Effective Field Theory for Plasmas at all Temperatures and Densities

The solution of the plasmon problem and the subsequent development of an effective field theory approach to ultrarelativistic plasmas are reviewed. The effective lagrangians that summarize collective effects in ultrarelativistic quark-gluon and electron-photon plasmas are presented. A generalization that describes an electromagnetic plasma at all temperatures and densities is proposed. Effective field theory methods have proven to be very powerful in treating plasmas at ultrarelativistic temperatures and densities. This approach was developed as a byproduct of the solution of the long-standing plasmon problem of high temperature QCD. It has opened the way for the systematic calculation of many fundamental properties of ultrarelativistic plasmas that were not feasible with previous methods. The effective field theory approach has recently been summarized compactly in the form of elegant effective lagrangians. For many applications, it would be useful to have a unified approach that works at all temperatures and densities. One promising approach is to generalize the effective field theory that describes the ultrarelativistic regime. The effective lagrangians for ultrarelativistic quarkgluon and electron-photon plasmas will be reviewed below and a generalization that describes an electromagnetic plasma at any temperature and density will be proposed. The plasma problem of high temperature QCD was first posed by Kalashnikov and Klimov in 1980 and by Gross, Pisarski, and Yaffe in 1981 [1]. The problem was that a presented at the Workshop on Finite Temperature Field Theory, Winnipeg, July 20-25, 1992. 1-loop calculation of the gluon damping rate, which is proportional to the imaginary part of the gluon self energy, gives a gauge dependent answer. Over the next 10 years, there were about a dozen published attempts to calculate the gluon damping rate, with almost as many different answers. In 1989, Pisarski pointed out that a 1-loop calculation of the damping rate is simply incomplete [2]. A consistent calculation to leading order in the QCD coupling constant gs must include contributions from all orders in the loop expansion. He was able to carry out the necessary resummation explicitly for the damping rate of a heavy quark. The resummation consisted of replacing the gluon propagator in the 1-loop diagram for the heavy quark self-energy by an effective gluon propagator obtained by summing up the hard thermal loop corrections (the terms proportional to g sT ) to the gluon self-energy. This effective propagator was first calculated by Klimov and by Weldon [3], who used it to study the propagation of gluons and the screening of interactions in the high temperature limit of the quark-gluon plasma. The problem of the gluon damping rate is a little more complicated. It is not enough to replace the gluon propagators in the 1-loop gluon self-energy diagrams by effective propagators, because there are also vertex corrections that are not suppressed by any powers of gs. In particular, the 3-gluon vertex has hard thermal loop corrections proportional to g 3 sT 2 which contribute at the same order as the bare vertex of order gs. Similarly, the 4-gluon vertex has hard thermal loop corrections proportional to g sT , which contribute at the same order as the bare vertex of order g s . A thorough diagrammatic analysis of the damping rate [4] by Pisarski and me revealed that the hard thermal loop corrections to the gluon propagator, the 3-gluon vertex, and the 4-gluon vertex are the complete set of diagrams that need to be resummed in order to calculate the damping rate to leading order in gs. The result of this resummation was proved to be gauge invariant, thus solving the plasmon problem. We were also able to calculate the damping rate explicitly [5], thus demonstrating that the resummation could be used as a practical calculational tool. The damping rate of a quark has also been calculated [6] and subtleties in the proof of gauge invariance of the damping rates have been carefully analyzed [7]. The resummation required to solve the plasmon problem has a simple interpretation in terms of an effective field theory. The complete damping rate to leading order in gs is given by the imaginary part of the 1-loop gluon self-energy diagrams, with the gluon propagators replaced by effective propagators and with the 3-gluon and 4-gluon vertices replaced by effective vertices obtained by adding the hard thermal loop corrections to the bare vertices. This is equivalent to calculating 1-loop diagrams in an effective field theory whose propagator is the effective gluon propagator of Klimov and Weldon and whose vertices are the effective 3-gluon and 4-gluon vertices that Pisarski and I introduced. These propagators and vertices are related by gauge invariance, just like their counterparts in the QCD lagrangian. Our analysis [4] revealed however that hard thermal loop corrections also appear in the effective n-gluon amplitudes for n = 5, 6, 7, .... They are related to the lower n-gluon amplitudes by gauge invariance, but they have no counterparts in the QCD lagrangian. While Pisarski and I were able to calculate all these amplitudes explicitly, we were unable to find an effective action which generated them. In 1990, Taylor and Wong [8] succeeded in writing down such an action in closed form, but it was very cumbersome. Finally in 1991, Frenkel and Taylor and, independently, Pisarski and I, succeeded in writing down an elegant effective action for the gluon amplitudes [9]. The lagrangian density which summarizes the effective field theory for a quark-gluon plasma at ultrarelativistic temperature or density has the form Leff = LQCD + Lgluon + Lquark . (1) The first term is the usual lagrangian density for QCD: LQCD = − 1 2 tr GμνG μν + i ∑ ψ̄γDμψ , (2) where Gμν = G a μνT a is the gluon field strength contracted with generators T a that satisfy tr(T T ) = δ/2. The sum is over nf flavors of massless quarks. The second term in (1) is the thermal gluon term: Lgluon = 3 2 m2g tr Gμα 〈 P P β (P ·D)2 〉