A Representation Transformation of Parametric Feynman Integrals

TL;DR

The paper introduces a reciprocal transformation on homogeneous polynomials that connects the parametric and Baikov representations of Feynman integrals. Using this transformation, it constructs a dual pair of integrals $I_x(\lambda)$ and $I_u(\lambda')$, proving their equivalence up to constants and providing explicit mappings such as $\mathcal{G}=\frac{1}{\mathcal{F}}$ and $P(v)= -\mathcal{G}_-(v)$. The Baikov representation is shown to be equivalent to the dual parametric form via a correspondence between the Gram-determinant polynomial and the dual function, unifying two central approaches to perturbative quantum field theory and enabling joint analysis of Landau equations and infrared structures. Overall, the work offers a rigorous bridge between parametric and Baikov frameworks, with practical implications for integral-reduction methods and the systematic study of singular regions in multiloop Feynman integrals.

Abstract

A transformation on homogeneous polynomials is proposed, which is further applied to parametric Feynman integrals. The two representations related through this transformation are dual to each other. It naturally leads to dualities of Landau equations and linear integral relations between the two representations. For integrals with momentum-space correspondences, the dual representation is equivalent to the Baikov representation.