Multiple Series Representations of N-fold Mellin-Barnes Integrals

Area of Science:

• Mathematics
• Theoretical Physics
• Computational Physics

Background:

• Mellin-Barnes (MB) integrals are fundamental in various scientific fields, including hypergeometric functions, quantum field theory, and solid-state physics.
• Despite extensive study for over a century, systematic computational techniques for N-fold MB integrals (N>2) remain underdeveloped.
• Existing methods lack a unified approach for handling both resonant (logarithmic) and nonresonant cases of multiple MB integrals.

Purpose of the Study:

• To develop a systematic computational technique for N-fold Mellin-Barnes integrals, particularly for N>2.
• To provide a unified method applicable to both resonant and nonresonant MB integral cases.
• To simplify the analysis and numerical evaluation of complex MB integrals.

Main Methods:

• A novel method based on geometrical analysis using conic hulls is employed.
• The technique generates series representations, which can be either convergent or asymptotic.
• A Mathematica implementation is provided for practical application and verification.

Main Results:

• A systematic computational technique for N-fold MB integrals (N>2) has been established.
• The method successfully handles both resonant and nonresonant cases, yielding convergent or asymptotic series.
• The first evaluations of hexagon and double box conformal Feynman integrals with unit propagator powers are presented.

Conclusions:

• The conic hull method offers a significant advancement in the systematic computation of multiple Mellin-Barnes integrals.
• The technique simplifies convergence studies and numerical checks through the identification of 'master series'.
• This work opens new avenues for evaluating complex integrals in theoretical and computational physics.