ar X iv : h ep - p h / 97 07 53 8 v 1 3 1 Ju l 1 99 7 hep - ph / 9707538 TUW 97 - 14 IMPROVED HARD - THERMAL - LOOP EFFECTIVE ACTIONS

Hard thermal loop effective actions furnish the building blocks of resummed thermal perturbation theory, which is expected to work as long as the quantities under consideration are not sensitive to the nonperturbative (chromo-)magnetostatic sector. A different breakdown of perturbation theory occurs whenever external mo-menta are light-like, because the hard thermal loops themselves develop collinear singularities. By additionally resumming asymptotic thermal masses for hard modes these singularities are removed while leaving the gauge invariance of the effective action intact. Precisely at (and below) those energy/momentum scales where perturbation theory of quantum field theories at non-zero temperature 1 predicts new effects from collective behaviour like screening, plasma frequencies, Landau damping etc., the conventional loop expansion ceases to be reliable. The infrared singularity of the Bose-Einstein distribution n B (k) = 1 e k/T − 1 ∼ T k for k → 0 (1) causes an increasing sensitivity to the infrared as the loop order increases. Eventually, this gives rise to infrared divergences which inhibit further refinements of the results or which even may prevent one from determining the leading terms. However, the very same nontrivial screening masses, plasma frequencies, and Landau damping effects in most cases provide the remedy for perturbation theory, because once taken into account they provide a cutoff to otherwise infrared divergent loop integrals. One of the simplest cases where one can observe this mechanism of self-healing is a massless scalar field theory with quartic self-interactions g 2 φ 4. At finite temperature, one finds for the thermal contribution to the one-loop scalar