Multiple Mellin-Barnes integrals and triangulations of point configurations (2309.00409v1)

Abstract: We present a novel technique for the analytic evaluation of multifold Mellin-Barnes (MB) integrals, which commonly appear in physics, as for instance in the calculations of multi-loop multi-scale Feynman integrals. Our approach is based on triangulating a set of points which can be assigned to a given MB integral, and yields the final analytic results in terms of linear combinations of multiple series, each triangulation allowing the derivation of one of these combinations. When this technique is applied to the computation of Feynman integrals, the involved series are of the (multivariable) hypergeometric type. We implement our method in the Mathematica package MBConicHulls.wl, an already existing software dedicated to the analytic evaluation of multiple MB integrals, based on a recently developed computational approach using intersections of conic hulls. The triangulation method is remarkably faster than the conic hulls approach and can thus be used for the calculation of higher-fold MB integrals as we show here by computing triangulations for highly complicated objects such as the off-shell massless scalar one-loop 15-point Feynman integral whose MB representation has 104 folds. As other applications we show how this technique can provide new results for the off-shell massless conformal hexagon and double box Feynman integrals, as well as for the hard diagram of the two loop hexagon Wilson loop.

• The paper introduces a novel triangulation approach for evaluating multifold Mellin-Barnes integrals, deriving efficient series representations from geometric configurations.
• It implements the method via a Mathematica package, MBConicHulls.wl, which outperforms previous conic hull techniques in reducing computation time.
• The approach successfully computes complex Feynman integrals, including off-shell massless scalar one-loop 15-point cases, offering significant theoretical and practical advances.

Overview of "Multiple Mellin-Barnes Integrals and Triangulations of Point Configurations"

This paper introduces a new method for the analytic evaluation of multifold Mellin-Barnes (MB) integrals, which are integrals commonly encountered in theoretical physics, especially in the context of evaluating multi-loop multi-scale Feynman integrals. The technique presented is based on the triangulation of point configurations associated with a given MB integral, offering an efficient and systematic approach to derive analytic results in terms of linear combinations of multiple series solutions.

Key Contributions

1. Triangulation Approach: The authors demonstrate a novel approach using triangulations of point configurations to evaluate MB integrals. Each triangulation corresponds to a series representation of the MB integral, allowing the method to derive these representations efficiently.
2. Computational Implementation: The technique has been implemented in a Mathematica package, MBConicHulls.wl, complementing the existing conic hull intersection method. The triangulation method surpasses previous computational approaches in efficiency, enabling the evaluation of integrals with a higher number of folds.
3. Applications to Complex Integrals: The paper showcases the application of this technique to compute new results for complex Feynman integrals, such as the off-shell massless scalar one-loop 15-point Feynman integral. It also revisits and provides improved series representations for previously studied integrals like the conformal hexagon and double box Feynman integrals.
4. Performance and Practical Utility: The presented method significantly reduces computation times compared to prior methods based on conic hull intersections. This enhancement enables the handling of MB integrals with up to 104 folds, which were previously intractable.

Numerical Results and Claims

• The authors report substantial gains in computational speed. For instance, a nine-fold MB representation for the conformal hexagon integral can be computed much faster using the triangulation method.
• They derive new, simpler series solutions for the conformal double-box and hexagon integrals, showcasing the technique's capability to explore and simplify the space of potential solutions.

Theoretical Implications

The paper's method promises potential insights into the theory of hypergeometric functions and provides a robust tool for evaluating Feynman integrals. The bijective correspondence between the regular triangulations and conic hull intersections offers a new perspective on these mathematical structures, fostering further exploration in both theoretical and applied mathematics.

Future Developments

The technique's improved computational capabilities pave the way for tackling even more complex integrals and exploring broader classes of problems in high-energy physics and related fields. Furthermore, the development of analytic tools like the presented algorithm could influence how future theoretical physics models employ MB integrals for precision calculations.

In conclusion, this paper presents a significant computational advancement in the analytic evaluation of Mellin-Barnes integrals. By integrating concepts from geometry and computational algebra, the authors provide a scalable and efficient approach, markedly impacting the calculation of multifold integrals in theoretical physics. This work holds substantial promise for advancing both theoretical insights and practical computations in particle physics and beyond.