The Massive Thermal Basketball Diagram

The “basketball diagram” is a three-loop vacuum diagram for a scalar field theory that cannot be expressed in terms of one-loop diagrams. We calculate this diagram for a massive scalar field at nonzero temperature, reducing it to expressions involving three-dimensional integrals that can be easily evaluated numerically. We use this result to calculate the free energy for a massive scalar field with a phi^4 interaction to three-loop order.