Growth estimates for Dyson-Schwinger equations

Abstract

Dyson-Schwinger equations are integral equations in quantum field theory that describe the Green functions of a theory and mirror the recursive decomposition of Feynman diagrams into subdiagrams. Taken as recursive equations, the Dyson-Schwinger equations describe perturbative quantum field theory. However, they also contain non-perturbative information. Using the Hopf algebra of Feynman graphs we will follow a sequence of reductions to convert the Dyson-Schwinger equations to the following system of differential equations, \[ γ_1^r(x) = P_r(x) - \sgn(s_r)γ_1^r(x)^2 + (\sum_{j \in \mathcal{R}}|s_j|γ_1^j(x)) x \partial_x γ_1^r(x) \] where $r\in \mathcal{R}$, $\mathcal{R}$ is the set of amplitudes of the theory which need renormalization, ${\gamma }_1^r$ is the anomalous dimension associated to $r$, $P_r(x)$ is a modified version of the function for the primitive skeletons contributing to $r$, and $x$ is the coupling constant. Next, we approach the new system of differential equations as a system of recursive equations by expanding ${\gamma }_1^r(x)={\sum }_{n\ge 1}{\gamma }_{1,n}^r{x}^n$. We obtain the radius of convergence of $\sum {\gamma }_{1,n}^r{x}^n/n!$ in terms of that of $\sum P_r(n)x^n/n!$. In particular we show that a Lipatov bound for the growth of the primitives leads to a Lipatov bound for the whole theory. Finally, we make a few observations on the new system considered as differential equations.

86 pages, the author's PhD thesis

References in corpus (2)

• Hopf algebras in renormalization theory: Locality and Dyson-Schwinger equations from Hochschild cohomology
• Recursion and growth estimates in renormalizable quantum field theory