Three-Loop Effective Potential of O(N) $ϕ^4$ Theory

The three-loop effective potential of the massless O(N) $ϕ^4$ theory is calculated analytically using techniques of dimensional regularization. We see a complete agreement between our result and Jackiw's result obtained only up to two-loop order using a different regularization (cutoff regularization) method, but the same renormalization conditions. For an easy check of the mutual cancellation of all the dangerous pole terms in each loop order, we give the $ε$-expanded loop integrals in full detail.