Some Algebraic and Arithmetic Properties of Feynman Diagrams

  • Yajun ZhouEmail author
Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)


This article reports on some recent progresses in Bessel moments, which represent a class of Feynman diagrams in 2-dimensional quantum field theory. Many challenging mathematical problems on these Bessel moments have been formulated as a vast set of conjectures, by David Broadhurst and collaborators, who work at the intersection of high energy physics, number theory and algebraic geometry. We present the main ideas behind our verifications of several such conjectures, which revolve around linear and non-linear sum rules of Bessel moments, as well as relations between individual Feynman diagrams and critical values of modular L-functions.



This research was supported in part by the Applied Mathematics Program within the Department of Energy (DOE) Office of Advanced Scientific Computing Research (ASCR) as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4).

My work on Bessel moments and modular forms began in 2012, in the form of preliminary research notes at Princeton. I thank Prof. Weinan E (Princeton University and Peking University) for running seminars on mathematical problems in quantum field theory at Princeton, and for arranging my stays at both Princeton and Beijing.

I am grateful to Dr. David Broadhurst for fruitful communications on recent progress in the arithmetic studies of Feynman diagrams [9, 10, 11, 12, 14]. It is a pleasure to dedicate this survey to him, in honor of his 70th birthday.


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Authors and Affiliations

  1. 1.Program in Applied and Computational Mathematics (PACM)Princeton UniversityPrincetonUSA
  2. 2.Academy of Advanced Interdisciplinary Studies (AAIS)Peking UniversityBeijingPeople’s Republic of China

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