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We calculate the pressure of massless $\phi^4$-theory to order $g^8\log(g)$at weak coupling. The contributions to the pressure arise from the hardmomentum scale of order $T$ and the soft momentum scale of order $gT$.Effective field theory methods and dimensional reduction are used to separatethe contributions from the two momentum scales: The hard contribution can becalculated as a power series in $g^2$ using naive perturbation theory with barepropagators. The soft contribution can be calculated using an effective theoryin three dimensions, whose coefficients are power series in $g^2$. Thiscontribution is a power series in $g$ starting at order $g^3$. The calculationof the hard part to order $g^6$ involves a complicated four-loop sum-integralthat was recently calculated by Gynther, Laine, Schr\"oder, Torrero, andVuorinen. The calculation of the soft part requires calculating the massparameter in the effective theory to order $g^6$ and the evaluation offive-loop vacuum diagrams in three dimensions. This gives the free energycorrect up to order $g^7$. The coefficients of the effective theory satisfy aset of renormalization group equations that can be used to sum up leading andsubleading logarithms of $T/gT$. We use the solutions to these equations toobtain a result for the free energy which is correct to order $g^8\log(g)$.Finally, we investigate the convergence of the perturbative series.

Pressure to order $g^8*log(g)$ in $\phi^4$-theory at weak coupling