Pith. sign in

REVIEW 4 minor 3 cited by

On the Gravitational Origin of the QCD Axion

T0 review · 0 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read A gravitational torsion scalar cannot be the QCD axion if its only matter coupling is derivative and exactly shift-symmetric.

desk verdict A small, rigorous no-go against derivative-coupled gravitational axions, with a clearly labeled speculative positive proposal; worth refereeing. read the letter →

arxiv 2506.11836 v1 pith:QC56KPFD submitted 2025-06-13 hep-th gr-qchep-ph

Georgios K. Karananas , Mikhail Shaposhnikov , Sebastian Zell This is my paper · ORCID
classification hep-thgr-qchep-ph
keywords strongCPproblemQCDaxionEinstein-CartangravitytorsionpseudoscalarshiftsymmetrychiralanomalyWeylinvarianceevanescentoperators
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

The paper takes aim at the proposal that the QCD axion could come purely from gravity, specifically from the pseudoscalar torsion mode of Einstein-Cartan gravity. It argues that the standard argument for such a gravitational axion is invalid: the coupling of that mode to fermions is a derivative coupling $\partial_\mu \varphi J_A^\mu$, which is exactly shift-symmetric, and an exact shift symmetry means no potential is generated, so the field cannot relax the strong CP angle. The apparent coupling $\varphi \operatorname{Tr} G_{\mu\nu} \tilde{G}^{\mu\nu}$ obtained by integrating by parts and using the chiral anomaly is cancelled by the fermion-mass term in the anomaly. The paper then maps the conditions under which a gravitational axion could still work: the shift symmetry must be explicitly broken, the induced QCD coupling must dominate over gravitational contributions, and Weyl-invariant Einstein-Cartan gravity is singled out as the promising setting where quantum Weyl invariance generates the necessary couplings through evanescent operators.

What carries the argument

The central object is the derivative coupling $\partial_\mu \varphi J_A^\mu$ between the gravitational pseudoscalar, the axial torsion mode of Einstein-Cartan gravity, and the axial fermionic current $J_A^\mu = \bar{\Psi} \gamma^5 \gamma^\mu \Psi$. Its defining property is an exact shift symmetry $\varphi \to \varphi + c$, which carries the whole argument: any operator with this symmetry cannot generate a potential, and the chiral-anomaly step that would turn it into $\varphi \operatorname{Tr} G \tilde{G}$ fails because the anomaly's fermion-mass term cancels the gluon term. The secondary machinery is the generalized Einstein-Cartan Lagrangian with $\varphi$-dependent $Z(\varphi)$ couplings to both axial and vector currents and to current-current operators, which breaks the shift symmetry and opens the loop-level route to Yukawa couplings. In the Weyl-invariant case, the decisive mechanism is the appearance of evanescent operators, factors of $(D-4)$ multiplying the interaction strength, whose poles in $D-4$ are compensated by loop divergences, yielding finite radiative corrections with coefficients not forced to cancel.

What would settle it

In a concrete Einstein-Cartan model with a propagating torsion pseudoscalar and massive fermions, evaluate the one-loop effective potential for $\varphi$ at zero momentum: the paper predicts an exact cancellation between the gluon-anomaly triangle diagram and the fermion-mass insertion, so any residual $\varphi$-dependent term proportional to $\operatorname{Tr} G \tilde{G}$ or any nonzero potential would falsify the central claim.

Watch

Extended reading notes

Core claim

The central claim is that a would-be axion arising from Einstein-Cartan gravity cannot solve the strong CP problem if its only matter interaction is the derivative coupling to the axial fermionic current, because this operator is exactly invariant under $\varphi \to \varphi + c$. The apparent topological coupling $\frac{\alpha}{4\pi} \varphi \operatorname{Tr} G_{\mu\nu} \tilde{G}^{\mu\nu}$ is obtained by integrating the derivative coupling by parts and using the chiral anomaly, but the same divergence also contains the fermion-mass term $2i \bar{\Psi} m_\Psi \gamma_5 \Psi$, and these two pieces cancel in the effective potential, as required by the unbroken shift symmetry. Existing gravitational-axion constructions rely on this invalid step, identifying the would-be topological coupling with the true axion coupling while neglecting the fermion-mass contribution. The no-go is evaded if the coupling to fermions explicitly breaks the shift symmetry, as in the general Einstein-Cartan Lagrangian with $\varphi$-dependent $Z(\varphi)$ functions; then loop corrections can generate Yukawa-type couplings, but naive power counting requires severe fine-tuning to keep gravitational contributions below $\Lambda_{\mathrm{QCD}}^4$, while a scale-invariant regularization renders the couplings too weak. Weyl-invariant Einstein-Cartan gravity is the remaining viable route: quantum Weyl invariance forces evanescent $(D-4)$-suppressed operators that, combined with loop poles, generate the needed axion couplings with uncorrelated coefficients.

Load-bearing premise

The proof assumes that the gravitational scalar's only interaction with matter is the derivative coupling to the gauge-invariant axial current, so if a UV completion of Einstein-Cartan gravity produces a direct $\varphi \operatorname{Tr} G \tilde{G}$ coupling or a $\varphi$-dependent quark mass, the no-go does not apply.

Editorial extensions

If this is right

  • The operator $\partial_\mu \varphi J_A^\mu$ does not produce an axion potential, so any gravitational-axion model whose only coupling is a derivative coupling to fermionic currents cannot solve the strong CP problem.
  • A genuine gravitational axion requires explicit shift-symmetry breaking in the fermionic coupling; without it, there is no potential and no dynamical relaxation of $\bar{\theta}$.
  • If the UV completion follows naive power counting, generating a large enough QCD coupling forces gravitational corrections far above $\Lambda_{\mathrm{QCD}}^4$ unless coefficients are fine-tuned, while a dimensional-regularization-style removal of power divergences makes the coupling practically zero.
  • In Weyl-invariant Einstein-Cartan gravity, quantum Weyl invariance generates higher-dimensional operators through evanescent $(D-4)$ factors, so the axion couplings to QCD can appear at one loop with independent coefficients.
  • The viability of a purely gravitational axion therefore rests on the specific Weyl-invariant Einstein-Cartan framework rather than on torsion alone.

Reading between the lines

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

  • Editorial extension: the no-go argument likely generalizes beyond Einstein-Cartan gravity, since any scalar whose only matter interaction is a total-derivative current coupling shares the exact shift symmetry; extra-dimensional or Stueckelberg axions built on the same pattern face the same obstacle unless a direct topological coupling is present.
  • Editorial extension: a sharp testable direction is to compute the full effective potential for $\varphi$ in the Weyl-invariant model; if the evanescent-operator mechanism accidentally restores the shift-symmetric cancellation at higher orders, the last viable route would close.
  • Editorial extension: because the proof relies on gauge invariance of $J_A^\mu$, non-perturbative gravitational effects such as gravi-scalar instantons could break the symmetry without reintroducing fermion masses; the paper leaves this possibility open, and it would be a natural next calculation.
  • Editorial extension: if the Weyl-invariant construction is realized, the axion decay constant and coupling pattern would be fixed by the Weyl-breaking spurion sector, making the model more predictive than conventional Peccei-Quinn axions and potentially distinguishable in axion search experiments.
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

0 major / 4 minor

Summary. The paper examines proposals in which the QCD axion emerges as a pseudoscalar degree of freedom of Einstein-Cartan gravity (torsion). It first shows that the operator J_A^mu ∂_mu phi, the effective interaction of the torsion pseudoscalar with fermions, is exactly shift-symmetric, so it cannot generate a potential for phi and cannot relax the strong-CP theta-angle. The apparent coupling phi Tr G Gtilde obtained by integration by parts is cancelled by the quark-mass term in the chiral anomaly. The paper then generalizes to couplings Z(phi) that explicitly break the shift symmetry, estimates the induced f(phi) and potential, finds a generic fine-tuning problem under naive power counting, and argues that in Weyl-invariant Einstein-Cartan gravity evanescent operators can produce the required couplings. The paper concludes that previous gravitational axion proposals are invalid and identifies the Weyl-invariant framework as a promising, though not fully constructed, avenue.

Significance. The central no-go is a clean, exact statement: a scalar whose only interaction with matter is a derivative coupling to a gauge-invariant fermionic current possesses an exact shift symmetry, and therefore a flat potential. The contrast between the gauge-invariant current J_A^mu and the non-gauge-invariant Chern-Simons current K^mu is the key physical point and is handled correctly. The cancellation between the gluon and fermion-mass contributions, eqs. (9)-(11), is standard and clearly presented. The paper is self-contained for this result and does not rely on the authors' previous work. The power-counting estimates (17)-(18) are explicitly heuristic but sufficient for the paper's purpose of identifying a generic problem. The positive Weyl-invariant scenario is presented as a promising mechanism rather than a complete model; this modesty is appropriate, although the claim that it 'enables' the solution is stronger than a proof. Overall the paper settles an important question in the gravitational axion literature and redirects the discussion toward viable mechanisms.

minor comments (4)
  1. [Proof to the contrary] The statement 'No effect, perturbative or non-perturbative, can break this symmetry' is too categorical, because quantum gravity is widely expected to break global symmetries and the later discussion correctly allows for non-perturbative gravitational effects; please qualify the statement to the EFT setting of eq. (7).
  2. [References] Reference [79] is a placeholder ('arXiv:xxxx.xxxx') and must be completed before publication.
  3. [Appendix B, eq. (B15)] The field redefinition leading to phi requires c_TT < 0 for a real phi; this domain condition is never stated and should be mentioned to avoid confusion about the parameter space.
  4. [Weyl-invariant R^2 gravity] The sentence stating that the mechanism is 'enabling the gravitational solution of the strong CP problem' goes slightly beyond what is demonstrated, since the function F in eq. (19) is arbitrary and the generated coefficients are not computed; wording such as 'may enable' or 'opens a viable avenue' would better match the evidence.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the no-go proof is self-contained; the Weyl-invariant proposal is explicitly conditional and does not feed back into the central result.

full rationale

The central no-go result (Section 'Proof to the contrary') rests entirely on the manifest exact shift symmetry of eq. (7), the anomalous-current identity in eq. (9), and the chiral rotation in eqs. (10)-(11). This is a parameter-free argument: no quantity is fitted and no result is imported from the authors' prior work. The apparent coupling to TrG Gtilde obtained by integration by parts is shown to be cancelled by the fermion-mass contribution; the contrast with the non-gauge-invariant Chern-Simons form (eq. (12)) is made explicitly. The positive Weyl-invariant proposal is transparently conditional: Section 'Weyl-invariant R^2 gravity' states that the induced couplings depend on an arbitrary function F that 'cannot be fixed by a symmetry principle', and the evanescent-operator mechanism is cited from [80] rather than derived anew. The paper presents this direction as 'promising' (Abstract, Outlook), not as a proven prediction, so no fitted parameter is renamed as a prediction. The self-citations to [39] and [80] are load-bearing only for the secondary, explicitly tentative proposal, not for the paper's main critical claim. The limitation passage 'The results depend on the function F... It cannot be fixed by a symmetry principle' is an admitted model-dependence, not a circular step. No equation in the paper reduces to its own input by construction.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The no-go result uses no free parameters. The positive proposal depends on free torsion-fermion couplings and on an arbitrary Weyl-invariant regulator function F. All entities (the torsion pseudoscalar φ) are pre-existing in the EC literature.

free parameters (2)
  • Nonminimal fermion-torsion couplings ζV, ζA, ζ̃V, ζ̃A
    Free constants in the EC fermion Lagrangian (eq B9). They determine the coupling functions Z(φ) in eqs (B12)-(B14) and hence the size of the induced axion-fermion and axion-potential interactions. Not fitted to data; chosen by model building.
  • Weyl-invariant regulator function F(a/χ, h/χ)
    Appears in eq (19) to make the D-dimensional action Weyl-invariant. It controls the finite radiative corrections and therefore the generated axion-quark couplings. The paper states it cannot be fixed by a symmetry principle.
assumptions (6)
  • standard math The axial current J_A is gauge-invariant, so the derivative coupling ∂φ J_A is a total derivative and the shift symmetry is exact (eq 7).
    Foundation of the no-go proof.
  • standard math The chiral anomaly has the standard form including the fermion mass contribution: ∇_μ J^μ_A = α/(4π) Tr G G̃ + 2i ψ̄ m γ5 ψ (eq 9).
    Required for the cancellation argument.
  • standard math A chiral rotation trades the derivative coupling for a mass phase plus an anomaly term with exactly canceling potential contributions (eqs 10-11).
    Underpins the conclusion that no potential is generated.
  • domain assumption Power-counting estimates (17)-(18) assume a universal UV cutoff with O(1) coefficients for all divergent diagrams.
    Used to conclude that generic shift-symmetry-breaking couplings are either fine-tuned or too weak.
  • domain assumption The scale-invariant (Weyl-preserving) regularization scheme and the evanescent operator mechanism are valid and produce finite higher-dimensional operators.
    Basis of the positive Weyl-invariant proposal; relies on refs [77,78,80].
  • domain assumption Non-perturbative gravitational contributions to the axion potential are negligible or do not spoil the mechanism.
    The paper states the gravitational axion remains viable only if this holds (Outlook).

how reviews work

0 comments
Cite this review

Pith. "Pith review of On the Gravitational Origin of the QCD Axion." pith.science (2026). https://pith.science/paper/QC56KPFD

@misc{pith2026250611836,
  author       = {Pith},
  title        = {Pith review of: On the Gravitational Origin of the QCD Axion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QC56KPFD}},
  note         = {Machine review of arXiv:2506.11836}
}
read the original abstract

Gravity can give rise to (pseudo)scalar fields, for instance due to torsion. In particular, axions of gravitational origin have been proposed as a minimal and compelling solution to the strong CP problem. In this work, we critically examine the feasibility of this approach. We demonstrate that models in which the scalar field couples to fermionic currents only through derivatives do not yield a satisfactory axion. Moreover, we identify the necessary conditions for generating a gravitational axion through quantum effects, highlighting Weyl-invariant Einstein-Cartan gravity as a promising theoretical setting.

Figures

Figures reproduced from arXiv: 2506.11836 by the authors.

Figure 1
Figure 1. FIG. 1. Exemplary Feynman diagrams generating Yukawa [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Exemplary two-loop Feynman diagrams, originat [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Contribution to the axion-fermion coupling from [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 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.

  3. Torsion induced current-scalaron coupling in Einstein-Cartan gravity

    hep-ph 2025-07 conditional novelty 6.0 of 10

    In Einstein-Cartan gravity, torsion-current couplings induce scalaron-current and scalaron Chern-Simons interactions, with explicit decay constants and a classification of their asymptotic behavior.

Reference graph

Works this paper leans on

86 extracted references · 14 canonical work pages · cited by 3 Pith papers

  1. [1]

    Aadet al.(ATLAS), Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys

    G. Aadet al.(ATLAS), Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B716, 1 (2012), arXiv:1207.7214 [hep-ex]

    arXiv 2012

  2. [2]

    Chatrchyanet al.(CMS), Observation of a New Bo- son at a Mass of 125 GeV with the CMS Experiment at the LHC, Phys

    S. Chatrchyanet al.(CMS), Observation of a New Bo- son at a Mass of 125 GeV with the CMS Experiment at the LHC, Phys. Lett. B716, 30 (2012), arXiv:1207.7235 [hep-ex]

    arXiv 2012

  3. [3]

    R. D. Peccei and H. R. Quinn, CP Conservation in the Presence of Instantons, Phys. Rev. Lett.38, 1440 (1977)

    1977

  4. [4]

    Weinberg, A New Light Boson?, Phys

    S. Weinberg, A New Light Boson?, Phys. Rev. Lett.40, 223 (1978)

    1978

  5. [5]

    Wilczek, Problem of StrongPandTInvariance in the Presence of Instantons, Phys

    F. Wilczek, Problem of StrongPandTInvariance in the Presence of Instantons, Phys. Rev. Lett.40, 279 (1978)

    1978

  6. [6]

    A. Hook, TASI Lectures on the Strong CP Problem and 5 Before spontaneous Weyl symmetry breaking, the Lagrangian only contains terms quadratic in the scalar and pseudoscalar cur- vatures and reads LW = 1 f 2 R2 + 1 ˜f 2 ˜R2 ,(C5) where for simplicity we omitted the cross-termR˜R; its inclusion leads to a shifted potential for the axion [68, 86], and is irr...

    arXiv 2019

  7. [7]

    Di Luzio, M

    L. Di Luzio, M. Giannotti, E. Nardi, and L. Visinelli, The landscape of QCD axion models, Phys. Rept.870, 1 (2020), arXiv:2003.01100 [hep-ph]

    arXiv 2020

  8. [8]

    Ehretet al.(ALPS), Resonant laser power build- up in ALPS: A ’Light-shining-through-walls’ experiment, Nucl

    K. Ehretet al.(ALPS), Resonant laser power build- up in ALPS: A ’Light-shining-through-walls’ experiment, Nucl. Instrum. Meth. A612, 83 (2009), arXiv:0905.4159 [physics.ins-det]

    arXiv 2009

Show all 86 references
  1. [9]

    S. J. Asztaloset al.(ADMX), A SQUID-based microwave cavity search for dark-matter axions, Phys. Rev. Lett. 104, 041301 (2010), arXiv:0910.5914 [astro-ph.CO]

    2010 arXiv

  2. [10]

    I. G. Irastorzaet al., Towards a new generation axion helioscope, JCAP06, 013, arXiv:1103.5334 [hep-ex]

    arXiv

  3. [11]

    Auneet al.(CAST), CAST search for sub-eV mass solar axions with 3He buffer gas, Phys

    S. Auneet al.(CAST), CAST search for sub-eV mass solar axions with 3He buffer gas, Phys. Rev. Lett.107, 261302 (2011), arXiv:1106.3919 [hep-ex]

    2011 arXiv

  4. [12]

    Budker, P

    D. Budker, P. W. Graham, M. Ledbetter, S. Rajendran, and A. Sushkov, Proposal for a Cosmic Axion Spin Pre- cession Experiment (CASPEr), Phys. Rev. X4, 021030 (2014), arXiv:1306.6089 [hep-ph]

    2014 arXiv

  5. [13]

    Armengaudet al., Conceptual Design of the Inter- national Axion Observatory (IAXO), JINST9, T05002, arXiv:1401.3233 [physics.ins-det]

    E. Armengaudet al., Conceptual Design of the Inter- national Axion Observatory (IAXO), JINST9, T05002, arXiv:1401.3233 [physics.ins-det]

    arXiv

  6. [14]

    Y. Kahn, B. R. Safdi, and J. Thaler, Broadband and Resonant Approaches to Axion Dark Matter Detection, Phys. Rev. Lett.117, 141801 (2016), arXiv:1602.01086 [hep-ph]

    2016 arXiv

  7. [15]

    B. M. Brubakeret al., First results from a microwave cav- ity axion search at 24µeV, Phys. Rev. Lett.118, 061302 (2017), arXiv:1610.02580 [astro-ph.CO]

    2017 arXiv

  8. [16]

    Caldwell, G

    A. Caldwell, G. Dvali, B. Majorovits, A. Millar, G. Raf- felt, J. Redondo, O. Reimann, F. Simon, and F. Steffen (MADMAX Working Group), Dielectric Haloscopes: A New Way to Detect Axion Dark Matter, Phys. Rev. Lett. 118, 091801 (2017), arXiv:1611.05865 [physics.ins-det]. 9

    2017 arXiv

  9. [17]

    Brouweret al.(DMRadio), Projected sensitivity of DMRadio-m3: A search for the QCD axion below 1µeV, Phys

    L. Brouweret al.(DMRadio), Projected sensitivity of DMRadio-m3: A search for the QCD axion below 1µeV, Phys. Rev. D106, 103008 (2022), arXiv:2204.13781 [hep- ex]

    2022 arXiv

  10. [18]

    C. B. Adamset al., Axion Dark Matter, inSnowmass 2021(2022) arXiv:2203.14923 [hep-ex]

    2022 arXiv

  11. [19]

    J. E. Kim, Weak Interaction Singlet and Strong CP In- variance, Phys. Rev. Lett.43, 103 (1979)

    1979

  12. [20]

    M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Can Confinement Ensure Natural CP Invariance of Strong In- teractions?, Nucl. Phys. B166, 493 (1980)

    1980

  13. [21]

    A. R. Zhitnitsky, On Possible Suppression of the Axion Hadron Interactions. (In Russian), Sov. J. Nucl. Phys. 31, 260 (1980)

    1980

  14. [22]

    M. Dine, W. Fischler, and M. Srednicki, A Simple Solu- tion to the Strong CP Problem with a Harmless Axion, Phys. Lett. B104, 199 (1981)

    1981

  15. [23]

    Kallosh, A

    R. Kallosh, A. D. Linde, D. A. Linde, and L. Susskind, Gravity and global symmetries, Phys. Rev. D52, 912 (1995), arXiv:hep-th/9502069

    1995 arXiv

  16. [24]

    Kamionkowski and J

    M. Kamionkowski and J. March-Russell, Planck scale physics and the Peccei-Quinn mechanism, Phys. Lett. B 282, 137 (1992), arXiv:hep-th/9202003

    1992 arXiv

  17. [25]

    Holman, S

    R. Holman, S. D. H. Hsu, T. W. Kephart, E. W. Kolb, R. Watkins, and L. M. Widrow, Solutions to the strong CP problem in a world with gravity, Phys. Lett. B282, 132 (1992), arXiv:hep-ph/9203206

    1992 arXiv

  18. [26]

    S. M. Barr and D. Seckel, Planck scale corrections to axion models, Phys. Rev. D46, 539 (1992)

    1992

  19. [27]

    Witten, Some Properties of O(32) Superstrings, Phys

    E. Witten, Some Properties of O(32) Superstrings, Phys. Lett. B149, 351 (1984)

    1984

  20. [28]

    Reece, Extra-Dimensional Axion Expectations, (2024), arXiv:2406.08543 [hep-ph]

    M. Reece, Extra-Dimensional Axion Expectations, (2024), arXiv:2406.08543 [hep-ph]

    2024 arXiv

  21. [29]

    Dvali, Three-form gauging of axion symmetries and gravity, (2005), arXiv:hep-th/0507215

    G. Dvali, Three-form gauging of axion symmetries and gravity, (2005), arXiv:hep-th/0507215

    2005 arXiv

  22. [30]

    Dvali, S

    G. Dvali, S. Folkerts, and A. Franca, How neutrino protects the axion, Phys. Rev. D89, 105025 (2014), arXiv:1312.7273 [hep-th]

    2014 arXiv

  23. [31]

    Dvali, Topological Origin of Chiral Symmetry Break- ing in QCD and in Gravity, (2017), arXiv:1705.06317 [hep-th]

    G. Dvali, Topological Origin of Chiral Symmetry Break- ing in QCD and in Gravity, (2017), arXiv:1705.06317 [hep-th]

    2017 arXiv

  24. [32]

    Dvali, Strong-CPwith and without gravity, (2022), arXiv:2209.14219 [hep-ph]

    G. Dvali, Strong-CPwith and without gravity, (2022), arXiv:2209.14219 [hep-ph]

    2022 arXiv

  25. [33]

    E. W. Mielke and E. S. Romero, Cosmological evolution of a torsion-induced quintaxion, Phys. Rev. D73, 043521 (2006)

    2006

  26. [34]

    Mercuri, Peccei-Quinn mechanism in gravity and the nature of the Barbero-Immirzi parameter, Phys

    S. Mercuri, Peccei-Quinn mechanism in gravity and the nature of the Barbero-Immirzi parameter, Phys. Rev. Lett.103, 081302 (2009), arXiv:0902.2764 [gr-qc]

    2009 arXiv

  27. [35]

    Mercuri and V

    S. Mercuri and V. Taveras, Interaction of the Barbero- Immirzi Field with Matter and Pseudo-Scalar Perturba- tions, Phys. Rev. D80, 104007 (2009), arXiv:0903.4407 [gr-qc]

    2009 arXiv

  28. [36]

    Lattanzi and S

    M. Lattanzi and S. Mercuri, A solution of the strong CP problem via the Peccei-Quinn mechanism through the Nieh-Yan modified gravity and cosmological implica- tions, Phys. Rev. D81, 125015 (2010), arXiv:0911.2698 [gr-qc]

    2010 arXiv

  29. [37]

    Castillo-Felisola, C

    O. Castillo-Felisola, C. Corral, S. Kovalenko, I. Schmidt, and V. E. Lyubovitskij, Axions in gravity with torsion, Phys. Rev. D91, 085017 (2015), arXiv:1502.03694 [hep- ph]

    2015 arXiv

  30. [38]

    G. K. Karananas, On the strong-CP problem and its ax- ion solution in torsionful theories, Eur. Phys. J. C78, 480 (2018), arXiv:1805.08781 [hep-th]

    2018 arXiv

  31. [39]

    G. K. Karananas, M. Shaposhnikov, and S. Zell, Weyl-invariant Einstein-Cartan gravity: unifying the strong CP and hierarchy puzzles, JHEP11, 146, arXiv:2406.11956 [hep-th]

    arXiv

  32. [40]

    Dvali, C

    G. Dvali, C. Gomez, and S. Zell, A Proof of the Axion?, (2018), arXiv:1811.03079 [hep-th]

    2018 arXiv

  33. [41]

    Dvali, The role of gravity in naturalness versus con- sistency: strong-CP and dark energy, Phil

    G. Dvali, The role of gravity in naturalness versus con- sistency: strong-CP and dark energy, Phil. Trans. Roy. Soc. Lond. A382, 20230084 (2023)

    2023

  34. [42]

    C. G. Callan, Jr., R. F. Dashen, and D. J. Gross, The Structure of the Gauge Theory Vacuum, Phys. Lett. B 63, 334 (1976)

    1976

  35. [43]

    Jackiw and C

    R. Jackiw and C. Rebbi, Vacuum Periodicity in a Yang- Mills Quantum Theory, Phys. Rev. Lett.37, 172 (1976)

    1976

  36. [44]

    Abelet al., Measurement of the Permanent Electric Dipole Moment of the Neutron, Phys

    C. Abelet al., Measurement of the Permanent Electric Dipole Moment of the Neutron, Phys. Rev. Lett.124, 081803 (2020), arXiv:2001.11966 [hep-ex]

    2020 arXiv

  37. [45]

    Di Vecchia and G

    P. Di Vecchia and G. Veneziano, Chiral Dynamics in the Large n Limit, Nucl. Phys. B171, 253 (1980)

    1980

  38. [46]

    Grilli di Cortona, E

    G. Grilli di Cortona, E. Hardy, J. Pardo Vega, and G. Villadoro, The QCD axion, precisely, JHEP01, 034, arXiv:1511.02867 [hep-ph]

    arXiv

  39. [47]

    Vafa and E

    C. Vafa and E. Witten, Parity Conservation in QCD, Phys. Rev. Lett.53, 535 (1984)

    1984

  40. [48]

    Heisenberg, A systematic approach to generalisations ofGeneralRelativityandtheircosmologicalimplications, Phys

    L. Heisenberg, A systematic approach to generalisations ofGeneralRelativityandtheircosmologicalimplications, Phys. Rept.796, 1 (2019), arXiv:1807.01725 [gr-qc]

    2019 arXiv

  41. [49]

    Beltrán Jiménez, L

    J. Beltrán Jiménez, L. Heisenberg, and T. S. Koivisto, The Geometrical Trinity of Gravity, Universe5, 173 (2019), arXiv:1903.06830 [hep-th]

    2019 arXiv

  42. [50]

    Rigouzzo and S

    C. Rigouzzo and S. Zell, Coupling metric-affine gravity to a Higgs-like scalar field, Phys. Rev. D106, 024015 (2022), arXiv:2204.03003 [hep-th]

    2022 arXiv

  43. [51]

    Cartan, Sur une généralisation de la notion de cour- bure de Riemann et les espaces à torsion, Comptes Ren- dus, Ac

    É. Cartan, Sur une généralisation de la notion de cour- bure de Riemann et les espaces à torsion, Comptes Ren- dus, Ac. Sc. Paris174, 593 (1922)

    1922

  44. [52]

    Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie), inAn- nales scientifiques de l’École normale supérieure, Vol

    É. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie), inAn- nales scientifiques de l’École normale supérieure, Vol. 40 (Gauthier-Villars, 1923) pp. 325–412

    1923

  45. [53]

    41 (Gauthier-Villars, 1924) pp

    É.Cartan,Surlesvariétésàconnexionaffine, etlathéorie de la relativité généralisée (première partie)(suite), in Annales scientifiques de l’École Normale Supérieure, Vol. 41 (Gauthier-Villars, 1924) pp. 1–25

    1924

  46. [54]

    42 (Gauthier-Villars, 1925) pp

    É.Cartan,Surlesvariétésàconnexionaffine, etlathéorie de la relativité généralisée (deuxième partie), inAn- nales scientifiques de l’École normale supérieure, Vol. 42 (Gauthier-Villars, 1925) pp. 17–88

    1925

  47. [55]

    Utiyama, Invariant theoretical interpretation of inter- action, Phys

    R. Utiyama, Invariant theoretical interpretation of inter- action, Phys. Rev.101, 1597 (1956)

    1956

  48. [56]

    T.W.B.Kibble,Lorentzinvarianceandthegravitational field, J. Math. Phys.2, 212 (1961)

    1961

  49. [57]

    D. W. Sciama, On the analogy between charge and spin in general relativity, inRecent developments in general relativity(Pergamon Press, Oxford, 1962) p. 415

    1962

  50. [58]

    Rigouzzo and S

    C. Rigouzzo and S. Zell, Coupling Metric-Affine Grav- ity to the Standard Model and Dark Matter Fermions, (2023), arXiv:2306.13134 [gr-qc]

    2023 arXiv

  51. [59]

    Barker and S

    W. Barker and S. Zell, Consistent particle physics in metric-affine gravity from extended projective symmetry, (2024), arXiv:2402.14917 [hep-th]

    2024 arXiv

  52. [60]

    G. K. Karananas, M. Shaposhnikov, A. Shkerin, and S. Zell, Matter matters in Einstein-Cartan gravity, Phys. 10 Rev. D104, 064036 (2021), arXiv:2106.13811 [hep-th]

    2021 arXiv

  53. [61]

    D. E. Neville, Gravity Theories With Propagating Tor- sion, Phys. Rev. D21, 867 (1980)

    1980

  54. [62]

    Sezgin and P

    E. Sezgin and P. van Nieuwenhuizen, New Ghost Free Gravity Lagrangians with Propagating Torsion, Phys. Rev. D21, 3269 (1980)

    1980

  55. [63]

    R. D. Hecht, J. M. Nester, and V. V. Zhytnikov, Some PoincaregaugetheoryLagrangianswithwellposedinitial value problems, Phys. Lett. A222, 37 (1996)

    1996

  56. [64]

    G. K. Karananas, The particle spectrum of parity- violating Poincaré gravitational theory, Class. Quant. Grav.32, 055012 (2015), arXiv:1411.5613 [gr-qc]

    2015 arXiv

  57. [65]

    Beltrán Jiménez and F

    J. Beltrán Jiménez and F. J. Maldonado Torralba, Re- visiting the stability of quadratic Poincaré gauge gravity, Eur. Phys. J. C80, 611 (2020), arXiv:1910.07506 [gr-qc]

    2020 arXiv

  58. [66]

    Pradisi and A

    G. Pradisi and A. Salvio, (In)equivalence of metric-affine and metric effective field theories, Eur. Phys. J. C82, 840 (2022), arXiv:2206.15041 [hep-th]

    2022 arXiv

  59. [67]

    Mikura, V

    Y. Mikura, V. Naso, and R. Percacci, Some simple the- ories of gravity with propagating torsion, Phys. Rev. D 109, 104071 (2024), arXiv:2312.10249 [gr-qc]

    2024 arXiv

  60. [68]

    G. K. Karananas, Geometrical origin of inflation in Weyl- invariant Einstein-Cartan gravity, Phys. Lett. B862, 139343 (2025), arXiv:2501.16416 [gr-qc]

    2025 arXiv

  61. [69]

    Barker, G

    W. Barker, G. K. Karananas, and H. Tu, The par- ticle spectra of parity-violating theories: A less radi- cal approach and an upgrade ofP SALT er, (2025), arXiv:2506.02111 [hep-th]

    2025 arXiv

  62. [70]

    Gasser and H

    J. Gasser and H. Leutwyler, Chiral Perturbation The- ory: Expansions in the Mass of the Strange Quark, Nucl. Phys. B250, 465 (1985)

    1985

  63. [71]

    Alvarez-Gaume and E

    L. Alvarez-Gaume and E. Witten, Gravitational Anoma- lies, Nucl. Phys. B234, 269 (1984)

    1984

  64. [72]

    J. E. Kim and G. Carosi, Axions and the Strong CP Problem, Rev. Mod. Phys.82, 557 (2010), [Erratum: Rev.Mod.Phys.91, 049902(2019)],arXiv:0807.3125[hep- ph]

    2010 arXiv

  65. [73]

    ’t Hooft, Symmetry Breaking Through Bell-Jackiw Anomalies, Phys

    G. ’t Hooft, Symmetry Breaking Through Bell-Jackiw Anomalies, Phys. Rev. Lett.37, 8 (1976)

    1976

  66. [74]

    ’t Hooft, Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle, Phys

    G. ’t Hooft, Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle, Phys. Rev. D14, 3432 (1976), [Erratum: Phys.Rev.D 18, 2199 (1978)]

    1976

  67. [75]

    Shaposhnikov, Is there a new physics between elec- troweak and Planck scales?, inAstroparticle Physics: Current Issues, 2007 (APCI07)(2007) arXiv:0708.3550 [hep-th]

    M. Shaposhnikov, Is there a new physics between elec- troweak and Planck scales?, inAstroparticle Physics: Current Issues, 2007 (APCI07)(2007) arXiv:0708.3550 [hep-th]

    2007 arXiv

  68. [76]

    Farina, D

    M. Farina, D. Pappadopulo, and A. Strumia, A modified naturalness principle and its experimental tests, JHEP 08, 022, arXiv:1303.7244 [hep-ph]

    arXiv

  69. [77]

    Englert, C

    F. Englert, C. Truffin, and R. Gastmans, Conformal In- variance in Quantum Gravity, Nucl. Phys. B117, 407 (1976)

    1976

  70. [78]

    Shaposhnikov and D

    M. Shaposhnikov and D. Zenhausern, Quantum scale in- variance, cosmological constant and hierarchy problem, Phys. Lett. B671, 162 (2009), arXiv:0809.3406 [hep-th]

    2009 arXiv

  71. [79]

    Shaposhnikov, Progress in Einstein-Cartan gravity, (2025), arXiv:xxxx.xxxx [hep-th]

    M. Shaposhnikov, Progress in Einstein-Cartan gravity, (2025), arXiv:xxxx.xxxx [hep-th]

    2025

  72. [80]

    M. E. Shaposhnikov and F. V. Tkachov, Quantum scale- invariant models as effective field theories, (2009), arXiv:0905.4857 [hep-th]

    2009 arXiv

  73. [81]

    Shaposhnikov and A

    M. Shaposhnikov and A. Shkerin, Conformal symmetry: towardsthelinkbetweentheFermiandthePlanckscales, Phys.Lett.B783,253(2018),arXiv:1803.08907[hep-th]

    2018 arXiv

  74. [82]

    Shaposhnikov and A

    M. Shaposhnikov and A. Shkerin, Gravity, Scale In- variance and the Hierarchy Problem, JHEP10, 024, arXiv:1804.06376 [hep-th]

    arXiv

  75. [83]

    Shaposhnikov, A

    M. Shaposhnikov, A. Shkerin, and S. Zell, Standard Model Meets Gravity: Electroweak Symmetry Break- ing and Inflation, Phys. Rev. D103, 033006 (2021), arXiv:2001.09088 [hep-th]

    2021 arXiv

  76. [84]

    G. K. Karananas, M. Michel, and J. Rubio, One residue to rule them all: Electroweak symmetry breaking, infla- tion and field-space geometry, Phys. Lett. B811, 135876 (2020), arXiv:2006.11290 [hep-th]

    2020 arXiv

  77. [85]

    G. K. Karananas, M. Shaposhnikov, A. Shkerin, and S. Zell, Scale and Weyl invariance in Einstein- Cartan gravity, Phys. Rev. D104, 124014 (2021), arXiv:2108.05897 [hep-th]

    2021 arXiv

  78. [86]

    I. D. Gialamas and K. Tamvakis, Inflation in Weyl- invariant Einstein-Cartan gravity, Phys. Rev. D111, 044007 (2025), arXiv:2410.16364 [gr-qc]

    2025 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.