Effective field theory for highly ionized plasmas

Abstract

We examine the equilibrium properties of hot, non-relativistic plasmas. The partition function and density correlation functions of a plasma with several species are expressed in terms of a functional integral over electrostatic potential distributions. This is a convenient formulation for performing a perturbative expansion. The theory is made well-defined at every stage by employing dimensional regularization which, among other virtues, automatically removes the unphysical (infinite) Coulomb self-energy contributions. The leading order, field-theoretic tree approximation automatically includes the effects of Debye 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 plus energy and number densities is shown to be described by a field-theoretic anomaly. The effective Lagrangian method that we employ yields a 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. The presentation is pedagogical and self-contained. The results for thermodynamic quantities at three-loop [or O( n 5/2)] order, and for the leading long-distance forms of correlation functions, agree with previous results in the literature, but they are obtained in a novel and simple fashion using the effective field theory. In addition to the new construction of the effective field theory for plasma physics, we believe that the results we report for the explicit form of correlation functions at two-loop order, as well as the determination of higher-order leading-logarithmic contributions, are also original.