Medium corrections to the $CP$-violating parameter in leptogenesis

In two recent papers [4,5], it has been demonstrated that one can obtain quantum-corrected Boltzmann kinetic equations for leptogenesis using a top-down approach based on the Schwinger-Keldysh/Kadanoff-Baym formalism. These “Boltzmann-like” equations are similar to the ones obtained in the conventional bottom-up approach but differ in important details. In particular there is a discrepancy between the $CP$-violating parameter obtained in the first-principle derivation and in the framework of thermal field theory. Here we demonstrate that the two approaches can be reconciled if causal $n$-point functions are used in the thermal field theory approach. The new result for the medium correction to the $CP$-violating parameter is qualitatively different from the conventional one. The analogy to a toy model considered earlier enables us to write down consistent quantum-corrected Boltzmann equations, for thermal leptogenesis in the standard model extended by three right-handed neutrinos, which include quantum statistical terms and medium-corrected expressions for the $CP$-violating parameter.

Figure 1

Tree-level and one-loop contributions to the heavy Majorana neutrino decay ${\psi }_i\to \alpha \beta$. The asymmetry, at lowest order, is due to the interference of these contributions.

Figure 2

Momentum flow in the vertex and the self-energy loop.

Figure 3

Circling rules for a generic theory used for the computation of $F_{>}$ in momentum space. The rules for the computation of $F_{<}$ differ by interchange of ${\Delta }^{+}(p)$ and ${\Delta }^{-}(p)$. The ${\Delta }^{\pm }$ propagators connecting circled and uncircled vertices may be interpreted as cut propagators. In vacuum they correspond to the cut propagators in the Cutkosky rules.

Figure 4

Circlings contributing to $\mathrm{Im}\{i^{-1}\mathcal{F}(1,1,1)\}$ for the vertex loop. At one-loop level the circlings can be interpreted as cuts, as indicated, by the lines separating circled from uncircled regions [11]. The contributions from diagrams involving cuts through the $x_2$-$x_3$ line are suppressed relative to the others in the hierarchical limit.

Figure 5

Circlings contributing to $\mathrm{Im}\{i^{-1}\mathcal{F}(1,1;z)\}$ for the self-energy loop. The graphs (b) and (c) vanish since ${\psi }_i$ and ${\psi }_j$ cannot be on-shell simultaneously. Note that we consider only the diagrams with ${\psi }_i$ in the external and ${\psi }_j$ in the internal line ($i\ne j$) because these are the only ones which contribute to ${ϵ}_i$.

Figure 6

Circlings contributing to $\mathrm{Im}\{i^{-1}{\mathcal{F}}_A^{(1)}(x_1,x_2,x_3)\}$ for the vertex loop. The advanced three-point function involves a difference of $F_{>}$ and $F_{<}$ contributions which differ by the replacement ${\Delta }^{\pm }\leftrightarrow {\Delta }^{\mp }$ in the circling rules. Since the finite temperature contributions (terms proportional to $f^{\mathrm{eq}}$) to the latter are the same, all contributions quadratic in the distribution functions cancel.

Figure 7

Circlings contributing to $\mathrm{Im}\{i^{-1}{\mathcal{F}}_A^{(1)}(x_1,x_2;z)\}$ for the self-energy loop. Graph (b) vanishes since ${\psi }_i$ and ${\psi }_j$ cannot be on-shell simultaneously.

Figure 8

Temperature dependence of the $CP$-violating parameter in the Majorana neutrino decay relative to its vacuum value. Shown are the thermal average $⟨{ϵ}_1^{\mathrm{th}}⟩/{ϵ}_1^{\mathrm{vac}}$ (solid red line) and the values for various momentum modes ${ϵ}_1^{\mathrm{th}}/{ϵ}_1^{\mathrm{vac}}$ (dotted red lines) corresponding to $|\mathbf{q}|=T$, $-1\le \sin ({\delta }^{\prime })\le +1$. For comparison we also show the conventional results $⟨{ϵ}_1^{\mathrm{th},\mathrm{conv}}⟩/{ϵ}_1^{\mathrm{vac}}$ (dashed black line), where the leading effects cancel as described in the text. Equilibrium distribution functions for bosons and fermions with negligible chemical potentials are assumed. Note that the shown behavior can be modified if thermal masses are included, since the decay $N_1\to \ell \varphi$ (and the conjugate process) becomes kinematically forbidden if the thermal Higgs mass becomes too large. At even higher temperature the process $\varphi \to N_1\ell$ becomes relevant instead [2].

Figure 9

Medium correction to the $CP$-violating parameters in the MSSM. The lines correspond to the thermal averages $⟨{ϵ}_1^{\mathrm{th}}⟩/{ϵ}_1^{\mathrm{vac}}$, and the shaded regions illustrate the momentum dependence of ${ϵ}_1^{\mathrm{th}}/{ϵ}_1^{\mathrm{vac}}$ for $0.5\le |\mathbf{q}|/T\le 4$ and ${\delta }^{\prime }=0$. Note also that in the weighted sum of ${\tilde{N}}_1\to \varphi \tilde{\ell }$ and ${\tilde{N}}_1\to \tilde{\varphi }\ell$ processes the cancellation of the medium contributions, observed in the earlier publications, does not occur anymore.