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arxiv: 2606.04229 · v1 · pith:ZIT3IBHAnew · submitted 2026-06-02 · ✦ hep-ph

A Deep Dive into Baryon Asymmetry -- the C2HDM

Margarete M\"uhlleitner , Johann Plotnikov , Rui Santos , Jo\~ao Viana This is my paper

Pith reviewed 2026-06-28 08:52 UTC · model grok-4.3

classification ✦ hep-ph
keywords baryon asymmetryC2HDMelectroweak phase transitiontransport equationsWKB approximationBSMPTCP violationgravitational waves
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0 comments X

The pith

A new BSMPT implementation computes baryon asymmetry using generalized transport equations with arbitrary moments in the C2HDM.

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

The paper introduces an updated version of the BSMPT code that calculates the baryon asymmetry produced during the electroweak phase transition. It generalizes the WKB-based transport equations to handle any number of moments and provides two truncation methods. The vacuum expectation value profile is obtained from the equations of motion instead of a simple kink approximation. The method is validated on a benchmark and then applied to the CP-violating two-Higgs-doublet model to examine how the asymmetry depends on parameters such as wall velocity, phase transition strength, and CP violation. This allows for uncertainty estimates and can be extended to other extended Higgs sectors without the collision term.

Core claim

The central discovery is a new implementation of baryon asymmetry computation in BSMPT based on the WKB ansatz that generalizes the transport equations to an arbitrary number of moments, with two truncation schemes implemented and the VEV profile derived from the equations of motion. Validation on a simple benchmark confirms the approach, and a detailed analysis in the C2HDM reveals dependencies on the number of moments, truncation scheme, wall velocity, VEV profile, phase transition strength, and CP violation, including an uncertainty analysis and connection to gravitational wave signals at LISA. The implementation applies to any extended Higgs sector with arbitrary VEV directions barring t

What carries the argument

The WKB ansatz generalized to an arbitrary number of moments in the transport equations, along with two truncation schemes and a VEV profile solved from the equations of motion.

If this is right

  • Baryon asymmetry calculations become possible for Higgs sectors with more than two doublets or arbitrary VEV directions.
  • The dependence of the asymmetry on the number of moments and truncation scheme can be systematically studied.
  • The generated asymmetry can be correlated with the gravitational wave signal observable at LISA.
  • Uncertainty estimates from the analysis provide a basis for constraining model parameters using cosmological data.

Where Pith is reading between the lines

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

  • Future work could incorporate the collision term to improve accuracy in dense plasma regimes.
  • The method might help identify which extended Higgs models can explain the observed baryon asymmetry while satisfying other constraints.
  • Testing against full numerical solutions of the transport equations would quantify the approximation errors.

Load-bearing premise

The WKB ansatz and the chosen truncation schemes for the transport equations remain valid approximations when the collision term is omitted and the derived VEV profile accurately represents the phase transition dynamics.

What would settle it

A full numerical solution of the Boltzmann transport equations including the collision term that yields a significantly different baryon asymmetry value for the same C2HDM benchmark point would falsify the results.

Figures

Figures reproduced from arXiv: 2606.04229 by Jo\~ao Viana, Johann Plotnikov, Margarete M\"uhlleitner, Rui Santos.

Figure 1
Figure 1. Figure 1: Computed chemical potentials of the particles involved in the fluid network as a [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The normalized BAU ¯η ≡ η/ηobs over the number n of moment equations. Left: the results for the collision network defined in Eq. (4.23) and the rates presented in Sec. 4.2; right: results for the collision network and rates used in Ref. [28]. The different colors represent different truncation scheme choices, where R = 0, −vw correspond to the constant truncation choice for two different truncation values,… view at source ↗
Figure 3
Figure 3. Figure 3: The normalized BAU ¯η as a function of R in the constant truncation scheme for different wall velocities. The lines represent the largest number n of moment equations taken into account, and the vertical dashed blue line marks the truncation choice R = −vw. The blue box marks the base scenario of the benchmark model. If the BAU converges to a constant result, the value of η err n should become zero, η err … view at source ↗
Figure 4
Figure 4. Figure 4: The normalized BAU ¯η over the R-value used in the constant truncation scheme for different values of LwTn. The lines represent the largest number n of moment equations taken into account. The blue box marks the base scenario of the benchmark model. the WKB method is based on the assumption that LwTn ≫ 1. Although for example Ref. [60] suggests that LwTn ≳ 2 is sufficient to satisfy this assumption, this i… view at source ↗
Figure 5
Figure 5. Figure 5: The normalized BAU ¯η over the R-value used in the constant truncation scheme for different values of ξn. The lines represent the largest number n of moment equations taken into account. The blue box marks the base scenario of the benchmark model. ⋄ the truncation scheme, where we will apply the constant truncation scheme, with R = −vw and the variance truncation; ⋄ the number n of moment equations with n … view at source ↗
Figure 6
Figure 6. Figure 6: Contour plot of the error quantity η err 46 defined in Eq. (7.2) in terms of vw and LwTn for different transition strengths ξn. The light yellow regions mark the areas where the error is larger than 10%. Here, we choose R = 1 in the constant truncation scheme. ⋄ wall velocity vw = 0.5; ⋄ field profile. We will study the impact these options by varying them one at a time, while keeping the others equal to t… view at source ↗
Figure 7
Figure 7. Figure 7: Scatter plots of the absolute value of the normalized baryon asymmetry for different [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Scatter plot of the relative difference between [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Scatter plot of the normalized baryon asymmetry for [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Scatter plot of the BAU |η2 | calculated at the critical (blue points) and the percolation (orange points) temperature for the C2HDM type I, for the wall velocities vw = 0.1, vw = 0.5 and vw = 0.9 from left to right. and vw = 0.5, the calculated BAU using Tp is larger than the BAU calculated using Tc but, for vw = 0.9, using Tp yields smaller results. In [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Scatter plot of the BAU |η2 | calculated at the critical (blue points) and the percolation (orange points) temperature for the C2HDM type II, for the wall velocities vw = 0.1, vw = 0.5 and vw = 0.9 from left to right [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Scatter plot of the normalized baryon asymmetry [PITH_FULL_IMAGE:figures/full_fig_p028_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Scatter plot of the normalized baryon asymmetry [PITH_FULL_IMAGE:figures/full_fig_p029_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Scatter plot of the normalized baryon asymmetry [PITH_FULL_IMAGE:figures/full_fig_p029_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: BP1: VEV profiles for the kink (blue) and the field (orange) solution as a function of z for ω1(z) (left), ω2(z) (middle) and ωCP(z) (right). In [PITH_FULL_IMAGE:figures/full_fig_p030_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: BP1: Left panel: absolute value of the top mass |mt | (solid lines) and the top mass phase θ (dashed lines) as a function of z for the kink solution (blue) and the field solution (orange); right panel: normalized baryon asymmetry η as a function of the number n of transport equations for the kink solution (blue) and the field solution (orange). n = 2 + 4k, k ∈ N, both for the kink solution (blue) and the … view at source ↗
Figure 17
Figure 17. Figure 17: Same as Fig. 15 but for [PITH_FULL_IMAGE:figures/full_fig_p032_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Same as Fig. 16, but for [PITH_FULL_IMAGE:figures/full_fig_p033_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Same as Fig. 15 but for [PITH_FULL_IMAGE:figures/full_fig_p033_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Same as Fig. 16, but for [PITH_FULL_IMAGE:figures/full_fig_p033_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: BP4: Left panel: η as a function of the number n of moment equations for the kink solution (blue) and the field solution (orange). right panel: the Euclidian action S3 T as a function of the temperature, where the red dots are the action calculated by BSMPTv3, the blue line is the cublic spline used to interpolate the action, the horizontal dashed red line is S3 T = 140 (which can be used to estimate the … view at source ↗
Figure 22
Figure 22. Figure 22: Feynman diagram of the process that leads to the top Yukawa rate. [PITH_FULL_IMAGE:figures/full_fig_p037_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Imaginary part of tU divided by m2 f at T = 1 over k0. On the left our result and on the right the result obtained in Ref. [73] is displayed. In both plots, the different lines shown from left to right denote the respective result for a fixed value of k = 0.5, 1, 1.5. The real parts of the above expressions agree with the results obtained in Ref. [73], in which the imaginary time formalism was used. Howev… view at source ↗
read the original abstract

In this paper, we present our new implementation of the computation of the baryon asymmetry in the code BSMPT. It is based on the WKB ansatz generalizing the transport equations to an arbitrary number of moments. Two different truncation schemes are implemented, and the profile of the vacuum expectation value (VEV) is derived from the equations of motion in addition to the modeling with the kink profile. We validate our implementation with a simple benchmark model and perform a detailed analysis within the CP-violating 2-Higgs-Doublet Model (C2HDM). Barring the collision term, however, our implementation can readily be applied to any extended Higgs sector with an arbitrary number of VEV directions. We study in detail the dependencies of the baryon asymmetry on the number of moment equations, the applied truncation scheme, the wall velocity, the wall velocity times wall width, the VEV profile, the strength of the phase transition, and the amount of CP violation in the model and present a detailed uncertainty analysis. We investigate the interplay of the generated baryon asymmetry and the gravitational waves signal at LISA. Our results guide the way for future improvements in the computation of the baryon asymmetry and give directions for model building. The uncertainty analysis is the basis for any investigation aiming at deducing model parameters from cosmological processes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript presents a new implementation in the BSMPT code for calculating the baryon asymmetry using a WKB-based generalization of the transport equations to an arbitrary number of moments, with two truncation schemes and VEV profiles obtained from the equations of motion (in addition to the kink ansatz). It validates the code on a benchmark model, performs a detailed parameter study in the CP-violating 2HDM including dependencies on moment count, truncation, wall velocity, phase transition strength, and CP violation, provides an uncertainty analysis, and examines the interplay with gravitational wave signals at LISA. The implementation is stated to apply to arbitrary extended Higgs sectors barring the collision term.

Significance. If the underlying approximations remain controlled, the work supplies a flexible, extensible tool for baryon asymmetry computations in multi-scalar models together with systematic uncertainty quantification. The arbitrary-moment capability and EOM-derived VEV profiles, combined with the C2HDM exploration, could usefully guide model-building and future cosmological interpretations.

major comments (2)
  1. [Abstract] Abstract: the central claim that the implementation 'can readily be applied to any extended Higgs sector with an arbitrary number of VEV directions' (barring the collision term) rests on the assumption that the collisionless WKB moment hierarchy plus chosen truncations remains a controlled approximation to the full Boltzmann system; no quantitative estimate of truncation error or comparison to solutions that retain the collision term is supplied for the C2HDM benchmark.
  2. [Validation and C2HDM analysis sections] Validation and C2HDM analysis sections: because the baryon asymmetry is sourced at the wall and propagated through the moment hierarchy, the omission of the collision term directly scales the final result; the paper should demonstrate that the reported dependencies and uncertainty bands are robust under this approximation, for example by showing convergence with increasing moment number against a reference calculation that includes damping.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. Below we respond point-by-point to the two major comments. Our responses clarify the scope of the work while acknowledging where additional discussion can be added.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the implementation 'can readily be applied to any extended Higgs sector with an arbitrary number of VEV directions' (barring the collision term) rests on the assumption that the collisionless WKB moment hierarchy plus chosen truncations remains a controlled approximation to the full Boltzmann system; no quantitative estimate of truncation error or comparison to solutions that retain the collision term is supplied for the C2HDM benchmark.

    Authors: The abstract already qualifies the claim with the explicit caveat 'barring the collision term'. Within the collisionless WKB framework the truncation error is quantified by the explicit convergence studies versus moment number and the two truncation schemes presented in the C2HDM analysis. A direct numerical comparison to a calculation that retains the collision term lies outside the present implementation, which is deliberately restricted to the collisionless case. We will revise the abstract to make this limitation even more prominent. revision: partial

  2. Referee: [Validation and C2HDM analysis sections] Validation and C2HDM analysis sections: because the baryon asymmetry is sourced at the wall and propagated through the moment hierarchy, the omission of the collision term directly scales the final result; the paper should demonstrate that the reported dependencies and uncertainty bands are robust under this approximation, for example by showing convergence with increasing moment number against a reference calculation that includes damping.

    Authors: The reported dependencies and uncertainty bands are obtained and shown to converge inside the collisionless approximation; the moment-number scans already provide a quantitative handle on truncation error within that framework. Adding damping (collision) terms would require a substantial extension of the transport-equation solver that is beyond the scope of this work. We will insert a concise paragraph in the validation section that explicitly states the limitation, references the standard use of the collisionless approximation in the literature, and notes that the present results are to be understood within that controlled setting. revision: partial

Circularity Check

0 steps flagged

No circularity detected in WKB transport implementation

full rationale

The paper's derivation chain consists of a numerical implementation extending the standard WKB ansatz to arbitrary moment count and VEV directions, with two explicit truncation schemes and VEV solved from the equations of motion (or kink profile). Validation occurs on an external benchmark model, followed by parameter scans in the C2HDM. No equation or step reduces a claimed prediction to a fitted input by construction, nor does any load-bearing premise collapse to a self-citation whose content is itself unverified. Self-citations for the BSMPT framework are infrastructural and do not render the asymmetry computation tautological. The reported dependencies and uncertainty analysis are direct numerical outputs, not definitional identities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review yields limited visibility into parameters and assumptions; the listed items are inferred directly from the abstract text.

axioms (2)
  • domain assumption WKB ansatz remains a valid approximation for the transport equations in the C2HDM
    Invoked to generalize the equations to arbitrary moments
  • domain assumption Two truncation schemes suffice to close the moment hierarchy
    Implemented without further justification in the abstract

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discussion (0)

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Reference graph

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