the statement that these variables 'produce a finite set of Feynman rules' lacks accompanying explicit vertex expressions or cross-checks against known amplitudes, leaving the correctness of the rewriting unverified in the manuscript.

    Authors: The Cheung-Remmen polynomial variables recast the Einstein-Hilbert action such that only a finite number of interaction vertices appear. The package generates these vertices automatically; the manuscript describes the implementation without reproducing the full (lengthy) expressions. We will incorporate explicit cross-checks of low-point amplitudes against established results from the literature in the revised manuscript. revision: yes

  3. Referee: Overall package description: no test suite, reproducible code snippets, or verification examples are supplied to allow independent confirmation that the BRST and Cheung-Remmen modules do not reintroduce infinite or incorrect interactions, directly bearing on the soundness of the reported updates.

    Authors: We agree that documented verification examples improve usability and confidence in the results. The package contains internal consistency checks, but these were not illustrated in the text. In the revised version we will include a dedicated section with reproducible code snippets that generate the finite vertex sets for both the BRST and Cheung-Remmen modules and compare them to expected outcomes. revision: yes

Circularity Check

0 steps flagged

No circularity: software update paper reports implementation changes without any derivation or self-referential reduction

full rationale

The manuscript describes updates to the FeynGrav package, including a revised BRST implementation claimed to yield finite ghost-graviton rules and a Cheung-Remmen variable rewriting claimed to produce finite Feynman rules. No equations, fitted parameters, predictions, or first-principles derivations are presented that could reduce to their own inputs by construction. The central claims rest on the correctness of the coded modules rather than on any mathematical chain that loops back to itself. Self-citations, if present, are not load-bearing for any derivation because none exists. This is a standard software-release note whose validity is externally checkable via the released code and is therefore scored as having no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The package rests on the standard BRST quantization procedure and the algebraic equivalence of Cheung-Remmen variables to the Einstein-Hilbert action; no new free parameters, ad-hoc axioms, or postulated entities are introduced beyond those already present in the referenced quantum-gravity literature.

axioms (2)
  • domain assumption BRST symmetry is preserved under the chosen gauge-fixing procedure for both GR and quadratic gravity
    Invoked when claiming the new implementation yields a finite ghost-graviton vertex set.
  • domain assumption Cheung-Remmen variables produce an exactly polynomial action equivalent to the Einstein-Hilbert term
    Required for the claim that the resulting Feynman rules are finite.

pith-pipeline@v0.9.0 · 5635 in / 1396 out tokens · 31485 ms · 2026-05-18T06:31:25.100035+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages · 14 internal anchors

  1. [1]

    Bryce S. DeWitt. Quantum Theory of Gravity. 1. The Canonical Theory.Phys. Rev., 160:1113–1148, 1967. doi:10.1103/PhysRev.160.1113

    work page doi:10.1103/physrev.160.1113 1967

  2. [2]

    Bryce S. DeWitt. Quantum Theory of Gravity. 2. The Manifestly Covariant Theory.Phys. Rev., 162:1195– 1239, 1967.doi:10.1103/PhysRev.162.1195

    work page doi:10.1103/physrev.162.1195 1967

  3. [3]

    Bryce S. DeWitt. Quantum Theory of Gravity. 3. Applications of the Covariant Theory.Phys. Rev., 162:1239–1256, 1967.doi:10.1103/PhysRev.162.1239

    work page doi:10.1103/physrev.162.1239 1967

  4. [4]

    Y. Iwasaki. Quantum theory of gravitation vs. classical theory. - fourth-order potential.Prog. Theor. Phys., 46:1587–1609, 1971.doi:10.1143/PTP.46.1587

    work page doi:10.1143/ptp.46.1587 1971

  5. [5]

    M. J. G. Veltman. Quantum Theory of Gravitation.Conf. Proc. C, 7507281:265–327, 1975

    work page 1975

  6. [6]

    General relativity as an effective field theory: The leading quantum corrections

    John F. Donoghue. General relativity as an effective field theory: The leading quantum corrections.Phys. Rev. D, 50:3874–3888, 1994.arXiv:gr-qc/9405057,doi:10.1103/PhysRevD.50.3874. 22

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.50.3874 1994

  7. [7]

    Gravitational interaction to one loop in effective quantum gravity

    Arif A. Akhundov, S. Bellucci, and A. Shiekh. Gravitational interaction to one loop in effective quantum gravity.Phys. Lett. B, 395:16–23, 1997.arXiv:gr-qc/9611018,doi:10.1016/S0370-2693(96)01694-2

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0370-2693(96)01694-2 1997

  8. [8]

    Gravity-Matter Feynman Rules for any Valence.Class

    David Prinz. Gravity-Matter Feynman Rules for any Valence.Class. Quant. Grav., 38(21):215003, 2021. arXiv:2004.09543,doi:10.1088/1361-6382/ac1cc9

    work page doi:10.1088/1361-6382/ac1cc9 2021

  9. [9]

    FeynGrav: FeynCalc extension for gravity amplitudes.Class

    Boris Latosh. FeynGrav: FeynCalc extension for gravity amplitudes.Class. Quant. Grav., 39(16):165006, 2022.arXiv:2201.06812,doi:10.1088/1361-6382/ac7e15

    work page doi:10.1088/1361-6382/ac7e15 2022

  10. [10]

    FeynGrav 2.0.Comput

    Boris Latosh. FeynGrav 2.0.Comput. Phys. Commun., 292:108871, 2023.arXiv:2302.14310,doi: 10.1016/j.cpc.2023.108871

    work page doi:10.1016/j.cpc.2023.108871 2023

  11. [11]

    FeynGrav 3.0.Comput

    Boris Latosh. FeynGrav 3.0.Comput. Phys. Commun., 310:109508, 2025.arXiv:2406.14872,doi: 10.1016/j.cpc.2025.109508

    work page doi:10.1016/j.cpc.2025.109508 2025

  12. [12]

    Boris N. Latosh. FeynGrav.https://github.com/BorisNLatosh/FeynGrav, 2024

    work page 2024

  13. [13]

    FeynCalc 10: Do multiloop integrals dream of computer codes?Comput

    Vladyslav Shtabovenko, Rolf Mertig, and Frederik Orellana. FeynCalc 10: Do multiloop integrals dream of computer codes?Comput. Phys. Commun., 306:109357, 2025.arXiv:2312.14089,doi:10.1016/j.cpc. 2024.109357

    work page doi:10.1016/j.cpc 2025

  14. [14]

    FeynCalc 9.3: New features and improvements

    Vladyslav Shtabovenko, Rolf Mertig, and Frederik Orellana. FeynCalc 9.3: New features and improvements. Comput. Phys. Commun., 256:107478, 2020.arXiv:2001.04407,doi:10.1016/j.cpc.2020.107478

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.cpc.2020.107478 2020

  15. [15]

    New Developments in FeynCalc 9.0

    Vladyslav Shtabovenko, Rolf Mertig, and Frederik Orellana. New Developments in FeynCalc 9.0.Comput. Phys. Commun., 207:432–444, 2016.arXiv:1601.01167,doi:10.1016/j.cpc.2016.06.008

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.cpc.2016.06.008 2016

  16. [16]

    Mertig, M

    R. Mertig, M. Bohm, and Ansgar Denner. FEYN CALC: Computer algebraic calculation of Feynman amplitudes.Comput. Phys. Commun., 64:345–359, 1991.doi:10.1016/0010-4655(91)90130-D

    work page doi:10.1016/0010-4655(91)90130-d 1991

  17. [17]

    Becchi, A

    C. Becchi, A. Rouet, and R. Stora. Renormalization of the Abelian Higgs-Kibble Model.Commun. Math. Phys., 42:127–162, 1975.doi:10.1007/BF01614158

    work page doi:10.1007/bf01614158 1975

  18. [18]

    Becchi, A

    C. Becchi, A. Rouet, and R. Stora. The Abelian Higgs-Kibble Model. Unitarity of the S Operator.Phys. Lett. B, 52:344–346, 1974.doi:10.1016/0370-2693(74)90058-6

    work page doi:10.1016/0370-2693(74)90058-6 1974

  19. [19]

    Becchi, A

    C. Becchi, A. Rouet, and R. Stora. Renormalization of Gauge Theories.Annals Phys., 98:287–321, 1976. doi:10.1016/0003-4916(76)90156-1

    work page doi:10.1016/0003-4916(76)90156-1 1976

  20. [20]

    I. V. Tyutin. Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism. 1975. arXiv:0812.0580

    work page internal anchor Pith review Pith/arXiv arXiv 1975

  21. [21]

    L. D. Faddeev and V. N. Popov. Feynman Diagrams for the Yang-Mills Field.Phys. Lett. B, 25:29–30, 1967.doi:10.1016/0370-2693(67)90067-6

    work page doi:10.1016/0370-2693(67)90067-6 1967

  22. [22]

    L. D. Faddeev and V. N. Popov. Covariant quantization of the gravitational field.Usp. Fiz. Nauk, 111:427– 450, 1973.doi:10.1070/PU1974v016n06ABEH004089

    work page doi:10.1070/pu1974v016n06abeh004089 1973

  23. [23]

    Some remarks on high derivative quantum gravity

    M. Asorey, J. L. Lopez, and I. L. Shapiro. Some remarks on high derivative quantum gravity.Int. J. Mod. Phys. A, 12:5711–5734, 1997.arXiv:hep-th/9610006,doi:10.1142/S0217751X97002991

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0217751x97002991 1997

  24. [24]

    Leonardo Modesto, Les law Rachwa l, and Ilya L. Shapiro. Renormalization group in super-renormalizable quantum gravity.Eur. Phys. J. C, 78(7):555, 2018.arXiv:1704.03988,doi:10.1140/epjc/ s10052-018-6035-2

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epjc/ 2018

  25. [25]

    Clifford Cheung and Grant N. Remmen. Hidden Simplicity of the Gravity Action.JHEP, 09:002, 2017. arXiv:1705.00626,doi:10.1007/JHEP09(2017)002

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep09(2017)002 2017

  26. [26]

    L. D. Landau and E. M. Lifschits.The Classical Theory of Fields, volume Volume 2 ofCourse of Theoretical Physics. Pergamon Press, Oxford, 1975

    work page 1975

  27. [27]

    J. N. Goldberg. Conservation Laws in General Relativity.Phys. Rev., 111:315–320, 1958.doi:10.1103/ PhysRev.111.315

    work page 1958

  28. [28]

    D. M. Capper, G. Leibbrandt, and M. Ramon Medrano. Calculation of the graviton selfenergy using dimensional regularization.Phys. Rev. D, 8:4320–4331, 1973.doi:10.1103/PhysRevD.8.4320

    work page doi:10.1103/physrevd.8.4320 1973

  29. [29]

    I. l. Buchbinder and I. l. Shapiro. One Loop Calculation of the Polarization Operator of Gravitons in the First Order Formalism.Soviet Journal of Nuclear Physics, 37:248–252, 1983. 23

    work page 1983

  30. [30]

    D. G. C. McKeon. The First order formalism for Yang-Mills theory.Can. J. Phys., 72:601–607, 1994. doi:10.1139/p94-077

    work page doi:10.1139/p94-077 1994

  31. [31]

    M. Yu. Kalmykov and P. I. Pronin. The One loop divergences and renormalizability of the minimal gauge theory of gravity.Gen. Rel. Grav., 27:873–886, 1995.arXiv:hep-th/9412177,doi:10.1007/BF02113069

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/bf02113069 1995

  32. [32]

    Feynman rules and beta-function for the BF Yang-Mills Theory

    Maurizio Martellini and Mauro Zeni. Feynman rules and beta function for the BF Yang-Mills theory.Phys. Lett. B, 401:62–68, 1997.arXiv:hep-th/9702035,doi:10.1016/S0370-2693(97)00379-1

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0370-2693(97)00379-1 1997

  33. [33]

    F. T. Brandt and D. G. C. McKeon. Perturbative Calculations with the First Order Form of Gauge Theories.Phys. Rev. D, 91(10):105006, 2015.arXiv:1503.02598,doi:10.1103/PhysRevD.91.105006

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.91.105006 2015

  34. [34]

    F. T. Brandt and D. G. C. McKeon. Radiative Corrections and the Palatini Action.Phys. Rev. D, 93(10):105037, 2016.arXiv:1601.04944,doi:10.1103/PhysRevD.93.105037

    work page doi:10.1103/physrevd.93.105037 2016

  35. [35]

    D. G. C. McKeon, J. Frenkel, S. Martins-Filho, and F. T. Brandt. Consistency Conditions for the First- Order Formulation of Yang-Mills Theory.Phys. Rev. D, 101(8):085013, 2020.arXiv:2003.06819,doi: 10.1103/PhysRevD.101.085013

    work page doi:10.1103/physrevd.101.085013 2020

  36. [36]

    F. T. Brandt, J. Frenkel, S. Martins-Filho, and D. G. C. McKeon. Structural identities in the first order formulation of quantum gravity.Phys. Rev. D, 102(4):045013, 2020.arXiv:2007.04841,doi:10.1103/ PhysRevD.102.045013

    work page arXiv 2020

  37. [37]

    F. T. Brandt, J. Frenkel, S. Martins-Filho, and D. G. C. McKeon. On the equivalence of first and second order formulations of the Einstein-Hilbert theory. 10 2025.arXiv:2510.17615

    work page arXiv 2025

  38. [38]

    Ruth Britto, Riccardo Gonzo, and Guy R. Jehu. Graviton particle statistics and coherent states from classical scattering amplitudes.JHEP, 03:214, 2022.arXiv:2112.07036,doi:10.1007/JHEP03(2022)214

    work page doi:10.1007/jhep03(2022)214 2022

  39. [39]

    The Becchi-rouet-stora Transformation for the Gravitational Field.Prog

    Kazuhiko Nishijima and Masanori Okawa. The Becchi-rouet-stora Transformation for the Gravitational Field.Prog. Theor. Phys., 60:272, 1978.doi:10.1143/PTP.60.272

    work page doi:10.1143/ptp.60.272 1978

  40. [40]

    E. S. Fradkin and I. V. Tyutin. S matrix for Yang-Mills and gravitational fields.Phys. Rev. D, 2:2841–2857, 1970.doi:10.1103/PhysRevD.2.2841

    work page doi:10.1103/physrevd.2.2841 1970

  41. [41]

    BRST and Anti-BRST Symmetries in Perturbative Quantum Gravity

    Mir Faizal. BRST and Anti-BRST Symmetries in Perturbative Quantum Gravity.Found. Phys., 41:270– 277, 2011.arXiv:1010.1143,doi:10.1007/s10701-010-9511-6

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s10701-010-9511-6 2011

  42. [42]

    General Procedure of Gauge Fixings and Ghosts.Phys

    Nobuyoshi Ohta. General Procedure of Gauge Fixings and Ghosts.Phys. Lett. B, 811:135965, 2020. arXiv:2010.11314,doi:10.1016/j.physletb.2020.135965

    work page doi:10.1016/j.physletb.2020.135965 2020

  43. [43]

    Shapiro.The Background Information About Perturbative Quantum Gravity

    Ilya L. Shapiro.The Background Information About Perturbative Quantum Gravity. 2023.arXiv:2210. 12319,doi:10.1007/978-981-19-3079-9_8-1

    work page doi:10.1007/978-981-19-3079-9_8-1 2023

  44. [44]

    Spivak.A Comprehensive Introduction to Differential Geometry

    M. Spivak.A Comprehensive Introduction to Differential Geometry. Number v. 1 in A Comprehensive Introduction to Differential Geometry. Publish or Perish, Incorporated, 1979

    work page 1979

  45. [45]

    Choquet-Bruhat, C

    Y. Choquet-Bruhat, C. DeWitt-Morette, and M. Dillard-Bleick.Analysis, Manifolds and Physics Revised Edition. Number v. 1 in Analysis, Manifolds and Physics Revised Edition. Elsevier Science, 1982

    work page 1982

  46. [46]

    Julve and M

    J. Julve and M. Tonin. Quantum Gravity with Higher Derivative Terms.Nuovo Cim. B, 46:137–152, 1978. doi:10.1007/BF02748637

    work page doi:10.1007/bf02748637 1978

  47. [47]

    E. S. Fradkin and Arkady A. Tseytlin. Renormalizable Asymptotically Free Quantum Theory of Gravity. Phys. Lett. B, 104:377–381, 1981.doi:10.1016/0370-2693(81)90702-4

    work page doi:10.1016/0370-2693(81)90702-4 1981

  48. [48]

    N. H. Barth and S. M. Christensen. Quantizing Fourth Order Gravity Theories. 1. The Functional Integral. Phys. Rev. D, 28:1876, 1983.doi:10.1103/PhysRevD.28.1876

    work page doi:10.1103/physrevd.28.1876 1983

  49. [49]

    Van Nieuwenhuizen

    P. Van Nieuwenhuizen. On ghost-free tensor lagrangians and linearized gravitation.Nucl. Phys. B, 60:478– 492, 1973.doi:10.1016/0550-3213(73)90194-6

    work page doi:10.1016/0550-3213(73)90194-6 1973

  50. [50]

    Accioly, S

    A. Accioly, S. Ragusa, H. Mukai, and E. C. de Rey Neto. Algorithm for computing the propagator for higher derivative gravity theories.Int. J. Theor. Phys., 39:1599–1608, 2000.doi:10.1023/A:1003632311419

    work page doi:10.1023/a:1003632311419 2000

  51. [51]

    K. S. Stelle. Renormalization of Higher Derivative Quantum Gravity.Phys. Rev. D, 16:953–969, 1977. doi:10.1103/PhysRevD.16.953. 24

    work page doi:10.1103/physrevd.16.953 1977

  52. [52]

    K. S. Stelle. Classical Gravity with Higher Derivatives.Gen. Rel. Grav., 9:353–371, 1978.doi:10.1007/ BF00760427

    work page 1978

  53. [53]

    Enrique Alvarez, Jesus Anero, and Carmelo P. Martin. One loop analysis of the cubic action for gravity. JHEP, 04:117, 2025.arXiv:2502.08434,doi:10.1007/JHEP04(2025)117

    work page doi:10.1007/jhep04(2025)117 2025

  54. [54]

    An Algorithm to Simplify Tensor Expressions

    R. Portugal. An algorithm to simplify tensor expressions.Comput. Phys. Commun., 115:215–230, 1998. arXiv:gr-qc/9803023,doi:10.1016/S0010-4655(98)00117-9

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0010-4655(98)00117-9 1998

  55. [55]

    Consistent Spin-Two Coupling and Quadratic Gravitation

    Ahmed Hindawi, Burt A. Ovrut, and Daniel Waldram. Consistent spin two coupling and quadratic gravi- tation.Phys. Rev. D, 53:5583–5596, 1996.arXiv:hep-th/9509142,doi:10.1103/PhysRevD.53.5583. 25

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.53.5583 1996