Radial expansion for spinning conformal blocks

This paper develops a method to compute any bosonic conformal block as a series expansion in the optimal radial coordinate introduced by Hogervorst and Rychkov. The method reduces to the known result when the external operators are all the same scalar operator, but it allows to compute conformal blocks for external operators with spin. Moreover, we explain how to write closed form recursion relations for the coefficients of the expansions. We study three examples of four point functions in detail: one vector and three scalars; two vectors and two scalars; two spin 2 tensors and two scalars. Finally, for the case of two external vectors, we also provide a more efficient way to generate the series expansion using the analytic structure of the blocks as a function of the scaling dimension of the exchanged operator.

Anomalous dimensions of scalar operators in CFT

Alessandro Vichi

We present a recently developed method to constrain the anomalous dimension of scalar operators in a general Conformal Field Theory (CFT). Using a consistency condition derived from four-point correlation functions it is possible to birid the anomalous dimension of a composite operator phi(2). The result has also consequences on model building.

Elsevier2009

A New Method to Explore Conformal Field Theories in Any Dimension

Alessandro Vichi

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.

EPFL2011

Momentum-space conformal blocks on the light cone

Marc Gillioz

We study the momentum-space 4-point correlation function of identical scalar operators in conformal field theory. Working specifically with null momenta, we show that its imaginary part admits an expansion in conformal blocks. The blocks are polynomials in the cosine of the scattering angle, with degree corresponding to the spin of the intermediate operator. The coefficients of these polynomials are obtained in a closed-form expression for arbitrary spacetime dimension d > 2. If the scaling dimension of the intermediate operator is large, the conformal block reduces to a Gegenbauer polynomial f on the contrary the scaling dimension saturates the unitarity bound, the block is different Gegenbauer polynomial These results are then used as an inversion formula to compute OPE coefficients in a free theory example.

SPRINGER2018

A New Method to Explore Conformal Field Theories in Any Dimension

Alessandro Vichi

EPFL2011