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Vector Space of Feynman Integrals and Multivariate Intersection Numbers.

Hjalte Frellesvig1,2, Federico Gasparotto1,2, Manoj K Mandal1,2

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We present a general algorithm for constructing multivariate intersection numbers for Feynman integrals. This method enables integral reduction to master integrals and derivation of functional equations, advancing multiloop calculations.

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Area of Science:

  • Theoretical Physics
  • Quantum Field Theory
  • Mathematical Physics

Background:

  • Feynman integrals are crucial in quantum field theory calculations.
  • Intersection numbers are key to understanding linear relations among Feynman integrals.
  • Integral reduction to master integrals is a fundamental challenge.

Purpose of the Study:

  • To develop a general algorithm for constructing multivariate intersection numbers.
  • To demonstrate the application of these numbers in Feynman integral reduction.
  • To derive functional equations for master integrals directly.

Main Methods:

  • Algorithm for constructing multivariate intersection numbers.
  • Projection method for integral reduction.
  • Derivation of functional equations using intersection numbers.

Main Results:

  • A general algorithm for multivariate intersection number construction is presented.
  • The algorithm successfully reduces Feynman integrals to master integrals via projections.
  • Functional equations for master integrals are directly derived.
  • The method is applied to one- and two-loop integrals.

Conclusions:

  • The developed algorithm provides a novel approach to Feynman integral reduction.
  • This method offers a direct route to deriving functional equations for master integrals.
  • The approach has potential applications for generic multiloop integrals and special functions.