A tale of scale, conformal, and superconformal invariance

This dissertation consists of two parts. In the first, we study the possibility of recurrent trajectories in renormalization-group flows of unitary four-dimensional gauge theories, and the relation of scale and conformal invariance. We carry out three-loop computations of the beta function in dimensional regularization, and we establish that the beta-function vector field of four- dimensional gauge theories admits recurrent trajectories. It is then demonstrated that theories that live on these trajectories are conformal. Along the way, we construct a perturbative proof that scale implies conformal invariance in relativistic unitary quantum field theories in four spacetime dimensions. We also point out that the beta function of supersymmetric theories does not admit limit cycles in perturbation theory. The second part of this dissertation pertains to theories that are superconformal, or approximately superconformal. We use the constraints of superconformal symmetry to illustrate features of two- and three-point correlators involving conserved-current insertions. This is motivated by "general gauge mediation'' where two-point current-correlators parametrize the soft masses of the minimal supersymmetric standard model. We show that the superconformal symmetry and current conservation are enough to fix the operator products of descendants in terms of those of the primaries. Subsequently we consider soltly broken superconformal symmetry and we study analyticity properties of these correlators, e.g. their discontinuities. We then use the optical theorem to relate them to total scattering cross sections from visible to hidden sector states. We also discuss how the current- current OPE can be truncated to the first few terms to get a good approximation to the soft masses. Finally, we demonstrate our techniques in several examples, both at weak and strong coupling. Among them, we introduce a new framework where supersymmetry-breaking arises both from a hidden sector and dynamicall