Module 6 · Weeks 10–11 · Research

Electroweak Baryogenesis: Bubble Walls & Sphalerons

Electroweak baryogenesis (EWBG) proposes that the matter asymmetry is generated at the electroweak scale T ~ 100 GeV, using sphalerons for B violation and bubble walls from a first-order phase transition for out-of-equilibrium dynamics. It requires new physics beyond the Standard Model, is testable at the LHC and gravitational-wave observatories, and is one of the most actively pursued scenarios.

1. Finite-Temperature Higgs Effective Potential

The one-loop thermal effective potential for the Higgs field \(\phi\) at finite T is

\[V_{eff}(\phi, T) = D(T^2 - T_0^2)\phi^2 - E\, T\, \phi^3 + \frac{\lambda(T)}{4}\phi^4,\]

where

\(D = \frac{1}{8v^2}(2m_W^2 + m_Z^2 + 2 m_t^2)\) — mass parameter
\(E = \frac{1}{4\pi v^3}(2m_W^3 + m_Z^3)\) — cubic coefficient, crucial for first-order transition
\(T_0\) — temperature where quadratic coefficient vanishes

The cubic term \(-E T\phi^3\) arises only from bosonic loops (the Matsubara zero-mode). It creates a barrier between \(\phi=0\) and \(\phi\neq 0\) minima, enabling a first-order phase transition.

Critical temperature & strength

Two minima are degenerate at the critical temperature

\[T_c = \frac{T_0}{\sqrt{1 - E^2/(\lambda D)}}, \qquad \phi_c(T_c) = \frac{2 E\, T_c}{\lambda(T_c)}.\]

To preserve baryon asymmetry inside bubbles, sphalerons must be out of equilibrium in the broken phase:

\[\frac{\phi_c}{T_c} \gtrsim 1 \quad \text{(strong first-order transition)}.\]

In the SM with \(m_h = 125\) GeV, one finds \(\phi_c/T_c \ll 1\) and actually no first-order transition at all (crossover). BSM physics with extra bosonic degrees of freedom (e.g. light stops, scalar singlet) is needed to get a strong enough transition.

2. The Higgs-Mass Obstruction in the SM

Sequential lattice calculations (Kajantie, Laine, Rummukainen, Shaposhnikov 1996) show that the SM EW transition is first-order only for

\[m_h \lesssim 72~\mathrm{GeV} \quad (\text{strong enough for baryogenesis}),\]

well below the measured \(m_h = 125\) GeV. In the SM the transition is a crossover, producing no bubble walls, no out-of-equilibrium dynamics, and therefore no baryogenesis.

BSM scenarios that can rescue EWBG:

Light stop MSSM: ruled out by LHC
Two-Higgs-doublet (2HDM): extended Higgs sector, new CP phases
Scalar singlet: adds one SM-singlet scalar coupling to Higgs
Higgs portal / composite: new strong dynamics modifying the potential

3. Bubble Walls and CP-Violating Transport

As the universe cools through \(T_c\), bubbles of the broken phase nucleate and expand. Inside the wall the Higgs field profile \(\phi(z)\) interpolates from 0 to v over a wall width \(L_w\). Particles scatter off the CP-violating mass terms at the wall, producing chiral asymmetries in the symmetric phase plasma.

\[\phi(z) = \frac{v_c}{2}\bigl[1 + \tanh(z/L_w)\bigr], \qquad L_w \sim \text{few}/T_c.\]

The complex top Yukawa coupling y_t(z) becomes space-dependent through the wall; its phase introduces CP violation that generates a chiral asymmetry \(\mu_L\) in left-handed quarks just ahead of the wall:

\[\mu_L(z) \sim v_w\, \mathrm{Im}(y_t^* y_t')\,\phi(z)^2\, L_w,\]

where \(v_w\) is the wall velocity. Sphalerons in the symmetric phase ( \(\Gamma_{sph} \sim \alpha_W^5 T^4\)) convert this chiral asymmetry to a net baryon number that is then absorbed into the expanding bubble:

\[\frac{n_B}{s} \sim \frac{\alpha_W^5\, \mu_L\, \tau_{diff}}{T_c}.\]

4. Diagram: Bubble Wall Mechanism

A bubble of true vacuum expands into the symmetric plasma. CP-violating reflection at the wall builds up a chiral asymmetry outside, which active sphalerons convert to baryons. The baryons then diffuse into the broken phase, where sphalerons are exponentially suppressed and the asymmetry is preserved.

5. Collider and Gravitational-Wave Probes

EWBG is one of the most testable baryogenesis scenarios:

Higgs self-coupling \u03BB_hhh

A strong first-order transition typically requires \(\lambda_{hhh}\) to differ from the SM value by \(\sim\)20–50%. HL-LHC will probe this at the 30% level; future colliders (FCC-hh, ILC) to ~5%.

EDMs & CP-violating couplings

Required BSM CP phases generate fermion electric dipole moments at loop level. Current electron EDM bound (ACME III, HfF+) already excludes significant parameter space.

Gravitational waves from bubble collisions

First-order transitions at T~100 GeV produce stochastic GW background peaked at milliHz; in LISA sensitivity range. A GW detection could confirm the bubble-wall mechanism.

Direct BSM searches

Scalar singlets, 2HDM, supersymmetric partners — all probed directly at the LHC. A viable EWBG model must evade current bounds while being detectable at HL-LHC.

Simulation: Higgs Effective Potential, Bubble, and Baryon Transport

The simulation (a) plots \(V_{eff}(\phi, T)\) for several temperatures showing the evolution from symmetric to broken vacuum, (b) computes \(\phi_c/T_c\)vs Higgs mass illustrating why the SM fails, (c) integrates the wall profile, and (d) solves a simplified transport equation for the baryon yield vs wall velocity.

Python

script.py163 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import matplotlib.gridspec as gridspec
from scipy.optimize import brentq

plt.rcParams['font.family'] = 'monospace'
BG='#0a0a1a'; INDIGO='#818cf8'; PURPLE='#c4b5fd'; WARN='#ef4444'; GOLD='#f59e0b'; GREEN='#34d399'

# SM parameters (GeV)
v0 = 246.0
mW = 80.4; mZ = 91.2; mt = 173.0

# Finite-T effective potential  V(phi, T) = D(T^2-T0^2)phi^2 - E T phi^3 + lam phi^4 / 4
def build_coeffs(mh):
    D = (2*mW**2 + mZ**2 + 2*mt**2) / (8*v0**2)
    E = (2*mW**3 + mZ**3) / (4*np.pi*v0**3)
    lam0 = mh**2 / (2*v0**2)
    # Approximate lam(T) by lam0 (neglect log corrections)
    B_const = (3*(2*mW**4 + mZ**4 - 4*mt**4)) / (64*np.pi**2*v0**4)
    T0 = np.sqrt(mh**2 / (4*D))   # approximate
    return D, E, lam0, T0

def V(phi, T, D, E, lam, T0):
    return D*(T**2 - T0**2)*phi**2 - E*T*phi**3 + lam*phi**4/4

def phi_min(T, D, E, lam, T0):
    # Nontrivial minimum solution of 2D(T^2-T0^2) phi - 3 E T phi^2 + lam phi^3 = 0
    # => phi (2D(T^2-T0^2) - 3 E T phi + lam phi^2) = 0
    a = lam; b = -3*E*T; c = 2*D*(T**2 - T0**2)
    disc = b**2 - 4*a*c
    if disc < 0: return 0
    phi1 = (-b + np.sqrt(disc))/(2*a)
    phi2 = (-b - np.sqrt(disc))/(2*a)
    candidates = [p for p in [phi1, phi2] if p > 0]
    return max(candidates) if candidates else 0

fig = plt.figure(figsize=(14, 12), facecolor=BG)
fig.suptitle('Electroweak baryogenesis: potential, wall, and yield',
             color='white', fontsize=16, fontweight='bold', y=0.98)
gs = gridspec.GridSpec(3, 2, figure=fig, hspace=0.45, wspace=0.32)

# ---- Plot 1: V_eff for several T (mh = 70 GeV, BSM-like) ------
mh_BSM = 70
D, E, lam, T0 = build_coeffs(mh_BSM)
Tc_est = T0 / np.sqrt(max(1e-6, 1 - E**2/(lam*D)))
phic = 2*E*Tc_est/lam

ax1 = fig.add_subplot(gs[0, 0], facecolor=BG)
phi_arr = np.linspace(0, 250, 400)
for T, col, lab in [(Tc_est*1.2, WARN, f'T > Tc'),
                     (Tc_est, GOLD, f'Tc ~ {Tc_est:.0f}'),
                     (Tc_est*0.8, GREEN, 'T < Tc'),
                     (0.01,      PURPLE, 'T ~ 0')]:
    Vvals = V(phi_arr, T, D, E, lam, T0)
    Vvals -= V(0.0, T, D, E, lam, T0)
    ax1.plot(phi_arr, Vvals/1e8, lw=2, color=col, label=lab)
ax1.axhline(0, color='white', lw=0.5, alpha=0.5)
ax1.set_xlabel('Higgs field φ [GeV]', color=PURPLE)
ax1.set_ylabel('V_eff / 1e8', color=PURPLE)
ax1.set_title(f'V_eff for m_h = {mh_BSM} GeV (first-order)', color='white', fontsize=11)
ax1.legend(fontsize=9, facecolor='#0f172a', labelcolor='white', edgecolor='#312e81')
ax1.tick_params(colors=PURPLE)
ax1.grid(True, alpha=0.15, color=PURPLE)
for sp in ax1.spines.values(): sp.set_color('#312e81')

# ---- Plot 2: V_eff for mh = 125 GeV (SM, crossover) -----
mh_SM = 125
D2, E2, lam2, T02 = build_coeffs(mh_SM)
Tc_SM = T02 / np.sqrt(max(1e-6, 1 - E2**2/(lam2*D2)))
ax2 = fig.add_subplot(gs[0, 1], facecolor=BG)
for T, col, lab in [(Tc_SM*1.2, WARN, 'T > Tc'),
                     (Tc_SM, GOLD, f'Tc ~ {Tc_SM:.0f}'),
                     (Tc_SM*0.8, GREEN, 'T < Tc'),
                     (0.01, PURPLE, 'T ~ 0')]:
    Vvals = V(phi_arr, T, D2, E2, lam2, T02)
    Vvals -= V(0.0, T, D2, E2, lam2, T02)
    ax2.plot(phi_arr, Vvals/1e8, lw=2, color=col, label=lab)
ax2.axhline(0, color='white', lw=0.5, alpha=0.5)
ax2.set_xlabel('Higgs field φ [GeV]', color=PURPLE)
ax2.set_ylabel('V_eff / 1e8', color=PURPLE)
ax2.set_title(f'V_eff for m_h = {mh_SM} GeV (SM → crossover)', color='white', fontsize=11)
ax2.legend(fontsize=9, facecolor='#0f172a', labelcolor='white', edgecolor='#312e81')
ax2.tick_params(colors=PURPLE)
ax2.grid(True, alpha=0.15, color=PURPLE)
for sp in ax2.spines.values(): sp.set_color('#312e81')

# ---- Plot 3: phi_c/T_c vs m_h ----
mh_vals = np.linspace(30, 200, 80)
ratio = []
for mh in mh_vals:
    Di, Ei, lami, T0i = build_coeffs(mh)
    denom = 1 - Ei**2/(lami*Di)
    if denom <= 0:
        ratio.append(np.nan); continue
    Tci = T0i/np.sqrt(denom)
    phici = 2*Ei*Tci/lami
    ratio.append(phici/Tci)
ax3 = fig.add_subplot(gs[1, 0], facecolor=BG)
ax3.plot(mh_vals, ratio, color=INDIGO, lw=2.5)
ax3.axhline(1, color=GOLD, lw=1.5, ls='--', label='Baryogenesis threshold')
ax3.axvline(125, color=WARN, lw=1.5, ls=':', label='Measured m_h')
ax3.fill_between(mh_vals, 0, 1, color='red', alpha=0.08)
ax3.fill_between(mh_vals, 1, 5, color='green', alpha=0.08)
ax3.set_xlabel('Higgs mass m_h [GeV]', color=PURPLE)
ax3.set_ylabel('φ_c / T_c', color=PURPLE)
ax3.set_title('SM fails: m_h = 125 gives φ_c/T_c << 1', color='white', fontsize=11)
ax3.set_ylim(0, 2.5)
ax3.legend(fontsize=9, facecolor='#0f172a', labelcolor='white', edgecolor='#312e81')
ax3.tick_params(colors=PURPLE)
ax3.grid(True, alpha=0.15, color=PURPLE)
for sp in ax3.spines.values(): sp.set_color('#312e81')

# ---- Plot 4: Wall profile phi(z) ----
Lw_vals = [3, 5, 10, 15]
z = np.linspace(-40, 40, 400)
ax4 = fig.add_subplot(gs[1, 1], facecolor=BG)
for Lw, col in zip(Lw_vals, [GREEN, INDIGO, GOLD, WARN]):
    phi_z = 0.5*v0*(1 + np.tanh(z/Lw))
    ax4.plot(z, phi_z, lw=2, color=col, label=f'L_w = {Lw}/T')
ax4.axvline(0, color='white', lw=0.5, alpha=0.3)
ax4.set_xlabel('z · T  (dimensionless)', color=PURPLE)
ax4.set_ylabel('φ(z) [GeV]', color=PURPLE)
ax4.set_title('Bubble-wall Higgs profile tanh(z/L_w)', color='white', fontsize=11)
ax4.legend(fontsize=9, facecolor='#0f172a', labelcolor='white', edgecolor='#312e81')
ax4.tick_params(colors=PURPLE)
ax4.grid(True, alpha=0.15, color=PURPLE)
for sp in ax4.spines.values(): sp.set_color('#312e81')

# ---- Plot 5: Baryon yield vs wall velocity ----
vw_vals = np.linspace(0.01, 0.99, 200)
# Rough parametric form: Y_B peaks at intermediate v_w
# Y_B ~ (v_w tau_diff) / (v_w^2 + alpha_W^5/T) times CP phase
# Simplified: Y_B ~ v_w exp(-v_w^2/sigma)
YB = 9e-11 * 10 * vw_vals * np.exp(-(vw_vals-0.1)**2/0.02)
ax5 = fig.add_subplot(gs[2, :], facecolor=BG)
ax5.plot(vw_vals, YB, color=INDIGO, lw=2.5)
ax5.axhline(8.7e-11, color=GOLD, lw=1.5, ls='--', label='Observed Y_B')
ax5.fill_between(vw_vals, 7e-11, 1.1e-10, color=GOLD, alpha=0.20)
ax5.set_xlabel('wall velocity v_w [c]', color=PURPLE)
ax5.set_ylabel('Y_B (simplified model)', color=PURPLE)
ax5.set_title('Baryon yield vs wall velocity (schematic): peaks at v_w ≈ 0.1c',
              color='white', fontsize=11)
ax5.legend(fontsize=9, facecolor='#0f172a', labelcolor='white', edgecolor='#312e81')
ax5.tick_params(colors=PURPLE)
ax5.grid(True, alpha=0.15, color=PURPLE)
for sp in ax5.spines.values(): sp.set_color('#312e81')

# --- console output ---
print('=== Potential parameters ===')
print(f'  D = {D:.4f}   E = {E:.4f}   lambda = {lam:.4f}')
print(f'  T0 = {T0:.2f} GeV   Tc = {Tc_est:.2f} GeV   phi_c = {phic:.2f} GeV')
print(f'  phi_c/T_c = {phic/Tc_est:.3f}  for m_h = {mh_BSM} GeV')
print()
print(f'For SM m_h = 125 GeV:')
print(f'  D = {D2:.4f}   E = {E2:.4f}   lam = {lam2:.4f}')
print(f'  T0 = {T02:.2f}  Tc = {Tc_SM:.2f}')
print(f'  phi_c/T_c = {2*E2*Tc_SM/lam2/Tc_SM:.4f}  <<  1  => crossover')

plt.savefig('output.png', dpi=110, bbox_inches='tight', facecolor=BG)
print('\nSaved output.png')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

6. BSM Extensions Enabling EWBG

Three leading BSM frameworks can restore a first-order EW transition with m_h = 125 GeV:

Two-Higgs-Doublet Model (2HDM)

Adding a second Higgs doublet enlarges the scalar potential and can produce extra bosonic loop contributions to the cubic term. Type-II 2HDM (as in MSSM) provides additional CP phases in the Higgs sector, sourcing EWBG. Collider constraints on charged Higgs ( \(m_{H^+} > 580\) GeV in Type-II from flavor) leave an allowed window.

Scalar singlet extension

A real scalar singlet S with a \(\lambda_{hs} |H|^2 S^2\) coupling can trigger a tree-level barrier at the EW transition. Constraints from Higgs signal strength ( \(\lambda_{hs} < 1.5\)) give testable predictions for di-Higgs production.

Light stop (historical)

Pre-LHC, light supersymmetric stops ( \(m_{\tilde t} \lesssim 150\) GeV) were the classic EWBG model. Their 12 bosonic degrees of freedom boosted the cubic term, ensuring first-order transition. LHC has excluded this region.

6a. Connection to Electron EDM

The new CP-violating couplings that EWBG requires inevitably generate an electron EDM at the two-loop Barr-Zee level. For a CP-violating top-Higgs coupling of phase \(\delta_t\):

\[d_e \sim \frac{e\,\alpha_{em}}{(4\pi)^3}\,\frac{m_e\,\sin\delta_t}{v^2}\, \log\frac{m_t^2}{m_h^2} \approx 10^{-27}\,\sin\delta_t\text{ e\u00B7cm}.\]

Current ACME bound \(|d_e| < 4\times 10^{-30}\) e\u00B7cm implies \(|\sin\delta_t| < 10^{-3}\), severely restricting EWBG parameter space. This is why EDM experiments are the strongest laboratory test of electroweak baryogenesis — closing the window, or discovering BSM physics, within the next decade.

6b. Daisy Resummation and Thermal Masses

The naive one-loop thermal effective potential has an IR divergence from the bosonic zero-mode. Resumming the dominant (“daisy”) diagrams leads to thermal mass corrections \(m^2 \to m^2 + \Pi(T)\), where

\[\Pi_{W,L}(T) = \frac{11}{6}g^2 T^2, \quad \Pi_{Z,L}(T) = \frac{11}{6}(g^2+g'^2) T^2/2, \ldots\]

Including these effects softens the effective potential cubic term and partially washes out the first-order transition in the SM. Careful resummation is essential for accurate predictions of \(\phi_c/T_c\) in BSM models.

7. Gravitational Wave Signature of First-Order PT

Bubble collisions, MHD turbulence in the plasma, and sound waves from bubble expansion all produce stochastic gravitational waves. The peak amplitude and frequency depend on:

Strength \(\alpha\): ratio of latent heat to radiation energy
Inverse duration \(\beta/H\): how fast the transition completes
Wall velocity \(v_w\): subsonic, supersonic, or detonation
Reheating temperature, setting the redshifted frequency today

A first-order EW transition at \(T\sim 100\) GeV produces GW peaked near \(f_{\mathrm{peak}} \sim 1\) mHz today, right in the LISA sensitivity band. A detection would be smoking-gun evidence for EWBG.

\[\Omega_{GW}(f) h^2 \sim 10^{-10}\, \alpha^2\, (\beta/H)^{-2}\, v_w^3\, \text{[spectrum shape]}.\]

7. Closed-Time-Path Formalism & Transport Equations

A rigorous treatment of baryon number generation at a moving bubble wall requires the closed-time-path (CTP) / Kadanoff-Baym formalism. The key result is a set of coupled transport equations for particle chemical potentials \(\mu_i(z)\) in the plasma rest frame, sourced by the wall's CP-violating interactions:

\[v_w \frac{d\mu_i}{dz} - D_i \frac{d^2\mu_i}{dz^2} = S_i^{CP}(z) - \sum_j \Gamma_{ij}\,\mu_j.\]

Solving these equations yields the baryon asymmetry. The sphaleron conversion rate \(\Gamma_{sph}\) enters as an overall coefficient. See Riotto (1998) and Carena-Quiros (1996) for pedagogical derivations.

References

Kuzmin, V. A., Rubakov, V. A. & Shaposhnikov, M. E., Phys. Lett. B 155, 36 (1985).
Cohen, A. G., Kaplan, D. B. & Nelson, A. E., “Progress in electroweak baryogenesis”, Ann. Rev. Nucl. Part. Sci. 43, 27 (1993).
Kajantie, K., Laine, M., Rummukainen, K. & Shaposhnikov, M., “Is there a hot electroweak phase transition at m_H \u2273 m_W?”, Phys. Rev. Lett. 77, 2887 (1996).
Cline, J. M., “Electroweak phase transition and baryogenesis”, hep-ph/0609145 (2006).
Morrissey, D. E. & Ramsey-Musolf, M. J., “Electroweak baryogenesis”, New J. Phys. 14, 125003 (2012).
Caprini, C. et al., “Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions”, JCAP 04 (2016) 001.

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