Multiple Mellin-Barnes integrals with straight contours

TL;DR

This paper extends the conic hull method and updates the Mathematica package MBConicHulls.wl to efficiently evaluate multifold Mellin-Barnes integrals with straight contours, aiding in Feynman integral calculations.

Contribution

The authors adapt the conic hull method for straight contours and enhance the MBConicHulls.wl package for automatic multivariable series representations of these integrals.

Findings

Successfully applied to the epsilon-expansion of a massless one-loop pentagon integral.

Provides an efficient computational tool for integrals with straight contours.

Demonstrates the method's effectiveness in practical Feynman integral evaluations.

Abstract

We show how the conic hull method, recently developed for the analytic and non-iterative evaluation of multifold Mellin-Barnes (MB) integrals, can be extended to the case where these integrals have straight contours of integration parallel to the imaginary axes in the complex planes of the integration variables. MB integrals of this class appear, for instance, when one computes the $\epsilon$-expansion of dimensionally regularized Feynman integrals, as a result of the application of one of the two main strategies (called A and B in the literature) used to resolve the singularities in $\epsilon$ of MB representations. We upgrade the Mathematica package MBConicHulls.wl which can now be used to obtain multivariable series representations of multifold MB integrals with arbitrary straight contours, providing an efficient tool for the automatic computation of such integrals. This new feature of the package is presented, along with an example of application by calculating the $\epsilon$-expansion of the dimensionally regularized massless one-loop pentagon integral in general kinematics and $D=4-2\epsilon$.