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REVIEW 3 major objections 5 minor 103 references

Dark matter in scale-invariant gravity with hidden-sector condensation

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A classically scale-invariant quadratic gravity with a hidden confining sector can generate the Planck and electroweak scales, drive Starobinsky inflation, and produce the observed dark matter via scalaron decays, all without input mass…

desk verdict Same-group extension with a systematic three-model DM study; the Planck-mass sign assumption is openly admitted but load-bearing, and the DM masses are fits, not predictions. read the letter →

arxiv 2608.12456 v1 pith:EOEEIY6P submitted 2026-08-12 hep-ph

Juan P. Garcés , Jisuke Kubo , Manfred Lindner , Markus Reinig This is my paper · ORCID
classification hep-ph
keywords classicalscaleinvariancequadraticgravityhiddenstrongdynamicsdimensionaltransmutationStarobinskyinflationinducedelectroweakgravitationalfreeze-incompositedarkmatter
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

This paper seeks to show that four long-standing puzzles—the size of the electroweak scale, the existence of the Planck scale, cosmic inflation, and dark matter—can be accounted for by a single classically scale-invariant Lagrangian. A hidden QCD-like SU(Nc) sector confines and condenses, generating the Planck mass by dimensional transmutation; because the hidden and visible sectors are orthogonal, the breaking reaches the Standard Model only gravitationally and induces the Higgs mass. The scalar degree of freedom of the $R^{2}$ term acts as the Starobinsky inflaton and reheats both sectors through its universal coupling to the trace of the energy-momentum tensor, so hidden-sector relics are produced by gravitational freeze-in. The paper derives the relic formula (4.8) and, in three concrete hidden-sector models, finds viable dark matter masses.

What carries the argument

The engine is the classically scale-invariant quadratic-gravity action with a dimensionless $R^2$ term, a Weyl-squared term, and a non-minimal Higgs-gravity coupling; the scalaron is the scalar degree of freedom hidden in the $R^2$ term and becomes the Starobinsky inflaton. The Planck mass is not put in by hand but appears from the hidden gluon condensate, while Eq. (2.8), obtained from the one-loop gravitational portal, converts that scale into the Higgs mass; in the semi-conformal limit $\xi_H \to -1/6$, the observed 125 GeV Higgs mass fixes $\kappa \simeq 5\times10^{14} C^{1/2}$, with all non-perturbative corrections absorbed in an unknown coefficient $C$ of order 0.1 to 10. All subsequent abundance calculations rest on the trace-anomaly interaction $\chi T^\alpha{}_\alpha/(\sqrt{6}M_{\rm Pl})$, which gives the scalaron partial widths into hidden states and, together with the Boltzmann system (4.1)–(4.3), yields the dark matter and dark radiation relic abundances and the key formula Eq. (4.8).

What would settle it

A lattice computation of the induced Einstein-Hilbert term in the hidden SU(Nc) theory with one fundamental Dirac fermion would settle the matter: if the chiral-condensate contribution has the wrong sign or cancels the gluon contribution, Planck-mass generation fails. Observationally, a tensor-to-scalar ratio that is not suppressed as strongly as predicted by the Weyl-squared ghost, or a dark radiation abundance $\Delta N_{\rm eff}$ incompatible with the model's predictions, would falsify the proposed models.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that a single scale-invariant framework replaces the usual hierarchy of input scales with one dynamical sequence: hidden confinement generates the Planck mass; the same condensation, acting through the non-minimal term $\xi_H H^\dagger H R$, induces the electroweak scale; the $R^2$ term supplies Starobinsky inflation; and scalaron decays replenish the Universe, producing dark matter gravitationally. The operative formula is Eq. (4.8), $\Omega_{\rm DM}h^2 \simeq 0.12\, (m_{\rm DM}/m_\phi)(B_{\rm DM}/2.9\times10^{-10})(T_{\rm RH}^{\rm SM}/{\rm GeV})$, which ties the relic abundance to the scalaron branching ratio, the dark matter mass, and the reheating temperature of the visible sector. Applied to a hidden $\eta'$ meson, a hidden vector from a gauged SU(2)L, and hidden charged pions, the requirement $\Omega h^2 = 0.12$ fixes the dark matter mass as a function of the inflationary e-folds; for the composite candidates of models I and III, viable masses lie in the range $10^8$ to $10^{11}$ GeV.

Load-bearing premise

The whole chain of scale generation rests on the assumption that the hidden gluon condensate, not the chiral condensate, dominates the induced Einstein-Hilbert term with the correct sign, and that the Higgs-gravity coupling stays at the semi-conformal value $\xi_H = -1/6$ with an unknown coefficient $C$ of order 0.1 to 10 absorbing all non-perturbative corrections.

Editorial extensions

If this is right

  • If the framework is correct, no explicit mass terms are needed in the fundamental Lagrangian: the Planck scale appears as a hidden-sector condensate and the electroweak scale as its gravitational by-product.
  • Inflation is Starobinsky-like, with the spectral index and tensor-to-scalar ratio controlled by the number of e-folds; the Weyl-squared ghost suppresses the tensor ratio by orders of magnitude.
  • Dark matter is a gravitational freeze-in product of the same reheating phase, and for the composite hidden mesons of models I and III the required masses are $10^8$ to $10^{11}$ GeV.
  • Hidden radiation from massless states contributes $\Delta N_{\rm eff}$, and the current bound $\Delta N_{\rm eff} \lesssim 0.12$ restricts the hidden gauge couplings, such as $e_{A'}$ in model III.
  • In model I, the hidden $\eta'$ can decay into gravitons, so requiring its lifetime to exceed the age of the Universe places an upper bound on $m_{\eta'}$ for each inflationary history.

Reading between the lines

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

  • One step beyond the paper's claims, the universal trace coupling suggests that any hidden species with a small enough branching ratio inherits the same parametric abundance law as Eq. (4.8), so the observed dark matter density may constrain ratios of hidden masses and decay widths rather than any specific particle identity.
  • A nonperturbative computation fixing the unknown coefficient $C$ would convert the condition $\kappa \simeq 5\times10^{14} C^{1/2}$ from a placeholder into a genuine prediction for the spin-two ghost mass and hence for the tensor-to-scalar ratio.
  • Because the hidden sector is gravitationally secluded, the model points toward cosmological rather than collider tests: a precise measurement of $\Delta N_{\rm eff}$ or of $r$ could discriminate among the three realizations even if no hidden particle is ever produced in the laboratory.
  • A lattice computation of the sign and size of the induced Einstein-Hilbert term in a hidden SU(Nc) gauge theory with chiral fermions would test the gluon-dominance premise directly, before any cosmological comparison is made.
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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 / 5 minor

Summary. The manuscript proposes a classically scale-invariant framework in which quadratic gravity plus a hidden confining SU(N_c) gauge sector generates the Planck and electroweak scales via hidden condensates, realizes Starobinsky inflation through the R^2 scalaron, and reheats both the visible and hidden sectors through scalaron decay. The central result is the gravitational freeze-in formula in Eq. (4.8), which relates the present dark matter abundance to the scalaron branching ratio, the dark matter mass, and the visible reheating temperature. Three hidden-sector dark matter realizations are studied: a hidden eta-prime meson (Model I), hidden vector bosons from a gauged SU(2)_L (Model II), and charged hidden pions together with massless dark radiation (Model III). The authors derive dark matter and dark radiation relic abundances and quote viable composite dark matter masses in the range 10^8 to 10^11 GeV for Models I and III, together with constraints on hidden gauge couplings and on Delta N_eff.

Significance. If the underlying scale-generation premises can be substantiated, the framework would be a noteworthy unification of the hierarchy problem, inflation, dark matter, and dark radiation in a single classically scale-invariant structure. The paper's strengths are its careful derivation of the freeze-in abundance and its agreement with Ref. [92], its explicit acknowledgment of the conditions needed for single-field inflation in footnote 7, and its treatment of three qualitatively different dark matter stability mechanisms. The analysis is nonetheless conditional on two unproven premises: the dominance and sign of the hidden gluon condensate over the chiral condensate in inducing the Einstein-Hilbert term, and the validity of the single-field reduction in the parameter regions used for the dark matter mass windows. Because these premises support the central claim, the manuscript needs substantive revision rather than minor polish.

major comments (3)
  1. [Sec. 2.1] The induced Planck scale is assumed, not derived. Equations (2.4), (2.8), and the subsequent determination of kappa in Eq. (3.34) rely on the hidden gluon condensate inducing an Einstein-Hilbert term with the correct sign and dominating the chiral condensate. The paper states, in Sec. 2.1, that 'one must assume that the gluon contribution dominates.' The supporting beta-function argument controls asymptotic freedom, not the curved-space matrix element of <F^2> relative to <psi-bar psi>; moreover, Ref. [53] is a lattice computation for pure Yang-Mills, whereas all models in Sec. 5 contain chiral fermions so that both condensates are present. Please provide a computation or a lattice-motivated estimate of the two contributions to the induced R coefficient for N_f > 0 and demonstrate the required sign and magnitude. Without this, the Planck scale, the induced Higgs mass, and every abundance computed from Eq. (4.8) rest on an untested premise.
  2. [Sec. 3.1, footnote 7] The reduction to single-field Starobinsky inflation is asserted rather than verified in the regions of parameter space used for the dark matter predictions. The valley condition gives the location sigma_v(phi) = v_sigma, but as footnote 7 correctly notes, single-field dynamics also requires m^2_{sigma,perp} >> H^2 and a negligible turning rate in field space. These conditions are not demonstrated for the parameter points that produce the mass windows in Figs. 5, 7, and 8. Figure 3 is only a representative illustration. In regions where the hidden condensate is shallow or the trajectory bends, Eq. (3.27) and the reheating relation (3.48) would not apply. Please compute the orthogonal mass and the turning rate from the NJL potential and state the allowed ranges of the hidden-sector parameters in which the single-field approximation is valid.
  3. [Secs. 5.1, 5.3, and 6] The quoted 'viable composite dark matter masses' are not predictions in the usual sense. In Model I the text says, 'Since m_eta' is not calculable, we regard it as a free parameter,' and in Model III the charged pion mass is fixed by inverting Eq. (4.8) while Delta in Eq. (5.30) is treated as an independent parameter. The mass windows are therefore the values required to reproduce Omega_DM h^2 = 0.12 for a chosen reheating history, not independent outputs of the framework. The summary in Sec. 6 should distinguish 'required mass' from 'predicted mass' and should state the number of free parameters in each model before the relic abundance is imposed.
minor comments (5)
  1. [Fig. 3] The axis labels in the left panel are ambiguous; the sigma/v_sigma and phi/M_Pl axes appear to be interchanged or incompletely labelled, which makes it hard to verify the claimed valley behaviour.
  2. [Fig. 5 caption] The caption and surrounding text contain notation slips such as m'_eta for m_eta' and f_eta for f_eta'; the ratio being plotted should also be defined explicitly.
  3. [Appendix B, Eq. (B.7)] The displayed factor contains a repeated polarization index, epsilon^{(s)*}_mu(k1) epsilon^{(s)*}_mu(k1), which is inconsistent with the subsequent helicity sum; please correct the index structure.
  4. [Sec. 3.3] The symbol for the mean equation of state is written as \(\bar\omega\) after Eq. (3.43) but as \(\bar w\) in Eq. (3.40); use one notation consistently.
  5. [References] Reference [31] is cited as a 2025 manuscript without journal information, yet it appears to contain several ingredients of the scale-generation argument; its status should be clarified or updated.

Circularity Check

0 steps flagged · score 0.0 of 10

No construction-level circularity: the DM mass windows are inverse constraints rather than predictions, and the key assumptions are unverified premises rather than tautological inputs.

full rationale

I find no circular step that reduces a claimed prediction to an input by construction. The scale-generation chain rests on the explicit gluon-dominance assumption of Sec. 2.1; that is a physical premise whose failure would break the mechanism, but it is not a tautology and is flagged by the authors as an assumption. The Higgs sector is not presented as a prediction: Eq. (3.34) fixes kappa by requiring Eq. (2.8) to give 125 GeV, and C is an explicitly absorbed O(0.1-10) uncertainty. The inflationary reduction is presented with footnote 7's caveat that single-field Starobinsky dynamics requires a heavy orthogonal mode; this is an unverified condition, not a circular input. Most importantly, the dark-matter masses in Models I and III are not obtained by predicting Omega_DM; the paper declares m_eta' and Delta to be free parameters and then inverts Eq. (4.8) with Omega=0.12, using language such as 'value ... needed to produce the observed dark matter relic abundance today, namely Omega_eta' h^2 = 0.12'. The quoted 10^8-10^11 GeV range is therefore an inverse constraint, not a fitted parameter renamed as a prediction. Self-citations to companion and earlier work (Refs. [28,31,62,64]) exist, but the central equations (3.27), (4.8), (5.7), (5.24) and (5.32) are displayed and derived in the text; no load-bearing result reduces to an unverified self-citation chain or to a uniqueness claim imported from the authors. The external benchmarks (Planck amplitude, Omega_DM, Delta_N_eff) are used as constraints rather than as outputs of a tautological derivation, so the paper is self-contained in the sense relevant to circularity.

Assumptions & free parameters 9 free parameters · 8 assumptions · 6 invented entities

The framework's predictive content is partly paid for by assumptions: the hidden-sector existence and condensate dominance, the NJL description, the semi-conformal limit, the instantaneous-reheating approximation, and the trace-anomaly effective couplings. The dark matter and Higgs mass numbers are then fitted to observations through free parameters (m_eta', Delta, g_Z', f_pi', C, kappa). The ledger above lists these inputs.

free parameters (9)
  • R^2 coefficient gamma = [4.5, 6.4] x 10^8 for N_e in [49,59]
    Fixed by matching the CMB scalar amplitude A_s at horizon exit (Eqs. 3.29 to 3.32); sets the scalaron mass m_phi and the inflation-normalized potential.
  • Weyl^2 coefficient kappa = about 5.0 x 10^14 C^(1/2)
    Chosen so the induced Higgs mass matches m_H = 125 GeV in the semi-conformal limit (Eq. 3.34 applied to Eq. 2.8); this is a fit to the measured Higgs mass.
  • Non-perturbative coefficient C = assumed O(0.1) to O(10)
    Introduced in Eq. (2.8) to absorb all non-perturbative corrections to the induced Higgs mass; its value is not computed, and kappa is normalized through it.
  • Hidden eta' mass m_eta' (Model I) = about 10^8 to 10^11 GeV depending on N_e (Fig. 5)
    Declared non-calculable and free (Sec. 5.1); solved from Eq. (4.8) with Omega_eta' h^2 = 0.12.
  • Hidden eta' decay constant f_eta' or ratio m_eta'/f_eta' (Model I) = varied: 1, 1e-1, 1e-2 curves in Fig. 5
    Free parameter controlling the eta' to gg lifetime constraint (Eq. 5.16) and thus the allowed mass window.
  • Hidden SU(2)_L gauge coupling g_Z' (Model II) = varied O(10^-5) to O(1) in Fig. 6
    Together with f_pi' sets M_Z' via Eq. (5.20); the combination g^5_Z' f_pi'/M_Pl is fixed by Omega h^2 = 0.12.
  • Hidden pion decay constant f_pi' (Model II) = varied through Fig. 6 with M_Z' up to about 1e14 GeV
    Free parameter; enters M_Z' and the decay width in Eq. (5.24).
  • NJL parameter Delta (Model III) = implicitly set by m_pi'+/- about 10^8 to 10^11 GeV (Fig. 7)
    Defines m^2_pi'+/- = e^2_A' Delta^2/(4 pi^2) (Eq. 5.30); treated as an independent parameter, and m_pi'+/- is then fitted to Omega h^2 = 0.12.
  • Hidden U(1)_A' coupling e_A' (Model III) = bounded by Delta_Neff < 0.12 (Fig. 8)
    Free parameter of the model; used to fix the dark radiation abundance via Eq. (4.17) against the Neff bound.
assumptions (8)
  • domain assumption The theory is classically scale invariant and all mass scales arise from spontaneous breaking of scale invariance via dimensional transmutation.
    Motivates the framework; invoked throughout Sec. 2 and the Introduction; relies on the existence of massless theories in perturbation theory, cited to Refs. [45,46].
  • domain assumption The hidden gauge gluon condensate dominates over the chiral condensate in inducing the Einstein-Hilbert term, with the correct sign.
    Needed for a positive Newton constant (Sec. 2.1); the paper states 'one must assume that the gluon contribution dominates', citing the opposite sign from the chiral condensate.
  • domain assumption The NJL effective potential in the mean-field approximation describes the hidden strong dynamics sufficiently well for the inflationary valley.
    Used to compute the hidden-sector potential (Eqs. 3.1 to 3.3); the location of the valley and the heavy orthogonal mode condition m^2_sigma,perp >> H^2 are not demonstrated explicitly (footnote 7).
  • domain assumption The semi-conformal limit xi_H = -1/6 holds in the relevant energy range with negligible corrections to the induced Higgs mass.
    Required to suppress the gamma^-1 term in Eq. (2.8); corrections from running are stated to be negligible (Sec. 2.2).
  • domain assumption The spin-2 ghost in quadratic gravity does not spoil inflation, and perturbation theory remains trustworthy.
    The paper notes the debate on the ghost and proceeds (footnote 8); the suppression of r in Eq. (3.33) depends on this assumption.
  • domain assumption The reheating phase is approximately instantaneous, with the scalaron decay width identified with the Hubble rate at the end of reheating and negligible production before or during inflation.
    Used to derive the relic abundance formulas in Secs. 3.3 to 4; the dilution argument and Eq. (4.11) encode this assumption.
  • domain assumption For the NGB sector at xi = -1/6, the trace of the energy-momentum tensor reduces to T^mu_mu = M^2 pi^2 to quadratic order (Eq. A.20), and the scalaron coupling is L_chi = -(1/sqrt(6))(chi/M_Pl) T^alpha_alpha.
    This is the key effective coupling used for all decay width computations (Eqs. 5.5 and 5.31); derived in Appendix A within the nonlinear sigma model but relies on the conformal value xi = -1/6.
  • standard math The gravitational axial anomaly coefficient for one Dirac fermion in SU(N_c) is N_c/(192 pi^2) with the stated conventions.
    Underlies the eta' to gg width estimate in Eq. (5.16); the paper notes the sign and factors of two depend on conventions.
invented entities (6)
  • Hidden SU(N_c) gauge sector with gluon condensate
    purpose: Provides dimensional transmutation to generate M_Pl and, through the gravitational portal, the electroweak scale; also hosts dark matter candidates.
    Postulated; no direct detection channel. Its existence is the core new physics of the paper.
  • Hidden fermions psi and Psi
    purpose: Form chiral condensates and composite states; the fermion content determines the dark matter candidates in the three models.
    No independent evidence; their number and flavor content are chosen by model.
  • Hidden eta' meson (Model I)
    purpose: Dark matter candidate: lightest hidden hadron, stable against strong decays, with gravitational-anomaly decay to gravitons constraining the mass.
    Mass is a free parameter fitted to the relic abundance (Fig. 5); only the lifetime constraint uses the age of the universe.
  • Hidden Z' vector bosons (Model II)
    purpose: Dark matter candidate stabilized by custodial SU(2)_V.
    The mass and coupling combination are fitted to Omega h^2 = 0.12 (Fig. 6).
  • Charged hidden pions pi'+/- (Model III)
    purpose: Dark matter candidate stabilized by unbroken U(1)_A'.
    Mass is set by the free parameter Delta and fitted to the relic abundance (Fig. 7).
  • Massless hidden gauge boson A' and neutral pion pi'0 (Model III) independent evidence
    purpose: Contribute to dark radiation Delta_Neff.
    The model yields a calculable Delta_Neff (Eq. 4.21) compared against the Planck and BAO bound (Eq. 4.22); a future measurement of extra relativistic degrees of freedom could detect or exclude these massless states, so a falsifiable handle exists in principle, though the coupling e_A' remains adjustable.

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Pith. "Pith review of Dark matter in scale-invariant gravity with hidden-sector condensation." pith.science (2026). https://pith.science/paper/EOEEIY6P

@misc{pith2026260812456,
  author       = {Pith},
  title        = {Pith review of: Dark matter in scale-invariant gravity with hidden-sector condensation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EOEEIY6P}},
  note         = {Machine review of arXiv:2608.12456}
}
abstract

The origin of the electroweak scale, cosmic inflation, and dark matter are often treated as independent problems beyond the Standard Models of particle physics and cosmology. In this work, we explore the possibility that they instead arise from a common underlying framework based on classically scale-invariant quadratic gravity coupled to a strongly interacting hidden sector. The $R^2$ term naturally realizes Starobinsky inflation, while confinement in the hidden sector dynamically generates the Planck scale and triggers electroweak symmetry breaking through a gravitationally induced Higgs mass generation mechanism. The scalar degree of freedom associated with the $R^2$ term subsequently reheats both the visible and hidden sectors through universal couplings to the energy-momentum tensor, leading to the gravitational freeze-in production of hidden-sector states. We investigate three representative realizations of the hidden sector in which the dark matter candidate is either a hidden $\eta'$ meson, a hidden vector boson, or charged hidden pions, and derive the corresponding dark matter and dark radiation relic abundances.

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Works this paper leans on

103 extracted references · 10 canonical work pages

  1. [92]

    Allahverdi and M

    R. Allahverdi and M. Drees,Production of massive stable particles in inflaton decay,Phys. Rev. Lett.89(2002) 091302 [hep-ph/0203118]. – 35 –

    arXiv 2002

  2. [53]

    Donoghue and G

    J.F. Donoghue and G. Menezes,Inducing the einstein action in qcd-like theories,Physical Review D97(2018)

    2018

  3. [1]

    Coleman and E.J

    S.R. Coleman and E.J. Weinberg,Radiative Corrections as the Origin of Spontaneous Symmetry Breaking,Phys. Rev. D7(1973) 1888

    1973

  4. [2]

    Gildener and S

    E. Gildener and S. Weinberg,Symmetry Breaking and Scalar Bosons,Phys. Rev. D13(1976) 3333

    1976

  5. [3]

    Hempfling,The Next-to-minimal Coleman-Weinberg model,Phys

    R. Hempfling,The Next-to-minimal Coleman-Weinberg model,Phys. Lett. B379(1996) 153 [hep-ph/9604278]

    arXiv 1996

  6. [4]

    Meissner and H

    K.A. Meissner and H. Nicolai,Conformal Symmetry and the Standard Model,Phys. Lett. B 648(2007) 312 [hep-th/0612165]

    arXiv 2007

  7. [5]

    R. Foot, A. Kobakhidze and R.R. Volkas,Electroweak Higgs as a pseudo-Goldstone boson of broken scale invariance,Phys. Lett. B655(2007) 156 [0704.1165]

    arXiv 2007

  8. [6]

    Chang, J.N

    W.-F. Chang, J.N. Ng and J.M.S. Wu,Shadow Higgs from a scale-invariant hidden U(1)(s) model,Phys. Rev. D75(2007) 115016 [hep-ph/0701254]

    arXiv 2007

Show all 103 references
  1. [7]

    R. Foot, A. Kobakhidze, K.L. McDonald and R.R. Volkas,Neutrino mass in radiatively-broken scale-invariant models,Phys. Rev. D76(2007) 075014 [0706.1829]

    2007 arXiv

  2. [8]

    R. Foot, A. Kobakhidze, K.L. McDonald and R.R. Volkas,A Solution to the hierarchy problem from an almost decoupled hidden sector within a classically scale invariant theory,Phys. Rev. D 77(2008) 035006 [0709.2750]

    2008 arXiv

  3. [9]

    S. Iso, N. Okada and Y. Orikasa,Classically conformalB − L extended Standard Model,Phys. Lett. B676(2009) 81 [0902.4050]

    2009 arXiv

  4. [10]

    Alexander-Nunneley and A

    L. Alexander-Nunneley and A. Pilaftsis,The Minimal Scale Invariant Extension of the Standard Model,JHEP09(2010) 021 [1006.5916]

    2010 arXiv

  5. [11]

    Holthausen, J

    M. Holthausen, J. Kubo, K.S. Lim and M. Lindner,Electroweak and Conformal Symmetry Breaking by a Strongly Coupled Hidden Sector,JHEP12(2013) 076 [1310.4423]

    2013 arXiv

  6. [12]

    Englert, J

    C. Englert, J. Jaeckel, V.V. Khoze and M. Spannowsky,Emergence of the Electroweak Scale through the Higgs Portal,JHEP04(2013) 060 [1301.4224]

    2013 arXiv

  7. [13]

    Kubo, K.S

    J. Kubo, K.S. Lim and M. Lindner,Electroweak Symmetry Breaking via QCD,Phys. Rev. Lett. 113(2014) 091604 [1403.4262]

    2014 arXiv

  8. [14]

    Kubo, K.S

    J. Kubo, K.S. Lim and M. Lindner,Gamma-ray Line from Nambu-Goldstone Dark Matter in a Scale Invariant Extension of the Standard Model,JHEP09(2014) 016 [1405.1052]

    2014 arXiv

  9. [15]

    Lindner, S

    M. Lindner, S. Schmidt and J. Smirnov,Neutrino Masses and Conformal Electro-Weak Symmetry Breaking,JHEP10(2014) 177 [1405.6204]

    2014 arXiv

  10. [16]

    Kubo and M

    J. Kubo and M. Yamada,Scale and electroweak first-order phase transitions,PTEP2015 (2015) 093B01 [1506.06460]

    2015 arXiv

  11. [17]

    Hatanaka, D.-W

    H. Hatanaka, D.-W. Jung and P. Ko,AdS/QCD approach to the scale-invariant extension of the standard model with a strongly interacting hidden sector,JHEP08(2016) 094 [1606.02969]

    2016 arXiv

  12. [18]

    Ahmed, J.P

    A. Ahmed, J.P. Garc´ es and M. Lindner,Radiative symmetry breaking with a scale invariant seesaw mechanism,Phys. Rev. D112(2025) 035026 [2504.13243]

    2025

  13. [19]

    Bardeen,On naturalness in the standard model, inOntake Summer Institute on Particle Physics, 8, 1995

    W.A. Bardeen,On naturalness in the standard model, inOntake Summer Institute on Particle Physics, 8, 1995

    1995

  14. [20]

    Stelle,Renormalization of Higher Derivative Quantum Gravity,Phys

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

    1977

  15. [21]

    Fradkin and A.A

    E.S. Fradkin and A.A. Tseytlin,Renormalizable asymptotically free quantum theory of gravity, Nucl. Phys. B201(1982) 469. – 32 –

    1982

  16. [22]

    Salvio,Quadratic Gravity,Front

    A. Salvio,Quadratic Gravity,Front. in Phys.6(2018) 77 [1804.09944]

    2018 arXiv

  17. [24]

    Einhorn and D.R.T

    M.B. Einhorn and D.R.T. Jones,Naturalness and Dimensional Transmutation in Classically Scale-Invariant Gravity,JHEP03(2015) 047 [1410.8513]

    2015 arXiv

  18. [25]

    Whitt,Fourth order gravity as general relativity plus matter,Phys

    B. Whitt,Fourth order gravity as general relativity plus matter,Phys. Lett. B145(1984) 176

    1984

  19. [26]

    Maeda,Inflation as a Transient Attractor in R**2 Cosmology,Phys

    K.-i. Maeda,Inflation as a Transient Attractor in R**2 Cosmology,Phys. Rev. D37(1988) 858

    1988

  20. [27]

    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

  21. [28]

    M. Aoki, J. Kubo and J. Yang,Inflation and dark matter after spontaneous Planck scale generation by hidden chiral symmetry breaking,JCAP01(2022) 005 [2109.04814]

    2022 arXiv

  22. [29]

    Hur and P

    T. Hur and P. Ko,Scale invariant extension of the standard model with strongly interacting hidden sector,Phys. Rev. Lett.106(2011) 141802 [1103.2571]

    2011 arXiv

  23. [30]

    Kubo and M

    J. Kubo and M. Yamada,Scale and confinement phase transitions in scale invariantSU(N) scalar gauge theory,JHEP10(2018) 003 [1808.02413]

    2018 arXiv

  24. [31]

    de Boer, J

    T. de Boer, J. Kubo, M. Lindner and M. Reinig,Gravity and the hierarchy problem, 2025

    2025

  25. [32]

    Kribs and E.T

    G.D. Kribs and E.T. Neil,Review of strongly-coupled composite dark matter models and lattice simulations,Int. J. Mod. Phys. A31(2016) 1643004 [1604.04627]

    2016 arXiv

  26. [33]

    Nussinov,TECHNOCOSMOLOGY: COULD A TECHNIBARYON EXCESS PROVIDE A ’NATURAL’ MISSING MASS CANDIDATE?,Phys

    S. Nussinov,TECHNOCOSMOLOGY: COULD A TECHNIBARYON EXCESS PROVIDE A ’NATURAL’ MISSING MASS CANDIDATE?,Phys. Lett. B165(1985) 55

    1985

  27. [34]

    Ackerman, M.R

    L. Ackerman, M.R. Buckley, S.M. Carroll and M. Kamionkowski,Dark Matter and Dark Radiation,Phys. Rev. D79(2009) 023519 [0810.5126]

    2009 arXiv

  28. [35]

    Weinberg,Goldstone Bosons as Fractional Cosmic Neutrinos,Phys

    S. Weinberg,Goldstone Bosons as Fractional Cosmic Neutrinos,Phys. Rev. Lett.110(2013) 241301 [1305.1971]

    2013 arXiv

  29. [36]

    Gorbunov and A.G

    D.S. Gorbunov and A.G. Panin,Scalaron the mighty: producing dark matter and baryon asymmetry at reheating,Phys. Lett. B700(2011) 157 [1009.2448]

    2011 arXiv

  30. [37]

    Garny, M

    M. Garny, M. Sandora and M.S. Sloth,Planckian Interacting Massive Particles as Dark Matter, Phys. Rev. Lett.116(2016) 101302 [1511.03278]

    2016 arXiv

  31. [38]

    Tang and Y.-L

    Y. Tang and Y.-L. Wu,Pure Gravitational Dark Matter, Its Mass and Signatures,Phys. Lett. B 758(2016) 402 [1604.04701]

    2016 arXiv

  32. [39]

    Craig,Naturalness: past, present, and future,Eur

    N. Craig,Naturalness: past, present, and future,Eur. Phys. J. C83(2023) 825 [ 2205.05708]

    2023 arXiv

  33. [40]

    Peskin,What is the Hierarchy Problem?,Nucl

    M.E. Peskin,What is the Hierarchy Problem?,Nucl. Phys. B1018(2025) 116971 [2505.00694]

    2025 arXiv

  34. [41]

    Garc´ es, F

    J.P. Garc´ es, F. Goertz, M. Lindner and ´A. Pastor-Guti´ errez,The quantum criticality of the Standard Model and the hierarchy problem,JHEP10(2025) 134 [2506.15919]

    2025

  35. [42]

    Wells,The Intrinsic and Extrinsic Hierarchy Problems,Found

    J.D. Wells,The Intrinsic and Extrinsic Hierarchy Problems,Found. Phys.56(2026) 29 [2506.05472]

    2026 arXiv

  36. [43]

    Callan, Jr.,Broken scale invariance in scalar field theory,Phys

    C.G. Callan, Jr.,Broken scale invariance in scalar field theory,Phys. Rev. D2(1970) 1541

    1970

  37. [44]

    Symanzik,Small distance behavior in field theory and power counting,Commun

    K. Symanzik,Small distance behavior in field theory and power counting,Commun. Math. Phys. 18(1970) 227

    1970

  38. [45]

    Lowenstein and W

    J.H. Lowenstein and W. Zimmermann,Infrared Convergence of Feynman Integrals for the Massless a**4 Model,Commun. Math. Phys.46(1976) 105

    1976

  39. [46]

    Lowenstein and W

    J.H. Lowenstein and W. Zimmermann,The Power Counting Theorem for Feynman Integrals with Massless Propagators,Commun. Math. Phys.44(1975) 73. – 33 –

    1975

  40. [47]

    Callan, Jr., S.R

    C.G. Callan, Jr., S.R. Coleman and R. Jackiw,A New improved energy - momentum tensor, Annals Phys.59(1970) 42

    1970

  41. [48]

    Chanowitz and J.R

    M.S. Chanowitz and J.R. Ellis,Canonical Trace Anomalies,Phys. Rev. D7(1973) 2490

    1973

  42. [49]

    Chanowitz and J.R

    M.S. Chanowitz and J.R. Ellis,Canonical Anomalies and Broken Scale Invariance,Phys. Lett. B40(1972) 397

    1972

  43. [50]

    Tristram et al.,Improved limits on the tensor-to-scalar ratio using BICEP and Planck data, Phys

    M. Tristram et al.,Improved limits on the tensor-to-scalar ratio using BICEP and Planck data, Phys. Rev. D105(2022) 083524 [2112.07961]

    2022 arXiv

  44. [51]

    Rubio,Higgs inflation,Front

    J. Rubio,Higgs inflation,Front. Astron. Space Sci.5(2019) 50 [1807.02376]

    2019 arXiv

  45. [52]

    Adler,Einstein gravity as a symmetry-breaking effect in quantum field theory,Rev

    S.L. Adler,Einstein gravity as a symmetry-breaking effect in quantum field theory,Rev. Mod. Phys.54(1982) 729

    1982

  46. [54]

    Hill and D.S

    C.T. Hill and D.S. Salopek,Calculable nonminimal coupling of composite scalar bosons to gravity,Annals of Physics213(1992) 21

    1992

  47. [55]

    Inagaki, T

    T. Inagaki, T. Muta and S.D. Odintsov,Nambu-Jona-Lasinio model in curved space-time,Mod. Phys. Lett. A8(1993) 2117 [hep-th/9306023]

    1993 arXiv

  48. [56]

    Inagaki, T

    T. Inagaki, T. Muta and S.D. Odintsov,Dynamical symmetry breaking in curved space-time: Four fermion interactions,Prog. Theor. Phys. Suppl.127(1997) 93 [hep-th/9711084]

    1997 arXiv

  49. [57]

    Salvio and A

    A. Salvio and A. Strumia,Agravity,JHEP06(2014) 080 [1403.4226]

    2014 arXiv

  50. [58]

    Ema,Higgs Scalaron Mixed Inflation,Phys

    Y. Ema,Higgs Scalaron Mixed Inflation,Phys. Lett. B770(2017) 403 [1701.07665]

    2017 arXiv

  51. [59]

    Pi, Y.-l

    S. Pi, Y.-l. Zhang, Q.-G. Huang and M. Sasaki,Scalaron fromR 2-gravity as a heavy field, JCAP05(2018) 042 [1712.09896]

    2018 arXiv

  52. [60]

    Salvio,Inflationary Perturbations in No-Scale Theories,Eur

    A. Salvio,Inflationary Perturbations in No-Scale Theories,Eur. Phys. J. C77(2017) 267 [1703.08012]

    2017 arXiv

  53. [61]

    Gundhi and C.F

    A. Gundhi and C.F. Steinwachs,Scalaron-Higgs inflation,Nucl. Phys. B954(2020) 114989 [1810.10546]

    2020 arXiv

  54. [62]

    J. Kubo, M. Lindner, K. Schmitz and M. Yamada,Planck mass and inflation as consequences of dynamically broken scale invariance,Phys. Rev. D100(2019) 015037 [1811.05950]

    2019 arXiv

  55. [63]

    Enckell, K

    V.-M. Enckell, K. Enqvist, S. Rasanen and L.-P. Wahlman,Higgs-R 2 inflation - full slow-roll study at tree-level,JCAP01(2020) 041 [1812.08754]

    2020 arXiv

  56. [64]

    J. Kubo, J. Kuntz, M. Lindner, J. Rezacek, P. Saake and A. Trautner,Unified emergence of energy scales and cosmic inflation,JHEP08(2021) 016 [2012.09706]

    2021 arXiv

  57. [65]

    Cecchini, M

    C. Cecchini, M. De Angelis, W. Giar` e, M. Rinaldi and S. Vagnozzi,Testing scale-invariant inflation against cosmological data,JCAP07(2024) 058 [2403.04316]

    2024 arXiv

  58. [66]

    Salvio,Quasi-Conformal Models and the Early Universe,Eur

    A. Salvio,Quasi-Conformal Models and the Early Universe,Eur. Phys. J. C79(2019) 750 [1907.00983]

    2019 arXiv

  59. [67]

    Salvio,Dimensional Transmutation in Gravity and Cosmology,Int

    A. Salvio,Dimensional Transmutation in Gravity and Cosmology,Int. J. Mod. Phys. A36 (2021) 2130006 [2012.11608]

    2021 arXiv

  60. [68]

    Kannike, G

    K. Kannike, G. H¨ utsi, L. Pizza, A. Racioppi, M. Raidal, A. Salvio et al.,Dynamically Induced Planck Scale and Inflation,JHEP05(2015) 065 [1502.01334]

    2015 arXiv

  61. [69]

    Nambu and G

    Y. Nambu and G. Jona-Lasinio,Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity II,Phys. Rev.124(1961) 246

    1961

  62. [70]

    Nambu and G

    Y. Nambu and G. Jona-Lasinio,Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity I,Phys. Rev.122(1961) 345. – 34 –

    1961

  63. [71]

    Mukhanov and G.V

    V.F. Mukhanov and G.V. Chibisov,Quantum Fluctuations and a Nonsingular Universe,JETP Lett.33(1981) 532

    1981

  64. [72]

    Starobinsky,The Perturbation Spectrum Evolving from a Nonsingular Initially De-Sitter Cosmology and the Microwave Background Anisotropy,Sov

    A.A. Starobinsky,The Perturbation Spectrum Evolving from a Nonsingular Initially De-Sitter Cosmology and the Microwave Background Anisotropy,Sov. Astron. Lett.9(1983) 302. [73]Planckcollaboration,Planck 2018 results. VI. Cosmological parameters,Astron. Astrophys. 641(2020) A6 ...

    1983 arXiv

  65. [74]

    Planckcollaboration,Planck 2018 results. X. Constraints on inflation,Astron. Astrophys.641 (2020) A10 [1807.06211]

    2020 arXiv

  66. [75]

    Deruelle, M

    N. Deruelle, M. Sasaki, Y. Sendouda and A. Youssef,Lorentz-violating vs ghost gravitons: the example of Weyl gravity,JHEP09(2012) 009 [1202.3131]

    2012 arXiv

  67. [76]

    Myung and T

    Y.S. Myung and T. Moon,Primordial massive gravitational waves from Einstein-Chern-Simons-Weyl gravity,JCAP08(2014) 061 [1406.4367]

    2014 arXiv

  68. [77]

    Myung and T

    Y.S. Myung and T. Moon,Scale-invariant tensor spectrum from conformal gravity,Mod. Phys. Lett. A30(2015) 1550172 [1501.01749]

    2015 arXiv

  69. [78]

    Ghilencea,Weyl R 2 inflation with an emergent Planck scale,JHEP10(2019) 209 [1906.11572]

    D.M. Ghilencea,Weyl R 2 inflation with an emergent Planck scale,JHEP10(2019) 209 [1906.11572]

    2019 arXiv

  70. [79]

    Anselmi, E

    D. Anselmi, E. Bianchi and M. Piva,Predictions of quantum gravity in inflationary cosmology: effects of the Weyl-squared term,JHEP07(2020) 211 [2005.10293]

    2020 arXiv

  71. [80]

    Bianchi and M

    E. Bianchi and M. Gamonal,Precision predictions of Starobinsky inflation with self-consistent Weyl-squared corrections,Phys. Rev. D112(2025) 124006 [2506.10081]

    2025

  72. [81]

    Kubo and J

    J. Kubo and J. Kuntz,Primordial gravitational waves in quadratic gravity,JCAP05(2025) 093 [2502.03543]

    2025 arXiv

  73. [82]

    Deruelle, M

    N. Deruelle, M. Sasaki, Y. Sendouda and A. Youssef,Inflation with a Weyl term, or ghosts at work,JCAP03(2011) 040 [1012.5202]

    2011 arXiv

  74. [83]

    Ivanov and A.A

    M.M. Ivanov and A.A. Tokareva,Cosmology with a light ghost,JCAP12(2016) 018 [1610.05330]

    2016 arXiv

  75. [84]

    De Felice, R

    A. De Felice, R. Kawaguchi, K. Mizui and S. Tsujikawa,Starobinsky inflation with a quadratic Weyl tensor,Phys. Rev. D108(2023) 123524 [2309.01835]

    2023 arXiv

  76. [85]

    Liddle and S.M

    A.R. Liddle and S.M. Leach,How long before the end of inflation were observable perturbations produced?,Phys. Rev. D68(2003) 103503 [astro-ph/0305263]

    2003 arXiv

  77. [86]

    Martin and C

    J. Martin and C. Ringeval,First CMB Constraints on the Inflationary Reheating Temperature, Phys. Rev. D82(2010) 023511 [1004.5525]

    2010 arXiv

  78. [87]

    Lozanov and M.A

    K.D. Lozanov and M.A. Amin,Self-resonance after inflation: oscillons, transients and radiation domination,Phys. Rev. D97(2018) 023533 [1710.06851]

    2018 arXiv

  79. [88]

    Turner,Coherent Scalar Field Oscillations in an Expanding Universe,Phys

    M.S. Turner,Coherent Scalar Field Oscillations in an Expanding Universe,Phys. Rev. D28 (1983) 1243

    1983

  80. [89]

    Shtanov, J.H

    Y. Shtanov, J.H. Traschen and R.H. Brandenberger,Universe reheating after inflation,Phys. Rev. D51(1995) 5438 [hep-ph/9407247]

    1995 arXiv

  81. [90]

    Chung, E.W

    D.J. Chung, E.W. Kolb and A. Riotto,Production of massive particles during reheating,Phys. Rev. D60(1999) 063504 [hep-ph/9809453]

    1999 arXiv

  82. [91]

    Kolb and M.S

    E.W. Kolb and M.S. Turner,The Early Universe, vol. 69, Taylor and Francis (5, 2019), 10.1201/9780429492860

    2019 doi

  83. [93]

    Steigman,Neutrinos And Big Bang Nucleosynthesis,Adv

    G. Steigman,Neutrinos And Big Bang Nucleosynthesis,Adv. High Energy Phys.2012(2012) 268321 [1208.0032]

    2012 arXiv

  84. [94]

    Anchordoqui, H

    L.A. Anchordoqui, H. Goldberg and G. Steigman,Right-Handed Neutrinos as the Dark Radiation: Status and Forecasts for the LHC,Phys. Lett. B718(2013) 1162 [1211.0186]

    2013 arXiv

  85. [95]

    Bennett, G

    J.J. Bennett, G. Buldgen, P.F. De Salas, M. Drewes, S. Gariazzo, S. Pastor et al.,Towards a precision calculation ofN eff in the Standard Model II: Neutrino decoupling in the presence of flavour oscillations and finite-temperature QED,JCAP04(2021) 073 [2012.02726]

    2021 arXiv

  86. [96]

    Witten,Current Algebra Theorems for the U(1) Goldstone Boson,Nucl

    E. Witten,Current Algebra Theorems for the U(1) Goldstone Boson,Nucl. Phys. B156(1979) 269

    1979

  87. [97]

    Araki, T

    T. Araki, T. Kobayashi, J. Kubo, S. Ramos-Sanchez, M. Ratz and P.K.S. Vaudrevange, (Non-)Abelian discrete anomalies,Nucl. Phys. B805(2008) 124 [0805.0207]

    2008 arXiv

  88. [98]

    Weinberg,Implications of Dynamical Symmetry Breaking,Phys

    S. Weinberg,Implications of Dynamical Symmetry Breaking,Phys. Rev. D13(1976) 974

    1976

  89. [99]

    Susskind,Dynamics of Spontaneous Symmetry Breaking in the Weinberg-Salam Theory, Phys

    L. Susskind,Dynamics of Spontaneous Symmetry Breaking in the Weinberg-Salam Theory, Phys. Rev. D20(1979) 2619

    1979

  90. [100]

    Lane,Two Lectures on Technicolor,hep-ph/0202255

    K. Lane,Two Lectures on Technicolor,hep-ph/0202255

    arXiv

  91. [101]

    Witten,An SU(2) Anomaly,Phys

    E. Witten,An SU(2) Anomaly,Phys. Lett. B117(1982) 324

    1982

  92. [102]

    K.D. Lane,An Introduction to technicolor, inTheoretical Advanced Study Institute (TASI 93) in Elementary Particle Physics: The Building Blocks of Creation - From Microfermius to Megaparsecs, 6, 1993, DOI [hep-ph/9401324]

    1993 arXiv

  93. [103]

    Dashen,Chiral SU(3) x SU(3) as a symmetry of the strong interactions,Phys

    R.F. Dashen,Chiral SU(3) x SU(3) as a symmetry of the strong interactions,Phys. Rev.183 (1969) 1245

    1969

  94. [104]

    Hill and D.S

    C.T. Hill and D.S. Salopek,Calculable nonminimal coupling of composite scalar bosons to gravity,Annals Phys.213(1992) 21

    1992

  95. [105]

    Dmitrasinovic, R.H

    V. Dmitrasinovic, R.H. Lemmer and R. Tegen,The Mass difference between the charged and neutral pions in the Nambu-Jona-Lasinio model,Phys. Lett. B284(1992) 201. – 36 –

    1992

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