Thermal Leptogenesis

• Thermal leptogenesis is a framework in early Universe cosmology where CP-violating decays of heavy Majorana neutrinos create a lepton asymmetry, later partially converted to baryon asymmetry.
• It employs finite-temperature quantum field theory and Boltzmann equations to model out-of-equilibrium decays, incorporating detailed flavor resolution, spectator, and washout processes.
• Variations such as resonant, supersymmetric, and triplet leptogenesis extend the model, linking the seesaw mechanism for neutrino masses to observable cosmological relics.

Thermal leptogenesis is a dynamically rich scenario in early Universe cosmology in which the decays of heavy singlet (sterile) neutrinos, thermally produced in the hot plasma, generate a CP-violating lepton asymmetry. This lepton asymmetry is then partially converted to a baryon asymmetry by rapid Standard Model sphaleron transitions. The theoretical development of thermal leptogenesis incorporates quantum field theory at finite temperature, detailed kinetic equations, and the interplay of flavour, spectator, and washout processes. The scenario is deeply intertwined with the seesaw origin of neutrino masses and offers a framework that connects high-scale physics to observable cosmological relics such as the baryon-to-photon ratio.

1. Production and Out-of-Equilibrium Decay of Heavy Neutrinos

In thermal leptogenesis, heavy Majorana neutrinos $N_i$ (labelled such that $N_1$ is the lightest) are postulated as gauge singlets, as motivated by the type-I seesaw mechanism. In the early Universe—at temperatures $T \gtrsim M_1$—these neutrinos are produced in a thermal bath primarily via scattering processes involving Standard Model particles, which are assumed to remain in kinetic equilibrium because of their rapid gauge interactions.

Their number density is expressed in terms of the ratio $Y_{N_1}$ to the entropy density, with the equilibrium density decaying exponentially as the Universe cools $(z = M_1/T)$: $$n^{\rm eq} = \frac{g\,T^3}{2\pi^2}z^2 K_2(z)\,,\quad z \equiv \frac{M_1}{T}$$ As the temperature drops below $M_1$, the equilibrium abundance falls, and $N_1$ decouples from the plasma. The out-of-equilibrium decay channels are

$$N_1 \to \phi\,\ell_\alpha, \qquad N_1 \to \overline{\phi}\,\overline{\ell}_\alpha$$

where $\phi$ is the Higgs doublet and $\ell_\alpha$ is a lepton doublet labeled by flavour $\alpha$.

CP violation enters through the asymmetry

$$\epsilon_{\alpha\alpha} \equiv \frac{\Gamma(N_1 \to \phi\,\ell_\alpha) - \Gamma(N_1 \to \overline{\phi}\,\overline{\ell}_\alpha)}{\Gamma(N_1 \to \phi\,\ell) + \Gamma(N_1 \to \overline{\phi}\,\overline{\ell})}$$

arising from the interference between tree-level and one-loop (vertex and self-energy) diagrams. Summing over flavours gives $\epsilon = \sum_\alpha \epsilon_{\alpha\alpha}$.

CP-violating decays are possible if there are at least two heavy neutrino generations with complex Yukawa couplings; the required out-of-equilibrium condition is naturally realized for $N_1$ decay rates smaller than the Hubble expansion rate at $T\sim M_1$.

2. Finite-Temperature Corrections and Quantum Field Theory Effects

At temperatures $T\gtrsim M_1$, finite-temperature effects are crucial:

• Thermal Masses: The Higgs boson $\phi$ develops a thermal mass $m_\phi(T) \sim 0.4\,T$, while lepton masses are smaller but nonzero. These masses modify decay kinematics; at high $T$, decays $N_1 \to \ell\phi$ can become kinematically forbidden, impacting the available phase space and changing which decay/inverse decay dominates.
• Statistical Factors and Propagator Modifications: Within real-time finite-temperature field theory, the fermion and boson propagators acquire additional terms proportional to their thermal occupation numbers ($n_\ell(p)$, $n_\phi(p)$). When evaluating the absorptive parts of one-loop diagrams relevant for CP violation, statistical (Bose-enhancement for bosons, Pauli-blocking for fermions) effects nearly cancel if decay products are massless, but imperfect thermal mass splitting leads to nontrivial, temperature-dependent CP asymmetry. As $m_\ell(T) + m_\phi(T)\to M_1$, the on-shell contributions and thus the CP asymmetry vanish.
• Motion Relative to the Plasma: Decaying $N_1$ are not at rest; their motion induces directional emission anisotropies that mostly cancel out but could induce small subleading corrections to the net asymmetry.

3. Boltzmann Equations and Washout Processes

The evolution of $N_1$ and the lepton asymmetry are governed by Boltzmann equations (with overdots denoting differentiation with respect to cosmic time): $\dot{Y}_{N_1} = - (y_{N_1} - 1)\gamma_{N_1\to 2}$

$$\dot{Y}_{\Delta L} \simeq \dot{Y}_{N_1}\,\epsilon - W\, Y_{\Delta L}$$

where $y_{N_1} = Y_{N_1}/Y_{N_1}^{\rm eq}$, $\gamma_{N_1\rightarrow 2}$ is the thermally averaged $N_1$ decay rate, and $W$ collects washout rates due to inverse decays ($\ell\phi\rightarrow N_1$) and $2\to2$ scatterings (e.g., involving gauge bosons and top quarks). Finite-temperature corrections alter both decay rates and washout terms, and $W$ includes contributions from processes that can erase the generated asymmetry.

A central result for the generated baryon asymmetry is: $$Y_{\Delta B} \simeq \frac{135\zeta(3)}{4\pi^4g_*}\sum_{\alpha}\epsilon_{\alpha\alpha}\,\eta_\alpha\,C$$ where $g_*$ counts effective relativistic degrees of freedom, $\eta_\alpha$ are efficiency factors incorporating washout, and $C$ encodes spectator effects (see below).

4. Spectator Processes and Chemical Equilibrium

Many Standard Model interactions, notably strong and electroweak gauge scatterings and various Yukawa interactions, are much faster than the cosmic expansion rate. These "spectator processes" do not change $B-L$ but redistribute chemical potential (density) asymmetries among particle species. For instance, they relate Higgs asymmetries ($\Delta y_\phi$) and lepton doublet asymmetries ($\Delta y_\ell$) to the conserved charges via coefficients $c_\phi$ and $c_\ell$: $$\Delta y_\ell = -c_\ell \frac{Y_{\Delta_\alpha}}{Y^{\rm eq}},\qquad \Delta y_\phi = -c_\phi \frac{Y_{\Delta_\alpha}}{Y^{\rm eq}}$$ The values of $c_\ell$ and $c_\phi$ depend on which interactions are in equilibrium at a given temperature, with their values tabulated for various $T$ ranges. Accurate modeling of these spectator effects is required, as they can significantly suppress or enhance the final baryon asymmetry.

5. Flavour Effects and the Physical Flavour Basis

Resolution of lepton flavours is essential for quantitative leptogenesis:

• Flavour Decoherence: Charged-lepton Yukawa interactions (for $\tau$ and then $\mu$) equilibrate at $T\lesssim 10^{12}$ GeV and $\lesssim 10^9$ GeV, respectively. Below these temperatures, lepton asymmetries in different flavours decohere, and Boltzmann equations must be written for each flavour: $$Y_{\Delta B} \sim 10^{-3}\sum_\alpha \epsilon_{\alpha\alpha}\,\eta_\alpha$$ The efficiency factors $\eta_\alpha$ differ flavour-by-flavour since washout is flavour-dependent and, depending on the orthogonality of the flavour states, can either dilute or enhance the net asymmetry.
• Enhanced Asymmetry: In cases where washout is weaker for some flavours, the summed asymmetry can be enhanced by as much as the number of resolved flavours, relative to the unflavoured case.
• Physical Basis: Care is required to choose the appropriate physical basis, as washout processes act in the interaction (flavour) eigenstate basis, not necessarily in the basis that diagonalizes the input Yukawa matrices or the density matrix at production.

6. Variations and Generalizations

Variants of the basic thermal leptogenesis framework are outlined:

• Supersymmetric Leptogenesis: In supersymmetric realizations (and for reheat temperatures $T_{RH}$ above $\sim 10^9$ GeV), one must consider additional $O(1)$ corrections but must also account for the gravitino problem, which limits $T_{RH}$.
• Resonant Leptogenesis: When two heavy neutrinos ($N_1, N_2$) are nearly degenerate, self-energy corrections to the decay amplitude generate a potentially large (resonant) enhancement of the CP asymmetry.
• Dirac Leptogenesis: If neutrinos are Dirac rather than Majorana, an asymmetry can be produced between left- and right-handed neutrinos; since right-handed states equilibrate only late, a left-handed lepton asymmetry survives to be transferred to baryons by sphalerons.
• Leptogenesis from Scalar or Fermion Triplets: Decays of $\rm{SU}(2)_L$ triplet states (scalar or fermion) with different gauge charges are also viable, having distinctive equilibrium properties and decay/washout patterns.

7. Quantitative Summary and Dependence on Microphysics

The detailed treatment, combining kinetic equations, finite-temperature corrections, spectator effects, and flavour physics, demonstrates that the final baryon asymmetry from thermal leptogenesis is highly sensitive to:

• The temperature-dependent CP asymmetry in $N_1$ decays as modified by finite-temperature field theory,
• The washout dynamics, notably the strength and flavour dependence of inverse decays and scatterings,
• Redistribution among degrees of freedom by fast spectator processes enforcing chemical equilibrium,
• The resolution and evolution of lepton flavour asymmetries.

Key numerical and analytic results (including the explicit formulae for $Y_{\Delta B}$ and the dependence on thermal masses, statistical distributions, etc.) show that successful leptogenesis is not only possible in the minimal seesaw scenario but is controlled by a complex interplay of microphysical and plasma effects. Variations (supersymmetry, flavour structure, resonance conditions, triplet decays) extend the range of viable models and further connect leptogenesis predictions to experimental observables such as neutrino mass scale, CP violation in the lepton sector, and constraints from collider and cosmological data (0802.2962).