Spontaneous baryosynthesis with large initial phase

Maxim Khlopov; Maxim Krasnov; Ufuk Aydemir

REVIEW 3 major objections 4 minor 12 references

Spontaneous baryosynthesis with large initial phase

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper claims that in spontaneous baryogenesis, the baryon asymmetry grows as the cube of the initial misalignment angle only for small angles; once the angle approaches π, particle production saturates, so large angles do not yield sub

desk verdict The paper's central result rests on a false delta-function identity in Eq. (19), so the damped-pendulum equation (20) and the saturation claim are not established. read the letter →

arxiv 2512.11011 v3 pith:3UBTDVDZ submitted 2025-12-11 hep-ph hep-th

Maxim Krasnov , Maxim Khlopov , Ufuk Aydemir This is my paper · ORCID

classification hep-phhep-th

keywords spontaneousbaryogenesispseudo-Nambu-GoldstonebosonlargemisalignmentanglebaryonasymmetryMinkowskispacetimedampedpendulumfermionbackreactioncosinepotential

topics Matter-Antimatter Asymmetry

open problems Matter-Antimatter Asymmetry

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 spontaneous baryogenesis driven by a pseudo-Nambu-Goldstone boson whose initial phase θ_i can be large, motivated by inflationary fluctuations that make patches with θ_i ≈ π statistically certain. Using a damped-pendulum equation of motion with a dimensionless decay rate Γ, it numerically computes the baryon asymmetry in Minkowski spacetime. It confirms the known θ_i³ scaling for small angles and finds that for larger angles the growth saturates as θ_i approaches π. The intended upshot is that large misalignment angles do not substantially alter the baryon asymmetry from small-angle predictions, despite the naive expectation of dramatic enhancement.

What carries the argument

The central object is the dimensionless damped-pendulum equation θ'' + Γθ' + sinθ = 0 (Eq. 21), obtained by approximating the fermion backreaction integral with a delta function via Eq. (19). The damping coefficient Γ = g²ωf/(Λ²π) is treated as a free parameter. The asymmetry is read off from integrals that convert the rolling phase history θ(t) into baryon versus antibaryon number densities. The saturation near θ_i = π is attributed to the oscillation period lengthening relative to the harmonic approximation.

What would settle it

Integrate Eq. (13) or Eq. (15) numerically with a finite cutoff ω ≈ f and no delta-function shortcut, and compare θ(t) and Δn_B(θ_i) with Eq. (21). The saturation result is falsified if the backreaction term does not reduce to a simple Γθ′ damping or if Δn_B keeps growing beyond θ_i ≈ 1.

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Extended reading notes

Core claim

The paper's central claim is that the baryon asymmetry generated in spontaneous baryogenesis depends on the initial phase θ_i in a piecewise way: Δn_B ∝ θ_i³ for small θ_i, but the growth decelerates and saturates as θ_i approaches π. This saturation is demonstrated numerically by solving a damped-pendulum equation of motion, Eq. (21), for a range of damping parameters Γ, and then evaluating the baryon number densities through the Fourier-type integrals of Eqs. (23)–(24). The authors conclude that, in Minkowski spacetime, the effects of a large misalignment angle are not substantially different from those predicted by the small-angle approximation, even though the oscillation period becomes

Load-bearing premise

The whole saturation claim rests on Eq. (19), which replaces an oscillatory integral by πωδ(t′); but that left-hand side is a total derivative whose integral is zero, so if the replacement fails the damped-pendulum equation — and with it the numerical results — do not follow.

Editorial extensions

If this is right

If correct, the common small-angle approximation for spontaneous baryogenesis remains quantitatively reliable even for initial phases near the top of the cosine potential.
Regions with θ_i ≈ π, which the paper argues are statistically guaranteed to exist in the observable universe, would not produce anomalously large baryon excesses; the asymmetry stays within the same order of magnitude.
The cubic θ_i³ scaling is confirmed but only as a small-angle limit, so extrapolating it to θ_i ≳ 1 would overestimate particle production.
The dimensionless decay rate Γ controls the overall amplitude of the asymmetry but does not change the qualitative saturation behavior.

Reading between the lines

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

If the delta-function approximation in Eq. (19) is not valid — and the left-hand side is exactly d/dt'[sin²(ωt')/t'], whose integral over t' vanishes — then Eq. (21) may not describe the backreaction; the saturation could be an artifact of that approximation.
One testable extension: integrate the original integro-differential equation Eq. (15) directly with a finite cutoff ω ∼ f and check whether damping survives; this would settle whether the pendulum equation is a faithful reduction.
In an expanding universe, Hubble friction adds an extra 3Hθ' term; whether saturation persists depends on how the damping term competes with expansion, which the Minkowski calculation cannot decide.
The near-π patches form domain walls separating the two minima; the paper's Minkowski treatment starts with θ_i = 3.1 rather than exactly π, so the domain-wall dynamics at exactly θ_i = π remain an open edge case.

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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 paper's central result is not established because the key identity in Eq. (19) is false. The bracket on the left-hand side is exactly d/dt'[sin²(ωt')/t'], so its integral over any interval containing zero is zero, not πω. As a distribution it does not blow up with ω; the leading ω terms cancel. Therefore Eq. (20) does not follow from Eq. (16), and all subsequent numerical work solves an unjustified equation.

What is new and good: the question is worth asking. Large initial misalignment angles are physically plausible, and the probability argument in Sec. 3 is reasonable. The paper checks itself against the known θ_i³ scaling at small angles, and the numerical implementation of the asymmetry formula is straightforward. If the equation of motion were properly derived, the saturation would be a useful observation.

The soft spots are serious. The delta-function approximation is simply wrong, and no alternative derivation of the damping term is offered. The small-angle cubic check in Fig. 2 only confirms that the asymmetry formula (23) is consistent with a damped pendulum; it does not validate the pendulum itself. The free parameter Γ is treated as arbitrary, so even the numerical results are illustrative, not predictive. There is also a minor inconsistency: the fermions are called massless in Sec. 4, but the baryon integrals assume a nonzero threshold m_Q + m_L > 0.

The paper is not a waste of time—the motivation and numerical framework are fine—but the central claim rests on a bad equation. The authors would need to redo the derivation or cite a valid one before the saturation result can be believed. For that reason I would not cite it in its current form, but I would send it to a referee: the mistake is substantive, a good referee would catch it, and the topic is legitimate. For a reading group, it's a useful example of how a delta-function approximation can hide a total-derivative error. My recommendation: don't take the result at face value; if you referee it, ask for a correct derivation of Eq. (20).

Referee Report

3 major / 4 minor

Summary. The paper studies spontaneous baryogenesis mediated by a pseudo-Nambu-Goldstone boson, focusing on large initial misalignment angles θ_i in Minkowski spacetime. It converts a nonlocal backreaction integral into a local damping term, solves the resulting damped-pendulum equation for θ(t), and computes the baryon asymmetry using a numerical implementation of Eq. (23). The authors report that the small-angle θ_i^3 scaling is reproduced, but that the asymmetry saturates as θ_i approaches π. The central conclusion rests on Eq. (20), which is obtained from Eq. (16) via the approximation in Eq. (19). I find that Eq. (19) is mathematically incorrect, so the damped-pendulum equation is not derived and the saturation claim is unsupported.

Significance. If the result were correct, the paper would make a useful step beyond the small-angle approximation in spontaneous baryogenesis: the saturation of the generated asymmetry for θ_i near π is a concrete, falsifiable prediction that differs from the naive θ_i^3 extrapolation. The manuscript is clearly written and the numerical setup is easy to follow; the small-angle check in Fig. 2 is a nice practical consistency test within the assumed framework. However, the main claim depends on an invalid distributional identity and on an unconstrained free parameter Γ. These issues are load-bearing, not cosmetic, and I cannot recommend publication in the present form.

major comments (3)

[Section 4, Eq. (19)] Equation (19) is mathematically false. The bracket B(t') = ω sin(2ωt')/t' − sin²(ωt')/t'² is exactly d/dt'[sin²(ωt')/t']. For a smooth compactly supported test function φ, ⟨B,φ⟩ = −∫ sin²(ωt')/t' φ'(t') dt'. As ω→∞, sin²(wt') averages to 1/2, so the limit is a finite functional independent of ω (for example, with φ(t')=t' e^{-t'^2} the integral tends to a constant), not πω φ(0). Therefore B cannot be replaced by πω δ(t'). Consequently Eq. (20) does not follow from Eq. (16). Since every numerical result in Figs. 1–3 is obtained by solving Eq. (20)/(21), the central saturation claim is not established.
[Section 5, Figs. 2 and 3] The validation in Fig. 2 does not test the derivation of Eq. (20). It shows only that, given a solution θ(t) of the damped-pendulum equation, the numerical evaluation of Eq. (23) reproduces the known θ_i^3 scaling for small angles. This checks the asymmetry integral, not the backreaction equation. Thus the agreement in Fig. 2 cannot compensate for the failure of Eq. (19), and the large-angle saturation in Fig. 3, which is entirely produced by Eq. (20), remains an artifact of the assumed local damping.
[Section 4, Eq. (22); Section 6] The dimensionless damping rate Γ is introduced as a completely free parameter: the paper states that 'there is no established relation between ω and g, Γ can assume any positive value.' All numerical outcomes are therefore scans over an unconstrained one-parameter family. Even if Eq. (20) were correct, this would make the saturation result a property of the chosen damped-pendulum equation rather than a quantitative prediction of the spontaneous-baryogenesis Lagrangian. The paper would need to fix Γ from microphysics, or at least show that the claimed conclusion is insensitive to the physically motivated range of Γ, before it can connect the calculation to the observed baryon asymmetry.

minor comments (4)

[Section 3, Eq. (11)] The numerical probability estimate is incorrect. With σ′ = √60/(2π) ≈ 1.23, P(|θ_i−θ_u|>π) = 1 − erf(π/(√2 σ′)) ≈ 1.1×10^{-2}, not 10^{-5}. The qualitative conclusion that many Hubble patches have large fluctuations still holds, but the quoted number should be corrected.
[Eqs. (8)–(9)] The Gaussian normalization is written as '1/√(2π), σ' in the text; it should be 1/(√(2π) σ).
[Section 4, after Eq. (12)] The statement that θ_i=π with θ˙_i=0 is 'not physically meaningful' is confusing, because Fig. 1 starts from θ_in=3.1 with zero velocity. Please clarify what distinguishes the two cases.
[References] Reference [11] is incomplete (missing volume and page information), and the journal abbreviation 'EPJC9' should be expanded appropriately.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the central quantitative claim is a numerical consequence of a free-parameter equation, benchmarked against an external reference.

full rationale

The derivation chain is: Eq. (13) (from Ref. [6]) is reformulated as Eq. (14), manipulated into Eq. (16), integrated by parts in Eq. (17), approximated via Eq. (19) to produce the damped-pendulum Eq. (20); θ(t) is then inserted into the baryon-density formula Eq. (23) (from Ref. [7]) to obtain Δn_B. The small-angle cubic behavior is checked against Ref. [7] as an external benchmark (Fig. 2, 'This serves to validate our methodology'); it is not a prediction obtained from fitted parameters that are then renamed as output. The large-angle saturation is a numerical outcome of solving Eq. (20) with Γ treated as a free parameter, not a fit to the data points in Fig. 3. The only external inputs, Ref. [6] and Ref. [7], are independent prior works by Dolgov/Freese et al., not self-citations, and no parameter is fitted to the target quantity Δn_B. The paper therefore contains no constructed circularity. A separate, non-circular concern is that Eq. (19) appears unsupported: its left-hand side is exactly d/dt'[sin²(ωt')/t'], whose distributional limit is finite and does not contain the factor πω δ(t'); if so, Eq. (20) does not follow from Eq. (16). That is a derivation/correctness flaw, not a reduction of the prediction to its own inputs, so it should not raise the circularity score under the stated rubric.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper contributes a numerical scan using the standard spontaneous baryogenesis model; Γ and θ_i are adjustable inputs. No new entities are introduced. The derivation of Eq. (20) relies on a flawed delta-function approximation, which is the main burden.

free parameters (2)

Γ (dimensionless decay rate) = scanned values: 0.2, 0.4, 0.6, 0.8, 1, 2, 5, 10, 15
Γ = g²ωf/(Λ²π) is treated as a free parameter because the paper states there is no established relation between ω and g. The saturation behavior depends on Γ, so it is a key input.
θ_i (initial misalignment angle) = scanned from 0 to ~3 (up to π)
The central claim is a function of θ_i; the paper scans initial phases to map the asymmetry and find saturation.

assumptions (4)

domain assumption Semiclassical approximation: θ is classical, fermions Q and L are quantum
Used to derive Eqs. (13)–(14); the paper notes this imposes restrictions, e.g., θ_i = π with zero velocity is excluded as unphysical.
domain assumption Effective potential V(θ) = Λ⁴(1−cosθ) and derivative coupling ∂_μθ Qγ^μ Q
This is the standard spontaneous baryogenesis model from Refs. [6,7]; the paper adopts it without modification.
domain assumption Minkowski spacetime neglects Hubble expansion
The paper solves in a static universe and argues this is valid when Γ >> H, but no expanding-universe results are given.
domain assumption Gaussian distribution for θ_i from Fokker-Planck, with σ′ ≈ 1.23
Section 3 assumes N≈60 e-folds and f≈H⋆, leading to the stated width; this justifies studying θ_i near π as statistically relevant.

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Cite this review

Pith. "Pith review of Spontaneous baryosynthesis with large initial phase." pith.science (2026). https://pith.science/paper/3UBTDVDZ

@misc{pith2026251211011,
  author       = {Pith},
  title        = {Pith review of: Spontaneous baryosynthesis with large initial phase},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3UBTDVDZ}},
  note         = {Machine review of arXiv:2512.11011}
}

abstract

We numerically investigate particle production by a pseudo-Nambu-Goldstone boson (pNGB) in spontaneous baryogenesis, focusing on large initial misalignment angles. Our analysis confirms the established cubic dependence of the baryon asymmetry on the initial phase for small angles. However, this scaling breaks down for larger angles, with particle production saturating as the initial phase approaches ${\pi}$ in Minkowski spacetime.

Figures

Figures reproduced from arXiv: 2512.11011 by the authors.

**Figure 2.** 9 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 2.** Baryon asymmetry in Minkowski space for a small initial phase [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

Figures from the paper (1 more)

**Figure 3.** Figure 3: Baryon asymmetry ∆nB as a function of the initial phase in Minkowski space-time. Particle production increases rapidly until θi ≈ 1, after which the rate decelerates considerably, tending toward saturation as θi approaches π. The curve’s behavior is not strongly influe…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references

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Dolgov, K

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Akrami et al: Planck2018 results: X

Y. Akrami et al: Planck2018 results: X. Constraints on inflation, A&A 641(2020) A10

2020

[2] [2]

A. D. Sakharov: Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe, Pisma Zh. Eksp. Teor. Fiz.5(1967) 32–35

1967

[3] [3]

Kuzmin: CP-noninvariance and baryon asymmetry of the uni- verse, JETP Letters12(1970) 228–230

V.A. Kuzmin: CP-noninvariance and baryon asymmetry of the uni- verse, JETP Letters12(1970) 228–230

1970

[4] [4]

A. G. Cohen, D. B. Kaplan: Thermodynamic Generation of the Baryon Asymmetry, Phys. Lett. B199(1987) 251–258

1987

[5] [5]

A. G. Cohen, D. B. Kaplan: SPONTANEOUS BARYOGENESIS, Nucl. Phys. B308(1988) 913–928

1988

[6] [6]

Dolgov, K

A. Dolgov, K. Freese: Calculation of particle production by Nambu- Goldstone bosons with application to inflation reheating and baryoge- nesis, Phys. Rev. D51(1995) 2693–2702

1995

[7] [7]

Dolgov, K

A. Dolgov, K. Freese, R. Rangarajan, M. Srednicki: Baryogenesis dur- ing reheating in natural inflation and comments on spontaneous baryo- genesis, Phys. Rev. D56(1997) 6155–6165

1997

[8] [8]

Freese, A

K. Freese, A. J. Frieman, A. V. Olinto: Natural inflation with pseudo - Nambu-Goldstone bosons, Phys. Rev. Lett.65(1990) 3233–3236

1990