The $l$-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods

TL;DR

This paper employs GKZ systems and relative Calabi-Yau periods to compute multi-loop banana amplitudes with mass dependence, extending known results up to three loops.

Contribution

It introduces a novel method using GKZ and Calabi-Yau periods to solve multi-loop banana amplitudes with general mass dependence.

Findings

Computed mass dependencies for up to three-loop banana amplitudes.

Validated results against known special mass cases.

Provided a new framework for multi-loop amplitude calculations.

Abstract

We use the GKZ description of periods and certain classes of relative periods on families of Barth-Nieto Calabi-Yau $(l-1)$-folds in order to solve the $l$-loop banana amplitudes with their general mass dependence. As examples we compute the mass dependencies of the banana amplitudes up to the three-loop case and check the results against the known results for special mass values.