Springer Theses Recognizing Outstanding Ph.D. Research Michael Borinsky Graphs in Perturbation Theory Algebraic Structure and Asymptotics Springer Theses Recognizing Outstanding Ph.D. Research Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. 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More information about this series at http://www.springer.com/series/8790 Michael Borinsky Graphs in Perturbation Theory Algebraic Structure and Asymptotics Doctoral Thesis accepted by the Humboldt-Universität zu Berlin, Germany 123 Author Supervisor Dr. Michael Borinsky Prof. Dirk Kreimer Departments of Physics and of Mathematics Departments of Physics and of Mathematics Humboldt-Universität zu Berlin Humboldt-Universität zu Berlin Berlin, Germany Berlin, Germany ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-030-03540-2 ISBN 978-3-030-03541-9 (eBook) https://doi.org/10.1007/978-3-030-03541-9 Library of Congress Control Number: 2018960214 © Springer Nature Switzerland AG 2018 This work is subject to copyright. 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Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Supervisor’s Foreword Perturbation theory is one of the cornerstones of theoretical and mathematical physics, in particular for quantum field theory through its famed expansion in terms of Feynman diagrams. It expands the solution of a non-linear fixed point equation, the Dyson–Schwinger equation in the case of quantum fields, in terms of a Taylor series in a small coupling parameter. Even low orders reveal fascinating connec- tions to algebraic and arithmetic geometry, which have been at the forefront of recent research through the clarification of the nature of periods seen in the com- putation of each single Feynman diagram in terms of the mathematical theory of motives. To gain insights though into the all-orders behaviour of quantum field theory amplitudes, an understanding of the summation of such diagrams at high orders is needed. For a start, one has to simply understand the asymptotics of graph counting. The fact that such perturbative expansions are generically divergent series—the magnitude of the coefficients is of rapid growth—necessitates to compute with formal power series. Whilst this study of counting Feynman graphs is a time-honoured problem, many questions have been left unanswered, in particular with regard to the counting of the skeleton diagrams which are the driving integral kernels in the Dyson–Schwinger equations. This thesis studies recently developed novel structures in perturbation theory combining graph-theoretic, combinatorial and Hopf algebraic formalisms. It derives computational methods which allow to calculate asymptotic properties of pertur- bative expansions in a generic manner. Michael Borinsky has developed a new mathematical tool for the handling of asymptotic series: he defines a map from the coefficients of a divergent series to the coefficients appearing in their large-order behaviour. This map has distinguished mathematical properties: it is in fact a derivation, with Leibniz and chain rules. It also gives a well-defined ring structure on such formal series. This result, in its generality, will find applications far beyond problems in quantum field theory allowing to study series at high order, where before results were only accessible at very low orders and by much more primitive and labour-intensive methods. v vi Supervisor’s Foreword As examples, Borinsky calculates the large-order asymptotics of the generating function of connected chord diagrams and simple permutations. He uses the Hopf algebraic structure in perturbative quantum field theory in the context of general graphs, and here the main result for physical applications is the correspondence between bridgeless graphs and the Legendre transformation. The outcome is a complete combinatorial dictionary for zero-dimensional quantum field theory, which is the basic underlying theory for the enumeration problem in quantum field theory. It is exhibited in the study of the most prominent quantum field theories, including the full Standard Model of particle physics. The resulting computations would be inaccessible by conventional textbook perturbative methods. To summarize, Borinsky provides us with a complete analysis of the asymptotics of graph counting for renormalizable quantum fields and at the same time gives a very detailed and pedagogical introduction to new techniques of asymptotic anal- ysis applicable in many other areas of science. Berlin, Germany Dirk Kreimer October 2018 Abstract This thesis provides an extension of the work of Dirk Kreimer and Alain Connes on the Hopf algebra structure of Feynman graphs and renormalization to general graphs. Additionally, an algebraic structure of the asymptotics of formal power series with factorial growth, which is compatible with the Hopf algebraic structure, is introduced. The Hopf algebraic structure permits the explicit enumeration of graphs with constraints for the allowed subgraphs. In the case of Feynman diagrams, a lattice structure, which will be introduced, exposes additional unique properties for physical quantum field theories. The differential ring of factorially divergent power series allows the extraction of asymptotic results of implicitly defined power series with vanishing radius of convergence. Together, both structures provide an alge- braic formulation of large graphs with constraints on the allowed subgraphs. These structures are motivated by and used to analyse renormalized zero-dimensional quantum field theory at high orders in perturbation theory. As a pure application of the Hopf algebra structure, an Hopf algebraic inter- pretation of the Legendre transformation in quantum field theory is given. The differential ring of factorially divergent power series will be used to solve two asymptotic counting problems from combinatorics: the asymptotic number of connected chord diagrams and the number of simple permutations. For both asymptotic solutions, all order asymptotic expansions are provided as generating functions in closed form. Both structures are combined and applied to zero-dimensional quantum field theory. Various quantities are explicitly given in the zero-dimensional version of u3, u4, QED, quenched QED and Yukawa theory with their all order asymptotic expansions. vii List of Publications Journal Articles [1] M Borinsky. “Renormalized asymptotic enumeration of Feynman diagrams”. In: Annals of Physics 385 (2017), pp. 95–135. [2] M Borinsky. “Generating asymptotics for factorially divergent sequences”. In: Manuscript accepted for publication in the Electronic