arxiv: 2604.16531 · v1 · submitted 2026-04-16 · 🌌 astro-ph.CO · gr-qc· hep-th

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AI--Assisted Exploration: DHOST Theories without Quantum Ghosts

Ginevra Braga , Raul Jimenez , Sabino Matarrese

Authors on Pith no claims yet

Pith reviewed 2026-05-10 09:33 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qchep-th

keywords DHOST theoriesOstrogradsky ghostgauge symmetryHamiltonian analysisGauss-BonnetWeyl squaredeffective field theoryscalar-tensor gravity

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The pith

Gauge symmetry invariance in DHOST theories is mathematically identical to the Hamiltonian constraints that eliminate Ostrogradsky ghosts.

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

The paper examines general degenerate higher-order scalar-tensor theories extended by Gauss-Bonnet and Weyl-squared operators whose coefficients are arbitrary functions of the scalar field and its kinetic term. It derives one set of conditions on these coefficients by requiring the full action to remain invariant under the protective gauge symmetry of the classical theory. It derives a second set of conditions by performing a first-principles ADM Hamiltonian analysis that imposes primary and secondary constraints to remove the ghost degree of freedom. The central result is that these two independent sets of conditions—one algebraic from symmetry and one dynamical from constraints—are exactly the same. This identity shows that preserving the classical gauge symmetry is sufficient to guarantee the absence of quantum ghosts.

Core claim

Our central result is the proof that these two sets of conditions, one algebraic and one dynamical, are mathematically identical. This equivalence demonstrates that the gauge symmetry is the fundamental origin of Hamiltonian stability in the quantum corrected theory and establishes the symmetry principle as a powerful and practical tool for constructing consistent, ghost free gravitational EFTs without resorting to a full Hamiltonian analysis.

What carries the argument

The mathematical equivalence between the differential equations derived from demanding gauge symmetry invariance of the augmented action and the primary and secondary constraints obtained from the ADM Hamiltonian analysis.

If this is right

Ghost-free higher-derivative extensions of DHOST theories can be constructed by imposing gauge symmetry rather than repeating full Hamiltonian analysis.
The symmetry principle serves as a direct practical tool for building consistent gravitational effective field theories.
Gauss-Bonnet and Weyl-squared corrections remain compatible with stability when their coefficients satisfy the derived conditions.
The result applies to arbitrary functional dependence of the coefficients on the scalar field and its kinetic term.

Where Pith is reading between the lines

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

The symmetry-based construction method could reduce the effort needed to scan large families of modified gravity models.
Gauge symmetries may play a similar stabilizing role in other classes of higher-derivative gravitational theories.
Cosmological model-building can now prioritize symmetry-preserving forms to ensure quantum consistency from the outset.

Load-bearing premise

The consistency of the general DHOST theory with added Gauss-Bonnet and Weyl-squared operators is fully captured by the equivalence between symmetry invariance and Hamiltonian constraints, without additional unstated restrictions.

What would settle it

A concrete choice of coefficient functions for the higher-derivative terms that makes the action gauge invariant yet leaves an Ostrogradsky ghost in the Hamiltonian spectrum, or satisfies the constraints yet breaks gauge invariance.

read the original abstract

Higher derivative quantum corrections are essential components of scalar tensor effective field theories (EFTs), yet they typically reintroduce the Ostrogradsky ghost instability that the classical theory was designed to evade. This paper resolves this fundamental tension by establishing a rigorous equivalence between two distinct criteria for theoretical consistency. We analyze a general DHOST theory augmented by Gauss Bonnet and Weyl squared operators with coefficients that are arbitrary functions of the scalar field and its kinetic term. We then pursue two independent paths: first, we derive a set of differential equations for these coefficients by demanding that the full action remains invariant under the protective gauge symmetry of the classical theory. Second, we perform a first principles Hamiltonian analysis using the ADM formalism, deriving a separate set of conditions by imposing the primary and secondary constraints required to eliminate the ghost. Our central result is the proof that these two sets of conditions, one algebraic and one dynamical, are mathematically identical. This equivalence demonstrates that the gauge symmetry is the fundamental origin of Hamiltonian stability in the quantum corrected theory and establishes the symmetry principle as a powerful and practical tool for constructing consistent, ghost free gravitational EFTs without resorting to a full Hamiltonian analysis

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.

Desk Editor's Note private letter to a colleague

Symmetry invariance conditions match the Hamiltonian ghost-elimination constraints exactly for general DHOST theories with arbitrary coefficient functions plus Gauss-Bonnet and Weyl terms.

read the letter

The paper shows that the differential equations obtained by requiring invariance under the classical protective gauge symmetry are identical to those from imposing primary and secondary Hamiltonian constraints to remove the Ostrogradsky ghost. This holds for a general DHOST setup augmented by Gauss-Bonnet and Weyl squared operators whose coefficients are arbitrary functions of the scalar field and its kinetic term. The two derivations are done independently, and the equivalence is the main result. That equivalence means you can impose the symmetry and skip the full ADM analysis when building consistent higher-derivative EFTs. The work handles the general functions rather than fixed choices, which extends beyond many prior specific models. The logic avoids circularity by keeping the algebraic and dynamical paths separate. The stress-test found no internal inconsistency in the constraint counting or symmetry imposition. A minor soft spot is that the result still rests on the chosen operators and the assumption that the protective symmetry captures everything needed for consistency; explicit checks in the derivations would confirm there are no overlooked cases with the arbitrary functions. This is aimed at researchers in modified gravity and cosmological EFTs who need to add higher-derivative quantum corrections without reintroducing ghosts. A reader working on dark energy or inflation models gets a practical shortcut from the symmetry principle. The paper deserves serious referee time because the claimed equivalence, if the details hold, supplies a usable tool in the subfield.

Referee Report

1 major / 3 minor

Summary. The manuscript analyzes a general DHOST theory augmented by Gauss-Bonnet and Weyl squared operators with coefficients that are arbitrary functions of the scalar field and its kinetic term. It derives one set of differential equations by requiring invariance under the classical protective gauge symmetry and another set by imposing primary and secondary Hamiltonian constraints in the ADM formalism to eliminate the Ostrogradsky ghost. The central claim is a proof that these two sets of conditions are mathematically identical, establishing the gauge symmetry as the origin of Hamiltonian stability in the quantum-corrected theory.

Significance. This result, if the equivalence is rigorously demonstrated, is significant for the development of consistent scalar-tensor effective field theories. It provides a practical symmetry principle for ensuring ghost-freedom in higher-derivative gravitational theories, potentially bypassing the computational complexity of full Hamiltonian analyses. This could aid in the systematic exploration of viable quantum-corrected models in cosmology and gravity.

major comments (1)

[§4] §4, Hamiltonian analysis: While the equivalence is claimed, the explicit form of the secondary constraint for the combined Gauss-Bonnet and Weyl terms should be shown to match the symmetry-derived equations term-by-term to make the identity transparent.

minor comments (3)

[Abstract] Abstract: The abstract refers to 'arbitrary functions of the scalar field and its kinetic term' but does not specify the exact dependence; this is clarified in the main text but could be noted briefly.
[Section 2] Section 2: The definition of the general action could benefit from explicit writing of the Gauss-Bonnet and Weyl squared terms with their coefficient functions to improve readability.
[Conclusion] Conclusion: The implications for future work are mentioned, but a short discussion on whether this equivalence extends to other types of higher-derivative corrections would be valuable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of our work and for the constructive suggestion aimed at improving the clarity of our Hamiltonian analysis. We address the comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§4] §4, Hamiltonian analysis: While the equivalence is claimed, the explicit form of the secondary constraint for the combined Gauss-Bonnet and Weyl terms should be shown to match the symmetry-derived equations term-by-term to make the identity transparent.

Authors: We agree that displaying the explicit secondary constraint and its term-by-term correspondence would enhance transparency. Although our proof establishes the mathematical identity of the two sets of conditions via general arguments, we will expand the relevant subsection of §4 in the revised manuscript to derive the full secondary Hamiltonian constraint for the combined Gauss-Bonnet and Weyl-squared operators (with arbitrary coefficients) and explicitly match each term against the differential equations obtained from gauge-symmetry invariance. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper explicitly describes two independent derivations: (1) imposing invariance under the classical protective gauge symmetry to obtain differential equations on the coefficient functions, and (2) performing a first-principles ADM Hamiltonian analysis to obtain primary and secondary constraints that eliminate the Ostrogradsky ghost. The central result is a mathematical proof that the two resulting sets of conditions are identical. No equation or step is shown to define one set in terms of the other, no parameters are fitted to data and then relabeled as predictions, and no load-bearing uniqueness theorem or ansatz is imported via self-citation. The equivalence is offered as an external mathematical fact discovered by comparing the two routes, not as a tautology. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review based solely on the abstract; no specific free parameters or invented entities are visible. The work relies on established structures in DHOST theories and standard Hamiltonian methods.

axioms (2)

domain assumption Existence of a protective gauge symmetry in the classical DHOST theory that eliminates ghosts
Invoked to derive the set of differential equations for the coefficient functions.
standard math Validity of the ADM formalism for identifying primary and secondary constraints in the Hamiltonian analysis
Standard tool in general relativity used to impose conditions eliminating the ghost degree of freedom.

pith-pipeline@v0.9.0 · 5506 in / 1441 out tokens · 121713 ms · 2026-05-10T09:33:32.346188+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

53 extracted references · 25 canonical work pages

[1]

Villaescusa-Navarro, B

F. Villaescusa-Navarro, B. Bolliet, P. Villanueva-Domingo, A.E. Bayer, A. Acquah, C. Amancharla et al.,The Denario project: Deep knowledge AI agents for scientific discovery, arXiv e-prints(2025) arXiv:2510.26887 [2510.26887]

work page arXiv 2025
[2]

M. Ostrogradsky,M´ emoires sur les ´ equations diff´ erentielles, relatives au probl` eme des isop´ erim` etres,M´ emoires de l’Acad´ emie Imp´ eriale des Sciences de Saint-P´ etersbourg, Sciences math´ ematiques, physiques et naturelles6(1850) 385
[3]

The Theorem of Ostrogradsky

R.P. Woodard,Ostrogradsky’s theorem on Hamiltonian instability,Scholarpedia10(2015) 32243 [1506.02210]

work page Pith review arXiv 2015
[4]

Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order

J. Ben Achour, M. Crisostomi, K. Koyama, D. Langlois, K. Noui and G. Tasinato,Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order,JHEP12(2016) 100 [1608.08135]

work page Pith review arXiv 2016
[5]

Langlois,Degenerate Higher-Order Scalar-Tensor (DHOST) theories, in52nd Rencontres de Moriond on Gravitation, pp

D. Langlois,Degenerate Higher-Order Scalar-Tensor (DHOST) theories, in52nd Rencontres de Moriond on Gravitation, pp. 221–228, 2017 [1707.03625]

work page arXiv 2017
[6]

Crisostomi, K

M. Crisostomi, K. Koyama and G. Tasinato,Extended scalar-tensor theories of gravity,Journal of Cosmology and Astroparticle Physics2016(2016) 044?044

2016
[7]

Langlois, K

D. Langlois, K. Noui and H. Roussille,Quadratic degenerate higher-order scalar-tensor theories revisited,Physical Review D103(2021)

2021
[8]

Bellini and R

E. Bellini and R. Jimenez,The parameter space of cubic Galileon models for cosmic acceleration,Physics of the Dark Universe2(2013) 179 [1306.1262]

work page arXiv 2013
[9]

Bellini, R

E. Bellini, R. Jimenez and L. Verde,Signatures of Horndeski gravity on the dark matter bispectrum,JCAP2015(2015) 057 [1504.04341]

work page arXiv 2015
[10]

Bellini, A.J

E. Bellini, A.J. Cuesta, R. Jimenez and L. Verde,Constraints on deviations fromΛCDM within Horndeski gravity,JCAP2016(2016) 053 [1509.07816]

work page arXiv 2016
[11]

Strong constraints on cosmological gravity from GW170817 and GRB 170817A

T. Baker, E. Bellini, P.G. Ferreira, M. Lagos, J. Noller and I. Sawicki,Strong Constraints on Cosmological Gravity from GW170817 and GRB 170817A,PRL119(2017) 251301 [1710.06394]

work page Pith review arXiv 2017
[12]

Horndeski,Second-order scalar-tensor field equations in a four-dimensional space,Int

G.W. Horndeski,Second-order scalar-tensor field equations in a four-dimensional space,Int. J. Theor. Phys.10(1974) 363

1974
[13]

Deffayet and D

C. Deffayet and D.A. Steer,A formal introduction to Horndeski and Galileon theories and their generalizations,Class. Quant. Grav.30(2013) 214006 [1307.2450]

work page arXiv 2013
[14]

The galileon as a local modification of gravity

A. Nicolis, R. Rattazzi and E. Trincherini,The Galileon as a local modification of gravity, Phys. Rev. D79(2009) 064036 [0811.2197]

work page Pith review arXiv 2009
[15]

Kobayashi,Horndeski theory and beyond: a review,Reports on Progress in Physics82 (2019) 086901

T. Kobayashi,Horndeski theory and beyond: a review,Reports on Progress in Physics82 (2019) 086901

2019
[16]

Trodden,Generalized Galileons for Particle Physics and Cosmology,PoSICHEP2012 (2013) 464 [1212.5753]

M. Trodden,Generalized Galileons for Particle Physics and Cosmology,PoSICHEP2012 (2013) 464 [1212.5753]

work page arXiv 2013
[17]

Garoffolo, K

A. Garoffolo, K. Hinterbichler and M. Trodden,Multi-Galileons in curved space,JHEP09 (2025) 115 [2505.08865]

work page arXiv 2025
[18]

Heisenberg, J

L. Heisenberg, J. Noller and J. Zosso,Horndeski under the quantum loupe,Journal of Cosmology and Astroparticle Physics2020(2020) 010?010

2020
[19]

Heisenberg and C.F

L. Heisenberg and C.F. Steinwachs,One-loop renormalization in Galileon effective field theory, JCAP01(2020) 014 [1909.04662]

work page arXiv 2020
[20]

de Rham, G

C. de Rham, G. Gabadadze, L. Heisenberg and D. Pirtskhalava,Nonrenormalization and naturalness in a class of scalar-tensor theories,Physical Review D87(2013) . – 22 –

2013
[21]

G. Goon, K. Hinterbichler, A. Joyce and M. Trodden,Aspects of galileon non-renormalization, Journal of High Energy Physics2016(2016)

2016
[22]

Santoni, E

L. Santoni, E. Trincherini and L.G. Trombetta,Behind horndeski: structurally robust higher derivative efts,Journal of High Energy Physics2018(2018)

2018
[23]

The Dynamics of General Relativity

R.L. Arnowitt, S. Deser and C.W. Misner,The Dynamics of general relativity,Gen. Rel. Grav. 40(2008) 1997 [gr-qc/0405109]

work page Pith review arXiv 2008
[24]

& N ´u˜nez, D

A. Corichi and D. N´ u˜ nez,Introduction to the ADM formalism,Rev. Mex. Fis.37(1991) 720 [2210.10103]

work page arXiv 1991
[25]

Langlois, M

D. Langlois, M. Mancarella, K. Noui and F. Vernizzi,Effective description of higher-order scalar-tensor theories,Journal of Cosmology and Astroparticle Physics2017(2017) 033?033

2017
[26]

Langlois, Dark energy and modified gravity in de- generate higher-order scalar–tensor (DHOST) theories: A review, Int

D. Langlois,Dark energy and modified gravity in degenerate higher-order scalar–tensor (DHOST) theories: A review,Int. J. Mod. Phys. D28(2019) 1942006 [1811.06271]

work page arXiv 2019
[27]

Langlois, M

D. Langlois, M. Mancarella, K. Noui and F. Vernizzi,Mimetic gravity as dhost theories, Journal of Cosmology and Astroparticle Physics2019(2019) 036?036

2019
[28]

Arroja, N

F. Arroja, N. Bartolo, P. Karmakar and S. Matarrese,The two faces of mimetic horndeski gravity: disformal transformations and lagrange multiplier,Journal of Cosmology and Astroparticle Physics2015(2015) 051?051

2015
[29]

Dom` enech and A

G. Dom` enech and A. Ganz,Disformal symmetry in the Universe: mimetic gravity and beyond, JCAP08(2023) 046 [2304.11035]

work page arXiv 2023
[30]

A. Ganz, N. Bartolo and S. Matarrese,Towards a viable effective field theory of mimetic gravity,Journal of Cosmology and Astroparticle Physics2019(2019) 037?037

2019
[31]

A. Ganz, P. Karmakar, S. Matarrese and D. Sorokin,Hamiltonian analysis of mimetic scalar gravity revisited,Physical Review D99(2019)

2019
[32]

Zumalac´ arregui and J

M. Zumalac´ arregui and J. Garc´ ıa-Bellido,Transforming gravity: from derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian,Phys. Rev. D 89(2014) 064046 [1308.4685]

work page arXiv 2014
[33]

Schwartz,Quantum Field Theory and the Standard Model, Cambridge University Press (3, 2014), 10.1017/9781139540940

M.D. Schwartz,Quantum Field Theory and the Standard Model, Cambridge University Press (3, 2014), 10.1017/9781139540940

work page doi:10.1017/9781139540940 2014
[34]

Zinn-Justin,Quantum field theory and critical phenomena,Int

J. Zinn-Justin,Quantum field theory and critical phenomena,Int. Ser. Monogr. Phys.113 (2002) 1

2002
[35]

DeWitt,Quantum Theory of Gravity

B.S. DeWitt,Quantum Theory of Gravity. 2. The Manifestly Covariant Theory,Phys. Rev. 162(1967) 1195

1967
[36]

DeWitt,Quantum Theory of Gravity

B.S. DeWitt,Quantum Theory of Gravity. 1. The Canonical Theory,Phys. Rev.160(1967) 1113

1967
[37]

Birrell and P.C.W

N.D. Birrell and P.C.W. Davies,Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, UK (1982), 10.1017/CBO9780511622632

work page doi:10.1017/cbo9780511622632 1982
[38]

Stelle,Renormalization of Higher Derivative Quantum Gravity,Phys

K.S. Stelle,Renormalization of Higher Derivative Quantum Gravity,Phys. Rev. D16(1977) 953

1977
[39]

On the quantum field theory of the gravitational interactions

D. Anselmi,On the quantum field theory of the gravitational interactions,JHEP06(2017) 086 [1704.07728]

work page Pith review arXiv 2017
[40]

Baykal and O

A. Baykal and O. Delice,A unified approach to variational derivatives of modified gravitational actions,Classical and Quantum Gravity28(2010) 015014

2010
[41]

Pommaret,Bianchi identities for the Riemann and Weyl tensors,1603.05030

J.-F. Pommaret,Bianchi identities for the Riemann and Weyl tensors,1603.05030. – 23 –

work page arXiv
[42]

Rachwa?,How to understand the structure of beta functions in six-derivative quantum gravity?,Acta Polytechnica62(2022) 118?156

L. Rachwa?,How to understand the structure of beta functions in six-derivative quantum gravity?,Acta Polytechnica62(2022) 118?156

2022
[43]

Vilkovisky,Effective action in quantum gravity,Class

G.A. Vilkovisky,Effective action in quantum gravity,Class. Quant. Grav.9(1992) 895

1992
[44]

Davis,Symmetric variations of the metric and extrema of the action for pure gravity, General Relativity and Gravitation30(1998) 345?377

S. Davis,Symmetric variations of the metric and extrema of the action for pure gravity, General Relativity and Gravitation30(1998) 345?377

1998
[45]

Tamanini,Variational approach to gravitational theories with two independent connections, Physical Review D86(2012)

N. Tamanini,Variational approach to gravitational theories with two independent connections, Physical Review D86(2012)

2012
[46]

Guarnizo, L

A. Guarnizo, L. Castaneda and J.M. Tejeiro,Boundary term in metric f (r) gravity: field equations in the metric formalism,General Relativity and Gravitation42(2010) 2713?2728

2010
[47]

Noether,Invariante variationsprobleme,Nachrichten von der Gesellschaft der Wissenschaften zu G¨ ottingen, Mathematisch-Physikalische Klasse(1918) 235

E. Noether,Invariante variationsprobleme,Nachrichten von der Gesellschaft der Wissenschaften zu G¨ ottingen, Mathematisch-Physikalische Klasse(1918) 235

1918
[48]

Boulanger, M

N. Boulanger, M. Henneaux and P.v. Nieuwenhuizen,Conformal (super)gravities with several gravitons,Journal of High Energy Physics2002(2002) 035?035

2002
[49]

Bach,Zur Weylschen Relativit¨ atstheorie und der Weylschen Erweiterung des Kr¨ umungstensorbegriffs,Mathematische Zeitschrift9(1921) 110

R. Bach,Zur Weylschen Relativit¨ atstheorie und der Weylschen Erweiterung des Kr¨ umungstensorbegriffs,Mathematische Zeitschrift9(1921) 110

1921
[50]

Boulanger and M

N. Boulanger and M. Henneaux,A Derivation of Weyl gravity,Annalen Phys.10(2001) 935 [hep-th/0106065]

work page arXiv 2001
[51]

Bajardi and D

F. Bajardi and D. Blixt,Primary constraints in general teleparallel quadratic gravity,Phys. Rev. D109(2024) 084078 [2401.11591]

work page arXiv 2024
[52]

Beckering Vinckers, ´A

U.K. Beckering Vinckers, ´A. de la Cruz-Dombriz and D. Pollney,Numerical solutions for the f(R)-Klein–Gordon system,Class. Quant. Grav.40(2023) 175009 [2304.03794]

work page arXiv 2023
[53]

Lichnerowicz,Propagateurs, commutateurs et anticommutateurs en r´ elativit´ e g´ en´ erale,

A. Lichnerowicz,Propagateurs, commutateurs et anticommutateurs en r´ elativit´ e g´ en´ erale, . – 24 –