Cracking a century-old mathematical problem

An important advance has been made towards the full solution of the old mathematical problem of constructing series representations of Mellin-Barnes (MB) integrals of arbitrary complexity. These are a special type of integral whose integrand consists primarily of ratios of products of Euler Gamma functions with complex number arguments.

MB integrals are ubiquitous in physics, engineering, and a variety of quantitative disciplines, and their evaluation ‒ particularly important for quantum field theory and hypergeometric functions theory ‒ has been an unsolved problem for over a century. In the present work, geometrical objects (conic hulls) and multivariate complex analysis have been combined in a new approach to make a breakthrough, allowing us to evaluate MB integrals when the number of integrations is any fixed positive integer.

The technique is simple enough to be appreciated and applied by anyone with a firm grasp of school-level geometry. A computer package automating this method has also been produced and provided by the authors, adding a powerful tool to the armoury of physicists and mathematicians. A potentially large number of applications could follow from these results.

The study was carried out by B Ananthanarayan and Sumit Banik from the Centre for High Energy Physics, along with collaborators in Europe.

Reference:

1. https://journals.aps.org/prl/accepted/6e07cY17N7d12462a9f587a9dad8ce029d73c6e7c (Will be updated after publication)
2. https://doi.org/10.1103/PhysRevD.103.096008
3. https://doi.org/10.1103/PhysRevD.102.091901