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Heavy particle production from colliding bubbles is governed by on-shell scatterings of wall quanta, not off-shell decays of the background.

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2026-08-01 23:40 UTC pith:E7IDMUO3

load-bearing objection The off-shell bubble-collision estimate is genuinely unphysical and overestimates hard production; the partonic replacement is a plausible ansatz that still needs a controlled derivation or lattice test before it replaces the old numbers. the 2 major comments →

arxiv 2607.15279 v1 pith:E7IDMUO3 submitted 2026-07-16 hep-ph astro-ph.COhep-th

Particle production from bubble collisions

Anish Ghoshal , Pratyay Pal , Alessandro Strumia This is my paper
classification hep-ph astro-ph.COhep-th
keywords bubble collisionsfirst-order phase transitionparticle productionpartonic approximationfree passagedark matterleptogenesisgravitational waves
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper argues that the standard way of calculating heavy particle production from cosmological bubble collisions, treating the colliding scalar field as off-shell quanta that decay, overestimates the yield and depends on unphysical choices of gauge and field coordinates. The correct physical picture in the ultra-relativistic limit is that the two Lorentz-contracted walls pass almost freely through each other, and heavy particles are produced only by ordinary on-shell scatterings among the quanta that make up the walls. The paper packages this as a partonic description, analogous to high-energy hadron collisions, and derives new rates for scalar, fermion, and vector production. These rates are parametrically smaller in the hard regime, changing forecasts for dark matter, leptogenesis, and gravitational waves.

Core claim

The central claim is that the previous 'off-shell' formalism, based on Fourier-transforming the colliding wall profile and convoluting it with the imaginary part of the scalar propagator, is not a valid approximation for bubble collisions. It inherits all the unphysical gauge and field-reparameterization dependence of off-shell quantities, and its assumption that the walls reflect elastically and push their quanta far off-shell is wrong at large boost. In the ultra-relativistic limit the walls undergo nearly free passage, with corrections of order 1/gamma^2, so hard production can only come from rare microscopic 2-to-N scatterings among quasi-real wall quanta. The resulting rates have the st

What carries the argument

The central object is the wall parton luminosity dL_ss/dshat, built from the Fourier transform of the wall rest-frame profile s0(k). For a tanh wall it behaves roughly constant up to sqrt(shat) ~ gamma m_s and falls exponentially beyond, replacing the 1/shat^2 tail assumed in the off-shell approach. Convolving this luminosity with gauge-invariant partonic cross sections gives the number of produced particles per unit area. The free-passage approximation, justified by the short overlap time gamma-suppressed, is the physical input that makes the decomposition into on-shell quanta valid.

Load-bearing premise

The entire calculation assumes ultra-relativistic bubble walls can be treated as free-streaming, incoherent beams of quasi-real on-shell quanta, with corrections only of order 1/gamma^2; if coherent collective fields during the brief wall overlap drive quanta far off-shell, the hard production rates would be larger than claimed.

What would settle it

A real-time lattice simulation of two ultra-relativistic wall collisions, with resolution finer than the Lorentz-contracted wall thickness and enough dynamic range to separate the first impact from later rollback, could measure the heavy-particle yield as a function of gamma and invariant mass. The partonic prediction is a factorized convolution with 1/shat cross sections and a gamma-independent shape in appropriate variables; a yield that grows unsuppressed with gamma or that violates the factorization would falsify the central claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Heavy-particle yields from bubble collisions are parametrically smaller than the off-shell estimates in the hard regime, because partonic cross sections fall as 1/shat.
  • Dark matter can still be produced in the observed abundance, but only with larger couplings or in regions where the DM mass is not far above the symmetry-breaking scale.
  • Leptogenesis from right-handed neutrinos made in bubble collisions works only for phase transitions at high scale w, with correspondingly high reheating temperature; supercooling cannot rescue low-scale leptogenesis.
  • Gravitational waves sourced by the motion of scattered particles are suppressed by an extra 1/gamma^2 relative to the off-shell claim, making them negligible at large boost.
  • Direct graviton production becomes significant only when collision energies approach the Planck scale and would appear at high frequencies around 10^10 Hz times gamma m_s/T_reh.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The partonic factorization implies a sharp, testable prediction: the hard-particle yield at fixed gamma should scale as the product of the two wall spectral densities integrated against the 2-to-N cross section. Any residual coherent contribution would show up as an excess over that convolution.
  • If collective or saturation effects in the soft, highly occupied tail are important, the logarithmic part of the luminosity could be modified, but this would mostly change the normalization of soft production, not the hard-rate suppression.
  • The same free-passage logic should apply to collisions of other boosted extended objects whose constituents are weakly coupled, suggesting a unified partonic description of soliton collisions.
  • The paper's negative result for leptogenesis hints that any viable low-reheating leptogenesis from bubble collisions would require a mechanism that restores coherent off-shell fields, which would itself be a new high-energy production process.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper argues that previous treatments of heavy-particle production from ultra-relativistic bubble-wall collisions, which model the collision as a classical background generating off-shell scalar quanta that decay via Im Π(ŝ), are parametrically wrong: they overestimate hard production and are not invariant under field redefinitions or gauge choices. The authors propose an alternative 'on-shell partonic' formalism in which each wall is represented as a coherent state of quasi-real scalar quanta and production is computed by convoluting a wall parton luminosity with ordinary on-shell 2→2 cross sections. They compute the parton luminosity for a tanh wall profile and give cross sections for scalar, fermion, vector, and graviton production, then use these rates to revisit dark matter, leptogenesis, and gravitational wave signatures.

Significance. If the on-shell partonic picture is correct, this is an important revision of a sizable literature: it replaces the off-shell decay rates used in Refs. [1–13] with parametrically smaller, gauge-invariant partonic rates and changes conclusions about heavy DM, leptogenesis, and GW signals. The paper contains concrete, computable ingredients: an explicit closed-form parton luminosity for the tanh wall, Eqs. (18)–(27), and analytic cross sections for scalars, fermions, and vectors. The paper is also explicit about its own limitations, notably in Sec. 3.2 where saturation corrections to the soft tail are acknowledged to be significant. However, the central factorization—coherent wall modes as an incoherent parton gas—is an ansatz whose quantitative error is not estimated. The significance of the phenomenological conclusions therefore depends on an unproven step.

major comments (2)
  1. [Sec. 3, Eqs. (18)–(20)] The factorization of the production rate as a convolution of wall parton luminosities with on-shell 2→2 cross sections is asserted rather than derived. The wall is a coherent, highly occupied classical field (occupation number n_k∼1/λ at k∼m_s, Sec. 3.2), not a dilute gas. The free-passage argument is supported by Refs. [25–27], but the step from 'the classical field is a sum of two Lorentz-contracted walls' to 'hard quanta are produced by independent 2→2 scatterings of the wall's Fourier modes' is a new model. This step is the basis of all quantitative predictions in Secs. 4–7. I would ask for: (i) a derivation from a field-theoretic expansion with explicit corrections (e.g., an estimate of coherent/interference effects in the hard regime), or (ii) a controlled numerical test, e.g., a classical lattice simulation with a heavy spectator field that compares the full collision with the par
  2. [Sec. 3.2, Eq. (27) and following paragraph] The paper concedes that 'saturation-type corrections are expected to be significant' for the log-enhanced low-ŝ tail and that the logarithmic enhancement is 'an estimate rather than a precision prediction.' This is a key caveat, because many phenomenological applications integrate over this tail: DM with M only moderately above m_s, and the low-ŝ part of the vector production rate used in Sec. 5.2, are precisely in this regime. The claimed parametric suppression of off-shell rates relative to on-shell rates is therefore not demonstrated for those applications. Please either (a) quantify the saturation corrections (e.g., by a dense-dense or dilute-dense model as in Ref. [31]), or (b) restrict the central claims to the hard regime ŝ≫m_s² and explicitly state which phenomenological conclusions in Secs. 5–7 rely on the unhardened soft tail.
minor comments (4)
  1. [Eq. (5)] The formula for f_PE(ŝ) has an ambiguous parenthesis structure in the logarithm and an unexplained factor ℓ^2; please rewrite it with clear parentheses and define ℓ explicitly in the same equation or just before.
  2. [Eq. (50)] The expression 'ΩDMh² = 0.1 β/H M w (100 TeV)² ...' appears dimensionally inconsistent. Presumably 'M w' should be 'M/w' or a division is missing; please check the full formula and its derivation.
  3. [Sec. 4.4, Fig. 6] The text refers to 'the four Feynman diagrams in Fig. 6' but does not enumerate them. Please label the contact, s-channel, t-channel, and u-channel diagrams in the figure caption or in the text, especially since the t/u-channel discussion is used to explain the constant high-energy limit of Eq. (43).
  4. [Sec. 3.2, Eq. (27)] The acknowledgment that the logarithmic low-ŝ tail is not a precision prediction is important and should be stated earlier in Sec. 3, perhaps directly after Eq. (26), so that readers do not mistake the numerical luminosity curve in Fig. 2 for a rigorous prediction in that region.

Circularity Check

0 steps flagged

No significant circularity: the on-shell rates are explicit convolutions of a wall Fourier profile with partonic cross sections, and the only self-citations are illustrative or descriptive.

full rationale

The central derivation chain is self-contained rather than circular. The production rates in Eqs. (18)-(20), (27), (36), (40) and (44) are constructed by (i) choosing an explicit tanh wall profile with fixed parameters w and m_s, (ii) computing its Fourier transform, Eq. (24), (iii) building the parton luminosity dL/dŝ in Eqs. (18)-(19), and (iv) convolving this luminosity with explicit gauge-invariant 2-to-2 cross sections computed in Section 4. No parameter is fitted to the quantity being predicted, and the ratios of on-shell to off-shell rates in Eqs. (29)-(31) are algebraic comparisons of independently defined formulas, not a redefinition of one quantity as another. The two self-citations of possible relevance are [19] (Salvio-Strumia-Vitti, used as the illustrative field-redefinition example behind Eq. (6)) and [12] (Ghoshal-Pal, cited only to characterize the earlier off-shell leptogenesis computation). Neither is load-bearing for the paper's new results: Eq. (6) is an explicit one-loop calculation whose stated assumptions (a free scalar with a field reparametrization) do not include the paper's conclusions, and [12] is descriptive rather than evidential for the new on-shell formalism. The free-passage premise is cited to independent classical soliton studies [25]-[27]. The paper also flags its own limitation in Sec. 3.2, noting that saturation corrections are significant in the soft tail and that the partonic treatment neglects coherence effects; this is an unproven approximation and a genuine correctness risk, but it is not a circular reduction, since the on-shell rates would simply be wrong if factorization failed rather than being identical to an input by construction. No fitted-input-called-prediction, imported uniqueness, smuggled ansatz, or renaming of a known result was found.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

The central claim rests on the free-passage approximation and the partonic factorization ansatz, both imported from soliton-collision literature or introduced by analogy; no new particles, forces, or conserved quantities are introduced. The calculation is otherwise self-contained.

free parameters (2)
  • Wall Lorentz factor at collision gamma = 10 to gamma_run ~ 3.7e17 GeV (benchmark)
    Controls the maximal kinematic reach and the normalization of the wall parton luminosity; not predicted by the paper, and subject to large theoretical uncertainty from friction. All phenomenological conclusions depend on it.
  • Benchmark phase-transition parameters (alpha, beta/H, c_V) = alpha=1, beta/H=10, c_V=0.1
    Taken from [7] and used for numerical illustrations in the dark-matter and leptogenesis sections. Not fitted to data, but chosen by hand.
axioms (4)
  • standard math The imaginary part of the quantum effective action gives the particle production probability, P = 2 Im Gamma.
    Invoked in Eq. (1) and used to frame the critique of the off-shell approximation.
  • domain assumption Ultra-relativistic bubble walls pass through each other freely, with O(1/gamma^2) corrections.
    Taken from [25-27]; load-bearing for the claim that the perfectly-elastic ansatz overestimates off-shell modes (Sections 2.3 and 3).
  • ad hoc to paper The wall can be treated as a coherent state of quasi-real on-shell partons, and hard production is an incoherent sum of partonic 2-to-2 scatterings.
    Central ansatz of Section 3 (Eqs. 13-20); justified by analogy to the parton model, not derived.
  • ad hoc to paper Saturation and coherence corrections are negligible for hard production.
    Stated in Section 3.2; the paper notes such corrections are 'expected to be significant' for the soft tail, and proceeds by ignoring them.

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read the original abstract

Collisions of ultra-relativistic bubbles during cosmological phase transitions can produce particles much heavier than the transition scale. Previous analyses modelled this process as the off-shell decay of the scalar background. We show that its results parametrically overestimate hard particle production and depend on the gauge choice and the coordinate choice in field space. We propose an alternative formalism, analogous to the partonic description of high-energy collisions. In the ultra-relativistic limit, the colliding bubbles undergo nearly free passage and hard production arises from on-shell scatterings among the quanta constituting the Lorentz-contracted walls. We apply this approach to heavy scalar, fermion, and vector particle production, and study the implications for dark matter, leptogenesis, graviton production and primordial gravitational waves.

Figures

Figures reproduced from arXiv: 2607.15279 by Alessandro Strumia, Anish Ghoshal, Pratyay Pal.

Figure 1
Figure 1. Figure 1: The left panel shows two incoming bubble walls, in their center-of-mass frame. The other panels show a schematic comparison of the two pictures. According to [2–13] the walls bounce back, sending all their quanta far off-shell, allowing abundant heavy particle production (middle panel). In our computation the ultra-relativistic walls undergo nearly free passage to leading order, while hard production arise… view at source ↗
Figure 2
Figure 2. Figure 2: Example of wall partonic luminosity s dˆ Lss/dsˆ (gray) compared to its low-energy (blue) and high-energy (red) asymptotics. The dotted curve is the parton luminosity equivalent to the problematic off-shell approach, arbitrarily assuming Im Π(ˆs) = 2ˆσswˆ 2 , see eq. (29). The Fourier transform of the vacuum jump gives the 1/k tail, starting from k ∼ ms . This is a general feature: the Fourier transform of… view at source ↗
Figure 3
Figure 3. Figure 3: Feynman diagrams for scalar ϕ pair production from collisions of two scalars s. 4.1 Scalar self-production The self-scattering amplitude is A (ss → ss) = 6iλ  1 + 3m 2 s sˆ− m 2 s + 3m 2 s tˆ− m 2 s + 3m 2 s uˆ − m 2 s  . (32) The cross section is ˆσ ≃ 9λ 2 /8πsˆ in the hard regime ˆs ≫ m 2 s , where the s-channel diagram is negligible. This means that wall collisions produce Ns/A ∼ m 4 s/sˆ quanta with … view at source ↗
Figure 4
Figure 4. Figure 4: The functions Is , If , IV that control as in eq. (36), (40), (44) the production rates of scalars, fermions, vectors with mass m from collisions of walls made of scalars with mass ms boosted to a Lorentz factor γ ≫ 1. where R = q 1 − 4m 2 ϕ/sˆ is the ϕ velocity. In the limit ˆs ≫ m 2 ϕ the purely quartic diagram dominates and ˆσ ≃ λ 2 sϕ/32πsˆ. Near threshold, by contrast, the t- and u-channel exchange te… view at source ↗
Figure 5
Figure 5. Figure 5: Feynman diagrams for fermion f pair production from collisions of two scalars s. s s V V s s V V s s s V V V s s V V V [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Gauge-invariant set of Feynman diagrams for vector V production from collisions of two scalars s. possible mass term unrelated to symmetry breaking [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Assuming the phase transition parameters of eq. (51), we show the contour values of the coupling λsϕ of scalar (green, left) and y of fermionic (blue) DM to the wall scalar needed to match the observed DM abundance assuming that wall collisions and no other process contributes to DM production. In the shaded region the DM is lighter than the contribution to its mass from the symmetry breaking. Above the ho… view at source ↗
Figure 8
Figure 8. Figure 8: Parameter space of the two models of dynamical EW symmetry breaking in section 5.2, as function of the dilaton mass ms and of the DM mass. The observed DM abundance is reproduced by thermal freeze-out along the darker green curve. Wall collisions would reproduce the DM abundance along the red dashed curves with the indicated Lorentz factor γ (from 10 up to the maximal γrun). However in these models the DM … view at source ↗
Figure 9
Figure 9. Figure 9: We show contour values of the right-handed neutrino coupling y to the wall scalar needed to match the observed baryon asymmetry. The left panel assumes the phase transition parameters of eq. (51), the right panel assumes the indicated more favourable values. The grey region approximates where M1 is so light to be dominantly produced by freeze-in via scattering of SM particles. The red region indicates wher… view at source ↗

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Forward citations

Cited by 3 Pith papers

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  1. Can the universe be matter-dominated after a supercooled first-order phase transition?

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