Analytic Continuations and Numerical Evaluation of Multivariable Hypergeometric Functions for Feynman Integrals

Core Concepts

The author explores the analytic continuations and numerical evaluation of multivariable hypergeometric functions, focusing on their application to Feynman integrals.

Abstract

The content delves into the investigation of Appell F1, F3, Lauricella F (3) D, and Lauricella-Saran F (3) S series. These functions are crucial in mathematical physics, especially in evaluating Feynman integrals. The study aims to provide comprehensive analytic continuations and practical packages for efficient numerical evaluations. Various strategies are employed to enhance the accuracy and efficiency of computations, including selecting suitable analytic continuations and optimizing summation techniques. The article also discusses the method of Olsson for deriving these analytic continuations and presents detailed algorithms for implementing these functions in Mathematica.

Stats

"AppellF1.wl : 28" "AppellF3.wl : 24" "LauricellaFD.wl : 96" "LauricellaSaranFS.wl : 102"

Quotes

Key Insights Distilled From

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

by Souvik Bera,... at arxiv.org 03-05-2024

https://arxiv.org/pdf/2403.02237.pdf

Deeper Inquiries

How do non-generic parameter values impact the evaluation of hypergeometric functions

Non-generic parameter values can significantly impact the evaluation of hypergeometric functions. When dealing with non-generic parameter values, where the difference between parameters is an integer, special care needs to be taken as these cases can lead to divergent results or premature termination of series. To address this issue, proper limiting procedures must be applied to ensure that any divergences among the series cancel out effectively. This requires a thorough understanding of how different parameter configurations affect the convergence properties of hypergeometric functions.

What are the implications of selecting different summation indices on computational efficiency

The selection of different summation indices plays a crucial role in determining computational efficiency when evaluating hypergeometric functions. By strategically choosing which index to sum over, it is possible to simplify complex double or triple summations into single-fold sums involving simpler hypergeometric functions like 2F1 or 3F2. This reduction in complexity leads to faster evaluations and reduces the number of operations required for computation. For example, by applying techniques such as rewriting double summations into single-fold sums using appropriate properties like Eq. (38), one can decrease the number of operations needed for finite summation significantly. Choosing an optimal strategy for selecting summation indices based on the simplicity and convergence properties of resulting hypergeometric functions can greatly enhance computational efficiency.

How can the findings in this study be applied to other areas beyond Feynman integrals

The findings from this study on analytic continuations and numerical evaluation methods for multivariable hypergeometric functions have broad applications beyond Feynman integrals in various areas within mathematical physics and theoretical physics: Conformal Field Theory: Hypergeometric functions frequently appear in conformal field theory calculations due to their analytical properties and symmetry relations. The techniques developed here could streamline computations involving these functions in conformal field theories. String Theory: In string theory calculations, especially when dealing with higher-dimensional spaces or complex geometries, efficient evaluation methods for multivariable hypergeometric functions are essential. The strategies outlined in this study could improve computational accuracy and speed in string theory models. Quantum Field Theory: Many quantum field theory problems involve intricate integrals that can be simplified using hypergeometric function representations similar to those studied here. Implementing optimized numerical evaluation techniques could enhance calculations related to particle interactions and quantum phenomena. 4 .Statistical Physics: Hypergeometric functions often arise in statistical physics models describing complex systems' behavior at equilibrium or near-critical points. These findings offer valuable insights into improving numerical evaluations across diverse scientific disciplines beyond just Feynman integral analysis.