Bounding scalar operator dimensions in 4D CFT

TL;DR

This paper derives a universal inequality relating the dimensions of a scalar operator and its square in 4D conformal field theories, providing bounds that are checked against known fixed points and models.

Contribution

It introduces a theory-independent bound on scalar operator dimensions in 4D CFTs using conformal bootstrap techniques, with numerical computation of the bounding function.

Findings

The bound f(d) approaches 2 as d approaches 1.

Weakly coupled fixed points satisfy the bound.

Wilson-Fisher fixed points violate the bound by a constant factor.

Abstract

In an arbitrary unitary 4D CFT we consider a scalar operator \phi, and the operator \phi^2 defined as the lowest dimension scalar which appears in the OPE \phi\times\phi with a nonzero coefficient. Using general considerations of OPE, conformal block decomposition, and crossing symmetry, we derive a theory-independent inequality [\phi^2] \leq f([\phi]) for the dimensions of these two operators. The function f(d) entering this bound is computed numerically. For d->1 we have f(d)=2+O(\sqrt{d-1}), which shows that the free theory limit is approached continuously. We perform some checks of our bound. We find that the bound is satisfied by all weakly coupled 4D conformal fixed points that we are able to construct. The Wilson-Fischer fixed points violate the bound by a constant O(1) factor, which must be due to the subtleties of extrapolating to 4-\epsilon dimensions. We use our method to derive an analogous bound in 2D, and check that the Minimal Models satisfy the bound, with the Ising model nearly-saturating it. Derivation of an analogous bound in 3D is currently not feasible because the explicit conformal blocks are not known in odd dimensions. We also discuss the main phenomenological motivation for studying this set of questions: constructing models of dynamical ElectroWeak Symmetry Breaking without flavor problems.