Symmetry breaking in de Sitter: a stochastic effective theory approach

Gianrocco Lazzari; Tomislav Prokopec

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Symmetry breaking in de Sitter: a stochastic effective theory approach

Gianrocco Lazzari, Tomislav Prokopec

High Energy Physics - Theory · Submitted 1 Apr 2013

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Making use of Starobinsky's stochastic inflation we prove that deep infrared scalar modes cannot form a condensate -- and hence they see an effective potential that allows no phase transition.

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We consider phase transitions on (eternal) de Sitter in an O(N) symmetric scalar field theory. Making use of Starobinsky's stochastic inflation we prove that deep infrared scalar modes cannot form a condensate -- and hence they see an effective potential that allows no phase transition. We show that by proving convexity of the effective potential that governs deep infrared field fluctuations both at the origin as well as at arbitrary values of the field. Next, we present numerical plots of the scalar field probability distribution function (PDF) and the corresponding effective potential for several values of the coupling constant at the asymptotic future timelike infinity of de Sitter. For small field values the effective potential has an approximately quadratic form, corresponding to a positive mass term, such that the corresponding PDF is approximately Gaussian. However, the curvature of the effective potential shows qualitatively different (typically much softer) behavior on the coupling constant than that implied by the Starobinsky-Yokoyama procedure. For large field values, the effective potential as expected reduces to the tree level potential plus a positive correction that only weakly (logarithmically) depends on the background field. Finally, we calculate the backreaction of fluctuations on the background geometry and show that it is positive.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider phase transitions on (eternal) de Sitter in an O(N) symmetric scalar field theory.	background 0.84	0.54	0.03	asserted 0.56
s1	Making use of Starobinsky's stochastic inflation we prove that deep infrared scalar modes cannot form a condensate -- and hence they see an effective potential that allows no phase transition.	result 0.97	0.92	0.83	asserted 1.00
s2	We show that by proving convexity of the effective potential that governs deep infrared field fluctuations both at the origin as well as at arbitrary values of the field.	method 0.46	0.90	0.17	asserted 0.97
s3	Next, we present numerical plots of the scalar field probability distribution function (PDF) and the corresponding effective potential for several values of the coupling constant at the asymptotic future timelike infinity of de Sitter.	contribution 0.74	0.95	0.04	asserted 0.99
s4	For small field values the effective potential has an approximately quadratic form, corresponding to a positive mass term, such that the corresponding PDF is approximately Gaussian.	result 0.95	0.86	0.04	asserted 0.91
s5	However, the curvature of the effective potential shows qualitatively different (typically much softer) behavior on the coupling constant than that implied by the Starobinsky-Yokoyama procedure.	result 0.95	0.81	0.07	asserted 0.88
s6	For large field values, the effective potential as expected reduces to the tree level potential plus a positive correction that only weakly (logarithmically) depends on the background field.	result 1.00	0.86	0.05	asserted 0.99
s7	Finally, we calculate the backreaction of fluctuations on the background geometry and show that it is positive.	result 0.87	0.96	0.08	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.96.

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The paper

Original abstract

We consider phase transitions on (eternal) de Sitter in an O(N) symmetric scalar field theory. Making use of Starobinsky's stochastic inflation we prove that deep infrared scalar modes cannot form a condensate -- and hence they see an effective potential that allows no phase transition. We show that by proving convexity of the effective potential that governs deep infrared field fluctuations both at the origin as well as at arbitrary values of the field. Next, we present numerical plots of the scalar field probability distribution function (PDF) and the corresponding effective potential for several values of the coupling constant at the asymptotic future timelike infinity of de Sitter. For small field values the effective potential has an approximately quadratic form, corresponding to a positive mass term, such that the corresponding PDF is approximately Gaussian. However, the curvature of the effective potential shows qualitatively different (typically much softer) behavior on the coupling constant than that implied by the Starobinsky-Yokoyama procedure. For large field values, the effective potential as expected reduces to the tree level potential plus a positive correction that only weakly (logarithmically) depends on the background field. Finally, we calculate the backreaction of fluctuations on the background geometry and show that it is positive.

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References

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Infrared Correlations in de Sitter Space: Field Theoretic vs. Stochastic Approach
hep-th 2013-10 conditional novelty 6.0 of 10

A two-loop closed-time-path calculation of infrared-corrected scalar correlations in de Sitter reproduces, diagram by diagram, the result of the stochastic Starobinsky approach.
Infrared dynamics in de Sitter space from Schwinger-Dyson equations
hep-th 2013-05 accept novelty 6.0 of 10

An exact analytic solution of the Schwinger-Dyson equations shows that de Sitter infrared logarithms resum into power laws with a renormalized mass and field strength, reducing in the deep infrared to a free massive field.
IR flow of stochastic power spectra for interacting spectator scalars in the inflationary de Sitter spacetime
hep-th 2026-10 conditional novelty 5.0 of 10

The IR (Wetterich) flow of one-loop spectator-scalar effective potentials in de Sitter drives symmetry restoration, but the claimed deep-IR spectral-index enhancement is not supported by the paper's own eigenvalue table.

How the literature uses it

25 arXiv papers cite this in 48 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 31 sentences without a confident label

Uses its method or tool 1 paper

Stochastic approach to inflation [ref] is used by [ref] to take into account, in a non-perturbative manner, quantum corrections to the same model as previous works cited here.

Non-equilibrium evolution of quantum fields during inflation and late accelerating expansion · Introduction · 0.87

Compares with it 3 papers

The mass obtained in the limit $κ=0$ is $m^2=1.11805e-3 H^2$ is to be compared with the late-time mass using the stochastic formalism [ref] , $m^2=1.11868e-3 H^2$.

Renormalization group approach to scalar quantum electrodynamics on de Sitter · Scalar field theory · 1.00
Again, this mirrors the QFT results of [ref] .

Functional renormalization group in stochastic inflation · Introduction · 0.91
The NPRG analysis shows that such gravitationally induced symmetry restoration occurs for any value of $N$ and in any space-time dimension, in agreement with the results of Ref. [ref] in the stochastic approach.

Nonperturbative renormalization group for scalar fields in de Sitter space: beyond the local potential approximation · Introduction · 0.94

Cites it as support 2 papers

A variety of approaches, many using stochastic methods, are converging to this conclusion [ref] which is strongly supported by the results of this work.

Thermal Interpretation of Infrared Dynamics in de Sitter · Discussion · 0.96
Different approaches also suggest that such infrared divergences can indeed tamed in de Sitter and physically interpreted [ref] .

Fluctuation-dissipation and equilibrium for scalar fields in de Sitter · Discussion · 0.97

Cites it as background 6 papers

Interactions of quantum fields in de Sitter (or nearly de Sitter) space-time have been the focus of important studies [ref] which show strong infrared and secular effects.

Fermionic influence (action) on inflationary fluctuations · Introduction · 0.89
This kind of fields have been considered by several authors in different frameworks; for instance, in the stochastic approach [ref] , in the context of Euclidean de Sitter space [ref] , using dynamical renormalization group resummations [ref] , or other nonperturbative approximation schemes [ref] .

The Hartree approximation in curved spacetimes revisited I: the effective potential in de Sitter · Introduction · 0.94
A more interesting and challenging question is whether there is a de Sitter invariant quantum state for an interacting scalar field, and it is this question that has drawn the interest of a number of authors who have addressed the problem using a wide range of methods [ref] .

Infrared Correlations in de Sitter Space: Field Theoretic vs. Stochastic Approach · Introduction · 0.87

What it cites · 13 sentences without a confident label

Builds on 1 paper

The generalization of ([ref]) to the $O(N)$ case is (cf. Ref. [ref] )

Symmetry breaking and the Goldstone theorem in de Sitter space · The $O(N)$ model · 0.95

Compares with 2 papers

In fact, it is worth making a detailed comparison of the mass implied by the effective action in figure [ref] and that implied by the Starobinsky-Yokoyama procedure [ref] which, when generalized to a finite background field $ϕ_b$, yields,

On "dynamical mass" generation in Euclidean de Sitter space · Analytical approximations for $V_eff$ · 0.99
There is a very intriguing connection of the result ([ref]), with the mean field (one loop) result for the mass [ref] .

Symmetry breaking and the Goldstone theorem in de Sitter space · Analytical approximations for $V_eff$ · 1.00

Cites as support 1 paper

The effective theory approach advocated here becomes even better motivated when one recalls the observed equivalence between the zero mode partition function $Z_E(ϕ_0E)$ on Euclidean de Sitter space and the PDF in stochastic approach at asymptotically late times $ρ(ϕ,t→∞)$ [ref] ,

On the proper treatment of massless fields in Euclidean de Sitter space · Introduction · 1.00

Cites as background 22 papers

Recently, a more sophisticated resummation scheme based on a large $N$ expansion in an $O(N)$ symmetric model has been utilized in Refs. [ref] to study deep infrared correlators on de Sitter space.

Nonperturbative resummation of de Sitter infrared logarithms in the large-N limit · Introduction · 0.93
Recently, a more sophisticated resummation scheme based on a large $N$ expansion in an $O(N)$ symmetric model has been utilized in Refs. [ref] to study deep infrared correlators on de Sitter space.

Nonperturbative infrared enhancement of nonGaussian correlators in de Sitter space · Introduction · 0.93
Furthermore, based on the Euclidean approach to de Sitter, progress has been also made towards understanding mass generation in an interacting scalar field theory with a quartic self-interaction and one real scalar field (the $O(1)$ model) [ref] .

On "dynamical mass" generation in Euclidean de Sitter space · Introduction · 0.95

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