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REVIEW 3 major objections 4 minor 2 cited by

Progress in Einstein-Cartan gravity

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper argues that a Weyl-invariant Einstein-Cartan gravity, where the torsion vector acts as the Weyl gauge field, yields exactly one new axion-like scalar that can solve the strong CP problem and traces the smallness of the…

desk verdict A competent and honest invited review of a bold program, but the headline claims—anomaly-free Weyl symmetry and smallness-from-tiny-couplings—are asserted rather than demonstrated in this paper. read the letter →

arxiv 2506.11847 v1 pith:OITCBF5J submitted 2025-06-13 hep-th gr-qchep-ph

Mikhail Shaposhnikov This is my paper
classification hep-thgr-qchep-ph MSC 83D0583F0581T2083C45
keywords Einstein-CartangravityWeylinvariancetorsionvectoraxion-likeparticlestrongCPproblemhierarchyinflationheavyneutralleptons
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

This paper reviews a research program arguing that gravity is best understood not as a property of the metric alone but as the gauge theory of the Lorentz group, formulated with a tetrad and a spin connection, i.e. Einstein-Cartan gravity. Its central proposal is that making this theory Weyl-invariant, invariant under spacetime-dependent rescalings of fields, removes a quantum obstruction that blocks Weyl symmetry in ordinary metric gravity, because the vector part of the torsion acts as a Weyl gauge field. If this survives quantization, the Standard Model coupled to this gravity automatically contains exactly one new scalar particle, an axion-like particle, that can solve the strong CP problem, and the smallness of the cosmological constant, the Higgs mass, and heavy neutrino masses all trace back to tiny dimensionless gauge couplings. The paper is careful that the quantum theory is not yet developed and is not perturbatively renormalizable, so the construction is a candidate route rather than a finished prediction. The reason to care is that it ties three open problems, strong CP, the cosmological-constant and hierarchy puzzle, and the dark-matter neutrino mass, to one geometric mechanism.

What carries the argument

The load-bearing object is the torsion vector $v_\mu = T^\nu_{\mu\nu}$, the vector irreducible component of the torsion tensor. Its transformation law $v_\mu \mapsto v_\mu + 3 q^{-1}\partial_\mu q$ in four dimensions is exactly that of a gauge field for local Weyl rescalings, so derivatives can be promoted to Weyl-covariant derivatives $D_\mu = \partial_\mu + \frac{1}{3}v_\mu$, and curvature invariants such as $F$ and $\tilde F$ transform covariantly. The argument then uses the standard auxiliary-field trick: the quadratic curvature action is rewritten as $F\chi^2 + \tilde F a^2 - \frac12\alpha\chi^4 - \frac12\beta a^4 + \gamma\chi^2 a^2$, with the couplings $\alpha,\beta,\gamma$ determined by $f,\tilde f,f_m$, and local Weyl symmetry is used to fix a gauge such as $\chi=M_P$. This reduces the spectrum to one physical scalar $a$, whose canonically normalized field $\tilde a$ acquires the potential displayed in the paper, and the same Weyl-covariant structure carries the fermionic, Higgs, and current couplings that make $a$ an axion-like particle.

What would settle it

A one-loop computation of the trace anomaly, or of the renormalized divergence of the Weyl current, in the $D=4-2\epsilon$ Einstein-Cartan continuation with Standard Model matter would settle the central claim: if the anomaly does not vanish after renormalization, the claimed quantum consistency fails. Observationally, if precision Higgs invisible-decay or fifth-force experiments detect a massless dilaton-like scalar with strong couplings, the claimed single-massive-scalar spectrum would be ruled out.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that replacing metric gravity by Einstein-Cartan gravity changes the status of Weyl symmetry. In metric gravity a local scale (Weyl) invariance is destroyed by the Weyl anomaly, and a local Weyl-invariant action is essentially the Weyl-squared theory with ghosts. In the EC formulation the vector part of the torsion, $v_\mu$, transforms as a Weyl gauge field, making curvature invariants Weyl-covariant and allowing a Weyl-invariant, ghost-free quadratic action $S_{EC,4} = \int \det e\,(\frac{1}{f^2}F^2 + \frac{1}{\tilde f^2}\tilde F^2 + \frac{2}{f_m^2}F\tilde F)$. Coupled to the Standard Model, after gauge-fixing the Weyl symmetry, for instance by setting $\chi=M_P$, one is left with the graviton, the Standard Model fields, and one extra scalar, the field $a$, an axion-like particle. The paper's conclusions state that this ALP has all properties needed to solve the strong CP problem, and that the smallness of the cosmological constant and of the ALP, Higgs, and heavy neutral lepton masses results from tiny values of the dimensionless Lorentz gauge couplings $f,\tilde f,f_m$. The paper also claims that the EC formulation permits anomaly-free quantum Weyl-invariant theories, while acknowledging that the full quantum theory remains to be constructed.

Load-bearing premise

The whole scheme rests on the assumption that in the Einstein-Cartan formulation the Weyl symmetry survives quantization, with the torsion vector $v_\mu$ acting as a Weyl gauge field and removing the Weyl anomaly that blocks this symmetry in metric gravity; if that anomaly cancellation fails, the single-scalar spectrum, the ALP solution to strong CP, and the smallness-from-tiny-couplings mechanism lose their quantum consistency, and the paper itself notes that the quantum theory remains to be developed and is not perturbatively renormalizable.

Editorial extensions

If this is right

  • The Standard Model plus Weyl-invariant Einstein-Cartan gravity predicts exactly one new light scalar, an ALP, and no massless dilaton, so the theory avoids the fifth-force and invisible-Higgs-decay problems of globally scale-invariant models.
  • The ALP provides a solution to the strong CP problem without introducing a new global symmetry by hand; its couplings to fermionic currents are part of the Weyl-invariant action.
  • The smallness of the cosmological constant, the Higgs mass, the ALP mass, and the heavy neutral lepton masses are all controlled by the same tiny Lorentz gauge couplings, making these numbers in principle computable from nonperturbative gravitational effects.
  • In the parameter region $\gamma \gg \beta$, the same single scalar drives inflation with predictions close to those of Starobinsky and Higgs inflation.
  • The Einstein-Cartan torsion also provides a production mechanism for heavy neutral leptons in the early universe, allowing the lightest one to be the dark matter.

Reading between the lines

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

  • If the radiative stability claim holds, the ALP's mass and couplings are tied to $f,\tilde f,f_m$ and are not arbitrarily tunable; I infer this makes the model directly testable by axion searches even before the quantum theory is fully formulated.
  • The same torsion-vector-as-Weyl-gauge-field mechanism could be applied in other settings where metric-formulation Weyl anomalies force nonlocal effective actions; a natural extension, not taken in the paper, would be to compute one-loop trace anomalies in the EC and related formulations and compare them.
  • The paper leaves open the magnitude and sign of the nonperturbative instanton effects that are supposed to generate the Higgs and heavy lepton masses; until that calculation exists, the hierarchy mechanism is a plausibility argument rather than a quantitative prediction.
  • I infer that if the ALP solves strong CP through its gravitational mass being below the QCD-induced mass, the theory predicts a specific relation between the ALP mass and the QCD scale, so measuring the ALP mass and coupling relation would distinguish it from generic axion models.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This invited overview presents the author's program for Weyl-invariant Einstein-Cartan gravity. It reviews how gravity arises from gauging the Lorentz group, the equivalence of the low-dimension Einstein-Cartan action to metric general relativity, the ghost-free quadratic extension with auxiliary scalars, and the restriction of the scalar spectrum by local Weyl symmetry. The paper argues that combining the Standard Model with Weyl-invariant Einstein-Cartan gravity yields exactly one new scalar axion-like degree of freedom and that tiny values of the Lorentz gauge couplings can explain the smallness of the cosmological constant and the Higgs, ALP, and heavy neutral lepton masses. The concluding bullets advertise this as a unified solution to the strong CP and hierarchy problems, while Section 7 explicitly states that the quantum theory remains to be developed.

Significance. If the central claims were established, this would be a substantial contribution to the search for a natural ALP solution to strong CP and for a gravitational origin of mass scales. The paper is useful as a concise exposition of the classical algebraic structure, and the derivations shown for Eqs. (13)-(14), (18), and (52)-(56) are internally consistent. However, the advertised conclusions go beyond what is demonstrated: the anomaly-free nature of the quantum theory is asserted rather than derived, the hierarchy explanation is a parametric restatement in terms of free couplings, and the strong CP solution is carried by citation [16]. The manuscript is best read as a research program summary, and its central physical claims are not yet established within this paper.

major comments (3)
  1. [Section 7, Eqs. (52)-(56)] The conclusion that Einstein-Cartan gravity 'allows the construction of anomaly-free quantum Weyl-invariant theories' is not derived in this manuscript. The D-dimensional continuation is explicitly non-unique: Eq. (56) and the preceding paragraph allow multiplication by an arbitrary function rho(a/chi) with rho -> 1 for D -> 4, and the text concedes that different choices lead to different quantum physics (refs. [73,74]). No loop computation is shown that would demonstrate cancellation of the 1/(D-4) poles in the Euler density and Weyl-squared terms. The local Weyl invariance of the classical action does not by itself establish anomaly freedom. The manuscript should either provide the calculation or explicitly present this as an open conjecture, and the Conclusions should be weakened accordingly.
  2. [Section 6, Eqs. (45)-(47)] The proposed explanation of the smallness of the cosmological constant, the ALP mass, and the Higgs mass reduces, by the paper's own equations, to choosing tiny values of the free parameters alpha, beta, and gamma. Since these are input parameters, Eqs. (45)-(47) are a parametric restatement of the hierarchy, not an explanation. The subsequent discussion of computability via fixed points and non-perturbative effects (Sections 4 and 6) is speculative and no concrete computation is presented. The text should clearly separate the algebraic statement that masses are proportional to small couplings from the unproven claim that these couplings can be computed from first principles.
  3. [Section 6, ALP discussion; Conclusions] The claim that the theory 'automatically contains just one extra scalar degree of freedom (ALP) with all properties to solve the strong CP problem' is not supported within this paper. The strong CP solution is asserted by reference to [16], with no analysis of the axion potential, the QCD-induced mass, or phenomenological constraints. The text itself qualifies the statement, noting that the ALP solves strong CP 'provided its gravitational mass is small compared with the QCD induced mass'. The Conclusions should indicate which properties are established in [16] and which depend on yet-unverified assumptions, especially given the unresolved quantum issues in Section 7.
minor comments (4)
  1. [Eq. (19) and Section 7] The symbol D is used for both the parameter combination in Eq. (19), D = 1/(f^2 tilde f^2) - 1/f_m^4, and for the spacetime dimension in Section 7. This notational collision is confusing and should be resolved, for example by renaming the parameter combination.
  2. [Eq. (45)] The minimum with a^2 = M_P^2 gamma/beta and h^2 = 2 xi_h M_P^2 (alpha beta - gamma^2)/(lambda beta) requires positivity conditions on beta, gamma, and alpha beta - gamma^2. These conditions should be stated explicitly, since they are not automatic for arbitrary couplings.
  3. [Section 7, Eq. (55)] The Weyl covariant derivative of the Higgs field is written as D_mu = partial_mu + v_mu/(D-1) in Eq. (55), which reduces to Eq. (39) for D=4. Because the transformation law for v_mu in Eq. (54) also depends on D, it would be helpful to state how the Weyl weights of matter fields are fixed in the D-dimensional continuation.
  4. [Throughout] Several central statements are attributed to the author's own work [16] without independent derivation or commentary. While self-citation is natural in an invited review, a reader would benefit from an explicit statement of which results are proven elsewhere and which are currently conjectural, especially for the strong CP and hierarchy claims.

Circularity Check

2 steps flagged · score 6.0 of 10

The classical one-scalar spectrum is derived internally, but the advertised hierarchy explanation is just the definitional dependence on tiny free couplings, and the strong-CP/ALP conclusion is imported from a same-author citation.

  1. self definitional [Section 6, Eqs. (45)-(47); Conclusions bullet 3]
    "If the Lorentz gauge couplings f, ˜f and f_m (or, what is the same, α, β and γ) are tiny and of the same order of magnitude, the tree, classical masses of the Higgs boson, the field a, and the cosmological constant are also small, as all these quantities are proportional to them."

    Eqs. (45)-(47) express h^2, a^2, ε_vac, m_h^2 and m_a^2 directly through the free parameters α, β, γ (via Eq. (19)). The conclusion that the smallness of these masses 'results from' tiny values of the couplings is therefore a restatement of the parameter dependence already present in the action, not a derived prediction. The paper itself defers any actual computation of α, β, γ to nonperturbative effects and states that the quantum theory 'remains to be developed'.

  2. self citation load bearing [Section 6, after Eq. (47); Conclusions bullet 2]
    "The field a – an axion-like particle (ALP) has a non-trivial coupling to the axial and vector fermionic currents, and has all the requisites to solve the strong CP-problem, provided its “gravitational” mass is small compared with the QCD induced mass [16]."

    The central conclusion that the Weyl-invariant EC+SM theory automatically contains an ALP with all properties to solve strong CP is not derived in this manuscript; the only cited support is [16], by G. K. Karananas, M. Shaposhnikov and S. Zell, i.e. the same group as the present paper. The quoted condition on the gravitational mass is not established in this text, so the load-bearing 'all requisites' claim rests on a self-citation rather than on a derivation shown here.

full rationale

The paper's own classical construction is largely self-contained: the counting of degrees of freedom, the Weyl gauge choice χ=M_P, the potential and the mass formulas (Eqs. (34)-(47)) are derived in the text. The Section 7 anomaly-free claim is not circular but underived: the D-dimensional Weyl-invariant extension is explicitly non-unique and the paper admits the quantum theory is unfinished; that is a correctness risk, not a self-referential reduction. The two genuinely circular/self-supporting moves are the hierarchy 'explanation' (small masses are just the small free couplings α, β, γ by Eqs. (45)-(47)) and the strong-CP/ALP bullet, which is imported from same-author ref. [16] without an in-text derivation. These are central to the advertised unification, so partial circularity is present; nevertheless the one-extra-scalar classical result remains independent content.

Assumptions & free parameters 9 free parameters · 6 assumptions · 1 invented entities

The program's explanatory power is parametric: masses are proportional to a small set of free couplings, and the couplings themselves are not derived. The strongest assumptions are the quantum survival of Weyl symmetry and the existence of nonperturbative mass generation, both unproven.

free parameters (9)
  • f (Lorentz gauge coupling, F^2 term) = assumed tiny
    Enters equation (17) and (19); tiny f is used to make scalar masses and the cosmological constant small in Section 6.
  • tilde f (gauge coupling, tilde F^2 term) = assumed tiny
    Same role as f; part of D = 1/(f^2 tilde f^2) - 1/f_m^4 in equation (19).
  • f_m (mixing coupling, F tilde F) = assumed tiny
    Controls the mixing term and enters the axion mass formula in equation (47).
  • alpha, beta, gamma (derived combinations) = tiny by assumption
    Define the potential (35) and the masses (45)-(47); no independent derivation is given.
  • xi_h (non-minimal Higgs-curvature coupling) = not specified
    Appears in (38) and in the mass formulas (45) and (47); free parameter of the Weyl-invariant Higgs action.
  • zeta_V^a, zeta_A^a, zeta_Z (fermion-torsion couplings) = not specified
    Appear in (41) and (44); set the ALP couplings to fermions, relevant for strong CP and dark matter.
  • c_v, c_av, zeta_vh, zeta_ah (explicit Weyl-breaking couplings) = not specified
    Appear in S_ECWB, equation (40); free parameters of the scale-invariant theory with matter.
  • lambda (Higgs self-coupling) = SM value
    Standard Model input used in the scalar potential (26) and in the flat direction (27).
  • rho(a/chi) (arbitrary function in D dimensions) = arbitrary
    Any function with rho -> 1 at D=4 is allowed in equation (56); introduces regularization ambiguity.
assumptions (6)
  • standard math Gauging the global Lorentz group gives the tetrad and spin connection as gravitational gauge fields; the resulting Einstein-Cartan theory is equivalent to GR on shell.
    Section 1, refs [3-9]; standard construction.
  • domain assumption The quadratic action (17) with only F^2, tilde F^2, and F tilde F terms is free of ghosts.
    Section 3, footnote 4: non-linear effects may jeopardize ghost freedom; the assumption is not fully proven.
  • domain assumption The torsion is auxiliary and can be integrated out algebraically; the tensor component vanishes on shell.
    Section 3, equations (13)-(15); standard for EC gravity, but the full non-linear dynamics is assumed to follow the linear treatment.
  • domain assumption In EC gravity the torsion vector v_mu acts as a Weyl gauge field, allowing Weyl-invariant actions and avoiding the metric-formulation Weyl anomaly.
    Section 7, equations (52)-(56); this is the key assumption that makes the quantum program viable.
  • ad hoc to paper Non-perturbative gravitational effects generate the Higgs and HNL masses, with no concrete computation.
    Sections 6 and 8: 'the order of magnitude of this effect remains obscure and has never been computed.'
  • domain assumption The field a receives a QCD axion-like mass and solves strong CP, provided its gravitational mass is small enough.
    Section 6, sentence about 'all the requisites to solve the strong CP-problem'; the condition is not established quantitatively.
invented entities (1)
  • Axion-like scalar a (from the auxiliary field associated with the Holst-squared curvature)
    purpose: Single new scalar degree of freedom; candidate to solve strong CP and provide dark matter.
    No mass or coupling range is predicted in this paper; existence is inferred from the Weyl-invariant EC framework, not from observation.

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Cite this review

Pith. "Pith review of Progress in Einstein-Cartan gravity." pith.science (2026). https://pith.science/paper/OITCBF5J

@misc{pith2026250611847,
  author       = {Pith},
  title        = {Pith review of: Progress in Einstein-Cartan gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OITCBF5J}},
  note         = {Machine review of arXiv:2506.11847}
}
read the original abstract

It is well-known that the gravitational force can be obtained by gauging the Lorentz group, which puts gravity on the same footing as the Standard Model fields. The resulting theory - Einstein-Cartan gravity - has several crucial advantages. I will overview the construction of the Weyl-invariant version of this theory and discuss its applications in particle physics and cosmology, in particular for inflation and the strong CP problem.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Infrared foundations for quantum geometry II: Catalogue of all torsion-like theories including new ghost-tachyon-free cases

    hep-th 2025-07 conditional novelty 7.0 of 10

    A systematic catalogue of symmetric pair-antisymmetric rank-three field theories yields 22 ghost-tachyon-free models, all propagating vector torsion and none propagating scalar or pseudoscalar torsion.

  2. Weyl-invariant Einstein-Cartan gravity with a heavy ALP: Higgs Inflation and $\alpha$-attractors

    hep-ph 2025-07 conditional novelty 6.0 of 10

    In a Weyl-invariant Einstein-Cartan gravity theory with the SM Higgs and a heavy gravitational ALP, tuning two nonminimal couplings reproduces metric Higgs inflation and α-attractor-like inflation with ns≈1−2/N and r≈12/N^2.

Reference graph

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