⚛️ phenomenology

Baryon number freeze-out in the Standard Model, precisely

This paper derives a Boltzmann equation to precisely calculate the baryon number freeze-out abundance across the electroweak crossover, yielding updated sphaleron conversion factors for both $B-L$ and flavored lepton asymmetries by incorporating higher-order corrections to the partition function and Higgs expectation value.

Original authors: Cristina Benso, Dietrich Bödeker, Kohei Kamada, Kyohei Mukaida, Laura Sagunski, Philipp Schicho, Kai Schmitz

Published 2026-09-09

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Cristina Benso, Dietrich Bödeker, Kohei Kamada, Kyohei Mukaida, Laura Sagunski, Philipp Schicho, Kai Schmitz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the earliest moments of the universe, a fraction of a second after the Big Bang, the cosmos was a seething, superheated soup of fundamental particles. Among these particles were protons and neutrons, the building blocks of all matter we see today, but they were not yet formed. Instead, the universe was filled with a plasma of quarks and other elementary particles, all moving at incredible speeds. A central mystery in cosmology is why this universe contains matter at all. According to the laws of physics, the Big Bang should have produced equal amounts of matter and antimatter, which would have annihilated each other instantly, leaving behind a universe filled only with light. Yet, we exist. Something must have tipped the scales, creating a tiny surplus of matter over antimatter that survived the annihilation to form stars, planets, and life.

To understand how this surplus arose, scientists look to a specific process that occurred as the universe cooled. As the temperature dropped, the universe underwent a phase transition, much like water turning into ice, but involving the fundamental forces that govern particles. During this transition, a mechanism known as the sphaleron process acted as a cosmic converter. This process could change the number of leptons, a family of particles that includes electrons, into the number of baryons, the family that includes protons and neutrons. Crucially, this converter did not act randomly; it followed strict rules, preserving a specific difference between the total number of baryons and leptons while allowing the individual counts to shift. If the universe started with a slight imbalance in the types of leptons, this process would translate that imbalance into the matter we see today. However, the exact efficiency of this conversion has been a subject of debate for decades, with previous calculations relying on simplified snapshots of the universe's state rather than a continuous, realistic view of the cooling process.

A new study by a team of theoretical physicists has now provided the most precise calculation to date of how this conversion works. The researchers focused on the specific moment when the universe cooled enough for the Higgs field, which gives particles their mass, to become active. They realized that previous estimates had treated this transition as if it happened instantly at a fixed temperature, ignoring the fact that the universe was actually cooling down continuously. By developing a new mathematical framework that tracks the evolution of particle densities as the temperature changes, the team was able to simulate the freeze-out of the baryon number with unprecedented accuracy. They accounted for complex interactions between particles and the changing strength of the forces involved, moving beyond the static approximations of the past.

The team's work involved deriving a detailed equation that describes how the number of baryons changes over time as the universe expands and cools. They incorporated high-level corrections to the behavior of the Higgs field and the rates at which particles interact, ensuring that their model reflected the true thermal history of the early universe. Instead of assuming the conversion happened at a single, unchanging temperature, they followed the process as it unfolded across a range of temperatures. This dynamic approach revealed that the efficiency of the conversion is slightly different from what was previously thought. The researchers found that the process is governed by two key factors: one that converts the overall difference between baryons and leptons, and another that depends on the specific types of leptons present, weighted by how strongly they interact with the Higgs field.

The results of this study provide two specific numbers that define the outcome of this cosmic conversion. For the overall difference between baryons and leptons, the conversion factor is approximately 0.3328. For the specific contribution from the different types of leptons, the factor is about 0.0279. These values are precise, with very small margins of error, and they sit between the values predicted by older, simplified models. The older models suggested values of roughly 0.354 for the symmetric phase and 0.324 for the broken phase, but the new calculation shows that the reality is a nuanced blend of these states, determined by the continuous cooling of the universe. The study confirms that the standard values used for decades are slightly off, misestimating the surviving matter by a few percent.

One of the most significant findings is that even if the universe started with no overall difference between baryons and leptons, a baryon asymmetry could still be generated if there were imbalances in the specific types of leptons. This mechanism, known as leptoﬂavorgenesis, relies on the fact that the different types of leptons interact with the Higgs field with different strengths. The researchers showed that if the universe had a surplus of tau leptons, or a specific imbalance between muon and electron leptons, the conversion process would still produce the observed amount of matter. This opens up new possibilities for understanding the origins of the universe, suggesting that the specific flavor of the initial lepton imbalance is just as important as the total amount.

The study also addressed the reliability of these numbers by testing how they change with different theoretical assumptions. The researchers found that the largest source of uncertainty comes from the choice of energy scale in their calculations, rather than from the experimental data on particle interactions. This indicates that while the current results are the most precise available, further theoretical refinements could reduce the uncertainty even more. The team also noted that if there were strong magnetic fields present during this era, they could have influenced the process, but their primary calculation focused on the standard scenario without these additional fields.

Ultimately, this work provides a solid foundation for understanding how the matter in our universe came to be. By replacing static, idealized models with a dynamic, continuous description of the early universe, the researchers have clarified the rules of the game. They have shown that the conversion of lepton asymmetry into baryon asymmetry is a precise, temperature-dependent process that can be calculated with high accuracy. The findings suggest that the universe's matter content is the result of a delicate interplay between particle flavors and the cooling of the cosmos, governed by laws that are now understood with greater clarity than ever before. This level of precision is essential for testing theories about the very first moments of existence and for determining whether the observed matter in the universe can be fully explained by the known laws of physics.

Technical Summary: Baryon Number Freeze-out in the Standard Model, Precisely

Problem Statement
The origin of the baryon asymmetry of the Universe (BAU) relies on processes that violate baryon number ( $B$ ) in the Standard Model (SM) plasma at high temperatures. In the electroweak sector, non-perturbative sphaleron transitions convert lepton asymmetries into baryon asymmetries while conserving $B-L$ and individual flavored charges $\Delta_\alpha = B/n_G - L_\alpha$ . The quantitative relationship between initial flavored asymmetries and the surviving baryon number is encoded in sphaleron conversion factors, $C_{sph}$ (flavor-blind) and $F_{sph}$ (flavor-dependent).

Standard calculations typically assume instantaneous freeze-out in either the symmetric ( $T > T_c$ ) or broken ( $T < T_{sph}$ ) phases, yielding constant values $C_{sph}^{sym} \approx 0.3544$ and $C_{sph}^{bro} \approx 0.3243$ . However, these equilibrium approximations neglect the continuous evolution of the Higgs expectation value and the sphaleron rate across the electroweak crossover. Furthermore, existing calculations often lack higher-order perturbative corrections necessary for precision cosmology, particularly regarding the flavor dependence induced by charged-lepton Yukawa couplings.

Methodology
The authors perform a precision analysis of baryon number violation across the electroweak crossover using a multi-step approach:

Transport Equation Derivation: Using linear-response theory, the authors derive a Boltzmann transport equation for the baryon number density $n_B$ . This equation describes the evolution of $n_B$ as a function of time (or temperature) in an expanding Universe, driven by the deviation from equilibrium:
$\partial_t n_B + 3H n_B = S_B - \Gamma_B(\phi/T) [n_B - n_B^{eq}(\phi/T)]$
Here, $\Gamma_B$ is the washout coefficient, and $n_B^{eq}$ is the equilibrium baryon number density determined by the conserved charges and the Higgs vacuum expectation value (VEV) $\phi$ .
Perturbative Computation of Equilibrium Quantities: To determine $n_B^{eq}$ and the washout coefficient $\kappa$ (where $\Gamma_B = \kappa \Gamma_{ws}/T^3$ ), the authors employ high-temperature dimensional reduction. They integrate out hard ( $k \sim \pi T$ ) and soft ( $k \sim gT$ ) momentum scales to construct a three-dimensional effective field theory (3d EFT).
- Hard Contributions: Calculated at $O(\mu^2 T^2)$ and relative $O(g^2)$ , including QCD corrections and top/lepton Yukawa effects.
- Soft Contributions: Computed via the effective potential of the 3d EFT. This includes tree-level terms and one-loop corrections to the effective potential, accounting for Debye masses and mixing between temporal gauge fields and the Higgs.
- Order of Accuracy: The calculation includes corrections up to Next-to-Leading Order (NLO) in the strong coupling and Next-to-Next-to-Leading Order (N2LO) in the soft sector for the equilibrium conversion factors. The Higgs VEV $\phi(T)$ is determined perturbatively up to N2LO in the broken phase.
Numerical Solution: The transport equation is solved numerically using updated lattice results for the sphaleron diffusion rate $\Gamma_{ws}$ across the crossover. The authors vary the renormalization scale $\bar{\Lambda}_{ref}$ to estimate theoretical uncertainties from missing higher-order terms.

Key Contributions

First Precision Analysis: This work provides the first precision calculation of baryon number freeze-out in the SM that accounts for the continuous thermal history across the electroweak crossover, rather than relying on step-function approximations between phases.
Higher-Order Corrections: The authors derive the equilibrium conversion factors and washout coefficients including $O(g^2)$ corrections (NLO) and soft one-loop corrections (N2LO). This includes the first fully perturbative determination of the Higgs VEV at N2LO in the broken phase.
Flavor Dependence: The study explicitly calculates the flavor-dependent conversion factor $F_{sph}$ , which arises from charged-lepton Yukawa couplings. This is crucial for scenarios where the total $B-L$ asymmetry vanishes but flavored asymmetries exist (Leptoﬂavorgenesis).
Self-Contained Derivation: The paper offers a complete derivation of the transport equation from first principles using linear-response theory, clarifying the connection between the grand canonical ensemble, chemical potentials, and the sphaleron rate.

Results
The authors determine the sphaleron conversion factors at the asymptotic freeze-out temperature ( $T_{asym} < T_{sph}$ ):

Flavor-blind factor: $C_{sph} = 0.3328 \pm 0.0005$
Flavor-dependent factor: $F_{sph} = 0.0279 \pm 0.0019$

These values differ from the standard equilibrium limits ( $28/79$ and $12/37$ ) by several percent, confirming that the continuous evolution across the crossover significantly impacts the final baryon abundance. The authors identify the baryon-number freeze-out temperature as $T_B \simeq 138.2$ GeV.

Regarding flavor effects:

For non-zero total $B-L$ , flavor redistribution among generations changes the final baryon number only at the level of $O(10^{-6})$ , which is negligible compared to the theoretical uncertainty.
For vanishing total $B-L = 0$ $B - L = 0$ , the baryon asymmetry is generated solely by the flavor-dependent term. The authors quantify the required initial lepton flavor asymmetries to reproduce the observed BAU:
- $\tau$ -philic case (asymmetry in $\tau$ ): $n_{\Delta\tau}/s \simeq 3.1 \times 10^{-5}$ .
- $\tau$ -phobic case (asymmetry in $\mu$ ): $n_{\Delta\mu}/s \simeq 8.9 \times 10^{-3}$ .

Significance and Claims
The paper claims that the previously used equilibrium values for sphaleron conversion are insufficient for precision cosmology, as they misestimate the surviving baryon asymmetry by several percent. The new values $C_{sph}$ and $F_{sph}$ provide the necessary precision for testing baryogenesis and leptogenesis scenarios.

The authors emphasize that the dominant uncertainty in their results stems from the variation of the 4D renormalization scale (theoretical uncertainty), which exceeds the uncertainty propagated from the lattice fit of the sphaleron rate by an order of magnitude. Consequently, they argue that further improvements depend on higher-order calculations of the soft-scale effective potential and the Higgs VEV evolution, rather than on reducing the statistical errors of the sphaleron rate fit.

The paper also notes that while flavor asymmetries in the $\tau$ -phobic scenario could potentially affect Big Bang Nucleosynthesis (BBN) via electron neutrino asymmetries, the required values are currently just below observational constraints, suggesting a potential link between primordial lepton flavor asymmetries and the observed helium abundance. However, the authors do not claim a definitive resolution to the helium abundance problem, noting that the magnitude of such effects remains a subject of debate.

Technical Summary: Baryon Number Freeze-out in the Standard Model, Precisely

More like this