Quantum aspects of classically conformal theories in four and six dimensions

Abstract

Perturbative quantum field theory is our most developed framework to accurately analyse many physical phenomena. The standard tool is Feynman diagrams, but at one-loop the heat kernel is also a powerful technique. It is however difficult to go beyond perturbation theory, and symmetry is a key factor. In particular, conformal symmetry strongly restricts the correlators, and have been combined with the average null energy condition (ANEC) to derive the Hofman-Maldacena bounds on the anomaly coefficients in four dimensions. In this thesis we study different problems in perturbative quantum field theory. First, we study the Weyl anomaly for a non-conformal free scalar in a four-dimensional curved spacetime. We diagrammatically understand the definition of the anomaly without classical symmetry, and we precisely interpret the well-known heat kernel calculation. Then, we study higher-derivative gauge theories in six dimensions. These theories are the natural candidate to perturbatively construct non-unitary conformal theories. The calculation is done with the heat kernel method and we derive the general expression of the relevant coefficient, which was previously unknown. Supersymmetry or the addition of a Yang-Mills term are also considered. Finally, we initiate the study of the consequence of the ANEC on non-conformal field theories with the example of the self-interacting scalar in four dimensions. The energy flux of a state with a single field insertion is computed. Starting from the perturbative momentum-space Euclidean correlators, we construct the relevant Wightman function to evaluate the energy flux. The calculation is considerably complicated, but we recover the expected result, opening the possibility of studying more interesting states.