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

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Cited by in corpus (13)

• Algebraic structures in quantum gravity
• Non-Perturbative Completion of Hopf-Algebraic Dyson-Schwinger Equations
• The QED beta-function from global solutions to Dyson-Schwinger equations
• Renormalization and Mellin transforms
• Feynman graphs, rooted trees, and Ringel-Hall algebras
• Diffeomorphisms of quantum fields
• Approximate Differential Equations for Renormalization Group Functions in Models Free of Vertex Divergencies
• Semiclassical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations: ${\varphi }^3$ QFT in 6 Dimensions
• Some combinatorial aspects of quantum field theory
• Dyson-Schwinger Equations in Minimal Subtraction
• Algebraic Structures in local QFT
• Algebra for quantum fields
• Riemann-Hilbert Problem and Quantum Field Theory: Integrable Renormalization, Dyson-Schwinger Equations