Bounds in 4D conformal field theories with global symmetry

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 (φφφ †φ†, where φ 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 ⊗ R and R ⊗ R̄. The coefficients in these sum rules are related to the Fierz transformation matrices for the R ⊗ R ⊗ R̄ ⊗ R̄ 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 φ × φ† OPE. We prove the existence of an upper bound on the dimension of this scalar. The bound depends on the conformal dimension of φ and approaches 2 in the limit dim(φ) → 1. For several small groups, we compute the behavior of the bound at dim(φ) > 1. We discuss implications of our bound for the conformal technicolor scenario of electroweak symmetry breaking. © 2011 IOP Publishing Ltd.