Finite Cutoff CFT's and Composite Operators

TL;DR

This paper constructs finite-cutoff conformal field theories with well-defined composite operators, calculating their scaling dimensions and demonstrating their consistency with known results, thus advancing the understanding of Wilson-Fischer fixed points.

Contribution

It introduces a method to define composite operators at finite cutoff in CFTs, including their scaling dimensions and operator mixing, extending previous work on fixed points with finite UV cutoff.

Findings

Composite operators are well-defined at all momenta with finite cutoff.

Operator dimensions match known results up to second order in coupling.

Operators mix with higher order irrelevant operators at finite cutoff.

Abstract

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.