Divergences in Real-Time Classical Theories
Bert-Jan Nauta
Time-dependent correlation functions at soft momenta can be calculated
in (effective) classical theories. I discuss the problem of classical
divergences.
At one-loop it is known that linear divergences occur from classical
HTL's.
Subleading logarithmic divergences are absent at one-loop [1]. To deal
with higher-loop diagrams a general argument is presented. Starting with
the assumption that the quantum self-energy is proportinal to T^2 in the
high-T limit, it is argued that the superficial degree of divergence of
classical L-loop diagrams is 2-L. In [1] it is verified that the degree
of divergence of two-loop self-energy diagrams is indeed only logarithmic.
An important consequence of this estimate for the degree of divergence
is that no new divergences are introduced beyond two-loops. And that
three-loops
and higher are IR-dominated (except for one- and two-loop subdivergences)
such that a classical calculation should give the quantum result in leading
order (after matching the subdivergences).
In the second part of my talk I discuss the introduction of counterterms
for the linear divergences on a lattice [2]. Since the linear divergences
are given by classical HTL's, we extend the classical equations of motion
to the HTL (or Vlasov) equations. The counterterms can then be introduced
by a subtraction in the (equilibrium) distribution, in this way gauge
invariance
and conservation of energy and phase-space measure are preserved. Matching
to the continuum and keeping the energy of the system positive requires
an exponentially small coupling.
references
[1] Gert Aarts, Bert-Jan Nauta and Chris G. van Weert, Divergences
in Real-Time Classical Field Theories at Non-Zero
Temperature,
hep-ph/9911463
[2] Bert-Jan Nauta, Counterterms for Linear Divergences in Real-Time
Classical Gauge Theories at High Temperature, hep-ph/9906389