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

Using the method of Chetyrkin, Misiak, and Münz 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 $O\left(N\right)\mathrm{}\hspace{0.5em}{\phi }^4$ theory. All pole terms in $\epsilon =4\mathrm{}-D\hspace{0.5em}(D$ 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 10, 2491 (1974)].