Reduction of a Class of Three-Loop Vacuum Diagrams to Tetrahedron Topologies

We obtain finite parts (as well as $\epsilon$-pole parts) of massive three-loop vacuum diagrams with three-point and/or four-point interaction vertices by reducing them to tetrahedron diagrams with both massive and massless lines, whose finite parts were given analytically in a recent paper by Broadhurst. In the procedure of reduction, the method of integration-by-parts recurrence relations is employed. We use our result to compute the $\bar{\rm MS}$ effective potential of the massive $\phi^4$ theory.