Finding new relationships between hypergeometric functions by evaluating Feynman integrals

Abstract

Several new relationships between hypergeometric functions are found by comparing results for Feynman integrals calculated using different methods. A new expression for the one-loop propagator-type integral with arbitrary masses and arbitrary powers of propagators is derived in terms of only one Appell hypergeometric function F_1. From the comparison of this expression with a previously known one, a new relation between the Appell functions F_1 and F_4 is found. By comparing this new expression for the case of equal masses with another known result, a new formula for reducing the F_1 function with particular arguments to the hypergeometric function _3F_2 is derived. By comparing results for a particular one-loop vertex integral obtained using different methods, a new relationship between F_1 functions corresponding to a quadratic transformation of the arguments is established. Another reduction formula for the F_1 function is found by analysing the imaginary part of the two-loop self-energy integral on the cut. An explicit formula relating the F_1 function and the Gaussian hypergeometric function _2F_1 whose argument is the ratio of polynomials of degree six is presented.

14 pages, 3 figures

References in corpus (4)

• Hypergeometric representation of the two-loop equal mass sunrise diagram
• A massive Feynman integral and some reduction relations for Appell functions
• Geometrical methods in loop calculations and the three-point function
• Numerical calculation of one-loop integration with hypergeometric functions

Cited by in corpus (16)

• Evaluation of conformal integrals
• Hypergeometric Series Representations of Feynman Integrals by GKZ Hypergeometric Systems
• Massive Conformal Symmetry and Integrability for Feynman Integrals
• From positive geometries to a coaction on hypergeometric functions
• NumExp: Numerical epsilon expansion of hypergeometric functions
• Yangian Bootstrap for Massive Feynman Integrals
• A new approach to the epsilon expansion of generalized hypergeometric functions
• Functional reduction of one-loop Feynman integrals with arbitrary masses
• On integral representations and asymptotics of some hypergeometric functions in two variables
• Analytic continuations and numerical evaluation of the Appell $F_1$, $F_3$, Lauricella $F_D^{(3)}$ and Lauricella-Saran $F_S^{(3)}$ and their Application to Feynman Integrals
• $\text{AlgRel.wl}$: Algebraic Relations for the Product of Propagators in Feynman integrals
• Two-loop Feynman integrals for ${\varphi }^4$ theory with long-range correlated disorder
• A Feynman integral in Lifshitz-point and Lorentz-violating theories in $R^D\oplus R^m$
• Feynman integral in ${\mathbb{R}}^1\oplus {\mathbb{R}}^m$ and complex expansion of ${}_2{F}_1$
• Linear relations for Lauricella $F_D$ functions and symmetric polynomial
• The diagrammatic coaction beyond one loop