Pressure to order $g^8\ast log(g)$ in ${\varphi }^4$-theory at weak coupling

Abstract

We calculate the pressure of massless ${\varphi }^4$-theory to order $g^8\log (g)$ 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 $g^2$ 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 $g^2$. This contribution is a power series in $g$ starting at order $g^3$. The calculation of the hard part to order $g^6$ 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 $g^6$ and the evaluation of five-loop vacuum diagrams in three dimensions. This gives the free energy correct up to order $g^7$. 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 $g^8\log (g)$. Finally, we investigate the convergence of the perturbative series.

29 pages and 12 figs. New version: we have pushed the calculations to g^8*log(g) using the renormalization group to sum up log(g) from higher orders. Published in JHEP

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