- The paper presents a novel algorithm that systematically expands hypergeometric functions about half-integer parameters in Feynman integrals.
- It employs a basis function approach with harmonic polylogarithms to achieve all-order series expansions for complex multi-loop calculations.
- Its enhanced HypExp package improves precision in QCD and electroweak computations, facilitating advanced analyses in theoretical particle physics.
Expansion of Hypergeometric Functions about Half-Integer Parameters
This paper presents a comprehensive paper on expanding hypergeometric functions around half-integer parameters, a topic specifically relevant to theoretical particle physics, where such functions feature prominently in higher-loop calculations of Feynman diagrams. The authors Tobias Huber and Daniel Maître introduce a novel algorithm that enhances the functionality of the pre-existing Mathematica package, HypExp, to handle expansions of hypergeometric functions of various types, including those with half-integer and integer parameters, within Feynman integrals.
Methodology and Implementation
The central advancement in this paper is the development of a new algorithm that systematically expands hypergeometric functions about half-integer parameters. The approach involves expressing a given hypergeometric function (HF) as a series of differential and integral operators acting on a simpler, canonical form of HF, referred to as the 'basis function.' The choice of basis functions is critical, as they form the backbone of the expansion process. The paper delineates a process for constructing the basis functions for several classes of hypergeometric functions, distinctly identifying their expansion characteristics at different orders.
The authors emphasize a mathematically robust strategy to find all-order expansions of the basis functions and provide several examples of practical applications of Feynman diagrams up to four loops. Their method offers a way to extend such expansions to arbitrary order, facilitated by the usage of harmonic polylogarithms (HPLs) of both integer and complex arguments. The mathematical constructs used, such as HPLs and their complex counterparts, are critical for handling the mathematical intricacies involved in such expansions.
Practical Applications and Theoretical Implications
The utility of this method is exemplified in its application to complex Feynman diagram calculations that arise in perturbative quantum field theory. These results are particularly insightful for integrals involving massive particles in loop calculations, a common scenario in precision computations of particle interactions.
The extended functionality of the HypExp package now accommodates hypergeometric functions essential for multi-loop integrals in quantum chromodynamics (QCD) and electroweak calculations, among other areas. It complements systematic approaches implemented in other computer algebra systems, namely GiNaC and FORM, but expands the scope to incorporate classes of hypergeometric functions rich in half-integer parameters.
Future Prospects
The advances discussed lay groundwork for further developments in the symbolic computation of multi-loop Feynman integrals, which are computationally intensive bottlenecks in high-energy physics phenomenology. Researchers can now more efficiently perform these expansions, potentially paving the way for more complex calculations involving hypergeometric functions beyond the classes covered in the present work. Additionally, the methodology could stimulate further research into the generalization of these techniques to other types of special functions appearing in theoretical physics.
Through this research, the authors contribute significantly to the computational toolkit available to physicists, allowing for more precise and efficient evaluations of fundamental particle interactions in scenarios where traditional analytical methods falter. This enhancement in computational capabilities is indispensable for the ongoing efforts at collider experiments where precise theoretical predictions are measured against experimental data.
