Module 7 · Weeks 12–13 · Research

Leptogenesis: From Right-Handed Neutrino Decays to Baryons

Leptogenesis (Fukugita & Yanagida 1986) elegantly connects the observed baryon asymmetry to the smallness of neutrino masses. Heavy right-handed Majorana neutrinos decay out of equilibrium, producing a lepton asymmetry that sphalerons reprocess into baryons. The same Yukawa couplings generate the seesaw neutrino masses measured in oscillation experiments.

1. The Fukugita-Yanagida Idea (1986)

Add three right-handed Majorana neutrinos \(N_{R,1}, N_{R,2}, N_{R,3}\) to the Standard Model. They are SM singlets, so their Majorana mass M_R is unprotected and can be very large ( \(\sim 10^{9}\)– \(10^{14}\) GeV). The relevant Lagrangian:

\[\mathcal L \supset -y_\nu^{\alpha i}\,\bar L_\alpha \tilde H\, N_{R,i} - \tfrac{1}{2} M_i\, \overline{N_R^c}_{,i} N_{R,i} + \mathrm{h.c.}\]

Two consequences:

After EW symmetry breaking ( \(\langle H\rangle = v\)), integrating out N yields the seesaw formula \(m_\nu = -m_D^T M_R^{-1} m_D\) with \(m_D = y_\nu v\). Taking \(y_\nu \sim 1\) and \(M_R \sim 10^{14}\) GeV gives \(m_\nu \sim 0.05\) eV — matching oscillation data.
The Yukawas \(y_\nu\) are complex \(3\times 3\) matrices, containing physical CP-violating phases. N_1 decays \(N_1 \to \ell H\)and \(N_1 \to \bar\ell H^*\) can have unequal partial rates.

All three Sakharov conditions are naturally satisfied: (1) L violation from Majorana mass, (2) CP violation from complex Yukawas, (3) out-of-equilibrium from N_1 decay with \(\Gamma_{N_1} < H\).

2. The Seesaw Mechanism

In the ( \(\nu_L, N_R^c\)) basis the full neutrino mass matrix is

\[\mathcal M_\nu = \begin{pmatrix} 0 & m_D \\ m_D^T & M_R \end{pmatrix}.\]

For \(M_R \gg m_D\), diagonalization gives two widely separated mass scales:

\[m_{\nu,\text{light}} \approx -\frac{m_D^2}{M_R}, \qquad M_{\nu,\text{heavy}} \approx M_R.\]

Thus the small observed neutrino masses and the large scale M_R that powers leptogenesis are two sides of the same mechanism. This is the key aesthetic appeal of leptogenesis.

Numerical consistency

With \(m_\nu \approx 0.05\) eV and electroweak-scale Dirac masses \(m_D \sim 100\) GeV, we get \(M_R \sim (100)^2/0.05\cdot 10^{-9}\)GeV \(\approx 2\times 10^{14}\) GeV. The Davidson-Ibarra lower bound below will fix \(M_1 \gtrsim 10^9\) GeV independently.

3. CP Asymmetry in N_1 Decay

The CP asymmetry from tree\u2013loop interference is (Covi, Roulet, Vissani 1996):

\[\varepsilon_1 = \frac{\Gamma(N_1\to \ell H) - \Gamma(N_1\to \bar\ell H^*)}{\Gamma(N_1\to \ell H) + \Gamma(N_1\to \bar\ell H^*)} = \frac{1}{8\pi}\,\sum_{j\neq 1}\frac{\mathrm{Im}[(y_\nu^\dagger y_\nu)_{1j}^2]}{(y_\nu^\dagger y_\nu)_{11}}\,f\!\left(\frac{M_j^2}{M_1^2}\right),\]

with vertex+self-energy loop function

\[f(x) = \sqrt x\left[\frac{2-x}{1-x} - (1+x)\ln\frac{1+x}{x}\right] \xrightarrow{x\gg 1} -\frac{3}{2\sqrt x}.\]

For hierarchical heavy neutrinos \(M_1 \ll M_{2,3}\), the formula simplifies. The resulting CP asymmetry is controlled by the Yukawa couplings and Majorana phases; it can in principle be of order \(10^{-6}\)– \(10^{-4}\).

4. The Davidson-Ibarra Bound

Combining the CP-asymmetry formula with seesaw constraints, Davidson and Ibarra (2002) derived a model-independent upper bound on \(|\varepsilon_1|\):

\[|\varepsilon_1| \leq \frac{3}{16\pi}\,\frac{M_1\,\Delta m_{atm}^2}{v^2\, m_{\nu,1}},\]

where \(\Delta m_{atm}^2 \approx 2.5\times 10^{-3}\) eV \(^2\) and v = 174 GeV. For thermal leptogenesis to reproduce \(Y_B^{obs} \approx 9\times 10^{-11}\) with maximum efficiency, this bound implies

\[\boxed{M_1 \gtrsim 10^9~\mathrm{GeV}}\]

This lower bound on the lightest right-handed neutrino mass is a sharp prediction: successful thermal leptogenesis requires right-handed neutrinos that are too heavy to ever observe at colliders. Probing this mechanism therefore requires indirect tests: neutrinoless double-beta decay, neutrino CP phase \(\delta_{CP}\), and absolute neutrino mass measurements.

5. Diagram: Seesaw, N Decay, Sphaleron Conversion

Leptogenesis is a two-stage process: N_1 decays produce a lepton asymmetry, which sphalerons then convert to a baryon asymmetry. All three Sakharov conditions appear naturally.

6. Washout and Efficiency Factor

The decay parameter

\[K = \frac{\Gamma_{N_1}(T=0)}{H(T=M_1)} = \frac{\tilde m_1}{m_*}, \qquad m_* \approx 1.1\times 10^{-3}~\mathrm{eV},\]

where \(\tilde m_1 = (m_D^\dagger m_D)_{11}/M_1\). The efficiency factor \(\kappa(K)\), parametrized fits (Buchmuller-Di Bari-Plumacher):

Weak washout ( \(K \ll 1\)): \(\kappa \sim 0.1 K^2\) (N_1 never thermalizes, inefficient)
Optimal ( \(K \sim 1\)): \(\kappa \sim 0.01\)
Strong washout ( \(K \gg 1\)): \(\kappa \sim 1/K\) (efficient thermalization, moderate washout)

Finally, using Harvey-Turner,

\[Y_B \approx \frac{28}{79}\,\kappa(K)\,\frac{\varepsilon_1}{g_*}.\]

Matching the observed \(Y_B\) requires \(\varepsilon_1 \sim 10^{-6}\)at moderate K — entirely natural for seesaw models with Yukawas of order \(y \sim 10^{-2}\).

7. Resonant Leptogenesis

The Davidson-Ibarra bound \(M_1 \gtrsim 10^9\) GeV is derived under hierarchical \(M_1 \ll M_2\). When \(M_2 - M_1 \sim \Gamma_1\), the self-energy loop becomes resonant and \(\varepsilon_1\) can be O(1).

Resonant leptogenesis (Pilaftsis & Underwood 2005) can then operate at \(M_1\)as low as a few TeV, potentially accessible at future colliders. This requires finely tuned near-degenerate masses, but naturally arises in some symmetry-based extensions of the neutrino sector.

Simulation: Leptogenesis Boltzmann System

The simulation numerically integrates the coupled Boltzmann equations for \(N_{N_1}(z)\) and \(N_{B-L}(z)\), extracts the efficiency \(\kappa(K)\), and then combines with the Davidson-Ibarra bound to constrain the viable \((M_1, m_\nu)\) parameter space.

Python

script.py154 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import matplotlib.gridspec as gridspec
from scipy.integrate import odeint
from scipy.special import kv

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

def Neq(z):
    return 0.75 * z**2 * kv(2, z) / 2

def D_W(z, K):
    B = kv(1, z)/kv(2, z)
    D = K * z * B
    W = 0.25 * K * z**3 * kv(1, z)
    return D, W

def solve(K, eps):
    z = np.linspace(0.1, 30, 3000)
    def rhs(y, z):
        NN, NBL = y
        D, W = D_W(z, K)
        return [-D*(NN - Neq(z)), eps*D*(NN - Neq(z)) - W*NBL]
    y0 = [Neq(z[0]), 0]
    sol = odeint(rhs, y0, z)
    return z, sol

fig = plt.figure(figsize=(14, 12), facecolor=BG)
fig.suptitle('Leptogenesis: Boltzmann evolution, efficiency, Davidson-Ibarra bound',
             color='white', fontsize=16, fontweight='bold', y=0.98)
gs = gridspec.GridSpec(3, 2, figure=fig, hspace=0.45, wspace=0.32)

# ---- Plot 1: N_N1 evolution for several K ----
ax1 = fig.add_subplot(gs[0, 0], facecolor=BG)
for K, col in zip([0.01, 0.1, 1, 10, 100], [GREEN, INDIGO, PURPLE, GOLD, WARN]):
    z, sol = solve(K, 1e-6)
    ax1.plot(z, sol[:,0], lw=2, color=col, label=f'K={K}')
z = np.linspace(0.1, 30, 500)
ax1.plot(z, [Neq(zz) for zz in z], color='white', lw=1.3, ls='--', label='N_eq')
ax1.set_xlabel('z = M₁/T', color=PURPLE); ax1.set_ylabel('N_N₁', color=PURPLE)
ax1.set_yscale('log'); ax1.set_ylim(1e-6, 2)
ax1.set_title('N_1 abundance', color='white', fontsize=11)
ax1.legend(fontsize=8, 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: B-L asymmetry ----
ax2 = fig.add_subplot(gs[0, 1], facecolor=BG)
for K, col in zip([0.01, 0.1, 1, 10, 100], [GREEN, INDIGO, PURPLE, GOLD, WARN]):
    z, sol = solve(K, 1e-6)
    ax2.plot(z, np.abs(sol[:,1]), lw=2, color=col, label=f'K={K}')
ax2.set_xlabel('z = M₁/T', color=PURPLE); ax2.set_ylabel('|N_(B-L)|  (ε=10⁻⁶)', color=PURPLE)
ax2.set_yscale('log')
ax2.set_title('B-L asymmetry vs K (washout)', color='white', fontsize=11)
ax2.legend(fontsize=8, 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: Efficiency kappa(K) ----
ax3 = fig.add_subplot(gs[1, 0], facecolor=BG)
K_vals = np.logspace(-2, 3, 80)
kappa = []
for K in K_vals:
    _, sol = solve(K, 1.0)  # eps=1 to extract kappa directly
    kappa.append(abs(sol[-1,1]))
kappa = np.array(kappa)
ax3.loglog(K_vals, kappa, color=INDIGO, lw=2.5, label='κ_num')
# analytic approximations
ax3.loglog(K_vals, 0.3*K_vals**2, color=GREEN, lw=1.2, ls='--', alpha=0.7, label='κ ~ K² (weak)')
ax3.loglog(K_vals, 0.5/K_vals,    color=WARN,  lw=1.2, ls='--', alpha=0.7, label='κ ~ 1/K (strong)')
ax3.axvline(1, color=GOLD, lw=1, ls=':')
ax3.set_xlabel('K (decay parameter)', color=PURPLE); ax3.set_ylabel('κ(K)', color=PURPLE)
ax3.set_title('Efficiency factor κ(K)', color='white', fontsize=11)
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: Davidson-Ibarra bound in (M1, m_nu) plane ----
ax4 = fig.add_subplot(gs[1, 1], facecolor=BG)
# |eps_1| <= (3/(16 pi)) * M1 * Delta_atm^2 / (v^2 * m_nu)
# Required eps: Y_B_obs = 28/79 * kappa * eps / g_star  =>  eps >= Y_B g_star / (0.35 kappa)
g_star = 106.75
YB_obs = 8.7e-11
v = 174  # GeV
Dm2_atm = 2.5e-3 * 1e-18  # eV^2 -> GeV^2

mnu_vals = np.logspace(-3, -1, 50)   # eV
M1_vals  = np.logspace(7, 15, 60)    # GeV
M1_grid, mnu_grid = np.meshgrid(M1_vals, mnu_vals)

# Max allowed epsilon (Davidson-Ibarra)
eps_max = (3/(16*np.pi)) * (M1_grid*1e9) * Dm2_atm / ((v*1e9)**2 * mnu_grid)   # bit sloppy on units

# Assume kappa ~ 0.01 (near optimum)
kappa_assumed = 0.01
eps_required = YB_obs * g_star / (0.35 * kappa_assumed)
# Region where eps_max > eps_required -> viable
viable = eps_max > eps_required
cm = ax4.contourf(np.log10(M1_vals), np.log10(mnu_vals),
                  np.log10(np.maximum(eps_max/eps_required, 1e-8)),
                  levels=np.linspace(-4, 2, 25), cmap='viridis')
ax4.contour(np.log10(M1_vals), np.log10(mnu_vals),
            eps_max/eps_required, [1.0], colors='red', linewidths=2.5)
ax4.axvline(np.log10(1e9), color=GOLD, lw=1.5, ls='--', label='M₁ = 10⁹ GeV')
cbar = fig.colorbar(cm, ax=ax4)
cbar.set_label('log10 ε_max/ε_required', color=PURPLE)
ax4.set_xlabel('log₁₀ M₁ [GeV]', color=PURPLE)
ax4.set_ylabel('log₁₀ m_ν [eV]', color=PURPLE)
ax4.set_title('Davidson-Ibarra viable region (red contour = boundary)', color='white', fontsize=11)
ax4.legend(fontsize=9, facecolor='#0f172a', labelcolor='white', edgecolor='#312e81')
ax4.tick_params(colors=PURPLE)
for sp in ax4.spines.values(): sp.set_color('#312e81')

# ---- Plot 5: Final yield vs K for different eps ----
ax5 = fig.add_subplot(gs[2, :], facecolor=BG)
for eps, col, lab in [(1e-4, GREEN, 'ε=10⁻⁴'),
                      (1e-6, INDIGO, 'ε=10⁻⁶'),
                      (1e-8, PURPLE, 'ε=10⁻⁸')]:
    YB_arr = []
    for K in K_vals:
        _, sol = solve(K, eps)
        YB_arr.append(abs(sol[-1,1]) * 28/79 / g_star)  # apply HT and g_*
    ax5.loglog(K_vals, YB_arr, lw=2.2, color=col, label=lab)
ax5.axhline(YB_obs, color=GOLD, lw=2, ls='--', label='Y_B obs')
ax5.fill_between(K_vals, 0.9*YB_obs, 1.1*YB_obs, color=GOLD, alpha=0.2)
ax5.set_xlabel('K', color=PURPLE); ax5.set_ylabel('Y_B', color=PURPLE)
ax5.set_title('Y_B(K, ε) vs observation: which parameter combos work?',
              color='white', fontsize=11)
ax5.legend(fontsize=10, 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
print('=== Leptogenesis efficiency kappa(K) ===')
for K in [0.01, 0.1, 1, 10, 100, 1000]:
    _, sol = solve(K, 1.0)
    print(f'  K = {K:<8}  kappa = {abs(sol[-1,1]):.4e}')

print('\n=== Target yield for different parameter choices ===')
for K, eps in [(1, 1e-6), (10, 1e-5), (100, 1e-4)]:
    _, sol = solve(K, eps)
    YB = abs(sol[-1,1]) * 28/79 / g_star
    print(f'  K={K}, eps={eps:.0e}  ->  Y_B = {YB:.2e}  (obs = {YB_obs:.2e})')

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

8. Flavor Effects in Leptogenesis

The standard leptogenesis treatment traces a single “lepton” quantum number. But below temperatures where charged-lepton Yukawa interactions become fast, the three lepton flavors ( \(e, \mu, \tau\)) evolve independently. Flavor effects modify the efficiency factor by up to O(1) and open new parameter space where leptogenesis succeeds with relatively small CP asymmetry.

The transition temperatures:

\(T \gtrsim 10^{12}\) GeV: unflavored regime, one lepton number
\(10^9\) GeV \( \lesssim T \lesssim 10^{12}\) GeV: two-flavored (\u03C4 vs \u03C4\u22A5)
\(T \lesssim 10^9\) GeV: three-flavored (e, \u03BC, \u03C4)

For moderate M_1 ( \(\sim 10^{10}\) GeV), flavored leptogenesis can evade the Davidson-Ibarra bound because different flavor asymmetries can add coherently while washout acts flavor-by-flavor.

7b. Types of Seesaw

The seesaw mechanism comes in three main flavors, classified by the nature of the mediator:

Type-I

Mediator: fermionic singlet N_R (Majorana). Standard leptogenesis framework. Yields \(m_\nu = m_D^2/M_R\).

Type-II

Mediator: scalar triplet \(\Delta\). Yields \(m_\nu = y_\Delta v_\Delta\). Can accommodate leptogenesis via scalar decays.

Type-III

Mediator: fermionic triplet \(\Sigma\). Yields similar seesaw formula. Colored-triplet variant gives color sphaleron washout.

8. Casas-Ibarra Parametrization

A convenient way to parametrize the neutrino Yukawa matrix consistent with measured light-neutrino observables is the Casas-Ibarra form:

\[y_\nu = \frac{\sqrt{M_R}\,\mathcal R\,\sqrt{m_\nu}\,U_{PMNS}^\dagger}{v},\]

where \(\mathcal R\) is an arbitrary complex orthogonal matrix ( \(\mathcal R^T \mathcal R = 1\)) parametrized by three complex angles \(\omega_{12}, \omega_{13}, \omega_{23}\). The Casas-Ibarra form makes explicit that leptogenesis CP asymmetries depend on the parameters of \(\mathcal R\), which are orthogonal to the PMNS matrix. Hence discovering \(\delta_{CP}\) in PMNS does not directly predict \(\varepsilon_1\), unless extra symmetries link the two sectors.

8b. Connection to Low-Energy Observables

The PMNS CP-violating phase \(\delta_{CP}\) (measured by T2K, NOvA, DUNE) is generally distinct from the Majorana phases that enter 0 \(\nu\beta\beta\) and leptogenesis CP asymmetries. But in specific flavor models (e.g. \(\mu\tau\)symmetry, SO(10) unified), these phases can be related.

In the most optimistic scenario:

Discover non-zero 0 \(\nu\beta\beta\): confirm Majorana nature, measure \(m_{\beta\beta}\)
Measure \(\delta_{CP}\) at DUNE: confirm CP violation in leptons
Global fit of light-\u03BD masses + \(\delta_{CP}\) + mixing angles: reconstruct \(y_\nu\) and Majorana phases within the chosen model
Predict \(\varepsilon_1\) and compare to \(Y_B^{obs}\)

This program, if executed, will effectively test leptogenesis indirectly by constraining the model space in which a viable CP asymmetry is achievable.

9. Variants: Dirac Leptogenesis & ARS

Dirac leptogenesis (Dick, Lindner, Ratz, Wright 1999)

If neutrinos are Dirac (no Majorana mass), standard leptogenesis is impossible. Dirac leptogenesis uses an out-of-equilibrium decay producing equal and opposite lepton asymmetries in left- and right-handed neutrinos. Since only the left-handed part interacts with sphalerons, a net baryon asymmetry results. This is disfavored by current neutrino mass bounds but remains viable in carefully-tuned models.

Akhmedov-Rubakov-Smirnov (ARS) mechanism

Uses oscillations of GeV-scale sterile neutrinos instead of decays. ARS can operate at M_1 ~ O(GeV), directly accessible at future intensity-frontier experiments like SHiP and FCC-ee. Combined with the \u03BDMSM model (Shaposhnikov), this provides a candidate theory for baryogenesis, neutrino mass, and dark matter in one framework.

References

Fukugita, M. & Yanagida, T., “Baryogenesis without grand unification”, Phys. Lett. B 174, 45 (1986).
Davidson, S. & Ibarra, A., “A lower bound on the right-handed neutrino mass from leptogenesis”, Phys. Lett. B 535, 25 (2002).
Covi, L., Roulet, E. & Vissani, F., “CP violating decays in leptogenesis scenarios”, Phys. Lett. B 384, 169 (1996).
Buchmuller, W., Di Bari, P. & Plumacher, M., “Leptogenesis for pedestrians”, Annals Phys. 315, 305 (2005).
Pilaftsis, A. & Underwood, T. E. J., “Resonant leptogenesis”, Nucl. Phys. B 692, 303 (2004).
Davidson, S., Nardi, E. & Nir, Y., “Leptogenesis”, Phys. Rept. 466, 105 (2008).
Minkowski, P. (1977); Gell-Mann, Ramond & Slansky (1979); Yanagida, T. (1979); Mohapatra & Senjanovic (1980) — type-I seesaw originators.

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