Effective-field-theory methods are used to separate the free energy for a nonabelian gauge theory at high temperature T into the contributions from the momentum scales T , gT , and g 2 T , where g is the coupling constant at the scale 2πT . The effects of the scale T enter through the coefficients in the effective lagrangian for the 3-dimensional effective theory obtained by dimensional reduction. These coefficients can be calculated as power series in g 2 . The contribution to the free energy from the scale gT can be calculated using perturbative methods in the effective theory. It can be expressed as an expansion in g starting at order g 3 . The contribution from the scale g 2 T must be calculated using nonperturbative methods, but nevertheless it can be expanded in powers of g beginning at order g 6 . We calculate the free energy explicitly to order g 5 .We also outline the calculations necessary to obtain the free energy to order g 6 .

An effective field theory approach is developed for calculating the thermodynamic properties of a field theory at high temperature T and weak coupling g. The effective theory is the three-dimensional field theory obtained by dimensional reduction to the bosonic zero-frequency modes. The parameters of the effective theory can be calculated as a perturbation series in the running coupling constant g 2 (~) .The free energy is separated into the contributions from the momentum scales T and gT, respectively. The first term can be written as a perturbation series in g 2 (~) .If all forces are screened at the scale gT, the second term can be calculated as a perturbation series in g(T) beginning at order g3. The parameters of the effective theory satisfy renormalization group equations that can be used to sum up leading logarithms of T/(gT). We apply this method to a massless scalar field with a a4 interaction, calculating the free energy to order g6 lng and the screening mass to order g5 lng.PACS number(s): 11.1O.W~ -EFFECTIVE FIELD THEORY APPROACH TO HIGH-. . . 6993

The free energy for QCD at high temperature $T$ is calculated to order $g^5$ using effective-field-theory methods to separate the contributions from the momentum scales $T$ and $gT$. The effects of the scale $T$ enter through the coefficients in the effective lagrangian for the 3-dimensional effective theory obtained by dimensional reduction. The perturbation series for these coefficients seem to be well-behaved if the running coupling constant is sufficiently small: $\alpha_s(2 \pi T) \ll 1$. For the contribution to the free energy from the scale $gT$, the perturbation series is well-behaved only if $\alpha_s(2 \pi T)$ is an order of magnitude smaller. The implication for applications of perturbative QCD to the quark-gluon plasma are briefly discussed.Comment: 12 pages, LaTe

Quantum corrections to the properties of a homogeneous interacting Bose gas at zero temperature can be calculated as a low-density expansion in powers of ρa 3 , where ρ is the number density and a is the S-wave scattering length. We calculate the ground state energy density to second order in ρa 3 . The coefficient of the ρa 3 correction has a logarithmic term that was calculated in 1959. We present the first calculation of the constant under the logarithm. The constant depends not only on a, but also on an extra parameter that describes the low energy 3 → 3 scattering of the bosons. In the case of alkali atoms, we argue that the second order quantum correction is dominated by the logarithmic term, where the argument of the logarithm is ρa ℓ 2 V , and ℓ V is the length scale set by the van der Waals potential.

The low-density expansion for a homogeneous interacting Bose gas at zero temperature can be formulated as an expansion in powers of ρa 3 , where ρ is the number density and a is the S-wave scattering length. Logarithms of ρa 3 appear in the coefficients of the expansion. We show that these logarithms are determined by the renormalization properties of the effective field theory that describes the scattering of atoms at zero density. The leading logarithm is determined by the renormalization of the pointlike 3 → 3 scattering amplitude.