Four-loop pressure of massless O(N) scalar field theory

Abstract

Inspired by the corresponding problem in QCD, we determine the pressure of massless O(N) scalar field theory up to order g^6 in the weak-coupling expansion, where g^2 denotes the quartic coupling constant. This necessitates the computation of all 4-loop vacuum graphs at a finite temperature: by making use of methods developed by Arnold and Zhai at 3-loop level, we demonstrate that this task is manageable at least if one restricts to computing the logarithmic terms analytically, while handling the ``constant'' 4-loop contributions numerically. We also inspect the numerical convergence of the weak-coupling expansion after the inclusion of the new terms. Finally, we point out that while the present computation introduces strategies that should be helpful for the full 4-loop computation on the QCD-side, it also highlights the need to develop novel computational techniques, in order to be able to complete this formidable task in a systematic fashion.