Algebraic Structure of Cut Feynman Integrals and the Diagrammatic Coaction

Area of Science:

• Theoretical Physics
• Quantum Field Theory
• Mathematical Physics

Background:

• Feynman integrals are crucial in quantum field theory for calculating scattering amplitudes.
• Understanding their algebraic and analytic structure is essential for theoretical advancements.
• Existing methods for analyzing these integrals can be complex and limited in scope.

Purpose of the Study:

• To develop a novel mathematical operation, termed a coaction, for analyzing Feynman integrals.
• To generalize the coaction beyond multiple polylogarithms to other functions like hypergeometric functions.
• To establish a diagrammatic representation of the coaction for one-loop Feynman integrals.

Main Methods:

• Proposing a new coaction that maps an integral to pairs of integrals from a master integrand and contour.
• Demonstrating the coaction's applicability to hypergeometric functions and generic one-loop Feynman integrals.
• Developing a diagrammatic representation using graph operations (contractions and cuts) for the coaction.

Main Results:

• The proposed coaction generalizes known coactions on multiple polylogarithms.
• The coaction is applicable to a broader class of functions, including hypergeometric functions.
• A diagrammatic representation of the coaction for one-loop Feynman integrals was established.
• The coaction provides direct access to iterated discontinuities and simplifies the derivation of differential equations.

Conclusions:

• The developed coaction offers a powerful and general tool for studying Feynman integrals.
• Its diagrammatic representation simplifies analysis and facilitates the derivation of differential equations for one-loop integrals.
• This work advances the understanding of the algebraic and analytic structure of Feynman integrals in quantum field theory.