Abstract
Systems of integration-by-parts identities play an important role in simplifying the higher-loop Feynman integrals that arise in quantum field theory. Solving these systems is equivalent to reducing integrals containing numerator products of irreducible invariants to a small set of master integrals. I present a new approach to solving these systems that finds direct reduction equations for numerator terms of a given Feynman integral. As a particular example of its power, I show how to obtain reduction equations for arbitrary powers of irreducible invariants, along with their solutions.
Highlights
The computation and simplification of Feynman integrals play a central role in the evaluation of higher-loop scattering amplitudes, form factors, and correlation functions in quantum field theory
Finding linear relations between Feynman integrals plays a key role in higher-loop calculations in quantum field theory
Integration by parts has become the method of choice for finding such relations, but the conventional approach to using them leads to equations involving many unwanted integrals with doubled propagators
Summary
The computation and simplification of Feynman integrals play a central role in the evaluation of higher-loop scattering amplitudes, form factors, and correlation functions in quantum field theory. The first question was answered affirmatively by Gluza et al [13], through the introduction of so-called generating vectors These avoid introducing higher powers of propagators into the system of equations, terms which would later disappear during Gaussian elimination to solve the system. These generating vectors have links to algebraic geometry [14,15] and have seen further development [16] and applications [6,17] recently. Given a set of vectors, an infinite tower of IBP equations can be generated by multiplying them by polynomials in Lorentz invariants of the loop momenta, PHYS. One can and must deal with the resulting simpler topologies to obtain a complete reduction to a basis of integrals
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