Leptogenesis: Interactive Origin of Matter Explorer

SECTION 1 • Observational Cosmology

The Matter-Antimatter Asymmetry Paradox

Standard Big Bang cosmology asserts that equal quantities of matter ($q$) and antimatter ($\bar{q}$) were created during the hot early universe. If symmetry were preserved exactly, matter and antimatter would have completely annihilated into photons during early expansion, leaving a dynamic "radiation universe" with virtually zero residual matter. Instead, precision measurement of the Cosmic Microwave Background (CMB) by Planck and Big Bang Nucleosynthesis (BBN) yields a small but critical surplus:

$$ \eta_B \equiv \frac{n_B - n_{\bar{B}}}{n_\gamma} = (6.14 \pm 0.25) \times 10^{-10} $$

For every ~10 billion antimatter particles created in the early universe, there were 10,000,000,001 matter particles. That single surplus particle survived annihilation to form all stars, planets, and galaxies.

Explore Cosmic Epochs & Annihilation

Adjust the cosmological timeline to see how matter and antimatter densities evolved during the universe's first second.

Cosmic Era / Temperature: T ~ 10¹³ GeV (GUT Scale)

GUT (10¹³) Leptogenesis EW (100 GeV) Freezeout Present

GUT Scale (T ~ 10¹³ GeV): Universe is symmetric. Matter and antimatter exist in equal densities ($n_q = n_{\bar{q}}$). Right-handed neutrinos ($N_1$) are in thermal equilibrium.

Particle Density relative to Photons vs Temperature

SECTION 2 • Theoretical Foundations

The Sakharov Conditions Diagnostic Matrix

In 1967, Andrei Sakharov proved that any microphysical theory capable of dynamically generating a baryon asymmetry from an initially symmetric universe must satisfy three strict requirements. Below, compare why the Standard Model (SM) fails alone, and how Leptogenesis successfully satisfies each criterion.

Condition 1 ⚖️

Baryon (or Lepton) Number Violation

Must have reactions where total B (or L) is not conserved ($\Delta L \neq 0$). Otherwise, starting from $L=0$ keeps $L=0$ forever.

✓ SM: Sphaleron non-perturbative transitions violate $B+L$.

✓ Leptogenesis: Majorana neutrino decays $N_1 \to l H$ and $N_1 \to \bar{l} \bar{H}$ explicitly violate $L$ by 2 units.

Condition 2 🪞

C and CP Violation

Charge conjugation (C) and Charge-Parity (CP) symmetries must be broken so rates for $N_1 \to l H$ and $N_1 \to \bar{l} \bar{H}$ are asymmetric.

✗ SM Fails: CKM matrix CP phase is far too small ($\eta_B^{\text{SM}} < 10^{-18}$).

✓ Leptogenesis: Complex phases in Yukawa coupling matrix ($Y_\nu$) yield robust CP violation in loop diagrams.

Condition 3 🌡️

Departure from Thermal Equilibrium

If decay processes occur in equilibrium, inverse decays re-erase any generated asymmetry ($\Gamma_1 < H$).

✗ SM Fails: Electroweak crossover for $m_h = 125\text{ GeV}$ is smooth, not 1st-order.

✓ Leptogenesis: Expansion rate $H(T)$ exceeds $N_1$ decay width at $T \sim M_1$, freezing out asymmetry.

SECTION 3 • Particle Mechanics

Type-I Seesaw & CP Asymmetry Calculator

The Type-I Seesaw Mechanism introduces right-handed, SM-singlet Majorana neutrinos $N_i$ with heavy mass $M_i$. This elegantly explains why observed light neutrinos have sub-eV masses: as heavy mass $M_1$ increases, light neutrino mass $m_\nu$ decreases according to $m_\nu \approx \frac{y^2 v^2}{M_1}$.

Seesaw Parameter Controls v = 174 GeV

Heavy Neutrino Mass ($M_1$): 10¹¹ GeV

10⁹ GeV (Davidson-Ibarra limit) 10¹⁵ GeV

Yukawa Coupling ($y_\nu$): 1.0 × 10⁻²

10⁻⁵ (Electron-like) 1.0 (Top-like)

Effective CP Phase ($\sin\delta$): 0.50

0.0 (No CPV) 1.0 (Max CPV)

Light Neutrino Mass ($m_\nu$): 0.030 eV

CP Asymmetry ($\epsilon_1$): 3.2 × 10⁻6

Predicted $\eta_B$: 6.4 × 10⁻¹⁰

Formula: $\epsilon_1 \approx \frac{3}{16\pi} \frac{M_1 m_{\text{top}}}{v^2} \sin\delta$. Notice how $M_1 \ge 10^9 \text{ GeV}$ is required to yield $\eta_B \sim 10^{-10}$ (Davidson-Ibarra bound).

Light Neutrino Mass $m_\nu$ vs Heavy Mass $M_1$ (Seesaw Balancing)

SECTION 4 • Cosmological Dynamics

Sphaleron Conversion: From Leptons to Baryons

Generating a lepton asymmetry ($\Delta L \neq 0$) is only step one. How does excess lepton number turn into protons and neutrons? Non-perturbative Standard Model field configurations known as sphalerons act as thermal bridges at $T > 132 \text{ GeV}$. They conserve $B-L$ while converting $L$ asymmetry into $B$ asymmetry.

SM Sphaleron Redistribution Formula

$$ B = \left( \frac{28}{79} \right) (B - L) = -\frac{28}{51} L $$

Sphalerons enforce $\Delta B = \frac{1}{3} \Delta L$ per generation, locking final baryon number to primordial $B-L$.

Cosmological Rates vs Temperature $T$ (Log-Log Scale)

Key Rate Interactions

1. Expansion Rate $H(T)$:

Hubble parameter $H \propto T^2 / M_{\text{Planck}}$. Governs cosmic expansion speed.

2. Sphaleron Rate $\Gamma_{\text{sph}}(T)$:

$\Gamma_{\text{sph}} \approx 25 \alpha_w^5 T$. Active above Electroweak Phase Transition ($T_{EW} \approx 132 \text{ GeV}$). Sharp cutoff below $T_{EW}$.

3. Decay Freezeout:

When $N_1$ decay rate falls below expansion ($\Gamma_1 < H$), $N_1$ decays out of equilibrium, locking in $\Delta L$.

SECTION 5 • Experimental Probes

Experimental Proof: Neutrinoless Double Beta Decay ($0\nu\beta\beta$)

Because the mass scale of right-handed neutrinos ($10^9\text{ GeV}$) is far higher than the Large Hadron Collider (13 TeV), testing Leptogenesis relies on proving neutrinos are Majorana particles. The definitive experimental test is Neutrinoless Double Beta Decay ($0\nu\beta\beta$): $(A, Z) \to (A, Z+2) + 2e^-$.

$0\nu\beta\beta$ Energy Spectrum Simulator

Select Isotope Core:

Detector Resolution (FWHM): 0.2%

Hypothetical Signal Scale: Moderate Peak

Key Signature: Standard $2\nu\beta\beta$ produces a broad continuous spectrum due to escaping neutrinos. A discovery $0\nu\beta\beta$ signal produces a sharp monochromatic peak exactly at $Q_{\beta\beta}$.

Electron Sum Energy ($E_{1} + E_{2}$) Spectrum

Current & Next-Generation $0\nu\beta\beta$ Experiments

Experiment	Target Isotope	Active Mass	Current $T_{1/2}^{0\nu}$ Limit	Effective Majorana Mass $\langle m_{\beta\beta} \rangle$	Status
LEGEND-200 / 1000	⁷⁶Ge	200 kg → 1000 kg	> 1.8 × 10²⁶ yrs	< 36 - 156 meV	Running / Construction
nEXO	¹³⁶Xe	5000 kg	> 1.35 × 10²⁷ yrs (Proj)	< 5 - 15 meV	Planned
CUORE / CUPID	¹³⁰Te / ¹⁰⁰Mo	206 kg	> 2.2 × 10²⁵ yrs	< 90 - 300 meV	Running
KamLAND-Zen 800	¹³⁶Xe	745 kg	> 2.3 × 10²⁶ yrs	< 36 - 156 meV	Running

Mathematical Summary & Formula Reference

Core analytical relations underlying Thermal Leptogenesis

1. CP Decay Asymmetry Parameter ($\epsilon_1$)

$$ \epsilon_1 = \frac{\Gamma(N_1 \to l H) - \Gamma(N_1 \to \bar{l} \bar{H})}{\Gamma(N_1 \to l H) + \Gamma(N_1 \to \bar{l} \bar{H})} \approx -\frac{3}{16\pi} \sum_{j \neq 1} \frac{M_1}{M_j} \frac{\text{Im}[(Y_\nu^\dagger Y_\nu)_{1j}^2]}{(Y_\nu^\dagger Y_\nu)_{11}} $$

2. Washout Parameter ($K$)

$$ K \equiv \frac{\Gamma_1}{H(T=M_1)} = \frac{\tilde{m}_1}{m_*}, \quad m_* \approx 1.08 \times 10^{-3} \text{ eV} $$

3. Davidson-Ibarra Bound

$$ | \epsilon_1 | \le \frac{3}{16\pi} \frac{M_1 (m_3 - m_1)}{v^2} \implies M_1 \ge 10^9 \text{ GeV} $$

4. Final Baryon Asymmetry ($\eta_B$)

$$ \eta_B = d \cdot C_{\text{sphaleron}} \cdot \kappa(K) \cdot \epsilon_1 \approx 0.96 \times 10^{-2} \kappa(K) \epsilon_1 $$

Explaining the Universe's Missing Antimatter