Scalar field effective potentials in de Sitter spacetime

Arttu Rajantie; Lucas Vicente Garc\'ia-Consuegra

arxiv: 2511.23076 · v2 · pith:APWOVWYCnew · submitted 2025-11-28 · ✦ hep-th · astro-ph.CO· gr-qc· hep-ph

Scalar field effective potentials in de Sitter spacetime

Lucas Vicente Garc\'ia-Consuegra , Arttu Rajantie This is my paper

Pith reviewed 2026-05-21 18:16 UTC · model grok-4.3

classification ✦ hep-th astro-ph.COgr-qchep-ph

keywords de Sitter spacetimeeffective potentialinfrared divergencestochastic inflationscalar fieldperturbation theoryStarobinsky-Yokoyama

0 comments

The pith

The constraint effective potential avoids the infrared divergence that blocks perturbative calculations of the standard effective potential for light scalar fields in de Sitter spacetime.

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

The paper compares two definitions of the effective potential for a scalar field in de Sitter spacetime: the standard textbook version and the constraint effective potential introduced in 1986. These definitions coincide in flat Minkowski space but produce distinct results in de Sitter geometry. Explicit one-loop computations show that the standard potential develops an infrared problem for light fields and cannot be evaluated perturbatively, whereas the constraint version remains finite and computable. The authors provide evidence that the constraint effective potential is the appropriate choice for matching to the stochastic Starobinsky-Yokoyama approach to inflation.

Core claim

We compute both the standard and constraint effective potentials at one-loop order and demonstrate that the constraint effective potential does not suffer from the infrared divergences that prevent perturbative evaluation of the standard effective potential when the scalar field is light. This difference arises because the two potentials are inequivalent in de Sitter spacetime. Comparison with the stochastic formalism indicates that the constraint effective potential is the correct one to employ in the Starobinsky-Yokoyama theory.

What carries the argument

The constraint effective potential, which incorporates a fixed field value constraint and differs from the standard effective potential in de Sitter due to the spacetime geometry, allowing perturbative computation without infrared divergences.

Load-bearing premise

The one-loop perturbative results for the constraint effective potential remain reliable and physically meaningful when extrapolated to the regime of very light fields where the standard potential diverges.

What would settle it

A non-perturbative calculation or lattice simulation of the scalar field effective potential in de Sitter spacetime that yields results inconsistent with the extrapolated one-loop constraint potential would falsify the claim that it is the correct choice for the stochastic theory.

read the original abstract

We investigate two different definitions of a scalar field effective potential in quantum field theory in de Sitter spacetime: the standard textbook definition, and the constraint effective potential proposed by O'Raifeartaigh et al. in 1986. While these definitions are equivalent in Minkowski spacetime, they differ significantly in de Sitter. We demonstrate this by computing them both explicitly at one-loop order in perturbation theory. It is well known that the perturbative expansion of the standard effective potential fails converge for light fields. In contrast, the constraint effective potential does not suffer from this infrared problem, and it can therefore be computed using perturbation theory. We discuss the physical interpretation of the two effective potentials. In particular, we provide evidence supporting an earlier conjecture that the constraint effective potential is the correct one to use in the stochastic Starobinsky-Yokoyama theory.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

This paper shows at one loop that the constraint effective potential stays finite for light scalars in de Sitter while the standard one diverges, and offers some evidence for its use in stochastic inflation.

read the letter

The main takeaway is that this paper computes the standard effective potential and the constraint effective potential at one-loop order in de Sitter and shows the constraint version remains finite for light scalars while the standard one suffers from infrared divergences. It also gives evidence supporting the conjecture that the constraint potential should be used in the stochastic Starobinsky-Yokoyama theory. The calculation is the strongest part. By doing the explicit one-loop integrals for both definitions, the authors make the difference concrete and confirm that the constraint effective potential sidesteps the infrared problem that has long plagued perturbative calculations for light fields in de Sitter. This is useful technical work that directly addresses a known limitation in the field. The softer aspect is the physical interpretation and the link to stochastic inflation. The support for the conjecture comes from matching the one-loop results to the stochastic formalism, but the paper does not provide two-loop corrections or non-perturbative validations to check whether the one-loop picture holds when the scalar is very light. The extrapolation to m/H approaching zero therefore rests on the assumption that higher orders stay under control, which is plausible but not verified here. This paper is for researchers working on quantum field theory in curved spacetime or on stochastic approaches to inflation. Someone already aware of the infrared issues in de Sitter effective potentials will appreciate the direct comparison and the discussion of which definition makes sense physically. It deserves a serious referee because the core perturbative results are solid and the question it raises has implications for cosmological modeling, even if the conclusions on the stochastic side could use more backing. I recommend sending it to peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript computes the standard effective potential and the constraint effective potential for a scalar field in de Sitter spacetime at one-loop order in perturbation theory. It shows that the standard potential develops infrared divergences for light fields while the constraint effective potential remains finite, and argues that the latter is the appropriate input for the stochastic Starobinsky-Yokoyama formalism, thereby supporting an earlier conjecture.

Significance. If the central results hold, the work addresses a persistent technical obstacle in applying effective potentials to de Sitter backgrounds, with implications for stochastic inflation and the computation of equilibrium distributions. The explicit one-loop calculations constitute a concrete strength, as they furnish verifiable expressions and directly identify the infrared issue in the standard case without relying on fits or self-referential definitions.

major comments (1)

[Physical interpretation and comparison with stochastic theory] Physical interpretation and comparison with stochastic theory: the claim that the one-loop constraint effective potential remains reliable and is the correct choice for the stochastic Starobinsky-Yokoyama equation when m/H → 0 rests on the unverified assumption that higher-order corrections stay small. No two-loop computation, non-perturbative check, or direct comparison of the resulting Fokker-Planck equilibrium against known lattice or exact results is reported to confirm this extrapolation to the deep infrared.

minor comments (2)

[Abstract] The abstract states that evidence is provided for an earlier conjecture; citing the specific reference or conjecture number already in the abstract would improve immediate clarity for readers.
[Notation and equations] Notation for the Hubble scale and scalar mass should be checked for consistency between the one-loop integrals and the stochastic-equation discussion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for recognizing the value of our explicit one-loop calculations in identifying the infrared issue. We address the major comment below.

read point-by-point responses

Referee: Physical interpretation and comparison with stochastic theory: the claim that the one-loop constraint effective potential remains reliable and is the correct choice for the stochastic Starobinsky-Yokoyama equation when m/H → 0 rests on the unverified assumption that higher-order corrections stay small. No two-loop computation, non-perturbative check, or direct comparison of the resulting Fokker-Planck equilibrium against known lattice or exact results is reported to confirm this extrapolation to the deep infrared.

Authors: We agree that our analysis is performed at one-loop order and does not include two-loop computations, non-perturbative checks, or direct comparisons with lattice or exact results for the Fokker-Planck equilibrium. The manuscript demonstrates that the constraint effective potential remains finite as m/H approaches zero at this perturbative order, in contrast to the infrared-divergent standard effective potential. This explicit difference furnishes evidence supporting the conjecture that the constraint effective potential is the appropriate choice for the stochastic Starobinsky-Yokoyama formalism, as it avoids the infrared problems that appear already at one loop in the standard definition. We do not claim that higher-order corrections necessarily remain small in the deep infrared; rather, the one-loop result shows that the constraint definition yields a well-defined potential where the standard one fails. We will revise the manuscript to clarify the perturbative scope of our evidence and to note that non-perturbative confirmation remains an important open question. revision: partial

Circularity Check

0 steps flagged

One-loop perturbative computations are independent and do not reduce to inputs or self-citations

full rationale

The paper's core derivation consists of explicit one-loop calculations of both the standard and constraint effective potentials in de Sitter spacetime, demonstrating finiteness of the latter without IR logarithms. These are direct perturbative results from the action and propagators, not fits, self-definitions, or renamings. The discussion of the stochastic Starobinsky-Yokoyama conjecture is presented as an interpretation supported by the new calculations rather than an assumption or prior self-citation that the results depend on. No load-bearing step reduces by construction to the paper's own inputs or unverified self-citations; the work is self-contained against external benchmarks like standard QFT perturbation theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard assumptions of quantum field theory in curved spacetime (de Sitter metric, perturbative expansion around a constant background) and on the validity of the 1986 constraint effective potential definition. No new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Standard perturbative QFT techniques apply to scalar fields in de Sitter spacetime at one-loop order
Invoked when stating that the constraint potential can be computed perturbatively while the standard one cannot.
domain assumption The stochastic Starobinsky-Yokoyama formalism requires a specific effective potential that can be matched to the constraint version
Central to the physical interpretation section and the conjecture support.

pith-pipeline@v0.9.0 · 5678 in / 1471 out tokens · 34710 ms · 2026-05-21T18:16:05.307825+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We demonstrate this by computing them both explicitly at one-loop order in perturbation theory... the constraint effective potential does not suffer from this infrared problem
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat embedding and J-cost positivity unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

p(¯ϕ)∝exp(−8π²VC(¯ϕ)/3H⁴)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages · 17 internal anchors

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S. R. Coleman and E. J. Weinberg, Phys. Rev. D7, 1888 (1973)

work page 1973

[2] [2]

Goldstone, A

J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev.127, 965 (1962)

work page 1962

[3] [3]

B. S. DeWitt, Conf. Proc. C630701, 585 (1964)

work page 1964

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Jona-Lasinio, Nuovo Cim.34, 1790 (1964)

G. Jona-Lasinio, Nuovo Cim.34, 1790 (1964)

work page 1964

[5] [5]

Jackiw, Phys

R. Jackiw, Phys. Rev. D9, 1686 (1974)

work page 1974

[6] [6]

G. M. Shore, Annals Phys.128, 376 (1980)

work page 1980

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Allen, Nucl

B. Allen, Nucl. Phys. B226, 228 (1983)

work page 1983

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Cosmological phase transitions: From perturbative particle physics to gravitational waves,

P. Athron, C. Bal´ azs, A. Fowlie, L. Morris, and L. Wu, Prog. Part. Nucl. Phys.135, 104094 (2024), arXiv:2305.02357 [hep-ph]

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Cosmological Aspects of Higgs Vacuum Metastability

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B. Allen and A. Folacci, Phys. Rev. D35, 3771 (1987)

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A. Folacci, Phys. Rev. D46, 2553 (1992), arXiv:0911.2064 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 1992

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On Resumming Inflationary Perturbations beyond One-loop

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work page internal anchor Pith review Pith/arXiv arXiv

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Effective potential for quantum scalar fields on a de Sitter geometry

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work page internal anchor Pith review Pith/arXiv arXiv 2011

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Feynman Diagrams for Stochastic Inflation and Quantum Field Theory in de Sitter Space

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Quantum scalar fields in de Sitter space from the nonperturbative renormalization group

M. Guilleux and J. Serreau, Phys. Rev. D92, 084010 (2015), arXiv:1506.06183 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2015

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Effective long wavelength scalar dynamics in de Sitter

I. Moss and G. Rigopoulos, JCAP05, 009, arXiv:1611.07589 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv

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The 1-loop effective potential for the Standard Model in curved spacetime

T. Markkanen, S. Nurmi, A. Rajantie, and S. Stopyra, JHEP06, 040, arXiv:1804.02020 [hep-ph]

work page internal anchor Pith review Pith/arXiv arXiv

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Backreaction of superhorizon scalar field fluctuations on a de Sitter geometry : a renormalisation group perspective

G. Moreau and J. Serreau, Phys. Rev. D99, 025011 (2019), arXiv:1809.03969 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2019

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C´ espedes, A.-C

S. C´ espedes, A.-C. Davis, and D.-G. Wang, JHEP04, 004, arXiv:2311.17990 [hep-th]

work page arXiv

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Di Pietro, V

L. Di Pietro, V. Gorbenko, and S. Komatsu, Cosmological correlators at finite coupling (2023), arXiv:2312.17195 [hep-th] . – 25 –

work page arXiv 2023

[30] [30]

Huenupi, E

J. Huenupi, E. Hughes, G. A. Palma, and S. Sypsas, Phys. Rev. D110, 123536 (2024), arXiv:2406.07610 [hep-th]

work page arXiv 2024

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V. Nath, K. Roy, and S. Bhattacharya, Phys. Rev. D112, 025017 (2025), arXiv:2504.16578 [gr-qc]

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J. E. Camargo-Molina, M. Carrillo Gonz´ alez, and A. Rajantie, Phys. Rev. D107, 063533 (2023), arXiv:2204.03480 [gr-qc]

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O’Raifeartaigh, A

L. O’Raifeartaigh, A. Wipf, and H. Yoneyama, Nucl. Phys. B271, 653 (1986)

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Gibbons, S

G. Gibbons, S. Hawking, and M. Perry, Nuclear Physics B138, 141 (1978)

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V. S. Adamchik, Journal of Computational and Applied Mathematics100, 191 (1998)

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S. W. Hawking and I. G. Moss, Phys. Lett. B110, 35 (1982)

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Markkanen, A

T. Markkanen, A. Rajantie, S. Stopyra, and T. Tenkanen, JCAP08, 001, arXiv:1904.11917 [gr-qc]

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Cable and A

A. Cable and A. Rajantie, Phys. Rev. D104, 103511 (2021), arXiv:2011.00907 [gr-qc]

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Cable and A

A. Cable and A. Rajantie, Phys. Rev. D106, 123522 (2022), arXiv:2209.02545 [gr-qc]

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Cable and A

A. Cable and A. Rajantie, Phys. Rev. D109, 045017 (2024), arXiv:2310.07356 [gr-qc]

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