REVIEW
Where is the conformal coupling in Palatini gravity?
General Relativity and Quantum Cosmology · Submitted 12 Oct 2026 · 44 references
Pith's verdict: Clear resolution of conformal coupling in Palatini gravity, with honest quantum side.
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Referee report
The paper
The paper claims
In Palatini gravity, there is no distinguished conformal coupling value because the scalar-curvature term is separately Weyl invariant, and the scalar propagates even when the kinetic coefficient vanishes.
The reading
Extended reading notes
Core claim
Tuning the curvature coupling cannot restore Weyl invariance in a Palatini theory with an ordinary scalar kinetic term because, for an independent and inert connection, the scalar-curvature term is separately Weyl-invariant, and no finite curvature coupling cancels the Weyl variation of the kinetic term. (From abstract and Section II.B.)
Load-bearing premise
The Weyl variation argument in Section II.B assumes the independent connection is inert under Weyl rescalings (it does not transform). This is standard in the Palatini formulation but is a structural choice; if the connection were allowed to transform through nonmetricity, the conclusion about the absence of a distinguished coupling might not hold, as the paper acknowledges in Section II.C. The one-loop calculations further assume a fixed metric background and perturbative regime, which the authors honestly state as limitations.
Original abstract
We study the conformal coupling in the torsion-free Palatini formulation with an ordinary scalar kinetic term. Tuning the curvature coupling cannot restore Weyl invariance in a Palatini theory with an ordinary scalar kinetic term because, for an independent and inert connection, the scalar-curvature term is separately Weyl-invariant, and no finite curvature coupling cancels the Weyl variation of the kinetic term. This can be demonstrated by evaluating the energy-momentum trace on the scalar equation of motion and noting that it is explicitly independent of the Palatini coupling. Eliminating the connection, we construct a map between the curvature coupling and the effective coupling for the equivalent metric theory, and demonstrate that the scalar continues to propagate even if the reduced kinetic coefficient vanishes. We also show that it becomes a Weyl compensator with a scale-invariant potential only in the rescaled infinite-coupling limit, which in four dimensions is directly related to Palatini ${\cal R}^2$ gravity. We calculate the running of the couplings at one loop, and we find that the conformal shift in the metric-curvature coupling's beta function generates a metric-curvature coupling on a fixed metric-affine background, whereas the independent-curvature coupling runs homogeneously. Furthermore, we follow an alternate prescription in which the connection is allowed to respond to the scalar fluctuations, which results in a different scalar Hessian and therefore different beta functions. Nonetheless, we find that both prescriptions admit the formal infrared limit $(\xi_{eff},\lambda_{eff})\to(1/6,0)$ through different limits of the original couplings in the case of positive perturbative quartics. Finally, we note that trajectories starting on the positive gravitational branch encounter a vanishing gravitational coefficient at a finite scale before this endpoint.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We study the conformal coupling in the torsion-free Palatini formulation with an ordinary scalar kinetic term. Tuning the curvature coupling cannot restore Weyl invariance in a Palatini theory with an ordinary scalar kinetic term because, for an independent and inert connection, the scalar-curvature term is separately Weyl-invariant, and no finite curvature coupling cancels the Weyl variation of the kinetic term. This can be demonstrated by evaluating the energy-momentum trace on the scalar equation of motion and noting that it is explicitly independent of the Palatini coupling. Eliminating the connection, we construct a map between the curvature coupling and the effective coupling for the equivalent metric theory, and demonstrate that the scalar continues to propagate even if the reduced kinetic coefficient vanishes. We also show that it becomes a Weyl compensator with a scale-invariant potential only in the rescaled infinite-coupling limit, which in four dimensions is directly related to Palatini ${\cal R}^2$ gravity. We calculate the running of the couplings at one loop, and we find that the conformal shift in the metric-curvature coupling's beta function generates a metric-curvature coupling on a fixed metric-affine background, whereas the independent-curvature coupling runs homogeneously. Furthermore, we follow an alternate prescription in which the connection is allowed to respond to the scalar fluctuations, which results in a different scalar Hessian and therefore different beta functions. Nonetheless, we find that both prescriptions admit the formal infrared limit $(\xi_{eff},\lambda_{eff})\to(1/6,0)$ through different limits of the original couplings in the case of positive perturbative quartics. Finally, we note that trajectories starting on the positive gravitational branch encounter a vanishing gravitational coefficient at a finite scale before this endpoint.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the conformal coupling in the torsion-free Palatini formulation with an ordinary scalar kinetic term. | contribution 0.94 | 0.90 | 0.04 | unclear 0.51 |
| s1 | Tuning the curvature coupling cannot restore Weyl invariance in a Palatini theory with an ordinary scalar kinetic term because, for an independent and inert connection, the scalar-curvature term is separately Weyl-invariant, and no finite curvature coupling cancels the Weyl variation of the kinetic term. | result 0.99 | 0.70 | 0.82 | asserted 1.00 |
| s2 | This can be demonstrated by evaluating the energy-momentum trace on the scalar equation of motion and noting that it is explicitly independent of the Palatini coupling. | method 0.88 | 0.86 | 0.10 | asserted 0.99 |
| s3 | Eliminating the connection, we construct a map between the curvature coupling and the effective coupling for the equivalent metric theory, and demonstrate that the scalar continues to propagate even if the reduced kinetic coefficient vanishes. | result 0.56 | 0.95 | 0.12 | asserted 1.00 |
| s4 | We also show that it becomes a Weyl compensator with a scale-invariant potential only in the rescaled infinite-coupling limit, which in four dimensions is directly related to Palatini ${\cal R}^2$ gravity. | result 0.90 | 0.90 | 0.14 | asserted 1.00 |
| s5 | We calculate the running of the couplings at one loop, and we find that the conformal shift in the metric-curvature coupling's beta function generates a metric-curvature coupling on a fixed metric-affine background, whereas the independent-curvature coupling runs homogeneously. | result 0.93 | 0.96 | 0.08 | asserted 1.00 |
| s6 | Furthermore, we follow an alternate prescription in which the connection is allowed to respond to the scalar fluctuations, which results in a different scalar Hessian and therefore different beta functions. | method 0.51 | 0.92 | 0.04 | asserted 1.00 |
| s7 | Nonetheless, we find that both prescriptions admit the formal infrared limit $(\xi_{eff},\lambda_{eff})\to(1/6,0)$ through different limits of the original couplings in the case of positive perturbative quartics. | result 1.00 | 0.95 | 0.16 | asserted 1.00 |
| s8 | Finally, we note that trajectories starting on the positive gravitational branch encounter a vanishing gravitational coefficient at a finite scale before this endpoint. | result 0.98 | 0.89 | 0.08 | asserted 0.99 |
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Figures reproduced from arXiv:2610.16954 by the authors.
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Discussion
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