Discontinuities of banana integrals in dispersion relation representation

TL;DR

This paper develops a method to analyze discontinuities of banana integrals using dispersion relations, introduces identities to simplify their calculation, and proposes a hypothesis for generalized dispersion relations to facilitate their evaluation.

Contribution

It introduces a novel approach to derive and simplify banana integrals' discontinuities via dispersion relations and identities, including a hypothesis for generalized relations.

Findings

Derived the discontinuities of banana integrals using dispersion relations.

Established identities linking parameterized discontinuities of banana integrals.

Proposed a hypothesis for generalized dispersion relations and p-DOBIs.

Abstract

We derive the discontinuities of banana integrals using the dispersion relation iteratively. We find a series of identities between the parameterized discontinuities of banana integrals (p-DOBIs). Similar to elliptic integrals, these identities enable the reduction of various p-DOBIs to be a linear combination of some fundamental ones. We present a practical application of p-DOBIs for deriving Picard-Fuchs operator. Then we establish the expression of generalized dispersion relation, which enables us to obtain the dispersion relation representation of arbitrary banana integrals. Moreover, we propose a hypothesis for generalized dispersion relation and p-DOBIs, which provides a simple way to calculate the discontinuities and transform dispersion relation representation to p-DOBIs.