One-loop finite potential for N-2 scalars from N quantum fields

We study the one-loop effective potential induced from quantum fluctuation of a finite number of fields. A series expansion in terms of the modified Bessel functions is useful to evaluate the one-loop effective potential. We find that at most $N-2$ scalars parameterize the one-loop finite potential and the explicit parameterization is shown. The structure of the potential for N=4 is investigated as the simplest case. The implication of the model is discussed.