CP asymmetry factor in decays at finite temperature

TL;DR

The paper addresses the leading-order CP asymmetry factor in finite-temperature decays involving Majorana neutrinos, a key input for leptogenesis. It develops a full thermal one-loop evaluation for the CP asymmetry in the decays $N_i\to \phi+L$ and $\phi\to N_i+L$ within equilibrium thermal field theory, accounting for self-energy and vertex corrections and employing a causal (Kobes) construction for the imaginary parts. The authors provide explicit expressions for the imaginary parts of the stripped amplitudes, perform the necessary loop integrations (with the self-energy integrals yielding closed forms and vertex integrals reduced by frame choices), and carry out the thermal averaging to obtain the complete $\epsilon_{a\to b+c}$. They compare the full causal, thermal results with previous approaches (Maxwell–Boltzmann, original causal prescriptions, and mixed treatments) across realistic and heavy Majorana-mass scenarios, highlighting regimes where purely thermal vertex cuts and causal amplitudes significantly alter the predicted CP violation. The findings improve the reliability of CP-violation inputs in Boltzmann-equation analyses of leptogenesis, particularly at high temperatures or near mass degeneracy, and have implications for models such as the Superweak extension of the Standard Model.

Abstract

We present in the conventional equilibrium approach to leptogenesis the complete leading order prediction for the CP asymmetry factor in finite-temperature decays involving Majorana neutrinos. As thermal effects are generally not negligible, the knowledge of the high temperature behavior of the underlying particle physics model, in particular that of mass generation, is required for reliable estimates of matter-antimatter asymmetry through the mechanism of leptogenesis. We present all necessary information needed to obtain the full one-loop evaluation for the thermal CP asymmetry factors for the processes $N_i\to φ+ L$ and $φ\to N_i+L$ at temperatures where they are kinematically allowed. We present a numerical comparison with previous formulae given in the literature.

Figures (3)

• Figure 1: One-loop cut diagrams contributing to the CP asymmetry factor. a) The cut neutrino self-energy (bubble) diagram with intermediate lepton and scalar fields in the loop. b) The three cuts of the one-loop vertex correction (triangle) diagram. The particle types of the incoming and outgoing states as well as the internal lines depend on the particular process considered, $N_i(P_1)\to \phi(P_2) + L(P_3)$ or $\phi(P_1)\to N_i(P_2) + L(P_3)$.
• Figure 2: CP asymmetry factor in the realistic mass scenario of $N_i\to\phi+L$ and $\phi\to N_i+L$ decays. The solid black lines give our full result. At low temperature the contribution due to the purely thermal cuts of the vertex function is shown explicitly (green, dash-dotted). Above $T_{\rm c}$ the system is in the symmetric phase and $m_i\propto T$ for all particles.
• Figure 3: CP asymmetry factor in the heavy mass scenario of $N_i\to\phi+L$ and $\phi\to N_i+L$ decays. The solid black lines give our full result, while the others correspond to different predictions from the literature as explained in the main text.