Critical points and syzygies for Feynman integrals

Area of Science:

• Theoretical Physics
• High-Energy Physics
• Mathematical Physics

Background:

• Integration-by-parts (IBP) relations are crucial for simplifying Feynman integrals in quantum field theory.
• Existing methods for computing IBP relations can be computationally intensive.
• Understanding the underlying theoretical structure is key to developing more efficient methods.

Purpose of the Study:

• To investigate a novel theoretical structure for computing integration-by-parts relations.
• To introduce and analyze the concept of

Main Methods:

• Syzygy-based methods
• Intersection theory
• Analysis of the large-ϵ limit of dimensional regularization
• Identification of the critical locus of the logarithm of the Baikov polynomial

Main Results:

• Total derivatives vanish on the critical locus in the large-ϵ limit.
• "Critical syzygies" are identified as a key subset of syzygies.
• Critical syzygies generate sufficient total derivatives for integral reduction when the critical locus is isolated.
• Analytical study at one loop and numerical construction at two loops.

Conclusions:

• Critical syzygies provide a powerful tool for integral reduction.
• The study offers a novel geometric perspective on integration-by-parts relations.
• This approach is applicable to complex, cutting-edge two-loop calculations.