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 dynamically

We carry out a three-loop computation that establishes the existence of scale without conformal invariance in dimensional regularization with the MS scheme in unitary theories in d = 4 - ε spacetime dimensions. We also comment on the effects of scheme changes in theories with many couplings, as well as in theories that live on non-conformal scale-invariant renormalization group trajectories. Stability properties of such trajectories are analyzed, revealing both attractive and repulsive directions in a specific example. We explain how our results are in accord with those of Jack & Osborn on a c-theorem in d = 4 (and d = 4-ε) dimensions. Finally, we point out that limit cycles with turning points are unlike limit cycles with continuous scale invariance. © 2012 SISSA.

We present the theoretical underpinnings of scale without conformal invariance in quantum field theory. In light of our results the gradient-flow interpretation of renormalization-group (RG) flow is challenged, due to deep connections between scaleinvariant theories and recurrent behaviors in the RG. We show that, on scale-invariant trajectories, there is a redefinition of the dilatation current that leads to generators of dilatations that generate dilatations. Finally, we develop a systematic algorithm for the search of scale-invariant trajectories in perturbation theory. © SISSA 2012.

We give an explicit example of a model in D=4-ε space-time dimensions that is scale but not conformally invariant, is unitary, and has finite correlators. The invariance is associated with a limit cycle renormalization group (RG) trajectory. We also prove, to second order in the loop expansion, in D=4-ε, that scale implies conformal invariance for models of any number of real scalars. For models with one real scalar and any number of Weyl spinors we show that scale implies conformal invariance to all orders in perturbation theory. © us-gov 2011.

We present an example of a limit cycle, i.e., a recurrent flow-line of the beta-function vector field, in a unitary four-dimensional gauge theory. We thus prove that beta functions of four-dimensional gauge theories do not produce gradient flows. The limit cycle is established in perturbation theory with a three-loop calculation which we describe in detail. © 2012 SISSA, Trieste, Italy.

The running of quantum field theories can be studied in detail with the use of a local renormalization group equation. The usual beta-function effects are easy to include, but by introducing spacetime-dependence of the various parameters of the theory one can efficiently incorporate renormalization effects of composite operators as well. An illustration of the power of these methods was presented by Osborn in the early 90s, who used consistency conditions following from the Abelian nature of the Weyl group to rederive Zamolodchikov's c-theorem in d = 2 spacetime dimensions, and also to obtain a perturbative a-theorem in d = 4. In this work we present an extension of Osborn's work to d = 6 and to general even d. We compute the full set of Weyl consistency conditions, and we discover among them a candidate for an a-theorem in d = 6, similar to the d = 2, 4 cases studied by Osborn. Additionally, we show that in any even spacetime dimension one finds a consistency condition that may serve as a generalization of the c-theorem, and that the associated candidate c-function involves the coefficient of the Euler term in the trace anomaly. Such a generalization hinges on proving the positivity of a certain "metric" in the space of couplings. © SISSA 2013.

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. © 2013 SISSA.

We explore phenomenological consequences of coupling a non-conformal scale-invariant theory to the standard model. We point out that, under certain circumstances, non-conformal scale-invariant theories have oscillating correlation functions which can dramatically modify standard model processes. We dub this scenario cyclic unparticle physics, or simply cyclunparticle physics. We compute phase spaces and amplitudes associated with final state cyclunparticle and cyclunparticle exchange, respectively. We show detailed formulae in a simple example. © 2012.

The possibility of a strong a theorem in six dimensions is examined in multiflavor ϕ^{3} theory. Contrary to the case in two and four dimensions, we find that, in perturbation theory, the relevant quantity a[over ˜] increases monotonically along flows away from the trivial fixed point. a[over ˜] is a natural extension of the coefficient a of the Euler term in the trace anomaly, and it arises in any even spacetime dimension from an analysis based on Weyl consistency conditions. We also obtain the anomalous dimensions and beta functions of multiflavor ϕ^{3} theory to two loops. Our results suggest that some new intuition about the a theorem is in order.

We use the background-field method and the heat kernel to obtain all counterterms to two-loop order of conformally coupled multiflavor Pdbl3 theory in six spacetime dimensions, defined in curved spacetime and with spacetime-dependent couplings. We also include spacetime-dependent mass terms for completeness. We use these results to write a general expression for the trace anomaly. With the use of Weyl consistency conditions, we are able to show that the strong a-theorem for a certain natural candidate quantity a is violated in this theory and obtain a three-loop expression for the coefficient a of the Euler term in the anomaly.