Finite Cutoff CFT's and Composite Operators

Recently a conformally invariant action describing the Wilson-Fischer fixed point in $D=4-\epsilon$ dimensions in the presence of a {\em finite} UV cutoff was constructed \cite{Dutta}. In the present paper we construct two composite operator perturbations of this action with definite scaling dimension also in the presence of a finite cutoff. Thus the operator (as well as the fixed point action) is well defined at all momenta $0\leq p\leq \infty$ and at low energies they reduce to $\int_x \phi^2$ and $\int _x \phi^4$ respectively. The construction includes terms up to $O(\lamda^2)$. In the presence of a finite cutoff they mix with higher order irrelevant operators. The dimensions are also calculated to this order and agree with known results.