Electroweak Baryogenesis with BARYONET: a self-contained review of the WKB approach

TL;DR

This work provides a comprehensive, self-contained treatment of electroweak baryogenesis in the semiclassical WKB framework, deriving gauge-invariant kinematics, CP-odd transport sources, and a model-independent collision closure. It unifies three decades of formalisms into a single language and delivers BARYONET, an open-source pipeline that translates wall profiles into the baryon asymmetry η_B with benchmark validations across SM extensions. The paper also revisits diffusion constants, helicity/helicity-flip rates, and sphaleron rates with updated inputs, demonstrating robust cross-model consistency and reproducibility. Together, these elements offer a transparent, reproducible bridge between microscopic CP violation and cosmological BAU predictions, enabling systematic exploration of EWBG in diverse new physics scenarios.

Abstract

We present a comprehensive, self-contained pedagogical computation of the baryon asymmetry of the Universe within electroweak baryogenesis (EWBG), from the derivation of the semiclassical, CP-dependent force to the formulation and solution of the transport equations obtained from the Boltzmann equations$-$all implemented in the open-source code BARYONET. Our analysis follows the semiclassical WKB approach, where spatially varying complex masses across expanding bubble walls feel CP-violating forces that bias plasma transport. Starting from the stationary Boltzmann equation in the wall frame and projecting onto a hierarchy of velocity moments, we derive a compact, fluid-like system of coupled differential equations for chemical potentials and velocity perturbations. After obtaining the solutions, one can define the left-handed baryon chemical potential, which acts as the source term for the weak sphalerons. These processes generate the baryon asymmetry in front of the wall, which is subsequently frozen once it passes through it. We validate the framework against established formalisms and provide benchmarks in representative scenarios, including singlet extensions of the Standard Model, two-Higgs-doublet models, and Higgs$-φ^6$ constructions. The resulting BARYONET implementation delivers an automated, reproducible pipeline for WKB-based baryogenesis studies, connecting formal derivations with phenomenological applications. In parallel, we revisit standard EWBG ingredients$-$diffusion constants, Yukawa/helicity-flip rates, and strong/weak sphaleron rates$-$to clarify conventions, update numerical inputs, and present a pedagogical derivation, ensuring transparent reproducibility.

Figures (10)

• Figure 1: Schematic illustration of EWBG transport across a planar bubble wall. The red curve is the Higgs background $h(z)$, interpolating from the broken phase (left, $\left\langle h \right\rangle \neq 0$) to the symmetric EW phase (right, $\left\langle h \right\rangle =0$). The shaded strip marks the wall of width $\pm L_w$; the particles in the wall frame are moving with velocity $v_w$ to the left. CP–violating semiclassical forces $F_\theta$ (blue arrows) act with opposite sign on particles and antiparticles, biasing left–handed quarks $q_L$ and antiquarks $\bar{q}_L$. This generates a left–baryon chemical potential $\mu_{B_L}(z)$ (black), peaked near the wall, which diffuses into the symmetric phase where weak sphalerons are active (grey + green region) and is converted into baryon number. The resulting baryon density $\eta_B$ (blue dashed) accumulates ahead of the wall and is preserved once the wall passes and sphalerons are quenched in the broken phase. Curves are illustrative and not to scale.
• Figure 2: Weighted source multiplets for the top sector as functions of the dimensionless wall coordinate $zT$. The left panel shows the CP–odd sources $\mathcal{S}^{t}_{\ell}$ for odd $\ell$, while the right panel shows those for even $\ell$. All curves are localised near the interface, and the even sources suffer from velocity suppression in comparison to the odd ones. This reproduces Fig. 1 of Kainulainen:2024qpm.
• Figure 3: From wall profiles to $\eta_B$ with BARYONET.Top row: Higgs profile $h(z)/v_{\rm EW}$, CP phase $\theta(z)$, and weak–sphaleron suppression $f_{\rm sph}(z)$. Gray shade region indicate $|z|<L_w$. Second row: CP sources $S_t^{(1)}(z)$ and $S_t^{(2)}(z)$ obtained from the WKB/gradient expansion and thermally averaged with the fluid weights. Third row: transport solution for $\mu_{t_L}/T$, $\mu_{b_L}/T$, $\mu_{t_R}/T$, and $\mu_h/T$. Fourth row: baryon–left chemical potential $\mu_{B_L}(z)/T$ constructed from the heavy-sector potentials with thermal weights. Last row: cumulative $\eta_B(z)/\eta_B^{\rm exp}$ versus $zT$; the plateau at negative $zT$ is the frozen asymmetry (here $\eta_B/\eta_B^{\rm exp}\simeq 2.4$). Benchmark parameters as in \ref{['eq:BM1']}.
• Figure 4: Model–independent benchmark in the FH04 framework: minimal CP–phase excursion $\Delta\Theta_t$ required to match the observed $\eta_B$ as a function of $\xi$, shown for several wall thicknesses $L_wT_c\in\{2,4,6,8\}$. Curves are generated by solving the transport equations for each $(\xi,L_wT_c)$, constructing $\mu_{B_L}(z)$, and using an overshoot method to find the phase reproducing the BAU's observed value. Benchmark parameters: $v_w=0.01,\ y_t=0.7,\ y_b=0.0$. This reproduce Fig. 1 of Espinosa:2011eu and Fig. 3 of Konstandin:2013caa.
• Figure 5: xSM benchmark,$\eta_B/\eta_B^{\rm obs}$ versus $v_w$. Blue: scheme with spin–$s$ source (CK–$V_s$); Orange: scheme with helicity–$h$ source (CK–$V_h$); Green: CK–$V_s$ with $K_0=1$; Red: CK–$V_h$ with $K_0=1$; Purple: FH06 with the spin–$s$ source. Curves coincide at small $v_w$, while FH06 drops near $c_s$ and the scheme remains smooth; the $s$–vs–$h$ difference is small. This reproduce Fig. 3 of Cline:2020jre.