There is a widely held belief that conformal field theories (CFTs) require zero beta functions. Nevertheless, the work of Jack and Osborn implies that the beta functions are not actually the quantites that decide conformality, but until recently no such behavior had been exhibited. Our recent work has led to the discovery of CFTs with nonzero beta functions, more precisely CFTs that live on recurrent trajectories, e.g., limit cycles, of the beta-function vector field. To demonstrate this we study the S function of Jack and Osborn. We use Weyl consistency conditions to show that it vanishes at fixed points and agrees with the generator Q of limit cycles on them. Moreover, we compute S to third order in perturbation theory, and explicitly verify that it agrees with our previous determinations of Q. A byproduct of our analysis is that, in perturbation theory, unitarity and scale invariance imply conformal invariance in four-dimensional quantum field theories. Finally, we study some properties of these new, "cyclic" CFTs, and point out that the a-theorem still governs the asymptotic behavior of renormalization-group flows.

Metastable vacua in supersymmetric QCD in the presence of single and multitrace deformations of the superpotential are explored, with the aim of obtaining an acceptable phenomenology. The metastable vacua appear at one loop, have a broken R-symmetry, and a magnetic gauge group that is completely Higgsed. With only a single trace deformation, the adjoint fermions from the meson superfield are approximately massless at one loop, even though they are massive at tree level and R-symmetry is broken. Consequently, if charged under the standard model, they are unacceptably light. A multitrace quadratic deformation generates fermion masses proportional to the deformation parameter. Phenomenologically viable models of direct gauge mediation can then be obtained, and some of their features are discussed.

It's well known that in conformal theories the two-and three-point functions of a subset of the local operators-the conformal primaries-suffice, via the operator product expansion (OPE), to determine all local correlation functions of operators. It's less well known that, in superconformal theories, the OPE of superdescendants is generally undetermined from those of the superprimaries, and there is no universal notion of superconformal blocks. We recall these and related aspects of 4d (S)CFTs, and then we focus on the super operator product expansion (sOPE) of conserved currents in 4d N = 1 SCFTs. The current-current OPE J(x)J(0) has applications to general gauge mediation. We show how the superconformal symmetry, when combined with current conservation, determines the OPE coefficients of superconformal descendants in terms of those of the superconformal primaries. We show that only integer-spin real superconformal-primary operators of vanishing R-charge, and their descendants, appear in the sOPE. We also discuss superconformal blocks for fourpoint functions of the conserved currents.July 2011 arXiv:1107.1721v2 [hep-th] 30 Apr 2014(1.4) with notation reviewed in section 3.1. For now we will just say that X −X = 4iΘΘ, withto superconformal primary components, but do contribute for superdescendants. Explicitly, in (1.3), the f abc term is a descendant coefficient that is unrelated to the kd abc primary coefficient. In (1.4) the Θ dependence is at least determined by G symmetry. For general operators, the Θ dependence is ambiguous, not fully determined by the symmetries.We will here study the general constraints of superconformal symmetry on the twoand three-point functions relevant for the J(x)J(0) sOPE, and how the sOPE coefficients are obtained from these correlators. We will do this both using the superspace results of Osborn [11] for the relevant two-and three-point functions, and also directly from the superconformal algebra. As we'll discuss, the fact that the currents are conserved here allows the superspace Θ dependence to be completely fixed. Thus, the coefficients of the superconformal primaries in the J(x)J(0) OPE suffice to fully determine all OPE coefficients of all descendants. We will discuss the contributions on the RHS of the J(x)J(0)OPE from integer-spin real U (1) R -charge-zero superconformal primaries, O µ 1 ...µ , and their superdescendants. 3The paper is organized as follows: section 2 briefly reviews the aspects of the OPE in 4dCFTs that we will use in the following discussion. Section 3 discusses superconformal theories, and the constraints of superconformal symmetry on two-and three-point functions and the OPE. The superspace formalism of [11], and the recent results about chiral-chiral and 3 Note added (April 2014 revision): as was later found in [15], there are additional contributing Lorentz representations. This revised version will also correct a couple of errors in our original version's coefficients, as pointed out to us by the authors of [15] and [16]; see these papers for further ...