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arxiv: 2512.16613 · v2 · submitted 2025-12-18 · ✦ hep-th

Effective potential in SO(N) symmetric scalar field theories in curved spacetime

V.A. Filippov , R.M. Iakhibbaev , D. M. Tolkachev This is my paper

Pith reviewed 2026-05-16 21:27 UTC · model grok-4.3

classification ✦ hep-th
keywords effective potentialSO(N) symmetrycurved spacetimerenormalization grouplarge N limitleading logarithmic correctionsJordan frameinflationary cosmology
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The pith

Recurrence relations organize leading-logarithmic corrections to the effective potential in SO(N) scalar theories on curved backgrounds.

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

The paper derives recurrence relations that generate the leading logarithmic quantum corrections at every loop order for an SO(N)-symmetric scalar field theory with an arbitrary potential in curved spacetime. These relations are then used to obtain a closed system of renormalization-group equations that govern the effective potential in the large-N limit. As an illustration the relations are applied to power-like potentials written in the Jordan frame, where the running can be connected to the dynamics of inflationary models.

Core claim

Recurrence relations for the leading logarithmic all-loop quantum corrections are derived for an SO(N) symmetric scalar theory with an arbitrary potential in curved spacetime. On this basis a system of renormalization group equations for the effective potential is obtained in the large N limit.

What carries the argument

Recurrence relations for leading-logarithmic all-loop corrections that close into a system of RG equations for the effective potential at large N.

If this is right

  • The effective potential receives a complete leading-log resummation for any starting potential.
  • The large-N RG equations close and determine the scale dependence of the potential in curved backgrounds.
  • Power-law potentials in the Jordan frame admit explicit analysis with direct relevance to slow-roll inflation.

Where Pith is reading between the lines

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

  • The RG system may track the potential across the end of inflation into reheating without additional assumptions.
  • Comparison of predicted spectral indices with cosmological data could test the large-N resummation.
  • Adding next-to-leading logarithms would quantify the truncation error of the leading-log scheme.

Load-bearing premise

The leading logarithmic approximation remains valid through all loop orders and the large-N limit accurately captures the dynamics without higher-order curvature terms becoming dominant.

What would settle it

A direct two-loop or three-loop perturbative calculation of the effective potential for a specific potential and nonzero curvature that deviates from the recurrence prediction would falsify the relations.

Figures

Figures reproduced from arXiv: 2512.16613 by D. M. Tolkachev, R.M. Iakhibbaev, V.A. Filippov.

Figure 1
Figure 1. Figure 1: One- and two-loop diagrams giving contributions to [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Modified Feynman rules Let us again consider the theory with a quartic potential (ϕaϕa) 2/4!. The one-loop correction W1 takes the form: W1 = 1 4 ϕ 2 ˆξR log  µ 2 m2 1  + Nˆ 12 ϕ 2 ˆξR log  µ 2 m2 2  , (21) which corresponds to [7]. The two-loop contribution W2 is given by: W2 = 1 4 ϕ 2 ˆξR log2  µ 2 m2 1  + Nˆ 8 ϕ 2 ˆξR log  µ 2 m2 1  log  µ 2 m2 2  + Nˆ 2 72 ϕ 2 ˆξR log2  µ 2 m2 2  . (22) To … view at source ↗
Figure 3
Figure 3. Figure 3: One- and two-loop diagrams giving contributions to [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: R′ operation for leading divergences of an n-loop diagram Following the approach of [8,9], recurrence relations can be obtained for the coefficients at the leading poles of an arbitrary diagram: n∆Vn = 1 4 Xn−1 k=0  ∂ 2∆Vk ∂ϕ2 · ∂ 2∆Vn−k−1 ∂ϕ2 + 4Nˆ ∂∆Vk ∂(ϕ2 ) · ∂Vn−k−1 ∆∂(ϕ2 )  , (26) 7 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Effective potential of the model with p = 4 for ξ = 0, various numbers of fields N and curvature values R = {RC1, RC2}. Critical curvature values RC1 ≃ 50·µ 2 , RC2 ≃ 70·µ 2 are selected for the effective potential with N = 100. Of particular interest is the transition case with the appearance of an additional min￾imum in the effective potential, in which a flat plateau is formed locally in a natural way. … view at source ↗
Figure 6
Figure 6. Figure 6: Dependence of the effective potential on various [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Solutions f1(x) and f2(x) for various numbers of fields N It is also possible to obtain the effective potential using expressions (31) and (33). The result is shown in [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Effective potential of the model with p = 6 for ξ = 0, various numbers of fields N and curvature values R equal to R ∼ −10µ 2 and R ∼ 10µ 2 , respectively. Parameters for illustrative purposes are chosen as λ ∼ 1, µ ∼ 1 [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Effective potential of the model with p = 6 for ξ = 0, N = 200, λ ∼ 1 with various values of µ and fixed R ∼ 60 (left), and also various values of curvature R and fixed µ ∼ 1 (right). following form: SJ = Z d 4x √ −g  − 1 2 F(ϕ)R + 1 2 gµν∂ µϕa∂ νϕa − V(ϕ)  , (43) where g = det(gµν) and the function F(ϕ) is defined by: F(ϕ) = 1 + 2 R W(ϕ). (44) Under a conformal transformation of the metric tensor g˜µν =… view at source ↗
Figure 10
Figure 10. Figure 10: The (ns, r)-diagram for effective potentials with p = 4, p = 6 and observed data. 6 Conclusion We have derived the RG equation in a general approach that enables one to calculate quantum corrections to an arbitrary classical potential in SO(N) symmetric scalar theo￾ries within the leading logarithmic approximation, taking into account the non-minimal coupling to gravity. We analysed cases of power-like po… view at source ↗
read the original abstract

We derive recurrence relations for leading logarithmic all-loop quantum corrections in the case of $SO(N)$ symmetric scalar theory with an arbitrary potential in curved spacetime. On this basis, a system of renormalisation group (RG) equations in the general is obtained approach for the effective potential in the large $N$ limit. As a simple illustration, we analyse the case of power-like potentials in the Jordan frame and discuss their application to inflationary cosmology.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The paper derives recurrence relations for the leading-logarithmic all-loop quantum corrections to the effective potential of an SO(N)-symmetric scalar field theory with arbitrary potential in curved spacetime. From these relations it constructs the corresponding renormalization-group system in the large-N limit. As an illustration it examines power-law potentials in the Jordan frame and comments on possible applications to inflationary cosmology.

Significance. If the central derivation holds, the work supplies a systematic route to resum leading-log corrections in curved space for large-N models. This is potentially useful for cosmological model-building, where the effective potential controls slow-roll parameters. The extension of heat-kernel methods to the Jordan frame while retaining curvature terms at the required order, followed by the large-N limit, is a clear technical strength that could enable reproducible calculations for specific potentials.

major comments (1)
  1. The recurrence relations are stated to follow from the heat-kernel expansion, but the manuscript does not provide an explicit check that they reduce to the known flat-space leading-log result for the SO(N) model when the curvature is set to zero. This verification is load-bearing for the claim that the curved-space generalization is under control.
minor comments (3)
  1. Abstract: the phrase 'in the general is obtained approach' is ungrammatical and should be rephrased.
  2. The large-N limit is taken after the recurrence is established; this ordering should be stated explicitly in the introduction so that the suppression of higher-curvature terms is clear to the reader.
  3. For the power-law examples, the manuscript should report at least one numerical comparison with a known one-loop result to illustrate the improvement gained by the all-loop resummation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for the recommendation of minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: The recurrence relations are stated to follow from the heat-kernel expansion, but the manuscript does not provide an explicit check that they reduce to the known flat-space leading-log result for the SO(N) model when the curvature is set to zero. This verification is load-bearing for the claim that the curved-space generalization is under control.

    Authors: We agree that an explicit reduction check is necessary to confirm consistency. In the revised manuscript we will add a dedicated paragraph (in Section 3) demonstrating that, upon setting all curvature scalars and their derivatives to zero, the recurrence relations for the leading-logarithmic corrections recover the known flat-space results for the SO(N) model, matching the coefficients obtained from standard flat-space methods in the literature. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The manuscript derives recurrence relations for leading-logarithmic all-loop corrections via heat-kernel methods in curved spacetime, then obtains the large-N RG system directly from those relations. No step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the recurrence is established first and the RG equations follow as a consequence. The large-N limit is taken after the recurrence is in hand, preserving parametric suppression of higher-curvature terms. The derivation is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on the validity of the leading-log resummation and the large-N limit in curved spacetime; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Leading logarithmic approximation captures the dominant all-loop quantum corrections
    Invoked to obtain the recurrence relations for the effective potential
  • domain assumption Large N limit simplifies the RG flow of the effective potential
    Used to close the system of RG equations

pith-pipeline@v0.9.0 · 5372 in / 1216 out tokens · 39990 ms · 2026-05-16T21:27:48.259983+00:00 · methodology

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Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    Coleman and Erick J

    Sidney R. Coleman and Erick J. Weinberg. Radiative Corrections as the Origin of Spontaneous Symmetry Breaking.Phys. Rev. D, 7:1888–1910, 1973

    work page 1910

  2. [2]

    R. Jackiw. Functional evaluation of the effective potential.Phys. Rev. D, 9:1686, 1974

    work page 1974

  3. [3]

    Kastening

    Boris M. Kastening. Renormalization group improvement of the effective potential in massive phi**4 theory.Phys. Lett. B, 283:287–292, 1992

    work page 1992

  4. [4]

    Ford and D

    C. Ford and D. R. T. Jones. The Effective potential and the differential equa- tions method for Feynman integrals.Phys. Lett. B, 274:409–414, 1992. [Erratum: Phys.Lett.B 285, 399 (1992)]

    work page 1992

  5. [5]

    J. M. Chung and B. K. Chung. Three-loop effective potential of o(n)ϕ 4 theory, 1998

    work page 1998

  6. [6]

    I. L. Buchbinder and S. D. Odintsov. Asymptotical behavior of the effective potential of the composite fields in curved space-time.EPL, 4:147–152, 1987

    work page 1987

  7. [7]

    I. L. Buchbinder, S. D. Odintsov, and I. L. Shapiro.Effective Action in Quantum Gravity. Routledge, 10 2017

    work page 2017

  8. [8]

    V. A. Filippov, R. M. Iakhibbaev, and D. M. Tolkachev. All-loop effective potential for arbitrary scalar models in curved space-time.Classical and Quantum Gravity, 42(14):145006, July 2025

    work page 2025

  9. [9]

    D. I. Kazakov, R. M. Iakhibbaev, and D. M. Tolkachev. Leading all-loop quantum contribution to the effective potential in general scalar field theory.JHEP, 04:128, 2023

    work page 2023

  10. [10]

    D. I. Kazakov, R. M. Iakhibbaev, and D. M. Tolkachev. Leading all-loop quantum contribution to the effective potential in the inflationary cosmology.JCAP, 09:049, 2023

    work page 2023

  11. [11]

    N. N. Bogoliubow and O. S. Parasiuk. ¨Uber die multiplikation der kausalfunktionen in der quantentheorie der felder.Acta Mathematica, 97:227–266, 1957

    work page 1957

  12. [12]

    K. Hepp. Proof of the Bogolyubov-Parasiuk theorem on renormalization.Commun. Math. Phys., 2:301–326, 1966

    work page 1966

  13. [13]

    Zimmermann

    W. Zimmermann. Convergence of Bogolyubov’s method of renormalization in mo- mentum space.Commun. Math. Phys., 15:208–234, 1969. 16

    work page 1969

  14. [14]

    Akrami et al

    Y. Akrami et al. Planck 2018 results. X. Constraints on inflation.Astron. Astrophys., 641:A10, 2020

    work page 2018

  15. [15]

    Gravitational effect on effective potential.Phys

    Kiyoshi Ishikawa. Gravitational effect on effective potential.Phys. Rev. D, 28:2445, 1983

    work page 1983

  16. [16]

    T. S. Bunch and L. Parker. Feynman Propagator in Curved Space-Time: A Momen- tum Space Representation.Phys. Rev. D, 20:2499–2510, 1979

    work page 1979

  17. [17]

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

    work page 1982

  18. [18]

    Petrov.Einstein Space

    A.Z. Petrov.Einstein Space. Pergamon, Oxford, 1969

    work page 1969

  19. [19]

    Ribeiro, and Ilya L

    Flavia Sobreira, Baltazar J. Ribeiro, and Ilya L. Shapiro. Effective Potential in Curved Space and Cut-Off Regularizations.Phys. Lett. B, 705:273–278, 2011

    work page 2011

  20. [20]

    R. M. Iakhibbaev and D. M. Tolkachev. Effective potential in leading logarithmic approximation in non-renormalisableSO(N) scalar field theories.Int. Mod. Phys. J. A, 2025

    work page 2025

  21. [21]

    N. N. Bogoliubov and D.V. Shirkov.Introduction To The Theory Of Quantized Fields. Nauka, Moscow, 1957. English transl.:Introduction to the Theory of Quantized Fields, 3rd ed., New York, Wiley, 1980

    work page 1957

  22. [22]

    Ivanov, P

    P. Ivanov, P. Naselsky, and I. Novikov. Inflation and primordial black holes as dark matter.Phys. Rev. D, 50:7173–7178, Dec 1994

    work page 1994

  23. [23]

    Primordial black hole production in critical higgs inflation.Physics Letters B, 776:345–349, January 2018

    Jose Mar´ ıa Ezquiaga, Juan Garc´ ıa-Bellido, and Ester Ruiz Morales. Primordial black hole production in critical higgs inflation.Physics Letters B, 776:345–349, January 2018

    work page 2018

  24. [24]

    Primordial black hole dark matter from single field inflation.Physical Review D, 97(2), January 2018

    Guillermo Ballesteros and Marco Taoso. Primordial black hole dark matter from single field inflation.Physical Review D, 97(2), January 2018

    work page 2018

  25. [25]

    Daniel Frolovsky and Sergei V. Ketov. Fitting power spectrum of scalar perturbations for primordial black hole production during inflation.Astronomy, 2(1):47–57, 2023

    work page 2023

  26. [26]

    Encyclopædia Inflation- aris: Opiparous Edition.Phys

    Jerome Martin, Christophe Ringeval, and Vincent Vennin. Encyclopædia Inflation- aris: Opiparous Edition.Phys. Dark Univ., 5-6:75–235, 2014

    work page 2014

  27. [27]

    f(r) theories.Living Reviews in Relativity, 13(1), June 2010

    Antonio De Felice and Shinji Tsujikawa. f(r) theories.Living Reviews in Relativity, 13(1), June 2010

    work page 2010

  28. [28]

    Starobinsky

    A.A. Starobinsky. A new type of isotropic cosmological models without singularity. Physics Letters B, 91(1):99–102, 1980

    work page 1980

  29. [29]

    Slow-roll inflation in the Jordan frame.Phys

    Mindaugas Karˇ ciauskas and Jos´ e Jaime Terente D´ ıaz. Slow-roll inflation in the Jordan frame.Phys. Rev. D, 106(8):083526, 2022. 17

    work page 2022

  30. [30]

    Jin-Ook Gong and Ewan D. Stewart. The power spectrum for a multi-component inflaton to second-order corrections in the slow-roll expansion.Physics Letters B, 538(3–4):213–222, July 2002. 18

    work page 2002