Interactions of the scalaron dark matter in $f (R)$ gravity

Yurii Sheiko; Yuri Shtanov

REVIEW 2 major objections 4 minor 2 cited by

Interactions of the scalaron dark matter in $f (R)$ gravity

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read In f(R) gravity the scalaron can be all of dark matter; its one-loop decay to two photons has a definite rate and thermal production is negligible.

desk verdict Careful re-derivation of the scalaron-to-two-photon rate with useful new cosmological applications, but the central claim to have eliminated the Jacobian ambiguity is not airtight. read the letter →

arxiv 2505.00324 v2 pith:NBLSFNKO submitted 2025-05-01 astro-ph.CO gr-qchep-ph

Yuri Shtanov , Yurii Sheiko This is my paper · ORCID

classification astro-ph.COgr-qchep-ph

keywords f(R)gravityscalarondarkmatterone-loopdecaytwo-photonphotonbackgroundthermalproductionmodified

topics Dark Matter

open problems Dark Matter

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 scalaron of $f(R)$ gravity—the scalar degree of freedom hidden in the modified metric—can account for all of the dark matter if its mass lies between a few meV and about 1 MeV. This paper re-derives the scalaron's interaction with the Standard Model and settles a disputed one-loop result: its decay to two photons has the definite rate $\Gamma_{\phi\to\gamma\gamma}=\alpha^2 m^3 |F|^2/(2^{10}\pi^3 M^2)$, with $\Gamma\approx 5.2\times 10^{-30}(m/\mathrm{MeV})^3\,\mathrm{s}^{-1}$ near $m=2m_e$. The paper computes the cosmological photon background this decay would create and shows that scalarons produced thermally in the early universe contribute at most one part in $10^{12}$ of the dark-matter density. These results support the original picture in which scalaron dark matter is a coherently oscillating classical field rather than a thermal relic.

What carries the argument

The load-bearing object is the one-loop effective interaction $$\mathcal{L}_{\phi\gamma\gamma}=\frac{\$\alpha$}{16\pi}F(m)\frac{\phi}{M}F_{\mu\nu}$F^{{\mu\nu}}$,$$ where $F(m)$ is the same Standard Model loop form factor that governs $H\to\gamma\gamma$ ($W$-boson and charged-fermion loops). It carries the argument because the scalaron enters each diagram at a single vertex with the replacement $\chi/v\to -\phi/2M$, so the entire disputed question reduces to whether the spinor field redefinition used to diagonalise the kinetic term generates extra contributions. The paper argues it does not, at one loop: redefinition-dependent kinetic terms are proportional to the free spinor equation of motion and vanish inside the triangle diagram, while the Jacobian is a renormalisation-scheme artefact; a ghost-field regularisation shows the Jacobians cancel between fields of opposite statistics. Only the mass couplings survive, making the amplitude unambiguous.

What would settle it

Compute the coefficient of $\phi F_{\mu\nu}F^{\mu\nu}$ in the one-loop effective action directly from the fermion determinant in a background $\phi$ and electromagnetic field, without any field redefinition; if the coefficient differs from $(\alpha/16\pi)F(m)/M$, the rate formula is wrong. Observationally, a dark-matter halo should show a two-photon line at $E_\gamma=m/2$ with the intensity and spectrum derived here, and a meaningful upper limit below that prediction would exclude the scenario once halo modelling is included.

Watch

Extended reading notes

Core claim

The paper's central claim is that the ambiguity in the scalaron–photon vertex is an artifact of regularisation-dependent Jacobians and disappears when the one-loop amplitude is computed directly. Because the scalaron couplings to fermion mass terms and to $W^\pm$ and $Z^0$ boson masses have the same form as the Higgs couplings with $\chi/v\to -\phi/2M$, the Standard Model $H\to\gamma\gamma$ form factor carries over. The spinor field redefinition needed to remove $\phi$ from kinetic terms produces no net contribution: the kinetic part of the redefined Lagrangian is proportional to the free spinor equation of motion, so it cannot contribute to the triangle diagram, and the Jacobian is a gauge-choice artifact rather than a physical term. The resulting decay width, $\Gamma_{\phi\to\gamma\gamma}=\alpha^2 m^3 |F|^2/(2^{10}\pi^3 M^2)$, is real for $m<2m_e$ and gives $\Gamma\approx 5.2\times10^{-30}(m/\mathrm{MeV})^3\,\mathrm{s}^{-1}$ near threshold; with this rate, the diffuse background from cosmological decays has a calculable spectrum and the thermal scalaron fraction is $n_b/n_s\lesssim 10^{-12}$.

Load-bearing premise

The calculation assumes the spinor field redefinition $\psi=e^{3\phi/4M}\tilde{\psi}$ leaves the S-matrix unchanged at one loop, so the Jacobian and kinetic-term contributions to $\phi\gamma\gamma$ vanish and only the mass couplings matter; if a Jacobian term survived, the decay rate and background predictions would shift.

Editorial extensions

If this is right

If the scalaron is all the dark matter, its decays produce a diffuse photon background with maximum energy $E_{\rm max}=m/2$ and a spectrum that rises toward that maximum; near the upper allowed mass bound the line intensity scales as $(m/\mathrm{MeV})^3$, making that regime the most visible.
The 511-keV positron-annihilation bound on scalaron decay to $e^+e^-$ confines a dark-matter scalaron to roughly $1.04\,\mathrm{MeV}\lesssim m\lesssim 1.15\,\mathrm{MeV}$ if it makes all of the dark matter.
Thermal scalarons from the hot plasma contribute at most $10^{-12}$ of the dark-matter number density, so the dark matter is a coherent condensate rather than a thermal relic in this model.
The rate is real and unambiguous for $m<2m_e$, so the formula stays valid across the whole allowed mass window and can be used as a prediction for line searches.

Reading between the lines

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

The paper does not work out the corresponding gluon final state, but the same one-loop matching applies directly: scalaron decay into two gluons should have an equally definite rate built from quark-loop contributions, which would extend the model's observability into cosmic-ray and gamma-ray channels.
The Jacobian-cancellation claim is a statement about the fermion determinant in the background $\phi F_{\mu\nu}F^{\mu\nu}$; a regulator-independent evaluation of that determinant would separate the physical amplitude from scheme-dependent artifacts and is a natural independent check.
Because the decay rate is fixed once $m$ is known and the mass window is narrow, the model is directly falsifiable by a line search at $E_\gamma=m/2$; a detection would measure $m$ and test the minimal-coupling assumption at the same time.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a solid, honest paper. The decay rate itself is not new — it confirms Cembranos — but the analytic extragalactic background spectrum (Eq. 4.13) and the estimate that thermal scalarons are negligible (Eq. 5.18) are genuinely useful additions. The loop calculation is careful, and the authors deserve credit for explicitly reversing their earlier preprint position and for flagging the limitations of their own argument.

Where I would push: the whole edifice rests on Appendix B, where the authors argue that derivative-coupling contributions from the spinor kinetic term vanish and no Jacobian anomaly survives. The dimensional-regularization argument involves a shift in a quadratically divergent integral; the Pauli–Villars construction uses one subtraction and the authors themselves note it breaks down beyond that. If a surface term survives, the effective interaction acquires an undetermined piece of the form (B.3). Near m = 2m_e, |F| ≈ 4.2, so an O(1) F_an changes the decay rate by an order-unity factor and all the Section 4 rates and bounds shift correspondingly. So the paper is not wrong in an obvious way, but its central claim to have resolved the literature discrepancy is asserted more strongly than the proof warrants.

The thermal section is order-of-magnitude, as the authors admit; that is fine because the conclusion — thermal scalarons are a negligible fraction of the dark matter — is robust to the large uncertainties. The self-citations [15,16] supply initial-condition premises and do not prop up the loop calculation, so I do not see a citation problem.

Who this is for: anyone working on scalaron or modified-gravity dark matter, and on line or continuum photon signals from f(R) models. A referee should spend time on Appendix B and ask for a more rigorous treatment of the Jacobian — or an explicit statement that the result is regularization-dependent within the allowed renormalization freedom. If that step is fixed or honestly downgraded, the paper is publishable; the cosmological spectrum and thermal bound are useful additions.

Referee Report

2 major / 4 minor

Summary. This paper studies the scalaron of f(R) gravity as a dark-matter candidate. It computes the one-loop scalaron decay into two photons, obtaining Γ_{φ→γγ} = α² m³ |F|²/(2^10 π³ M²) (Eq. A.7), argues that a direct loop calculation with dimensional and Pauli–Villars regularization removes the Jacobian ambiguities associated with spinor field redefinitions, derives the resulting cosmological photon background spectrum, and estimates the thermal scalaron abundance from the hot early universe, finding n_b/n_s ≲ 10^-12 (Eq. 5.18). The intended message is that the model is predictive with essentially one parameter, the scalaron mass, and that the coherent-condensate picture is self-consistent.

Significance. If Eq. (A.7) is accepted, the paper resolves a genuine literature discrepancy and provides a closed-form, falsifiable prediction for scalaron decay into photons, with concrete observational targets in Sec. 4. The cosmological radiation calculation is careful and transparent, and the thermal bound (5.18) is robust to many orders of magnitude of uncertainty in the cross-section estimates. The authors are also honest about the heuristic character of the regularization argument in Appendix B, which is the main risk to the central claim.

major comments (2)

[Appendix B, Eqs. (B.2)–(B.8)] The central claim that the derivative-coupling and Jacobian contributions to the φγγ amplitude vanish is not established rigorously. The integral (B.2) is quadratically divergent in d=4, and the cancellation in dimensional regularization depends on the validity of the momentum shift, which is exactly the regulator-sensitive step. The Pauli–Villars construction uses a single subtraction, and the paper itself concedes in the last paragraph of Appendix B that the argument breaks down when more than one subtraction is needed. Since |F(2m_e)|≈4.2, an omitted O(1) F_an in Eq. (B.3) would change Γ in Eq. (A.7) by an order-unity factor and would shift all Sec. 4 background predictions, including Eq. (4.1) and the intensity spectrum (4.13). Please provide a regulator-independent proof, or an explicit calculation with at least two independent regulators demonstrating that the surface term vanishes, or alternatively quantify the residual uncertainty and propagate it into the mass bounds and background estimates.
[Section 5.1, Eqs. (5.1)–(5.3)] The cross-sections used for the thermal scalaron abundance are asserted as order-of-magnitude estimates with no derivation or error estimate, and the prefactors 10^-3 and 10^-1 are not explained. The final bound (5.18) is so small that an O(1) or even O(100) uncertainty does not affect the qualitative conclusion, so this is not a fatal issue. However, the text should state more explicitly that these prefactors are not derived from the Lagrangian and that the conclusion assumes no exponentially larger production mechanism beyond the processes considered.

minor comments (4)

[Section 1 and Section 6] The statement that the model has 'only one essential free parameter' is overstated: the thermal production estimates in Sec. 5 depend on the reheating temperature T_i and on the undetermined prefactors in Eqs. (5.1)–(5.3). The sentence should be qualified to refer to the scalaron's tree-level couplings and decay rates.
[Section 5.4, Eq. (5.14)] The ratio n^(b)/n^(f) is said to be 'typically much larger than unity'; it would be helpful to give the explicit numerical value for the fiducial parameters T_i=10^15 GeV, α_*=0.1, and m_f=m_e, so the reader can see the margin directly.
[Figure 5] The ratio R_Γ is plotted over a range that includes the W and top thresholds; a brief mention of where these thresholds occur would make the flatness of the ratio easier to interpret.
[Eq. (4.13)] The denominator in the final expression is written without parentheses; adding parentheses would improve readability, e.g., 1/√(Ω_m (E_max/E)^3 + 1-Ω_m).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the scalaron-photon decay rate is derived from the standard Higgs-loop amplitude with an explicit field-redefinition check, benchmarked against external results, and not repackaged from fitted inputs.

full rationale

The paper's central quantitative claim, the scalaron decay rate into two photons in Eq. (A.7), is obtained by a self-contained derivation rather than by fitting or by importing the desired conclusion. The interaction vertices (3.7) and (3.8) follow from the conformal transformation of the minimally coupled Standard Model Lagrangian, and Appendix A reduces the scalaron loop amplitude to the known Standard Model Higgs-to-two-photon amplitude (A.1)-(A.6) via the substitution chi/v -> -phi/2M, with the form factors taken from the independent literature [36,37]. Appendix B then directly addresses the only delicate step, namely whether the derivative couplings produced by the spinor redefinition (3.3) contribute to the phi-gamma-gamma loop; the paper argues by explicit diagrammatic cancellation in dimensional regularization and gives a Pauli-Villars construction, rather than assuming the vanishing kinetic-term contribution. The final rate is benchmarked against the independent result of Cembranos [3,4]. No fitted parameter is renamed as a prediction: the scalaron mass m is scanned over a range that is externally constrained by fifth-force bounds and the 511 keV line, and the cosmological photon spectrum uses that same independently derived rate together with standard background cosmology. The self-citations [15,16] concern initial conditions for the scalaron condensate and are auxiliary to the loop calculation; they do not supply the numerical value of the decay rate or the thermal fraction. The admitted limitation that the Pauli-Villars argument breaks down beyond one subtraction is a regularization-robustness concern, not circularity, because the central derivation does not reduce to that argument alone.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The model has one free parameter, the scalaron mass m, which ranges over the meV to MeV window and is constrained by external observations. The cross-section prefactors in Sec. 5 are hand-set order-of-magnitude estimates. The main axioms are the f(R) action, minimal coupling in the Jordan frame, the validity of the field-redefinition equivalence at one loop, and standard cosmology with Planck parameters. The scalaron itself is not an invented entity: it is the well-known scalar degree of freedom of f(R) gravity.

free parameters (3)

Scalaron mass m = meV to MeV, with bounds m ≥ 2.7 meV (fifth-force) and 1.04 MeV ≲ m ≲ 1.15 MeV (511 keV line constraint)
Central parameter of action (2.1); all predictions are functions of it, constrained by external observations.
Fermion annihilation cross-section prefactor = 10^-3
Order-of-magnitude coefficient in Eq. (5.1); set by hand without detailed derivation; affects the fermionic thermal production estimate.
Boson annihilation cross-section prefactor = 10^-1
Order-of-magnitude coefficient in Eq. (5.3); set by hand; affects the bosonic thermal production estimate, which dominates the thermal fraction.

assumptions (7)

domain assumption The theory is f(R) gravity with f(R) = R + R^2/(6m^2) (Eq. 2.10), with scalaron mass m as the model parameter.
The entire paper studies this specific modification of general relativity; other f(R) forms yield different high-field behavior but the quadratic term is the dark-matter-relevant part.
domain assumption Standard Model fields are minimally coupled to the Jordan-frame metric g_{μν} (Section 3).
This assumption generates the scalaron couplings (3.7)-(3.8). Other coupling choices, such as non-minimal coupling, would change the decay rates.
ad hoc to paper The spinor field redefinition ψ = e^{3φ/4M} ψ̃ is an equivalence transformation at one loop, so the S-matrix is invariant and the Jacobian contributes nothing (Appendix B).
This is the paper's central technical assumption. It is argued heuristically and the authors note the argument breaks down for more than one Pauli-Villars subtraction.
standard math The universe is homogeneous and isotropic with standard FLRW evolution: radiation domination in the early universe and matter plus cosmological constant at late times.
Used in Secs. 4 and 5 for redshift, expansion, and the Boltzmann equation.
domain assumption Cosmological parameters take the Planck 2018 values, Ω_m ≈ 0.3, Ω_φ h^2 ≈ 0.24, H0 ≈ 70 km/s/Mpc.
External input from [34]; the final extragalactic background amplitude scales with Ω_φ.
domain assumption The scalaron field oscillates with small amplitude around the quadratic minimum, so the mass m is constant and the potential is harmonic (justified by Eq. 2.17).
The dark matter condensate is treated as a coherent harmonic oscillator, which underpins the decay-rate and abundance calculations.
domain assumption The reheating temperature T_i can be as high as ~10^15 GeV ≈ 10^-3 M, used for the upper estimate of thermal scalaron production.
This bound sets the maximal thermal production; if reheating were hotter, the fraction would increase, but the estimate would still be small.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Interactions of the scalaron dark matter in $f (R)$ gravity." pith.science (2026). https://pith.science/paper/NBLSFNKO

@misc{pith2026250500324,
  author       = {Pith},
  title        = {Pith review of: Interactions of the scalaron dark matter in $f (R)$ gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NBLSFNKO}},
  note         = {Machine review of arXiv:2505.00324}
}

abstract

In $f(R)$ gravity, the scalaron -- a scalar degree of freedom arising from modification of General Relativity -- could account for all dark matter in the universe if its mass lies in the meV--MeV range. In this work, we revisit the scalaron's interactions with Standard Model particles, assuming their minimal coupling to gravity. In particular, we provide a detailed calculation of the scalaron's decay rate into two photons -- a one-loop process of significant interest that has been the subject of discrepancies in the literature. We demonstrate that a direct evaluation of loop diagrams with appropriate regularisation eliminates the ambiguities inherent in methods relying on Jacobians from field redefinitions. Assuming the scalaron constitutes all of dark matter, we calculate the average cosmological background radiation produced by its decays into photons. We also estimate the contribution of primordial scalarons emitted in the hot early universe to the present dark matter density and find it to be negligible. Our results support all key aspects of the original scenario, in which scalaron dark matter behaves as a coherently oscillating field.

Discussion (0). Continue with ORCID to comment.

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

Effective scalaron--photon interaction in $f(R)$ gravity
hep-th 2026-06 unverdicted novelty 7.0 of 10

In the Jordan-frame treatment of f(R) gravity the scalaron-photon effective coupling vanishes for m much less than loop-particle masses because the classical-trace diagrams cancel the Fujikawa anomaly term.
Scalaron dark matter dynamics: effects of Higgs non-minimal coupling to gravity
hep-ph 2025-09 unverdicted novelty 6.0 of 10

Adding Higgs non-minimal coupling ξ modifies the scalaron-Higgs trilinear term, yielding scalaron dark matter masses in 2.7 meV–0.7 MeV (misalignment case) or 3.6 meV–770 meV (interaction-dominated cases) plus an LHC-...

Reference graph

Works this paper leans on

42 extracted references · 7 canonical work pages · cited by 2 Pith papers

[1]

Capozziello, V.F

S. Capozziello, V.F. Cardone and A. Troisi,Dark energy and dark matter as curvature effects,JCAP08(2006) 001 [astro-ph/0602349]

arXiv 2006
[2]

Nojiri and S.D

S. Nojiri and S.D. Odintsov,Dark energy, inflation and dark matter from modified F(R)gravity,TSPU BulletinN8(110)(2011) 7 [0807.0685]

arXiv 2011
[3]

Cembranos,Dark matter fromR 2 gravity,Phys

J.A.R. Cembranos,Dark matter fromR 2 gravity,Phys. Rev. Lett.102(2009) 141301 [0809.1653]

arXiv 2009
[4]

Cembranos,Modified gravity and dark matter,J

J.A.R. Cembranos,Modified gravity and dark matter,J. Phys. Conf. Ser.718(2016) 032004 [1512.08752]. – 19 –

arXiv 2016
[5]

Corda, H.J

C. Corda, H.J. Mosquera Cuesta and R. Lorduy G´ omez,High-energy scalarons inR 2 gravity as a model for Dark Matter in galaxies,Astropart. Phys.35(2012) 362 [1105.0147]

arXiv 2012
[6]

Katsuragawa and S

T. Katsuragawa and S. Matsuzaki,Dark matter in modified gravity?,Phys. Rev. D95 (2017) 044040 [1610.01016]

arXiv 2017
[7]

Katsuragawa and S

T. Katsuragawa and S. Matsuzaki,Cosmic history of chameleonic dark matter in F(R)gravity,Phys. Rev. D97(2018) 064037 [1708.08702]

arXiv 2018
[8]

Yadav and M.M

B.K. Yadav and M.M. Verma,Dark matter as scalaron inf(R)gravity models,JCAP 10(2019) 052 [1811.03964]

arXiv 2019

Show all 42 references

[9]

Parbin and U.D

N. Parbin and U.D. Goswami,Scalarons mimicking dark matter in the Hu-Sawicki model off(R)gravity,Mod. Phys. Lett. A36(2021) 2150265 [2007.07480]

2021 arXiv
[10]

Shtanov,Light scalaron as dark matter,Phys

Y. Shtanov,Light scalaron as dark matter,Phys. Lett. B820(2021) 136469 [2105.02662]

2021 arXiv
[11]

Kumar Sharma and M.M

V. Kumar Sharma and M.M. Verma,Unifiedf(R)gravity at local scales,Eur. Phys. J. C82(2022) 400 [2201.01058]

2022 arXiv
[12]

Sotiriou and V

T.P. Sotiriou and V. Faraoni,f(R)theories of gravity,Rev. Mod. Phys.82(2010) 451 [0805.1726]

2010 arXiv
[13]

De Felice and S

A. De Felice and S. Tsujikawa,f(R)theories,Living Rev. Rel.13(2010) 3 [1002.4928]

2010 arXiv
[14]

Nojiri and S.D

S. Nojiri and S.D. Odintsov,Unified cosmic history in modified gravity: FromF(R) theory to Lorentz non-invariant models,Phys. Rept.505(2011) 59 [1011.0544]

2011 arXiv
[15]

Shtanov,Initial conditions for the scalaron dark matter,JCAP10(2022) 079 [2207.00267]

Y. Shtanov,Initial conditions for the scalaron dark matter,JCAP10(2022) 079 [2207.00267]

2022 arXiv
[16]

Shtanov,Scalaron dark matter and the thermal history of the universe,JCAP12 (2024) 028 [2409.05027]

Y. Shtanov,Scalaron dark matter and the thermal history of the universe,JCAP12 (2024) 028 [2409.05027]

2024 arXiv
[17]

Starobinsky,A new type of isotropic cosmological models without singularity, Phys

A.A. Starobinsky,A new type of isotropic cosmological models without singularity, Phys. Lett. B91(1980) 99

1980
[18]

Vilenkin,Classical and quantum cosmology of the Starobinsky inflationary model, Phys

A. Vilenkin,Classical and quantum cosmology of the Starobinsky inflationary model, Phys. Rev. D32(1985) 2511. [19]Planckcollaboration,Planck 2018 results. X. Constraints on inflation,Astron. Astrophys.641(2020) A10 [1807.06211]

1985 arXiv
[20]

Wald,General Relativity, University of Chicago Press, Chicago (1984), 10.7208/chicago/9780226870373.001.0001

R.M. Wald,General Relativity, University of Chicago Press, Chicago (1984), 10.7208/chicago/9780226870373.001.0001

1984
[21]

Shtanov, V

Y. Shtanov, V. Sahni and S.S. Mishra,Tabletop potentials for inflation fromf(R) gravity,JCAP03(2023) 023 [2210.01828]

2023 arXiv
[22]

Dicke,Mach’s principle and invariance under transformation of units,Phys

R.H. Dicke,Mach’s principle and invariance under transformation of units,Phys. Rev. 125(1962) 2163

1962
[23]

Faraoni and S

V. Faraoni and S. Nadeau,(Pseudo)issue of the conformal frame revisited,Phys. Rev. D75(2007) 023501 [gr-qc/0612075]. – 20 –

2007 arXiv
[24]

Shtanov,On the conformal frames inf(R)gravity,Universe8(2022) 69 [2202.00818]

Y. Shtanov,On the conformal frames inf(R)gravity,Universe8(2022) 69 [2202.00818]

2022 arXiv
[25]

Stelle,Classical gravity with higher derivatives,Gen

K.S. Stelle,Classical gravity with higher derivatives,Gen. Rel. Grav.9(1978) 353

1978
[26]

Kapner, T.S

D.J. Kapner, T.S. Cook, E.G. Adelberger, J.H. Gundlach, B.R. Heckel, C.D. Hoyle et al.,Tests of the gravitational inverse-square law below the dark-energy length scale, Phys. Rev. Lett.98(2007) 021101 [hep-ph/0611184]

2007 arXiv
[27]

Adelberger, B.R

E.G. Adelberger, B.R. Heckel, S.A. Hoedl, C.D. Hoyle, D.J. Kapner and A. Upadhye, Particle-physics implications of a recent test of the gravitational inverse-square law, Phys. Rev. Lett.98(2007) 131104 [hep-ph/0611223]

2007 arXiv
[28]

Perivolaropoulos and L

L. Perivolaropoulos and L. Kazantzidis,Hints of modified gravity in cosmos and in the lab?,Int. J. Mod. Phys. D28(2019) 1942001 [1904.09462]

2019 arXiv
[29]

Chisholm,Change of variables in quantum field theories,Nucl

J.S.R. Chisholm,Change of variables in quantum field theories,Nucl. Phys.26(1961) 469

1961
[30]

Kamefuchi, L

S. Kamefuchi, L. O’Raifeartaigh and A. Salam,Change of variables and equivalence theorems in quantum field theories,Nucl. Phys.28(1961) 529

1961
[31]

Divakaran,Equivalence theorems and point transformations in field theory,Nucl

P.P. Divakaran,Equivalence theorems and point transformations in field theory,Nucl. Phys.42(1963) 235

1963
[32]

Arzt,Reduced effective lagrangians,Phys

C. Arzt,Reduced effective lagrangians,Phys. Lett. B342(1995) 189 [hep-ph/9304230]

1995 arXiv
[33]

Cohen, M

T. Cohen, M. Forslund and A. Helset,Field redefinitions can be nonlocal,2412.12247. [34]Planckcollaboration,Planck 2018 results. VI. Cosmological parameters,Astron. Astrophys.641(2020) A6 [1807.06209]

2020
[35]

Shtanov and Y

Y. Shtanov and Y. Sheiko,Interactions of the scalaron dark matter inf(R)gravity, 2505.00324v1

arXiv
[36]

Shifman, A.I

M.A. Shifman, A.I. Vainshtein, M.B. Voloshin and V.I. Zakharov,Low-energy theorems for Higgs boson couplings to photons,Sov. J. Nucl. Phys.30(1979) 711

1979
[37]

Marciano, C

W.J. Marciano, C. Zhang and S. Willenbrock,Higgs decay to two photons,Phys. Rev. D85(2012) 013002 [1109.5304]

2012 arXiv
[38]

Goldberger, B

W.D. Goldberger, B. Grinstein and W. Skiba,Distinguishing the Higgs boson from the dilaton at the Large Hadron Collider,Phys. Rev. Lett.100(2008) 111802 [0708.1463]

2008 arXiv
[39]

Fujikawa and H

K. Fujikawa and H. Suzuki,Path Integrals and Quantum Anomalies, Clarendon Press, Oxford, UK (2004), 10.1093/acprof:oso/9780198529132.001.0001

2004
[40]

Larue, J

R. Larue, J. Quevillon and R. Zwicky,Trace anomaly of Weyl fermions via the path integral,JHEP12(2023) 064 [2309.08670]

2023 arXiv
[41]

Larue, J

R. Larue, J. Quevillon and R. Zwicky,Gravity-gauge anomaly constraints on the energy-momentum tensor,JHEP05(2024) 307 [2312.13222]

2024 arXiv
[42]

Larue, J

R. Larue, J. Quevillon and R. Zwicky,Model-independent results on parity violation in the trace anomaly, in17th Marcel Grossmann Meeting: On Recent Developments in Theoretical and Experimental General Relativity, Gravitation, and Relativistic Field Theories, 11, 2024 [2411.005...

2024 arXiv
[43]

Bonora and S.G

L. Bonora and S.G. Giaccari,Something anomalies can tell about Standard Model and gravity,Symmetry17(2025) 273 [2412.07470]

2025
[44]

Schwartz,Quantum Field Theory and the Standard Model, Cambridge University Press, Cambridge (2014)

M.D. Schwartz,Quantum Field Theory and the Standard Model, Cambridge University Press, Cambridge (2014). – 22 –

2014

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Capozziello, V.F

S. Capozziello, V.F. Cardone and A. Troisi,Dark energy and dark matter as curvature effects,JCAP08(2006) 001 [astro-ph/0602349]

arXiv 2006

[2] [2]

Nojiri and S.D

S. Nojiri and S.D. Odintsov,Dark energy, inflation and dark matter from modified F(R)gravity,TSPU BulletinN8(110)(2011) 7 [0807.0685]

arXiv 2011

[3] [3]

Cembranos,Dark matter fromR 2 gravity,Phys

J.A.R. Cembranos,Dark matter fromR 2 gravity,Phys. Rev. Lett.102(2009) 141301 [0809.1653]

arXiv 2009

[4] [4]

Cembranos,Modified gravity and dark matter,J

J.A.R. Cembranos,Modified gravity and dark matter,J. Phys. Conf. Ser.718(2016) 032004 [1512.08752]. – 19 –

arXiv 2016

[5] [5]

Corda, H.J

C. Corda, H.J. Mosquera Cuesta and R. Lorduy G´ omez,High-energy scalarons inR 2 gravity as a model for Dark Matter in galaxies,Astropart. Phys.35(2012) 362 [1105.0147]

arXiv 2012

[6] [6]

Katsuragawa and S

T. Katsuragawa and S. Matsuzaki,Dark matter in modified gravity?,Phys. Rev. D95 (2017) 044040 [1610.01016]

arXiv 2017

[7] [7]

Katsuragawa and S

T. Katsuragawa and S. Matsuzaki,Cosmic history of chameleonic dark matter in F(R)gravity,Phys. Rev. D97(2018) 064037 [1708.08702]

arXiv 2018

[8] [8]

Yadav and M.M

B.K. Yadav and M.M. Verma,Dark matter as scalaron inf(R)gravity models,JCAP 10(2019) 052 [1811.03964]

arXiv 2019