Effective Field Theory for Quasi - Classical Plasmas

We examine the equilibrium properties of hot, dilute, non-relativistic plasmas. The partition function and density correlation functions of a classical plasma with several species are expressed in terms of a functional integral over electrostatic potential distributions. This is a convenient formulation for performing a well-defined perturbative expansion. The leading order, field-theoretic tree approximation automatically includes the effects of De-bye screening. (No further partial resummations are needed for this effect.) Subleading, one-loop corrections are easily evaluated. The two-loop corrections , however, have ultraviolet divergences. These correspond to the short-distance, logarithmic divergence which is encountered in the spatial integral of the Boltzmann exponential when it is expanded to third order in the Coulomb potential. Such divergences do not appear in the underlying quantum theory — they are rendered finite by quantum fluctuations. We show how such divergences may be removed and the correct finite theory obtained by introducing additional local interactions in the manner of modern effective quantum field theories. We compute the two-loop induced coupling by exploiting a non-compact su(1, 1) symmetry of the hydrogen atom. This enables us to obtain explicit results for density-density correlation functions through two-loop order and thermodynamic quantities through three-loop order. The induced couplings are shown to obey renormalization group equations, and these equations are used to characterize all leading logarithmic contributions in the theory. A linear combination of pressure and energy and number densities is shown to be described by a field-theoretic anomaly. The effective Lagrangian method that we employ yields a very simple demonstration that, at long distance , correlation functions have an algebraic fall off (because of quantum effects) rather than the exponential damping of classical Debye screening. We use the effective theory to compute, easily and explicitly, this leading long distance behavior of density correlation functions.