REVIEW
On the proper treatment of massless fields in Euclidean de Sitter space
High Energy Physics - Theory · Submitted 6 Aug 2010 · 18 references
Pith's verdict: The exact zero-mode treatment in §III is correct and worth citing; the loop section mislabels its own result because an O(√λ) tadpole dominates the O(λ) term it computes.
T0 · AI review
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Referee report
The paper
The paper claims
The infrared divergence of a massless scalar on the Euclidean de Sitter sphere is an artifact of quantizing the constant mode with the free action alone; treating that one mode exactly gives a finite two-point function that scales as $1/2/3
The reading
What carries the argument
The mechanism is an exact treatment of the zero mode, the $1/2/3 spherical harmonic on the $1/2/3-dimensional sphere. Everything else in the theory is Gaussian and can be integrated out mode by mode, but the constant mode has no quadratic term in the massless limit, so its fluctuations are governed entirely by $1/2/3. That makes the zero-mode path integral a one-dimensional integral of the form $1/2/3, whose moments $1/2/3 and $1/2/3 are elementary Gamma-function ratios. This is what converts an apparent $1/2/3 divergence into a $1/2/3 contribution and produces an effective mass gap of order $1/2/3 for the constant mode, which the author compares to a gap generated by resummation rather than by a mass counterterm.
Extended reading notes
Core claim
The paper's central claim is that the massless limit of $1/2/3 theory on the Euclidean de Sitter sphere is finite once the constant mode is quantized correctly. Expanding the field as $1/2/3, the author observes that the quadratic action for $1/2/3 is proportional to $1/2/3 and disappears as $1/2/3, so the free-theory treatment of that mode is invalid; the quartic term $1/2/3 must be kept. Because only one mode is involved, the resulting path integral over $1/2/3 is an ordinary single-variable integral, and the exact two-point function becomes $1/2/3 with $1/2/3, a finite result as $1/2/3. The author then treats the mass as a perturbation and resums it, quoting $1/2/3 for the propagator and $1/2/3 for its massless limit. The paper further argues that interactions between the constant mode and the higher harmonics are perturbative, and computes what it calls the first nontrivial loop correction to the nonzero-mode propagator, an $1/2/3 shift of the pole in eq. (26).
Load-bearing premise
The argument assumes that the cross terms $1/2/3 can be treated as small perturbations, even though contracting the zero-mode pair in that vertex shifts the nonzero-mode propagator by an amount of order $1/2/3, which is larger than the $1/2/3 correction the paper presents as the first nontrivial loop effect.
Editorial extensions
If this is right
- The massless propagator on the Euclidean sphere is finite, so tree-level correlators built from products of propagators are finite as well.
- The perturbative expansion parameter for the constant mode changes from $1/2/3 to $1/2/3, which keeps the series under control for small coupling even when $1/2/3.
- The constant mode acquires a gap of order $1/2/3 that is not a renormalization of the bare mass, so de Sitter invariance is not broken by the resummation in this model.
- Loop corrections to nonzero-mode correlators can be organized as nonzero-mode propagators convolved with exact zero-mode four-point functions, giving corrections suppressed by powers of $1/2/3.
- In the Lorentzian in-in formalism a continuum of long-wavelength modes is strongly coupled rather than one, so the same exact treatment is not available and only qualitative statements about a dynamically generated mass are possible.
Reading between the lines
- If the cross term $1/2/3 really generates an $1/2/3 shift of every nonzero-mode pole, then the harmonic spectrum is not $1/2/3 but $1/2/3, and the paper's $1/2/3 correction in eq. (26) is not the leading effect; a reader should check whether a mass counterterm is implicitly absorbing this tadpole.
- The same one-mode-exact strategy should be testable numerically: a Monte Carlo evaluation of the $1/2/3 path integral on a discretized sphere would show directly whether the $1/2/3 propagator scales as $1/2/3 or picks up a $1/2/3 gap.
- The argument suggests a general criterion: infrared divergences are harmless when the strongly coupled sector is finite-dimensional, and intractable when it is a continuum, which would explain why the Euclidean and in-in answers differ here and would predict trouble for gravitons and for massless fields coupled to other light species.
- Because the gap is generated by self-interaction rather than by a mass term, one might look for an analogous phenomenon in other compact Euclidean backgrounds where a single constant mode decouples from the kinetic operator.
Original abstract
We analyze infrared divergences arising in calculations involving light and massless fields in de Sitter space. We show that these arise from an incorrect treatment of the constant mode of the field, and show that a correct quantization leads to a well-defined and calculable perturbation expansion. We illustrate this by computing the first nontrivial loop correction in a theory of a massless scalar field with a quartic interaction.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We analyze infrared divergences arising in calculations involving light and massless fields in de Sitter space. We show that these arise from an incorrect treatment of the constant mode of the field, and show that a correct quantization leads to a well-defined and calculable perturbation expansion. We illustrate this by computing the first nontrivial loop correction in a theory of a massless scalar field with a quartic interaction.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We analyze infrared divergences arising in calculations involving light and massless fields in de Sitter space. | background 0.54 | 0.65 | 0.05 | asserted 0.76 |
| s1 | We show that these arise from an incorrect treatment of the constant mode of the field, and show that a correct quantization leads to a well-defined and calculable perturbation expansion. | result 0.81 | 0.95 | 0.89 | asserted 1.00 |
| s2 | We illustrate this by computing the first nontrivial loop correction in a theory of a massless scalar field with a quartic interaction. | contribution 0.56 | 0.95 | 0.07 | asserted 0.99 |
What was examined, and what was not
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References
How the literature uses it
55 arXiv papers cite this in 88 sentences
Papers citing it · 54 sentences without a confident label
Builds on it 3 papers
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We generalize to the $O(N)$ model the method developed in [ref] and [ref] for a single field, for which the constant zero modes $ϕ_0a$ play a crucial role.
Massless Interacting Scalar Fields in de Sitter space · $O(N)$ model in Euclidean de Sitter space · 0.99
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Generalizing the work of Rajaraman [ref] , we obtained the dynamical mass $m^2∝√(λ) H^2$ of the scalar for non-vanishing Lagrangian masses, as well as the first perturbative quantum correction in the massless case, and developed the rules of a systematic perturbative expansion, which after treating the zero-mode non-perturbatively, goes in powers of $√(λ)$.
On "dynamical mass" generation in Euclidean de Sitter space · 8 Conclusion · 1.00
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We extended the approach of Refs. [ref] to the $O(N)$ model.
$O(N)$ model in Euclidean de Sitter space: beyond the leading infrared approximation · Conclusions · 1.00
Uses its method or tool 4 papers
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Note that, as the zero modes are constant on the sphere, their kinetic terms vanish, and $Z_0[J_0]$ involves only ordinary integrals, which allows it to be exactly computed in several interesting cases following Ref. [ref] .
Massless Interacting Scalar Fields in de Sitter space · $O(N)$ model in Euclidean de Sitter space · 0.83
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This part gives the leading infrared contribution and can be exactly computed in several interesting cases following Ref. [ref] .
$O(N)$ model in Euclidean de Sitter space: beyond the leading infrared approximation · Euclidean de Sitter Space · 0.90
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In order to find a perturbation expansion of the correlators of a deSitter state in the interacting theory, one can use a modification of the standard perturbative setup [ref] which takes into account the zero mode, as is also often done for perturbations of massless fields in 2-dimensional flat space.
Massless interacting quantum fields in deSitter spacetime · Introduction · 0.91
Compares with it 3 papers
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The expansion parameter turns out to be $√(λ)$ instead of the coupling $λ$ of the standard perturbation expansion. * For the massive scalar field we obtain the dependence of the dynamical mass on the Lagrangian mass $m$; for the massless field the leading $O(√(λ))$ correction. * We show that the dynamical mass can be obtained from the loop expansion of the self-energy after summing a divergent series to all orders in the loop expansion. * It seems to have gone unnoticed that the Euclidean dynamical mass [ref] agrees with the result from the stochastic approach.
On "dynamical mass" generation in Euclidean de Sitter space · 1 Introduction · 0.80
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At the leading IR order the interaction between the zero and UV modes in Eq. [ref] can be neglected, [equation] For $N = 1$, we recover the result of [ref] , which is also the one from the stochastic approach [ref] .
Massless Interacting Scalar Fields in de Sitter space · $O(N)$ model in Euclidean de Sitter space · 0.95
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The approximation strategies employed in the present work appear related to those of Ref. [ref] for a scalar field on a four-dimensional sphere.
Self Regulation of Infrared Correlations for Massless Scalar Fields during Inflation · Discussion and Conclusions · 0.91
Cites it as support 2 papers
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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] ,
Symmetry breaking in de Sitter: a stochastic effective theory approach · Introduction · 1.00
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In this massless limit, the Bunch-Davies state distributes the Euclidean zero mode uniformly along $φ$'s field space [ref] .
Compact Scalars at the Cosmological Collider · The Euclidean Path Integral · 0.98
Cites it as background 18 papers
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(See also Refs. [ref] and [ref] .) Despite the several efforts [ref] , the debate regarding the possibility of the IR divergence has not been settled.
Natural selection of inflationary vacuum required by infra-red regularity and gauge-invariance · Introduction · 0.94
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Several nonperturbative approaches have been considered in the literature to address this problem, including stochastic formulations based on [ref] , the so-called Hartree approximation [ref] , the $1/N$ expansion [ref] , renormalization group equations [ref] , or the connection to the theory formulated on the sphere (i.e., on the Euclidean version of dS space) [ref] .
Quantum backreaction of $O(N)$-symmetric scalar fields and de Sitter spacetimes at the renormalization point: renormalization schemes and the screening of the cosmological constant · Introduction · 0.90
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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
What it cites · 14 sentences without a confident label
Compares with 1 paper
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In many ways, this result is similar to the analysis of the gap equation in [ref] ; however we have succeeded in obtaining the gap without the need for any approximation because only one mode is strongly coupled.
On Resumming Inflationary Perturbations beyond One-loop · Discussion and Conclusions · 1.00
Cites as support 5 papers
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This can be shown to be due to a failure of the standard in-out formalism of field theory [ref] .
A conformally coupled massive scalar field in the de Sitter expanding universe with the mass term treated as a perturbation · Scalar Field Theory in de Sitter space · 0.99
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This suggests that the late time divergences are unphysical, and arise from the breakdown of the in-out formalism in Lorentzian de Sitter space in global coordinates [ref] .
Decay of the free-theory vacuum of scalar field theory in de Sitter spacetime in the interaction picture · Introduction · 0.99
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Correlation functions are therefore infrared finite, and can be continued back to the Lorentzian theory to obtain well defined results [ref] .
The IR stability of de Sitter: Loop corrections to scalar propagators · Introduction · 0.97
Cites as background 4 papers
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There are many possible coordinatizations of de Sitter space [ref] .
Les Houches Lectures on De Sitter Space · Scalar Field Theory in de Sitter space · 1.00
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There has been much recent interest in the possibility of probing nongaussianities in the CMB at the WMAP and PLANCK experiments [ref] .
Constraints on local primordial non-Gaussianity from large scale structure · Introduction · 0.99
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There has been much recent interest in the possibility of probing nongaussianities in the CMB at the WMAP and PLANCK experiments [ref] .
Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation · Introduction · 0.99
Discussion
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