Calculation of a class of three loop vacuum diagrams with two different mass values

Using the method of Chetyrkin, Misiak, and M{umlt u}nz we calculate analytically a class of three-loop vacuum diagrams with two different mass values, one of which is one-third as large as the other. In particular, this specific mass ratio is of great interest in relation to the three-loop effective potential of the O(N) {phi}{sup 4} theory. All pole terms in {epsilon}=4{minus}D (D being the space-time dimensions in a dimensional regularization scheme) plus finite terms containing the logarithm of mass are kept in our calculation of each diagram. It is shown that a three-loop effective potential calculated using the three-loop integrals obtained in this paper agrees, in the large-N limit, with the overlap part of the leading-order (in the large-N limit) calculation of Coleman, Jackiw, and Politzer [Phys. Rev. D {bold 10}, 2491 (1974)]. {copyright} {ital 1999} {ital The American Physical Society}