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.

[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