Breaking of scaling symmetry by massless scalar on de Sitter

D. Glavan; R. P. Woodard; S. P. Miao; T. Prokopec

REVIEW 3 major objections 4 minor 19 references

Breaking of scaling symmetry by massless scalar on de Sitter

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A static point charge in de Sitter space forces the surrounding massless scalar field to break scaling symmetry, even though the field equation is scale invariant and the resulting energy-momentum tensor is not.

desk verdict A clean, correct little paper whose core symmetry-breaking claim holds up once you accept the regularity criterion; the abstract overstates what is shown about backreaction. read the letter →

arxiv 1908.11113 v2 pith:N2BXWWL7 submitted 2019-08-29 gr-qc hep-th

D. Glavan , S. P. Miao , T. Prokopec , R. P. Woodard This is my paper · ORCID

classification gr-qchep-th PACS 04.50.Kd95.35.+d98.62.-g

keywords deSitterspacemasslessminimallycoupledscalardilatationsymmetryscalingbreakingpointchargeretardedGreen'sfunctionnon-inheritanceinfraredeffects

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

The paper studies a massless, minimally coupled scalar field (one with no mass and no direct coupling to curvature) in de Sitter space, sourced by a static point charge. It argues that although the sourced field equation is invariant under dilatations, every solution respecting that symmetry has a logarithmic singularity at the Hubble horizon, so the physical field profile necessarily breaks scaling symmetry. The physical profile instead contains a term proportional to the logarithm of the scale factor, while the associated energy-momentum tensor remains dilatation-invariant. This gives a classical, cosmological example of symmetry non-inheritance: the field breaks a symmetry that its energy-momentum tensor preserves.

What carries the argument

The load-bearing object is the retarded Green's function (17), which combines the usual light-cone delta function with a de Sitter tail term proportional to the step function times $H^2/2$. Integrating this Green's function against the point source gives the causal response (19), and matching the initial data to the interior bracket extends it to the global profile (20). A second structural ingredient is the small-mass expansion of the massive scalar solution (12), whose $1/m^2$ divergence and logarithmic terms explain why the massless scaling solution cannot be obtained as a regular limit and why the horizon singularity of the scaling solution (5) is unavoidable.

What would settle it

Directly integrate the retarded Green's function with vanishing initial data and compare with (19); then check whether any homogeneous solution can be added to the scaling profile (5) to make the energy-momentum tensor (6) finite at $X=1$ while keeping the profile a function of $X$ alone. The paper's claim is that no such regular scaling solution exists, so exhibiting one would falsify the necessity claim.

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Extended reading notes

Core claim

The central claim is that the physical response of a massless minimally coupled scalar to a static point charge in de Sitter space is the field profile $\Phi = -\frac{\lambda H}{4\pi X} - \frac{\lambda H}{4\pi}\ln\left(\frac{a}{1+X}\right)$, which is regular away from the charge but breaks dilatation symmetry through the $\ln(a)$ term. The scaling-symmetric solution $\Phi(X)$, which depends only on the scale-invariant combination $X = aHr$, develops a logarithmic singularity at the Hubble horizon $X=1$, and its energy-momentum tensor diverges quadratically there. The same obstruction appears in the small-mass limit of the massive scalar, where the regular solution's coefficient diverges as $1/m^2$, showing that the massless scaling solution is not a regular limit. Using the retarded Green's function with initial data chosen from the interior of the causal domain yields the globally valid broken-symmetry profile, whose energy-momentum tensor is finite outside the origin and, in contrast to the field, respects dilatation symmetry. The pattern persists in arbitrary spacetime dimension.

Load-bearing premise

The conclusion rests on two choices: that the physical solution must be regular away from the charge, and that the scalar's initial data are taken from inside the light cone rather than set to zero; if the initial data are instead taken to vanish, the causal solution becomes the step-function profile (19) and the 'necessarily' claim becomes conditional on that choice.

Editorial extensions

If this is right

The field produced by a static scalar charge grows as $\frac{\lambda H}{4\pi}\ln(Hr)$ at large comoving distances and late times.
The gravitational backreaction, computed from the energy-momentum tensor (22), does not break the de Sitter scaling symmetry, so the metric response can remain scale invariant even though its scalar source does not.
The argument generalizes to $D$ spacetime dimensions: the naive scaling solution has a logarithmic horizon singularity, so the physical solution must break scaling in any dimension.
The massless theory is discontinuous from the small-mass limit, in that the coefficient of the regular massive solution diverges as $1/m^2$ as $m\to 0$.
The authors suggest the solution may help model how point charges during inflation affect temperature fluctuations in the cosmic microwave background.

Reading between the lines

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

If the energy-momentum tensor inherits the symmetry while the field does not, then observers who only probe gravitational effects will see a scale-invariant source; detecting the symmetry breaking requires measuring the scalar field directly, for instance through its coupling to matter.
The step-function solution obtained with vanishing initial data suggests the strength of the logarithmic tail may depend on how the source is switched on; an extension would be to compute the response for a smooth, adiabatic switch-on and see whether the $\ln(a)$ coefficient changes.
The horizon-regularity condition functions as an imposed boundary condition rather than a derived one; alternative regularizations, such as admitting distributional sources at the horizon or different homogeneous additions, could be tested to see whether a regular scale-invariant profile can be recovered.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core of this paper is right: for a massless minimally coupled scalar with a static point source on de Sitter, the scaling solution (5) has a logarithmic singularity at the Hubble horizon, so no regular solution respecting the remaining four isometries exists. The exact profile (20) was already obtained by Akhmedov, Roura and Sadofyev, and the authors say so. The genuinely new content is the demonstration that the scaling solution is unavoidably singular and that the regular physical profile necessarily breaks dilatation symmetry. That argument is clean and, with one caveat, I think it holds up.

What the paper does well: the differential equations are worked out carefully, the massive-to-massless limit is instructive, and the connection to the de Sitter scalar propagator is natural. The self-citation to [10] only fixes a constant that cancels in the retarded Green's function, so I see no circularity. The paper is also honest about the overlap with [18].

The caveats are two. First, the abstract says "the gravitational backreaction ... does respect dilatation symmetry." That goes beyond what is computed. The paper shows the energy-momentum tensor (22) is scale-invariant, but it does not solve the Einstein equations. A scale-invariant source does not guarantee a scale-invariant metric perturbation; the linearized Einstein equations can break scaling through boundary conditions or the graviton propagator's infrared behavior. This is an overclaim, and the body is more careful than the abstract. The abstract should be reworded to say "the energy-momentum tensor" instead of "the gravitational backreaction," or the backreaction calculation should actually be done.

Second, the selection of (20) over (19) is a choice of initial data. The paper says it is "more natural" to take the values from inside the bracket, which is reasonable but not a theorem. That said, the scale-breaking conclusion is robust: even the causal step-function solution (19) contains the same ln(a) term inside the light cone, and taking the initial time to the infinite past makes (19) reduce to (20). So the ambiguity does not undermine the central claim, but the word "necessarily" is a bit strong because it rests on deeming the horizon singularity unphysical. That is a physical judgment, not a mathematical impossibility.

Minor: the "wall of fire" phrase is unnecessary flourish.

Overall this is a solid, honest contribution. It deserves a serious referee; the main fix is the abstract. I would be happy to cite it for the symmetry non-inheritance story, and it would make a good reading-group paper precisely because the abstract invites discussion of what backreaction really requires.

Referee Report

3 major / 4 minor

Summary. This paper studies a classical massless minimally coupled scalar field coupled to a static point charge in the Poincaré patch of de Sitter space. The equation of motion (3) is invariant under spatial rotations and dilatations. The authors first exhibit the scaling-invariant solution (5), which has a logarithmic singularity at the Hubble horizon X=1 and a correspondingly divergent stress tensor (6)-(7), and they argue this solution is unphysical. They support this by studying the massive scalar, whose regular solution (10) has a small-mass limit (12) that is singular in 1/m^2 and cannot be reduced to a regular massless scaling solution. Using the de Sitter retarded Green's function (17), they obtain the causal response (19) and, after a particular choice of initial data, the globally defined profile (20), which depends on ln a and hence breaks dilatation invariance. The associated stress tensor (22) depends only on X, so it respects dilatation symmetry; the authors interpret this as classical symmetry non-inheritance.

Significance. The paper gives an explicit, self-contained demonstration that the natural classical response of a massless minimally coupled scalar to a static charge in de Sitter breaks the scaling symmetry of the dynamical equations, while the associated stress tensor remains scale-invariant. The horizon-singularity argument against the scaling profile is clean and the retarded Green's function tail is transparent. If accepted, the result provides a simple classical analogue of infrared-driven symmetry breaking and a concrete example of symmetry non-inheritance in a gravitational context. The manuscript does not fit parameters to data and all computations are explicit, which is a strength. The main caveat is that the advertised gravitational-backreaction claim in the abstract is not actually computed in the body of the paper.

major comments (3)

[Abstract and 'Breaking of dilatation symmetry' (Eq. (22))] The abstract's claim that 'the gravitational backreaction ... does respect dilatation symmetry' is not supported by the body of the paper. What is computed in Eq. (22) is the stress-energy tensor of the scalar solution (20); the metric perturbation sourced by this T^mu_nu is never constructed. Scale invariance of the stress tensor is necessary but not sufficient for the linearized Einstein equation solution to be dilatation-invariant, since boundary conditions, the graviton's infrared behaviour, and explicit ln(a) terms in retarded solutions can break the symmetry. The summary section correctly speaks only of the energy-momentum tensor respecting dilatation; the abstract should be changed to match, or the backreaction computation should be added.
[Breaking of dilatation symmetry, Eq. (19) vs (20)] The selection of the physical solution rests on an unstated initial-data choice. The authors state that 'it is more natural' to take the initial values at eta0 from the bracket in (19), but the standard retarded solution with vanishing initial data is (19) itself, with the step function. The two profiles differ outside the forward light cone. The conclusion that a regular solution must break scaling does not depend on this choice, because no regular scaling solution exists (5); however the identification of the unique physical profile with (20), and hence the statement that this particular field is 'regular everywhere except at the origin', is not derived. The authors should either justify the choice (for example by taking eta0 -> -infinity, in which case (19) reduces to (20), or by invoking a regularity condition at the horizon), or soften the claim of uniqueness.
[Point particle and scaling solution, Eqs. (5)-(7)] The argument that (5) is unphysical relies on the criterion that the stress tensor must be regular at the Hubble horizon (7). This is a physical regularity assumption, not a consequence of the field equations. It is a reasonable criterion for a classical solution sourced by a point charge, but the paper should state it explicitly as an assumption when claiming that the solution 'necessarily' breaks scaling.

minor comments (4)

[Point particle and scaling solution, below Eq. (5)] The phrase 'wall of fire' is informal and should be replaced with a precise statement about the horizon divergence of the energy-momentum tensor.
[Breaking of dilatation symmetry, Eq. (19)] For clarity, state that (19) is the response to a source switched on at eta0 with vanishing initial data; the step discontinuity at the light cone is then expected. This would make the subsequent initial-data discussion in (20) easier to follow.
[Summary and discussion, Eq. (21)] The term 'regular everywhere except at the origin' should be qualified: (20) grows logarithmically at large r, as shown in (21), so 'regular' here means free of curvature-type singularities rather than bounded.
[References, [18]] The authors already note that some results were obtained in [18]; it would be helpful to state explicitly which parts of (20) and which aspects of the symmetry-breaking interpretation are new relative to that work.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivation is self-contained, and the only self-cited constant cancels in the retarded Green's function.

full rationale

The derivation is self-contained. The scaling ODE (4) is solved explicitly, and its general solution (5) is shown to diverge logarithmically at the Hubble horizon, with the associated energy-momentum tensor (6)-(7) diverging quadratically. The massive-scalar limit (12) independently indicates that the massless scaling solution is singular, so the conclusion that no regular scaling solution exists is supported by the paper's own equations rather than by any fitted input. The physical solution (20) is then constructed from the retarded Green's function (17), which is obtained from the Wightman function (16); the only self-citation, the constant fixed from the authors' earlier paper [10], enters the Wightman function in a symmetric, time-independent way and cancels in the antisymmetric combination defining the retarded Green's function, so it is not load-bearing. The choice between the step-function solution (19) and the globally continued solution (20) is an initial-data selection, not a fit, and in either case a late-time observer inside the forward light cone sees the same scale-breaking bracket of (20). No parameter is fitted to data, no uniqueness theorem is imported from the authors' prior work, and no known result is renamed. The abstract's claim that the gravitational backreaction respects dilatation symmetry goes beyond what is actually computed, since only the scalar stress tensor (22) is shown to be scale-invariant, but that is an unsupported extrapolation or correctness concern, not a circular reduction of the paper's central derivation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numerical fits appear anywhere in the paper. The only inputs are H and lambda, both physical constants of the background and source. The load-bearing choices are the regularity criterion at the horizon and the particular initial-data selection that promotes (20) over (19); both are domain assumptions, not fitted parameters.

assumptions (5)

domain assumption The de Sitter background is described in Poincare coordinates with scale factor a = -1/(H eta), and the point charge is modeled as a delta-function source without self-interaction.
Equation (3) treats the charge as a distribution and ignores its own gravitational and scalar self-interaction, which is standard for a probe source but is still an idealization.
standard math The retarded Green's function (17) is the correct classical response kernel for the massless minimally coupled scalar, with the tail term H^2/2 theta(Delta eta - ||Delta x||) inherited from the massless propagator (16).
The Green's function is asserted after citing the spatially homogeneous and isotropic massless propagator; the non-universal constant in (16) is fixed by the authors' earlier paper [10].
domain assumption A physically acceptable scalar field solution must be regular away from the point charge, in particular at the Hubble horizon X=1.
This criterion is used to reject the scaling solution (5) and to select the homogeneous mode in (20). If horizon singularities are deemed acceptable, the scaling-invariant solution is admissible.
ad hoc to paper Initial data at eta0 are chosen to match the interior bracket, so the causal step-function solution (19) is replaced by the globally valid expression (20).
The paper describes this choice as 'more natural' rather than deriving it from the initial-value problem. Different initial data would give different solutions, some of which do not have the same scaling behavior.
standard math The small-mass expansion of the massive hypergeometric solution (10), including the singular 1/m^2 term, correctly describes the massless limit.
Equation (12) is used to argue that no regular scaling limit exists; the expansion is a standard asymptotic analysis of the hypergeometric functions.

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Pith. "Pith review of Breaking of scaling symmetry by massless scalar on de Sitter." pith.science (2026). https://pith.science/paper/N2BXWWL7

@misc{pith2026190811113,
  author       = {Pith},
  title        = {Pith review of: Breaking of scaling symmetry by massless scalar on de Sitter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N2BXWWL7}},
  note         = {Machine review of arXiv:1908.11113}
}

read the original abstract

We study the response of a classical massless minimally coupled scalar to a static point scalar charge on de Sitter. By considering explicit solutions of the problem we conclude that -- even though the dynamics formally admits dilatation (scaling) symmetry -- the physical scalar field profile necessarily breaks the symmetry. This is an instance of symmetry breaking in classical physics due to large infrared effects. The gravitational backreaction, on the other hand, does respect dilatation symmetry, making this an example of symmetry non-inheritance phenomenon.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [10]

Gravito n Loop Corrections to Vacuum Polarization in de Sitter in a General Covarian t Gauge,

D. Glavan, S. P. Miao, T. Prokopec and R. P. Woodard, “Gravito n Loop Corrections to Vacuum Polarization in de Sitter in a General Covarian t Gauge,” Class. Quant. Grav. 32 (2015) no.19, 195014 doi:10.1088/0264- 9381/32/19/195014 [arXiv:1504.00894 [gr-qc]]

arXiv 2015

[2] [18]

Classical radiatio n by free-falling charges in de Sitter spacetime,

E. T. Akhmedov, A. Roura and A. Sadofyev, “Classical radiatio n by free-falling charges in de Sitter spacetime,” Phys. Rev. D 82 (2010) 044035 doi:10.1103/PhysRevD.82.044035 [arXiv:1006.3274 [gr-qc]]. 9

arXiv 2010

[3] [1]

Quantum theory of scalar ﬁe lds in de Sitter space-time,

N. A. Chernikov and E. A. Tagirov, “Quantum theory of scalar ﬁe lds in de Sitter space-time,” Ann. Inst. H. Poincare Phys. Theor. A 9 (1968) 109

work page 1968

[4] [2]

Scalar Field Fluctuations in Expanding Universe and th e New Inﬂationary Universe Scenario,

A. D. Linde, “Scalar Field Fluctuations in Expanding Universe and th e New Inﬂationary Universe Scenario,” Phys. Lett. 116B, 335 (1982). doi:10.1016/0370-2693(82)90293-3 which exhibits a logarithmic singularity at the horizon, Φ( X) X→ 1 ∼ − λH D− 3 Γ ( D− 1 2 ) 4π D− 1 2 ln ( 1−X ) . 7

doi:10.1016/0370-2693(82)90293-3 1982

[5] [3]

Dynamics of Phase Transition in the New Inﬂa tion- ary Universe Scenario and Generation of Perturbations,

A. A. Starobinsky, “Dynamics of Phase Transition in the New Inﬂa tion- ary Universe Scenario and Generation of Perturbations,” Phys. Le tt. 117B, 175 (1982). doi:10.1016/0370-2693(82)90541-X

doi:10.1016/0370-2693(82)90541-x 1982

[6] [4]

Gravitational Eﬀects upon Cos- mological Phase Transitions,

A. Vilenkin and L. H. Ford, “Gravitational Eﬀects upon Cos- mological Phase Transitions,” Phys. Rev. D 26 (1982) 1231. doi:10.1103/PhysRevD.26.1231

doi:10.1103/physrevd.26.1231 1982

[7] [5]

Quantum Instability of De Sitter Space-time,

L. H. Ford, “Quantum Instability of De Sitter Space-time,” Phys. Rev. D 31 (1985) 710. doi:10.1103/PhysRevD.31.710

work page doi:10.1103/physrevd.31.710 1985

[8] [6]

Vacuum States in de Sitter Space,

B. Allen, “Vacuum States in de Sitter Space,” Phys. Rev. D 32 (1985)

work page 1985