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Critical points and syzygies for Feynman integrals.

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Summary

We introduce critical syzygies for calculating integration-by-parts relations in Feynman integrals. These critical syzygies simplify integral reduction, offering a new geometric viewpoint in quantum field theory calculations.

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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.