  We study the effective potential for composite operators. Introducing a
source coupled to the composite operator, we define the effective potential by
a Legendre transformation. We find that in three or fewer dimensions, one can
use the conventionally defined renormalized operator to couple to the source.
However, in four dimensions, the effective potential for the conventional
renormalized composite operator is divergent. We overcome this difficulty by
adding additional counterterms to the operator and adjusting these order by
order in perturbation theory. These counterterms are found to be
non-polynomial. We find that, because of the extra counterterms, the composite
effective potential is gauge dependent. We display this gauge-dependence
explicitly at two-loop order.
