On Laplace-Borel resummation of Dyson-Schwinger equations

On Laplace-Borel Resummation of Dyson-Schwinger Equations Abstract: In this work we conduct a complex analytic study of Dyson-Schwinger equations, the quantum equations of motion. Focusing on a particular family of those functional equations we consider the class of formal solutions G(x,L), series expansions in the coupling constant x and the energy scale L whose formal and real analytic aspects have already been studied. Taking the point of view of complex analysis we are able to shed some new light on the structure of these solutions and provide useful tools to consider asymptotic questions. This thesis is built around two functions. The anomalous dimension $\gamma_1$ which is closely tied to the energy scaling properties of quantum field theory and the Green function G(x,L), the actual solution to the Dyson-Schwinger equation. We study their dual aspects as formal power series and analytic functions in the variable x and L. Our tool of choice is the Laplace-Borel resummation method which proves suitable to take care of the divergent series occurring naturally in quantum field theory. Our main results consist in: i) conducting a Laplace-Borel analysis of the anomalous dimension $\gamma_1$; ii) constructing a Laplace-Borel solution to our Dyson-Schwinger equation by using the renormalization group equation.