Pith. sign in

REVIEW 4 major objections 4 minor 17 references

One loop analysis of the cubic action for gravity

T0 review · 4 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A one-loop analysis shows that the cubic first-order action for gravity is BRST-consistent only with finite counterterms beyond the MS scheme, and that the connection–graviton two-point function, zero at tree level, is generated…

desk verdict Solid one-loop results for the Cheung-Remmen cubic action, but the Section 4 first-order/second-order reconciliation is not proven—send to a referee with that caveat. read the letter →

arxiv 2502.08434 v1 pith:M2N6NSCR submitted 2025-02-12 hep-th gr-qc

Enrique Alvarez , Jesus Anero , Carmelo P. Martin This is my paper · ORCID
classification hep-thgr-qc MSC 83C4581T1581T7083C05
keywords cubicgravityactionfirst-orderPalatiniformalismone-loopgravitonpropagatorSlavnov-TayloridentityBRSTinvariancecompositeoperatorsdimensionalregularizationfinitecounterterms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

At one loop around flat space, this paper studies the cubic action for pure gravity proposed by Cheung and Remmen, a first-order Palatini formulation in which the metric and the connection are independent fields. The central result is that the dimensionally regularized one-loop graviton two-point function and the one-loop BRST-insertion function satisfy the Slavnov-Taylor identity, the quantum Ward identity of gauge invariance, but the minimal-subtraction (MS) renormalized versions do not at $n=4$. The failure is not an anomaly: explicit finite counterterms, constrained by equation (36), restore the identity. The paper also finds that the mixed $h$-$B$ two-point function, zero at tree level, is generated at one loop and cannot be removed by a local field redefinition. To reconcile this with second-order gravity, it identifies first-order connection-field Green functions with normal products of a second-order composite operator, equation (45), so the apparent discrepancy is a statement about composite operators rather than about the spectrum of the theory.

What carries the argument

The argument is carried by completing the square in the Gaussian path integral over the connection field. The quadratic kernel $\Delta$ in Eq. (48) is algebraic, and its inverse $G$ in Eq. (53) is also algebraic, so the determinant $\det G^{-1/2}$ is local and the source coupling becomes a composite-operator source, Eq. (54). This is the step that converts first-order connection-field Green functions into Zimmermann normal products of the composite operator $B^{c}[g]$ in a second-order Hilbert-type action, and it is the mechanism behind the operator equality (45). On the BRST side, the load-bearing identity is Eq. (30), the one-loop linearized Slavnov-Taylor identity, whose coefficient matching in the tensor basis $T^{(i)}$ fixes the allowed finite counterterms.

What would settle it

Compute the one-loop mixed two-point function $\langle T B(x) h(y)\rangle$ directly from the second-order Hilbert action using the normal product $N[B]$ and compare with the first-order result in Eq. (45); any difference beyond the local contact term $\langle G\rangle$ in Eq. (55) would refute the claimed equality. Independently re-evaluating the pole parts of $\Gamma^{(hh)}$ and $\Gamma^{(RHO)}$ in strictly $n$ dimensions would also settle the point, since the paper's need for finite counterterms rests on those pole parts failing Eq. (30).

Watch

Extended reading notes

Core claim

The paper's central claim is that the dimensionally regularized one-loop graviton self-energy $\Gamma^{(hh)}_{\mu\nu\rho\sigma}(p)$ and the one-loop BRST insertion $\Gamma^{(RHO)\mu\nu}_{\lambda}(p)$ satisfy the Slavnov-Taylor identity (30) in $n$ dimensions, but after MS renormalization in $n=4$ they do not; the identity can be restored by choosing finite constants $F_i$ and $G_i$ subject to equations (36), so the theory has no BRST anomaly, only a scheme mismatch. It further claims that the mixed $h$-$B$ two-point function vanishes at tree level yet receives a one-loop radiative correction that cannot be removed by a local field redefinition. The proposed reconciliation with second-order gravity is the identity (45): the first-order Green function of two connection fields equals the second-order Green function of Zimmermann normal products of the composite operator $B$, because integrating out the connection in the first-order path integral turns the connection source into a composite-operator source, equation (54).

Load-bearing premise

The argument assumes that formally integrating out the connection by completing the square is valid, including the $\det G^{-1/2}$ factor and the application of Zimmermann normal products to a theory that is not renormalizable by power counting.

Editorial extensions

If this is right

  • A pure MS subtraction scheme is not BRST-compatible for the cubic first-order action, so a correct one-loop renormalization must include finite counterterms.
  • The one-loop nonzero $h$-$B$ propagator implies that the connection field and the graviton mix radiatively, and no local field redefinition removes the mixing.
  • The first-order connection two-point function should be read as a normal-product correlator in the second-order theory, not as an independent-field correlator.
  • The one-loop $B$-$B$ divergence requires counterterms outside the tree-level action, consistent with power-counting nonrenormalizability.

Reading between the lines

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

  • If Eq. (45) survives at higher loops, the first-order/second-order equivalence may be understood as an operator equivalence under the quantum equations of motion, so the $h$-$B$ radiative mixing would correspond to anomalous-dimension mixing of the composite operator $B$ with the graviton.
  • The finite-counterterm structure found here may be a generic feature of Palatini-like gravity, meaning other first-order calculations should state their subtraction scheme explicitly rather than quoting MS results as scheme-independent.
  • A two-loop check of the graviton self-energy in the cubic action, compared with the second-order result after applying Eq. (45), would locate the first place where the formal completion-of-square argument fails if the two disagree.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper studies the cubic first-order (Palatini-type) gravity action proposed by Cheung and Remmen. After a field redefinition to variables hμν and Bλμν, it fixes a gauge, derives BRST transformations and Feynman rules, and computes the one-loop 1PI two-point functions for hh, hB, and BB using dimensional regularization with FORM and Mathematica. It verifies that the dimensionally regularized graviton propagator satisfies the Slavnov-Taylor identity (30) together with the one-loop BRST insertion, that the MS renormalization scheme violates this identity at n=4, and that finite counterterms parametrized by constants Fi and Gi can restore it. It also finds a one-loop hB propagator despite a vanishing tree-level value and a divergent BB propagator whose counterterms are not in the tree action. Section 4 attempts to resolve the apparent conflict with second-order gravity by identifying first-order connection-field Green functions with normal products in the second-order action.

Significance. If correct, the paper provides a useful explicit one-loop data point for first-order quantum gravity. The Slavnov-Taylor check is nontrivial, self-contained, and presented with enough detail that the algebraic steps can be reproduced; the discussion of n-dimensional eta contractions and the MS-scheme violation is clear and valuable. The paper also ships complete coefficient lists and tensor decompositions, which is a strength. However, the claimed resolution of the first-order/second-order paradox in Section 4 is not established by the derivation given, and the 'no anomaly' conclusion is stronger than the single identity that is checked. The central loop computations appear sound, but the interpretive claim needs substantial work.

major comments (4)
  1. [Section 4, Eq. (45)] Equation (45) is stated for the B field, but the derivation in Eqs. (46)-(55) integrates out A and computes the A-A Green function in an ungauged path integral. Since B is related to A and h by Eq. (11), the B-B Green function contains h-B and h-h contributions whose matching to N[B]N[B] in the second-order theory is never shown. This is the central resolution of the first-order/second-order paradox, so the claim needs either a real derivation for B or a clear restriction of the statement to A.
  2. [Section 4, Eqs. (47) and (54)] The path integral in Eq. (47) contains no gauge-fixing term or ghost action, whereas the Green functions in Section 3 are computed in the specific gauge (12) with ghosts. A first-order/second-order equality must specify a common gauge-fixing procedure. Also, the determinant det G^{-1/2} in Eq. (54) is a functional of the metric and is moved outside the g integral without stated justification; in dimensional regularization it may be harmless for an algebraic kernel, but this needs to be explained rather than assumed.
  3. [Section 3.1, around Eq. (36)] The conclusion that 'no anomaly arises' is stronger than what is demonstrated. The restoration of the Slavnov-Taylor identity is shown only for the graviton-propagator identity (30); the h-B and B-B 1PI functions computed in Sections 3.2 and 3.3 are not checked against the linearized ST identity, and the finite constants Fi and Gi are not shown to correspond to local BRST-invariant counterterms. The paper should either check the remaining identities or qualify the claim to the specific identity that was verified.
  4. [Section 3.2] The statement that the one-loop h-B propagator 'cannot be removed by a local field redefinition' is asserted without proof. Since the computed coefficients contain log(p^2/mu^2) terms, a short nonlocality argument would suffice, but the argument should be included because this claim appears in the abstract and conclusions as a central result.
minor comments (4)
  1. [Section 4 and reference [15]] The text writes 'Zimmerman' in Section 4, while the reference list uses 'Zimmermann'; please correct the spelling.
  2. [Eq. (38)] In the definition of the tensor t^8 there appears a typo 'pnu2' instead of a properly typeset p_nu2; the full set of tensor definitions should be checked for similar typos.
  3. [Eqs. (26) and (41)] Some displayed formulas contain unbalanced parentheses or missing closing brackets, which makes verification harder; a careful proofreading pass over the Feynman-rule and tensor-basis equations is recommended.
  4. [Section 2] The sentence after Eq. (25) stating that sB has no contribution depending only on c_mu is important for the ST identity but is not demonstrated; a one-line explanation would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the explicit one-loop amplitudes and Slavnov-Taylor check are self-contained, with only a minor self-cited equivalence premise and an unsupported B-B normal-product identification that is a gap, not a circular reduction.

full rationale

The core loop computations in Sections 3.1-3.3 are direct diagrammatic evaluations in dimensional regularization. The Slavnov-Taylor check (Eqs. 30-36) verifies the computed Γ(hh) and Γ(RHO) against the external BRST identity; the finite counterterms Fi and Gi are free parameters chosen to satisfy that identity, not fitted to a quantity that is then called a prediction. No fitted input is relabeled as a prediction, and no ansatz is smuggled in via citation. The one-loop h-B propagator result is an explicit calculation, and its non-removability by local field redefinitions follows from the presence of nonlocal log terms, not from a circular definition. The paper does cite the authors' own prior work (ref. [6]) for the claim that first-order and second-order effective actions match at one loop, and this claim motivates the paradox and Section 4. However, that self-cited equivalence is not used to derive the explicit propagators or the Slavnov-Taylor check, so it is a minor, non-load-bearing self-citation. One flagged issue is that Eq. (45) asserts the first-order B-B propagator equals a second-order normal-product Green function, but the derivation in Section 4 (Eqs. 47-55) establishes the corresponding identity only for A-A, while B and A are related by derivative terms in Eq. (11). This is an omitted derivation or a gap in the claimed resolution, not a circular reduction by construction, and it does not affect the independence of the explicit one-loop results.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

All items the analysis depends on that are not derived in the paper. No parameters are fitted to data; the only free constants are finite renormalization constants needed to satisfy the ST identity. The main domain assumptions come from prior first-order/second-order equivalence claims and from formal path-integral manipulations in Section 4.

free parameters (1)
  • Finite renormalization constants F_i and G_i = Undetermined; constrained by equations (36)
    These constants are introduced in Section 3.1 to restore the Slavnov-Taylor identity in n=4 dimensions. They are renormalization-scheme choices rather than data fits, but they are free parameters in the central claim that no anomaly arises.
assumptions (3)
  • standard math Dimensional regularization with n=4+2epsilon and the Breitenlohner-Maison scheme is a valid regulator for the one-loop gravity computations.
    Used throughout Section 3; the Slavnov-Taylor identity in Eq. (30) explicitly relies on n-dimensional contractions such as eta_mu_nu eta^mu_nu = 4+2epsilon.
  • domain assumption The first-order Palatini and second-order Einstein-Hilbert actions are equivalent at one loop in the background field gauge, as claimed in refs. [5] and [6].
    This is the premise of Section 4's comparison and of the statement that first-order results can be mapped to composite operators in second order; it relies on prior work by Buchbinder-Shapiro and by the present authors.
  • ad hoc to paper The path integral over the connection field A can be performed exactly, and the resulting determinant det G^{-1/2} does not affect the identification in Eq. (45).
    In Section 4, Eqs. (47)-(55), the generating functional is reduced by completing squares over A; this formal manipulation is not fully justified for a non-renormalizable theory.

how reviews work

0 comments
Cite this review

Pith. "Pith review of One loop analysis of the cubic action for gravity." pith.science (2026). https://pith.science/paper/M2N6NSCR

@misc{pith2026250208434,
  author       = {Pith},
  title        = {Pith review of: One loop analysis of the cubic action for gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M2N6NSCR}},
  note         = {Machine review of arXiv:2502.08434}
}
read the original abstract

We analyze some aspects of the cubic action for gravity recently proposed by Cheung and Remmen, which is a particular instance of a first order (Palatini) action. In this approach both the spacetime metric and the connection are treated as independent fields. We discuss its BRST invariance and compute explicitly the one-loop contribution of quantum fluctuations around flat space, checking that the corresponding Slavnov-Taylor identities are fulfilled. Finally, our results on a first order action are compared with the existing ones corresponding to a second order action.

Figures

Figures reproduced from arXiv: 2502.08434 by the authors.

Figure 1
Figure 1. Graviton propagator 1-loop diagrams and C p1bq 1 “ i κ 2 16π 2 ´ 1 60ǫ ` 1 60 ln ” ´ e γ p 2 4πµ2 ı ´ 77 1800 ¯ ` Opǫq, C p1bq 2 “ C p1bq 1 , C p1bq 3 “ i κ 2 16π 2 ´ 13 120ǫ ` 13 120 ln ” ´ e γ p 2 4πµ2 ı ´ 56 225 ¯ ` Opǫq, C p1bq 4 “ i κ 2 16π 2 ´ 1 240ǫ ` 1 240 ln ” ´ e γ p 2 4πµ2 ı ´ 23 1800 ¯ ` Opǫq, C p1bq 5 “ i κ 2 16π 2 ´ 7 15ǫ ` 7 15 ln ” ´ e γ p 2 4πµ2 ı ´ 157 225 ¯ ` Opǫq, The fact that the propagator of … view at source ↗
Figure 2
Figure 2. Diagram with one ρµν insertion This is the very same result that it is obtained by substituting (31) in the right hand side of (30). We thus conclude that the regularized expressions in (28) and (31) are consistent with BRST invariance. We have seen that there is not a clash between the dimensionally regularized theory and BRST symmetry. But what about the renormalized theory? It has long been known that when the Wa… view at source ↗
Figure 3
Figure 3. Graviton-B propagator 1-loop diagram By working out the Feynman integrals we have obtained the following results Γ p3aqλ2 µ1ν1µ2ν2 ppq “ ř10 i“1 Ei t λ2 i µ1ν1µ2ν2 ppq Γ p3bqλ2 µ1ν1µ2ν2 ppq “ 0, with E1 “ ´ κ 2 16π 2 ´ 13 24ǫ ` 13 24 ln ” ´ e γ p 2 4πµ2 ı ´ 113 72 ¯ ` Opǫq, E2 “ ´ κ 2 16π 2 ´ ´ 1 2ǫ ´ 1 2 ln ” ´ e γ p 2 4πµ2 ı ` 2 3 ¯ ` Opǫq, E3 “ ´ κ 2 16π 2 ´ ´ 1 48ǫ ´ 1 48 ln ” ´ e γ p 2 4πµ2 ı ` 1 18 ¯ ` Opǫq, E… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Bλ µν propagator 1-loop diagrams A final comment. Notice that the counterterms needed to renormalize Γλ1λ2 µ1ν1µ2ν2 ppq in (43) are not part of the tree level action. Indeed, the theory is not renormalizable by power￾counting. Again, this is similar to what happens whe…
Figure 5
Figure 5. Figure 5: Free propagators 20 [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: Vertices 21 [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 11 canonical work pages

  1. [1]

    Hamiltonsches Prinzip und allgemeine Relativit¨ ats theorie

    Albert Einstein, “Hamiltonsches Prinzip und allgemeine Relativit¨ ats theorie”, Preuss. Akad.Wiss Berlin Sitzber 1111 (1916)

    work page 1916

  2. [2]

    Die Grundlagen der Physik

    David Hilbert, D. (1915) “Die Grundlagen der Physik”, Konigl. Gese ll. d. Wiss. G¨ ottin- gen, Nachr. Math.-Phys. Kl. 395

    work page 1915

  3. [3]

    Deduzione invariantiva delle equazioni gravitazionali d al principio di Hamil- ton,

    A. Palatini, “Deduzione invariantiva delle equazioni gravitazionali d al principio di Hamil- ton,” Rend. Circ. Mat. Palermo 43 (1919) no.1, 203-212 doi:10.1007/BF03014670

    doi:10.1007/bf03014670 1919

  4. [4]

    Physical content of Quadratic Gravity

    E. Alvarez, J. Anero, S. Gonzalez-Martin and R. Santos-Garcia , “Physical content of Quadratic Gravity,” Eur. Phys. J. C 78 (2018) no.10, 794 doi:10.1140/epjc/s10052-018- 6250-x [arXiv:1802.05922 [hep-th]]

    work page Pith review arXiv 2018

  5. [5]

    One loop counterterm deriva tion in the first order quantum gravity formalism,

    I. L. Buchbinder and I. L. Shapiro,“One loop counterterm deriva tion in the first order quantum gravity formalism,” Acta Phys. Polon. B 16 (1985), 103-107

    work page 1985

  6. [6]

    One-loop divergences in first order Einstein-Hilbert gravity

    E. Alvarez, J. Anero and R. Santos-Garcia, “One-loop divergen ces in first order Einstein- Hilbert gravity,” [arXiv:1706.02622 [hep-th]]. E. ´Alvarez and J. Anero, “Covariant techniques in Quantum Field Theor y,” Springer, 2022, ISBN 978-3-031-13725-9, 978-3-031-13726- 6 doi:10.1007/978-3-031- 13726-6 [arXiv:2203.11292 [hep-th]]

    work page Pith review arXiv 2022

  7. [7]

    One loop divergencies in the theo ry of gravitation,

    G. ’t Hooft and M. J. G. Veltman, “One loop divergencies in the theo ry of gravitation,” Ann. Inst. H. Poincare A Phys. Theor. 20 (1974), 69-94

    work page 1974

  8. [8]

    Twofold Symmetries of the Pure Gravity Action

    C. Cheung and G. N. Remmen, “Twofold Symmetries of the Pure Gr avity Action,” JHEP 01 (2017), 104 doi:10.1007/JHEP01(2017)104 [arXiv:1612.03927 [hep- th]]. “Hidden Simplicity of the Gravity Action,” JHEP 09 (2017), 002 doi:10.1007/JHEP09(2017)002 [arXiv:1705.00626 [hep-th]]

    work page Pith review arXiv 2017

Show all 17 references
  1. [9]

    Schwarzschild Metric from Sca ttering Am- plitudes to All Orders in GN,

    S. Mougiakakos and P. Vanhove, “Schwarzschild Metric from Sca ttering Am- plitudes to All Orders in GN,” Phys. Rev. Lett. 133 (2024) no.11, 111601 doi:10.1103/PhysRevLett.133.111601 [arXiv:2405.14421 [hep-th]]

    2024 arXiv

  2. [10]

    FORM version 4.2,

    B. Ruijl, T. Ueda and J. Vermaseren, “FORM version 4.2,” [arXiv:17 07.06453 [hep-ph]]

  3. [11]

    Wolfram Research, Inc, Mathematica 12.3, Champaign, IL (202 1)

  4. [12]

    Dimensional Renormalization an d the Action Princi- ple,

    P. Breitenlohner and D. Maison, “Dimensional Renormalization an d the Action Princi- ple,” Commun. Math. Phys. 52 (1977), 11-38 doi:10.1007/BF01609069

    1977 doi

  5. [13]

    Twenty years of the Weyl anomaly,

    M. J. Duff, “Twenty years of the Weyl anomaly,” Class. Quant. G rav. 11 (1994), 1387- 1404 doi:10.1088/0264-9381/11/6/004 [arXiv:hep-th/9308075 [he p-th]]. 24

    1994 arXiv

  6. [14]

    Calculatio n of the gravi- ton selfenergy using dimensional regularization,

    D. M. Capper, G. Leibbrandt and M. Ramon Medrano, “Calculatio n of the gravi- ton selfenergy using dimensional regularization,” Phys. Rev. D 8 (1973), 4320-4331 doi:10.1103/PhysRevD.8.4320

    1973 doi

  7. [15]

    Composite operators in the perturbation the ory of renormalizable interactions,

    W. Zimmermann, “Composite operators in the perturbation the ory of renormalizable interactions,” Annals Phys. 77 (1973), 536-569 doi:10.1016/0003-4916(73)90429-6

    1973 doi

  8. [16]

    Ward Identities in Gauge Theories,

    A. A. Slavnov, “Ward Identities in Gauge Theories,” Theor. Math . Phys. 10 (1972), 99-107 doi:10.1007/BF01090719 J. C. Taylor, “Ward Identities and Charge Renormalization of the Ya ng-Mills Field,” Nucl. Phys. B 33 (1971), 436-444 doi:10.1016/0550-3213(71)90297-5

    1972 doi

  9. [17]

    Introduction to BRS symmetry,

    C. Becchi, “Introduction to BRS symmetry,” [arXiv:hep-th/960 7181 [hep-th]]. 25

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.