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Updated: Aug 18, 2025

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Determining Feynman Integrals with Only Input from Linear Algebra.

Zhi-Feng Liu1, Yan-Qing Ma1,2

  • 1School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China.

Physical Review Letters
|December 9, 2022
PubMed
Summary
This summary is machine-generated.

All Feynman integrals (FIs) can be determined using linear relations, transforming FI computation into a linear algebra problem. This provides a powerful new method for calculating quantum field theory corrections.

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Area of Science:

  • Quantum Field Theory
  • High Energy Physics
  • Theoretical Physics

Background:

  • Feynman integrals (FIs) are essential for calculating scattering amplitudes in quantum field theory.
  • Current methods for computing FIs can be computationally intensive, especially for higher loop orders.

Purpose of the Study:

  • To demonstrate that all Feynman integrals can be determined through linear relations.
  • To reframe Feynman integral computation as a linear algebraic problem.
  • To develop a powerful method for calculating perturbative corrections.

Main Methods:

  • Identifying and utilizing linear relations between Feynman integrals.
  • Applying linear algebra techniques to solve for unknown FIs.
  • Verifying the method with examples up to five loops.

Main Results:

  • All Feynman integrals, regardless of loop number, are fully determined by provided linear relations.
  • Feynman integral computation is shown to be equivalent to a linear algebraic problem.
  • A novel and powerful method for calculating perturbative corrections is obtained.

Conclusions:

  • The proposed linear algebraic approach offers a significant conceptual shift in Feynman integral computation.
  • This method simplifies the calculation of perturbative corrections in quantum field theory.
  • The approach is validated by successful application to complex, multi-loop integrals.