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The Standard Model in a Weak Gravitational Background. Dilatons, Scale Anomalies and Conformal Methods

The principal goal of the physics of the fundamental interactions is to provide a consistent description of the nature of the subnuclear forces, which manifest in our universe, together with the gravitational force, in a unified framework. This attempt, which is far from being complete, is characterized by two milestones, the Standard Model of the elementary particles and the Einstein's theory of General Relativity. The coupling of a quantum field theory, such as the Standard Model, to a weak gravitational background provides significant information concerning the coupling of matter to gravity and allows to study in a systematic way the origin of the conformal anomaly. For this reason, the computation of correlation functions in a weak gravitational background is of remarkable interest and the consequences of this analysis are also of phenomenological relevance. For instance, they concern the appearance in the spectrum of the theory of a composite state, the dilaton, which is identified, in perturbation theory, by an infrared coupled anomaly pole. The study of some applications of the methods discussed so far is presented in the second part of this thesis. In particular it is shown that the analysis of the classical metric perturbations in gravity can be related to some specific field theories via a holographic mapping. This requires the computation of correlators of energy-momentum tensors in $d=3$ dimensions and plays a role in the study of the non-gaussianities of the cosmic background radiation in the early universe. Finally, we mention an application in momentum space of the constraints derived from the conformal symmetry in the characterization of some correlation functions. This relies on the solution of a system of second order partial differential equations for a certain class of generalized hypergeometric functions of two variables.