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:
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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.
-
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.
-
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:
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Real-time Dynamics with Fermions on a Lattice, Nucl. Phys. B555
(1999) 355, hep-ph/9812413
-
Particle Production and Effective Thermalization in Inhomogeneous Mean
Field Theory, to appear in Phys. Rev D (2000), hep-ph/9906538
-
Dynamics of Fermions and Inhomogeneous Bose Fields on a Real-Time Lattice,
in proceedings of Strong and Electroweak Matter '98, Kopenhagen,
Denmark, December 2-5, 1998, hep-ph/9902231
Gert Aarts
aarts@thphys.uni-heidelberg.de