Particle Production and Effective Thermalization

Gert Aarts (Heidelberg)
work done in collaboration with Jan Smit (Amsterdam)

Our motivation to study dynamics in nonequilibrium quantum field theory comes from both electroweak baryogenesis (e.g. sphaleron transitions in a nonequilibrium situation) and the end of inflation (preheating and thermalization).
As a toy model we consider the abelian Higgs model in 1+1 dimensions with Nf fermion flavours in the large Nf limit. In the approximate dynamical equations, inhomogeneous classical (mean) Bose fields are coupled to quantized fermion fields. The dynamics (and the inhomogeneous backreaction) of the fermions is calculated with a mode function expansion. Note that the effective equations of motion imply e.g. Coulomb scattering, due to the inhomogeneous gauge field. We solve the equations numerically on a lattice in space and time, and I presented three numerical results:

  1. Renormalization: the effective equation of motion for the scalar field contains an ultraviolet divergent contribution, in the backreaction of the fermions. This divergence is the usual logarithmic divergence (in 1+1 dimensions) and it can be renormalized by adjusting the bare parameters in the proper way. 
  2. Baryogenesis: starting from a nonequilibrium initial state with many sphaleron transitions initially and fewer later on, we follow both the fermion number (which is the axial charge in our formulation) and the Chern-Simons number in real-time and we show that the anomaly equation (which relates a change in the Chern-Simons number to a change in fermion number) is approximately satisfied. 
  3. Thermalization: during time evolution  there is energy transfer from the Bose fields to the fermion degrees of freedom, which leads to fermionic particle creation. To analyse this, we define time dependent fermion particle numbers with the help of the single-time Wigner function, and compare the particle numbers with the (equilibrium) Fermi-Dirac distribution parametrized by a time dependent temperature and chemical potential, coupled to the axial charge. We find that the fermions approximately thermalize locally in time.
Open questions concern: a possible thermalization of the complete system (i.e. do the Bose fields thermalize as well, on what time scale, and to what kind of equilibrium?), an extension of the approximation to include Bose field 'fluctuations' also, and an extension of the inhomogeneous mode functions method to 3+1 dimensions.

This talk is based on the following papers:


Gert Aarts