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REVIEW 4 major objections 6 minor 28 references

Grand Unified Theories in Renormalisable, Classically Scale Invariant Gravity

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read An SO(12) gauge theory with adjoint and fundamental scalars, coupled to classically scale-invariant gravity, can be ultraviolet complete and break to SU(6)×U(1) by dimensional transmutation.

desk verdict A serious, explicit SO(12) GUT in scale-invariant quadratic gravity with a concrete UV fixed point and DT points, but the central asymptotic-freedom claim hinges on a contested sign in the gravitational beta functions that the paper itself flags. read the letter →

arxiv 1908.01400 v3 pith:KXFLYYTJ submitted 2019-08-04 hep-th gr-qchep-ph

Martin B Einhorn , D R Timothy Jones This is my paper · ORCID
classification hep-thgr-qchep-ph
keywords asymptoticfreedomdimensionaltransmutationclassicallyscaleinvariantgravityrenormalisablequantumSO(12)grandunifiedtheoryultravioletfixedpointnonminimalcouplinghigher-derivative
topics Quantum Gravity
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 argues that a renormalisable, classically scale-invariant gravity theory coupled to an SO(12) grand unified gauge theory with both adjoint and fundamental scalars can be ultraviolet complete: every dimensionless coupling runs to an ultraviolet fixed point at high energies. The construction uses 52 two-component fermions in the fundamental representation to make the one-loop gauge $\beta$ function as small as possible, $b_g = 1/6$, which the analysis finds is the minimum needed for a fixed point. For a region of parameter space the same model undergoes dimensional transmutation: the adjoint scalar acquires a vacuum expectation value that breaks $SO(12)$ to $SU(6)\otimes U(1)$ and generates an Einstein-Hilbert term for gravity at low energies. The authors exhibit points at which the transmutation minimum is locally stable and lies on renormalisation-group trajectories that flow back to the ultraviolet fixed point. This matters because it is a concrete prototype in which quantum gravity and a grand unified theory are both ultraviolet complete within ordinary renormalisable quantum field theory.

What carries the argument

The load-bearing object is the set of reduced couplings, $a = a/\alpha$, $x_i = \lambda_i/\alpha$, and $x = b/a$, together with the shifted nonminimal couplings $\xi'_i = \xi_i + 1/6$. Dividing every dimension-four coupling by the gauge coupling $\alpha = g^2$ converts the renormalisation-group flow into equations for the ratios, and the vanishing of the reduced $\beta$ functions defines the ultraviolet fixed point. The gravitational corrections to the matter $\beta$ functions, Eqs. (3.5)-(3.6), are claimed to have a universal form depending only on $a$, $x$, and $\xi'_i$; they shift the fixed point only slightly because $a_{\rm FP} = 1/354$ is small. For dimensional transmutation, the key identity is $B_1 = \sum_i \beta_{\lambda_i}\,\partial S_{\rm cl}/\partial\lambda_i = 0$, supplemented by the condition $\varpi_2 > 0$; Eq. (10.23) expresses $B_1^{(\rm os)}$ in terms of the gravitational $\beta$-function coefficients $b_1, b_3$ and the combination $z_1 = \zeta_1/\alpha$, and Appendix A gives the explicit formula for $\varpi_2$. These relations carry the argument because they convert the existence of a stable scale-breaking vacuum into algebraic conditions on the running couplings at the transmutation scale.

What would settle it

Recompute the one-loop gravitational contribution to the $\beta$ function of the $R^2$ coupling $b$ (equivalently, the coefficient $b_3$ in Eq. (3.7b)) in an independent renormalisation scheme; if $\beta_b$ is positive rather than negative at small $b$, the fixed point of Eq. (4.3) does not exist.

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Extended reading notes

Core claim

The central claim is that in the $SO(12)$ case with 312 two-component fermions in the fundamental representation, giving $b_g = 1/6$, the reduced couplings $a = a/\alpha$, $x_i = \lambda_i/\alpha$, $x = b/a$, and $\xi'_i = \xi_i + 1/6$ jointly have an ultraviolet fixed point, numerically $x_1 = 0.263283$, $x_2 = 0.111708$, $x_3 = 0.377518$, $x_4 = 0.104565$, $x_5 = 0.582159$, $\xi'_1 = -1.41379\times 10^{-6}$, $\xi'_2 = -2.00257\times 10^{-6}$, $x = 153.548$, and $a = 1/354$. Because the gauge coupling is asymptotically free and the fixed-point value $a_{\rm FP} = b_g/b_2$ is small, the fixed point is perturbatively accessible. On the dimensional-transmutation side, the paper uses the condition $B_1 = 0$ with $\varpi_2 > 0$ and exhibits points (Table 5) at which $\xi_1 > 0$; there the adjoint vacuum expectation value breaks $SO(12)\to SU(6)\otimes U(1)$, the nonminimal coupling generates a low-energy Einstein term, and the couplings flow from the transmutation scale to the ultraviolet fixed point. The paper also reports that in the large-$N$ limits of both $SO(N)$ and $SU(N)$, with either of the two natural rescalings of the gravitational couplings, the flat-space ultraviolet fixed point is destabilised by gravitational corrections.

Load-bearing premise

The argument rests on the disputed sign of the $R^2$ coupling $b$ in the gravitational action: with the opposite sign, $b$ is not asymptotically free, and the ultraviolet fixed point and the dimensional-transmutation scenario are not established.

Editorial extensions

If this is right

  • The SO(12) model is a perturbatively ultraviolet-complete quantum-gravity-plus-GUT prototype: above the transmutation scale all couplings remain small and flow to the fixed point of Eq. (4.3).
  • At the transmutation scale the low-energy theory automatically contains an Einstein-Hilbert term whose coefficient is set by the adjoint vacuum expectation value, giving a field-theoretic origin for Newton's constant.
  • The minimum gauge group for asymptotic freedom with this scalar content is SO(12) (or SU(9)), so SO(10) and SU(5) cannot be made ultraviolet complete in this framework.
  • In both large-N limits examined for SO(N) and SU(N), gravitational corrections destabilise the flat-space ultraviolet fixed point, so the mechanism is a finite-N phenomenon.
  • Two open corollaries follow from the construction: the electroweak scale is not generated in the model, and the unitarity of the higher-derivative gravity sector remains to be settled.

Reading between the lines

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

  • If the sign of the $R^2$ coefficient $b$ were the opposite one, the ultraviolet fixed point and the dimensional-transmutation scenario would fail; an independent recalculation of $\beta_b$ in another scheme is the most direct test of the programme.
  • The near-cancellation $b_g = 1/6$ forces a very large fermion sector (312 two-component fermions), so a realistic embedding of the Standard Model would have to account for many extra fermions, a phenomenological cost the paper does not address.
  • The fixed-point values of $\xi'_i$ are tiny and negative while $\xi_1 > 0$ is required at the transmutation scale, which ties the generation of Einstein gravity to the running of the nonminimal couplings; one could search for the same pattern in other scale-invariant models as a robustness check.
  • The large-N instability suggests that model-builders in this class should work at finite N rather than relying on large-N approximations, because the ultraviolet-complete cases are isolated.
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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

4 major / 6 minor

Summary. This paper studies SO(N) and SU(N) gauge theories with scalar fields in the adjoint and fundamental representations, coupled to renormalisable, classically scale invariant gravity (RQG). The authors extend their previous SO(10) analysis by adding a fundamental scalar, and they identify SO(12) as the minimal case where a UV fixed point in all couplings can exist, with bg=1/6 achieved by 52 two-component fermions in the fundamental representation. They report a UVFP (Eq. (4.3)) with a small reduced gravitational coupling a=1/354 and a large value x=b/a = 153.548, and they show that similar fixed points exist for SU(N) with N=9. They then study dimensional transmutation (DT) in the SO(N) model, assuming the adjoint scalar alone acquires a vacuum expectation value, breaking SO(12) to SU(6)⊗U(1) and generating an Einstein-Hilbert term. They derive the on-shell condition B1=0 and the stability criterion ϖ2>0 (Secs. 10 and 11), and present two points in Table 5 that satisfy these conditions and are claimed to lie in the catchment basin of the UVFP. The large-N limit is analyzed under two rescalings; in both cases the flat-space UVFP is destabilised by gravitational corrections. The paper concludes that this provides a prototype UV-complete GUT with quantum gravity, while acknowledging in Sec. 12 that the asymptotic-freedom claim is controversial because of disagreement over the sign of the R^2 coupling b.

Significance. If the beta functions and the sign convention adopted in Sec. 3.2 are correct, the paper provides a concrete, explicit example of a classically scale-invariant, renormalisable gravity plus GUT model that is asymptotically free in all couplings and can undergo dimensional transmutation, thereby giving a possible UV completion of Einstein gravity together with grand unification. The authors give explicit fixed-point values, a compact formula for B1, and a very detailed closed-form expression for ϖ2 in Appendix A, with supplementary Mathematica/Maple files. The paper is self-contained in its fixed-point search: within the stated beta functions, the numbers in Sec. 4 and Table 5 are internally consistent tests of the stated conditions. However, the central physical claim rests on an externally referenced and explicitly contested ingredient — the universal gravitational corrections inferred from Ref. [20], especially the sign of b — and the DT claim is supported by only two hand-picked points under the assumption that the fundamental scalar does not condense.

major comments (4)
  1. [Sec. 3.2, Eqs. (3.5)–(3.7), footnote 5, Sec. 12] The asymptotic-freedom result and the existence of the UVFP (4.3) depend critically on the sign and the universal form of the gravitational corrections to matter beta functions, in particular the +5x^2/12 term in b3 of Eq. (3.7b), which drives β_x = a(b2 x − b3) to a fixed point at x ≈ 153.5. These corrections are not derived in the paper but are inferred from Ref. [20], whose journal and arXiv versions differ according to footnote 5. Section 12 itself concedes that the asymptotic-freedom claim is controversial because of disagreement over the sign of b, and the path-integral-convergence argument given there is an argument, not a derivation. This is a load-bearing premise for the UV-complete GUT prototype and for the DT basin in Table 5. I ask the authors to either provide an independent one-loop derivation of b3 and of the matter-correction terms, or to perform an explicit analysis of the opposite sign showing what happens to the fixed point and the DT surface. Without this, the central claim is not established within the manuscript.
  2. [Sec. 10, first paragraph; Eq. (10.2); Sec. 11, Table 5] The DT analysis is performed under the explicit 'crucial assumption' that only the adjoint scalar acquires a vev, with the fundamental scalar χ having zero vev. The full scalar potential (2.5) contains couplings λ4 and λ5 that directly couple χ to Φ, so the χ=0 direction is not automatically stable; the paper does not check that no fundamental-vev direction leads to an extremum with comparable or lower action. If χ condenses, the breaking pattern SO(12)→SU(6)⊗U(1) and the generation of the Einstein-Hilbert term through ξ1>0 are not established. This assumption is not a harmless simplification: it is essential to the claimed low-energy symmetry-breaking pattern, and it is not demonstrated to be the preferred vacuum direction of the effective potential.
  3. [Sec. 11, Table 5; abstract] The abstract claims that dimensional transmutation occurs 'for a region of parameter space,' but the paper demonstrates only two isolated points (Table 5) satisfying B1=0 and ϖ2>0, with several couplings fixed at selected values (e.g., x2=x4=0.25, x3 and x5 held at their UVFP values). No neighborhood of these points is examined, and no scan over the DT surface is reported; the statement that the points are in the 'catchment basin' of the UVFP is not supported by shown RG trajectories. To support the region claim, the authors should either map an open subset of the B1=0 surface on which ϖ2>0 and ξ1>0 simultaneously hold, or soften the abstract and the conclusions to say 'two examples.'
  4. [Appendix A, Eq. (A.2); Sec. 10, Eq. (10.21)] The stability criterion ϖ2 is central to the DT claim, but the closed form in Eq. (A.2) is extremely complicated, and the important identity ϖ2 = ½ β_i^(1) ∂ B1/∂λ_i (Eq. (10.21)) is said to be explained 'elsewhere' in an unpublished paper. The reader cannot easily verify that the two points in Table 5 indeed give ϖ2>0 without re-implementing the supplemented files. I recommend either publishing the derivation of Eq. (10.21) in the appendix or giving a clear reference to a published source; as it stands, an essential step of the argument is deferred to a future publication.
minor comments (6)
  1. [Abstract] The abstract contains a duplicated phrase: 'the quantum field theory can be can be asymptotically free.'
  2. [Sec. 5.1] There is a typo, 'There was an calculational error in Ref. [26],' which should read 'a calculational error.'
  3. [Sec. 3.2, footnote 5] Since the journal and arXiv versions of Ref. [20] differ, the paper should state explicitly which version was used in Eqs. (3.5)–(3.7), and list any consequences for the coefficients.
  4. [Sec. 11, Eq. (11.2) and surrounding text] The notation z4 is defined as x4 + x5/24 for N=12, while earlier in Sec. 10 the general definition is x4 + x5/(2N); the paper should state the N=12 specialization explicitly at first use to avoid confusion.
  5. [Sec. 4, Eq. (4.3)] The fixed-point values are quoted to six significant figures, but no numerical precision or iteration error is mentioned; a brief statement of the numerical method and tolerance would be helpful.
  6. [Sec. 12] The final paragraph lists the electroweak-scale generation and unitarity as unresolved issues; these are important and could be listed as an explicit outlook subsection, but the current phrasing is acceptable for a conclusion.

Circularity Check

0 steps flagged · score 1.0 of 10

No equation-level circularity; the UVFP and DT points are solved numerically from stated beta functions, though the R^2 sign convention is a contested external input.

full rationale

The paper's derivation chain is: choose fermion content to set bg=1/6 (Sec. 4), solve the reduced beta functions (3.11)-(3.15) for the UVFP (4.3), then solve B1=0 and ϖ2>0 for the DT points in Table 5. These are polynomial/numerical root conditions, not identities; e.g., beta_x=0 is solved at x=153.548, and B1=0 fixes x1 for the tabulated points. There is no self-definitional step, no fitted parameter renamed as a prediction, and no uniqueness theorem invoked from same-author work. The gravitational beta functions (3.5)-(3.7) are imported from Ref. [20] with a stated sign convention; Sec. 12 flags this as controversial ('the correct sign of the coefficient b of the R2 term'), and footnote 5 warns that the journal and arXiv versions of Ref. [20] differ. That is an external-dependency/correctness risk, not circularity. The flat-space beta functions from Ref. [26] and the B1/ϖ2 formalism from Refs. [21,24,25] are same-author results, but they are parameter-free with stated assumptions that do not include the SO(12) UVFP/DT claim, so under the review rules they count as independent support rather than load-bearing self-citation. No specific reduction can be exhibited, so the appropriate finding is no significant circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The central results require: (1) the disputed universal gravitational corrections, (2) an explicitly assumed vacuum direction, and (3) matter content chosen to set bg=1/6. There are no numerically fitted observables; the fixed point is solved from beta functions, and the DT points are examples. The main burden is the unverified gravitational sector, not circular fitting.

free parameters (3)
  • bg (gauge beta-function coefficient) = 1/6
    Tuned by choosing 52 two-component fundamental fermions; the UVFP exists only for small bg as explained in Sec. 4 and footnote 3.
  • Fermion content = 52 two-component fermions in the SO(12) fundamental, N_{1/2}=312
    Model-building choice with no independent evidence; it sets b2=59 and a=1/354 at the fixed point (Sec. 4).
  • Table 5 DT sample couplings = Row 1: x1=0.373073, x2=0.25, x3=0.377518, x4=0.25, x5=0.582159, x=148.271, ξ1=0.190359, ξ2'=0.042969; Row 2…
    Hand-picked points used to verify B1=0, ϖ2>0, and flow to the UVFP; no continuous region is mapped.
assumptions (5)
  • domain assumption Renormalizable quantum gravity with action (2.2) is a valid framework, and the path integral converges for the chosen sign of b.
    Underlies all beta functions; the sign of b is disputed (Sec. 12).
  • ad hoc to paper Gravitational corrections to matter beta functions have the universal form of Eqs. (3.5) to (3.6) as inferred from Ref. [20].
    Inferred rather than derived; footnote 5 warns that the arXiv and journal versions of Ref. [20] differ.
  • ad hoc to paper Dimensional transmutation proceeds via an adjoint vev only, with the fundamental scalar χ having zero vev.
    Called 'the crucial assumption' in Sec. 10; if χ condenses, the quoted breaking pattern is not established.
  • domain assumption The background during dimensional transmutation is de Sitter with constant scalar field, Eq. (10.1).
    Used to compute the off-shell action and stability conditions in Sec. 10.
  • domain assumption One-loop and two-loop perturbative beta functions remain reliable at the fixed point despite x being around 150.
    Authors argue b=ax is O(1), but the large value of x means higher-order corrections are not shown to be small.
invented entities (1)
  • 52 Weyl fermions in the SO(12) fundamental (312 two-component degrees of freedom)
    purpose: To make bg=1/6 so the gauge and scalar couplings can be simultaneously asymptotically free
    The fermions have no specified couplings to known particles or any predicted spectra; they are scaffolding to tune the beta function (Sec. 4).

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Pith. "Pith review of Grand Unified Theories in Renormalisable, Classically Scale Invariant Gravity." pith.science (2026). https://pith.science/paper/KXFLYYTJ

@misc{pith2026190801400,
  author       = {Pith},
  title        = {Pith review of: Grand Unified Theories in Renormalisable, Classically Scale Invariant Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KXFLYYTJ}},
  note         = {Machine review of arXiv:1908.01400}
}
abstract

We analyze $SO(N)$ and $SU(N)$ gauge theories with scalars in adjoint and fundamental representations, coupled to renormalisable, classically scale invariant gravity. In the specific case of $SO(12),$ we show that the quantum field theory can be can be asymptotically free in all couplings (hence ultra-violet complete). For a region of parameter space, Dimensional Transmutation occurs, with the adjoint vacuum expectation value breaking $SO(12) \to SU(6)\otimes U(1)$ and producing a Low Energy Effective Theory having Einstein-Hilbert gravity. We verify that certain minima are locally stable and lie within the catchment basin of the ultraviolet fixed points.

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Works this paper leans on

28 extracted references · 17 canonical work pages

  1. [20]

    Agravity,

    A. Salvio and A. Strumia, “Agravity,” JHEP 1406 (2014) 080, [arXiv:1403.4226 [hep-ph]]

    arXiv 2014

  2. [1]

    Renormalization of Higher Derivative Quantum Gravity,

    K. S. Stelle, “Renormalization of Higher Derivative Quantum Gravity,” Phys. Rev. D 16 (1977) 953

    work page 1977

  3. [2]

    Renormalizable Asymptotically Free Quantum Theory of Gravity,

    E. S. Fradkin and A. A. Tseytlin, “Renormalizable Asymptotically Free Quantum Theory of Gravity,” Phys. Lett. B 104 (1981) 377

    work page 1981

  4. [3]

    Renormalizable asymptotically free quantum theory of gravity,

    E. S. Fradkin and A. A. Tseytlin, “Renormalizable asymptotically free quantum theory of gravity,” Nucl. Phys. B 201 (1982) 469

    work page 1982

  5. [4]

    Asymptotic Freedom In Higher Derivative Quantum Gravity,

    I. G. Avramidi and A. O. Barvinsky, “Asymptotic Freedom In Higher Derivative Quantum Gravity,” Phys. Lett. B 159 (1985) 269

    work page 1985

  6. [5]

    Effective action in quantum gravity,

    I. L. Buchbinder, S. D. Odintsov and I. L. Shapiro, “Effective action in quantum gravity,” Bristol, UK: IOP (1992)

    1992

  7. [6]

    Heat kernel and quantum gravity,

    I. G. Avramidi, “Heat kernel and quantum gravity,” Lect. Notes Phys. M 64 (2000) 1

    work page 2000

  8. [7]

    Higgs Phenomena in Asymptotically Free Gauge Theories,

    T. P. Cheng, E. Eichten and L. F. Li, “Higgs Phenomena in Asymptotically Free Gauge Theories,” Phys. Rev. D 9 (1974) 2259

    work page 1974

Show all 28 references
  1. [8]

    The Stability of Asymptotic Freedom in Grand Unified Models Coupled to R2 Gravity,

    I. L. Buchbinder, et al., “The Stability of Asymptotic Freedom in Grand Unified Models Coupled to R2 Gravity,” Phys. Lett. B 216 (1989) 127

    1989

  2. [9]

    Asymptotically free grand unification models with quantum R**2 gravitation,

    I. L. Buchbinder, et al., “Asymptotically free grand unification models with quantum R**2 gravitation,” Sov. J. Nucl. Phys. 49, 544 (1989) [Yad. Fiz. 49, 876 (1989)]

    1989

  3. [10]

    Ultraviolet divergences in quantum theories of gravitation

    S. Weinberg, “Ultraviolet divergences in quantum theories of gravitation”, Chpt. 16 in S. W. Hawking, W. Israel (eds.). General Relativity: An Einstein centenary survey. Cambridge University Press (1979), pp. 919

    1979

  4. [11]

    Gravitational Fixed Points from Perturbation Theory,

    M. R. Niedermaier, “Gravitational Fixed Points from Perturbation Theory,” Phys. Rev. Lett. 103 (2009) 101303

    2009

  5. [12]

    Gravitational fixed points and asymptotic safety from perturbation theory,

    M. Niedermaier, “Gravitational fixed points and asymptotic safety from perturbation theory,” Nucl. Phys. B 833 (2010) 226

    2010

  6. [13]

    Status of the asymptotic safety paradigm for quantum gravity and matter,

    A. Eichhorn, “Status of the asymptotic safety paradigm for quantum gravity and matter,” Found. Phys. 48 (2018) no.10, 1407, [arXiv:1709.03696 [gr-qc]]

    2018 arXiv

  7. [14]

    Asymptotic safety of gravity with matter,

    N. Christiansen, D. F. Litim, J. M. Pawlowski and M. Reichert, “Asymptotic safety of gravity with matter,” Phys. Rev. D 97 (2018) no.10, 106012, [arXiv:1710.04669 [hep-th]]

    2018 arXiv

  8. [15]

    Aspects of asymptotic safety for quantum gravity,

    K. G. Falls, D. F. Litim and J. Schrder, “Aspects of asymptotic safety for quantum gravity,” Phys. Rev. D 99 (2019) no.12, 126015, [arXiv:1810.08550 [gr-qc]]

    2019 arXiv

  9. [16]

    An Introduction to Covariant Quantum Gravity and Asymptotic Safety,

    R. Percacci, “An Introduction to Covariant Quantum Gravity and Asymptotic Safety,” . doi:10.1142/10369, World Scientific, 2017

    2017 doi

  10. [17]

    Asymptotic Freedom In Extended Conformal Supergravities,

    E. S. Fradkin and A. A. Tseytlin, “Asymptotic Freedom In Extended Conformal Supergravities,” Phys. Lett. 110B (1982) 117

    1982

  11. [18]

    Conformal Anomaly in Weyl Theory and Anomaly Free Superconformal Theories,

    E. S. Fradkin and A. A. Tseytlin, “Conformal Anomaly in Weyl Theory and Anomaly Free Superconformal Theories,” Phys. Lett. 134B (1984) 187

    1984

  12. [19]

    Conformal Supergravity,

    E. S. Fradkin and A. A. Tseytlin, “Conformal Supergravity,” Phys. Rept. 119 (1985) 233. – 25 –

    1985

  13. [21]

    Naturalness and Dimensional Transmutation in Classically Scale-Invariant Gravity,

    M. B. Einhorn and D. R. T. Jones, “Naturalness and Dimensional Transmutation in Classically Scale-Invariant Gravity,” JHEP 1503 (2015) 047 [arXiv:1410.8513 [hep-th]]

    2015 arXiv

  14. [22]

    Radiative Corrections as the Origin of Spontaneous Symmetry Breaking,

    S. R. Coleman and E. J. Weinberg, “Radiative Corrections as the Origin of Spontaneous Symmetry Breaking,” Phys. Rev. D 7 (1973) 1888

    1973

  15. [23]

    Softened Gravity and the Extension of the Standard Model up to Infinite Energy,

    G. F. Giudice, G. Isidori, A. Salvio and A. Strumia, “Softened Gravity and the Extension of the Standard Model up to Infinite Energy,” JHEP 1502 (2015) 137 [arXiv:1412.2769 [hep-ph]]

    2015 arXiv

  16. [24]

    Induced Gravity I: Real Scalar Field,

    M. B. Einhorn and D. R. T. Jones, “Induced Gravity I: Real Scalar Field,” JHEP 1601 (2016) 019 [arXiv:1511.01481 [hep-th]]

    2016 arXiv

  17. [25]

    Induced Gravity II: Grand Unification,

    M. B. Einhorn and D. R. T. Jones, “Induced Gravity II: Grand Unification,” JHEP 1605 (2016) 185, [arXiv:1602.06290 [hep-th]]

    2016 arXiv

  18. [26]

    Asymptotic freedom in certain SO(N) and SU (N) models,

    M. B. Einhorn and D. R. T. Jones, “Asymptotic freedom in certain SO(N) and SU (N) models,” Phys. Rev. D 96 (2017) no.5, 055035, [arXiv:1705.00751 [hep-ph]]

    2017 arXiv

  19. [27]

    Renormalizable, asymptotically free gravity without ghosts or tachyons,

    M. B. Einhorn and D. R. T. Jones, “Renormalizable, asymptotically free gravity without ghosts or tachyons,” Phys. Rev. D 96 (2017) no.12, 124025 [arXiv:1710.03795 [hep-th]]

    2017 arXiv

  20. [28]

    A First-Order Formalism for Renormalizable Quantum Gravity,

    M. B. Einhorn and D. R. T. Jones, “A First-Order Formalism for Renormalizable Quantum Gravity,” (in preparation) – 26 –

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