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UCCS MATHEMATICS
COLLOQUIUM

FEYNMAN DIAGRAMS, INTEGRALS, AND SPECIAL FUNCTIONS

Abstract: Perturbative quantum field theory visually and systematically describes higher order corrections by means of Feynman diagrams.  In many cases the corresponding multidimensional Feynman integrals may be reduced all the way to one dimension.  In recent years remarkable both rigorously and empirically based expressions for these integrals in terms of values of special functions have appeared.    In this presentation we provide an introduction to this subject, and mention the large number of corresponding open problems in analysis and special function theory.  Particular special functions of interest include the generalized hypergeometric functions, the Clausen functions, and the polylogarithms.  Results to date indicate connections between multiple subjects including hyperbolic geometry, analytic number theory, special functions, and Feynman diagrams.  A very recent example [1] evaluates a highly symmetric Feynman integral in terms of a remarkably compact difference of Clausen function Cl2 values, proving a numerically based conjecture that was open for approximately ten years

[1] M. W. Coffey, J. Math. Physics 49, 043510 (2008).