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

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

TL;DR

This work develops a first-principles, order $\hbar$ gradient expansion toward Boltzmann transport equations for massive chiral fermions and scalars using the Schwinger-Keldysh formalism and a 2PI effective action. It demonstrates a consistent on-shell (quasiparticle) reduction, identifies CP-violating sources arising from complex mass terms and mixing, and shows that in planar walls the scalar flow term lacks a CP-violating source at this order while fermions acquire semiclassical CP-violating forces in the flow term. The analysis establishes a mass-eigenbasis decoupling that is robust away from near-degeneracy, resolving basis-dependence questions that have clouded prior semiclassical approaches. The results apply to MSSM and NMSSM transport, providing explicit expressions for CP-violating sources in chargino and scalar sectors and lay the groundwork for the collision terms and fluid equations treated in Part II. Overall, the paper delivers a principled kinetic-theory framework for electroweak baryogenesis driven by CP-violating transport of mixing fermions and scalars.

Abstract

This is the first in a series of two papers. We use the Schwinger-Keldysh formalism to derive semiclassical Boltzmann transport equations for massive chiral fermions and scalar particles. Our considerations include complex mass terms and mixing fermion and scalar fields, such that CP-violation is naturally included, rendering the equations particularly suitable for studies of baryogenesis at a first order electroweak phase transition. In part II we discuss the collision terms.

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

TL;DR

This work develops a first-principles, order gradient expansion toward Boltzmann transport equations for massive chiral fermions and scalars using the Schwinger-Keldysh formalism and a 2PI effective action. It demonstrates a consistent on-shell (quasiparticle) reduction, identifies CP-violating sources arising from complex mass terms and mixing, and shows that in planar walls the scalar flow term lacks a CP-violating source at this order while fermions acquire semiclassical CP-violating forces in the flow term. The analysis establishes a mass-eigenbasis decoupling that is robust away from near-degeneracy, resolving basis-dependence questions that have clouded prior semiclassical approaches. The results apply to MSSM and NMSSM transport, providing explicit expressions for CP-violating sources in chargino and scalar sectors and lay the groundwork for the collision terms and fluid equations treated in Part II. Overall, the paper delivers a principled kinetic-theory framework for electroweak baryogenesis driven by CP-violating transport of mixing fermions and scalars.

Abstract

This is the first in a series of two papers. We use the Schwinger-Keldysh formalism to derive semiclassical Boltzmann transport equations for massive chiral fermions and scalar particles. Our considerations include complex mass terms and mixing fermion and scalar fields, such that CP-violation is naturally included, rendering the equations particularly suitable for studies of baryogenesis at a first order electroweak phase transition. In part II we discuss the collision terms.

Paper Structure

This paper contains 29 sections, 241 equations, 5 figures.

Table of Contents

  1. Introduction
  2. From the 2PI effective action to kinetic theory
  3. Green Functions
  4. Lagrange density
  5. Two-particle-irreducible (2PI) effective action
  6. Wigner representation and gradient expansion
  7. Sum rules and Equilibrium Green functions
  8. Reduction to the on-shell limit
  9. Propagator equation
  10. Kinetic equation in the on-shell limit
  11. Kinetics of scalars: tree-level analysis
  12. Boltzmann transport equation for CP-violating scalar densities
  13. Applications to the stop sector of the MSSM
  14. Kinetics of fermions: tree-level analysis
  15. Spin conservation and the fermionic Wigner function
  16. ...and 14 more sections

Figures (5)

  • Figure 1: The complex time contour for the Schwinger-Keldysh non-equilibrium formalism.
  • Figure 2: The one-loop diagrams contributing to the 2PI effective action (\ref{['2PI_effective_action']}).
  • Figure 3: Integration contour for the retarded propagators $\Delta^r$ and $S^r$ in (\ref{['Delta-ra:free']}-\ref{['S-ra:free']}).
  • Figure 4: Integration contour for the advanced propagators $\Delta^a$ and $S^a$ in (\ref{['Delta-ra:free']}-\ref{['S-ra:free']}).
  • Figure 5: The integrals ${\cal I}_a$ and ${\cal I}_b$ in Eq. (\ref{['parametric-flow-source']}) as a function of the mass. We scaled ${\cal I}_a$ with $|m|^2$ because this is the way it appears in the source term, analogously ${\cal I}_b$ is scaled with $|m|^4$. Note that these contributions enter the flow source with different signs.