Four-loop screened perturbation theory

We study the thermodynamics of massless φ⁴-theory using screened perturbation theory. In this method, the perturbative expansion is reorganized by adding and subtracting a thermal mass term in the Lagrangian. We calculate the free energy through four loops expanding in a double power expansion in $m/T$ and g², where m is the thermal mass and g is the coupling constant. The expansion is truncated at order g⁷ and the loop expansion is shown to have better convergence properties than the weak-coupling expansion. The free energy at order g⁶ involves the four-loop triangle sum-integral evaluated by Gynther, Laine, Schr\"oder, Torrero, and Vuorinen using the methods developed by Arnold and Zhai. The evaluation of the free energy at order g⁷ requires the evaluation of a nontrivial three-loop sum-integral, which we calculate by the same methods.