Pressure to orderg8loggof massless ϕ4theory at weak coupling

Abstract

We calculate the pressure of massless ϕ4-theory to order g8log (g) at weak coupling. The contributions to the pressure arise from the hard momentum scale of order T and the soft momentum scale of order gT. Effective field theory methods and dimensional reduction are used to separate the contributions from the two momentum scales: The hard contribution can be calculated as a power series in g2 using naive perturbation theory with bare propagators. The soft contribution can be calculated using an effective theory in three dimensions, whose coefficients are power series in g2. This contribution is a power series in g starting at order g3. The calculation of the hard part to order g6 involves a complicated four-loop sum-integral that was recently calculated by Gynther, Laine, Schröder, Torrero, and Vuorinen. The calculation of the soft part requires calculating the mass parameter in the effective theory to order g6 and the evaluation of five-loop vacuum diagrams in three dimensions. This gives the free energy correct up to order g7. The coefficients of the effective theory satisfy a set of renormalization group equations that can be used to sum up leading and subleading logarithms of T/gT. We use the solutions to these equations to obtain a result for the free energy which is correct to order g8log (g). Finally, we investigate the convergence of the perturbative series.