Electroweak Baryogenesis

1. The Electroweak Phase Transition

The finite-temperature effective potential for the Higgs field receives thermal corrections. At one loop, the potential takes the form

$$V_{\rm eff}(\phi, T) = D(T^2 - T_0^2)\phi^2 - ET\phi^3 + \frac{\lambda_T}{4}\phi^4\,,$$

where $D$, $E$, and $\lambda_T$ are determined by the particle content. The cubic term $-ET\phi^3$ arises from bosonic thermal loops and is responsible for the barrier between the symmetric ($\phi = 0$) and broken ($\phi = v(T)$) phases. A first-order phase transition occurs when the two minima are degenerate at the critical temperature $T_c$:

$$V_{\rm eff}(0, T_c) = V_{\rm eff}(v_c, T_c)\,,\qquad v_c = \frac{2ET_c}{\lambda_{T_c}}\,.$$

2. The Sphaleron Washout Condition

After the transition, the generated baryon asymmetry must not be erased by residual sphaleron processes inside the broken-phase bubbles. The sphaleron rate in the broken phase is

$$\Gamma_{\rm sph}^{\rm broken} \propto e^{-E_{\rm sph}(T)/T}\,,\qquad E_{\rm sph}(T) = \frac{4\pi v(T)}{g}\,B(\lambda/g^2)\,,$$

where $B$ is a slowly varying function of order unity. The washout condition requires the sphaleron rate to drop below the Hubble rate immediately after the transition:

$$\frac{v_c}{T_c} \gtrsim 1\,.$$

This is the key criterion: the VEV at the critical temperature must be comparable to$T_c$ itself, indicating a strongly first-order phase transition.

3. Failure of the Standard Model

In the SM, the cubic coefficient is generated by the $W$ and $Z$bosons:

$$E_{\rm SM} = \frac{1}{4\pi v^3}\left(2m_W^3 + m_Z^3\right) \approx 6.9\times10^{-3}\,.$$

The resulting ratio $v_c/T_c$ depends critically on the Higgs mass. Lattice studies show that for $m_H \gtrsim 72$ GeV the transition becomes a smooth crossover — no phase boundary, no bubble nucleation. With $m_H = 125$ GeV, the SM electroweak transition is definitively a crossover, and EWBG in the SM is excluded.

4. BSM Scenarios for a First-Order Transition

5. CP Violation at Bubble Walls

During a first-order transition, bubbles of broken phase nucleate and expand into the symmetric phase. The bubble wall profile interpolates between the two phases over a thickness$L_w \sim 5/T$. CP-violating interactions of particles with the spatially varying Higgs field create a chiral asymmetry in front of the bubble wall.

The transport equations for the chemical potentials $\mu_i$ of particle species$i$ in the plasma frame take the diffusion form:

$$-D_i\,\mu_i'' + v_w\,\mu_i' + \sum_j \Gamma_{ij}\,\mu_j = S_i^{\rm CP}\,,$$

where $D_i$ is the diffusion coefficient, $v_w$ is the wall velocity,$\Gamma_{ij}$ encode relaxation rates, and $S_i^{\rm CP}$ is the CP-violating source localized at the wall. The left-handed quark chemical potential$\mu_{q_L}$ that diffuses ahead of the wall biases sphalerons:

$$\frac{dn_B}{dt} = -\frac{3\Gamma_{\rm sph}}{T}\sum_i \mu_i\,.$$

6. Gravitational Wave Signatures

A first-order phase transition produces a stochastic gravitational wave background through three mechanisms: bubble collisions, sound waves in the plasma, and magnetohydrodynamic turbulence. The peak frequency today (redshifted from $T \sim 100$ GeV) is

$$f_{\rm peak} \sim 10^{-3}\;\text{Hz}\left(\frac{T_*}{100\;\text{GeV}}\right)\left(\frac{\beta}{H_*}\right)\,,$$

where $\beta/H_*$ characterizes the inverse duration of the transition. The gravitational wave energy density spectrum is parametrized by

$$\Omega_{\rm GW}(f)\,h^2 = \Omega_{\rm sw}\,h^2\,S_{\rm sw}(f) + \Omega_{\rm turb}\,h^2\,S_{\rm turb}(f)\,,$$

where the sound-wave contribution typically dominates:

$$\Omega_{\rm sw}\,h^2 \sim 2.6\times10^{-6}\left(\frac{H_*}{\beta}\right)\left(\frac{\kappa_v\,\alpha}{1+\alpha}\right)^2\left(\frac{100}{g_*}\right)^{1/3}\,.$$

Here $\alpha$ is the ratio of the vacuum energy released to the radiation energy density, and $\kappa_v$ is the fraction of energy going into bulk fluid motion. The peak frequency falls squarely in the LISA sensitivity band ($10^{-4}\text{--}10^{-1}$ Hz), making LISA a powerful probe of electroweak-scale phase transitions.

7. Bubble Nucleation and Wall Dynamics

The nucleation rate of broken-phase bubbles per unit volume per unit time is given by

$$\Gamma_{\rm nuc}(T) = T^4\left(\frac{S_3}{2\pi T}\right)^{3/2}e^{-S_3(T)/T}\,,$$

where $S_3$ is the three-dimensional bounce action. Nucleation becomes efficient when $\Gamma_{\rm nuc}/H^4 \sim 1$, defining the nucleation temperature$T_n < T_c$. The amount of supercooling is parametrized by$\beta = -dS_3/dt|_{T_n}$, with faster transitions (larger $\beta/H_*$) producing weaker gravitational wave signals.

The bubble wall velocity $v_w$ is determined by the balance between the vacuum pressure driving expansion and the friction from particles gaining mass as they cross the wall:

$$\Delta p_{\rm vac} = \epsilon(T_n) \approx \Delta V_{\rm eff}(T_n) + T_n\,\Delta s\,,$$

where $\Delta s$ is the entropy density difference. For strong transitions, the wall can reach relativistic velocities ($v_w \to 1$), entering the detonation regime. For weaker transitions, friction from top quarks and $W$bosons limits $v_w \sim 0.1\text{--}0.6$ (deflagration regime), which is actually preferred for efficient baryon production.

8. Connection to Electric Dipole Moments

The new CP-violating phases required for EWBG generically contribute to electric dipole moments (EDMs) of the electron, neutron, and atoms. The current bounds are:

$$|d_e| < 1.1\times10^{-29}\;\text{e}\cdot\text{cm}\,,\qquad |d_n| < 1.8\times10^{-26}\;\text{e}\cdot\text{cm}\,.$$

Many EWBG scenarios predict EDMs within one or two orders of magnitude of current limits, making next-generation EDM experiments (ACME III, n2EDM) powerful probes of the CP violation needed for baryogenesis.

9. Collider Tests at the LHC and Beyond

A strongly first-order EW phase transition modifies the Higgs self-coupling. The triple Higgs coupling receives corrections:

$$\lambda_3 = \lambda_3^{\rm SM}\left(1 + \delta_3\right)\,,\qquad |\delta_3| \sim 10\text{--}60\%$$

in viable EWBG models. The HL-LHC can probe $|\delta_3| \gtrsim 50\%$ via di-Higgs production ($gg \to hh$), while future lepton colliders (ILC, FCC-ee, CLIC) could reach percent-level precision. Additional signatures include new scalar bosons from extended Higgs sectors, accessible through $pp \to H/A \to t\bar{t}$ or$H^+ \to tb$ channels.