Aspects of trace anomaly in perturbation theory and beyond

In this thesis we study the connection between conformal symmetry breaking and the the renormalization group. In the first chapter we review the main properties of conformal field theories (CFTs), Wilsonian RG and describe how renormalization induces a flow between different CFTs. The prominent role is given to the trace of energy-momentum tensor (TEMT) as a measure for conformal symmetry violation. Scaling properties of supersymmetric gauge theories are also reviewed . In the second chapter the quantum action principle is introduced as a scheme for renormalizing composite operators. The framework is then applied to derive conditions for UV finiteness of two-point correlators of composite operators with special emphasis on TEMT. We then proceed to discuss the application of the Feynman-Hellmann theorem to evaluate gluon condensates. In the third chapter the basic elements the Trace anomaly on curved space are examined. The finiteness results from Chapter 2 are given physical meaning in relation with the RG flow of the geometrical quantity ~ d (coefficient of □R in the anomaly). The last chapter is dedicated to the a-theorem. First we apply some of the results derived in Chapter 3 to extend the known perturbative calculation for the flow of the central charge βa for gauge theories with Banks-Zaks fixed point. In the last part we review the main ideas of the recent proof of the a-theorem by Komargodski and Schwimmer and apply their formalism to re-derive the known non-perturbative formula for ∆ βa of SUSY conformal window theories