Systematic approach to leptogenesis in nonequilibrium QFT: Self-energy contribution to the $CP$-violating parameter

In the baryogenesis via leptogenesis scenario the self-energy contribution to the $CP$-violating parameter plays a very important role. Here, we calculate it in a simple toy model of leptogenesis using the Schwinger-Keldysh/Kadanoff-Baym formalism as starting point. We show that the formalism is free of the double-counting problem typical for the canonical Boltzmann approach. Within the toy model, medium effects increase the $CP$-violating parameter. In contrast to results obtained earlier in the framework of thermal field theory, the medium corrections are linear in the particle number densities. In the resonant regime quantum corrections lead to modified expressions for the $CP$-violating parameter and for the decay width. Most notably, in the maximal resonant regime the Boltzmann picture breaks down and an analysis in the full Kadanoff-Baym formalism is required.

Figure 1

Tree-level and one-loop self-energy diagrams of the decay process ${\psi }_i\to bb$.

Figure 2

Two-loop contribution to the 2PI effective action and the corresponding contributions to the self-energies of the real and complex fields.

Figure 3

Effective self-energy $CP$-violating parameter ${ϵ}_1$ in medium obtained from the Kadanoff-Baym formalism. The shaded areas correspond to the range $0.25\le |\mathbf{p}|/T\le 4$ of momenta $|\mathbf{p}|$ of the decaying particle ${\psi }_1$ with respect to the rest-frame of the medium. Here we assumed a thermal Bose-Einstein (BE) or Maxwell-Boltzmann (MB) distribution for $b/\overline{b}$ with vanishing chemical potential. In the low-temperature limit (NR), the vacuum value is approached. In the high-temperature limit (UR), the $CP$-violating parameter is enhanced within the toy model. We also show the thermally averaged $CP$-violating parameter $⟨{ϵ}_1⟩$ for the BE (red long-dashed line) and MB (blue dashed line) cases.

Figure 4

Qualitative behavior of the components of the full and diagonal spectral functions $G_{\rho }^{ij}$ and ${\mathcal{G}}_{\rho }^{ii}$ for ${\Gamma }_j\ll M_j$ and $M_j{\Gamma }_j\ll |\Delta M_{ij}^2|$. For illustration we choose $M_2/M_1=2$, $|g_1|/M_1=0.5$ and $|g_2|/M_2=0.8$. The diagonal components are down-scaled by a factor of 50.

Figure 5

Qualitative behavior of the diagonal and off-diagonal components of full and diagonal spectral functions $G_{\rho }^{ij}$ and ${\mathcal{G}}_{\rho }^{ii}$ in the maximal resonant limit $M_j^2\gg |\Delta M_{ij}^2|\lesssim M_j{\Gamma }_j$. For illustration we choose $M_2/M_1=1.02$, $|g_1|/M_1=0.5$ and $|g_2|/M_2=0.8$. The diagonal components are not scaled.

Figure 6

Corrections to the effective self-energy $CP$-violating parameter ${ϵ}_1$ in the resonant case obtained from the Kadanoff-Baym formalism (for $|g_1|=|g_2|=0.01M_1$, ${\delta }_{CP}=\pi /8$, and $|\mathbf{p}|=T$; the mass $M_2$ is determined by the value of $R$).

Figure 7

Dependence of the vacuum (dotted line), hierarchical (dashed line), and resonant (solid line) approximations for the $CP$-violating parameter on the degeneracy parameter $R$ calculated at $T=M_1$ (for the same parameter values as in Fig. 6). The dot-dashed line shows the approximate expression for the resonant case from the last line of Eq. (63). The Boltzmann approximation requires $R\gg 1$ and therefore the results for the $CP$-violating parameter are not applicable in the gray shaded region. We show them only for comparison with the conventional result.

Figure 8

Ratio of the in-medium decay width of the lightest toy-Majorana to its tree-level value as a function of the dimensionless inverse temperature calculated for various values of the degeneracy parameter $R$.

Figure 9

Ratio of the in-medium decay width of the lightest toy-Majorana to its tree-level value as a function of the degeneracy parameter $R$ calculated at $T=M_1$.

Figure 10

The distribution function for the massive species $f_{{\psi }_1}$ can deviate significantly from equilibrium for smaller washout factors (here $\kappa \simeq 0.01$).

Figure 11

The ratio $⟨{ϵ}_1⟩/{ϵ}_1^{\mathrm{vac}}$. The curves flatten for small $M_1/T$ because the initial conditions involve a finite chemical potential.

Figure 12

Dependence of the final asymmetries and the relative quantum correction $(\eta -{\eta }^{\mathrm{vac}})/{\eta }^{\mathrm{vac}}$ on the washout factor $\kappa$. The cases $a$, $b$, $c$, $d$, $e$, $f$ correspond to washout factors 0.01, 0.1, 0.366, 1, 10, 100.

Figure 13

Resummed diagrams contributing to the resonant part of the $2\to 2$ scattering amplitude of the process $bb\to \overline{b}\overline{b}$.

Figure 14

Closed real-time path $\mathcal{C}$.