Bounds in 4D Conformal Field Theories with Global Symmetry

Abstract

We explore the constraining power of OPE associativity in 4D Conformal Field Theory with a continuous global symmetry group. We give a general analysis of crossing symmetry constraints in the 4-point function <Phi Phi Phi* Phi*>, where Phi is a primary scalar operator in a given representation R. These constraints take the form of 'vectorial sum rules' for conformal blocks of operators whose representations appear in R x R and R x Rbar. The coefficients in these sum rules are related to the Fierz transformation matrices for the R x R x Rbar x Rbar invariant tensors. We show that the number of equations is always equal to the number of symmetry channels to be constrained. We also analyze in detail two cases - the fundamental of SO(N) and the fundamental of SU(N). We derive the vectorial sum rules explicitly, and use them to study the dimension of the lowest singlet scalar in the Phi x Phi* OPE. We prove the existence of an upper bound on the dimension of this scalar. The bound depends on the conformal dimension of Phi and approaches 2 in the limit dim(Phi)-->1. For several small groups, we compute the behavior of the bound at dim(Phi)>1. We discuss implications of our bound for the Conformal Technicolor scenario of electroweak symmetry breaking.