Classical Colored Particles -- Unified Description of HTL and Small x
Co-workers: Jamal Jalilian-Marian (U of Ariz.), Raju Venugopalan (BNL) and Jens Wirstam (U of Stockholm)
The HTL system and small x system shares a common feature -- the hard modes have big momenta and act like classical particles and the soft modes have big occupation number and hence act like classical field. Kelly, Liu, Lecchesi and Manuel [KLLM] showed that the HTL action results from corresponding classical kinetic theory and McLerran and Venugopalan [MV] showed that basically the same idea can be used to calculate gluon density in the very small x regime. In both cases, the starting point is the collision-less Vlasov equation where hard modes are represented by classical particle degrees of freedom and the soft modes are represented by classical field degrees of freedom.
As explained in a recent work [MJVW], this can be understood from QCD one-loop action. One first separates the hard and soft modes. Integrating out the hard modes to one-loop yields effective action for the soft modes. At zero temperature, Strassler showed how to rewrite the one-loop effective action using world line particles. The generalization to many body system, however, is non-trivial. One needs to pay attention to the boundary conditions that enter into the calculation of the functional determinant. In this way, we obtain a fully effective action for the soft modes in terms of a many-body quantum mechanical path integral.
In particular, we study hot QCD and obtain the classical transport equations which, as Litim and Manuel have shown, reduce in the appropriate limit to the non--Abelian Boltzmann--Langevin equation first obtained by Bodeker. In the Vlasov limit, the classical kinetic equations are those that correspond to the hard thermal loop effective action. In the equilibrium limit, this should be equivalent to Blaizot and Iancu's result.
The inclusion of noise and damping terms to give the Boltzmann--Langevin equation, goes beyond hard thermal loops and allows one to address physics at the softer scale of g**2 T. The potential advantage of the formalism explained in this talk is that one has a formalism at the level of the effective action. It may therefore facilitate computations of effects beyond those of classical transport theory. Its form suggests that it is amenable to a ``dynamical renormalization group'' treatment. Similar ideas have been used previously in small x physics to develop a Wilsonian renormalization group for wee partons (for instance see [JKW]) exact connection to which is currently being considered by the authors of the above paper.
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