Baryon asymmetry from electroweak tachyonic preheating

The thing worth knowing: this is the first 3+1D lattice simulation of CP violation during a tachyonic electroweak transition, and it does produce a net Chern-Simons asymmetry that grows with the CP-violating parameter k. The authors' earlier work was 1+1D, and the one prior 3+1D attempt found nothing. So the qualitative claim — a cold quench with an effective CP term generates an asymmetry proportional to k — is a real, new, simulation-backed result. That part I believe. The initial-rise analytic estimate in section 5.4 (eq. 5.20) agreeing with the measured bump is a genuinely nice cross-check, and the lattice machinery in the appendices — the symmetrized F F-tilde, the implicit solver, the Gauss-constraint sampling — is careful work. The paper is also honest about its limits in a way that is unusual and welcome: it says outright that it lacks statistics to pin the linear regime and that the effective operator is a crude placeholder for (E)SM fermion CP violation. The wash-out discussion flags the large chemical potentials from their prior work as a possible enhancement rather than waving it away. Credit for all of that. Where it gets soft is the headline number. The stress-test concern lands: for m_H = m_W the two fits give (0.32 ± 0.15) and (0.79 ± 0.31) × 10⁻⁵ k — a factor 2.5 apart, with χ²/d.o.f. = 2.4 on the through-origin fit — and the final δ_CP ≈ 0.7 × 10⁻⁵ uses eq. 5.7, whose error is not independently verified. With 45 configurations per point and the asymmetry living in integer-shift probabilities, the quoted errors probably understate the discreteness. The authors say as much. So the

The central quantitative claim is the slope of ⟨N_CS⟩/(L m_W)^3 vs k, which sets δ_CP ≈ 0.7×10^-5 (eq. 6.4). The paper's own numbers show this slope is not stably determined. For m_H = m_W, a fit through the origin gives (0.32 ± 0.15)×10^-5 k (eq. 5.6, χ²/d.o.f = 2.4), while a fit through the two lowest-k points gives (0.79 ± 0.31)×10^-5 k (eq. 5.8) — a factor 2.5 apart. For m_H = √2 m_W, the analogous numbers are (0.79 ± 0.36)×10^-5 k (eq. 5.7) and ≃1.4×10^-5 k (eq. 5.9, no error quoted). The final extrapolation (eq. 6.2) uses eq. 5.7, whose ±0.36 is not independently verified. The ensembles are 45 configurations (300 only for two k=0/3 thermal cases), and the asymmetry is a difference in the fraction of trajectories that shift by an integer N_CS (Fig. 2 shows a full ±1 shift for a single trajectory). With 45 samples, the statistical error on a probability difference of order 10^-5 is large, and the quoted errors may not capture the discreteness of the shift events. The authors acknowledge 'We lack sufficient statistics to draw a conclusion' and 'it is unclear where it ends' (the linear regime), so the qualitative claim (asymmetry ∝ k) stands, but the quantitative 'below J' claim does not.

The reader's CONDITIONAL verdict is appropriate and I would not change it. I agree with the general thrust: the qualitative result (a net asymmetry proportional to k) is well supported, and the quantitative extrapolation is an order-of-magnitude estimate. I disagree partially with the reader's identification of the weakest assumption. The reader points to the effective operator κφ†φ Tr[F F̃] as a proxy for SM CP violation (eqs. 1.5-1.6). That is a real modeling choice, but the authors explicitly acknowledge it ('crude representation,' 'approximate nature of the model'), so it is not a new finding. The more actionable soft spot is the statistical robustness of the slope, which the authors also partly acknowledge but which directly determines the headline δ_CP number. Two fitting methods on the same data give slopes differing by a factor of 2.5 (eqs. 5.6 vs 5.8), and the final extrapolation uses a slope whose error is not independently verified. I also note a factual slip in the reader's rationale: the reader says 'the best-estimate slope is taken from the fit with the worse chi-squared,' but eq. 5.7 (the best estimate) has χ²/d.o.f = 0.25, the better fit; eq. 5.6 has the worse χ²/d.o.f = 2.4. Separately, eq. 6.3 quotes κ ≃ 2.2×10^-51 TeV², which is dimensionally impossible (κ has mass dimension -2); it should read ≈2.2×10^-5 TeV^-2. This is almost certainly a typo and does not affect the dimensionless δ_CP = 0.7×10^-5, but it is an error in the final result and should be corrected. None of these points changes the verdict; they sharpen the rationale for CONDITIONAL.

This is, to my knowledge, the first 3+1 dimensional lattice study of CP-violation-induced Chern-Simons production in a tachyonic electroweak transition, extending the authors' earlier 1+1 dimensional work and complementing their particle-number study of the same scenario. The lattice formulation is careful: the implicit equations of motion are iterated to double precision, the Gauss constraints are imposed on the initial Higgs and gauge fields with an explicit construction (Appendix B), and the Chern-Simons and winding numbers are implemented with a symmetrized plaquette. The analytic estimate of the initial rise (eq. 5.20) matching the simulated bump is a genuine and non-trivial check. The paper is also candid about its statistical limitations. The result is a parameter-free-in-form prediction of the sign and rough magnitude of the asymmetry, and it yields a falsifiable statement about the size of δ_CP required for the observed baryon asymmetry.

The paper asks whether the universe's baryon asymmetry could have been made during a cold, non-thermal electroweak transition rather than a hot first-order phase transition. The setup is a quench: the Higgs effective mass-squared flips sign suddenly, its low-momentum modes grow exponentially (tachyonic preheating), and the resulting rolling and rescattering of the gauge fields can change Chern-Simons number, which the electroweak anomaly converts into baryon number. The authors add a CP-violating term of the form $kappaphi^daggerphi,mathrm{Tr}[F^{munu}tilde F_{munu}]$ to the SU(2)-Higgs model and follow the Chern-Simons number in 3+1 dimensional lattice simulations over an ensemble of classical initial conditions. The payoff, if the modelling holds, is a numerical conversion factor between the CP-violation strength and the final asymmetry, which lets one ask how much CP violation is actually needed. Their answer is that the observed asymmetry could be reached with a CP-violating coefficient below the size the Standard Model is usually credited with, which is the reason the scenario is worth pursuing.

What carries the argument. The machinery is a classical lattice simulation of the SU(2)-Higgs model in 3+1 dimensions with an added CP-odd operator $kappaphi^daggerphi,mathrm{Tr}[F^{munu}tilde F_{munu}]$, which is inert whenever $phi^daggerphi$ is constant (it becomes a total derivative) and therefore acts only while the Higgs condensate is changing. The quench is modelled as an instantaneous flip of the sign of $mu^2_{rm eff}$, with initial conditions drawn from a Gaussian ensemble built from the free-field quantum two-point functions of the unstable modes, split into 'Just-the-half' and 'Thermal' variants, and with the Gauss constraints imposed globally by solving for the homogeneous momentum components. Because the CP term makes the equations of motion implicit, each timestep is solved by iteration. The observable is the change in Chern-Simons number, computed from the lattice $mathrm{Tr}[Ftilde F]$ density, and cross-checked against the Higgs winding number, which should track it once the gauge field is near pure-gauge.

Core claim. The central claim is that a tachyonic electroweak quench with an effective CP-violating operator produces a net Chern-Simons number whose ensemble average grows approximately linearly with the CP-violation parameter $k = 16pi^2kappa m_W^2$, at least for small $k$. Fitting the difference between runs at $k=0$ and small non-zero $k$ that started from identical initial configurations, the authors report $langle N_{CS}rangle/(Lm_W)^3 = (0.79 pm 0.31)times 10^{-5},k$ for $m_H = m_W$ and about $1.4times 10^{-5},k$ for $m_H = sqrt{2},m_W$ (eqs. 5.8, 5.9), with the second quoted as the preferred slope. Converting through the anomaly equation $B = 3langle N_{CS}rangle$ and an entropy density fixed by an estimated reheat temperature of about 43-74 GeV, they find that the observed $n_B/n_gamma approx 6.5times 10^{-10}$ requires $kappa approx 1.4times 10^{-7},m_W^{-2}$, i.e. $delta_{CP} approx 0.7times 10^{-5}$ if the mass scale in the operator is set to $m_W$. That number sits below the Standard-Model Jarlskog factor $J approx 3times 10^{-5}$, which is the paper's headline: at electroweak-scale CP violation the cold tachyonic route is not obviously out of reach. The authors also give an analytic estimate of the early-time rise, eq. (5.20), which reproduces the size of the initial bump in the data, and they argue that the post-transition effective temperature and the measured effective sphaleron rate leave no room for wash-out on the timescales simulated.

The premise it rests on. The load-bearing premise is that a single hand-added operator, $kappaphi^daggerphi,mathrm{Tr}[F^{munu}tilde F_{munu}]$ with its mass scale set to $m_W$, adequately represents the CP violation of the Standard-Model fermions; if it does not, the derived value of $delta_{CP}$ says nothing about whether the real Standard Model can do the job.

The observed baryon-to-photon ratio can be reproduced with an effective CP-violation coefficient $delta_{CP} approx 0.7times 10^{-5}$, roughly a factor of four below the Standard-Model Jarlskog invariant, so weak-scale CP violation need not be ruled out for this scenario on magnitude grounds alone.

The asymmetry scales approximately linearly in the CP-violation strength for small $k$, giving a definite conversion factor that any competing model of cold electroweak baryogenesis must reproduce or beat.

The final effective temperature after the quench, estimated at 43-74 GeV for a Higgs mass near 114 GeV, is low enough that equilibrium sphaleron processes should not erase the generated asymmetry.

The generated asymmetry depends on the Higgs-to-W mass ratio, with the $m_H = sqrt{2},m_W$ case giving a larger slope than $m_H = m_W$, so the scenario makes a (weak) statement about Higgs mass dependence.

The early-time rise of the Chern-Simons density can be predicted analytically from the homogeneous-field approximation and matches the simulation, which gives the mechanism a semi-quantitative explanation rather than a purely numerical one.

If the effective operator is a faithful stand-in for integrated-out fermions, the result suggests the usual lore that Standard-Model CP violation is far too weak for electroweak baryogenesis deserves re-examination at the level of operator coefficients rather than Jarlskog factors alone.

The linear-in-$k$ regime is established only for small $k$ and with limited statistics; a natural follow-up is a dedicated scan at $k lesssim 3$ with several hundred configurations to pin down where linearity ends and whether the apparent curvature is physical.

Because the CP-odd term is only active while $phi^daggerphi$ changes, a finite-rate quench (rather than the instantaneous limit used here) would likely reduce the asymmetry, so the quoted $delta_{CP}$ should be read as a lower bound on the CP violation required in any realistic realisation.

Extending the simulation to include the U(1) hypercharge and the top Yukawa would test whether the operator-based estimate survives when the fermionic sector that supposedly generates the operator is treated dynamically.

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