The Matter-Antimatter Asymmetry

1. The Sakharov Conditions

Any dynamical mechanism for baryogenesis must satisfy three conditions simultaneously:

2. Sphalerons and B+L Violation

The electroweak vacuum has a periodic structure labeled by the Chern-Simons number$N_{\rm CS}$. Transitions between adjacent vacua change both baryon and lepton number:

$$\Delta B = \Delta L = 3\,\Delta N_{\rm CS}\,,\qquad \Delta(B-L) = 0\,.$$

At zero temperature, these transitions proceed via quantum tunnelling with an exponentially suppressed rate:

$$\Gamma_{\rm tunnel} \propto e^{-4\pi/\alpha_W} \sim e^{-164} \approx 0\,.$$

At high temperatures $T \gtrsim 100$ GeV, the transitions proceed classically over the energy barrier (the sphaleron configuration) with a rate

$$\Gamma_{\rm sph} \sim \alpha_W^5\,T^4\,e^{-E_{\rm sph}/T}\,,\qquad E_{\rm sph} = \frac{8\pi v}{g} \approx 9\;\text{TeV}\,.$$

Above the electroweak phase transition ($T > T_{\rm EW} \approx 160$ GeV), the sphaleron rate is unsuppressed:

$$\Gamma_{\rm sph} \sim \alpha_W^5\,T^4 \gg H\,T^3\,,$$

so $B+L$ violating processes are in thermal equilibrium, while $B-L$is exactly conserved by sphalerons.

3. Thermal Leptogenesis

The seesaw mechanism introduces heavy right-handed Majorana neutrinos $N_i$ with masses $M_i \gg v_{\rm EW}$. The light neutrino mass matrix is

$$m_\nu = -m_D\,M_R^{-1}\,m_D^T\,,$$

where $m_D = Y_\nu v/\sqrt{2}$ is the Dirac mass matrix. The out-of-equilibrium decay of the lightest heavy neutrino $N_1$ can generate a lepton asymmetry through interference between tree-level and one-loop diagrams:

$$N_1 \to \ell_\alpha + H\,,\qquad N_1 \to \bar\ell_\alpha + H^\dagger\,.$$

The CP asymmetry in $N_1$ decays is

$$\epsilon_1 = \frac{\sum_\alpha[\Gamma(N_1\to\ell_\alpha H)-\Gamma(N_1\to\bar\ell_\alpha H^\dagger)]}{\sum_\alpha[\Gamma(N_1\to\ell_\alpha H)+\Gamma(N_1\to\bar\ell_\alpha H^\dagger)]}\,.$$

At one loop (vertex + self-energy corrections), this evaluates to

$$\epsilon_1 = -\frac{1}{8\pi}\frac{1}{(Y_\nu^\dagger Y_\nu)_{11}}\sum_{j\neq 1}\text{Im}\left[(Y_\nu^\dagger Y_\nu)_{1j}^2\right]\,g\!\left(\frac{M_j^2}{M_1^2}\right)\,,$$

where $g(x) = \sqrt{x}[1/(1-x)+1-(1+x)\ln((1+x)/x)]$. In the hierarchical limit$M_1 \ll M_2, M_3$, this simplifies to

$$\epsilon_1 \approx -\frac{3}{16\pi}\frac{M_1}{(Y_\nu^\dagger Y_\nu)_{11}}\sum_{j\neq 1}\text{Im}\left[(Y_\nu^\dagger Y_\nu)_{1j}^2\right]\frac{1}{M_j}\,.$$

4. From Lepton to Baryon Asymmetry

The lepton asymmetry $Y_{B-L}$ generated by $N_1$ decays is partially converted to a baryon asymmetry by sphalerons. In the SM with $n_g$ generations, the conversion factor is

$$Y_B = \frac{8n_g + 4}{22n_g + 13}\,Y_{B-L}\,.$$

For $n_g = 3$:

$$Y_B = \frac{28}{79}\,Y_{B-L} \simeq -\frac{28}{79}\,Y_{B-L}\,,$$

where the sign reflects the convention that a positive lepton asymmetry generates a positive baryon asymmetry. Including washout effects parametrized by an efficiency factor$\kappa_f < 1$:

$$Y_B \simeq -\frac{28}{79}\,\epsilon_1\,\kappa_f\,\frac{1}{g_*}\,.$$

5. The Davidson-Ibarra Bound

There is an upper bound on $|\epsilon_1|$ from unitarity and the seesaw relation, which in the hierarchical limit reads

$$|\epsilon_1| \leq \frac{3}{16\pi}\frac{M_1\,m_3}{v^2}\,,$$

where $m_3$ is the heaviest light neutrino mass and $v = 174$ GeV. Requiring successful leptogenesis ($Y_B \sim 10^{-10}$) then implies a lower bound on $M_1$:

$$M_{N_1} \gtrsim 10^9\;\text{GeV}\,,$$

assuming thermal initial abundance and a normal neutrino mass hierarchy with$m_3 \sim 0.05$ eV. This bound has profound implications: it requires a very high reheat temperature after inflation, $T_{\rm RH} \gtrsim 10^9$ GeV, which can conflict with gravitino overproduction in supersymmetric theories.

6. Washout and the Efficiency Factor

The lepton asymmetry generated by $N_1$ decays is partially erased by inverse decays and scattering processes. The washout parameter is defined as

$$K_1 \equiv \frac{\Gamma_{N_1}}{H(T=M_1)} = \frac{\tilde{m}_1}{m_*}\,,$$

where $\tilde{m}_1 = (Y_\nu^\dagger Y_\nu)_{11}v^2/M_1$ is the effective neutrino mass and $m_* \approx 1.1\times10^{-3}$ eV is the equilibrium neutrino mass. Three regimes emerge:

7. GUT Baryogenesis

In grand unified theories, heavy gauge or Higgs bosons ($X$, $Y$) with masses $M_X \sim 10^{16}$ GeV mediate baryon-number-violating interactions. Their out-of-equilibrium decays at $T \sim M_X$ can generate a baryon asymmetry:

$$Y_B \sim \epsilon_X\,\frac{n_X}{s}\bigg|_{T\sim M_X} \sim \frac{\epsilon_X}{g_*}\,,$$

where $\epsilon_X$ is the CP asymmetry in $X$ decays. A serious challenge for GUT baryogenesis is that sphalerons, active at $T \gg T_{\rm EW}$, can wash out any $B+L$ asymmetry. Only the $B-L$ component survives, so the GUT must generate a net $B-L$ asymmetry to be viable.

8. Affleck-Dine Mechanism

In supersymmetric theories, scalar fields carrying baryon number (flat directions of the scalar potential) can develop large field values during inflation. When the Hubble rate drops below their mass, they begin oscillating and decay. CP-violating A-terms in the potential rotate the field trajectory in the complex plane, generating a net baryon charge:

$$n_B \propto \text{Im}\left(A\,\phi^{*n}\right)\,|\phi|^2\,,$$

where $A$ is the soft SUSY-breaking A-term coefficient. This mechanism can produce the observed asymmetry for a wide range of parameters and is particularly efficient in scenarios with a low reheat temperature.