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Rearranging Dyson-Schwinger equations
About this Title
Karen Yeats, Department of Mathematics, Simon Fraser University
Publication: Memoirs of the American Mathematical Society
Publication Year:
2011; Volume 211, Number 995
ISBNs: 978-0-8218-5306-1 (print); 978-1-4704-0612-7 (online)
DOI: https://doi.org/10.1090/S0065-9266-2010-00612-4
Published electronically: October 5, 2010
Keywords: Dyson-Schwinger equations,
Feynman diagrams,
recursive equations
MSC: Primary 81T18
Table of Contents
Chapters
- Foreword
- Preface
- 1. Introduction
- 2. Background
- 3. Dyson-Schwinger equations
- 4. The first recursion
- 5. Reduction to one insertion place
- 6. Reduction to geometric series
- 7. The second recursion
- 8. The radius of convergence
- 9. The second recursion as a differential equation
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, \[ \gamma _1^r(x) = P_r(x) - \mathrm {sign}(s_r)\gamma _1^r(x)^2 + \left (\sum _{j \in \mathcal {R}}|s_j|\gamma _1^j(x)\right ) x \partial _x \gamma _1^r(x) \] 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 $\gamma _1^r(x) = \sum _{n \geq 1}\gamma ^r_{1,n} x^n$. We obtain the radius of convergence of $\sum \gamma ^r_{1,n}x^n/n!$ 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. In particular in the case of quantum electrodynamics we find a distinguished physical solution and find the possibility of avoiding a Landau pole.
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