Convex effective potential of $O(N)$-symmetric $phi^4$ theory for large $N$

We obtain effective potential of $O(N)$-symmetric $\phi^4$ theory for large $N$ starting with a finite lattice system and taking the thermodynamic limit with great care. It is globally real-valued and convex in both the symmetric and the broken phases. In particular, it has a flat bottom in the broken phase, which is totally consistent with numerical simulations or an argument based on the Maxwell construction. Taking the continuum limit, we discuss renormalization effects to the flat bottom and exhibit the effective potential of the continuum theory in three and four dimensions.