Foam Diagram Summation at Finite Temperature

We show that large-$N$ $\phi ^4$ theory is not trivial if one accepts the presence of a tachyon with a truly huge mass, and that it allows exact calculation. We use it to illustrate how to calculate the exact resummed pressure at finite temperature and verify that it is infrared and ultraviolet finite even in the zero-mass case. In 3 dimensions a residual effect of the resummed infrared divergences is that at low temperature the leading term in the interaction pressure becomes independent of the coupling. In 4 dimensions the pressure is well-defined provided that the temperature is below the tachyon mass and its expansion in powers of the coupling contains terms missed by previous authors. We examine how rapidly this expansion converges and use our analysis to suggest how one might reorganise perturbation theory to improve the calculation of the pressure for the QCD plasma.