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Transport equations for chiral fermions to order \hbar and electroweak baryogenesis: Part II

T. Prokopec, M. G. Schmidt, S. Weinstock

TL;DR

This work provides a first-principles, order-$\hbar$ treatment of CP-violating collision terms in transport for chiral fermions with space-time dependent masses, using a Yukawa model and one-loop self-energies. Building on Paper I's spin-quasiparticle framework, it derives CP-violating sources in both scalar and fermionic collision terms, computes relaxation rates, and formulates fluid equations that couple CP-violating densities to diffusion. The analysis shows that collisional sources exist and are qualitatively distinct from semiclassical flow sources, generally subdominant unless diffusion is strong, and subject to mass thresholds and Yukawa suppression. It also clarifies the relationship between spin- and helicity-based descriptions in the relativistic limit and discusses how two-loop self-energies would be required for quantitative, massless-limit equilibration and robust baryogenesis predictions. Overall, the paper provides a principled, first-principles framework for assessing CP-violating transport and its implications for electroweak baryogenesis within a controlled truncation.

Abstract

This is the second in a series of two papers. While in Paper I we derive semiclassical Boltzmann transport equations and study their flow terms, here we address the collision terms. We use a model Lagrangean, in which fermions couple to scalars through Yukawa interactions and approximate the self-energies by the one-loop expressions. This approximation already contains important aspects of thermalization and scatterings required for quantitative studies of transport in plasmas. We compute the CP-violating contributions to both the scalar and the fermionic collision term.

Transport equations for chiral fermions to order \hbar and electroweak baryogenesis: Part II

TL;DR

This work provides a first-principles, order- treatment of CP-violating collision terms in transport for chiral fermions with space-time dependent masses, using a Yukawa model and one-loop self-energies. Building on Paper I's spin-quasiparticle framework, it derives CP-violating sources in both scalar and fermionic collision terms, computes relaxation rates, and formulates fluid equations that couple CP-violating densities to diffusion. The analysis shows that collisional sources exist and are qualitatively distinct from semiclassical flow sources, generally subdominant unless diffusion is strong, and subject to mass thresholds and Yukawa suppression. It also clarifies the relationship between spin- and helicity-based descriptions in the relativistic limit and discusses how two-loop self-energies would be required for quantitative, massless-limit equilibration and robust baryogenesis predictions. Overall, the paper provides a principled, first-principles framework for assessing CP-violating transport and its implications for electroweak baryogenesis within a controlled truncation.

Abstract

This is the second in a series of two papers. While in Paper I we derive semiclassical Boltzmann transport equations and study their flow terms, here we address the collision terms. We use a model Lagrangean, in which fermions couple to scalars through Yukawa interactions and approximate the self-energies by the one-loop expressions. This approximation already contains important aspects of thermalization and scatterings required for quantitative studies of transport in plasmas. We compute the CP-violating contributions to both the scalar and the fermionic collision term.

Paper Structure

This paper contains 23 sections, 166 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Collision term
  3. Scalar collision term
  4. The scalar collision term with no mixing
  5. The scalar collision term with mixing
  6. Discussion of scalar collisional sources
  7. Fermionic collision term
  8. Fermionic collision term with no mixing
  9. Fermionic collision term with mixing
  10. Relaxation to equilibrium
  11. Boltzmann collision term
  12. Relaxation rates
  13. Fluid equations
  14. Currents
  15. Spin or helicity?
  16. ...and 8 more sections

Figures (12)

  • Figure 1: The two-loop diagram corresponding to the Yukawa interaction (\ref{['Yukawa_part2']}) contributing to the 2PI effective action (\ref{['Gamma2_part2']}).
  • Figure 2: The one-loop scalar vacuum polarization diagram (\ref{['Pi-<>']}) from the Yukawa interaction (\ref{['Yukawa_part2']}), which contributes to the collision term of the scalar kinetic equation (\ref{['Wigner-space:scalar_eom_part2']}).
  • Figure 3: The integral in the CP-violating collisional source (\ref{['C-phi0-g']}) of the scalar continuity fluid equation as a function of the mass parameters, $M/T$ and $|m|/T$.
  • Figure 4: The one-loop collision term in the scalar kinetic equation in a theory with the Yukawa interaction (\ref{['Yukawa_part2']}). The interactions of the scalar propagator with the scalar condensate are of the same nature as the tree-level interactions discussed in section \ref{['Kinetics of scalars: tree-level analysis']}. Just like in the tree-level case, at order $\hbar$, scalars do not feel the condensate.
  • Figure 5: The collision term in the scalar kinetic equation as in figure \ref{['fig:scalar-1loop']}. The interactions of the fermionic (loop) propagators with the scalar field condensate that can contribute in CP-odd manner at the order $\hbar$ to the scalar kinetic equation.
  • ...and 7 more figures