A New Method to Explore Conformal Field Theories in Any Dimension

The thesis represents an investigation into Conformal Field Theories (CFT's) in arbitrary dimensions. We propose an innovative method to extract informations about CFT's in a quantitative way. Studying the crossing symmetry of the four point function of scalar operators we derive consistency constraints on the CFT structure, in the form of functional sum rules. The technique we introduce allows to address the feasibility of the sum rule and translate it into restrictions on the CFT spectrum and interactions. Our analysis only assumes unitarity of the CFT, crossing symmetry of the four point function and existence of an Operator Product Expansion (OPE) for scalars. We demonstrate that a CFT satisfying the above hypothesis and containing a scalar operator is not compatible with arbitrary spectra of the operators nor with arbitrary large OPE coefficients. More specifically we prove two main results. First, the spectrum of the CFT must contain a second scalar operator with dimension smaller than a given value. Second, the value of the three point function of two scalars with equal dimension and a third arbitrary operator is bounded from above. As an application of the fist statement we present the bound on the smallest dimension operator entering the OPE of a real scalar with itself. We perform the analysis in two and four dimensions. The comparison of the two dimensional case with exactly solvable models shows a saturation of the bound. We repeat for CFT's with global symmetries and superconformal field theories in four dimensions. As a demonstration of the second result we provide a lower bound on the central charge for CFT in two and four dimensions without global symmetries and for superconformal field theories. We also discuss in the potentialities of the method and possible future research lines. Finally, we discuss possible implications for model building beyond the Standard Model of particle physics.