Finite Number Density Corrections to Leptogenesis

TL;DR

The paper develops a first-principles treatment of leptogenesis using the Closed Time Path formalism to derive and solve the Kadanoff-Baym equations, explicitly addressing real intermediate state subtraction within a quantum-statistical framework.A key result is that no lepton asymmetry is generated in equilibrium, in line with CPT, while finite-density loop corrections induce a CP-violating source that includes Bose enhancement from Higgs distributions and Fermi suppression from leptons.The CP asymmetry arises from both wave-function and vertex corrections, which are computed with full KMS-consistent finite-density propagators, and RIS subtraction emerges naturally within the CT(P) approach.Numerical studies show that these finite-density corrections are modest in the strong-washout regime but can be sizable in weak-washout scenarios, depending on initial conditions, and can even affect the sign of the final asymmetry.Overall, the work provides a self-consistent, quantum-statistical framework for leptogenesis that extends the conventional Boltzmann approach and lays groundwork for including flavor effects and resonant phenomena.

Abstract

We derive and solve kinetic equations for leptogenesis within the Closed Time Path (CTP) formalism. It is particularly emphasised how the procedure of real intermediate state subtraction familiar from the Boltzmann approach is realised within the CTP framework; and we show how in time-independent situations, no lepton asymmetry emerges, in accordance with the CPT-theorem. The CTP approach provides new quantum statistical corrections from evaluating the loop integrals. These lead to an enhancement of the asymmetry that is originating from the Bose statistics of the Higgs particles. To quantify this effect, we define and evaluate an effective CP-violating parameter. We also solve the kinetic equations and show explicitly that the new quantum statistical corrections can be neglected in the strong washout regime, while, depending on initial conditions, they can be very sizable for weak washout.

Figures (5)

• Figure 1: Diagrammatic representation of the lepton-number violating contribution to ${\Sigma\!\!\!/}_\ell^{{\rm wf}<,>}$. The solid lines with arrows represent the lepton $\ell$, the solid lines without arrows the neutrinos $N_{1,2}$ and the dashed lines with arrows the Higgs boson $\phi$. The solid (light grey/orange) double line represents the cut that corresponds to the subtracted real intermediate states when finite density corrections to $CP$ violation are neglected. The dashed (dark-grey/blue) double line represents the cut that corresponds to decays and inverse decays when finite density corrections are neglected.
• Figure 2: Diagrammatic representation of ${\Sigma\!\!\!/}_\ell^{{\rm v}<,>}$. The solid lines with arrows represent the lepton $\ell$, the solid lines without arrows the neutrinos $N_{1,2}$ and the dashed lines with arrows the Higgs boson $\phi$. The solid (light-grey/orange) double line represents the cut that corresponds to the subtracted real intermediate states when finite density corrections to $CP$ violation are neglected. The dashed (dark-grey/blue) double line represents the cut that corresponds to decays and inverse decays when finite density corrections are neglected.
• Figure 3: The ratio of the $CP$-violating parameter when taking effects of finite density in the loop into account over the $CP$-violating parameter at $T=0$.
• Figure 4: The absolute value of lepton-to-entropy ratio $Y_\ell$ over $z=M_1/T$. The parameters are $M_1=10^{13}\,{\rm GeV}$, $M_2=10^{15}\,{\rm GeV}$, $Y_1=2\times10^{-2}$, $Y_2=10^{-1}$ and a maximal $CP$ phase. Both, thermal initial conditions (dark grey/blue) and vanishing initial conditions (light grey/red) for $N_1$ are chosen. The solid lines correspond to solutions where finite density corrections in the loop are taken into account, the dotted lines to solutions where these are omitted.
• Figure 5: The absolute value of lepton-to-entropy ratio $Y_\ell$ over $z=M_1/T$. The parameters are $M_1=10^{13}\,{\rm GeV}$, $M_2=10^{15}\,{\rm GeV}$, $Y_1=5\times10^{-2}$, $Y_2=10^{-1}$ and a maximal $CP$ phase in the upper panel (strong washout) and $Y_1=10^{-2}$, $Y_2=10^{-1}$ in the lower panel (weak washout). Both, thermal initial conditions (dark grey/blue) and vanishing initial conditions (light grey/red) for $N_1$ are chosen. The solid lines correspond to solutions where finite density corrections in the loop are taken into account, the dotted lines to solutions where these are omitted.